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

Spatial Transformations in Simplicial Chord Spaces

In this article, we present a set of musical transformations based on chord spaces representations derived from the Tonnetz. These chord spaces are formalized as simplicial complexes. A piece is represented in such a space by a trajectory. Spatial transformations are applied on these trajectories and induce a transformation of the original piece. These concepts are implemented in two applicatio...

Topological Structures in Computer-Aided Music Analysis

We propose a spatial approach to musical analysis based on the notion of a chord complex. A chord complex is a labelled simplicial complex which represents a set of chords. The dimension of the elements of the complex and their neighbourhood relationships highlight the size of the chords and their intersections. Following a well-established tradition in set-theoretical and neo-Riemannian music ...

Vertex Decomposable Simplicial Complexes Associated to Path Graphs

Introduction Vertex decomposability of a simplicial complex is a combinatorial topological concept which is related to the algebraic properties of the Stanley-Reisner ring of the simplicial complex. This notion was first defined by Provan and Billera in 1980 for k-decomposable pure complexes which is known as vertex decomposable when . Later Bjorner and Wachs extended this concept to non-pure ...

Cohen-Macaulay-ness in codimension for simplicial complexes and expansion functor

In this paper we show that expansion of a Buchsbaum simplicial complex is $CM_t$, for an optimal integer $tgeq 1$. Also, by imposing extra assumptions on a $CM_t$ simplicial complex, we provethat it can be obtained from a Buchsbaum complex.

On Isomorphism of Simplicial Complexes and Their Related Algebras