Page  597 ï~~ FRACTAL DEPTH STRUCTURE OF TONAL HARMONY Thomas Noll Technical University Berlin, Research Center for Semiotic Studies Tel: +4930 314 25403, E-Mail: noll@cs.tu-berlin.de ABSTRACT: The paper presents an algebraic/geometric approach for the morphological study of Western Harmony. Within the framework of Mathematical Music Theory, chords are studied by means of affine endomorphisms of the 12-tone system, which we call fractal tones. The consideration of fractal chords allows an intensional study of harmonic functions. In order to signify a tonal function, the fractal closure of a given chord must contain significant fractal tones. The classical contrapuntal consonance/dissonance dichotomy of intervals gives rise to a similar(!) dichotomy of fractal tones. Harmonic thinking in just intervals and the 12-tone system are related to each other through an"uncertainty relation" in the symmetry group of the Euler Algebra. We present the hypothesis that the commutators of this symmetry group SL2(Z) describe the cognitive counterparts of what musicians call commata. 1. Fractal chords and tonal functions. In this first paragraph chords are considered to be simply subsets of the 12-tone system, which is described by the ring T = Z/12Z of integers Z modulo 12. There are 144 affine endomorphisms of the 12-tone system A= {[a, b]: T->TI [a, b](t)=at+b; (a,be T)) which we call fractal tones, since endomorphisms represent self-similarities. The composition of two fractal tones is given by the formula [a2,b2][ai,bi]=[a2a1, a2bi+b2]. Semigroups of fractal tones are called fractal chords. For any (nonempty) chord XcT we consider the fractal closure A(X) = {$ E Al $(X) c X). This notion makes sense because,,ordinary tones" - the elements of T - may be identified with the constant (fractal) tones of type [0,b]. Besides the constant tones, the halfconstant tones [6,b] and the four types of symmetry tones [1,b], [11,b], [5,b], [7,b] there are two more types of fractal tones, which are essential for the intensional study of tonal functions: minus tones [a,b] with a = 3, 9 and plus tones [a,b] with a = 2, 4, 8, 10. The significants F of tonal functions are fractal chords F = <a, 3>, generated by a minus tone a and a plus tone $. The evaluation of whether a given chord X may signify the tonal function F is characterized by the position of the intersection A(X)nF within the Hasse-diagramm of all subsemigroups of F - the intension of F. In the case of the Tonic in C-Major we have a = [3,1] and $ = [8,4]. F = <a, Â~> contains three constant tones: c = [0,0], g = [0,1], e = [0,4]. The Parallel- and the,,Gegenklang" functions correspond to <D>= {D, $2} and <a> = {a, a2}, the mediants correspond to a2 and to $2. The paradigmatic relation of two chords X and Y may be studied by considering the set A(Y,X) = ($ e Al $(Y) c X) of all fractal tones mapping Y into X. In the case of the,,fifth-fall" of any major chord X (say X = (1,2,5), Y = (0,1,4)) this set A(Y,X) generates fractal chord KONS = {[a,b] e A I a e {0,1,3,4,8,9} }of 72- elements, which we call the consonant fractal tones. As a surprising fact one should notice, that {0,1,3,4,8,9) represent the consonant intervals according to the Fuxian dichotomy (cf. Mazzola 1990). I C M C P R O C E E D I N G S 19 9 5 597

Page  598 ï~~ 2. Enharmonicity. The description of harmonic thinking in just octaves, fifths and thirds is based on the study of a 3-dimensional lattice E spanned by o =1n2, q = 1n3 and t = 1n5 over the ring Z of integers - the Euler module. Within E one has the comma-sublattice K, spanned by o, p = 12q and s = 40 - 4q + t. Enharmonic and octave identification is then described by the projection map enh: E-+E/K, where E/K is a finite cyclic lattice of order 12. Since E is a dense subset of the reals R, the traditional descriptive or normative characterisation of commata as accoustic compromizes lacks explanatory power. The following considerations suggest a plausible way to explain the phenomenon of enharmonicity. With respect to the basis o, q and r = 4o -5q +t we identify the Euler module E with the additive structure of the special linear algebra s12(Z) = {[mkkmneZ} by setting: o 1 00q= I r [= []i The Lie bracket [x y]:= xy - yx then also induces a Lie algebra structure on E, which then shall be called Euler algebra. Let F denote the sublattice of E spanned by q and r. The projection proj: E->F correspondes to the identification of octave classes. On F we define the exponential function exp: F-*SL2(Z) (Special Linear Group ) exp([0-1 = [1 and expr 0 = -111 0 0 0 1 L 0 0 For arbitrary x = nq + mr we set exp(x):= exp(q)nexp(r)m. The elements of G = SL2(Z) may be interpreted as inner symmetries of the Euler algebra. This symmetry group G is non commutative. It's commutator subgroup [G,G] generated by the commutators [a, $] = afIa~1$~1 is of index 12 (cf. Noll 1994: 119ff.). Via the exponential function commata (and only commata) correspond to commutators of G. Hence the abstract 12-tone system appears as the homology group of G, i.e. as a group of symmetry classes of the Euler algebra due to the uncertainty relation of non commutative symmetries. The latter somehow may be related to the processuality of cognitive operations. 3. Remark: The implementation of this theory within the Zurich RUBATOÂ~-project (cf. Mazzola 1993/94) will provide a powerfull tool for the experimental study of harmony in relation to performance. 4. References: Mazzola, Guerino (1990), Geometrie der Tone. Birkh"user Basel. Mazzola, Guerino (1993/94), Geometry and Logic of Musical Performance I/Il. University of Zurich Mazzola, Guerino & Thomas Noll (1995), Detecting harmonic functions: fractal harmony and the theory of analytical weights. (to appear) Noll, Thomas (1994), Morphologische Grundlagen der abendlandischen Harmonik. Dissertation: Technical University of Berlin Noll, Thomas (1994),,,Form und Prozessualitut" in: Klaus Robering(ed.), Zur Semi otik von Worn, Ton und Karte. Working papers in Linguistics: Institut for Linguistics. 598 I C M C P R O C E E D I N G S 1 9 9 5