Shape Computation Lab

Soundshapes

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01. Olivier Messiaen's 7 modes of limited transposition. Different colors denote different symmetry classes

02. Permutation of the vertices of the regular dodecagon

03. Complete computation of all coefficients in the expansion of the cycle index of the permutation group of the dodecagon.

04. A visual catalogue of the 224 non-equivalent scales based the equal temperament tuning (1-8): a) 1 0-note scale; 1 1-note scale; 6 2-note scales

05. A visual catalogue of the 224 non-equivalent scales based the equal temperament tuning: Tritonic scales/trichords (1-8 out of 12)

06. A visual catalogue of the 224 non-equivalent scales based the equal temperament tuning: 12 Tritonic scales/trichords (9 -12); 29 Tetratonic scales / Tetrachords (1-4)

07. A visual catalogue of the 224 non-equivalent scales based the equal temperament tuning: 29 Tetratonic scales / Tetrachords (5-12)

08. A visual catalogue of the 224 non-equivalent scales based the equal temperament tuning: 29 Tetratonic scales / Tetrachords (13-20)

09. A visual catalogue of the 224 non-equivalent scales based the equal temperament tuning: 29 Tetratonic scales / Tetrachords (21-28);

10. A visual catalogue of the 224 non-equivalent scales based the equal temperament tuning: 29 Tetratonic scales / Tetrachords (29); 38 Pentatonic scales / Pentachords (1-7)

11. A visual catalogue of the 224 non-equivalent scales based the equal temperament tuning: 38 Pentatonic scales / Pentachords (8-15)

12. A visual catalogue of the 224 non-equivalent scales based the equal temperament tuning: 38 Pentatonic scales / Pentachords (16-23)

13. A visual catalogue of the 224 non-equivalent scales based the equal temperament tuning: 38 Pentatonic scales / Pentachords (24-31)

14. A visual catalogue of the 224 non-equivalent scales based the equal temperament tuning: 38 Pentatonic scales / Pentachords (32-38); 50 Hexatonic scales / Hexachords (1)

Athanassios Economou

2006

Keywords: Cube; Hexahedral group; Permutation groups; Cycle index; Symmetry; Configuration;

“[The] charm of impossibilities… at once voluptuous and contemplative, resides particularly in certain mathematical impossibilities of the modal and rhythmic domains. Modes which cannot be transposed beyond a certain number of transpositions, because one falls again into the same notes; rhythms which cannot be used in retrograde, because in such a case one finds the same order of values again” (Messiaen, The Technique of My Musical language, 1956). The symmetry properties of the equal temperament scale are computed here using Polya’s theorem of enumeration of non-equivalent configurations with respect to the permutation group of the vertices of the dodecagon. All scales are diagrammatically represented as colored polygons within the regular dodecagon and accompanied by their respective representation in typical western notation in their first transposition. The color in the first set of diagrams classifies the scales in terms of their number of notes and in the second set of diagrams in terms of their symmetry properties.

Acknowledgments. The illustration and sonification of the patterns has been done by Haldoun Kececigil and Heath Washburn.