Algebraic tunings

Carl P Dettmann *, Liam Taylor-West

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

Abstract

We propose an approach to tuning systems in which octave doubling ratio is replaced by a suitable algebraic unit $\tau$, and note frequencies are proportional to a subset of the ring $\mathbb{Z}[\tau]$. Thus, many difference tones correspond to frequencies in the tuning. After outlining more general principles, we consider in detail some natural examples based on the golden ratio $\phi=(1+\sqrt{5})/2$, limited by norm or by the number of digits in the greedy $\beta$-expansion. We discuss additive and multiplicative properties, implementation and composition using these tunings. The Online Supplement contains MIDI and websynths files to implement the tuning $S_\beta^5(\phi)$ (based on $\beta$-expansions to $\phi^{-5}$) on websynths.com and a composition {\em Three Places}.
Original languageEnglish
Pages (from-to)203-216
Number of pages14
JournalJournal of Mathematics and Music
Volume18
Issue number2
Early online date24 Jul 2023
DOIs
Publication statusE-pub ahead of print - 24 Jul 2023

Bibliographical note

Publisher Copyright:
© 2023 The University of Bristol. Published by Informa UK Limited, trading as Taylor & Francis Group.

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