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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
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Abstract |
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Motivated by the music-theoretical work of Richard Cohn and David Clampitt on late-nineteenth century harmony, we mathematically prove that the -group of a hexatonic cycle is dual (in the sense of Lewin) to its /-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 /-group and -group. We also discuss how some ideas in the present paper could be used in the proof of /- 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
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Mathematical Subject Classification 2010
Primary: 20-XX
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Milestones
Received: 18 February 2016
Revised: 3 January 2017
Accepted: 24 January 2017
Published: 17 September 2017
Communicated by Joseph A. Gallian
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Authors |
| Cameron Berry | |
| Department of Mathematics Michigan State University East Lansing, MI United States |
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| Thomas M. Fiore | |
| Department of Mathematics and
Statistics University of Michigan-Dearborn Dearborn, MI United States |
NWF I - Mathematik Universität Regensburg Regensburg Germany |
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