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

Acknowledgments

We would like to thank Ulrich Kockelkorn for proof reading and thorough advice in correspondence analysis, Mark Lindley and Hans-Peter Reutter for instruction about temperament, and Thomas Noll for inspiring discussions.

6 Appendix

6.1 Technical Details of Correspondence Analysis

In this appendix we provide some more technical details relating singular value decomposition and correspondence analysis (Greenacre, 1984; Kockelkorn, 2000). The following theorem is crucial for analyzing the co-occurrence matrix in correspondence analysis:

Theorem 1 (Generalized Singular Value Decomposition) Let $A$ be a positive definite symmetric $m × m$ matrix and $B$ a positive definite symmetric $n × n$ matrix. For any real-valued $m × n$ matrix $F$ of rank $d$ there exist an $m × d$ matrix $U = (u1,...,ud),$ a $d× n$ matrix $V = (v1,...,vd)'$ with $U'AU = V'BV = Id$ , and a diagonal $d× d$ matrix $D = (dij)$ so that:

sum d F = UDV'= dkkukv'k. (6) k=1

Cf. Greenacre (1984) for a proof. For $A = Im,B = In$ , Theorem 1 yields the ordinary singular value decomposition. If furthermore $F$ is symmetric, we get the familiar eigendecomposition.

The columns $uk$ of $U$ can be viewed as the column factors with singular values $dkk$ . Vice versa the rows $vk$ of $V$ are the row factors with the same singular values $dkk$ . The magnitude of $F$ in each of the $d$ dimensions in the co-ordinate system spanned by the factors $uk$ is then given by $dkk$ .

For the matrix of relative frequencies $P,K FP,K = (fij )$ and positive definite diagonal matrices $(FP,P)-1$ and $(FK,K)-1$ with the inverted relative frequencies of row and column features, respectively, on their diagonal, Theorem 1 yields: