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 analysis, we present the family of T/I complexes which represent classes of chords which are transpositionally and inversionally equivalent and which relate to the notion of Generalized Tonnetze. A musical piece is represented by a trajectory within a given chord complex. We propose a method to compute the compactness of a trajectory in any chord complex. Calculating the trajectory compactness of a piece in T/I complexes provides valuable information for music analysis and classification. We introduce different geometrical transformations on trajectories that correspond to different musical transformations. Finally, we present HexaChord, a software package dedicated to computer-aided music analysis with chord complexes, which implements most of the concepts discussed in this chapter.

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