Divisions without hierarchy: four-dimensional modeling of submeter and its use in empirical analysis of the musics of the new complexity

Central to virtually all scholarship of meter is a notion of beat hierarchy. However, when beat units and/or tempi are in constant flux, defining a hierarchy becomes nearly impossible. Such impulse structuresâ€"common to music of the New Complexityâ€"resist traditional scholarship of meter/subdivision-based hierarchical layers, yet are also more structured than simple rhythms. This dissertation observes how these structures relate to more traditional definitions of meter and rhythm, and categorizes them as submeter. The number of pulse layers defines each of the three categories. Rhythm is shown to require only a single pulse layer, while meter requires three or more. Submeter, containing the beat unit subdivision of meter without its associated beat hierarchy, falls between rhythm and meter, requiring exactly two pulse layers. After establishing this definition of submeter, this dissertation focuses on a four- dimensional formalization of the defining characteristics of submeter. The formalization models beat unit, subdivision, tempo, and duration as orthogonal dimensions. Through an analysis of Brian Ferneyhough’s Unsichtbare Farben, the empirical modeling and its potential are demonstrated. Comparing the values of each dimension shows that, beneath the notational complexity, there are a highly limited number of subdivisions and beat units, and that tempo has the highest rate of variance. In addition, by analyzing frequency distribution and transitional probabilities of beat units and subdivisions, it is demonstrated that while the majority of cases do not impart statistically significant transitional syntax, beat units can be categorized into three functional groups based on their transitions. The dissertation concludes by comparing the shifts in tempo to pitch transformations. Using the submetric units of the first 22 measures (comprising the first section) of Unsichtbare Farben, the tempo shifts are used to project a series of pitches. These pitches are then processed through Robert Morris’s contour reduction algorithm, with the prime layer projecting the trichord {F, F-sharp, A}. This trichord fits into two possible groups with the trichords presented as the closing pitch material of the opening section, demonstrating the relationship between the submeter and the pitch material in Unsichtbare Farben.

DIVISIONS WITHOUT HIERARCHY: FOUR-DIMENSIONAL MODELING OF SUBMETER AND ITS USE IN EMPIRICAL ANALYSIS OF THE MUSICS OF THE NEW COMPLEXITY by Aaron J. Kirschner 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 May 2017 Copyright © Aaron J. Kirschner 2017 All Rights Reserved THE UNIVERSITY OF UTAH GRADUATE SCHOOL STATEMENT OF DISSERTATION APPROVAL The dissertation of Aaron J. Kirschner has been approved by the following supervisory committee members: Steven T. Roens 4/29/15 , Chair date Miguel Basim Chuaqui , Member 4/29/15 date Robert L. Baldwin Michael W. Chikinda Abby S. Kaplan and by Steven T. Roens, , Member 4/29/15 , Member 4/29/15 date date 4/29/15 , Member Area Chair date Music Composition and by David B. Kieda, Dean of The Graduate School. ! ABSTRACT Central to virtually all scholarship of meter is a notion of beat hierarchy. However, when beat units and/or tempi are in constant flux, defining a hierarchy becomes nearly impossible. Such impulse structures-common to music of the New Complexity-resist traditional scholarship of meter/subdivision-based hierarchical layers, yet are also more structured than simple rhythms. This dissertation observes how these structures relate to more traditional definitions of meter and rhythm, and categorizes them as submeter. The number of pulse layers defines each of the three categories. Rhythm is shown to require only a single pulse layer, while meter requires three or more. Submeter, containing the beat unit subdivision of meter without its associated beat hierarchy, falls between rhythm and meter, requiring exactly two pulse layers. After establishing this definition of submeter, this dissertation focuses on a fourdimensional formalization of the defining characteristics of submeter. The formalization models beat unit, subdivision, tempo, and duration as orthogonal dimensions. Through an analysis of Brian Ferneyhough's Unsichtbare Farben, the empirical modeling and its potential are demonstrated. Comparing the values of each dimension shows that, beneath the notational complexity, there are a highly limited number of subdivisions and beat units, and that tempo has the highest rate of variance. In addition, by analyzing frequency distribution and transitional probabilities of beat units and subdivisions, it is demonstrated that while the majority of cases do not impart statistically significant transitional syntax, beat units can be categorized into three functional groups based on their transitions. The dissertation concludes by comparing the shifts in tempo to pitch transformations. Using the submetric units of the first 22 measures (comprising the first section) of Unsichtbare Farben, the tempo shifts are used to project a series of pitches. These pitches are then processed through Robert Morris's contour reduction algorithm, with the prime layer projecting the trichord {F, Fs, A}. This trichord fits into two possible groups with the trichords presented as the closing pitch material of the opening section, demonstrating the relationship between the submeter and the pitch material in Unsichtbare Farben. iv TABLE OF CONTENTS ABSTRACT ........................................................................................................................................ iii ACKNOWLEDGEMENTS ........................................................................................................... vii INTRODUCTION ............................................................................................................................ 1 Chapters 1 DEFINITIONS ............................................................................................................................ 5 Impulse ..................................................................................................................................... 5 Pulse and Pulse Layers .......................................................................................................... 7 Beat and Beat Unit; Prime and Origin .............................................................................. 10 Rhythm, Meter, and Submeter .......................................................................................... 11 Rhythm ........................................................................................................................... 12 Meter ............................................................................................................................... 13 Submeter ........................................................................................................................ 15 2 FOUR-DIMENSIONAL MODELING ................................................................................ 22 Dimension (x)-Beat Unit ................................................................................................. 23 Dimension (y)-Subdivision .............................................................................................. 24 Dimension (z)-Tempo ..................................................................................................... 25 Dimension (w)-Duration ................................................................................................. 26 Basic Modeling ..................................................................................................................... 27 Rallentandos and Accelerandos ......................................................................................... 31 3 APPLICATIONS OF SUBMETRIC MODELING DEMONSTRATED THROUGH AN ANALYSIS OF FERNEYHOUG'S UNSICHTBARE FARBEN ........................... 36 Partitioning the First Five Measures in Submetric Units ............................................... 38 Analysis of the First Section Using Submetric Ordered Quadruples ......................... 48 Transitional Probabilities of Dimensions x and y .......................................................... 53 4 AFTERWORD: SUBMETER AS PITCH ............................................................................ 76 Notated Tempo Changes and Pitch Transformations .................................................. 77 Submetric Beat-Length Alterations as Pitch Transformations ..................................... 80 Appendices A: LIST OF VARIABLES AND FORMULAS .......................................................................... 95 B: COMPLETE FOUR-DIMENSIONAL SUBMETRIC CODING OF UNSICHTBARE FARBEN.....................................................................................................100 C: PITCH MATERIAL PROJECTED IN THE FIRST 22 MEASURES OF UNSICHTBARE FARBEN AT EACH CONTOUR LAYER ...................................... 110 D: REPRODUCTION OF "FOUR-DIMENSIONAL MODELING OF RHYTHM AND METER" ..........................................................................................................................117 SELECTED BIBLIOGRAPHY ................................................................................................. 121 vi ACKNOWLEDGEMENTS I cannot properly express my gratitude to the many faculty, friends, and family members who have helped me to realize this dissertation. I hesitated to even include these acknowledgements, as I worry of omitting any of the multitude of inspirational and encouraging people who brought me this far. However, I also have far too much gratitude to conclude this project without expressing at least a fraction of it. To my many advisors and teachers, this would never have been possible without your encouragement and wisdom. Ted Powell (my director at Yankton High School) first directed me towards a career in musicology and theory; I vividly remember our lunch conversation in my sophomore year when he directed my mind down this path. At the University of Iowa, Maurita Murphy Mead opened my eyes to professional musicianship. While we may never have seen eye-to-eye, her clarinet lessons were monumental in my development and I hold her in the utmost esteem as a pedagogue and musician. I would be remiss to speak of my time at Iowa without offering the most sincere thanks to my composition teachers, David Gompper, Lawrence Fritts, Christopher Gainey, and John Eaton (may he rest in peace). I also would be nowhere without the support of Martin Amlin, J. Rodney Lister, Ketty Nez, and John H. Wallace of Boston University. This dissertation would, of course, be nothing without the wonderful support of my committee; Rob Baldwin, Michael Chikinda, Miguel Chuaqui, and Abby Kaplan, I am indebted to all of you. Without minimizing anyone, I would like to direct special thanks to two people from my time at Utah. First, Steve Roens, as my committee chair, has been instrumental in my development as a composer and theorist. His tutelage allowed my compositions to develop into a much more mature sound and process, and he pushed me to better refine and communicate my theoretical ideas. Second, thanks to Bruce Qugalia, without whom I would not have redoubled my focus on music theory. Bruce's mentorship was one of the most meaningful experiences of my doctorate. He constantly pushed me to work my abstract theories into communicable messages, and helped me to further become a theorist. I cannot put into words how much I owe him. To my colleagues and friends, I cannot express enough how much your support brought me through this tiring process. Thank you for being there, not simply to help me create this treatise, but also to provide camaraderie and humor beyond it. Specific thanks to Nathan Wilks, Larry Spell, and George Marie; when we finally lifted that toast upon this defense, know that it was in large part due to how many previous toasts we had lifted as friends. I must, of course, also express my love and gratitude to Alex Schumacher. Alex, not only were you happy to assist in this project, but your encouragement and friendship throughout this writing and our academic careers has been invaluable to me. You once told me that I was someone who you knew could not fail. I hope you realize-if that statement is indeed true-it is only because of your friendship. Finally, I must express my love for my family and their tremendous support over the years of this project. To my wife Maureen, I cannot ever express how much it meant to have you by my side. You were there to build me up when I doubted myself, and you never let me doubt the importance of my work. All of the trials and difficulties of writing this dissertation were made easier with you, and it was doubly sweet to celebrate its completion viii with you. I would never wish it any other way. To my brother, Ken, and sister-in-law, Elizabeth, thank you for never letting me give in to my self-doubt. Your continuous belief in me has sustained me through my pursuits. Ken, after you received your Ph.D. when I was 11, I had wanted to get mine to not allow my big brother to one-up me. Now, I only hope that you know that my pride in completing this dissertation is dwarfed by my pride in calling you my brother. To my mother, I can never thank you enough for all of your time and energy in supporting my music education. You were always there for me, and always willing to help me better myself. I love you, Mom, and will always appreciate your support through these years. Of all the people who have made this dissertation possible, the one who will always be the most important to me is my father. He passed away during the early stages of this research, and I have never spent a day not wishing he could read this. His support for me was always unwavering, and no matter how difficult this writing became, I could always remember his encouragement. I still remember how happy and proud he was when I began my Ph.D., and I wish I had been able to see that again when I finished. Instead, I must dedicate this writing to his memory, knowing that his love and support is what has made this possible. I cannot express my sorrow that he will never read this, but I console myself in remembering all the joy that he brought into my family's life. I love you and miss you, Dad. viii ix INTRODUCTION Since before the introduction of the term, music of the so-called New Complexity has been as controversial as it is enigmatic. Assigning criteria for categorizing music as New Complexity has proven even more difficult than for many other styles and schools. The music is most readily recognized by its extreme detail and specificity in notation. However, while possibly the most quickly observable characteristic, defining this music simply by the detail of notation disregards many of the underlying processes of its construction and emergent properties of its sound. The complexity of notation is not a defining characteristic of the music, but rather an emergent feature necessitated through its construction. In his entry in the Grove Dictionary of Music, Christopher Fox makes this distinction clearer, describing the music of the New Complexity as: …a complex, multi-layered interplay of evolutionary processes occurring simultaneously within every dimension of the musical material. Since composers within the New Complexity usually chose to realize their music through acoustic instrumental resources, their scores necessarily pushed the prescriptive capacity of traditional staff notation to its limits, with a hitherto unprecedented detailing of articulation.1 Fox's definition rightly describes the notation as a byproduct of larger processes that create New Complexity music, rather than itself as a defining characteristic. The multitude of processes that form this music has become a rich analytic well for composers and theorists (be they supporter or critic) alike. Moreover, because virtually all composers associated with 1 Christopher Fox, The Grove Dictionary of Music and Musicians s.v. "New Complexity" (Oxford: Oxford University Press, 2001) accessed April 15, 2015, http://www.oxfordmusiconline.com/ subscriber/article/grove/music/51676. 2 this movement are living and teaching, their processes (no matter how inscrutable) can be verified and discussed. As the composers are easily contacted (and many regularly speak openly about their music in masterclasses and at Summer festivals), there is little incentive to treat the music as an artifact. Rather than take the existing music and observe the aurally apparent properties, the majority of scholarship on music of the New Complexity is focused on its creation. However, while this focus on process has produced a wealth of valuable scholarship, it has also often neglected the final product. Compositional process, while a valuable analysis, is not the only possibility for research into the New Complexity. Focus limited only to the direct actions taken by the composer-especially in such a complicated music-will often lead to details so subtle as to be unnoticeable,2 and will likewise overlook obvious emergent structures. In one of the more scathing criticisms of the New Complexity, Erik Ulman took umbrage with music defined simply by the complexity of creation process and notation, rather than by its aural value: Several times I have examined a score and been excited by its apparent vitality, only to hear what had looked so powerful in the notation shrivel to banality. I do not disparage the claim advanced by composers as different as Ferneyhogh and Feldman that the score is a powerful object which inflects performance in many ways that are difficult to verbalize… However, if notation can signify richness and multivalency, may it not also conceal their absence? Sometimes the "complex" score becomes an intimidation mechanism, staving off critical scrutiny by cultivating incomprehension, substituting colorful notational and verbal detail for musical detail, and depending on an inevitable inaccuracy of interpretation for either a genuinely improvisatory performance or a final excuse for the failure of the material to present itself audibly.3 Ulman's criticism is something of a point of departure for this writing. While 2 Brian Ferneyhough, Unscichtbare Farben (New York: Edition Peters, 2002), i. 3 Erik Ulman, "Some Thoughts on the New Complexity," Perspectives of New Music 32 (1994): 204-5. 3 Ulman's criticism is of music of deep compositional and/or notational complexity yet lacking any aural value, there is a similar parallel to much of the exiting scholarship on New Complexity. While rich in detail of process, it often overlooks many of the emergent structures. To overlook the most salient aural details of a work simply because they are emergent from other processes is hardly defensible. The goal of this text is to create an empirical analytical model of the impulse structures that emerge from the complex rhythmic/metric processes at play in New Complexity. In creating this model, it quickly became apparent that traditional metrical theories would not wholly suffice. Notions of metric dissonance would hardly be worthwhile in a context where virtually every grouping could be considered dissonant. 4 More importantly, many of the projected metric units change too rapidly to impart any sense of beat hierarchy. Without beat hierarchy or a sense of metric consonance, what possibilities remained for metric analysis? Slowly, the realization came that these units were neither rhythm nor meter in the traditional sense. Rather, they were something in between, something that will be called submeter. This text is divided into four chapters. In the first, the notions of rhythm, meter, and submeter are explored through building up from the smallest levels of musical stimuli. The interactions of different cardinalities of pulse layers are identified as the primary difference between each category, and the analytical applications of each is demonstrated. The second chapter focuses on a four-dimensional formalization of submetric units. The formalization models beat-unit, subdivision, tempo, and duration in orthogonal dimensions. Using this model, the third chapter presents an analysis of Brian Ferneyhoug's Unsichtbare Farben. The analysis demonstrates the model's value in analyzing the development of each dimension of 4 Harald Krebs, personal communication, October 31, 2013 4 submeter throughout the piece. The model is also used to track the distribution frequency and transitional probabilities of the beat-unit and subdivision dimensions, eventually demonstrating (through transitional probability analysis) that beat-units in Unsichtbare Farben can be categorized into functional groups based on a first-order Markov Chain. The final chapter observes the similarities between tempo shifts in submetric units and pitch transformations. The text concludes by comparing the relationships between the pitch material of the first twenty-two measures of Unsichtbare Farben and the projected pitch transformations of the tempo of the submetric units in those measures. CHAPTER 1 DEFINITIONS In preparation for discussion and analysis within this dissertation's model of rhythm, meter, and submeter, certain terms must be defined. Much of the prior research on this subject involves divergent-and often even contradictory-definitions of pulse, impulse, rhythm, meter, beat-unit, etc. Comparative analysis of each of these terms as defined by preceding authors is beyond the immediate purview of this document. However, in the interest of providing cognitive arguments for these definitions, this section will, as appropriate, compare the merits of some prior definitions. Furthermore, the exploration of the mathematical postulates and models that comprise the structure of four-dimensional analysis of submeter is left for Chapter 2. Many terms are defined more broadly in this section than their later usage in this document will suggest, allowing and, hopefully, encouraging, future research to apply different concepts within this framework. Impulse An impulse will be defined as the smallest single unit of temporal demarcation. In the context of time, it is a point (i.e., a unit of zero-dimension) having only position without duration or direction. Moreover, an impulse does not have implied accent pattern. It simply exists as a single unit. All other temporal units and structures are built from impulses. 6 To be understood as an impulse, a sound must have a clear and noncontinuous point of change from the surrounding sound(s). The most obvious-and commonly analyzed- example of this is attack-onset. The vast majority of prior research (as well as most of this dissertation's analysis), has focused on attack-onset-interval (or inter-onset-interval [IOI] as discussed by Justin London) 5; such an approach in effect treats attack-onset as the only form of impulse.6 While this is quite often a perfectly acceptable approach, other forms of impulse can be identified. Provided that there is a clearly demarcated change in sound, attack-onset need not be the only form of impulse. Possibly the most obvious second candidate for impulse is cutoff. However, unlike attack-onset, cutoff is not always a clear temporal demarcation. Compare the three examples in Figure 1.1. The first simply places a single impulse on each quarter-note. The staccatos merge the nature of attack-onset and cutoff-the listener does not perceive a separate cutoff impulse. The second example, by contrast, clearly demarcates both the attack-onset and cutoff, again on each quarter-note. By means of a sharp cutoff following a crescendo, the second example creates a clear change in sound not only at the start of each note, but also at the end. The final example, however, places impulses only on each half-note. Because of the diminuendo to niente, the cutoff is the end of a continuous change in sound, and thus not an impulse. In fact, the majority of cases will consider a cutoff to be part of a natural, continuous decay-and thus not an impulse-except in the cases of notation or performance in the manner of the second example. Likewise, in the case of a crescendo from niente, the attack could not be considered an impulse, while the associated cutoff would 5 Justin London, Hearing in Time: Psychological Aspects of Musical Meter (New York: Oxford University Press, 2004), 4. 6 Gert ten Hoopen, et al., "Time Shrinking and Categorical Temporal Ratio Perception: Evidence for a 1:1 Temporal Category," Music Perception, An Interdisciplinary Journal 24 (2006): 1-10, 18. 7 instead constitute the relevant impulse. 7 Any clear, temporally demarcated change in sound can be understood as an impulse; attack-onset and cutoff need not be the only possibilities. 8 If a specific piece, or body of music, consistently presents different types of impulses, a concept of "impulse-class" can be drawn. In this manner, not only can larger structures be formed from the series of all impulses, but different types of impulses can be grouped into their own series. Moreover, one could expand the notion of impulse-class to include changes in texture, tempo, and other larger structures. While it is undoubtedly strange to think of a zero-dimensional unit representing structures that, by definition, require time and space, it is not inconsistent with how pitch is understood as an abstract concept.9 Pulse and Pulse Layers A pulse is a series of regularly repeated impulses, with or without accentuation that could imply some impulses as more important than others. This definition will accept two 7 While this definition fits cognitively with how sound and music are understood, it does present an unintuitive corner case. When a sound is "attacked" and "cutoff " by crescendo and diminuendo to niente, the sound would not be considered to have any impulse (especially when there is no internal sustain of a dynamic). The most logical place to infer an impulse would be when the sound reaches its maximum volume and begins the diminuendo. In acoustic music, the performer will often provide some semblance of an accent at this point, making understanding this point as an impulse a reasonable inference. However, in the case of computer music, where a sound can have a perfectly logarithmic dynamic curve without accent, such an inference becomes less reasonable. Other possible points for impulse would be when the dynamic crosses over the boundary of human hearing, yet this would run afoul of the previous understanding of crescendi and diminuendi as continuous changes in sound (and thus not creating impulses). Thus, the local maximum(s) of such a sound would be the most logical and cognitively consistent places to infer an impulse. However, as many of these sounds are utilized specifically to remove the sense of temporal demarcation (or are part of larger sound-masses that mark time in other manners), the specific point of impulse in these sounds is rarely of analytic or perceptual relevance. 8 Martin Boykan, Silence and Slow Time (Lanham, MD: Scarecrow Press, Inc., 2004), 139. 9 Karlheinz Stockhausen and Elaine Barkin, "The Concept of Unity in Electronic Music," Perspectives of New Music 1 (1962): 40-42. 8 regular structures for pulse, each determined by the ratio of duration between two adjacent inter-onset-intervals (IOI). When the duration is the same between all impulses, the pulse is 1:1. When there are two alternating IOIs with one being twice the length of the other, the pulse is 2:1. The recurrence of impulses need not be perfectly regular; research by ten Hoppen, et al. has shown that listeners will group impulses "categorically" into regular pulses (of ratios of 1:1 or 2:1), provided the difference in IOI is within 80ms.10 11 In live performance, it is almost impossible for there to be a perfectly regular pulse, yet the listener smooths over these "micro-timings" and infers one of the two categorical structures. 12 Also included in an understanding of pulse are implied impulses. A repeated series of eighth-notes with the occasionally inserted quarter note need not indicate a change in the perceived eighth-note pulse. Implicit in such an idea is the understanding of pulse layers. Figure 1.2 shows a simple example; there are two implied pulse layers from the pattern of impulses. Even where quarter notes appear, the listener does not completely lose the sense of an eighth-note pulse.13 Were it impossible to understand multiple layers of pulse, 14 the listener would have to infer a constantly shifting pulse rather than simply impulses 10 ten Hoopen, et al., "Time Shrinking," 2-4. 11 From this point forward, this text will only concern itself with pulses constructed of impulses in 1:1 initial onset interval. The curious reader can easily convert virtually all further definitions to account for 2:1 IOI pulses. However, the coding described for dimension y in Chapter 2 would require a more nuanced conversion. 12 Mitchell Ohriner, "Grouping Hierarchy and Trajectories of Pacing in Performances of Chopin's Mazurkas," Music Theory Online 18 (2012): §§9, 17, accessed April 8, 2015, http://www.mtosmt.org/ issues/mto.12.18.1/mto.12.18.1.ohriner.php. 13 Arthur Olaf Andersen, Geography and Rhythm (Tucson, AZ: The University of Arizona Bulletin, 1935), 34-36. 14 Ève Poudrier and Bruno H. Repp, "Can Musicians Track Two Different Beats Simultaneously?", Music Perception: An Interdisciplinary Journal 30 (2013): 369-372, 385-389. 9 articulating different pulse levels. 15 Another aspect of this example shows the hierarchy created between certain pulse layers. The two layers in Figure 1.2 are obviously related by a multiple of 2, and an even higher layer could be created at, say, every fourth eighth-note. 16 Figure 1.3 demonstrates the multitude of pulse layers created by the rather simple polyrhythm of 2 against 3. Both dotted and regular eighth-notes create their own pulse layers (shown as pulse layers 2.1 and 2.2), which again can be continuously inferred even when longer note values are interspersed. A third pulse layer (labeled layer 1) is shared by every two dotted eighths and every three regular eighths. This layer is hierarchical to both layer 2.1 and 2.2, as well as marking their meeting point. Also created is a "composite" pulse, mapping the minimum number of regular impulses to match up with every impulse in the pattern.17 By definition, each pulse layer is hierarchical to a composite layer to which it contributes. Once the dotted eighth-notes split into dotted sixteenths, an additional composite pulse is created, and the first composite layer is also hierarchical to this layer. 18 While impulses are zero-dimensional points, pulse layers are one-dimensional lines. Because accentuation plays no role in determining a pulse (although it can be used to imply an additional higher pulse layer), there is no inherent beginning or end to an abstract pulse layer. Musically, these lines are segmented as part of the structure of a work, and can range in length from just a few seconds to the entirety of the movement. 19 15 Christopher F. Hasty, Meter as Rhythm (New York: Oxford University Press, 1997), 115. 16 James Morgan Thurmond, Note Grouping: A Method for Achieving Expression and Style in Musical Performance (Camp Hill, PA: JMT Publications, 1982), 29-33. 17 Poudrier and Repp, "Can Musicians Track Two Different Beats," 384-386. 18 Harald Krebs, Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann (New York: Oxford University Press, 1999), 39-44. 19 Hasty, Meter as Rhythm, 14. 10 Beat and Beat-Unit; Prime and Origin A beat is a regular accentuation in a pulse layer that implies at least one additional hierarchical pulse layer. 20 By definition, a pulse layer created from beats must have less impulse (in a given unit of time) than the layer from which it is created. The beat-unit is the notational value ascribed to a given beat. A pulse layer without beats will be called the prime layer, and a pulse layer that is not created from the beat of a lower layer will be called the origin. Neither of these layers, prime or origin, need be musically relevant; it is quite plausible for the most important pulse layers in a work to lie somewhere in between. The notational value ascribed to the beat-unit is, in general, simply a matter of convenience. Obviously, a beat-unit of a quarter note is only relevant in the context of a given tempo, and a doubled tempo will, of course, make a quarter note beat-unit equivalent to the previous eighth. While there would most likely be some standard to what notations would be ascribed by trained musicians to different rates of beat,21 for this text, beat-unit notation will simply be taken from the score in question. This prevents the psychological discontinuity created by different stimuli representing the same value. 22 When coded into numbers (rather than represented in traditional Western music notation), beat-units will be described by number of pulses from the smallest origin layer in the corpus. 20 Thurmond, Note Grouping, 27. 21 Bengt Edlund, Performance and Perception of Notational Variants (Stockholm: Almqvist & Wiksell International, 1985), 132-134. 22 David M. Egaleman, and Vani Pariyadath. "Is Subjective Duration a Signature of Coding Efficiency?", Philosophical Transactions: Biological Sciences 364 (2009): 1841-43. 11 Rhythm, Meter, and Submeter In the same way that pulse layers are structures created from impulses, there are three structures created by interactions of pulse layers (and impulses). Two of these, rhythm and meter, are common concepts, although these definitions will approach them slightly differently from prior scholarship. Submeter is a new item. 23 Each of these labels is defined by the number of pulse layers needed to accurately represent the salient accentuations in the structure. Should only one pulse layer suffice (i.e., the origin and prime are the same layer), the structure is a rhythm. 24 Where there are three or more pulse layers (at least one of which will, by definition, be neither origin nor prime), the structure is a meter. 25 Submeter is created by the separation of origin and prime layers without the insertion of any other layers between them. Thus, one pulse layer constitutes a rhythm, two a submeter, and three or more a meter. It is important to note that one structure can be described in multiple ways- discussion of a structure as a rhythm does not preclude it from also being understood as part of a larger meter or submeter, but only means that a single pulse layer will suffice to describe some salient feature. The following subsections will explore each of these concepts in greater detail and compare them to other definitions. 23 Christopher Hasty briefly remarked that certain layers could be "demoted to the status of the…‘submetrical'" but gave no specific indication of what this would mean. His context also implied that it could only occur within a system that already contained a meter. This document will instead explore examples of submeter that exist without the presence meter, and those that exist above meter(s) projected at lower pulse layers. Hasty, Meter as Rhythm, 14. 24 Grosvenor Cooper and Leonard B. Meyer, The Rhythmic Structure of Music (Chicago, IL: The University of Chicago Press, 1960), 6-7. 25 Fred Lerdahl and Ray Jackendoff, A Generative Theory of Tonal Music (Cambridge, MA: The Massachusetts Institute of Technology Press, 1983), 17-24. 12 Rhythm As rhythm is described only by a single pulse layer, it is inherently an impulse structure, rather than-as in meter and submeter-a relationship between beat-units and pulse layer hierarchy. Figure 1.4 shows a randomly generated rhythm-each impulse is separated by one to seven sixteenth notes without any implied accentuation (the beamgrouping by four sixteenth notes is simply for ease of reading).26 The pulse layer shown above is both the origin and the prime layer for this structure. As there are no units smaller than a sixteenth note, those are the smallest units needed for an even spacing where each impulse lines up with one pulse unit.27 The consistency of accentuation within the impulses precludes the inference of a larger salient structure, thus making the origin also the prime layer. 28 This definition of rhythm significantly limits the number of impulse structures that can be described only as such. An impulse structure created entirely from randomly-spaced, like units would be quite unusual, and even then local hierarchies could possibly emerge throughout a long work. In general, structures that can be described only as rhythm will be either short units (themselves possibly part of larger meters and submeters) or very large structures that describe the formal construction of a work (the latter of which will be shown to frequently constitute submeter in the majority of pre-twentieth-century Western art musics). 26 Chosen at random by generating a random sequence of integers between 1 and 7, using the atmospheric noise random number generator at http://www.random.org on January 5, 2015. 27 London, Hearing in Time, 56-58. 28 Cooper and Meyer, Rhythmic Structure, 7-10. 13 Meter Meter requires at least three pulse layers: the origin, prime, and at least one other layer in between. Thus, in direct contrast to rhythm, meter is an inherently hierarchical structure, understood by interaction between pulse layers. 29 The requirement of three or more salient pulse layers implies at least two levels of accentuation-one level creating a hierarchical structure from the origin, and another (or many others) further accentuating specific impulses within that structure. 30 This is wholly consistent with traditional definitions of meter.31 The accentuation of the lower layers (i.e., the origin) is analogous to subdivision, while the accentuation of upper layers is the equivalent of beat grouping. 32 A very simple example is shown in Figure 1.5, demonstrating the meter of the opening of Mozart's famous C-major sonata (K. 545). Even in this short example, seven pulse layers are projected. Any three adjacent layers can be understood as a meter. The most common understanding of this passage would be to consider it a quadruple meter with quarter note beat-unit and simple subdivisions into two eighths per quarter. This understanding is projected in pulse layers one to three. However, another perfectly valid metric analysis could look at pulse layers two through four and see duple meter at the half note, subdivided into two quarters. Moreover, an analysis of the hyper meter could rely on the topmost pulse layers and find the double-whole note articulated into duple meter with simple subdivision. By defining meter as the interaction of three or more pulse layers, it 29 Lerdahl and Jackendoff, Generative Theory of Tonal Music, 18-21. 30 David Temperley, The Cognition of Basic Musical Structures (Cambridge, MA: The Massachusetts Institute of Technology Press, 2001), 30-39. 31 Cooper and Meyer, Rhythmic Structure, 4-12. 32 Barbara 115. R. Barry, Musical Time: The Sense of Order (Stuyvesant, NY: Pendragon Press, 1990), 109- 14 becomes quite easy to show not only hierarchy of pulse layers creating a single meter, but also a hierarchy of meter itself. 33 For the vast majority of musical analysis, meter is the primary tool. The notions of hierarchy are well entrenched in the prevailing metrical theories and pedagogy. 34 The importance of this particular definition is the stated requirement of at least two levels of hierarchy (i.e., three pulse layers). While this is not in and of itself a new definition-most theories understand both patterns of beats and beat subdivision-it does bring into question certain prior metrical analyses that forgo either subdivision and/or beat hierarchy. These are most noticeable in four areas, each of which will be examined in the coming sections: 1) the highest level of hyper-meter (subdivision without hierarchy) 35; 2) the smallest of rhythmic units (hierarchy without subdivision) 36; 3) ostinati37; and 4) frequent tempo and/or time signature changes.38 By defining meter as requiring at least three pulse layers, each of these aspects can be more correctly analyzed in terms of their lack of traditional meter-namely what will be called submeter. 33 Lerdahl and Jackendoff. Generative Theory, 25-29. 34 Cooper and Meyer, Rhythmic Structure, 4-11. 35 Lerdhal and Jackendoff, Generative Theory, 99. 36 Gert ten Hoopen, et al., "A New Illusion of Time Perception-II" Music Perception: an Interdisciplinary Journal 11 (1993): 15-18. 37 Gregory R. McCandless, "Rhythm and Meter in the Music of Dream Theater" Ph.D. diss., Florida State University, 2009, 25-28. 38 Wing Lau, "The Expressive Role of Meter Changes in Brahms's Lieder" (paper presented at the Society for Music Theory Annual Meeting, Charlotte, NC, October 31, 2013). 15 Submeter Submeter is any impulse structure consisting of exactly two pulse layers-the origin and the prime. In such a structure, the origin layer can be shown to group upwards directly into the prime layer. There are no interceding layers in a submeter. By definition, the origin cannot have further subdivision and the prime cannot have implied beat hierarchy. Thus, a submeter consists of one layer (prime) that has subdivision but not beat grouping, while the other layer (origin) has beat grouping without subdivision. 39 These are two of the four aforementioned categories troublesome to prior metrical analysis. To demonstrate submeter's definition and use as an analytical tool, the first (the highest levels of hyper-meter) and third (ostinati) will be explored briefly. The second and fourth will be introduced as well, and are the main tools for the analysis of Ferneyhough's Unisicthbare Farben, comprising the bulk of this paper. Figure 1.6 reproduces the theme from Handel's "Larghetto, e piano" (the second movement of his Concerto Grosso in B-minor), complete with all implied pulse layers. At face value, this is a rather simple example of hyper-meter. 40 Each half-note groups into a whole, each whole-note into a two-bar group, and each two-bar group into a four-bar phrase. However, the form of the "Larghetto" makes this the upper limit of the implied hierarchy. 41 While an argument could be made for eight-bar groupings, this would not be musically relevant, given that measures 5-12 comprise a period independent of measures 1- 39 This directly contrasts Hasty's notion of a pulse layer being "demoted…to the status of the…‘submetrical.'" Submeter, in this formalization, is not an inferior level of meter, but rather a different structure entirely, defined by the existence of two and only two pulse layers. Hasty, Rhythm as Meter 14. 40 Temperley, 41 Lerdahl Cognition, 60-64. and Jackendoff, Generative Theory, 99-101. 16 4. 42 Thus, beats occurring every four bars defines the prime layer for the "Larghetto," with the origin of this movement defined at the eighth-note. This highest level of hyper-meter (and all hyper-meters that do not reduce to a single beat) is best understood as a submeter. The submetric designation allows for the description of its regular duple subdivision, but acknowledges the lack of any regular musical beats above the four-bar impulses.43 For the second demonstration, attention will be turned to the famous snare drum ostinato from Ravel's Bolero, reproduced with pulse layers in Figure 1.7. Unpitched ostinati such as this are one of the best examples of submeter. While the hyper-meter of the Handel example gradually progressed into a submeter, the Bolero ostinato achieves this in only two measures. No repetition of the two bar phrase in the snare drum part is accented more than any other. This is not to say that the Bolero as a whole is purely a two-bar repeating submeter, 44 but there is legitimate analytical interest in the fact that such a large structure is based around such a limited submetrical unit. Much of the hypnotic quality of the Bolero is derived from the steady repetition of the snare drum part below more metrically complex music. The most analytically rich examples of submeter occur at the smallest rhythmic levels and/or with frequent shifts in tempo and notated time-signature. Observe Figure 1.8a, 42 The repeat makes this even more true. Should the repeat not be taken, there is an argument that it is an eight bar grouping beginning on the upbeat. However, with the repeat, there is no correction for the upbeat before the repeat, and it would simply imply metric truncation. Finally, even if one only wishes to consider the material after the repeat and argue for eight bar groupings, by doubling the beat layers described, this demonstration still works. 43 In Hasty's discussion of submeter as a "demotion" of meter, he includes hyper-meter as another candidate for demotion. In such a manner, he implies an understanding of the lack of hierarchy at at least one layer of hyper-meter. One major advantage in this definition of submeter is that it allows the structure of this type of hyper-meter to be labeled not by its inferiority to typical meter, but simply by its difference in the number of meaningful pulse-layers. Hasty, Rhythm as Meter, 14. 44 The level at which the whole piece's hyper-meter reaches submetric levels is significantly higher than the Handel Larghetto (Figure 1.6). 17 the first bar of Fernyhough's Unsicthbare Farben. While there is a notated meter of 4 8 , there is no meaningful accentuation of such a regular meter. Rather, there are four distinct units where the impulses appear related to some (different) hierarchical pulse layer. Figure 1.8b shows each of these divisions. While each unit can be shown to subdivide a beat-unit at some tempo, the remarkably short nature of each beat coupled with the constant implied tempo changes make any determination of beat hierarchy a quixotic endeavor. It is these units that provide the most musically meaningful examples of submeter. They demonstrate a clear sense of beat subdivision without any sense of beat hierarchy. They are something less than meter yet greater than rhythm. 18 j j j j j j j 4 j 2 œ. ‰ œ. ‰ œ. ‰ œ. ‰ œ. ‰ œ. ‰ œ. ‰ œ. ‰ sfz sempre 4 2œ fp 4 2œ fp j œ. ‰ œ j œ. ‰ œ j œ. ‰ œ j œ. ‰ sfz fp sfz fp sfz fp sfz n fp n fp n fp n œ œ j œ ‰ œ j œ ‰ œ j œ ‰ œ j œ ‰ Figure 1.1: Three Impulse Structures Pulse Layer 1 œ œ œ œ Pulse Layer 2 œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ Impulses œ œ œ œ œ œ œ œ œ J œ J Figure 1.2: Simple Pulse Layer Demonstration œœ œ œ œ œœ œ œ œ œ œœ œ œ œœ œ œ œ œœ œ œ œœ œ œ œ œœ œ œ œœ œ Composite Pulse Pulse Layer 1 Pulse Layer 2.1 Pulse Layer 2.2 Impulses œ œ œ œ œ œ œ œ œ œ œ œœ ™ œ œ ™œ œ œ œ œ œ œ œ œ œ œ œ œ œœ ™ œ œ ™ œ œ œ œ œ œ œ œ œ œœ ™ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ j j œ̇ ™ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œœ ™ œ ™œ œ ™ œ œ ™ œœ ™ œ œ ™ œ œœ ™™ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ™ Figure 1.3: Composite Pulse Layer Demonstration Pulse layer Impulses œ œœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœ œ ‰™ ‰™ œ ‰™ œ ‰ œ ≈ ‰™ œ≈œœ≈≈œ‰ ‰ œœ‰ œ≈‰ œ≈≈œ‰ Figure 1.4: "Rhythm" With Associated Pulse Layer 19 œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœœœœœœœœœœœ Prime Pulse Layer 5 Pulse Layer 4 Pulse Layer 3 Pulse Layer 2 Pulse Layer 1 œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœœœœœœœœœœœ œœœœœœœœœœœœœœœœ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœœœœœœœœœœœ œœœœœœœœœœœœœœœœ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœœœœœœœœœœœ Origin Allegro œ ˙ œ œ œœœ œ™ &c˙ œ œ Ÿœ œ œ œ œ œ œœœ Œ Œ œœœœœœœœ œœ œ œœœœœ œœœœœœ œœœ p Piano ? &c { œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœœœœœœœœœœœ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœœœœœœœœœœœ œœœœœœœ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœœœœ œ œ œ œ œœœœœœœœœœœœ Œ œœ œ œ Œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœœœœœœœœœœœ Œ Œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœœœœœœœœœœœ 7 & œ œœœœ œœœœœœœœ œœ œ œœœœœ œœœœœœ . . œ nœ œ œ œ œ œ œ œ œ œ œ œ œ œœœœ œ œ œ œ. œ œ. œ. œ. œ œ. œ. œ œ œ œ œ œ œ œ œ#œ œ œ œ œ. . œœœ Œ œ. cresc. f ? œœ { Œ Œ œ œ œ œ Œ Œ œœ ww œ™ œ œ™ J #œ J œœ œœ œœœ œœœ œœœ œ œ œ œ. Figure 1.5-Mozart, Sonata in C-Major, K.545 (with pulse layers added) œ. Œ œ. 20 œ œ œ œ œ œ œ œ œ œ œœœœœœ œœœœœœ Prime Pulse Layer 3 Pulse Layer 2 Pulse Layer 1 œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœ œœœœœœ œœœœœœ œœœœœœ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœ œœœœœœ œœœœœœ Origin Larghetto, e piano Violino Viola Bassi ˙ ˙™ ° #### 3 & 4˙ œ B # # # # 43 œ œ œ œ œ ? #### 3 œ œ œ 4 œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ #œ œ œ œ œ œ œ œ ™™ œ ™™ œ ™™ œ œ ™™ œ œ œ œ œ œ ™™ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ ™™ ™™ ™™ œ œ œ œ œ œ ™™ œ œ œ œ œ œ œ œ œ œ ™™ œ œ œ œ œ ¢ œ œ œ œ œ œ œ œœœœœœ œœ œ #˙ ˙™ œ œ œ œ ˙™ ˙™ œ œ œ œ œ œ œ œ œ œœœœœœ œœœœœœ œ œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœ œœœœœœ 10 ˙™ ° #### œ #œ œ & ˙ œ ? #### ˙ œ œ #œ œ œ œ ¢ œœ J œœœ ˙ J œ ™™ ™™ œ ˙™ J œ™ B #### #œ œ œ œ™ ˙™ œ œ™ œ œ œ ™™ ™™ Œ œ œ œ ™™ ™™ Œ œ œ œ œ œ œ œ œ œ œ œ™ œœ J #œ œ œ ˙ œ œ œ œ œ œ œ œ œ œ ˙ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœ œœœœœœ œ œ œ œ œ œ œ œ œ œ œœœœœœ œœœœœœ œ œ œ œ œ œ œ œ œ œœœœœœ œœœœœœ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœ œœœœœœ œœœœœœ œœœœœœ ™™ ™™ ™™ ™™ ™™ 19 ˙ ° #### œ œ œ ™ & B #### ˙ œ ˙™ J œ œ œ #œ ? #### œ œ ¢ œ œ œ ˙™ œ œ œ œ œ #œ œ ˙™ œ ™™ œ ˙™ ™™ œ œ œ œ ™™ œ œ œ™ Œ œ #œ œ œœœ œ ˙ œ œ œ œ ˙ ˙ œ #œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ j œ ˙™ ˙ œ œ œ Figure 1.6-Handel, "Larghetto, e piano", (with pulse layers added) ˙ œ œ 21 ™™ ° œ Prime œ ™™ ™™ œ Pulse Layer 1 œ œ œ œ œ œ Pulse Layer 2 ™™ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ ™™ œ Pulse Layer 3 Origin œ œ œ œ œ œ œ œ œ œ œ Tempo di bolero 3 Snare Drum 3 ¢/ 4 œ 3 œœœœ 3 œœœœ œ œ œœœœ 3 3 3 ™ œœœœœœœœœ ™ Figure 1.7-Ravel, Bolero, snare drum part (repeat implied) e = 54 4 & 8® n >œ ™ µ >œ ™ n>œ 3 Bœ > nœ œ n œ B >œ n>œ B œ n>œ > > > µ>œ 2 8 œ™ > Bœ > nœ œ n œ B >œ n>œ B œ n>œ > > > µ>œ 2 8 œ™ > 5 Figure 1.8a-Ferneyhough, Unsicthbare Farben, mm. 1 e = 54 4 & 8® n >œ ™ µ >œ ™ n>œ 3 5 Figure 1.8b-Ferneyhough, Unsicthbare Farben, mm. 1 showing submetric units CHAPTER 2 FOUR-DIMENSIONAL MODELING Having identified the existence of submetric units, it becomes important to devise some method of describing them. Meter is easily described through conventional notions of beat subdivision (simple, compound, complex) and hierarchy (duple, triple, etc.). As submeter lacks beat hierarchy, these categorizations will not work; the notions of duple and triple meter are impossible outside the realm of beat hierarchy. Using the traditional notions of beat subdivision is equally problematic. Without beat hierarchy, there cannot be a sense of complex meter. Moreover, it is more common for the origin layer of submeters to be quadruple, quintuple, or even higher partitioning of the associated prime layer. In place of the conventional descriptions of meter, submetric units can be described by an ordered quadruple (x, y, z, w). In this model, x defines the beat-unit, y the subdivision, z the tempo of the beat-unit, and w the duration of the submetric unit. For the purposes of this theory, each dimension is postulated as orthogonal to all others. This allows a submetric unit to be described by any number of these dimensions, and allows easy comparison. Moreover, equivalence between submeters on a dimension imparts some sense of musical relationship; equivalence on multiple dimensions will result in even higher degrees of similarity. 45 45 Aaron J. Kirschner, "Four-Dimensional Modeling of Rhythm and Meter" (presentation at the Society for Music Theory Annual Meeting, Milwaukee, WI, November 6, 2014). 23 Dimension x-Beat-Unit The first dimension on which submeters are labeled is the duration of their beatunit. This is represented numerically in the number of impulses in the origin layer required to complete one beat-unit. Thus, if the origin layer represents pulses of thirty-second notes, a beat-unit of an eighth-note beat-unit would be represented as x = 4 . Because the beat-unit is, by definition, a hierarchical grouping of the origin layer, x will always be greater than one. 46 The beat-unit will be represented as a whole number in the vast majority of cases; in very rare cases (generally involving beat-units of doubly-dotted notes), x may be a positive fraction greater than one. The labeling for a beat-unit should be kept constant throughout a piece. This allows easy comparison between submeters, as all submeters with equivalent values for x can be easily seen as related. In coding, this will be represented at the beginning of the code as x(unit ) = k , where k represents the note value of one impulse at the origin layer. For an origin layer of a quarter-note k = 4 , for an eighth-note k = 8 , for a sixteenth-note k = 16 , and so on.47 Thus, in our previous example of an origin layer of thirty-second notes, the coding would open with x(unit ) = 32. Beyond allowing internal consistency, by allowing one to always be the unit of the origin pulse duration, it becomes more meaningful in comparison between pieces. If one piece has an origin layer at the sixty-fourth note and another at the sixteenth-note, by coding one as x(unit ) = 64 and the other x(unit ) = 16 , all further codings of submeter will show not an absolute beat-unit but one related to the deep structure of the piece. Moreover, if one 46 Edlund, 47 Notational Variants, 157-159. Should the smallest note-values present be triplets (or other tuplets), the x(unit) values is assumed to be the next smallest non-tuplet note-value (i.e., triplet thirty-second notes at the lowest level impart x(unit ) = 64 ). 24 does wish to compare beat-units in the absolute sense, as the values of k are in powers of two, a simple doubling (as many times as needed) of the x values of one piece can bring them to the equivalent x(unit) of another.48 Dimension y-Subdivision The nature of a submeter is subdivision without beat hierarchy. 49 The cardinality of subdivision is modeled in dimension y by the number of equally spaced impulses that subdivide the beat-unit x. This will always be a whole number of two or greater. A submeter must impart some level of subdivision to be understood as more than simple rhythm; thus, a case of y = 1 would not imply submeter. In general, y will be some value between two and five, and occasionally seven.50 Values of two and three will be by far the most common, as bipartite and tripartite divisions of the beat are the most easily perceived. At slower tempi, most subdivisions besides two and three sound as either multiple statements of a two or three part subdivision (i.e., four sounding as two groups of two, six as two groups of threes, etc.), or two beats of different length subdivided by different values (i.e., three-plus-two). However, at sufficiently fast tempi, groupings of four, five, and seven without an understanding of further subdivision become quite plausible. 51 Groupings of six are more unlikely (but not impossible) to be understood without subdivision, and higher 48 The smaller of the two values of k should be doubled, rather than halving the larger value. This prevents the needless complexity of many fractional values of x. 49 The term "subdivision" is technically incorrect in the context of submeter. As there is no beat hierarchy, the division of a beat-unit is not "sub-" to anything. The term, however, is used for consistency with the prevailing metric theories. A more proper term (and one that would work in cases of beat hierarchy as well) would be beat-division. 50 Jay Reise, Rhythmic Garlands (Bryn Mawr, PA: Merion Music, Inc., 1994), i, 2-5. 51 London, Hearing in Time, 100-115. 25 cardinality groupings almost always impart some implication of further subdivision. 52 A basic example would be to take a sixteenth-note subdivided into triplet thirtysecond notes (thus x(unit ) = 64 ). The (x, y) model for this would be (4, 3). While the x value is taken from the number of pulses from the origin, the value for y is dependent only on the number of subdivisions of the beat-unit, not their duration. Thus, should the note-values of this example be doubled or halved (eighth subdivided by triplet sixteenths, sixteenth subdivided by triplet thirty-seconds), the ordered pairs would be (8, 3) and (2, 3), respectively. By defining y as the cardinality of the subdivisions (rather than their absolute length), all submeters with like subdivisions are easily grouped together. Dimension z-Tempo The tempo coded into dimension z is the frequency of the beat-unit. There are two ways that this can be represented: an absolute value in beats-per-minute, or a categorical value based on the tempo gradations of a specific piece or corpus. For both of these, the most important aspect of z is that it describes the tempo of the relevant beat-unit x. Thus, if at one point an eighth, subdivided into two sixteenths, is repeating 74 times per minute, the coding would be (8, 2, 74) [ x(unit ) = 64 ]. Should a later submeter imply dotted eighths, subdivided into three sixteenths, also at 74 beats-per-minute, the coding would be (12, 3, 74). Dimension z is not concerned with the change in tempo of the eighth-note between the two examples, but rather the relationship between the beat-unit. 53 This paper will primarily focus on an absolute value of z, but categorization and 52 ibid. 125-141. 53 For the dotted-eighth to have a tempo of 74 beats-per-minute, the eighth must have a tempo of 111 beats-per-minute. However, as the eighth-note is no longer the beat-unit, this does not affect dimension z. 26 smoothing of the values is quite an easy proposition. In both cases, however, it is generally preferable to begin with an absolute labeling. To categorize z, beginning with the absolute values, a graph of all values of z can quickly show if there are obvious clusters of similar tempi. A computer program could also easily be written to identify the emergent clustering. Once the values of z are grouped into categories, these categories can be represented by the median or mean tempo of each category, or simply by ordinal ranking ascending from slowest to fastest. Another option for simplifying z is to smooth out all tempi within the just noticeable difference of human perception (all tempi whose IOIs differ by less than 80-100ms),54 or between standard metronome markings for a more rough estimate. This would fall somewhere between categorization and absolute value, as it would reduce the number of total tempi while retaining more micro-level changes. 55 An additional clustering method, based on pitch transformations, 56 is briefly explored in Chapter 4. Dimension w-Duration Dimension w describes the duration of a specific submeter in seconds. While the previous dimensions could be described in an abstract, nontemporal sense, duration requires a specific temporal context for modeling. 57 If a submeter of (8, 3, 74) ( x(unit ) = 64 ) exists for three-and-a-half eighth-notes, the duration is 2.84 seconds, 58 thus resulting in an ordered 54 London, Hearing in Time, 27-30. 55 Martin F. McKinney and Dirk Moelants, "Ambiguity in Tempo Perception: What Draws Listeners to Different Metrical Levels?", Music Perception: An Interdisciplinary Journal 24 (2006): 155-57. 56 Andrew Mead, "On Tempo Relations," Perspectives of New Music, 45(2007): 64-109. 57 Kirschner, "Four-Dimensional Modeling." 58 The formula for determining duration of a given submetric unit is shown in Appendix A. 27 quadruple of (8,  3,  74,  2.84). Because the nature of submeter precludes beat hierarchy, durational values will generally be rather low. Longer durations would doubtless impart a sense of beat hierarchy. Basic Modeling To begin, take the first simple example in Figure 2.1, a quarter-note subdivided into four sixteenths. Because the smallest note value is a sixteenth, let x(unit ) = 16 . The quarternote duration of the beat is four sixteenth, so x = 4 . The quadruple subdivision gives y = 4 . Thus, in two dimensions, this submetric unit can be modeled as the (x, y) metric double of (4, 4). An immediately apparent implication is that these models do not need to represent all four dimensions to be analytically relevant. The (x, y) metric double of (4, 4) accurately describes many submetric units of differing tempi and duration, each related by the nature of a beat-unit created from four units of the origin layer subdivided into four. Because all dimensions are orthogonal, each maintains analytical meaning even in the absence of any or all others. For notational clarity, it is important to express what dimensions are being modeled (when less than four are being used) by stating the relevant dimensions and the x(unit). For extreme clarity, the above example should be referred to as "the (x, y) metric duple of (4, 4) where x(unit ) = 16 ." Once a tempo is added to this unit (also Figure 2.1), the dimension z can be modeled, resulting in the (x, y, z) metric triple (4, 4, 60). The z = 60 dimension refers to the tempo of the quarter note, and the x and y values are calculated as above. When submetric triples are used, the (x, y, z) variety will be the most common. Finally, for a relevant example of submetric quadruples, it becomes helpful to 28 integrate changes in the submeters. Figure 2.2 demonstrates how three submetric quadruples are assigned to a short passage. The calculations here are rather simple. The smallest note value is the sixteenth note, thus x(unit ) = 16 . Each submetric unit is based on beat-units of quarter notes, so for all units x = 1 . As the tempo is constant, each of the z = 60 for all units. The only dimensions that change between units are y and w. For the first unit, the quarter-note is divided into four sixteenths ( y = 4 ). The second is divided into two ( y = 2 ), and the third into three ( y = 3 ). The durations are two seconds, two seconds, and one second, respectively (thus w = 2, 2,1 ). This is a rather simple example, and generally not one that submeter would be used to describe (meter is probably preferable, as additional levels of hierarchy could be argued), 59 but it serves to demonstrate the process. A more complex example is shown in Figure 2.3. The first unit is calculated in the same manner as the first unit in Figure 2.2. However, the second unit implies a dotted-eighth beat-unit ( x = 3 ) subdivided into threes ( y = 3 ). The tempo (z) of this dotted eighth is 80 beats-per-minute ( ( 43 )(60) = 80 ), and this submetric unit exists for three seconds ( 3( 60 )= 3 60 60 and 4( 80 ) = 3 ). The final two units are slightly more complicated. Even though both are grouped under the larger quintuplet bracket, the nested tuplets impart different pulse rates, and thus require separate submetric units. The triplets are a tripartite subdivision of an eighth-note, thus x = 2 and y = 3 . While the beat-unit is now an eighth-note, this eighth-note is contained within a quintuplet. 60 Thus, its beats-per-minute are calculated by doubling the quarter note value, then multiplying by 59 5 4 to account for the quintuplet resulting in Barry, Musical Time, 110. 60 Brian Ferneyhough, "Duration and Rhythm as Compositional Resources" in Brian Ferneyhough: Collected Writings, ed. James Boros and Richard Troop (Amsterdam: Harwood Academic Publishers, GmbH., 1989), 53. 29 60 z = 60(2)( 45 ) = 150 . The duration of the single beat-unit is calculated as w = 150 = 0.4 . The second submetric unit divides a dotted-eighth into four, thus x = 3 , y = 4 . 61 The tempo is now calculated as the rate of a dotted-eighth quintuplet; z = 60( 43 )( 45 ) = 100 . 60 Duration is again that of a single beat-unit, thus w = 100 = 0.6 . From each of these values, the submetric units of Figure 2.3 can be represented by the collection of ordered quadruples (4, 4, 60, 2); (3, 3, 80, 3); (2, 3, 150, 0.4); (3, 4, 100, 0.6). To aid in further calculation, generalized formulas for z and w can be derived. For a general calculation of z, a new unit called z(base) will be introduced. This unit is used mainly for convenience and human markup (although some long-range coding comparison can make use of it), and is notated whenever the notated tempo marking changes. The z(base) allows for easy comparison between the notated tempo and beat-unit and any submetric unit(s). It is defined as z(base ) ≡ x mark : z mark where x mark is the notated beat-unit and z mark is the notated tempo. In Figure  2.3, the notated q = 60 would impart z(base ) ≡ 4 : 60 .62 While helpful for markup, the primary use of z(base) is calculating the tempo of any given submetric unit. The formula for calculating tempo is z = (z mark )( x mark x )(u ) where u is the product of all relevant tuplet ratios. For a tuplet ratio to be relevant, it must affect the duration of the beatunit, not simply the subdivision. For example, in Figure 2.4, the triplet over the first eighthnote value are not considered a "relevant" tuplet as they affect only the subdivision of the beat-unit, not the beat-unit itself. As there are no relevant tuplets (u = 1) and the beat-unit an eighth-note x mark x = 44 = 1 , the tempo is simply the given value of 84. However, in the second 61 When the tuplet affects only the subdivision (not the beat-unit), tuplets of 4:3 (and 2:3, 8:6, etc.) should always be treated as subdivisions of a dotted note value. Moreover, if the final tuplet in Figure 2.3 were rewritten as dotted thirty-seconds, the submetric unit label would not change. 62 When coded in raw text files, the notation can be simplified to <zbase 4:60> 30 unit, the beat-unit is shifted from an eighth-note to a quarter, and the 8:7 tuplet is acting on the beat-unit. There are now meaningful values for all elements of the formula: and u = 8 7 x mark x = 84 = 1 2 , therefore z = (84)( 12 )( 87 ) = 48 . The final submetric unit requires an even more detailed calculation. Here, the beat-unit is a dotted-eighth ( x = 3 ) and it is subdivided into four ( y = 4). Moreover, the dotted-eighth beat-unit is nested inside both the 8:7 and 5:4 tuplets. Thus, x mark x = 43 and u = ( 87 )( 45 ) . Therefore, z = (84)( 43 )(( 87 )( 45 )) = 160 .63 The duration of a unit of known tempo can be calculated by the formula w = ( 60z )(c ) where c is the number of cycles of the tempo. 64 For the first unit in Figure 2.4, the tempo is indicated as 84 BPM and there is a single cycle; thus, w = 60 84 = 57 ≈ 0.71 . The second unit includes three beat-units at a tempo of 48 BPM, which would imply w = ( 60 )(3) = 154 = 3.75 . 48 However, the eighth-rest under the quintuplet must also be accounted for in the duration. Thus,  its  duration must be added to the three cycles of 8:7 quarters, resulting in )(3)) + ( 84( 608 )( 5 ) ) = 174 = 4.25 . The final submetric unit is again a single cycle, giving w = (( 60 48 7 4 60 w = 160 = 83 ≈ 0.38 .65 Having calculated all of the dimension, the coding of Figure 2.4 reads as follows: <xunit=32> <zbase 4:84> (4, 3, 84, .71); (8, 2, 48, 3.75); (3, 4, 160, .38) 63 The extra set of parentheses in the previous equation are simply for demonstration (clearly showing the value of u); as multiplication is commutative, they are wholly unnecessary. 64 Formula adapted from Morris, (2006). 5. 65 From this point forward, all tempi (in beats-per-minute) and durations (in seconds) will be rounded to the nearest hundredth. Moreover, those equations whose values are rounded will forgo approximately equals signs (≈), as there is no chance that the rounding would impart change outside of human just-noticeable-difference. See, McKinney and Moelants, (2006). 155-157. 31 Rallentandos and Accelerandos Two special cases, rallentandos and accelerandos, also require generalized formulas. For the purposes of these models, rallentandos and accelerandos are treated as "terraced," with each submetric unit sequentially shorter or longer than the preceding unit. There are two ways to treat these terraces-ordinal or proportional. An ordinal method would alter each submetric unit by the same relative amount. A proportional method would ascribe an alteration based on the durational size of each unit relative to the total duration of the tempo shift. These are best demonstrated through their formulas. Before applying these formulas to account for rallentandos and accelerandos, the submetric units should be coded as if they were at the starting tempo. For an ordinal rallentando, each submetric unit is altered by a linearly decreasing percentage. As a rallentando affects tempo (and not beat-unit nor subdivision), this alteration affects only dimensions z and w (w being, in the context of actual music, a function of z). The tempo is assumed to decrease by the same amount at the start of each unit inside the rallentando. The value of this decrease is defined as h and is calculated as h = P− Q n where P is the starting tempo, Q is the goal tempo, and n is the number of submetric units in the rallentando. Using h, each submetric unit of the rallentando will be altered by a variable b, calculated as b = P −( h⋅s i ) P where s i is the ordinal position of the submetric unit within the rallentando. Note that h is constant for all submetric units within a rallentando, while b differs. Having calculated the variable to alter each submetric unit, the tempo is multiplied by b and the duration divided. The formula on the next page summarizes this method. 32 rall (P → Q) P− Q n =h P −( h⋅s i ) P =b z rall = z ⋅b w rall = w b An ordinal accelerando is calculated in much the same way, except that the formula must account for an increase in tempo, rather than decrease. To do so, h is now defined as h= Q−P n and b is defined as b = P +( h⋅s i ) P . All other calculations remain the same. accel (P → Q) Q−P n =h P +( h⋅s i ) P =b z accel = z ⋅b w accel = w b A proportional rallentando or accelerando would decrease or increase the tempo of each submetric unit by an amount proportional to the duration of each unit. To do so, the sum total of durations implied without the tempo shift must be calculated (defined as n w whole = ∑ w i where w whole is the total duration inside the tempo shift, and w i is the duration of i =1 an individual submetric unit). From here, the percentage of each submetric unit's duration relative to the whole is multiplied by change in tempo, and then converted into the variable shift in tempo. The formulas for a proportional rallentando and accelerando are as follows: rall _ prop(P → Q) accel _ prop(P → Q) n n w whole = ∑ w i w whole = ∑ w i i =1 i =1 wi P − ((P − Q)( wwhole )) z rall P = z ⋅b w rall = w b w =b P + (( Q − P )( wwholei )) z rall P = z ⋅b w rall = =b w b The process for making all of these calculations can be automated. Most coding for corpus analysis can simply use the original tempo marking and a command for the tempo 33 shift. Where the tempo shift ends, a new z(base ) is entered, and the material continues. If Figure 2.4 were to depict the three units slowing from e = 84 to e = 56, the coding would only need to add in a beginning and ending for the rallentando. <xunit=32> <zbase 4:84> <rall 84-56> (4, 3, 84, .71); (8, 2, 48, 3.75); (3, 4, 160, .38) <zbase 8:56> which would automate the following calculations: h= b1 = 84−54 3 = 84−( 28 3 ) 84 28 3 = 8 9 b2 = 84−2( 28 3 ) 84 = 7 9 b3 = 84−3( 28 3 ) 84 = 2 3 zrall = 84( 89 ) = 74.67 zrall = 48( 97 ) = 37.33 zrall = 160( 23 ) = 106.67 wrall = 0.71( 98 ) = 0.80 wrall = 3.75( 97 ) = 4.82 wrall = 0.38( 23 ) = 0.57 resulting in the following ordered quadruples: (4, 3, 74.67, .80); (8, 2, 37.33, 4.82); (3, 4, 106.67, .57) 34 q = 60 >œ œ œ œ °¢™™ œ œ œ œ ™™ ü† > (4, 4) (4, 4, 60) Figure 2.1: Two- and Three-Dimensional Modeling of Four Sixteenth-Notes q = 60 3 œœœœœœœœœ œ œ œ œ œ œ (4, 4, 60, 2) (4, 2, 60, 2) (4, 3, 60, 1) Figure 2.2: Simple Four-Dimensional Modeling q = 60 œ œ œ œ œ œ œ œ œ™ (4, 4, 60, 2) 3 5:4 4:3 œ œ œ œ œ œ™ œ œ œ œ œ œ œ (3, 3, 80, 3) (3, 4, 100, .6) (2, 3, 150, .4) Figure 2.3: Complex Four-Dimensional Modeling 35 e = 84 8:7 5:4 4:3 3 >œ œ œ >œ >œ œ >œ ‰ œœ œ > Figure 2.4: Tuplets of Different Relevance to Tempo Calculations ∑ ∑ ∑ ∑ ∑ CHAPTER 3 APPLICATIONS OF SUBMETRIC MODELING DEMONSTRATED THROUGH AN ANALYSIS OF FERNEYHOUGH'S UNSICHTBARE FARBEN # qœ= 120 œ # œ isœthe rather œ œcurious œ case œ of œbeat œsubdivision œ œwithout œ beat œ hierarchy, œ its œ As submeter œ œ œ & application is seemingly limited. The vast majority of Western art and popular musics are Piano # œthrough œ more œ traditional œ œ metrical œ œ theories. œ œSubmeter œ isœ mostœ applicable œ œ to the œ better handled œ œ œ #œ œ œ œ œ œ œ œ œ ¬ated meter nothing more than a phantom beat. In general, the notated meter is simply for œ œ { & music of the so-called "New Complexity." The frequent time signature changes and extreme use of tuplets in this music remove any sense of beat hierarchy. In place of beat hierarchy, the division of otherwise œ unaccented beat-units is the primary impulse structure. Even in bœ n œwhere #œ œ œ 3 cases œ there is a morebconsistent time signature, the rapid change in tuplets makes the œ 3 convenience and ensemble coordination, rather than a musical idea that should be 3 3 3 œ ? &perceptible to# œthe listener. 66 & # œ œ #œ #œ œ œ œ œ œ œ œ œ œ { #œ œ œ œ To demonstrate some of the potential for submetric analysis, Brian Ferneyhough's violin solo Unsichtbare Farben will be considered. More so than almost any other composer, Ferneyhough's works epitomize submeter. Nearly every bar has a different b œ time# œsignature # œ bœ œ nœ œ 5 œbœ œbœ bœ œb œ œ #œ œ œ bœ #œ (often irrational), and those that do not are generally grouped by different tuplets. Within #œ œ & each of these 7 bars, more tuplets are nested, creating an almost constant change in speed. In 5 7 66 { 5 Jason Eckardt, (lecture given at New Music on the Point. Leister, VT, June 6, 2013) & bœ œœ #œ œ #œ œ œ œ œ œ #œ #œ #œ #œ œ œ 37 the preface to his solo piccolo work Superscriptio, Ferneyhough stated that his "bar lines should be invariably regarded as instantaneous alteration of beat-length."67 However, the alterations often happen on even smaller levels than the bar line. The majority of his tuplets are not simply different groupings or divisions of a steady pulse, but rather shifts in beatunit and/or tempo. This analysis of Unsichtbare Farben will be primarily concerned with mapping shifts of beat-unit, subdivision, and tempo. Using the four-dimensional model of submeter, each submetric unit can be easily related. Where only one dimension changes, it is readily apparent in the coding, even if it is not as clear from the score. 68 The analysis will begin by demonstrating how each submetric unit in the first five measures is identified and modeled. Following this, the majority of the analysis will focus on the first section of the piece (through measure 22), before concluding with observations on the patterns of beat-unit and subdivision that emerge form the submetric modeling of the entire work. One aspect this analysis will not be concerned with is Ferneyhough's compositional process. While this is an undoubtedly rich area of scholarship, it is not of immediate concern to submetric analysis. 69 As Mikhïal Malt demonstrated, Ferneyhough began by creating two impulses with durations in ratios from 67 1 1 to 1 11 and then splitting the second impulse into Brian Ferneyhough, Superscriptio (New York: Edition Peters, 1982), i. 68 This analysis will take the printed score as a "ground truth" and assume that an exact replication is aurally perceived. The published score from which this analysis is completed includes a note that "this score is a facsimile of the composer's manuscript, reflecting the editorial work and correction as of 03 07 02." While this indicates that the possibility exists of changes to the score, the author's particular copy was ordered from Edition Peters in late 2013, indicating there were no published changes in the decade prior to this writing. Ferneyhough's catalogue lists the date of composition as 1998. Ferneyhough, Unsitchbare Farben. 69 The majority of the existing pitch analysis vis a vis Ferneyhough's process is concerned with the imbedding of a plain chant melody. Louis Fitch, Brian Ferneyhough (Chicago, IL: The University of Chicago Press, 2013), 89. 38 equal divisions of 1-6.70 This system, however, is infiltrated by many further layers of rhythmic processes, such that the resultant material does not aurally exhibit the regular construction Malt demonstrated. Instead, aurally apparent submetric units emerge from this deep structure.71 These are the aspects that this analysis will consider. Ferneyhough himself noted the lack of aural appearance of his compositional process: In a sense, Unsichtbare Farben might be seen as the "tip of the iceberg", to the extent that the vast preponderance of materials that went into its preparation appears nowhere in the musical phenomenon itself, having been suppressed by a formal filtering operation selecting and interleaving structurally equivalent elements from a relatively large number of through-composed layers. Correspondingly, the unfolding of the work's argument is characterised primarily by a series of rhetorical ruptures as short fragments of otherwise impalpable processes are abruptly invoked and, equally suddenly, abandoned. 72 Partitioning the First Five Measures into Submetric Units At the end of Chapter 1, the first measure of Unsichtbare Farben was shown to contain four distinct submetric units. (Figure 3.1 reproduces this partitioning.) These units are determined by observing groups of equally spaced impulses, and/or groups where the spacing is halved or doubled. Wherever adjacent IOIs differ-except by a multiple of two- a submetric shift occurs. The dotted-thirty-second notes impart the first set of equally spaced impulses. 73 Immediately upon attacking the Dn, the spacing changes to triplet eighth70 Mikhïal Malt, "Some Considerations on Brian Ferneyhough's Musical Language Through his use of CAC: Part I-Time and Rhythmic Structures," in The Open Music Composer's Book Vol. 2, ed. John Bresson, et al. (Sampzon, France: Editions-Delatour, 2008). 71 Thomas Reiner, Semiotics of Musical Time (New York: Peter Lang Publishing, 2000), 56-8. 72 Ferneyhough, Unsitchbare Farben. 73 Bruno H. Repp and Yi-Huang Su, "Sensorimotor Synchronization: A Review of Recent Research (2006-2012)," Psychological Bulletin & Review 20 (2013): 416-18. 39 notes, with the final duration curtailed to make way for the quintuplet thirty-seconds that follow.74 The last submetric unit contains the sixty-fourth notes (and larger groupings) that lead into measure two. There are three important observations from this partitioning. First, note that while the sixty-fourth notes at the end of the bar are grouped with the sixteenth and thirty-second notes that follow, they are not grouped with the preceding quintuplet sixteenth and thirtyseconds. For impulses to be grouped together into a submetric unit, their durations (or, more specifically, attack-onset-intervals) must be either equivalent, related by some integer power of two (i.e., eighths and sixteenths), or the result of combining smaller impulses in the specific subdivision (i.e., dotted-eighth and sixteenth, where a quarter is divided into four). 75 Second, observe how the duration of the second submetric unit is curtailed to make way for the third. This will not be an uncommon occurrence. Moreover, "out-of-submeter" pick-ups will be encountered (the first occurrence-in measure four-will be used to demonstrate how they act). Finally, while two of these submetric units started on the notated "beats," this has absolutely no meaning. A submetric unit can begin anywhere in a notated measure, and the true beats are those implied by the unit(s), not the notated time-signature. The notated time-signature is treated simply as a convenience. 76 Having identified the submetric units in the opening measure, their ordered quadruples must now be calculated. As the smallest note values are sixty-fourth notes and the tempo is e  =  54, x(unit ) = 64 and z(base ) ≡ 8 : 54 . The first submetric unit features a 74 While the duration of the second eighth in this submetric unit is curtailed, the beat-unit is still labeled as an eighth. The change in IOI from the previous submetric unit imparts this consideration. 75 London, Hearing in Time, 60-79. 76 Eckardt, "lecture." 40 dotted-sixteenth note as the beat-unit (six sixty-fourth notes, thus x = 6 ) subdivided into two dotted-thirty-second notes ( y = 2 ). The tempo of a dotted-sixteenth is calculated by multiplying the ratio of eighth-notes to dotted-sixteenth notes (i.e., z = z mark ( x mark x x mark x in the formula )(u )) by the notated tempo of the eighth-note. Performing these calculations gives z = 54( 86 ) = 72 . Finally, to calculate the duration of the submeter, the tempo of the beat-unit is divided by 60 and multiplied by the number of beats (in this case only one), resulting in w = 60 72 = 0.83 . The ordered quadruple for the first submetric unit is (6, 2, 72, 0.83). The second submetric unit divides a eighth-note ( x = 8 ) into two ( y = 2 ), but this eighth-note is transformed by the triplet. Except for tuplets that affect only subdivision (c.f. Figure 2.4), all tuplets will be treated as instantaneous transformations of tempo ("beatlength alterations" in Ferneyhough's terminology). Thus, while the beat-unit is still taken from the notation as a quarter note, the triplet will act upon the tempo. The tempo is now calculated as z = 54( 32 ) = 81 , with z = z mark ( x mark x 3 2 representing the tuplet ratio (u in the formula )(u ) ). Calculating the duration of this unit is a slightly more challenging procedure. In cases where the duration of the submeter is curtailed and/or there is an "outof-submeter" pick-up, the simple formula w = 60z (c ) will not entirely suffice. Instead, the duration must be calculated through a summation of multiple units, each following the same formula (much the same as how total duration is calculated in accelerandi and rallentandos). When these cases arise, it is often helpful to calculate the tempo of the following unit(s) before calculating the duration of the unit in question. Occasionally, such calculations will even require calculating tempo(s) that are unrelated to any salient submetric unit.77 77 In the case of this specific submetric unit, it is obviously possible to simply calculate the duration as w = ( 5460 )( 65 ) = 1.33 (the triplet fills one eighth-note; the quintuplet-thirty-second one-fifth of the next). A shortcut such as this will not always be available and/or as intuitive. The method described in the text will work for all cases, is easily repeatable, and minimizes human error in calculation. 41 Moving to the third submetric unit, a sixteenth-note ( x = 4 ) is divided into two thirty-seconds ( y = 2 ). This sixteenth-note is transformed by the quintuplet, giving z = 54(2)( 45 ) = 135 . This tempo can be used to calculate the duration of both the second and third submetric units. The third is the easier of the two. It is comprised of simply two 60 cycles at 135, thus w = ( 135 )(2) = 0.89 . For the second unit, two different products must be added. The triplet-eighth-note at 81 BPM exists for one-and-a-half cycles and the unit is curtailed after holding for an additional half of a cycle at 135 BPM.78 The duration of the second unit  is the  sum  of the durations of each of these fractional cycles, calculated as 60 60 w = (( 54 )( 32 )) + (( 135 )( 12 )) = 1.33 . Having calculated both durations and tempi, the ordered quadruples for the second and third submetric units are, respectively, (8,  2,  81,  1.33) and (4, 2, 135, 0.89). The last unit of this measure is rather simple to calculate. The beat-unit is reduced to the thirty-second note ( x = 2 ), it is subdivided into two ( y = 2 ), and the tempo is not transformed by a tuplet ( z = 54(4) = 216 ). The unit exists for five-and-a-half cycles, giving a duration of w = ( ) = 1.53 . The relevant ordered quadruple is (2, 2, 216, 1.53). 60 11 216 2 The second measure (Figure 3.2) presents only a single submetric unit. Here, the beat-unit is a thirty-second note ( x = 2 ), and it is subdivided in two ( y = 2). The tempo remains untransformed, giving z = 54(4) = 216 .79 Calculating the duration will require another sum, as the unit is extended through the triplet-sixteenth note in measure three. The 78 Although the quintuplet-thirty-second is one-fifth of the notated beat-unit, it is one-half of the submetric unit to which it belongs. The relevant ratio is to the submetric beat-unit, not the notated beat-unit. 79 It is legitimate to question why this is not considered the same submetric unit as the end of the first measure. The consistent accents coupled with the 1+2+1 grouping to begin the new unit make the first thirty-second note (not the dotted-thirty-second note) sound as a beat, not a subdivision. To consider these as the same submetric group would require one of them to be considered "syncopated." 42 tempo of the triplet-sixteenth can simply be calculated in the denominator of its fraction. 60 Doing so shows w = (( 216 )( 132 )) + ( 54( 602 )( 3 ) ) = 2.18 and results in the ordered quadruple 2 (2, 2, 216, 2.18). The third measure (Figure 3.3) presents the first nested tuplet in the piece (which become increasingly common). However, the nested triplet is one that affects only the subdivision and does not transform the tempo of the beat-unit. Thus, the beat-unit is an eighth-note ( x = 8 ) subdivided into three ( y = 3 ). While the nested triplet does not affect the beat-unit, the larger triplet does, resulting in z = 54( 32 ) = 81 . Only a singled cycle is projected ( w = 60 81 = 0.74 ); the submetric unit is represented by the ordered quadruple (8, 3, 81, 0.74). The second submetric unit occurs inside the septuplet, and projects a dotted-eighthnote ( x = 12 ) subdivided into two ( y = 2 ). The dotted eighth is transformed by the septuplet, requiring the tempo to be calculated as z = 54( 23 )( 47 ) = 63 . The submetric unit exists for only a single cycle ( w = 60 63 = 0.95 ) resulting in the ordered quadruple of (12, 3, 63, 0.95). The third and final submetric unit in measure three provides the first case of an "out-of-submeter" pick-up. The last thirty-second note of the septuplet acts as a pick-up into this submetric unit, but does not fit into the beat-unit or subdivision projected. It is, in effect, a grace-note that neither curtails the previous note's duration nor delays the following note's arrival (while fitting into a specific duration). 80 The submetric unit itself is a sixteenth note ( x = 4 ) divided in four ( y = 4 ), without any tuplet alteration to the notated beat-length ( z = 54(2) = 108 ). In addition to the pick-up, the submetric unit extends into the following 80 Jason Eckardt, Echoes' White Veil (New York: Carl Fischer, 1996) 5. 43 bar for the first notated sixteenth note in 5 12 time. 81 The duration, therefore, will be the sum of three different products: the length of the septuplet pick-up, the submetric  unit  itself, and the 5 12 sixteenth note hold. The calculation gives 60 )( 16 )) + (( 108 )(2)) + ( 54( 602 )( 3 ) ) = 1.64 and models this submetric unit as (4, 4, 108, 1.64). w = (( 60 63 2 The irrational time signature of measure four (Figure 3.4) requires an extra multiplicand when calculating tempo, but is otherwise similar to the process for the first three measures. The dotted-thirty-second notes at the measure's outset split a dotted sixteenth in two, giving x = 6 and y = 2 . The tempo is calculated in the same manner as the previous dotted sixteenth tempo, and the 5 12 time-signature is accounted for by adding the requisite multiplicand, resulting in z = 54( 43 )( 32 ) = 108 . There is a single cycle, and no pick60 ups or holds; w = 108 = 0.56 and the ordered quadruple is (6,  2,  108,  0.56). The following submeter uses the same beat-unit ( x = 6) but subdivides it in three ( y = 3 ) rather than two. 60 Moreover, the tempo ( z = 54( 43 )( 32 ) = 108 ) and duration ( w = 108 = 0.56 ) are the same, giving an ordered quadruple with only one altered dimension: (6, 3, 108, 0.56). The triplet division of the next sixteenth note begins the third submetric unit, and defines it as a sixteenth note ( x = 4 ) subdivided in three ( y = 3 ). This is another case where the tuplet only affects the subdivision, and not the beat-unit. Therefore, the ratio of the triplet subdivision is not multiplied into the product to calculate the tempo (although the 5 12 time signature must still be accounted for), and z = 54(2)( 32 ) = 162 . Four sixteenth 60 notes project four cycles of this submeter with total duration w = ( 162 )(4) = 1.48 . The metric quadruple is (4, 3, 162, 1.48). 81 The "irrational" time signatures are quite common in Unsichtbare Farben. They are invariably treated as large tuplets, and simply add one additional fraction to be multiplied when calculating tempo. For those with a denominator that is a multiple of three, a multiplicand of 32 is included in the product; for those based on a multiple of five, 45 is the relevant multiplicand. 44 The final submetric unit in measure four shifts the beat-unit to an eighth-note ( x = 8 ) and the subdivision to four ( y = 4). The tempo is only altered by the 5 12 time signature, resulting in z = 54( 32 ) = 81 . While there is a single cycle at 81 BPM, there is also the hold into the next measure to consider (Figure 3.5). The sixteenth note under a 5:3 tuplet in a 3 20 measure is the beat-unit for the following submeter (it is subdivided in three, giving x = 4 , y = 3 ); calculating its tempo will allow calculation of the duration of the last unit in measure four as well. The 3 20 time signature requires a multiplicand of 5 4 to be included in the product, resulting in z = 54(2)( 53 )( 45 ) = 225 . Using this tempo, the duration of the last 60 submetric unit in measure four is calculated as w = ( 60 ) + ( 225 ) = 1.01 and the duration of the 81 60 first submetric unit in measure five is w = ( 225 )(2) = 0.53 . The submetric units, respectively, are (8, 4, 81, 1.01) and (4, 3, 225, 0.53). The final submetric unit in these first five bars divides an eighth-note ( x = 8 ) in two ( y = 2 ). The beat-length, however, is altered by three elements: the 3 20 time signature, the bar-length 5:3 tuplet, and the nested 5:4 tuplet. Each of these must be factored into the tempo product, resulting in z = 54( 45 )( 45 )( 53 ) = 140.63 . Calculating the duration requires adding the duration of one-and-a-quarter cycles of the submetric unit to the length of the triplet-thirty-second note in the next (still in 3 20 ) bar. The relevant equation is 60 )( 45 )) + ( 54( 604 )( 5 ) ) = 0.76 and the ordered quadruple is (8, 2, 140.63, 0.76). w = (( 140.63 4 45 In the first five bars alone, there are 14 distinct submetric units. The coding for these measures appears below. <xunit=64> <zbase = 8:54> (6, 2, 72, .83); (8, 2, 81, 1.33); (4, 2, 135, .89); (2, 2, 216, 1.53); (2, 2, 216, 2.18); (8, 3, 81, .74); (12, 2, 63, .95); (4, 4, 108, 1.80); (6, 2, 108, .56); (6, 3, 108, .56); (4, 3, 162, 1.48); (8, 4, 81, 1.01); (4, 3, 225, .53); (8, 2, 140.63, .76); One major advantage of this four-dimensional coding of submetric units is the immediate appearance of number patterns. While the notated music becomes increasingly complex-with nested tuplets in measures of "irrational" time-signatures-the actual beatunits projected are restricted to the rather manageable thirty-second through dotted-eight note values, with dotted values only on sixteenth and eight notes. The subdivisions are even more restricted; beats are only divided into two, three, or four impulses. From this coding, it is readily apparent that the main dimension traversed is tempo. Notating the submetric units with the tempo always related to the beat-unit has the added advantage of clearly showing that a thirty-second note in 3 is the same length as a dotted-thirty-second in 5 .82 83 12 8 It is in combining these elements that the true value of these units is seen. While 82 This is especially helpful for Ferneyhough's notation, where he omits any indication of the highest level tuplet in his "irrational" time-signatures. The reader is expected to perform their own calculations, understanding that meters with denominators of 6, 12, 24, etc. impart a triplet at the highest level, and those based on denominators of 10, 20, etc. impart a quintuplet. The majority of other composers who use such time-signatures will include a partial tuplet bracket, making the comparisons slightly easier (although it is still confusing to equate triplet-dotted notes with their undotted-duple counterparts). 83 Morris, (2006). 3. 46 only the tempo of the beat is necessary,84 by relating it to a specific beat-unit, the transformational process of the music is shown. In the case of the final submetric unit in measure five (8, 2, 140.63, 0.76), it is not enough to simply say that a beat has a tempo of 140.63. This does not account for the precise way in which this beat-unit is attained. Rather than through simple notation of q = 140.63 (which would be an impossible tempo for a performer to find), the beat transformation occurs first through the irrational time signature of 3 , then again transforming the entire bar with a 5:3 tuplet, and finally by altering this 2 specific unit with a 5:4 tuplet. This is not a meaningless bit of arithmetic, but a crucial process in arriving at this tempo, both in the process of the music and the practicality of performance. Moreover, observe the third submetric unit (4, 2, 135, 0.89). In this unit, the tempo of the sixteenth note is slower than the eighth-note in the final unit. Through combining beat-unit and tempo, these submetric ordered quadruples show not only the different speeds of each submeter, but also their larger relationship and transformations. The concept of beat-length transformation is key to understanding how Ferneyhough's music develops. It is worth revisiting how Ferneyhough prefaces Superscriptio: "Bar lines should invariably be regarded as marking instantaneous alterations of beatlength."85 Putting aside the requirement that these shifts occur at bar lines,86 Ferneyhough 84 Dimension x could be completely removed and still allow exact construction of impulse-rate simply by implying a tempo change to a given note-value at each new submetric unit. Such a construction, however, would not impart the notion of changing beat-unit, but rather project beatlength alteration. Moreover, it would greatly complicate performance, replacing easily understood triplet- and dotted-eighth-note beat-units with tempo changes of e = 81 and e = 72, respectively. 85 Ferneyhough, Superscriptio, iv. 86 In Superscriptio, there are fewer submeters per bar, and often each bar acts as its own submeter. Applying this quote to Unsichtbare Farben would entail regarding the majority of tuplets (in addition to bar lines) as instantaneous alterations of beat-length. (There is no preface included with the score for Unsicthbare Farben, and the prefatory materials available through Edition Peters online catalogue only speak to the inspiration for the music, not the execution of performance.) ibid. 47 does not refer to a change in the overall tempo, but in the length of whatever beat-unit is taken for that measure. There is an inherently acknowledged difference between a beat-unit of a sixteenth note and one of an eighth, even if the eighth-note unit is altered in such a way as to make it faster than the sixteenth note. 87 In discussing the process of learning Ferneyhough's solo percussion work Bone Alphabet, Steven Shick relayed Fereneyhough's aversion to performers simply treating every poly-rhythm and tuplet as a tempo change. 88 While both Shick89 and Irvine Arditti90 (for whom Unsichtbare Farben was written) have used implied tempo changes as rehearsal and learning techniques, 91 Shick is very quick to assert that the notated tempo must be understood by the performer, and the beat-length altered in proportional relation. To simply treat each submetric shift as a new tempo of an arbitrary beat-unit would, according to both Shick and Ferneyhough, "imply a reorientation of the overall metric point of view." Rather, by understanding each submetric unit as an alteration of beat and beat-length, the fundamental difference between "change of speed and change of meter" is realized.92 87 This concept even exists in more traditional music, when a triplet eighth is further subdivided into triplet sixteenths inside the triplet eighths. These triplet-in-triplet sixteenths are, in fact, shorter in duration than thirty-second notes, yet are intuitively obvious as subdivision of an altered eighth. 88 Steven Shick, "Developing and Interpretive Context: Learning Brian Ferneyhough's Bone Alphabet," Perspective of New Music 32 (1994): 139. 89 ibid. 90 Paul Archbold, "Performing Complexity: a pedagoical resource tracing the Arditti Quartet's preparations for the première of Brian Ferneyhough's Sixth String Quartet," (2011), http:// events.sas.ac.uk/uploads/media/Arditti_Ferneyhough_project_documentation.pdf. 91 Fitch, Ferneyhough, 40. (see picture of Arditti's part for Ferneyhough's String Quartet № 3). 92 Shick, Interpretive Context, 139-41. 48 Analysis of the First Section Using Submetric Ordered Quadruples The first section of Unsichtbare Farben comprises the opening 22 measures. Although sectional divisions are rather weak and somewhat blurred, there is a clear understanding of closure in measure 22 (Figure 3.6). The extremely high and quiet sounds contrast with the violent mid-to-low range material presented for the majority of the opening, while the speed of the material greatly slows down. Also notable, this is the first measure (and one of very few in the entire work) that includes no microtonal pitch material. 93 The combination of register, dynamic, speed, and pitch contrast clearly demarcate this measure as a sectional division. Taking these first 22 measures as the opening section, the submetric ordered quadruples will be analyzed for distribution frequency of beat-unit and beat subdivision, and used to graph tempo changes.94 Below is the coding for the 73 submetric units presented in the first 22 measures. 95 <xunit=64> <zbase = 8:54> (6, 2, 72, .83); (8, 2, 81, 1.33); (4, 2, 135, .89); (2, 2, 216, 1.53); (2, 2, 216, 2.18); (8, 3, 81, .74); (12, 2, 63, .95); (4, 4, 108, 1.80); (6, 2, 108, .56); (6, 3, 108, .56); (4, 3, 162, 1.48); (8, 4, 81, 1.01); (4, 3, 225, .53); (8, 2, 140.63, .76); (4, 2, 202.5, .64); (4, 4, 135, .54); (7.5, 5, 86.4, .82); (4, 2, 162, .65); (2, 2, 324, .37); (8, 5, 121.5, 1.23); (3, 3, 324, 1.05); (4, 2, 108, .56); (3, 2, 144, .42); (4, 2, 108, .56); 93 Aaron J. Kirschner, "New Applications (and Operations) for Hyper-Transformations in K-Net Theory," (presentation given at the University of Utah Salt Lake City, UT, March 20, 2015). 94 The probability is calculated using a proprietary algorithm written in Python by R. Alexander Schumacher and the author, 2014-15. 95 This coding is, in practice, placed in a raw text file (.txt) for computer analysis. The parentheses and semicolons delimit each submetric ordered quadruple. Angled brackets contain any information not directly related to the ordered quadruples. 49 (2, 2, 216, .83); (3, 2, 144, .42); <zbase = 8:47.25> (10, 5, 37.8, 2.70); (2, 2, 189, 1.11); (8, 4, 77.96, 1.64); (6, 3, 103.95, 1.15); (4, 4, 155.92, 1.49); (4, 2, 137.81, 1.94); (4, 4, 94.5, .63); (8, 2, 70.88, .85); (2, 2, 189, .63); (4, 2, 94.5, .63); (8, 2, 88.59, .67); (4, 4, 141.75, 1.65); (4, 4, 157.50, .63); (4, 2, 196.88, .51); (4, 2, 248.06, .24); (6, 2, 110.15, .45); (12, 3, 59.06, 1.51); (8, 4, 115.17, 1.64); (8, 4, 93.52, 1.12); (6, 2, 266, .38); (9, 3, 66.5, 1.17); (4, 2, 224.44, .33); (8, 2, 70.88, 4.23); (2, 2, 413.44, .22); (4, 2, 241.17, .31); (4, 4, 241.17, .56); (2, 3, 275.63, .22); (4, 4, 122.85, 3.42); (12, 2, 103.91, 1.35); (5, 5, 124.69, .96); (4, 4, 170.1, 1.15); (4, 4, 113.4, .53); (2, 2, 255.15, 1.06); (4, 4, 155.93, 1.35); (6, 3, 104.5, 1.72) <end 1st page> (4, 4, 157.5, .76); (2, 2, 275.63, .44); (2, 3, 275.63, .22); (2, 2, 275.63, .44); (4, 2, 226.8, 1.23); (3, 3, 567, .25); (6, 3, 283.5, .56); (4, 2, 425.25, .29); (8, 2, 198.45, .45); (8, 3, 113.4, 1.06); (32, 2, 55.37, 1.63); (3, 3, 369.14, 1.63); <double bar mm 23, end 1st section>96 In the first five measures, it was observed-despite the notational complexity-a limited amount of projected beat-units. Figure 3.7 shows the beat-units used in this section, ordered by rate of occurrence.97 While the opening section contains more beat-units than only thirty-second through dotted-eighth-notes, the total number is significantly more manageable than the notation might suggest. The beat-units projected in the first five measures comprise the vast majority of the beat-units for this section, with the dotted-thirtysecond also becoming a more common structure. More significantly, the non-dotted sixteenth, thirty-second, and eighth values (in order of occurrence) clearly are the main 96 Kirschner, "Four-Dimensional Modeling." 97 The percentages (in this and all subsequent examples) are rounded to the nearest basis point (hundredth of a percent), and thus sum to slightly over 100%. There are 73 total submetric units; each percentage is an approximation of 73n . 50 structures, comprising a super-majority of the beat-units. The dotted versions of each of these notes comprise most of the remaining beat-units, with only 6.85% of the values not being a thirty-second, sixteenth, or eighth-note or their dotted variations. More importantly, of this anomalous 6.85%, each value has only one occurrence. There is even less variation in the subdivisions of beat-unit. Only four subdivisions are used in the opening section, dividing the beat-unit into two to five equal parts. 98 Duple subdivision is by far the most common, itself comprising a majority (52.10%) of the total subdivisions. Quadruple subdivision occurs in another 20.55% of the submetric units, showing a clear preponderance of duple/quadruple subdivisions. Virtually all of the remaining subdivisions are triple (21.92%), with quintuple subdivision occurring in only 5.48% of submetric units. Such an ordering of subdivisions accurately reflects the relative occurrence in most music, with duple and triple being by far the most common, quadruple (understood not simply as a hyper-meter of duple) following in prevalence, and true quintuple subdivision (i.e., not 2+3 or 3+2) occurring much less frequently.99 Two very valuable observations emerge from this data. First, duple (or quadruple) subdivision of non-dotted beat-units is clearly the most prevalent category of submeter. This reinforces the notion that tuplets and irrational time signatures act to alter beat-length, rather than completely change the meter. Once again, this harkens to Shick's assertion that there is a fundamental difference (in Fernyhough's music) between a change in meter and a change in speed. 100 This also underlines a fundamental difference between Fernyhough's music and music of other composers of the "New Complexity." Whereas many other 98 Beat-Units subdivided into 2:1 or 1:2 IOIs are considered triple subdivisions for the purpose of this analysis. 99 London, Hearing in Time, 109-115. 100 Shick, Interpretive Context, 131. 51 composers layer tuplets to articulate shifting groupings of a more static beat, 101 Ferneyhough maintains a rather static set of groupings (and, to a lesser extent, beat-units) while continuously altering the beat-length. 102 The second important observation focuses on how the prevalence of duple/ quadruple subdivision eclipses even that of non-dotted note values. Non-dotted thirtysecond, sixteenth, and eighth-notes comprise 68.7% of the section, yet duple and quadruple subdivisions account for 72.65% of the total subdivisions. Especially factoring in the occasional cases of non-dotted note-value beat-units exhibiting triple subdivisions, it is obvious that numerous dotted-note beat-units-which would typically imply triple subdivision-exhibit duple/quadruple subdivision. This establishes duple and quadruple subdivisions (and, specifically, duple subdivision alone) as the single most consistent dimension of the submetric units in this section. By observing the rate of occurrence of both beat-unit and subdivision, the opening section of Unsichtbare Farben is characterized by the highest level of stasis in the beat subdivision dimension, relative stasis of the beat-unit dimension, and (contrastingly) regular beat-length alterations. Given the patterns in beat-unit and subdivision occurrence, it is a natural step to look for patterns in tempo. This, however, is a more difficult task from a cognitive perspective. While the small number of categories of beat-unit and subdivision make those dimensions easy to relate over longer periods of time, remembering a specific beat-per- 101 Jason Eckardt, Paths of Resistance (New York: Carl Fischer, 1995), i. 102 Archbold, (Performing Complexity), 4-6. 52 minute over the course of a section would appear much more difficult.103 For instance, while a tempo of 81 beats-per-minute occurs three times in the first twelve submetric units, it is unlikely to be as readily recalled at each return as a duple subdivision would be. Moreover, the interspersing of other similar tempi (i.e., 72 and 108 beats-per-minute) makes it more likely for it only to be understood as something slower or faster, by various degrees, from what comes before and after.104 Rather than search for regularly occurring specific tempi, it is worthwhile to look at the total trajectory of the tempo over the course of the section. Figure 3.8 graphs the tempo of each submetric in the first section (each point along the x-axis is ordinal, not proportional to the submetric unit's duration). The most obvious aspect of the tempo graph is the contour and widening of the range. In the most general sense, the graph shows the contrast between the small expansion of the lowest value (72-37.8) against the large expansion of the uppermost values (72-567). Because of the exponential nature of tempo, percentage relations are more indicative of categorical relationships than absolute beats-per-minute. The slowest tempo is 52.5% of the opening, while the largest is 787.5% of the opening. The large upward expansion against minimal downward expansion mimics the register of the pitch-material. Beginning on B3, the violin has little additional downward range, contrasted with the significant expansion of the upper register (all the way to Bf7 as a harmonic, Fs7 as a fundamental pitch). 103 While previous experiments have shown that humans can memorize the "correct" tempo of particular melodies to within 2%, this requires great familiarity and repetition of the associated melody (i.e., listeners could identify the correct tempo of the default Nokia ring tone only after Nokia's market penetration had become such that the average listener would regularly encounter such a melody). In the context of a specific piece, it is highly unlikely that a listener would remember a tempo that existed for less than five seconds, especially when the associated melodic material changes at each occurrence. Poudrier and Repp, Coding Efficiency, 384-386. 104 ten Hoopen, et al., One-to-One, 1. 53 While there is an observable pattern to the registral expansion, the complete plot of the tempo is resistant to a highly correlated trend line, based on beats-per-minute (z) as a function of ordinal position of submetric unit (s). The linear function with the highest correlation to the plot is z(s ) = 1.7368s + 105.34 , but the coefficient of determination is only R 2 = 0.1465 . The coefficient of determination can be improved slightly (to R 2 = 0.1856 ) by assigning the quadratic function z(s ) = 0.0476s 2 − 1.7858s + 149.37 . Increasing the polynomial order to six only improves the coefficient of determination to R 2 = 0.2183   while  modeling the tempo plot with the extremely awkward function z(s ) = −(1.245 ⋅10 −7 )s 6 + (2.601⋅10 −5 )s 5 − 0.0021s 4 + 0.0891s 3 − 1.9837s 2 − 21.247s + 68.809 . Between the low coefficient of determination and the extremely low multiples of the higher order powers of s, it can be concluded that tempo is not a function of ordinal position of submetric unit.105 The plot of the tempi in the first section has an observable contour pattern, but there is no function of ordinal position that accurately correlates to tempo. In effect, the beat-length alterations are shown to be observable patterns in noise. Transitional Probabilities of Dimensions x and y In total, there are 469 submetric units in Unsichtbare Farben. The complete coding is reproduced in Appendix B. From these 469 data points, focus will be placed on the x and y dimensions; specifically, the transitional probabilities between each. 106 The rate of 105 Logarithmic, exponential, and power functions impart even lower correlations. Logarithmic: z(s ) = 32.437ln(s ) + 61.508 ; R 2 = 0.0945 Exponential: z(s ) = 108.88e 0.0082 s ; R 2 = 0.106 Power: z(s ) = 85.473s 0.1634 ; R 2 = 0.0783 106 The lack of a meaningful regression in the plot of dimension z continues. The coefficient of determination of the most appropriate sixth order polynomial function actually decreases to R 2 = 0.043 when the entire list of tempi are plotted. This furthers the conclusion that tempo is, mathematically speaking, noise. 54 occurrence for beat-units and subdivisions is largely consistent with the projections in the first section. Figures 3.9 a&b present graphs showing the occurrence of each beat-unit and subdivision. Beat-Units primarily consist of sixteenth notes, with eighth, thirty-second, dotted-thirty second, and dotted-sixteenth notes comprising meaningful minorities. Two is the primary subdivision; three and four are about equally likely, and five is a significant minority. The only meaningful differences when comparing the frequencies observed in the opening section are the significantly higher occurrence of eighth-note beat-units relative to thirty-second note, and the almost equal amount of dotted-sixteenth and thirty-second notes. In the first section, thirty-second notes are the second most common beat-unit, 107 yet they are only the fourth most common across the entire piece. The thirty-second note is now clustered with its dotted counterpart, as well as the dotted-sixteenth, dotted-eighth, and quarter notes. Moreover, there is a large chasm between the eighth-note and all other units. Thus, the beat-units can be divided into four distinct tiers. The top two tiers each contain only a single note: the topmost tier contains the sixteenth note, and the second to top contains the eighth. Combined, these tiers alone contain 51% of the beat-units in Unsichtbare Farben. The third tier groups the thirty-second, dotted-thirty-second, dottedsixteenth, dotted-eighth, and quarter notes (making up 41% of total beat-units). The remaining 8% of the beat-units are considered the lowest tier. While categorizations can be easily made from the clustering of instances of occurrence, there is also the possibility of categorization based on transitional probability. 108 107 Kirschner, "Four-Dimensional Modeling." 108 Ian Quinn and Christopher Wm. White, "Expanding Notions of Harmonic Function Through a Corpus Analysis of the Bach Chorales," (paper presented at the Society for Music Theory Annual Meeting Charlotte, NC, November 2, 2013). 55 To this end, each significant 109 beat-unit and subdivision will be examined for the probability of the beat-unit and subdivision following. Specifically, if submetric units are treated as a Markov Chain, are there values (n) of x and/or y for which x i = n impacts the probability of the value of x i +1 and/or y i +1 ?110 111 To begin, each subdivision from two through five will be analyzed based on the transitional probability of the next subdivision. As these are treated as first-order Markov Chains, only the current value of y is considered; prior values are not relevant. Figures 3.10  a-d present pie charts showing the transitional probability of each subdivision (i.e., given y i = n , the pie charts show the probable values of y i +1 ). The transitional probabilities of subdivision based on subdivision directly mirror the total distribution of subdivisions. This is a telling statistic, as it indicates that there is little to 109 The top three tiers of beat-units ( x = 2,3, 4,6,8,12,16 ), and subdivisions of two through five ( y = 2,3, 4,5 ). 110 Transitional probability is defined as the rate of occurrence for each possible value of x i +1 given x i = n . For example, take the number series {1, 3, 2, 1, 5, 7, 4, 7, 3, 7, 2, 1, 3, 2, 1, 4, 3, 4, 6, 5, 4, 2, 1, 3}. While this series is apparently random (and the majority of it is derived from a randomnumber generator), there are three transitional patterns. First, no number is immediately repeated; there is a 100% transitional probability given x i = n that x i +1 ≠ n . The other two patterns relate to specific numbers. The number 2 is always followed by 1, and 1 is followed by a number greater than two. Thus, given x i = 2 , then x i +1 = 1 . And, given x i = 1 , then x i +1 ≥ 3 . Moreover, each of these patterns relies only on the value of x i ; as a first-order Markov chain, the value of x i −1 is not needed-nor advantageous-to predict the value of x i +1 . 111 The transitional probabilities were calculated using the proprietary program written by Kirschner and Schumacher. All transitional probabilities output by the program were then checked by hand. Two vectors of cardinality 469, one consisting of the ordinal x values and one of y values, were constructed. From these vectors, four 468×2 matrices were created such that all permutations of the first column being the original x or y vector (excluding the final value) and the second being the last 468 values (i.e., excluding the opening value of x or y) were explored. The matrices were then rearranged such that the values in the first column progressed in ascending order (maintaining their associated value in the second column). For each discreet, significant value in the first column, the frequency distribution of all associated second column values was tabulated. Second column values that had only a single association with a discreet first column value were tabulated together as singletons. 56 no correlation between the value of y i and the probability of a given value of y i +1 . While probability of y i +1 = 2 is consistently the highest, this is simply because there are more total values of y = 2 , rather than indicating any impact of y i . There is a slight correlation between the higher the subdivision and a reduced probability of y i +1 = 2 (culminating with an equal probability of y i +1 = 2,3 when y i = 5 ), but as the total number of data points decreases for each successively greater subdivision, this is likely to be statistically insignificant. Thus, there is no observable impact of subdivision of one submetric unit on the submetric unit that follows. The next consideration will be given to the impact that subdivision has on the following beat-unit. Given y i = n , what are the probable values of x i +1 and do these probabilities impart a significant pattern? Figures 3.11 a-d present the probable beat-units following a submetric unit of a given subdivision. In this case, there are more meaningful observations than subdivision to subdivision. While the large number of sixteenth note beatunits imparts a high likelihood of a sixteenth note beat-unit following any submetric unit, there is a clear negative correlation between a triple or quintuple subdivision and a sixteenth note beat-unit following. Sixteenth note beat-units are almost twice as common as eighthnotes (and about 32% of the total beat-units), yet they are equally (or even slightly less) common than eighth-notes following triple or quintuple subdivision. Thus, duple and quadruple subdivision are characterized by their primary transitions being to the sixteenth note (the most common of beat-units), while triple and quintuple subdivisions are characterized by an increased likelihood of transition to the eighth-note (the second most common beat-unit) and an increased avoidance of the sixteenth-note. When beat-unit is taken as the marker for transitional probability, there is minimal impact on the probability of the following subdivision. On the whole, the probabilities of 57 the subdivisions following any given beat-unit are roughly equivalent to the general distribution of subdivisions. All but two of the beat-units impart an approximately 50% probability of being followed by a duple subdivision (which has a 46% general occurrence rate). The two that do not are the thirty-second note (40%) and the eighth-note (43%), yet transition to duple subdivision is still the clear plurality in both cases. More interestingly, each of these impart a different second most likely transitional probability, the thirty-second note moving to quadruple subdivision in 33% of occurrences and the eighth-note transitioning to triple subdivision in 31% of occurrences. The transition from the thirty-second note to quadruple subdivision is most interesting in that it would also imply-given how infrequent one-hundred-twenty-eighth-notes appear-a shift to a larger beat-unit. Figures 3.12 a-f summarize the transitional probabilities for each beat-unit not in the lowest tier. The transition to quadruple subdivision from the thirty-second note beat-unit is a clue that the beat-unit may also affect the transitional probability to the following beat-unit. As there are so few one-hundred-twenty-eighth-notes, a thirty-second note transitioning to a quadruple subdivision would require a corresponding transition to a larger beat-unit. This is borne out in the data, as the thirty-second note transitions to the sixteenth note in 49% of its occurrences (the expected probability from the general distribution is 32%). When all beat-units of sixteenth note and above are considered, the transitional probability jumps to 82%. The remaining 18% of the time, the thirty-second note transitions to itself or its dotted counterpart with equal frequency. When all transitional probabilities from one beat-unit to the next are considered, three categories naturally emerge based on the length of the beat-unit (in sixty-fourth notes). The thirty-second and dotted-thirty-second note fall into the category of "less-thanfour" (<4), the sixteenth note comprises its own category of "four" (4), and all larger values 58 are categorized as "greater-than-four" (>4).112 The "less-than-four" category is characterized by its strong tendency (about 50%) to move to "four," with the remaining transitions equally split between transitioning to itself and transitioning to "greater-than-four." The sixteenth note ("four") is similarly categorized by its likelihood to transition to longer note values. When it does not transition upward, it is slightly more likely to transition to itself than downward to the "less-than-four" category. The "greater-than-four" category is strongly correlated with transitioning to itself, with occasional shifts to "four" and incredibly rare transitions to "less-than-four." These categories impart a certain function to each beat-unit, directing each towards a specific class of unit for the next. 113 While sixteenth notes are by far the most common single beat-unit, these categories show that there is a stronger pull towards note values greater than four. As the note values increase, there is a stronger tendency for the category to transition to itself, and the two lower categories both exhibit tendencies to transition upward. Thus, there is a tension between the most common beat-unit (the sixteenth) and the regular pull for the beat-unit to transition to beat-units larger than a sixteenth. Figure 3.13 shows the categories and how they transition and expand upon themselves. A thicker line indicates a higher likelihood of transition. Figure 3.14 shows the pie chart for category "four," Figure 3.15 a&b show the "less-than-four" category, and Figures 3.16 a-d show the "greater-than-four" category. 112 The dotted sixteenth note is a slightly awkward case, as it is approached as if it were >4, but its transitional probability is closer to the 4 category. In this analysis, it is grouped as >4, but it could also be considered a subcategory of 4. 113 Quinn & White, "Expanding Notions." 59 e = 54 4 & 8® n >œ ™ µ >œ ™ n>œ 3 Bœ > nœ œ n œ B >œ n>œ B œ n>œ > > > 5 µ>œ Figure 3.1: Measure 1 of Unsichtbare Farben 2 2 & 8 œ ™ n œ B œ nœ n œ > > > > > 3 3 8 nœ n >œ >œ >œ >œ >œ > Figure 3.2: Measure 2 of Unsichtbare Farben 3 3 7 3 5 12 & 8 nœ nœ nœ n>œ ™ Ù B œ ™ nœ œ œ œ ™ µ œ B œ ™ > > > > > 3 Figure 3.3: Measure 3 of Unsichtbare Farben 4 3 5 3 20 & 12 n œ ™ Ù µ œ ™ ™ µ>œ B œ n œ B œ B >œ >œ µ œ n œ  œ > > n >œ > > >œ >œ > Figure 3.4: Measure 4 of Unsichtbare Farben 2 8 œ™ > Opening section (mm 1-22) of Unsictbare Farben, coded into ordered quadruples 5 3 & 20 5:3 3 , 60 3 Bœ ® n œ œ n >œ > > µ œ- B œ n œ 5:4 œ Figure 3.5: Measure 5 of Unsichtbare Farben 15:8 15:12 . bb Oœ ^ nn Oœ Violin 15:12 . . # œ n œ ™ . nœ . ™b œ. ™ b œ . bb Oœ . . . . . . . 4 & 10 ≈ ≈ ≈ ≈ ≈ Œ sub. pppp Single-dimensional analytical possibilities Figure 3.6: Measure 22 of Unsichtbare Farben. End of first section. 2 & Perceived ∑ beat units∑ (x), by probability ∑ Beat sub-divisions ∑ (y), by probability x Kr q 38.56% 2! 52.10% 16.44% 3! 21.92% 13.70% 4! 20.55% x. Kr q. 10.96% 5! 5.48% e e. ∑ 6.85% 6.85% All others 6.85% (one occurrence each) Figure 3.7: Beat-Unit Probability (dimension x) the tempo of the perceived beat unit (i.e. value of dimension z) through the first section earch possibilities currently being explored include relationships between dimensions, ∑ 61 600 500 400 300 200 100 0 Figure 3.8: Tempo (in beats-per-minute) of Beat-Units in the First Section 62 5 7% Others 2% 4 19% 2 51% 3 20% Figure 3.10a: Transitional Probability of y from y=2 63 Others 5 2% 3% 4 22% 2 47% 3 27% Figure 3.10b: Transitional Probability of y from y=3 64 5 8% Others 1% 4 20% 2 42% 3 28% Figure 3.10c: Transitional Probability of y from y=4 65 5 9% Others 3% 2 34% 4 19% 3 34% Figure 3.10d: Transitional Probability of y from y=5 66 16 10% Others 6% 2 11% 3 6% 12 5% 8 17% 4 34% 6 12% Figure 3.11a: Transitional Probability of x from y=2 67 Others 2 8% 2% 16 4% 3 15% 12 8% 4 25% 8 27% 6 11% Figure 3.11b: Transitional Probability of x from y=3 68 Others 2 16 4% 7% 7% 12 3% 3 9% 8 15% 6 8% 4 45% Figure 3.11c: Transitional Probability of x from y=4 69 Others 8% 2 12% 16 15% 3 8% 12 4% 4 23% 8 27% 6 4% Figure 3.11d: Transitional Probability of x from y=5 70 5 Others 5% 2% 2 40% 4 33% 3 21% Figure 3.12a: Transitional Probability of y from x=2 2 51% 3 20% 2 47% 2 47% Figure 3.12b: Transitional Probability of y from x=4 3 24% 4 22% Others 5 5%1% Figure 3.12b: Transitional Probability of y from x=6 4 24% 5Others 4%4% Figure 3.12b: Transitional Probability of y from x=3 3 26% 4 13% 5 10% 71 3 31% 2 43% Others 3% 3 19% 5 16% 5Others 4%4% 2 50% 2 49% Figure 3.12f: Transitional Probability of y from x=12 3 23% 4 19% Figure 3.12g: Transitional Probability of y from x=16 4 14% Figure 3.12e: Transitional Probability of y from x=8 4 17% 5Others 7% 2% 72 73 4 <4 Figure 3.13: Three Categories of Beat-Unit by Transitional Function >4 Figure 3.15a: Transitional Probability to x from x=2 Figure 3.15b: Transitional Probability to x from x=3 Figure 3.14: Transitional Probability to x from x=4 74 Figure 3.16b: Transitional Probability to x from x=8 Figure 3.16c: Transitional Probability to x from x=12 Figure 3.16a: Transitional Probability to x from x=6 75 CHAPTER 4 AFTERWORD: SUBMETER AS PITCH Through the process of partitioning Unsichtbare Farben into submetric units and calculating their tempi, a rather curious pattern emerged. The vast majority of the tuplets encountered are ratios of very low numbers, often only single digits. This is not, in and of itself, unusual. However, even the more complex beat-length alterations were created not by tuplets of larger-order ratios (c.f., Ferneyhough's In Nomine a 3), but though combining many tuplets of low-number ratios.114 These ratios (i.e., 3:2, 5:4, 7:4, 5:3, etc.) are immediately recognizable as the frequency ratios of intervals in just intonation. Thus, the alteration of a beat-unit of an eighth-note placed inside two quintuplets in a 3 12 measure is analogous to the pitch transformation of transposition by two major thirds (5:4) and a perfect fifth (3:2). 115 To show the potential of this conversion, two brief analyses will be presented. The first will chart the relationships and transformations between the notated tempo changes (i.e., changes in z(base ) ). This will provide a long-range view of the role of beat-length alteration within the larger metric orientation. 116 The second analysis will focus on the first 114 Fitch, Ferneyhough, 141. 115 Mead, Tempo Relations, 64-67. 116 Shick, Interpretive Context, 131. 77 section (through measure 22), and will notate each submetric unit as a pitch. The resulting pitch-collection will then be subjected to Robert Morris's contour smoothing algorithm, and compared to the pitch material of the closing measure of the first section. 117 In both cases, e = 54 will be treated as middle C. Notated Tempo Changes as Pitch Transformations The notated tempo changes in Unsichtbare Farben are not as frequent as beat-length alteration by tuplet, but still occur regularly. To compound the matter, rather than being based on traditional metronome markings, all but e = 54 are decimal fractions (in fourths).118 The hyper-specificity of these metronome markings invites the assumption that they are somehow based on specific ratios to the opening e = 54. 119 Besides e  =  54, there are six additional tempo markings used in Unsichtbare Farben: (from slowest to fastest) e  =  38.5, e = 42.5, e = 47.25, e = 60.75, e = 67.5, e = 74.25. When related to e = 54, the four largest of these tempo reduce to ratios of small numbers, and all ratios have analogues in the pitch world. 120 The two simplest are e  =  60.75 and e  =  67.5. Beginning with e  =  60.75, the ratio reduces as 60.75 54 = 89 . This ratio ( 89 ) is analogous to a the frequency ratio of two pitches a just major second apart; taking e = 54 as C, e = 60.75 becomes D. The tempo of e = 67.5 also 117 Mustafa Bor, "Contour Reduction Algorithms: A Theory of Pitch and Durational Hierarchies for Post-Tonal Music" (Ph.D. diss., The University of British Columbia, 2009), 29. 118 In Elliot Carter's music, metronome markings requiring decimal values are occasionally encountered when complex metric modulations occur. Indeed, there is a strong correlation between Ferneyhough's beat-length alteration and Carter's metric modulation. 119 David Lewin, Generalized Musical Intervals and Transformations (New York: Oxford University Press, 1980), 67-9. 120 Kyle Gann, "Anatomy of an Octave," Just Intonation Explained, (2008), accessed March 24, 2015 http://www.kylegann.com/tuning.html. 78 reduces to a regularly encountered frequency ratio; 67.5 54 = 45 . The 5 4 ratio is the frequency ratio of a just major third, resulting is e = 67.5 being treated as E.121 Two other tempi produce ratios of small numbers when divided by 54, but these require microtonal understandings. The tempo e = 47.25 relates to the e = 54 tempo by the ratio 47.25 54 = 7 8 . Doubling this ratio (giving 7 4 ) represents the frequency interval of the harmonic seventh. Thus, e = 47.25 projects the harmonic second below the C represented by e = 54, specifically BD. The tempo of e = 74.25 (which exists only as the starting point of a rallentando) gives a ratio of 74.25 54 = 118 , representing the undecimal (11-limit) tritone. The pitch representation of this tempo is F+. 122 The two remaining tempi do not reduce to ratios of small numbers when divided by e = 54. Both are based on denominators of 108: 38.5 54 77 = 108 and 42.5 54 85 123 = 108 . Neither of these are relationships that are reasonable for a performer to directly understand, but by relating 121 Both of these (and all subsequent) pitch labels assume just intonation to C. The 89 D is about four cents sharp compared to equal temperament; the 45 E is about 14 cents flat compared to equal temperament. Using just intonation for these relationship is significantly easier than using equal temperament, as the small-number ratios are much simpler for the performer to compute. Were these tempi related by equal temperament, the equivalent of D is 54( 6 2 ) and the equivalent of E is 54( 3 2 ) , both of which (by the nature of being based on roots of two) are irrational numbers. As there is no relevant metric modulation, nor can there be the expectation that any performer would have a conception of multiplications by roots of two, the use of just intonation frequency ratios allows the pitch relationships to be more easily understood and performed. 122 These quarter-tones are again approximations. The ratios 47 and 118 refer to the seventh and eleventh harmonic, respectively. The eleventh harmonic fits almost perfectly into 24-tone equal temperament, deviating by less than two cents (within human just noticeable difference) from equal temperament. The seventh harmonic is farther from its 24-tone equal temperament approximation, sounding about eighteen cents above the quarter-tone approximation. For hyperspecificity, Bob Johnston's notation of B7b would be preferable to the BD approximation. 123 108 approximates three tritones: the lower septimal tritone ( 75 ), the low Pythagorean tritone 45 ), and the high 5-limit tritone ( 32 ). Of these, the 75 tritone is probably the most logical tempo/pitch relationship (not withstanding the relationship shown in the main text). It may have simply been a conscious choice by Ferneyhough to simply notate 38.5 to continue the pattern of only writing the tempo in fractions of fourths of a beat-per-minute ( 54( 57 ) = 38.571428 ≈ 38.5 ). Nevertheless, 57 is by far the easiest relationship for the performer to process. Gann, "Anatomy." 77 ( 1024 729 79 these tempi to others in the piece, the large-scale relationships are made clearer. Beginning with e = 42.5, the ratio of the smallest number is between it and e = 67.5, giving The ratio 27 17 67.5 42.5 = 27 17 . almost perfectly approximates (within a cent) the equal tempered minor sixth 27 ( 3 2 2 ≈ 17 ). Therefore, the tempo e = 42.5 can be defined as a minor sixth below e = 67.5. Having established e = 67.5 as E, the minor sixth transformation projects e = 42.5 as Gs/ A f. The slowest tempo can be related to the largest as 74.5 38.5 27 27 = 14 . This ratio ( 14 ) is the septimal (7-limit) major seventh. The "sharpness" of this interval (about 37 cents sharp compared to equal temperament) makes Gf the result of transposing F+ down by a 27 14 major seventh. For clarity, each of these intervals will be assigned a different pitch name and represented in equal-temperament. 124 This results in the whole-tone collection {Gf, Af, Bf, C, D, E} with the F+ acting as a microtonal (neutral second) upper neighbor to E. Figure 4.1 presents the notated tempo changes as pitch transformations. Each half note represents a notated tempo. Glissandi indicate rallentandos and accelerandos, and grace-notes represent tempi that only exist as the start or end of a rallentando or accelerando (i.e., they are not sustained before being subjected to a tempo shift). The figure does not represent the duration of any given tempo notation. Note the tritone descent (C-Gf) over the course of the piece. This is a rather common treatment of the whole-tone collection. However, by using the upper limit and the (approximately) 7 5 tritone for the lower limit (creating a 11 8 27 14 tritone for the major seventh F+ exists only as the start of a rallentando, and is thus kept as a microtone. The Gf, Af, D, and E are all within 14 cents of equal temperament, making the conversion of just intonation to equal temperament trivial. The only major alteration is converting the BD (or B7f) to Bf. This, however, is the notion of considering the flat 7! as a harmonic seventh, a practice not wholly foreign. 124! The 80 rather than an octave), 125 the projected tritone is no longer a symmetrical division of an octave. Rather, the tritone relationship is more indicative of the bottom note of the two distinct whole-tone trichords {Gf, Af, Bf} and {C, D, E}. Thus, the tempi are divided into two groups: e = 38.5, e = 42.5, e = 47.25; and e = 54, e = 60.75, e = 67.5. That e = 74.25 exists only as the start of a rallentando further demonstrates its removal from the wholetone relationship. Submetric Beat-Length Alterations as Pitch Transformations The second analysis of submeter as pitch will look at beat-length alterations as pitch transformations in the first section of Unsichtbare Farben. The analysis will begin by demonstrating the methodology of converting the submetric units in the first five measures (those discussed in the opening of Chapter 3) to pitch material. The entirety of the opening section will then be presented with submetric units converted to pitches. The pitches in this section will then be processed via Robert Morris's contour smoothing algorithm, and the prime contour layer will be compared to the pitch material in the final measure of the section. To convert a submetric unit to a pitch, it is necessary to return to the calculation of that unit's z dimension, and to the assignment of intervalic transformations to beat-length alterations. The notated tempo is the pitch to be transformed, and the multiplicands (both from tuplets and from change in beat-unit) in the calculation of the submetric unit's tempo are the pitch transformations. For example, consider the notated tempo of e  =  54 and a 125 Were the fastest notated tempo simply twice the slowest, it would give e = 77. The frequency ratio 77 of 54 approximates (to within four cents) the 107 tritone (the octave about the 75 lower tritone implied by the e  =  38.5). Moreover, the ratio of 107 is, for all intents and purposes, as easy to process as the 118 ratio projected by e  =  74.25, meaning the lack of octave duplication is not simply for practicality of performance. 81 beat-unit of a sixteenth inside a 7:4 tuplet. The relevant calculation for the tempo is z = 54(2)( 47 ) . For this analysis, e = 54 has been established as middle C.126 The multiplicands in the tempo calculation are 2 (for the beat-unit transformation from eighth to sixteenth) and 7 4 (for the 7:4 tuplet). These result in transposing the C4 (e = 54) up an octave (2) and a minor seventh ( 47 ), 127 resulting in Bf5. 128 There are seventeen intervals implied by the submetric units in the first section; Table 4.1 summarizes these ratios. Using these ratios, the calculations for the tempo of the submetric units in the first five measures can be revisited and used to convert them into pitch material. The relevant z calculations in measure one are z = 54( 86 ) = 54( 43 ) , z = 54( 32 ) , z = 54(2)( 45 ) , and z = 54(4). Each of these is based on the notated e  =  54, so C4 is the pitch to be transformed. 129 The first submetric unit is altered by the 4 3 ratio (a perfect fourth), giving F4. The second is taken up a fifth ( 32 ), resulting in G4. The third is transposes C4 up an octave plus a major third, producing E5. 130 The last submetric unit simply transposes the pitch up two octaves to C6. As this is the same tempo as the submetric unit in measure two, that unit 126 This is simply for convenience in this particular analysis. The assignment of some tempo to middle C is required for any submeter as pitch analysis is required, and is usually an obvious decision. 127 Henry Cowell, New Musical Resources (Cambridge: Cambridge University Press, 1930), 68-73. 128 This analysis will use 24-tone equal temperament, with the harmonic seventh tempered up to the 12 10 2 minor seventh. 129 This will hold for all of the examples shown in this section. In measure nine, the tempo changes to e = 47.25. Each of those pitches are calculated by transforming Bb3. 130 In this case, it would be equally simple to first multiply the fractions (giving 2 ⋅ 45 = 52 ) and understand the transformation as an octave plus a fifth up. However, in many cases, it is simpler is an almost impossible to treat the multiplicands as successive transformations. For example, 245 64 frequency ratio to understand as an interval (it is the 245th harmonic), but when represented as 7 7 5 ⋅ ⋅ it is clearly understood as two minor sevenths and a major third. 4 4 4 82 is also represented as C6. 131 The equations for tempo in the third measure are as follows: z = 54( 32 ) , z = 54( 23 )( 47 ) , z = 54(2) = 108 . The first of these transposes up a fifth ( 32 ) to G4. This is followed by a transposition down a fourth ( 23 ) then up a minor seventh ( 47 ), resulting in F4. Finally, the doubling of the tempo results in an octave transformation to C5. Measure four uses the following equations to calculate tempo: z = 54( 43 )( 32 ) = 108 , z = 54( 43 )( 32 ) = 108 , z = 54(2)( 32 ) = 162 , z = 54( 32 ) = 81 . The first two units project an upward transposition of a perfect fourth plus a perfect fifth, combining to an octave up to C5. The third submetric unit transforms the pitch up an octave plus a perfect fifth, resulting in G5. 132 The last unit simply transposes up a fifth to G5. The tempi of the two submetric units in the fifth measure are calculated as z = 54(2)( 53 )( 45 ) and z = 54( 45 )( 45 )( 53 ) . The first of these transforms C4 up an octave plus a major sixth plus a major third, resulting in Cs6. The second unit replaces the octave transfer with another major third (in total two major thirds plus a major sixth), giving Es5. In whole, the pitch material projected by the submetric units in the first five measures is {F4, G5, E5, C6, C6, G4, F4, C5, C5, G5, Cs6, Es5}. This methodology is repeated for each submetric unit in the first section. Figure 4.2 shows the pitch material of the submetric units, with the durations represented rhythmically. A cross note-head (X) indicates an "out-of-submeter" pick-up. Observe how the pitch material projected by the submetric units becomes more 131 When shown graphically, such repetition of tempo (i.e., pitch) between two submetric units should be shown as a "rearticulation." This clearly shows a submetric shift, but the beat-unit transformation is arrived at differently and/or the subdivision is different. 132 By separating the multiplicands, this examples demonstrates the somewhat unintuitive nature of just intonation ratios, whereby (in pitch-class space) 12 = 12 and 12 = 41 , but 13 = 32 and 13 ≠ 41 . 83 complex as the section progresses. The opening projection of {F,  G,  E} rotates and transposes the plain chant {B,  D, C} embedded as the only non-microtonal pitches within the opening melody (Figure 4.3).133 The first four measures project only the pitch-classes C, E, F, and G. Gradually, more pitches are added in the second system, but within the first notated tempo, there are no microtonal implications. The pitch-class set for the first tempo is {C, Cs, D, E, F, G, Gs, A, B}. The chromatic infiltration of the Cs and Gs into the otherwise diatonic collection mirrors the microtonal infiltration of the plain-chant melody. After the tempo change to e  =  47.25, microtones begin to infiltrate the projected pitch-material of the submeter. The microtones reflect the use of more complex tuplet relationships. As ratios such as 11:8 and 7:6 begin to appear, the pitch analogue can no longer remain in the chromatic space and must move into 24-tone equal-temperament. Thus, an increase in the amount of microtonal pitches indicates an increase in the complexity of beat-length alterations. 134 Figure 4.4 shows the prime contour layer of the submeter as pitch graph. This is found by applying Robert Morris's contour smoothing algorithm until the lowest possible level is reached. 135 Appendix C contains graphs for each layer. The contour reduction algorithm reduces the pitch material to only four points and three pitch classes {F, Fs, A}, with the highest pitch-class (Fs) doubled in octaves 6 and 7. Compare this pitch-class set with the pitches in measure 22 (the end of the first section [Figure 4.5]). The final pitch (and highest non-harmonic pitch) of the section is Fs7. In the 133 Fitch, Ferneyhough, 90. 134 Mead, Tempo Relations, 68 135 Morris's contour reduction algorithm filters out all contour values that are not peaks or valleys, then repeats the process for peaks-within-peaks and valleys-within-valleys until all points are local minimums and maximums both within the overall line, and the line of local minimums or maximums. Bor, Contour, 29-31. 84 prime layer of the submeter as pitch, Fs serves as the highest pitch (Fs7) as well as the closing pitch (Fs6). Thus, the conclusion of the section projects a symmetry of pitch and submeter. In addition to the symmetry of ending pitches, the trichord from the prime layer can be compared to the three trichords of the closing measure. While the opening melody embeds {B, D, C}, measure 22 (plus pick-up) is the only part of the opening section where discreet non-microtonal trichords are projected. These three trichords can be compared to the {F, Fs, A} trichord projected in the prime contour layer of the submeter. Both the first trichord in measure 22 {Gs, Gn, Bf} and the submetric trichord share dyad-class (01), although its projection in the pitch material is as a major seventh and in the submeter a minor second. Thus, the two trichords can be related by the hyper-transformation 〈T8I〉. 136 The second two trichords of measure 22 ({Bf, Df, Af} and {Bn, En, Fs}) share dyad-class (05), again projected in inversions. The hyper-transformation between these trichords inverts the hyper-transformation of the prior transformation while retaining the remapping-namely 〈T4I〉.137 Thus, the first trichord of the ending section pairs with the prime layer of the submetric contour to relate, by the hyper-hyper-transformation 〈〈T0I〉〉, to the final two trichords. 138 Notably, the only section of non-microtonal material is related to the rest of the section not only by pitch, but through the projection of beat-length alterations throughout the opening section. These hyper- and hyper-hyper transformations are shown in Figure 4.6. 136 David Lewin, "Klumpenhouwer Networks and some Isographies that Involve Them," Music Theory Spectrum 12 (1990): 89. 137 Kirschner, "K-Net Theory." 138 Dimitri Tymoczko, "Recasting K-nets," Music Theory Online 13 (2007), §§14-15. accessed April 11, 2015, http://www.mtosmt.org/issues/mto.07.13.3/mto.07.13.3.tymoczko.html 85 The second and third trichords can also be related to the first and the submetric trichords by hyper-transformations. However, because the pairs of trichords do not share a dyad-class, hyper-transposition and inversion alone will not suffice. Instead, the dyad-classes (01) and (05) are related via the hyper-M5 and hyper-M7 transformations. 139 Each trichord in each hyper-inversionally related pair relates to the two other trichords by one hyper-M5 transformation and one hyper-M7 transformation. Moreover, the transformations alternate, such that each trichord is related to all others by one each of hyper-inversion, hyper-M5, and hyper-M7. While these are admittedly remote relationships, they map the symmetry of dyadclass transformation from each trichord projected in the closing material of the first section. Figure 4.7 shows the Klumpenhouwer Network of the three trichords of measure 22 and the trichord projected in the prime contour layer of the submetric pitch projections. Both of these analyses have demonstrated that there is a wealth of analytical potential in the comparison of submetric beat-length alterations and pitch transformations. Moreover, the nature of the calculations required in submetric analysis allows easy conversion into the pitch realm. Such an analysis also demonstrates that, far from being complex, simple tuplets such as 5:4, 7:4, and 9:8 represent the easily understood intervals of a major third, a minor (harmonic) seventh, and a major second. 140 As the pitch relationships become more complex and dissonant (i.e., 11:8 representing the undecimal tritone, a microtonal interval), so too does the cognitive relationship between the beat-lengths. 141 Another advantage of pitch representation of submetric shifts comes from the association 139 Kirschner, "K-Net Theory." 140 Mead, Tempo Relations, 69. 141 Gann, "Anatomy." 86 that many musicians naturally have with the staff. 142 Octaves, for example, readily show doubled tempi that may be buried in tuplets and irrational time signature in the score. Further refinements to this system could be used to show additional dimensions beyond the tempo and duration. Various shapes of note-heads, for example, could be used to denote subdivisions, and color is just one option to indicate the beat-unit. When tempered, as these examples were, the pitch representation also provides an easy method of clustering similar tempi. If an eighth-note at e = 54 is transformed once by a 10:9 tuplet and later by a 9:8 tuplet, the resulting tempi would differ by 0.75 beats-perminute ( 54( 89 ) = 60.75 and 54( 109 ) = 60 ).143 While a skilled performer might well be able to realize the difference by splitting the requisite beat-units, it is highly doubtful that even the most well-attuned listener could perceive the difference in tempo.144 By converting the beatlength alterations to pitch transformations, both are represented as a major second, thus indicating their extreme similarity in speed.145 Whatever refinements are made to the system to relay additional information, the use of pitch notation to show submetric shifts has great potential. It not only clarifies the submetric relationships by placing them in a context more easily grasped by most musicians, but also provides an easy way of showing duration. Moreover, the temperament of intervals provides a musically relevant grouping of tempo clusters. All of these advantages are 142 Mead, Tempo Relations, 72-73. 143 There is, of course, the slight chance that the slight difference between 10:9 and 9:8 transformations crosses a cognitively and/or mathematically important categorical boundary. For this reason, it would be useful to run a clustering algorithm on the absolute (untempered) values before rushing to group all 10:9 and 9:8 ratios into a single category. 144 Should the tempi be notated via constantly shifting metronome markings (rather than tuplets), the performer might well fail to infer and project the subtle beat-length alterations. 145 Mead, Tempo Relations, 68. 87 realized even before the ability for comparisons between pitch mapping of submeter and the pitch material of the work. 88 Table 4.1: Table of Transformations Ratio 25 24 16 15 11 10 9 8 8 7 7 6 6 5 5 4 13 10 4 3 11 8 3 2 8 5 13 8 5 3 7 4 15 8 Interval Name Pitch Above C minor 5-limit half-step C+ minor second Db neutral second Dd major second Dn harmonic second D+ septimal minor third ED minor third Eb major third En augmented third E+ perfect fourth Fn undecimal tritone F+ perfect fifth Gn minor sixth Ab neutral sixth Ad major sixth An minor seventh Bb major seventh Bn 89 ˙ b˙ b˙ ˙ b˙ b˙ ˙ b˙ j µ œ œj B ˙ B ˙ ˙ ˙ B ˙ b˙ j bœ b˙ ˙ b˙ j œ ˙ ˙ j bœ ˙ b˙ b˙ b˙ ˙ ˙ b˙ b˙ ˙ ˙ ˙ b˙ j œ b˙ b˙ b˙ b˙ Figure 4.1: Notated Tempo Changes in Unsichtbare Farben, Represented as Pitches b˙ j bœ j bœ œ œ e = 47.25 #œ j œ 5 œ J bœ™ œ J 3 5 12 B œ 33 œ œ œ œ ‹ œ œ œ Bœ Figure 4.2: 3 nœ ™ & 20 R 20 3 & 20 œ 17 13 5 12 œ 5 ™™ 8 µœ œ œ J j Bœ œ œ 3 8 œ 3 8:5 œ œ J 7:6 3 œ Bœ ™ 20 œ œ™ ‹ œ J 8:6 19:12 7:4 25:24 3 œ J nœ ‹  œ ™™ œ ™ µ œ œ 3 ? œ 3 8 œ 3 3 bœ J bœ J œ J 5 15:8 & #œ 6:5 Bœ J œ 11:8 œ B œ ™™ œ J œ b œ œ œ œ n œJ 5 3 œ™ œ œ™ œ œ œ bœ™ r œ œ œ™ 5œ 8 b œ œ nœ 3 œ 5 12 œ œ ™ 4 10 œ ™ ‹ bœ J Analysis Copyright © 2014 by Aaron J. Kirschner 30:16 œ 7 œ 83 œ œ ‹b œ œ œ b œ ™œ ™b œ ™œ ™ œ b œ œ 125 œ ™ RÔ Bœ œ ™ J n œ ™™ 13:10 13:8 11:10 B œ ™™ 5 j 12 # œ œ™ # œ ™™ #œ œ œ œ # œ ™™ bœ™ R 2 8 Submetric Units in Measures 1-22 Represented as Pitches 7:6 Bœ ™Bœ Bœ 7:4 7:6 b œ œ ™b œ œ B œ 4:3 ‹ nœ bœ bœ œ œ œ œ 3 R 5 & 20 R R 12 œ 7 3 & 9 5:3 #œ 3 œ & 20 5 4 & 8 ® œ™ e = 54 in 24-tone equal temperment plus allowance for harmonic sevenths (7:4) Unsichtbare Farben-Meter as Pitch œ R 3 20 3 20 3 20 3 20 œ œ™ 3 20 90 91 e = 54 4 & 8® n >œ ™ µ >œ ™ n>œ 3 Bœ > nœ œ n œ B >œ n>œ B œ n>œ > > > Figure 4.3: Embedded {B, D, C} in Measure 1 5 µ>œ 2 8 œ™ > œ™ 5 12 œ 5 8œ 5 12 œ j œ™ 5 12 œ œ œ œ™ œ™ œ™ 30:16 # Xœ œ ™ 5 12 œ 2 8œ 8:5 ˙ œ™ 3 8 œ™ 3 8 œ™ 3 j 20 œ ™ œ ? Figure 4.4: Prime Contour Layer of Submetric Tempi in the First Section 3 & 20 œj™ 20 3 & 20 œj™ 17 3 & 20 œj™ 13 & œ™ 9 e = 47.25 3 & 20 œj™ 5 4 & 8 ® œ™ e = 54 N 4 10 œ ™ 5 12 œ 5 8œ 5 12 œ r œ œ™ 3 8 œ™ 5 15:8 & #œ œ™ œ™ œ™ œ œ R 3 20 3 20 3 20 3 20 3 20 92 93 15:8 15:12 . b O bœ 8:5 3 Violin n Oœ^ n r 4 & #>œ 10 . . . . . . . . ™b œ. ™ b œ . bb Oœ ≈ ≈ ≈ ≈ 15:12 . # œ. n œ ™ . nœ ≈ Œ sub. pppp f Figure 4.5: Unsichtbare Farben, mm.22 (and pick-up). End of First Section 3 ∑ ∑ ∑ ∑ & Fs Af T7 T1 7 I3 Fn ∑ I6 Df I2 & An Bf 〈〈T0I〉〉 〈T8I〉 〈T4I〉 Gn Fs I5 T11 I10 I5 Bf En I6 Gs ∑ I11 T5 Bn Figure 4.6: One Possible K-net of Projected Trichord and Last Three Trichords ∑ 94 〈T11M7〉 〈T5M5〉 Gn I5 T11 Fs T7 I6 Bf 〈T4I〉 Af I5 Df I6 I10 En I11 Gs Bf T5 Bn 〈T8I〉 〈T9M7〉 〈T7M5〉 Fs T1 I3 Fn I2 An Figure 4.7: K-Net Relating Each Trichord in Measure 22 to the Projected Trichord APPENDIX A LIST OF VARIABLES AND FORMULAS List of Variables, in alphabetical order b Used for the adjustment of tempo and durations in rallentando and accelerando. Scales (ordinally or proportionally) for each submetric unit in a given tempo shift, and thus is always used in practice as bi . Defined as bi = P −( h⋅s i ) P . c Represents the number of cycles of a beat-unit at a given tempo. h The beats-per-minute shifted in each terrace of an ordinal rallentando or accelerando. Constant for a given tempo shift. Defined as h = P− Q n for rallentandos and h = Q−P n for accelerandos, where n is the total number of submetric units in the tempo shift. i Dummy variable for representation of ordinal position of other variables. For example, if x i is the value of x in a given submetric unit, x i +1 is the value of x in the submetric unit immediately following. k Represents the note value that the beat-unit is calculated in (i.e., the origin layer). Always represented as a power of two; k = 4 represents a quarter note, k = 8 an 96 eighth-note, k = 16 a sixteenth, etc. (This text primarily uses k = 64 .) n Ad hoc variable. P The starting tempo of a rallentando or accelerando. Q The ending (goal) tempo of a rallentando or accelerando. s Represents a submetric unit, irrespective of its x, y, z, or w values. Primarily used (as  s i ) to denote ordinal position of submetric unit. u Product of all tuplet ratios affecting the beat-unit of a given submetric unit. If there are multiple tuplets affecting a beat-unit in a given submetric unit, u is defined as n u = ∏ u i where u1 is the most nested tuplet and u n is the highest level tuplet. i =1 w Duration of a given submetric unit, represented in seconds. w accel Adjusted duration based on accelerando. Defined as w accel = wb . w rall Adjusted duration based on rallentando. Defined as w rall = wb . n w whole Total duration for a given series of submetric units s 1 to s n . Defined as w whole = ∑ w i i =1 x Beat-Unit of a given submetric unit. Represented numerically number of x(unit ) vales 97 to complete one beat-unit. If x(unit ) = 64 and the beat-unit is a sixteenth note, x = 4 . Always two or greater; generally (but not necessarily) a whole number. x mark Beat-Unit represented in the notated metronome marking. If the score is notated e = 54 and x(unit ) = 64 , then x mark = 8 . x(unit ) Base unit for x = 1 . Defined as x(unit ) = k . y Subdivision of a given submetric unit. Always a whole number two or greater. Most often two, three, four, or five. z Denotes the tempo of a given submetric unit, usually in beats-per-minute. Can alternatively represent categories of tempo. z(base ) Notated metronome marking, comprising both x mark and z mark . Represented as z(base ) ≡ x mark : z mark z accel Adjusted tempo based on accelerando. Defined as z accel = z ⋅b . z mark Tempo (in beats-per-minute) in notated metronome marking. If the score is notated e = 54, then z base = 54 . z rall Adjusted tempo based on rallentando. Defined as z rall = z ⋅b . 98 Formulas x mark x Calculate tempo of a given beat-unit z = z mark ⋅ Calculate duration with only a single beat-unit w= Calculate duration of multiple beat-units w = ∑ ( 60 ⋅c i ) zi 60 z ⋅u ⋅c n i=1 rall (P → Q) P− Q n Calculate adjustments for ordinal rallentando =h P −( h⋅n i ) P = bi z rall = z i ⋅bi w rall = wi bi accel (P → Q) Q−P n Calculate adjustments for ordinal accelerando =h P +( h⋅n i ) P = bi z accel = z i ⋅bi w accel = wi bi 99 rall _ prop(P → Q) n w whole = ∑ w i i =1 w Calculate adjustments for proportional rallentando P − ((P − Q)( wwholei )) z rall P = z i ⋅bi w rall = = bi wi bi accel _ prop(P → Q) n w whole = ∑ w i i =1 w Calculate adjustments for proportional accelerando P + (( Q − P )( wwholei )) z accel P = z i ⋅bi w accel = wi bi =b APPENDIX B COMPLETE FOUR-DIMENSIONAL SUBMETRIC CODING OF UNSICHTBARE UNSCHITBARE FARBEN FARBEN # qœ= 120 #œ <xunit=64> œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ & <zbase = 8:54> Piano # œ1.33);œ (4, œ2, 135,œ .89);œ (6, 2, 72, .83); (8, 2, 40.5, œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ { & (2, 2, 216, 1.53); (2, 2, 216, 2.18); (8, 3, 81, .74); (12, 2, 63, .95); (4, 4, 108, 1.80); (6, 2, 108, .56); (6, 3, 108, .56); (4, 3, 162, 1.48); œ(8, 4, 81, 1.01); bœ (4, 3, 225,3.53);n œ(8,œ 2, 140.63, .76); (4, b2,œ202.5, .64); #œ œ œ #œ œ (4, 4, 135, & .54); (7.5, 5, 86.4, .82); (4, 2, 162, .65); 3 (2, 2, 324, .37); (8, 5, 121.5,31.23); (3, 3, 324, 1.05); 3 3 ?144, .42); (4, 2, 108, .56); (4, 2, 108, & .56); (3, 2, œœ & # œ #œ { œ œ œ œ #œ œ œ œ œ œ œ œ #œ #œ (2, 2, 216, .83); (3, 2, 144, .42); <zbase = 8:47.25> (10, 5, 37.8, 2.70); (2, 2, 189, 1.11); (8, 4, 77.96, 1.64); bœ bœ #œ #œ 5 1.15); œ n œ œ 1.49); (4,œ b2,œ 137.81, 1.94);œ (4, œ4, 94.5, .63); (8, bœ (6, 3, 103.95, œ b œ(4, 4, 155.92, #œ œ b œ2, 70.88, .85); bœ œ #œ #œ (2, 2, 189, & .63); 7 (4, 2, 94.5, .63); (8, 2, 88.59, .67); (4, 4, 141.75, 7 1.65); 5 5 #œ (4, 4, 157.50, & .63); (4,b œ2, 196.88, .51); (4, 2, 248.06, .24); (6, 2, 110.15, # œ (12, 3, 59.06, 1.51); œ œ .45); #œ œ œ { #œ œ œœ #œ œ (8, 4, 115.17, 1.64); (8, 4, 93.52, 1.12); (6, 2, 266, .38); (9, 3, 66.5, 1.17); #œ œ bœ œ 7 œ œ œœ œbœ œ œ 101 (4, 2, 224.44, .33); (8, 2, 70.88, 4.23); (2, 2, 413.44, .22); (4, 2, 241.17, .31); (4, 4, 241.17, .56); (2, 3, 275.63, .22); (4, 4, 122.85, 3.42); (12, 2, 103.91, 1.35); (5, 5, 124.69, .96); (4, 4, 170.1, 1.15); (4, 4, 113.4, .53); (2, 2, 255.15, 1.06); (4, 4, 155.93, 1.35); (6, 3, 104.5, 1.72) <end 1st page> (4, 4, 157.5, .76); (2, 2, 275.63, .44); (2, 3, 275.63, .22); (2, 2, 275.63, .44); (4, 2, 226.8, 1.23); (3, 3, 567, .25); (6, 3, 283.5, .56); (4, 2, 425.25, .29); (8, 2, 198.45, .45); (8, 3, 113.4, 1.06); (32, 2, 55.37, 1.63); (3, 3, 369.14, 1.63); <double bar mm 23, end 1st section> (4, 2, 118.5, .51); (6, 2, 79.17, .76); <zbase = 8:60.75> (8, 2, 97.2, 1.23); <rit 60.75-54> (4, 2, 145.8, 1.23); (5, 5, 48.6, 1.78); (4, 2, 218.7, 1.80); (6, 2, 81, 3.09); (4, 2, 121.5, .47); (6, 2, 87.75, 1.25); <zbase = 8:54> <rit 54-47.25> (8, 2, 54, 3.33); (8, 3, 54, 1.11); (4, 2, 135, .44); (4, 3, 135, .44); (2, 3, 270, .22); (8, 4, 67.5, 1.48); (3, 2, 180, .96); (2, 2, 315, .67); <zbase = 47.25> (4, 2, 100.41, 1.20); (2, 2, 200.81, 1.34); (4, 4, 141.75, 1.06); (2, 3, 283.4, .21); (4, 2, 94.5, .63); (4, 4, 165.38, 1.27); (4, 4, 94.5, 1.27); (4, 2, 141.75, 2.54); (4, 2, 198.45, 60); (6, 3, 132.3, .76); (4, 2, 170.1, 1.85); (3, 3, 226.8, .71); (4, 2, 198.45, 1.06); (4, 2, 141.75, 1.57); (16, 2, 59.06, 1.02); (8, 2, 131.25, 1.89); (14, 2, 75, 102 3.89); <accel 47.25-54> (18, 2, 21, 3.11); (16, 2, 47.25, 2.96); (16, 2, 35.44, .85); (6, 3, 105, 1.27); (4, 2, 94.5, 1.90); <zbase = 8:54> (16, 4, 31.5, 3.81) <end 2nd page> (4, 4, 170.1, 2.77); (24, 3, 37.8, 1.59); (48, 2, 22.05, 2.72); <rall 54-47.25> (4, 4, 162, 3.68); (24, 2, 39.27, 1.53); (24, 3, 39.27, 1.53); (4, 3,, 162, .74); (24, 2, 18, 3.33); <zbase 8:47.25> (8, 4, 47.25, 2.54); (8, 3, 47.25, 1.27); (8, 2, 59.06, 1.27); (8, 3, 47.25, 1.27); <double bar mm 42, end 2nd section> (16, 3, 47.25, 1.90); (12, 2, 63, 1.14); (4, 4, 135, 1.18); (8, 2, 84.46, .80); (16, 2, 32.48, 2.20); (8, 2, 84.46, .80); <zbase 8:67.5> (2, 2, 385.71, .98); (7, 7, 119.53, 1.22); (2, 2, 385.71, 1.07); (3, 3, 176.79, 1.24); (6, 3, 88.39, 1.36); (4, 2, 265.18, .91); (4, 2, 337.5, .36); (8, 4, 67.5, 2.34); (16, 2, 63.28, .95); (4, 2, 168.75, 2.75); (32, 2, 39.55, 1.69); (6, 2, 112.5, 1.24); <rall 67.5-47.25> (6, 3, 112.5, .89); (2, 2, 270, .89); (10, 5, 94.5, .63); (4, 2, 236.25, .65); (5, 5, , 121.5, .79); (8, 3, 101.25, .59); (12, 2, 78.75, .76); (4, 4, 135, 1.34); (2, 4, 385.71, .35); (2, 3, 385.71, .47); (4, 4, 192.86, .54); (4, 6, 192.86, .73); (4, 3, 289.29, .21); (4, 2, 202.5, 1.19); (4, 4, 303.75, .77); <zbase 8:47.25> (4, 5, 170.1, .71); (2, 2, 340.2, .25); (4, 4, 212.63, .28); 103 (4, 4, 170.1, 1.76); (4, 4, 231.53, .26); (3, 3, 308.7, .53); (2, 2, 264.6, .34); (2.5, 5, 211.68, .68); (2, 2, 264.6, .28); (4, 7, 132.3, .45); (4, 4, 165.38, .57); <end 3rd page> (4, 2,318.94, .92); (4, 3, 141.75, .84); (12, 3, 105, .97); (8, 3, 75.94, .79); (6, 2, 101.25, 1.19): (8, 4, 101.25, 1.19); <zbase 8:54> (6, 2, 72, .83); (8, 3, 54, 1.11); (8, 4, 90, .83); (16, 2, 64.8, 1.85); (16, 2, 72.9, 1.85); (4, 2, 180, 1.33); (5, 5, 100.8, .60); (3, 3, 168, 1.44); (8, 3, 81, .74); (4, 3, 108, .56); (16, 2, 64.8, 4.26); <rall 54-42.5> (4, 2, 108, 1.71); (4, 2, 162, .93); <zbase 8:42.5> (4, 4, 85, 1.79); (6, 3, 99.17, 1.48); (8, 2, 111.56, 1.18); (6, 6, 85, .81); (4, 4, 143.44, 1.25); (4, 4, 191.25, .63); (8, 4, 63.75, 1.10); (12, 3, 76.5, .78); (6, 3, 118.06, 1.02); (4, 4, 177.08, 1.03); (6, 3, 56.67, 1.53); (8, 2, 63.75, 1.17); (16, 2, 33.20, 1.89); (4, 4, 85, 1.94); (4, 5, 265.63, .68); (4, 3, 265.63, .68); (4, 3, 177.08; .34); (4, 4, 127.5, 1.25); (3, 2, 170, 2.33); (8, 2, 52.13, 1.27); (2, 2, 170, 1.59); (8, 2, 42.5, 2.41); (3, 3, 170, .35); (4, 3, 127.5, .71); (12, 2, 42.5, 1.89); (6, 3, 167.90, .99); <zbase 8:47.25> (4, 4, 94.5, 1.27); (3, 2, 160, .38); (2, 2, 189, .32); (3, 2, 160, .38); (10, 5, 37.8, 1.59); (6, 2, 63, .95); (4, 4, 94.5, 1.27); <end 4th page> 104 (3, 3, 126, .48); (4, 4, 94.5, .63); (3, 2, 126.1, .19); (4, 4, 94.5, 1.43); (4, 2, 118.13, .38); (6, 2, 72.19, .42); (4, 2, 184.28, .81); (6, 2, 122.85, 1.30); (9, 9, 63, .95); (4, 3, 141.75, .42); (3, 2, 189, .74); (4, 3, 196.88, .30); (4, 4, 319.92, 1.71); <zbase 8:60.75> (16, 2, 37.13, 1.61); (12, 2, 48.89, 1.43); (8, 2, 74.25, 1.41); <rall 60.75-47.25> (12, 3, 40.5, 1.45); (6, 2, 94.5, .79); (4, 2, 162, 1.02); (4, 2, 212.63, .41); (3, 3, 202.5, 1.04); (4, 2, 121.5, .91); (2, 2, 364.5, .82); (4, 3, 121.5, .49); (12, 3, 48.6, 2.01); (16, 4, 85.05, 1.94); (4, 2, 253.13, 1.19); (4, 4, 212.63, .49); (2, 2, 455.63, .33); (4, 4, 205.03, 1.88); <zbase 8:47.25> (4, 2, 196.86, .68); (2, 2, 189, .63); (4, 5, 94.5, .63); (5, 5, 85.05, .71); (4, 4, 106.31, .95); (4, 2, 118.13, .89); (8, 2, 47.25, 2.54); <zbase 8:74.25> (8, 3, 74.25, .81); <rall 74.25-60.75> (16, 2, 37.13, 1.55); (8, 2, 74.25, 1.12); <zbase 8:54> (8, 2, 67.5, 2.09); (2, 2, 297, .51); (4, 4, 198, .30); <rall 54-38.5> (4, 4, 180, 1.33); (4, 4 202.5, .67); (8, 5, 54, 2.89); <zbase 8:42.5> 105 (16, 4, 51, 6) <end 5th page> (6, 3, 99.17, 1.59); (3, 2, 113.33, .53); (4, 2, 85, .91); (6, 4, 99.17, 2.09); (12, 3, 68, .88); (6, 3, 136, .88); (4, 3, 204, .29); (4, 4, 229.5, .59); (16, 2, 51, 1.47); (8, 2, 68, .99); (4, 4, 153, .96); <zbase 8:54> (16, 2, 43.88, 2.05); (2, 2, 324, 1.11); (4, 2, 189, 1.57); (12, 3, 63, 1.32); <rall 54-38.5> (3, 3, 216, .74); (4, 4, 175.5, 1.11); (6, 3, 108, .56); (3, 3, 216, 1.17); (8, 2, 170.1, .44); (10, 5, 113.4, 1.06); (16, 3, 56.7, 2.91); (10, 5, 113.4, .83); <zbase 8:38.5> (3, 3, 269.5, .40); (8, 2, 86.63, .64); (3, 3, 269.5, .40); (4, 3, , 115.5, .52); (2, 2, 433.13, 1.04); (4, 2, 160.42, 2.24); (4, 4, 280.73, .75); (8, 3, 80.21, 1.27); <accel 38.5-54> (4, 3, 115.5, .52); (8, 4, 57.75, 1.30); (4, 3, 115.5, 2.47); (12, 3, 36.96, 2.73); (6, 3, 128.33, .78); (12, 4, 38.5, 1.87); (2, 2, 288.75, 1.73); (8, 2, 102.67, 1.46); (6, 3, 61.6, 4.71); <zbase 8:54> (16, 2, 60.75, 5.69); (16, 2, 57.86, 1.04); (8, 2, 92.57, 2.08); (9, 2, 96, .95); (6, 2, 185.14, .32); (6, 2, 185.14, .65); <zbase 8:38.5> (3, 2, 171.11, .70); (3, 4, 299.44, .30); (3, 3, 205.33, 1.00); <end 6th page> (6, 3, 190.58, .31); (12, 2, 79.41, 1.15); (4, 2, 243.63, 1.14); (6, 2, 154, .78); (16, 2, 54.14, 3.39); <accel 38.5-47.25> 106 (2, 2, 192.5, .31); (6, 2, 106.95, .56); (4, 4, 160.42, .89); (24, 2, 21.39, 4.68); (8, 4, 57.75, 3.64); <zbase 8:47.25> (16, 3, 28.35, 2.12); (16, 6, 28.35, 4.80); (8, 5, 53.16, 1.69); (8, 4, 53.16, 1.69); (8, 5, 53.16, 2.58); (5, 5, 127.58, .83); (12, 3, 103.36, .97); (16, 4, 40.5, 2.96); (12, 4, 31.5, 1.90); (4, 5, 165.38, .91); (12, 7, 31.5, 1.90); (8, 4, 47.25, 1.26); <zbase 8:60.75> <rall 60.75-54> (3, 3, 378, .99); (8, 2, 121.5, .73); (6, 3, 167.06, 1.08); (24, 2, 45.56, 2.63); <zbase 8:54> (4, 2, 263.25, 4.44); (16, 3, 33.75, 1.78); (6, 2, 157.5, .76); (4, 2, 236.25, .83); (6, 2, 112.5, 1.59); (6, 5, 115.71, .52); (6, 3, 115.71, 1.52); (8, 4, 90, 1.33); (20, 2, 32.4, 4.44); <end 7th page> (2, 2, 260.36, 1.24); (4, 2, 144.64, 1.23); (4, 4, 188.04, .56); (8, 2, 81, 1.56); (8, 3, 81, .97); <accel 54-60.75> (16, 4, 64.8, 2.55); (8, 2, 81, 1.48); (8, 3, 81, 1.21); (8, 3, 96.43, 3.42); (8, 5, 96.43, 2.06); <zbase 8:60.75> <rall 60.75-38.5> (8, 3, 69.43, 1.37); (4, 2, 138.86, 1.51); (8, 6, 69.43, 2.59); (4, 3, 121.5, 1.48); <zbase 8:38.5> (4, 2, 134.75, .89); (8, 3, 67.36, 1.96); (8, 2, 168.44, .53); (8, 2, 112.29, 1.07); (8, 3, 67.36, 1.56); <zbase 8:60.75> (8, 2, 104.14, 1.30); (16, 3, 52.07, 1.84); <rall 60.75-47.25> (12, 3, 86.79, 1.90); (8, 2, 104.14, 2.02); 107 <zbase 8:47.25> <zbase 8:54> (16, 2, 46.29, 2.31); (4, 2, 246.86, .73); (8, 2, 102.86, 1.38); (12, 2, 61.71, 1.30); (4, 2, 216, .97); (8, 3, 61.71, 1.45); <rall 54-42.5> (4, 2, 189, .84); (8, 3, 157.5, .95); (8, 2, 157.5, .61); (8, 2, 94.5, .95); (8, 4, 141.75, 1.42); (4, 2, 135, 1.33); <zbase 8:42.5> <zbase 8:47.25> (24, 3, 70.88, 1.52); (16, 4, 59.06, 1.02); (8, 4, 56.7, 1.65); (8, 2, 66.15, .91); (4, 5, 113.5, 4.97); (2, 3, 206.72, 2.03); (8, 5, 59.06, 3.73); (4, 3, 177.19, .34); <end 8th page> <accel 47.25-54> (32, 2, 20.25, 5.93); <zbase 8:54> (2, 3, 388.8, .93); (12, 4, 64.8, .93); (8, 4, 121.5, .62); (4, 4, 194.5, 1.28); (24, 2, 30.99, 4.42); (12, 2, 48.21, 1.24); <rall 54-47.25> (16, 2, 54, 2.94); (6, 2, 112, .88); <zbase 8:47.25> (20, 2, 22.68, 4.84); (12, 2, 37.8, 1.59); (16, 15, 23.63, 5.08); (12, 4, 31.5, 1.90); (20, 5, 18.9, 9.17); <zbase 8:42.5> (4, 2, 197.32, 1.98); (16, 2, 45.54, 2.13); (16, 2, 81.54, 1.84); (4, 2, 197.32, 1.98); (6, 2, 125.93, 2.54); (4, 5, 177.08, 1.22); (6, 2, 68, 3.96); (16, 2, 39.84, 1.88); (4, 4, 229.5, 1.30); <zbase 8:38.5> 108 (4, 2, 77, 2.34); (32, 2, 19.25, 6.23); (4, 4, 148.38, 1.25); (3, 3, 192.5, .31); (5, 5, 61.6, .97); (5, 5, 107.8, 1.13); (4, 4, 134.75, .45); (4, 4, 115.5, 1.56); (4, 2, 134.75, .89); (2, 2, 269.5, .68); (4, 2, 115.5, .52); (3, 3, 154, 2.34); (2, 2, 115.5, .84); <end 9th page> <zbase 8:60.75> (8, 2, 78.11, 2.30); <rall 60.75-47.25> (8, 5, 78.11, 1.28); (16, 2, 87.87, 1.64); (8, 4, 78.11, 3.72); <zbase 8:47.25> (6, 2, 63, 1.32); (12, 3, 47.25, 2.17); (4, 2, 94.5, 3.54); <accel 47.25-67.5> (8, 4, 40.5, 1.98); (4, 4, 121.5, .99); (3, 3, 148.5, .81); (4, 4, 111.38, .54); (3, 3, 121.5, 1.62); (3, 3, 162, 1.48); (4, 4, 121.5, 1.11); (3, 3, 162, .37); (3, 3, 177.19, .34); (4, 4, 132.89, .68); (3, 3, 157.5, 1.43); (2, 2, 270, .90); (4, 4, 118.13, 1.33); <zbase 8:67.5> <rall 67.5-47.25> (8, 4, 67.5, 2.35); (4, 2, 236.25, 1.22); (6, 2, 174.38, .34); (4, 2, 261.56, .46); (6, 2, 174.38, .51); (4, 4, 219.38, .89); <zbase 8:47.25> (2, 2, 433.13, .42); (3, 4, 157.5, .84); (3, 2, 257.63, .44); (3, 2, 157.5, .38); (2, 2, 236.25, 1.29); <zbase 8:38.5> 109 (5, 5, 82.5, 3.78); (8, 2, 96.25, 2.18); (4, 4, 137.5, 1.58); (16, 5, 41.25, 2.98); (8, 4, 98.26, .79) <end piece> APPENDIX C PITCH MATERIAL PROJECTED BY SUBMETRIC UNITS IN THE FIRST 22 MEASURES OF UNSICHTBARE FARBEN , # qœ= 120AT EACH CONTOUR #œ œ œ œ œ œ œ REDUCTION & œ œ œ œ œ œ œ œ œ LAYER Piano #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 3 nœ œ #œ bœ œ œ œ œ & 3 3 3 3 œ ? { & & œ œ œ œ #œ œ #œ œ œ œ œ œ œ #œ # œ #œ bœ œ nœ œ 5 œbœ œbœ bœ œb œ œ #œ œ bœ bœ #œ #œ #œ & 7 5 7 5 #œ { & bœ œœ #œ œ #œ œ œ œ œ #œ œ œ bœ œ 7 œ #œ œ œœ œbœ œ œ j bœ œ œ e = 47.25 5:3 #œ œ #œ j œ 5 œ J bœ™ œ J 3 5 12 B œ 33 œ œ œ œ ‹ œ œ œ Bœ 3 n œR ™ & 20 20 3 & 20 œ 17 13 7:6 Bœ ™Bœ Bœ 7:4 7:6 5 12 b œ œ ™b œ œ B œ 4:3 œ 5 ™™ 8 µœ ‹ nœ bœ bœ œ œ œ œ 3 R 5 & 20 R R 12 œ 7 3 & 9 3 & 20 5 4 & 8 ® œ™ e = 54 j Bœ œ œ 3 8 œ 3 8:5 œ œ J 7:6 3 œ Bœ ™ 20 œ œ™ ‹ œ J 8:6 19:12 7:4 25:24 3 œ J nœ ‹  œ ™™ œ ™ µ œ œ 3 ? œ 3 8 œ 3 3 bœ J bœ J œ J 5 15:8 & #œ 6:5 Bœ J B œ ™™ œ J œ 11:8 œ œ b œ œ œ œ n œJ 5 3 œ™ œ œ™ œ œ œ bœ™ r œ œ œ™ 5œ 8 b œ œ nœ 3 œ 5 12 œ œ ™ 4 10 œ ™ ‹ bœ J Analysis Copyright © 2014 by Aaron J. Kirschner 30:16 œ 7 œ 83 œ œ ‹b œ œ œ b œ ™œ ™b œ ™œ ™ œ b œ œ 125 œ ™ RÔ Bœ œ ™ J n œ ™™ 13:10 13:8 11:10 B œ ™™ 5 j 12 # œ œ œ J # œ ™™ #œ œ œ œ # œ ™™ bœ™ R 2 8 œ™ Unreduced in 24-tone equal temperment plus allowance for harmonic sevenths (7:4) Unsichtbare Farben-Meter as Pitch œ R 3 20 3 20 3 20 3 20 œ œ™ 3 20 111 N j bœ X Bœ 3 & 20 œR ™ 20 3 & 20 œ 17 5:3 # Xœ œ N œ 7:4 7:6 7:6 B Xœ ™ B œX B Xœ N N nœ 5 12 b œ œ ™ 7b œ œ B Xœ 7 7X b Xœ ™ œ J œ™ J bœ 5 12 J œ œ e = 47.25 3 j & 20 œ ™ 13 & 9 3 & 20 œ ™ 5 4 & 8 ® œ™ e = 54 5 8 œ ™™ N 5 12 B œ œ J X 13:8 X B Xœ œ ™ J œ X N bœ 7 13:10 8:5 3 N N 3 œ™ 20 J N œ J 8:6 19:12 25:24 ‹  œ ™™ œ # Xœ ™™ # Xœ ˙ œ œ # œ ™™ 30:16 N œ œ 3J 8 Reduction Layer 1 œ 3 5œ 8J œ X 5 12 œ 3 8 ? 3 N 4 10 œ ™ ‹ b Xœ RÔ J N N bœ J 3 N œ N r œ œ b Xœ œ œ X X ™ ™ œ 83 œ œ ‹b œ œ œ b œ œ b œ ™œ ™ œ b œ œ 125 œ ™ RÔ N 11:10 œ ™™ œ œ J X j Bœ œ N 5 12 # œ 2 8 œ™ 5 15:8 & # Xœ 6:5 b Xœ J N j œ N B œ ™™ œ J N 3 j œ œ œ N 11:8 œ œ 5 œ œ X œ ™ œ œ ™ œ œJ X œ X œ R 3 20 3 20 3 20 3 20 3 20 112 N j bœ 3 & 20 œJ ™ 20 3 & 20 œ 17 7:6 B œX ™ œ œ œ e = 47.25 3 & 20 œj™ 13 & 9 3 & 20 œj™ 5 4 & 8 ® œ™ e = 54 œ ™™ 5 12 œ 5 8 bœ 5 J 12 7X b Xœ ™ œ J j œ™ N ˙ 5 12 œ J X œ œ œ œ 30:16 #œ œ ™ X 5 12 œ 2 8 8:5 13:10 ˙ N ‹  œ ™™ œ™ œ 25:24 œ 3 8 œ ™™ œ 3J 8 œ ? Kr œ ˙ 8:6 19:12 bœ ™ œ ™ X œ™ 3J 20 œœ œ X N bœ 7 Reduction Layer 2 N 4 10 œ ™ N œ™ 5 12 œ ™ œ 5J 8 5 12 œ œ 3 8 N œ r œ 3 œ 5 # Xœ œ 15:8 & 6:5 N œ œ™ œ œ N œ B œ ™™ œJ 11:8 œ j œ œ R 3 20 3 20 3 20 3 20 3 20 113 N bœ™ e = 47.25 3 & 20 œj™ 20 3 & 20 œj™ 17 3 & 20 j œ™ 13 & 9 3 & 20 œj™ 5 4 & 8 ® œ™ e = 54 œ™ 5 12 œ 5 8œ bœ 5 J 12 7X j œ™ N ˙ 5 12 œ œ œ œ™ œ™ 30:16 # Xœ œ ™ 5 12 œ 2 8œ 8:5 ˙ œ 3 8 œ™ 3 j 8œ œ X N bœ 7 œ 3 20 j œ™ œ Reduction Layer 3 ? N 4 10 œ ™ œ 5 12 œ 5 8 5 12 œ r œ œ™ 3 8 œ™ 5 15:8 & # Xœ œ™ œ™ œ™ œ œ R 3 20 3 20 3 20 3 20 3 20 114 N bœ™ e = 47.25 3 & 20 j œ™ 20 3 & 20 j œ™ 17 3 & 20 j œ™ 13 & 9 3 j & 20 œ ™ 5 4 & 8 ® œ™ e = 54 œ™ 5 12 5 12 5 8 œ œ œ j œ™ 5 12 œ œ œ œ™ œ™ œ™ 30:16 # Xœ œ ™ 5 12 œ 2 8œ 8:5 ˙ 3 8 3 8 œ™ œ™ œ œ X œ 3 20 j œ™ Reduction Layer 4 œ ? N 4 10 œ ™ 5 12 5 8 œ œ 5 12 œ œ™ 3 8 r œ œ™ 5 15:8 & # Xœ œ™ œ™ œ™ œ œ R 3 20 3 20 3 20 3 20 3 20 115 3 j & 20 œ ™ 20 3 & 20 œj™ 17 3 & 20 œj™ 13 & œ™ 9 e = 47.25 3 & 20 œj™ 5 4 & 8 ® œ™ e = 54 œ™ 5 12 œ 5 8œ 5 12 œ j œ™ 5 12 œ œ œ œ™ œ™ œ™ 30:16 # Xœ œ ™ 5 12 œ 2 8œ 8:5 ˙ œ™ 3 8 œ™ 3 8 œ™ 3 j 20 œ ™ œ ? Reduction Layer 5 (Prime) N 4 10 œ ™ 5 12 œ 5 8œ 5 12 œ r œ œ™ 3 8 œ™ 5 15:8 & #œ œ™ œ™ œ™ œ œ R 3 20 3 20 3 20 3 20 3 20 116 APPENDIX D REPRODUCTION OF "FOUR DIMENSIONAL MODELING OF RHYTHM AND METER" This appendix reproduces the author's poster from the 2014 Society for Music Theory Annual Meeting. This was the first academic presentation of the four-dimensional model described in this dissertation. The original material was presented in three columns on a three foot by four foot poster; these columns are reproduced in grayscale, one per page. The poster is reproduced exactly. In cases where the poster differs from the main text, the main text should be taken as the author's editorial correction. Specifically, this poster did not make a distinction between meter and submeter. Where the poster describes "metric quadruples" or "metric units," the correct terms are "submetric quadruples" and "submetric units." The poster is primarily reproduced to document the history of this project. Virtually all material that would be referenced from the poster is included, often edited and refined, in the text of this dissertation. However, should the reader wish to cite the poster in specific, the author recommends the following citation: Aaron J. Kirschner, "Four Dimensional Modeling of Rhythm and Meter." (presentation at the Society for Music Theory Annual Meeting, Milwaukee, WI, November 6, 2014). Four-Dimensional Mod 118 Partitioning Meter and Rhythm into Four Dimensions Partition Ferneyhoug Any rhythmic unit can be modeled in four dimensions, as defined: (each dimension is orthogonal to all others) x! =!beat-unit, represented by number of smallest pulses y! =!sub-division of beat unit (x) z! =!tempo of beat-unit (x), in beats-per-minute w!!=!duration of rhythmic structure (x,y,z), in seconds e = 54 4 & 8® n >œ ™ µ >œ ™ n>œ (6, 2, 72, .83̅) (8, 2 2 2 & 8 œ ™ n œ Bœ > > > Beginning with only dimensions x and y: >œ œ œ œ (4, 4) x=q=4 Beat unit is one quarter note, smallest pulse is sixteenth notes Beat unit is divided into four sixteenth notes y = 4×x = 4 3 Tempo is added, introducing dimension z: q = 60 ° ¢™™ >œ œ œ œ ™™ ü† (4, 4, 60) (12, 3, 91.125, . (12, 2, 7 By adding z (60 BPM), this ordered metric triple now describes tempo- but not duration-of the preceding ordered metric double. 4 5 & 12 n œ ™ Ù µ œ ™ ™ µ > > n >œ By placing the rhythmic units in context, w can be analyzed: (6, 2, 108, .5 (6, q = 60 3 œœœœœœœœœ œ œ œ œ œ œ (4, 4, 60, 2) 3 3 3 & 8 nœ nœ nœ > (4, 2, 60, 2) 5 3 & 20 œ (4, 3, 60, 1) A more complex example, with shifting tempi: q = 60 œ œ œ œ œ œ œ œ œ™ (4, 4, 60, 2) 3 5:4 4:3 œ œ œ œ œ œ™ œ œ œ œ œ œ œ (3, 3, 80, 3) (3, 4, 100, .6) (2, 3, 112.5, .4) N.B.! A lead rhythm an out- ensional Modeling of Rhythm a 119 Rhythm Partitioning the Opening of Ferneyhough's Unsichtbare Farben nsions, as defined: of smallest pulses Potential f Com Opening s e = 54 4 & 8® n >œ ™ µ >œ ™ n>œ (6, 2, 72, .83̅) r-minute y,z), in seconds 3 Bœ > nœ œ n œ B >œ n>œ B œ n>œ > > > 5 (8, 2, 81, 1.3̅) (4, 2, 135,. 8̅) µ>œ (2, 2, 216, 1.527̅)* 2 3 2 & 8 œ ™ n œ B œ nœ n œ > > > > > x=q=4 2 8 œ™ > 3 8 nœ œ œ n >œ > >œ > >œ > (2, 2, 216, 2.175̅9̅2̅) y = 4×x = 4 3 3 Si 7 3 3 5 12 & 8 nœ nœ nœ n>œ ™ Ù ™ B œ œ n>œ œ ™ µœ œ B œ ™ > > > > (12, 3, 91.125, .7̅4̅0̅) Per (x x Kr q *(4, 4, 108, .771) (12, 2, 70.875, .77̅1̅4̅2̅8̅5̅) rdered mpo- ceding 4 e x Kr q 3 5 3 & 12 n œ ™ Ù µ œ ™ n œ ™ µ>œ B œ n œ B œ B >œ >œ µ œ n œ  œ œ 20 œ > > > > > > > > (6, 2, 108, .5̅) (4 3, 162, 1.4̅8̅1̅) (6, 3, 108, .5̅) œ 5 5:3 3 , (4, 2, 162, 1.90) 3 & 20 œ n œ B >œ n œ ® µ œ B œ nœ > > , 1) 5:4 (one Graphing the tempo of the per 600 500 3 5:4 œ (4, 3, 225, .53̅) (12, 3, 105.46875, .68̅1̅4̅) 3 All BPM be analyzed: e 400 300 200 100 0 Further research possibilities cu transitional probabilities (betw developments within each dimen 4:3 œ™ œ œ œ œ œ œ œ (3, 4, 100, .6) (2, 3, 112.5, .4) N.B.! A leading asterisk indicates an out-ofrhythm pick-up; a trailing asterisk indicates an out-of-rhythm hold. Ferneyhough, Brian. "Duration and Rhyt Boros and Richard Toop. Amsterdam lecture by the author in 1989, at the N _________. Unsichtbare Farben. New York _________. Performance Notes for Super London, Justin. Hearing in Time: Psychologic g of Rhythm and Meter 120 pening of tbare Farben nœ B >œ n>œ > ) µ>œ Potential for Further Research in Computational Musicology Opening section (mm 1-22) of Unsictbare Farben, coded into ordered quadruples 2 8 œ™ > (2, 2, 216, 1.527̅)* 3 3 œ> >œ œ 8 n>œ > 5̅9̅2̅) Single-dimensional analytical possibilities 5 œ ™ µ œ 12 n>œ ™ Ù > Perceived beat units (x), by probability , 4, 108, .771) 38.56% 2! 52.10% 16.44% 3! 21.92% 13.70% 4! 20.55% x. Kr q. 10.96% 5! 5.48% (4, 2, 162, 1.90) 6.85% All others 6.85% (one occurrence each) Graphing the tempo of the perceived beat unit (i.e. value of dimension z) through the first section 500 3 œ 105.46875, .68̅1̅4̅) out-ofndicates 6.85% 600 5:4 œ nœ e. BPM 1̅) x Kr q e 3 µ>œ n>œ  œ œ 20 œ > > Beat sub-divisions (y), by probability 400 300 200 100 0 Further research possibilities currently being explored include relationships between dimensions, transitional probabilities (between and within each dimension), algorithmic modeling of developments within each dimension, computer-assisted composition, and corpus studies. Selected Bibliography Ferneyhough, Brian. "Duration and Rhythm as Compositional Resources." In Brian Ferneyhough-Collected Writings. Ed. James Boros and Richard Toop. Amsterdam: Harwood Academic Publishers GmbH, 1995. 51-65. Print. First presented as a lecture by the author in 1989, at the National Percussion Conference in Nashville. _________. Unsichtbare Farben. New York: Edition Peters. 1999. _________. Performance Notes for Superscriptio. New York: Edition Peters. 1981. London, Justin. Hearing in Time: Psychological Aspects of Musical Meter. New York: Oxford University Press. 2004. SELECTED BIBLIOGRAPHY Andersen, Arthur Olaf. Geography and Rhythm. Tucson, AZ: The University of Arizona Bulletin, 1935. Archbold, Paul. "Performing Complexity: a pedagoical resource tracing the Arditti Quartet's preparations for the première of Brian Ferneyhough's Sixth String Quartet." 2011, a c c e s s e d A p r i l 1 1 , 2 0 1 5 , h t t p : / / e ve n t s. s a s. a c. u k / u p l o a d s / m e d i a / Arditti_Ferneyhough_project_documentation.pdf. Barry, Barbara R. Musical Time: The Sense of Order. Stuyvesant, NY: Pendragon Press, 1990 Bor, Mustafa. "Contour Reduction Algorithms: A Theory of Pitch and Durational Hierarchies for Post-Tonal Music." Ph.D. diss., The University of British Columbia, 2009. Boykan, Martin. (2004). Silence and Slow Time. 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