Makams And Maqamat V

: Zalzal, al Farabi, 87-EDO, Margo Schulter

We've seen 24-EDO neutral seconds and we've seen 7-limit neutral seconds . We've seen 53-EDO neutral seconds and neutral second made by an extended Pythagorean spiral and neutral seconds made by 5-limit adjustments with syntonic commas. We've even seen 17-limit neutral second inspired by Archytas. What else could there be? (We've seen neutral thirds and sixths and sevenths in all those varieties.)

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, who will be even more important in this section.

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 invented some tetrachords using these neutral seconds - tetrachords being little scales of four tones extending from P1 to P4. All of al-Farabi's tetrachords hit P1, Pythagorean M2, some intermediate neutral third, and P4.

[P1, M2, ?, P4]

Here are the tetrachords, given with relative intervals on the left and absolute on the right:

[9/8, 11/10, 320/297] : [1/1, 9/8, 99/80, 4/3]

[9/8, 12/11, 88/81] : [1/1, 9/8, 27/22, 4/3]

[9/8, 13/12, 128/117] : [1/1, 9/8, 39/32, 4/3]

[9/8, 14/13, 208/189] : [1/1, 9/8, 63/52, 4/3]

Al-Farabi's tetrachords look a little funny at first, but they're kind of beautiful if you squint. In the first tetrachord based on 11/10, the wonky guy at the end of the relative intervals, 320/297, is really close to 14/13. The difference is like 1 cent, which is too small for humans to hear.

Likewise in the last tetrachord based on 14/13, the wonky guy at the end of the relative intervals, 208/189, is really close to 11/10. Only off by one cent. The same one cent, actually:

(320/297) / (14/13) = 2080/2079

(208/189) / (11/10) = 2080/2079

The other two tetrachords have a similar relationship: The wonky 88/81 in the second tetrachord based on 12/11 is basically the same as 13/12. And the wonky 128/117 in the third tetrachord based on 13/12 is basically the same as 12/11. The difference is larger here, at 5 cents, but that's still basically the smallest thing that humans can hear:

(128/117) / (12/11) = 352/351

(88/81) / (13/12) = 352/351

It would be nice if we could just use the simple fractions all throughout, right? We could say things like:

(9/8) * (11/10) * (14/13) ~= (4/3)

It's not like there aren't error bars on these things. There are limits to the fidelity of our recognition and our performance. So what system will let us do impressionistic arithmetic that expresses al-Farabi's tetrachords in terms of the Zalzalian neutral seconds, without any wonky bits? One solution is to find a new fine-grained EDO in which e.g.  (320/297) and (14/13)  are tuned to the same step. Then we can describe the tetrachord in terms of EDO steps instead of ratios.

The simplest EDO that has distinct steps for all of the Zalzalian neutral seconds is 45-EDO, but it doesn't put them in ascending order and doesn't .  After that we have 71-EDO, 78-EDO, 85-EDO, all well formed. 

But here's the list of EDOs that tune the Zalzalian seconds to distinct ascending step while also tempering out the intervals that are justly associated with (2080/2079) and (352/351): 

(87, 94, 99, 111, 128, 133, 135, 140, 145, 157, 174, 181, 186, 198, 205, 210, 227, 232, 244, 251, 268, 269, ...]-EDO.

In 87-EDO, 

{9/8 * 11/10 * 14/13 ~= 4/3} 

{9/8 * 12/11 * 13/12 ~= 4/3} 

are explained as 

(15 + 12 + 9 = 36) steps

(15 + 11 + 10 = 36) steps

respectively. In 94-EDO, the explanations are (16 + 13 + 10 = 39) and (16 + 12 + 11 = 39), respectively.

The people on the Xenharmonic Alliance discord point out that 87-EDO is a "Parapyth" EDO, identified by Margo Schulter for use in analyzing middle eastern music just like this. I got scooped. Many of the other higher EDOs mentioned above are Parapyth EDOs as well.

Ozan Yarman is a Turkish music theorist who also participates in the Xenharmonic community a little. He often gives just analyses of tetrachords which are literally false but perhaps still useful in this program of impressionistic arithmetic. I don't think he ever actually drawn attention to the fact that most of his work contains errors of normal arithmetic, but I'm pretty sure he knows and just doesn't care. But we don't have to pretend that {9/8 * 11/10 * 14/13 ~= 4/3}! We can just say that (15 + 12 + 9 = 36) steps of 87-EDO. I think that's a good way to elevate his work to mathematical validity.

And 87-EDO is great for this; it also tempers out the intervals justly associated with 256/255, 406/405, and 154/153, which all show up as deviations from truth when Yarman does arithmetic. Although 94-EDO tempers out the interval justly associated with 225/224, which Yarman also ignores. I think I still prefer 87-EDO, partly because it's the simplest one that works with Al-Farabi, but I admdit that I haven't investigated the merits of the higher EDOs in detail.

When we did a combined 24-EDO + 53-EDO analysis to just neutral thirds for Arabic and Turkish tradition, three of the four Arabic frequency ratios (16/13, 21/17, 27/22) that we found are justly associated with intervals that are tuned to 26 steps of 87-EDO. Both of the Turkish frequency ratios (56/45, 68/55) and one of the Arabic ones (99/80) get tuned one step higher to 27 steps of 87-EDO. So in today's  music, a tetrachord phrased with absolute intervals of the form [P1, M2, n3, P4] will look like

[0, 15, 26 or 27, 36]

when tuned in 87-EDO. This is the modern baseline that we'll use for looking at al-Farabi's medieval tetrachords.

The 11/10 tetrachord of al-Farabi is actually modern, since  (9/8) * (11/10) = (99/80) was one of our modern Arabic ratios in the combined 24 & 53 analysis. Here's the tetrachord, relative on the left, absolute on the right:

[15, 12, 9] : [0, 15, 27, 36]

A fine modern Turkish Rast. 

The 12/11 tetrachord of al-Farabi is also modern, since (9/8) * (12/11) = (27/22), a ratios we've seen which leads us to

[15, 11, 10] : [0, 15, 26, 36]

A fine modern Arabic Rast.

The 13/12 tetrachord of al-Farabi is flatter on the third than a modern Rast, as (9/8) * (13/12) = (39/32), which gives us

[15, 10, 11] : [0, 15, 25, 36]

The 14/13 of al-Farabi is flatter still than that on the third, as (9/8) * (14/13) = (63/52), which gives us

[15, 9, 12] : [0, 15, 24, 36]

as a tetrachord. 

I'm of the opinion that these last two are not Rast tetrachords at all, since Rast, in relative intervals, has a larger neutral second followed by a smaller neutral second (or at least two equal neutral seconds in 24-EDO), whereas these last two tetrachords of al-Farabi have a smaller neutral second followed by a larger one. That is to say, [T, K, S] is Rast, while these last two tetrachords of al-Farabi are [T, S, K]. Margo Schulter sometimes calls this Rast Jadid ("New Rast") or Mustaqim.

...

: Ozan Yarman on Just Tunings of Turkish Makams

Ozan Yarman is a Turkish music theorist who has written at length about the just tuning of Turkish makams. That sounds amazing, right? It's definitely cool, but sometimes his work seems a little sloppy to me. I'll start with some of his data from his doctorate thesis, "79-Tone Tuning & Theory For Turkish Maqam Music", and then I'll pick it apart a bit.

Rast (ascends and descends the same way, tonic on C): [1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1].

Acemli Rast (rises the same as Rast but descends as follows, tonic on C): [2/1, 16/9, 5/3, 3/2, 4/3, 5/4, 9/8, 1/1].

Mahur (ascending), tonic on C: [1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2/1].

Mahur (descending): [2/1, 15/8, 27/16, 3/2, 4/3, 5/4, 9/8, 1/1]. # Descends just like Rast.

Pencgah (ascends and descends the same way), tonic on C: [1/1, 9/8, 5/4, 7/5, 3/2, 27/16, 15/8, 2/1].

Nihavend (ascending), tonic on C: [1/1, 9/8, 6/5, 4/3, 3/2, 8/5, 32/17, 2/1].

Nihavend (descending): [2/1, 9/5, 8/5, 3/2, 4/3, 6/5, 9/8, 1/1].

All of those had C as a tonic. When we get makams with different tonics, we see that Ozan Yarman still notates the frequency ratios relative to C, so that the first note isn't tuned to 1/1.

Hicaz (ascending), tonic on D: [9/8, 6/5, 7/5, 3/2, 27/16, 50/27, 2/1, 9/4]. If we divide through by 9/8, we get [1/1, 16/15, 56/45, 4/3, 3/2, 400/243, 16/9, 2/1].

Hicaz (descending), tonic on D: [9/4, 2/1, 9/5, 27/16, 3/2, 7/5, 6/5, 9/8]. Dividing through by 9/8 to re-root gives [2/1, 16/9, 8/5, 3/2, 4/3, 56/45, 16/15, 1/1]. 

Huseyni (ascends and descends the same way), tonic on D: [9/8, 21/17, 4/3, 3/2, 27/16, 63/34, 2/1, 9/4]. Dividing through by 9/8 gives this rooted makam: [1/1, 56/51, 32/27, 4/3, 3/2, 28/17, 16/9, 2/1].

Segah (ascending), tonic on E: [(20/17), 5/4, 4/3, 3/2, 5/3, 15/8, 2/1, 40/17, 5/2] # The 20/17 note in parentheses is a leading tone below the tonic. Dividing through by 5/4 gives [(16/17), 1/1, 16/15, 6/5, 4/3, 3/2, 8/5, 32/17, 2/1].

Segah (descending): [40/17, 9/4, 2/1, 30/17, 5/3, 3/2, 4/3, 5/4]. Dividing through by 5/4 gives [32/17, 9/5, 8/5, 24/17, 4/3, 6/5, 16/15, 1/1]. # This doesn't descend from the octave and that's how Yarman has it written.

Huzzam (ascending), tonic on a slightly flat E: [(7/6), 36/29, 4/3, 3/2, 48/29, 54/29, 2/1, 7/3, 72/29] # The (7/6) note in parentheses is a leading tone below the tonic. Dividing through by 36/29 gives [(203/216), 1/1, 29/27, 29/24, 4/3, 3/2, 29/18, 203/108, 2/1]

Huzzam (descending): [40/17, 9/4, 2/1, 30/17, 48/29, 3/2, 4/3, 36/29]. Dividing through by 36/29 gives [290/153, 29/16, 29/18, 145/102, 4/3, 29/24, 29/27, 1/1]. # Like Segah makam (descending), this doesn't descend from the octave and that's how Yarman has it written.

Saba (ascending), tonic on D: [9/8, 21/17, 27/20, 16/11, 27/16, 11/6, 2/1, 9/4]. Dividing through by 9/8 gives [1/1, 56/51, 6/5, 128/99, 3/2, 44/27, 16/9, 2/1].

Saba (descending): [8/3, 5/2, 32/15, 2/1, 25/14, 27/16, 10/7, 4/3, 11/9, 9/8]. Dividing through by 9/8 gives [64/27, 20/9, 256/135, 16/9, 100/63, 3/2, 80/63, 32/27, 88/81, 1/1].

Now these are mostly fine. It's a little weird that there are a lot of fractions with factors of 17 (all in the denominators): [16/17, 20/17, 21/17, 24/17, 28/17, 30/17, 32/17, 40/17, 56/51, 63/34, 145/102] and a bunch of fractions with factors of 29 (mostly in the numerators): [29/16, 29/18, 29/24, 29/27, 36/29, 48/29, 54/29, 72/29, 203/108, 203/216, 290/153], but maybe that regularity is a consequence of some regular construction. I'm open to that. Like if you want 5-limit major intervas to be 60 cents sharper, you multiply them all by (30/29). Whatever.

What's weird is that Yarman gives tetrachord glosses on top of the makams, and the math doesn't work out. A bunch of the tetrachords don't exactly form a perfect fourth, but that might be excusable since Yarman's dissertation is about constructing Turkish microtones using weird irregular chains of tempered perfect fifths; if your P5s are all messed up, then your P4s will be messed up too. The weirder part to me is that he gives tetrachord glosses on top of frequency ratios that are obviously different from the tetrachords.

For example, Rast starts out with a dead simple 5-limit scale: [1/1, 9/8, 5/4, 4/3], which has relative intervals of 

[9/8 * 10/9 * 16/15].

But Yarman's notes on top say that the makam starts out with the "tempered Rast" tetrachord:

[28/25 * 28/25 * 17/16]

which would actually form 

[1/1, 28/25, 784/625, 833/625]

This is flat of 4/3 by a tiny 2500/2499 at 0.7 cents.  So maybe Ozan Yaramn uses 2500/2499 as part of his tempering scheme: I don't know. But if he does, that means that he thinks Rast makam starts 

[1/1, 28/25, 784/625, 833/625]

and he should notate it that way. He definitely shouldn't have it notated such that the first two relative steps differ by 22 cents and then in a parenthetical remark say "Never mind, they're actually equal to each other at 28/25".

Okay, so: weird frequency ratios, don't bother me, but they befuddle me. Tetrachords that don't form P4, bother me a little, but there's a chance that's on purpose. The inconsistent descriptions between the scale steps and the tetrachords bother me a lot. There are many more of those which I haven't mention. Let's do one more. If we divide his Huseyni through by 9/8 so that it's rooted on unison, the opening tetrachord is:

[1/1, 56/51, 32/27, 4/3]

with relative intervals of

[(56/51) * (68/63) * (9/8)]

However the gloss above the scale says the tetrachord is

[(11/10) * (13/12) * (9/8)]

which in absolute intervals is

[1/1, 11/10, 143/120,  429/320]

and it can't be both. The number 56/51 does not equal 11/10. He doesn't comment on this anywhere in the section. He just makes errors of arithemtic all over the place. Maybe he explains it in the section with the irregular tempering. I wouldn't know. Anyway,  the 429/320 is sharp of 4/3 by a factor of 1287/1280 at 9 cents. So if he is mistuning his tetrachords on purpose, then it looks like he sometimes goes flat and sometimes goes sharp.

There other inconsistencies not related to the tetrachords. For example, the Segah genus on page 137 has frequency ratio (40/7) indicated as a D# in the ascending form, and then it's notated as an Eb with some Saggital Notation accidentals in the descending form. Am I supposed to believe that his tuning system tunes both of those intervals over C to the same just ratio, and that one of the pitches happens to occur in ascending form of Segah and the other pitch occurs in the descending form?

It would make more sense if Yarman was just thinking in terms of some high-division EDO and then finding simple frequency ratios that were associated with each step, but he's clearly not doing that or he wouldn't have factors of 17 in only the numerators and it wouldn't be the case that he had factors of 17 and 29 but not factors of 13, 19 or 23.

Anyway, as much as his inconsistencies and math errors annoy me, it's still useful data for characterizing Turkish tuning. So let's go over his tetrachords. If he has a pentachord ending in (9/8), I just lop that off and call it a tetrachord:

Pure Rast tetrachord: (9/8) * (10/9) * (16/15)

Tempered Rast tetrachord: (28/25) * (28/25) * (17/16)

Mahur tetrachord: (9/8) * (9/8) * (256/243)

Pencgah tetrachord: (9/8) * (10/9) * (28/25)

Nihavend tetrachord: (9/8) * (16/15) * (10/9)

Segah tetrachord: (16/15) * (9/8) * (10/9)

Hicaz tetrachord: (16/15) * (7/6) * (15/14)

Wide Hicaz tetrachord: (16/15) * (20/17) * (17/16)

Huzzam tetrachord: (15/14) * (9/8) * (32/29)

Huseyni tetrachord: (11/10) * (13/12) * (9/8)

Ussak tetrachord: (12/11) * (12/11) * (9/8)

Çargah tetrachord: (15/14) * (13/11) * (55/52)

Saba pentachord: (11/10) * (12/11) * (13/12) * (15/13)

Here's a summary of their factor structure:

3-limit:

(9/8) * (9/8) * (256/243): Mahur tetrachord

5-limit:

(9/8) * (10/9) * (16/15): Pure Rast tetrachord

(9/8) * (16/15) * (10/9): Nihavend tetrachord

(16/15) * (9/8) * (10/9): Segah tetrachord

7-limit:

(9/8) * (10/9) * (28/25): Pencgah tetrachord

(16/15) * (7/6) * (15/14): Hicaz tetrachord

11 and 13 limit:

(11/10) * (13/12) * (9/8): Huseyni tetrachord

(12/11) * (12/11) * (9/8): Ussak tetrachord

(15/14) * (13/11) * (55/52): Çargah tetrachord

(11/10) * (12/11) * (13/12) * (15/13): Saba pentachord

17-limit:

(16/15) * (20/17) * (17/16): Wide Hicaz tetrachord

(28/25) * (28/25) * (17/16): Tempered Rast tetrachord

29-limit:

(15/14) * (9/8) * (32/29): Huzzam tetrachord

Remember the good old days when we though Turkish music theory was 53-EDO and it only had [Rast, Çârgâh, Bûselik, Kürdî, Uşşâk, Hicaz] as tetrachords? 

Actually, let's tune Yarman's tetrachrods to 53-EDO and see how they compare to the standard Turkish ajnas  and the ajnas of Alsiadi. I'll phrase them all in relative steps.

Yarman's Hicaz tetrachord and his Wide Hicaz tetrachord are both tuned to [5, 12, 5] , which is a normal Turkish Hicaz intonation.  An intonation at [4, 13, 5] would also have been standard. 

What he calls "Mahur" is tuned to [9, 9, 4] steps of 53-EDO. This is normally called Çargâh in Turkish theory and Ajam in Arabic theory. "Mahur" is the Persian name for this tetrachord, so that's not crazy.

In contrast, Ozan Yarman's Çargâh tetrachord is tuned to [5, 13, 4] steps of 53-EDO, which is an Arabic intonation for Hijaz. Weird.

Yarman's Pure Rast is [9, 8, 5] and this is the standard 53-EDO Turkish tuning for Rast. His Tempered Rast tetrachord is [9, 9, 5], which adds up to 23 steps, not 53-EDO's tuned P4 at 22 steps.

Yarman's Nihavend is tuned to [9, 5, 8] steps of 53-EDO, which is like a permutation of traditional Turkish Rast at [9, 8, 5] or Turkish Uşşak at [8, 5, 9]. I'd like to point out that Nihavend is an alternative spelling of Nahawand, which is a city in present day Iran. In Arabic music theory, the Nahawand or Busalik tetrachord is [9, 4, 9]. In standard Turkish music theory based on 53-EDO, the [9, 4, 9] tetrachord is called Bûselik. This [9, 4, 9] intonation is just a Pythagorean [M2, m2, M2], like the first four notes of a rank-2 minor scale. Ozan Yarman's Nihavend is justly tuned to [(9/8) * (16/15) * (10/9)], which are the just tunings for the rank-3 intervals [AcM2, m2, M2], i.e. the start of the rank-3 minor scale. So I think of Yarman's tetrachord as a refreshing confirmation that some of Turkish music's tetrachordal structure is pushing toward scales of 5-limit just intonation in a way that Arabic music isn't.

Both Yarman's Huzzam and Segah tetrachords are tuned to [5, 9, 8] which, again, is like a permutation of traditional Turkish Rast at [9, 8, 5] or Uşşak at [8, 5, 9].  Yarman's Segah is justly tuned in 5-limit while his Huzzam had a factors of 7 and 29. The Arabic Maqam Huzam starts on an E half flat. Yarman's Huzzam starts on an E somewhat flat, at (1200 * log_2(36/29) = ) 374 cents relative to C. I think this is fantastic. Ozan Yarman really wants you to know that Huzzam should be played with an Ed at 374 cents over C, and he introduced a factor of 29 into this and only this tetrachord to make it happen. Want some weirdly specific insight into the regional intonation of microtonal middle eastern scales? Ozan Yarman has got you covered, and he's packing a factor of 29 for coverage.

Ozan Yarman's Uşşak is tuned to [7, 7, 9], which overshoots P4 by a step like his Tempered Rast genus did. Perhaps he could tune the second intervals a little flatter.

Yarman's Huseyni tetrachord is tuned to [7, 6, 9]. This is like a permutation of Arabic Rast at [9, 7, 6].

His Pencgah tetrachord is [9, 8, 9]. This is hugely sharp of P4 at 26 steps of 53-EDO instead of 22. I don't know anything about Pencgah.

His Saba pentachord is [7, 7, 6, 11] at 31-steps of 53-EDO. This is actually a perfect fifth! Well done, Ozan.

So that's a 53-EDO analysis of the tetrachord glosses that Ozan Yarman puts on diagrams of his makams. Next let's look at the actual frequency ratios between steps of his makams, which should be equivalent to the notated tetrachords, or at least equivalent up to tempering, but rarely are.

First let's write all of Ozan Yarman's makams in terms of relative frequency ratios between steps. I'll ignore leading tones and write desending makam forms as though they were ascending, so that all of the fequency ratios will be larger than 1/1.

Rast: [9/8, 10/9, 16/15] + [9/8] + [9/8, 10/9, 16/15] # Rast + T + Rast

Acemli Rast (descending): [9/8, 10/9, 16/15] + [9/8, 10/9, 16/15] + [9/8] # Rast + Rast + T

Mahur (ascending): [9/8, 9/8, 256/243] + [9/8] + [9/8, 9/8, 256/243] # Mahur + T + Mahur

Mahur (descending) [9/8, 10/9, 16/15] + [9/8] +  [9/8, 10/9, 16/15] # Rast + T + Rast

Pencgah: [9/8, 10/9, 28/25, 15/14] + [9/8, 10/9, 16/15] # Pencgah pentachord + Rast

Nihavend (ascending): [9/8, 16/15, 10/9, 9/8] + [16/15, 20/17, 17/16] # Nihavend pentachord + Wide Hicaz

Nihavend (descending): [9/8, 16/15, 10/9, 9/8] + [16/15, 9/8, 10/9] #  Nihavend pentachord + Segah

Hicaz (ascending): [16/15, 7/6, 15/14] + [9/8] + [800/729, 27/25, 9/8] # Hicaz + T + Huseyni

Hicaz (descending): [16/15, 7/6, 15/14] +  [9/8, 16/15, 10/9, 9/8] # Hicaz + Nihavend pentachord

Huseyni: [56/51, 68/63, 9/8, 9/8] + [56/51, 68/63, 9/8] # Huseyni pentachord + Huseyni

Segah (ascending): [16/15, 9/8, 10/9] + [9/8] + [16/15, 20/17, 17/16] # Segah + T + Wide Hicaz

Segah (descending): [16/15, 9/8, 10/9] + [18/17] + [17/15, 9/8, 160/153] # Segah + 18/17 + "Mahur" + 17/16

Huzzam (ascending): [29/27, 9/8, 32/29] + [9/8] + [29/27, 7/6, 216/203] # Huzzam + T + "Hicaz"

Huzzam (descending): [29/27, 9/8, 32/29] + [145/136] + [17/15, 9/8, 160/153] # Huzzam + "16/15" + "Mahur"

Saba (ascending): [56/51, 153/140, 320/297, 297/256] + [88/81, 12/11, 9/8] # Saba pentachord + Ussak

Saba (descending): [88/81, 12/11] + [15/14, 189/160, 200/189] + [28/25] + [16/15, 75/64, 16/15]  # The first trichord isn't given a gloss, but it looks a lot like the start of Ussak from Saba ascending. Above that we have Çargah + "9/8" + "Hicaz".

There's this crazy thing with Ozan Yarman's Rast, where the given frequency ratios for each scale degree are the same ascending and descending, but the notated tetrachords and the EDO-like steps of his tempered 79-tone tuning system both differ in the ascending and descending forms. It's on page 134 of his doctoral thesis.

I looked up Pencgah a little. It seems to be cognate with the Persian dastgah scale "Rast Panjgah". The scale is a lot like Rast, with the main difference being that the fourth scale degree is sharpened by a factor of 21/20, which, at 84 cents, is a little larger than the 5-limit augmented unison, 25/24 at 71 cents. So, rooted on C, it's like we take the F of Rast up to an F# plus a little more. Although it's still less than a 24-EDO augmented unison. In so much as Turkish Rast looks like a push from Pythagorean major toward 5-limit major, Pencgah looks a lot like a 5-limit Locrian scale.

The Hicaz makam (ascending) ends with a Huseyni tetrachord, (800/729 * 27/25 * 9/8). But later we see the Huseyni tetrachord given as (56/51 * 68/63 * 9/8) in the Huseyni makam, and the Huseyni pentachord given as (56/51 * 68/63 * 9/8 * 9/8)  within the same makam. Both intonation of the Huseyni tetrachord are tuned to [7, 6, 9] steps of 53-EDO. The 56/51 is sharp of 800/729 by a factor of 1701/1700 at 1 cent. The 68/63 is flat of 27/25 by the saem 1 cent. So they're perceptually indistinguishable. The version from the Hicaz makam has shorter fractions, but a factor of 17. The version from the Huseyni makam has more complicated fractions, but it's 5-limit.

Segah (descending) has the tetrachord [17/15, 9/8, 160/153], which Ozan Yarman glosses as Mahur, instead of his previous Pythagorean version [9/8, 9/8, 256/243].  The first ratio, (17/15), is sharp of the usual 9/8 by 136/135 at 13 cents. And the last ratio, (160/153), is flat of the usual (256/243) by the same amount. I think 13 cents is enough of a difference that he might have given this tetrachord intonation a different name, like "17-limit mahur" or something. Segah (descending) also has some weird stuff going on where there's an 18/17 in the middle and a 17/16 up top to reach the octave. I don't feel qualified enough on Turkish makam intonation to comment on this.

Huzzam (descending) has the same 17-limit intonation of Mahur that we saw in Segah (descending). It also has a step of 145/136 between the tetrachords, which is 77 cents, but it's glossed as 16/15, which is also 77 cents. If you forget that and just put in the regular Pythagorean major second, (9/8), between the tetrachords, then Huzzam reaches the octave. I don't feel qualified enough on Turkish makam intonation to comment on this.

Huzzam (ascending) has a third intonation for the Hicaz tetrachord,  [29/27, 7/6, 216/203]. We'd previously been given Wide Hicaz as [16/15, 20/17, 17/16] and normal Hicaz as [16/15, 7/6, 15/14]. All three of these are just tuned to [5, 12, 5] in 53-EDO. In the Huzzam intonation, the first ratio, (29/27), is sharp of the usual 16/15 by a factor of 145/144 at 12 cents. The last ratio in the Huzzam intonation, (216/203) is flat of (17/16) by this same 12-cents. In the glosses, Yarman gave a version of a Huzzam tetrachord that didn't equal a justly tuned P4: 

(15/14 * 9/8 * 32/29) != 4/3

This is sharp of 4/3 by a factor of 406/405 at 4 cents. But if you look at the just tunings of the scale degrees, the tetrachord works out fine:

(29/27 * 9/8 * 32/29) = 4/3

In Saba (descending) ozan Yarman has a gap of 9/8 notated, but the actual relative step ratio is 28/25. This is flat of 9/8 by 225/224 at 8 cents. The Çargah tetrachord he gave right before that was sharp of a just P4 by that same 8 cents.

(15/14 * 189/160 *  200/189) / (225/224) = 4/3

I really don't know with Ozan Yarman whether these things that seem to be error of arithmetic are intentional or not. It you want a 7-limit Çargah tetrachord that equals 4/3, I'd reccomend flattening the last ratio, giving

(15/14 * 189/160 *  256/243) 

Saba (descending) has one more intonation for Hicaz.  This one, [16/15, 75/64, 16/15], is also tuned to [5, 12, 5] in 53-EDO.

So these are the tetrachords and pentachords we have infer from the justly tuned scale degrees:

Rast tetrachord: (9/8 * 10/9 * 16/15)

Mahur tetrachord: (9/8 * 9/8 * 256/243)

Mahur tetrachord from Segah:  (17/15 * 9/8 * 160/153)

Pencgah pentachord: (9/8 * 10/9 * 28/25 * 15/14)

Nihavend pentachord: (9/8 * 16/15 * 10/9 * 9/8)

Wide Hicaz tetrachord: (16/15 * 20/17 * 17/16)

Normal Hicaz tetrachord: (16/15 * 7/6 * 15/14)

Hicaz tetrachord from Huzzam: (29/27 * 7/6 * 216/203)

Hicaz tetrachord from Saba: (16/15 * 75/64 * 16/15)

Segah tetrachord: (16/15 * 9/8 * 10/9)

Huseyni tetrachord: (56/51 * 68/63 * 9/8) or (800/729 * 27/25 * 9/8)

Huzzam tetrachord: (29/27 * 9/8 * 32/29)

Saba pentachord: (56/51 * 153/140 * 320/297 * 297/256)

Ussak tetrachord: (88/81 * 12/11 * 9/8) 

Çargah tetrachord: (15/14 * 189/160 * 200/189) or maybe (15/14 * 189/160 *  256/243) if you want it to hit P4

In contrast to the glosses

Pure Rast tetrachord: (9/8) * (10/9) * (16/15)

Tempered Rast tetrachord: (28/25) * (28/25) * (17/16)

Mahur tetrachord: (9/8) * (9/8) * (256/243)

Pencgah tetrachord: (9/8) * (10/9) * (28/25)

Nihavend tetrachord: (9/8) * (16/15) * (10/9)

Segah tetrachord: (16/15) * (9/8) * (10/9)

Hicaz tetrachord: (16/15) * (7/6) * (15/14)

Wide Hicaz tetrachord: (16/15) * (20/17) * (17/16)

Huzzam tetrachord: (15/14) * (9/8) * (32/29)

Huseyni tetrachord: (11/10) * (13/12) * (9/8)

Ussak tetrachord: (12/11) * (12/11) * (9/8)

Çargah tetrachord: (15/14) * (13/11) * (55/52)

Saba pentachord: (11/10) * (12/11) * (13/12) * (15/13)

It's kind of crazy how often he has two expressions that are almost equal, but differ by a complex super particular ratio. Ozan Yarman's math would make a lot more sense in a system that tempered out the intervals justly associated with the frequency ratios [145/144, 225/224, 243/242, 406/405, 1701/1700, 2500/2499] . No EDO tempers out all of these. The only EDOs that temper out all but 1 are:

12-EDO, which tunes  243/242 to -1 step.

41-EDO, which tunes 2500/2499 to -1 step.

60-EDO, which tunes 243/242 to -1 step.

The members of the set [10, 19, 29, 53, 72, 82, 96]-EDO also do a decent job, tempering out all but 2 of Ozan Yarman's problematic commas.

:: Yarman Spelled by Seconds

I went over Yarman's intonation for Turkish makams again, and worked really hard to clean them up. In particular, I wanted the scales to be alphabetical in pitches: i.e. increasing by 2nd intervals. This took more than a week to accomplish.

These makams were already spelled alphabetically as presented by Yarman:

Rast (ascending & descending): [P1, AcM2, M3, P4, P5, AcM6, M7, P8] [1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1]

Acemli Rast (ascends as Rast, descends as follows): [P8, Grm7, M6, P5, P4, M3, AcM2, P1] [2/1, 16/9, 5/3, 3/2, 4/3, 5/4, 9/8, 1/1]

Mahur (ascending): [P1, AcM2, AcM3, P4, P5, AcM6, AcM7, P8] [1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2/1]

Mahur (descending): [P8, M7, AcM6, P5, P4, M3, AcM2, P1] [2/1, 15/8, 27/16, 3/2, 4/3, 5/4, 9/8, 1/1]

Nihavend (descending): [P8, m7, m6, P5, P4, m3, AcM2, P1] [2/1, 9/5, 8/5, 3/2, 4/3, 6/5, 9/8, 1/1]

Segah (descending): [Hbm10, AcM9, P8, Hbm7, M6, P5, P4, M3] [40/17, 9/4, 2/1, 30/17, 5/3, 3/2, 4/3, 5/4]

His Saba (ascending) was misspelled:

[AcM2, HbSbAcd4, Ac4, De5, AcM6, AsGrm7, P8, AcM9] [9/8, 21/17, 27/20, 16/11, 27/16, 11/6, 2/1, 9/4]

Here it is fixed:

My Saba (ascending): 

[AcM2, Asm3, Ac4, Prd5, AcM6, AsGrm7, P8, AcM9] # [9/8, 99/80, 27/20, 117/80, 27/16, 11/6, 2/1, 9/4]

His Saba (descending) was misspelled: 

[AcM2, AsGrm3, P4, SpA4, AcM6, SpA6, P8, m9, M10, P11] # [9/8, 11/9, 4/3, 10/7, 27/16, 25/14, 2/1, 32/15, 5/2, 8/3]

Here it is fixed:

My Saba (descending):

[AcM2, AsGrm3, P4, DeSb5, M6, Grm7, P8, Prm9, M10, P11] [9/8, 11/9, 4/3, 140/99, 5/3, 16/9, 2/1, 13/6, 5/2, 8/3]

His  Huseyni (ascending and descending) was misspelled:

[AcM2, HbSbAcd4, P4, P5, AcM6, HbSbAcd8, P8, AcM9] [9/8, 21/17, 4/3, 3/2, 27/16, 63/34, 2/1, 9/4]

Here it is fixed:

My Huseyni (ascending and descending):

[P1, Asm2, Grm3, P4, P5, Asm6, Grm7, P8] # [1/1, 11/10, 32/27, 4/3, 3/2, 33/20, 16/9, 2/1]

His Pencgah (ascending and descending) was misspelled:

[P1, AcM2, M3, Sbd5, P5, AcM6, M7, P8] [1/1, 9/8, 5/4, 7/5, 3/2, 27/16, 15/8, 2/1]

Here it is fixed:

My Pencgah (ascending and descending):

    [AcM2, M2, AcM2, m2] + [AcM2, M2, m2] # [9/8, 10/9, 9/8, 16/15] + [9/8, 10/9, 16/15]

[P1, AcM2, M3, AcA4, P5, AcM6, M7, P8] # [1/, 9/8, 5/4, 45/32, 3/2, 27/16, 15/8, 2/1]

His Hicaz (ascending) was misspelled:

[AcM2, m3, Sbd5, P5, AcM6, GrM7, P8, AcM9] [9/8, 6/5, 7/5, 3/2, 27/16, 50/27, 2/1, 9/4]

Here it is fixed:

My Hicaz (ascending):

[m2, AcA2, m2, AcM2, GrM2, Acm2, AcM2] # [16/15, 75/64, 16/15] + 9/8 +  [800/729, 27/25, 9/8]

[AcM2, m3, AcA4, P5, AcM6, GrM7, P8, AcM9] # [9/8, 6/5, 45/32, 3/2, 27/16, 50/27, 2/1, 9/4]

His Hicaz (descending) was misspelled:

[AcM9, P8, m7, AcM6, P5, Sbd5, m3, AcM2] [9/4, 2/1, 9/5, 27/16, 3/2, 7/5, 6/5, 9/8]

Here it is fixed:

My Hicaz (descending):

[m2, AcA2, m2] + [AcM2, m2, M2, AcM2] # [16/15, 75/64, 16/15] * [9/8, 16/15, 10/9, 9/8]

[AcM2, m3, AcA4, P5, AcM6, m7, P8, AcM9] # [9/8, 6/5, 45/32, 3/2, 27/16, 9/5, 2/1, 9/4]

His Nihavend (ascending) was misspelled: 

[P1, AcM2, m3, P4, P5, m6, Hbd8, P8] [1/1, 9/8, 6/5, 4/3, 3/2, 8/5, 32/17, 2/1]

Here it is fixed:

My Nihavend (ascending):

[AcM2, m2, M2, AcM2] + [m2, AsSpM2, DeSbAcM2] # [9/8, 16/15, 10/9, 9/8] * [16/15, 33/28, 35/33]

[P1, AcM2, m3, P4, P5, m6, AsSpGrm7, P8] [1/1, 9/8, 6/5, 4/3, 3/2, 8/5, 66/35, 2/1]

His Segah (ascending) was misspelled:

[(Hbm3), M3, P4, P5, M6, M7, P8, Hbm10, M10] [(20/17), 5/4, 4/3, 3/2, 5/3, 15/8, 2/1, 40/17, 5/2]

Here it is fixed:

My Segah (ascending): 

(DeSbAcM2) + [m2, AcM2, M2] + AcM2 + [m2, AsSpM2, DeSbAcM2] # (35/33) * [16/15, 9/8, 10/9] * 9/8 * [16/15, 33/28, 35/33]

[(AsSpM2), M3, P4, P5, M6, M7, P8, AsSpM9, M10] # [(33/28), 5/4, 4/3, 3/2, 5/3, 15/8, 2, 33/14, 5/2]

His Huzzam (ascending) and (descending) were by far the hardest ones to solve. They included 29-limit ratios, which I don't even have interval names for. Even ignoring those, the intervals that I can name were also misspelled, e.g. he used a low 10th interval as a ninth and a 3rd interval as an low ornamental 2nd interval.

His Huzzam (ascending) was misspelled: 

[(Sbm3), ?, P4, P5, ?, ?, P8, Sbm10, ?] [(7/6), 36/29, 4/3, 3/2, 48/29, 54/29, 2/1, 7/3, 72/29]

Here it is fixed:

My Huzzam (ascending):

[DeM2, AcM2, Asm2] + AcM2 + [DeM2, AsAcM2, m2] # [320/297, 9/8, 11/10] * 9/8 * [320/297, 297/256, 16/15]

[Asm3, P4, P5, Asm6, Asm7, P8, AsAcM9, Asm10] # [99/80, 4/3, 3/2, 33/20, 297/160, 2/1, 297/128, 99/40]

His Huzzam (descending) was misspelled: 

[Hbm10, AcM9, P8, Hbm7, ?, P5, P4, ?] [40/17, 9/4, 2/1, 30/17, 48/29, 3/2, 4/3, 36/29]

Here it is fixed:

My Huzzam (descending):

[DeM2, AcM2, Asm2] + m2 + [DeAcA2, AcM2, AsGrd2] # [320/297, 9/8, 11/10] + 16/15 + [25/22, 9/8, 704/675]

[Asm3, P4, P5, Asm6, AsGrd7, P8, AcM9, AsGrd10] # [99/80, 4/3, 3/2, 33/20, 44/25, 2/1, 9/4, 176/75]

The low ornamental Sbm3 should be replaced with some kind of second around

    1200 * log_2(7/6) = 267c

I'm not very particular on what it is. An AcA2 justly tuned to 75/64 seems fine. It's only 8 cents sharp of Yarman's Sbm3.

These intonations are very close in sound to Yarman's intonations, and use Zalzalian seconds (for historicity) or 5-limit seconds (a nod to contemporary Turkish theory) as much as possible.

In the process of cleaning up Yarman's intonations, I settled on some intonations for the various tetrachords and pentachords, which were often inconsistent or arithmetically malformed in Yarman's analysis. Here's a summary:

...

:: Margo Schulter Reports on Tetrachords

Margo Schulter is a great asset to the microtonal community. She has gotten access to lots of historical sources on makams and maqamat, and she shares her analysis and summaries with the community.

One thing she's done that I value very highly is to take a lot of mathematically defined tetrachords in terms of Pythagorean and Zalzalian frequency ratios and describe which ones are suitable intonations for different ajnas/genera in Arab, Turkish, and Persian musical traditions. Let's have a look:

Current Arab `Ajam or Persian Mahur:

[1/1, 9/8, 81/64, 4/3] _ [9/8, 9/8, 256/243]

Current Arab Nahawand or Persian Nava:

[1/1, 9/8, 32/27, 4/3] _ [9/8, 256/243, 9/8]

Current Arab or Turkish Kurdi:

[1/1, 246/243, 32/27, 4/3] _ [256/243, 9/8, 9/8]

Systematist Isfahan:

[1/1, 13/12, 13/11, 14/11, 4/3] _ [13/12, 12/11, 14/13, 22/21]

Higher septimal Shur, Bayyati, or Ushshak:

[1/1, 13/12, 7/6, 4/3] _ [14/13, 13/12, 8/7]

Lower septimal Shur, Bayyati, or Ushshak:

[1/1, 14/13, 7/6, 4/3] _ [14/13, 13/12, 8/7]

Higher Buzurg, Avaz-e Bayat-e Esfahan, or Byzantine Soft Chromatic:

[1/1, 13/12, 26/21, 4/3] _ [13/12, 8/7, 14/13]

or

[1/1, 14/13, 16/13, 4/3] _ [14/13, 8/7, 13/12]

Higher Mustaqim, Dastgah-e Afshari, Gushe-ye Shekaste:

[1/1, 9/8, 39/32, 4/3] _ [9/8, 13/12, 128/117]

Higher Mustaqim or Arab Rast Jadid:

[1/1, 9/8, 11/9, 4/3] _ [9/8, 88/81, 12/11]

Medium-high Arab Rast, Low Turkish Rast:

[1/1, 9/8, 16/13, 4/3] _ [9/8, 128/117, 13/12]

Arab Rast, Byzantine Diatonic:

[1/1, 9/8, 27/22, 4/3] / [9/8, 12/11, 88/81]

Moderate Shur, Arab Bayyati, Turkish Ushshak:

[1/1, 13/12, 32/27, 4/3] _ [13/12, 128/117, 9/8]

Moderate Arab Bayyati:

[1/1, 88/81, 32/27, 4/3] _ [88/81, 12/11, 9/8]

Moderate Arab Huseyni, Low Turkish Huseyni:

[1/1, 128/117, 32/27, 4/3] _ [128/117, 13/12, 9/8]

Moderate Arab Huseyni:

[1/1, 12/11, 32/27, 4/3] _ [12/11, 88/81, 9/8]

Persian Old Esfahan:

[1/1, 13/12, 39/32, 4/3] _ [13/12, 9/8, 128/117]

Arab `Iraq:

[1/1, 88/81, 11/9, 4/3] _ [88/81, 9/8, 12/11]

Medium Persian Esfahan:

[1/1, 128/117, 16/13, 4/3] _ [128/117, 9/8, 13/12]

or

[1/1, 12/11, 27/22, 4/3] _ [12/11, 9/8, 88/81]

Low Mustaqim, Afshari, or Shekaste; possibly High Turkish Nihavend:

[1/1, 9/8, 63/52, 4/3] _ [9/8, 14/13, 208/189]

or

[1/1, 9/8, 40/33, 4/3] _ [9/8, 320/297, 11/10]

High Syrian Rast, or Medium Ottoman Rast:

[1/1, 9/8, 26/21, 4/3] _ [9/8, 208/189, 14/13]

or

[1/1, 9/8, 99/80, 4/3] _ [9/8, 11/10, 320/297]

Low Shur, Lebanese Folk Bayyati, or Turkish Ushshak:

[1/1, 14/13, 32/27, 4/3] _ [14/13, 208/189, 9/8]

or

[1/1, 320/297, 32/27, 4/3] _ [320/297, 11/10, 9/8]

Turkish Huseyni or High Arab Huseyni:

[1/1, 208/189, 32/27, 4/3] _ [208/189, 14/13, 9/8]

or

[1/1, 11/10, 32/27, 4/3] _ [11/10, 320/297, 9/8]

Low Arab `Iraq, High Turkish Segah, Low Persian Old Esfahan:

[1/1, 14/13, 63/52, 4/3] _ [14/13, 9/8 208/189]

or

[1/1, 320/297, 40/33, 4/3] _ [320/297, 9/8, 11/10]

Possible High Persian Segah:

[1/1, 208/189, 26/21, 4/3] _ [208/189, 9/8, 14/13]

or

[1/1, 11/10, 99/80, 4/3] _ [11/10, 9/8, 320/297]

Pretty handy!

:: The finger of Navid Goldrick

As long as we're talking about just tuning, Navid Goldrick of Oud For Guitarists described the geometry of his fingering for oud, and that's enough to figure out his just tuning. Here it is:

Navid tells us that we can divide the string length into fractions and place markers there to learn fingering. When you divide a string length by {x}, you get a frequency ratio that is 1/(1-(1/x)) (or equivalently (x/(x-1)) times the open frequency of the string.

Divide by 3, you get a perfect fifth, 3/2 at 702 cents.

Divide by 3.4, you get a 17/12 tritone at 603 cents. 

Divide by 4, you get a perfect fourth, 4/3 at 498 cents.

Divide by 5, you get a just major third, tuned to 5/4 at 386 cents.

Divide by 6.5, you get 13/11 at 289 cents, an imperceptible 5 cents below a Pythagorean minor third at 32/27. You'd divide by 6.4 exactly to get the Pythagorean value.

Divide by 9, you get a Pythagorean major second at 9/8 at 204 cents.

Divide by 16.7, you get 167/157 at 107 cents, an imperceptible 5 cents below a just minor second at 16/15. You'd divide by 16 to get the Pythagorean value.

Next Navid places some markers between the already placed ones. When you a marker between two white ones at (1/x) and (1/y), that's the same as placing it at (((1/x) + (1/y)) / 2) of the string length, or equivalently, (x + y) / (2 * x * y) of the string length.

If you add a marker between the markers for 9/8 and 4/3, that's a string proportion of (9 + 4) / (2 * 9 * 4) = 13/72, so you're dividing the string into 72/13 ~= 5.5, and you get a frequency ratio of 72/59 at 345 cents, indistinguishable from the Zalzalian neutral third 11/9.

If you add a marker between the markers for 167/157 and 9/8, that's like dividing by 3006/257 ~ 11.7, and you get a frequency ratio of 3006/2749 at 155 cents. Alternatively, if you place your minor second marker at 1/16 of the string length, then the midpoint with the major second marker has a string ratio of 25/288, which gives you a frequency ratio of 288/263 at 157 cents. Either one has the sound of the Zalzalian neutral 2nd ratio 12/11. So you could also divide the string by 12 to get that marker.