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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 ).
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