Just Tuning Of Arabic Ajnas

:: Just Arabic Intonation Of Ajnas Via Margo Schulter

Margo Schulter shares some tunings for maqam scale fragments in "The Zalzalian Diatonic Tetrachords of Ibn Sina". We'll talk more about them in the post on Turkish makam intonation, but for now, let's look at the Arabic intonations. We have three Pythagorean tetrachords, which differ in the placement of the Grm2:

Arab 'Ajam: [9/8, 9/8, 256/243]

Arab Nahawand: [9/8, 256/243, 9/8]

Arab Kurdi: [256/243, 9/8, 9/8]

She gives several more tetrachords that include a Pythagorean major second AcM2, along with a Zalzalian neutral second, i.e. an interval justly tuned to one of {14/13, 13/12, 12/11, 11/10}.

If the AcM2 is the first of three tetrachord intervals, we could have jins Rast:

Arab Rast: [9/8, 12/11, 88/81] _ [204, 151, 143] cents

Medium-High Arab Rast: [9/8, 128/117, 13/12] _ [204, 156, 139] cents

High Syrian Rast: [9/8, 208/189, 14/13] _ [204, 166, 128] cents

These two of thse intonation are clearly very close. They all have a larger neutral second followed by a smaller neutral second. If we reverse the order so that a small neutral second precedes a large neutral second, then we get intonations for jins Mustaqim or jins Rast Jadid (which means New Rast).

Low Mustaqim: [9/8, 14/13, 208/189] or [9/8, 320/297, 11/10]

Higher Mustaqim: [9/8, 13/12, 128/117]

Arab Rast Jadid: [9/8, 88/81, 12/11]

You can see that the highest (/Syrian) intonation for jins Rast, when the relative neutral seconds are re-arranged, becoms the lowest intonation for Mustaqim / Rast Jadid, as we'd expect.

For tetrachord ajnas with the Pythagorean Major second in the second position, Schulter gives 

Moderate Arab Huseyni: [128/117, 13/12, 9/8] or [12/11, 88/81, 9/8]

High Arab Huseyni: [208/189, 14/13, 9/8] or [11/10, 320/297, 9/8]

and also

Low Arab 'Iraq: [14/13, 9/8 208/189] or [320/297, 9/8, 11/10]

Arab 'Iraq: [88/81, 9/8, 12/11]

And finally for tetrachord ajnas with the Major second in the last position, Schulter gives 

Arab Bayyati: [13/12, 128/117, 9/8]

Moderate Arab Bayyati: [88/81, 12/11, 9/8]

Lebanese Folk Bayyati: [14/13, 208/189, 9/8]

In other sources, Schulter notes that "Folk" in "Lebanese Folk Bayyati" is meant to suggest a common/popular/rural intonation.

These are all quite useful! These tetrachords have a historical basis in medieval eastern music theory and we could not have arrived at them just by looking at the 24-EDO tunings of maqamat. These tetrachords are all 13-limit: they have factors of [2, 3, 5, 7, 11, 13], but no higher primes.

: Other Ajnas

Schulter gave us intonations for roughly eight ajnas ('Ajam, Nahawand, Kurdi, Rast, Mustaqim, Huseyni, 'Iraq, Bayyati). Those are a huge help if you want to figure out the just intonation of Arabic maqmat: we can now build up an intonation for any maqam made from those ajnas. There are however some other ajnas to look at. We'll start with some ajnas from MaqamWorld.com. They're presented as fractions of a "tone", i.e. a major second, but we'll multiply through by 4 to get steps of 24-EDO.

: The Just Intonation Of Jins Hijaz

Let's start with jins Hijaz, because it's very important and very tricky. I could write a chapter on jins Hijaz alone. It has a highly variable intonation, both between geographic regions, and between players, and even for a single player it will be/should be intoned differently from one maqam to another.

In 24-EDO steps, jins Hijaz is written [2, 6, 2]. In absolute terms this is

[0, 100, 400, 500] cents

and in relative terms this is

[100, 300, 100] cents

This looks perfectly tonal, and if we expect Arabic music to use Pythagorean diatonic intervals, then our intonation would be this in absolute terms:

[P1, Grm2, AcM3, P4] # [1/1, 256/243, 81/64, 4/3]  _ [0, 90, 408, 498] cents

or this in relative terms:

[Grm2, AcAcAcA2, Grm2] # [256/243, 19683/16384, 256/243] _ [90, 318, 90] cents

This looks a little funny, but it's simply the Pythagorean version of [m2, A2, m2].

There's so much variation in the intonation of Hijaz that it's hard to call these wrong exactly, but I'm not a fan of these in most contexts. MaqamWorld isn't either: they explictly say that the notation doesn't match practice: 

"The interval between the 2nd and 3rd degrees is usually played smaller than notated by raising the 2nd a little and lowering the 3rd a little."

We're not told how much to raise and lower things to match Arabic practice, but if it was as much as 50 cents, we'd have reached the next step of 24-EDO and the jins wouldn't be approximately described as [2, 6, 2]\24. Thus, alterations somewhere between 5 and 30 cents seem plausible to me. Let's try to make that more specific.

Navid Goldrick of Oud For Guitarists describes the most common jins Hijaz intonation as falling in this range:

[0, 125 to 135, 375 to 385, 498] cents

If we want to spell that with second intervals, a nearby option is:

[P1, ReSbAcM2, M3, P4] # [1/1, 14/13, 5/4, 4/3] _ [0, 128, 386, 498] cents

[ReSbAcM2, PrSpM2, m2] # [14/13, 65/56, 16/15] _ [128, 258, 112] cents

You can see that the absolute intervals are adjusted in the direction that MaqamWorld prescribes, with the 2nd interval raised (28 cents over 24-EDO's m2 at 100 cents) and the third interval lowered (14 cents down from 24-EDO's M3 at 400 cents), and the adjustments are by plausibly small amounts. I like this a lot.

Navid also tells us that maqam Nahawand has a form that uses jins Hijaz and that the intonation of jins Hijaz in maqam Nahawand is quite different, falling in this range:

[0, 80 to 85, 390 to 395, 498] cents

If we want to spell that with second intervals, a nearby option is:

[P1, SbAcm2, AsSpGrm3, P4] # [1/1, 21/20, 44/35, 4/3] _ [0, 84, 396, 498] cents

[SbAcm2, AsSpSpGrM2, DeSbAcM2] # [21/20, 176/147, 35/33] _ [84, 312, 102] cents

I like Navid a lot as a presenter, teacher, and performer, and I take these two intonations seriously. The just tunings are obviously my own, but there are hundreds of plausible just tunings for jins Hijaz, and the reason I'm presenting these is that I trust Navid's intonation enough to paint a just intonation portrait of it.

Let's look at a few other intonations of Hijaz for comparison.

Margo Schulter tells us in "MET-24: A Milder Extended Temperament" that the oldest intonation we have for Hijaz comes from 13th century Persian polymath Qutb al-Din al-Shirazi. Here is his tetrachord:

[P1, DeAcM2, DeSbAc4, P4] # [1/1, 12/11, 14/11, 4/3] _ [0, 151, 418, 498]

[DeAcM2, Sbm3, AsSpGr1] # [12/11, 7/6, 22/21] _ [151, 267, 81] cents

Things have changed a bit in 800 years, but this is still recognizably a Hijaz intonation. The first relative interval of this Hijaz is fairly wide; it's indistinguishable from a 24-EDO neutral second at 150 cents.

Let's look at a few more intonations for Hijaz that people have proposed, to get a feel for its variety.

I've seen this intonation of Hijaz attributed to Jaques Dudon:

[P1, Prm2, M3, P4] # [1/1, 13/12, 5/4, 4/3] _ [0, 139, 386, 498]

[Prm2, ReAcA2, m2] # [13/12, 15/13, 16/15] _ [139, 248, 112] cents

This is just Navid's standard Hijaz tetrachord with the first interval widened from 14/13 to 13/12.

I've seen this jins Hijaz intonation attributed to Turkish music theorist Ozan Yarman:

[1/1, 16/15, 56/45, 4/3] _ [0, 112, 379, 498] cents

[16/15, 7/6, 15/14] _ [112, 267, 119] cents

I don't love that this isn't spelled by 2nd intervals, but Yarman has a discerning ear and I trust that his Hijaz is at least sometimes sized like this, though I think maybe we should call this a Turkish intonation rather than Arabic. In the chapters on Turkish music theory, we'll discuss Yarman's intonations at great length. The Turkish version of jins Hijaz is usually spelled "Hicaz", and I'm not sure if the tetrachord above is Yarman's attempt to describe an Arabic intonation (or a hybrid Arabic/Turkish intonation) or if the label is just misspelled.

I've seen this intonation of jins Hijaz floating around the internet without attribution

[P1, ReSbAcM2, ReM3, P4] # [1/1, 14/13, 16/13, 4/3] _ [0, 128, 359, 498] cents

[ReSbAcM2, SpM2, Prm2] # [14/13, 8/7, 13/12] _ [128, 231, 139] cents

and I think it's very beautiful numerically,  but that SpM2 interval in the middle strains my credibility a little with its just tuning at a mere 231 cents. Remember that the equal tempered version of Hijaz has the middle relative interval at 300 cents. Still, if someone told me that a prominent music theorist or performer of middle eastern music stands by this tuning of jins Hijaz, I wouldn't bat an eye.

Mohamed A. Alsiadi is a Syrian music theorist who describes Arabic maqamat in 53-EDO instead of 24-EDO. He variably uses two intonations of jins Hijaz, namely

[4, 14, 4]\53 _ [91, 317, 91] cents

[5, 13, 4]\53 _ [113, 294, 91] cents

The first of these is actually consistent with and aurally indistinguishable from the Pythagorean Hijaz intonation. It's also consistent with this intonation:

[HbAcm2, ExExGrA2, HbAcm2] # [18/17, 289/243, 18/17] _ [99, 300, 99] cents

which is aurally indistinguishable from the 24-EDO intonation. There are dozens of moderately low-complexity intonations consistent with [5, 13, 4] steps of 53-EDO and I don't have any strong reasons to prefer one or another as a detempering, but maybe you'll agree that this one is nice aesthetically:

[P1, m2, ExGrM3, P4] # [1/1, 16/15, 34/27, 4/3] _ [0, 112, 399, 498]

[m2, ExA2, HbAcm2] # [16/15, 85/72, 18/17] _ [112, 287, 99] cents

In Turkish music theory, the Hicaz genus is cognate with the Arabic jins Hijaz and it's often described in 53-EDO as

[5, 12, 5]\53

[4, 13, 5]\53

with both of these being used in different makams or even different places in a single makam.

In Persian music, the cognate tetrachord is called "dang Chahārgāh", and they have their own intonation, and indeed they have a different intonation for each Dastgah (mode) where it appears: Homayoun, Chahargah, Avaz-e Esfahan, Mokhalef-e Segah.

In short, jins Hijaz is hugely variable and people aren't going to agree precisely on how it sounds, but I really think that Navid Goldrick's intonation

[ReSbAcM2, PrSpM2, m2] # [14/13, 65/56, 16/15] _ [128, 258, 112] cents

is a good starting point for describing Arabic practice. Even though that 5-limit 16/15 interval looks a little suspiciously Turkish.

Jins Hijaz is an especially difficult case, but I hope you can see now some of the difficulty in figuring out just tunings that describe Arabic intonation better than 24-EDO.

: The Just Tuning Of Jins Jiharkah.

MaqamWorld's description of the pentachord jins Jiharkah is equivalent to these steps if 24-EDO:

[4, 4, 2, 4]

This looks tonal, but they note that the third and fourth absolute degrees are flattened relative to what is notated.

Navid of OudForGuitarists specifies this intonation: 

[0, 204, 380 to 390, 460 to 465, 702]

Spelling this alphabetically, a moderately-low complexity rational version of this would be:

[P1, AcM2, M3, DeAc4, P5] # [1/1, 9/8, 5/4, 72/55, 3/2] _ [0, 204, 386, 466, 702] cents

[AcM2, M2, DeAcm2, AsM2] # [9/8, 10/9, 288/275, 55/48] _ [204, 182, 80, 236] cents

but if we adjust the 4th scale degree up just six cents, then we get this tremendously beautiful pentachord: 

[P1, AcM2, M3, SbAc4, P5] # [1/1, 9/8, 5/4, 21/16, 3/2] _ [0, 204, 386, 471, 702] cents

[AcM2, M2, SbAcm2, SpM2] # [9/8, 10/9, 21/20, 8/7] _ [204, 182, 84, 231] cents

which is low in its prime-limit and has all super particular ratios. I find that this is my preferred intonation of jins Jiharkah, even if it's a little different from Navid's.

: The Just Tuning Of Jins Saba

Jins Saba describes the first few notes of Maqam Saba, and it's not used in many places, so sometimes you'll see it called a four, five, or six note jins. The jins, as it is described on MaqamWorld, outlines six notes and is equivalent to [3, 3, 2, 6, 2] steps of 24-EDO. In pitch classes, they describe it as

[D, Ed, F, Gb, A, Bb]

In 53-EDO, Alsiadi describes the full maqam Saba in ascending form as

[6, 7, 4, 14, 4, 9, 4]

The first two tones look Zalzalian to me and the rest looks Pythagorean, so I'm suggesting this detempering:

[P1, Prm2, Grm3, GrGrd4, P5, Grm6, Grm7, GrGrd8] # [1/1, 13/12, 32/27, 8192/6561, 3/2, 128/81, 16/9, 4096/2187] _ [0, 139, 294, 384, 702, 792, 996, 1086] cents

[Prm2, ReM2, Grm2, AcAcAcA2, Grm2, AcM2, Grm2] # [13/12, 128/117, 256/243, 19683/16384, 256/243, 9/8, 256/243] _ [139, 156, 90, 318, 90, 204, 90] cents

He also describes the Maqam as starting out with a Bayati trichord [6, 7] and a Hijaz tetrachord [4, 14, 4], so he doesn't even recognize a Saba pentachord, but in so far as it can be found in his intonation, it would be [6, 7, 4, 14, 4]\53.

Mauro Braunstein at the website OffTonic describes Arabic maqamat as they're used in Syrian Jewish liturgical music. I'm never sure how much information I can draw about Arabic intonation from that, but for comparison at least, their maqam Saba looks like this in 53-EDO: [6, 7, 5, 12, 5, 9, 4]. This is compatible with

[P1, Prm2, Grm3, Grd4, Gr5, Grm6, Grm7, GrGrd8] # [1/1, 13/12, 32/27, 512/405, 40/27, 128/81, 16/9, 4096/2187] _ [0, 139, 294, 406, 680, 792, 996, 1086] cents

[Prm2, ReM2, m2, AcA2, m2, AcM2, Grm2] # [13/12, 128/117, 16/15, 75/64, 16/15, 9/8, 256/243] _ [139, 156, 112, 275, 112, 204, 90] cents

You'll notice that it reaches a Gr5 at 40/27 instead of a perfect fifth P5. I don't know if this is a mistake or not. When I look at Turkish descriptions of Saba in 53-EDO, some reach P5 and some don't. I think it's common for Arabic music to reach P5 though.

For further comparison, the Turkish website Kazimyigiter presents makam Saba with 53-EDO steps (or equivalent symbols) as

[8, 5, 5, 13, 4, 9, 5, 12, 5]\53

This does reach P5 and is consistent with:

[P1, M2, Grm3, Grd4, P5, Grm6, Grm7, Grd8, M9, Grm10] # [1/1, 10/9, 32/27, 512/405, 3/2, 128/81, 16/9, 256/135, 20/9, 64/27] _ [0, 182, 294, 406, 702, 792, 996, 1108, 1382, 1494] cents

[M2, m2, m2, AcAcA2, Grm2, AcM2, m2, AcA2, m2] # [10/9, 16/15, 16/15, 1215/1024, 256/243, 9/8, 16/15, 75/64, 16/15] _ [182, 112, 112, 296, 90, 204, 112, 275, 112] cents

The Turkish version starts with [8, 5, ...]\53 instead of [6, 7, ...]\53. Hugely different.

You can see that this Turkish description of the makam falls short of P8, only reach Grd8, but then also extends past it up to Grm10. This is normal in Arabic music as well. Maqam Saba has two forms: starting at the minor sixth degree, you can finish off with jins Ajam, [9/8, 9/8, 256/243], forming an octave with the root of the maqam, or you can finish off with jins Nikriz, which is a pentachord formed by first playing AcM2 and then a Hijaz tetrachord, i.e. [4, 2, 6, 2]/24. My favorite intonation of Hijaz comes from Navid Goldrick as

[14/13, 65/56, 16/15]

so I'd intone jins Nikriz as

[9/8, 14/13, 65/56, 16/15]

Easy. When you use Nikriz, you fall short of the octave at first and then also extend past it. That's the second form of Saba.

In terms of notes, jins Saba is basically jins Bayati with a flattened 4th degree, although jins Bayati has its ghammaz (an emphasized note besides the tonic, often the starting note) on its 4th degree, while jins Saba has its ghammaz on the third or sixth degree.

So, the Arabic jins Saba has two neutral seconds followed by jins Hijaz, and the two neutral tones should probably go [larger neutral second, smaller neutral second, ...]. No one ever agrees on the intonation of jins Hijaz, but if your neutral tones outline Grm3 at 32/27, then the first two notes of your jins Hijaz will have to outline an AcM3 at 81/64 to hit P5. And you probably should hit P5, at least for an Arabic intonation, and probably also for Turkish and Syrian Jewish intonation. Sadly this means that my favorite Hijaz intonation from Navid Goldrick:

[14/13, 65/56, 16/15]

isn't suitable for use within jins Saba, since its first two intervals outline a just major third at 5/4 instead of a Pythagorean AcM3. Also, frankly, 14/13 is too low a neutral third for Saba: the neutral seconds in its relative formulation should be sized like those of jins Rast and jins Bayati, closer to 140 cents and 150 cents. You may be noting that now I've prescribed a common intonation of Hijaz, and an intonation for use in maqam Nahawand, and an intonation for use in the Saba tetrachord/pentachord/hexachord. That's how these things go.

We've done a lot so far. From what I've seen, Turkish and Persian modal music have fewer named modes and fewer named scale fragments than Arabic music theory. I'd say that the remaining ajnas of Arabic practice are less important and we'll hopefully spend less time on them.

: The Just Intonation Of All The Rest Of The Ajnas

Jins Sikah is notated on MaqamWorld as [3/4, 1/1] in fractions of a whole tone, i.e. [3, 4] steps of 24-EDO. The lowest note is the tonic, the highest note is the ghammaz. The name Sikah comes from "Se-gah" in Persian, meaning third place, because the jins and the maqam both start on the third note of maqam Rast, which is like the basic generating mother scale of all maqamat. All of the notes in middle eastern music have names like "the seventh one" and "the one from Boston" and "Wilbur's favorite note". Each one is called a "perde". Perde Sikah is a neutral third over perde Rast.

This might be a good time to learn a little Persian. The numbers one through five are "yek, do, seh, chehar, panj", and across middle eastern musical traditions, we will see notes and scale fragments and modes called yekah/yegah, dugah, sikah/segah, chahargah/jiharkah, and panjgah. There are historical reasons why you won't always see these words in their normal numerical order, but it's good to keep an eye out for them in case their logical structure has survived through the weird human history.

Anyway, Sikah has a neutral third. If it derives from Rast, we could intone it like Rast. The scale built on the second degree of Rast is Bayati, and we could also look at the tuning of maqam Bayati for hints.

Unfortunately, this already gives us four options. Let's look back at the Arabic jins intonations relayed by Margo Schulter for Rast and Bayati.

Arab Bayyati: [13/12, 128/117, 9/8]

Moderate Arab Bayyati: [88/81, 12/11, 9/8]

Arab Rast: [9/8, 12/11, 88/81] # [204, 151, 143]c

Medium-High Arab Rast: [9/8, 128/117, 13/12] # [204, 156, 139]c

From jins Rast, our first note for Sikah would be higher than perde Rast by one of  

(9/8) * (12/11) = 27/22 _ 355 cents

with the starting interval being of the jins being 88/81, or the perde Sikah would be placed a factor of

(9/8) * (128/117) = 16/13 _ 359 cents

over perde Rast, with the first interval of the jins being 13/12. These look different in their ratios, but they really only differ by like 4 cents. If we use the "High Syrian" intonation of jins Rast, then we'd have something genuinely different though, with a tonic at

(9/8) * (208/189) = 26/21 _ 370 cents

If instead we take cues from jins Bayati, perde Sikah might be tuned to

9/8 * 13/12 = 39/32

over perde Rast, with the first interval of jins Sikah tuned to 128/117, or perde Sikah might be tuned to

9/8 * 88/81 = 11/9

over perde Rast, with the first interval of the jins tuned to 12/11.

You might find this disheartening. What's the point of finding just tunings for ajnas and maqamat if there are going to be so many intonation options like this when we expect a logical relationship to hold between maqamat? Why not just stick to 24-EDO? I have four responses.

Schulter's intonation might suggest four different tonics for maqam Sikah, but that doesn't mean that she's confused about the tonic, or that middle eastern musicians are confused about the tonic. Maybe the [Rast, Bayati, Sikah, ...] modulation scheme held exactly in historic times but the tunings have since been adjusted. I'm pretty sure this is true. Maybe maqam Sikah has one or two intonations these days, and they're not derivable from maqam Rast or maqam Bayati, but they're still used consistently and can be shared and transmitted and taught.

Alternatively, even if everyone is confused or in disagreement about maqam intonation, it still might sound better to use low-complexity just frequency ratios on harmonic instruments than it would sound to use irrational 24 equal temperament frequency ratios; people could disagree about what just ratios to use and still agree that just ratios are better than equal tempering.

Even if Schulter's intonation seems inconsistent between related maqamat, yours doesn't have to be: you could unify your intonation for maqamat Rast, Bayati, and Sikah to use a small set of common tones. I think this is actually less accurate to Arabic practice. The tones do vary between maqamat, and it's better for us to figure out how they vary than to ignore the variation. And in particular, I think the second note of maqam Bayati is widely agreed to be lower than the third note of maqam Rast. My guess is that the tonic of maqam Sikah is the same as the third note of maqam Rast, however you intone maqam Rast, but I'm not srue. We'll keep looking into it.

Even when musicians and theorists don't agree on intonation, they can agree that 24-EDO intonation is wrong and that we need to adjust some of the steps of our scales away from it. MaqamWorld had notes to this effect on both jins Hijaz and jins Jiharkah. Even if we can't agree on exact frequency ratios, we can agree to do better than 24-EDO.

I planned to next talk about 53-EDO intonations of jins Sikah as relayed by Alsiadi and Braunstein, but I just found a pretty amazing source of data on Arabic intonation that might help us to understand jins Sikah and many others ajnas beside it. There's an American violinist and scholar of middle eastern music named Sami Abu Shumays who plays Arabic maqamat in an Egyptian style/intonation (and also he studied some in Syria, but Syrian and Egyptian music have similar intonational practices, and both styles fall within the variation of intonations of Arab-speaking world, rather than Turkish or Persian intonation). I'm going to guess that "Abu Shumays" is his last name, not a middle name and a last name. Abu Shumays writes in "The Fuzzy Boundaries of Intonation in Maqam" that he has identified "at least 12 notes between [his] lowest E flat and [his] highest E natural", and then he lists these {E}s in order, as they arise from maqamat on different tonics. Here's his list:

1. Low E flat in Bayati on G

2. E flat in Nahawand on C

3. E flat in Kurd on D

4. E flat in Hijaz on D

5. E half-flat in Saba on D

6. E half-flat in Bayati on D

7. E half-flat in Rast on C

8. E half-flat in Sikah on E half-flat

9. E half-flat in Sikah Beladi on C

10. Low E natural in Jiharkah on C

11. E natural in Hijaz on C

12. E natural in Ajam on C

These are listed with flatter {E}s at the top and sharper {E}s at the bottom. Abu Shumays expressed doubt that just intonation can capture all of this variation (twelve whole intervals!), but I think he simply rejects ratios too readily as being overly numerically complex to use, especially when you consider that, like, al-Farabi was describing lute fingerings that were geometrically constructed to have frequency ratios as complex as {162/149, 54/49, and 81/68} way back in ~ 920 AD, and a hundred years early Mansour Zalzal was using ratios like [208/189, 128/117, 88/81, 320/297]. There's a long history of using complex ratios to describe middle eastern music, and there are certainly more than 12 microtonal third intervals with just tunings less complex than ratios considered by al-Farabi and Zalzal.

I don't know exactly how Abu Shumays intones any of these {E}s, but if we want Grm3 and m3 to be associated with E flat, and if we want M3 and AcM3 to be associated with E natural, then things start falling into place.

Here's my first guess as to a just tuning consistent with his ordering of {E}s. I have them almost evenly spaced, so that they increase by about 10.3 cents from one interval to the next.

1. Low E flat in Bayati on G :: Hbm3 # 20/17 _ 281c

2. E flat in Nahawand on C :: Grm3 # 32/27 _ 294c

3. E flat in Kurd on D :: HbAcm3 # 81/68 _ 303c

4. E flat in Hijaz on D :: m3 # 6/5 _ 316c

5. E half-flat in Saba on D :: PrGrm3 # 65/54 _ 321c (or AsSbm3 # 77/64 _ 320c)

6. E half-flat in Bayati on D :: DeM3 # 40/33 _ 333c

7. E half-flat in Rast on C :: AsGrm3 # 11/9 _ 347c

8. E half-flat in Sikah on E half flat :: DeAcM3 # 27/22 _ 355c

9. E half-flat in Sikah Beladi on C :: GrM3 # 100/81 _ 365c

10. Low E natural in Jiharkah on C :: PrSpGrm3 # 26/21 _ 370c

11. E natural in Hijaz on C :: M3 # 5/4 _ 386c

12. E natural in Ajam on C :: AcM3 # 81/64 _ 408c

This could be significantly off from Abu Shumays's intonation, but it seems plausible to me, and I'm very grateful that Abu Shumays described the order of his {E}s. I can follow his order even without knowing his intonation, and even if he disagrees with the intonation I present here, there's nothing stopping me from following his general order of intervals even when our intonations differ by, oh, 15 cents or whatever.

I will say that I wish I knew his upper and lower bounds. Like, if his Low E flat in Bayati is as low as (Sbm3 # 7/6 _ 267c), then I should instead have spaced the justly tuned intervals out by more like 12.8 cents.

Now, Abu Shumays says that Ed of Sikah is the next degree higher in his order than the Ed of Rast, and I'm going to guess that a degree higher means 10 to 13 cents. Schulter describes a typical Arabic intonation of jins Rast that has its third note at

(9/8) * (12/11) = 27/22 _ 355 cents

or

(9/8) * (128/117) = 16/13 _ 359 cents

meaning that, if we use her intonation and the order of {E}s from Abu Shumays, then we could place the tonic of maqam Sikah somewhere in (365 to 372) cents and probably more like (365 to 369) cents. For low complexity ratios in that range that my interval naming system would call third-type intervals, we have just tunings at [100/81, 68/55, 99/80, 26/21].

Instead of spreading cents evenly over Abu Shumays's ordering of {E}s, we could do a similar analysis all over: we could try to pair what we have already learned about the intonation of jins and maqamat and then fill in gaps evenly when we have no better information. This could be very instructive. For example, if both [13/12, 128/117] and [88/81, 12/11] are consistent intonations for the first two relative intervals of jins Saba and jins Bayati, but Abu Shumays says that Saba has a lower Ed than Bayati, well, let's give Saba an Ed at

(9/8) * (13/12) = 39/32 _ 342 cents

and give jins Bayati an Ed at

(9/8) * (88/81) = 11/9 _ 347 cents

Alternatively, since we only got our intonation for jins Saba from 53-EDO commas ([6, 7]\53 per both Alsiadi and OffTonic), we could find lower tunings consistent with those commas, now that Abu Shumays has told us we should be lower than the Ed of jins Bayati, which Schulter tells us can be 39/32 or 11/9. If people present ajnas to the nearest 53-EDO comma, that just means that the tuned interval 6\53 just falls in the range 5.5\53 to 6.5\53, so we have half a 53-EDO step of wiggle room to find a lower or higher intonation for the Ed of jins Saba. If the Ed of jins Saba is somewhere in 5.5\53 to 6.0\53, that puts us in a range of (204 + 125 =) 329 cents to (204 + 136 =) 340 cents over perde Rast. And the next {E} below this is the Eb of jins Hijaz on D, which, based on a recommendation from Navid Goldrick, I like to put at (9/8 * 14/13 = ) 63/52 @ 332 cents, so we've closed in moderately tightly from both sides. The Ed of Saba should be noticeably higher than 332 cents and noticeably lower than (342 to 347) cents. Some options for this include

...

Put another way, we want a 2nd interval with a just tuning between 1) 14/13 and 2) (13/12 or 88/81) which we can place over 9/8. Decent options for this include:

HbAsm2 # 55/51 _ 131c

Acm2 # 27/25 _ 133c

Alternatively we could simply give Saba the 13/12 and Bayati the 88/81.

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Let's redo it all. Let's figure out the values of of Abu Shumays's order of {E}s as well as we can with all our information. I believe in us.

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There's an American-Palestinian violinist named Sami Abu Shumays who has created Scala tuning files describing Arabic modal intonation in units of cents so that people can play Arabic maqamat precisely and accurately. His intonation data is very extensive, including some intonations that are localized to specific regions like Egypt, Syria, and Lebanon. Let's look at his intonation and try to provide a rational and intervallic analysis.

We'll look at his intonation in terms of ajnas, i.e. scale fragments, including both the moral tonal ajnas (jins Ajam, Ajam Murassaa, Hijaz, Hijaz Murassaa, Hijazkar, Kurd, Nahawand, Nikriz, Saba Zamzama) and the more microtonal ajnas (jins Bayati, Jiharkah, Rast, Upper Rast, Saba, Saba Dalanshin, Sikah, and Sikah Baladi). 

For completeness, we'll also talk about a few ajnas that Abu Shumays doesn't touch on, namely (Athar Kurd, Lami, Mukhalif Sharqi, Musta'ar, and Sazkar).

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