Construction and interpretation of equal-tempered scales using frequency ratios, maximally even sets, and P-cycles.

Abstract

Using recent developments in music theory, which are generalizations of the well-known properties of the familiar 12-tone, equal-tempered musical scale, an approach is described for constructing equal-tempered musical scales (with "diatonic" scales and the associated chord structure) based on good-fitting intervals and a generalization of the modulation properties of the circle of fifths. An analysis of the usual 12-tone equal-tempered system is provided as a vehicle to introduce the mathematical details of these recent music-theoretic developments and to articulate the approach for constructing musical scales. The formalism is extended to describe equal-tempered musical scales with nonoctave closure. Application of the formalism to a system with closure at an octave plus a perfect fifth generates the Bohlen-Pierce scale originally developed for harmonic properties similar to traditional chords but without the perceptual biases of these familiar chords. Subsequently, the formalism is applied to the group-theory-based 20-fold microtonal system of Balzano. It is shown that with an appropriate choice of nonoctave closure (6:1 in this case), determined by the formalism combined with continued fraction analysis, that this group-theoretic-generated system may be interpreted in terms of the frequency ratios 21:56:88:126. Although contrary to the spirit of the group-theoretic approach to generating scales, this analysis may be applicable for discovering the ratio basis of unusual tunings common in non-Western music.