Hexatonic Systems and Dual Groups in Mathematical Music Theory

TL;DR

This paper explores the mathematical structure of hexatonic systems in music theory, proving a duality between certain transformation groups inspired by late-19th-century harmony analysis.

Contribution

It establishes a new duality result between the PL-group of a hexatonic cycle and its T/I-stabilizer, extending theoretical understanding of musical symmetries.

Findings

Proves PL-group of hexatonic cycle is dual to its T/I-stabilizer

Uses symmetry group of the 12-gon and Cohn's concepts of maximal smoothness

Provides insights that could aid in proving T/I-PLR duality

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 point of departure is 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--Satyendra.