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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
    |  |
 F--C--G
 | 
 D♭-A♭

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 in some traditions[which?] the second tone of the scale is 3/4 tones flattened (i.e. 1/4 tone above the tonic) and the seventh tone of the scale is raised by a quarter tone (i.e. 3/4 tone below 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♮).

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    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) Jul 4, 2021 at 22:34

2 Answers 2

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TL;DR - yes, your intuition is basically right and if you use 5-limit ratios for this you will be pretty close.

Your basic idea of 5 limit JI is approximately correct for Arabic and Turkish music as performed today. However, the actual intonation for both these traditions is largely arbitrary and based on aural tradition, not ratios. Many of the intervals do not correspond to JI ratios, and change depending on the region and era you are looking at.

While some very old Eastern theory texts did indicate that these notes were so-called "quarter tones" (actually a misnomer in this case because it was conceptially a Pythagorean whole step divided in 9 komas, yielding 5/9 and 4/9 large and small half steps, so 1/4 tones were not the theoretical basis), they generally have not been played that way in modern practice.

Since the default of a M3 and M7 are close to the 5-Limit ratios, these don't particularly need adjustment (though sometimes they are played even a bit lower).

Turkish sheet music notates the lower of these pairs (Db and Ab in this case) as being a koma (1/9) higher than the regular flat (which are approximately a Pythagorean 256/243 m2 and 128/81 m6 here). Which corresponds roughly with using the 5-limit versions (16/15 and 8/5) for the higher versions. In Arabic music, it is usually not indicated in the notation, one is just supposed to know that when encountering the A2 interval that the intervals are raised/lowered appropriately.

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  • Your answer made me revisit my question and I noticed I had read the excerpted Wikipedia page incorrectly. I updated my question, but it doesn't affect your answer. Where you say "conceptially a Pythagorean whole step divided in 9 komas", I assume this is the 53-tone equal temperament approximation to Pythagorean.
    – Theodore
    Jan 24 at 17:56
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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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