Hexatonic systems and dual groups in mathematical music theory
Copyright policy )Hexatonic systems and dual groups in mathematical music theory
Motivated by the music-theoretical work of Richard Cohn and David Clampitt on late-nineteenth century harmony, we mathematically prove that the PL-group of a hexatonic cycle is dual (in the sense of Lewin) to its T/I-stabilizer. Our point of departure is Cohn's notions of maximal smoothness and hexatonic cycle, and the symmetry group of the 12-gon; we do not make use of the duality between the T/I-group and PLR-group. We also discuss how some ideas in the present paper could be used in the proof of T/I-PLR duality by Crans--Fiore--Satyendra.
16 pages. To appear in Involve: A Journal of Mathematics, published by Mathematical Sciences Publishers
- Michigan State University United States
- University of Regensburg Germany
- University of Michigan–Ann Arbor United States
- University of Michigan–Flint United States
dual groups, Permutation groups, 20-XX, triad, PLR-group, 00A65, 20B35 (Primary), 20-01 (Secondary), Mathematics and music, Group Theory (math.GR), centralizer, maximally smooth cycle, inversion, mathematical music theory, transposition, FOS: Mathematics, Parsifal, neo-Riemannian group, transformational analysis, simple transitivity, Mathematics - Group Theory, hexatonic cycle
dual groups, Permutation groups, 20-XX, triad, PLR-group, 00A65, 20B35 (Primary), 20-01 (Secondary), Mathematics and music, Group Theory (math.GR), centralizer, maximally smooth cycle, inversion, mathematical music theory, transposition, FOS: Mathematics, Parsifal, neo-Riemannian group, transformational analysis, simple transitivity, Mathematics - Group Theory, hexatonic cycle
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