Topologizing Sound Synthesis Via Sheaves

In recent years, a range of topological methods have emerged for processing digital signals. In this paper we show how the construction of topological filters via sheaves can be used to topologize existing sound synthesis methods. I illustrate this process on two classes of synthesis approaches: (1) based on linear-time invariant digital filters and (2) based on oscillators defined on a circle. We use the computationally-friendly approach to modeling topologies via a simplicial complex, and we attach our classical synthesis methods to them via sheaves. In particular, we explore examples of simplicial topologies that mimic sampled lines and loops. Over these spaces we realize concrete examples of simple discrete harmonic oscillators (resonant filters), and simple comb filter based algorithms (such as Karplus-Strong) as well as frequency modulation.