Why topology?

Abstract

Music theorists have modeled voice leadings as paths through higher-dimensional configuration spaces. This paper uses topological techniques to construct two-dimensional diagrams capturing these spaces' most important features. The goal is to enrich set theory's contrapuntal power by simplifying the description of its geometry. Along the way, I connect homotopy theory to "transformational theory," show how set-class space generalizes the neo-Riemannian transformations, extend the Tonnetz to arbitrary chords, and develop a simple contrapuntal "alphabet" for describing voice leadings. I mention several compositional applications and analyze short excerpts from Gesualdo,

Generalizing the Orbifold Model for Voice Leading

“…The possibility of associating multiple, distinct members of the groupoid π orb (X/G) to the same pair of voicings in X is one of the main differences between the orbispace path approach to voice leading and other approaches. As noted in [17], this possibility is occasionally counterintuitive, but it is a natural, musically relevant consequence of the idea of "continuous transformations" articulated in [4], and adopted in [5]. We demonstrate this with an example.…”

“…The orbispace path groupoid can be regarded as a refinement of the fundamental groupoid of a space, analogous to how the orbifold fundamental group is a refinement of the ordinary fundamental group of a space. We use the term "orbispace path groupoid" instead of "orbifold fundamental groupoid" to help prevent inaccurate conflation of terms of the sort that occurs in [17] with "fundamental group" and "orbifold fundamental group".…”

“…Remarkably, the mathematically motivated characterization of elements of π orb (X/G) as products of the form ((x, y), 1) * ((y, y), g) in T(X/G) G is the mathematical formulation of the musically motivated method described on pages 122, 130, 135, and elsewhere in [17] (which in turn references earlier work of Roederer) for dealing with voice leadings between different chords. In that method, one chooses an arbitrary, fixed voice leading between the two chords, and factors any given voice leading between them as the chosen voice leading followed by a voice leading from the second chord to itself.…”

“…A specific choice of basepoint x0 ∈ X for a chord x ∈ X/G is a choice of a specific voicing of the chord x-that is, a specific ordering (part assignment) of the pairs in x and, for each pair, a choice of octave (register). As noted above, it is an interesting and important fact that a collection of such choices for every element of X/G is the same as a choice of a fundamental domain for the action of G on X. Insofar as fundamental domains are used in [16,17] to study quotient spaces, basepoint selection should appear as a natural and familiar step. It is also important to note that basepoint selection is to be done once and remain unchanged throughout a discussion.…”

Characterisation of the Degree of Musical Non-Markovianity

“…They are used both as an aid for musical analysis, and to investigate some aspects of samples of compositions (Mazzola, 2007;Mazzola, 2002). In particular, these methods have been applied to analyse: musical theory in a geometrical way (M. et al, 2006); gesture in music (Mazzola & Andreatta, 2007;Mannone, 2018a;Arias, 2018); counterpoint; interpretation; harmony (Mazzola, 2002); common patterns in sound amplitude (Mendes et al, 2011); chord-sequences and voice-leading in the light of topology (Tymoczko, 2006;Tymoczko, 2020); and symmetries between chords in lattices called Tonnetze (Amiot, 2017;Jedrzejewski, 2019b). Formal tools also include using the abstract power of category theory in order to formalise musical structures, including relationships within music theory and between music theory and musical performance (Popoff et al, 2019;Mannone, 2018b;Mazzola, 2002;Mannone, 2018a;Arias, 2018;Jedrzejewski, 2019a).…”