Arabic Maqamat In 53-EDO Per Alsiadi

: Intro

There's a Syrian-born oud player named Mohamed Alsiadi with a nice website listing *Arabic* maqamat with *Turkish* 53-EDO commas. It's great.

Rast is probably the most important scale in middle eastern music, and Alsiadi gives us a version of Rast very similar to Turkish, but the middle third and middle seventh are flattened by an additional comma each relative to the Turkish version, which gets us closer to 24-TET neutral tones. The Turkish Rast makam is not so far from a major scale if you're tone deaf, but Alsiadi's version definitely deviates from major in a way that any western ear will notice.

I've transcribed the descending maqamat as though they were ascending for easier comparison with ascending forms. I kind of regret it, but the fact is at least indicated as "(descending)" in the name.

: Alsiadi's 53-EDO Arabic Maqamat

Maqam Rast (ascending): [C, D, E-, F, G, A, B-, C] : [9, 7, 6, 9, 9, 7, 6]. # Rast tetrachord + 9/8 + Rast tetrachord.

Maqam Rast (descending with Nahawand ending): [C, D, E-, F, G, A, Bb, C]. : [9, 7, 6, 9, 9, 4, 9]. # Rast tetrachord (9, 7, 6) + 9/8 + Nahawand tetrachord (9, 4, 9).

Maqam Basandida (ascending): [C, D, Eb, F#, G, A, Bb, C] : [9, 4, 14, 4, 9, 4, 9] # Nakriz or Nawa Athar pentachord + Busalik tetrachord

Maqam Basandida (descending): [C, D, E-, F, G, A, B-, C] : [9, 7, 6, 9, 9, 7, 6] # Rast tetrachord + 9/8 + Rast tetrachord. These are the same pitch classes that Rast uses to ascend.

Maqam Dalansheen (ascending): [C, D, E-, F, G, A, B-, C, Db, E, F] : [9, 7, 6, 9, 9, 7, 6, 4, 14, 4]. # Compound maqam. It's like Rast with an ornamentation on top that derives from an overlap with Saba (starting on the A).

Maqam Dalansheen (descending): [C, D, E-, F, G, A, Bb, C, D, E-, F] : [9, 7, 6, 9, 9, 4, 9, 9, 7, 6].

Mahur (ascending): [C, D, E, F, G, A, B, C] : [9, 9, 4, 9, 9, 9, 4].

Mahur (descending): [C, D, E-, F, G, A, Bb, C] : [9, 7, 6, 9, 9, 4, 9].

Maqam Nishaburk (ascending): [D, E, F+, G, A, B-, C, D] : [9, 7, 6, 9, 6, 7, 9].

Maqam Nishaburk (descending): [D, E, F+, G, A, Bb, C, D] : [9, 7, 6, 9, 4, 9, 9].

Maqam Suzdilara ascending (Jaharka ending): [C, D, E, F, G, A, Bb, C] : [9, 9, 4, 9, 9, 4, 9].

Maqam Suzdilara ascending (Bayati ending): [C, D, E, F, G, A B-, C] : [9, 9, 4, 9, 9, 6, 7].

Maqam Suzdilara descending: [C, D, E-, F, G, A, Bb, C] : [9, 7, 6, 9, 9, 4, 9] # The scale has Bb written, but then there are accidentals after the scale in the style of a key signature which has a half flat on the B line. So maybe you can do Bb or B- descending? But the notated jins above the scale says Jaharkah, which wouldn't gvie a microtone. Whatever. These scales are all the same. This is just Rast descending with the Nahawand ending.

Maqam Suznak ascending: [C, D, E-, F, G, A-, B, C] : [9, 7, 6, 9, 5, 13, 4].

Maqam Suznak descending: [C, D, E-, F, G, A, Bb, C] : [9, 7, 6, 9, 9, 4, 9] # Still just Rast descending with Nahawand ending.

Maqam Yakah: [G, A, B-, C, D, E-, F, G] : [9, 7, 6, 9, 6, 7, 9].

Maqam Nahawand (ascending): [C, D, Eb, F, G, Ab, Bb, C] : [9, 4, 9, 9, 4, 9, 9].

Maqam Nahawand (descending): [C, D, Eb, F, G, Ab, B, C] : [9, 4, 9, 9, 4, 14, 4] # this is the Hijaz ending for Nahawand per MaqamWorld

Maqam Al-Sinbulah (ascending) / Nahawand Murassah (ascending): [C, D, Eb, F, Gb, A, Bb, C] : [9, 4, 9, 4, 14, 4, 9]

Maqam Al-Sinbulah (descending) / Nahawand Murassah (descending): [C, D, Eb, F, G, Ab, B, C] : [9, 4, 9, 9, 4, 14, 4]

Maqam Farah Fazah (ascending): [G, A, Bb, C, D, Eb, F, G] : [9, 4, 9, 9, 4, 9, 9].

Maqam Farah Fazah (descending): [G, A, Bb, C, D, Eb, F#, G] : [9, 4, 9, 9, 4, 14, 4].

Maqam Hisar: [D, E-, F, G#, A, Bb, C#, D] : [6, 7, 14, 4, 4, 14, 4].

Maqam Nahawand Kabir (ascending): [C, D, Eb, F, G, Ab, B, C] : [9, 4, 9, 9, 4, 14, 4].

Maqam Nahawand Kabir (descending): [C, D, Eb, F, G, A, Bb, C] : [9, 4, 9, 9, 9, 4, 9].

Maqam Nakriz (ascending): [C, D, Eb, F#, G, A, Bb, C] : [9, 4, 14, 4, 9, 4, 9].

Maqam Nakriz (descending): [C, D, Eb, F#, G, A, B-, C] : [9, 4, 14, 4, 9, 7, 6].

Maqam Nawa Athar (ascending): [C, D, Eb, F#, G, Ab, B, C] : [9, 4, 14, 4, 4, 14, 4].

Maqam Nawa Athar (descending): [C, D, Eb, F#, G, Ab, Bb, C] : [9, 4, 14, 4, 4, 9, 9].

Maqam Sultani Yakah: [G, A, Bb, C, D, Eb, F#, G] : [9, 4, 9, 9, 4, 14, 4].

Maqam Hijaz Kar Kurd: [C, Db, Eb, F, G, Ab, Bb, C] : [4, 9, 9, 9, 4, 9, 9].

Maqam Ajam Kurdi: (page under construction)

Maqam Shawq Tarab (ascending): [A, Bb, C, D, E-, F, Gb, A] : [4, 9, 9, 6, 7, 4, 14].

Maqam Shawq Tarab (descending): [A, Bb, C, D, Eb, F, G, A] : [4, 9, 9, 4, 9, 9, 9].

Maqam Kurdi: [D, Eb, F, G, A, Bb, C, D] : [4, 9, 9, 9, 4, 9, 9].

Maqam Tarz Nawayn: [C, Db, Eb, F, Gb, A, Bb, C] : [4, 9, 9, 4, 14, 4, 9].

Maqam Hijaz (ascending): [D, E-, F#, G, A, B-, C, D] : [5, 13, 4, 9, 7, 6, 9] # Hijaz tetrachord + Rast pentachord

Maqam Hijaz (descending): [D, E-, F#, G, A, Bb, C#, D] : [5, 13, 4, 9, 4, 14, 4] # Hijaz tetrachord + 9/8 + Hijaz tetrachord

Maqam Hijaz Kar (ascending): [C, Db, E, F, G, Ab, B, C] : [4, 14, 4, 9, 4, 14, 4] # Hijaz tetrachord + 9/8 + Hijaz tetrachord

Maqam Hijaz Kar (descending): [C, Db, E, F, G, Ab, Bb, C] : [4, 14, 4, 9, 4, 9, 9] # Hijaz tetrachord + 9/8 + Kurd tetrachord. Alsiadi has "Busalik tetrachord + 9/8 + Kurd tetrachord" notated in text, but the commas and pitch classes don't support that.

Maqam Shad Araban (ascending): [G, A-, B, C, D, Eb, F#, G] : [5, 13, 4, 9, 4, 14, 4] # Hijaz tetrachord + Nakriz pentachord. This one is a little weird. The  pitch classes suggest a Nakriz pentachord [9, 4, 14, 4], but the commas written in originally by Alsiadi were [9, 7, 6, 9] for the pentachord, which is Rast pentachord. Since the pitch classes and the pentachord label match each other, and the commas don't have any corroboration, I changed the commas for consistency. My version with the changed commas has the same intervals as Maqam Hijaz, but here rooted on G instead of D. I've read on Oud For Guitarists that "maqams Hijaz kar, Shad Araban, Suzidil, and Shahnaz"  have the same intervals, so that's encouraging. Alsiadi seems to agree that all of these are mad of Hijaz tetrachord + 9/8 + Hijaz tetrachord (or equivalently Hijaz tetrachord + Nakriz pentachord), but they're not all the same for him since he uses two different Hijaz tetrachords in different places, which I believe is intentional on his part.

Maqam Shad Araban (descending): [G, A-, B, C, D, Eb, F, G] : [5, 13, 4, 9, 4, 9, 9] # Hijaz tetrachord + Nahawand pentachord

Maqam Shahnaz: [D, Eb, F#, G, A, Bb, C#, D] : [4, 14, 4, 9, 4, 14, 4] # Hijaz tetrachord + 9/8 + Hijaz tetrachord

Maqam Suzdal: [A, Bb, C#, D, E, F, G#, A] : [4, 14, 4, 9, 4, 14, 4] # Hijaz tetrachord + Nakriz pentachord

Maqam Zinkulah: [C, Db, E, F, G, A, Bb, C] : [4, 14, 4, 9, 9, 4, 9] # Hijaz tetrachord + Ajam pentachord

Maqam Bayati ascending: [D, E-, F, G, A, B-, C, D] : [6, 7, 9, 9, 6, 7, 9] # Bayati tetrachord + 9/8 + Bayati tetrachord

Maqam Bayati descending: [D, E-, F, G, A, Bb, C, D] : [6, 7, 9, 9, 4, 9, 9] # Bayati tetrachord + 9/8 + Kurd tetrachord

Maqam Qarjighar (Bayati Shuri) (ascending): [D, E-, F, G, Ab, B, C, D] : [6, 7, 9, 4, 14, 4, 9] # Bayati tetrachord + hijaz pentachord

Maqam Qarjighar (Bayati Shuri) (descending): [D, E-, F, G, A, Bb, C, D] : [6, 7, 9, 9, 4, 9, 9] # Bayati tetrachord + busalik pentachord

Maqam Husayni (ascending): [D, E-, F, G, A, B-, C, D] : [6, 7, 9, 9, 6, 7, 9] # Bayati pentachord + Bayati tetrachord

Maqam Husayni (descending): [D, E-, F, G, A, Bb, C, D] : [6, 7, 9, 9, 4, 9, 9] # Bayati pentachord + Kurd tetrachord

Maqam Bayati Ushayran (ascending): [A, B-, C, D, E-, F, G, A] : [6, 7, 9, 6, 7, 9, 9] # Bayati tetrachord + Bayati pentachord

Maqam Bayati Ushayran (descending): [A, B-, C, D, E, F, G, A] : [6, 7, 9, 9, 4, 9, 9] # Bayati tetrachord + Busalik pentachord

Maqam Husayni Ushayran (ascending): [A, B-, C, D, E-, F, G, A] : [6, 7, 9, 6, 7, 9, 9] # Bayati tetrachord + Bayati pentachord

Maqam Husayni Ushayran (descending): [A, B-, C, D, Eb, F#, G, A] : [6, 7, 9, 4, 14, 4, 9] # Bayati tetrachord overlapping with Nakriz hexachord

Maqam Saba (ascending): [D, E-, F, Gb, A, Bb, C, Db] : [6, 7, 4, 14, 4, 9, 4] # Alsiadi describes this as having a Bayati trichord on "Re", a Hijaz tetrachord starting on "Fa", and a Hijaz trichord starting on the high "Do". As notated, the "Re" is the low "D" note though, as though C is the tonic of the scale. Also there is no third note written above the Db to complete the Hijaz trichord: there's just blank space. I don't know what a Hijaz trichord is in order to infer the pitch above "Db" or the simge-comma-integer after "4". Perhaps 14?

Maqam Saba (descending): [D, E-, F, Gb, A, Bb, C, D] : [6, 7, 4, 14, 4, 9, 9] # Bayati trichord + Hijaz tetrachord + Ajam trichord.

Maqam Saba Zamzamah: [D, Eb, F, Gb, A, Bb, C, D] : [4, 9, 4, 14, 4, 9, 9] # Kurd trichord + Hijaz tetrachord + Ajam trichord

Maqam Sikah (ascending): [E-, F, G, A, B-, C, D, E-] : [6, 9, 9, 7, 6, 9, 7] # Sikah trichord + Rast tetrachord + Rast trichord

Maqam Sikah (descending): [E-, F, G, A, Bb, C, D, E-] : [6, 9, 9, 4, 9, 9, 7] # Sikah trichord + Nahawand tetrachord + Rast trichord. Alsiadi's comments make it seem like he thinks the tonic is C rather than E-. That's fine. He can think what he likes.

Maqam Huzam: [E-, F, G, Ab, B, C, D, E-] : [6, 9, 4, 14, 4, 9, 7] # Sikah trichord + Hijaz tetrachord + Rast trichord.

Maqam Mayah: [E-, F, G, A, Bb, C, D, E-] : [6, 9, 9, 4, 9, 9, 7] # Sikah trichord + Nahawand tetrachord + Rast trichord

Maqam Musta'ar: [E-, F#, G, A, Bb, C, D, E-] : [11, 4, 9, 4, 9, 9, 7] # Musta'ar trichord + Nahawand tetrachord + Rast trichord

Maqam Jaharkah: [F, G, A, Bb, C, D, E-, F] : [9, 9, 4, 9, 9, 7, 6] # Jaharkah tetrachord + whole tone + Rast tetrachord

Maqam Ajam Ushayran: [Bb, C, D, Eb, F, G, A, Bb] : [9, 9, 4, 9, 9, 9, 4] # Ajam tetrachord + whole tone + Ajam tetrachord

Maqam Shawq Afza (ascending): [Bb, C, D, E-, F, Gb, A, Bb] : [9, 9, 6, 7, 4, 14, 4] # Ajam trichord (9, 9) + Saba pentachord (6, 7, 4, 14) overlapping with Hijaz tetrachord (4, 14, 4).

Maqam Shawq Afza (descending):  [Bb, C, D, Eb, F, G, A, Bb] : [9, 9, 4, 9, 9, 9, 4] # Ajam tetrachord (9, 9, 4) + whole tone + Jaharkah tetrachord (9, 9, 4).

Maqam Iraq (ascending): [B-, C, D, E-, F, G, A, B-] : [6, 9, 6, 7, 9, 9, 7] # Iraq trichord (6, 9) + Bayati tetrachord (6, 7, 9) + Rast trichord (9, 7)

Maqam Iraq (descending): [B-, C, D, E-, F, G, A, Bb] : [6, 9, 6, 7, 9, 9, 4] # Iraq trichord (6, 9) + Bayati tetrachord (6, 7, 9) + Busalik trichord (9, 4). This one has B- at the bottom and Bb at the top, and it doesn't form an octave. Really weird.

Maqam Awj Ara: [B-, C, D#, E-, F#, G, A#, B-] : [6, 14, 2, 11, 4, 14, 2] # Awj tetrachord (6, 14, 2) + Mustaar pentachord (11, 4, 14, 2)

Maqam Bastah Nikar: [B-, C, D, E-, F, Gb, A, Bb] : [6, 9, 7, 7, 4, 14, 4] # Iraq trichord (6, 9) + Saba tetrachord (7, 7, 4) + 14 commas + Kurd dichord (4).  On Alsiadi's website, the Iraq trichord is annotated here with commas [7, 9].  But the Iraq trichord with commas [6, 9] is listed multiple times in other places and the [7, 9] version only shows up here. Also, the first three pitch classes are [Bd, C, D] which is described with [6, 9] commas everywhere else, even when an Iraq trichord is not listed. So I changed it, and I stand by it.

Maqam Farahnak (ascending): [B-, C, D, E, F+, G, A, B-] : [6, 9, 9, 7, 6, 9, 7] # Sikah trichord (6, 9) + Rast tetrachord (9, 7, 6) + Rast trichord (9, 7)

Maqam Farahnak (descending): [B-, C, D, E, F+, G, A, Bb] : [6, 9, 6, 7, 6, 9, 4] # Sikah trichord (6, 9) + Rast tetrachord (9, 7, 6) + Busalik trichord (9, 4). This one has both B- and Bb and it doesn't form an octave. Weird.

Maqam Rahit Al Arwah (ascending): [B-, C, D, Eb, F#, G, A, B-] : [6, 9, 4, 14, 4, 9, 7] # Iraq trichord (6, 9) + Hijaz tetrachord (4, 14, 4) + Rast Trichord (9, 7)

Maqam Rahit Al Arwah (descending): [B-, C, D, Eb, F#, G, A, Bb] : [6, 9, 4, 14, 6, 9, 4] # Iraq trichord (6, 9) + Hijaz tetrachord (4, 14, 4) + Busalik Trichord (9, 4). Alsiadi has the Hijaz tetrachord annotated as [4, 14, 6] but that's not consistent with his pitch classes or how he quantifies Hijaz anywhere else, so I fixed it.

: Alsiadi's Ajnas

Alsiadi notates some ajnas. I'll try to share them in families:

[9, 7]: Rast trichord // [9, 7, 6]: Rast tetrachord // [9, 7, 6, 9]: Rast pentachord

[4]: Kurd dichord // [4, 9] : Kurd trichord // [4, 9, 9]: Kurd tetrachord

[9, 9]: Ajam trichord // [9, 9, 4]: Ajam tetrachord // [9, 9, 4, 9]: Ajam pentachord

He also has a "Jaharkah" tetrachord which is notated the same as the Ajam tetrachord in 53-EDO commas. 

 [9, 9, 4]: Jaharkah tetrachord 

I believe that in practice the third note of Jaharkah is a little lower than the third note of Ajam, but not so much lower that it shows up at this resolution. Perhaps we could pretend that Jaharkah is  [9, 8.5, 4.5] commas of 53-EDO, just so that we can start putting some of this nebulous unnotated practice into notation. Although even more complicated, I think Jaharkah is usually a pentachord, and both the third and the fourth notes are a little flat, so that the tetrachord made by removing the last note of the pentachord wouldn't quite reach P4. Perhaps the pentachord could be written [9, 8.75, 3.75, 9.5].

Alsiadi's Hijaz comes in two intonations:

[4, 14, 4]: Hijaz tetrachord

[5, 13, 4]: Hijaz tetrachord

They're both common across his transcriptions and sometimes occur within a single maqam. This is like how Hicaz had a variable intonation in traditional Turkish music theory.

[9, 4, 14, 4]: Nakriz pentachord // [9, 4, 14, 4, 9]: Nakriz hexachord

I don't think there can be a Nakriz tetrachord. It seems to me that Nakriz is just one intonation of Hijaz on top of a Pythagorean major second, so cutting it off to include the M2 and part of Hijaz won't work. You have to cut reality at the joints.

[6, 7]: Bayati trichord // [6, 7, 9]: Bayati tetrachord // [6, 7, 9, 9]: Bayati pentachord

[11, 4]: Musta'ar trichord // [11, 4, 14, 2]: Musta'ar pentachord

The Musta'ar tetrachord doesn't make an appearance, but maybe it's [11, 4, 14]. Although that doesn't add up to a 22-comma P4. The pentachord does reach a 31-comma P5 though.

Alsiadi regularly interchanges the names "Busalik" and "Nahawand", which I believe is in standard in the middle east, but a little unfortunate because I would have preferred to list the commas just once.

[9, 4]: Busalik trichord // [9, 4, 9]: Busalik tetrachord // [9, 4, 9, 9]: Busalik pentachord

[9, 4, 9]: Nahawand tetrachord // [9, 4, 9, 9]: Nahawand pentachord

He also uses the same commas for the Iraq trichord and the Sikah trichord:

[6, 9]: Iraq trichord

[6, 9]: Sikah trichord

I can deal with that. I don't think it's an error.

The Awj tetrachord was only used once, but it sums to a 22-comma P4, so I'll take it.

[6, 14, 2]: Awj tetrachord

Finally he lists both a Saba tetrachord and Saba pentachord, but the tetrachord doesn't sum to a P4 of 22 commas and the pentachord doesn't sum to a P5 of 31 commas. That's seems okay to me: Saba is weird like that. But also the pentachord isn't an extension of the tetrachord, which seems wrong:

[7, 7, 4]: Saba tetrachord // [6, 7, 4, 14]: Saba pentachord

He only uses each of these once, and neither of them are used in the actual Maqam Saba. But Maqam Saba start out [6, 7, 4, 14], so if I had to guess which if either of these is in error and which is correct, I'd say that the Saba pentachord [6, 7, 4, 14] is correct while the tetrachord should be [6, 7, 4]. The [7, 7, 4] tetrachord showed up in Maqam Bastah Nikar, which Alsiadi originally wrote with commas [7, 9, 7, 7, 4, 14, 4]. This sums to a P8 of 53-commas, but (as discussed above) it should start [6, 9] based on the notated Iraq trichord and based on the starting pitch classes, [Bd, C]. Fixing Iraq brought us down to 51 commas, and if we alter the Saba tetrachord, then the maqam only reaches as high as 50 commas. I still think it's an improvement that way though, honestly. Partly I think a 50-comma tall maqam is okay because Maqam Saba itself falls flat of the octave. Maybe it infects everything that it touches.

: Comparing Alsiadi's Ajnas With Turkish Ajnas

The Turkish Pythagorean ajnas have exact counterparts in Arabic tradition:

Bûselik tetrachord: [9, 4, 9] <-> Arabic Busalik/Nahawand: [9, 4, 9]

Kürdi tetrachord: [4, 9, 9] <-> Arabic Kurd: [4, 9, 9]

Çargâh tetrachord: [9, 9, 4] <-> Arabic Ajam: [9, 9, 4]

Alsiadi's Arabic Rast tetrachord has the same same P1, M2, and P4, but a lower and more neutral third interval: 

Rast tetrachord: [9, 8, 5] <~> Arabic Rast: [9, 7, 6]

Turkish tradition has a permutation of Rast tetrachord called Uşşak/Hüseyni  [8, 5, 9]. If we did the same with The Arabic Rast, we would get this for an Arabic Uşşak/Hüseyni [7, 6, 9]. This doesn't show up in Alsiadi's maqamat, but I've heard from Margo Schulter that [7, 6, 9] commas of 53-EDO described an Arabic Huseyni tetrachord, so let's say that there's a correspondence here too.

Arabic tradition has a permutation of its Rast tetrachord called Bayati [6, 7, 9]. If we did the same with the Turkish Rast, we would get this for a Turkish Bayati: [5, 8, 9].

Finally, Arabic Hijaz and Turkish Hicaz are clearly related:

Arabic Hijaz: [4, 14, 4] <~> Turkish Hicaz: [4, 13, 5]

 Arabic Hijaz [5, 13, 4] <~> Turkish Hicaz:[5, 12, 5]

It's not totally obvious how to put them in one-to-one correspondence, but I think I support the pairing that you see above. Putting Arabic [4, 14, 4]  with Turkish [5, 12, 5] instead would also be attactive for the palindrome symemtry, but then the other pair makes less sense, in my judgement.

I've heard that in Persian tradition, the Bûselik/Busalik/Nahawand: [9, 4, 9] tetrachord is called Nava. And that the Kürdi/Kurd: [4, 9, 9] tetrachord is still called Kurdi. And that the Çargâh/Ajam: [9, 9, 4] is called Mahur. And that the [smaller neutral second, larger neutral second, major second] Bayati is called Shur.

I've also heard that Persian tradition has a tetrachord called Esfahan, which can be notated with 53-EDO commas as [7, 9, 6], a permutation of Arabic Rast. We'll see about that.

: Combined EDO Analysis

 I had a cute idea. Suppose we take 7 steps of 24-EDO and (9 + 7 =) 16 steps of 53-EDO as portraits of the neutral third in the Arabic Rast. What intervals are compatible with both of those facts? Here are some simpler intervals in the 17-limit prime harmonic basis that fit the bill, along with their just tunings:

(4, 0, 0, 0, 0, -1, 0) # 16/13 at 359c. Flat of Pythagorean M3 (81/64) by 1053/1024. Sharp of Pythagorean m3 (32/27) by (27/26).

(0, 1, 0, 1, 0, 0, -1) # 21/17 at 366c. Flat of Pythagorean M3 by 459/448. Sharp of Pythagorean m3 by 567/544.

(-1, 3, 0, 0, -1, 0, 0) # 27/22 at 355c.

(-4, 2, -1, 0, 1, 0, 0) # 99/80  at 369c.

Any of those are good just frequency ratios for an Arabic Rast neutral third, so far as the 24-EDO and 53-EDO descriptions convey. Pretty cool? I think it's cool. The Arabic Rast third should thus be about 362 cents.

There are fewer simple options for the neutral third of Turklish Rast. If we want the interval to be tuned to 7 steps of 24-EDO and 17 steps of 53-EDO, the only such intervals justly tuned to short fractions are:

(3, -2, -1, 1, 0, 0, 0) # 56/45 at 379c. Flat of Pythagorean major 3rd (81/64) by a factor of 3645/3584. Sharp of Pythagorean minor third (32/27) by 21/20.

(2, 0, -1, 0, -1, 0, 1) # 68/55 at 376c. Flat of Pythagorean major 3rd (81/64) by a factor of 4455/4352. Sharp of Pythagorean minor third (32/27) by 459/440.

I know it's only two data points, but I enjoy the tight grouping. An average neutral third of Turkish Rast might be 377.5 cents, based just on this joint 24- and 53-EDO analysis with low complexity determpering.

The Xenharmonic wiki page on 56/45 mentions that Turkish musicologist Ozan Yarman has identified 56/45 as a good "segah perde", i.e. a good rational value for the neutral third over G (as used in the Turkish makam Rast which is rooted on G), but I know Yarman's work and he usually gives like six possible frequency ratios for each named Turkish tone, so I bet this one wasn't singled out for special attention.

I've got a table for you. It shows intervals that are neutral 2nds, 3rds, 6ths, and 7ths (according to 24-EDO) and also the numbers of tuned steps for the interval in 53-EDO. The first datum in each line shows the number of steps for the interval in 24-EDO & 53-EDO.

Neutral seconds:

3&5 : 55/51 HbAsm2 _ 131 cents

3&6 : 95/88 LfDeAcM2 _ 133 cents

3&6 : 92/85 NbHbAcm2 _ 137 cents

3&6 : 13/12 Prm2 _ 139 cents

3&6 : 38/35 LfSpm2 _ 142 cents

3&6 : 88/81 AsGrm2 _ 143 cents

3&6 : 25/23 DsM2 _ 144 cents

3&7 : 12/11 DeAcM2 _ 151 cents

3&7 : 35/32 SbAcM2 _ 155 cents

3&7 : 57/52 LfReAcM2 _ 159 cents

3&7 : 11/10 Asm2 _ 165 cents

3&8 : 21/19 LwSbAcM2 _ 173 cents

3&8 : 72/65 ReAcM2 _ 177 cents

3&8 : 51/46 DsExM2 _ 179 cents

Neutral thirds:

7&14 : 65/54 PrGrm3 _ 321 cents

7&14 : 76/63 LfSpGrm3 _ 325 cents

7&15 : 40/33 DeM3 _ 333 cents

7&15 : 95/78 LfReM3 _ 341 cents

7&15 : 39/32 Prm3 _ 342 cents

7&15 : 11/9 AsGrm3 _ 347 cents

7&16 : 92/75 Nbm3 _ 354 cents

7&16 : 27/22 DeAcM3 _ 355 cents

7&16 : 70/57 LwSbM3 _ 356 cents

7&16 : 16/13 ReM3 _ 359 cents

7&16 : 85/69 DsExGrM3 _ 361 cents

7&17 : 68/55 ExDeM3 _ 367 cents

7&16 : 99/80 Asm3 _ 369 cents

7&17 : 81/65 ReAcM3 _ 381 cents

Neutral sixths:

17&36 : 55/34 HbAsm6 _ 833 cents

17&37 : 13/8 Prm6 _ 841 cents

17&37 : 57/35 LfSpm6 _ 844 cents

17&37 : 44/27 AsGrm6 _ 845 cents

17&37 : 75/46 DsM6 _ 846 cents

17&38 : 18/11 DeAcM6 _ 853 cents

17&38 : 64/39 ReM6 _ 858 cents

17&38 : 33/20 Asm6 _ 867 cents

17&39 : 63/38 LwSbAcM6 _ 875 cents

Neutral sevenths:

21&45 : 92/51 NbHbm7 _ 1021 cents

21&45 : 65/36 PrGrm7 _ 1023 cents

21&45 : 38/21 LfSpGrm7 _ 1027 cents

21&46 : 20/11 DeM7 _ 1035 cents

21&46 : 95/52 LfReM7 _ 1043 cents

21&46 : 64/35 SpGrm7 _ 1045 cents

21&46 : 11/6 AsGrm7 _ 1049 cents

21&47 : 46/25 Nbm7 _ 1056 cents

21&47 : 81/44 DeAcM7 _ 1057 cents

21&47 : 35/19 LwSbM7 _ 1058 cents

21&47 : 24/13 ReM7 _ 1061 cents

21&47 : 85/46 DsExGrM7 _ 1063 cents

This table is goes up to 19-limit frequency ratios with numerators less than 100. All the intervals are tuned by 24-EDO to one of {3, 7, 17, or 21} steps, i.e. the steps for the neutral 2nd, 3rd, 6th, and 7th. Also I've only shown e.g. 2nd intervals in the neutral second category, even though 24-EDO tunes some sharp 1st and some flat 3rds to 3 steps.

This table gives us a quick guide for combined-EDO-detempering. If one middle eastern music theorist says that a mode has a scale degree tuned to 3\24 and another says that the same scale degree is tuned to 8\53, then we immediately have some low-complexity candidates for the rational identification of that scale degree:

3&8 : 21/19 LwSbAcM2 _ 173 cents

3&8 : 72/65 ReAcM2 _ 177 cents

3&8 : 51/46 DsExM2 _ 179 cents

Not too shabby.