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Computation and Visualization of Musical Structures in Chord-Based Simplicial Complexes

Louis Bigo 1, * Jean-Louis Giavitto 1, 2 Moreno Andreatta 1 Olivier Michel 3 Antoine Spicher 3
* Corresponding author
1 STMS - Sciences et Technologies de la Musique et du Son
2 MuTant - Synchronous Realtime Processing and Programming of Music Signals
Inria Paris-Rocquencourt, UPMC - Université Pierre et Marie Curie - Paris 6, IRCAM, CNRS - Centre National de la Recherche Scientifique
3 LACL - Laboratoire d'Algorithmique Complexité et Logique
Abstract : We represent chord collections by simplicial complexes. A temporal organization of the chords corresponds to a path in the com- plex. A set of n-note chords equivalent up to transposition and inversion is represented by a complex related by its 1-skeleton to a generalized Ton- netz. Complexes are computed with MGS, a spatial computing language, and analyzed and visualized in Hexachord, a computer-aided music anal- ysis environment. We introduce the notion of compliance, a measure of the ability of a chord-based simplicial complex to represent a musical object compactly. Some examples illustrate the use of this notion to characterize musical pieces and styles.
Keywords : chord spaces music theory simplicial complexes generalized Tonnetze compli- ance Hexachord MGS musical transformation harmonization counterpoint self-assembly Tonnetz lattice simplicial complex formalization of music
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Louis Bigo, Jean-Louis Giavitto, Moreno Andreatta, Olivier Michel, Antoine Spicher. Computation and Visualization of Musical Structures in Chord-Based Simplicial Complexes. MCM 2013 - 4th International Conference Mathematics and Computation in Music, Jun 2013, Montreal, Canada. pp.38-51, ⟨10.1007/978-3-642-39357-0_3⟩. ⟨hal-00925748⟩

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