The Group-theoretic Description of Musical Pitch Systems Marcus Pearce
[email protected] 1 Introduction Balzano (1980, 1982, 1986a,b) addresses the question of finding an appropriate level for describing the resources of a pitch system (e.g., the 12-fold or some microtonal division of the octave). His motivation for doing so is twofold: “On the one hand, I am interested as a psychologist who is not overly impressed with the progress we have made since Helmholtz in understanding music perception. On the other hand, I am interested as a computer musician who is trying to find ways of using our pow- erful computational tools to extend the musical [domain] of pitch . ” (Balzano, 1986b, p. 297) Since the resources of a pitch system ultimately depend on the pairwise relations between pitches, the question is one of how to conceive of pitch intervals. In contrast to the prevailing approach which de- scribes intervals in terms of frequency ratios, Balzano presents a description of pitch sets as mathematical groups and describes how the resources of any pitch system may be assessed using this description. Thus he is concerned with presenting an algebraic description of pitch systems as a competitive alternative to the existing acoustic description. In these notes, I shall first give a brief description of the ratio based approach (§2) followed by an equally brief exposition of some necessary concepts from the theory of groups (§3). The following three sections concern the description of the 12-fold division of the octave as a group: §4 presents the nature of the group C12; §5 describes three perceptually relevant properties of pitch-sets in C12; and §6 describes three musically relevant isomorphic representations of C12.