122.1 The Torus and Music 

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The Constant Q Transform


(from B. Blankertz, H. Purwins, and K. Obermayer. Toroidal models of inter-key relations in tonal music. In VI. International Conference on Systematic and Comparative Musicology. Oslo, 1999) Emergence of ToMIR from Psychoacoustic Experiments and Analysis of Western Tonal Music

The torus on the left is a geometric representation of the inter-key relations of all major and minor keys, derived from psychoacoustic experiments by (Krumhansl and Kessler 1982) (shown in 3-dimensional representation). The torus on the right was derived solely from Alfred Cortot's recording of the Chopin Preludes, op. 28, by the technique of constant Q profiles. (shown in 2-dimensional representation: identify left and right border and upper and lower border). Deriving a ToMIR from Basic Harmonic Relations.

Strip of key, derived from the charts of key regions in (Schoenberg 1969 ).

First Step: bending the strip three times.

Second step: Double appearances of minor keys are brought to accordance.

The borders of the tube show keys that are enharmonically equal. These are united by bending the tube and gluing together its ends. This gives a toroid surface around which the circles of fifths for major and minor keys wrap three times before joining themselves.


Krumhansl, C.L. and Kessler, E.J. 1982. Tracing the dynamic changes in perceived tonal organization in a spatial representation of musical keys, In Psychological Review 89(4): 334-368.

Schoenberg, A. 1969. Structural functions of harmony. Norton, New York. 2. ed.



Impossible Correspondence Index

Copyright. Robert Grace. 2002