Andreas Nestke

Technical University of Berlin

Research Group KIT-MaMuTh for Mathematical Music Theory

nestke@t-online.de

Financed by the Volkswagen-Foundation in its »Young Research Groups at the Universities« programm.

Abstract Paradigmatic Motivic Analysis (PMA) aims at finding, classifying and, thereby, helping to understand structural relations within the variety of subsets of a single or between those of different musical pieces. According to Mathematical Music Theory as developed by G. Mazzola, a paradigm is determined by the geometric transformations from a suitably chosen subgroup of the affine group acting on the ambient space, describing the so-called Gestalt, in which the piece is modeled. The particular point of view taken here reduces a given musical melody to a (finite) set of rational points in the affine plane (the components of a point being onset and pitch), which are strictly ordered by their first components. Moreover, choosing the positive generator of the additive group generated by all onset differences of the notes occurring in the melody to be analyzed as a unit, the points in the plane can be assumed to have integer coordinates. Then, for a single melody the PMA investigates the inner symmetries of a finite set of points with integer coordinates in the affine plane, not two of which lie in a vertical line. An inner symmetry with respect to a given subgroup of the planar affine group is constituted by a pair of subsets of the melody, each containing at least two points, for which there exists a transformation in the group mapping one subset onto the other. Starting from this set of symmetry pairs we introduce several numerical and other characteristics, each intended to grasp pieces of information concerning symmetry and measure its extent within a melody.

1 Introduction

One of the fundamental convictions, lying at the basis of mathematical investigations into nature as well as art, is that the notion of symmetry plays a central role in understanding the laws and rules governing both. According to H. Weyl (see Weyl, 1981) symmetry is “that kind of concordance of several pieces through which they unite to form a whole”. A. Speiser (see Speiser, 1945) defined it as “the coherence of different pieces of a whole”. In Mazzola and Zahorka (1996) described Paradigmatic Motivic Analysis (PMA) as a mathematical approach to the study of symmetries in music based on the notions of Gestalt, distance and topology. In