## Dissertation Index

 Author: Carey, Norman, A Title: Distribution Modulo 1 and Musical Scales Institution: University of Rochester Begun: July 1996 Completed: February 1998 Abstract: This dissertation examines the relationships between the mathematics of distribution modulo 1 and the theory of well-formed scales. Distribution modulo 1 concerns the distribution of real numbers between 0 and 1. In particular, finite sets of real numbers have been studied with respect to the Steinhaus Conjecture, proven by Sós and others. Well-formed scales, first introduced in Carey and Clampitt 1989, are generated by iterations of a given musical interval modulo the octave, the standard musical interval of periodicity. An introductory survey of ten scale theorists provides a context in which to understand the properties of the well-formed scale. A scale is well-formed if each generic interval comes in two specific sizes, or if it consists of equal step intervals. The structure of the well-formed scale is a function of the continued fraction representing the log ratio of the generator ("fifth") and the interval of periodicity ("octave"). The diatonic scale in Pythagorean tuning serves as the prototype: the generator is the overtone fifth (3:2) and the interval of periodicity is the octave (2:1). The diatonic is a member of an infinite hierarchy of well-formed scales, recursively generated by the continued fraction of Log 2 (3/2). This hierarchy also includes the pentatonic and chromatic collections. In general, the well-formed scale belongs to a hierarchy determined by the continued fraction of, Log I (G), where I is the frequency ratio of the interval of periodicity and G is the frequency ratio of the generator. Five theorems are presented that characterize well-formed scales, their hierarchies, and the patterns of step intervals they exhibit. The step patterns themselves form the basis for a secondary system of well-formed scale classification. The conditions on "coherence" for well-formed scales are fully characterized. Also discussed are applications and extensions of the theory, including tuning theory, rhythmic analysis, and composition. Keywords: scale theory, well-formed, maximally even, Myhill's Property, diatonic, coherence, microtonal, rhythm, distribution modulo 1, continued fractions TOC: I Diatonic Theory A Introduction B Foundational and Structural Properties C Definitions D Diatonic Theory - Antecedents E Three Diatonic Theories II Well-formed Scales A Diatonic Theory and Well-formed Scales B The Theory of Well-formed Scales III Five Theorems Concerning Well-formed Scales A Introduction B Distribution Modulo 1: The Three-Gap Theorem C Well-formed Scales and the Multiplicative Permutation D Symmetry and Closure E Well-formed Scales and Myhill's Property F The Well-formed Scale Sequence G Generic Ordering ("Coherence") IV Applications and Extensions A Microtonalism and Well-formed Scales B Rhythm and Well-formedness C Diatonic Theory and Compostion D Conclusion and Prospects Contact: Eastman School of Music nac@theory.esm.rochester.edu