In musical traditions where something like the double-harmonic scale (e.g. C D♭ E F G A♭ B C) is used, and tuned with some kind of just intonation, what just intervals are preferred for the scale degrees in those traditions?

Considering it as two harmonic tetrachords, there seem to be obvious 5-limit choices for all the tones: Perfect 5ths at F-C, C-G, and D♭-A♭; Just major 3rds at C-E, D♭-F, and G-B. Is this how instruments might be tuned in some traditions?

    E  B
    |  |

There also several other ways we could think of these pitches in terms of diatonic modes with raised or lowered tones, which might imply different just relationships.

An uncited statement in the Wikipedia article indicates that the second and seventh tones of the scale in some traditions[which?] are separated by a quarter-tone from the tonic:

Double-harmonic with quarter-tones

This is a little more of a puzzle. Are they in fact quarter tones, or only approximately? What would the relationships be?

I suppose it could be that these "quarter" tones tend only occur in passing on a fretless string or other continuous-pitch instrument, and are not ever precisely tuned.

I also imagine D♭ might be tuned a chromatic semitone above C (25:24 or about 71¢). This would give a just minor D♭-E; B might be tuned similarly with respect to A♭ (however it is derived). Of course, that's just how they seem harmonically-useful to the tonality in my head.

Do any traditions go above 5-limit? My personal inclination, also not rooted in any existing tradition, is to use the 5-limit relationships I listed above, except for B which would be tuned to a septimal minor 7th above D♭. This would give the D♭7 chord a nice harmonic 7th, and that chord is a tritone substitution for the G7 which we lack in the scale (for want of a D♮).

  • 1
    You might find further research into these scales by noticing the fact that (I might have permutated) these double harmonic tetrads (or tetrachords as I like to spell it out) are connected with Roma/Gypsy and Balkan folk traditions... (which in turn are influenced by Bulgarian/Byzantine/Turkish mixed bag of nuts)
    – Hatebit
    Jul 4 '21 at 22:34

This isn't a direct answer, but I wonder if Archytas' Enharmonic might be something to investigate

gamut = { 1/1, 28/27, 16/15, 4/3, 3/2, 14/9, 8/5, 2/1 }

G, G 1/4#, Ab, C, D, D 1/4#, D, Eb, G

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