The Harmonic Pattern Function: A Mathematical Model Integrating Synthesis of Sound and Graphical Patterns

The current landscape of parametric techniques for synthesis of digital sound waveforms and graphical curves and shapes is vast, but it is largely an incongruous mixture of closed and highly specialized mathematical equations. While much of this can be attributed to the independent development of synthesis techniques within each field, upon closer examination it is clear that there exist common mathematical bases between the modalities. From a more general mathematical context, it is possible to develop a language of unified audio/visual synthesis principles so that many of the existing paradigms, regardless of modality, can be understood from a single vantage point.

This dissertation substantiates the thesis that a large portion of known sound and graphical synthesis techniques can be unified through a rational function of inverse discrete Fourier transforms and that symmetry, invariance under transformation, plays an important role in understanding the patterns that it produces. The author calls this newly proposed audio/visual synthesis model the harmonic pattern function. A survey of a wide assortment of historic mechanical and electronic devices and computational systems used for generating sonic and visual patterns in art and science reveals that their underlying mathematical descriptions are special cases of this synthesis function.

The contributions of this dissertation include the introduction of a simple mathematical function, the harmonic pattern function, that is capable of generating a wide assortment of both known and previously unknown patterns useful for sound and/or visual synthesis; a simplified notation for specifying the complex sinusoids composing such patterns; and a thorough analysis of general themes and specific instances of patterns producible from the harmonic pattern function.

Lance Putnam, various curves generated from the harmonic pattern function, 2012–2016.

(© Lance Putnam)