Algebraic tunings

We propose an approach to tuning systems in which octave doubling ratio is replaced by a suitable algebraic unit τ, and note frequencies are proportional to a subset of the ring Z[τ]. Then it is possible for many difference tones between notes in the tuning to also appear in the tuning. After outlining more general principles, we consider in detail some natural examples based on the golden ratio ϕ=(1+5)/2, limited by norm or by the number of digits in the greedy β-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 Sβ5(ϕ) (based on β-expansions to ϕ−5) on websynths.com and a composition Three Places.