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

.

...

Tura's Baglama Scale

Follow up to Archytas' Harmonic Means

On Wikipedia, I found a really interesting description of the fret placement on the Turkish Baglama, a kind of lute. The frets are almost 24-EDO quarter tones. I'm going to call the frequency ratios associated with the frets a scale. The scale is 17-limit and due to Yalçın Tura, but I haven't read the original source.

I like this scale because it's justly tuned, it has fairly small frequency ratios, the intervals between the scale steps are repeated a few times (suggesting a regular construction and the possibility of an intervallic analysis), and the subset of covered 24-EDO values seems like an important hint about which microtones are actually of use in middle eastern music. I'm also curious if the slightly deviations from 24-EDO are reproduced by players of fretless lutes like the oud. I also don't know much about the principled use of frequency ratios with factors of 17, so there's another interesting thing about the scale.

First, I tried making my own 17-limit scale that approximates 24-EDO in order to get a really good understanding of what choices Tura made and why. In comparing the two scales, I figured out how Tura made his, and it's very simple and actually non-intervallic. I feel a little sad deleting all the work I did on my scale, but the reader shouldn't have to suffer through my work just because I did.

So here directly is the 17-limit Baglama tuning given by Yalçın Tura, which is also approximately a subset of 24-EDO:

Fret 0: 1/1 ~ 0c
.
Fret 1: 18/17 ~ 100c
Fret 2: 12/11 ~ 150c
Fret 3: 9/8 ~ 200c
.
Fret 4: 81/68 ~ 300c
Fret 5: 27/22 ~ 350c
Fret 6: 81/64 ~ 400c
.
Fret 7: 4/3 ~ 500c
.
Fret 8: 24/17 ~ 600c
Fret 9: 16/11 ~ 650c
Fret 10: 3/2 ~ 700c
.
Fret 11: 27/17 ~ 800c
Fret 12: 18/11 ~ 850c
Fret 13: 27/16 ~ 900c
.
Fret 14: 16/9 ~ 1000c
.
Fret 15: 32/17 ~ 1100c
Fret 16: 64/33 ~ 1150c
Fret 17: 2/1 ~ 1200c

It has just four distinct step-wise interval differences: (34/33 and 33/32 at about 50 cents) and (256/243 and 18/17) at 90 to 100 cents.

0 to 1 : 18/17
1 to 2 : 34/33
2 to 3 : 33/32
3 to 4 : 18/17
4 to 5 : 34/33
5 to 6 : 33/32
6 to 7 : 256/243
7 to 8 : 18/17
8 to 9 : 34/33
9 to 10 : 33/32
10 to 11 : 18/17
11 to 12 : 34/33
12 to 13 : 33/32
13 to 14 : 256/243
14 to 15 : 18/17
15 to 16 : 34/33
16 to 17 : 33/32

Interestingly, the scale fails to respect octave-complementation in several places.

In total, Tura has frets for all of the 12-EDO pitches (in hundreds of cents), and also half-flat frets for (150c, 350c, 650c, 850c, and 1150c). If we root these on a C natural, the microtones are a neutral second (D-), a neutral third (E-), a half-flat fifth (G-) which is crazy to me, a neutral sixth (A-), and a half-flat octave (C-) rather than a neutral seventh, which is also crazy. I guess the open string isn't the tonic. Maybe some permutation of those makes sense. The thing I expected was was (150c, 350c, 850c, 1050c).

Ok, yes! There are two cyclic permutations of Tura's Baglama scale which give us neutral seconds, thirds, sixths, and sevenths.

If we start on fret-7 (500 c), we get a scale with neutral 2nds, 3rds, 6ths, 7ths, and also an interval at 650 cents that's like a half-flat P5. The frequency ratios rooted from here are:

    [1/1, 18/17, 12/11, 9/8, 81/68, 27/22, 81/64, 4/3, 24/17, 16/11, 3/2, 27/17, 18/11, 27/16, 243/136, 81/44, 243/128, 2/1]

If we start on fret-14 (1,000 cents), we get a scale with neutral 2nd, 3rds, 6ths, 7ths, and also an interval at 550 cents (half sharp from P4). The frequency ratios rooted from here are:

    [1/1, 18/17, 12/11, 9/8, 81/68, 27/22, 81/64, 729/544, 243/176, 729/512, 3/2, 27/17, 18/11, 27/16, 243/136, 81/44, 243/128, 2/1]

This has a pretty messed up P4 (729/544), but a lot in common with the scale rooted on fret-7 actually.

Okay, time to explain where the scale really comes from. We can construct a Major scale from major seconds and minor seconds. The usual format is

    [M2, M2, m2, M2, M2 M2 m2]

These are the intervals between successive scale degrees. We can accumulate them to get the intervals for each scale step relative to the tonic:

    [P1, M2, M3, P4, P5, M6, M7, P8]

This can be obtained by a cycle of pure fifths, starting on P4 and going up to M7. Then we add on P8 for closure. For some reason, Tura starts with a cyclic permutation of this, the mixolydian mode, but we'll just work with the major mode. In Pythagorean tuning, the M2 is tuned to 9/8 and the m2 is tuned to 256/243. So, we have 

    [9/8, 9/8, 256/243, 9/8, 9/8, 9/8, 256/243]

as the tuned intervals between steps of a Pythagorean major scale. 

We can accumulate these frequency ratios (multiplicatively) to get a the frequency ratios for each step of the major scale in Pythagorean tuning:

P1: 1/1 - 0c
M2: 9/8 - 203c
M3: 81/64 - 407c
P4: 4/3 - 498c
P5: 3/2 - 701c
M6: 27/16 - 905c
M7: 16/9 - 996c
P8: 2/1 - 1200c

A nice familiar major scale. But now for something exotic. Instead of making a chromatic scale by continuing the cycle of fifths, we'll use harmonic means!

The 9/8 value for the tuned major second (that showed up so frequently between scale degrees) can be split, roughly, into rational multiplicative-halves (i.e. approximate square roots) using the harmonic mean. More precisely, we take the harmonic mean of the ratio, 9/8 in this case, with 1/1 (as Archytas did, see my recent post on Archytas means and complements). This harmonic mean and its Archytas complement (given by dividing 9/8 by its harmonic mean) gives us two frequency ratios that are roughly minor seconds, in the sense of being tuned to about 100 cents:

    9/8 → 18/17 and 17/16

They're 99 cents and 105 cents, respectively. The 17/16 one is a little bigger. We'll split all the tuned major seconds (9/8s) of the major scale 

    [9/8, 9/8, 256/243, 9/8, 9/8, 9/8, 256/243]

in this way, with 18/17 coming first:

    [(18/17, 17/16), (18/17, 17/16), 256/243, (18/17, 17/16), (18/17, 17/16), (18/17, 17/16), 256/243]

I've included parentheses just to help show the grouping. This gives us intervals that are like minor Nths below the major Nths of the major scale. Also, we happen to introduce below the P5 an interval that's like a diminished fifth, since there was a gap of 9/8 between P4 and P5 and this also became more fine-grained through division.

If we multiplicatively accumulate these interval between scale degrees, we get a running product that is a fine chromatic scale approximating 12-EDO:

P1: 1/1 - 0c
m2: 18/17 - 98c
M2: 9/8 - 203c
m3: 81/68 - 302c
M3: 81/64 - 407c
P4: 4/3 - 498c
d5: 24/17 - 596c
P5: 3/2 - 701c
m6: 27/17 - 800c
M6: 27/16 - 905c
m7: 243/136 - 1004c
M7: 243/128 - 1109c
P8: 2/1 - 1200c

It looks like the major seventh has the most deviation from 12-EDO, being 9 cents sharper, and that's just due to it being unmodified by this procedure: the Pythagorean M7 is a little sharp, relative to both 12-EDO and 5-limit just intonation.

So that's a nice chromatic scale. To get Turkish microtones, we're just going to break up one of the intervals of our (now chromatic) scale one more time. All of the 17/16 ratios that take us from a minor Nth to a major Nth are going to get split into harmonic means and complements also. This will give us neutral intervals on the 2nds, 3rds, 6ths, and 7ths. It will also add a new little guy between the diminished fifth and the perfect fifth. A half-flat fifth we might call it.

To do this, we first break up the ratio 17/16 into its Archytas mean and complement:

    17/16 → 34/33 and 33/32

and then we perform that replacement all throughout our chromatic scale, again with the small harmonic mean (34/33) being placed first and the larger Archytas complement (33/32) being placed second:

    [1/1, 18/17, (34/33, 33/32), 18/17, (34/33, 33/32), 256/243, 18/17, (34/33, 33/32), 18/17, (34/33, 33/32), 18/17, (34/33, 33/32), 256/243]

If we accumulate the frequency ratios between scale degrees multiplicatively, we get this for the tuned steps of each scale degree relative to the tonic:

     P1: 1/1 - 0c

m2: 18/17 - 98c

n2: 12/11 - 150c

M2: 9/8 - 203c

m3: 81/68 - 302c

n3: 27/22 - 354c

M3: 81/64 - 407c

P4: 4/3 - 498c

d5: 24/17 - 596c

-5: 16/11 - 648c

P5: 3/2 - 701c

m6: 27/17 - 800c

n6: 18/11 - 852c

M6: 27/16 - 905c

m7: 243/136 - 1004c

n7: 81/44 - 1056c

M7: 243/128 - 1109c

P8: 2/1 - 1200c

where an "n7" is a neutral seventh and "-5" is a half flat fifth. This scale above is exactly Tura's Baglama scale rooted on the 7th fret (4/3 at 500 cents). The scale doesn't obey octave complementation, because it wasn't made from regular interval arithmetic, like the Pythagorean major and chromatic scales, but was rather made from subdivisions of frequency ratios into harmonic means.

Now I kind of want to define a system now where intervals are named based on subdivision by harmonic means. That could be cool.

If we take octave complements of Tura's scale, we find that there are multiple frequency ratios associated with a few of the 24-EDO steps:

^0: 1/1 ~ 0c

^1: 33/32 ~ 50c

^2: 18/17, 17/16 ~ 100c

^3: 12/11 ~ 150c

^4: 9/8 ~ 200c

^5: ?

^6: 81/68, 32/27 ~ 300c

^7: 27/22, 11/9 ~ 350c

^8: 81/64, 34/27 ~ 400c

^9: ?

^10: 4/3 ~ 500c

^11: 11/8 ~ 550c

^12: 24/17, 17/12 ~ 600c

^13: 16/11 ~ 650c

^14: 3/2 ~ 700c

^15: ?

^16: 27/17, 128/81 ~ 800c

^17: 18/11 ~ 850c

^18: 27/16, 136/81 ~ 900c

^19: ?

^20: 16/9 ~ 1000c

^21: 11/6 ~ 1050c

^22: 32/17 ~ 1100c

^23: 64/33 ~ 1150c

^24: 2/1 ~ 1200c

.

The commas between ratios on a single step are these (below 3/2):

     (17/16) / (18/17) = 289/288, "semitonisma"

(81/68) / (32/27) = 2187/2176, "septendecimal schisma"

(27/22) / (11/9) = 243/242, "rastma"

(81/64) / (34/27) = 2187/2176

(17/12) / (24/17) = 289/288

You might wonder what EDO tempers out all three of those. I checked every EDO up to 100. The simplest one is 24-EDO, and then we get integer multiples of 24-EDO too, e.g. (48, 72, 96, ...)-EDO. That's it.

If we go the other way and just look for intervals on each step of 24-EDO that are justly tuned to low complexity frequency ratios, then comparing that to its octave complement we get:

^0: 1/1

^1: 21/20, 28/27

^2: 16/15

^3: 11/10, 12/11

^4: 9/8, 10/9

^5: 7/6, 8/7

^6: 6/5

^7: 11/9

^8: 5/4

^9: 9/7

^10: 4/3

^11: 7/5

^12: 25/18, 36/25

^13: 10/7

^14: 3/2

^15: 14/9

^16: 8/5

^17: 18/11

^18: 5/3

^19: 7/4, 12/7

^20: 9/5, 16/9

^21: 11/6, 20/11

^22: 15/8

^23: 27/14, 40/21

^24: 2/1

which has tempered commas below 3/2 of:

(21/20) / (28/27) = 81/80

(11/10) / (12/11) = 121/120

(9/8) / (10/9) = 81/80

(7/6) / (8/7) = 49/48

(36/25) / (25/18) = 648/625

The first three unique ones of these are square super particulars:

    S9 = 81/80

    S11 = 121/120

    S7 = 49/48

and the last one is the justly tuned acute diminished second, Acd2. So there.

Here are just names for the Tura fractions and their octave complements:

     ~0c: P1 = [0, 0, 0, 0, 0, 0, 0] # 1/1

~50c: As1 = [-5, 1, 0, 0, 1, 0, 0] # 33/32

~100c: HuAcm2 = [1, 2, 0, 0, 0, 0, -1] # 18/17

~100c: ExA1 = [-4, 0, 0, 0, 0, 0, 1] # 17/16

~150c: DeAcM2 = [2, 1, 0, 0, -1, 0, 0] # 12/11

~200c: AcM2 = [-3, 2, 0, 0, 0, 0, 0] # 9/8

~250c: ?

~300c: HuAcm3 = [-2, 4, 0, 0, 0, 0, -1] # 81/68

~300c: Grm3 = [5, -3, 0, 0, 0, 0, 0] # 32/27

~350c: DeAcM3 = [-1, 3, 0, 0, -1, 0, 0] # 27/22

~350c: AsGrm3 = [0, -2, 0, 0, 1, 0, 0] # 11/9

~400c: AcM3 = [-6, 4, 0, 0, 0, 0, 0] # 81/64

~400c: ExGrM3 = [1, -3, 0, 0, 0, 0, 1] # 34/27

~450c: ?

~500c: P4 = [2, -1, 0, 0, 0, 0, 0] # 4/3

~550c: As4 = [-3, 0, 0, 0, 1, 0, 0] # 11/8

~600c: Hud5 = [3, 1, 0, 0, 0, 0, -1] # 24/17

~600c: ExA4 = [-2, -1, 0, 0, 0, 0, 1] # 17/12

~650c: De5 = [4, 0, 0, 0, -1, 0, 0] # 16/11

~700c: P5 = [-1, 1, 0, 0, 0, 0, 0] # 3/2

~750c: ?

~800c: HuAcm6 = [0, 3, 0, 0, 0, 0, -1] # 27/17

~800c: Grm6 = [7, -4, 0, 0, 0, 0, 0] # 128/81

~850c: DeAcM6 = [1, 2, 0, 0, -1, 0, 0] # 18/11

~900c: AcM6 = [-4, 3, 0, 0, 0, 0, 0] # 27/16

~900c: ExGrM6 = [3, -4, 0, 0, 0, 0, 1] # 136/81

~950c: ?

~1000c: Grm7 = [4, -2, 0, 0, 0, 0, 0] # 16/9

~1050c: AsGrm7 = [-1, -1, 0, 0, 1, 0, 0] # 11/6

~1100c: Hud8 = [5, 0, 0, 0, 0, 0, -1] # 32/17

~1150c: De8 = [6, -1, 0, 0, -1, 0, 0] # 64/33

~1200c: P8 = [1, 0, 0, 0, 0, 0, 0] # 2/1

Hu1 is the humbled unison, justly tuned to 50/51, and Ex1 is the exalted unison, justly tuned to 51/50.

The commas tempered out in the system that equates Tura's fretting with its octave complement are:

    ExExGrA0 = [-5, -2, 0, 0, 0, 0, 2] # 289/288

    HuAcAc1 = [-7, 7, 0, 0, 0, 0, -1] # 2187/2176

    DeDeAcAcA1 = [-1, 5, 0, 0, -2, 0, 0] # 243/242

I should try starting with rank-7 space, tempering out those three commas to produce a rank-4 space, and then seeing which simple intervals have frequency ratios on 250c, 450c, 750c, 950c.

I notice that none of the frequency ratios have factors of 5, 7, or 13. Maybe that suggests a few more commas to temper out. Also, ratios in the complementized Tura fretting have factors of 11 or 17 or neither, but not both. Also, the powers of 11 and 17 are in (-1, 0, 1).

Using those constraints, we find that these are very natural additions to Tura's complementized Baglama scale:

    ~250c: AsAcM2 = [-8, 3, 0, 0, 1, 0, 0] # 297/256

    ~450c: De4 = [7, -2, 0, 0, -1, 0, 0] # 128/99

    ~750c: As5 = [-6, 2, 0, 0, 1, 0, 0] # 99/64

    ~950c DeGrm7 = [9, -3, 0, 0, -1, 0, 0] # 512/297

Now all the holes are filled in. These happen to just Pythagorean intervals raised or lowered by by Johnston's 11 limit-comma. No factors of 17 are involved.

...

The commas that arose when we tried mixing Tura's scale with its complement were

(17/16) / (18/17) = 289/288, "semitonisma"

(81/68) / (32/27) = 2187/2176, "septendecimal schisma"

(27/22) / (11/9) = 243/242, "rastma"

The second one contains the third one as a factor. If we remove it, then we get 1089/1088, with prime interval [-6, 2, 0, 0, 2, 0, -1]. 

I'm not sure this is an iron-clad proof, but here's an argument that tempering out those three comas and purely tuning the octave produces 24-EDO: 

    determinant of [[-6, 2, 2, -1], [-1, 5, -2, 0], [-5, -2, 0, 2], [1, 0, 0, 0]] = 24

I just removed the slots of the prime intervals corresponding to powers of 5, 7, and 13.

Whether you find that convincing or not, it seems probable to me and very nice that we can define 24-edo on the 2.3.11.17 JI subspace in terms of pure octaves and three intervals to temper out that are justly tuned to superparticular ratios.

Two of the ratios are also square super-particulars:

1089/1088 = (33/32) / (34/33)

289/288 = (17/16) / (18/17)

Isn't life wonderful?

Some more linear algebra that might not be convincing to you, perhaps in part because I don't know any linear algebra:

If we take the inverse of our comma + octave matrix in the 2.3.11.17 subspace

    inverse of [[-6, 2, 2, -1], [-1, 5, -2, 0], [-5, -2, 0, 2], [1, 0, 0, 0]]

then WolframAlpha tells us we get

    1/24 * [[0, 0, 0, 24], [4, 4, 2, 38], [10, -2, 5, 83], [4, 4, 14, 98]]

Suppose we take the 2.3.11.17 subsection of two intervals that were about (350c/50c = ) 7 steps of 24-EDO, e.g. 

~350c: DeAcM3 = [-1, 3, 0, 0, -1, 0, 0] # 27/22

~350c: AsGrm3 = [0, -2, 0, 0, 1, 0, 0] # 11/9

Multiply the 2.3.11.17 parts by the inverse of the comma+octave matrix and we get:

     [-1, 3, -1, 0] * (1/24 * [[0, 0, 0, 24], [4, 4, 2, 38], [10, -2, 5, 83], [4, 4, 14, 98]]) = (1/12, 7/12, 1/24, 7/24)

    [0, -2, 1, 0] * (1/24 * [[0, 0, 0, 24], [4, 4, 2, 38], [10, -2, 5, 83], [4, 4, 14, 98]]) = (1/12, -5/12, 1/24, 7/24)

The 7/24ths at the end of each 4-component vector are powers of the octave, and the other coordinates are powers of the commas, but those comma powers become irrelevant when we temper out the commas. So tempering the three commas and tuning the octave purely means that intervals justly associated with 27/22 and 11/9 will both be tuned to a frequency ratio of 2^(7/24). As well they should in 24-EDO.

And all of this goes to show that Tura's 17-limit baglama tuning, when combined with its octave complement, implies 24-EDO. And this is a pretty cool way of defining 24-EDO in which simple just intervals before mis-tuning are basically already 24-EDO to begin with, so maybe those 17-limit just intervals have some explanatory merit in the interpretation of 24-EDO music.