Hexatonic systems and dual groups in mathematical music theory

Abstract

Motivated by the music-theoretical work of Richard Cohn and David Clampitt on late-nineteenth century harmony, we mathematically prove that the $\mathit{PL}$-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 $\mathit{PLR}$-group. We also discuss how some ideas in the present paper could be used in the proof of $T$/$I$-$\mathit{PLR}$ duality by Crans, Fiore, and Satyendra (Amer. Math. Monthly 116:6 (2009), 479–495).