While skimming a back issue of Mathematics Magazine, I came across Jordan Schettler’s article Wendy Carlos’s Xenharmonic Keyboard which describes a continued fraction derivation of an idea by Wendy Carlo—whereas “traditional” tuning ideas are based on preserving the octave (2:1 frequency) ratio, we might alternatively just try to find equal tempered scales that give a good approximation to the perfect fifth (3:2) and major third (5:4) and forget about this giving an octave…

Here’s a short 6-minute video describing and sonifying Carlo’s alpha, beta, and gamma scales…

Benson’s Music: A Mathematical Offering Section 6.6 gives a lucid description of Carlo’s approach. You can listen to some example notes on the alpha, beta, and gamma scale wikipedia pages.

Carlos used the alpha and beta scales on an album Beauty in the Beast. These scales have some interesting properties—for example, alpha splits the minor third exactly into halves (and quarters), and beta splits the perfect fourth into halves.

Closely related is the Bohlen-Pierce scale starts from the foundational choice of the tritave (3:1)—aka the perfect twelth or an octave higher than a perfect fifth (3:2)—and divides this into 13 parts. This is discussed in Section 6.7 in Benson. Apparently this idea is called EDONOI—“equal divisions of non-octave intervals.”

More fundamentally, why do we priviledge the octave in the first place? Keuler writes about the octave identity paradox, noting that the octave identity is a historical byproduct, and that many musically untrained listeners will mistake fourths and fifths for octaves.