: Persian Dastgāh
I'm still figuring out Persian modal scales, called dastgāh. I've got four sources of data on them so far. There are two sources of pitch classes that disagree with each other, one of which aslo relates the dastgāh to makams and maqamat. There's also a source that gives dastgāh in terms of cents, and another that give them in 60-EDO steps.
The 60-EDO data source is "Recognition Of Dastgah And Maqam For Persian Music With Detecting Skeletal Melodic Models" by Darabi, Azimi, and Nojumi. They claim their data is based on and slightly altered from the work of Hormoz Farhat.
Converting the 60-EDO steps to cents, their scales are:
Shur, Dashti, Bayate Kord, Abu Ata: [0, 120, 300, 500, 700, 800, 1000, 1200]
Bayate Tork: [0, 200, 400, 500, 700, 900, 1040, 1200]
Afshari: [0, 200, 320, 500, 700, 900, 1000, 1200]
Nava: [0, 200, 300, 500, 700, 820, 1000, 1200]
Homayun: [0, 120, 400, 500, 700, 800, 1000, 1200]
Esfahan: [0, 200, 300, 500, 700, 820, 1100, 1200]
Mokhalefe Segah, Esfahane Ghadim: [0, 200, 300, 500, 700, 820, 1040, 1200]
Bidad: [0, 200, 300, 500, 700, 820, 1100, 1200]
Ist -e- Dovom –e- Homayun: [0, 200, 320, 600, 700, 900, 1060, 1200]
Segah: [0, 140, 340, 480, 640, 840, 1040, 1200]
Chahargah: [0, 140, 400, 500, 700, 820, 1100, 1200]
My source that was already phrased in cents was "Iranian Traditional Music Dastgah Classification" by Sajjad Abdoli. Abdoli claims that his frequency ratios (phrased in cents) come "Karimi’s Radif and Farhat", and also remarks that Shur and Nava have the same intervals, and also Mahur and Rast-panjgah have the same intervals.
Chahargah: [0, 134, 397, 497, 634, 888, 994, 1200]
Homayun: [0, 100, 398, 502, 715, 800, 990, 1200]
Mahur & Rast-panjgah: (0, 208, 397, 497, 702, 891, 994, 1200]
Segah: (0, 198, 352, 495, 707, 826, 1013, 1200]
Shur & Nava: (0, 149, 300, 500, 702, 783, 985, 1200]
Now, these are obviously more fine grained than the 60-EDO ones, and I'd be tempted to only use them, except that many of the 60-EDO ones don't have more precise intervals in Abdoli, and also the 60-EDO ones clearly aren't just rounding the Abdoli ones to the nearest 20 cents, since e.g. Chahargah's second to last interval is 1100c in the 60-EDO source but below 1000c in Abdoli. Also, Abdoli says Shur and Nava have the same intervals, while Darabi et al. list both separately and these differ on the second and sixth scale degrees. Also Abdoli has some dastgāhs that aren't in Darabi et al., so it seems there's no way to avoid using both sources. For two sources that presumably both draw on Farhat, there isn't a ton of consistency. Honestly there isn't a single dastgāh that's consistent between the two to the nearest 20 cents.
Maybe Oud for Guitarists will clear things up? That's one of the music education projects of Navid Goldrick. Navid tells us that Dastgah Mahur and Dastgah Rast-Panjgah just C major scales, like Maqam Ajam.
Mahur: [C, D, E, F, G, A, B, C]
Rast-Panjgah: [C, D, E, F, G, A, B, C]
Abdoli also said that Mahur and Rast-Panjgah have the same intervals, although Abdoli gave us a seventh scale degree more like a Bb.
Navid has his own accidental for a quarter flat, namely "qb". Navid related Dastgah Shur and Dastgah Abu-Atta to Maqam Bayati, which he gives as:
Maqam Bayati: [D, Eqb, F, G, A, Bb, C, D]
If the second scale degree had been an E half flat, Ed, then this would be an Arabic Bayati with the normal a Nahawand ending. But I guess that a quarter flat is more like a Turkish Beyâti makam?
[K, S, T, T, B, T, T] # Uşşak tetrachord [K, S, T] + Bûselik pentachord [T, B, T, T]
Although I notated that with a half flat second instead of a quarter flat, just like Arabic.
[A, Bd, C, D, E, F, G, A]
Anyway, dastgah Abu-atta has the same pitch classes as Navid's Maqam Bayati:
Abu-Atta: [D, Ed, F, G, A, Bb, C, D]
except that Navid tells us A quarter-flat can be an occasional substitution for the A natural.
In the same family, with the same starting tetrachord, Navid relates Dastgah Shur as:
Shur: [D, Eqb, F, G, (Aqb) (Ab), Bb, C, D]
and says "Shur alternates between A natural and A quarter-flat after the opening tetrachord". I don't know what he means by that. Is there a regular alternation, like one used ascending and one descending, or is either A an acceptable option at any time? Perhaps unfortunately, the versions of Shur from Darabi et al. and Abdoli are consistnet with each other on the fifth scale degree being ~700 cents, which is just a A natural, not A flat or A half flat or A quarter flat. Also those two Dastgahs Shur differ by a quarter-tone on the sixth scale degree, which Navid gives as Bb. So this is great, right? Just layer after layer of disagreement. Lots of confusion to wade through.
...
Here are the Dastgah-s as they appear at fis-iran.org, "The Foundation For Iranian Studies":
Shur: [G Ad Bb C Dd Eb F G] and its four derivatives.
Avaz-e Abu-Ata: [G, Ad, Bb, C, D, Eb, F, G] - "C" underlined
Avaz-e Bayat-e Tork: [F, G, Ad, Bb, C, D, Eb, F] - "Bb" underlined
Avaz-e Afshari: [F, G, Ad, Bb, C, D(d) Eb, F] - "C" underlined
Avaz-e Dashti: [G, Ad, Bb, C, D(d) Eb, F, G] - "C" underlined
Homayun: [G, Ad, B, C, D, Eb, F, G]
Avaz-e Bayat-e Esfahan: [G, Ad, B, C, D, Eb, F, G] - "C" underlined
Segah: [F, G, Ad, Bd, C, Dd, Eb, F] - "F" underlined
Chahargah: [C, Dd, E, F, G, Ad, B, C] - "C" underlined
Mahur: [C, D, E, F, G, A, B, C] - "C" underlined
Rast-Panjgah: [F, G, A, Bb, C, D, E, F] - "F" underlined
Nava: [D, Ed, F, G, A, Bb, C, D] - "G" underlined
One difference: I've used a "d" as an accidental instead of a "p" when they indicate a half-flat. I've also added indentation before the "Avaz" scales, which I believe are supposed to be variations on the preceding non-Avaz scales. They also remark that "The underlined letters have approximately the function of a tonic". I don't know why not every scale has a tonic.
Here are Dastagh's from wikipedia:
Abu-ata: [C, D, Eb, F, G, Ad, Bb, C] or [C, D, Eb, F, G, Ad, Bd, C]
Afshari: [C, D, Eb, F, G, A, Bb, C] or [C, D, Eb, F, G, Ad, Bb, C]
Bayat-e-Esfahan: [C, D, Ed, F#, G, A, Bb, C]
Bayat-e-kord: [C, D, Eb, F, G, Ad, Bb, C]
Bayat-e-tork: [C, D, Ed, F, G, A, Bb, C]
Chahargah: [C, Dd, E, F, G, Ad, B, C]
Dashti: [C, D, Eb, F, G, A, Bb, C] or [C, D, Eb, F, G, Ad, Bb, C]
Homayun: [C, D, Eb, F, G, Ad, B, C]
Mahur: [C, D, E, F, G, A, B, C]
Nava: [C, D, Ed, F, G, A, Bb, C]
Rast-Panjgah: [C, D, E, F, G, A, Bb, C]
Segah: [C, D, Ed, F, G, Ad, Bb, C] or [C, Dd, Ed, F, G, Ad, Bb, C]
Shur: [C, D, Ed, F, G, A, Bb, C] or [C, D, Ed, F, G, Ad, Bb, C]
The originals had lots of superscript letters that weren't explained on the page. I've removed those.
Ooh, here's another source: iranicaonline.org, the "Encyclopaedia Iranica". They've got a system of intervals written between the notes!
N: ~170 cents, large neutral second
n: ~130 cents, small neutral second
N: ~204 cents, Pythagorean major second
It looks to me like they're using Zalzalian neutral seconds of 11/10 and 14/13. That's great.
They give Afšārī as
[M, n, N, M, n, N, M] :: [C, D, Ed, F, G, Ad, Bb, C]
with some other comments about how you can dwell on certain notes with different functions, and also "Bd" is a possible decoration below the low C.
Their article on Čahārgāh does not have a diagram with intervals, but it does mention that it is "like a Western major scale, except that the second and sixth degrees are lowered a quarter-tone". So if we root on C, then we have:
[C, D, Ed, F, G, Ad, B, C]
The article on Homāyun clarifies that G is the tonic, and gives us for a scale:
[D, Ed, F, G, Ad, B, C, D, Eb, F] or [D, Ed, F, G, Ap, B, C, D, Ed, F]
But the page on Bīdād, a melody within Homāyun, says that Homāyun has an F#.
Here's their Bayāt-e Kord: [G, Ad, Bb, C, D, Eb, F, G]. They tell us that "The "recitation tone" (šāhed) is D, the initial pitch (āḡāz) is C, and the cadential pitch (īst) is Bb." I don't know what any of that means.
Here's their Bayāt-e Tork: [F, G, Ad, Bb, C, D, Eb, F]. The "primary reference pitch" (šāhed) is Bb. So a primary reference pitch and a recitation tone are the same thing, "šāhed".
They talk about Daštī but it makes no sense to me. It's made of two descending pentachords: (G, F, Eb, D, C, Bb) and (Dd, C, Bb, Ad, G). But you'll notice the first scale fragment has six notes, so it's a hexachord, not a pentachord. The fundamental note of the hexachord is "D". Fine. Next they say that, "Like all āvāzes it begins in the upper register of the scale (D) and finishes in the lower register (G). That makes no sense to me. If you link up those two scale fragments, you obviously are going to do merge them at the common note of G, giving a descending scale of [Dd, C, Bb, Ad, G, F, Eb, D, C, Bb] But this doesn't have D in the upper register, or even in the upper pentachord. On the same page they describe a scale fragment with a name that's a lot like our old friend "Uşşâk":
'Oššāq: [D, Ed, F, G, Ad, Bb, (C)]
Great.
They give Ḥejāz as [(Eb), D, C, Bb, A, G, F, Ed, D] which would be fine, but then they say it begins with A and ends with D.
The site describes a motifc called "Ḥesār" which they claim has one form when used with Čahārgāh rooted on C:
[Ed, F#, G, Ad, B, C]
(in which the motif has a melodic/tonic center on the G note), and also a form used with Segāh rooted on Ed:
[G, A, Bd, C, D]
(in which the melodic/tonic center is Bd). I would be fine with all of that, except that they say the two forms have the same "melodic content", which I think means "the same intervals". They don't though. I can deal with inconsistencies between sources, but I don't think I can deal with this particular website anymore.
Here's new source: "Classic Music Of Iran", compiled and edited by Ella Zonis for Folkways Records, 1966:
Mahour: [C, D, E, F, G, A, B, C]
Shour: [C, Dd, Eb, F, G, Ab, Bb, C]
Avez of Afshari: [C, D, Ed, F, G, A(d), Bb, C]
Avaz of Bayate Tork: [C, D, E, F, G, A, Bd, C]
Avaz of Abu Ata: [C, Dd, Eb, F, G, Ab, Bb, C]
Chahrgah: [C, Dd, E, F, G, Ad, B, C]
Homayoun: [C, Dd, E, F, G, Ab, Bb, C]
Avez of Esfahan: [C, D, Eb, F, G, Ad, B(d), C]
Segah: [C, D, Ed, F, G, Ad, Bb, C] # Ed is the tonic
Rast-panjgah: [C, D, E, F, G, A, B, C]
Nava: [C, Dd, Eb, F, G, Ab, Bb, C]
Dashti: [C, Dd, Eb, F, G, Ab, Bb, C]
...
Most Persian music theorists claim there are four basic tetrachords or "dang-s". Ali-Naqi Vaziri gave 24-EDO descriptions of them in "Dastur-e Tàr", published 1913. Below I give the name of the dang, the cents, the steps in 24-EDO, and pitch classes rooted on C:
Shur: [150, 150, 200] : [3, 3, 4] :: [C, Dd, Eb, F]
Chahargah: [150, 250, 100]: [3, 5, 2] :: [C, Dd, E, F]
Dashti: [200, 100, 200] : [4, 2, 4] :: [C, D, Eb, F]
Mahoor: [200, 200, 100] : [4, 4, 2] :: [C, D, E, F]
We can see that Dang Shur is the same as Jins Bayati so far as 24-EDO distinguishes, with a [n2, n2, M2].
Dang Dashti is Jins Nahawand/Busalik, with a [M2, m2, M2].
Dang Mahoor is Jins Çargâh/Jaharkah/'Ajam, with [M2, M2, m2].
Fascinatingly, Dang Chahargah is not Jins Çargâh, despite the similarity of the names. Instead Dang Chahargah looks more like a Hicaz or Hijaz, with a [narrow, very wide, narrow] thing going on, although this is a new intonation. The 24-EDO arabic Hijaz was [2, 6, 2] in comparison to Dang Chahargah at [3, 5, 2]. The Persian tetrachord is more microtonal, with a bigger first jump, a smaller second jump, and the same minor second to finish.
That was all from Vaziri. Thanks, Vaziri. But everyone hated it, because Persian music isn't and wasn't in 24-EDO edo.
In 1995, Iranianc omposer Dariush Talai gave more precise intervals for the four main Dang-s in "Honare musiqi-ye sonnati-ye Irani: Radif -e Mirzā Abdullāh".
Shur: [140, 140, 220]c : [C, Dd, Eb, F]
Chahargah: [140, 240, 120]c : [C, Dd, E, F]
Dashti: [200, 80, 220]c : [C, D, Eb, F]
Mahoor: [200, 180, 120]c : [C, D, E, F]
Same pitch classes. I got these from "An Investigation On The Value Of Intervals In Persian Music" by Farshad Sanati (2020). These frequency ratios are all multiples of 20 cents, so I wouldn't be surprised if Talai used 60-EDO.
In Sanati's 2020 paper, he also presents frequency ratios measurements for the four Dang-s from six different performers. And weirdly, most of the tetrachords don't quite reach a just P4 of 1200 * log_2(4/3) cents ~ 498 cents or the 500 cents that you could find in 12/24/60-EDO. I thought that was silly, so I scaled all of the measurments to maintain P4 purity before averaging the six different players' data. Rounding to the nearest cent, this gives:
Shur: [143, 145, 211]
Chahargah: [142, 238, 118]
Dashti: [200, 99, 199]
Mahoor: [212, 182, 104]
The measured dang Shur is pretty exactly intermediate between Vaziri's 24-EDO dang and Talai's 60-EDO dang.
The measured dang Chahargah is Talai's 60-EDO thing to within human perception.
The measured dang Dashti is Vaziri's 24-EDO thing to within human perception.
The measured dang Mahoor is different from both gives dang-s, being sharper on the first interval, then matching Talai on the second and matching Vaziri on the third.
The good news is that there's a wide range of stuff you can play and still be called an expert Persian musician. The bad news is that it's going to be really hard for us to nail down the correct intervals or frequency ratios, so far as they can be said to exist.
I wonder if I made a mistake scaling all of the measured dang-s to reach a just P4. What if P4 is generally flat in Persian music because it's supposed to be? In most western temperaments, P5 is flattened slightly, making P4 slightly flat relative to just intonation, but middle eastern music can be different.
For example, 87-EDO, which Margo Schulter and I both identified for its value in modelling middle eastern tuning, has a P5 that's sharp of the just value by:
(1200 * 51/87) - (1200 * log_2(3/2)) ~ 1.5 cents
I thought 94-EDO also had utility but wasn't as good, and it has a P5 that's sharp by
(1200 * 55/94) - (1200 * log_2(3/2)) ~ 0.2 cents
So if the theory says a sharp P5 is fine, which implies a flat P4, and the measurements say that a flat P4 is fine, then maybe we should accept that. The measurements were nothing like 1.5 cents flat, they were off from just by like 20 cents, but still.
In general, if you're looking at rank-2 intervals and you want them to be tuned such that the natural intervals maintain the natural order of 12-TET, then a P5 tuned between 2^(7/12) and 2^(10/17) will look Pythagorean in how it orders the once augmented and once diminished intervals, and a tuned P5 between 2^(10/17) and 2^(13/22) induces another ordering of the once modified intervals, and 2^(13/22) < t(P5) < 2^(3/5), induces a third second order. Since the upper bound of the Pythagorean order, 2^(10/17), is a whole
1200 * 10/17 - (1200 * log_2(3/2)) ~ 3.9 cents
sharp of a just P5, the EDOs compatible with middle eastern tuning are still squarely Pythagorean regime. And probably not perceivable in their difference from Pythagorean tuning until you get out to severally-times modified intervals, I'd guess.
Here's a fun idea: remember when we did combined 24-EDO and 53-EDO analysis the middle third of Turkish and Arabic Rast? Let's do a combined analysis like that using 24-EDO and 60-EDO Persian dang-s.
Suppose we want an interval that is tuned to 3 steps of 24-EDO (i.e. 150 cents) and 7 steps of 60-EDO (i.e. 140 cents). Here are some candidates, presented in the rank-8 (or 19-limit) prime harmonic basis, that are also justly tuned to simple fractions:
[2, 1, 0, 0, -1, 0, 0, 0] # 12/11
[-2, -1, 0, 0, 0, 1, 0, 0] # 13/12
[-5, 0, 1, 1, 0, 0, 0, 0] # 35/32
[0, -1, 1, 0, 1, 0, -1, 0] # 55/51
[2, -2, 0, -1, 0, 0, 1, 0] # 68/63
[-1, -1, 1, 0, 0, -1, 1, 0] # 85/78
Nice. Two of those are Zalzalian neutral seconds, good for use in an al-Farabi tetrachord.
For an interval that's tuned to 4 steps of 24-EDO (at 200 cents) and 11 steps of 60-EDO (at 220 cents), we have:
[0, -1, -1, 0, 0, 0, 1, 0] # 17/15
[2, -1, 0, 0, 1, -1, 0, 0] # 44/39
[0, 2, -1, 1, -1, 0, 0, 0] # 63/55
[-2, 0, 0, 1, 1, 0, -1, 0] # 77/68
[-4, 0, -1, 1, 0, 1, 0, 0] # 91/80
[5, 1, -1, 0, 0, 0, -1, 0] # 96/85
which look weirder, but the 44/39 one is actually what you need for the Farabi tetrachord:
(13/12) * (12/11) * (44/39) = (4/3)
to work out exactly rather than impressionistically. I wonder if I should limit my search to intervals which have 0 in the M17 coordinates, since I have yet to really see any factors of 5 being useful in just analysis of middle eastern music for some reason.
For an interval that's tuned to 4 steps of 24-EDO (at 200 cents) and 9 steps of 60-EDO (at 180 cents), we have few options:
[1, -2, 1, 0, 0, 0, 0, 0] # 10/9
[1, 3, 0, -2, 0, 0, 0, 0] # 54/49
[-2, 0, 1, 0, 0, 0, 1, -1] # 85/76
But I like the first two quite a bit. The second one is a famous ratio called "the middle finger of Zalzal", in reference to placement of the fingers on a lute's neck. It's only about 168 cents, but apparently 24-EDO and 60-EDO both tune it fairly sharp. So maybe 54/49 is an option for the just tuning of the second realtive interval of Mahoor. I think this is working surprisingly well. I really didn't expect it to work. I checked another weird frequency ratio associated with lute geometry, (162/149), called the "Persian middle finger". It's weird in thatthe 149 in the denominator is a pretty high prime. The interval justly associated with this ratio is tuned to 3 steps of 24-EDO and 7 steps of 60-EDO, i.e. 150 cents and 140 cents, just likethe intervals for 12/11 and 13/12.
For an interval that's tuned to 2 steps of 24-EDO (at 100 cents) and 4-steps of 60-EDO (at 80 cents), we have:
20/19 # [2, 0, 1, 0, 0, 0, 0, -1]
35/33 # [0, -1, 1, 1, -1, 0, 0, 0]
51/49 # [0, 1, 0, -2, 0, 0, 1, 0]
81/77 # [0, 4, 0, -1, -1, 0, 0, 0]
85/81 # [0, -4, 1, 0, 0, 0, 1, 0]
95/91 # [0, 0, 1, -1, 0, -1, 0, 1]
I confess that I don't know the historic or numeric importance of any of these, so maybe the second interval of dang Dashti is still a little mysterious.
We've now done a lot to n arrow down the intonation of tetrachords from which Persian scales are built, but I've never seen outright statements of which Persian Dastgahs are built from which tetrachords. That seems soluble to me. We can see what scales result from different combinations of the tetrachords and see when the theoretical possibilities match any of the scales as they're reported in different sources.
I'm going to venture a guess that
Dastgah Mahur = Dang Mahur + M2 + Dang Mahur
Dastgah Chahargah = Dang Chahargah + M2 + Dang Chahargah
Dastgah Shur = Dang Shur + M2 + Dang Shur
And more speculatively, it might be the case that:
Humayun = (Chahargah + M2 + Shur)
Avez e Esfahan: (Dashti + M2 + Chahargah)
In so far as Abu-ata, Afshari, Bayat-e-kord, and Dashti can be played rooted on C with [C, D, Eb, F, G, Ad, Bb, C], they can also be represented as: (Dashti + M2 + Shur).
In so far as Afshari and Dashti look like [C, D, Eb, F, G, A, Bb, C], they can also be represented as: (Dashti + M2 + Dashti), or perhaps (Dashti + Mahur + M2)
So far as Chahargah looks like [C, Dd, E, F, G, Ad, B, C], it's also (Chahargah + M2 + Chahargah).
So far as Homayun and Esfahan look like [C, D, Eb, F, G, Ad, B, C], they're also (Dashti + M2 + Chahargah).
So far as Mahur and Rast-panjgah look like [C, D, E, F, G, A, B, C], they are also (Mahoor + M2 + Mahoor).
So far as Rast-Panjgah instead looks like [C, D, E, F, G, A, Bb, C], it can be represented as (Mahoor + M2 + Dashti) or (Mahoor + Mahoor + M2).
So far as Shur, Avaz of Abu Ata, Nava, and Dashti look like [C, Dd, Eb, F, G, Ab, Bb, C], they are also (Shur + Dashti + M2).
A bunch of Persian scales from one source or another had "C, D, Ed, F, G, A, Bb, C" as description. They were Bayat-e-tork, Nava, Shur, and Afshari. This is Arabic Rast with Nahawand ending, but there wasn't a persian Dang that looked like Rast with [M2, n2, n2]. So....what if we try a cyclic permutation? If we pretend that D is the tonic instead of C and then move everythign down a M2 so that it's rooted on C again:
[D, Ed, F, G, A, Bb, C, D] -> [C, Dd, Eb, F, G, Ab, Bb, C]
then this is (Shur + Dashti + M2).
Segah, Shur, and Afshari were all described in one place or another as looking like [C, D, Ed, F, G, Ad, Bb, C]. Again, we don't have a rast tetrachord to make this. But it's called Makam Nairuz in Arabic theory, a transposition of the older Makam Yakah on G. Yakah is a [G, A, Bd, C, D, Ed, F, G], which is a cyclic permutation of Rast with upper Rast ending.
Of the remaining Persian scale descriptions rooted on C that I've seen, I have no explanation for:
Esfahan: [C, D, Eb, F, G, Ad, Bd, C]
Abu-ata: [C, D, Eb, F, G, Ad, Bd, C]
Esfahan: [C, D, Ed, F#, G, A, Bb, C]
Segah: [C, Dd, Ed, F, G, Ad, Bb, C]
Bayat-e Tork: [C, D, E, F, G, A, Bd, C]
Homayoun: [C, Dd, E, F, G, Ab, Bb, C]
The first Esfahan and Abu-ata here have the same pitch classes. They're both [M2, m2, M2] + M2 + [n2, M2, n2].
The second Esfahan has an F# and overshoots P4. The interval between Ed and F# is ...halfway between M2 and A2, and I don't even have a name for that unless we go to like a septimal analysis with SpM2 or SbA2.
The Segah here is [n2, M2, n2] + M2 + [n2, n2, M2].
The Bayat-e Tork remaining is [M2, M2, m2] + M2 + [M2, n2, n2] . That's actually pretty normal? You've got Dang Mahoor aka Jins Çargâh/Jaharkah/'Ajam, with [M2, M2, m2], and then a M2, and then Rast, [M2, n2, n2]. How is this not another scale anywhere? If we do a cyclic permutation so that G is the root and then drop it down P5, we get
[G, A, Bd, C, D, E, F, G] -> [C, D, Ed, F, G, A, Bb, C]
which is Maqam Rast with the Nahawand ending, and also a different description we've seen for Bayat-e Tork.
The last one here is Homayoun from Ella Zonis. We also have Homayun listed elsewhere as [G, Ad, B, C, D, Eb, F, G], which has the same intervals so that may be encouraging. If we do a cyclic permutation of this so it starts on C, then we get [C, D, Eb, F, G, Ad, B, C], which is a thing we've already accounted for as (Dashti + M2 + Chahargah).
I have no idea what I'm doing.
Ooh, this looks good. Dastgahs from Kees van den Doel at persianney.com:
He's explicit about the "finalis" note being the tonicand he names the tonic for each scale, so I've rotated things to have the tonic at the start and end of each scale. He also have very a nice notation key:
b = flat
# = sharp
p = koron (60 cent flat)
> = sori (40 cent sharp)
Here we go! The dastgahs Mahur and Rast-panjgah:
First position (finalis C): [C D E F G A B C]
Second position (finalis F): [F G A Bb C D E F]
Third position (finalis G): [G A B C D E F# G]
Fourth position (finalis D): [D E F# G A B C# D]
Nice. Everything is transposing regularly.
The dastgahs Shur, Abuata, Afshari, Bayat-e-Tork, Dashti, Nava:
1st position (finalis = D): [D Ep F G A(p) Bb C D]
2nd position (finalis = A): [A Bp C D E(p) F G A]
3rd position (finalis = G): [G Ap Bb C D(p) Eb F G]
Looks good to me
The dastgahs Homayoun and Esfahan:
First position (finalis Homayoun = D, Esfahan = G): [D Ep F# G A Bb C D] and [G A Bb C D Ep F# G].
Second position (finalis Homayoun = G, Esfahan = C): [G Ap B C D Eb F G] and [C D Eb F G Ap B C].
Third position (finalis Homayoun = A, Esfahan = D): [A Bp C# D E F G A] and [D E F G A Bp C# D].
We're on a roll. But all of the other dastags fall apart a little bit.
?Dastgah Segah:
?First position (finalis = Ep): [Ep F G Ap Bb(p) C D Eb]
?Second position (finalis = Bp): [Bp C D Ep Fb(>) G A Bp]
If the first position is notated correctly, then the second position Segah should have "F(p)", not "Fb(>)". He mentions that the optional F> is fingered as F# in a certain important melodic motif called "mokhalef-segah", which again makes me think that it should be F(p) not Fb(>). You're not going to have options of [flat, ???, half sharp, and full sharp ] without having an option of a natural F. Also if the second position were written correctly, the first position should be [Ep F G Ap Bbb(>) C D Ep], with a Bbb. Seems fake.
The other problem, which I didn't notice at first, is that the first position Segah ends in Ep instead of Ed. This thing doesn't reach the octave. Which actually isn't crazy in middle eastern music - neither does Arabic maqam Saba, but like ...It's really hard to figure out from a single source whether something like this is a typo or not. I'll check with other sources. ...
Navid of Oud For Guitarists give Segah as:
[Ed, F, G, Ad, Bb, C, D, Ed]
He actually uses quarter flats instead of half flats, but he's the only Persian music theorist I've seen do so, and to translate his notation for correspondence with the notation of others, it's convenient to pretend that he wrote half-flats.
Wikipedia gives Segah as:
[Ed, F, G, Ad, Bb, C, D(d), Ed]
and attributes this to Mirza Abdollah. Interesting option of a half flat on the D but not the B.
Ella Zonis gives Segah as:
[Ed, F, G, Ad, Bb, C, D, Ed]
after permutation to the tonic, which she indicates as Ed.
The foundation for Iranian studied gives Segah as
Segah: [F, G, Ad, Bd, C, Dd, Eb, F] - "F" underlined
which I thought was consistent with other things somehow, but I've been working on Persian music theory all day and my brain would be protesting if it weren't also melting. Anyway, I'm gonna go with "The first position is correct or correct enough and the second position is wrong." Kind of weird that he broke out his only half-sharp sori accidental and used it incorrectly? Moving on.
Dastgah Segah:
First position (finalis = Ep): [Ep F G Ap Bb C D Ep]
Second position (finalis = Bp): [Bp C D Ep F G A Bp]
Dastgah Chahargah:
First position (finalis = D): [D Ep F# G A Bp C# D]
Which looks fine. He tells us that the second position is rooted on G. We'd expect that to transpose as:
Second position (finalis = G): [G Ap B C D Ep F# G]
but the thing he writes is
Second position (finalis = G): C# D Ep F# G Ap B C D Ep
which I don't know how to close up into a ring because it's not consistent on either end of the scale. There's a C# on the low end and a C natural on the high end. Usually the two ends of his scales match. If we assume that the first position scale is correct, then obviously we would have C natural, but I'd like to confirm that with another source.
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Woo! Wikipedia and the Foundation For Iranian studied both give Chahargah as [C, Dd, E, F, G, Ad, B, C], with "C" underlined to indicate the tonic. This transposes to [D Ep F# G A Bp C# D] and [G Ap B C D Ep F# G]. It's possible "C#" is an ornament that can be played in a lower register, but for the middle of the scale Dastgah Chahargah, in the second ney position, it should be a C natural. Nice.
And that's it! Pretty great source, I'd say. Thanks, Kees van den Doel. If I do a rank 4 analysis and then remove all the "acutes" and "graves" in interval names so it's like we're using the 7-limit Johnston comma for a neutral tone and otherwise only deviating from the Pythagorean spiral of fifths, then we get these relative intervals for the simpler dastgahs of Doel:
Mahur: [M2, M2, m2, M2, M2, M2, m2] # Also Rast-panjgah.
Homayoun: [SbM2, SpM2, m2, M2, m2, M2, M2]
Esfahan: [M2, m2, M2, M2, SbM2, SpM2, m2]
Chahargah: [SbM2, SpM2, m2, M2, SbM2, SpM2, m2]
You can see that Homayoun and Esfahan are permutations of each other, as they should be. As for the dastgah Shur that has an optional half flat, we have:
Shur: [SbM2, Spm2, M2, M2, m2, M2, M2] or [SbM2, Spm2, M2, SbM2, Spm2, M2, M2] # Also Abuata, Afshari, Bayat-e-Tork, Dashti, and Nava
Segah: [Spm2, M2, SbM2, Spm2, M2, M2, SbM2] or ... [....]
Oh! I think I know why Doel's notated Segah gave me some trouble: there's an ambiguity of notation! If you see Bb(p), does it mean the pitch could be (Bb or Bd) or does it mean that it could be (Bb or Bbd)? The first option looks more natural in that it has fewer accidentals, but in other cases, like if we had F(d), we see that the option of a half flat accidental should lower the F, not raise it, as we would be doing when we go from Bb to Bd. I think I'm going to solve the notational ambiguity by appealing to other sources, which all had Bb and not Bb(d). Therefore, Segah shall be simply:
Segah: [Spm2, M2, SbM2, Spm2, M2, M2, SbM2]
Nice. You can see that lots of the Persian dastgahs have the fairly wide "super major second" microtone. Love it.
We can accumulate relative intervals between scale degrees to get absolute intervals of scale degrees relative to the tonic:
Mahur: [P1, M2, M3, P4, P5, M6, M7, P8] // Also Rast-panjgah.
Homayoun: [P1, SbM2, M3, P4, P5, m6, m7, P8]
Esfahan: [P1, M2, m3, P4, P5, SbM6, M7, P8]
Chahargah: [P1, SbM2, M3, P4, P5, SbM6, M7, P8]
Shur: [P1, SbM2, m3, P4, P5, m6, m7, P8] or [P1, SbM2, m3, P4, Sb5, m6, m7, P8] // Also Abuata, Afshari, Bayat-e-Tork, Dashti, Nava.
Segah: [P1, Spm2, Spm3, P4, Spd5, Spm6, Spm7, P8]
Also nice. And here they are all rooted on C:
Mahur: [C, D, E, F, G, A, B, C]
Homayoun: [C, Dd, E, F, G, Ab, Bb, C]
Esfahan: [C, D, Eb, F, G, Ad, B, C]
Chahargah: [C, Dd, E, F, G, Ad, B, C]
Shur: [C, Dd, Eb, F, G, Ab, Bb, C] or [C, Dd, Eb, F, Gd, Ab, Bb, C]
Segah: [C, Dbt, Ebt, F, Gbt, Abt, Bbt, C]
What a great source this site was.
Here are the dastgas as they are rendered in "Music and Song in Persia" by Lloyd Miller:
Shur: [C, Dp, Eb, F, G, Ap, Bb, C]
Abu 'Ata: [C, D, Eb, F, G, Ap, Bb, C]
Bayat-e Tork: [Bb, C, D, Eb, F, G, Ap, Bb]
Afshari: [C, D(p), Eb, F, G, Ap, Bb, C]
Dashti: [C, D(p), Eb, F, G, Ap, Bb, C]
Homayun: [C, D, Eb, F, G, Ap, B, C]
Segah: [F, G, Ap, Bb, C, Dp, Ep, G, F]
Chahargah: [C, Dp, E, F, G, Ap, B, C]
Mahur: [C, D, E, F, G, A, B, C]
Rastpanjgah: [F, G, A, Bb, C, D, E, F]
Nava: [G, A, Bb, C, D, Ep, F, G]
I've rotated them so that indicated tonics begin and end each dastgah.
Here are 7 dastgahs from "Transcultural Music" by Alireza Ostovar
Shur: [A, Bd, C, D, E, F, G, A]
Homayun: [A, Bd, C#, D, E, F, G, A]
Mahur: [A, B, C#, D, E, F#, G#, A]
Segah: [A, Bd, C, D, Ed, F, G, A]
Chahargah: [A, Bd, C#, D, E, Ft, G#, A]
Nava: [A, Bd, C, D, E, F, G, A]
Rast Panjgah: [A, B, C#, D, E, F#, G#, A]
All rooted on "A" for convenience I guess. Ostovara als ogives the dastgahs in terms of fractional multiples of a whole steps, which we can multiply through by a factor of 4 to get 24-EDO steps. Here are the names, the absolute steps of 24-EDO, and the relative steps of 24-EDO.
Shur/Nava : [0, 3, 6, 10, 14, 16, 20, 24] : [3, 3, 4, 4, 2, 4, 4]
Homayun : [0, 3, 8, 10, 14, 16, 20, 24] : [3, 5, 2, 4, 2, 4, 4]
Mahur/Rastpanjgah : [0, 4, 8, 10, 14, 18, 22, 24] : [4, 4, 2, 4, 4, 4, 2]
Segah : [0, 3, 6, 10, 13, 16, 20, 24] : [3, 3, 4, 3, 3, 4, 4]
Chahrgah : [0, 3, 8, 10, 14, 17, 22, 24] : [3, 5, 2, 4, 3, 5, 2]
I see two ways we could decompose these into 4 tetrachords, one of them using an arabic Kurd tetrachord [2, 4, 4] instead of the dashti tetrachord [4, 2, 4], but I trust that Vaziri is breaking the dastgahs up into dang-s correctly, in a way that suggests and explains melodic fragments. Using Vaziri's dang-s, there is no analysis available to us save for:
Shur/Nava: Shur + Dashti + T.
Homayun: Chahargah + Dashti + T.
Mahur/Rastpanjgah: Mahur + T + Mahur.
Segah: Shur + Shur + T.
Chahargah: Chahargah + T + Chahargah.
I think we can also say that Homayun's permutation Esfahan has got to be
Esfahan : Dashti + T + Chahargah.
even though Ostovara didn't give a 24-EDo analysis of Esfahan.
...
You know what are some other great sources? The websites Oud For Guitarists and Majnuun Music And Dance, both with musical articles by Navid Goldrick. I had started looking at those and got distracted. Let's go through all of his stuff, comparing it to the other sources, especially persianney.com.
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: Theoretical Gamuts Of Persian Tuning
Earlier we talked about a 2020 paper by Farshad Sanati, "An investigation on the value of intervals in Persian music". In addition to relaying the 60-EDO dang-s and the 24-EDO dang-s and the measured dang-s, Sanati also describes some tunings for scales in which we start with 24-EDO pitch classes and then tune successive steps to values other than 50 cents, such as 30 or 70 cents. Multiple Persian music theorists have given alterations of 24-EDO like this, and I'll try to post those in an organized way that makes more insights than confusions.
The "gamut" of pitch classes that comes from Ali-Naqi Vaziri is just 24-EDO and not worth posting. The gamut of Farhat, as I saw it, had separations in terms of cents between different pitch classes, but it was five cents short of an octave.
These are the pitch classes: [C, Db, Dd, D, Eb, Ed, E, F, Ft, Gd, G, Ab, Ad, A, Bb, Bd, B, C].
These are the original separations: [90, 45, 70, 90, 45, 70, 90, 65, 60, 70, 90, 45, 70, 90, 45, 70, 90].
Turning the 65 into a 70 is the obvious way to fix this to reach a sum of 1200: [90, 45, 70, 90, 45, 70, 90, 70, 60, 70, 90, 45, 70, 90, 45, 70, 90].
Accumulate the step-wise intervals and zip together with the pitch classes to get an association: [(C, 0), (Db, 90), (Dd, 135), (D, 205), (Eb, 295), (Ed, 340), (E, 410), (F, 500), (Ft, 570), (Gd, 630), (G, 700), (Ab, 790), (Ad, 835), (A, 905), (Bb, 995), (Bd, 1040), (B, 1110), (C, 1200)]
The same source gave a gamut of tunings for pitch classes from Dariush Talai. Talai uses these pitch classes: [C, Dd, D, Eb, Ed, E, F, F#, Gd, G, Ad, A, Bb, Bd, B, C]
With these separations: [140, 60, 80, 70, 30, 120, 80, 60, 60, 140, 60, 80, 70, 30, 120].
And that already sums to a 1200 cent octave, so we can just accumulate and zip them together immediately: [(C, 0), (Dd, 140), (D, 200), (Eb, 280), (Ed, 350), (E, 380), (F, 500), (F#, 580), (Gd, 640), (G, 700), (Ad, 840), (A, 900), (Bb, 980), (Bd, 1050), (B, 1080), (C, 1200)]
Farhat and Talai are usually within 5 cents of agreement on the neutral microtones and weirdly differ by like 40 cents on many of the natural intervals:
0: C
90: Db # Farhat
130: Dd # Talai
135: Dd # Farhat
175: D # Talai
205: D # Farhat
250: Eb # Talai
295: Eb # Farhat
325: Ed # Talai
340: Ed # Farhat
370: E # Talai
410: E # Farhat
500: F
570: F# # Talai
570: Ft # Farhat
630: Gd
700: G
790: Ab # Farhat
830: Ad # Talai
835: Ad # Farhat
875: A # Talai
905: A # Farhat
950: Bb # Talai
995: Bb # Farhat
1025: Bd # Talai
1040: Bd # Farhat
1070: B # Talai
1110: B # Farhat
1200: C
So there you have it. Another set of options for tuning Dastgahs, straight from the Persian music theorists.
: Wavelength Calculations and Medieval Lute Geometry
The following section is based on "Musical Mathematics" by Cris Forster at chrysalis-foundation.org. He relates that in the "Kitab al-musiqi al-kabir", al-Farabi described the locations of frets on a middle eastern lute called an oud (or 'ud). These frets are simply strings tied around the neck of the instrument and can be moved laterally to accommodate different scales. Still, al-Farabi teaches us about common fret positions. Even on a fretless string instrument, this math still described the placement of fingers to achieve common tones. Forster gives the frequency ratios of al-Farabi, but I thought the constructions could be a lot clearer, so I've added a lot of exposition.
The index finger plays at a spot that is 1/9 of the string length away from the end of the lute that is far from the player's body, i.e. 1/9 of the string length away from the nut where the strings make contact with the neck. This divides the string into two unequal segments. The long segment is then plucked, which means that the vibrating string segment has a wavelength that is 8/9 of the full string length, and so it produces a tone whose fundamental frequency is
1 / (8/9) = (9/8)
times the frequency of the string played "open" without any fretting, i.e. it's higher by a Pythagorean majord second.
The middle finger is associated with a few locations and we'll come back to it.
The ring finger's fret is placed 1/9 of the distance between the index finger's fret and the near end of the string ("the bridge" as it is called). Thus the fret location is
1/9 + (1/9 * 8/9) = (9/81 + 8/81) = 17/81
And the wavelength ratio is
1 - (17/81) = 64/81
The frequency ratio is the inverse of this at (81/64), a Pythagorean major third M3 at 408 cents. You can also see that this is just (9/8) * (9/8), so I probably could have done the math more parsimoniously.
The little finger is associated with a point that is 1/4 of the full string length from the nut. This has wavelength 3/4 and frequency ratio 4/3, i.e. a perfect fourth, P4, over the open string frequency. Maybe lute players who have flat tetrachords just don't like stretching their pinkies. No shame in that.
The first option for the middle finger is the Pythagorean m3. I think the construction feels a little artificial, but we didn't really need a natural construction in terms of fret geometry to justify the Pythagorean intervals. Take the wavelength ratio associated with the little finger (playing P4), which is 3/4 the full string length, and divide it into eighths, giving 3/32. Subtract one of these eighth distances from the location of the little finger fret:
1/4 - 3/32 = 5/32
This is a fret location with a wavelength ratio at
1 - 5/32 = 27/32
and a frequency ratio of 32/27 at 294 cents.
Another option for the middle finger is the "Persian middle finger". This is fretted halfway between the index finger (Pythagorean M2)and the ring finger (Pythagorean M3), i.e. fretted at
((1/9) + 17/81) / 2 = 13/81
of the full string length, with a wavelength ratio of
1 - (13/81) = 68/81
and a frequency ratio of 81/68 at 303 cents.
Another option for the middle finger is "the middle finger of Zalzal", which is halfway between the Persian middle finger and the ring finger (Pythagorean M3):
((13/81) + (17/81)) / 2 = 15/81
with a wave length of
1 - 15/81 = 22/27
and a frequency ratio of 27/22 at 355 cents.
The operation of averaging wavelengths is equivalent to the operation of taking the harmonic mean, favored so highly by Archytas. I think these constructions are interesting, and they tell us exactly (within the precision of physical constructability) what frequency ratios were in common use at the time. Unfortunately, they don't tell us about changes in intonation between different maqamat that have nominally the same e.g. neutral third. But if neutral thirds fall in a range of like 40 cents, then we can be just play the pitches consistently across maqamat and not worry too much about being wrong. We'll be within 40 cents of right, perhaps.
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