relation are reflected by their spatial configuration. In particular, correspondence analysis can be thought of as a method that aims at finding a new co-ordinate system that optimally preserves the -distance between the frequency -, and -profiles, i.e., of columns and rows. For 12-dimensional pitch class frequency vectors and the -distance is defined by

(4)

with a generalized inner product defined by

where denotes the transpose of vector . The -distance is equal to the Euclidean distance in this example if all pitch classes appear equally often. The -distance weights the components by the overall frequency of occurrence of pitch classes, i.e., rare pitch classes have a lower weight than more frequent pitch classes. The -distance satisfies the natural requirement that pooling subsets of columns into a single column, respectively, does not distort the overall embedding because the new column carries the combined weights of its constituents. The same holds for rows.

We can explain correspondence analysis by a comparison to principal component analysis. In principal component analysis eigenvalue decomposition is used to rotate the co-ordinate system to a new one with the axes given by the eigenvectors.