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.

中文翻译：

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用于近对称六弦的 5 维 Tonnetz

标准的二维Tonnetz描述了大三和弦和小三和弦之间的简约语音领先连接，就像 3 维Tonnetz对主要七和弦和半减七和弦所做的那样。在本文中，引入了一个 5 维的Tonnetz，为神秘-沃采克属的近对称六和弦之间的简约语音引导提供了一个几何框架。该离散网格上的点的笛卡尔坐标、对应于神秘和沃采克和弦的 5-单纯形顶点的广义坐标集合以及选定和弦的几何最近邻都被导出。