Vol. 11, No. 2, 2018

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ISSN: 1944-4184 (e-only)
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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
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).

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