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Toroidal models of inter-key relations in tonal music

Title Toroidal models of inter-key relations in tonal music
Publication Type Conference Paper
Year of Publication 1999
Conference Name VI. International Conference on Systematic and Comparative Musicology
Authors Blankertz, B. , Purwins H. , & Obermayer K.
Conference Location Oslo
Abstract We show three different derivations of toroidal models of inter-key relations (ToMIR): (i) geometric explanation, (ii) emergence in the Self Organizing Feature Map (SOM, Kohonen 82 [2]) trained by Shepard cadences previously processed by an auditory model, (iii) emergence in a SOM trained by averaged constant Q (cq-)profiles of Chopin's preludes recorded by Cortot (1933/34). In method (iii) the cq-profiles are 12-dimensional vectors, each component referring to a pitch class. They can be employed to represent keys. Cq-profiles are calculated with the constant Q filter bank (Brown & Puckette 92 [1]). This filter bank gives equal resolution for all regions in the logarithmic frequency domain. The cq-profiles are also used for key recognition, and for investigating pitch use in Bach, Chopin, and Alkan. The chart of key regions in Schoenberg 69 [5] emphasizes dominant, subdominant, relative, and parallel relationships. A ToMIR is geometrically derived assuming every element in an arrangement of keys to be the center of its own chart of key regions (i), and to allow enharmonically equivalent keys to occupy the same position in the arrangement. These considerations are in accordance with a specially designed algorithm. The latter brings keys spatially close together in a configuration on a toroidal surface according to a given set of key pairs that have been previously determined to be closely related.
preprint/postprint document