: Persian Dastgāh
I'm still figuring out Persian modal scales, called dastgāh. I've got many sources on them and they all disagree, but I'm gradually coming to conclusions on which of them are better and we're going to be able to figure this out.
:: Dastgah In 60-EDO
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 Dastaghs from wikipedia:
Shur: (C D Ep F G (A | Ap) Bb C)
C is Āghāz, the starting pitch.
D is Finalis, the ending pitch.
A/Ap is Moteghayyer, a changeable pitch.
Bayat-e-tork: (C D Ep F G A Bb C)
C is Āghāz, the starting note, and Ist, an ending pitch for phrases other than the Finalis.
F is Finalis, the ending note, and Shāhed, a prominent pitch.
Dashti: (C D Eb F G (A | Ap) Bb C)
D is Finalis, the ending pitch.
A is Āghāz, the starting pitch.
A/Ap is both Moteghayyer, a changeable pitch, and Shāhed, a prominent pitch.
Abu-ata: (C D Eb F G Ap (Bb | Bp) C)
D is Finalis, the ending pitch.
Eb is Āghāz, a starting pitch, and Ist, an ending pitch for phrases other than the Finalis.
G is Āghāz, a starting pitch, and Shāhed, a prominent pitch.
Afshari: (C D Eb F G (Ap | A) Bb C)
C is Finalis, the ending pitch.
Eb is Ist, an ending pitch for phrases other than the Finalis.
G is Āghāz, the starting pitch, and Shāhed, a prominent pitch.
Segah: (C (D | Dp) Ep F G Ap Bb C)
Ep is Āghāz, the starting pitch, Finalis, the ending pitch, and Shāhed, a prominent pitch.
Nava: (C D Ep F G A Bb C)
Ep is Ist, an ending pitch for phrases other than the Finalis.
G is Finalis, the ending pitch.
Homayun: (C D Eb F G Ap B C)
Eb is Āghāz, the starting pitch.
F is Ist, an ending pitch for phrases other than the Finalis.
G is Finalis, the ending pitch.
Ap is Shāhed, a prominent pitch.
Bayat-e-Esfahan: (C D Ep F# G A Bb C)
Ep is Ist, an ending pitch for phrases other than the Finalis.
G is Āghāz, the starting pitch, Finalis, the ending pitch, and Shāhed, a prominent pitch.
Chahargah: (C Dp E F G Ap B C)
C is Finalis, the ending pitch.
Ap is Āghāz, the starting pitch.
Mahur: (C D E F G A B C)
C is Āghāz, the starting pitch, and Finalis, the ending pitch.
D is Shāhed, a prominent pitch.
Rast-Panjgah: (C D E F G A Bb C)
F is Āghāz, the starting pitch, and Finalis, the ending pitch.
Bayat-e-kord (C D Eb F G Ap Bb C)
G is Finalis?
No Finalis is listed for Bayat-e-kord on the Wikipedia page, but The Encyclopaedia Iranica describes Bayāt-e Kord as (G Ap Bb C D Eb F), and notes
The recitation tone (šāhed) is D,
The initial pitch (āḡāz) is C,
The cadential pitch (īst) is Bb.
The finalis is G.
Since this matches Wikipedia's Bayat-e-kord in in having (Ap, Bb, and Eb) for its key signature, I trust it fairly well. Kord has G as a Finalis.
I'm going to show each dastgah from Wikipedia rotated so that the the finalis is in the bass:
Abu-Ata: [D, Eb, F, G, Ap, (Bb | Bp), C, D]
Afshari: [C, D, Eb, F, G, (A | Ap), Bb, C]
Bayat-e-Esfahan: [G, A, Bb, C, D, Ep, F#, G]
Bayat-e-Kord : [G, Ap, Bb, C, D, Eb, F, G]
Bayat-e-Tork: [F, G, A, Bb, C, D, Ep, F]
Chahargah: [C, Dp, E, F, G, Ap, B, C]
Dashti: [D, Eb, F, G, (A | Ap), Bb, C, D]
Homayun: [G, Ap, B, C, D, Eb, F, G]
Mahur: [C, D, E, F, G, A, B, C]
Nava: [G, A, Bb, C, D, Ep, F, G]
Rast-Panjgah: [F, G, A, Bb, C, D, E, F]
Segah: [Ep, F, G, Ap, Bb, C, (D | Dp), Ep]
Shur: [D, Ep, F, G, (A | Ap), Bb, C, D]
This makes it easier for us to see how the pitches and accidentals relate to the tonic / Finalis, but this representation has a downside; I don't think actual dastgah are generally ordered from tonic to tonic. Like it's generally not even an option to play them that way. Instead, you have a pitch collection that spans about an octave, and the lowest and highest notes might not be the tonic/Finalis or the initial melodic note / Aghaz. The wikipedia page just started every dastgah on C, which I don't believe is true persian practice, so we'll have to keep an eye out for dastgah written on staves to figure out the actual highest pitch, lowest pitch, and general octave placement of pitches.
Lets' look a little more at The Encyclopaedia Iranica. It can be found at iranicaonline.org. They've got a system of intervals written between the notes!
N: ~170 cents, large neutral second
n: ~130 cents, small neutral second
M: ~204 cents, Pythagorean major second
For a neutral second around 130 cents, the Zalzalian super-particular ratio 14/13 is 128 cents. For a neutral second around 170 cents, the Zalzalian super particular ratio 11/10 is 165 cents. If you want to be able to get to a pure octave, it works better to use just one of these with along with a second ratio derived by dividing 32/27 by the first:
(32/27) / (14/13) = 208/189 at 166 cents
(32/27) / (11/10) = 320/297 at 129 cents
This is analogous to how the Pythagorean Grm2 and AcM2 sum to Grm3, and how the just m2 and M2 sum to Grm3. The (14/13 and 208/189) pair only differs from the (320/297 and 11/10) pair by one cent, which is inaudible, so if you have a preference for representing dastgahs with factors of 13 and 7 or with factors of 11 and 5, now's your chance to express yourself.
After giving us approximate sizes for intervals, they also give a description of dastgah 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. They also give describe the scale on a staff, which lets us put octave numbers on the pithes. I'll just put them on "C"s for brevity.
Afšārī : [(Bp), C4, D, Ep, F, G, Ap, Bb, C5]
C4 is the Finalis.
Ep is Ist, a temporary stopping point.
G is Sahed, the reciting tone, and Agaz, the starting tone.
Ap is motagayyer, a changeable tone. It can specifically change to A natural in ascending passages.
In Wikipedia's rendition of Afshari, the {Ep} was {Eb}. So we should keep that in mind. Which one is correct? Why don't the sources agree? Are both acceptable? I don't know. I hope we find out.
If you use the 14/13 version of the Zalzalian small neutral second, Afshari looks like this:
[P1, AcM2, ReSbAcM3, P4, P5, ReSbAcM6, Grm7, P8] :: [1, 9/8, 63/52, 4/3, 3/2, 21/13, 16/9, 2]
If you use the 11/10 version of the Zalzalian large neutral second, Afshari looks like this:
[P1, AcM2, DeM3, P4, P5, DeM6, Grm7, P8] :: [1, 9/8, 40/33, 4/3, 3/2, 160/99, 16/9, 2]
I didn't have a preference before, but now that I've looked at the intervals and ratios, I think I like the undecimal version better than the tridecimal version.
Let's learn more from the Encyclopaedia Iranica. 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 gives these pitch classes:
[D, Ep, F, G, Ap, B, C, D, (Eb | Ep), F]
with a finalis of G. Dastgah Bayat-e Esfahan is a sub-dastgah (avaz) of Homayun, and they give Bayat-e Esfahan's pitch classes as
[F, G, Ap, B, C, D, Eb]
where
C is šāhed, the recitation tone.
C or G may be āḡāz, the initial pitch.
C or Ap may be īst, the cadential pitch,
and G is the finalis, although historically it was F.
Their Esfahan is clearly related to their Homayun, as we're told from other sources that it should be, but their accounts differs quite a lot from the Wikipedia descriptions of Homayun and Esfahan. The Wikipedia descriptions, rooted on the finalis, look like this:
Homayun: [G, Ap, B, C, D, Eb, F, G]
Bayat-e-Esfahan: [G, A, Bb, C, D, Ep, F#, G]
The Wikipedia version of Homayun is different from but compatible with the Encyclopedia Iranica version; Wikipedia specifies only a high Eb instead of (Eb | Ep) as changeable tones, and Wikipedia fails to mention low tones of (D, Ep) at the bottom of the scale.
Esfahan is totally different between the accounts though! The Iranica version of Esfahan has F natural and Ap and B natural, while the Wikipedia version has F# and A natural and Bb. Wikipedia's version of Esfahan is closer to a Shur! What is happening?
I don't know, but maybe the page on Bīdād will help clear things up? Bidad is a melody or gūša of Homāyūn. The Iranica page on Bidad lists this as the scale of Homayun:
[A, Bp, C, D, Ep, F#, G, A, (Bb)]
where
Bp is Shaded, a prominent pitch.
D is Finalis, the final note.
Also C is notated functionally as "P" and Ep is notated functionally as "R". I don't know what those are; they're not abbreviations for any of the usual roles, [Finalis, Ist, Moteghayyer, Shāhed, Āghāz]. It would be nice to know, but we've already got what we need to solve the mystery: we've got two modulations: this version of Homayun with F# has finalis D instead of G. If we transposed the previous Iranica version of Homayun with G as Finalis up by P5, we'd get this scale:
[A, Bp, C, D, Ep, F#, G, A, (Bb | Bp), C]
which is close enough to the version on the Bidad page for me. The Wikipedia version of Bayat-e Esfahan is a subset of this, so we're doing well.
Here's Iranica's 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 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.
...
I came back to it, because I increasingly found that it was a good source. I was probably just misunderstanding things. Most Iranian and Persian modes (in so far as they're different) are variations on dastgah Shur. Quote:
"Thus, more than 50 percent of Persian music is based on Šur scale (parda), the rest using three other scales, namely, those of Māhur/Rāst, Homāyun/Eṣfahān, and Čahārgāh."
Encyclopedia Iranica has an amazing table describing how scales are based on Shur.
We start with a description of Shur. For certain transposing instruments, Shur may be written with its finalis at C, but more canonically Shur has its finalis on G. It looks like this:
[(D, Ep), F, G, Ap, Bb, C, (Dp | D), Eb, F, G, (Ab, Bb...)]
The outer parentheses are optional decorations I believe, while the (Dp | D) is a changeable pitch, a motagayyer/moteghayyer. Using this same set of pitch classes, but moving the finalis, we get many other Persian modes. The table showing this also lists (Dp) or (D) or (Dp | D) for each mode, so I think it's specifying that some of the modes make one choice of pitch for the moteghayyer and some don't.
* Shur has G as finalis and Dp is the more common moteghayyer choice.
* Abu Ata is Shur with Ap as finalis and D as the choice moteghayyer.
* Segah is Shur with Ap as finalis and Dp as the choice moteghayyer.
* Tork is Shur with Bb as finalis and D as the choice of moteghayyer.
* Afshari is Shur with C as finalis and (Dp | D) both used.
* Shanaz is Shur with C as finalis and Dp as the choice of moteghayyer.
* 'Ozzal is Shur with C as finalis and D as the choice of moteghayyer.
* Nava, like 'Ozzal, is Shur with C as finalis and D as the choice of moteghayyer.
* No Avaz of Shur has D aas finalis.
* Kord and Dashti are Shur with D as finalis and both (Dp | D) are used.
* Hejaz is also Shur with D as finalis and no choice of (Dp | D) is listed.
* Qaraca is Shur with Eb as finalis and Dp as the choice of moteghayyer.
* Kara, Kuchak, and Shekasta are all Shur with F as finalis and D as the choice of moteghayyer.
* Bozorg is Shur with G as finalis, but the high G, whereas the low G is the finalis of Shur. Bozorg also has "Ap" in parentheses, like that's the choice among (Dp | D). I'm not sure if that's a typo for Dp or if Ap is simply a distinguished pitch of the Avaz.
* Delkash is also Shur with high G as finalis, and this one used D as the choice among (Dp | D).
Let's try to write the scales out in pitches based on this information. We can compare it to pitch that other sources have given when we have them.
...
Here's the 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. But the Arabs adopted 24-EDO for notation. So that's weird.
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. Before we saw Zalzalian neutral tones, like a small neutral second at 130 cents and a large neutral second at 165 cents. That made sense to me historically and intervallically. These [120, 140, 200, 220, 240, 180] cent are quite baffling in comparison, but at least they give us more precise intonation for the dang-s than Vaziri I guess.
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, using frequency measurements of performed music or using EDO steps for Persian music theorists.
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 could 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 tunings 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.
Thwe 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 tonic and 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)
To me the Persian accidentals look like they have 60-EDO tunings (because they're both multiples of 20 cents, which is 1 step of 60-EDO). One somewhat sensible intonation consistent with this is that the koron is a SpM0, justly tune to (27/28) (i.e. the inverse of Sbm1, justly tuned to 28/27) and sori is a SpGr1, justly tuned to (64/63). These are actually -62 cents and +27 cents respectively, but 60-EDO tunes them to -60 cents and +40 cents exactly. This is a pretty weird intonation because the simplest septimal ratios aren't neutral tones like those used in middle eastern music, they're the tones between e.g. (AcM2 and Grm3) or between (AcM3 and P4). Here's a little table with tempered cents in 60-EDO:
0 : 1 _ P1
40 : 64/63 _ SpGr1
60 : 28/27 _ Sbm2
100 : 256/243 _ Grm2
140 : 243/224 _ SpAcA1
160 : 567/512 _ SbAcAcM2
200 : 9/8 _ AcM2
240 : 8/7 _ SpM2
260 : 7/6 _ Sbm3
300 : 32/27 _ Grm3
340 : 2048/1701 _ SpGrGrm3
360 : 896/729 _ SbGrd4
400 : 81/64 _ AcM3
440 : 9/7 _ SpM3
460 : 21/16 _ SbAc4
500 : 4/3 _ P4
540 : 256/189 _ SpGr4
560 : 112/81 _ SbGrd5
600 : 1024/729 _ GrGrd5 __ and __ 600 : 729/512 _ AcAcA4
640 : 81/56 _ SpAcA4
660 : 189/128 _ SbAc5
700 : 3/2 _ P5
740 : 32/21 _ SpGr5
760 : 14/9 _ Sbm6
800 : 128/81 _ Grm6
840 : 729/448 _ SpAcA5
860 : 1701/1024 _ SbAcAcM6
900 : 27/16 _ AcM6
940 : 12/7 _ SpM6
960 : 7/4 _ Sbm7
1000 : 16/9 _ Grm7
1040 : 1024/567 _ SpGrGrm7
1060 : 448/243 _ SbGrd8
1100 : 243/128 _ AcM7
1140 : 27/14 _ SpM7
1160 : 63/32 _ SbAc8
1200 : 2/1 _ P8
Alternatively, the {-60, +40} accidental might be just meeting in the middle, i.e. in this system there's only one neutral interval between a tempered m2 at 100c and a tempered M2 at 200 cents, and it is at 240 cents. If that's the case, then the +40 accidental isn't really doing any work, and we could flatten Pythagorean chromatic intervals by any other interval that has a tempered tuning of 60 cents, e.g.
60c : 25/24 _ A1
60c : 33/32 _ As1
60c : 27/26 _ ReAcA1
These would give us neutral seconds at
140c : 27/25 _ Acm2
140c : 12/11 _ DeAcM2
140c : 13/12 _ Prm2
respectively. The actual just tunings of those intervals A1, As1, ReAcA1 (with 60 cent tempered tunings in 60-EDO) are (71, 53, 65) cents respectively.
If you've got a better way to explain/justify/rationalize/detemper the -60 and +40 cent accidentals, I'm all ears. Anyway, regardless of the intonation, this is a good source for pitch classes. Let's see the scales
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, 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.
...
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.
...
: 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.
...
:: "Just Intonation, Dastgāh-Music, and Scordatura in Kupferteich" By Navid Bargrizan
I found another source describing Persian Dastgah! It's fairly regular and complete. I hope that means that it's correct. First up, it distinguishes among four interval sizes:
m: minor second, 90c
n: small neutral second, 135c
N: large neutral second: 160c
M: major second, 204c
P: plus tone, 270c
The minor second and major second are clearly Pythagorean in their tuning. The original source actually listed 240c instead of 204c, but I know a Pythagorean typo when I see one. I've got a clever analysis of the neutral seconds.
The obvious candidates for a small neutral second around 135 are the small Zalzalian seconds (14/13 and 13/12) at 128c and 139 cents. You can derive a neutral second that naturally pairs with each of these by dividing 32/27 (the just tuning of Grm3) by each Zalzalian second:
(32/27) / (14/13) = 208/189 at 166c
(32/27) / (13/12) = 128/117 at 156c
Bargrizan's small neutral second is between 14/13 and 13/12, and we might guess that his is some kind of an average in frequency space or log frequency space. Whether Persian music is better represented by different Zalzalian seconds in different contexts, or by an average of them used consistently in all contexts, I don't know, but we're going to try Bargrizan's way.
If we take the harmonic mean of (14/13 and 13/12) in order to geta ratio in between, we get
364/337 at 133 cents
and its 32/27 complement at
2696/2457 at 161 cents
These have factors of 337, which is nonsense. Instead of the harmonic mean, we could take the mediant:
(14 + 13) / (13 + 12) = 27/25 at 133 cents
which is the just tuning of Acm2. This has a complementary large neutral second interval of
(32/27) / (27/25) = 800/729 at 161c.
We could have also found this intervallically as
Grm3 - Acm2 = GrM2
The harmonic mean and the mediant gave us equally sized intervals, but the mediant gave us much simpler ratios. I think this is pretty funny: Zalzal gave us very precise 13-limit neutral seconds that differed by like 10 cents and told us to use them differently, but if we're not that precise in our hearing or our playing, then we lose 3-primes of precision (factors of 7, 11, 13) and go down to 5-limit intervals. I think it's worth working in this system! Maybe Persian music is 5-limit. I don't yet know what interval the "plus tone" of 270 cents will be: a septimal 7/6 is pretty close, but couldn't form an octave if combined with the other intervals. I had to check how P is used in the Dastgahs below to be sure, but I had a guess that P would be AcA2, justly tuned to 75/64, and indeed that works swimmingly.
This plus tone is a Pythagorean augmented unison, AcAcA1, larger than the large neutral tone:
GrM2 + AcAcA1 = AcA2
(800/729) * (2187/2048) = (75/64)
One way to think about this is that if you want to reach GrM3, you can add a large neutral second and a Pythagorean major second:
N + AcM2 = GrM3
Or you can add a plus tone and a Pythagorean minor second:
(N + AcAcA1) + m2 = GrM3
The GrM3 isn't going to be used as a scale tone (instead dastgah will use the small neutral third at Acm3), but sometimes the large neutral third will occur as a relative interval between alternating scale tones.
I don't know why, but some of the note heads in Bargrizan's dastgahs are filled solid and some are filled solid and some are hollow. I'll make notes of that in case it's important. I've also placed octave numbers on the C pitches, namely middle C at C4 and an octave up at C5.
Shur:
[N, M, n, N, M, M, m] :: [Bp, C4, D, Ep, F, G, A, Bb]
Final is D.
{A} has a {p} way above it instead of on the staff. Maybe that's an option for A? A or Ap?
[D, Ep, F, G] is in brackets like it's a special tetrachord.
Filled note heads: [Bp, C4, A, Bb]
Abu Ata:
[M, n, N, M, M, m, M] :: [C4, D, Ep, F, G, A, Bb, C5]
Final is D.
[D, Ep, F, G, A] is in brackets like it's a special pentachord.
Filled note heads: [C4, Bb, C5]
Dashti:
[M, n, N, M, M, m, M, M, m] :: [C4, D, Ep, F, G, A, Bb, C5, D, Eb]
Final is the D directly above C4.
[D, Ep, F, G, A, Bb, C5] is in brackets.
Filled note heads: [C4, Ep, D, Eb]
Bayat-e Tork:
[M, n, N, M, M, m, M] :: [C4, D, Ep, F, G, A, Bb, C5]
Final is F.
[C4, D, Ep, F] and [F, G, A, Bb] are both in brackets.
Filled note heads: [C5]
Afshari:
[M, n, N, M, n, N, M] :: [C4, D, Ep, F, G, Ap, Bb, C5]
Final is C4.
[Ep, F, G, Ap] is in brackets.
Filled note heads: [D, Bb, C5]
Segah:
[N, M, n, N, M, n, N] :: [Bp, C4, D, Ep, F, G, Ap, Bb]
Final is Ep.
[Ep, F, G, Ap] is in brackets.
Filled note heads: [Bp, D, Bb]
Chahargah:
[n, P, m, n, P, m] :: [G, Ap, B, C5, Dp, E, F]
Final is C5.
[G, Ap, B, C5, Dp, E] is in brackets.
Filled note heads: [F]
Homayun:
[N, M, n, P, m, M, m] :: [Ep, F, G, Ap, B, C5, D, Eb]
Final is G.
[Ep, F, G, Ap] is in brackets, overlapping with [G, Ap, B, C] in brackets.
Filled note heads: [D, Eb]
Bayat-e Isfahan:
[n, M, N, M, m, M] :: [D, Ep, F#>, G, A, Bb, C5]
Final is G.
The F appears to have a combination of > and # as its accidentals, and also # appears again high above the staff. My guess is that this indicates that F> and F# are both options, although other sources only have F# for Isfahan, so maybe I'm just misreading the accidental. It's pretty small and blurry.
[D, Ep, F>#, G, A, Bb] is in brackets.
Filled note heads: [C5]
Nava:
[n, N, M, M, m, M] :: [D, Ep, F, G, A, Bb, C5]
Final is G.
[D, Ep, F, G, A, Bb] is in brackets.
Filled note heads: [C5]
Mahur:
[M, M, m, M, M, m] :: [G, A, B, C5, D, E, F]
Final is C5.
[G, A, B, C5] and [C5, D, E, F] are in brackets.
Filled note heads: None.
Rast:
[M, M, m, M, M, m, M, M, m] :: [C4, D, E, F, G, A, Bb, C, D, Eb]
Final is F.
[C4, D, E, F] and [F, G, A, Bb] and [Bb, C, D, Eb] are all in brackets.
Filled note heads: [C4, Eb]
A lot of these don't span a full octave, and I'm okay with that.
Abu Ata, Afshari, and Bayat-e Tork all hit the octave. Isfahan, Mahur, Nava, and Chahargah hit Grm7, justly tuned to 16/9. Dashti and Rast both hit the octave but also go past it and land on Grm10, justly tuned to 64/27. Homayun, Segah, and Shur all land on GrGr8, justly tuned to 12800/6561.
Here's the full 5-limit intervallic analysis:
Abu Ata:
[P1, AcM2, Acm3, P4, P5, AcM6, Grm7, P8] :: [1, 9/8, 243/200, 4/3, 3/2, 27/16, 16/9, 2]
Afshari:
[P1, AcM2, Acm3, P4, P5, Acm6, Grm7, P8] :: [1, 9/8, 243/200, 4/3, 3/2, 81/50, 16/9, 2]
Bayat-e Isfahan:
[P1, Acm2, Acm3, P4, P5, Grm6, Grm7] :: [1, 27/25, 243/200, 4/3, 3/2, 128/81, 16/9]
Bayat-e Tork:
[P1, AcM2, Acm3, P4, P5, AcM6, Grm7, P8] :: [1, 9/8, 243/200, 4/3, 3/2, 27/16, 16/9, 2]
Chahargah:
[P1, Acm2, AcM3, P4, d5, AcM6, Grm7] :: [1, 27/25, 81/64, 4/3, 36/25, 27/16, 16/9]
Dashti:
[P1, AcM2, Acm3, P4, P5, AcM6, Grm7, P8, AcM9, Grm10] :: [1, 9/8, 243/200, 4/3, 3/2, 27/16, 16/9, 2, 9/4, 64/27]
Homayun:
[P1, GrM2, GrM3, P4, A5, GrM6, GrM7, GrGr8] :: [1, 800/729, 100/81, 4/3, 25/16, 400/243, 50/27, 12800/6561]
Mahur:
[P1, AcM2, AcM3, P4, P5, AcM6, Grm7] :: [1, 9/8, 81/64, 4/3, 3/2, 27/16, 16/9]
Nava:
[P1, Acm2, Grm3, P4, P5, Grm6, Grm7] :: [1, 27/25, 32/27, 4/3, 3/2, 128/81, 16/9]
Rast:
[P1, AcM2, AcM3, P4, P5, AcM6, Grm7, P8, AcM9, Grm10] :: [1, 9/8, 81/64, 4/3, 3/2, 27/16, 16/9, 2, 9/4, 64/27]
Segah:
[P1, GrM2, GrM3, P4, GrGr5, GrM6, Grm7, GrGr8] :: [1, 800/729, 100/81, 4/3, 3200/2187, 400/243, 16/9, 12800/6561]
Shur:
[P1, GrM2, GrM3, P4, GrGr5, GrM6, GrM7, GrGr8] :: [1, 800/729, 100/81, 4/3, 3200/2187, 400/243, 50/27, 12800/6561]
Next I want to see if I can rederive the pitches from the intervals (and the lowest note of each Dastgah).
...
These intervals, over C natural, have obvious pitch classes to me:
P1: C
Grm2: Db
Acm2: Dp
AcM2: D
Grm3: Eb
Acm3: Ep
AcM3: E
P4: F
P5: G
Grm6: Ab
Acm6: Ap
AcM6: A
Grm7: Bb
Acm7: Bp
AcM7: B
And that gets us most of the way to describing all the dastgahs.
Abu Ata: [C, D, Ep, F, G, A, Bb, C]
Afshari: [C, D, Ep, F, G, Ap, Bb, C]
Bayat-e Tork: [C, D, Ep, F, G, A, Bb, C]
Chahargah: [G, Ap, B, C, Dp, E, F]
Dashti: [C, D, Ep, F, G, A, Bb, C, D, Eb]
Homayun: [Ep, F, G, Ap, B, C, D, Eb]
Mahur: [G, A, B, C, D, E, F]
Nava: [D, Ep, F, G, A, Bb, C]
Rast: [C, D, E, F, G, A, Bb, C, D, Eb]
Segah: [Bp, C, D, Ep, F, G, Ap, Bb]
Shur: [Bp, C, D, Ep, F, G, A, Bb]
The one hiccup is Bayat-e Isfahan. It has an Acm3 over D natural, which is equivalent to AcAc4 over C, which is justly tuned to 2187/1600. After thinking about it, I'm pretty sure this is "F#p", since
1) The koron accidental {p} functions to lower Pythagorean ratios by the 5-limit Augmented unison (justly tuned to 25/24), and
2) The Pythagorean augmented fourth, AcAcA4 justly tuned to 729/512, minus A1, is AcAc4.
3) Thus, if we're taking natural interval names from Pythagorean intervals, then a Pythagorean A4 over C gives F#, which we then lower by {p} to get AcAc4 over C with pitch class F#p.
And that gives us all the pitch classes for our final Dastgah:
Bayat-e Isfahan: [D, Ep, F#p, G, A, Bb, C]
Those pitch classes were derived from Bargrizan's [N, n, M, m, P] intervals. When we compare to the pitches that Bargrizan gives separately, the only difference is that pesky altered F# in Isfahan, and I trust my analysis of that more than I trust a small blurry accidental in a textbook. We did it!
: The Anooshfar Modes
I found another person who independently settled on Acm and GrM qualities for Persian neutral intervals! Kind of. There's a file floating around the internet with a Persian scale attributed to Dariush Anooshfar, supposedly based on a post of his to "The Tuning List" mailing list. The scale of frequency ratios looks like this:
[1/1, 256/243, 27/25, 9/8, 32/27, 243/200, 81/64, 4/3, 25/18, 36/25, 3/2, 128/81, 81/50, 27/16, 16/9, 729/400, 243/128, 2/1]
These are the just tunings for this intervallic scale:
[P1, Grm2, Acm2, AcM2, Grm3, Acm3, AcM3, P4, A4, d5, P5, Grm6, Acm6, AcM6, Grm7, Acm7, AcM7, P8]
Anooshfar's scale has Pythagorean chromatic intervals, 5-limit/rank-3 neutral imperfect intervals, and a 5-limit A4 and d5. I'm kind of amazed that someone else found the same system of dastgah tuning that I did. I wish I could find more from Anooshfar.
Within this scale, Anooshfar gives Persian modes just by specifying scale degrees, like "skip 2 scale degrees, skip 1 scale degree, ...", but it's easy enough to translate that. Here are the rank-3 intervallic interpretations and 5-limit tunings of Persian modes according to Dariush Anooshfar:
Dastgah-e Mahur, Rast Panjgah: [P1, AcM2, AcM3, P4, P5, AcM6, AcM7, P8] # 1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2/1
Dastgah-e Shur: [P1, Acm2, Grm3, P4, P5, Grm6, Grm7, P8] # 1/1, 27/25, 32/27, 4/3, 3/2, 128/81, 16/9, 2/1
Naghmeh Abuata, Naghmeh Afshari: [P1, Acm2, Acm3, A4, d5, Acm6, Acm7, P8] # 1/1, 27/25, 243/200, 25/18, 36/25, 81/50, 729/400, 2/1
Naghmeh Bayat-e Tork, Naghmeh Dashti: [P1, AcM2, AcM3, P4, P5, AcM6, Acm7, P8] # 1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 729/400, 2/1
Dastgah-e Homayun: [P1, Acm2, AcM3, P4, P5, Grm6, Grm7, P8] # 1/1, 27/25, 81/64, 4/3, 3/2, 128/81, 16/9, 2/1
Naghmeh Esfahan: [P1, AcM2, Grm3, P4, P5, Acm6, AcM7, P8] # 1/1, 9/8, 32/27, 4/3, 3/2, 81/50, 243/128, 2/1
Dastgah-e Sehgah: [P1, Acm2, Acm3, P4, d5, Acm6, Acm7, P8] # 1/1, 27/25, 243/200, 4/3, 36/25, 81/50, 729/400, 2/1
Dastgah-e Mokhalif, Bayat-e Esfahan: [P1, AcM2, Grm3, P4, P5, Acm6, Acm7, P8] # 1/1, 9/8, 32/27, 4/3, 3/2, 81/50, 729/400, 2/1
Dastgah-e Chahargah: [P1, Acm2, AcM3, P4, P5, Acm6, AcM7, P8] # 1/1, 27/25, 81/64, 4/3, 3/2, 81/50, 243/128, 2/1
Dastgah-e Nava: [P1, AcM2, Grm3, P4, P5, Acm6, Grm7, P8] # 1/1, 9/8, 32/27, 4/3, 3/2, 81/50, 16/9, 2/1
These are cool and a little confusing. It's confusing that he uses the term "Naghmeh" which I hadn't heard of, but at least one website confirms that it is a concept in Persian music and tries to explain it. Other websites that mention it but don't explain the concept generally spell it "nagmeh". The source I read is not written in very clear English, but it seems that a naghmeh is a slight variation on a dastgah, possibly specifying different {Finalis, Ist, Moteghayyer, or Shāhed} tones. The site claims that a gusheh is more specific object than a naghmeh which is a more specific object than a dastgah, but then the author goes on to list gushehs as example naghmeh, so I don't know what to think. Ah, "naḡma" is the spelling used on Encyclopedia Iranica.
Another confusing this about Anooshfar's scales is that most of them don't quite match scales from other people, but I should be used to that by now. No one seems to agree on anything relating to dastgah.
Let's compare Anooshfar's scales with our 5-limit intervallic interpretations of Bargrizan's scales.
Anooshfar's Shur: [P1, Acm2, Grm3, P4, P5, Grm6, Grm7, P8] :: [1/1, 27/25, 32/27, 4/3, 3/2, 128/81, 16/9, 2/1]
Bargrizan's Shur: [P1, GrM2, GrM3, P4, GrGr5, GrM6, GrM7, GrGr8] :: [1, 800/729, 100/81, 4/3, 3200/2187, 400/243, 50/27, 12800/6561]
These only agree exactly on P1 and P4. They also agree to use a neutral second, but not the same one. What am I supposed to do with this? Probably ignore Anooshfar since anything scale you find .scl file is likely going to be garbage from some random internet person, but I really like that Anooshfar figured out the same 5-limit tuning as me. That has to count for something, right?
Anooshfar gives one scale for Mahur and Rast Panjgah, and that scale is a normal Pythagorean major scale. Bargrizan gives two different scales, with Mahur not reaching the octave, and Rast agreeing with Mahur as far as it goes, but then going past the octave for two more tones. Also Bargrizan uses Grm7 for both Mahur and Rast instead of the AcM7 in a Pythagorean major scale. And what do we call a major scale with a minor seventh? The grade school name is the "mixolydian mode", which is a cyclic permutation of a major scale, so maybe Anooshfar is giving scales rooted on their Finalis and Bargrizan isn't. I don't know.
Anooshfar's scale in root position for Abu Ata & Afshari doesn't match either of Bargrizan's scale for Abu Ata or for Afshari.
Anooshfar's Abuata & Afshari: [P1, Acm2, Acm3, A4, d5, Acm6, Acm7, P8] :: [1/1, 27/25, 243/200, 25/18, 36/25, 81/50, 729/400, 2/1]
Bargrizan's Abu Ata: [P1, AcM2, Acm3, P4, P5, AcM6, Grm7, P8] :: [1, 9/8, 243/200, 4/3, 3/2, 27/16, 16/9, 2]
Bargrizan's Afshari: [P1, AcM2, Acm3, P4, P5, Acm6, Grm7, P8] :: [1, 9/8, 243/200, 4/3, 3/2, 81/50, 16/9, 2]
I tested, and no cyclic permutation of Anooshfar's Abu Ata/Afshari matches either Bargrizan's Abu Ata or Afshari scale exactly. If you allow Acm intervals to substitue for GrM and vice versa, then we had a close call, but still no.
...
After looking at these scales a lot, I think I'm ready to commit to an identification of the tetrachords in terms of [m, n, N, M, P] pseudo-intervals:
[M, M, m] :: Dang Mahoor
[M, m, M] :: Dang Dashti
[n, N, M] :: Dang Shur
[n, P, m] :: Dang Chahargah
With these in hand, I'm going to use the pitch classes from Wikipedia and these pseudo-intervallic dangs to analyze almost all of the dastgahs.
For each dastgah, if it has a moteghayyer, a changeable note, and analyze both options separetely, and I'll call the version with the lower option of moteghayyer "version one", and the version with the higher option of moteghayyer will be "version two". If you have choices (Ab, Ap), then A flat is lower than A half-flat, so version one has Ab, and version two has Ap.
If we write the dastgahs so that the Finalis is at the starting and ending position, then we have these tetrachordal interpretations:
First the easy ones:
Mahur:
[M, M, m] + M + [M, M, m] __ Mahoor + Tone + Mahoor
Rast-Panjgah:
[M, M, m] + M + [M, M, m] __ Mahoor + Tone + Mahoor
Shur1:
[n, N, M] + [n, N, M] + M __ Shur + Shur + Tone
Shur2:
[n, N, M] + [M, m, M] + M __ Shur + Dashti + Tone
Nava:
[M, m, M] + M + [n, N, M] __ Dashti + Tone + Shur
Chahargah:
[n, P, m] + M + [n, P, m] __ Chahargah + Tone + Chahargah
Homayun:
[n, P, m] + [M, m, M] + M __ Chahargah + Dashti + Tone
Afshari1:
[M, m, M] + M + [n, N, M] __ Dashti + Tone + Shur
Afshari2:
[M, m, M] + M + [M, m, M] __ Dashti + Tone + Dashti
Bayat-e-Kord:
[n, N, M] + [M, m, M] + M __ Shur + Dashti + Tone
Next we have some scales where the finalis comes from the middle of a tetrachord. In my analysis:
Dashti1:
[..., m, M] + M + [n, N, M] + [M, ...] __ Dashti end + Tone + Shur + Dashti start
Dashti2:
[..., m, M] + M + [M, m, M] + [M, ...] __ Dashti end + Tone + Dashti start
Abu-Ata1:
[... m, M] + M + [n, N, M], [M, ...] __ Dashti end + Tone + Shur + Dashti start
Abu-Ata1 is the same as Dashti1, which is a little suspicious, but not crazy, and we can look at that more in a bit. Abu-Ata2 I haven't shown and I don't know how to analyze it in terms of our four tetrachords:
Abu-Ata2: [m, M, M, n, M, N, M]
Can't do it. Ridiculous dastgah. Fake. But then we continue on with some more scales that make sense, although still with the Finalis in the middle of a tetrachord:
Bayat-e-Tork:
[... M], [M, m, M] + M + [n, N, ...] __ Shur end + Dashti + Tone + Shur start
Bayat-e-Esfahan:
[M, m, M] + M + [n, P, m] __ Dashti + Tone + Chahargah
Segah2:
[..., N, M] [n, N, M] + M + [n, ... __ Shur end + Shur + Tone + Shur start
But then Segah1 doesn't make any sense in terms of tetrachords:
Segah1:
[N, M, n, N, M, n, M]
I'm using these pseudo-intervals because I think they will help us to explore differing intonations. For example, here's a 13-limit intonation of the pseudo-intervals:
m -> 256/243
n -> 13/12
N -> 128/117
M -> 9/8
P -> 243/208
I've adhered to these constraints in devising the intonation:
{m} and {M} are fixed to their Pythagorean tunings of Grm2 and AcM2.
{n} is about 60 cents less than M, and hopefully both {n} and {M - n} have fairly simple tunings. It's nice if {M - n} is tuned 3 steps of 60-EDO, but that's much less important. To have {n} less than {N} in the next step, it will turn out that {n} has to be tuned smaller than 1200 * log_2(32/27) / 2 = 147 cents.
N is found as {Grm3 - n}.
P is found as {N + AcAcA1}
And once again, for easy reference, AcAcA1 is justly tuned to
A1 + 2 * Ac1 = AcAcA1
(25/24) * (81/80)^2 = 2187/2048
If you do those things, you get a tuning of the dastgah system that has "neutral" which are more like 60 cents flat. Is this particular 13-limit intonation more accurate than the 5-limit version? I don't know. But it's neat and it has a Zalzalian neutral second and I like it.
Here's an 11-limit intonation based on Zalzal's second largest neutral second:
n -> 88/81
N -> 12/11
P -> 6561/5632
It's a fairly neutral intonation. If we use
Chahargah = [n, P, m, M, n, P, m]
then this intonation corresponds to
[1, 88/81, 81/64, 4/3, 3/2, 44/27, 243/128, 2]
which is the just tuning of
[P1, AsGrm2, AcM3, P4, P5, AsGrm6, AcM7, P8]
Let's look at another! Here's an intonation based on the other small Zalzalian neutral second:
m -> 256/243
n -> 14/13
N -> 208/189
M -> 9/8
P -> 1053/896
The frequency ratios of the pseudo-intervals here aren't so small, but the ratios on the scale degrees still look fine. For example,
Chahargah = [n, P, m, M, n, P, m]
is tuned as
[1/1, 14/13, 81/64, 4/3, 3/2, 21/13, 243/128, 2/1]
The interval names are a little complicated:
[P1, ReSbAcM2, AcM3, P4, P5, ReSbAcM6, AcM7, P8]
but you're simply going to have somewhat long names if you need to specify factors of 13 and 7 on top of a Pythagorean spine (and if your interval system's naming scheme prioritizes 5-limit ratios over 3-limit ratios, which I stand by in general, even if it works out a little less well for Persian music).
Let's look at a bunch of intonations very quickly.
(n, N, P) -> (14/13, 208/189, 1053/896) : {n} is smallest neutral second of Zalzal. 13-limit.
(n, N, P) -> (13/12, 128/117, 243/208) : {n} is second smallest neutral second of Zalzal. 13-limit.
(n, N, P) -> (27/25, 800/729, 75/64) : {n} is the mediant of (14/13) and (13/12). 5-limit.
(n, N, P) -> (88/81, 12/11, 6561/5632) : {N} is the second largest neutral second of Zalzal. 11-limit.
(n, N, P) -> (320/297, 11/10, 24057/20480) : {N} is the largest neutral second of Zalzal. 11-limit.
{n, N, P) -> (224/207, 23/21, 16767/14336) : {N} is the mediant of (12/11) and (11/10). 23-limit.
Here's a 7-limit intonation:
(n, N, P) -> (1024/945, 35/32, 76545/65536) : {N} is the mediant of (12/11) with (128/117) or the mediant of (12/11) with (23/21).
It's no so pretty in the ratios besides {N}, but I was almost as excited to find such a low prime-limit large neutral second by mediant operations as I was when I found the 5-limit intonation. Here's a 17-limit intonation:
(n, N, P) -> (68/63, 56/51, 5103/4352)
For this one, both {n} and {N} can be generated multiple ways by mediants from previously seen {n} and {N} intonations. Pretty cool.