Hexatonic systems and dual groups in mathematical music theory

Berry, Cameron; Fiore, Thomas M.

doi:10.2140/involve.2018.11.253

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Hexatonic systems and dual groups in mathematical music theory

Cameron Berry and Thomas M. Fiore

Vol. 11 (2018), No. 2, 253–270

DOI: 10.2140/involve.2018.11.253

Abstract

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 points of departure are 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, and Satyendra (Amer. Math. Monthly 116:6 (2009), 479–495).

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Keywords

mathematical music theory, dual groups, hexatonic cycle, maximally smooth cycle, triad, transposition, inversion, simple transitivity, centralizer, PLR-group, neo-Riemannian group, transformational analysis, Parsifal

Mathematical Subject Classification 2010

Primary: 20-XX

Milestones

Received: 18 February 2016

Revised: 3 January 2017

Accepted: 24 January 2017

Published: 17 September 2017

Communicated by Joseph A. Gallian

Authors

Cameron Berry
Department of Mathematics Michigan State University East Lansing, MI United States

Thomas M. Fiore
Department of Mathematics and Statistics University of Michigan-Dearborn Dearborn, MI United States	NWF I - Mathematik Universität Regensburg Regensburg Germany

Vol. 11, No. 2, 2018