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