Synthesis by Mathematical Models

Domenico Vicinanza; Vittorio Cafagna
DAFx-2004 - Naples
Sound synthesis methods can be interpreted, from a mathe­ matical point of view, as a collection of techniques of selecting and conceptually organizing elements of a Hilbert space. In this sense, mathematics, being a highly structured and sophisticated system of classification, modeling and categorization, seems to be the natural tool to describe existing synthesis methods and to pro­ pose new ones. Because, from this perspective, one can think of any available (or theoretically predictable, or imaginable) synthe­ sis method as a collection of procedures to deal with meaningful parameters, with the term ”synthesis by mathematical models” we mean an extensive use of the modeling and categorization power of mathematics applied to the world of sounds. In this paper we give a few examples of sound synthesis tech­ niques, based on mathematical models. After reviewing shortly FM synthesis and synthesis by nonlinear distortion, and suggest­ ing some, to our advice, interesting open problems, we propose two different new methods: synthesis by means of elliptic func­ tions and synthesis by means of nowhere (or almost­nowhere) dif­ ferentiable functions and lacunary series. The resulting waveforms have been produced using CSound as an audio engine, driven by Python scripts.