Explicit presentations of topological categories of gestures

Abstract

Thanks to Mazzola's notion of gestures on topological categories we can appreciate how the notion of gesture transcends its manifestation as the movement of the body's limbs and takes a more abstract form that blends diagrammatic (discrete gesturality) and bodily aspects (continuous gesturality). These two aspects are strongly related to two main branches of mathematical music theory, namely, a discrete branch and a continuous branch. The discrete branch corresponds to the diagrams of transformational theory, that is, to networks in musical analysis. The continuous branch corresponds to the movement of the musical performer's body. Informally, a gesture on a topological category is a diagram of continuous paths of morphisms in the category. This definition amounts to that of topological category of gestures, whose structure we study in this article. Specifically, we study the presentation of topological categories of gestures as suitable categories of topological functors and as suitable categories of sequences, and the explicit presentation of morphisms of a typical topological category of gestures. In particular, we present an exhaustive study, not included in previous publications, of the topological category of continuous paths of an arbitrary digraph. This article can be regarded as a continuation of a previous publication, in a previous issue of this journal, on the presentation of spaces of gestures as function spaces. We include an application of the theory to the variations in Mozart's Piano Sonata K. 331. We provide an Online Supplement, in which we include some technical passages.