Dissertation Index

Author: Carey, Norman, A

Title: Distribution Modulo 1 and Musical Scales

Institution: University of Rochester

Begun: July 1996

Completed: February 1998


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


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


Eastman School of Music

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