GeneralizedTonnetze

Abstract

Abstract. We study a generalization of the classical Riemannian Tonnetz to N -tone equally tempered scales (for all N ) and arbitrary triads. We classify all the spaces which result. The torus turns out to be the most common possibility, especially as N grows. Other spaces include 2-simplices, tetrahedra boundaries, and the harmonic strip (in both its cylinder and Möbius band variants). The final and most exotic space we find is something we call a 'circle of tetrahedra boundaries'. These are the Tonnetze for spaces of triads which contain a tritone. They are closely related to Peck's Klein bottle Tonnetz.

Filtration of Pitch-Class Sets Complexes

“…Geometrical simplicial complexes have been used to represent musical objects, in particular pitch-class sets [18,6,5] and largely applied to stylistic and transformational analysis [4].…”

“…We use labelled geometric simplices to represent pitch-class sets and labelled geometric simplicial complexes to represent collections of pitch-class sets. This approach goes back to Guerino Mazzola's Mathematical Music Theory [17,18] which has been enriched recently to study aspects of generalized Tonnetze [6,5]. A simplex of dimension n, or n-simplex, represents a pitch-class set of size n + 1.…”

“…The values taken by the cursor run from 0 to 1 to stick with the definition of the normalized cumulative duration of pitch-class sets. 6 https://gitlab.com/lbigo/pitchsalami 7 http://lbigo.gitlab.io/ChordComplexViz Filtration of Pitch-class sets complexes 13…”

Topology of musical data

“…where σ c ranges over all cyclic shifts and d N (·, ·) is the distance defined in (1). Under this measure, all three of the named rhythms in Figure 1 are the same since they are cyclical shifts of each other.…”

“…To name a few examples: musical scores present notes in a two-dimensional array, Euler's classical Tonnetz represents musical intervals in a lattice as do modern generalizations [1], Partch [2] draws the tonality diamond where the two axes represent significant musical intervals, Chew [3] visualizes musical progressions along a spiral array, and computerbased music visualizers [4] display moving patterns in real time as music progresses. Lewin [5] suggests the application of simplicial complexes and homology to music theory and Mazzola [6] considers actions by the symmetric group on n-tuples.…”