Index of content:
Volume 107, Issue 5, May 2000
- MUSIC AND MUSICAL INSTRUMENTS 
Construction and interpretation of equal-tempered scales using frequency ratios, maximally even sets, and P-cycles107(2000); http://dx.doi.org/10.1121/1.428658View Description Hide Description
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 generatingscales, this analysis may be applicable for discovering the ratio basis of unusual tunings common in non-Western music.