Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

Morphology of Chords

Thomas Noll and Monika Brand

Technical University of Berlin

Research Group KIT-MaMuTh for Mathematical Music Theory $*$

noll@cs.tu-berlin.de, mona.brand@gmx.de

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

Abstract

This article provides an introduction to basic morphological investigations of the 12-tone sytem and its subsets, the chords. The various definitions and results are intended to lay a theoretical basis for 12-tone-based explorative and empirical research on occidental harmony. The entire approach is motivated by the assumption that the 12 tones do not constitute an arbitrary 12-element set with arbitrary paradigmatic relations, but rather a discrete homogeneous space. Transpositions obviously are instances of paradigmatic relations, that are reflected by affine transformations. From the mathematical point of view, the article presents a straightforward elaboration of this observation. From the music theoretical point of view it is of speculative nature and asks for experimental evaluation in musical analysis. The Computer-Programm jMorph has been developed for such experiments.

1 Introduction

This article is based on an earlier condensed and purely mathematical paper, Noll (2002). This extension now includes considerations upon its music-theoretical motivation, by tracing some guiding ideas in the work of other authors and by indicating possibilities of applying its approach experimentally to musical analysis with the help of a software tool jMorph.

1.1 Homogeneity and Temperament

The point of departure of these investigations is the cyclic homogeneous twelve-tone-system. We prefer the attribute »homogeneous« instead of »equal-tempered« because the latter implies references to the physical description of tuning practice, which is not subject of this study. The music-theoretical applicability of a homogenous twelve-tone-system is a controversial issue, which cannot not be decided on the basis of tuning arguments: It is neither self-evident that a harmonic theory for piano music (written after 1830) must be based on a homogenous twelve-tone system, just because of the practice of equal temperament and likewise it is questionable to discard a homogeneous 12-tone approach for keyboard music (written