Involve: A Journal of Mathematics

  • Involve
  • Volume 11, Number 2 (2018), 253-270.

Hexatonic systems and dual groups in mathematical music theory

Cameron Berry and Thomas M. Fiore

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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).

Article information

Source
Involve, Volume 11, Number 2 (2018), 253-270.

Dates
Received: 18 February 2016
Revised: 3 January 2017
Accepted: 24 January 2017
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513775061

Digital Object Identifier
doi:10.2140/involve.2018.11.253

Mathematical Reviews number (MathSciNet)
MR3733956

Zentralblatt MATH identifier
06817019

Subjects
Primary: 20-XX

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

Citation

Berry, Cameron; Fiore, Thomas M. Hexatonic systems and dual groups in mathematical music theory. Involve 11 (2018), no. 2, 253--270. doi:10.2140/involve.2018.11.253. https://projecteuclid.org/euclid.involve/1513775061


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