Balzano and Zweifel: Another Look at Generalized Diatonic Scales.

I. INTRODUCTION

NEARLY THIRTY YEARS AGO, an alternative basis for the derivation of the Western diatonic scale was proposed by Budden (1972), and developed more recently by Balzano (1980). This alternative basis is derived from Group Theory, rather than just-intonation ratios. The theory also predicts that other "diatonic" scales exist within microtonal scales.

This paper approaches these and other "diatonic" scales from the viewpoint of a composer seeking new materials for creative work, rather than trying for rigorous mathematical proof. The mathematical basis for the theory that follows can be found in any of the referenced texts. Instead, I have concentrated on the "topology" of the scales produced. In so doing I discovered a number of other "diatonic" scales within other equal-tempered microtonal scales.

In addition to looking at these new tonal materials, I have also taken a look at ratio-based approaches to "diatonic" scale construction and related them to Balzano's work. This has provoked some interesting ideas on the creation of tonalities from basic musical intervals and perhaps also given an insight into the persistence of the 5-limit basis for Western tonality.

Evidently, the "diatonic" scales presented here (and the "pentatonic" scales that are not) all require detailed investigation. For reasons of space I have left out some topics of discussion; (1) inclusion of them all would have extended this paper to absurd length.

In the body of this paper, I use pitch-class number notation, which can be described simply as a means of using numbers to notate the different tones of a scale of any number of pitch classes to the "octave." These numbers begin with zero (0) and proceed by integer steps to a number that is one less than the number of pitch-classes in the scale, thus making the correct number of pitch classes in all.

The "Cn scale" is a generic label applied to all equal-tempered scales of n pitch-classes. Specific instances of such scales are referred to by giving n a value. Thus, the equal-tempered scale of twelve pitch-classes is referred to as the C12 scale. Wherever the C12 scale itself is used, I use the common names for the intervals, but not letter names for the pitch-class numbers. For example the sequence of pitch-classes 0 4 8 0 represents a cycle of major thirds, beginning with pitch-class 0.