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The Symmetry of the Equal Temperament Scale
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Mathematics and Design 98: Proceedings of the Second International Conference
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The University of the Basque Country, San Sebastian, Spain
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Musical scales, Equal temperament, Symmetry, Enumeration, Polya’s Theorem of Counting, Olivier Messiaen
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The structure of the dodecagon is postulated as a model for the structure and the symmetry properties of the equal temperament scale. The cycle index of the permutation group of the vertices of the dodecagon is used in Polya's theory of counting configurations non-equivalent with respect to a given permutation group; the numbers of structurally differentiated scales with n = 0,1,2,...,12 notes are specified. The symmetry properties of the regular n-gons for n≤6 are used in the decomposition of the scalar patterns. A complete enumeration of all scales, illustrated by a mapping of the scalar patterns on the plane, is presented in the end.
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