The Group-Theoretic Description of 12-Fold and Microtonal Pitch Systems

It is well-known that most of the musical instruments of the orchestra are constrained to produce 12 distinct pitch classes or categories per octave. This makes it rather easy to realize music based on a 12-fold octave division with such instruments but next to impossible to realize a piece of music based on any other pitch system. With the advent of the computer, the possibilities of exploring alternative microtonal systems of octave division broaden considerably. In the face of seemingly boundless freedom of choice, what is needed is a basis for selection that will tell us which systems offer the greatest resources and will thereby be the most likely to reward our exploration. In fact, there is a deeper question than this, and that is the question of how one might appropriately describe the resources of a pitch system. To be sure, the ultimate resources of a pitch system are some function of its intervals, the primitive pairwise relations between pitches. So the question really boils down to one of how to conceive of intervals. The commonly accepted answer is that the canonical definition of an interval is to be couched in terms of a frequency ratio, moreover a ratio of powers of small integers, a mathematical object of the form 23q5r. . . , with p, q, r ranging over positive and negative integers. The resources of an equal-tempered n-fold pitch system of octave division are then a function of the between the equal log-frequency grid of the system and some set of ratios (Mandelbaum 1961; Stoney 1970; von Hoerner 1974; 1976). Certain ratios may be set aside as special in the sense that it is particularly important to approximate them closely, for example, 2-13' (p5) or 2-251 (M3). In this paper I shall argue for another way of assessing the resources of a pitch system, one that is independent of ratio concerns and that considers the individual intervals as transformations forming a mathematical group. Every equal-tempered system of n-fold octave division, as well as every system of n ratios that can be approximated by an equal-tempered system, possesses the structure of the so-called cyclic group of order n, Ci. We will examine the structure of C12 and see that it possesses rather special properties that make the sets we call diatonic scales possible. Our first major result will thus be that diatonic scales may be profitably represented in terms of C12, without recourse to ratios. Inquiring into the specific nature of C12 that supports such pitch sets will suggest a method of generalization that yields a new family of microtonal systems. Unlike the goodness-of-fit approach, which leads to systems of size 9, 31, and 41 tones, our group-theoretic concerns will suggest systems of octave division based on 20, 30, and 42 tones. Further, it will be possible to specify diatonic scale analogs in each of these systems and to say Computer Music Journal, Vol. 4, No. 4, Winter 1980, 0148-9267/80/ 040066-19 04. 00/0 C 1980 Massachusetts Institute of Technology.