Following fish and fractals: self-affine patterns of hierarchical formal structures embedded in the contours of Rolf Wallin's ning

This dissertation is comprised of two parts: an analytical paper assessing the "fractal" quartet ning by composer Rolf Wallin, and an original composition for wind ensemble entitled Graven Images, based on the triptych of short stories with the same name by author Paul Fleischman. Chapter 1 is an expansion of a symposium lecture from 1989 by Wallin, following the same format and content, but with greater detail. It introduces the fractals and formulas that inspired Wallin, explains their mathematical functions, and presents the methodology by which he composed ning using two specific fractal formulas: the logistic equation, and a formula shared with Wallin by physicist Jan Frøyland. Chapter 2 takes an extensive look at Wallin's use of the logistic equation to create the formal parameters of ning, then provides a detailed overview of the piece's form, finding parallels with traditional sonata form. Chapter 3 explains how Wallin generated the "foreground melody" of ning (the mathematically constructed melodic line that serves as the piece's backbone, which Wallin then "composed out" into the quartet) using Frøyland's formula (as constrained by the limits of the formal parameters derived from the logistic equation), which in turn was modified by another fractal-inspired technique of his own devising: the crystal-chord technique. This chapter also emphasizes the work's inherent heterophonic construct, in which harmony and counterpoint appear only as derivative embellishments of the "foreground melody." Chapter 4 gives an in-depth analysis of the piece's melodic content, with pitch and rhythmic durations (and, to a lesser extent, amplitude) considered as separate (though mutually influential) factors. Contour theory is introduced as an analytical tool in this chapter, and is used to demonstrate embedded levels of hierarchy through which fractal-like self-affinity is shown to permeate the structure of the piece. Chapter 5 gives a brief treatment of the composer's iv subjective handling of the "foreground melody" through orchestration (including the counterpoint and harmony derived therefrom) and his use of motives. This is followed by general conclusions, which correlate the larger picture of sonata form structure with the fractal-like connections found in the contours of the "foreground melody" at its various depths of reduction.

FOLLOWING FISH AND FRACTALS: SELF-AFFINE PATTERNS OF HIERARCHICAL FORMAL STRUCTURES EMBEDDED IN THE CONTOURS OF ROLF WALLIN’S NING by Benjamin Robert Bidwell A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy School of Music The University of Utah December 2018 Copyright © Benjamin Robert Bidwell 2018 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL Benjamin Robert Bidwell The dissertation of has been approved by the following supervisory committee members: Steven T. Roens , Chair 09/21/2018 Date Approved Miguel B. Chuaqui , Member 09/21/2018 Date Approved Michael W. Chikinda , Member 09/21/2018 Date Approved D. Michael Cottle , Member 09/21/2018 Date Approved Matthew Cecil , Member 09/21/2018 Date Approved and by Miguel B. Chuaqui the Department/College/School of and by David B. Kieda, Dean of The Graduate School. , Chair/Dean of Music ABSTRACT This dissertation is comprised of two parts: an analytical paper assessing the “fractal” quartet ning by composer Rolf Wallin, and an original composition for wind ensemble entitled Graven Images, based on the triptych of short stories with the same name by author Paul Fleischman. Chapter 1 is an expansion of a symposium lecture from 1989 by Wallin, following the same format and content, but with greater detail. It introduces the fractals and formulas that inspired Wallin, explains their mathematical functions, and presents the methodology by which he composed ning using two specific fractal formulas: the logistic equation, and a formula shared with Wallin by physicist Jan Frøyland. Chapter 2 takes an extensive look at Wallin’s use of the logistic equation to create the formal parameters of ning, then provides a detailed overview of the piece’s form, finding parallels with traditional sonata form. Chapter 3 explains how Wallin generated the “foreground melody” of ning (the mathematically constructed melodic line that serves as the piece’s backbone, which Wallin then “composed out” into the quartet) using Frøyland’s formula (as constrained by the limits of the formal parameters derived from the logistic equation), which in turn was modified by another fractal-inspired technique of his own devising: the crystal-chord technique. This chapter also emphasizes the work’s inherent heterophonic construct, in which harmony and counterpoint appear only as derivative embellishments of the “foreground melody.” Chapter 4 gives an in-depth analysis of the piece’s melodic content, with pitch and rhythmic durations (and, to a lesser extent, amplitude) considered as separate (though mutually influential) factors. Contour theory is introduced as an analytical tool in this chapter, and is used to demonstrate embedded levels of hierarchy through which fractal-like self-affinity is shown to permeate the structure of the piece. Chapter 5 gives a brief treatment of the composer’s subjective handling of the “foreground melody” through orchestration (including the counterpoint and harmony derived therefrom) and his use of motives. This is followed by general conclusions, which correlate the larger picture of sonata form structure with the fractal-like connections found in the contours of the “foreground melody” at its various depths of reduction. iv TABLE OF CONTENTS ABSTRACT .............................................................................................................................. iii LIST OF TABLES.................................................................................................................... vii ACKNOWLEDGMENTS ...................................................................................................... viii Chapters 1 ROLF WALLIN’S APPROACH TO FRACTAL-BASED COMPOSITION...................... 1 Rolf Wallin’s Aesthetic Philosophy of Mathematics in Music Composition ...................... 1 The Mandelbrot Set ....................................................................................................... 3 The Logistic Equation..................................................................................................... 5 The Frøyland Formula.................................................................................................. 11 Harnessing the Equations for Compositional Application .............................................. 16 The Focus of This Paper ............................................................................................... 21 2 FORM IN NING ................................................................................................................. 23 Background of the Composition .................................................................................... 23 Taking the Data to the Score ........................................................................................ 34 Comments on Form ...................................................................................................... 45 ning Prior to the Golden Section: “Exposition” and “Development”............................... 49 ning Following the Golden Section: “Recapitulation” and “Coda” ................................. 54 Conclusions on Form in ning.......................................................................................... 69 3 CREATION AND IMPLEMENTATION OF THE FOREGROUND MELODY........... 78 Applying the Frøyland Formula .................................................................................... 79 The Crystal Chord Technique ...................................................................................... 85 Heterophony in ning ...................................................................................................... 94 Analytical Approaches ................................................................................................ 100 4 STRUCTURAL ANALYSIS OF NING IN TERMS OF CONTOUR ............................ 102 Sectional Comparisons from a Contour Perspective .................................................... 102 Contour Similarity................................................................................................ 112 Emerging Patterns and Connections ........................................................................... 120 Contour Middleground (Analytical Excerpt from the “Floating” Section) .................... 124 Contours of the Maxima and the Minima ............................................................. 128 Contour Reduction for Pitch ................................................................................ 135 Contour Reduction for Duration .......................................................................... 139 Middleground Interconnections and Conclusions ................................................. 146 Contour Background: Connecting the Large and the Small......................................... 148 5 STRUCTURE REFLECTED IN THE SCORE/CONCLUSIONS ............................... 159 Observations of the Composer’s Personal Inflections ................................................... 159 Derivative Counterpoint (in the “Floating” Section) .............................................. 160 Derivative Harmony (Pitch-Class Set Analysis) (in the “Floating” Section)............. 165 Use of the Motive to Highlight the Crystal Scales and Flesh-Out Contour Patterns ................................................................................................................ 172 General Conclusions ................................................................................................... 179 A Final Thought ................................................................................................... 185 Appendices A FACSIMILE AND EXPLANATIONS OF NING DATA TABLES.................................. 189 B EXCERPTS FROM AN INTERVIEW WITH ROLF WALLIN .................................... 202 C GRAVEN IMAGES, SYMPHONY FOR WIND ENSEMBLE ........................................... 220 SELECTED BIBLIOGRAPHY.............................................................................................. 224 vi LIST OF TABLES Table Page 2.1. Formal Parameters in ning ................................................................................................ 30 2.2. Formal Sections in ning ..................................................................................................... 36 2.3. Formal Layout of Thematic Labels Tied to the Pitch and Rhythmic Parameters of Each Section of ning ......................................................................................................... 47 2.4. Schematic for Sonata Form Structure Aligned with the Schematic for Sectional-Length Groupings in ning ............................................................................................................. 71 3.1. Formal Parameters for the Frøyland Formula and Pitch Collections in ning....................... 84 4.1. COM Matrices for the Thirteen Pitches in Comparison Between the Pitch Contours of Precompositional Sections 17 and 25 ............................................................................. 115 4.2. COM Matrices for the Thirteen Rhythmic Durations in Comparison Between the Duration Contours of Precompositional Sections 17 and 25 ........................................... 116 4.3. CSIM Matrices for the Contour Points Being Compared in Tables 4.1 and 4.2, Respectively ................................................................................................................... 117 4.4. Final Depth Reductions for the Contours of the Formal Parameters of ning .................... 155 ACKNOWLEDGMENTS The contributions of many individuals were paramount in the realization of this dissertation. My deepest thanks are extended to Rolf Wallin, who graciously provided me crucial precompositional source materials and took the time to patiently explain to me the intricate process of his compositional methods; without his generous help, this project could never have been realized. I also must thank the proponents of contour theory, particularly Robert Morris and Elizabeth West Marvin, for the groundwork they laid in codifying a theory that gave me an effective and applicable way for presenting the heart of my thesis. My gratitude also goes out to the members of my committee, Miguel Chuaqui, Michael Chikinda, M. David Cottle, and Matthew Cecil, whose counsel, advisement, guidance, and patience were critical to my professional development. I give particular recognition to my committee chair, Steve Roens, whose constancy and commitment to me through the long journey is appreciated on a very personal level; his influence on my craft and writing is indelible, and his vested support of my work has been a dominant force in seeing me through to completion. I deeply thank my parents, my parents-in-law, and my family members without whose love, encouragement, and unfailing belief I could not have managed to accomplish what I did. I must especially thank my wife, Katie, who made significant personal sacrifice to give me the space and time necessary to bring this project to its conclusion. Her gesture of support and true love deserves acclaim beyond my power to deliver. I truly honor and cherish her for making this all possible. Lastly, my deepest gratitude is given to God, who orchestrated everything to fall into place, kept me going through the most difficult of times, and never failed me. CHAPTER 1 ROLF WALLIN’S APPROACH TO FRACTAL-BASED COMPOSITION Rolf Wallin’s Aesthetic Philosophy of Mathematics in Music Composition In a lecture1 given at the Nordic Symposium for Computer Assisted Composition in Stockholm in 1989, Norwegian composer Rolf Wallin addressed a concern he shared with many in the established music community, namely whether the recent cultural obsession with fractal mathematics was merely an unsubstantiated passing fad, or a potential tool of yet-to-bedetermined value for both composers and theorists: Chaos, fractals, Mandelbrot Sets, strange attractors, Cantor dust—these and some dozens [of] other magical words have totally invaded popular science magazines during the 80’s, accompanied by glossy colourprints [sic] of strange, disturbingly organical [sic] computergenerated [sic] shapes….So how about Music? Will fractal mathematics come to rescue in a time of apparent stagnation, this time in the guise of simple mathematical algorithms that yield delicious patterns that can be further processed, either as [a] succession of musical events or as complex frequency spectra? The attempts to find an answer to this [have] barely begun. In the field of music [,] the answers are not as selfevident [sic] as for the visual field. But this can actually turn out to be an advantage, if not technically, then at least aesthetically.2 Wallin’s interest in mathematical and computer-generated data applications as resources for his compositional work had been molding his personal aesthetic over the years, so to him, fractals, under the broader umbrella of chaos theory, seemed to be an appealing and natural next 1Rolf Wallin, “Fractal Music – Red Herring or Promised Land? or ‘Just Another of those Boring Papers on Chaos,’” rolfwallin.org, accessed Feb. 24, 2018, http://www.rolfwallin.org/ articles/c_5165b8ec1e98c045d02be0b3/. The current chapter is an expansion of this lecture. Information in this chapter is presented in roughly the same order and format as in the lecture, but with a great deal more in-depth explanation of mathematical concepts and compositional procedures. This is done with the express written permission of the composer. 2Ibid. 2 step. Previously, his work in this ilk dealt mostly with data from stochastic programs, but while this was useful for mapping forms for whole pieces, Wallin often felt that the data fell short of generating satisfactory microstructures, obliging him to write out the surface details of the musical foreground intuitively.3 Such “impurity” bothered Wallin, which is why fractals, with sophisticated formations even at microscopic levels, encouraged him to believe that engaging melodies, rhythms, and expressions generated in the foreground could be mostly (if not wholly) contained in the mathematics describing them. But Wallin was cautious. In the 1980s, fractals had been in the public consciousness for just a few years, still a “fresh discovery” that could only recently be printed and disseminated due to technological developments.4 The new and sensational quality of the subject, with its beautiful, artistic whimsy and near psychedelic quality of certain colored renderings, caused Wallin to be apprehensive about falling prey to a popular craze while unintentionally ignoring true musical substance congruent with his aesthetic. Fractals had excited the imaginations of many other equally intrigued composers, including Charles Wuorinen, György Ligeti, and Robert Sherlaw Johnson, to name a few.5 Wallin was leery of being just one among many, so he closely 3Ibid. 4Though fractals are readily found in natural phenomena, mathematical conceptualizations for self-similar objects didn’t really begin until the late 19th Century with the work of Weierstrauss, Cantor, Klein, and Poincaré. The formulas describing objects like this grew more complex over the century through the efforts of Koch, Sierpiński, Julia, Lévy, and others, but accurate visual depictions of these more complex descriptions began to surface only when computer-based technology had advanced sufficiently, which coincidentally came not long after the fabled mathematician Benoît Mandelbrot coined the term “fractal” in 1975. Heinz-Otto Peitgen and Peter H. Richter, The Beauty of Fractals (Berlin, Heidelberg: Springer-Verlag, 1986), v; Anthony Barcellos, “Benoît Mandelbrot: In his own words,” Mathematical people: profiles and interviews, ed. Donald J. Albers and Gerald L. Alexanderson (Wellesley, MA: A.K. Peters, 2008), 214. 5Wuorinen’s Natural Fantasy for organ and Ligeti’s Études pour piano, Livre 1 are examples of pieces considered to some extent to be fractal-inspired, involving compositional processes that evoke the ideas of fractals and chaos theory in some way, but not necessarily with literal application of any specific fractal formulae. On the other hand, Johnson’s Fractal in A Flat (which later became the basis for the scherzo movement of his Northumbrian Symphony) was composed from a 3 investigated the math behind the marvel, looking for ways that the numbers could be construed sonically or be applied to musical form and structure, and still arrive at an end result with acceptable emotional (as well as logical) appeal. The Mandelbrot Set The most fabled fractal, the Mandelbrot set, was a logical starting point for Wallin (see Figure 1.1). This fractal is drawn on the complex plane, which is a two-dimensional layout of Cartesian coordinates (even though fractals are not technically two-dimensional) where the x axis a. b. Figure 1.1: The Mandelbrot Set. a. Colored representation of the set.6 b. The set relative to Cartesian coordinates.7 more deliberate manipulation of mathematical formulae with the aid of technology. Michael Bolstein, “Charles Wuorinen,” AllMusic Guide, AllMusic, accessed Feb. 24, 2018, https:// www.allmusic.com/artist/charles-wuorinen-mn0000684647/biography; Wallin, Fractal Music; Michael Frame and Amelia Urry, Fractal Worlds: Grown, Built, and Imagined (New Haven: Yale University Press, 2016), 99-105; Robert Sherlaw Johnson, “Composing with fractals,” in Music and Mathematics: From Pythagoras to Fractals, ed. John Fauvel, Raymond Flood, and Robin Wilson (Oxford University Press, 2003), 163-72. 6Image accessed July 28, 2017, https://stackoverflow.com/questions/10837244/ mandelbrot-set-vertical-display. 7Image by Geek3 - Own work, CC BY-SA 3.0, accessed July 28, 2017, https:// commons.wikimedia.org/w/index.php?curid=5702420. 4 represents real numbers in the usual fashion, while the y axis represents multiples of the imaginary number i, where i = √–1. All points on the plane are called complex numbers, denoted by the variable c and described as c = x + yi, plotted on the plane at (x, y) coordinates. The Mandelbrot set is determined via an iterated functional equation, that is, an equation where its result, calculated from some predetermined starting point of its principal variable (in this case, z), is then reinserted into its principal variable again for another new result, and then repeats this process infinitely. This set is the collection of all complex numbers (c) that are bounded under iterations of the equation: fc(z) = z2 + c (1.1) with iterations beginning at z = 0 (bounded meaning that the iterations will never diverge towards infinity, that they will stay within a given value range—in the case of the Mandelbrot set, |fc(z)| ≤ 2). The equation can be rewritten as an iterated function (and in terms of the complex number coordinates): zn+1 = z2n + x + yi (1.2) for z0 = 0. The Mandelbrot set is entirely contained between -2 and 2 on both axes, circumscribed by a circle with a radius of 2 centered at (0, 0) (Figure 1.1b). In colored visual depictions of the set (such as in Figure 1.1a), the black area indicates values of c within the set, but it is in the varying shades of color immediately surrounding this area that things get truly interesting. This “halo” constitutes values of c that are nearly part of the set, some of which may take upwards of hundreds of iterations to finally escape the bounds of |fc(z)| ≤ 2. Different shades of color are arbitrarily applied to different values of c to indicate the number of iterations required to escape the aforementioned bounds. And when a particular value of c is computed and visually scaled to an astonishing degree (magnifications of powers upwards of 10225 are available in online videos, and more keep coming out), the relative placements of these colors reveal breathtaking images of surreal and recurrent complexity: curled “seahorse tails,” swirly Julia-set islands, chandelier-like 5 whirlpools, nebulas, and several instances of smaller replicas of the larger Mandelbrot cartioid, bulb, and antenna (and myriad other replications of extremely sophisticated yet surprising form, self-similar in the way that fractals are meant to be). The topic gets even more interesting with variations on the Mandelbrot equation. “Multibrot” sets use larger exponential powers of z, transforming the number and orientation of the bulbous set regions. “Buddhabrot” images render the Mandelbrot set in such a way as to color points by how often they recur within the many iterations of different values of c in the iterative creation algorithm. And the “Mandelbulb” and “Mandelbox” expand the computations to a Cartesian space of three or more dimensions, making for curious coral-like pottery shapes (or cubes) that bend and twist through rendered video journeys zooming both in and through the object. Any of these fascinating variants might whet the appetite of composers wondering how such marvels could be replicated or at least approached sonically, or possibly lead theorists to conjecture whether such structures (seeing their ubiquity in mathematics and in the physical world) are found inherently in music. Credibility in this notion arises from the purveyance of shared attributes between fractal and musical structures; in fact, one must wonder whether fractals could be a subconscious guiding factor behind human creative thinking.8 The Logistic Equation Wallin recognized the remarkable potential of fractals for music, but was loathe to go in a direction already bedeviled by “computer freaks,” art hacks, and the worn-out aesthetics of tourist brochures and science fiction.9 Rather than pursue the Mandelbrot set directly, Wallin instead took a tangent to look at another tenant of chaos theory, closely related to the Mandelbrot set, but 8Nicoletta Sala, “Fractal Geometry in the Arts: An Overview Across the Different Cultures,” in Thinking in Patterns: Fractals and Related Phenomena in Nature, ed. Miroslav M. Novak (River Edge, NJ: World Scientific, 2004), 178-85. The idea that human thought processes are subconsciously governed by fractal-based principles is also promoted by the journal Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society. 9Wallin, “Fractal Music.” 6 with an altogether different purview: the bifurcation diagram of the logistic map. In 1838, Pierre François Verhulst published an equation intended to describe a model for population growth (or dynamics) under certain prescribed environmental conditions. This equation came to be known as the logistic equation, and like the equation describing the Mandelbrot set, it too is iterative in nature. The logistic map uses a simplified version of Verhulst’s original logistic equation to plot the behavior of population growth: xn+1 = rxn (1 – xn) (1.3) where xn represents the ratio of an existing population to a maximum possible population, with 0 < xn < 1, and r stands for the intrinsic growth rate.10 When the equation is iterated from some initial value of x for a given value of r, the value of x eventually converges onto a predictable value, set of values, or value range, regardless of the initial value of x. Such values are aptly called attractors. Depending on the value of r, iterations of x will eventually arrive at a population ratio that is either stable (a single attractor) or vacillating (consistently changing between multiple attractors), with vacillations themselves being either stable or chaotic. The parameter r (intrinsic growth rate) is generally only discussed for values 0 < r ≤ 4, since iterated values of x where r > 4 tend to go outside of the interval 0 < xn < 1 and diverge. The attractor values of x can be plotted graphically on the vertical axis against the r values on the horizontal axis, as in Figure 1.2. Note that for values 0 < r < 1 (not shown in Figure 1.2), the population eventually dies (i.e., the iterations for x approach 0), while for 1 ≤ r ≤ 3 (partially shown in Figure 1.2), the population ratio will come to the value of (r – 1)/r, which happens very quickly for 1 ≤ r < 2 and with more initial fluctuation for 2 ≤ r ≤ 3, being especially slow at r = 3. At this point, things begin to change: the values for x eventually settle on an oscillation between two values for 3 < r ≲ 3.44949. This split of values is the initial bifurcation, that is to say, a fundamental qualitative 10The original logistic equation given by Verhulst reads: dN/dt = rN (1 – N/K) where d indicates a derivative function, N(t) is the number of individuals present in a population at time t, r is the intrinsic growth rate, and K is the maximum capacity of the population’s environment. 7 Figure 1.2: Bifurcation diagram of the logistic map (partial view) showing self-similarity (in red). change in the behavior of the equation’s results due to a small shift in the value of the intrinsic growth rate. This split-of-two splits further into four, then eight, then sixteen oscillatory x values, with each split occurring more closely than the previous one.11 After r ≈ 3.56995, the splits become exponentialized at such a dense rate that the result is chaotic. Within the chaos (3.56995 < r ≤ 4), however, there are certain visible patterns and anomalies of note. Firstly, as the value of r increases, the outer edges for the values of x appear to widen in a consistent, nonchaotic way, nearly linearly in the upper limit, and with an apparent quadratic curve in the lower limit. Secondly, sinusoidal curves appear to emanate from the split- 11The ratio between the lengths of successive splits eventually approaches something known as the first Feigenbaum Constant, approx. 4.6692. This ratio is also relevant to the Mandelbrot set, where the radii of the circles appearing along the center axis of symmetry moving in the negative-x direction are likewise proportional, with each circle being a self-similar replica of the former getting infinitely smaller. 8 of-eight, which are more visually pronounced (darker) and clearly traceable against the chaotic backdrop due to value convergences (there also appear to be less-pronounced curves emanating from the split-of-sixteen, but these seem to eventually merge with the other bold curves from the split-of-eight). And finally, there are several slats, or “windows,” where the behavior briefly exhibits a moment of order.12 Under intense magnification, one can see within each of these slats that the oscillations occur at first between a few values, which then quickly split and double as seen before, though there is hardly any space between such splits to effectively make them out except in the largest slat (starting at r ≈ 3.82843). In this wide gap, the x values oscillate between three points for a significant amount of space, but then the values soon bifurcate again, splitting to six, then twelve, and so forth. The slats also exhibit a relationship to one another. Notice the three largest slats (or four, if the vertical gap found within the first slat is viewed as making two slats out of one) starting at r ≈ 3.63, r ≈ 3.73, and r ≈ 3.83. The first slat begins with six points of value-oscillation, or, put another way, three points of oscillation within each slat on either side of the gap. The second slat starts with five points, and the third (as already pointed out) begins with three. Now, by zooming in on either part of the first slat, top or bottom, similarity to the feature just described becomes apparent: Two smaller slats precede it, the first of which also has a gap, and both of which have the same points of value-oscillation as the larger picture, six and five points, respectively. In fact, the very shape of the figure found within the approximate range of 3.37 ≲ r ≲ 3.67 and 0.72 ≲ x ≲ 0.91 looks like the whole picture (shown in the large red box in Figure 1.2). And more of this type of replication can be found with further zooms (as seen in the smaller red boxes in Figure 1.2). Such self-similarity, then, makes the bifurcation diagram a fractal. Most interesting of all is the correlation of the bifurcation diagram with the Mandelbrot set (see Figure 1.3). The two images can be scaled and lined up with one another, if one or the 12These windows of stability are also called “islands of stability” or “Arnold tongues” (named for the Russian mathematician Vladimir Arnold). 9 Figure 1.3: The bifurcation diagram of the logistic map aligned with the Mandelbrot Set. other is rotated 180°. The bifurcation split-points concur perfectly with instances where the Mandelbrot set “pinches” around the x-axis.13 Wherever the diagram shows open space, the Mandelbrot values expand to incorporate a vast array of imaginary values for c (the circular “bulbs” on the horizontal axis); wherever it is thick with chaos, the Mandelbrot tends towards c values that approach real numbers converging on the horizontal axis (y approaches 0). The values of r from the diagram correspond to the real-number values of c directly via the transformation formula c = r/2(1 – r/2), for all values 1 ≤ r ≤ 4. The math of chaos theory attracted Wallin because of the way that it was revolutionizing science: “Fluid mechanics, geology, medicine, meteorology, in most fields many unsolvable 13Take care to note that x in the bifurcation diagram stands for the population ratio, plotted on the vertical axis, and is not to be confused with x in the complex plane of the Mandelbrot set, where x indicates real numbers along the horizontal axis. 10 problems have been solved by applying some part of this manyfaceted [sic] and bewildering bag of theories.”14 To work intimately with such math, he felt, was a more tenable and less reactionary pursuit than composing music simply “inspired” by “glossy colourprints [sic]” that had been dubiously manipulated for public appeal.15 And he felt that applying chaos theory to music could be done more legitimately than it had yet been done.16 Indeed, to Wallin’s thinking at the time, if a truly innovative and aesthetically satisfactory musical result could not come directly from the math and data of chaos and fractals (especially since such beautiful fractal images did), then “there [was] no point in using ‘fractal’ as a trendy tag on one’s music.”17 To Wallin, there was great potential in Verhulst’s logistic equation for both musical form and content. Enlisting the help of programmer Øyvind Hammer (who also deserves credit for these findings), Wallin began his investigations into the ways this and other chaotic equations could be interpreted and/or integrated into sound. His logical starting point for musically interpreting the equation was to have x represent pitch, with the range of 0 < x < 1 being scaled to some arbitrary range of pitches, and create a melodic sequence of pitches generated by iterations of the formula. Naturally, for this process, Wallin experimented with values for r within the “chaotic” ranges of the bifurcation diagram (see Figure 1.2), since values from other ranges would eventually create repetitive pitches and/or motives ad infinitum. Remarkably, the chaotic patterns turned out to be less chaotic than expected—the iterations only exhibited about three 14Wallin, “Fractal Music.” 15Ibid. 16Ibid. György Ligeti was one composer who pioneered the use of such ideas in music. But for all his admiration of Ligeti, Wallin felt somewhat underwhelmed to learn that Ligeti had settled on taking the underlying “idea” behind chaos theory as an inspiration for his composing, rather than attempting to stringently calculate his musical plotting (and this after a belabored and meticulous precompositional analysis of the math). Though Wallin acknowledged that the true value of Ligeti’s music is not to be judged by the means of its creation, this was still unsatisfactory to him. 17Ibid. 11 characteristic motivic ideas: an alternating volley between high and low pitches, pitches meandering or looping around a mid-range pitch, and moments of pitch repetition on a midrange pitch. The iterated melody would change from one of these general states of pitch-motion to another at differing intervals of iterations, never staying in any one state for very long.18 Though this created some short-term surprise, there never seemed to be any real musical development of interest due to the self-similar recursive nature of the fractal. The result was remarkably similar when attempting to alter the purity of the iterative process by changing the value of r along with each iteration by some increment (a more direct interpretation of graphical data from the bifurcation diagram into musical notation, with pitch vertical and time horizontal, albeit for only one pitch at a time). These experiments did not provide enough useful material for musical implementation (not to mention being devoid of rhythmic variety, dynamic contrast, and formal scope). Wallin felt that there was still more there to be explored and exploited from this equation toward reaching his aesthetic goal, but likely with some kind of additional external input. The Frøyland Formula Wallin turned then to another formula suggested to him by physicist Jan Frøyland from the University of Oslo, similar in nature to Verhulst’s logistic equation but with dimensional augmentation.19 In this version, the population ratio x is given two or more dimensions (Wallin 18Generally, the pitch repetition motive was the least common and shortest lived. This motive would quickly evolve into the meandering motive, which also did not last very long. In turn, the meandering motive would evolve into the high/low volley motive, which was by far the most common and longest lasting of the three. But the important thing to note is that in all three cases, the general pitch motion was usually alternating between ascent and descent (though for certain values of r [those closer to the gap where the attractor values oscillate between three points] there occasionally occurred moments with two consecutive ascending pitches). This behavior of iterative change in pitch will be revisited as a feature of structural significance later (see Chapter 4, pages 150-52). 19This formula will be referred to repeatedly throughout the text as the “Frøyland formula.” Though it is possible that this formula originated elsewhere, Jan Frøyland was the source from whom Wallin obtained the formula, and is the only source whom Wallin cites, so it will likewise be appropriated to him in this dissertation. 12 ultimately decided to use three: x, y, and z) that mutually feed back into one another in interlocked equations, while the parameter rate r is replaced by two parameters, c and d, where c < 0 and d > 0. All these equations computed simultaneously together constitute a single multidimensional iteration. A few other coefficients and operations are different in this formula as well: xn+1 = 2d(yn + zn) + 2xn(c + xn) (1.4) yn+1 = 2d(xn + zn) + 2yn(c + yn) zn+1 = 2d(xn + yn) + 2zn(c + zn) In this scenario, x, y, and z behave as three different interdependent populations, analogous to wolves, sheep, and grass within a hypothetical contained environment, where the population of each will grow or shrink depending on the present size of the other two, and which all depend on the initial sizes of each. The rates of growth/shrinkage are also affected by certain “environmental” factors described by the parameters c and d, analogous to climate conditions, such as humidity or temperature. As different values of r reflect different degrees of order or chaos in Verhulst’s logistic equation, different coordinates in the (c, d) plane of the Frøyland formula cause the iterated values of x, y, and z to shift by different degrees, triggering behaviors somewhere along the predicable/chaotic spectrum. All these factors provided a vastly more diverse body of data, and for Wallin, this was much more suitable as musical source material for three reasons: 1) extra variables meant that more foreground musical parameters could be defined by the math (Wallin elected for pitch, duration/rhythm [weighted exponentially], and dynamics to be controlled by x, y, and z, respectively), 2) the variety of formations plotted by the data provided more motivic possibilities, and 3) the flow of the computer’s sonic translation of the data is much more striking as “human,” nuanced with gestures and phrasing familiar to acoustic performance. Though there are several dimensions involved, two-dimensional graphical representations can show only a pair of parameters at a time, but a glance at just one example (provided by Wallin himself) shows a much richer and more intricate outlay of data (Figure 1.4). It is important 13 Figure 1.4: Example bifurcation diagram of part20 of the Frøyland formula.21 to note that here Wallin is using an asymmetrical version of the aforementioned multidimensional formula, in which the zn parameter is omitted in the first of the three equations: xn+1 = 2d(yn) + 2xn(c + xn) (1.5) yn+1 = 2d(xn + zn) + 2yn(c + yn) zn+1 = 2d(xn + yn) + 2zn(c + zn) This favors the y and z parameters with more processing and variability than the x parameter. At first, this was a mistake due to a programming error, but after hearing the “organic” results, Wallin decided that he liked it better. This is the version that he ultimately went with in crafting most of his fractal pieces. Just as the bifurcation diagram of the logistic map shows the attractor values of iterations of x for a given value for r (Figure 1.2), attractor values of iterations of x (as affected by the concurrent evolutions of y and z) appear in like fashion for a given value for c in the Frøyland formula (Figure 1.4), within the range –0.57 ≥ c ≥ –0.8, where d is constant at 0.2. What is not shown in Figure 1.4 is what is simultaneously going on with y and z, which one might imagine to 20The numerical reference scales along the axes are absent in this figure, but just as with the bifurcation diagram of the logistic map, the values for x are plotted along the vertical axis, and range between –0.36 and 1.25, with –0.36 at the bottom and 1.25 at the top, and 0.0 indicated by the solid horizontal line seen traversing the length of the figure. The values of c along the horizontal axis range between –0.57 and –0.8, with –0.57 at the left and –0.8 at the right (opposite of the normal left-to-right orientation of increasing value). 21Diagram created by Rolf Wallin. Wallin, “Fractal Music.” 14 have similar imagery but with completely different data sets. Again, similar patterns of chaos and slots of order appear, but with more complexity and intricacy. Near-self-similarity (that is, selfaffinity) is manifested around the line drawn through the center of Figure 1.4 (where x = 0), but the bottom portion is a sort-of “diminished” version of the upper portion. This is fascinating, but Wallin realized that there would be a far more dynamic outcome in the formations (and hence the corresponding sonic patterns) when d changed along with c. He plotted a linear progression for c and d between a chosen start point and a chosen end point within the (c, d) plane, along which the values for c and d were a good representative mix of different orderly and chaotic states. From this, Wallin could process a string of formula iterations, by either 1) selecting a single point along this line and iterating the formula with a stable set of c and d values, or by 2) causing c and d to change their values linearly between the start and end points along with each formula iteration, with the rate of change from iteration to iteration being determined relative to a proportional scaling for the value of y (exponentially weighted) in each iteration.22 Figure 1.5 presents three sample visual representations of points found along Wallin’s linear continuum of c and d values (each image corresponds to a separate, single set of values for c and d) that he might have selected for passages in his compositions. One might imagine how these forms would gradually morph into one another as the values of c and d shift across this continuum. These images capture possible attractor values for iterations of x, y, and z (using the asymmetrical version of the Frøyland formula, equation 1.5), and though they are only two-dimensional representations of a three-dimensional space as seen from just one vantage point, they each exhibit a wealth of self-affine formations (which meant a wealth of musical possibilities for Wallin to explore and extract from strings of iterations generated out of various seed values for x, y, and z). 22Wallin used the former approach for his compositions more often (such as with his pieces ning and Onda di ghiaccio), but once used the latter approach in the final “movement” of his piece for percussion, Stonewave. 15 a. b. c. Figure 1.5: Images of attractor values for x, y, and z in Cartesian space23 for three different combinations of values for c and d in the Frøyland formula.24 23For comparative purposes, these pictures are analogous to capturing the points of x along different vertical slices (separate, single values of r) in the bifurcation diagram (Figure 1.2). The scaling of values for x, y, and z in the images of Figure 1.5 is unknown, as are the exact values of c and d associated with each image. They are not intended to represent any specific settings used by Wallin in his compositions, but are merely presented here as a small sampling of the gamut of formations from possible c and d value combinations. However, Figures 1.5a and 1.5b are very similar to illustrations provided in the liner notes for Wallin’s album move (1994), presented there in conjunction with the pieces Stonewave and Onda di ghiaccio, respectively. Rolf Wallin, move, liner notes, Hemera Music HCD 2903. 24Still images were taken from video created by Simen Svale Skogsrud to accompany the final “movement” of Stonewave, in which the values of c and d are continually changing along a linear continuum. Used by permission. 16 Wallin also had to make other programming adjustments, including setting upper and lower limits for x, y, and/or z, “normalizing” the range of values for each variable to be between 0 and 1, and either using the absolute values of negative numbers or dismissing them entirely, all in order to avoid unconnected leaps between extreme pitches, durations, and amplitudes. Wallin experimented with many different initial values, iteration counts, and different formulaic constrictions, and found that most of the melodies generated by the computer were very intriguing, and could viably be used and transcribed as conventionally notated music for acoustic performance. Harnessing the Equations for Compositional Application Still, as raw material, the computer’s output for the asymmetrical Frøyland formula needed tempering, but in recognizing this, Wallin felt that the musical solution could still be obtained, at least primarily, from the math. He then developed a compositional process for harnessing the data systematically, in which he combined both the original logistic equation (equation 1.3) and the asymmetrical multidimensional formula (equation 1.5) in tandem, with the former controlling the overall form and the latter determining the minutiae of pitch, rhythm, and dynamics (as constrained by the parameters of the former). Wallin began by first determining a duration for the piece he was to compose, over the course of which he would set up the formal constraints to be produced with the logistic equation. The next step was to make preliminary decisions about the intrinsic growth rate parameter r in that formula: its beginning and ending values (the range to correspond with the piece’s duration), the direction of progression (positive or negative), and a methodology by which the value of r should change from iteration to iteration of x relative to the duration of the piece (for example, linear equidistant increments, proportional increments, or some other systematic change). Each new value of r would then represent a new formal section or division within the piece. 17 Wallin preferred to have the equation itself determine the changes in r. By scaling the range of 0 < x < 1 with some determined range of temporal durations (of his choosing, generally in terms of seconds), Wallin could choose an initial value of x to coincide with the starting value for r (generally, but not necessarily, from the pool of attractor values found there), which would then give him the location for the value of r for the next iteration of x. In other words, the temporal value corresponding with the initial value of x would be matched to a change in r proportionate to the piece’s total duration corresponding to the continuum of r values (between its predetermined beginning and ending values). Each subsequent iteration would then change the value of r accordingly, giving the piece sectional divisions of varied but directly related lengths. The number of sections (Wallin used the term “fields”) in the piece was therefore determined once the entire duration of the piece was “filled up” by the cumulative iterated values of x.25 Wallin then had to determine additional initial values of x in similar fashion to define other musical parameters scaled to x values in some way. These parameters (termed “criteria” by Wallin) might include centric melodic path, pitch range around the melodic path, pitch collections to be used, average note durations, rhythmic variability, global dynamic plans, timbre (orchestrational) plans, the degree of order/chaos in the musical foreground, and other parameters limiting the computed iterations of the asymmetrical Frøyland formula. Different initial x values meant that each of the parameters would follow unique iterative trajectories as r 25The description of the method for deriving sectional durations given in this paragraph is, according to Wallin, the manner in which the sectional durations for ning were derived. However, by following this prescription, the iterations for most initial values of x tend to follow a pattern of “increase-decrease-increase” at first, but soon give way to the tendency to simply “increasedecrease” as the values of r shift further into the chaotic regions of the bifurcation diagram. Given this behavior, the consistent “increase-decrease-increase” pattern seen in perpetuity with the sectional durations in ning (described in greater detail in Chapter 2) gives room for doubt to Wallin’s claim. It appears more likely that the sectional durations were derived from a series of iterations using a constant value of r near the “split-of-three” in the bifurcation diagram (probably 3.855, as insinuated later in this chapter) rather than a changing value. But this is nearly impossible to verify, since (due to the nature of chaos) the tiniest variances in initial values can have profoundly different outcomes a few iterations later. So, though it appears unlikely on the surface, Wallin’s claim could very well be correct after all. See Figure 2.2, footnote 42, Chapter 2, page 28. Also see pages 150-52 in Chapter 4, including Figure 4.30 and footnotes 151 and 155. 18 changed. At times, these would increase and decrease in phase with one another as iterated values converged and then at other moments slip out of phase as values diverged in opposite directions, due to the degree of chaos given by different values of r. Playing with different initial values in the computer to come up with a compelling sonic result would put Wallin through a tedious process of trial and error.26 As part of honing in on these choices, Wallin’s next step was to temper the data values by weighting each parameter in a way that met two objectives: first, so that the data could be useful in acoustic performance by avoiding impractical sonic extremes (such as pitches too high or dynamics too loud), and second, so that the general trajectory of the data from iteration to iteration could bias a logical, global shape for the form preconceived by the composer. This skewing of the results did not change the output of the formula iterations themselves directly, but rather changed the relationship of the iteration results relative to the scaling for the parameter in question. By adjusting the maximum and/or minimum possible values of the parameter’s scale to new values of x (maximum < 1 and/or minimum > 0) for certain values of r along the bifurcation diagram, the resultant value of x for such a given value of r could be pulled to a higher or lower corresponding value for the parameter. Over the breadth of the span of r preselected by the composer, this value skewing could be done in a gradual way, either by applying a logarithmic function or by setting flexible outer limits to the data set and scaling the results, thus “shaping” the graphical representations. While this procedure bent the “rules,” it was a necessary compositional decision for practical and logical (formal) reasons, but it nevertheless maintained the integrity of Wallin’s mathematically driven ethic. Figure 1.6 shows what happens in graphs undergoing such transformations. With the data determined, Wallin could then map out the formal sections of the piece, identifying each of his so-called “fields” by their signature sonic behaviors and marked differences 26Just as in the case of traditional composing, beginning the process is the hardest part. Wallin describes this part of the process in Appendix B, page 206. 19 a. b. Figure 1.6: Examples of skewed and reversed sections of the bifurcation diagram of the logistic map.27 (Compare with Figure 1.2 in the range 3.550 ≤ r ≤ 3.855.)28 a. A sinusoidal skew collapsing the values of x, maximally (by about half) near r ≈ 3.703, and minimally at r = 3.855 and r = 3.550. b. A sinusoidal skew collapsing the values of x, maximally (by about half) at r = 3.855 and r = 3.550, and minimally near r ≈ 3.666 27Diagrams 28In created by Rolf Wallin. Wallin, “Fractal Music.” both Figures 1.6a and 1.6b, the original bifurcation diagram of the logistic map has been flipped horizontally (to show a negative direction of progression through values of r) and truncated to only show values for r between 3.855 and 3.550 (from the widest slat to the point just beyond where the chaos meets bifurcation process). Note that these graphs contain the entire continuous gamut of attractors from this section of the bifurcation diagram, rather than showing a smattered path of points from successive iterations of x at incremental linear shifts in r (which would merely appear as a few plot points strewn across the graph, and reflect the results of only one initial x value rather than several). This allows the relationship to the bifurcation diagram and the realms of total possibility for the parameters to be seen much more clearly. If the span of r in both figures is thought to represent the duration of a hypothetical piece, it is easy to make general predictions about how a musical parameter might behave over the course of the piece. Figure 1.6a shows a hypothetical skew of values in which the values around the center of the piece tend toward lower values, which could translate into lower pitch tessitura, softer dynamics, or shorter rhythms (depending on which musical parameter is defined by this skew curve) around the middle of the piece as compared to either end. In like manner, a wide/high tessitura, large range of dynamics, and varied rhythmic durations might be possible around the golden section of a piece with the superimposed shape of Figure 1.6b. (In actuality, these figures are pulled from the precompositional materials for Wallin’s Onda di ghiaccio, with Figure 1.6a representing the durations of formal sections in the piece and Figure 1.6b representing the change in global dynamics throughout the piece.) 20 in character. Once such data maps were charted, and his layout of the piece’s form in place, he could then process the Frøyland formula through this form, skewing its results in turn (just as the general outlining shapes skewed the parameters of the logistic equation). After making the final necessary adjustments (as described on page 16), he could generate a fantastically intricate melodic tapestry, and then finally begin a transcription/interpolation of the data into standard musical notation, and compose out the finer details (not already predetermined by the parameters in either formula). This process, being what Wallin describes as “a continuous dialectic between systematic calculations and his own musical intuition,”29 gives his music a distinctive signature style: organic, varied, connected, and astonishingly “human” in its depth of expression.30 Wallin himself was in large part attracted to computer music because of the element of surprise, but was particularly gratified that, in the case of these formulas and others he toyed with from chaos theory, surprises arose from natural and consequential connections rather than randomness: What intrigues me about the dynamic systems is that the order and the disorder have the very same origin, not like in classical stochastical [sic] computer music where Euclidian geometry has been ‘ragged’ by a random generator, a prosess [sic] that can be likened to mixing oil and water. With chaos theory, the gap between total order and entropy is about to be filled in, and exactly this can turn out to be interesting for musicians. After all, this gap has been our playground for some thousands of years....31 29Hallgjerd Aksnes, “Rolf Wallin – Biography {extended},” rolfwallin.org (updated by Chester Music, 2002), accessed Feb. 24, 2018, http://www.rolfwallin.org/about/ c_515a0b4f1e98c00ef89091af/. 30It is important to note here that the ideal of composing completely with mathematics was never fully attained (nor entirely desired) by Wallin. To a degree, he exercised the “impurity” of his own intuition in writing his pieces, both in setting up his precompositional constructs (in his choices of initial values and mathematical manipulations for formal constraints) and in “fleshingout” the raw musical material generated by the mathematical output (in his orchestration, embellishment, harmonic thickening, and other kinds of musical coloring). Even so, Wallin was very careful to not disregard the mathematical output, always allowing it to speak through his music. Any alterations made to it in the notation stage were as few and as slight as possible, done only according to certain preconceived “styles” that he permitted himself to use. 31Wallin, “Fractal Music.” 21 Wallin has implemented this process in many of his pieces: Onda di ghiaccio (written for the Oslo Sinfonietta in 1989), ning (for oboe/string quartet, 1989/91), Stonewave (for percussion, 1990), Chi (for orchestra, 1991, later withdrawn), Solve et coagula (for large ensemble, 1992), and Boyl (for chamber orchestra, 1995). Other fractal/mathematical influences are evidenced in other works (of particular note is the relatively recent composition for orchestra Many Worlds (2010), which features an integrated video depicting multidimensional Mandelbulb formations as part of an immersive experience). From the formal integrity, rich detail, and surprising emotional impact demonstrated by each of these works (as well as their garnered critical acclaim32), it appears that Wallin feels he has successfully demonstrated the usefulness of fractal geometry and chaos theory as compositional tools, not merely as sources of inspiration or simply because “everyone else is doing it.”33 The Focus of This Paper The usefulness of mathematics in music has been unquestionable since time immemorial. It continues to be compelling, though, because the breadth of mathematics continues to expand, revealing new understandings and descriptions of our universe and ourselves. It is curious to note 32Geir Johnson, “Rolf Wallin,” rolfwallin.org, accessed Feb. 24, 2018, http:// www.rolfwallin.org/articles/c_516588e71e98c02697ada306/; Hild Borchgrevink, “Rolf Wallin’s Playful Languages,” rolfwallin.org (Huddersfield Contemporary Music Festival, 2010), accessed Feb. 24, 2018, http://www.rolfwallin.org/articles/c_51704ec91e98c0265da0786e/. 33Wallin admits that his compositional technique has evolved since his “fractal period” of the early 1990s, and confesses to have written pieces more intuitively in later years, which, in principle, runs counter to the aesthetic premise he initially sought so vigorously to purport. This is not to say that he grew disenfranchised or bored by fractals, but that he felt, as many artists do, that there are other facets of life to discover, explore, and upon which to experiment. Yet even with such relinquishment, he continues to regularly draw upon the principles and formulas he learned then, still employing mathematical processes in his precompositional and conceptual sketchings…it proved to be successful enough that he never totally abandoned it. It was, after all, a big part of his musical identity. As an addendum, however, it should be noted that Wallin has once again taken up the compositional techniques from his fractal period with his new violin concerto Whirld (2018). 22 the ways that new mathematical discoveries are shaping not only the ways that music is composed, but also the ways in which it is understood. In an effort to approach a deeper understanding of such applications, a detailed look at a sampling from Wallin’s oeuvre may prove useful. It is the scope of this paper to analyze one of the aforementioned works, ning, by both traditional and innovative means, making connections with well-known musical structures (including form and structural hierarchy) as well as with fractal geometry. Significant consideration will be given to the structural patterns inherent in both the logistic equation and the Frøyland formula, and how such structure is manifest in both the raw and the final musical material that make up this particular piece. The goal is to come to conclusions about how successful Wallin’s compositional process is from a structural as well as an aesthetic standpoint (depending on how well both musical function and fractal-connections can be demonstrated in ning), and along the way shed some light on deeper, more universal questions about the relevance of fractals and chaos in the creation of music and in music theory applications. CHAPTER 2 FORM IN NING Background of the Composition When the Ensemble Borealis of Oslo commissioned ning in 1991, Wallin elected to use only four of its eight musicians: an oboe/English horn player pitted against a violin, viola, and cello. Wallin’s decision for timbre, uncommon but not atypical, was based primarily on his personal taste and on the extended techniques available from the instruments.34 The striking contrast between the sounds of the double reed and those of the strings leads the listener to believe that most of the time the strings work as a single entity, i.e., as a single counterpoint or accompaniment to the oboe. This idea may not have been Wallin’s initial intention (granted there are several moments where the voices are either all four distinctive or all four homogenous), but even he, listening to it in hindsight, believes the work to function like a concerto, in much the same way as many of Mozart’s quartets.35 The concerto-like behavior of ning provides a commentary on the mathematics behind its musical structure: the different iterations of dimensional feedback (through the process of the tripartite Frøyland formula described in Chapter 1) create musical patterns that are highly suggestive of (and readily lend themselves to) behaviors associated with concerti. Notions such as call and response, repetitions, solo with accompaniment, tutti unisons, polyphony, and 34See 35In Appendix B, page 216. fact, Wallin intends to rewrite the piece as a concerto for oboe and string orchestra, not just because that’s how it works, but also “because it’s so complex…[it hasn’t been] played successfully without a conductor, [which] looks kind of silly [with a chamber group].” See Appendix B, pages 216-17. 24 motifs/motivic development seem to pop out of the fractal iteration sequences, and Wallin composes the layered voices of the piece in direct response to these cues. A noteworthy point to stress in light of such “traditional” formations is that Wallin is not trying to purposely evoke the past in the same way as neoclassic or neoromantic composers do. Rather, these behaviors are by and large embedded in the calculations themselves, and his work as composer is simply to extrapolate them.36 Wallin states: …in my work with the mathematical phenomena called fractals I have encountered similar ghosts emanating from my pure algorithmic clockwork. The small, innocent fractals do not have the slightest idea about which musical patterns that have existed in various styles throughout the centuries. Still some phrases occur that make the puritan in me blush and turn on all the alarms. Many times I’ve felt like an explorer crossing jungles and mountains just to end up in a place that disturbingly resembles his own back yard. The lighthearted consonance of my oboe quartet ning, one of my most ‘scientific’ and predeterminative [sic] works so far, is a far cry from the dissonant angularity of the 50’ies and 60’ies [sic]. Yet this is, in my opinion, just a superficial difference. Independently of its surface I consider ning a truly modernistic piece, heavily dependent on techniques and concepts developed in ‘classic’ modernism.37 Changes in the x, y, and z variables of the Frøyland formula occur in tandem, and the values of each weave in and out with one another between moments of synchronicity and independence. After being tempered (by the parameters fashioned from the logistic equation), melodic sequences from this formula exhibit shifts between conjunctive and disjunctive behaviors. This opens the way for interpolating the data into the four voices of the quartet by reflecting these behaviors accordingly, switching from total harmonious congruency to complete voice autonomy and varied degrees in between. Like the details of this piece’s inceptive fractal images, there is a coherence of form and interconnection tying the different parts together neatly, behavior naturally befitting a concerto, and captured very smartly by the title (if one is familiar with its origin): 36Wallin’s approach to extrapolation is explained more fully in Chapter 3 under the heading “Heterophony in ning,” pages 94-97. 37Rolf Wallin, “Avant-garde, Ghosts, and Innocence,” rolfwallin.org, accessed Feb. 24, 2018, http://www.rolfwallin.org/articles/c_5165b8481e98c045d1c4f3a7/. 25 The title and concept behind the piece comes from David Grossmann's novel “see under: love,” in which he among other things describes a man who joins a school of salmon on its way through the oceans. He swims, eats and sleeps together with them, and gradually he understands the forces that make this large group of fish swim almost as one body. One of these forces is a cohesive joining force called ning. In my oboe quartet, the four instruments act very much like fish or birds in a group, sometimes moving fast and close to each other, with one clear direction, then all of a sudden the group dissolves to four separate individuals, finding their own paths within a wider area, but still dependent on one another. ‘Ning’ is the fourth in a series of pieces where I use so called fractal mathematics in the process of composition.38 If the data came first, it is logical to assume that Wallin chose the title in reaction to what he heard from his computer. But did he? There is some reason to believe that the idea of salmon shoals may have played a part in the precompositional process. Recall that Wallin sometimes uses upper and lower limits to tame the extremes of his sonic parameters (see Chapter 1, pages 16, 18, and 20, and Figure 1.6, page 19). In his initial implementations of the logistic equation, he had tried constraining each parameter by using a simple pair of constant, invariable outer limits through the length of the piece (as in Onda di ghiaccio or Solve et Coagula), but found that this made for either too much homogeneity (pitches “getting stuck” for long stretches of time) or too much disconnect (leaps between extreme highs and lows) in the synthesizer’s audio output.39 So in ning, Wallin opted to scale his results by superimposing fluctuating outer limits on two of these parameters (sectional durations and pitch), which changed gradually as the piece progressed. This data skewing (defined by its own predetermined set of rules—in this case, golden mean proportions) crunches the numbers and molds the form of the piece. Figure 2.1 diagrams these upper and lower limit lines with regard to pitch in time (spread over a total duration of twelve minutes, the original predetermined length for ning). Notice that the shape drawn by the outer limits in the first part of the diagram curiously resembles a fish (there even appears to be a mouth 38Rolf Wallin, “English Programme Notes for ning,” rolfwallin.org (1991), accessed Feb. 24, 2018, http://www.rolfwallin.org/works/chamber-music/; David Grossmann, See Under: Love, trans. Betsy Rosenberg (New York: Farrar Straus Giroux, 1989), 118. 39Wallin, “Fractal Music.” See also Appendix B, page 213. 26 Figure 2.1: Facsimile of the original computer graphic rendering of the formal diagram for the melodic trajectory (pitch path and range) of ning as it evolves in time, with superimposed sectional divisions (in red).40 40This diagram, which Wallin created in 1989, comes from his personal notes and sketches. Like standard musical notation, pitch is on the vertical axis (the different octaves of C are shown on the left*) and time is on the horizontal axis (minutes are shown across the top). The 27 on it!). Coincidence? Perhaps. Wallin did intend to superimpose a general gesture guiding the form through the space defined by these high and low pitch limits, beginning with a narrow range, expanding outward to a wide range, then coming back inward to a narrow range again, and then following this by reiterating the same gesture, inverted, reversed, and scaled in proportion to the first gesture by the golden ratio.41 Such mirroring makes for sound formal structure, but to the imaginative eye, it suggests a pair of fish, one smaller and upside-down, apparently overlapping at the tail with its larger counterpart. Of course, making more of something than is actually there is to tread in dangerous theoretical waters (pun intended), so the notion should be taken with a degree of skepticism. But considering the programmatic aspect of the title, one must wonder. Such conjecture aside, it is now compulsory to pick apart the diagram for what it does in fact show. The beginning of each formal section in the piece (determined by the process described in Chapter 1, pages 16-17) is indicated with a vertical red line in Figure 2.1 (thirty-two sections in total). These correspond to the values of r to be used for each iteration of the logistic equation, whereby the values at the beginning of each formal section are determined for every other formal parameter (including parameters not dealing with pitch, which are not represented in the diagram portion of Figure 2.1). These r values were in turn derived from the calculation of the sectional durations, established under their own set of outer limit constraints, as shown in Figure 2.2. minimum and maximum values (0 and 1) for the x variable (iterations of which indicate the centric pitch at the beginning of each section) are repositioned onto the upper and lower skew curves, in turn repositioning the iterated values of x itself relative to these outer limits. These skewed values translate into pitches according to the absolute range seen on the diagram, between C–1 and C5. (*Note: The octaves of C as labeled here use the German/Scandinavian system native to the composer. To convert these to standard scientific (American) labeling, add three [i.e., C1 is actually equivalent to C4, or middle C].) The path followed by the centric melodic pitch between sectional divisions (and the pitch range around it) is explained in Appendix A. The numbers shown below the diagram represent the values of other parameters (“Shape,” “Q,” “Speed,” “Rvekt” [rhythmic weight], “Skala” [scale], and “Posi” [position of coordinate (c, d) values for the Frøyland formula]). These are vertically aligned with the beginning of each corresponding formal section of the piece. Of these additional parameters, only “Q” and “Shape” are reflected in the imagery of the diagram itself. These numbers are presented more clearly later in Table 2.1 (see also footnote 44, page 30), and thoroughly explained in Appendix A. 41See Appendix B, page 213. 28 Figure 2.2: Diagram plot of sectional durations in ning.42 With the explanation Figure 2.2 provides for the relative spacing between sections in Figure 2.1 (red vertical lines), it becomes clear that the iterated values of the different sonic parameters found in Figure 2.1 (listed underneath the diagram) are not spaced erratically, but are spaced to align vertically with the beginning of each formal section. (Note that not all of the parameters listed underneath the diagram are represented in the diagram proper, but are included there for the sake of corollary convenience.) Most of these values are presented more 42This figure shows the shape of the outer limits used to skew the sectional duration results (and thereby determine the iterative values of r for the other formal parameters using the logistic equation—see pages 16-17 in Chapter 1, including footnote 25). As with the diagrams represented in Figure 1.6 (Chapter 1, page 19), r is represented horizontally on this graph, decreasing from 3.855 to 3.55, and matched up to scale with the predetermined duration of twelve minutes. The minimum and maximum values (0 and 1) for the x variable (iterations of which indicate sectional durations) are repositioned onto the upper and lower skew curves, in turn repositioning the iterated values of x itself relative to these outer limits (as shown by the scatterplot in between). These skewed values are then interpolated by comparing them against an absolute range of duration times estimated to be between 4.0 seconds and 60.0 seconds, as shown along the vertical axis. Therefore, the interpolated value of each iteration is also equal to the relative horizontal space leading up to the following iteration (the correlation being shown on the horizontal axis). 29 clearly in Table 2.1, in which the parameters for ning are laid out in each column (many of the same parameters described on page 17 in Chapter 1, including the parameter for the sectional durations themselves shown in Figure 2.2 [termed “Section Length” in Table 2.1]). The rows in Table 2.1 show how each parameter changes from iteration to iteration as they unfurl in time with each new formal section of the piece.43 However, the diagram itself of Figure 2.1 only shows details for those parameters dealing with pitch/melody. The undulating dotted and solid lines appearing in between the solid “fish shape” lines (upper and lower limits) constitute the resultant output from the logistic equation for these pitch-related parameters (as circumscribed and sheared by the outer limits). These parameters include the centric melodic path (termed “Median Pitch” in Table 2.1, not listed below the diagram), the pitch range around the melodic path (termed “Q” below the diagram, renamed “Pitch Spread (Range)” in Table 2.1), and another parameter causing a secondary skew of path curvature between points of data for the other two parameters (termed “Shape” both below the diagram and in Table 2.1). Together these three parameters create what shall henceforth be called the melodic trajectory. Notice in Figure 2.1 that each time the melodic trajectory exhibits a change in behavior (direction, tessitura, ambit of the pitch spread, and/or degree of curvature) it coincides exactly with a red sectional division line; ergo each section has a consistent melodic fingerprint, making it so that any musically detectable shifts in melodic behavior will clearly convey the instances of the piece’s formal parsing. In some sections, there appears to be a steady ambit of pitch with very 43The remaining parameters not listed in Table 2.1 can be found in Table 3.1 in Chapter 3, page 84. Facsimiles of the original printouts used by Wallin containing his data tables for the form of ning, which Tables 2.1 and 3.1 represent, are found in Figure A.1, Appendix A. These facsimiles show the composer’s handwritten notes and changes, which are discussed in the text in conjunction with Tables 2.2 and 2.4, as well as with the implementation of the Frøyland formula addressed in Chapter 3. Detailed explanations for the meaning of the data found in each column of data tables (i.e., each sonic parameter) are given in Appendix A, which the reader is strongly encouraged to consult in order to make sense of the way that Table 2.1 translates into the diagram of Figure 2.1. 30 Table 2.1: Formal Parameters in ning44 Section Number Section Length 1 Section Start Location 0.00 2 34.43 59.66 3 94.09 11.68 4 105.77 29.48 5 135.25 6 190.88 55.64 (55.63*) 9.37 7 200.25 8 227.35 9 275.86 10 279.87 11 297.31 27.09 (27.10*) 48.52 (48.51*) 4.00 (4.01*) 17.45 (17.44*) 34.72 12 332.03 5.10 13 337.13 16.62 14 353.75 27.75 44Table 34.43 Pitch Lower Limit 0.30 (A#3) 0.19 (D3) 0.12 (A2) 0.11 (G#2) 0.09 (F#2) 0.05 (E2) 0.04 (D#2) 0.03 (D2) 0.00 (C2) 0.00 (C2) 0.01 (C2) 0.04 (D#2) 0.05 (E2) 0.09 (F#2) Pitch Upper Limit 0.50 (C5) Median Pitch 0.61 (G#5) 0.68 (C#6) 0.69 (C#6) 0.71 (D#6) 0.75 (F#6) 0.76 (F#6) 0.77 (G#6) 0.79 (A6) 0.77 (G6) 0.73 (E6) 0.68 (C#6) 0.68 (C#6) 0.66 (C5)†6 0.97 (G5) 0.01 (A2) 0.37 (C3)†4 0.91 (B5) 0.07 (G2) 0.65 (C#5) 0.48 (D#4) 0.77 (G#5) 0.26 (D#3) 0.93 (C#6) 0.04 (F2) 0.52 (D#4) 0.69 (B4) 0.32 (D4) Pitch Spread (Range) 4.89 Average Note Duration 0.36 Rhythmic Weight Shape 0.12 7.88 1.03 3.82 0.49 1.04 2.33 0.10 0.01 3.14 4.77 0.51 0.19 7.79 1.11 3.36 0.46 1.10 2.94 0.10 0.03 3.62 3.97 0.87 0.31 7.41 2.17 2.02 0.27 1.47 4.74 0.29 0.34 5.46 1.13 2.28 0.22 4.74 2.99 0.10 0.41 5.92 3.83 0.57 0.09 3.87 2.39 1.09 0.44 7.01 4.52 0.29 0.05 1.98 notes: Times in the “Section Start Location” and “Section Length” columns are given in seconds. The times shown in the “Section Length” column are those found in the composer’s notes, but are occasionally off from the difference between “Section Start Location” times by ±0.01, due to rounding (corrected differences are noted parenthetically with asterisks [*]). The pitches given in parentheses that correspond to the values in the “Pitch Lower Limit,” “Pitch Upper Limit,” and “Median Pitch” columns are those found in the composer’s notes, converted into standard scientific (American) pitch labels. Occasionally, however, the octave number is incorrect by –1 on pitch class C (so noted with a dagger [†] followed by the corrected octave number). The numerical values of all other columns are explained thoroughly in Appendix A. Note that the values in the “Average Note Duration” column (which match those of the “Speed” column in Figure A.1, Appendix A, pages 190-91) differ somewhat from the numbers for “Speed” found in Figure 2.1. The reason for this disagreement is unclear, nor is it clear which group of values Wallin ultimately used in the final version of ning. However, both sets of values increase and decrease in tandem with one another, representing very similar trajectories. 31 Table 2.1 (continued) Section Number Section Length 15 Section Start Location 381.50 16 386.14 13.71 17 399.85 23.98 18 423.83 4.08 19 427.91 11.04 20 438.95 21.07 21 460.02 22 464.52 4.49 (4.50*) 12.59 23 477.11 24.53 24 501.64 25 505.67 26 519.14 27 549.91 28 555.31 29 578.78 30 627.01 4.04 (4.03*) 13.46 (13.47*) 30.78 (30.77*) 5.39 (5.40*) 23.48 (23.47*) 48.22 (48.23*) 10.58 31 637.59 29.69 32 667.28 52.72 4.64 Pitch Lower Limit 0.16 (B2) 0.17 (C3) 0.22 (E3) 0.31 (A#3) 0.32 (B3) 0.37 (D#4) 0.39 (E4) 0.38 (D#4) 0.37 (D4) 0.34 (C4) 0.33 (C3)†4 0.31 (A#3) 0.21 (D#3) 0.20 (D#3) 0.23 (E3) 0.28 (G#3) 0.29 (A3) 0.33 (C3)†4 Pitch Upper Limit 0.64 (A#5) 0.64 (A#5) 0.63 (A5) 0.61 (G#5) 0.61 (G#5) 0.60 (G5) 0.71 (D#6) 0.73 (F6) 0.81 (A#6) 0.92 (F#7) 0.93 (G7) 0.97 (A#7) 1.00 (C7)†8 1.00 (C7)†8 0.97 (A#7) 0.92 (F#7) 0.91 (F7) 0.87 (D7) Median Pitch 0.40 (C#4) 0.84 (F5) 0.15 (G#3) 0.83 (E5) 0.17 (D#4) 0.85 (F5) 0.13 (G4) 0.78 (B5) 0.23 (A#4) 0.89 (C#7) 0.08 (D#4) 0.65 (F6) 0.45 (E5) 0.76 (A#6) 0.26 (G4) 0.87 (C7) 0.10 (C#4) 0.68 (D6) Pitch Spread (Range) 1.34 Average Note Duration 1.34 Rhythmic Weight Shape 0.36 6.70 3.82 0.11 0.19 2.48 2.39 0.65 0.44 7.25 4.49 0.47 0.06 1.63 1.37 0.69 0.38 5.83 3.88 0.33 0.14 3.94 2.28 0.96 0.45 6.81 4.53 0.16 0.03 2.27 1.33 1.19 0.31 6.96 3.71 0.12 0.27 2.02 2.54 1.05 0.33 6.60 4.26 0.29 0.22 2.57 1.66 2.20 0.39 7.14 4.33 0.13 0.13 1.76 1.55 1.55 0.44 5.95 4.13 0.72 0.05 3.63 1.83 2.42 0.35 6.83 4.35 0.28 0.19 2.16 32 little movement up or down, while at other times, there appears to be divergence or convergence of pitch-spread around a kind of centric pathway that moves in a slant or curve. When patterns recur in regular ways, larger formations and formal associations appear. A clear example of this occurs with the repetition of the formations seen in sections 1, 2, and 3 happening again in sections 4, 5, and 6, respectively: sections 1 and 4 have a narrow pitch ambit and a linear melodic path over a medium-sized length of time, 2 and 5 have a wide-spread pitch ambit moving to converge along a downward melodic slope over a long length of time, and 3 and 6 have a tightly clustered pitch ambit moving up a steep melodic curve over a brief length of time (to suggest a rough analogy with traditional sonata form, these two groups-of-three are like the first and second themes of an exposition).45 The idea continues in sections 7, 8, and 9, but apparently in a sort of inversion. Similar groupings of shapes and lengths can be found throughout the diagram in different variations and orderings.46 The notion of sectional duration alone is intriguing. As already observed, the first three sections together form a medium-long-short ordering of lengths relative to one another. This pattern of medium-long-short consistently perpetuates through the whole course of the piece at every three sections. Each group of three also gets progressively smaller approaching the golden section of the piece (in section 20), then progressively larger from there to the end.47 These 45The comparison with sonata form is used pervasively throughout this chapter, since the piece exhibits many formal parallels with it. However, in reality the piece does not follow a prescribed traditional formal plan; the only plan that the form strictly follows comes from the results of Wallin’s data output. Hence, the associations made with sonata form in this text are typically given in quotation marks to deemphasize any misconstrued notions that the composer purposefully adhered to it. (The piece is far longer than the typical first movement of a sonata would be anyway.) That said, it is rather remarkable just how similar the formal design of ning is to sonata form, which is why this idea is the principal theme of this chapter. 46These groupings are succinctly diagrammed later in this chapter as a formal plan that uses musical cues from the score, in Table 2.4, pages 71-72. 47This convergence on the golden section is reminiscent of a similar (but not by any means identical) formal patterning Wallin used in writing his Clarinet Concerto (1995). In the concerto, formal sections get progressively larger moving outward from the midpoint, and are 33 observations raise the question: Is there compelling evidence of this same durational association embedded in the local material of the piece (medium-long-short rhythmic patterns)? This question is further warranted by considering these ordered sectional lengths in their totality with a passing comment made by the composer: “…each column [in the table] is a kind of melody then, that is, ‘Length’ is a melody, the ‘Medium [Median] Pitch’ is a melody…”48 Are there embedded likenesses of these large-scale formal constructions (rhythmic, pitch, etc.) found in the surface activity of the piece, or even at some middleground level of analysis? (The question is particularly relevant considering the implications of fractal self-similarity suggested in the form and its origins.) These ideas shall be revisited and considered further on (see Chapter 4, pages 148-58). Meanwhile, in viewing Figure 2.1, one should be careful not to infer too much about other musical parameters (rhythm, dynamics, articulation, timbre, etc.) from the diagrammed portion alone, as it deals only with pitch in time.49 Although there is nothing in the diagram itself showing specifics of any other parameters, data from Table 2.1 for the rhythmic parameters is given below the diagram, aligned with the beginning of each formal section for ease of correlation. Like the two melodic parameters, the two rhythmic parameters describe corresponding paths for a central proportionate to one another by the golden mean. In ning, formal sections get progressively larger moving outward from the golden section of the whole piece, but formal sections (or groups of sections in threes) do not exhibit any specific ratio to one another. Rather, in ning, the golden proportions guide the behavior of the precompositional outer limits for sectional length (as seen in Figure 2.2), which has a direct bearing on the lengths of formal sections. The role of the golden mean in the form will be discussed further on pages 49, 55-56, and 63. 48See Appendix B, pages 209-10. 49There is especially a temptation to see rhythmic and/or articulative activity in the layout of the upper and lower melodic paths (which indicate the “Pitch Spread (Range)” of the melodic trajectory) because these appear as dotted lines, but caution: the dots do not indicate the distinct articulations that they might suggest. They are actually spaced at equidistant time intervals horizontally (at about every two seconds). Similarly, stretches of continuous solid lines along the median pitch path must not be construed as long, sustained rhythmic durations. Why Wallin’s graphic printout features these different line types for the median pitch path and the outer limits of the pitch spread is not entirely clear, but it most likely has to do with the way that the computer distinguished the two parameters from each other (without being able to produce colors, it was forced to use a monochromatic approach to differentiate the parameters). 34 rhythmic duration/baseline tempo (termed “Speed” below the diagram, renamed “Average Note Duration” in Table 2.1) and the degree to which rhythms can deviate from this baseline tempo (termed “Rvekt” below the diagram, renamed “Rhythmic Weight” in Table 2.1).50 Also like the two melodic parameters, these rhythmic parameters are affected by the skew of the “Shape” parameter, with all three parameters combining to create a rhythmic trajectory (in the same fashion that the melodic trajectory is crafted). One might imagine a diagram for these rhythmic parameters that looks something akin to the one shown for pitch in Figure 2.1, but with one notable exception: Unlike pitch, the rhythmic parameters are not constrained by any changing outer limit boundaries—the maximum and minimum rhythmic values are fixed (flat) for the entire length of the piece.51 These rhythmic values, together with the pitch information shown within the diagram, give an idea of the general musical behavior found within any given section, thus providing a glimpse of the precompositional framework that lead the composer to his choices of stylistic nuance in the final product.52 Taking the Data to the Score Wallin initially conceived ning with a twelve-minute total duration. The timings provided in the “Section Start Location” and “Section Length” columns of Table 2.1 correspond with this original plan, so that there is visual agreement with Figure 2.1. However, after some scrutiny, the composer determined that the timing for the piece’s events was too rushed, so he decided to 50The remaining parameters “Skala” and “Posi” appearing below the diagram of Figure 2.1 are addressed later in Table 3.1 (Chapter 3, page 84) as “Scale (Pitch Collection)” and “Position Number for c and d paired values,” respectively, and are explained at length in Chapter 3. 51Again, the reader is encouraged to refer to Appendix A for a more detailed explanation of these parameters and how they function in terms of the form of the piece. 52A summarized interpretation of the values in Table 2.1 are to be found later, in Table 2.3, pages 47-48. 35 lengthen the piece from twelve to fifteen minutes, scaling the events proportionately by multiplying each length time by 1.25. Table 2.2 again shows the 32 formal sections with their start and length times (in seconds) adjusted accordingly, and then correlates the sections as closely as possible with the music in the score itself.53 This helps to elucidate the way in which the individual raw formations of the formal diagram in Figure 2.1 get “translated” by the composer into actual musical thematic ideas and form. Interestingly, there is not always a clear one-to-one correspondence between Wallin’s precompositional “fields” (in the “Section Number” column of Table 2.2, imported from Table 2.1) and the formal demarcations that actually occur in the score (“Expressive Marking” and “Tempo Marking” columns of Table 2.2, which both often indicate beginnings of new sections of the piece). Seemingly more often than not, the sections of the score do not agree precisely with the precompositionally calculated timings. On one hand, sometimes a single self-contained musical unit in the score will consist of multiple precompositional sections. Even with attempts to seek out other subtler partitions (like caesurae, texture alterations, or stylistic changes) within the score, the shifts between these precompositional sections often remain well-disguised, even questionable at times (moments that are thus unclear are so noted “§” in the “Start Measure” and “General Characteristics” columns of Table 2.2). Conversely, there might be multiple formal markers given in the score that take place within the musical space of a single precompositional section. This calls into question the reasons for which the composer felt compelled to add additional partitions where none were called for by the data. And finally, some formal markers are not quite aligned between the score and the data, shifted slightly from one another in time. This raises questions about why the composer “misaligned” formal markers in the score. 53Please note that the “Adjusted Length” results given here are those figured by Wallin himself, and that they exhibit a margin of error mostly within ± 0.01 as they did in Table 2.1, with two exceptions: Wallin’s value for section 7 is off by –0.02, still a reasonable error, but his value for section 3 is off by a more drastic +0.33 (for reasons unknown). All these errors are duly noted in Table 2.2. 36 Table 2.2: Formal Sections in ning54 All Formal Demarcations in the Music (as indicated by both the score and the sections of Table 2.1) Demarcations Given in the Score Start Measure General Characteristics Expressive Marking Tempo Marking Length in Quarter Notes Length Section in Number Seconds 1 unison; active rhythmic accents, trills, embellishments “Rustic, Vigorous” q = 60 43.00 (45.50 to 47.00) 1 0.00 43.04 0.04 (–2.46 to –3.96) 13 English horn riff gestures, string reinforcement transition to rhythmic regularity driving, straight sixteenth-note rhythms; accents followed by diminuendo tight pitch cluster; active rhythmic accents, trills, embellishments English horn riff gestures, tremolo string reinforcement transition to rhythmic regularity driving, straight sixteenth-note rhythms; accents followed by diminuendo “Transparent” 43 (45.5 to 47 counting the introductory gesture in m.1) 56 56.00 2 43.04 74.58 –1.06 12 4 12.00 3.42 8 20 11 4.22 8.00 6.57 3 117.61 14.93 (14.60*) 0.36 (0.03*) 40 36.92 4 132.21 36.85 –0.07 48 44.31 5 169.06 69.55 (69.54*) 6.81 (6.80*) 12 11.08 4 3.21 8 10 14 4.14 4.00 7.69 6 238.60 11.72 (11.71*) 0.03 (0.02*) 27 31 32 34 39 41, beat 4 52 64 67 68 70 72, beat 3 54Table “Onward” molto accelerando “Motoric” q = 150 molto rallantando “Rustic, Vigorous” “Transparent” “Onward” “Motoric” q = 65 (Tempo is given as q = 60 at m.52 in the draft score.) molto accelerando q = 150 molto rallantando Demarcations Given from Precompositional Calculations (from Table 2.1 with durations multiplied by 1.25) Timing Discrepancy (“Adjusted Length” Adjusted Adjusted column less “Length in Start Length Seconds” Time column) notes: As in Table 2.1, times noted parenthetically with asterisks (*) are adjusted to correct for rounding errors in the original times given by the composer in his notes (the handwritten times in the facsimile of Figure A.1, Appendix A). Also, some of the times listed include the fermata symbol (!) to indicate that a fermata occurs at this spot in the score, and the given time may be slightly longer that noted. Instances where a sectional demarcation is indicated in the precompositional sections but are difficult to confidently detect and accurately locate in the score are noted with the symbol “§” in the “Start Measure” column. If the symbol is followed by a question mark (?), the demarcation is particularly questionable. (This symbol is also used in Tables 2.3 and 2.4.) 37 Table 2.2 (continued) All Formal Demarcations in the Music (as indicated by both the score and the sections of Table 2.1) Start General Measure Characteristics Demarcations Given in the Score Expressive Marking Tempo Marking Length in Quarter Notes 76 “Rustic, Vigorous” (implied at m. 76, beat 2) q = 72 80 87 104, beat 4 §? 106, beat 3 §? 113 116 136 141 142, beat 3 §? 147 154 160, beat 4 §? English horn changed to oboe; tight pitch cluster; active rhythmic accents, trills, embellishments transition to rhythmic regularity straight sixteenth-note rhythms; tenuto; hairpin dynamics Cello “solo” of articulated downward glissandi (105, beat 3 §?) oboe drops out (106, beat 4 §?) transition with harmonics, tremolo light, staccato, with grace notes; rhythmic interplay transition: becomes more legato; note durations lengthen broad glissandi; cello “solo” feature short cello embellishment figure; texture thins; grand pause in m.145 §? driving eighth notes; articulate embellishments longer note durations viola enters §? Length in Seconds Demarcations Given from Precompositional Calculations (from Table 2.1 with durations multiplied by 1.25) Section Adjusted Adjusted Number Start Length Time Timing Discrepancy (“Adjusted Length” column less “Length in Seconds” column) 15 12.5 7 250.31 33.86 (33.88*) –0.72 (–0.70*) “Softening” 26.5 22.08 “Floating” 99 (!) 82.50 (!) 8 284.19 60.65 (60.64*) 0.71 (0.70*) (!) 9 344.83 5.00 (5.01*) 10 349.84 21.81 (21.80*) 11 371.64 43.40 1.65 12 415.04 6.37 0.14 (0.15*) 13 421.41 20.77 (20.78*) 14 442.19 34.69 15 476.88 5.80 8.5 4.25 “Playful” 66 33.00 “Broadening” 17.5 8.75 27 27.00 q = 120 q = 60 “Sharp” “Broadening” q = 105 26.5 15.14 39 (!) 22.29 (!) 3.06 (!) 38 Table 2.2 (continued) All Formal Demarcations in the Music (as indicated by both the score and the sections of Table 2.1) Start General Measure Characteristics Demarcations Given in the Score Expressive Marking Tempo Marking Length in Quarter Notes 163 “Playful, Innocent” q = 108 167 171 178, beat 3 § 179, beat 4 §? 183 187 189, beat 3 §? 191 195 197 202, beat 4 § 203, beat 4 § light, oscillatory gestures with tremolo background transition to flautando gestures flautando, quick scale-run gestures transition to accents with embellishments; ord. bowing; violin enters § cello glissandi pizzicati §? unison; active rhythmic accents, trills, embellishments accents dissipate into sustained trills brief cello articulation §? straight sixteenth-note rhythms; light, sul ponticello, staccato, gracenote figures transition to longer durations and tremolo oboe riff gestures with tremolo string reinforcement strings play harmonics; rhythmically active § caesura followed by the oboe riff gestures with tremolo string reinforcement used previously § “Broadening” “Transparent, Evaporating” “Rustic, Vigorous” q ≈ 60 (Note the approximation, indicating flexibility of tempo.) Length in Seconds Demarcations Given from Precompositional Calculations (from Table 2.1 with durations multiplied by 1.25) Section Adjusted Adjusted Number Start Length Time Timing Discrepancy (“Adjusted Length” column less “Length in Seconds” column) 13 7.22 16 482.68 17.14 1.03 16 8.89 49 ≈ 49.00 17 499.81 29.97 (29.98*) ≈ –0.13 (–0.12*) 18 529.79 5.10 19 534.89 13.80 20 548.69 26.34 21 575.03 22 580.64 5.62 (5.61*) 15.74 (15.75*) 23 596.39 30.66 24 627.05 5.05 (5.04*) 25 632.09 16.83 (16.84*) (Tempo is given as a more absolute q = 60 at m.183 in the draft score.) 15 ≈ 15.00 16 ≈ 16.00 q = 90 16 10.67 “Broadening” 8.5 5.67 “Transparent” q = 45 39 (!) 52.00 (!) “Softening” “Sharp” ≈ 0.96 (0.95*) –0.60 (–0.59*) 0.54 (0.54*) (!) 39 Table 2.2 (continued) All Formal Demarcations in the Music (as indicated by both the score and the sections of Table 2.1) Start General Measure Characteristics Demarcations Given in the Score Expressive Marking Tempo Marking Length in Quarter Notes 207 “Rustic, Vigorous” q = 74 216 219 §? 220, beat 4 235 237 250, beat 2 §? unison; active rhythmic accents, trills, embellishments quick scale-run gestures introduced behavior modifies slightly to accelerating rhythmic gestures; brief harmonics; dies off §? light, staccato; rhythmic interplay, later transforming to steady eighthnote triplets transition, triplets dissipate thin texture; decelerating rhythmic gestures with quick scale-run gestures; grand pause in m.244 transition with strings playing high harmonics “Softening, Yielding” “Playful” “Evaporating” §? 254 263 loud unisons, “Strong” attacked and embellished; drastic crescendi/dynamic contrasts unisons fancifully “Playful” articulated and embellished Length in Seconds Demarcations Given from Precompositional Calculations (from Table 2.1 with durations multiplied by 1.25) Section Adjusted Adjusted Number Start Length Time Timing Discrepancy (“Adjusted Length” column less “Length in Seconds” column) 33 26.76 26 648.93 38.47 (38.46*) 1.42 (1.42*) (!) 21 (!) 17.03 (!) 27 687.39 6.74 (6.75*) 28 694.14 29.35 (29.34*) 0.15 (0.14*) 29 723.48 60.28 (60.29*) 3.51 (3.52*) 30 783.76 13.23 q = 150 57 22.80 q = 75 8 6.40 q = 60 70 70.00 (Tempo given as q = 50 at m. 251 in the draft score.) q = 74 32.5 32.5 31 796.99 37.12 (37.11*) 4.62 (4.61*) 81 65.68 32 834.10 65.90 0.22 40 Though these incongruities of timing cast doubt upon the accuracy of Table 2.2, there is nevertheless much evidence giving credence to its layout, especially if one becomes less concerned with precision and takes on a more generalized associative perspective (as can well be assumed the composer himself did in molding the raw data to his own aesthetic).55 With this perspective, general tendencies come into view that greatly help to answer the questions posed above and settle the difficulties with timing (mentioned in the previous paragraph). In cases where many musical partition markers occur within a single precompositional section, the answer is always because there is a transitional change occurring within the behavior of the section leading up to the next section, denoted by the markers “Onward”56 (always followed by “molto accelerando”), “molto rallantando,” “Softening,” “Broadening,” or a new tempo marking.57 Taking a look at the sections in question as they show up in Figure 2.1, there are even visual indicators suggested in the pitch behavior that demonstrate transitions into subsequent sections (sharp dives or rises in the melodic trajectory curve), with perhaps the exception of section 7, in which the transition “Softening” behaves rather similarly to the initial “Rustic, Vigorous.”58 Wallin was perhaps justified in putting in additional expression markings after all, especially if he perceived an alteration of character aurally as well as visually (from the computer’s graphical rendering). 55Aksnes, “Rolf Wallin.” Also see Figure A.1 in Appendix A for Wallin’s suggestive markings in grouping musical sections. 56For clarification purposes, expressive markings from the score are cited in the text with capitalized first letters and quotation marks (the same way that column headings for tables are cited in the text). In the score, these markings are found without capitalization or quotation marks, but are instead boxed like rehearsal marks. 57Sections containing transitional expressive marks are 2, 3, 5, 6, 7, 11, 16, and 22. Sections 10 and 28 contain transitional tempo changes, though section 10 is more like a misalignment, as it is itself more logically grouped with sections 8 and 9. 58In this case, the “Softening” passage functions as it does at other occurrences of the term (mm. 187-190 and mm. 216-220, beat 3), where it works as a modified continuation of the previous expressive marking. See also footnote 60. 41 In the case of single musical sections containing multiple precompositional sections, it is clear that many of these sections were grouped together as single ideas by Wallin (indicated by the hand-drawn brackets around section groups in the facsimile in Appendix A).59 The same is true of instances where precompositional sections seem to misalign with sections in the score: The misalignments are contained within the bracket.60 But why did he consider these to be single ideas? Perhaps this is because some of these sections (9, 12, 15, 18, 21, 24, and 27) are too short to really assert themselves as independent ideas, working instead more like transitional or linking 59Musical partitions containing sections bracketed together by Wallin include “Floating” with its contained transitional tempo change, mm. 87-115 (sections 8, 9, and 10), and tempo mark " = 60, mm. 141-146 (sections 12 and 13). Other musical partitions containing bracketed sections include “Transparent, Evaporating,” mm. 171-182 (sections 17, 18, and 19), “Transparent,” mm. 197-206 (sections 23, 24, and 25), and “Evaporating,” mm. 237-253 (sections 29 and 30), though in these cases, the brackets are not comprehensive: 17 and 18 are bracketed together while 19 is bracketed alone, 23 is bracketed alone while 24 and 25 are bracketed together, and 29 and 30 are each bracketed alone. It is not entirely clear what Wallin meant by creating separate brackets in these cases, but there is musical evidence of subtle demarcations (not explicit) at these moments: at section 19, there is a shift of character in the cello part, at 24, the strings begin playing harmonics, and at 30, the musical material begins to transition into the character of the following “Strong” section. 60Bracketed sections with misalignments include 14 and 15 (“Sharp” and “Broadening” in the score, mm. 147-162), 20 and 21 (“Rustic, Vigorous” and “Softening” in the score, mm. 183190), and 26 and 27 (“Rustic, Vigorous” and “Softening, Yielding” in the score, mm. 207-220, beat 3). While there are explicit markings for precompositional sections 15 and 21 found in Wallin’s draft score (section 27 is not marked there), the placements of sections 15, 21, and 27 relative to the final score are musically vague in each case, as there are no truly substantial formal demarcations found in the score at these timings, explicit or nonexplicit (though some attempt is made in Table 2.2 to account for them [marked “§?”]). Thus, it seems better to simply think of these bracketed groups as single ideas (which it seems clear that Wallin intended) that, like sections 2, 3, 5, 6, 7, 11, 16, 22, and 28, simply contain a transitional indicator within them. See also footnote 58. There is also an apparent misalignment of the tempo change marking in “Floating,” mm. 87-115, with precompositional section 10 (falling within the bracketed group of 8, 9, and 10) in the score, where section 10 begins on the third beat of m. 106, while the tempo change marking happens in m. 113. Though marked in the draft score, the placements of both sections 9 and 10 are again vague in the context of the final score, their corresponding formal demarcations (as presented in Table 2.2) being quite subtle. From observation of the score alone, it almost seems as if Wallin could have flipped sections 9 and 10 in order to align the shorter section with the tempo change marking. However, not only does the draft score prove this theory false, but such a reversal would ruin the consistent medium-long-short ordering of sections. The shift of tempo in the middle of precompositional section 10 must therefore be due to something innate within the musical material of the section itself. 42 passages to longer adjoining sections. This is reasonable to assume, since this is just the way these parts seem to behave musically in the context of the larger formal design. The composer has integrated them quite seamlessly into the fabric of the surrounding material. Though it seems that the explanations just pointed out for the alignment and quantity disagreements are reasonable, there still remains a problem of durational disagreement. In many cases, difficulty arises from different length times between the score (“Length in Seconds” column) and the precompositional calculations (“Adjusted Length” column), as shown in the “Timing Discrepancy” column of Table 2.2. The vast majority of these timing comparisons show the precompositional sectional durations to be longer than those of the score, giving the score a total length of about 877.69 seconds, 22.67 seconds shorter than the anticipated 900.00 seconds (fifteen minutes).61 Granted, most discrepancies are reasonably forgivable, falling within a couple of seconds, a margin of error imperceptible in practicality, and likely to be approximated at best in actual performance anyway.62 These can reasonably be ignored, but parts here demonstrating a particularly great discrepancy of several seconds beg for further investigation. These include precompositional sections 5 (mm. 52-69), 14 and 15 (mm. 147-162), 29 and 30 (mm. 237-253), and 31 (mm. 254-262), with discrepancies of 6.81, 3.06, 3.51, and 4.62 seconds, respectively. 61The score length given here does not account for the “introductory” measure (as the pitch material suggested by the initial values of Table 2.1 does not happen until measure 2) or the extra indeterminate length of fermatas in the piece (fermata occurrences are noted in Table 2.2). Taking these into consideration, the length extends to somewhere between 881.69 seconds (which includes the rests at the beginning) and an estimated 893.69 seconds (giving a maximum of 3.00 seconds to each of the four fermatas), which is still deficit by at least 6.31 seconds. Likewise, the total precompositional duration suggested by Wallin’s numbers in the “Adjusted Length Time” column of Table 2.2 comes out to be 900.36 seconds. When adjusted for rounding errors (timings marked “*”), the total comes to an imperfect but forgivable 900.03 seconds, the 0.33 second difference being exactly the timing error of section 3. 62For example, the recording of ning by BIT20—The Bergen Ensemble of 20th Century Music (featured on Wallin’s album move [1994]) ends up being nearly two minutes longer than expected, at about 16:50. Wallin, move, Hemera Music HCD 2903. 43 Why did Wallin choose to deliberately make the score shorter in these spots? Or could it be that they aren’t quite as egregiously short as they seem? The portion of the score corresponding with section 5 contains three measures of “molto accelerando,” calculated at a duration of 7.35 seconds (assuming a graded linear increase in tempo throughout the duration), but could potentially last a couple of seconds longer if the rate of tempo change is disproportionately distributed toward the end of the passage. The part corresponding to sections 14 and 15 contains a fermata, which could possibly eek out the needed three extra seconds. But the score for sections 29 through 31 exhibits no such temporal buffers, and the case for section 5 is a bit of a stretch anyhow. There are three other fermatas occurring elsewhere in the piece, which can help to extend the overall duration of the work, but as outliers, they do not address the sectional deficits in a direct, convincing way. No, these things are not enough to compensate for the missing seconds. But a look at Wallin’s rough draft of the score provides some additional insight into these large temporal discrepancies. In the first place, some additional tempo markings are penciled into the draft that are absent in the score, as those seen in the excerpts shown in Figure 2.3: " = 60 appears at the beginning of section 5, m. 52 (Figure 2.3a), and " = 50 appears in m. 251, near the beginning of section 30, with the tempo carrying over into and through section 31 (Figure 2.3b). By adjusting for these tempo differences, the time spans of the sections in question come much more closely into agreement with the “Adjusted Length” times in Table 2.2: Approximately 5.30 additional seconds are added to section 5 (including adjusted time for the “molto accelerando”), about 2.00 additional seconds are added to sections 29 and 30 (together as a single unit), and an additional 6.50 seconds are added to section 31. For section 5, the discrepancy is reduced to about 1.51 seconds. For sections 29 and 30, the discrepancy becomes 1.51 seconds, while for section 31, the discrepancy overcompensates, and becomes a –1.88 seconds. Furthermore, the exact starting point for section 31 is curiously not specified in the draft score, which means that the location of the sectional demarcation is debatable to a degree. If the durations for sections 30 and 31 are 44 a. b. c. Figure 2.3: Facsimiles of excerpts from the draft score of ning (with temporal issues shown in red). a. mm. 51-53. b. mm. 249-251. c. mm. 235, b. 4-236. 45 adjusted for the new tempo, and the actual starting point for section 31 is moved later by about 1.8 quarter notes, the timing discrepancies in these sections are nearly eliminated (leaving just –0.37 seconds of discrepancy in sections 29, 30, and 31 altogether). Clearly the composer did not intend for these large discrepancies in his first draft. So, what is the reason for them in the final score? The simplest answers, though highly subjective, are the most probable: 1) the composer thought those sections needed to be shorter through intuitively feeling the timing of events, 2) he determined that the changes of tempo were perhaps a bit too slight or imperceptible in practicality, making for awkwardness of ensemble coordination, or 3) he made an outright error/oversight. Whatever the case, there were some durational calculations that got lost in translation between the composer’s initial interpretation of the fractal formula data and the final inked score. As a final observation regarding these sections with large durational discrepancies, there is one other major temporal issue found in the draft score that is worth pointing out. Prior to m. 237 (the beginning of section 29), Wallin appears to have added an extra measure of duration. However, the amount of this duration is confusing, as the notation seems to show only one additional quarter note, while time signatures of #$ and %$ are both written by it (see Figure 2.3c). This measure is omitted in the final score, making it the only instance where an entire measure from the draft is missing in the final score. The reasons for this are unknown, but one possible explanation is that this is not a true measure, but just a “gap” in the composer’s draft for writing his personal thoughts. In any case, the presence of this measure does not help with any notable timing discrepancies, as there are not really any to be spoken of at this point in the music. Comments on Form With the association established between the precompositional sectional plan and the finished score, a basic formal scheme can be drawn out, which describes the melodic and rhythmic trajectories and general behavior of the music in each precompositional section, as 46 shown in Table 2.3. This table borrows elements from Tables 2.1 and 2.2, in essence translating portions of Table 2.1 into common musical parlance.63 Then, with each precompositional section matched with a corresponding thematic label from the score (“Expressive Markings” from Table 2.2),64 the general attributes for pitch and rhythmic duration of each section become associated with a specific thematic idea. The table thereby gives a blueprint that describes how the thematic ideas evolve (are developed) with each recurrence through the piece, in terms of pitch range, melodic direction, general rhythmic patterns, and variances thereof.65 Other thematic characteristics (dynamics, timbre, motives, embellishments, etc.) are not presented in Table 2.3, but will be given more attention in the formal descriptions in the following sections of this chapter. The thematic labels of the piece and their associated thematic characteristics (which, in part, are described in Table 2.3) constitute what shall be called a thematic idea. Although thematic ideas recur frequently in ning, no musical theme is ever repeated per se in the course of the piece, exactly or even partially, which is a signature characteristic of Wallin’s personal style.66 This presents a bit of a challenge to making formal connections with confidence, but nevertheless, just as patterns emerge out of chaotic material in fractals, similarities of recurring ideas with clear associations arise out of the piece’s independent formal partitions. 63The “Starting Tessitura,” “Starting Pitch Variance,” “Starting Average Note Duration,” and “Starting Rhythmic Variance” columns in Table 2.3 correlate to the “Median Pitch,” “Pitch Spread (Range),” “Average Note Duration,” and “Rhythmic Weight” columns in Table 2.1, respectively. Each “Change” column in Table 2.3 reflects the way in which the “Shape” parameter from Table 2.1 affects each of the other parameters, respectively, and describes how the initial parameters gradually alter throughout the course of each section. 64“Tempo Markings” and other musical sectional demarcations pointed out in Table 2.2 are also included where necessary in Table 2.3 for convenience. 65Generally, the changes that are given in the respective “Changes” columns of Table 2.3 progress to the point in the next row of the respective “Starting” column, but sometimes there is a break in this progress between sections, in which the starting point in the next row suddenly shifts away from the destination implied in the “Changes” columns. Whenever this happens, the “Precompositional Section Number” is accompanied by an asterisk (*). 66See Appendix B, pages 215-16. “Rustic, Vigorous” “Transparent” “Onward” “Motoric” “Rustic, Vigorous” “Softening” “Floating” (cello solo feature §?) (harmonics §?) “Playful” “Broadening” (q = 60) (texture thins §?) “Sharp” “Broadening” (viola enters §?) 4 5* 7 8* 9 10 11 12 13 16* 15 14* 6 “Playful, Innocent” “Broadening” “Transparent” “Onward” “Motoric” 2* 3 “Rustic, Vigorous” 1 Thematic Demarcations PrecompoSectional sitional Demarcations Section (thematic labels in Number the score) 163 167 147 154 160, beat 4 142, beat 3 141 116 136 104, beat 4 (105, beat 3) 106. beat 3 76 80 87 52 64 68 (70) 41, beat 4 13 27 32 (34) 1 Starting Measure high mid-range mid-high mid-range low high mid-low high mid-high high low high mid-range low high mid-range minimally ascends gradually descends minimally ascends starkly descends slightly-thenstarkly descends starkly ascends no detectable change gradually ascends starkly descends slightly-thenstarkly ascends minimally descends gradually descends starkly ascends starkly ascends minimally descends gradually descends Melodic Trajectory Starting Changes Tessitura narrow ambit wide ambit medium-wide ambit narrow ambit narrow ambit narrow ambit wide ambit mediumnarrow ambit mediumnarrow ambit medium-wide ambit narrow ambit wide ambit mediumnarrow ambit narrow ambit wide ambit narrow ambit Pitch Variance Starting Pitch Variance gradually widens gradually widens no change no change starkly widens gradually narrows slightly-thenstarkly narrows gradually narrows starkly widens gradually narrows gradually narrows no change gradually narrows no change gradually narrows no change Changes very short medium mediumshort short short very short mediumlong mediumlong short short very short very long short very short very long short gradually lengthens gradually lengthens no change gradually shortens starkly lengthens slightly-thenstarkly shortens slightly-thenstarkly lengthens starkly lengthens no change gradually shortens gradually lengthens no change gradually lengthens no change gradually shortens no change Rhythmic Trajectory Starting Changes Average Note Duration Table 2.3: Formal Layout of Thematic Labels Tied to the Pitch and Rhythmic Parameters of Each Section of ning moderately variant slightly variant invariant slightly variant very variant very variant moderately variant moderately variant moderately variant moderately variant invariant moderately variant very variant invariant moderately variant very variant gradually becomes more variant starkly becomes more variant no change slightly-thenstarkly becomes less variant starkly becomes more variant no change gradually becomes more variant gradually becomes less variant gradually becomes more variant gradually becomes invariant gradually becomes more variant no change gradually becomes more variant no change gradually becomes invariant no change Rhythmic Variance Starting Changes Rhythmic Variance 47 (cello §?) “Rustic, Vigorous” “Softening” (cello §?) “Sharp” “Broadening” “Transparent” (harmonics §) (caesura §) “Rustic, Vigorous” “Softening, Yielding” (harmonics §?) “Playful” (q = 75) “Evaporating” (grand pause) (harmonics §?) “Strong” “Playful” 18* 19 20 21 22* 24* 25 26* 28* 30 31 32* 29 27 23 “Transparent, Evaporating” (accents §) 17 Thematic Demarcations PrecompoSectional sitional Demarcations Section (thematic labels in Number the score) 263 254 250, beat 2 220, beat 4 235 237 244, beat 4 219 207 216 203, beat 4 202, beat 4 191 195 197 189, beat 3 183 187 179, beat 4 178, beat 3 171 Starting Measure high mid-range very high mid-range very high mid-high high mid-range very high mid-range high mid-range high mid-range high mid-low slightly-thenstarkly ascends starkly descends minimally ascends minimally descends no detectable change gradually descends slightly ascends starkly descends slightly-thenstarkly ascends slightly-thenstarkly descends no detectable change gradually descends no detectable change starkly descends minimally descends gradually descends Melodic Trajectory Starting Changes Tessitura narrow ambit wide ambit narrow ambit wide ambit narrow ambit wide ambit narrow ambit medium ambit narrow ambit wide ambit narrow ambit medium ambit narrow ambit wide ambit narrow ambit medium ambit Pitch Variance Starting Pitch Variance no change no change starkly widens slightly-thenstarkly narrows gradually widens no change gradually widens slightly-thenstarkly widens gradually widens gradually widens no change no change gradually widens slightly-thenstarkly narrows starkly widens no change Changes Table 2.3 (continued) short mediumshort long medium very short long mediumshort short mediumshort very short mediumshort very short short mediumshort short short no change slightly-thenstarkly shortens starkly lengthens no change gradually lengthens no change gradually lengthens gradually lengthens no change gradually lengthens no change no change gradually lengthens gradually lengthens gradually shortens no change Rhythmic Trajectory Starting Changes Average Note Duration slightly variant slightly variant very variant very variant slightly variant very variant moderately variant moderately variant moderately variant moderately variant invariant very variant slightly variant slightly variant very variant very variant no change starkly becomes more variant no change gradually becomes less variant gradually becomes more variant no change gradually becomes more variant no detectable change no change gradually becomes more variant no change no change gradually becomes more variant gradually becomes more variant gradually becomes less variant no change Rhythmic Variance Starting Changes Rhythmic Variance 48 49 In the remaining portion of this chapter, thematic relationships are connected to formal function by means of a comparison with traditional sonata form. Though sonata form is not at all what the composer intended (at least not overtly), there are enough similarities between this classical format and the design of ning to make it useful for comparative reasons, and to help make the case for when certain themes emerge, and how and why they are transformed.67 The main purpose here is twofold: to point out recurrences of thematic signatures and from thence draw formal patterns and larger groupings, and to show exactly how the precompositional diagram of Figure 2.1 plays out in the score. In this comparison, precompositional sections 1 through 6 are likened to an “exposition” with two theme groups, sections 7 through 19 make up a “development,” sections 20 through 30 represent a “recapitulation” (two theme groups with some alteration compared to the “exposition”), and sections 29 through 32 constitute a “coda” (with the boundary between “recapitulation” and “coda” being somewhat disputable). Important things such as the use of the golden mean and the connotation that sonata form may have with fractal patterns are addressed as needed in this paper. Please note that a large measure of the discussion will characterize different themes through descriptive (rather than analytical) means, in essence providing a walk-through of the contents of Tables 2.2 and 2.3. Analysis of thematic material (specifically identifying thematic features of pitch and rhythm) is reserved for later chapters (mostly Chapter 4), but the discussion there merits the groundwork of connective associations established in the present chapter in order to provide meaningful context to analytical observations. ning Prior to the Golden Section: “Exposition” and “Development” As pointed out previously, the music is laid out in the beginning as an “exposition”: first there is a “Rustic, Vigorous” idea with heavily accented kinetic activity but very little change in pitch ([a], mm. 1-12), followed by a thinly textured but more floridly embellished “Transparent” 67See footnote 45, page 32. 50 section with its transitional “Onward” passage ([b], mm. 13-31), which then speeds up into the “Motoric” engine driven by insistent sixteenth notes ([c], mm.32-41).68 The thematic scheme of these first three sections is restated in the next three (mm. 41-75), creating something like two theme groups, that is, in formal abbreviation, [A] ([a-b-c]) and [A'] ([a'-b'-c']). While the texture and style of both “Rustic, Vigorous” and both “Motoric” sections are essentially the same, the “Transparent” sections here differ in a more noticeable way: both feature the English horn as the prominent soloist, but the solo is backed by accelerating/decelerating articulated notes on a single pitch interspersed with hints of narrow trills in the former (mm. 13-31), while in the latter, the accompaniment takes on a tremolo guise—a similar level of background activity but with wider oscillation (mm. 52-67). Throughout the remainder of the piece, frequent thematic connections hearken back to the material laid out in the opening passages. The plan beyond this “exposition” of sorts (continuing with the sonata-form analogy) moves into something akin to a development, marked by a distinctive switch from English horn to oboe (and remains oboe for the rest of the piece). It is initially similar to the “Rustic, Vigorous” music ([a''], mm. 76-86), but then gradually diverges through a “Softening” transition (starting in m. 80) into a “Floating” section that resembles the even rhythmic minimalism of the earlier “Motoric” sections, but which is lighter and less emphatic ([d (c)], mm. 87-115). It glides with connected swells and ebbs, like dynamic breathing, where the timbres of the oboe and strings intertwine, blending with one another in a way that at times makes it difficult to perceive who is playing and who is not. Pitches stay rather steady here, until hairpins start to occur in rapid succession, transitioning into nothingness behind a descending glissando feature with harmonics in the cello part ([d (c)-trans], starting in m. 105, beat 3). This feature is subsidiary to the larger “Floating” section, but the solo cello is worth pointing out because it resurfaces later, suggesting another formal connection (see pages 55-56). 68Theme and theme group labels are given with brackets in the text. These labels are presented in their totality later, in Table 2.4, pages 71-72. 51 When the steady rhythms have died, the lilting scamper of the “Playful” section emerges ([e], mm. 116-146). This is a distinctively new theme, and gives the impression of a flock of seagulls, with the high quick leaps and grace-note embellishments of the upper voices against the harmonic “lifts” and “sighs” in the lower strings. The “Playful” idea relaxes gradually through the “Broadening” transition (starting m. 136) into a second cello solo feature marked with a slowing of tempo and faint, whispy glissandi in the background (starting in m. 141). Though thematically distinct from the earlier cello solo, it functions as an appendage to the “Playful” section in the same way that the former did with the “Floating” section. The oboe forms a sort of duo with this cello part, but both are brought to a grand pause (m. 145), which acts as a precursory formal marker, a bit prior to the exact moment of formal division (the first of two such grand pauses in the piece, making for another possible formal connection—see page 65). At this point, it becomes imperative to diverge a bit from the descriptive discussion of the formal sections and revisit the medium-long-short ordered patterning of the sections in Figure 2.1 (as mentioned on page 32). After the six expository sections, section 7 begins the “development” by evoking the theme of sections 1 and 4, correlating with the “Rustic, Vigorous” theme already observed (mm. 76-86). But from there, the material moves into new territory, and consequently so does the regular association of the medium-long-short ordering. At section 8, Wallin’s handwritten grouping brackets begin to appear in his precompositional table, continuing through the remainder of the table.69 The formal patterning gets offset: Now it seems that formal blocks begin with the longer section, forming long-short-medium groups. The short sections now act as a sort of glue between principal thematic material of the long sections and the transition of the medium sections into subsequent groups. The ordering hasn’t changed, but the grouping implied at the start of the piece has now shifted, and the formal role of the short sections has diminished...not surprising, since the general length of all sections is becoming more concise as 69For 190-91. a facsimile of Wallin’s precompositional table, see Figure A.1 in Appendix A, pages 52 the golden section approaches. The “Floating” and “Playful” sections already observed follow this new long-short-medium association (mm. 87-146). Returning now to describing the score, this brings the discussion to the contrastive “Sharp” section ([f (c)], mm. 147-162). As another new theme, it draws upon familiar elements from the “Motoric” thematic idea, with the reappearance of the constant pulsating rhythm, this time as steady eighth notes. It also draws heavily upon the “Rustic, Vigorous” thematic idea. In this case, the music gives the impression of a folk-ish dance with a tambour drum-beat, brought about through the pulsating col legno gettato in the cello and the detached playing of the upper strings. The shawm-like pitch scooping of the oboe transmutes the dance into an exotic fancy, but the idea crumbles (sadly, too soon) as another “Broadening” transitional indicator slowly increases rhythmic durations and softens the articulations (starting m. 154), until an expressive oboe line boosted by the sustained tremolo resonance of the strings emerges (fully present by m. 157). This notion was seen before in the second “Transparent” section ([b'], mm. 52-67), which suggests a reversal of the “Transparent” and its transitional “Onward” with the process found here in “Sharp” and its transitional “Broadening.” There is a visual correlation with this behavior in the pitch plots of Figure 2.1, comparing the corresponding sections 5 and 14: Both sections have a downward trajectory, but the pitch ambit of section 5 begins wide and then converges, while that of section 14 begins narrow and then spreads. “Sharp,” it appears, is a variation of the “Transparent” idea, which further serves the purposes of the “development.”70 “Sharp” and its “Broadening” together consist of a long-short grouping, and through a persistence of sustained tremolos into the music beyond, make the expected connection with the following medium-length section. However, though this medium-length section seems to function 70In this paragraph, “Sharp” has been likened to all three of the expository thematic ideas: “Rustic, Vigorous,” “Transparent,” and “Motoric.” As pointed out here, the character seems to be an amalgamation of “Rustic, Vigorous” with “Motoric,” but the functional purpose is more closely akin to “Transparent.” When the “Sharp” idea returns later in the piece, again it relates similarly to these expositional themes, in both character and function (see page 62). 53 principally as an extension/transition of the “Sharp” material, it claims its own distinction with its tempo micro-adjustment and “Playful, Innocent” label ([e'], mm. 163-170). There is reminiscence of the previous “Playful” idea here, but it seems to be a mere shadow of its former incarnation, focusing its activity only in the oboe and violin parts. The lower voices stay out of the way, and immediately change at the onset of yet another “Broadening” transition marker (starting m. 167) into fleeting, flautando scalar licks, foreshadowing the upcoming “Transparent, Evaporating” section. The return of “Playful” implies (with some imaginative stretching) a traditional formation nested within the greater work, with the sections “Playful,” “Sharp,” and “Playful, Innocent” ([ef-e']) being similar to the parts of a rounded-binary form, both in function and in the proportionate sizes of the sections (theme group [B], mm. 116-170, precompositional sections 11 through 16). The following “Transparent, Evaporating” section ([g], mm. 171-182) does not markedly resemble the other instances of “Transparent” in the piece, but the connection with “Transparent” implied in the label will return again as a point of discussion in conjunction with the more closelyrelated “Evaporating” section later on (see pages 64-65). The two labels occur in tandem only this once in the piece, and as the words together imply, the texture of this passage is very thin, with melodic fragments interspersed with breaths of silence. The strings play quick scalar runs, flautando sul ponticello, sometimes spiccato, with the violin ascending, the cello descending, and the viola alternatively ascending and descending. The oboe retains a more prominent role with its sustained pitches, though it also participates in the scale gestures. This subtle passage rather abruptly bursts into loud dynamics and accented trilling (starting in m. 178, [g-trans]), with distinctive cello downward pizzicati glissandi a few beats later (starting in m. 179, beat 4), a more deliberate and succinct execution of its previous gesture. All this together constitutes another 54 long-short-medium ordered grouping of precompositional sections (17 through 19), the final drive into the golden section.71 ning Following the Golden Section: “Recapitulation” and “Coda” It is at this point the familiar “Rustic, Vigorous” makes a return ([a'''], mm. 183-190). The character here is similar to that of its earlier renditions, though the attitude of the cello is more like what was seen in the latter part of the “Floating” section, with the articulated downward glissandi. (Apparently, the cello’s behavior here is carried over from its soloistic role in the previous section, which in turn connects reasonably with the other cello solo instances in the piece.) The subsequent “Softening” transition of the passage ([a'''-trans], starting m. 187) once again brings the cello part forward as the other voices back away. The discussion of thematic content shall now be put on hold again in order to return attention to Figure 2.1. At previous occurrences of the “Rustic, Vigorous” marker, there is a rather flat melodic trajectory (sections 1, 4, and 7), but here the plotted path is more of a curve, sloping slightly upward before plunging downward (section 20).72 Why does Wallin choose to associate this very different curvature with a thematic idea that was always flat before? Perhaps it is because he felt that this moment, being the golden section, needed to be made significant. In the figure, the outer pitch limits converge at this point to their narrowest span since the beginning 71Technically, the golden section of the piece according to the exact length of the score in seconds falls between beat 4 of m. 183 and beat 2 of m. 184, depending on how much of the introductory measure at the beginning of the piece is counted. At this specific moment, two things happen: the cello begins uttering a glissando motive (as spoken of in the following paragraph), and there is a small sixteenth-note’s worth moment of silence. Neither of these is particularly convincing as a salient event, so it is probably safer to simply approximate the golden section at the downbeat of m. 183, in conjunction with the formal marker “Rustic, Vigorous.” 72Section 20 is conjoint with section 21 according to Wallin’s brackets. Section 21 actually does exhibit a flat plotting, but its duration is very brief, and by this point in the score (mm. 189190), the character of “Rustic, Vigorous” has already dissipated away through the “Softening” transition. Having no real bearing on the preceding music, these few seconds are not enough to confidently posit a connection with the other flat precompositional instances of “Rustic, Vigorous” in Figure 2.1. 55 of the piece, so it makes sense to bring back the same musical idea found at that other narrow span. And indeed, there appears to be merit to this idea when drawing (once again) on the theoretical comparison with sonata form: At this stage of the piece’s progression, a “recapitulation” is due, and is here signified with a resounding return to the beginning musical idea of the work. This of course implies that the distinct change at m. 178 within the “Transparent, Evaporating” section ([g-trans], sections 18 and 19 in the figure) likewise serves in the role of “retransition.” But knowing that the pitch trajectory here is so different leads to a query about how Wallin does in fact translate such a melodically static theme into a dynamic one. Looking at this section in the score causes bewilderment at first blush, because the melodic activity apparently is stuck on one or two pitches, just as in the other cases of “Rustic, Vigorous.” It is true that the pitches are not expected to descend until the latter bit of the section, but the change in tessitura around m. 187 is so slight that it does not alter the feeling of this thematic idea substantially. With this observation, it is now an opportune moment to learn a bit about Wallin’s personal touch, that is, the way he takes raw data and creates music from it that is genuinely his own. Amidst the accentuated activity of the passage, the cello line here is the odd fellow. Each downward glissando is implicative of the shape found in section 20 of Figure 2.1—a quick falling gesture. This leads to the supposition that Wallin’s technique here is to take the overall shape of the precompositional version of this section, and somewhat loosely use that to guide individual gestures within the section, in addition to having the melody move through the precompositional shape’s path over the course of the whole section.73 Wallin’s choice of the “Rustic, Vigorous” theme pitted against the cello’s falling glissandi is further justified by simply considering golden ratio proportions themselves. Recall that the length of the whole piece is proportional by the golden ratio to the length of the piece up to this point, and that the composer purposefully manipulated the register (the outer-limit values for 73This technique for deriving motivic gestures from the shapes in the formal diagram is revisited in Chapter 5, page 176. 56 pitch through the course of the piece) to come together in a narrow span here, essentially dividing the piece into two parts (the two “fish” of Figure 2.1, which, of course, are also proportional to one another by the golden ratio).74 But there is also an approximately equal proportionality found within each of these two parts, with self-contained golden sections marked by the moments where the outer pitch limits are at their widest spans. The formal implications here are too compelling to ignore, and indeed, the previous instance of the cello glissando feature occurs precisely at this first wide point (section 9 in Figure 2.1). Wallin is thus evoking two prior significant formal instances to mark the arrival of a third significant formal instance, by relating the beginning of the first part (the narrowest point in the diagram of Figure 2.1) to the golden section of the first part (the widest point) at the golden section of the whole piece (which is the beginning of the second part). From this, it is reasonable to posit that themes and formal associations found prior to the golden section will be revisited throughout the rest of the piece in a corresponsive way (the “recapitulation”). But within these revisited themes and forms, alterations should be equally expected. In Figure 2.1 alone, it is visually clear that the general tessitura of the outer pitch limits is shifted higher, and that the image of the last part (sections 20 through 32) is a sort of inverted/retrograde mirror (mostly with regard to the outer pitch limits) of the first part (sections 1 through 19). Given this association, the compositional treatment here ought to be a variation of what happens in the first part, in which thematic ideas are now reflexively related and integrated. The discussion of form will now be resumed to see how this theory is substantiated. 74By way of reminder and reiteration for clarification, the “Pitch Lower Limit” and “Pitch Upper Limit” columns of Table 2.1 constitute the outer lines found in Figure 2.1, outlining the “fishes.” These values were not obtained from the data output of the logistic equation (equation 1.3, page 6, Chapter 1), but rather were imposed upon the other pitch parameters using the logistic equation (“Median Pitch” and “Pitch Spread (Range)”) in order to shear (skew) their results, to make practical ranges for performance and provide additional melodic interest. Wallin did, therefore, manipulate the data, but to stay true to the mathematical appeal of his aesthetic, his manipulations were not done by unrestrained whimsy, but by a logically purposed governing principle (the golden ratio). It makes sense that Wallin used this principle to shape the trajectory of the outer limits, maybe in tandem with (but not merely by) the idea of the fish shape. See pages 25-27. 57 The ordered sectional grouping of long-short-medium extends from the golden section’s “Rustic, Vigorous” passage (long-short) into a new “Sharp” section (medium) ([f' (c)], mm. 191196). It comes as little surprise that the “Motoric” rhythmic idea returns, since this happened in the previous “Sharp” section, though this time the attacks are more emphatic as staccato sixteenth notes. In “Motoric,” however, the custom is to have full harmonic chords played in rhythm, but in the present case, the idea is presented simply in unison. This is brief (understandably, since the sections near the golden section are concise), and quickly transforms through a “Broadening” transition (mm. 195-196) into the following half-tempo “Transparent” section ([b''], mm. 197-206). Here a very familiar treatment of “Transparent” is seen, with the oboe part saliently featured as a solo with tremolo string accompaniment, like that found earlier in mm. 52-63 (also in the “Broadening” transition of the “Sharp” section [f (c)], mm. 157-162, which borrows this theme from “Transparent”). This is the general behavior of the whole passage, though with a couple of nuances suggested by correlated precompositional formal divisions found midway, which include a brief moment when the violin and viola play harmonics in a rhythmic articulation mimicking the oboe (starting in m. 202, beat 4) followed by a short caesura (m. 203, beat 4). This makes the whole “Transparent” passage here another long-short-medium ordered grouping of corresponding precompositional sections (sections 23 through 25). The thematic connections of the “recapitulation” thus far with the “exposition” are quite clear, but the ordering of themes is slightly different. In this case, the “Transparent” and “Motoric” ideas of the original [A] and [A'] theme groups have now flipped positions, with the “Motoric” theme (in the guise of the “Sharp” section) now landing in the middle of the theme group. And the transition passages found within themes are different as well: Where in the [A] and [A'] groups, only the “Transparent” section had a transition (“Onward”), here the “Rustic, Vigorous” and “Sharp” themes bear transitions (“Softening” and “Broadening,” respectively), while “Transparent” has none. In spite of these differences, the many connections with [A] and [A'] are enough to label this theme group in kind, [A'']. 58 Making yet another quick comparison with Figure 2.1, resemblance appears between the melodic-trajectory shapes of the first theme group (sections 1 through 3) and those of this later group (sections 20 through 25), though in a different order and with some alteration. What was formerly flat, gradually sloping downward, and steeply jutting upward in the first three sections (corresponding to “Rustic, Vigorous,” “Transparent,” and “Motoric,” respectively) is now mostly gradually sloping downward, steeply sloping downward, and mostly flat (corresponding to “Rustic, Vigorous,” “Sharp/Broadening,” and “Transparent,” respectively). Seeing these differences is not completely surprising; after all, change is to be expected in a piece fashioned from fractal formula iterations with continually evolving outputs, and whose writer has made clear his avoidance of exact repetition.75 Still, the relationships between the “expositional” and “recapitulated” themes in light of these changes are interesting to observe. For instance, the two steeply-sloping curves mentioned here concur with “Motoric” in the first case and “Sharp” in the second. These sections correspondingly relate to one another thematically by their rhythms and relative lengths. But the thematic ideas are nevertheless unique, and their difference is also reflected in the opposing directions of their respective slopes in Figure 2.1. Then there are the “Rustic, Vigorous” and “Transparent” passages of the “recapitulation” (sections 20 and 21, and 23 through 25, respectively), which have exchanged their melodic trajectories from what was found in the “exposition” (sections 1 and 2, respectively). However, each of these passages contains within itself a hint of the original shapes: section 21 is actually flat, and section 24 is actually a brief downward slope. This ties the themes with their melodic roots without being exact reproductions (or so it appears precompositionally in Figure 2.1). Actually, the “expositional” and “recapitulated” versions of both the “Rustic, Vigorous” and “Transparent” sections exhibit a far greater likeness as played from the score than is suggested by the diagram of 75See Appendix B, page 216. 59 Figure 2.1 alone, the thematic connections being easily discerned aurally (see Figures 2.4 and 2.5 for comparative examples of both versions of each thematic idea). Finally, the relative duration of this theme group should be borne in mind. In sonata form, the restatement of this theme group in the recapitulation is expected to be close to the same length as in the exposition. While the initial theme group is made up of three distinct precompositional sections (sections 1 through 3, with the medium-long-short ordered grouping, [A]), here it contains six precompositional sections (sections 20 through 25, making two long-short-medium ordered groupings, [A'']). However, given that the precompositional sections are smaller at this point in the music (being near the golden section), these two pairs of groupings can be lumped together into an apparent medium-short-long association, as suggested by Wallin’s brackets (sections 20 and 21 = “Rustic, Vigorous” = medium; section 22 = “Sharp” = short; sections 23 through 25 = “Transparent” = long). This makes for a better comparison with the original theme group, both in overall duration and in the individual durations of each of the three themes, and can thus be heard as a return of the theme group. Even so, the question of why the themes are reordered (from medium-long-short to medium-short-long) remains. Before addressing this, it is best to keep in mind that a dogged oneto-one superimposition of traditional sonata form on this piece is fallacious and bound to fall apart. That was never the purpose to begin with. Rather, the purpose of the comparison is simply to derive a formal organization from the likenesses of thematic associations, and so far, the comparison has been remarkably successful. The case that an original theme group returns at all, albeit in an altered ordering, makes the comparison useful enough, so there is no need to furrow brows over minutiae.76 That said, there is one more possible connection here that is worth consideration, because it suggests another idea about why the themes in the group have been reordered. Recalling 76Sonata form itself is not completely codified after all. The recapitulation can restate the expositional theme groups in reverse order, or drop one of them altogether. 60 a. b. Figure 2.4: Comparison of excerpted passages from the “exposition” and “recapitulation” versions of the “Rustic, Vigorous” thematic idea. a. mm. 7-10. b. mm. 183-86. 61 a. b. Figure 2.5: Comparison of excerpted passages from the “exposition” and “recapitulation” versions of the “Transparent” thematic idea. a.mm. 17-20. b.mm. 197-200. 62 observations made about the earlier occurrence of the “Sharp”/“Broadening” labels in the middle of the “development” ([f (c)], mm. 147-162), it was likened to being a flipped variation of “Transparent”/“Onward” ([b'], mm. 52-67) (see page 52). So even though “Sharp”/“Broadening” has a different character from its predecessor this time around, the connotations associated with the label itself imply that the role of “middle theme” in the theme group has been transferred from “Transparent” in the “exposition” to “Sharp” in the “recapitulation,” by means of what happened in the intervening “development.” But on the other hand, can a case also be made for the new “Transparent” section here (mm. 197-206) as the last theme of this theme group? In the “exposition,” the “Motoric” section filled this role, which, given the (albeit forced) superimposition of the sonata form, seems to function like the modulatory transition between the two theme groups in [A] and as the codetta of [A']. In a traditional recapitulation, the transition between the theme groups is expected to occur without modulating. Here it simply does not occur: “Transparent” changes abruptly into “Rustic, Vigorous” (mm. 207-220) of the supposed second theme group, deeming a transition unnecessary. Moving forward with the sonata form comparison, a second theme group in the “recapitulation” should now be expected, reminiscent of the same found in the “exposition” ([A']), but this time functioning in the capacity of a resolution. Considering how the first theme group ([A]) was treated in its “recapitulated” state ([A'']) might lead to the expectation that new innovations and connections along the same lines should happen in the second. Returning to the score, the second theme group appears to fall between m. 207 and m. 253.77 It begins unsurprisingly with “Rustic, Vigorous” again ([a'''']), but takes a curious turn with the appellations of “Playful” at m. 221 ([e'']) and “Evaporating” at m. 237 ([g']), giving the theme group enough differences from [A''] to label it distinctly as [C]. These differences are worth exploring, especially 77Measure 253 is given here because it is what appears to be the ending measure of the second theme group due to the placement of formal thematic markers in the score. An argument is made later (on pages 64-66) for ending this theme group earlier at m. 244, because of the grand pause. See also footnote 78. 63 considering how they might act as resolutions to earlier presentations of thematic material in the work. The “Rustic, Vigorous” theme ([a''''], mm. 207-220), has an active and articulated character in the high tessitura countered by the cello, as in the previous “Rustic, Vigorous” section ([a'''], mm. 183-190), though here the cello remains melodically static on a (relatively) low steady pitch with crescendo swells. The anticipated falling motive appears a few measures into the passage (starting m. 211) as the music merges into the “Softening, Yielding” transition ([a''''-trans], starting m. 216), though this time the motive occurs in the upper three voices, and appears more like the gestures seen in “Transparent, Evaporating” ([g], mm. 171-178) with written pitches rather than simple glissando marks. This probably alludes to the upcoming “Evaporating” section, making a connection that binds the theme group together. The conclusion of this passage, with the oboe and cello in enharmonic unison (D♭4/C♯4 in mm. 219-220), marks the golden section of the last part (mm. 183-286—see the discussion about embedded golden sections on pages 55-56). There is something about the pitch played at this formal juncture that is particularly significant, due to the fact that the composer dwells on this pitch a long while after the active material of the following section has begun, well into m. 221. Not only is such a sustained emphasis drawing attention to the fact that this spot is another instance of the embedded golden section, but the pitch itself relates back to the beginning of the piece yet again. Of the limited scope of five pitches found in the beginning “Rustic, Vigorous” section, D♭4/C♯4 was articulated the most frequently, giving this pitch a strong significance that reemerges throughout the piece. This spot is a particularly poignant usage of the pitch, emerging starkly from the din of the surrounding activity, making no mistake that this is an important point in the piece’s formal design. The subsequent “Playful” ([e''], mm. 220-236) seems to be an unexpected character here at first glance, since this theme originated in the “development,” having nothing to do with the initial theme group. But consider the implications of its placement as the middle theme of group 64 [C]—there has already been a connection formed with the middle theme of group [A''], “Sharp,” back in the “development.” Recall that “Playful” and “Playful, Innocent” occurred there as bookends to “Sharp,” forming something akin to a self-contained rounded binary form (theme group [B], mm. 116-170). The ties of “Playful” to “Sharp” are further strengthened upon noting that the “Sharp” theme’s steady rhythmic identity (the connection already made with the “Motoric” theme in the original theme group) surfaces again midway through the “Playful” passage (initiated by the oboe, starting in m. 225), as its signature lilting oscillatory “bird calls” gradually stabilize into constant triplets with very restrained melodic shifting (which sounds very much like the “Motoric” theme, though in a high tessitura). While these connections seem to help bring resolution to the otherwise ancillary “Playful” idea, instrumental color might also be an important feature here. The unique usage of the oboe in this passage is rather new, suggesting that there is yet more reckoning to do. It seems to have precedence principally in the earlier “Transparent, Evaporating” section between m. 173 and m. 178, where the oboe plays sustained pitches intermingled with short scalar gestures. This might foreshadow the upcoming character in the subsequent “Evaporating” section as a way to thematically tie the passages together. But this is a lukewarm association at best. Rather it seems that a sort of middleground stepwise descent of the line is the thing at play here, which perhaps implies a connection with the process found in the final occurrence of “Playful” at the end of the piece (part of the “coda,” [e'''], mm. 263-286). Finally, the [C] group rounds out with “Evaporating” ([g']), constituting mm. 237-253.78 The section works functionally like the “Transparent” section of [A''] ([b'''], mm. 197-206) as the final theme of the group, but is in its melody and character like the “Transparent, Evaporating” section of the “development” ([g], mm. 171-182). This intersection of function and theme 78This “Evaporating” passage is arguably not completely contained by the [B] theme group, but instead dovetails after m. 244 with the following sections as part of the “coda,” as discussed in the following two paragraphs. 65 provides an explanation for the earlier combination of the two expressive terms (see also pages 5354). The credibility of this connection is substantiated by the fact that the profiles for the precompositional melodic trajectories conflate with the musical themes present in the score: in Figure 2.1, the shape found in sections 17 and 18 (equivalent to “Transparent, Evaporating” in the score) is more like sections 23 through 25 (“Transparent” in the score), but the score itself shows “Evaporating” ([g']) to be closer in character to “Transparent, Evaporating” ([g]) than “Transparent” ([b''']). The big formal issue at this moment in the piece is what to make of the grand pause in m. 244. It apparently occurs out of the blue, midsection, and the material on both sides of it is more or less the same. “Evaporating” is constituted of two precompositional sections (sections 29 and 30), but the timings assigned to them place the approximated sectional split in or around m. 250, making the actual placement of the grand pause formally illogical.79 And when it is compared with the previous grand pause, the issue does not make any more sense. As has been observed, the grand pause in m. 145 seems to anticipate the immediately upcoming formal marker, “Sharp,” in m. 147 (see page 51). In the present case, however, the grand pause is located nowhere near another formal division; in fact, it happens exactly in the middle of the “Evaporating” section. Still, even with the doubt this causes, a formal division at this moment of prolonged silence seems logical. While it does mimic the general goings-on in the surrounding texture, which contains several “breaths” of silent caesurae, it is too lengthy to simply be counted as such itself. Furthermore, by making a case of proportionality with the theme groups of the “exposition” compared to the theme groups of the “recapitulation,” the grand pause is favored as the cutoff to the second theme group over the actual end of the “Evaporating” section. The ratio of the lengths of the two theme groups of the “exposition,” [A] and [A'], is approximately 1.12. In the 79The part of the score correlating with precompositional section 30 is differentiated from the part corresponding to section 29 by the appearance of high harmonics in the strings, essentially fading into the background behind a solo oboe line. 66 “recapitulation,” the ratio of comparing [A''] with [C] more closely corresponds with that of the “exposition” when [C] ends with m. 244 (also approximately 1.12) than when [C] ends with m. 253 (approximately 1.48). From this, it appears that, in the context of the superimposed sonata structure, mm. 245-253 ought to be considered separate from [C]. An additional comment should be made again here concerning the groupings of the precompositional sections. As was demonstrated for the first theme group of the “recapitulation” ([A''], mm. 183-206), the three themes contained therein occurred in a medium-short-long ordering of their relative lengths, created by combining the smaller groups contained within (see page 59). The second theme group of the “recapitulation” ([C], mm. 207-244) exhibits the same type of ordering for its three themes, with “Rustic, Vigorous” as medium ([a''''], mm. 207-220, combining corresponding precompositional sections 26 and 27), “Playful” as short ([e''], mm. 220236, corresponding to precompositional section 28), and “Evaporating” as long ([g'], mm. 237244, consisting of most of precompositional section 29). Even when excluding the remaining measures after the grand pause, “Evaporating” is still the longest part of this second theme group. The remainder of the score from m. 245 onward can be chalked up to the “coda.” In traditional sonata form, a coda generally occurs after the final structural cadence of the piece. In the present context, however, the music just prior to the grand pause, though fairly cadential, hardly appears to be conclusive or have structural finality. But on close examination, it functions as a sort of anticlimax, and brings the “recapitulation” to a close in a rhetorically ironic way. This anticlimax comes in anticipation of (and in counterbalance to) the piece’s true climax, the “Strong” section ([h], mm. 254-262). The continuation of the “Evaporating” theme after the grand pause ([g'-trans], mm. 244-253) provides something of a runway into this climax, and might be thought of as a prolongation of the “final cadence” of the “recapitulation.” (It is noteworthy to point out that [g-trans], mm. 178, b.3-182, and [g'-trans], mm. 244-253, are not only thematically related, but also have a similar formal function, where the former operates as the retransition of the development into the recapitulation and the latter is the transition into the piece’s climax.) 67 The purpose of a coda in traditional sonata form is to prolong the thematic resolutions of the recapitulation and/or bring about the resolutions of any elements of the piece that have yet to be completely resolved. As for ning, it appears that its thematic ideas have all come full-circle already within its “recapitulation.” However, the lack of any clear structural climax up to this point is still problematic, and begs to be addressed in the “coda.” And it makes sense that, on some level, this climax should pull together ideas from the other themes of the piece. The interesting thing here is that “Strong” as a label is unprecedented, occurring only at this spot in the piece. As such, it is unique as a theme (which sometimes happens in a sonata-form coda), marked by several forte-piano accents followed by rapid crescendi and wide melodic leaps (initially, at least). However, it does borrow thematic ideas from other sections, and thereby serves to tie them together nicely. Here, the voices attack several successive pitches in unison, differentiating from one another only by articulation or embellishment, causing the pitches to be active and vibrant while melodic changes are kept minimal, a characteristic taken from the “Rustic, Vigorous” thematic idea. The decelerating articulated notes on a single pitch are also a prominent feature, hearkening back to the very first “Transparent” section ([b], mm. 13-31) and to the more recent employ of the motive overlapping “Playful” and “Evaporating” (mm. 235-240). And the special descending gettato glissando motive, which appeared in the cello part at the conjunctions of golden sections (mm. 105-112 and mm. 183-188), surfaces again at the end of “Strong,” though this time in all three string voices, offset from one another by a half note’s duration. There is another observation to point out. Of all the expressive demarcations used in the piece, “Floating” is the only other one to be used only once besides “Strong.”80 The character of these two sections could not be more contrastive (except for the shared descending glissando 80Technically, “Innocent” and “Yielding” also appear only once in the piece, but they do so in tandem with other markers (“Playful, Innocent” at m. 163 and “Softening, Yielding” at m. 216). The terms simply act as modifying words rather than as distinct sections or thematic ideas of their own merit. 68 motive), and yet there may be reason to think that these passages share a common trait, not in their content, but in the uniqueness of their functional roles in the overall formal scheme (as compared to the functions of the other thematic markers that make explicit returns). Whereas the unique function of “Floating” appears to be as transformative catalyst, developing and morphing the “Motoric” thematic idea into its “Sharp” guise (as has already been pointed out), the unique function of “Strong” is to converge several thematic ideas (as pointed out in the previous paragraph) into the climax of the piece. Although each of these passages is capable of being a theme on its own merits, in terms of function, they each serve other thematic ideas as a development or a resolution, and at rather significant junctures of the piece (near the beginning of the “development” and near the end, respectively). Yet “Strong” is not the ultimate resolution of all things. The piece closes with a final “Playful” section ([e'''], mm. 263-286), which, in terms of function, works as a bookendcounterpart to the opening section of the piece, the initial “Rustic, Vigorous,” resembling it melodically with its gradual unison descent through a tiny range of pitches. Both the “Rustic, Vigorous” and “Playful” thematic ideas have received a considerable amount of treatment and development, each having prominent appearances throughout the piece. In fact, the two thematic ideas have much in common stylistically. It seems that here both ideas are brought to bear. Taking another look at Figure 2.1, the sections associated with “Playful” prior to this moment all had rather curved melodic trajectories (precompositional sections 11, 16, and 28), but the flat pattern found in this last section is more often associated with “Rustic, Vigorous” (as in precompositional sections 1, 4, and 7). With this information, the connection of the two themes is clearly revealed, showing how the characteristic idiosyncrasies of “Playful” superimposed on the “Rustic, Vigorous” melodic plan bring both ideas full circle to finish the piece. And since “Rustic, Vigorous” exhibited a change in its precompositional melodic trajectory when presented in the “recapitulation” (curved more like that typical of “Playful,” occurring in precompositional sections 20 and 26), it makes sense too that “Playful” should experience a similar change in kind. 69 By the end of the piece, both “Rustic, Vigorous” and “Playful” have become very much like one another, their distinctions merging into a single thematic identity. Taken all together, the ordered grouping association of the three themes in the coda fall into a short-medium-long ordering of relative lengths. This ordering has not emerged previously, and is interesting to note since the notion suggests another completion of sorts. The ordering found in the theme groups of the “exposition” was medium-long-short, but after a stand-alone medium section (“Rustic, Vigorous,” [a''], mm. 76-86), the ordering shifted to long-short-medium grouping associations. This perpetuated through the “development” and into the “recapitulation,” where larger ordered groupings containing the smaller groupings were observed in its theme groups: medium-short-long orderings containing embedded long-short-medium orderings. With one last shift occurring with a long section (“Evaporating,” [g'], mm. 237-244, which doubles as the final long section of the second large-scale medium-short-long grouping), the remainder of the piece occurs with the last remaining rotation of the ordered groupings of relative lengths, short (“Evaporating” after the grand pause, [g'-trans], mm. 244-253), medium (“Strong,” [h], mm. 254262), and then long (“Playful,” [e'''], mm. 263-286). All three orderings for the groups have thus been used: medium-long-short, long-short-medium, and short-medium-long.81 Conclusions on Form in ning After seeing how the themes of the piece unfurl, the correlation with sonata form is compelling. There are two similar theme groups at the beginning of the piece, which are then developed and integrated with new thematic material in the middle, then restated with transformation before finally coming to a climax and conclusion at the end. The principal 81This progression through the “full circle” of ordered grouping rotations also brings the perceptual idea of finality and closure to the formal layout of ning as much as does the idea of return suggested in the “recapitulation” of the superimposed sonata form. The perceptual psychology of completion with musical patterns is treated at length in Leonard Meyer’s book, Emotion and Meaning in Music. Leonard B. Meyer, Emotion and Meaning in Music (Chicago: University of Chicago Press, 1956), 129-30, 151. 70 thematic ideas of the beginning theme groups evolve through the course of the piece into other ideas: “Rustic, Viogrous” finds its counterpart in “Playful,” the “Transparent” theme associates with the “Evaporating” theme, and “Motoric” reemerges as “Sharp” after passing through a metamorphic “Floating” section. The piece is also concerned with the grouping of thematic sections according to their relative lengths, with the exposition featuring medium-long-short ordered groupings, the development and recapitulation featuring long-short-medium ordered groupings, the recapitulation also featuring large-scale medium-short-long ordered groupings (associating the themes with those of the exposition), and the coda with a short-medium-long ordered grouping. Table 2.4 gives a succinct summary of how the sonata form structure aligns with the themes of the piece (identified by the sectional demarcations) and the relative lengths of the precompositional sections and their ordered groupings. The natural question that consequently arises is why this comparison works at all. It seems that in making the two-part form around the golden mean, the composer created a fertile bed for seeding thematic restatement and variation between the two parts, which is akin to the notion of sonata form (although this might not have been Wallin’s conscious intention).82 And the full-circle course of this piece (with its implicit return to origins, seen in the recurrence of melodic shapes throughout the expanding/contracting path represented in Figure 2.1) and its relationship to sonata form itself (with themes coming back after a journey through a developmental process) together suggest a narrative not unlike the life cycle of the very salmon evoked by the music (as implied in the title), travelling from their freshwater birthplaces out to sea and then back again to spawn and begin the cycle anew. 82Sonata form is a two-part form itself, being a large version of rounded binary form. The form places the exposition in the first part and the development and recapitulation together in the second part, or an |:A:|:BA':| thematic layout. The two parts of ning, however, reverse this layout, with the exposition and development together in the first part, and the recapitulation (with coda) in the second, i.e., AB|A'. 71 Table 2.4: Schematic for Sonata Form Structure Aligned with the Schematic for Sectional-Length Groupings in ning Starting Measure Sectional Demarcations Formal Groupings 1 a 13 27 “Rustic, Vigorous” “Transparent” “Onward” 32 (34) 41, beat 4 52 64 “Motoric” “Rustic, Vigorous” “Transparent” “Onward” 68 (70) 76 “Motoric” “Rustic, Vigorous” “Softening” 80 87 104, beat 4 (105, beat 3) 106. beat 3 116 136 141 142, beat 3 147 154 160, beat 4 163 167 171 178, beat 3 179, beat 4 “Floating” (cello solo feature §?) Relative Length Designation Ordered Groupings of Relative Lengths 1 Medium 2 Long MediumLongShort 3 4 Short Medium 5 Long 6 [7] Short Medium [8 9 Long Short 10] Medium [11] Long [12 13] Short Medium [14 Long 15] Short [16] Medium — [17 Long Retransition 18] Short [19] Medium A b btrans. c a' b b' b'trans. c' a'' b' a''trans. d (c) d (c)trans. Precompositional Section Number including Wallin’s groupings (brackets) A' a'' — d (c) (harmonics §?) “Playful” “Broadening” (q = 60) (texture thins §?) “Sharp” “Broadening” (viola enters §?) e etrans. e f (c) f (c)trans. f (c) “Playful, Innocent” “Broadening” e' e' e'trans. g g “Transparent, Evaporating” (accents §) (cello §?) gtrans. E X P O S I T I O N D E V E L O P M E N T B MediumLongShort Medium (Shift of Ordered Groupings) LongShortMedium LongShortMedium LongShortMedium LongShortMedium 72 Table 2.4 (continued) Starting Measure Sectional Demarcations Formal Groupings 183 “Rustic, Vigorous” “Softening” (cello §?) a''' “Sharp” “Broadening” f' (c) f' (c)trans. b'' 187 189, beat 3 191 195 197 202, beat 4 203, beat 4 207 216 219 220, beat 4 235 “Transparent” (harmonics §) a''' A'' a'''trans. f' (c) (caesura §) Ordered Groupings of Relative Lengths [20 Long LongShortMedium LargeScale Medium 21] Short [22] Medium [23] [24 Long Short LongShortMedium LargeScale Short LargeScale Long 25] Medium [26 Long LongShortMedium LargeScale Medium a'''' e'' e'' 27] [28] Short Medium (q = 75) e''trans. g' g' [29] Long “Evaporating” 244, beat 4 250, beat 2 254 263 (grand pause) a''''trans. g'trans. (harmonics §?) “Strong” “Playful” C Relative Length Designation “Rustic, Vigorous” “Softening, Yielding” (harmonics §?) “Playful” 237 a'''' R E C A P I T U L A T I O N Precompositional Section Number including Wallin’s groupings (brackets) h e''' — C O D A [30] Short [31] [32] Medium Long LargeScale MediumShortLong LargeScale MediumShortLong LargeScale Short Long (Shift of Ordered Groupings) ShortMediumLong LargeScale Long LargeScale Short Medium Long (LargeScale) ShortMediumLong 73 Knowing that the piece was generated from fractal formulae, the connections made with sonata form (and, by extension, other traditional forms involving thematic development or restatement) strongly suggest that its underlying patterns can be associated with the self-similar patterns of fractals. Metaphorically, the superimposition of traditional musical form onto a fractalbased piece is much like using Euclidian geometry to generically describe fractals themselves. Benoît Mandelbrot said that using Euclidian shapes to describe natural phenomena (mountains, coastlines, organisms, etc.) is to grossly oversimplify the reality of the formations, while fractals do a much better job of it.83 And yet it is through the process of simplification that classifications, associations, and parallels best emerge to cognitive perception. After all, even the most classic examples of movements in sonata form never completely conform to its schematic, but the form is nevertheless useful as a tool for identifying and grouping themes and musical processes in a clear and logical way. So, while the complexities of fractal mathematics may describe pieces like ning with greater accuracy than traditional formal analysis, idealized and simplified forms are still essential for interpreting and grasping these complexities. Yet while “purified” simplification represents one kind of ideal, another (and quite opposite) kind of ideal aspires to the rigor of precision in mathematical exactness. With the pendulum swinging to this other extreme, the music again falls short, belying the ways and the extent to which the composer was simplifying things himself. The music must not only be regarded from the perspective of simplification for the sake of comprehension, but must also be regarded from the perspective of detailed precision for the sake of arriving at an accurate understanding of the composer’s intentions. The music lies between these two ideals, and therefore, both perspectives are necessary for a complete picture of how it works. It is ironic to note that, in comparing an object against its ideals (whether it be the simplified one or the painstakingly accurate one), it becomes evident that the object itself is 83Cited in Peitgen and Richter, The Beauty of Fractals, v. 74 actually preferable to the ideals. The imperfections found in music (or any other art) are what provide interest, intellectual engagement, and aesthetic fulfillment, far more than a “perfect” ideal ever would. The comparison with the standard put forth by an ideal thus not only provides greater comprehension, but also exposes deviations that provoke questions and lead to further exploration. With ning, themes are never explicitly restated, but are repeatedly evoked with transformation.84 By analyzing the way that these thematic ideas are transformed with regard to both the ideal formal structure (laid forth in this chapter) and the notions propagated by the fractal mathematics (which generated the raw musical material from which the themes emerge), the deviations from these two ideals are revealed. This in turn provides commentary on the extent to which the composer used form and formula versus his own artistic intuitions and subjective aesthetic choices.85 In the present chapter, the formal connections drawn between the themes of ning suggest certain ideas about Wallin’s overarching conceptualization of the work: 1) Melodic shapes plotted precompositionally to determine the melodic trajectory of a section (start and end points for pitch, and the melodic path followed between them, as seen in Figure 2.1) are sometimes also used as the shape of individual motives found within a section. This depends on the thematic idea (that is, the musical “character”) which the composer imposes upon a given section of the piece. 84Constant thematic transformation or variation is not a foreign concept to sonata form. For example, the Chopin Ballade in F Minor, Op. 52, is classified as sonata form, but its themes are never restated verbatim, but are continually varied throughout the piece. 85Generally, form is thought to emerge naturally as a byproduct of the process of composition, so to say that the composer uses form to compose is usually an inaccurate representation. But in the case of ning, the composer indeed began with formal constraints in mind, prior to (or, more accurately, as a part of) the process of composition. The form itself, as the composer designed it, drove the other musical elements into being. However, the final outcome resulting from the composer’s intuitive response to these processes (his “fleshing-out” of the mathematically constructed skeleton) is the naturally emerging byproduct that is sonata form. 75 2) Thematic ideas are represented, not so much by a specific melody line (as in a true theme), but by their general character. There is no explicit repetition in thematic restatement, only continual development and variation with each appearance, though each reappearance of a thematic idea is similar enough to be recognizable as such, which in turn makes it possible for formal connections to be drawn between sections of the piece. There is a balance between variation and similarity with each return of a thematic idea.86 3) Thematic ideas can occasionally commingle, either by overlapping one another in transitional passages, or in places where a formal connection needs to “relate back” with something presented previously. In the case of the latter, often a motive generally associated with one thematic idea might appear in the context of another thematic idea’s “character” (for example, there is an allusion to the “Playful” thematic idea contained within a portion of a “Transparent” passage, found in m. 203). Thematic commingling allows for different thematic ideas to play similar formal roles in the context of the larger scheme (for example, “Transparent,” mm. 197-206, and “Evaporating,” mm. 237-244 [up to the grand pause], have the same role in their respective theme groups that was made possible through their previously established association in the “Transparent, Evaporating” section, mm. 171-182). 4) Connections between two sections with differing thematic ideas can also occur as the result of having similar formal parameters (those calculated with the logistic equation). (For example, the downward curving slope of the melodic trajectory of the “Playful” thematic idea as presented in the “development” [as in precompositional sections 11 and 16] appears in conjunction with the “Rustic, Vigorous” thematic idea in the 86This piece essentially uses the developing variation technique described by Arnold Schoenberg. Arnold Schoenberg, “Bach, 1950,” in Style and Idea: Selected Writings of Arnold Schoenberg, ed. Leonard Stein, trans. Leo Black (London: Faber and Faber, 1975), 397; See also Appendix B, pages 215-16. 76 “recapitulation” [as in precompositional sections 20 and 26]. Conversely, the flat melodic trajectory of the “Rustic, Vigorous” thematic idea as originally presented in the “exposition” [precompositional sections 1 and 4] is taken up by the “Playful” thematic idea at the very end of the piece [precompositional section 32].) 5) Thematic ideas come together to form theme groups. Theme groups reemerge in the piece (making it possible to identify a “recapitulation”), but the order of the thematic ideas within a group is flexible, and which themes occur as part of a group may change according to how formal associations have been established previously (such as the relationship of “Playful” with “Sharp” or “Transparent” with “Evaporating,” for example). 6) There is also a balancing act going on between the thematic ideas and the ordered relative lengths of sections (or lengths of section groups). Some precompositional sections are grouped together to represent a single thematic idea or formal section in the final score, which creates new relative section lengths in the theme groups, and embeds smaller ordered patterns within larger ordered patterns in the overall formal scheme. Now that comparisons of ning as a whole have been made against the “simplified” ideal of sonata form, it remains to do the same with the piece’s surface details against the opposing ideal, i.e., the standard of mathematically exact fractal formations. The next chapter is a review of the methods (mathematical and subjective) by which the foreground details of pitch, rhythmic duration, and amplitude were produced. From this, it becomes clear that the analytical approach to the surface details must necessarily be melodically (not harmonically) driven. Because the surface events are generated by sequential mathematical formula iterations, the structural content of musical events is consequently also sequential (and not due to simultaneously-sounding pitches). Therefore, in Chapter 4, the minutiae of pitch, rhythmic duration, and (to a lesser extent) amplitude are addressed from a melodic (sequential) standpoint, in accordance with (and compared against the ideal of) the way the mathematics function. From this analytical perspective, 77 the underlying structural patterns inherent in the melodic behavior of the music emerge, allowing for both thematic connections (between different passages of the piece) and hierarchical connections (between surface structures and the background form) to be derived. Indeed, it is now time to “zoom in” on the fractal, and spy the elements of the overall form lurking in the details. Such structural relationships, should they be found, will lend credibility to the theory that the same kind of self-similarity intrinsic to fractals also plays a role in the hierarchy of the piece, and, by extension, make connections with other existing theories about hierarchy87 (in which surface material reflects the larger form). 87The hierarchical analysis given in this paper draws parallels with ideas from Schenkerian analysis (albeit without a tonal basis), but also owes somewhat to ideas from developing variation and motivic design in structural levels (such as with Allen Forte’s analysis of Brahms’s String Quartet in C Minor), among other theories of hierarchical structure. Allen Forte, “Motivic Design and Structural Levels in the First Movement of Brahms’s String Quartet in C Minor,” The Musical Quarterly 69, no. 4 (1983): 471-502. CHAPTER 3 CREATION AND IMPLEMENTATION OF THE FOREGROUND MELODY So far, the discussion of the contents of ning has been devoid of any reference to specific pitches or pitch collections, and has merely touched on rhythmic patterning strictly in conjunction with form. Themes and treatment of voices (counterpoint, chords, etc.) have only been described generically, and usage of accent and dynamics has barely received attention at all. Even with the extensive treatment linking the piece with sonata form in the previous chapter, the discourse was entirely bereft of any mention of harmony, something that normally would be a prominent and crucial part of sonata form’s functionality. It is high time to give these things their due, and by so doing make some deeper connections. However, before exploring these specific aspects in depth (in Chapters 4 and 5), some additional preliminary items need first to be addressed. In the previous chapter, the formal aspects that were surveyed—sectional lengths, the melodic trajectory (including direction and curvature), melodic spread, the rhythmic trajectory (average rhythmic durations), rhythmic variability, general dynamic levels, and orchestration choices (all of which gave rise to certain thematic ideas, so identified by their general musical behaviors)—were by and large (with the exception of the latter two aspects) derived through the logistic equation (see equation 1.3, Chapter 1, page 6). There, pitch and section-length parameters were iterated within the context of preexisting formal shapes with golden mean proportions. In turn, the current chapter will focus on the approaches to and results of the other formula, the asymmetrical version of the tripartite set of equations from Jan Frøyland (see equation 1.5, Chapter 1, page 14), used under the formal constraints created with the first formula to generate the basic melodic lines from which each section of the piece is 79 composed out. These generated melodic lines will be referred to in this paper as foreground melodies, or simply the foreground melody of the piece (referring to them cumulatively). Applying the Frøyland Formula In order to make sense of how this second formula was used to create the foreground melodic structures, it is important to differentiate between elements that resulted strictly from the math and those that came about by the composer’s manipulations, including both those he imbued into the formula (his preliminary choices) and those he wrote directly into the music itself (his artistic/interpretive choices) (see Chapter 1, pages 16-20). Recall that there are three interdependent variables at work in this formula, x, y, and z, in which the initial value of each has direct bearing on the subsequent iterated values of the others. The values of these three variables represent the dimensions of a single note: pitch, duration, and amplitude, respectively.88 The succession of iterated values therefore determines the succession of melodic notes. All three variables are, in turn, affected by the “master control” elements c and d, which basically govern the extent of variability that the iterations of x, y, and z can exhibit, that is, how chaotically they change from iteration to iteration (see Chapter 1, pages 12-15). As with the x and r parameters of the logistic equation (see Chapter 1, pages 6 and 16-17), Wallin had to make preliminary choices about the initial x, y, and z values for the Frøyland formula, as well as determine values and a methodology for changing values for c and d. Where 88In most musical analysis, individual musical events are generally dubbed as “pitches” rather than as “notes,” for two reasons: 1) pitch and pitch relationships (harmony, progression, hierarchy, etc.) are usually the main focus of analysis, and 2) the term “note” normally indicates duration (quarter note, half note, etc.), which is subliminal (and in some cases irrelevant) to the discussion of pitch. However, “note” is also used colloquially to refer to any and all attributes of an individual musical event taken together. In this and subsequent chapters, analysis cannot be limited to pitch alone, since duration (and meter, where applicable) and amplitude (articulations and/or accentuations) play equally important parts in the conception of the musical material in ning, so simply calling musical events “pitches” would negate these other aspects. Because “pitch” fails to capture the whole essence of an individual musical event, the term “note” will be used in this discussion, with “pitch” being used only in the specific sense. 80 Wallin’s approaches to implementing the logistic equation were fairly similar (though unique) amongst his fractal pieces, his manner of using the Frøyland formula differed from piece to piece. For instance, in the third section of his ternary-form piece Stonewave (1990), Wallin used a linear progression to change the values of c and d over the duration of the section, with each incremental change in c and d being relative to the duration inferred by the value of y from the previous iteration. Wallin simplified the implementation of c and d in ning. Borrowing from the succession of values he had used in Stonewave, Wallin selected five points along the linear progression that, to him, seemed to be good representative samples of five general states of orderliness/chaos. He called them “positions” (meaning coordinate locations on the (c, d) plane) and assigned a cardinal number to each point (position 1, 2, 3, 4, and 5). For each formal section of ning, the (c, d) position number was determined just as the other parameters, through iterations of the logistic equation, though in this case, the five discreet values were assigned to five equal divisions of the total range for x (that is 0 < x < 1, with position 1 for 0 < x ≤ 0.2, position 2 for 0.2 < x ≤ 0.4, and so forth).89 Wallin’s decisions about the initial values for x, y, and z in the Frøyland formula, however, were a bit more subjective. Because each section of the piece had its own character (timing, tessitura, range, tempo, rhythmic variability, and melodic shape, i.e., the parameters defined by the logistic equation), he determined that it was desirable to have more control on the iterative output of each section separately, rather than use a single set of initial values to spur the iterations through the length of the entire piece. 89The exact c and d values associated with each position number are unavailable from either the composer’s notes or memory, but they are ranked in ascending order, 1 to 5, from most orderly (meaning x, y, and z iterate stable or regularly oscillating results) to most chaotic (meaning varied and unexpected iterated results). To get a good idea of some of the possible states the composer might have chosen for his positions, see Figure 1.5, Chapter 1, page 17. 81 Then, to simplify the process for choosing initial values for x, y, and z in each section, Wallin created five lists, of some 2,000 iterations each, for each of the five (c, d) positions.90 Each iteration in these lists offered a possible set of initial values in itself, and because of the nature of iterations, the successive iterations after any given starting point would be exactly as found in the list (i.e., no new x, y, and z values diverging from the list would ever be introduced).91 Though Wallin limited himself to using only the results of the iterations found in these lists, he nevertheless exercised a great deal of liberty in deciding what portions of the list to use in each section of the piece. Of course, the actual playback of the computer interpreting these lists into sequences of musical notes was not so straightforward. Every x, y, and z variable had to first be matched against some kind of gradated scale for pitch, duration, and amplitude respectively in order to render a sonic representation. Naturally, for Wallin, the referential scales for pitch and duration were to be established by the parameters derived from the logistic equation (“Median Pitch” and “Pitch Spread (Range)” for x,92 “Average Note Duration” and “Rhythmic Weight” for y, and “Shape” for both x and y—see Table 2.1, Chapter 2, pages 30 and 31), while the referential scale for amplitude would come from practical synthesizer programming restrictions (MIDI velocity 90It is unknown whether each of the five lists had unique initial values for x, y, and z, or whether they all had the same initial values in common, but even if the latter were the case, the subsequent iterations among the lists would still end up being starkly unalike from one another due to the different multiplicative factors of c and d for each. 91In other words, imagine five strings of 2,000 notes each, where each note has its own pitch, duration, and amplitude, and where each note is created by the formula on the basis of the value of the previous note. An arbitrary chunk from a string could thus be taken and used as a musical material for a section of a piece. 92Technically, the “Scale (Pitch Collection)” parameter also effects how the x variable is scaled, or rather, bent (meaning the value of x gets pushed downward to the nearest pitch in the current pitch collection, which is not necessarily the exact pitch to which x is scaled). This parameter is given in Table 3.1 and explained under the heading “The Crystal Chord Technique,” pages 85-94. 82 values for z).93 Because the pitch and duration parameters change dynamically in real time throughout the piece (according to the formal maps described in Chapter 2, pages 27-29 and 3334, and Figure 2.1, page 26), the referential parameter scales (into which the Frøyland variables x and y are made to fit) expand or contract accordingly right alongside each iteration of the Frøyland formula (specifically, at the interval of time determined by each adjusted iteration of the y variable: The y value is first exponentially weighted (squared so as to reflect a logarithmically associated scaling), then adjusted by its immediate “Average Note Duration” and “Rhythmic Weight” parameter values, which in turn are affected by their current “Shape” value).94 To get this to really work, however, there were still a couple of kinks to iron out. Recall that Wallin implements an asymmetrical version of the Frøyland formula (see equation 1.5, Chapter 1, page 13), in which the x variable is not determined by the value of z (at least not directly, though it is affected indirectly vis-à-vis the value of y in subsequent iterations). This resulted in a mathematical bias that caused the iterative possibilities (or attractor values) of the variables to fall out of balance between positive and negative values (such as manifested in Figure 1.4, Chapter 1, page 13, in which there are more possible positive attractor values [0 < x < 1.25] than negative [-0.36 < x < 0]). 93Since Wallin opted to not have any global trajectories for dynamics in ning (and hence no corresponding parameter created from the logistic equation), the amplitude readings derived from the iterations of z were produced at the same general scale throughout the piece: MIDI velocity values between 0 and 127. Rather than writing out these iterations as quirky vacillations between different loud and soft dynamic markings from note to note, the composer used his own intuition of musical dynamic phrasing and expression as the “referential scale” into which these iterative values were contextualized. Wallin then interpreted the output values of z merely as a blueprint for relative accentuation (marcato), articulation, and expression within the melodic line. 94In other words, the constraining parameters for each and every iterated “note” of the Frøyland formula will be different (usually slightly, occasionally starkly), changing from one to the next. Therefore, because of this additional processing through these fluctuating parameters, none of the lists of iterations (created by Wallin with the Frøyland formula) will ever sound in the actual music exactly as they were originally generated. 83 Knowing that he would need to crunch the formula’s results through the framework of the aforementioned formal parameters (those of Table 2.1), Wallin had to work out a solution for any values of x and y95 less than 0 or greater than 1 (the range of the logistic equation x variable defining the pitch and rhythm parameters, so that the values wouldn’t “explode” out of bounds in that context). In some of his pieces, he tackled the issue of negative values by using the absolute value of the iterations, but with ning, he decided instead to ignore the negative iterated values (which ended up almost being every other iteration, as positive and negative values tend to switch off with one another). As for managing values above 1, he took to normalizing the positive values linearly, so that all such values were proportionately scaled between the absolute range of 0 and 1. Once these Frøyland formula lists of iterations had thus been skewed and processed to fit within the constraints of the parameters from the logistic equation, the next step was to determine which segments from his iteration lists to use in each section of the piece. This meant that Wallin had to spend a great deal of time trying out different starting points for the Frøyland formula, pitting them against different combinations of values for the parameters from the logistic equation, processing them, and deciding whether he liked the results. In Table 3.1, the (c, d) position numbers determined for each section (see page 80) are shown next to the beginning iteration numbers from their respective iteration lists that Wallin ultimately decided upon, that is, the iteration corresponding to the first note of each section. Notice that some of Wallin’s earlier iteration choices are also provided in Table 3.1 for reference (struck through), giving insight into some of the composer’s struggles with getting at a satisfactory musical result. (The final column of Table 3.1 has not yet been fully addressed, but will be in the next section of this chapter.) It is interesting to investigate the choices that Wallin made with respect to each section. Although he could only draw from the list corresponding with the (c, d) position assigned to each 95Though constrained by global dynamic parameters in other pieces, in ning, the z variable was not limited by any parameters derived from the logistic equation (as already mentioned), and therefore is not mentioned along with x and y in this sentence. 84 Table 3.1: Formal Parameters for the Frøyland Formula and Pitch Collections in ning Section Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Position Number for c and d Paired Values 2 5 1 3 5 1 4 2 5 1 3 4 2 5 1 4 2 5 1 4 2 5 1 4 2 5 1 4 3 4 2 5 Starting Iteration of the Frøyland Formula (Handwritten) 261 261 511 412 201 166 111 140 511 415 184 482 571 111 611 172 275 177 41, 35 217, ? 415, 20? ?, 72 161, 222 555, 512 271 372, 211 444 375, 371, 512 481, 402, 406 276, 150, 88 176, 122, 70, 71 545, 441, 244, 141, 236 Scale (Pitch Collection) 0 1 4 0 1 4 0 2 4 1 4 0 2 3 2 3 2 4 0 4 0 3 1 4 0 3 2 3 1 4 0 3 section, there are inferences to be drawn based on the starting points he chose. For instance, some sections have iteration numbers that are the same (sections 1 and 2; 3 and 9; 7 and 14; 10 and 21; and 24 and 28) or within a relatively close range of one another (such as sections 4, 10, 21, and 29; 16 and 18; 17, 25, and 30; 20 and 26; 22 and 31; 20 and 26; and other possibilities), with a few of these also sharing the same (c, d) position number. In such cases, an increased chance of similar melodic patterns appearing between sections is strongly implied, even though other 85 parameters may differ. Then there is also the possibility of making connections between entirely different iterative starting points (especially if they share position numbers 1 or 2, i.e., the more “orderly” (c, d) positions, in which repetitions or regularly oscillating values are more likely).96 These kinds of correlations could be inferred by comparing those sections that are already known to have thematic material in common. Additionally, there may be something to be said about the iteration values that Wallin discarded, questions about which deeper analysis may provide plausible answers. Some of the connections suggested here are addressed in Chapter 4, where the melodic details and thematic ideas of select formal sections in ning are associated with the starting iterations and the (c, d) positions of the Froyland formula used in (and shared between) these sections. The Crystal Chord Technique The processes explained thus far are together quite sufficient for generating a foreground melodic tapestry from which to derive a written composition. This was certainly the case for pieces such as Onda di ghiaccio and Stonewave, and could have potentially been so for many more hypothetical compositions. But Wallin, ever the explorer, liked to see if he could add any additional ideas to his mix of methods, hopefully improving upon his technique (or, at the very least, “shake things up a bit”). Ning proved to be a particularly pivotal composition in Wallin’s technical development, because he devised for it a brand-new method for creating speciallyformed pitch collections, which he dubbed “crystal chords” (or the derivative, “crystal scales”). This new technique continued with Wallin into his later compositional period (long after he had set aside the fractal formulae) because it gave him interesting and useful pitch combinations in a systematic way, similar to the Messiaen modes.97 96See footnote 89, page 80. 97See Appendix B, page 218. 86 The crystal chord method forms pitch collections by taking a single pitch (usually a pitch somewhat close to C4, at a comfortable mid-range on the keyboard) and building intervals outward from it in three “dimensions,” creating an illusory cubical figure (hence the moniker) when diagrammed in the two-dimensional space of musical staves.98 One dimension appears vertical on the diagram (analogous to height), one appears diagonal going up-right/down-left (analogous to width), and one appears diagonal going up-left/down-right (analogous to length). Each dimension is identified by its distinct “germ” interval, and multiples of that interval occur within each dimension “stacked” above one another. These “stackings” and the cubical figures they form have the appearance of chords or chord-like arrays (which led to the name Wallin gave them), which implies one way in which the collection might be used harmonically. The original pitch from which these dimensions spring forms the center of the cubical figure, or “crystal,” which (depending on the multiples used for each dimension) may or may not actually occur as part of the pitch collection itself.99 Because this last fact can lead to some confusion, it is simpler for illustrative purposes to imagine constructing a crystal figure from its lowest pitch (which forms the “lower corner”) for the time being. From this low pitch, the outer edges of the crystal are easily built upward/outward, which in turn makes filling in the remainder of the cubical figure a simple process. Figure 3.1a gives one example of this technique in which each of the three dimensions is multiplied by two, that is, two intervals occur in each dimension by the specified interval size for that dimension. 98Wallin remarked that he was reminded of salt or pyrite, which, like fractals, create the same shape (cubical) when split into smaller pieces, i.e., self-similarity. Rolf Wallin, “Lobster Soup,” Nordic Sounds 1 (1998), rolfwallin.org, accessed Feb. 24, 2018, http://www.rolfwallin.org/ articles/c_5165891b1e98c026984eb382/. 99If all three dimensions have even-numbered multiples (2, 4, 6, etc.), then the originating center pitch will appear as part of the collection. But if any dimension has an odd-numbered multiple (1, 3, 5, etc.), this will cause the originating center pitch to not appear as part of the collection. However, it is possible that an enharmonic duplicate of the originating center pitch could occur elsewhere in the cubical figure. 87 a. b. c. Figure 3.1: Example of a crystal chord formation built on intervals of m7, M10, and M14. a. A 2x2x2 crystal array centered on G♯4. b. The same array in ascending scalar form. c. The same array in ascending scalar form enharmonically simplified. 88 Starting from a very low pitch of G0, other pitches are generated with one dimension built on the interval of the minor seventh (generating F1 above G0, and then E♭2 above F1, built upwards in what looks to be the “height”), another on the major tenth (B1, then D♯3, shown upwards to the left, or “length”), and the third on the major fourteenth (F♯2, then E♯4, shown upwards to the right, or “width”).100 The result of a “cube” created by multiplying the three dimensions by two (two m7 units, two M10 units, and two M14 units, or 2x2x2) is 27 distinct pitch points.101 These pitches are shown in normal ascending order in Figure 3.1b (and again in Figure 3.1c, adjusted for simplified enharmonic spellings using naturals and sharps only), constituting the collection (and changing a “crystal chord” into its corresponding “crystal scale”). Note that the pitches listed are particular to their register; no octave equivalencies or pitch classes are to be inferred from this collection. Figure 3.2 shows another collection (in ascending order) created from a crystal formation with the same interval dimensions as those of Figure 3.1a, but this time multiplied by four.102 This formation springs from an extremely low pitch of G♭–4 (G♭ negative 4, well below audio range at a frequency of approximately 1.4453 Hz) and expands to a high pitch of G♯#12 (A♯12, also well out of audio range with a frequency of approximately 119,337.92 Hz). Yet even though many of 100The major seventh/fourteenth (and by similar token the perfect fifth/twelfth) was a personal favorite of Wallin’s because it would avoid pitch-class repetitions as intervals stacked upward (as opposed to stacking octaves or tritones, where pitch classes recur repeatedly when stacked). See Appendix B, page 218. 101Of course, 27 is the mathematical result of 3x3x3, not 2x2x2. The 2x2x2 refers to the number of intervals (not pitches), that is, the illusory “boxes” created in the crystal formation in Figure 3.1a, generated between the pitches. The pitches occur at the “corners” of these eight “boxes,” which are themselves arranged in a 3x3x3 array. 102The collection shown in Figure 3.2 appears according to the way in which Wallin himself charted the pitches, eliminating inaudible extremes and enharmonic duplicates that would unnecessarily clutter the chart: the 20 lowest and 23 highest pitches found in the 4x4x4 crystal are missing, as well as 21 additional enharmonic duplicates. Because a diagram of such a crystal (with its 125 distinct pitch points [i.e., 5x5x5 pitches]) would be dense and visually confusing, it is left out here for practicality’s sake. 89 Figure 3.2: Partial example of a 4x4x4 crystal chord formation built on intervals of m7, M10, and M14, shown in ascending scalar form. the pitches in this collection are beyond practical use, the audible range shown is much more densely populated with pitches than is the 2x2x2 formation, thus providing a richer palate of melodic and harmonic options. The explanation provided above for building crystal figurations (starting from the lowest pitch and working upward) is the most straightforward way to construct their dimensions. But as has already been pointed out, these collections are generated outward from a center pitch rather than from the bottom up, and there is strong evidence to support this idea of pitch-collection centricity. Figures 3.1 and 3.2 are specifically related to one another around the originating center pitch in this way, where the collection for the 4x4x4 crystal contains all the pitches of the 2x2x2 crystal (in other words, the 2x2x2 collection is a subset of the 4x4x4 collection). If a complete 4x4x4 crystal diagram were provided here, the 2x2x2 crystal would be found exactly at the center, completely enveloped on all sides by the larger crystal formation.103 The pitch found in the very middle of both formations is G♯4, from which the same quantity of intervals spring out in all dimensional directions of the crystal formations: up and down, up-left and down-right, and up- 103For similar reasons of clutter (as mentioned in the previous footnote), the 4x4x4 crystal is not included here. 90 right and down-left, with one interval each way in the 2x2x2 array, and two intervals each way in the 4x4x4 array. Consequently, when the pitches are written in ascending order (Figures 3.1b, 3.1c, and 3.2), an intervallic palindrome spells outward from the center pitch of G♯4. The palindrome exists both with and without the enharmonic pitch duplicates (where, if they were listed, would show unison-equivalent intervals). Together with the equidistance of intervals from this center pitch in both sizes of crystal figures, the appearance of the palindrome not only confirms the centric role of this single pitch, but strongly suggests that the center pitch may also have some kind of centric role to play with regard to the piece in which these collections are used. Intriguing possibilities emerge from this pitch geometry, especially considering fractal-like correlations that could be drawn therefrom. Indeed, it is quite clear that Wallin’s special technique for generating pitch collections was fractal-inspired. There are a few more observations to point out regarding subset crystal arrays within larger crystal formations. First, because the same intervals are used in the respective dimensions of each of the array sizes, the intervallic content at the high and low extremes of both arrays is identical to a point, until the differing density in the mid-range takes over. Second, because the intervals in each of the dimensions are consistent, the 2x2x2 array can be shifted around within the scope of the 4x4x4 array by way of transposition. And third, other hypothetical sizes of cubical arrays could of course be extracted from the large crystal formation at many different levels of transposition (3x3x3, 1x1x1, or even boxes of different dimensional lengths: 1x2x3, 3x1x3, etc.). This is all well and good, and certainly in the pieces where Wallin used the crystal chord technique, he had at his disposal several ways to implement the collections he created, such as using subsets (as described above), extracting chords from notes connected along dimensional “lines,” or using centric symmetry in his melodic lines (from the ordered scales derived from the crystals), among many other possibilities. But in ning, where the technique was still new and yet to 91 be fully exploited in these ways, Wallin simply restricted his use of the collections to the pitch input from the two fractal formulas discussed in this paper. For ning, Wallin crafted four different scales derived from crystal formations, presented in Figure 3.3.104 Each of these scales has unique intervallic properties. Scale 1 is mostly comprised of half steps, with a skip of a major third occurring between every two or three half steps in a regular repeating sequential pattern: two half steps—one major third—three half steps—one major third— three half steps—one major third. If this scale were expanded to include the inaudible extremes of the corresponding crystal formation, this pattern would eventually break down as it approaches the extremes, but within the range given in Figure 3.3, the pattern is fully consistent. It is highly chromatic, but with noticeable gaps. Scale 2 is perhaps the most interesting of the four scales because of its greater variety of intervallic patterning (it is also the collection presented for illustration in Figure 3.2). The scale is more or less made up of half steps and whole steps, though traces of intervals beginning to widen near the extremes (as expected) occur in the range of this scale (minor thirds appear near the top and bottom). Often whole steps alternate with half steps, but this is occasionally offset by groups of two, four, or five half steps occurring consecutively. This means that in some moments, the scale is chromatic, while at others, it more closely resembles an octatonic pattern. Scale 3 exhibits an extremely homogenous pattern, alternating groups of four whole steps with groups of three half steps: mostly a whole-tone scale with chromatic spots. The regularity of the pattern makes it difficult to pinpoint the center generating pitch of the scale, which could potentially be D3, C♯4, or C5 (C♯4 is the most plausible, being closest to middle C).105 Scale 4 is (1992). 104Wallin 105Wallin used these same pitch collections in another fractal-based piece, Solve et coagula himself said that he strove to keep the centric generating pitch as close to middle C as possible, as this would allow most of the outwardly expanding pitches of the crystal formation to fall in a usable range. Rolf Wallin, e-mail message to author, March 12, 2017. 92 œ œ œ œ # œ œ & œ œ #œ œ œ œ # œ #œ œ #œ œ œ œ # œ # œ & #œ œ œ bœ œ œ # œ b œ œ ? œ #œ œ #œ œ œ œ # œ œ ? #œ œ œ œ #œ œ œ Scale 1 15 15 œ #œ œ œ œ # œ œ œ # œ œ #œ #œ œ # œ œ # œ œ # œ œ #œ œ #œ œ bœ b œ œ œ œ #œ œ œœœ œ # œ œ œ #œ œ #œ œ #œ # œ œ # œ b œ œ œ bœ œ #œ #œ œ œ œ # œ # œ #œ œ œ œ Scale 2 15 & & ? ? 15 œ œ œ #œ œ œ œ # œ œ œ bœ bœ bœ œ # œ # œ œ œ #œ œ œ œ œ œ b œ œ bœ bœ bœ œ #œ œ œ œ # œ # œ œ œ œ b œ œ œ # œ œ œ bœ bœ œ # œ # œ # œ œ œ œ Scale 3 15 & & ? ? 15 #œ œ œ # œ œ œ # œ #œ œ œ œ œ œ œ œ œ œ œ b œ œ œ b œ œ bœ œ bœ bœ œ b œ b œ b œ œ bœ bœ œ œ # œ # œ œ œ #œ œ œ œ # œ œ œ œ Scale 4 15 & & ? ? 15 Figure 3.3: The four crystal-scale pitch collections used in ning. 93 fairly homogenous as well, mostly made up of whole steps, interrupted with an occasional half step occurring between groups of three whole steps. However, this pattern is thrown off by a couple of groups of four whole steps appearing in the scale, which seem to occur after every three groups of three whole steps with half steps in between: one half step—three whole steps—one half step—three whole steps—one half step—three whole steps—one half step—four whole steps. This gives the scale an overall appearance similar to a diatonic scale, or something midway between a diatonic and a whole-tone scale. In ning, each precompositional section is assigned a different scale, as listed in Table 3.1 (in the “Scale (Pitch Collection)” column), with the numbers 1, 2, 3, and 4 referring to their respective scales, and 0 referring to the complete chromatic gamut (a pitch collection not derived from the crystal chord technique). Like the five numbers for the (c, d) positions (as mentioned on page 80), the numbers determining which scale is used in each section of the piece are derived from the logistic equation, with iterations falling into one of five equal divisions of the total range for x (0 < x <1 in the logistic equation) that correspond with each scale (with scale 0 for the range 0 < x ≤ 0.2, scale 1 for 0.2 < x ≤ 0.4, and so forth). Also like the (c, d) position numbers, the scale number is unchanging throughout the section’s duration. These crystal scales perform the final adjustment to the results of the x variable from the Frøyland formula. Once the iterated x values are normalized, with the negative values eliminated, and then processed to fit into the “Median Pitch” and “Pitch Spread (Range)” parameters, a pitch for each note can finally be obtained from the corresponding gradated scale for pitch between C2 and C8 (see the gradated scale given in Figure 2.1, Chapter 2, page 26). In sections using Scale 0, this final adjusted value for x is straightforwardly approximated to the nearest corresponding equal-tempered pitch. But in sections using one of the other four scales, the pitch must come from the scale. When the resulting pitch is found in the scale, that pitch is simply used, but when it is not found in the scale, the pitch “rounds down” to the nearest scale member below it. 94 Knowing which pitch collection is in use at any given time in the piece greatly aids with making certain analytical associations between sections with regard to their pitch content. In cases where the pitch parameters are similar and the starting iterations of the Frøyland formula are close, musical events (motives, pitch sets, and even long melodic phrases) with a certain relative affinity should appear among sections with the same scale number. Heterophony in ning At this point, it becomes necessary to address the “elephant in the room.” By now, from the observations made thus far about ning, it should be obvious that this piece was conceived in a horizontal (rather than vertical) fashion—Wallin’s precompositional processes were specifically designed to create a single monophonic sequence of notes.106 But what about harmony, or counterpoint? Given that ning is written for a quartet, how does the composer concoct four parts out of a single melody? Certainly, the score of ning appears to contain overlapping independent lines at times and blatant chords at other times. But there is no indication that Wallin made provisions for multiple voices in his precompositional fractal-formula schematic. Given his penchant for adhering to the rigor of mathematics, what, then, is to be made of this? Did the composer suddenly abandon his “prime directive,” and simply haphazardly thicken the texture by pure intuition? Surely, there must be a more reasonable explanation than this. That Wallin resorted to his intuition is only partly true. To better understand the composer’s approach to harmonization and counterpoint in ning, one must return to the original inceptive idea of the piece: the mysterious “force” keeping salmon together in their shoals. When the fish migrate, they swim as a unified body following an instinctual trajectory. But the fishes’ 106Of course, this fact alone comes as no surprise, since most Western music is melodically driven; harmony is either just the byproduct of multiple melodies working in counterpoint (polyphony), or the subsidiary accompaniment in which a prevailing melodic theme is couched (homophony), but in either case, the operative force at play is melody. Stephen G. Laitz, Instructor’s Manual to Accompany The Complete Musician: An Integrated Approach to Tonal Theory, Analysis, and Listening, 3rd ed. (New York: Oxford University Press, 2012), vi. 95 adherence to this trajectory is not strictly precise, as the fish must deal with multiple variables (current, obstacles, predators, etc.), so the shoal often deviates from its course, abruptly changing direction at times, scattering at others. Even still, the fish don’t entirely assert autonomy and swim off on their own. They maintain a certain comfortable space between one another, they all attempt to orient in the same general direction as one another, and they strive to swim in cohesion with one another.107 For Wallin, then, the trajectories he made with the logistic equation were analogous to the migration path followed by the fish, and the foreground melody generated with the Frøyland formula was similarly descriptive of the general way in which the fish would swim along this path. But with the requirements for these constraints met, Wallin was left with a degree of personal expression that he could write into the specifics of his score. After all, even though each fish swims in more or less the same way, there are still glimmers of individuality in the details of their immediate behavior: the timings of their tailfin swishes, nudging ahead or lagging behind the group, varying motions up and down or side to side, etc. Often the deviations of one fish relative to another are subtle, gradual, and nuanced, but occasionally, are stark and sudden (which usually effects the behavior of the other fish in turn). This sense of individuality embedded in a larger picture of unity is precisely the way in which Wallin thought of the different voices working together to create the textures of the quartet. Certainly, from a first glance at the score, the profile of each part appears quite individualistic, with a multiplicity of rhythmic figurations and pitches, differing from one part to another at almost any given time (with the exception of the rhythmic design of the “Motoric,” “Floating,” 107An interesting algorithm called “boids” has been developed by programmer Craig Reynolds, which emulates a similar condition for a flock of birds. The algorithm uses the three parameters mentioned here: separation, alignment, and cohesion. Craig Reynolds, Boids (Flocks, Herds, and Schools: a Distributed Behavioral Model), last modified July 30, 2007, accessed Feb. 24, 2018, http://www.red3d.com/cwr/boids/. 96 and “Sharp” themes, where all parts are found playing steady sixteenth notes). But a close examination reveals that the surface activity is merely embellishing a single underlying foreground melodic line, and the uniqueness of each part comes, not from an independence of pitch (in a true homophonic or polyphonic sense), but simply from the independence of its embellishing figurations and the way that they change in time relative to one another. It is as though each part, when viewed individually, seems to operate with a sort of restrained chaos (stochasticallycontrolled randomness), but when put together, the parts reveal that they have all been doing more or less the same thing. This makes ning an unusual case in the Western music repertoire. The four parts of the quartet do not function together in a typically polyphonic or homophonic fashion. By and large, explicit harmony is evaded: when chords do appear, they are merely the result of melodic embellishment. This piece is therefore correctly categorized as heterophonic, with moments of polyphonic or homophonic texture interspersed, appearing mainly as a byproduct of the heterophony.108 And because of this, Wallin had no real need to devise additional mathematic schemes to justify the use of plural voices. Wallin did, however, lay out three of his own “ground rules” for writing in heterophonic style, to keep the composition from straying too far from the foreground melody: 1) Once a note from the melody is initiated, it can last as long as desired, as long as it doesn’t impede the timing of the entrance of the next note in the sequence; 2) Any number of grace notes (or similarly quick notes), trills, or other similar ornamentations can come before or after a true melody note; 3) Any 108Although heterophony describes a single melody with embellishment and variation, this is not to say that in this case the heterophonic texture is completely devoid of a harmonic component. Given that differing embellishments between the four parts of the quartet can and do align vertically to create chords, harmonic analysis cannot be entirely dismissed (harmony is given treatment in Chapter 5, pages 165-172). However, harmonic set analysis (of the chord types and harmonic intervals used) is less relevant/useful to understanding what is really happening in ning without also contextualizing these pitch sets (and subsets) relative to the melody. But most of the analysis in this paper will only take a melodic point of view, since this more accurately reflects structure. See also page 101. 97 number of “echo” (or “pre-echo”) notes on the same pitch can come after (or before) the true entrance of a note from the melody.109 Consequently, the ensemble executes the foreground melody as a conglomerate in one of two basic ways (which occasionally overlap a bit): The melody either passes subtly from part to part (i.e., like a Schoenbergian klangfarben Melodie), or is shared by some or all of the parts at once in unison. A look at an excerpt from Wallin’s draft score for ning reveals his methodology for fleshing out his precomopsitional material into the final product of the score (see Figure 3.4). At the top of each system is the composer’s transcription of the foreground melody.110 Below this is the composer’s orchestration of the melody into the four parts of the quartet. The parts are enlivening and enriching the underlying melodic line, providing surface activity in the form of trills, tremolos, glissandi, special techniques, and other sundry articulations and auxiliary decorative notes. The parts are so closely bound to the foreground melody that even their registers are within a few steps of one another. And even though there is a great deal of surface activity, the composer has taken care not to completely obscure the actual notes of the melody. By the same token, the underlying foreground melodic line can be somewhat reasonably deduced from the score by reduction of the four parts into a single staff. The main challenge in making such a reduction comes in trying to sift out the composer’s embellishments from the actual elements of the foreground melody. For some sections of the piece, a reasonable approach is to eliminate ornamentations (grace notes, trills, very quick riffs, etc.) and focus on the notes with 109Rolf 110It Wallin, e-mail message to author, March 19, 2017. is important to point out that the melody as originally provided from Wallin’s computations was not tied to any meter or tempo. The rhythmic durations were given in more absolute terms (seconds), leaving Wallin to interpret the best relative tempo and metric fit. Hence, there is a small degree of approximation in the rhythmic values, but Wallin strived to be as faithful to the original melody as possible in his transcription, with the aid of quantization from his computer/synthesizer. It will be noted that the meter and/or tempo found in the final score differs in many places from that of the draft score, presumably with no alteration to the original rhythmic durations (though the timing discrepancies in Table 2.2 [Chapter 2, pages 36-39] has already cast a small amount of doubt on this). 98 Figure 3.4: Example from the draft score of ning, demonstrating one of Wallin’s techniques for orchestrating/embellishing the foreground melody (mm.237 b.3-243). 99 a weightier presence (durations of sixteenth notes or longer [in some contexts, eighth notes or longer], pitches duplicated between parts, notes with accented or emphatic articulation markings, etc.). However, this process is imperfect, and can result in a faulty representation, especially in cases where the actual foreground melody contains short rhythmic durations. In other words, some “ornamentations” in the score may actually be part of the generated sequence of notes, while other notes from the foreground melody might be absent in the score (due to a number of possible factors, most likely stylistic, but occasionally simply because of human error). Also, in some cases, other personal fingerprints of the composer’s style muddles the clarity of the foreground melody with “decoys,” such as pitches borrowed from elsewhere in the melody for the purpose of harmonizing, or making overlapping allusions to the thematic material of other sections (such as anticipations to the melodic theme of the upcoming section, or causing the thematic idea of the previous section to linger into the following) in order to obscure sectional divisions (for example, the persistent sixteenth notes in the upper voices that extend the “Floating” thematic idea beyond into mm. 105-111 over the cello solo line, or the quick scalar runs in the cello starting in mm. 167 happening prior to the “Transparent, Evaporating” section at m. 171). Because such compositional liberties obfuscate the exact entrance moments for the “true” melodic notes, extrapolating the foreground melody from the score through this process of deduction simply leaves too much to guesswork. And knowing that heterophony is the principal machine at work in ning, and that most ideas about underlying musical structure should therefore principally derive from the single line of the foreground melody, the foreground melody to be examined must be as accurate as possible. Therefore, for the purposes of making a reliable hierarchical structural analysis, the foreground melody to be examined in this dissertation will come from original source material of the draft score, rather than be deduced from the final score. Ornamentation and orchestration as found in the score will serve mostly to provide clarification, 100 cursory commentary, or supportive evidence to structural claims, but the meat of the discussion will come from observations of the underlying melody directly. Analytical Approaches With the understanding of the way in which the foreground melody of the piece is constructed (vis-à-vis the Frøyland formula results as adjusted by the logistic equation results), the kinds of analytical approaches that are most useful for ning (which will yield the most fruit in terms of making connections, deriving structural hierarchy, and positing the intent of the composer) become more clearly manifest. In the first place, the visual diagrams of the formulae (such as Figures 1.2, 1.4, and 1.5 in Chapter 1, pages 7, 13, and 15, respectively) evidence fractal patterns (self-similarity and complexity) for the attractor values of their variables. From this, it is logical to posit that fractal patterns are also manifest in strings of iterations of the formulae, the sequences of which are evident in the pitches, rhythms, and articulations of the foreground melody. Since fractals are by definition self-similar embedded hierarchical structures, there should be some way to find an in-kind hierarchical structure within the melodic patterns. Because the music is not tonal, the analytical method by which the musical hierarchy is to be divined must consider more than pitch relationships alone; rhythmic durations and accentuation relationships must be accounted for as well, and the pitch relationships themselves may need to be scrutinized from a different perspective. There are a few possible approaches to addressing this issue, but the approach ultimately deemed best for the present piece is contour theory (explained and used in depth in Chapter 4), for two main reasons: 1) contour analysis is particularly well-suited to dealing with sequences of musical events such as the foreground melody of ning (versus harmonic/simultaneous events), and 2) it allows a way to account for all three of the different relationships (pitch, rhythmic duration, and accentuation) and cross-compare them. Though the analysis will borrow the concepts of foreground, middleground, and background hierarchies from traditional tonal (Schenkerian) analysis, these terms will embody an altogether 101 different meaning in the context of contour theory,111 and, as will be shown in the case of ning, the different levels of hierarchy found in the pitch, rhythmic durations, and accentuation of the music will all demonstrate a kinship with fractal self-affinity, where small/local patterns are mirrored in larger/global patterns. Although the structural analysis of ning is primarily derived from melodically driven material, harmony and counterpoint do have a minor role within the work, and, as part of Wallin’s composing-out and orchestration of the foreground melody, will also be given some consideration (in Chapter 5). While counterpoint can also be demonstrated with contour theory to some extent, harmony will require a pitch-class set analysis perspective. The difficulty with this type of analysis is that it is not as relevant to what is really going on in the piece, and so will receive only cursory treatment. But it is still useful for showing intervallic relationships within a passage (which reflects the intervals present in the crystal scale used therein), and gives insight into the ways in which the composer uses the harmonic palate to bring added emphasis to the underlying structure. Finally, there will be some commentary on the composer’s artistic fingerprints on the piece, regarding the embellishing figurations and surface motives where the precompositional formulae have little to no bearing (also in Chapter 5). Though many of these items were addressed in the previous chapter, there are still some further points worth addressing regarding motivic connections in Wallin’s personal interpretation of (and dialectic with) his mathematically generated foreground melody. 111There is a slight difference between Schenkerian viewpoints and those used in this dissertation, regarding the concept of the relationship of the foreground to the background. In Schenkerian thinking, the background “comes out” of the foreground. However, in the present context (as demonstrated with contour theory), the background is likened to the “whole picture” of a fractal image, while the foreground is likened to the complex images discovered upon “zooming in” on “details” of the fractal. Though the foreground can be shown to “reduce” to a background structure, in reality, it is the background form that “complicates” into the local details of the foreground. See Chapter 4, pages 157-58. CHAPTER 4 STRUCTURAL ANALYSIS OF NING IN TERMS OF CONTOUR Sectional Comparisons from a Contour Perspective As mentioned earlier, a logical approach for finding connections within ning is to look at the melodic reductions for sections that are hypothesized to have similar foreground melodies due to their similar or identical starting iterations and (c, d) position numbers (see Table 3.1, Chapter 3, page 84). Yet, because the other precompositional parameters between these “kindred” sections will vary, the connections are not likely to be obvious at first glance. The question, then, that this comparative analysis should answer is: “In what way(s) specifically do these otherwise different passages show their common thread of origin?” The answer to this question may provide insight into the deeper structural workings of the piece. The first two excerpts presented here for comparative analysis are the original foreground melodies for precompositional sections 17 and 25, which correspond to mm.171–178 and mm. 203–206 in the score, respectively. The melodic line for these sections comes from the iterative sequence for (c, d) position 2, with section 17 beginning on iteration 275 and section 25 beginning on iteration 271. Because section 25 in its entirety is quite short, only the first three measures of section 17 are shown here (mm. 171–173), which correspond to the stretch of iterated notes matching up with those in section 25 (see Figure 4.1). As anticipated, these two excerpts do not initially appear to have much in common at the surface. Though they occur within general pitch ranges nearby to one another, their pitches differ completely, lacking any clear direct transposition transformational relationships (no 1:1 intervallic 103 a. b. Figure 4.1: Excerpted foreground melody reductions. a. mm. 171–173 (the first three measures of precompositional section 17). b. mm. 203, b. 4–206 (precompositional section 25 in its entirety).112 consistency). The durations of notes are also completely unique in each section, both relatively and absolutely (due to each section having a different tempo marking). However, by looking at the placements of accent marks, a correspondence emerges. Ignoring the accent marks in braces shown in Figure 4.1 (“{>},” indicating accent marks written in the score but not in the original foreground melody), the first thirteen notes of section 17 share the same accent pattern as the last thirteen notes of section 25, with only one exception (on the second note of each series of thirteen).113 This coincides exactly with the iteration numbers: If the starting iteration for section 25 is 271, making its first note, then iteration 275 occurs four notes later, which matches against the first note of section 17, also iteration 275. Given that the values for the z variable in the Frøyland formula are not affected by the parameters of the logistic equation, the coincidence of these accent marks makes perfect sense. The one discrepancy (on the 112For this and all subsequent figures featuring foreground melody reductions of the score, all markings that occur in the original foreground melody that are absent in the final score are noted with parentheses (“( )”), and all markings that occur in the final score which differ from the original foreground melody are noted with braces (“{ }”). 113In the draft score, Wallin uses only two types of accent marks (“>” and “^”) in his transcription of the foreground melody. There are no other dynamic or articulation markings, so these marks (and the absence thereof) are the only way in which the z variable of the Frøyland formula iterations is represented. Although this greatly simplifies the actual numerical values for z, general patterns between high, medium, and low values are still apparent in this notational system. 104 notes corresponding to iteration 276, accented in section 25 but unaccented in section 17) is forgivable as a possible oversight in Wallin’s transcription of the foreground melody (especially since Wallin did include the accent mark in the final score for section 17). Knowing exactly how and where the two sections correlate to one another, note for note, the pitch and durational relationships now become more apparent. Given that the “Shape” parameter is high and the “Pitch Spread (Range)” parameter is moderate (almost identical) for both sections, the pitch paths for each should bear reasonably close resemblance, with only slight skew due to the melodic trajectory ascending in section 17 and descending in section 25. The “Rhythmic Weight” parameter differs a bit more between the sections, but not so much as to completely demolish any rhythmic resemblances. Acknowledging these “fun-house mirror” distortions, the two excerpts manage to retain evidence of their common origin in the form of their overall contour, with the ascent/descent of pitches and the lengthening/shortening of durations coinciding extremely well. This is best represented visually by arrows depicting the contour between each note (see Figures 4.2 and 4.3). Again, between the two sections, there is only one instance where the contours for pitch fail to agree (between iterations 280 and 281) and only one instance where the contours for duration fail to agree (between iterations 281 and 282). In both cases, the disagreement happens where the value in section 17 stays the same between the iterations where the value in section 25 changes. However, the changes at these spots in section 25 are miniscule (F♯4 to E4, and e."3e to #"e), which explains why the corresponding expected changes at these moments in section 17 3 may have gone undetected: the changes were so marginal that they did not perceptibly manifest at those points under the influence of their slightly different parameter skewings. With this in mind, it is important to note that the discrepancies are not contradictory (i.e., one pitch ascending while the other is descending, or one duration lengthening while the other shortens), so the matchup is satisfied. 105 a. b. Figure 4.2: Directional contour arrows for pitch and rhythmic duration represented below each excerpt from Figure 4.1. For pitch: = ascending, = descending, = common tone. For rhythmic duration: = increasing, = decreasing, = repeated duration. a. mm. 171–173. b. mm. 203, b. 4–206. Iteration: 271 275 281 287 Pitch: mm. 171–173: | || || || | ||| mm. 203, b.4–206: Duration: mm. 171–173: || | || | | | || | mm. 203, b.4–206: Figure 4.3: Directional contour arrows extracted from each excerpt of Figure 4.2, aligned with iteration numbers for comparison, with disagreements highlighted. 106 Indeed, the concept of contour is the take-away from this analysis, the “smoking gun” that proves the common origins of both excerpts. A look at another pair of excerpts reveals nearly the same story. Precompositional sections 24 (mm. 202–203) and 28 (mm. 220–236, excerpting the beginning two-and-a-half measures only for comparison) both begin on iteration 512 for (c, d) position 4 (see Figures 4.4 and 4.5). Once again there is a great deal of agreement in the accentuation and in the pitch and duration contours, enough to confidently confirm the melodic connection between these two sections. The four discrepancies in the accent marks do not pose much of a problem, since each anomaly happens at corresponding points where no accent mark is given in one of the sections, which again is likely due to oversight or perception differences in Wallin’s melodic transcription. And the pitch contours match up beautifully, disagreeing on only two out of twenty-eight contour points, neither of which contain real contradictions. The duration contour profiles are a bit murkier, though, disagreeing on eight out of twenty-eight points, one of which does appear to contradict itself (between iterations 522 and 523). However, the difference in change at this contradictory point is very subtle: the rhythmic value decreases for section 24 only by a triplet-thirty-second note’s worth of value, and increases for section 28 by the same small amount. Moreover, the trajectory of the “Rhythmic Weight” parameter changes more dramatically over the course of section 28 (moving from 0.13 to 0.44) than it does over the course of section 24 (moving from 0.27 to 0.33 [refer back to Table 2.1, Chapter 2, pages 30-31]), providing the conditions under which more differences between the rhythmic contours could occur. One more excerpt is worth featuring here for comparison: Precompositional sections 4 (mm.41–51, excerpting mm.41–48 only for comparison) and 29 (mm. 237–250) (see Figure 4.6). Both sections are fashioned from the iteration sequence for (c, d) position 3. Wallin claims that section 4 begins on iteration 412 and section 29 begins on iteration 406, but there is some room for doubt about these numbers. Because these excerpts call into question the exact starting 107 a. b. Figure 4.4: Excerpted foreground melody reductions with directional contour arrows for pitch and rhythmic duration. For pitch: = ascending, = descending, = common tone. For rhythmic duration: = increasing, = decreasing, = repeated duration. a. mm. 202, b. 4–203, b. 3 (precompositional section 24 in its entirety). b. mm. 220, b. 4–223, b.1 (the first 2.5 measures of precompositional section 28). Accent markings: --^-> - --^->>-^--^--^->^--^->> || | | || | | | | | | | | | | | | | | | | | | | mm. 220, b.4–223, b. 1: - - - - > > - - ^ - > > - ^ - - ^ - - ^ - - ^ - - - - > > mm. 202, b.4–203: Pitch: mm. 202, b.4–203: ||| || | | || | || | || || || || || ||| | || mm. 220, b.4–223, b. 1: Duration: mm. 202, b.4–203: || || || ||| | || || mm. 220, b.4–223, b. 1: Figure 4.5: Accent markings and directional contour arrows extracted from each excerpt of Figure 4.4, aligned for comparison, with disagreements and contradictions highlighted. || | 108 a. b. Figure 4.6: Excerpted foreground melody reductions with directional contour arrows for pitch and rhythmic duration. For pitch: = ascending, = descending, = common tone. For rhythmic duration: = increasing, = decreasing, = repeated duration. a. mm. 41, b. 4–48 (the first seven measures of precompositional section 4). b. mm. 237, b. 2–250 (precompositional section 29 in its entirety). 109 iterations of each section, they are more interesting to compare, and as an added bonus, the process of comparison also reveals certain consistencies in contour patterns, things that will have a bearing on deeper structural relationships (see under the heading “Emerging Patterns and Connections,” beginning on page 120). When the pitch and duration contours of section 4 are aligned with those of section 29 at the anticipated point for iteration 412 (seven notes into section 29), there is agreement for the most part, but not quite to the degree expected (see Figure 4.7): For pitch, there are ten discrepancies, three of which are contradictory, and for duration, there are also ten discrepancies, two of which are contradictory. While the parameters from the logistic equation are demonstrably different for these two sections, this does not entirely resolve the issue of the discrepancies. For instance, the melodic trajectory ascends considerably over the course of section 29, while it descends very slightly over Pitch: mm.41, b. 4–48: || | || || | | || || | | mm. 237, b. 2–250: mm.41, b. 4–48 (cont.): etc. | || | | || || | | | | mm. 237, b. 2–250 (cont.): Duration: mm.41, b. 4–48: | | || || || | | | | || mm. 237, b. 2–250: mm.41, b. 4–48 (cont.): etc. || | || | | || || | | | mm. 237, b. 2–250 (cont.): Figure 4.7: Directional contour arrows extracted from each excerpt of Figure 4.6, aligned for comparison (the first note of precompositional section 4 with the seventh note of precompositional section 29), with disagreements and contradictions highlighted. 110 the course of section 4. Because of this, the expectation is that if a contradiction occurs, section 4 should be tending downward against section 29 tending upward. But only one of the three discrepancies follows this logic. Even considering that the “Pitch Spread (Range)” is initially wide for section 29, there should still be no instances where the melody descends in section 29 against a corresponding ascending melodic fragment in section 4, since the “Pitch Spread (Range)” there is always relatively narrow, and wouldn’t allow room for the melody to move in the opposite direction. Likewise, the rhythmic trajectory moves gradually from longer note durations to shorter ones over the course of section 29, while it maintains a fairly even keel in section 4. Therefore, the expectation in the case of a rhythmic contradiction is that the rhythm should shorten in duration in section 29 against lengthening in section 4. But again, one of the two contradictions here fails to meet this expectation, and the differences in “Rhythmic Weight” between the two sections should still not be enough to have a result like this. This is problematic, and the problem is confirmed in the accent pattern alignments for the two sections. However, by looking at the accent patterns of each section individually, a very clear association emerges between them, suggesting that perhaps the two sections are not being compared with one another in the correct configuration. Just like comparing two strands of genetic code, the accent patterns should be realigned to a position of “best fit” with one another to confirm a positive match. This turns out to correlate the first note of section 4 with the fourth note of section 29, not the seventh (see Figure 4.8). And when the pitch and duration contours are accordingly realigned to this position, the contradictions disappear entirely (see Figure 4.9). Most of the remaining discrepancies are due to common tones and repeated rhythms occurring in section 4 (with only a couple of exceptions), which agrees with the melodic and rhythmic trajectories found there (both barely changing due to the high value of the “Shape” parameter). With such a satisfactory fit, it is therefore reasonable to conclude that there must have been some kind of miscalculation with the starting iteration of one section or the other. Most 111 mm.41, b. 4–48: mm. 237, b. 2–250: ^--^--^-->->>->--^--^->>-->->-| || | || | || | | | | | || || | | | || | | | || ^--^--^--^-->-> -->-- ---^- >- -->->-- ^--^->>-^--^--^-> | || | | | | | | || mm. 237, b. 2–250 (cont.): ^ - - ^ - > > - ^ - mm.41, b. 4–48 (cont.): Figure 4.8: Accent patterns for precompositional sections 4 and 29, aligned for best fit, with disagreements highlighted. Pitch: mm.41, b. 4–48: || | || || | || || | || || | mm. 237, b. 2–250: mm.41, b. 4–48 (cont.): etc. | | || | || | || | | || | mm. 237, b. 2–250 (cont.): Duration: mm.41, b. 4–48: | | || || || | | || | || || | mm. 237, b. 2–250: mm.41, b. 4–48 (cont.): etc. || | | || || || | | || mm. 237, b. 2–250 (cont.): Figure 4.9: Directional contour arrows extracted from each excerpt of Figure 4.6, aligned for comparison (the first note of precompositional section 4 with the fourth note of precompositional section 29), with disagreements highlighted. 112 likely, the problem lies with section 29, since there is evidence here that Wallin went through a few experiments with different starting iterations (see Table 3.1, Chapter 3, page 84). The starting iteration there should probably have read 409. Contour Similarity So far, though, the comparison of contours has dealt only with direction of change (increase, decrease, or equal). To bolster the assertion of contour similarity requires a look at contour specifics, taking into account the relative size of melodic intervals and rhythmic changes, in addition to direction. An excellent tool for systematically evaluating the comparison of contours in a quantifiable way draws upon techniques developed by Elizabeth West Marvin, specifically the CSIM (contour similarity index) function.114 Before jumping into the CSIM function, it is easier to first visualize the comparison of two contours graphically relative to one another, accounting for the different intervallic sizes proportionately. This clarifies how the content in the ensuing comparison is derived. The pitch and rhythmic contours for the six sections previously presented are given for consideration (see Figures 4.10 and 4.11). To systematically compare two contours, all of the points within each contour (in this case pitches and durations) must first be evaluated against one another internally by means of a COM 114Contour theory arose as a discipline in music theory between the late 1980s and early 1990s (coincidentally around the same time ning was written), as a method for comparing musical events in generalized terms. Contour theory, codified in large part by Robert Morris (1987, 1993), borrows much of its methodology and formality from mathematical set theory in nearly the same way that Allan Forte used it to develop pitch-class set theory, with many of the same transformational operations. Many others have contributed to the theory as well, including Michael L. Friedmann (1985, 1987), Larry Polansky and Richard Bassein (1987, 1992), and Ian Quinn (1997, 1999), among several others (see Selected Bibliography). Elizabeth West Marvin’s major contribution to the theory was the CSIM function, expounded in her dissertation, which in turn was an expansion of her work with Paul A. Laprade. Elizabeth West Marvin, “A Generalized Theory of Musical Contour: Its Application to Melodic and Rhythmic Analysis of Non-Tonal Music and Its Perceptual and Pedagogical Implications” (PhD diss., Eastman School of Music, University of Rochester, 1988), 81-87; Paul A. Laprade and Elizabeth West Marvin, “Relating Musical Contours: Extensions of a Theory for Contour,” Journal of Music Theory 31, no. 2 (1987): 234-40. 113 a. A5- G♯3- b. D7- c. B♭4- C7- G3- Figure 4.10: Contours for pitch of precompositional sections in comparison.115 a. Precompositional sections 17 (green) and 25 (red). b. Precompositional sections 24 (blue) and 28 (orange). c. Precompositional sections 4 (purple) and 29 (tan). 115The contours presented in Figures 4.10 and 4.11 are only the aligning portions of the precompositional sections in comparison: the first note of section 17 is aligned with the fifth note of section 25 (thirteen total notes); the first notes of both sections 24 and 28 are aligned (twentynine total notes); the first note of section 4 is aligned with the fourth note of section 29 (forty-two total notes). Note that in both Figures 4.10 and 4.11, horizontal distance between points is not indicative of duration; all points are spaced horizontally equidistantly. Points are vertically positioned relative between the highest and lowest points, whose values are given at the left of each contour set (pitches in Figure 4.10, relative durations in Figure 4.11). 114 a. 23- b. 0- 60- c. 47.2- 2.4- Figure 4.11: Contours for rhythmic duration of precompositional sections in comparison.116 a. Precompositional sections 17 (green) and 25 (red). b. Precompositional sections 24 (blue) and 28 (orange). c. Precompositional sections 4 (purple) and 29 (tan). (comparison) matrix. COM matrices show how each member (point) of a contour relates to every other member thereof in terms of ascent. Tables 4.1 and 4.2 show the COM matrices for the contours of precompositional sections 17 and 25, for pitch and rhythmic duration, respectively. Table 4.3 presents the CSIM matrices for both pitch and rhythmic duration, which show how the COM matrices relate to one another (the CSIM function) within Tables 4.1 and 4.2, respectively. Because the main diagonal of each COM matrix has contour members relating to themselves, 116Relative durations are converted from standard rhythmic notation into numerical equivalents, with 0 representing % (the grace note), and every increment representing one 3Æ (triplet-thirty-second note). Figure 4.11a ranges from % to #"3#" & ; Figure 4.11b ranges from % to e; Figure 4.11c ranges from 5& to °._3e_5e. . 115 Table 4.1: COM Matrices117 for the Thirteen Pitches in Comparison Between the Pitch Contours of Precompositional Sections 17 and 25 G♯3 A♯3 C♯4 G♯3 C♯4 A♯3 A♯3 G♯4 G♯3 A3 E4 D♯4 A♯4 A5 D4 A♯3 A♯3 G♯3 A♯3 C♯4 G♯3 C♯4 A♯3 A♯3 G♯4 G♯3 A3 D4 A♯3 A♯3 F♯4 B4 F♯4 E4 D4 D♯4 B4 E4 E4 E4 F♯4 B4 D♯4 A♯4 F♯4 E4 A5 D4 D♯4 B4 E4 E4 117Though the usual procedure in contour theory (using the so-called combinatorial model) is to normalize contours (also termed “translating” contours) by giving rankings to contour points (that is, distilling the relative values of the points to 0, 1, 2, and so on, in cardinal order without any skipped ranks [for the purpose of finding prime forms and set classes]), it is not necessary to do so in a similarity evaluation, as the simple ascent/descent relationships are the same either way. Besides, a normalized contour can sometimes oversimplify some crucial details about the actual contour, especially if there are extreme peaks or troughs. Therefore, the original contour points are retained for Tables 4.1 and 4.2. Also, conventional annotation in COM matrices uses “+” or “+1” for ascent, “–” or “ –1” for descent, and “=” or “0” for no change, but in Tables 4.1 and 4.2, the same directional arrows seen earlier in the chapter are used for consistency. 116 Table 4.2: COM Matrices for the Thirteen Rhythmic Durations in Comparison Between the Duration Contours118 of Precompositional Sections 17 and 25 0 12 6 3 23 0 12 12 0 6.4 19.2 2.4 9.6 3 13.5 8 6 20.5 4.5 13 14 3 6 22.5 3 7.5 0 12 6 3 23 0 12 12 0 6.4 19.2 2.4 9.6 3 13.5 8 6 20.5 4.5 13 14 3 6 22.5 3 7.5 118The numeric values for the rhythmic durations given in this table are the same as those used in Figure 4.11, with 0 equal to % (the grace note), and every whole increment equal to one 3Æ (triplet-thirty-second note). 117 Table 4.3: CSIM Matrices for the Contour Points Being Compared119 in Tables 4.1 and 4.2, Respectively Similarity between the contours for pitch of precompositional sections 17 and 25 (Table 4.1). 1 1 1 0.5 1 1 0.5 1 0.5 0 1 0.5 0.5 1 1 1 1 1 1 0.5 1 1 1 1 0.5 0.5 1 1 1 1 0.5 1 1 1 1 1 0.5 1 1 0.5 1 1 1 1 1 1 1 0.5 0.5 1 1 1 1 1 0.5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.5 1 1 1 1 0.5 0.5 0.5 0.5 1 1 1 0.5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.5 1 1 0.5 1 1 1 1 1 1 1 1 1 0 1 1 0.5 1 1 1 1 1 1 1 1 1 1 1 0.5 1 1 1 1 1 1 1 1 1 1 0.5 0.5 1 1 1 0.5 1 1 1 1 1 1 1 0.5 0.5 1 1 1 0.5 1 1 1 1 1 1 1 Similarity between the contours for duration of precompositional sections 17 and 25 (Table 4.2). 1 1 1 1 1 0.5 1 1 1 1 1 0.5 1 119For 1 1 1 1 1 1 0.5 0.5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0.5 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0.5 1 1 1 1 1 1 1 0.5 1 1 0 1 1 0.5 1 1 1 1 1 0.5 1 1 1 1 1 1 0.5 1 1 1 1 0.5 1 1 1 1 1 1 1 1 1 1 1 0.5 1 1 1 1 1 0.5 1 1 1 0 0.5 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0.5 1 1 1 1 0 1 1 0.5 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 the convenience of visual correlation, matrix cells are color coded in Tables 4.1, 4.2, and 4.3. Cells that comparatively agree (matching arrow directions, or “1”) are uncolored, cells that comparatively “half agree” (“ ” matched against “ ” or “ ,” or “0.5”) are colored yellow, cells that comparatively contradict (“ ” matched against “ ,” or “0”) are colored red, and (as mentioned in the text) cells that are not counted for the CSIM function are colored gray. 118 and everything below the main diagonal is simply a reflexive inversion of everything above (backwards relationships), only those parts above the main diagonal are considered for the CSIM function (the part of Tables 4.1 and 4.2 that has not been grayed out, also reflected in Table 4.3). The CSIM index gives the degree to which two contours are considered similar to one another in terms of a ratio, with numbers approaching 1.00 being highly similar and numbers approaching 0.00 being highly dissimilar. To obtain the CSIM index for the two contours being compared, the corresponding cells of each of their COM matrices must be matched up against one another (as shown in Table 4.3). Wherever they agree, “1” is given in the corresponding cell of the CSIM matrix; wherever they contradict each other (an ascent matched against a descent), “0” is given; wherever they disagree but also do not contradict (a “no change” matched against either an ascent or a descent, or where they “half agree”), “0.5” is given.120 The value of all cells above the diagonal are then tallied and added, and the total value is then divided by the total number of matrix cells above the diagonal. When the excerpted contours from precompositional sections 17 and 25 are compared in this way, their CSIM indexes result as approximately 0.8910 and 0.8974 for pitch and duration, respectively. These values are indeed relatively close to 1.00, quantifying the high degree of similarity that previously was merely suggested in the alignment of their ascents and descents (Figures 4.10 and 4.11). The only reason for any dissimilarity at all is in the different directional skews imposed on each section by the formal parameters from the logistic equation (which, in spite of everything, only manages to differ the contours from one another by about 11%). It is also gratifying to see how close the CSIM indexes for both pitch and duration are to one another, 120Using “0.5” is a modification on the CSIM function attributable to Ian Quinn. Marvin’s original CSIM function considers all disagreements to be “0,” but this fails to account for the fact that instances of “no change” are not exactly opposite to either ascent or descent. As per the argument being presented in this paper, moments of “no change” (common tone or repeated rhythm) are not considered contradictory, and therefore should be given their proper due. Ian Quinn, “Fuzzy Extensions to the Theory of Contour,” Music Theory Spectrum 19, no. 2 (1997): 255. 119 showing once again in no uncertain terms how closely each of these musical dimensions is tied to the other. But perhaps the greatest confirmation for this connection comes in the composer’s use of his thematic ideas: Both precompositional sections 17 and 25 appear in the context of the score as part of the “Transparent” thematic idea, linking their formal ties by their formulaic ties. The relationship of these two passages becomes more than just a stylistic likeness or aural association, but a literal recurrence of theme (at least for thirteen notes-worth), somewhat transposed and with some slight transformations through augmentation or diminution. CSIM indexes for precompositional sections 24 and 28, and 4 and 29, are also obtained in like manner as those for sections 17 and 25 (though the COM and CSIM matrices for these contours are not presented due to their larger sizes being difficult to represent visually—compare the sizes of Figures 4.10a and 4.11a with Figures 4.10b, 4.10c, 4.11b, and 4.11c, and consider the sizes of Tables 4.1, 4.2, and 4.3 with respect to the relative size of figures 4.10a and 4.11a). The sequences of twenty-nine notes being compared between sections 24 and 28 exhibit a moderatehigh degree of similarity, with CSIM indexes of 0.7352 and 0.7217 for pitch and duration respectively. Though these contours are not highly dissimilar, the commonality of their origin is much more obscured by the parameters from the logistic equation than are those of sections 17 and 25. This is particularly true in the case of the duration contours, where repeated rhythms and a limited assortment of rhythmic values tend to “white-wash” the real extent of contour activity originally generated in the formula iterations. As for pitch, the difference is largely due to the very different melodic trajectories of each section. But regardless, the CSIM index values are still above 50%, which means that the contours are more similar than they are dissimilar, a conclusion that is fairly obvious from the contour graphs alone. Moreover, there is a definite thematic connection between these two parts of the score, aurally perceptible. Although precompositional section 24 (m. 202, b.4-m. 203, b.4) 120 appears in a “Transparent” section of the piece, it functions thematically as an allusion to the upcoming “Playful” thematic idea, precompositional section 28 (starting m. 220, b.4). The CSIM indexes for precompositional sections 4 and 29 appear to fare better, though, with 0.7938 and 0.7898 for pitch and duration, respectively. This is quite remarkable given the forty-two-note span over which these contours have been matched up with one another (aligned as shown in Figure 4.9), but even more so when considering the extremely wide ranges of the pitches and rhythms in section 29 (mm. 238-250) in contrast with the restricted ranges for pitch and rhythm in section 4 (m. 41, b.4-m. 51). In spite of these differences in size, both sets of contours differ from one another only by about 21%; it is the background contour trajectories (that is, the melodic and rhythmic trajectories) that are the biggest contributing factors to this differentiation, with section 29 ascending overall in pitch and descending overall in duration, against the fairly steady contours of section 4. Thematically, however, it is precisely these large overall differences (in both size and direction) that result in two very contrasting formal sections of the piece. There is no indication in the score that Wallin was attempting to make a connection between them. Precompositional section 4 corresponds to the “Rustic, Vigorous” thematic idea—section 29 is the first part of the “Evaporating” passage. This is a case where, in spite of the high degree of contour similarity, the differences in the parameters of the logistic equation are drastic enough to obscure their connection aurally. But this was probably also a deliberate compositional choice (given Wallin’s intentional use of close starting iterations) by which he was seeking to demonstrate how the same original material can be varied in such strikingly unique ways. Emerging Patterns and Connections In spite of the thematic contrast, the melodic and rhythmic similarity of contour (rooted in their shared commonality) between precompositional sections 4 and 29 is very good, at least with the contours aligned as proposed in Figures 4.9, 4.10c and 4.11c. Still, even when the 121 contours for sections 4 and 29 were aligned at the mistaken presupposed position (see Figure 4.7, page 109), the amount of agreement was nevertheless remarkable in spite of being incorrect. In fact, the CSIM indexes for the contours when compared for this alignment are still quite good— that is, more similar than dissimilar—with indexes of 0.6862 and 0.7240 for pitch and rhythm, respectively (this compares only thirty-nine matched pairs of notes). It must be more than mere coincidence that the overall concurrence of the two sections’ contours is still mostly retained after being shifted by three notches. This strongly implies the presence of a regular pattern. This suspicion is further confirmed by moving the contour of section 4 by three notches yet again, aligning the first notes of both sections 4 and 29 with one another (see Figure 4.12). Once again, the similarity is very close and concurrence very high, with only a handful of contradictions. And the CSIM indexes for pitch and rhythm contours at this alignment are still Pitch: mm.41, b. 4–48: || | || || || | | || || || | | mm. 237, b. 2–250: mm.41, b. 4–48 (cont.): | | | || | || | || || mm. 237, b. 2–250 (cont.): Duration: mm.41, b. 4–48: | | || || | | | | | || | | | | || || | || | mm. 237, b. 2–250: mm.41, b. 4–48 (cont.): | | || || mm. 237, b. 2–250 (cont.): Figure 4.12: Directional contour arrows extracted from each excerpt of Figure 4.6, aligned for comparison at the beginning of both precompositional sections 4 and 29, with disagreements and contradictions highlighted. 122 good, even slightly better than those for the alignment shown in Figure 4.7: 0.6934 and 0.7247 respectively (comparing forty-five matched pairs of notes in this case). Since the contour comparisons bear such great resemblance when shifting at increments of three, the pattern must replicate at about every three notes, with only occasional disruptions in its regularity. In the case of pitch, the principal contour motif that is reiterated is a For duration, it is a pattern. pattern (which is merely a rotation of the motif for pitch). And even though the contour for amplitude is more difficult to discern (with such a limited set of accent marks used to represent a wide array of numerical values), it still bears the tell-tale evidence of two increases in value for every decrease. Instances of common tones or repeated rhythms (“ ”) are actually instances of increase or decrease in disguise (thrown off by logistic equation skew and/or iterative values that change within an imperceptible margin), and so do not truly interrupt the pattern.121 When the pattern is disrupted, however, a contour-motif grouping of either two or five increments is interjected between the regular incremental groupings of three, breaking up the normal repetition either by featuring two consecutive decreasing values, or one isolated increasing value. Generally, the groupings of two appear as isolated groupings of five appear as one another: the and and groups interrupt the or or contours, while the (all of which, again, are simply rotations of groups interrupt the regular pattern, while the regular pattern). For pitch, duration, and (as far as can be discerned) amplitude, these disruptions occur in roughly the same spots122: between 121The contours for amplitude tend to have many more instances of repeated (“ ”) accent marks (the lack of accent mark being the most common type) than its pitch or duration counterparts, due to the oversimplification of the z variable values. Often it is easy to logically guess the true direction (increase/decrease) associated with such moments, based on recognizable patterning, but pattern interruptions can make such guessing less reliable. 122The iteration numbers given here assume that 409 is the true starting iteration for section 29 (as stated on page 112), and are relative to the alignment in Figures 4.8 and 4.9. 123 iterations 421 and 429, between iterations 438 and 443, and 446 and 451123 (refer to Figures 4.8 and 4.9 for the correct iterative placements). The close proximity of pattern disruptions between the three dimensions (pitch, duration, and amplitude), together with the regularity of the pattern itself appearing in all three dimensions, jointly attest to the behavior of iteration values in the Frøyland formula. In this case, under the conditions set by the c and d values for position 3 (the position used for precompositional sections 4 and 29), the iterated values for all three variables of the formula have a propensity to increase twice for every decrease, but whenever there is an exception to the usual behavior, there is still a consistent expectation as to what the exception will be: either an isolated increase/decrease pair, or a double decrease with an increase coupled with an increase/decrease pair. In the observations made here, never does there occur an instance of three or more consecutive increases or decreases, and never is there an exception to the regular rule-of-three that consists of one, four, six or more contour increments—they only occur as groups of two or five. It is logical to say that these samples are representative of the contour patterns formed by all of the iterations for (c, d) position 3. Interestingly, though, the double-increase/decrease pattern and its related pattern-exceptions (groups of two and five) are also found in other (c, d) positions. This is true for position 4, as implied in the contours for precompositional sections 24 and 28 previously shown124 (see Figures 4.5 and 4.10). Position 2 also appears to have the same patterns 123More specifically, the disruptions in the pattern occur in each of the dimensions thus: for pitch, between iterations 424 and 429 ( ), 438 and 443 ( ), and 449 and 451 ( ); for duration, between iterations 424 and 426 ( ), 438 and 443 ( ), and 446 and 451 ( ); for amplitude (as far as can be discerned), between iterations 421 and 426 ( ), 438 and 440 ( ), and 449 and 451 ( ), with one additional questionable contour direction between iterations 434 and 435 ( ?—it does not throw off the regular grouping of three in the pattern, but it does point in the wrong direction…possibly human error?). 124These sections begin on iteration 512 with a rotation of the pattern in their pitch contours, with one pattern-exception of occurring between iterations 524 and 529. Their rhythmic contours begin on the rotation of the pattern, and seem to run into their pattern-exception of between iterations 521 and 526, though the contradiction between iterations 522 and 523 makes this less certain. 124 based on the contours of precompositional sections 17 and 25125 (see Figure 4.3 and 4.10). Both of these cases are affirmed in the continuations of sections 17 and 28 (mm. 174-178 and mm. 223236, respectively, not included in the figures) where the patterns perpetuate. In fact, the double-increase/decrease pattern (along with its two specific patternexceptions) is ubiquitous to a great extent in the foreground melody of the entire work. The only spots where the pattern falls apart (showing more than two contour increases or decreases in a row) are where the parameters from the logistic equation bend the melodic or rhythmic trajectories steeply and/or greatly limit the Pitch Spread (Range) or Rhythmic Weight, forcing the pattern’s normal alternations into a single direction or repetitiveness. Generally, the difference between the contours of the five (c, d) positions merely boils down to the frequency with which the double-increase/decrease pattern is interrupted by its pattern-exceptions. Naturally, the less chaotic positions result in fewer pattern-exceptions than the more chaotic positions. The beauty behind chaotic iterative processes is not only in their varied and unpredictable surface features (the actual pitches, rhythms, and accents generated), but in the regularity of their underlying tendencies to increase or decrease, pulled into certain gravitational orbits by strange attractors (the contours of pitch, rhythm, and accent). Contour Middleground (Analytical Excerpt from the “Floating” Section) The contours of the foreground melody are couched in the background of the contours of the form, that is, the melodic and rhythmic trajectories. But true to Schenkerian modes of thinking, there ought to be some sort of middleground contour representation that links the two extremes. If this can be properly identified, the patterns contained therein should provide a structural connection between the contour patterns of the surface and those of the deeper form. 125Though the excerpts in these figures are brief, the pitch and rhythmic contours both follow the pattern starting at iteration 275 (where the sections’ contours match up), with a pattern-exception of occurring in both contour excerpts: between iterations 278 and 283 for pitch, and between iterations 275 and 280 for rhythm. Other pattern-exceptions are hinted at in the extremities of the excerpts (where they do not match up). 125 As will be demonstrated, a sense of self-affinity will manifest through such analysis, showing that smaller patterns are embedded in larger ones, and affirming the existence of the fractal structure hypothesized to permeate this piece. The tricky part, however, is determining how exactly to derive a middleground contour. It must draw from structural cues, which are signaled in the foreground and which somehow tie in with the background trajectory, but in terms of contour, what are those? Because of the case being made for pattern detection, one approach might be to parse the contour by its pattern groupings, i.e., divide the contour into two-, three-, or five-increment pieces (based on whichever groupings are found in the foreground), and then trace some kind of underlying macro-contour with fragments126 that connect piece to piece (such as connecting the first point of each piece, for example). This is logical, but because the contour patterning in ning is susceptible to having differing parsing interpretations based on pattern rotation, this may not always lead to a consistent or reliable middleground contour representation. Another approach that may be more relevant to an actual musical sense of structure (and is therefore less dependent on patterning to make its case) takes into consideration the relative weightiness inherent in each point of the contour. Firstly, the first and last points of the contour should certainly be considered as structural since they encapsulate the essence of the trajectory, like a pair of bookends, or the proverbial “point A” and “point B” between which the contour takes its journey. That much is clear, but of the many meanderings that happen along the way, some legs of the journey seem to adhere closely to the trajectory, while others deviate away from it. Working out which points are more important than others is not so straightforward, because attempting to evaluate a point’s structural weight on the basis of its degree of conformity with the background trajectory (the straight line between the first and last points) is not only potentially subjective and unreliable, but also may not accurately describe structure in the most relevant way. 126“Fragment” denotes any two adjacent points within a contour referential to itself. If the contour is a reduction, its fragments may encapsulate several fragments of the original contour. 126 Ironically, in fact, it is the parts of the contour that diverge from the tendency of the trajectory that bear the most structural significance. In spite of leading the contour’s path momentarily away from its intended goal, these points are more relevant to the contour itself simply because they “stick out.” Just as with musical events that demand the attention of the listener simply by virtue of their being salient and atypical (which is a form of accentuation or highlighting), so too the abnormalities of the contour reflect noteworthy musical phenomena, such as pitches that are outside the general tessitura or rhythms that are much longer or shorter than most of the other surrounding notes. A middleground should say more about the contour itself than the generic trajectory it follows, lest it forsake the very attributes that give it its “contour-ness.” With a mere glance, it is easy to see those points of a contour that jut away from the majority, but trying to create a representative middleground contour from a simple visual assessment is subjective and somewhat vague. Therefore, a more systematic approach is necessary for this kind of contour reduction. As a preliminary step, the contour can be reduced to a simplified form that uses only its minima (trough points) and maxima (peak points) and eliminates any midpoints (which, by default, are not structurally significant outlying points). But in the case with ning, where the direction of the contour is almost always alternating between increase and decrease in the foreground, this does not reduce the contour very much at all, since only the occasional instances of two increases in a row (and the rarer two decreases in a row) are eliminated. However, this is only one depth of reduction. There are further depths of reduction that can follow this preliminary step, through a process outlined in an algorithm designed by Robert Morris.127 This entails considering the maxima and the minima as separate sets (that is, as “temporary contours” in their own right, with both the first and last points of the principal 127Robert Morris, “New Directions in the Theory and Analysis of Musical Contour,” Music Theory Spectrum 15 (1993): 212. 127 contour included in both groups de facto), and eliminating those points from each set that meet certain requirements. For the maxima set, all of the points that do not form internal maxima should first be eliminated (excepting the first and last points); for the minima set, the same applies to all of the points that do not form internal minima (excepting the first and last points). Then from the remaining points in each set, any instance of a string of two or more adjacent points with equal values should be reduced to a single point. Generally, it does not matter which point in such a string is chosen to remain (though in some circumstances, it can result in different middleground interpretations).128 However, if either the first point and/or the last point of the original contour are part of such a string of equal and adjacent points, all other points in this string must be eliminated to only leave the first and/or last point.129 Finally, the remaining points in both the maxima and minima sets are joined again into a single contour (with points in the same proper order as found in the original contour). All newly resulting midpoints (those not maxima or minima) in this new contour are then eliminated. Following this process once gives a contour reduction of two depths. Repeating this process with the newly reduced contour will provide the next depth of reduction, and can continue until no further reductions can be reached. Different depths of reduction will provide different amounts of information, like different degrees of middleground. But reduction like this has one major problem in the present context: By eliminating points in between maxima and minima (including within the maxima and minima sets themselves), a very crucial identifying element of the contour pattern is automatically removed. As stated before, the grouping of two increases for every decrease is the recognizable marker of the 128This is exactly the case with the reduction of the duration contour for precompositional section 8, which is analyzed in this paper for middleground (see pages 139-46). Here, several different reduced contours can result depending on which points in equal and adjacent strings of points are selected to remain as part of the reduction. 129To clarify, if both the first and last points are in the same string of equal and adjacent values, then the string is reduced to those two points only. 128 regular contour pattern, and moments with two consecutive decreases are the thumbprint of one of the two specific pattern-exceptions (the one with five increments). If all points in between maxima and minima are eliminated, this effectually creates a sequence of alternating increases and decreases, making the entire contour appear like a string of the other pattern-exception (the one with two increments). Further reduction depths simply exacerbate this issue, making any trace of the original recognizable pattern completely obscured. Reductions of this fashion are perhaps useful for comparing two contours with one another, or for determining the relative hierarchical weight of different contour points, but not necessarily for making connections in terms of patterning between the foreground, middleground, and background contours. However, the process to get from one depth of reduction to the next includes several steps, with each step eliminating further information from the previous step. If hierarchical pattern similarities are the genuine article to be sought after, then it may be beneficial to consider what the appearance of the middleground contour would be if it were formed at each step of reduction, rather than at each depth. Contours constructed from points prior to equal and adjacent strings of points being reduced (the first step of reduction), and then prior to midpoints being cut (the next step of reduction), would each likely contain instances of double increases and/or double decreases, and each give a different level of insight into pattern connections. The maxima and/or minima contours may also elucidate some connections in their own right. As a case in point for analysis, consider one of the lengthier sections of the piece, precompositional section 8, the first and largest subsection of the “Floating” passage (mm. 87-105, b.3) (see Figure 4.13). The foreground contours for pitch and duration are presented in Figure 4.14, with first-depth reductions shown superimposed on the foreground in Figure 4.15. Contours of the Maxima and the Minima By following the process through the different stages of reduction for both contours between the first and second depths, some very interesting associations emerge. First, consider the 129 Figure 4.13: Foreground melody for precompositional section 8 (mm. 87-105, b.3). 130 a. G5- C♯3- b. 42- 0- Figure 4.14: Contours of the foreground melody for precompositional section 8 (mm. 87-105, b.3).130 a. Contour for pitch. b. Contour for rhythmic duration. 130The contours in Figures 4.14 through 4.29 inclusive all reflect precompositional section 8, and all conform to the same vertical scaling between the lowest and highest contour points (C♯3 to G5 for pitch [at equally tempered chromatic increments], and % [0] to °_#. [42] for rhythmic duration [with 0 equal to %, and at increments the size of 3Æ ]), though the scales themselves are only provided as a reference in the present figure. Also, the contours are presented here with large images to facilitate visibility of the details, but subsequent figures will show these contours at smaller magnifications. As with Figures 4.10 and 4.11, the horizontal distance between points in these figures is not indicative of duration; all points are spaced horizontally equidistantly. Also, the contours of the foreground melody as given in the present figure reappear in later figures (4.15 through 4.29 inclusive) for convenience (and clarity) as grayed-out “shadows,” against which subsequent contour steps of reduction are referenced. 131 a. b. Figure 4.15: Contours from Figure 4.14 reduced to the first depth of reduction. a. First-depth reduction for pitch. b. First-depth reduction for rhythmic duration. contour patterns found within the contours of the maxima and minima contours themselves (see Figure 4.16). These contours are particularly relevant because they essentially parse the foreground contour by its pattern motifs (as was suggested on page 125 as one approach for middleground interpretation).131 For pitch, the contour of the maxima has some familiar ascent/descent patterning near the beginning. This is especially true when certain pitch points are discredited as true maxima (such as the 7th and 11th maximum points in Figure 4.16a—the 15th and 23rd pitches of the section) as seen in Figure 4.16c. Here, two three-increment groupings appear (the regular pattern, between the 1st and 4th maximum points [1st and 9th pitches], and the 9th and 12th maximum points [22nd and 31st pitches]), and one five-increment grouping appears, albeit in an unusual rotation compared to the regular pattern’s ordering (the pattern-exception, between the 4th and 9th maximum points [9th and 22nd pitches], if the first contour fragment “ ” is assumed as “ ”). But the remainder (and majority) of the pitches in the contour of the maxima for pitch (from the 12th point [31st pitch] to the end) lack ascent/descent alternation, 131Each contour fragment (in the contour of the maxima or the contour of the minima) contains either a three-increment regular pattern motif or a two-increment pattern-exception motif in the foreground (which may combine with another three-increment pattern-exception to make a five-increment pattern-exception in the foreground over two contour fragments). The contours of the maxima and of the minima parse the foreground pattern at different rotations to one another. 132 a. b. c. d. Figure 4.16: The contours of the maxima and the minima for the contours in Figure 4.14.132 a. Minima and maxima for pitch. b. Minima and maxima for rhythmic duration. c. Adjusted minima and maxima for pitch. d. Adjusted minima and maxima for rhythmic duration. with mostly common tones and an overall ascent trajectory. It is a similar case with the contour of the minima for duration, which starts off with clear increase/decrease patterns, but eventually evens out with very little change in rhythmic value, trending downward overall (as seen in Figures 4.16b and 4.16d). These two contours (pitch maxima and duration minima) seem to be more representative of background or early middleground structure (i.e., overall trajectory), rather than of a late middleground. Though there are some hints of familiar foreground patterns, there are 132To clarify, the contours of the maxima and the minima are the same for both the foreground and for the first depth of reduction, shown in Figures 4.16a and 4.16b. However, given the reality of the iterative processes at work behind the points in these contours, there is a slight difference between the contours in Figures 4.16a and 4.16b (made directly from the pitches and rhythms in the music) and those in Figures 4.16c and 4.16d (adjusting for the slight mathematical value differences between consecutive contour points that appear in the music to have equal value). Recall that (as stated on page 122) instances of “ ” in the foreground are really “ ” or “ ” in disguise, occurring only because of approximation or superimposed skewing. Therefore, though any instance of a “plateau” instead of a “peak” (or a “basin” instead of a “trough”) in the foreground may appear to be two consecutive points that are both maxima (or minima), only one of the two points is in fact the true maximum (or minimum). The other, by default, is actually a contour midpoint, because the original mathematical value associated with it is slightly less (or more) than the value of the adjacent maximum (or minimum), but the difference is not significant enough to show up in the musical translation of that value. 133 often too many consecutive contour lines moving in the same direction for any of these patterns to be clearly identified as true instances of such. The contours opposing these, however, seem to have a bit more variability, and turn out more pattern associations with the foreground than their counterparts. The appearance of these patterns (and the contours’ greater independence from the gravity of the overall contour trajectory) may be due to the fact that these contours capture the extreme outlying points, which (in terms of contour hierarchy) bear more structural importance. The contour of the minima for pitch exhibits several moments of the standard two-increment grouping throughout (the pattern-exception), with one instance of the three-increment grouping (the regular pattern, between the 13th and 16th contour points [the 32nd and 40th pitches of the section]). There are also a couple of instances that resemble the three-increment grouping, but with an extra ascent ( between the 7th and 11th points [16th and 26th pitches], and the 16th and 20th points [40th and 48th pitches]). Although the appearance of three consecutive ascents is a rather unusual event in the foreground itself (it does not happen at all in this particular section of the music), in the contour of the minima, this extra ascent seems to correlate with the foreground in two respects: 1) there are no contour midpoints found in the foreground within the span of this extra ascent, and 2) the descent immediately following the extra ascent in the contour of the minima approaches one of the low outlying (and thus structurally significant) pitches. This is quite consistent. Even when the ascent and descent tendencies are obscured by common tones in the latter third of the contour of the minima, there is still evidence of such an “extra” ascent appearing prior to a structural descent (between the 26th and 28th points [71st and 76th pitches] inclusive). The contour of the maxima for duration, on the other hand, is rife with familiar patterns, most of all the five-increment grouping ( , between the 2nd and 7th contour points [4th and 18th rhythms], the 10th and 15th points [26th and 40th rhythms], and the 19th and 24th points [51st and 65th rhythms]), and a couple of moments with two consecutive two-increment 134 groupings that emulate the five-increment grouping ( , between the 15th and 19th points [40th and 51st rhythms] and the 27th and the 31st points [73rd and 86th rhythms], with the latter being most similar to the five-increment arrangement based on the number of intervening points in the foreground under it). Curiously, though there are many moments of two consecutive decreases, there are no instances of two or more increases in a row within the contour of the maxima. The three-increment contour grouping does not appear here per se, but there are a couple of inversions thereof (rotated as between the 7th and 10th points [18th and 26th rhythms] and the 24th and 27th points [65th and 73rd rhythms]). In fact, the entire contour of the maxima could be alternatively interpreted as an inversion of the regular three-increment contour pattern, with interspersed two-increment pattern-exceptions. Regardless of the interpretation, there is a clear correlation between the contour of the maxima and the foreground contour for duration, demonstrating how the contour of the maxima functions as a reduction of the foreground. Where there are two decreases followed by an increase ( ) in the contour of the maxima, the second of the two decreases usually occurs over a span of foreground with no midpoints ( ), while the increase usually encapsulates the same pattern in the foreground.133 In other words, the five-increment pattern-exception in the foreground concurs to a high degree with the ending of the same pattern found in the contour of the maxima. This makes a strong case for embedded structure, that is, fractal-like self-affinity. 133There are some exceptions to this: 1) between the 17th and 19th points (46th and 51st rhythms) is a two-increment grouping in the contour of the maxima (not preceded by an additional decrease) over which the foreground does exhibit the same patterns; 2) another twoincrement grouping (also not preceded by an additional decrease) seems to have similar foreground patterning following the 31st maximum point (86th rhythm), though repeating rhythms (88th and 89th rhythms) make this difficult to say for certain; 3) under the span between the 25th and 27th points of the maxima (68th and 73rd rhythms), the foreground contour reads rather than , even though this is preceded by an additional decrease in the maxima. 135 Contour Reduction for Pitch Where structural connections between the foreground contours and their maxima or minima are clearly evident, these relationships also appear in the combined contours created from the maxima of the maxima and the minima of the minima (which is the first major step in the reduction process). For pitch, the resulting combined contour reads (see Figure 4.17). This can be broken up into contour segments with either the regular three-increment pattern grouping or the two-increment pattern-exception.134 In fact, the consistent alternation of the two patterns creates a sequence of three five-increment groupings in inversion ( ) as an alternative interpretation. Moreover, the combined contour itself gives structural weight to those descending fragments connecting maxima with minima, as these fragments exactly overlap with foreground contour fragments (between the 3rd and 4th combined contour points [6th and 7th pitches], 6th and 7th points [17th and 18th pitches], and 8th and 9th points [28th and 29th pitches]). When the excess equal and adjacent contour points are eliminated (where the fragments occur, which, in this case, only occur in the minima of the minima), the contour pattern is only minimally altered: a. (see Figure 4.18). The same b. Figure 4.17: The maxima of the maxima and the minima of the minima for the contour for pitch from Figure 4.14a. a. Contours of the maxima of the maxima and the minima of the minima for pitch. b. The contour formed from the combined points of the contours in Figure 4.17a. 134The two consecutive common-tone contour fragments are interpreted as based on the tendencies suggested by foreground contour (overall descent is more strongly implied between the 43rd and 54th pitches than between the 35th and 43rd pitches because of the large descent found between the 53rd and 54th pitches in the foreground). 136 Figure 4.18: The contour from Figure 4.17b with excess equal and adjacent points eliminated.135 recognizable pattern groupings are retained, showing that self-affine patterns occur at yet another level of middleground reduction, perpetuating the idea of embedded structure on multiple levels. With the next step of reduction (midpoint elimination), the second depth is finally reached (see Figure 4.19). As previously mentioned (see pages 127-28), the contour is strictly made up of alternating ascents and descents, providing essentially the same insight as Figure 4.18, except that it better highlights the structure by sifting out the less structurally important contour midpoints of Figure 4.18. The contours in both Figures 4.18 and 4.19 contain the same four structurally significant contour fragments (which exactly overlap with fragments from the foreground contour). The only curious feature of the contour of Figure 4.19 is that, since it is fully reduced to the second depth and therefore is no longer contingent on the maxima and minima of the first depth (as the contours of Figures 4.17b and 4.18 were), one of its maximum points (the 9th) is formed from what was formerly a minimum point in the foreground and first-depth contours (the 60th pitch). What this means is that whereas this pitch (B4) is locally a melodic low point (in the foreground as well as within the contour of the minima), it functions as a structural high point 135Because there are no intervening points in the contour of the maxima of the maxima occurring between the string of equal and adjacent points found in the contour of the minima of the minima, it technically does not matter which of these three points is selected to remain in this reduction. The contour pattern will be the same regardless. However, to stay consistent with the idea of making structural hierarchical connections on the basis of contour, the last of the three points has been selected here on the grounds that this point is connected to the largest trough in the foreground of the three points (that is, it is implied to be the most structurally significant of the three points because the shift to this pitch is a more dramatic leap downwards). 137 Figure 4.19: Contour for pitch from Figure 4.14a reduced to the second depth of reduction. within the context of the reduced (second-depth) contour. This pitch could be said to serve the general purpose of pulling the minima of the melodic trajectory up to and above where the maxima of the contour once were. The only reason that this phenomenon occurs is the abnormally low (and structurally significant) 76th pitch of the foreground contour (A4), which tugs against the trajectory’s ascending tendency. These two pitches (the 60th and the 76th) are assigned by the composer to the violin part, thus connecting the pitches in the score through structural voice leading, and causing these pitches to truly sound as a counterpoint against the reiterated high pitches of the rest of the ensemble.136 Beyond the second depth, the familiar patterns (well-established with the contours made from the steps between the first and second depths of reduction) become much less obvious, with each newly-reduced contour conforming more and more closely to the background trajectory (the line connecting the first and last contour points).137 Nevertheless, there are a couple of things worth pointing out from these further reductions. First, the combined contour formed from the maxima of the maxima and the minima of the minima from the second depth (see Figure 4.20) 136This 137At idea of contours and counterpoint is expanded upon in Chapter 5, pages 160-65. most steps of reduction for pitch beyond the second depth there appear only alternating ascending and descending contour fragments, or the two-increment pattern-exception ( ). The only times that allude to the regular pattern are 1) one version of the combined contour of the maxima of the maxima with the minima of the minima for the second depth, in which all excess equal and adjacent points have been eliminated, where two consecutive ascending contour fragments appear ( ), and 2) the next-step reduction of the same contour (a third-depth reduction), which appears like a rotation of the regular pattern ( ). 138 a. b. Figure 4.20: The maxima of the maxima and the minima of the minima for the second depth of reduction for pitch from Figure 4.19. a. Contours of the maxima of the maxima and the minima of the minima for the second depth of reduction for pitch. b. The contour formed from the combined points of the contours in Figure 4.20a. has two contour fragments in common with the foreground (between its 2nd and 3rd points [6th and 7th pitches] and 4th and 5th points [28th and 29th pitches]), showing that these fragments (and the points that make them) have the greatest structural clout of the four such fragments in the second-depth contour. But, due to the 6th and 28th pitches being equal and adjacent in the maxima of the maxima set from the second depth, this leads to two interpretations of the third depth: one that retains the 6th and 7th pitches (and the contour fragment between them shared with the foreground), and the other that retains 28th and 29th pitches (and the contour fragment between them shared with the foreground) as well as the 7th pitch (see Figure 4.21). Because the 7th pitch appears in both versions of the third-depth-reduction contour, and because it is the only pitch remaining (besides the first and last pitches) in the fourth-and-final depth of reduction (which is the same reduction for both versions of the third depth—see Figure 4.22), it distinguishes a. b. Figure 4.21: Contour for pitch from Figure 4.14a reduced to the third depth of reduction. a. Third depth, version 1. b. Third depth, version 2. 139 Figure 4.22: Contour for pitch from Figure 4.14a reduced to the fourth/final depth of reduction. itself as the most structurally significant pitch of the contour, and therefore of the passage (as its lowest pitch, C♯3).138 Contour Reduction for Duration Creating a combined contour from the minima of the minima with the maxima of the maxima from the duration contour, however, is a bit more complicated than it is for pitch. The reason for this has to do with whether to eliminate the extra equal and adjacent contour points from the contour of the minima of the minima before or after the formation of the combined contour. The normal procedure for reduction requires these to be eliminated prior to the formation of the combined contour, but because there are so many instances of equal and adjacent points in the contour of the minima of the minima, it becomes difficult to determine which points to keep and which to eliminate given the points’ positions in relationship to those of the contour of the maxima of the maxima (see Figure 4.23). Each of these could potentially create a different combined contour (there are twelve possible combinations, in fact).139 Because of this ambiguity, it may be better to first consider the appearance of the combined contour as it would be 138Once again, this pitch emerges as a significant structural marker (see page 63 for a discussion on how C♯3 also marks a golden section of the piece). 139There are twelve possible combinations assuming that equal and adjacent strings of points in the contour of the minima of the minima that have no intervening points from the contour of the maxima of the maxima occurring between them can be considered the same for the purpose of creating a reduced combined contour. That is, regardless of which of these points 140 Figure 4.23: The maxima of the maxima and the minima of the minima for the contour for rhythmic duration from figure 4.14b. with the equal and adjacent points retained (see Figure 4.24), and then with the excess equal and adjacent points eliminated after the formation of the contour (see Figure 4.25). The patterns evident in the combined contour reduction that retains all points from the maxima of the maxima and minima of the minima (Figure 4.24) are, by and large, reminiscent of the familiar formations of the foreground and of the contour of the maxima: . Still, instances of identical rhythms following one another (“ ”) tend to interrupt the patterning, rather than participate in it. Because this contour was formed prior to the elimination of the extra equal-and-adjacent points, it incorporates all equal and adjacent points from the minima of the minima on both sides of (before and after) every intervening maximum point.140 But the equal and adjacent points that occur within the contour itself (uninterrupted by an opposing maximum or minimum) can be reduced to single points without disrupting the essence of this contour.141 Removing these (so described) is ultimately chosen to remain in the reduction for creating the combined contour, the contour will be, for all intents and purposes, the same. 140Specifically, these instances (in Figure 4.24) include the 5th through 8th contour points (11th, 14th, 18th, and 19th rhythms), where the 7th point (18th rhythm) is a maximum; the 13th through 20th points (44th, 46th, 51st, 52nd, 55th, 56th, 57th, and 61st rhythms), where the 14th, 15th, and 19th points (46th, 51st, and 57th rhythms) are maxima; and the 26th through 31st points (80th, 83rd, 84th, 86th, 87th, and 90th rhythms), where the 29th point (86th rhythm) is a maximum. 141These include the following points from the combined contour (Figure 4.24): for contour minima, the 5th and 6th points (11th and 14th rhythms), the 16th, 17th, and 18th points 141 Figure 4.24: The contour formed from the combined points of the contours in Figure 4.23 (formed with equal and adjacent points retained). Figure 4.25: The contour from Figure 4.24 with excess equal and adjacent points eliminated after its formation (rather than prior to its formation).142 (52nd, 55th, and 56th rhythms), the 22nd and 23rd points (66th and 69th rhythms), the 26th, 27th, and 28th points (80th, 83rd, and 84th rhythms), and the 30th and 31st points (87th and 90th rhythms); for contour maxima, the 14th and 15th points (46th and 51st rhythms). 142As with Figure 4.18, because no intervening opposing points occur between strings of equal and adjacent points within the combined contour, it technically does not matter which point in a string is chosen to remain in the reduction. The contour pattern will be the same regardless. But just as with Figure 4.18, to stay consistent with making structural hierarchical connections, the point chosen to remain from each string was selected on the basis of the structural significance implied by the foreground contour surrounding it: Wherever an original maximum-minimum direct-point connection (single contour fragment) in the foreground could also be found in the combined contour, these points were retained. 142 particular excess points declutters and clarifies the patterns of the combined contour (Figure 4.25), namely that it is principally made up of much of the same stuff as the contour of the maxima,143 specifically the two-increment and five-increment pattern-exception groupings: . Perhaps another (and a bit more generous) way to reduce the combined contour of the maxima of the maxima and the minima of the minima for duration (Figure 4.24) is to only eliminate those extra equal and adjacent points from the contour that are also equal and adjacent in the original contours of the maxima and of the minima (see Figure 4.26). This may appear to fudge the rules (by removing excess equal and adjacent points from the maxima and the minima sets before forming the contours for the maxima of the maxima and the minima of the minima rather than after), but in fact, it makes only one difference with the contour in Figure 4.25: The equal and adjacent points found in the maxima of the maxima are both restored (the 45th and 50th rhythms of the original foreground contour). These two points which were equal and adjacent in the contour of the maxima of the maxima have an additional point in between them in the contour of the maxima, whereas none of the equal and adjacent points in the minima of the minima had any extra points in between them in the contour of the minima. This allows for one more five-increment grouping, joining together two two-increment groupings from Figure 4.25: Figure 4.26: The contour from Figure 4.24 with excess equal and adjacent points eliminated after its formation (rather than prior to its formation), eliminating only those points that are also equal and adjacent in the contours of Figure 4.16b (or 4.16d). 143Refer also to Figure 4.16, page 132. 143 [ ] . But this contour may be a bit contrived, and is not fully necessary to establish what has already been demonstrated in Figure 4.25, mainly that the two-increment and five-increment pattern-exceptions (or, alternatively interpreted, the inversion of the three-increment regular pattern and the two-increment patternexception) define the middleground structure of the duration contour, underpinning the foreground and linking it with its overall downward trajectory in the background. The fact that the above contours (Figures 4.24, 4.25, and 4.26) were derived by combining the maxima of the maxima with the minima of the minima before any strings of equal and adjacent points in the maxima of the maxima or the minima of the minima sets were pruned gives these middleground interpretations a closer relationship with the foreground contour for duration (that is, more information is retained) than contours created with reduction steps made in the standard order. But considering the contours formed by following the reduction steps in the proper order does not upset these well-established connections too greatly. The only difficulty with this (as already mentioned) is in determining which of the twelve possible contour arrays is the most appropriate. Each of these twelve configurations is technically equally valid, as there is no rule favoring one over another as the “most correct” or “most accurate” interpretation. However, there are some configurations that have less contour pattern anomalies than others, which might make them more favorable choices. But choosing a contour merely on the basis of how well it fits a preexisting hypothesis of patterning is to stand on shaky ground. In that case, perhaps a more logical and theoretically sound choice of contour would be the configuration with the most structural ties (shared contour fragments) to the foreground. By great fortune, though, it so happens that the most structurally relevant choice is also the best fit for the patterning (see Figure 4.27).144 144Actually, there are three other contours that fit the structural-relevance criterion as equally well as the one given in Figure 4.27, but these are not included here for the following reasons: 1) the first contour contains a contour fragment that connects two points from the 144 Figure 4.27: One contour formed from the combined points of the contours in Figure 4.23, formed after excess equal and adjacent points were eliminated from the contours of the maxima of the maxima and the minima of the minima in one of twelve possible configurations. The contour shown in Figure 4.27 is almost entirely comprised of the inverted form of the regular three-increment pattern motif ( , beginning with the second point of the contour, or, to take a different interpretation of pattern rotation, from the first point of the contour). Just as with the contours formed prior to extra equal and adjacent points being eliminated from the maxima of the maxima and the minima of the minima sets (Figures 4.24, 4.25, and 4.26), every pair of consecutive decreasing contour fragments occurs with the former appearing between two maxima points of the foreground contour for duration, and the latter appearing between a maximum and a minimum point. The only exception to this occurs between the 6th and 9th contour points (18th and 36th durations), where three consecutive decreasing contour fragments appear, with the first two fragments occurring between maxima points. But otherwise, the contour conforms closely with the ideal inverted pattern. In many respects, the contours of Figures 4.25 and 4.27 have much in common. Both contain the same maxima points (quantity and position), and the majority of the contour maxima of the maxima in an increasing formation, whereas most contour fragments connecting two points from the maxima of the maxima decrease (tending with the overall downward trajectory); 2) the second contour uses a structural contour fragment (shared with the foreground) that is less dramatic than the fragment used in the first contour; and 3) the final contour has both issues mentioned for the previous two contours. 145 fragments are shared between them.145 Many contour fragments that are identical with fragments in the foreground appear in each of these contours, so both effectively demonstrate where the most salient and important foreground events occur.146 And beyond the second depth of reduction, both contours reduce further in essentially the same way, to a third-and-final-depth contour of , i.e., a rotation of the three-increment regular pattern (see Figure 4.28).147 The essence of both methods of reduction, therefore, distills the contour in practically the same way, making a strong case for the existence of an inherent hierarchy permeating the rhythmic patterns, because the same overall patterns emerge equally well through differing approaches and perspectives. Also, they both show that ultimately (aside from the first and last rhythmic durations) the 4th duration (°_#.) and the 80th duration (or the 87th, which are both grace notes [%]) are the a. b. Figure 4.28: Contour for rhythmic duration from Figure 4.14b reduced to the third/final depth of reduction. a. Third depth, version 1 (further reduction of the contour in Figure 4.25). b. Third depth, version 2 (further reduction of the contour in Figure 4.27). 145The two contours mismatch between the 18th and 26th durations, the 40th and 51st durations, the 57th and 65th durations, and the 86th and 92nd durations, where a contour fragment joining two maxima points in Figure 4.27 corresponds to two contour fragments joining maxima with a minimum point in Figure 4.25. All other contour fragments are identical. 146All five of the contour fragments that are identical with the foreground that appear in Figure 4.27 also occur in Figure 4.25 (between the 3rd and 4th, 10th and 11th, 51st and 52nd, 65th and 66th, and 79th and 80th rhythms). The extra minima points present in Figure 4.25 only provide it with two additional contour fragments in common with the foreground (between the 18th and 19th, and 86th and 87th rhythms). 147Reducing the contour of Figure 4.25 by eliminating contour midpoints would technically not be considered an “official” reduction to the second depth as it is made from reduction steps performed out of order (and therefore, reductions to further depths are likewise not “official”). But as a contour in its own right, it does ultimately reduce to nearly the same the final reduction as that of the contour of Figure 4.27 (both of which are that shown in Figure 4.28). 146 most structurally significant durations according to contour, being the longest and shortest durations of precompositional section 8. Middleground Interconnections and Conclusions Analysis of the different steps of contour reduction through middleground layers (from the late middlegrounds of the contours of the maxima and of the minima, and the combined contours of the maxima of the maxima and the minima of the minima, to the early middlegrounds of the second and further depths of reduction) all demonstrate embedded self-similar pattern structures. The remarkable consistency with which these self-similar—or self-affine—patterns occur (in spite of the superimposed background skew of the melodic and rhythmic trajectories [i.e., those fashioned from the logistic-equation parameters]) suggests that they indelibly permeate the contours, and that fractal-like features are a primary driving force inherent in the contours’ structure. (In fact, the few embedded pattern-anomalies that do appear should be blamed on the superimposed trajectory skew of the logistic-equation parameters, because such anomalies are not inherent to the rather consistent way in which iteration values vacillate up and down [the regular patterns and pattern-exceptions] as driven by the strange attractors of chaos.) With the above middleground analysis of the “Floating” passage of ning, similar embedded patterns occur even between the contours of the different musical dimensions (pitch and duration, and probably [though not shown] amplitude as well) at deeper levels of reduction. A specific example of this is in the third depth of reduction, where both possibilities for duration (Figure 4.28) and one of the versions for pitch (Figure 4.21, page 138) all exhibit the same contour pattern: . Though the contours’ emphases are completely different (with the short ascent, the quick-but-stark descent, and the long dramatic ascent for pitch, versus the incidental increases bracketing the long sweeping central decrease for duration), it is nevertheless interesting to note 147 how the sequence of rise and fall in the early middleground unifies the structure of the two dimensions.148 Such ever-increasing uniformity between the musical dimensions’ contours at deeper levels of structure highlights their interdependent nature (and that of their formulaic origins) once again. In the foreground, the contour patterns for pitch, duration, and amplitude experience the regular pattern motif (three-increment) and the two pattern-exceptions (two-increment and fiveincrement) in tandem or close proximity with one another, due to the fact that all three variables in the Frøyland formula directly affect the iterated outcomes of one another (see page 12). With late middleground contours, each dimension seems to show different favoritisms to one pattern motif or another, which begins to show the effects that each dimension’s background trajectory (from the logistic equations superimposed parameters) has upon the underlying structure of the patterning. At this stage of reduction, the connections of the dimensions with one another in terms of patterning might be less clear, especially if the background trajectories differ (which they do in the case of the “Floating” passage). What does seem to be consistent, though, is that instances of two (or more) consecutive contour fragments moving in the same direction concur with the direction of the background trajectory. Therefore, where the background trajectories of two dimensions’ late middlegrounds are opposite (one ascending and one descending), the middleground patterns can perhaps be thought as having an inversional relationship. In the early middleground contours, the background trajectory is much more obvious, but the patterns insinuated in the foreground are still detectable, and, as shown in Figure 4.28, more alike between different musical dimensions than in the late middleground. Therefore, the interdimensional relationships of the foreground, in which the dimensions affect one another directly, effectually perpetuate themselves through the reduction process into the late and then 148This also highlights the usefulness of contour theory for comparison, making it very simple to relate very different musical dimensions with one another that otherwise would be difficult to compare. 148 early middleground, and (as will be discussed in the next section of the paper) into the background as well, in which the trajectories of each section of the piece form collective contour lines for each musical parameter. At its very essence, this whole process is analogous with a “fractal zoom-out,” and shows quite clearly that, in terms of the way a contour moves up or down, there is replication of the same pattern types at multiple degrees of “musical magnification.” (For a visual summary of the reduction process for both the pitch and rhythmic duration contours of precompositional section 8, please see Figure 4.29.) Contour Background: Connecting the Large and the Small The Frøyland formula (equation 1.5, Chapter 1, page 13) is more complex than the logistic equation (equation 1.3, Chapter 1, page 6), and so is well-suited to foreground activity. But both formulae operate on the same basic premise, so even though the iterative patterns of the logistic equation are simpler, they exhibit general tendencies of increase and decrease that are not far different from those of the Frøyland formula’s variables. Consequently, the background activity of ning (produced through the logistic equation) contains contour elements that resemble the foreground contours (and this in spite of the Frøyland formula having no bearing on the results of the logistic equation).149 Granted, Wallin treats the two formulas in different ways (for instance, holding the (c, d) values constant for the Frøyland formula iterations [within each section of the piece, but not from section to section] while changing the r value from iteration to iteration for the logistic equation [connecting from section to section]), but even so, some key contour relationships are still evident. Compare the patterns of increasing and decreasing for the different 149Of course, as already established, the logistic equation does affect the results of the Frøyland formula (postcomputationally), so it could be argued that the background patterns are imbued into the foreground patterns. Even so, the local increases and decreases of the foreground contours are principally due to the iterative behavior inherent in the Frøyland formula alone, and are only secondarily dependent on the logistic-equation parameters. The contour patterns that come about from each formula independently are, by happenstance, similar to one another. This similarity should not be mistakenly attributed to the fact that one formula affects the other in the context of the present piece. 149 a. foreground (0) first depth (1) min, max (1) min/min, max/max (1) min/min, max/max w/o excess points (1)* min/min plus max/max w/o excess points (1) second depth (2) min, max (2)* min/min, max/max (2) min/min, max/max w/o excess points (2), v.1 of 2* third depth (3), v.1 of 2† min/min, max/max w/o excess points (2), v.2 of 2* third depth (3), v.2 of 2† min/min, max/max (3), v.1 and v.2 of 2*† fourth/final depth (4)† foreground (0) first depth (1) min, max (1) min/min, max/max (1) min/min, max/max w/o excess points (1), v.1 of 12* min/min plus max/max w/o excess points (1), v.1 of 12 second depth (2), v.1 of 12* min/min, max/max w/o excess points (2), v.1 of 2*† third/final depth (3), v.1 of 2† min/min, max/max w/o excess points (1), v.2 of 12* min/min plus max/max w/o excess points (1), v.2 of 12* second depth (2), v.2 of 12* min/min, max/max w/o excess points (2), v.2 of 2*† min/min plus max/max w/o excess points (2), v.2 of 2* third/final depth (3), v.2 of 2 b. Figure 4.29: Reduction process summaries for the contours of precompositional section 8.150 a. Reductions for pitch. b. Reductions for rhythmic durations. 150Asterisks (*) indicate contours not included in previous figures in the text. Conversely, the contours from Figures 4.16a, 4.16b, 4.17b, 4.24, 4.25, and 4.26 are not shown here because they are not technically part of the official reduction process. Daggers (†) indicate contours that appear with the same result for two or more consecutive steps of reduction (to eliminate 150 formal parameters determined by the logistic equation (those from Table 2.1, Chapter 2, pages 30-31, and table 3.1, Chapter 3, page 84) in Figure 4.30. Every one of the parameter contours consists of both the three-increment regular pattern motif ( ) and the two-increment pattern-exception ( ), with the exception of the “Section Length” contour (made of only the three-increment regular pattern motif).151 The commonality of the contour pattern types generated by both formulas is the way in which the background form relates to the foreground melody. However, though the pattern motifs are shared traits, the background contours differ from those of the foreground in their overall scope. In this case, the two-increment patternexception is less an exception and more the norm, while the three-increment pattern seems to be much less common, confined mostly to the beginning third or so of each of the parameter contours (again, with “Section Length” excepted). This is the result of the way that the value of r in the logistic equation has been prearranged to decrease from iteration to iteration (explained in detail in Chapter 1, pages 16-17). Higher values of r, which are used at the beginning of the piece, are closer to the slot with three attractor values in the bifurcation diagram (see Figure 1.2, Chapter 1, page 7), and so iterations of x multiplied by such high r values will tend to alternate among these three attractor values: a medium value becomes a high value, a high value becomes a low value, and a low value becomes a medium value (which gives rise to the three-increment contour pattern). But as r progressively decreases through the piece, it enters the chaotic regions of the bifurcation diagram (again, refer to Figure 1.2), where the iterated values of x are more redundancy). In cases were multiple reduction versions occur, note that there may be more versions at some depths than at others. For the duration reductions, only two of the twelve second-depth reductions are presented here for demonstration, but all twelve eventually reduce to one of only two third-depth reductions, both of which are included here. 151The distinctly different appearance of the “Section Length” contour in comparison with the others sheds doubt on Wallin’s claim that all the parameters used the same iterative scheme (with incrementally decreasing values for r in the logistic equation). See the description of this method on page 17, including footnote 25, in Chapter 1, for a detailed explanation of why this parameter may have been derived differently from the others. 151 “Section Length”: “Median Pitch”: ( ) “Pitch Spread (Range)”: ( ) “Average Note Duration”: ( ) “Rhythmic Weight”: ( ) “Shape”: “Position Number for c and d paired values”: “Scale (Pitch Collection)”: Figure 4.30: Directional contour arrows152 for the contours of the formal parameters of ning as their values change from each precompositional section to the next.153 152For highlighted. visual convenience, pairs of consecutive increasing contour fragments are 153Because the final point of data given for each parameter in Tables 2.1 and 3.1 corresponds to the beginning of the final section (precompositional section 32), it does not technically indicate the final value of each parameter at the end of the piece. However, because there are no values for any parameter following this last set, the final values given in Tables 2.1 and 3.1 are perpetuated without change through the full duration of the final section, so by default, the final values given for each parameter are those at the end of the piece. Specifically, the values for of the “Section Length,” “Shape,” “Position Number for c and d paired values,” and “Scale (Pitch Collection)” parameters do not change through the duration of the final section, just as they do not change throughout the duration of any other section of the piece. The values for of the “Median Pitch,” “Pitch Spread (Range),” “Average Note Duration,” and “Rhythmic Weight” parameters, however, do change gradually over the course of each section of the piece toward the starting value of the following section, but whereas there are no additional values following the final section, these values have no alternative but to also stay stable and unchanging throughout the duration of the final section as the other parameters do (this is indicated by the additional contour fragment in parentheses at the end these parameters’ contours in Figure 4.30). Consequently, it appears that the “Shape” parameter has no effect on these in this case. The only parameter that appears to change superficially (as in the diagram of Figure 2.1, Chapter 2, page 26, or in the score itself) is the “Median Pitch,” since the pitch descends gradually over the course of the final section. In reality, though, this change is only due to the parameter value adjusting relative to the envelope created by the “Pitch Lower Limit” and “Pitch Upper Limit” parameters (which are changing gradually over the course of the final section). 152 difficult to forecast. Nevertheless, by moving away from the tendency to oscillate between the three attractor values, the iterations of x take on a different kind of behavior,154 in which the values simply increase and decrease in alternation (which gives rise to the two-increment patternexception motif).155 What is interesting to glean from these observations is the fact that, in spite of the parameters each having very different values with each iteration, the parameter contours are strikingly alike (at least in simple terms of increase/decrease). Some parameter contours are so similar that one might be tempted to think that the initial value of x (in the logistic equation) is the same between them. However, any contour differences, no matter how few, will still indicate that the initial values of x were indeed different (though perhaps not by much). Great swaths of the contours are the same for different parameters, but none of them are completely identical. There is something about the moment in which each contour shifts from the threeincrement pattern into the two-increment pattern that merits some attention. This moment of transition occurs for each parameter somewhere between precompositional sections 5 (“Transparent,” beginning in m. 52) and 14 (“Sharp,” beginning in m. 147), with most of them happening between precompositional sections 12 (midway through “Broadening,” beginning in m.141) and 14. Indeed, by precompositional section 13 (midway through “Broadening,” beginning in m.143), the contours of all the parameters (except “Section Length”) have settled into lock-step with one another until the end of the piece: “Median Pitch,” “Pitch Spread (Range),” “Position Number for c and d paired values,” and “Scale (Pitch Collection)” increase and decrease together as a group, and “Average Note Duration,” “Rhythmic Weight,” and “Shape” increase and decrease together as a group, with each group moving opposite each other. It is curious that 154See specific details about this iterative behavior in Chapter 1, on page 10-11, including footnote 18. 155Again, for these reasons, it is dubious that the “Section Length” parameter was conceived with a changing value for r (refer to footnote 151, page 150, and footnote 25, Chapter 1, page 17). 153 this moment (precompositional section 13) is marked compositionally by the first of the two grand pauses in the piece (m. 145), which creates a sort of formal division between the three-increment pattern (commingled with the two-increment pattern) contour behavior and the two-increment pattern contour behavior (strictly by itself). The similarity of these contours’ structures is also evident in their own reductions, in which the same three-increment and two-increment patterns also appear. Reducing these contours is essentially to find the background of the background, the ultimate contour Ursatz (as it were) for each parameter of the piece. The contour fragments between consecutive parameter values basically define the background of each formal division in the piece individually, but all the formal sections inclusive have their own hierarchy relative to one another on a macro scale, just as the contour points and fragments of the foreground melody internally within each section do. Because the values for each parameter are so very different from one another in terms of scaling, it is simpler (in this case) to normalize156 the contour point values in order to facilitate relating the parameter contours with one another (see Figure 4.31). Rather than going through a detailed walk-through of the reduction process for each parameter (such as was provided for precompositional section 8 in the preceding middleground analysis), the final reduction depths157 are presented, normalized, and summarized in Table 4.4, with only certain key details about the reduction process (along with corresponding observational analysis) provided in the text that follows. 156For reference, the process of contour normalization is explained in footnote 117, page 115. 157Some parameters have more than one version of the final contour reduction due to the presence of equal and adjacent contour points, but each version happens to have the same pattern regardless of which of these versions is used: “Average Note Duration” has two versions, “Position Number for c and d paired values” has seven versions, and “Scale (Pitch Collection)” has eight versions. At earlier depths of reduction, more versions exist: “Average Note Duration” has four versions at the second depth of reduction, “Rhythmic Weight” has two versions at the second depth of reduction, and “Position Number for c and d paired values” has forty-nine versions at the second depth of reduction. 154 “Section Length”: 25-31-10-22-30-7-20-28-0-15-26-5-14-21-4-13-18-2-9-16-3-11-19-1-12-24-6-17-27-8-23-29 “Median Pitch”: 10-29-0-11-27-2-17-15-21-9-28-1-16-19-13-23-6-22-7-24-5-12-8-26-3-17-14-20-9-25-4-18-(18) “Pitch Spread (Range)”: 30-0-11-29-1-14-20-9-28-2-15-18-12-27-4-17-12-25-5-19-10-26-3-16-13-22-7-23-6-21-8-24-(24) “Average Note Duration”: 7-26-0-9-25-0-14-21-5-23-0-10-17-5-19-1-11-8-12-6-15-2-18-2-16-5-22-3-20-13-24-4-(4) “Rhythmic Weight”: 5-22-0-8-21-1-11-10-13-9-18-4-19-2-15-8-19-3-16-7-20-1-11-10-12-9-17-6-19-2-14-8-(8) “Shape”: 31-0-11-30-1-12-29-2-17-16-19-14-26-5-22-9-28-3-18-15-23-8-25-6-21-10-27-4-20-13-24-7 “Position Number for c and d paired values”: 1-4-0-2-4-0-3-1-4-0-2-3-1-4-0-3-1-4-0-3-1-4-0-3-1-4-0-3-2-3-1-4 “Scale (Pitch Collection)”: 0-1-4-0-1-4-0-2-4-1-4-0-2-3-2-3-2-4-0-4-0-3-1-4-0-3-2-3-1-4-0-3 Figure 4.31: Normalized values of the contour points for the contours of the formal parameters of ning.158 158Colors of contour-point values in this figure correspond to the various depths of contour reduction as follows: Black = points in the original contour only. Green = points in the first depth and original contour. Blue = points in the second and first depths, and original contour. Purple = points in the third, second, and first depths, and original contour. Red = points in the fourth, third, second, and first depths, and original contour. For the second, third, and fourth depths: When a point is given in bold, it is consistently present in all versions of that depth. When a point is given in normal typeface, it is present in only some (often just one, occasionally more) versions of that depth. 155 Table 4.4: Final Depth Reductions for the Contours of the Formal Parameters of ning Formal Parameter Final Depth Final Depth Level Reduction Normalized Contour (Translated) Pattern “Section Length” third <25 31 0 29> <1 3 0 2> “Median Pitch” fourth <10 29 0 18> <1 3 0 2> “Pitch Spread (Range)” “Average Note Duration” “Rhythmic Weight” fourth <30 0 24> <2 0 1> fourth <7 26 0 4> <2 3 0 1> fourth <5 22 0 8> <1 3 0 2> “Shape” fourth <31 0 7> <2 0 1> <1 0 4> <1 0 2> <0 4 3> <0 2 1> “Position Number for third c and d paired values” “Scale (Pitch second Collection)” Most parameter contours are complex enough to require four depths of reduction. The exceptions to this are the “Position Number for c and d paired values” and “Scale (Pitch Collection)” parameters, where the values for the contour points have been considerably diluted to only five possibilities, and the “Section Length” parameter, which, as has already been pointed out, is made up of only one type of pattern motif. Of those parameters remaining, three ultimately reduce to the rotation of the three-increment pattern motif ( the two-increment pattern-exception ( ) while the other two reduce to ). In fact, of all eight parameters, the only one to have a unique final-depth reduction is the “Scale (Pitch Collection)” parameter, with its second-depth final reduction of an inverted two-increment pattern ( ). The degree of consistency amongst the parameters, although imperfect, is quite noteworthy. Another interesting feature that the parameter contours share with the contours of the foreground melody (and, by extension, what the logistic equation has in common with the Frøyland Formlua) is the penchant for retaining contour fragments from the original contour in 156 the further steps and depths of reduction beyond the first depth,159 particularly those fragments that descend.160 This means that the extreme high points in the contour almost universally lead directly to extreme low points. The derived contours of the maxima of the maxima and of the minima of the minima contain points that are consecutive to one another in the original contour. This is not necessarily typical of most hypothetical contours (nor of most music, since this represents very disjunct pitch motion with wide leaps and rhythms that move from the very long to the very short), but it is a hallmark of ning. In fact, there appears to be something of a correlation in the distance between consecutive maxima and minima points in these contours, where the higher that a maximum point is, the lower the following minimum point will tend to be. At the final depths of each of the parameter contours, they all have one or two of these socalled structural contour fragments occurring right at the beginning of the work (with the exception of the “Position Number for c and d paired values” and “Scale (Pitch Collection)” parameters [though had the values for these contours been more precise, they probably would have had this same feature]). What this means is that the biggest contrasts within each parameter (and hence the way that they manifest in the foreground melody) occur within the first three sections of the music, that is, the first theme group, giving the “Rustic, Vigorous,” “Transparent,” and “Motoric” thematic ideas their very distinctive flavors. As the piece progresses, these contrastive changes in contour become less dramatic, which in turn facilitates the merging and commingling of the initial thematic ideas in the musical development that follows. 159This is evidenced in Figure 4.31 by the abundance of pairs of adjacent contour points color-coded for the second, third, and/or fourth depths of reduction (pairs in red-red, purple-red, purple-purple, blue-purple, blue-blue, or blue-red combinations), which most often show the point with the greater value coming first in each pair. 160The principal exception to this greater-smaller ordering of structural contour-point pairs occurs in the very first contour fragment of most of the parameter contours, where the fragment ascends from a medium value to a high value (this is not the case for “Pitch Spread (Range),” “Shape,” and “Scale (Pitch Collection)”). 157 However, structural contour fragments present in the penultimate depth of reduction for some parameters (such as the third depth for “Median Pitch,” “Pitch Spread (Range),” “Rhythmic Weight,” and, to a lesser extent, “Shape”) occur close to the golden section of the piece, i.e., the “recapitulation,” in which the initial theme groups of the beginning reemerge in their transformed state. The appearance of these high-low contour contrasts at these moments of the music essentially ties these junctures in the form of the piece together in a logical hierarchical association. And it is also worth pointing out that the structural contour fragments of the second depth of reduction for most parameter contours occur (with the exception of the fragments at the very beginning, which also form part of the final depth of reduction) in and after the “development.” In a way, this suggests that the initial contour idea of the “exposition” (i.e., a large contrastive shift from a high value to a low value) is prolonged (so to speak) throughout the rest of the piece for all the parameters in the late middleground. But perhaps the most important thing to take away from these contour reductions of the background parameters is how they tie in with the contours found in the foreground melody. The final contour reductions of the parameters (the background of the background) as presented in Table 4.4 are not anything special or unique per se. For any contour (of any hypothetical shape, size, or combination of directional changes in the foreground), there are only a handful of possible normalized final-depth contour reductions,161 from which only six possible patterns can result: , , , , or , .162 But the remarkable thing about ning (as compared to other pieces of music that could undergo contour analysis) is not the background contours to which the foreground contours reduce, nor even the background of the background contours to which the background contours (i.e., the parameter contours) reduce, but rather the continual appearance of 161The same is true for most types of reductive analysis (e.g., Schenkerian analysis), where the ultimate reduction of most pieces of music results in one of only a few basic structure types or formal schemes. 162Technically, there is one other possible final-depth reduction pattern, only achieved if all points in a contour are equal, which is extremely rare. , but this one is 158 these ultimate background patterns that permeate all later levels of reduction, from the background through the middleground and into the foreground. Going through the process of reduction from foreground to background (as presented in this paper) might give the mistaken impression that the pattern motifs found in the foreground contours are the “primary elements” which replicate in the same direction as the reduction, toward deeper levels of structure. But in fact, the reverse is really what is going on here: the deepest levels of structure, the macro-sized contour patterns and overall shapes in the background of the background, are the “primary elements” replicating in the opposite direction of reduction, toward more and more complexity, through the early then late middlegrounds right up to the micro-sized contour patterns in the surface of the music. This is the very essence of fractal self-similarity, with the big replicating into the small. Indeed, from its conception to its structural underpinnings to its unifying single melody line, ning has a fractal quality about it through and through. CHAPTER 5 STRUCTURE REFLECTED IN THE SCORE/CONCLUSIONS Observations of the Composer’s Personal Inflections Up to this point, the analytical focus for ning has principally been on the structural underpinnings that lie beneath the surface of the music, from detailing the mathematics and precompositional processes that went into building the framework of the piece, to analyzing the form and the melodic trajectory, to the hierarchical analysis found in the contours of pitch and duration (and amplitude) that make up the foreground melody into the contours of the background parameters governing each formal section. While a suitable case for a robust structural theory has already been made, there is one significant issue that has yet to receive adequate attention: the way in which Wallin himself, through artistic interpretation, orchestration, and composing-out the foreground melody, has highlighted musical events. It is these signatory features that make the composition truly personal, and, from an analytical standpoint, provide commentary on the underlying structures that the composer’s inner ear must have perceived to be important. Some observations have already been made with regards to the musical (as opposed to structural) content of ning in order to draw thematic connections in the form (given throughout Chapter 2). Indeed, the musical features that most clearly identify thematic ideas are not overtly found in the foreground melody, but are simply embellishments thereof, completely of the composer’s (and not the mathematics’) aesthetic design. Consideration of a few more examples from the piece helps bring to light both the ways in which the foreground melody “spoke to” Wallin and the grander schematic picture that he was intending to paint. 160 Derivative Counterpoint (in the “Floating” Section) Precompositional section 8, the first subsection of the “Floating” passage (mm. 87-105, b.3), is worth revisiting once again in the present context. In the formal description of the passage, “Floating” was deemed to be a sort of “transformative catalyst” through which the “Motoric” thematic idea from the “exposition” was developed to reemerge later as the “Sharp” thematic idea (refer to Chapter 2, page 68). That “Floating” should have this formal function is actually the composer’s doing, a fact revealed by comparing the foreground melody (as presented in the draft score) of the “Floating” passage with that of the two “Motoric” passages (mm. 32-41, b.3 and 6875) and two “Sharp” passages (mm. 147-153 [162] and 191-194 [196]). In the final score, the principal thematic feature in common with these three passages is the invariable rhythm (constant sixteenth or eighth notes) with pitches receiving several repetitions, but in the draft score, the foreground melody only exhibits this characteristic for the “Motoric” and “Sharp” passages, not for “Floating.” As seen in the contours of the foreground melody for this section (refer to Figure 4.14, Chapter 4, page 130), there is very little rhythmic or pitch repetition until near the end of the section, but the final score is rife with a steady rhythmic drive and repeated pitches throughout the whole section. Certainly, there are a number of alternative ways that Wallin could have colored this passage of the music, but by using the “echo/pre-echo” technique (see Chapter 3, pages 96-97), Wallin’s ultimate decision here was to deliberately associate this passage with the “Motoric” and “Sharp” passages, even though there is no apparent connection precompositionally. Wallin must have felt that, in terms of compositional balance and thematic presentation, this interpretation was a necessary adjustment at this point in the music. By rendering the foreground melody in this way, however, Wallin was able to turn this single melodic line into a harmonic canvas with multiple lines in counterpoint. The melodic pitches are disseminated amongst the four parts of the quartet in a klangfarben Melodie style of orchestration (rather than a unison style), and their durations are stretched out, both before their original onsets and after their original cutoffs. The pitches gradually emerge from niente, crescendo 161 toward an accent, and then diminish to niente again, creating an artificial large acoustic space.163 The resulting pitch overlap of these crescendo/diminuendo hairpins can make up to seven pitches sounding simultaneously (with double stops in each string part), creating chords that gradually morph and change, shifting one pitch at a time. This derivative counterpoint and its byproduct harmony will both receive treatment in turn here. Because excerpting this entire passage (with all four parts from the score) in order to analyze the counterpoint would be spatially impractical, it is easier to see the distribution of the pitches amongst the parts by consolidating them graphically, superimposed over the familiar pitch contour for this passage taken from Figure 4.14a (see Figure 5.1). A couple of things become apparent immediately from this representation. First, Wallin is largely using good voice-leading principles to guide which instrument takes up which pitches, particularly in the first half of the passage. The foreground melody by itself is almost completely comprised of large leaps, but by keeping the low-ranged, mid-ranged, and high-ranged pitches in the same respective voice, the result is an audibly distinguishable set of relatively smooth lines playing against one another. There is a little bit of role swapping where the voices cross (between the violin and viola at the 17th and 18th pitches, and the 34th and 35th pitches, and between the viola and cello at the 4th and 5th pitches, and the 40th and 41st pitches), and after about the 44th pitch onward, the voices are almost constantly crossing (though here there are very few large leaps due to the closeness of the pitch range, so the interweaving of the counterpoint sounds perfectly natural). Otherwise, the instruments are playing for good chunks within a reasonably similar tessitura. Another thing that becomes apparent from Figure 5.1 are some of the minor changes that Wallin made in implementing the foreground melody in this section. There are some notes omitted (the 25th, 48th, and 90th), pitches altered (the 31st, from C5 to B4; the 69th, from F5 to 163The only means of detecting the actual durations and pitch change positions of the original foreground melody in this passage is by the accents, that is, each note at the commencement of a diminuendo marking. 162 Figure 5.1: Orchestrational distribution of the pitches from the foreground melody of precompositional section 8 (with parts color-coordinated). D5; and the 87th, from G5 to F5), and numerous shifts of rhythmic placement (not shown in Figure 5.1), partly to adjust to the constant pulsing of sixteenth notes, but sometimes to outright change the placement or emphasis of attacks relative to the surrounding sonic landscape. Though some of these might be due to human error, it is also likely that the composer made some deliberate editorial decisions for the sake of musicality according to his perception. This reveals that the composer’s adherence to the absoluteness of the math, although close, was not obstinately rigid, and provides information about those very spots in the foreground melody that he felt needed massaging. A most intriguing observation from Figure 5.1 is the corollary pattern of instrumentation, which emerges as a byproduct of distributing the pitches in specific registers to facilitate the voice leading. The patterns of ascent and descent in the foreground melody contour for pitch (and, though not shown, also for duration) are manifested in an artificial voicing contour (represented at the bottom of Figure 5.1, vertically aligned with the pitch and counterpoint contours above it). 163 Some caution must be taken not to conflate the ascents and descents of this new artificial contour with the ascent and descent of pitch (contour points here only indicate instrumentation, and not necessarily registeral pitch placement). Nevertheless, the regularity of the three-increment pattern still appears, at least in the first half of the section (where the three string instruments are mainly in use), with the melody consistently passing off from viola to cello to violin cyclically for the first thirteen pitches, then reverse cyclically (from violin to cello to viola) from the 15th pitch until the oboe enters (28th pitch), then in a sort of near palindrome configuration from the 29th to the 38th pitches (violin-cello-viola-violin-cello-violin-viola-cello-violin-viola), and again after an intermittent pitch in the oboe from the 40th to the 46th pitches (cello-viola-violin-cello-violinviola-cello), after which point a different contour pattern configuration takes over (involving all four instruments equally). The moments where the patterns break and/or shift cyclical direction coincide with both the crossing of voices in the counterpoint contours and two-increment patternexceptions in the foreground melody contour. The voicing contour for the remaining half of the section (starting with the 46th pitch) is remarkably regular, with an ordered cycle of cello-violin-viola-oboe repeating right up to just before the section’s end (the 91st pitch). This regularity is a dead giveaway that Wallin was using patterning to systematically assign the melodic pitches. Where the closeness of the pitch range meant that voice-leading alone (avoiding voice crossing) would be drab and musically uninteresting, Wallin clearly abandons this for a sonic environment where the voices weave with one another organically (this is a great spot for visualizing the subtle nuances of the swimming salmon mingling with one another). The resulting four-strand French braid, as it were, bears a coincidental if not altogether blatant resemblance to the contour behavior found in the parameters illustrated in Figures 4.30 and 4.31 (see Chapter 4, pages 151 and 154, respectively), most notably the contour for “Position Number for c and d paired values.” The cello-violin-violaoboe sequence in the second half of Figure 5.1 follows a <…0 2 1 3 0 2 1 3 …> contour point 164 pattern, which uncannily resembles the <…0 3 1 4 0 3 1 4 …> contour point pattern that is regularly found in the second half of this parameter’s contour in Figure 4.31.164 Could it be that Wallin derived a voicing pattern for orchestration in the same way as he derived the sequence of values for this parameter (and for other parameters for that matter), by using his method from the logistic equation? Evidence seems to suggest so. Even though it is possible that this is just a funny coincidence resulting from cycling through the four instruments in a straightforward, consistent way, the extra element of the section’s “voicing” contour changing its behavior midway through the section strongly implies a connection with the logistic equation process that rendered the parameter values. Most of the parameter contours transition from three-increment contour patterns to steady two-increment increase-decrease contour alternations, and it seems that the voicing contour for this section exhibits this same kind of transition. Of course, there are many more contour points in the voicing contour than there are in the parameter contours (those from Figures 4.30 and 4.31), so no one-to-one correlations can be drawn, but there is still an overall likeness that suggests, at the very least, a source of inspiration in Wallin’s orchestrational scheme. While the counterpoint contours for this passage provide great insight into the voicing strategies used by the composer, the harmonic content here supplies further commentary on the composer’s artistry, since the effect of the passage is less one of interacting melodic lines and more one of harmonic coloring in broad strokes. Indeed, the counterpoint is heavily disguised by being broken up with intervening rests and with notes that “breathe” organically (i.e., crescendo/ diminuendo), so that the listener, caught up in the moment of lush harmony and slowly evolving chords, is made to be almost unaware of the guiding melodic line (or lines) at work backstage. This style of shaping harmony makes this passage stand out, since it never really appears with 164This contour point pattern appears as <…1 4 2 5 1 4 2 5…> in the original values for the “Position Number for c and d paired values” parameter (the form prior to being normalized) as seen in Table 3.1. 165 quite the same effect anywhere else in the piece, and makes this aspect of it worth a little consideration. Because the pitches are perceived here as chords, a comparison of the pitch-class sets (and their subsets) formed by distinctive chord formations could offer a narrative of continuity that reflects the composer’s manipulations of his materials. Derivative Harmony (Pitch-Class Set Analysis) (in the “Floating” Section) The harmonic component of ning is perhaps more complex in precompositional section 8 than anywhere else in the work, and the sets (chords) created thereby are worth examination from this standpoint. There are two main problems to address before an effective harmonic analysis can take place. First, as was previously pointed out (in Chapter 3, page 93), the pitches that appear in any given section can only come from its corresponding crystal scale (for precompositional section 8, this is scale 2). So, to consider the notion of pitch-classes (which are the same for all octaves) is somewhat contrary to the pitch collection formed by the scale, because each pitch is specific only to itself, and does not replicate at each and every other octave. Therefore, when the pitches are labeled according to pitch class (for the purpose of making a pitch-class set), it gives the erroneous impression that such a set could be transposed (either at the octave or any level of transposition, including the prime form), because the same configuration of pitches may not be actually available in different pitch ranges of the scale. That said, labeling pitches according to their class can still provide a useful (although not wholly accurate) way to look at the harmonic makeup of pitches that are available within such a collection, since it facilitates the comparison of harmonic sets with one another. The second problem is one of judgment: By what criteria shall the sets in this section be segmented? Because there are practically no simultaneous pitch changes, the way in which to group pitches together into sets is tricky with no clear harmonic separations. This is especially troublesome when considering the placement of each pitch: Should pitches be considered respective to where they enter in foreground melody, or according to the “pre-echo” entrance of 166 the pitches as found in the score? Where pitches are constantly overlapping one another in the score, potential chords could pull from both types of pitch entrances. And where should the sets themselves be considered to overlap, and under what conditions should a given pitch be considered to be a member of two (or more) adjacent sets? The answers to these questions are subjective, and, depending on the line of reasoning, could result in very different harmonic readings of the same passage. Bearing these problems in mind, the approach taken here is to segment chords according to the appearance of pitches in the score (rather than by the ordering of the foreground melody), on the grounds that harmony in the present piece is not intrinsic to its specifically melodic structural makeup. This means that harmony will be considered for its own sake, by how it actually sounds in performance, and not on the basis of melodic note sequence. This more correctly reflects the reality of the presence of harmony in ning: the harmony is less the byproduct of an underlying melodic line, and more the result of the composer’s artistry. The order of the pitches is one thing, but the stretching-out and overlapping of those pitches (backward and forward) that makes the harmony possible is due completely to Wallin’s intuition. By the same token, the vertical segmentation of chords and the determination of which pitches will be shared between chords will also come from an intuitive approach, relying on visual and aural perception of chord structure. Though this type of analysis has these problematic issues (particularly for the section in question), there is still some useful information to be gleaned from it, for even a flawed approach to divining pitch-class sets still yields some insight. And though the specifics are difficult to pinpoint, a general harmonic trend or pattern should still emerge from the overall harmonic picture of this section. Figure 5.2a provides a pitch schematic for precompositional section 8 that eliminates the various entrances of the different voices of the quartet and accounts simply for the moments in time at which any particular pitch may be sounding. This makes chord segmentation 167 a. b. Figure 5.2: Harmonic makeup of precompositional section 8. a. Pitch distribution within the section segmented according to perception. b. The thirteen chords segmented from the section in order of occurrence. easier to visualize than with the score (chord partitions are included). Figure 5.2b then shows these chords in standard notation with their pitch-class sets and prime forms (set classes). The chords formed in this passage (according to intuitive, perceptual pitch-grouping) are tetrachords, pentachords, and hexachords. As complete chords, the only set class to occur more than once is class 5-10, and this only by inversion. But though the harmonic restatement of entire chords at any level of transposition is absent superficially, there is unmistakable resemblance 168 between the chords in their pitch-makeup and construct. A couple of things visibly emerge from Figure 5.2 about the harmonic content of this passage, including the purveyance of many common tones, and the clusters of pitches that get progressively tighter from third-related intervals to a preponderance of stepwise relationships. Most of all, these sets clearly exhibit common trichord and tetrachord subsets embedded within them. Some observations about specific pitch content are worth pointing out because they reflect the foreground melody and the harmonic possibilities of the crystal scale in use in this section. For instance, some pitch classes appear frequently, particularly B (E) and D (2), with G (7) also being quite common. This tends to give these pitch classes a sort of tonal gravity, favoring harmonies that build around a G-major sonority base, generally throughout the duration of the whole section. Conversely, some pitch classes are notably missing over large swaths of the section, such as F (5) in the first nine measures, E♭ (3) in the first twelve measures, F♯ (6) in the first sixteen measures, E (4) in the last nine measures, C♯ (1) in the last eleven measures, and A♯/B♭ (T) throughout the entire section. This is due to the gradual change in tessitura, and demonstrates the ranges in which certain pitches are and are not available for use from the crystal scale. This also points to the way in which the harmonic palate gradually shifts from a sort of E4/F4-oriented centricity (the first and second chords of Figure 5.2b), in which the F is sonically absent, to an E5/F5-oriented centricity based major/minor sonority (the twelfth and thirteenth chords of figure 5.2b), in which the E is sonically absent. As for set relationships, though, the chords share a certain intervallic consistency. Even though the spacing of the pitches is wide with intervals of a third or larger, as sets, the pitchclasses actually come together in a stepwise array. The first chord, set class 4-21 (0246), is fundamentally distilled as a group of whole steps, even though sonically they are more widely spaced. The sets of the chords thereafter begin to introduce half steps into their makeup bit by bit. While this in itself is not necessarily anything special (since the majority of set classes are comprised of mostly half and whole step relationships), what is noteworthy is the way in which the 169 actual sonorities seem to align their formation more and more closely with that of the set formations, as the intervals begin to compact together over time. The chords at and near the end of the passage, by and large, mirror the set distillations themselves. The common ground between chords lies in their subset relationships.165 Figure 5.3 provides a comprehensive inclusion lattice for the segmented sets in Figure 5.2, with the direct subset/superset interrelationships of these sets themselves highlighted. From this, it becomes apparent that all of these sets (as they have been segmented) are related somehow as subsets/supersets of one another, save for one, the fifth chord of Figure 5.2, set class 6-5 (012367). Given that the composer was not thinking at all in terms of sets or set relationships, it is nevertheless remarkable that the passage exhibits such a large degree of harmonic consistency, having only one chord out of thirteen that does not seem to fit with the harmonic palate of the rest of the passage. The lattice does show in a more abstract way, however, that there is resemblance between this “outlying” hexachord and the other segmented sets in terms of shared trichordal subsets. Of particular note is set 3-5 (016), the most pervasive of the trichords in the lattice, which occurs amongst all four segmented hexachords of the passage, and in three of the segmented pentachords, the fourth chord, set 5-32 (01469), and the tenth and eleventh chords, set 5-10 (01346). The comprehensive lattice also demonstrates another interesting point. Having such an array of tetrachords, pentachords, and hexachords in this passage, it is reasonable to expect that all twelve trichord subsets ought to appear in some guise somewhere amongst them, but curiously, set 3-12 (048) (i.e., the augmented triad) is conspicuously absent. While this is quite interesting, it is not necessarily meaningful, since this is less to do with Wallin’s harmonization and more to do 165Allen Forte called such relationships of subsets and supersets the “inclusion relation.” This is described in his seminal work about pitch-class set theory, The Structure of Atonal Music. Allen Forte, The Structure of Atonal Music (New Haven: Yale University Press, 1973), 25. (013469) 6-27 (023579) 6-33 (012479) 6-Z47 5-5 5-6 5-7 5-8 5-10 5-11 5-14 5-16 5-Z18 5-19 5-23 5-24 5-25 5-29 5-31 5-32 5-34 5-35 5-Z36 (012) 3-1 (014) 3-3 (015) 3-4 (016) 3-5 (024) 3-6 (025) 3-7 (026) 3-8 (027) 3-9 (036) 3-10 (037) 3-11 4-4 4-5 4-6 4-7 4-8 4-9 4-10 4-11 4-12 4-13 4-14 4-Z15 4-16 4-17 4-18 4-21 4-22 4-23 4-26 4-27 4-28 4-Z29 Figure 5.3: Comprehensive inclusion lattice for set classes segmented from precompositional section 8 (from Figure 5.2, shown in bold). (013) 3-2 4-1 4-2 4-3 (0123) (0124) (0134) (0125) (0126) (0127) (0145) (0156) (0167) (0235) (0135) (0236) (0136) (0237) (0146) (0157) (0347) (0147) (0246) (0247) (0257) (0358) (0258) (0369) (0137) 5-4 (01236) (01237) (01256) (01267) (02346) (01346) (02347) (01257) (01347) (01457) (01367) (02357) (01357) (02358) (01368) (01369) (01469) (02469) (02479) (01247) (012367) 6-5 170 171 with the sequence of pitches generated in the foreground melody. It is odd that the foreground melody for this section never really afforded Wallin the opportunity to bring two adjacent majorthirds into simultaneous sonority, but this only would have been possible if Wallin had used some alternative scheme for pitch change amongst the instruments of the quartet (other than that shown in Figure 5.1). This fact is made aurally evident by the way in which Wallin deliberately emphasized tertiary harmonies with the chords he fashioned in the beginning of the passage. On the other hand, some of the trichord and tetrachord set classes that do appear frequently as subsets embedded in the passage include 3-2 (013), 3-7 (025), 4-3 (0134), 4-10 (0235), and (as already mentioned) 3-5 (016) and 4-21 (0246), sets that emphasize stepwise relationships. The tetrachords especially make this point, as all three appear in the section as stand-alone harmonies (the first, third, and twelfth chords). Both 4-3 (0134) and 4-10 (0235) are made up of two sets of 3-2 (013) overlapped, and both also encapsulate third-related intervals (which are common to the makeup of all the chords), while 4-21 (0246) emphasizes whole steps. Musically relevant examples of these subset relationships occur when common tones forming these sets are shared between adjacent chords, a deliberate result of Wallin’s compositional input. For instance, the set class 4-10 (0235) is shared between the eighth, ninth, and tenth chords on common tones A4, B4, C5, and D5. Then there is the idea of a set recurring though a process of inversion, such as seen in the last chords of the section, where the B4, C5, D5, and E♭5 creating the subset 4-3 (0134) in the tenth chord gradually transform into the D5, E♭5, F5, and F♯5 in the twelfth chord. The trichord set 3-7 (025) is most obviously associated with the beginning of the section, its wider intervals reflective of the wider spacing found there. Many permutations of this set appear embedded in later chords (there are three found in the seventh chord, for example), but especially in instances with the tetrachord 4-10 (0235), of which the set 3-7 (025) is a subset (twice). This trichord therefore perpetuates throughout the section, though its presence dissipates by the eleventh chord. On the other hand, the trichord 3-2 (013) is also incredibly common, but is most obviously associated with the chords at the end of the section, becoming more and more prevalent 172 through the progression of the section. It also appears as part of the tetrachord subsets, but often in other arrangements independent of the tetrachords as well, with its final appearances symmetrically encapsulated in the final pentachord 5-8 (02346). The sort of “exchange” wrought between these two trichord subsets over the course of this section is representative of the intervallic collapse happening with the section as a whole, the set 3-2 (013) being a diminution of the set 3-7 (025). Once again, the concept of fractal self-affinity seems to appear, even in such basic harmonic building blocks as these, whose simultaneous sonorities are merely the result of arbitrary compositional decisions. The way in which the available pitches are combined and associated with one another harmonically provides insight into Wallin’s intuitive response. This section is one in which the wide spread of the pitch range is brought to a focal point bit by bit, where the outlying pitches are tied into the general center pathway of the melodic trajectory by means of harmonic connections. The harmonic coherence of this section is established via the regular appearance of subsets drawn from set classes 3-2 (013), 3-5 (016), 3-7 (025), 4-3 (0134), 4-10 (0235), and 4-21 (0246), set classes that form the basis of the half-step/whole-step melodic makeup found at the end of the section, the “goal” to which everything is headed. Indeed, by making these trichords and tetrachords (already present in the foreground melody) more emphatic via harmonically overlapping the pitches that create them, Wallin is effectually foreshadowing that the melody is soon going to be similarly tightly clustered. Though it feels at first like a harmonically wide space in which the “fishes” are spread out, it is not long before it becomes aurally evident that all the “fishes’ paths” are eventually going to converge as they move in the same direction, precisely because the harmony highlights this closeness by moving from open-spaced chords to close-spaced chords. Use of the Motive to Highlight the Crystal Scales and Flesh-Out Contour Patterns Because the foreground melody often contains wide leaps between its pitches (except where the “Pitch Spread (Range)” parameter has a relatively high value, which limits the melody 173 to only a narrow band of pitches), sometimes the essence of musical line and the scalar pitch collections that make up a melody get lost in the din of wide pitch space. Certainly, Wallin liked such a wild and interesting palate of sonic possibility, but must have also felt the need to stay grounded and counterbalance this with a healthy dose of simple stepwise motion, as evidenced by scalar embellishments (large groups of grace notes or very short rhythms) included in some passages of ning. Probably, too, having gone to all the trouble to construct his crystal chords and their derivative crystal scales, he thought it a shame not to feature these remarkable pitch collections in a clear way. These composed-out scalar embellishments can be considered to be motives, because they are small gestures that function as significant thematic traits identifiable with a thematic idea.166 These gestures seem to fit into certain motive types, or categories, defined less specifically by their pitches and intervals and more by their generic gesture and shape (i.e., overall contour). The reemergence of a particular motive type in the music causes the listener to draw upon memories of earlier occurrences of that gesture, and therefore make a thematic connection, even though there may not be an exact replication of the motive in the traditional sense. Several of these kinds of connections were already pointed out in Chapter 2 in conjunction with the form, but looking at the specific motivic gesture types again here provides the additional benefit of knowing how the composer intended to make connections, since these are now clearly understood to be purely intuitive compositional devices, and not the byproducts of the mathematical formulae. One motivic type is the long, unidirectional scale, either descending or ascending. This “scale-run” motive perhaps gives the flavor of the crystal scale being used in the passage better 166Because motives are small musical cells with a distinctive identity, they are traditionally excepted to be constituted of only two to four notes. In the case here, however, the scalar embellishment motives are made up of several grace and/or short notes, with most being upwards of five and many upwards of ten. The reason that so many notes are included in these motives is that they happen very quickly as a single musical gesture that really cannot (and should not) be further parsed. 174 than any other motive type, and therefore imbues the thematic idea with a sense of that scale’s unique character. This motive identifies with the “Evaporating” thematic idea, appearing in both the “Transparent, Evaporating” passage (beginning with the transition starting m. 167 through m. 178, b.3) and the “Evaporating” passage (mm. 237-253), as well as making an appearance in the “Softening, Yielding” transitional passage (the motive gradually emerges, staring in m. 211, appearing more and more frequently up until about m. 220). In both the “Transparent, Evaporating” section and in the “Softening, Yielding” passage, scales 3 and 2 are being used in succession, so the evocative power of the motive here creates a modality-of-sorts that is at first whole-tone-like, then octatonic in character. In the “Evaporating” section, scale 1 is in use most of the time, so the passage has a sense of small chromatic pitch-clusters broken up by intervening leaps of a third. The “scale-run” motive is most often initiated by the pitch derived from the foreground melody, with the remaining pitches that follow being a length of the crystal scale (see the excerpt in Figure 5.4). The composer, in determining the relative length of this motive at any given moment, evokes a certain kind of emphasis that is somewhat difficult to interpret. It is used rather consistently with almost all of the melodic notes in the “Transparent, Evaporating” and “Evaporating” sections, but the long versions of the motive tend to draw attention to the scalar motion and give the note a weightier presence, while the very short versions tend to emphasize the effect of silence. It becomes unclear which (the scale or the silence) the composer is evoking the most, but this interplay and exchange between the two uses of the motive give these passages a great deal of musical vibrancy and engagement. In fact, it seems these two uses of the motive are two sides of the same coin, and that the composer is using both effects in balance with one another. The flautando, detaché, and dolce expressive markings that often accompany this motive also add to its unique thematic character, and are almost as much a part of the motive as the gesture itself. 175 Figure 5.4: Excerpt from the “Transparent, Evaporating” section of ning, mm. 171-78. 176 When the composer uses this motive, the most telling thing is those pitches in the passages that are sustained against this backdrop of fleeting scalar gestures. These stand out in such a way as to be the pitches the composer deemed to be important, and especially so being assigned to the oboe. For instance, the E4 in m. 173, the B3 in m. 174, and the recurring C♯4 in mm. 175-177 all receive this extra emphasis (see Figure 5.4). As it goes, these notes are not necessarily structural in the sense of the pitch or durational contour-hierarchy for the foreground melody in this passage, though they do seem to carry some weight as far as amplitude (accentuation). What they are, then, is evidence of Wallin’s own sense of musical timing and shaping. They are not innately structural, but are made to seem so by the composer making them deliberately different than the rest of the surrounding sonic landscape created by the “scale-run” motive. Wallin also seems to take inspiration for motivic shape from the diagram of the form for pitch (Figure 2.1, Chapter 2, page 26). Some of these motives have already been pointed out in Chapter 2 (for example, the downward glissando motive in the cello part, mm. 183-188, which borrows from the shape of the melodic trajectory found at the corresponding spot in the diagram [see page 55]). In a way, by writing such musical shapes into the surface material of the music, Wallin is embedding the shape of the form within itself, i.e., fractal self-similarity. The case for the music’s fractal connection therefore stems not only from its inherent mathematical structure, but from the composer’s intuitive motivic responses to the forms created in that structure as well. Sometimes, the defining motives of one theme are alluded to in sections of a different prevailing theme (such as in the case of the “Softening, Yielding” passage mentioned above), but there are also some motives that are exclusive to a single thematic idea. The “Transparent” thematic idea contains an exclusive motive (primarily in the oboe/English horn, sometimes in the violin) that starts from a relatively high pitch, quickly moves downward and returns upward again to the high point in stepwise motion with occasional skips. This motive appears like a little “valley,” and as a motive type, it is particular to the “Transparent” sections of the piece (mm. 13-26, 52-63, and 197-206) and therefore functions as a thematic identifier because it does not really appear in 177 quite the same way in other sections. It does, however, draw some similarity from the “scale-run” motive by bringing awareness to the underlying crystal scale in a comparable way, which ties it in with the “Transparent, Evaporating” section (and thus the “Evaporating” thematic idea). Another motive type similar to the “valley” motive appears in the oboe part of the “Strong” section, though it is more of a “valley/peak” combination, looking like a sine-wave in the score. It gives a mild allusion to the “Transparent” thematic idea (with its “valley” motive), but it is nevertheless distinct from it, and is therefore a thematic identifier for “Strong.” In the “Strong” section, this motive exhibits a great deal of repetition, though with some fragmentation and interpolation, which makes “valley/peak” more of a motive in a literal sense (rather than just a generic motive type). Indeed, the motivic repetition and fragmentation seen in this section is a strikingly unique feature of the whole work, and is highly suggestive of fractal patterning in its ordered reiteration (see Figure 5.5). The ordered pitches of this motive as they first occur in m. 254 are B♭4-E♭4-A4-B4-F♯5G5-D5-A♭4-F4, the last of which is here sustained. In m. 256, the motive appears to be nearly identical, with a few extra pitches added (G4, D♭4) and one reordered (A4): G4-B♭4-A4-D♭4-E♭4B4-F♯5-G5-D5-A♭4-F4. In m. 257, a fragmented version appears, starting with E♭4 and eliminating F♯5 and G5: E♭4-B4-D5-A♭4-F4. Then in m. 259, it appears as it did in m. 256, but without the initial G4 and B♭4: A4-D♭4-E♭4-B4-F♯5-G5-D5-A♭4-F4. The final fragmented version in m. 260 is again unique but familiar: G5-F♯5-E♭5-D5-A♭4-F4. Other small grace-note groups in the oboe part in this section borrow from these pitch groups. Something similar happens in the violin and viola parts of this section. In these instruments, the quick-riff gestures are mostly unidirectional and mostly ascending, often utilizing open strings. The violin’s initial sequence of pitches in m. 254 is G3-D4-A4-B♭4-D♯5-E5. This appears again twice in m. 259, first without the B♭4 (G3-D4-A4-D♯5-E5), then completely, with E5 being a sustained pitch. Other remnants of this motive occur elsewhere: three grace notes in 178 Figure 5.5: Excerpt from the “Strong” section of ning, mm.254-59 (252-59). m. 258 (G3-D4-A4), and a retrograde version with a couple of alterations in m. 255 (C♯5-B♭4-A4E♭4-G3-D4). The viola’s motive appears fully formed in m. 257: G3-B♭3-F4-F♯4-A4-B4-D4-F4C♯5. A truncated version appears in m. 256 (G3-B♭3-F4-F♯4-A4), and another allusion occurs in the grace notes of m. 254 (G3-B♭3-D4). Such motivic repetition with fragmentation in each of the three upper voices is strikingly akin to the repetitive patterns found in the self-similar nature of fractals. Yet while this patterning is apparent and ties together the elements of the “Strong” 179 passage, it is the very avoidance of exact repetition that bears the fingerprint of the composer seeking to personalize the work. Both the “valley” motive type and the “valley/peak” motive type can also be categorized as pitch contours. The “valley” motive reduces to a basic “valley/peak” motive reduces to the familiar three-increment pattern formation, while the pattern formation, both with only one depth of reduction. Unlike the contours at play in the foreground melody and the background parameters (which were generated through the aid of mathematics), these pitch contours have several more pitches occurring in between maximum and minimum points, the tell-tale sign of the composer’s embellishing figurations. But even with the fleshing-out of these contour shapes, the basic contour patterns are still easy to see, and provide a sort of connection between the superficial musical activity and the deeper contour structures at work. There are several other motive types in the piece, including the characteristic two-note skip/leap of the “Playful” thematic idea, and the quick, tremolo-like three-to-four-note oscillation between two pitches separated by a large interval that appears in several thematic ideas as a sort of consistent element throughout the work. Each motive provides commentary on the composer’s style, nuance, and personal interpretation of the mathematical sequences of values that his computer/synthesizer generated for him. But in more general terms, the motives function to create formal connections between sections and the themes they contain, connections that are not manifested in the bare bones of the foreground melody alone. Indeed, without the composer’s motivic inventiveness (as well as his orchestrational style and compositional development processes), the piece’s distinctive musical form would be very difficult to discern, but with it, the form (namely sonata form, as presented in Chapter 2) emerges quite intuitively. General Conclusions From what has been observed, it is clear that Wallin was guided as much (or more) by intuition as by mathematical structure in composing ning. Wallin’s use of motivic devices and 180 orchestrational coloring to flesh out the monophonic foreground melody into a heterophonic (which occasionally is quasi-homophonic/polyphonic) musical narrative is evidence that the composer took most of the writing into his own hands, rather than merely trusting the final sonic result to the outcomes of formula iterations. And it must be remembered that even the foreground melody itself, as mathematically rigorously as it is constructed, was also selectively engineered and patched together by the composer, as were the initial precompositional decisions for the parameter values for formal constraints. Wallin’s hand is in the entire compositional process, and the mathematics that drive the underpinning structure (and provide the piece with its inherent hierarchy of contour patterning) were never left to randomness. He admits as much: …fractal mathematics [has] been an important component in…my instrumental music. These fascinating mathematical phenomena, simple equations from which strangely organic structures [emanate]…provided a means to generate a quite detailed skeleton on which I could sculpt the music. With this technique, there was a constant collaboration between my subjective, artistic choice and the computer: I gave the computer specific tasks, and chose the ‘best one’ out of the different solutions offered. When a satisfactory skeleton had been erected, I could let my imagination interact with it to shape the final score.167 However, knowing that Wallin’s “fractal music” is ultimately guided by intuition does not diminish the intricate role that the mathematics play in the music’s success, functionally and formally. Indeed, the mathematics were the primary source of inspiration for writing this piece, and certainly something inherent in the iterated results of the formulae themselves evoked a specific musical narrative imagery in Wallin’s mind. Clearly, Wallin’s intuition is not simply reactive or impressionistic (which was his chief criticism of the “fractal” music of Ligeti and others [see Chapter 1, page 10, including footnote 16]), but is always carefully bridled by a sturdy, mathematically fortified methodology. If mathematics has had such a significant influence on ning from a compositional perspective, then it stands to reason that the same is true for an analytical perspective. Therefore, 167Wallin, “Lobster Soup.” 181 the form superimposed on the piece, i.e., sonata form, not only comes about from intuitive deduction via analysis, but is also reinforced by mathematically based sectional divisions. The golden-mean proportion of the large-scale formal divisions (between the exposition/development and the recapitulation/coda) is also the same proportion by which the formal limits on pitch and the small-scale formal parsings are guided to shape the piece (such as seen in Figures 2.1 and 2.2, Chapter 2, pages 26 and 28, respectively). Every other formal parsing that appears in the piece is based on the calculated iterations for sectional length (from the logistic equation results skewed by the golden-mean proportioned outer limits), which are combinable into larger groupings based on ordered relative section lengths, first in medium-long-short groupings, then long-short-medium groupings, and then finally a short-medium-long grouping, with grouping shifts occurring at formal junctures of the piece (between the exposition and development, and between the recapitulation and coda). It is according to these mathematically determined formal parsings that the formula iterations guiding the other formal parameters (those from Tables 2.1 and 3.1, pages 30-31 and 84, Chapters 2 and 3, respectively) are derived. Therefore, every parameter is interconnected directly with the sectional lengths, and in turn, these parameters give each formal section its unique melodic and rhythmic trajectories. This is the fertile ground out of which each of the piece’s distinct thematic ideas spring correspondingly, as they emerge from the foreground melody planted therein (which itself is computed based on specific parameter values, namely the paired values for c and d in the Frøyland formula, and the special scales derived from Wallin’s unique technique of constructing crystal-like pitch formations). While all of this formal scaffolding was deliberately calculated by the composer, the association with sonata form was not. And yet sonata form arises organically from it, as if it were somehow intrinsic to the process. Normally in analysis, a piece’s formal structure is derived from the thematic changes that are found on the music’s surface, but in this case, it is vice-versa, i.e., it is the thematic changes appearing on the surface that derive from the formal structure. The form is 182 compositionally deliberate, and the interpretation of sonata form thereof is just as much the result of the precompositional planning as it is of the thematic material present in the final score, even if unintentional. While there is a great deal of mathematical precision that drives both the piece’s form and the foreground melody couched therein, the most remarkable aspect of ning is found through a less precise, more generalized means: the patterns of increasing and decreasing note values in terms of pitch, rhythmic duration, and (less precisely) amplitude, as best seen in representative contour figures. When pitch, durational, and amplitude contours of the foreground melodic line are examined in different formal sections, connections emerge where the sequence of ascents and descents match up. In some cases, these matching sequences confirm a formal association already well-established (they occur within two instances of the same thematic idea [e.g., precompositional sections 17 and 25]), while in others, they suggest new kinds of formal relationships (they occur within instances of differing thematic ideas, indicating either completely contrasting variations of the same melodic material [e.g., precompositional sections 4 and 29], or a merging of two thematic ideas, providing a referential link that associates the two instances in the context of the larger formal scheme [e.g., precompositional sections 24 and 28]). Whether the contours of the foreground melody in different formal sections can be sequentially matched with one another or not, they all exhibit a notably similar manner in which they alternate between increases (ascents) and decreases (descents). In spite of the formal parameters that govern and skew the foreground melody, as mentioned before, there are tell-tale patterns that consistently appear throughout the piece, which can be categorized into three types: a combination of two increases with one decrease in some rotation (i.e., , , or ) that constitutes the three-increment regular pattern (so named due to its predominance in the piece’s contours), a combination of one increase and once decrease in some rotation (i.e., or ) that constitutes the occasional two-increment pattern-exception, or a set of two decreases with one increase appearing in tandem with the two-increment pattern-exception in some 183 combined rotation (i.e., , , , , or ) that constitutes the less-frequent five-increment pattern-exception. Though the piece often exhibits instances where the contour fragments neither increase nor decrease (i.e., , indicating repeating pitches or rhythms from note to note), these do not compromise the consistent occurrence of the three aforementioned pattern types, but merely disguise them, happening only because of the parameter restrictions superimposed on the foreground melody.168 Very rarely does any other combination of increases and decreases (namely instances of three or more consecutive contour fragments moving in the same direction) appear in the pitch or rhythmic duration contours, which is due to moments of extraordinarily extreme skewing in the formal parameters of the melodic or rhythmic trajectories.169 Perhaps the most compelling feature of these patterns is that they also appear at various levels of contour reduction, at late middleground, early middleground, and background levels, which strongly makes the case for the piece’s fractal correlation, demonstrating that self-similar pattern replication occurs at various degrees of “magnification.” The patterns are present in the contours of the minima and/or the maxima, which are derived from the foreground melody contours for both pitch and rhythmic duration, though these patterns are often more obvious in one over the other (for contours whose trajectories are downward, the pattern association is clearer in the contour of the maxima; for contours whose trajectories are upward, the pattern 168Amplitude is very often the same from note to note in the foreground melody, due to the fact that the iterated values thereof are approximated into one of three basic outcomes, i.e., “>,” “^,” or the absence of a mark. Though the normal patterning is occurring in the mathematical values for this dimension in the Frøyland formula, it is very hard to accurately detect in an amplitude contour, since so much detail is lost. In any case, amplitude is not affected by the skew of the formal parameters in the way that pitch and rhythm are. 169Again, the parameters which skew the melodic and rhythmic trajectories do not affect the amplitude of each note, but whereas there are only three ways in which the amplitude of each note can be written in the score (i.e., “>,” “^,” or the absence of a mark), it would be impossible to have more than two increases or decreases in a row anyway. See footnote 113, Chapter 4, page 103. 184 association is clearer in the contour of the minima). This kind of reduction represents the general increase/decrease embedded within a contour from one pattern group to the next (as found in the foreground), i.e., it divides the contour by each two- or three-increment pattern grouping. The patterns also appear at the different stages of the reduction process between the first and second depths of reduction, including the combined contours of the maxima of the maxima with the minima of the minima, in forms containing equal-and-adjacent contour points and in forms without equal-and-adjacent contour points. In performing the operations of reduction, it becomes clear that many of the contour fragments in the foreground also connect maximum and minimum points in the contours made through the reduction process, implying that there are several structurally significant gestures (in terms of contour hierarchy) present in the note-to-note motion of the foreground: dramatic shifts of pitch from high to low, or stark contrast in rhythmic duration from long to short. This reinforces the pattern tendency at all steps and depths of reduction, because such contour fragments (and the musical motions they represent) lack any intermediary contour points that would otherwise frustrate the consistency of the patterns’ appearances in the contour. The pattern types, especially the two- and three-increment pattern types, appear in each step and depth of reduction right up through the final depth of the contour’s reduction. The remarkable thing is not in the reductions themselves (as these are typical final-depth reductions for many contours), but in the fact that these final-depth (early middleground) patterns are also present in the original (foreground) contour. This is true for both the contours of the foreground melody within a given section of the music, and the contours of the parameters that constitute the form of the whole work (i.e., the background, the reduction of which would constitute a “background of the background,” that is, an ultimate contour “Ursatz” of sorts). This in turn shows that the Frøyland formula (responsible for the foreground melody) and the logistic equation (responsible for the formal parameters) are akin to one another in the way that their respective iterated values exhibit oscillatory tendencies, either between high and low values (the two- 185 increment pattern type), or between high, low, and medium values (the three-increment pattern type). Though the exact values of the iterated results of the formulae are chaotic (unpredictable), the consistent way in which these values increase and decrease is intrinsic to the math, and manifests itself everywhere in the music, from the basic line of the foreground melody to the “structural skeleton” of the formal parameters, and all the hierarchical depths of reduction in between. Even on the surface of the fleshed-out score, such as in the orchestrational hand-off of the foreground melody and in a few of the superficial compositional embellishments (both of which are unrelated to the formulae), such contour patterns occasionally show up, belying the subliminal influence of these contour patterns upon the composer’s style and aesthetic. The music is not only fractal-inspired and constructed from fractal-related (chaotic) formulae, but is fractal in its very hierarchical structure, with self-similar contour patterns governing every level of the piece. A Final Thought The analysis of ning as given in this paper also gives further credibility to relatively new theoretical ideas, including contour theory and fractal-related applications to musical analysis. While many aspects of these ideas (especially fractal) still need some work to be more fully codified and gain general acceptance in the music theory community, the success of pieces like ning (as demonstrated through such theoretical constructs) certainly keeps open the discussion of the plausible notion that, just as fractals are inherent throughout nature itself, fractals may also have an innate (though hidden) presence in all music.170 Several composer/theorists have explored this 170Michael F. Barnsley, Fractals Everywhere, 2nd ed. (Cambridge, MA: Academic Press Professional, 1993), 1; Martin Gardner, “White, Brown, and Fractal Music,” in Fractal Music, Hypercards and More: Mathematical Recreations from Scientific American Magazine (New York: W. H. Freeman and Company, 1992), 3. See also Chapter 1, page 5, including footnote 8. 186 path, recognizing the potential of a fractal-based theory to broaden and/or deepen existing conceptualizations of music.171 It is clear that ning has a rigorously patterned structural underpinning, which is ratified by its fractal basis. But the piece is also successful as a musical representation of its programmatic subject, i.e., the migration of the salmon shoal, as evocative in its mood and soundscape of the fishes’ journey as it is in its metaphorical heterophonic construct. This aesthetic evidence favors Wallin’s claim, that one’s music can indeed “benefit from chaos theory in a truly innovative way.”172 Wallin’s other fractal pieces follow close to the same basic compositional processes as ning (with only a few minor tweaks), and yet they are completely different sonic experiences/expressions. Although much of their difference is due to the composer’s subjective artistry, it is nevertheless remarkable how common origins can evolve into such unique outcomes. For example, in the sketches for Wallin’s piece Stonewave for percussion, the familiar contour patterns of the three-increment, high-low-medium oscillation can be seen in the relative pitches of the principal line (see Figure 5.6a), which, through a gradual process of parameter change (a steady increase of the c and d values in the Frøyland formula over time), eventually turns into the two-increment, high-low oscillation (see Figure 5.6b). The likeness of this line with the structures of ning is undeniable, but the final product fashioned therefrom could not be more different. With ning and Stonewave (and, by extension, Wallin’s other fractal works [Onda di ghiaccio, Boyl, etc.]) as a case in point, if pieces of music can be built in roughly the same way from the 171For example, some who have explored these theoretical avenues and/or developed additional compositional methods involving fractals include mathematicians Harlan Brothers (2007, 2009, etc.) and Gordon Monro (1995); composers Stelios Manousakis (2009), Gustavo Diaz-Jerez (1999), and Dmitry Kormann (2010); physicists Richard Voss and John Clarke (1975, 1978); psychologist Daniel Levitin (2012); and amateur theorist Charles Madden (2007); as well those composers cited in footnote 5, Chapter 1, pages 2-3. Some of the writings of Iannis Xenakis (1992) also touch on fractal musical concepts. See Selected Bibliography for citations of these individuals’ contributions to the discipline. 172Wallin, “Fractal Music.” 187 a. b. Figure 5.6: Excerpts of the precompositional sketches for the foreground melody of Stonewave. a. Part 3, mm. 1-17, b.4. b. Part 3, mm. 105, b.3-123, b.2. 188 same structural materials and yet come out completely unique and distinct from one another, it is not unreasonable to suggest that within all music, in its limitless diversity, there is a common structural thread. Certainly, Schenkerian theory manages to come close to establishing this within the tonal repertoire of western music, but this scope is rather limited. In all likelihood, there could be something more fundamental, more essential, to musical structure, a musical “theory of everything” as it were, as innate to all music as mathematics are to the physical world, regardless of compositional or cultural origins. Music theorists (just like mathematicians) are pattern hunters by trade, seeking to classify and describe the diverse musical repertoire in a distilled set of terms, and ever questing for a more perfect understanding of music’s inner workings. It may very well be that fractals could provide them the way to such an ultimate answer, and that such fractal patterns may be couched within, among other possibilities, the contour patterns of the music.173 Though such a theoretical path is on the frontier, and carries no guarantees, the observations in ning not only point to concepts in common with all music, but also open up other theoretical possibilities of substance well worth investigating. 173Aside from contour theory, there are ample examples of analytical applications of fractal and chaotic mathematics for music already in wide use, such as, for example, statistical analysis, dimensional analysis, waveform analysis, 1/f noise and power laws, L-systems, etc. See Selected Bibliography for sources that explain and explore these ideas. APPENDIX A FACSIMILE AND EXPLANATIONS OF NING DATA TABLES This appendix provides a facsimile (Figure A.1) of the original data table for the form in ning, printed in 1989 and used by the composer. The information printed on the page reflects the iterations for the different parameters that govern the form of the piece, generated with the aid of the logistic equation (see equation 1.3, Chapter 1, page 6), and which in turn constrain the foreground melody generated by the tripartite Frøyland formula.174 The information provided in Tables 2.1 and 3.1 (pages 30-31 and 84, Chapters 2 and 3, respectively) are taken from this table. The facsimile and the ways in which it translates into Tables 2.1 and 3.1 and Figures 2.1 and 2.2 (Chapter 2, pages 26 and 28, respectively) require some detailed explanation. On the left side of the page is the cardinal number for each formal section and, hence, each row of the table (the “Section Number” column in Tables 2.1 and 3.1), which corresponds to the iteration number of the logistic equation. In the “Start” and “Lengde” columns (“Section Start Location” and “Section Length” columns, respectively, in Table 2.1—“Lengde” is Norwegian for “length”), the beginning point and length (duration) times of each section are respectively given in seconds.175 174All columns in the facsimile reflect iterated results from applying the logistic equation to calculate the parameters, except for the “Start,” “Min-pitch,” and “Max-pitch” columns, which are merely descriptive, and the handwritten “Start Tone” column, which reflects initial iterations of the Frøyland formula. 175“Lengde” is equal to the difference between the “Start” time of the following section and the “Start” time of the current section. Or, more accurately, the “Start” times are simply the cumulative value of all preceding “Lengde” times, because it is “Lengde” that is calculated by means of the logistic equation, and not “Start.” However, because values listed here are rounded to the nearest 0.01, sometimes the difference is off by ± 0.01. 190 Figure A.1: Facsimile of the original data tables for the form in ning. 191 Figure A.1 (continued). 192 Both columns were originally configured for a preconceived piece duration of twelve minutes. The original times printed in the “Lengde” column have been crossed out and adjusted by hand to the actual fifteen-minute duration of ning, with each value multiplied by 1.25 (as indicated by the handwritten asterisk above the column in the facsimile, and reflected in the “Adjusted Length” column of Table 2.2 [Chapter 2, pages 36-39]). The lengths were obtained through the iterative process of the logistic equation as described on pages 16-17 of Chapter 1,176 and as skewed by the outer limits described in Figure 2.2. The “Min-pitch” and “Max-pitch” columns (the “Pitch Lower Limit” and “Pitch Upper Limit” columns in Table 2.1 respectively) give the outer pitch limits found at the beginning of each section, high and low ends, respectively. In these two columns, the values are not derived from iterations of the logistic equation as they are in the other columns; rather, they are merely descriptive of values found on the limit lines (already put in place, formed previously on the basis of golden mean proportions—see Chapter 2, pages 27, 32-33, and 55-56). These values range between 0 and 1, shown here rounded to the nearest 0.01. The corresponding pitches listed with these values are derived by first equating 0 and 1 with C3 and C9, respectively,177 then linearly scaling178 the pitches in between with numerical values, thereby adjusting these values to the 176See also footnote 25, Chapter 1, page 17, for clarification on an alternative theory about how the numbers for “Lengde” were obtained. 177This is how the pitch octave labels in the original facsimile read, but it is different from the standard scientific (American) octave labelling system. It follows another labeling system established by either Roland or Yamaha (the manufacturers of the MIDI tools the composer used) in which middle C is called C5 (instead of the standard C4). This has been corrected in Table 2.1, where the standard labels are used. This matter is complicated further when attempting to compare either the facsimile or Table 2.1 with Figure 2.1, in which the octaves of C use the German/Scandinavian system. There, the octave numbers can be converted to standard scientific (American) labeling by adding 3 (i.e., C1 is actually equivalent to C4, or middle C). 178To clarify, this scaling is actually logarithmic in terms of pitch frequency, but pitch perception is generally considered as linear, which is what is meant here. Also, since equal temperament is used here, it is more straightforward to simply imagine the 73 pitches between C3 193 nearest pitch in equal temperament.179 Within the form of the piece, each value point of the “Min-pitch” and “Max-pitch” columns gradually progresses along the preexisting curve of the limit lines from the beginning of one section to the next, that is, they gradually change over the span of time listed in the “Lengde” column. Hence, the outer limit lines drawn in Figure 2.1 are fashioned in a kind of “connect-the-dots” sort of way.180 The “Mid-pitch” column (the “Median Pitch” column in Table 2.1) describes the initial median pitch of each section. Like the “Min-pitch” and “Max-pitch” columns, the values here also range between 0 and 1, rounded to the nearest 0.01, with 0.01 as the smallest value and 0.97 as the largest. As with “Lengde” and other columns, these values reflect iterations of the logistic equation. Though the values in this column appear to give a straightforward reading of the x variable because they fall perfectly within the acceptable range of x, this is a deception, for in reality, the iterated values cannot lie outside the plotted area of the bifurcation diagram (see Figure 1.2, Chapter 1, page 7), thereby making values such as 0.01 and 0.97 impossible as iterations. The way in which the true iterated values (from the logistic equation used for the “Median Pitch” parameter) are transformed into those found in the “Mid-pitch” column is not entirely clear,181 but Wallin has indicated that the values in this column (and other columns as and C9 spread equidistantly along the vertical axis of the diagram in Figure 2.1 than a continuous spectrum of microtones or pitches particular to other tuning systems. Some pitches listed in the facsimile are mistakenly listed with the wrong octave: In the “Min-pitch” column, C4 should read C5 for both sections 25 and 32; in the “Max-pitch” column, C6 should read C7 for section 14, and C8 should read C9 for both sections 27 and 28; in the “Mid-pitch” column, C4 should read C5 for section 4. (These corrections are given in the context of the octave labelling system being used in the facsimile, addressed in footnote 177, page 192.) 179 180The rendering of Figure 2.1 simplifies the smoothness of the curves originally associated with the outer pitch limits. In the figure, it appears that the values move in straight lines from the beginning of one section to the values at the beginning of the next, but this angularity is due to the computer’s limited graphical interpolation of these data points. 181Because chaotic mathematical systems are highly sensitive, two initial values with even the most miniscule difference between them will yield entirely different values only a few 194 well) are not only scaled but also reflect the exponential relationships present in human perceptions of pitch (and, in the case of durations, perceptions of time).182 The values in the “Mid-pitch” column are scaled proportionately to the limits imposed by the “Min-pitch” and “Max-pitch” columns, that is, the value of the “Min-pitch” and “Max-pitch” columns respectively represent 0 and 1 with respect to the value given in the “Mid-pitch” column (which is why the pitch names for the values given here do not match up with those of the previous two columns). This indicates the centric pitch around which the foreground melody183 of the section begins, but it may or may not be the actual pitch used at the beginning of the section as this is contingent on the “spread” around this pitch (determined by the value in the “Q” column, as explained in the following paragraph). This median pitch will move over time through the section toward the median pitch at the beginning of the next section, in similar fashion to the behavior of the “Min-pitch” and “Max-pitch” values. This is what determines the direction (ascending or descending) of the melodic trajectory in each section. Notice that this causes the melodic trajectory of each individual section to be unidirectional, changing direction only when a new section is reached.184 However, unlike the “Min-pitch” and “Max-pitch” paths, the melodic iterations later (such as when multiplied for values of r that fall within chaotic regions of the bifurcation diagram of Figure 1.2). Whereas the values of the “raw data” (as Wallin has described it) found in Figure A.1 are at different relative scales and are rounded to only the nearest hundredth, it becomes impossible to replicate any series of iterations without knowing the precise initial values. 182Wallin, e-mail message to author, March 19, 2017. 183To clarify, the term “foreground melody” as found in this sentence (and elsewhere) refers to the iterative output of the Frøyland formula. In the case here, it refers to the x variable in that formula, as this deals with pitch. 184It is also worth pointing out that, for the most part, the direction of the melodic trajectory alternates between ascending and descending from section to section. The exceptions to this are between sections 3 and 4, and sections 12 and 13, which all ascend (see Figure 4.30, page 151). Also, there are a few sections that appear to be going the opposite direction in Figure 2.1 than is suggested by the facsimile or Table 2.1: Sections 1, 4, and 25 appear to descend, and sections 23 and 27 appear to be stationary along the median pitch path, while the table suggests 195 trajectory from median pitch to median pitch is not always linear, but follows a path determined by the factor in the “Shape” column. The numbers in the “Q” column (the “Pitch Spread (Range)” column in Table 2.1) determine the initial melodic spread around the median pitch in each section. It works just like the Q factor (or quality factor) in a band-pass filter (such as in a digital audio workstation), which establishes a given range of frequencies (bandwidth) centered around a target frequency that are permitted to pass through digital processing into audio, with other frequencies outside this range being attenuated.185 Specifically, the Q factor is the ratio of the target frequency to the bandwidth. The greater the value of the Q factor, the more specific to the target frequency are the frequencies that are allowed to pass unfettered by the filter (the narrower the bandwidth). Likewise, when the value for “Q” is high in the case here, the associated maximum and minimum pitch values collapse in on the median pitch, restricting the foreground melody to only that pitch’s path. When “Q” has a low value, its associated maximum and minimum pitch values spread out symmetrically186 from the median pitch by a given degree. This allows the foreground melody, which is centered around the path of the median pitch, to experience greater flexibility (more pitch options and movement), mostly within the range marked by the confines of the “Q” they should all ascend. These apparent contradictions are in fact illusions resulting from both outer-pitch-limit skewing and high values in the “Shape” column. Further explanation is provided in the description of the “Shape” column, pages 197-99. 185Please note that band-pass filters can only attenuate frequencies outside the range of the Q factor, not completely prevent them from sounding at all. This fact carries over to the present application, and has bearing on the actual pitch output of the Frøyland formula. See footnote 187, page 196. 186Note that the width of the Q is intended to be symmetrical around the median pitch in theory, but it does not always appear this way in Figure 2.1. At times, it appears as though the gap between the maximum pitch path and the middle pitch path have a different ambit than the gap between the minimum pitch path and the middle path. The reason for this is, just as with other parameters at play here, the Q is also affected by both the skew of the outer pitch limits and the “Shape” column factor. 196 value.187 As with the “Mid-pitch” values, the values for “Q” dynamically change over the course of each section, headed toward the initial value of the following section, causing the melodic spread to either widen or narrow throughout each section accordingly. The values of the “Q” column range between 1.03 and 4.89 (apparently out of a scale of 1.00, maximally open, to 5.00, completely restrictive).188 The “Speed” column (the “Average Note Duration” column in Table 2.1) indicates the average duration of the notes at any given moment, given in seconds. This serves as a sort of general baseline tempo, or rhythmic trajectory, to which the rhythmic profile of the foreground melody adheres.189 These values range between a brief 0.10 seconds and a lengthy 3.82 seconds. And like the “Mid-pitch” and “Q” values, these values change gradually over the course of each section, headed toward the value at the beginning of the next section.190 187As mentioned in footnote 185, page 195, the Q factor only attenuates outlying pitches. In the case of using mathematical formulae, it works like a probability gauge: The likelihood of pitches occurring outside the maximum and minimum values of the pitch spread is very low, with the probability decreasing the further a pitch is from the median pitch value, but occasionally, an outlying pitch value will poke through and occur in the foreground melody (for example, in m.237-39, near the beginning of precompositional section 29 [see Figure 4.6b, Chapter 4, page 108], the melodic extremes of F3 and A6 both lie outside of the expected pitch spread (range) suggested in the corresponding part of the formal diagram of Figure 2.1). 188In terms of the Q factor, this range is relatively small, since the Q factor in a band-pass filter can range between 0.10 and 100.00 (being the ratio of the target frequency to the bandwidth). Normally, a Q factor of 5.00 does not completely collapse onto the target frequency, but in this case, it is the highest possible value for “Q.” The reason for why the composer scaled “Q” in this way is unclear. 189The term “rhythmic trajectory” is to duration as the term “melodic trajectory” is to pitch, that is, the rhythmic trajectory follows a path that rises and falls between long and short durational values just as the melodic trajectory rises and falls between high and low pitch values. The “rhythmic profile of the foreground melody” refers to the output of the y variable of the Frøyland formula. 190The diagram in Figure 2.1 only reflects the pitch parameters. Rhythmic/durational parameters (“Speed” and “Rvekt” [rhythmic weight]) are not graphically represented, but are still given as data points below the diagram. 197 The values in the “Rvekt” column (the “Rhythmic Weight” column in Table 2.1— “Vekt” is Norwegian for “weight”) indicate the allowed degree of rhythmic variation (or deviance) from the corresponding average duration given in the “Speed” column that the foreground melody’s rhythmic profile can take.191 These values appear to be scaled between 0.00 (meaning no rhythmic variation or deviance, i.e., all notes are restricted to same duration as that of “Speed”) and 0.50 (meaning maximal variation or deviance, making a complex and diverse rhythmic profile). The values here actually range between 0.01 and 0.49. Here again, the values change gradually throughout each section in similar fashion to the other aforementioned shifting parameters. The values of the “Shape” column (also “Shape” in Table 2.1) are influential on the melodic trajectory, affecting the behavior of the dynamic changes in value that occur in both the “Mid-pitch” and “Q” over the course of a section. In the “Shape” column, the number describes the curvature of the melodic path (as seen in the diagram of Figure 2.1). The values used here range between 1.04 and 7.88 (apparently out of a scale of 1.00 to 8.00), where lower values cause the melodic path to move in a more linear way from the initial median pitch of the section to that of the next, while higher values cause the melodic path to unfurl exponentially along a parabolic curve, i.e., the path will bend slightly (or not bend at all) near the beginning of a section, then more drastically near the end of a section.192 The minimum and maximum pitch trajectories derived from “Q” also bend in like manner. 191In other words, “Rvekt” is to “Speed” as what “Q” is to “Mid-pitch,” though with the caveat that the relationship of “Rvekt” to “Speed” is not necessarily symmetrical, and that the values of “Rvekt” function inversely and out of scale with those of “Q.” 192Please note that this drastic steepness never occurs at the beginning of a section, and the slightly bending/nonbending part never occurs at the end of a section. Midway through a section, the curvature can lean either way, depending on the value of “Shape”: Higher values cause the trajectory to plateau midway and force the curve to the section’s end, while lower values cause the curve to happen more gradually over the course of the section, so the trajectory will bend midway. 198 At this point, some explanation is warranted for the melodic trajectory’s appearance in Figure 2.1 with respect to corresponding “Shape” values, which can be confusing. Logically, the melodic path of the median pitch should be continuous through the entire diagram, without any breaks, just as with the paths for the upper and lower pitch limits (“Min-pitch” and “Max-pitch”). But the trajectory apparently breaks in many spots. And there are moments where the “Shape” value is very high (greater than 6.00) where a curve in the path is expected, but instead straight lines appear, sometimes flat, sometimes at an angle. These bewildering incongruities are actually not incongruities at all, but merely illusions that appear due to the limitations of print space and computer resolution (the figure was originally rendered with a Macintosh from 1989), as well as the threshold imposed by the composer for the behavior of high “Shape” values. The melodic trajectory is in fact completely connected. Notice that the apparent breaks in the path consistently occur between sections (never midsection), and always following a section comprised of a straight-lined trajectory.193 The straight lines could be thought of as the “plateaus” of very steep parabolic curves that bend only immediately before the beginning of the next section, rising or falling so steeply that no data points along them could be rendered with the computer’s and printer’s limited resolutions.194 The angle of these “plateaus” 193Sometimes the straight lines would be hard to see without the red vertical sectional partitions superimposed on Figure 2.1 (because of the section’s brevity). 194More accurately, the composer foisted a deliberate threshold of 6.00 upon the “Shape” parameter, where any value exceeding the threshold would behave identically. Therefore, all values 6.00 and above cause the other parameter values to “plateau” throughout the entirety of sectional durations to the same degree, then abruptly (in the composer’s words) “cut away” to the new values only right at the beginning of the following section. The composer even suggested that the curve is eliminated altogether for these high values, though there is evidence in the actual score of very brief connective material filling in these “cut-away gaps” at the tail end of such sections (thus avoiding extreme melodic leaps and angular rhythmic shifts). 199 has to do with the skew of the outer pitch limits rather than the curve direction, since in a couple of instances, they apparently angle in the opposite direction.195 “Shape” also effects the skew of the intrasectional shifts between data points in the “Speed” and “Rvekt” columns as well (though these are not graphically represented). As with the pitch parameters, the durational parameters also abruptly change from seemingly stable value “plateaus” to new values at sectional divisions for those sections where “Shape” values are above 6.00. The “Skala” (Norwegian for “scale”) and “Posi” columns are somewhat different from the other columns in the facsimile, which is why they have been separated from Table 2.1 into Table 3.1 in the text (the “Scale (Pitch Collection)” and “Position Number for c and d paired values” columns, respectively, in Table 3.1). Unlike the values in the other columns, the numbers here are cardinal, exhibiting no decimal divisions. Nevertheless, the numbers in these columns were derived through the same iterative process of the logistic equation as the other parameters, but in this case, the x values were pocketed into five equal divisions of the x scale (0 < x ≤ 0.2, 0.2 < x ≤ 0.4, 0.4 < x ≤ 0.6, 0.6 < x ≤ 0.8, and 0.8 < x < 1.0), with corresponding “Skala” numbers of 0, 1, 2, 3, and 4, or corresponding “Posi” numbers of 1, 2, 3, 4, and 5. For “Skala,” each number refers to a different pitch collection derived from Wallin’s crystal chord technique (described in detail in Chapter 3, pages 85-94), ranked in order from most chromatic (0) to most diatonic (4).196 For “Posi,” each number refers to a different pair of preselected values for the coefficients c and d in the Frøyland formula (also described in detail in Chapter 3, pages 79-85), ranked in order from 1 195See footnote 184, page 194. Though the melodic trajectory is technically unidirectional within each section, this fact is obscured in these cases. 196While the term “chromatic” in this sentence straightforwardly represents the chromatic gamut of equally tempered pitches, the term “diatonic” is not wholly accurate, since this last collection doesn’t favor one particular diatonic key. It only means that it more closely resembles a true diatonic collection than any of the other collections, featuring mostly whole steps with half steps usually occurring between about every three whole steps, but occasionally between every four whole steps. 200 as the most orderly (meaning x, y, and z iterate stable or regularly oscillating results) to 5 as the most chaotic (meaning x, y, and z iterate varied and unexpected results). Whereas the values for “Min-pitch,” “Max-pitch,” “Mid-pitch,” “Q,” “Speed,” and “Rvekt” change gradually over the course of each section, the values for “Skala” and “Posi,” like “Shape,” are constant through the duration of each section. Nor does “Shape” effect the results for “Skala” or “Posi” in any way; they are dependent only on the r values given by the “Lengde” parameter. The final column on the facsimile is the handwritten “Start-tone” column (the “Starting Iteration of the Frøyland formula (Handwritten)” column in Table 3.1). Whereas the other columns printed by the computer derive from the logistic equation, this column deals with the initial values (Wallin terms them “seed” values) for x, y, and z in the Frøyland formula (explained in detail in Chapter 3, pages 80-81 and 83-85). Though all the numbers generated here are arbitrarily chosen to a certain degree (based on initial values for the logistic equation), the numbers in this column are more arbitrary than the rest because each line represents a fresh sequence of iterations (rather than a continuous string of iterations that follow from section to section), giving the composer a greater degree of control over the outcome of the foreground melody. Note that in some cases, a few different initial iterations were tried (crossed out values) before one was ultimately decided upon in each section. The range of these values is not significant in the way the ranges for the other columns are (they are not scaled with respect to anything else). There are only two other things to observe with regard to the facsimile. First, along the left side are handwritten brackets showing how the composer grouped sections together in association. These groupings are dealt with at length in Chapter 2, with regard to the form of the piece. The composer neglected to add any grouping brackets for the first six sections of the piece, but if he had, he would have grouped sections 2 and 3 together, as well as sections 5 and 6 (due to the fact that there is no clear cadence dividing them—the thematic material of the former section 201 in each pair simply merges seamlessly into that of the next [by means of an accelerando and gradual change in rhythmic behavior in both cases]).197 Secondly, in other pieces Wallin wrote using these processes, he also accounted for the parameters of timbre and global dynamic levels. In ning, however, he left these parameters out of the mix, allowing himself to more freely make choices about orchestration and general dynamic levels throughout the piece, based on contextual artistic response to the results of the other parameters and the foreground melody.198 197Rolf 198Note Wallin, email message to author, March 12, 2017. that the z variable in the Frøyland formula determines the dynamics of the foreground melody, evidenced only in the accentuation and/or expressiveness of individual notes. These are contextualized into whatever general dynamic level and/or texture is employed at any given moment. APPENDIX B EXCERPTS FROM AN INTERVIEW WITH ROLF WALLIN The following is an edited transcription of a video-conference interview between the author and the composer Rolf Wallin, held on November 4, 2015. In this interview, many of the precompositional concepts and processes used in the creation of ning were first introduced and explained. Only certain excerpts of the interview (those relevant to ning) are included here, some of which are referenced/cited in the text (omitted passages are indicated with ellipses, and lengthy omitted passages are indicated by a short, centered line). Please note that nonverbal communication and interjected clarifying commentary are also included (italicized in brackets), as well as some added verbal edits for clarity (in brackets, not italicized). Note that “right” and “left” gestures correspond to the person speaking. Benjamin Bidwell: I’m very excited to get to talk to you just because I really enjoy your music… I really am intrigued by the mathematical aspect of how these things came together in addition to the music itself… Rolf Wallin: Yeah…looking back through this, it was quite remarkable that I was able to get everything together like that. I had this kind of…combined Bach and Stockhausen [thing]…like these amazing building blocks [that] work [interlocks fingers demonstratively] where everything is dependent upon each other. BB: …Can I start by asking you some questions? RW: Sure. 203 BB: To start off here, what was your source material for ning, the mathematical source? Was there an equation? RW: Yeah. It is this equation which I use all the way, which you see in that article [referring to the “Fractal Music” lecture]. There are actually two equations… The article was made about the first fractal piece [I wrote], which is called Onda di ghiaccio, but it’s the same rules, the same machinery, with one formal “shell,” using one formula, which is Verhulst’s equation… And then inside it’s this other 3-dimentional formula that I got from this Norwegian physics professor. BB: Oh, OK. Is there a visual representation of the equation that you know of that you could refer me to? RW: Yeah. In that article there is… you see that it also makes forms. If you plot the points then you get quite nice pictures. I think what’s in that article is just a sideways evolution thing where it goes like that [gestures a flat line with fingers] and then kind of meandering or intertwining [gestures with hands to show multiple line sprouting out from the initial line and wildly interweaving] from left to right. But…if you plot these on a computer screen you get quite nice, beautiful pictures… that also was a very important help…for me to find out where the interesting material is. Do you see the formula which has x, y, z and c and d? BB: Yes. RW: Because if you have this x, y, z, if you think of them as foxes, rabbits, and grass, in a valley, they will be interdependent; the compilation of them will grow and rise dependent on each other. So, if there is lots of grass, there will be more rabbits, and then there will be less grass because the rabbits will eat the grass, and then…there will be more foxes. But this will go in a kind of very strange, actually then, “fractal” way [gestures once again with hands and arms moving in an interweaving, undulating pattern]. So that’s x, y, z. So if you call c and d…[pauses to come up with the right metaphor], if you like…a more environmental thing, like how much humidity there is and how warm it is, 204 that’s c and d. And all these fluctuations will be dependent upon that more, kind of, “global” thing. So I use c and d as…[gestures as if turning knobs or dials] two parameter buttons, [which] together make what’s called a parameter plan…or space… BB: It’s creating your limits, then. RW: So if you have c there that way [indicates horizontal axis] and d there that way [indicates vertical axis], you can plot a place where the value[s] for c and d here will be just do-DA-do-do-DA-do-doDA-do-do-DA-do-do-DA-do-do-DA [indicates a regular rhythmic pattern with an accentuation in triple meter], and will go on forever in a loop with three [beats]. Other places it will do-do-DA-do-doDÁ-do-do-dó-do-dó-do-do-do-DA-do-dó-do-do-da [indicating a differentiation in the accent but with the same pulse and tempo], so it will go a little more [randomly]—but they will always snap into x, y, z values. That’s what’s called “strange attractors.” BB: Yes, right—in fractal terminology. RW: Yeah. That’s…kind of the stepping-stones where they can jump to. And the more chaotic, the more stepping-stones, the more— BB: —possibilities— RW: —places it can go to, and therefore it will be somewhat more chaotic… BB: So when you make your choices for c and d, are those consistent through the whole piece, or do you change them periodically? RW: Exactly… c goes from one…value to another… what a friend [of mine] did then, he just…[made] these different pictures… Actually, in the booklet of the CD with both Stonewave and ning on it, I used two different…pictures… BB: [Looking at the fractal picture for Stonewave in it] Oh, yes, very beautiful! 205 RW: Isn’t it? It’s quite amazing. BB: Quite striking… RW: Anyway…it’s a value of c and d which makes it quite complex…these pictures are plotted parallel with the evolution of c and d. So c and d are, first, in this place [gestures to the lower right], and then it’s in another place [gestures to the upper left] at the end… BB: OK, yes. RW: So what I did there was simply to let [the] formula—[gestures with a circular motion implying that the values of the formula changed linearly with time and that output of the rhythmic accentuations and alterations of the music simply followed the trajectory]. Of course, you put an initial value into the x, y, z, that’s kind of random, and then you just compute that again and again… ————————— BB: …Is anything like meter or tempo determined also? RW: Yeah… in ning…I use what I call “rhythmicicity” (or something like that)— BB: [laughs]. RW: —[that’s] how much this y factor influences the duration. BB: OK… ————————— RW: So…meter… That’s interesting because…what came out of the computer was the sounds, and there was—[makes “tick marks” in the air moving from right to left], and then I wrote out just these blobs…on the printout…just like whole notes, going like that [gestures right to left with a circular 206 arc]… at first I was thinking of writing everything in [snaps his fingers several times in tempo like a metronome] !", because that’s what I usually do when— BB: —when things get complicated. RW: Yeah, because it’s [regular], but here, it was evident that it had to be changing meters because it has very, very clear patterns. ————————— BB: Yes, yes. Did you go through several different experiments with different…points for your c and d? RW: Yeah, boy… that was…a lot of— BB: —Trial and error? RW: [Nods] Trial and error, patience… it’s very much like getting to know some strange animals. Because you see, say that if I put them there, then they do this, if I put them there, they freak totally out, then [in] most [places] in this c/d chart, it floats—the values just go [off] to eternity, and that’s it. And other places are totally barren, so you just only get one point. So…it’s a kind of mapping of that land where these animals have a good time, and where these animals, if you bring them from there to there, they tell a kind of interesting story. But you have to really work hard on that, on how it works. And…c and d [are]…between, I think, –0.4 and 1.2, or something like that, and that’s so abstract. So, it was really helpful to…[make] a kind of mapping [of] where it exploded (keep away from that), and then after a while I saw that I had that also charted…where interesting stuff is… So it took a long, long, long, long time. BB: Wow. Yes, a lot of painstaking stuff. RW: Absolutely. 207 ————————— BB: What was your difference in approach for ning vs. how you went into Stonewave? RW: Yes. Stonewave was a special thing because there I haven’t used…in all the three sections of Stonewave I’m doing something different from what I usually do. BB: Uh-huh. RW: So ning is typical form for these fractal pieces: Onda di ghiaccio, ning, Solve et coagula, and Boyl, and also a piece called Chi, which was an orchestra piece, but which I had to withdraw because I didn’t think it worked actually…[laughs]… So ning is the typical form—machinery, you can say— and there are slightly different details, but it’s more or less this big form where you see this fishshaped thing with lots of small things in it…those small things, they are melodies. BB: OK. Yeah, I’m looking at the charts for that right here… So those points then are determined. So, I see…and I noticed this in the score…where all of the players seem to tend to pull together to a unison, and then they spread apart a little bit— RW: Hm, exactly. BB: —and then they pull together… RW: Yes…the ning form picture…[is]…like it is [because of] the parameters you can see under [it]… first, if we say that every melodic line, or melody bit, there is a start note and then there is a stop note, then there’s the stop note of the next melodic bit… so the first one is a very centralized one…and then everybody goes up for the next, and the third point… they slide down to the bottom. So what you have got, the print of notes that you got from me, that is from the second and third melodies. That means that it goes iiiuuuop [makes a vocal glissando that descends then 208 ascends]… The thing that looks like something that goes down to the ground and up again before it flats out again…from half minute to one-and-a-half minutes, that’s in the ning form… BB: I think I understand. I’m looking across the top where there’s the little 0, 1, 2, 3, 4 across the top of the page there… RW: Those are minutes… I was first intending on making it a twelve-minute piece, and then it became a fifteen-minute piece. So…I’ve recalibrated everything on the…page…it has a lot of squigglings/corrections from how many seconds length… you see that “lengde”? That’s “length” in Norwegian. And everything is multiplied by 1.25 from the first— BB: —from the original values. RW: Yeah. So, everything you see up there fits into a piece of a duration of twelve minutes, but then I found out that it would go too fast, so then I changed it and made a fifteen-minute piece instead. BB: I see… RW: So…there you can see what came out of the computer also on that…page… First is the start point of that melody. The first melody starts on “zero” time and ends at thirty-four seconds. BB: OK…So these are different formal sections of the piece? RW: Yeah. So, every melody you have up there has a number… So, you can see there it goes minimum pitch and maximum pitch and medium pitch. Let’s see…so the first thing that was calculated was the start point… And then, when the start point has come it takes a fractal number from…Verhulst’s formula…. And [then I] said, “OK, do we have a long or a short section here now? Will this melody be long or short?”… And that comes down then into the length parameter. BB: OK. 209 RW: So that’s why we first have a medium-sized thing which has become forty-three seconds long, and then it’s a longer one and then a shorter one. So there’s a melody, in a way, in the length of the melodies [leans forward to emphasize the point]…if you understand. BB: So, the new lengths that have been calculated here will create new start points for subsequent… RW: Yeah, that’s simply because I wanted it 25% longer. Yeah, that’s the only reason. BB: Right, that makes sense. Your…pitch values were determined by what part of the equation? RW: Exactly. That’s the same equation again but with different seed values because this only gives out one—that formula, the Verhulst’s formula—just has one variance that goes like this [swipes finger downward a few times as if making tally marks or a waveform in the air, tracing the pattern of the iterations through the bifurcation diagram], the x goes just like this. BB: Right. RW: But with the different seed values they will have different trajectories through that space… that’s also very important, because that determines if you start high up with a very concentrated—[brings left hand up to meet right hand up high, as if reaching some kind of upper limit], or low with very low and very fast—[moves hands wildly], all these seed values are formally very important… For the form…call it a “form melody”… In a way, all these—what you see, if you read each column there, it is, in a way, a melody—I could have played that as a melody. BB: Yes, certainly. RW: So, it’s…medium-long-short, medium-long-short, and then it becomes a little more “muddily.” That’s also because I’m doing it more orderly first and more chaotic in the middle… anyway…each column is a kind of melody then, that is, length is a melody, the medium pitch is a 210 melody—that’s the second thing which is chosen where the initial pitch, the initial medium pitch of that—what shall we call the section there? BB: Each formal section, yes. RW: Yeah, yeah. So, that’s…low-high—and then you can see that also, you can go through that and you will see that that corresponds to what you see on the map. And then “min” and “max” pitch [of the Q factor] is simply just plus and minus…“the same,” so that they will always be the same distance over and under [the medium pitch]. So you see that in…the second formal section, it will start very [high and very wide]—[indicates high pitches with raised hands and a large gap between them], and it will go down and go very narrow [moves hands downward and closes the gap]… And why is that, why does this change? And that is what I call “shape”… if it’s very low [in value]…it will—if you take the medium pitch, for instance, that’s very easy to see—it will go almost linearly from that pitch to that pitch [indicates a point with each hand, the right hand a bit higher than the left, and draws a line in the air between them]. BB: OK. RW: If it’s all over a special, certain value—if it’s over 6.0 or 7.0, I don’t remember—then it just stays at the same pitch…and the same rhythm, the same everything… BB: OK. RW: So, therefore, you see in the map, you see that—[mutters something to himself]—in the picture, you see that some are broken and some are continuous… And some are more exponential than others. And when they are moving from one thing to another, it’s because it’s moving—let’s see [holds head in hand thinking]. With the first that is broken (alright?), it doesn’t (the break between the first and the second [sections]). But between the second and the third, and the third and the 211 fourth, it will slide all the values: speed, mid-pitch, range (what do you call it?) ambitus, and everything will change during the— BB: —the process. RW: —and the higher that shape number is, the more exponential it goes [makes a sinewave-shaped large swoop with left hand, from left to right], so if it’s low it just goes…[quickly moves left hand downward from left to right] linearly one point to another. BB: I see. So, there’s more of a curve to a higher number. RW: Yeah. So, the first is quite higher in the shape. [The] first shape [value] is so high that it breaks… [Looks at chart] That’s [7.88]. And then 1.04…the exponentiality is quite subtle. But I can see one which is quite exponential. It’s that, if you see…around five minutes in the chart…you see that it keeps very high for a very long time and then it just drops totally down in the basement. BB: Right…it just zooms down, yeah… RW: And then it goes up again there, but that’s a very exponential thing, whereas the one which is around four minutes is a quite linear thing… So, then it goes more gradually from the start point to the end point. BB: Right. I think if I compare this with the actual score of the piece, it looks to me like you assign a different “character” to [each melody]. So, it starts with “rustic, vigorous,” and then it moves to “transparent,” and then onward. And so that corresponds with this [indicating the chart] melody 1, 2, 3, and so forth. RW: Yeah… you can say that for each new formal section I’ve chosen a coloring of it. And that’s a totally subjective thing that I’ve done. BB: Ah-hum. Well, that’s cool. 212 RW: So what you see in the print out is just like a very, very long note: deeeeeeee [the pitch is D4, but that’s probably arbitrary], and then deeeeee [C3 tuned flat, again probably arbitrary] deeee-derrrdeeee-derrr-da-da-da-da-da-da [continues a sequence of non-descript sung pitches, indicating a fluctuation back and forth between higher and lower pitches with accelerando and an overall trajectory downward in pitch]— goes like that. And then, “OK, what’ll I do with that?” And that’s when me as a composer—as a musician [and as a colorist]—comes in. BB: Right… So, just for instance, it looks like in the very first section, the fist twelve measures (let’s take that for example): You focus on the pitch D (of course, which I see right here in the chart, D5), and it gradually goes from there down to a B by the time we get to measure 12. RW: Yeah. And why is that? Because (I said to you) that it actually should stay in place, and then jump to the next section. BB: Right. RW: But then you have to look at this fish-like form which is the—there’s a [draws the outline of the fish shape in the air, indicating the upper and lower limits shown on page one of the graph of the form]— everything happens within the shape. BB: So, yes, the outline of the fish. RW: So, actually…all the numbers which I get out of this, every little note—they are then squeezed into the amounts [uses some interesting flat-handed gestures, as if taking pain-staking effort to fit something very fluid into something else very rigid]—that goes for the pitches, so what you see there is the pitches. So, I get what I get because the lower frame is drooping, going down and down, and the melody, which relatively seen, is like this [indicates a flat line in the air]—the number, it should stay like that, but because it’s expanding [referring to the outer limits], it goes like that [slants the line downward]. 213 BB: OK. That makes sense. That makes sense because the overall parameters— RW: So, if I hadn’t made that frame, it [would] just keep like that [indicates flat line]. And actually [in] Solve et Coagula I don’t use that external frame. So there, everything is going like that [shows flat-lined upper and lower limits], so if it’s flat, it’s flat… NO, NO, NO! [Catches his error] I’m wrong, I’m wrong. That’s also the same. I think it was in my first piece, Onda di ghiaccio, where everything goes like that [shows flat-lined upper and lower limits one again]. I found out then that you will always go [talks and gestures with leaps between extremely high and extremely low pitches]. BB: [laughs]. RW: See [referring again to ning], I wanted to have kind of a trajectory where you start with the current layer, and everything’s quite dark, you go into a midsection—it’s actually at the golden section, if you measure it around a little after seven…minutes there, that’s where the narrowest section is—narrowest melody span is. BB: [I] see that, yes. RW: Yeah. And that’s the golden section of the whole piece. And then everything is mirroring, the form is then mirroring that in the— BB: OK, that makes sense… RW: Yeah—Oh! [Looking through old charts from old files on computer] Well, actually no. I was right: In Solve et Coagula…everything is going like that [shows parallel flat lines for upper and lower limits]. I don’t have that [shows the expansive curvature for the outer limits, such as is found in ning], that outer cylinder, outer— BB: —parameter. 214 RW: Yeah. So that’s the reason when—for your question—it starts at a D and goes/ends down there around the B. Yeah, that’s it. And why is it so narrow? That’s because there it is a low number for the span. BB: OK. So, if I can get a little bit more particular (and let’s just use these first twelve measures as an example): With all of the different articulations and variances, the kind of rhythmic characters that you give to it, how much of that is you just trying to, as a composer, bring a little bit of life to the process, or how much of it is actually determined by the calculations? RW: The calculations you can see…all the accent notes, or everything [shows indication of articulative markings/punctuations in the air]—that [is what] is determined by computer, and I haven’t moved it… Where either a pitch lies or [it’s] placing in the bar. BB: So, accents— RW: So, Ba Bum bada ba ba dum da dum da da dum [sings the rhythm of the passage adding accents for emphasis where applicable]…[the] tape with the melody that came out just as I had there, with a piano, see. Dum dada da da dadum da da dum dadum dadum [as if playing the melody with his index finger on an imaginary keyboard], so you get a kind of Balkan-y kind of “husking” rhythm… that kind of folksy Rum da dum da DA [mimics playing a violin], like that. BB: Uh-huh. And I see that that rhythm idea jumps between the different instruments. So, it’s heard coming through different instruments and changing around. Is that correct? RW: Yeah, yeah. So that’s simply like me finding a place for it, a kind of a picture, of trying to see it like a painting or something like that… And then, like, all my other rules that I apply, what kind of liberties I can take, will change from section to section… BB: Ah, excellent. 215 RW: Then…for instance, later there is one with lots of little…minimalist [things] where [there are] only sixteenth notes: ya da da da da da da da da da da da ya da da da da da da da da da da da [mimics violin again, sings with crescendo/diminuendo]. BB: Yes, I [know] that section… RW: And the only thing in there is you will then find the top point of the crescendo and the diminuendo…that [points to the point in the middle of the hairpin drawn in the air] will be the note, that will then be the focal note. So, I just simply said, “Here, I am allowed to have ‘pre-echoes’ and echoes of the note.” BB: I see. RW: So that’s how that came about… And I could have made the whole piece like that, you know, I could have just [swipes hand] made a whole piece with that, but I chose not to. BB: OK. So, different sections have all the different characters that you’ve given them— RW: Absolutely. BB: —so they’re decorated/embellished in those characters. RW: Yeah, exactly. And then you will hear some of the things repeating—the characters, like the first one (Da dadum…), which I call a Balkan kind of—husky and quite raw, shawm-like—that comes there, it comes again in the fourth section which is around two minutes in the chart (the next “flat” thing), and it also comes again (I think) in that…third minute there also. But there [the player is] using the oboe… So, that dodloori DAI da DA da da DA da da [glissando up to high oboe part, sings on B♭ (arbitrary, most likely)]—lots of trills and, you know, lots of Bartók pizzicato and things like that. BB: Yes, yes. 216 RW: Interesting thing that it never comes exactly the same, so it will not be a repetition of it. Both the melodies are different, and there might be a little, maybe…that bit is a little wider, so you will get different times [indicates differences between the two melodies pitted against one another], and maybe it’s a little slower also… So, it’s quite interesting to see how this same character is—just like in a symphonic work, you will get the return of certain characters…but it will never be exactly the same. BB: Right. RW: So, it’s not like a repetition in a classical sense… And also, that’s something I wanted…the balance between…similarity and variance, [the way] they vary… variation. That’s the very important thing, that I wanted to have some kind of—needs [control]—can’t go totally everywhere… At the same time, I want variation. BB: Uh-huh… I know that this was commissioned by the Ensemble Borealis, so they are made of the violin, viola, cello, and— RW: Yeah, they are a larger ensemble. It’s like…eight [members]—so we decided on [four of them]. BB: OK. Why was it this choice of the three strings and then the oboe/English horn? RW: I think, just simply, I liked it…that was just my choice. BB: I really do like the color of it. I wondered if there was— RW: You know that—have I told you that I’ll now actually (next year, when I have time again), I will make/change/rewrite it to concerto for oboe and string orchestra? BB: Oh! 217 RW: Because, in a way, as I listen to it in hindsight, in a way I feel that it’s a concerto for oboe and three strings. And that’s—well, in a way—all these Mozart [quartets] and all these are, in a way, concerti for— BB: Right. OK. RW: But also, because it’s so complex, and because it’s—I haven’t seen it…played successfully without a conductor, and it looks kind of silly with— BB: —with a small ensemble and a conductor [laughs]. RW: Yeah. The funny thing of it (just a little anecdote), it was played in Kiev, in Ukraine, just after the fall of the Soviet Union, and there was this festival for young European composers. And there was this really, very, very good oboe—the musicians were great. But I couldn’t come to any rehearsals—I just landed there, just before the concert. And I talked to them, “Oh, you don’t have a conductor?” “No,” [they replied], “we shouldn’t have that.” [Throws up hands in exasperation]. They were beautiful, they played everything, all the parts, beautiful, but they kind of drifted a couple of bars from each other [back and forth], and [I was] never happy with the result. BB: Oh dear… Well, it certainly looks very difficult. RW: But I’m now looking forward to actually spreading it out a little…and make it an oboe concerto. BB: OK. That’s very neat. RW: And then I will go, of course, back to, also, the original lines, and then spread out more. BB: OK, excellent. Well, I was just looking through the questions I had for you today and I think we addressed most of my questions for now, and I think this also gives a lot more to dig into as far as coming up with the theoretical explanation for things, and so— 218 RW: Yeah. And one thing, which is very important, is the skala, which means the scale. BB: OK. RW: And that’s the column almost to the right there. And that is—did I talk to you about my crystal chords? BB: No…tell me about that. RW: No? Because that’s a very important thing, which is actually the first time I used that at all. It is, actually, another fractal (in effect) system, which makes scales a little Messiaen-like, but they are made in another way, and they don’t repeat in the octaves [indicates octave increments moving upward with a flat hand] like the Messiaen modes are. BB: Oh, OK. RW: So they typically repeat on major seventh, so you’re going to—[moves over to piano and rolls a chord: C♯3, E3, A3, C4, E♭4, G♭4, B♭4, D5]—you get the same intervals coming again, but they will arrange in different ways through the octaves, which I have used always since then, and which I find very useful. And that’s why…some places, it will be very diatonic. And so, if…the value for the scale is zero, then it’s chromatic, and that’s why there in…[the] beginning, it’s chromatic… But other places it sounds very—almost tonal… And that’s because then it just—the melodic value—just goes down to the closest in that scale. BB: OK. That makes sense… RW: We can take that another time, but that’s what I call the crystal chords, which also is kind of a fractal thing because it is a very simple system which is quite unpredictable, but still it has a character… BB: OK. Well, I think that gives me a lot to go with. 219 RW: Yeah. Very good. BB: Well, thank you. This has been a great pleasure for me to speak with you and get some information, and I’ll be working on it hard, and we’ll be in touch… RW: Very good. APPENDIX C GRAVEN IMAGES, SYMPHONY FOR WIND ENSEMBLE The score for this music is not provided in this volume of the dissertation, but is available on ProQuest as a separate volume, under the file name “Bidwell Dissertation-Graven Images.” The program notes for the piece are provided here. Program Notes Graven Images, symphony for wind ensemble, is based upon and inspired by the book of the same name from author Paul Fleischman. The book is a collection of three short stories, which together form a triptych, with two grim narratives encapsulating a contrasting, lighthearted comedy of errors in the middle. Their plots all revolve around the common theme of effigies, with each serving as a cautionary tale against putting one’s faith in such graven images. The three movements of the symphony are abstract musical depictions of each of these three stories respectively, but are likewise all connected musically by the recurring presence of a motive representing the graven images, appearing in various guises and transformations (inversion, retrograde, augmentation, fragmentation, etc.). Set in a New England port town in the nineteenth century, The Binnacle Boy is a tale about the enigmatic demise of a ship’s crew, and the guarded secrets apropos these dreadful circumstances. When the ship, whose complement includes the town’s own sons inscribed among its ranks, is discovered meandering aimlessly near the town’s harbor, the ensuing investigation reveals that not a single surviving soul is aboard. How the crew came to this ghastly fate is a mystery, and the only “witness” of this horrific event is the wooden statue of a young sailor boy 221 holding the ship’s binnacle (the housing for the compass) perched on its deck. The statue is then transplanted to the middle of town as a memorial to the dead crew, and the citizens, after vainly wishing for the binnacle boy to somehow part his lips and bring closure to this tragedy, soon begin to privately confide their own deep secrets into the carved ears of this expert secret-keeper. Then a coterie of gossiping women, while offering condolence to the adoptive mother of one of the crew members, discover that they are able to “eavesdrop” on the confessors from the handy vantage point of her parlor by employing the aid of her reluctant lip-reading servant girl. Through repeated visits to “pay respects,” the women eventually learn that someone in town in fact knows that the crew was killed by tainted tea. Being unable to press the girl into divulging the confessor, the women throw aside the curtains and are shocked to see the girl’s sister whispering to the binnacle boy. They leap to the conclusion that the girl’s sister poisoned the crew, and full of righteous indignation, rally others in the town to chase and seize her, only to find that she has run away terrified and fallen off a cliff to her own death in the sea. It seems that the complete answers to the crew’s mystery have died with her, but in a final twist, another revelation comes to light as one last lingering confession enters the ears of the statue. The first movement begins with an introduction that presents the “graven images” motive, after which the music enters a widely spaced harmonic world that sets the mood and tone of the story’s setting. The loosely defined sense of meter and lack of tonal orientation evokes imagery of the foggy harbor, while shifts in harmonic density highlight the shocking discovery of the murdered crew, hints of metric pulse suggest the subdued funerary procession, and flighty open fifths coupled with a shanty tune in the harp conjure up secretive whisperings to the binnacle boy. The general flow and ambience established by these elements are occasionally interrupted by stark sounds in the high woodwinds, representing the gossiping women, which gives rise to the sound of crystal glass, representing the deaf servant girl. All these sonic elements converge to the climactic crescendo of the accusatory confrontation and fate of the girl’s sister, followed by the grim aftermath and the sinister final revelation. Musically, this piece takes much 222 inspiration from the work of Charles Ives, such as “Tom Sails Away” (No. 51 from his 114 Songs) and The Unanswered Question. Saint Crispin’s Follower recounts the misadventures of a love-struck and maladroit shoemaker’s apprentice. In his quest to woo the damsel of his dreams and ask her to the militia review and celebratory ball the following evening, the boy turns to the broken weathervane atop the cobbler’s shop for guidance, as it bears the image of Saint Crispin, the patron saint of shoemakers. The weathervane points him in various sundry directions, each leading him to a series of missteps and misunderstandings, from the misinterpretation of the hidden meanings behind gifted flowers, to a misplaced and wrongly delivered love letter. He makes frantic attempts to undo or redirect his missteps, but it quickly seems that all his efforts are either doomed by his own incompetence, or thwarted by his master’s plans and the interference of his inamorata’s spidery and sharp-tongued mistress. Feeling like a failure, he despondently turns to the weathervane one last time for guidance. This time, though, it seems that Saint Crispin smiles on him, as he crosses paths with his beloved, and after clearing up the earlier confusions of the day, finds (to the sweeping sounds of cannons and fireworks) that he may have a sporting chance with the girl after all. In stark contrast to the first movement, this piece has a driving pulse and a lilting, catchy tune, albeit with a touch of melodic/harmonic frenzy. Themes representing the apprentice and the girl of his dreams are presented, and then often interweave with one another, while the “graven images” motive is commingled in such a way as to steer the main theme into a mad frenzy, compounded by dissonant motivic ideas representing the shoemaker and the girl’s menacing mistress. After a hysterical climax, the boy’s final downhearted appeal is heard, followed by the ecstatic resolution (complete with cannons and fireworks). The Man of Influence tells the tale of a self-important sculptor who prides himself for having memorialized the most prominent and influential citizens of Genoa in stone monuments. But though he esteems himself as a member of elite circles, in reality business is slow, until he is 223 approached by a specter in the night who commissions a carving in his likeness for a substantial sum of money. The sculptor is not only astonished by the appearance of the ghost, but is also taken aback by his haggard and unsavory figure, and is skeptical that such a being should be worthy of capturing in stone. Nevertheless, after receiving a large down payment, and learning that the ghost was a man of “influence” during his lifetime, he agrees. The process of making such a repulsive figure leaves him greatly disturbed, but he persistently justifies making the statue, reassuring himself with each chisel that he is enshrining another important and influential person. When he finally delivers the finished commission to the ghost, he learns that the statue’s purpose is to bring the ghost peace and closure for the vile deeds of his life. The ghost then reveals that his connection of “influence” with Genoa’s high society was his role as a hired assassin. To his utter horror, the sculptor realizes that the prominent and influential persons he had immortalized and revered were equally vile, suddenly turning everything that he thought of himself and his world upside-down. This movement has elements from both of the two contrasting styles found in the other movements, and seems almost like two separate pieces meshed into one. In following the sculptor’s journey of self-discovery, the movement starts out with a reassured, stately theme, clearly tonal, which develops into the proud and arrogant music of the city and the influential subjects of his sculptures, represented by augmented appearances of the “graven images” motive. 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