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

We will first introduce the technique of correspondence analysis with a focus on the analysis of co-occurrences of keys and pitch-classes in Section 2. In Section 3 we will present the results of our correspondence analysis of inter-key relations in scores and recorded performances, that leads to the emergence of the circle of fifths and to a toroidal model of inter-key relations. We show how these results relate to a similar model from music theory (Chew, 2000) and to earlier experiments with a different cognitive model (Purwins et al., 2000a). In Section 4 we apply correspondence analysis to the problem of stylistic discrimination of composers based on their key preference. Finally, in Section 5 we point out some relations of our results to previous work and discuss potential application to other analysis tasks arising in music theory. Please note that we provide a more technical perspective on correspondence analysis in the Appendix, Section 6.

2 Analysis of Co-occurrence

Co-occurrence data frequently arise in various fields ranging from the co-occurrences of words in documents (information retrieval) to the co-occurrence of goods in shopping baskets (data mining). In the more general case, we consider the co-occurrence of two different features. One feature $K$ is described by a vector that contains the frequencies how often it co-occurs with each specification of the other feature $P$ and vice versa. Correspondence analysis aims at embedding the features $K$ in a lower-dimensional space such that the spatial relations in that space display the similarity of the features $K$ as reflected by their co-occurrences together with feature $P$ .

Co-occurrence Table. Consider, as our running example, the co-occurrence table

HK,P = (hK,P)1<i<24 ij 1<j<12

(1)

for keys ( $K$ ) and pitch classes ( $P$ ).

	c	$...$	$...$	b	$K h$

C	$hK,CP,c$	$...$		$hKC,,Pb$	$hKC$
$...$	$...$				$...$
B	$...$				$hK B$
Cm	$...$				$K hCm$
$...$	$...$				$...$
Bm	$hK,BPm,c$	$...$		$hKB,mP,b$	$hKBm$

$hP$	$hP c$	$...$		$hP b$	n

Table $K,P H$ reflects the relation between two sets $K$ and $P$ of features or events (cf. Greenacre, 1984), in our case $K = {C,...,B, Cm,...,Bm}$ being the set of different keys, and $P = {c,...,b}$ being the set of different pitch classes. Then an entry $K,P hij$ in the co-occurrence table would just be the number of occurrences of a particular pitch class $j (- P$ in musical pieces of key $i (- K$ . The frequency $hKi$ is the summation of occurrences of key $i$ across all pitch classes. The frequency of pitch class $j$ accumulated across all keys is denoted by $hPj$ . The sum of the occurrences of all pitch classes in all keys is denoted by $n$ . From a co-occurrence table one can