N-gon Waves - Audio Applications of the Geometry of Regular Polygons in the Time Domain

This paper presents an approach to the generation of periodic signals through the application of Euclidean geometry in the time domain. In particular, the content of this paper is focused on waveforms derived from regular polygons. It is an attempt to show how the geometrical relations and proportions of regular polygons and star polygons with respect to their Schläfli symbol can be made audible, and how these relations can be used in an acoustical or a musical context. A basic description is given of how to construct such geometrical waveforms and musical scales using the underlying geometry. In addition, this paper draws inspiration for its approach to synthesis and composition from experimental approaches to drawn graphical / ornamental sound. These include methods that came to prominence in Russia and Germany in the first part of the 20 century, such as those which used film and paper as primary media, and those that developed during the post-war period, including Oramics, and others. Most importantly, this paper describes a framework and examples that demonstrate processes whereby the geometry of regular polygons can be used to generate specific acoustic phenomena in frequency, timbre, phase, and metre.