Mathematics and the Twelve-Tone System: Past, Present, and Future

Mathematics and the Twelve-Tone System: Past, Present, and Future Robert Morris Introduction Certainly the first major encounter of non-trivial mathematics and non-trivial music was in the conception and development of the twelve-tone systemfrom the 1920s to the present. Although the twelve tone systemwas formulated byArnold Schoenberg, itwas Milton Babbitt whose ample but non-professional background inmathematics made it possible for him to identify the links between themusic of the Second Viennese School and a formal treatment of the system.To be sure, there were also important inroads inEurope aswell,1 but thesewere not often marked by the clarity and rigor introduced by Babbitt in his series of seminal articles from 1955 to 1973 (Babbitt 1955, 1960, 1962, 1974). Mathematics and the Twelve-Tone System 77 This paper has four parts. First, I will sketch a rational reconstruction of the twelve-tone system as composers and researchers applied mathematical terms, concepts, and tools to the composition and analysis of serial music. Second, I will identify some of the major trends in twelve-tone topics that have led up to the present. Third, I will give a very brief account of our present mathematical knowledge of the system and the state of this research. Fourth, Iwill suggest some futuredirections aswell as provide some open questions and unproven conjectures. But before I can start,we need to have a working definition of what the twelve-tone system is, ifonly tomake this paper's topic manageable. Research into the system eventually inspired theorists to undertake formal research into other types of music structure; moreover of late, such research has now actually identified twelve-tone music and structure as special cases of much more general musical and mathematical models, so that for instance serialmusic and Riemannian tonal theory are both aptly modeled by group theory, rather than demarked as fundamentally different or even bi-unique. Thus I will provisionally define the twelve-tone system as themusical use of ordered sets of pitch classes in the context of the twelve-pitch class universe (or aggregate) under specified transformations that preserve intervals or other features of ordered sets or partitions of the aggregate. Thus the row, while itonce was thought to be the nexus of the system, isonly one aspect of thewhole?even ifthe row embraces all of the characteristics I've mentioned: the aggregate, ordered pc sets and, not so obviously, partitions of the aggregate. Thus an object treated by the twelve-tone system can be a series or cycle of any number of pitch classes, with or without repetition or duplication, as well as multi dimensional constructs such as arrays and networks, or sets of unordered sets that partition the aggregate. The Introduction of Math into Twelve-Tone Music Research Schoenberg's phrase, "The unity of musical space," while subject to many interpretations, suggests that he was well aware of the symmetries of the system (Schoenberg 1975). In theoretic word and compositional deed he understood that therewas a singular two-dimensional "space" inwhich his music lived?that is, the space of pitch and time. Indeed, the basic transformations of the row, retrograde and inversion, plus ret rograde-inversion for closure (and P as the identity) were eventually shown to form a Klein four-group (see Example 1). That this space is not destroyed or deformed under these operations 78 PerspectivesofNew Music P R I RI R P RI I I RI P R RI I R P P! fe IJU _ m w_m\m * -*"* 01627934AB58 R! ? _ #t* .. * Ji-_ 85BA43972610 I: (| ^ ?j,?^ .^ ,fl? ^ p 0 B 6A5 3 9 8 2 1 7 4 RIs fe ^ ?* m\ -V+hm # 47128935A6B0 EXAMPLE 1 :THE SERIAL FOUR-GROUP gives itunity. Yet, from today's standpoint, the details of this symmetry are quite unclear. What kinds of pitch spaces? Pitch, or pitch-class, or merely contour? Is I mirror inversion or pitch-class inversion? Is RI a more complex operation than I or R alone? What about transposition's interaction with the P, I, R, RI group? And so forth.The lack of clarity, which is actuallymore equivocal than I've mentioned (because there isno acknowledgment of the different conceptions of intervals between...