Abstract:
My Xenharmonikon 18 paper _A Middle Path between just intonation and the equal temperaments_ shows how any “comma” or small JI interval can be “tempered out” and in the process define a whole (often unique) series of musical scales rich in consonant harmonic resouces. Of particular importance to Western musical history is the syntonic comma (81:80), the system in which it is tempered out (meantone), and the series of scales (pentatonic, diatonic, chromatic) that arise naturally when building said system. However, there are dozens of other viable choices for a comma to temper out, and many of these result in completely non-traditional series of scales nonetheless fairly rich in consonant harmony.
The BP diatonic scale and related constructs similarly arise when starting with a JI system that omits prime 2 and only uses primes 3, 5, and 7; tempering out the comma 245:243; and generating the resulting tuning system. My talk will illustrate a few particular, very slightly different ways of doing this. Along the way I’ll illustrate the harmonic resources of the BP diatonic scale with a lattice, and show how this lattice is “broken” when one does not temper out 245:243 — i.e., certain BP diatonic chords will fail to be consonant, or at a minimum will require multiple, differently-tuned renditions of some of the same letter-name pitches to be available (ruining the integrity of the scale), when a tuning system is used in which 245:243 does not vanish. One can easily use this lattice to find a chord progression based on the BP diatonic that, when rendered in strict JI (preserving all common tones from one chord to the next), the progression will drift in pitch by 245:243 each time it is cycled through. These are analogous to the classic Western 81:80 “comma problems” identified by Benedetti in the 16th century and oft-discussed since then, but deserve to be more widely known and understood in the BP context. For example, 31-EDO and 41-EDO are both well-known for their good approximations to consonant intervals based on frequency ratios built from the prime factors 3, 5, and 7. If one attempts to render BP diatonic music in 31-EDO, however, one may experience some severe problems indeed, as 245:243 comes out as a whopping 39 cents in 31-EDO! However 245:243 vanishes entirely in 41-EDO, so the latter makes a fine tuning system in which to perform or compose BP diatonic music without any danger to its integrity.
Other important papers can be found here.