A 5-dimensional Tonnetz for Nearly Symmetric Hexachords
The standard 2-dimensional Tonnetz describes parsimonious voice leading connections between major and minor triads as the 3-dimensional Tonnetz does for dominant seventh and half-diminished seventh chords. In this paper, a 5-dimensional Tonnetz is introduced, providing a geometric framework for parsimonious voice leading among nearly symmetric hexachords of the mystic-Wozzeck genus. Cartesian coordinates for points on this discretized grid, generalized coordinate collections for the vertices of the 5-simplices corresponding to mystic and Wozzeck chords, and the geometric nearest-neighbours of a selected chord are derived. Read here.
Dodecatonic Cycles and Parsimonious Voice-Leading in the Mystic-Wozzeck Genus
This paper develops a unified voice-leading model for the genus of mystic and Wozzeck chords. These voice-leading regions are constructed by perturbing symmetric partitions of the octave, and new Neo-Riemannian transformations between nearly symmetric hexachords are defined. The behaviors of these transformations are shown within visual representations of the voice-leading regions for the mystic-Wozzeck genus. Read here.