5 Discussion

In this paper we have shown how meaningful parameters in the complex structure of music can be visualized, by this revealing the inter relations of music looked upon in the perspective of a certain parameter. To demonstrate the high potential of this approach we have given examples in the domain of inter-key relations based on the perspective of looking at the frequency of pitch class usage and in the domain of stylistic categorization based on a view of the key preference of the different composers. The benefit of the method reveals since the approach is simple but non the less does require almost no assumptions, neither musical knowledge, nor special artificial data. The emergence of the circle of fifths has been observed in previous work on cognitive models. In Leman (1995) artificially generated cadential chord progressions constructed from Shepard tones are used as training data. Purwins et al. (2000a) used overall averaged digitized sound samples (Chopin’s Préludes op. 28) for training. In contrast, in the present work we used accumulated vectors of each single digitized recording of the pieces in WTC as training vectors. In both Leman (1995) and Purwins et al. (2000a) the circular key structure is implicitly stipulated by the training of a toroidal self-organizing feature map. In the simulation discussed here the circularity emerges from the data alone, without an implicit assumption of periodicity in the model. In this sense, our analysis can be viewed as discovering a model of circular structure rather than merely fitting such a model.

The available data has not been exhaustively analyzed. Projections to different sub-planes could be explored and interpreted. The method can be used to model an experienced listener exposed to a new piece of music, and the listening experience in the context of a body of reference pieces. In correspondence analysis this would correspond to embedding the new pieces in a co-ordinate system obtained from analyzing the reference data. As an example, Bach’s WTC has been used to generate a tonal co-ordinate system which then served to embed a number of other works including the Chopin’s Préludes Op.  28, Alkan’s Préludes, Scriabin’s Préludes, Shostakovich’s Préludes, and Hindemith’s »ludus tonalis«. In this way the method can be used to model how a listener who is familiar with Bach’s WTC would perceive these keys and pitches in these more recent works. In addition, concepts of inter-key relations underlying Hindemith and Scriabin may be discovered.

We would like to emphasize that the use of correspondence analysis is by no means limited to tonality analysis. The method is a universal and practical tool for discovering and analyzing correspondences between various musical parameters that are adequately represented by co-occurrences of certain musical events or objects. Examples include pitch classes, keys, instrumentation, rhythm, composers, and styles. Three-dimensional co-occurrence arrays, for instance of pitch classes, keys, and metric positions can be analyzed. In particular, it seems promising to extend our analysis to temporal transitions in the space of musical parameters.