The music-theoretic idea of a Tonnetz can be generalized at different levels: as a network of chords relating by maximal intersection, a simplicial complex in which vertices represent notes and simplices represent chords, and as a triangulation of a manifold or other geometrical space. The geometrical construct is of particular interest, in that allows us to represent inherently topological aspects to important musical concepts. Two kinds of music-theoretical geometry have been proposed that can house Tonnetze: geometrical duals of voice-leading spaces and Fourier phase spaces. Fourier phase spaces are particularly appropriate for Tonnetze in that their objects are pitch-class distributions (real-valued weightings of the 12 pitch classes) and proximity in these space relates to shared pitch-class content. They admit of a particularly general method of constructing a geometrical Tonnetz that allows for interval and chord duplications in a toroidal geometry. This article examines how these duplications can relate to important musical concepts such as key or pitch height, and details a method of removing such redundancies and the resulting changes to the homology of the space. The method also transfers to the rhythmic domain, defining Zeitnetze for cyclic rhythms. A number of possible Tonnetze are illustrated: on triads, seventh chords, ninth chords, scalar tetrachords, scales, etc., as well as Zeitnetze on common cyclic rhythms or timelines. Their different topologies – whether orientable, bounded, manifold, etc. – reveal some of the topological character of musical concepts.

Tonnetz的音乐理论概念可以在不同的层面上推广：作为通过最大交点关联的和弦网络，以顶点表示音符而简单表示弦的单纯形复形以及流形或其他几何空间的三角剖分。几何构造特别令人感兴趣，因为它使我们能够代表重要音乐概念的固有拓扑方面。已经提出了可以容纳Tonnetze的两种音乐理论几何形状：语音引导空间和傅立叶相位空间的几何对偶。傅里叶相空间特别适合于Tonnetze因为它们的对象是音调等级分布（12个音调等级的实值加权），并且在这些空间中的接近度与共享的音调等级内容有关。他们承认构造几何Tonnetz的一种特别通用的方法，该方法可以在环形几何体中进行音程和和弦重复。本文研究了这些重复如何与重要的音乐概念（例如键或音高）相关联，并详细介绍了一种消除此类冗余以及由此导致的空间同源性变化的方法。该方法还转移到节奏域，为循环节奏定义Zeitnetze。图中显示了许多可能的Tonnetze：在三合音，第七和弦，第九和弦，标量四和弦，音阶等上，以及Zeitnetze上常见的循环节奏或时间表。它们的不同拓扑结构-是否可定向，有界，多方面等-揭示了音乐概念的某些拓扑特征。