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Constant Q Profiles and Toroidal Models of Inter-Key Relations
Citation key Blankertz1999a
Author Blankertz, B. and Purwins, H. and Obermayer, K.
Title of Book Proceedings of CMI-99, Conference on Musical Imagery, VI. International Conference on Systematic and Comparative Musicology, Oslo
Year 1999
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 ) 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). 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. \\newline \\newline The chart of key regions in Schoenberg 69 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. Shepard tone cadences of the form I-IV-V7-I are filtered by an auditory model, autocorrelated, and then clustered by training a SOM (Leman 95). Again a ToMIR (ii) emerges. The constant Q transform is employed for deriving the ToMIR in (iii), and for key analysis. The algorithm is efficiently implemented by calculating the kernel of the constant Q filters in the frequency domain in advance and exploiting the sparsity of that kernel. The 12-dimensional cq-profile is the concentrated spectral information supplied by the constant Q transform. Each component of the profile indicates the strength of one pitch class. A cq-profile is calculated by summing up all values (bins) of the constant Q transform that belong to the same pitch class. Cq-profiles have the following advantages: (a) The 12 components correspond to values in probe tone ratings. In a psychological probe tone experiment a quantitative description of a key is derived from rating the different pitch classes within the tonal context of that key (Krumhansl \& Kessler 82). (b) Calculation is possible in real-time. (c) Stability is obtained with respect to sound quality. (d) Cq-profiles are transposable. Another application is automated modulation tracking.
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