Owen Wright is a scholar of medieval middle eastern music. In "The Modal System of Arabian and Persian Music 1250 to 1300", he writes extensively about music theorist Qutb al-Din, a prominent Systematist who wrote just after another prominent Systematist, writer Safi al-Din al-Urmawi, whose 17-tone Pythagorean gamut we've discussed before.
Qutb al-Din wrote extensively and he's a great source about medieval Persian and Arabic music of the time. In this post we'll be exploring his tetrachords (and pentachords and other ajnas) and modes, as relayed by Owen Wright.
Wright remarks that the modes of Qutb al-Din are notated twice, once in frequency ratios and once in pitches, and these are not consistent - clearly derived separately. The pitches weren't in Latin script, they were in Arabic I think, and they were a weird Pythagorean holdover notation from Qutb al-Din's predecessor Safi al-Din, but Owen Wright doesn't write the pitches in Arabic and neither will I. The fact that the frequency ratios and pitches are inconstent in Qutb al-Din's work makes it a little hard to be sure what al-Din is talking about, but it gets worse. Another complication is that the Systematists really like simple frequency ratios even when they weren't true-to-sound. It seems, from reading Wright, that they basically only use Pythagorean or super particular ratios as relative degrees in tetrachords. For example, the only super-particular frequency ratios between a just minor third 6/5 and a Pythagorean major second 9/8, are 7/6 and 8/7, and so when a Systematist needs an interval in that range, you can be pretty sure that they'll grab one of those two, regardless of whether it is sonically appropriate. There is a difference of 111 cents between 6/5 and 9/8, and if we only distinguish two ratios in that range, well, that's roughly a 37-cent granularity, which might be better than 24-EDO, but it's not amazing for nailing down intonation.
For some reason Wright shares the tetrachords in terms of cents instead of ratios, but since there aren't that many frequency ratios that are used by the Systematists, it's easy enough to figure out what arithmetic they're doing behind Wright's notation.
Here's a pentachord from Qutb al-Din as relayed by Wright. He calls it "24b shahnaz":
[G, Ad, Bb, Bd, C, Dd, D]
[139, 128, 49, 139, 128, 49] cents
The cents correspond to these relative frequency ratios:
[13/12, 14/13, 36/35, 13/12, 14/13, 36/35]
Wright points out that these frequency ratios don't form a perfect fifth, but that they should (based on the pitches and other facts, I'm sure). As another example, many ajnas have what would be a neutral third based on the pitch notation, but the frequency ratios have the third at a just major third, 5/4 at 386 cents. This is indeed lower than a Pythagorean major third and thus more in the direction of neutral. And indeed there are arguments that some middle eastern musicians have used 5/4 as a neutral third at some points in history, as many musicians currently do in Turkey. Owen Wright knows all the medieval manuscripts though and says that 5/4 doesn't make sense, and any time 5/4 is written, it should really be interpreted as a true neutral third, as the pitches indicate, i.e. more like 330 to 370 cents. So there's another case where the frequency ratios are probably not correct, and we're better off looking at the pitches, but what a shame because we'd really like a precise rational intonation.
Wright provides his own plausible ranges of cents for each scale degree of the ajnas, presented in little ruler graphics. By measuring the pixels and taking midpoints, I can tell you that this is a prototypical intonation for a medieval shahnaz pentachord according to Owen Wright:
[0, 147, 293, 348, 498, 642, 702]
And a decent representation of that might be
[1/1, 12/11, 32/27, 11/9, 4/3, 13/9 or 81/56, 3/2]
My pixel measuring process is a little bit labor intensive, so I'm not going to provide Wright's intonation everywhere just yet, but I do want to present the Systematist frequency ratios and pitch classes. Partly, these ratios are what was actually written, and I think there is some import to transmitting the history of the music theory veridically. Secondly, these frequency ratios still give us more clues about intonation than the naive 24-EDO interpretation of pitch classes, and I really want to know what the medieval modes sounded like and how they've evolved into modern ones.
...
* Zirafkand-i Kuchek (or Zirafkand, or Kuchek, or Mukhalifak).
Pitches: [G, Ad, Bb, Bd]. // G and Bd are prominent notes.
Absolute: [1/1, 13/12, 7/6, 6/5]
Relative: [13/12, 14/13, 36/35] _ [139, 128, 49]
// Given by Safi al-Din as [14/13, 13/12, 36/35].
* 'Iraq:
Pitches: [G, Ad, Bd] // G and Bd are prominent notes.
Absolute: [1/1, 10/9, 5/4]
Relative: [10/9, 9/8]
* Zawli:
Pitches: [G, A, Bd] // G and Bd are prominent notes.
Absolute: [1/1, 9/8, 5/4]
Relative: [9/8, 10/9]
* Rahawi:
Pitches: [G, Ad, Bb, B] // G is the only prominent pitch.
Absolute: : [1/1, 13/12, 7/6, 5/4]
Relative: [13/12, 14/13, 15/14] _ [139, 128, 119]
* 'Ushshaq:
Pitches: [G, A, B, C] // G is the only prominent pitch.
Absolute: [1/1, 9/8, 81/64, 4/3]
Relative: [9/8, 9/8, 256/243]
* Busalik:
Pitches: [G, Ab, Bb, C] // G is the only prominent pitch.
Absolute: [1/1, 256/243, 32/27, 4/3]
Relative: [256/243, 9/8, 9/8]
* Nawa:
Pitches: [G, A, Bb, C] // No prominent pitch listed.
Absolute: [1/1, 9/8, 32/27, 4/3]
Relative: [9/8, 256/243, 9/8]
* Rast:
[G, A, Bd, C] // G is the prominent pitch.
[1/1, 9/8, 5/4, 4/3]
[9/8, 10/9, 16/15] _ [204, 182, 112]
* Nawruz:
Pitches: [G, Ad, Bb, C] // G and C are prominent.
Absolute: [1/1, 16/15, 32/27, 4/3]
Relative: [16/15, 10/9, 9/8] _ [112, 182, 204]
* 'Iraq (or Ru-yi 'Iraq):
Pitches: [G, Ad, Bd, C] // G and C are prominent.
Absolute: [1/1, 10/9, 5/4, 4/3]
Relative: [10/9, 9/8, 16/15] _ [182, 204, 112]
* Isfahan:
Pitches: [G, Ad, Bb, B, C] // G and C are prominent.
Absolute: [1/1, 13/12, 7/6, 5/4, 4/3]
Relative: [13/12, 14/13, 15/14, 16/15] _ [139, 128, 119, 112]
Here Owen Wright stops us to say that the frequency ratios for jins Isfahan and and the related jins Rahawi are unbelievable, and that the pitch notation is more accurate. Further he notes that jiins Isfahan is clearly derived by adding a major third within jins Nawruz.
I think it's time we looked a little more closely at jins Isfahan, jins Rahawi, and jins Nawruz. From the frequency ratios
Rahawi has Bb at 7/6
Isfahan has Bb 7/6
Nawruz has Bb at 32/27
But from Owen Wright's ruler diagrams, it's clear he thinks that all three are Pythagorean on the chromatic intervals including the Bb, so Rahawi is:
[1/1, ?, 32/27, 4/3]
By eye, it's obvious that Wright's intonation on the second scale degree, the {Ad}, is not much more precise than "some kind of neutral second". I'm tempted to use 13/12, since it's the ratio of the second scale degree used by Qutb al-Din in both jins Isfahan and jins Rahawi. On the other hand, Qutb al-Din uses 16/15 for the second scale degree of Nawruz, which is quite a bit lower, and so maybe we should use 14/13 as a compromise. But I'm not really feeling that. Let's use 13/12 for all of them. Or at least for Rahawi and Isfahan.
Lots of modern middle eastern music theorists, at in the xenharmonic parts I frequent, are very pleased with the idea that Isfahan as a tetrachord is tuned to [12:13:14:15:16]. And since that's how Qutb al-Din described it, ... maybe that's a fine way to play it? I don't know for sure if it's a medieval intonation, but if someone in modern times plays Isfahan like that, I won't be upset. I don't think I've ever seen anyone extend it all the way to a pentachord as
[13/12 * 14/13 * 15/14 * 16/15 * 17/16 * 18/17]
In absolute terms that would be
[1/1, 13/12, 7/6, 5/4, 4/3, 17/12, 3/2]
Maybe no one adds in the tritone because they care more about history than about playing a harmonic series. Anyway, let's keep looking at more ajnas.
* Hijazi:
Pitches: [G, Ad, B, C] // G and C are prominent.
Absolute: [1/1, 12/11, 14/11, 4/3]
Relative: [12/11, 7/6, 22/21] _ [150, 267, 81]
Owen wright says that the major third was probably not sharper than major thirds of other genera, but otherwise he's happy with the ratios:
"This minor adjustment apart, Hijazi is one of the rare cases in which the ratios for a theoretical non-diatonic genus would seem to correspond exactly to intervals used in practice."
Super hot fire, Owen. Way to stick it to the Systematists frequency ratios. Although there are only three relative ratios in the tetrachord, and if we're changing the intonation of the 3rd scale degree, then we have to fiddle with two of them, so is this really any kind of praise? Even at his most congratulatory, Wright is basically saying "Way to get a single frequency ratio right, Qutb."
Anyway, I think this is what Owen Wright would condone for Hijazi:
[G, Ad, B, C]
[1/1, 12/11, 81/64, 4/3] _ [0, 151, 408, 498]
[12/11, 297/256, 256/243] _ [151, 257, 90]
This looks kind of weird to me as a modern intonation for jins Hijaz, but maybe it's a medieval one, sure.
Now we get some pentachords
* 'Ushshaq pentachord:
[G, A, B, C, D] // G is prominent.
[9/8, 9/8, 256/243, 9/8]
* Busalik pentachord:
[G, Ab, Bb, C, D] // G and D are prominent.
[256/243, 9/8, 9/8, 9/8]
Which just add an AcM2 onto the tetrachord of the same name. We also get a pentachord version of the Nawa tetrachord:
[G, A, Bb, C, D]
[9/8, 256/243, 9/8, 9/8]
This jins was mentioned in the "Kitab al-Adwar" by Safi al-Din but not in works by Qutb al-Din, who is normally the more comprehensive source. Apparently this jins doesn't have a historic name, but I think "Nawa Pentachord" suits it just fine. But also, who cares about pentachords that are just tetrachords with AcM2 added on the top. Boring.
* Rast pentachord:
[G, A, Bd, C, D]
Absolute: [1/1, 9/8, 5/4, 4/3]
Relative: [9/8, 10/9, 16/15, 9/8]
* Isfahan-i Asl pentachord (also called Mukhalif-i Rast):
[G, A, Bd, C, C#, D] // G and D are prominent.
Absolute: [1/1, 9/8, 39/32, 21/16, 45/32, 3/2]
Relative: [9/8, 13/12, 14/13, 15/14, 16/15]
Owen wright points that this is an Isfahan tetrachord with AcM2 added at the bottom instead of the top. Mukhalif (or mukhtalif) means "differing" in Arabic, like saying "the other/alternative Rast pentachord. It's worth noticing that these two scales having very different frequency ratios but very similar pitches. You can decide for yourself if this is another point against the frequency ratios or evidence of different intonation for the tones across different ajnas. I will say that 21/16 at 470 cents is noticeably flat of a normal C natural over G at 498 cents, and that the author might have added an accidental to the pitches to drawn attention to this 28 cent difference, which is not really a subtle thing. Although I did mention that Systematists sometimes only have like a 37-cent granularity. I still think someone would have mentioned an impure P4, since pure perfect fourths were maybe considered the highest consonance in medieval middle eastern music.
...
I'm sad and tired.
...
* Husayni pentachord
[G, Ad, Bb, C, D] # G and D are prominent notes.
[1/1, 16/15, 32/27, 4/3, 3/2]
[16/15, 10/9, 9/8, 9/8] _ [112, 182, 204, 204]
* Zirkesh Huseyni pentachord:
[G, Ad, Bb, B, C, D] # G and D are prominent notes.
[1/1, 13/12, 7/6, 5/4, 4/3, 3/2]
[13/12, 14/13, 15/14, 16/15, 9/8] _ [139, 128, 119, 112, 204]
* Buzurg of Safi al-Din (or Buzurg-i Asl of Qutb al-Din):
[G, Ad, (Bd), C, C#, D] # G and D are prominent notes.
[1/1, 14/13, 16/13, 4/3, 56/39, 3/2]
[14/13, 8/7, 13/12, 14/13, 117/112] _ [128, 231, 139, 128, 76]
* Variant of Buzurg:
[G, Ad, B, C, C#, D] # G and D are prominent notes.
[1/1, 12/11, 14/11, 4/3, 56/39, 3/2]
[12/11, 7/6, 22/21, 14/13, 117/112] _ [150, 267, 81, 128, 76]
...
I'm getting a little bit sad with these ajnas. The frequency ratios are all wrong. I should just list the pitches and be done.
There are pages and pages of more ajnas in the Owen Wright reference. But let's go over what we've already seen a bit.
Three Pythagorean tetrachords:
[M2, M2, m2]: 'Ushshaq // Modern jins 'Ajam
[M2, m2, M2]: Nawa // Modern jins Nahawand
[m2, M2, M2]: Busalik // Modern jins Kurdi
Three tetrachords that mix neutral seconds with a major second:
[M2, n2, n2]: Rast // Modern jins Rast
[n2, M2, n2]: Ru-yi 'Iraq // Modern jins Huseyni or jins 'Iraq
[n2, n2, M2]: Nawruz // Modern jins Bayyati
Two funny tetrachords:
[n2, n2, A1, m2]: Isfahan
[n2, M3 - n2, m2]: Hijazi
For ajnas that span less than P4, we have
[n2, M2]: 'Iraq
[M2, n2]: Zawli
[n2, n2, n3 - m3]: Zirafkand
[n2, n2, m2]: Rahawi
I don't have a name for intervals like [n3 - m3] in the shorthand where I use {n} for neutral intervals, but they happen to fall between P1 and A1 in just tuning, so I guess we could call them neutral unisons, {n1}. These can be generated by subtracting our favorite neutral seconds from a major second:
9/8 / (14/13) = 117/112
9/8 / (13/12) = 27/26
9/8 / (12/11) = 33/32
9/8 / (11/10) = 45/44
Almost all super particular. Nice. This notion of a neutral unison makes several other things more compact or sensible. I had written the middle relative degree of jins Hijazi as [M3 - n2], but we could also write it as [M2 + n1], which is more obviously a kind of 2nd interval. Also the 32/27 complements of Zalzalian neutral seconds can be succinctly written as [m2 + n1], e.g.
(256/243) * (27/26) = 128/117 (The 32/27 complement of 13/12)
(256/243) * (33/32) = 88/81 (The 32/27 complement of 12/11)
And I think that's nice too.
...
I think I'm going to skip to modes.
'Ushshaq: [G, A, B, C, D, E, F, G] : [M2, M2, m2] + [M2, M2, m2] + M2
Busalik: [G, Ab, Bb, C, Db, Eb, F, G] : [m2, M2, M2] + [m2, M2, M2] + M2
Nawa: [G, A, Bb, C, D, Eb, F, G] : [M2, m2, M2] + [M2, m2, M2] + M2
Rast: [G, A, Bd, C, D, Ed, F, G] : [M2, n2, n2] + [M2, n2 n2] + M2
'Iraq: [G, Ad, Bd, C, Dd, Ed, F, F#, G] : [n2, M2, n2] + [n2, M2, n2] [A1 + m2]
Isfahan: [G, A, Bd, C, D, Ed, F, F#, G] : [M2, n2, n2] + M2 + [n2, n2, A1, m2]
Rahawi: [G, Ad, Bd, C, Dd, Eb, F, G] : [n2, M2, n2] + [n2, n2, M2] + M2
Hijazi version 1: [G, Ad, Bb, C, Dd, Ed, F, G] : [n2, n2, M2] : [n2, M2, n2] + M2
and/or
Hijazi version 2: [G, Ad, Bd, C, Dd, Ed, F, G] : [n2, M2, n2] + [n2, M2, n2] + M2
Husayni: [G, Ad, Bb, C, Dd, Eb, F, G] : [n2, n2, M2] + [n2, n2, M2] + M2
Here are the same pitches with briefer tetrachord analyses:
'Ushshaq: [G, A, B, C, D, E, F, G] : Ushshaq + Ushshaq + T
Busalik: [G, Ab, Bb, C, Db, Eb, F, G] : Busalik + Busalik + T
Nawa: [G, A, Bb, C, D, Eb, F, G] : Nawa + Nawa + T
Rast: [G, A, Bd, C, D, Ed, F, G] : Rast + Rast + T
'Iraq: [G, Ad, Bd, C, Dd, Ed, F, F#, G] : 'Iraq + 'Iraq + [A1 + m2]
Isfahan: [G, A, Bd, C, D, Ed, F, F#, G] : Rast + T + Isfahan
Rahawi: [G, Ad, Bd, C, Dd, Eb, F, G] : 'Iraq + Nawruz + T
Hijazi version 1: [G, Ad, Bb, C, Dd, Ed, F, G] : Nawruz + 'Iraq + M2
Hijazi version 2: [G, Ad, Bd, C, Dd, Ed, F, G] : 'Iraq + 'Iraq + M2
Husayni: [G, Ad, Bb, C, Dd, Ed, F, G] : Nawruz + Nawruz + M2
Here are some weirder modes that I'm gradually analyzing intervallically:
Buzurg: [G, Ad, B (or Bd), C, C#, D, E, Ft, G] : [n2, M2 + n1 (or M2), m2 (or n2), A1, m2, M2, n2, n2] // Something like Hijazi + T + Rast, but with an extra C# and and also the weirdness about the variable third interval meaning that the first tetrachord might be 'Iraq.
Zankula: [G, A, Bd, C, Dd, Ed, F, (F#), G] : [M2, n2, n2, n2, M2, n2, M2 (or A1, m2)] // jins Rast + jins Iraq + T, and you can add in a leading tone at to the high octave if you want. This one isn't so crazy.
Zirafkand: [G, Ad, Bb, C, Dd, Eb, Ed, Ft, G] : [n2, n2, M2, n2, n2, n1, M2, n2]// This one is genuinely crazy.
Kardaniya: [G, A, Bd, C, C#, D, E, Ft, G] : [M2, m2, M2, A1, m2, M2, n2, n2] // Almost jins Nawa + T + jins Rast, except the tone is split into [A1, m2].
Kawasht: [G, Ad, Bd, C, Dd, Eb, Ed, Ft, G] : [n2, n2, M2, n2, n2, n1, M2, n2] // Genuinely crazy.
Nawruz: [G, Ad, Bb, C, Dd, Eb, F, ?(G)] : [n2, n2, M2, n2, n2, M2, ?(M2)] // jins Nawruz + jins Nawruz + ?(T). Not crazy at all, except I'm not sure that this one hits the octave.
Maya: [G, Bb, C, Eb, F] : [m3, M2, m3, M2] # Not interesting, not really a scale, maybe just someone's favorite melody?
Shahnaz: [G, Ad, Bb, Bd, Ab, G] : [n2, n2, n1, -(M2 + n1), -m2] // Ascends and descends, not really a scale.
Salmak: [G, A, B, Ct, D, Eb] : [M2, M2, n2, n2, m2] // Whatever.
Anyway, lots of conjunct tetrachords in the medieval modes.
...
Somehow I'm more curious which of these modes matches modern Persian Dastgah-ha than modern Arabic maqamat. Let's have a look. I'm going to hope that all of the modes presented by Owen Wright actually have the tonic on the starting note G.
Qutb's mode Ushshaq looks like Dastgah Rast-Panjgah if you just look at the intervals for each scale degree, although they have different tetrachord structures. Qutb's mode Busalik is a cyclic permutation of Rast-Panjgah and so is Qutb's mode Nawa. Qutb's mode Rast is a cyclic permutation of Dastgah Shur. Qutb's mode 'Iraq is very similar to Dastgah Chahargah, but they have different intonation, and Chahargah has a parallel disjunct tetrachord structure while 'Iraq is conjunct. Qutb's mode Isfahan has the same lower tetrachord and disjunct tone as the Persian Bayat-e Esfahan, but Qutb's version has a 5-note tetrachord at the top while Bayat-e Esfahan has a 4-note dang Chahargah as its upper tetrachord, which is more like the Hijazi tetrachord. Qutb's mode Rahawi does not seem to have any correspondence in the modern Persian Dastgah. Hijazi is also related to Chahargah. Qutb's mode Huseyni doesn't seem to correspond to anything in modern Persian Dastgah.
...
