How to Tune a Stiff Sawtooth
Emily Henderson and Jordan Schettler

Proceedings of Bridges 2022: Mathematics, Art, Music, Architecture, Culture
Pages 363–366
Short Papers

Abstract

The vibrations of a flexible string are well-modeled by the classical wave equation. However, when the string is sufficiently stiff, like those on a piano, an extra term needs to be added. This impacts the tuning of the instrument, where frequencies need to be stretched apart. This stretch is sometimes done by forcing the first pair of corresponding partials to coincide, but that approach does not work for a general stiff waveform like an inharmonic sawtooth wave produced by a synthesizer. We propose tuning via a tempering factor given by a certain weighted geometric mean.

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