GeneralizedTonnetzeandZeitnetze, and the topology of music concepts

Abstract

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 twelve 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. The present 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 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 a common types of cyclic rhythms or timelines. Their different topologies -whether orientable, bounded, manifold, etc. -reveal some of the topological character of musical concepts.

Wavescapes: A visual hierarchical analysis of tonality using the discrete Fourier transform

“…By symmetry, the coefficients for k ∈ false[ 1 , … , 5 false] are conjugate to the ones for k ∈ false[ 11 , … , 7 false] while the sixth coefficient is its own conjugate, X false[ k false] = true X ¯ false[ 12 − k false] for k ∈ false[ 1 , … , 11 false] . Therefore, we consider only the coefficients 1 to 6 in accordance with previous research (Amiot, 2007; Yust, 2019a).…”

“…It has also been applied to the study of changes in pitch-class distributions, both within pieces (Harding, 2020) and between compositions from different historical periods (Yust, 2019b). Moreover, several connections have been drawn between the DFT of pitch-class sets and geometric approaches to harmony such as voice-leading spaces (Hoffman, 2008;Tymoczko, 2008;Tymoczko & Yust, 2019) and the Tonnetz (Yust, 2019a).…”