Dissertation on MOS, Steinhaus, etc.

I found this ref to Norman Carey's dissertation and thought it might be
of interest. Carey was a student of John Clough in Buffalo.

Carey, Norman, A. "Distribution Modulo 1 and Musical Scales"

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 S�s 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