There's was a medieval music theorist named Safi al-Din al-Urmawi. I've also seen his name spelled Safiaddin Ormavi. He wrote a book called Kitab al-Adwār that supposedly outlined a system with 17 tones per octave for analyzing Arabic modal music (which music was derived from Persian Dastgāh).
I don't read modern or medieval Arabic and I don't know what the book says. But after a lot of searching, I found a description of the 17 tones in "The modal system of Arabian and Persian music" by Owen Wright. This was on page 41 and the whole thing is 500 pages. I'm pretty excited to read more. It looks really good.
Wright tells us that the 17 tone scale is Pythagorean in its tuning and he gives us the steps between notes in terms of the Pythagorean limma, i.e. the minor second with a tuned value of 256/243, and the Pythagorean comma, i.e. the augmented zeroth with a tuned value of 531441/524288. Adding up all the steps, we get these intervals:
m2 = (-5, 3) # 256/243
d3 = (-10, 6) # 65536/59049
M2 = (2, -1) # 9/8
m3 = (-3, 2) # 32/27
d4 = (-8, 5) # 8192/6561
M3 = (4, -2) # 81/64
P4 = (-1, 1) # 4/3
d5 = (-6, 4) # 1024/729
d6 = (-11, 7) # 262144/177147
P5 = (1, 0) # 3/2
m6 = (-4, 3) 128/81
d7 = (-9, 6) 32768/19683
M6 = (3, -1) 27/16
m7 = (-2, 2) # 16/9
d8 = (-7, 5) 4096/2187
d9 = (-12, 8) 1048576/531441
P8 = (0, 1) # 2/1
When the A1 component goes down by 5, that means we just added a m2. When the A1 component jumps up by 12, that means we just added an A0.
You might notice that there's no major seventh interval as written. If we do a cyclic permutation to start on the P4, then we get our usual chromatic scale and some other ornaments. The ornaments actually stay the same! The only thing the cyclic permutation changes is tuning the d9 into M7:
M7 = (5, -2) # 243/128
But as written by Owen Wright, Safi al-Din's scale has more of mixolydian feel.
If we do the cyclic permutation, then we have the usual chromatic pitch classes, (C, Db, D, Eb, E, F, Gb, G, Ab, A, Bb, B) and the five additional notes of (Ebb, Fb, Abb, Bbb, Cb). They're ordered like this:
[C, Db, Ebb, D, Eb, Fb, E, F, Gb, Abb, G, Ab, Bbb, A, Bb, Cb, B, C]
In the previous posts on maqams, we've discovered that diminished pitches in Pythagorean analysis outside the chromatic scale often correspond to pitches which modern Arab musicians call "half flat". In particular, you go backwards a letter name in the alphabet and call it half flat, so that Ebb is a D half flat, Fb is an E half fat, Abb is a G half flat, and so on. Safi al-Din thus has a chromatic scale plus (D-, E-, G-, A-, B-).
Again, all of hose can be arranged consecutively in a chain of fifths. Safi's system is just the Pythagorean spiral. The next ones to add would be Dbb and then Gbb.
(-12, 8) 1048576/531441 d9 | Dbb
(-11, 7) 262144/177147 d6 | Abb
(-10, 6) 65536/59049 d3 | Ebb
(-9, 6) 32768/19683 d7 | Bbb
(-8, 5) 8192/6561 d4 | Fb
(-7, 5) 4096/2187 d8 | Cb
(-6, 4) 1024/729 d5 | Gb
(-5, 3) 256/243 m2 | Db
(-4, 3) 128/81 m6 | Ab
(-3, 2) 32/27 m3 | Eb
(-2, 2) 16/9 m7 | Bb
(-1, 1) 4/3 P4 | F
(0, 0) 1 P1 | C
(1, 0) 3/2 P5 | G
(2, -1) 9/8 M2 | D
(3, -1) 27/16 M6 | A
(4, -2) 81/64 M3 | E
(5, -2) 243/128 M7 | B
So the 17-tone system for analyzing maqamat is basically taken verbatim from Pythagoras who lived 1700 years before Safi.
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