Perspectives of New Music

From a Categorical Point of View: K-nets as Limit Denotators Le?7tn GUERINO MAZZOLA AND Moreno Andreatta Introduction The shift from a set-theoretical to a transformational approach isone of the most intriguing and promising achievements of modern music theory.A study of the emergence of the group concept inmusic theory, analysis, and composition shows that the transformational per spective arises independently and almost contemporaneously in the American and European traditions in the theoretical writings of some mathematically inclined theorists and composers, in particular Wolfgang Graeser, Milton Babbitt, Iannis Xenakis and Anatol Vieru [1]. They all Froma Categorical PointofView 89 developed theoretical models of the truly musical duality between "tones" and "intervals" by considering the notion of an algebraic struc ture as a basic theoretical concept for explaining this duality. In fact, after Graeser's group-theoretical study of symmetries in Bach's Art of the Fugue, with Milton Babbitt's algebraic formalization of the twelve-tone system, Xenakis's sieve-theoretical constructions of musical (ordered) structures, and Vieru's group-theoretical description ofmodalism, music theory gradually shifts from the study of the properties of collections of elements to the study of the relations between these elements and, more crucially, to the operations upon them. Modern algebra is the natural theoretical framework for the new trans formational perspective since it enables a precise characterization of the concept of mathematical structure as a collection of elements together with (internal) relations and operations between them. The application of algebraic methods in twentieth-century music theory, analysis, and composition has therefore strong consequences in the evolution of trans formational) theory as an autonomous discipline in the music theoretical and analytical community. Such transformational methods were the basis of the categorical approach of one of the authors in his book with the programmatic titleGruppen und Kategorien in derMusik [16]. Already in these "early days" of applications ofmodern mathemat ics tomusic theory, a special process diagram typewas considered, giving rise to the so-called circle chords (theywere actually introduced by one of the authors in a university course in 1981 and published in [16]). A circle chord is a local composition1KG Z12 such that there is a not necessarily invertible affinemap g: Z12 -* Z12, the monoid (g) generated by g describing K as follows. We select a pitch class x and then successively apply all powers g to x. So we have the sequence xi _7*(A;):> Since the different entries in such a sequence are finite in number, we must have g (x) = g (x) for determined minimal k, t.This means thatwe have a diagram scheme2A 9 Vfc+1 Vk +t-2 y \ 9 9 / 9 * ^0 V \ Vk +Vk +t-1 terminating with a backwards arrow from vertex vk+t_l to vertex vk. The diagram D then associates themap g with each arrow, and we have the "network," i.e., an element of limD consisting of the given sequence 90 Perspectives of New Music s = (#,g(x\ /(#), ... /(x), ... /+t'\x)). The circle chords have been classified in [16] and yield only a small number of chords (isomorphism classes of pitch-class sets), which are very common in classical harmony, among others the major and minor triad; see Example 1. V2^1^ __ o Qo] | i m ___ t yi't W 'm ^ 'V / 4 ^tV \ > ;. It ^^3 si 6 \ ? \__ o * -?\ J^ it D_l^o ..1AM. 8 4?4 @, EXAMPLE 1: THE SIXTEEN CIRCLE CHORDS IN Z12 , TOGETHER WITH THEIR GENERATING SYMMETRY ACTION (ARROWS). THE NUMBERS REFER TO THE CLASSIFICATION OF CHORDS IN [19]. A LARGE ARROW FROM ONE CHORD TO ANOTHER MEANS THAT THE TARGET CHORD IS ISOMORPHIC TO A SUBCHORD OF THE START CHORD, AND THAT THE SUBCHORD IS GENERATED BY A SUBORBIT OF THE SAME SYMMETRY ACTION AS FOR THE SUPERCHORD This typeof transformational method is also well represented byDavid Lewin's algebraic formalizations of traditional set-theoretical concepts that lead to some constructions, such as the Generalized Interval System. These formalizations are progressively shifting the analytical perspective from the description of abstract collections of elements and their rela tions to the construction of conceptual spaces, where the properties of elements are described by the transformation groups acting...