The Persian/Iranian microtonal musical modes are called dastgahs. I was writing about them at length on microtonaltheory.com, but it got to be embarrassing how many sources I found that disagreed with each other about what the scales look like. I'm going to try figuring it all out and cleaning it all up here and then I'll repost an abbreviated form there.
There are only a few dastgahs, so this shouldn't be hard to codify. They are called:
Māhūr
Homāyūn
Chahārgāh
Segāh
Navā
Rāst-panjgāh
Bayāt-i Iṣfahān
Dashtī
Abū ‛aṭā’
Bayāt-i Tork
Afshārī
Bayāt-i Kord
They also have a concept of a scale derived from a dastgah, an "avaz", but I don't think they're really treated any different musically, so I'll just call them all dastgahs.
Supposedly a man named Ali-Naqi Vazir gave 24-EDO descriptions of the scales way back in 1913. I haven't found that. But I have found 24-EDO tetrachords / scale fragments attributed to him. There are only four tetrachords. Each tetrachord is called a "dang" and the dang-s are named after dastgahs. Below I show the name of a dang, the size of the relative intervals in cents, the size in steps of 24-EDO, and pitch classes rooted on C, using "d" as an accidental for half flats (and "t" as an accidental for half sharps, if that had come up).
Chahargah: [150, 250, 100]: [3, 5, 2] :: [C, Dd, E, F]
Dashti: [200, 100, 200] : [4, 2, 4] :: [C, D, Eb, F]
Mahur: [200, 200, 100] : [4, 4, 2] :: [C, D, E, F]
Here are descriptions of some scales in 24-EDO steps, described with absolute steps and then relative steps on each line:
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]
Chahargah : [0, 3, 8, 10, 14, 17, 22, 24] : [3, 5, 2, 4, 3, 5, 2]
These are derived from "Transcultural Music" by Alireza Ostovar. Using Vaziri's tetrachords, we can see how the dang-s combine into these dastgahs:
Shur/Nava: Shur + Dashti + T.
Homayun: Chahargah + Dashti + T.
Mahur/Rastpanjgah: Mahur + T + Mahur.
Segah: Shur + Shur + T.
Chahargah: Chahargah + T + Chahargah.
Ostovara didn't give a 24-EDO analysis of dastgah Esfahan, but from other sources, it's clearly a cyclic permutation of dastgah Homayun, and we can give it as
Esfahan : Dashti + T + Chahargah.
or as
Esfahan : [0, 4, 6, 10, 14, 17, 22, 24] :: [4, 2, 4, 4, 3, 5, 2]
Which dastgahs are left to characterize in 24-EDO? Dashti, Abu 'Aṭa, Bayat-i Tork, Afshari, and Bayat-i Kord. We're told by Kees van den Doel of persianney.com that all of these have the same intervals as Shur, except that he doesn't mention Bayat-i Kord. So maybe we only have one left. But Darabi, Azimi, and Nojumi say that Bayat-i Kord shares its intervals with Shur as well, so everything sure is Shur.
Are we done? Not at all. Every source disagrees with every other source and I won't feel satisfied till I have some kind of framework for understanding what they're all smoking. So this is going to be our baseline and we'll try to align the works of others with it as much as we can, to figure out when and how they're deviating. Kees gives us precise intonation for the persian accidentals, which I shall render as:
p = koron (60 cent flat)
> = sori (40 cent sharp)
I like that they're not equal in magnitude but that they do sum to a 12-EDO A1 or m2.
Mahur is just the C major scale without any microtones,
One deviation from the above that I suspect is true-to-practice is that Dastgahs normally just fill one octave, and below or above that octave you might have weird different notes as ornaments that you wouldn't expect from octave-repetition of the scale. So a dastgah might have both a B natural within its normal octave and a Bb outside the normal octave. Or something like that.
I believe there are also traditional variations within the octave, like if you'd play a B quarter flat in an ascending melodic line, there might be some good chance that you'd play it as a a B half-flat in a descending melodic line, for example. There are standardized deviations from the base scale.
We'll get to both kinds of ornaments in time, but first I just want to look at cases where sources seem to deviate totally on what the base scales are. We'll also look at sources on precise intonation of these scales, even though if you measure the frequency ratios of Persian musicians, there isn't very precise agreement.
...
I think the source that I found most regular and simple after the 24-EDo stuff was Kees van den Doel at persianney.com. Let's compare his stuff to the 24-EDO scales of Ostovar. Here's Mahur in terms of absolute and relative intervals:
Mahur: [P1, M2, M3, P4, P5, M6, M7, P8] :: [M2, M2, m2] + M2 + [M2, M2, m2] // Also Rast-panjgah.
which matches the 24-EDO scale of Ostovar perfectly.
Mahur/Rastpanjgah : [0, 4, 8, 10, 14, 18, 22, 24] :: [4, 4, 2] + 4 + [4, 4, 2]
I'm not sure whether Persian scales are associated with definite tonics, as they are in Arabic music, such that a transposed scale would have a new name. Kees suggests that this isn't the case, and gives the dastgahs in multiple keys, which was great because the multiple keys mostly agreed with each other and this was good confirmation of what he thought the scales really were, without typos.
Here's a 2.3.7 justintervallic analysis of Kees's pitch classes for Homayoun:
Homayoun: [P1, SbM2, M3, P4, P5, m6, m7, P8] :: [SbM2, SpM2, m2, M2, m2, M2, M2]
this is also consistent with the 24-EDO version from Ostovar,
Homayun : [0, 3, 8, 10, 14, 16, 20, 24] :: [3, 5, 2] + 4 + [2, 4, 4]
If you can't see it, just look at the relative intervals after the double colon, and think of m2 as 2 steps of 24 edo, M2 as 4 steps, a sub-major second or super-minor second as 3 steps, and a super-major second or sub-minor third as 5 steps. The "Sub" flattens by a step and the "Super" raises by a step. The 5-step interval is, I think, fairly characteristically Persian. Not something you see in Arabic or Turkish intonation. I think. Still figuring this out.
Here's Esfahan from Kees:
Esfahan: [P1, M2, m3, P4, P5, SbM6, M7, P8] :: [M2, m2, M2] + M2 + [SbM2, SpM2, m2]
which again has perfect agreement with the 24-EDO version from Ostovar:
Esfahan : [0, 4, 6, 10, 14, 17, 22, 24] :: [4, 2, 4, 4, 3, 5, 2]
This is going swimmingly. Here's Chahargah from Kees:
Chahargah: [P1, SbM2, M3, P4, P5, SbM6, M7, P8] : [SbM2, SpM2, m2, M2, SbM2, SpM2, m2]
Which perfectly matches the 24-EDO version from Ostovar:
Chahargah : [0, 3, 8, 10, 14, 17, 22, 24] : [3, 5, 2] + 4 + [3, 5, 2]
I knew I liked this guy for a reason.
But! Kees's dastgah Segah looks like this, in a 2.3.7 just interval analysis:
Segah: [P1, Spm2, Spm3, P4, Spd5, Spm6, Spm7, P8] : [Spm2, M2, SbM2, Spm2, M2, M2, SbM2]
This at first looks starkly different from the 24-EDO version of Ostovar. If we tune Kees's Segah in 24-EDO, we get these for absolute and relative steps:
Segah : [0, 3, 7, 10, 13, 17, 21, 24] :: [3, 4, 3, 3, 4, 4, 3]
In comparison, here's the version in 24-EDO from Ostovar:
Segah : [0, 3, 6, 10, 13, 16, 20, 24] :: [3, 3, 4, 3, 3, 4, 4]
We can now see they're cyclic permutations of each other! Look at the relative intervals in Kees' version of Segah, and shift the last 3 to the start. Now they're equal. I'm tempted to prefer Ostovar's version, since it can be constructed from Vaziri's tetrachordal dang-s. Although maybe there's a way that both can be right. Maybe the tonic center of the Shur dang isn't at the bottom of the tetrachord. Then Kees could be right about where the tonic of the scale is and Ostovar could be right in presenting the scales such that the underlying tetrachord structure is clear and contiguous.
Kees's dastgah Shur can be rendered in 2.3.7 just intervals as:
Shur: [P1, SbM2, m3, P4, P5, m6, m7, P8] :: [SbM2, Spm2, M2] + [M2, m2, M2] + M2 // [Shur + Dashti + T]
or
Shur: [P1, SbM2, m3, P4, Sb5, m6, m7, P8] :: [SbM2, Spm2, M2] + [SbM2, Spm2, M2] + M2 // [Shur + Shur + T]
According to Kees, dastgah Shur has an optional half flat on the 5th scale degree. How peculiar not not hit P5! To be clear, we're describing the Shur dastgah (scale) in terms of components dang-s (tetrachords), which include a dang called Shur.
Here's the dastgah Shur of Ostovar for comparison:
Shur/Nava : [0, 3, 6, 10, 14, 16, 20, 24] :: [3, 3, 4] + [4, 2, 4] + [4]
Which is the [Shur + Dashti + T] form, not the [Shur + Shur + T] form. This second form is actually Ostovar's Segah. So Shur with a half-flat 5th is a lot like Segah.
Remember how almost everything sure looks like Shur? It's a little confusing that Kees has two forms for dastgah Shur, because that also introduces uncertainty about (Nava, Dashti, Abu 'Aṭa, Bayat-i Tork, Afshari, and Bayat-i Kord), which supposedly have the same pitch classes as dastgah Shur. Is a septimal sub perfect fifth an option for all of them? I dunno. If it were, then most of the Persian scales, in addition to basically being Shur, would also be basically Segah. I wonder why they have so many names if they're all the same. They're probably not all the same.
Looking at lots of other sources that all have their own characteristic notations, ornaments, and perhaps typos, if I had to reconstruct the correct spelling of Segah from the ?misspellings, I'd write it as
[Ed, F, G, Ad, Bb, C, D, Ed]
This form was given exactly by Ella Zonis, and was given like this but with quarter-flats instead of half-flats by Navid Goldrick, and wikipedia relates this same form but with an option of a half flat on the D (the 7th scale degree) and attributes this to Mirza Abdollah.
Some other sources are quite different. I kind of glossed over that Kees Segah is ...I presented what I thought was a cleaned up version of his data, which wasn't self consistent.
He gives
First position: [Ed F G Ad Bb(d) C D Eb]
Second position: [Bd C D Ed Fb(t) G A Bd]
but these aren't equivalent through transposition. Like the first position doesn't reach an octave and the optional accidental on the fifth scale degree changes in a way that it shouldn't. But if we ignore Kees's ornametn on the 5th scale degree and Mirza's ornament on the seventh scale degree, and replace Navid's quarter-flats with half-flats, then it seems everyone things Segah is something like:
[Ed, F, G, Ad, Bb, C, D, Ed]
But this is the version of Segah I originally presented with Kees, not the cyclic permuted version that agrees with Ostovar! So it seems like multiple sources are disagreeing with Ostovar. Or maybe that the Shur dang doesn't have its tonal center at its bottom, but then that should also change the form of other dastgahs so that other sources disagree with Ostovar.
I think Navid's notation of a quarter flat E for the root of Segah is quite interesting. Using the intonation of Kees, the koron accidental is 60 cents flat (relative to ... Pythagorean, 12-EDO, meantone, whatever - no one specified), not 25 cents. And 35 cents difference between musicians is kind of a lot.
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