1 Introduction

The correspondence of musical harmony and mathematical beauty has fascinated mankind ever since the Pythagorean idea of the »harmony of the spheres«. Of course, there exists a long tradition of analyzing music in mathematical terms. Vice versa, many composers have been inspired by mathematics. In addition, psychophysical experiments have been conducted, e.g., by Krumhansl and Kessler (1982) to establish the relation between different major and minor keys in human auditory perception by systematic presentation of Shepard tones. The results of these experiments were visualized by a technique known as multidimensional scaling which allows to construct a two-dimensional map of keys with closely related keys close by. As a central result the circle of fifths (CoF) as one of the most basic tonal structures could be reproduced. In related work, a self-organizing feature map (Kohonen, 1982) of adaptive artificial neurons was applied to similar data, and showed, how the circle of fifths could be recovered by neural self-organization (Leman, 1995). In Leman and Carreras (1997) cadential chord progressions were embedded in a self-organizing feature map trained on Bach’s »Well-Tempered Clavier« (WTC I). Based on that work, a cognitive model consisting of an averaged cq-profile ¹

extraction (Purwins et al., 2000a) in combination with a self-organizing feature map revealed the circle of fifths after training on Alfred Cortot’s recording of Chopin’s Préludes.

In this paper we extend this general idea of embedding musical structure in two-dimensional space by considering the Euclidean embedding of musical entities whose relation is given in terms of a co-occurrence table. This general approach enables us not only to analyze the relation between keys and pitch-classes, but also of other musical entities including aspects of the style of composers. We can, for instance, exploit the fact that composers show strong preferences towards particular keys. This provides the basis for arranging the composers by correspondence analysis reflecting their stylistic relations.

According to Greenacre (1984), the interest in studying co-occurrence tables emerged independently in different fields such as algebra (Hirschfeld, 1935), psychometrics (Horst, 1935; Guttman, 1941), biometrics (Fisher, 1940), and linguistics (Benzécri, 1977). Correspondence analysis was discovered not only in distinct research areas but also in different schools, namely the pragmatic Anglo-American statistical schools as well as the geometric and algebraic French schools. Therefore, various techniques closely related to correspondence analysis have been discussed under various names, e.g., »reciprocal averaging«, »optimal (or dual) scaling«, »canonical correlation analysis of contingency tables«, »simultaneous linear regressions«.