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

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expect to gain information about both sets of features, $K$ and $P$ , and about the relation between features in $K$ and $P$ , i.e., between keys and pitch classes in the example above. The relative frequency of the entries is denoted by

fiKj,P = 1-hKij,P . n

(2)

It is the joint distribution of $K × P$ . The relative frequency of column $j$ is $fPj = n1hPj .$ It is the marginal distribution of $K,P fij$ . The diagonal matrix with $fP$ on the diagonal is denoted $FP,P$ . The conditional relative frequency is denoted by

P|K=i hKij,P- fj = hKi ,

(3)

in matrix notation: $FP|K = (f P| K=i)ji j$ .

Instead of co-occurrence tables $K,P H$ of frequencies of occurrences, in the sequel, we will also consider co-occurrence tables of overall symbolic durations (cf. Section 3.1) as well as co-occurrence tables of accumulated intensities (cf. Section 3.2).

Figure 1:

Co-occurrence table (cf. Table) of Bach’s »Well-Tempered Clavier«, Fugues of Book II recorded by Glenn Gould. The keys of the 24 Fugues are labeled on the vertical axis. For each fugue the intensities are accumulated for each pitch class, calculating cq-profiles (Purwins et al., 2000b). Light color indicates high intensity. Dark color indicates low intensity. This table is analyzed in Section 3.2.

Figure 2:

Relative frequency of pitch classes $fP$ and keys $fK$ of performed WTC II, Fugues, accumulated from the co-occurrence table (Figure 1). It is remarkable that the non-diatonic notes in C-Major are the most prominent notes, as if Bach would have wanted to oppose to the emphasis of C-Major in mean tone tuning. Upper: $fP$ is the normalized vector of pitch class intensities accumulated across all fugues in WTC II. Lower: $fK$ is the normalized vector of accumulated intensity of each fugue.

2.1 Correspondence Analysis

Given a co-occurrence table $HK,P ,$ for visualization purposes we aim at displaying the features $K$ and $P$ in a two-dimensional space, such that aspects of their tonal

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