Download 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.
Download Deforming the Oscillator: Iterative Phases Over Parametrizable Closed Paths Iterative phase formulations allow for the generalization of many oscillatory sound synthesis methods from circles to general parametrizable loops, with or without explicit geometric contexts. This paper describes this approach, leading to the ability to perform modulation, feedback and chaotic oscillations over deformed circles that can include ill-behaved geometries, while allowing modulations or feedback to be deformed as well.
Download Topology-Preserving Deformations of Digital Audio Topology provides global invariants for data as well as spaces of deformation. In this paper we discuss the deformations of audio signals which preserve topological information specified by sublevel set persistent homology. It is well known that the topological information only changes at extrema. We introduce box snakes as a data structure that captures permissible editing and deformation of signals and preserves the extremal properties of the signal while allowing for monotone deformations between them. The resulting algorithm works on any ordered discrete data hence can be applied to time and frequency domain finite length audio signals.