Tetrachords

A medieval Iranian musician and music theorist named Manṣūr Zalzal, who lived around 800 CE, is credited with introducing specific neutral seconds and neutral thirds as intervals in Arabic lute music. Credit comes from almost-as-old-Philosopher and music theorist al-Farabi.

If a minor second has a cent value around 100 cents and a major second has a cent value around 200 cents, then a neutral second is something like 150 cents.

Zalzal analyzed neutral seconds with four small super-particular ratios:

11/10 ~ 165 cents

12/11 ~ 155 cents

13/12 ~ 140 cents

14/13 ~ 130 cents

Following Zalzal, al-Farabi made some tetrachords with these neutral seconds - tetrachords being little scales of four tones extending from P1 to P4. Tetrachords can be used to make larger scales too. They're a big part of ancient Greek and medieval middle eastern music theory.

Mr. al-Farabi started his tetrachords with a unison of course, P1, and then a Pythagorean M2, tuned to 9/8. Between M2 and P4, there remains an interval of a Pythagorean minor third, tuned to 32/27, which al-Farabi would split into two roughly equal pieces. A minor third is about 300 cents, so splitting it into two pieces will give us parts of roughly 150 cents, i.e. neutral seconds. The actual cent value of 32/27

    1200 * log_2(32/27)

is closer to 294 than 300 even.

If you're a fan of Archytas, you might think to make a neutral third by dividing the Pythagorean m3, 32/27, into

    * its harmonic mean with 1/1, giving 64/59 at 141 cents, and 

    * its arithmetic mean with 1/1, giving 59/54 at 153 cents.

The product of these gives us back our original 32/27. That's not what al-Farabi did.

If you're a fan of Pythagoras, you might say, "You want an interval between m3 and M3? Try one of the intervals generated by extending a spiral of pure fifths with octave reduction, like A2, dddd6, ddd5, or d4." That's not what al-Farabi did. 

If you're a fan of Just Intonation, you might say, "Have you tried raising m3 or lowering M3 by a suitable number of syntonic commas, giving Acm3 or GrM3 or similar?" That's not what al-Farabi did.

Mr al-Farabi came up with tetrachords that used his super-particular ratios for neutral seconds. And they look a little funny, but they're kind of beautiful if you squint.

If you divide 32/27 by the largest of his neutral seconds 11/10, you get a ratio very close to 14/13, the smallest of his neutral seconds. But it's not actually equal. The ratio you get is 320/297. The difference between 14/13 and 320/297 is like one cent, and humans can only reliably identify about 5 cents of difference in a lab setting, so these frequency ratios are indistinguishable to the human aural system. Conversely, if you divide 32/27 by 14/13, you get something very close to 11/10 in terms of cents, but again, kind of weird: 208/189.

We'd like to say 11/10 * 14/13 = 32/27 to summarize what we hear with simple ratios, but obviously this is mistaken arithmetic: the product is 77/65. The difference is tiny! (32/27) / (77/65) = (2080/2079).

The situation is likewise with Zalzal's other two neutral seconds: diving 32/27 by 12/11 gives basically 13/12, but it's actually 88/81. Dividing by 13/12 gives roughly 12/11, but it's actually 128/117. We'd like to say that 12/11 * 13/12 = 32/27, in order to summarize what we hear with simple frequency ratios, but actually the product is 13/11. Now, 13/11 has a smaller numerator and and smaller denominator than 32/27, so you might call it less complex if you're ignoring which prime factors show up, but it's still not the minor third we were looking for. The difference here is not so small as before, (32/27) / (13/11) = (352/351), but it's still less than 5 cents.

N.B: I've just been showing the intervals of al-Farabi's tetrachords above the Pythagorean M2, not necessarily in the right order. 

I really like the idea that Zalzal's simple beautiful super-particular ratios should be enough to analyze his tetrachords. There should be a mathematical system where we can say that 12/11 * 13/12 might-as-well-equal 32/27. Perhaps we should say that frequency ratios have a width of some cents around their nominal magnitudes - the width of recognition, the width of performance - and allow ourselves to use these error-bars to bless those beautiful Zalzalian/al-Farabian relations that would otherwise be grade school mistakes of arithmetic.

Mr. al-Farabi himself didn't argue for any of this. In his time and the time of the ancient Greeks, it was normal to say "if two of the frequency ratios in a tetrachord are simple and beautiful, then the third one can be garbage". But I wish we could have them all be beautiful. And perhaps we can with some interval arithmetic. I haven't looking into that yet. In the mean time, how about some tables of tetrachord data?

Some simple tetrachords:

[102, 112, 284] [35/33, 16/15, 33/28]
[102, 142, 254] [35/33, 38/35, 22/19]
[102, 165, 231] [35/33, 11/10, 8/7]
[102, 187, 209] [35/33, 39/35, 44/39]
[105, 112, 281] [17/16, 16/15, 20/17]
[112, 119, 267] [16/15, 15/14, 7/6]
[112, 139, 248] [16/15, 13/12, 15/13]
[112, 155, 231] [16/15, 35/32, 8/7]
[112, 165, 221] [16/15, 11/10, 25/22]
[112, 182, 204] [16/15, 10/9, 9/8]
[119, 182, 196] [15/14, 10/9, 28/25]
[124, 151, 224] [29/27, 12/11, 33/29]
[124, 170, 204] [29/27, 32/29, 9/8]
[125, 182, 190] [43/40, 10/9, 48/43]
[128, 139, 231] [14/13, 13/12, 8/7]
[128, 182, 187] [14/13, 10/9, 39/35]
[133, 182, 182] [27/25, 10/9, 10/9]
[139, 151, 209] [13/12, 12/11, 44/39]
[142, 173, 182] [38/35, 21/19, 10/9]
[151, 165, 182] [12/11, 11/10, 10/9]

The block on the left of each line shows sizes in cents, the block on the right shows a set of three simple frequency ratios with a product of 4/3.

If we allow the complexity of frequency ratios to increase a little, we can also include these:

[102, 122, 274] [35/33, 44/41, 41/35]
[108, 151, 239] [33/31, 12/11, 31/27]
[112, 135, 251] [16/15, 40/37, 37/32]
[112, 144, 242] [16/15, 25/23, 23/20]
[112, 161, 225] [16/15, 45/41, 41/36]
[112, 176, 210] [16/15, 31/28, 35/31]
[115, 124, 259] [31/29, 29/27, 36/31]
[117, 169, 212] [46/43, 43/39, 26/23]
[122, 132, 244] [44/41, 41/38, 38/33]
[122, 151, 225] [44/41, 12/11, 41/36]
[122, 178, 198] [44/41, 41/37, 37/33]
[128, 157, 212] [14/13, 23/21, 26/23]
[135, 146, 217] [40/37, 37/34, 17/15]
[135, 165, 198] [40/37, 11/10, 37/33]
[139, 169, 190] [13/12, 43/39, 48/43]
[139, 170, 189] [13/12, 32/29, 29/26]
[144, 157, 196] [25/23, 23/21, 28/25]

And what the heck; I'm feeling sassy: Let's increase the complexity one more time so that we can see the Archytas-style neutral thirds:

[101, 182, 215] [53/50, 10/9, 60/53]
[102, 196, 200] [35/33, 28/25, 55/49]
[103, 128, 267] [52/49, 14/13, 7/6]
[103, 147, 248] [52/49, 49/45, 15/13]
[105, 131, 262] [17/16, 55/51, 64/55]
[105, 141, 252] [17/16, 64/59, 59/51]
[105, 162, 231] [17/16, 56/51, 8/7]
[105, 170, 223] [17/16, 32/29, 58/51]
[105, 193, 201] [17/16, 19/17, 64/57]
[107, 112, 279] [50/47, 16/15, 47/40]
[107, 195, 196] [50/47, 47/42, 28/25]
[108, 146, 244] [33/31, 62/57, 38/33]
[108, 182, 207] [33/31, 10/9, 62/55]
[109, 157, 231] [49/46, 23/21, 8/7]
[110, 112, 276] [65/61, 16/15, 61/52]
[112, 114, 272] [16/15, 47/44, 55/47]
[112, 125, 261] [16/15, 43/40, 50/43]
[112, 128, 258] [16/15, 14/13, 65/56]
[112, 131, 256] [16/15, 55/51, 51/44]
[112, 151, 236] [16/15, 12/11, 55/48]
[112, 159, 227] [16/15, 57/52, 65/57]
[112, 168, 219] [16/15, 65/59, 59/52]
[112, 172, 215] [16/15, 53/48, 60/53]
[112, 186, 200] [16/15, 49/44, 55/49]
[112, 189, 197] [16/15, 29/26, 65/58]
[114, 153, 231] [63/59, 59/54, 8/7]
[115, 160, 223] [31/29, 34/31, 58/51]
[119, 128, 250] [15/14, 14/13, 52/45]
[119, 147, 231] [15/14, 49/45, 8/7]
[119, 162, 217] [15/14, 56/51, 17/15]
[124, 143, 231] [29/27, 63/58, 8/7]
[128, 142, 227] [14/13, 38/35, 65/57]
[128, 175, 195] [14/13, 52/47, 47/42]
[131, 151, 217] [55/51, 12/11, 17/15]
[131, 160, 207] [55/51, 34/31, 62/55]
[139, 141, 219] [13/12, 64/59, 59/52]
[139, 159, 201] [13/12, 57/52, 64/57]
[141, 153, 204] [64/59, 59/54, 9/8]
[142, 146, 210] [38/35, 62/57, 35/31]
[142, 155, 201] [38/35, 35/32, 64/57]
[146, 160, 193] [62/57, 34/31, 19/17]
[147, 151, 200] [49/45, 12/11, 55/49]
[147, 168, 182] [49/45, 54/49, 10/9]
[156, 170, 172] [58/53, 32/29, 53/48]
[157, 162, 179] [23/21, 56/51, 51/46]
[160, 162, 176] [34/31, 56/51, 31/28]

I think my favorites are probably

[12/11, 11/10, 10/9]
[13/12, 12/11, 44/39]

[13/12, 57/52, 64/57]
[14/13, 13/12, 8/7]
[15/14, 14/13, 52/45]

[16/15, 10/9, 9/8]
[16/15, 13/12, 15/13]
[16/15, 15/14, 7/6]
[17/16, 16/15, 20/17]
[27/25, 10/9, 10/9]

[29/27, 32/29, 9/8]
[35/33, 39/35, 44/39]
[38/35, 35/32, 64/57]
[43/40, 10/9, 48/43]
[53/50, 10/9, 60/53]

[64/59, 59/54, 9/8]

for different reasons of numeric aesthetics.

It is done and it is done well. Goodnight, dear reader.

Update: I did a thing with EDOs!

I wanted to find an EDO that tempered out the intervals that are justly associated with (2080/2079) and (352/351). The sequence starts (7, 29, 34, 39, 41, 46, 48, 53, 58, 70, 87, 92, 94, 99, ...)-EDO. Of the ones listed, 87-EDO and 94-EDO are the only ones that actually tune all of the intervals justly associated with Zalzalian neutral seconds, (14/13, 13/12, 12/11, 11/10), to four different EDO steps. So it seems to me like those are the smallest EDOs suitable for analyzing medieval Persian music well 

For example, you can summarize that the steps 11/10 * 320/297 are perceptually equivalent to 208/189 * 14/13 and to the simpler 11/10 * 14/13 (which doesn't reproduce the Pythagorean minor third, 32/27, needed to close the gap between 9/4 and 4/3), by saying that these are all representable as M steps + N steps.

In 87-EDO, {9/8 * 11/10 * 14/13 ~= 4/3} is explained as (15 + 12 + 9 = 36) and {9/8 * 12/11 * 13/12 ~= 4/3} is explained as (15 + 11 + 10 = 36). In 94-EDO, the explanations are (16 + 13 + 10 = 39) and (16 + 12 + 11 = 39), respectively.

Cool find.

The list of EDOs that temper out both continues (..., 99, 111, 128, 133, 135, 140, 145, 157, 174, 181, 186, 198, 205, 210, 227, 232, 244, 251, 268, 269, ...]. All of these distinguish between the Zalzalian neutral seconds.

The people on the Xenharmonic Alliance discord point out that 87-EDO is (and many of the other ones mentioned are) a "Parapyth" EDO, identified by Margo Schulter for use in analyzing music just like this. Scooped.

I like that 94-EDO also tempers out the interval justly associated with the frequency ratio 225/224, which shows up when Ozan Yarman makes basic math errors does impressionistic arithmetic to describe tetrachords. Although 87-EDO tempers out the intervals justly associated with 256/255, which also shows up when Ozan Yarman does the thing. And 87-EDO also tempers out the intervals justly associated with 406/405 and 154/153, which show up when Ozan does the thing! 87-EDO also tempers out the interval justly associated with 768/767 = (72/59) / (39/32) (at 2 cents), but sadly not the interval justly associated with 649/648 = (11/9) / (72/59) at 3 cents. I think 87-EDO is the one for me. It also tempers out 

I've made a chart of 87-EDO steps up to the P4, showing 29-limit (undetrigintal) frequency ratios that are justly associated to intervals which 87-EDO tunes to each step size:

0: 1/1
1: 70/69, 77/76, 88/87, 91/90, 96/95
2: 49/48, 52/51, 55/54, 56/55, 58/57, 64/63, 65/64, 66/65, 69/68, 76/75, 78/77, 81/80, 85/84, 92/91, 99/98
3: 35/34, 39/38, 40/39, 45/44, 46/45, 50/49, 51/50, 57/56, 77/75, 87/85, 98/95
4: 28/27, 30/29, 33/32, 34/33, 36/35, 65/63, 88/85, 91/88, 95/92
5: 24/23, 25/24, 26/25, 27/26, 29/28, 57/55, 80/77, 91/87, 99/95
6: 20/19, 21/20, 22/21, 23/22, 51/49, 68/65, 85/81, 95/91
7: 18/17, 19/18, 35/33, 49/46, 55/52, 58/55, 81/77, 92/87, 96/91
8: 16/15, 17/16, 52/49, 69/65, 77/72, 81/76, 91/85
9: 14/13, 15/14, 29/27, 55/51, 99/92
10: 13/12, 25/23, 27/25, 49/45, 63/58, 68/63, 69/64, 88/81, 92/85, 95/88
11: 12/11, 35/32, 38/35, 56/51, 85/78, 87/80, 95/87, 99/91
12: 11/10, 21/19, 23/21, 32/29, 57/52, 75/68, 76/69
13: 10/9, 49/44, 51/46, 54/49, 72/65, 77/69, 85/77
14: 19/17, 28/25, 29/26, 39/35, 64/57, 65/58, 85/76, 91/81, 98/87
15: 9/8, 26/23, 44/39, 55/49, 77/68, 96/85
16: 17/15, 25/22, 33/29, 58/51, 65/57, 87/77, 91/80, 92/81, 95/84
17: 8/7, 39/34, 55/48, 57/50, 63/55, 87/76, 98/85
18: 15/13, 22/19, 23/20, 38/33, 52/45, 80/69
19: 7/6, 29/25, 51/44, 64/55, 65/56, 81/70, 99/85
20: 20/17, 27/23, 34/29, 57/49, 75/64, 76/65, 88/75, 90/77, 95/81
21: 13/11, 32/27, 33/28, 45/38, 46/39, 77/65, 85/72
22: 19/16, 25/21, 55/46, 58/49, 68/57, 69/58, 81/68, 91/76
23: 6/5, 35/29, 65/54, 77/64, 92/77, 98/81
24: 23/19, 28/23, 29/24, 40/33, 63/52, 76/63, 91/75
25: 11/9, 17/14, 39/32, 49/40, 70/57, 95/78
26: 16/13, 21/17, 27/22, 60/49, 85/69, 92/75
27: 26/21, 36/29, 56/45, 57/46, 68/55, 69/56, 95/77, 99/80
28: 5/4, 49/39, 64/51, 81/65, 87/70, 96/77
29: 24/19, 29/23, 34/27, 44/35, 63/50, 69/55, 91/72
30: 14/11, 19/15, 33/26, 65/51, 80/63, 81/64, 88/69
31: 23/18, 32/25, 49/38, 50/39, 51/40, 77/60, 87/68
32: 9/7, 22/17, 35/27, 58/45, 75/58, 84/65, 85/66
33: 13/10, 30/23, 57/44, 98/75, 99/76
34: 17/13, 21/16, 25/19, 38/29, 55/42, 64/49, 72/55, 91/69
35: 29/22, 33/25, 45/34, 46/35, 77/58, 95/72
36: 4/3, 65/49, 69/52, 85/64

I'm never going to get all of the frequency ratios that are used for middle eastern maqam analysis, like 128/117 and 162/149, but this chart is a good start, and it suggests simpler ratios that can stand in for those in arithmetically impressionistic analyses. There are only 476 frequency ratios of comparable length with numerator and denominator below 100 that are greater than P1 and less than P4. So.... that's a start. Woo.

Some examples. Ozan Yarman in his dissertation "79-Tone Tuning & Theory For Turkish Maqam Music" described the "Tempered Rast" Genus/Tetrachord as: 

    28/25 * 28/25 * 17/16. 

This doesn't equal 4/3, but the corresponding steps do: 

    {14 + 14 + 8 = 36}. 

He gives the "Wide Hicaz" Genus as: 

    16/15 * 20/17 * 17/16. 

This doesn't equal 4/3, but the 87-EDO steps do: 

    {8 + 20 + 8 = 36}.

He gives the "Huzzam" Genus as: 

    15/14 * 9/8 * 32/29, 

which we can explain as 

    {9 + 15 + 12 = 36}.

He gives the "Huseyni" Genus as: 

    11/10 * 13/12 * 9/8. 

This doesn't equal 4/3, ...and neither do the steps: {12 + 10 + 15 = 37}. Obviously we can fix this by replacing one of the ratios with a ratio that comes from one-step lower in 87-EDO. If we keep 9/8 in tact, then our options both will look familiar to you: 

    12/11 * 13/12 * 9/8 -> {11 + 10 + 15 = 36}

    11/10 * 14/13 * 9/8 -> {12 + 9 + 15 = 36}

Old friends! They're the impressionistic al-Farabi tetrachords that we made sure we'd be able to recover by tempering out (2080/2079) and (352/351)! I'm not sure which, if either of these, is actually a good representation of the modern Turkish form of Huseyni, but I know that they're both closer to being tetrachords than what Ozan Yarman gave.

Something that might be an actual challenge for 87-EDO's adequacy: Ozan Yarman gives the "Ussak" Genus as 

    12/11 * 12/11 * 9/8

which I would explain in 87-EDO as

    {11 + 11 + 15 = 37}.

The actual frequency ratio you get (from multiplying out his tetrachord fractions) is 162/121, which is 243/242 over 4/3, or about 7 cents sharp. If the Ussak Genus really has two perfectly equal steps at the start, then 87-EDO just can't represent it without ruining the 9/8 by replacing it with something ugly like 28/25 or 19/17. Although Ozan Yarman himself used 28/25 in his gloss of the Tempered Rast Genus, so maybe it's not *all that* ugly? If we can alter one of the 12/11s, then we could just do 13/12 * 12/11 * 9/8, which is one of the old al-Farabi tetrachords again. I suppose either option is okay, but I don't know enough about how Ussak is played to choose between them.

I'm really quite pleased with this. If 53-EDO isn't granular enough to represent the variety of middle seconds and thirds that are observed in maqam/makam practice, and if we have a long history of using bad arithmetic (up to a few cents) to gloss tetrachord relationships, let's use a more granular EDO which can explain these lossy arithmetic glosses. I found the simplest EDO that I think does a good job at it, and it happens to also do a great job of it, am I right? Hell yeah.

If I take all of the valid tetrachords that I generated way above and I simplify their frequency ratios by taking the simplest frequency ratio that has the same 87-EDO step, then there are only 25 distinct tetrachords (again ignoring the order of the steps). Five of them are still valid arithmetically:

[7, 13, 16] [18/17, 10/9, 17/15]
[8, 10, 18] [16/15, 13/12, 15/13]
[8, 13, 15] [16/15, 10/9, 9/8]
[9, 10, 17] [14/13, 13/12, 8/7]
[11, 12, 13] [12/11, 11/10, 10/9]

and the remainder are the impressionistic sort:

[7, 8, 21] [18/17, 16/15, 13/11]
[7, 9, 20] [18/17, 14/13, 20/17]
[7, 11, 18] [18/17, 12/11, 15/13]
[7, 12, 17] [18/17, 11/10, 8/7]
[7, 14, 15] [18/17, 19/17, 9/8]
[8, 8, 20] [16/15, 16/15, 20/17]
[8, 9, 19] [16/15, 14/13, 7/6]
[8, 11, 17] [16/15, 12/11, 8/7]
[8, 12, 16] [16/15, 11/10, 17/15]
[8, 14, 14] [16/15, 19/17, 19/17]
[9, 9, 18] [14/13, 14/13, 15/13]
[9, 11, 16] [14/13, 12/11, 17/15]
[9, 12, 15] [14/13, 11/10, 9/8]
[9, 13, 14] [14/13, 10/9, 19/17]
[10, 10, 16] [13/12, 13/12, 17/15]
[10, 11, 15] [13/12, 12/11, 9/8]
[10, 12, 14] [13/12, 11/10, 19/17]
[10, 13, 13] [13/12, 10/9, 10/9]
[11, 11, 14] [12/11, 12/11, 19/17]
[12, 12, 12] [11/10, 11/10, 11/10]

There are two more maybe-decent 87-EDO tetrachord glosses which didn't show up in descriptions of any of my generated tetrachords. They both have both a very small step and a very large step: 

[7, 10, 19] [18/17, 13/12, 7/6]
[7, 7, 22] [18/17, 18/17, 19/16]

and are not valid arithmetically but only impressionistically. Those have some potential too, I think.

...

I went through all of the historic middle-eastern tetrachords that Margo Schulter presents in https://www.bestii.com/~mschulter/Ibn_Sina-overview.txt. Almost every time that she presents two tetrachords as being basically the same, my 87-EDO analysis says that they are the same. 

One counter examples is "Higher Buzurg" with adjacent ratios of "13/12, 8/7, 14/13" or "14/13, 8/7, 13/12". My system distinguishes 13/12 from 14/13 and I stand by this.

Here are the 87-EDO commas for all of her tetrachords:

[9, 10, 17]: Lower septimal Shur, Bayyati, or Ushshak
[9, 12, 15]: Low Shur, Lebanese Folk Bayyati, or Turkish Ushshak
[9, 15, 12]: Low Arab 'Iraq (as theoretical jins), High Turkish Segah, Low Persian Old Esfahan
[9, 17, 10]: Higher Buzurg, Avaz-e Bayat-e Esfahan, or Byzantine Soft Chromatic, 8-13-7 steps of 68 in system of Chrysanthos of Madytos 2
[10, 9, 17]: Higher septimal Shur, Bayyati, or Ushshak
[10, 11, 15]: Moderate Arab Bayyati, 6-7-9 commas
[10, 11, 15]: Moderate Shur, Arab Bayyati, Turkish Ushshak
[10, 15, 11]: Arab 'Iraq, in theory, 6-9-7 commas
[10, 15, 11]: Persian Old Esfahan
[10, 17, 9]: Higher Buzurg, Avaz-e Bayat-e Esfahan, or Byzantine Soft Chromatic, 8-13-7 steps of 68 in system of Chrysanthos of Madytos 1
[11, 10, 15]: Moderate Arab Huseyni, 7-6-9 commas
[11, 10, 15]: Moderate Arab Huseyni, Low Turkish Huseyni
[11, 15, 10]: Medium Persian Esfahan
[11, 15, 10]: Medium Persian Esfahan, 7-9-6 commas
[12, 9, 15]: Turkish Huseyni or High Arab Huseyni
[12, 15, 9]: Possible High Persian Segah
[15, 9, 12]: Low Mustaqim, Afshari, or Shekaste; possibly High Turkish Nihavend
[15, 10, 11]: Higher Mustaqim or Arab Rast Jadid, 9-6-7 commas (cf. Rast, 9-7-6 commas)
[15, 10, 11]: Higher Mustaqim, Dastgah-e Afshari, Gushe-ye Shekaste
[15, 11, 10]: Arab Rast, Byzantine Diatonic (Chrysanthos) 9-7-6 commas
[15, 11, 10]: Medium-high Arab Rast, Low Turkish Rast
[15, 11, 10]: Medium-high Rast
[15, 12, 9]: High Rast
[15, 12, 9]: High Syrian Rast, or Medium Ottoman Rast
[17, 9, 10]: Septimal Rast, 16/13
[17, 10, 9]: Septimal Rast, 26/21

I feel it necessary to point out that Margo has a typo in her data: she gives "14:13 13:12 8:7" as the adjacent ratios for both higher septimal Shur and lower septimal Shur, but from looking at the cumulative scale degrees, we can see that higher septimal Shur should be "13:12 14:13 8:7", so that's how I've analyzed it.

All of the tetrachords end up being permutations of these three numerically sorted tetrachords:

[9, 10, 17]
[9, 12, 15]
[10, 11, 15]

Margo notates most of the tetrachords with Turkish simgeler (plural of "simge", meaning a sign/token/symbol). She uses "T" for a "tanîni", i.e. intervals of roughly a major second (perhaps 182 to 232 cents), which for her include lots of 9/8s and a few 8/7s. These are mapped to 15 and 17 steps of 87-EDO respectively. In occasional comments in the paper, outside the main data sections, she uses "B" for "bakiye", intervals of roughly a minor seconds (perhaps 80 to 112 cents). She also uses "J" extensively in the data section for the neutral seconds or "mücenneb", which get mapped to one of [9, 10, 11, 12] steps of 87-EDO. These fall in the range like (128 to 165) cents for the neutral seconds that she considers, but it's common for neutral seconds to fill to full range of (113 to 181) cents between m2 and M2 in other makam analysis, particularly Pythagorean / 53-EDO analysis where a small neutral second (kücük mücenneb, simge "S") and a large  neutral second (büyük mücenneb, simge "K") sit directly on those extremes: these are respectively a comma sharp relative to m2 and a comma flat relative to M2. She doesn't use "S" and "K" though, just "J" for all mücenneb. I'm probably using singular turkish words when I should be using plural. My apologies.

Next up: analyzing genera / ajnas from outside the main data sections of Margo's paper and also genera/ajnas from Yarman Ozman's various papers with explicit frequency ratios.

...

Margo gives:

A few modern makams in terms of just M2 and m2:

T T B (current Arab `Ajam or Persian Mahur, e.g. (9/8 * 9/8 * 256/243) 
T B T (current Arab Nahawand or Persian Nava, e.g (9/8 * 256/243 * 9/8)
B T T (current Arab or Turkish Kurdi, e.g. (256/243 * 9/8 * 9/8)

And a few more with neutral seconds:

T J J (current Arab or Turkish Rast, e.g. (9/8 * 12/11 * 88/81)
J J T (current Arab Bayyati or Persian Shur, e.g. (13/12 * 128/117 * 9/8)
J T J (Buzurg, current Persian Segah or Esfahan, e.g. (13/12 * 8/7 * 14/13) # "Buzurg" = current Turkish Büzürk?

J J J B (Systematist Isfahan, e.g. modern (13/12 * 12/11 * 14/13 * 22/21)

.

She also shares a chromatic Hijaz tetrachord from Qutb al-Din al-Shirazi as related by Owen Wright:

adjacent ratios: 12/11, 7/6, 22/21

which is a permutation of Ptolemy’s intense diatonic:

adjacent ratios: 22/21, 12/11, 7/6

.

Ptolemy's Syntonic Diatonic...

adjacent ratio: 10/9, 9/8, 16/15

also shows up in a permuated form: Margo relates that it was used by Qutb al-Din al-Shirazi to define the Rast tetrachord: 

adjacent ratios: 9/8, 10/9, 16/15 

And modern Turkish theory uses the nearly equivalent

adjacent ratios: 9/8, 65536:59049, 2187:2048

.

...