A topological approach of musical relationships
Résumé
The notion of space is often summoned in music theory, both for the composition of new pieces or for the analysis of existing pieces. The
spatial metaphor is instrumental to convey musical insights but it is also a very effective heuristics to develop new computer tools to assist
musicians in their creative processes.
This presentation explores some spatial representations of musical notions based on elementary concepts in algebraic topology: melodic and
harmonic transformations, all-interval series classification or generation, style recognition... The basic idea is to represent simple
musical objects (for example pitches, chords or intervals) by elementary spatial domains and their relationships (e.g., their co-occurrence or
their succession) as neighborhood relationships. The topological notions of incidence, path, boundary, obstruction, etc., are then used to
unravel the musical structure. For instance, a musical sequence can be represented as a path in a cellular complex materializing the chords
structure and the look of the path reveals some information concerning the musical strategies used by the composer. Applying geometrical
operations on trajectories leads to musical transformations of the initial piece. Etc.
We attempt to explain the "Unreasonable Effectiveness” of spatial representations in Music by some profound connections between the
organization of percepts and some topological relationships. In particular, we investigate the links between the topological approach
sketched and the Formal Concept Analysis developed in symbolic learning.