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)

1 STMS - Sciences et Technologies de la Musique et du Son
2 MuTant - Synchronous Realtime Processing and Programming of Music Signals
3 LACL - Laboratoire d'Algorithmique Complexité et Logique

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.

Mots clés: MGS; musical transformation; harmonization; counterpoint; self-assembly; Tonnetz; lattice; simplicial complex; formalization of music; music theory; simplicial complexes; generalized Tonnetze; compli- ance; Hexachord; chord spaces

Domaines: Mathématique discrète [cs.DM]; Multimédia [cs.MM]; Modélisation et simulation; Son [cs.SD]

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⟩