Pure-Tone Approximation to Equal-Temperament: How Close is Close?
Year: 2002 Authors: Jack Douthett; Richard Krantz
Core claim
ETS variations can quantify how close a musical scale is to equal temperament and support pure-tone approximations to equal-temperament.
Topics
generated tuning systems, well-formed musical scales, equal temperament, scale equivalence, musical tuning measures
Domains
music theory, combinatorics, discrete geometry, music, sound design
Methods
literature review, equivalence of scale properties, comparative scale analysis
Media
musical scales, equal-tempered tuning, pure tones
Paper text
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BRIDGES Mathematical Connections in Art, Music, and Science
Pure-Tone Approximation to Equal-Temperament: How Close is Close?
Jack Douthett Department of Arts and Sciences TVI Community College Albuquerque, NM 87106 email: jdouthett@tvi.cc.nm.us
Richard Krantz Department of Physics Metropolitan State College of Denver Denver, CO 80217 e-mail: krantzr@mscd.edu
We, briefly, review the role of generated tuning systems in music theory and discuss several scale properties inherent in these generated systems. First explored in the ground breaking works by Carey and Clampitt (1989, 1996), the seemingly unrelated properties of: 1) Dual Symmetry [Clampitt (1997)], 2) Myhill’s Property [Clough and Myerson (1982, 1983)], 3) Spectra Widths [Carey and Clampitt (1996), and Clampitt (1997)], and 4) the Three Gap Theorem [Swierczkowski (1958) and Sos (1958) among others] will be shown to give equivalent descriptions of, so-called, well-formed musical scales - which have been shown to provide a single structural basis in music theory. The scale properties of generated systems will be discussed in the context of the well-formed scales of Carey and Clampitt. We will, then, introduce and discuss equivalent measures, so-called equal-tempered system variations (ETS variations), based on the properties of well-formed scales. It will be shown that ETS variations determine the “closeness” of a musical scale to a given equal-tempered scale and may be used to generate pure-tone approximations to equal-temperament.