Makams and Maqamat

:: The Motivation: You know who really loves microtonal music?

The most widely practiced and loved microtonal music scene in the modern era is in the Middle East, Maghreb, and Mediterranean. Lots of musical traditions in these areas are based on basically the same microtonal modal scales. In Arabic-speaking nations, such a scale is called a maqam. In Turkey, it's makam. The Turkish Uyghurs (mostly in China) call it Muqam. In Azerbaijan, it's mugham. In Uzbekistan and Tajikistan, it's shashmaqam. In Greek Rebetiko music, which was influenced by the music of the Ottoman empire, such a scale is called adromoi. In Morocco, Algeria, Tunisia, and Libya, each such scale is called a tab' in the musical tradition Nubah. And all of these are actually derived from the ancient Persian modal system, where the scale is called a dastgah.

The Turkish and the Arabic musical traditions are the best documented online, and the Turkish tradition has more precise and logical interval analysis. I'll focus on those. For the Arabic scales I'll say "a maqam" or multiple "maqamat". The plural of the Arabic maqam is "maqamat". For Turkish scales, I'll just say makam and makams. The actual plural in Tukish is "makamlar", but Turkish people writing in English say "makams", so I will too.

Anyway, in a blog about microtonal music theory, I'm definitely going to talk makams/maqamat - the most widely practiced and loved microtonal music tradition in the modern era. I'm going to start by giving lots of raw data in a machine readable text format, rather than the usual images of scales with weird accidentals. I'll describe the scales as I've learned about them from  multiple sources, basically in the order that I found them: Wikipedia, The Turkish Sources (including Esendere Kültür Sanat Derneği and the Website of Dr. Hazim Gökçen and the publications of Dr. Ozan Yarman and some other things I've forgetten), then MaqamWorld, and Alsiadi, and the Kitāb al-Adwār by Safi al-Din al-Urmawi (as related in English by Owen Wright), and maybe Oud For Guitarists on youtube and makamlar.net, though I haven't transcribed data from them yet. Once I've shown all the scales, I'll do a lot of analysis to point out inconsistencies of description within a musical tradition, and to point out which scales are related across musical traditions (and how they differ), all in terms of intervals - with at least rank-2 and rank-3 analysis of everything, maybe higher ranks and prime limits if I'm feeling sassy.

It's a little frustrating to try to characterize makams and maqamat, since they vary by region and historic period and musician. It's a little like trying to construct the platonic ideal recipe for pie - not the best pie, but the one fundamental pie as it truly is, expressing itself in the recipes of all lands. You can't really do it, but you can gather lots of pie recipies and talk a little bit about the differences. That's what I'm here to do.

:: The Data

: Arabic Maqamat And 24-EDO (As Related By Wikipedia)

Arabic maqams half a few "half-flat notes", like "G half flat". A G hafl flat sounds somewhere between a G and a G flat. For notation, I'll use "d" to indicate a half-flat accidental, and "t" to indicate the rarer half-sharp accidental. Some other decent options in Unicode are ᵈ, as in Gᵈ, and ѣ as in Gѣ. It's easy to mix up "d"s and "b"s in reading, but those options are the easiest for me to type and render and such. If you think of maqamat as tuned-scales within 24-EDO, then a half-flat corresponds to one step of 24-EDO down in frequency. However, this blog takes seriously that we can represent music in terms of intervals, agnostic to tuning, so I say that that 24-EDO interpretation is right out. We'll talk more about it later, but one way to interpret Arabic scales that are nominally tuned to 24-EDO is to interpret the half flat accidental as a lowering things by a septimal super unison, Sp1, which is justly tuned to 36/35, and which 24-EDO tunes to one step, i.e. a frequency ratio of 2^(1/24). Remarkably, raising a Gb by Sp1 or lowering G by Sp1 produce two pitches which are tuned by 24-EDO to the same frequency ratio. There's also a weird temperament I found that starts with the just-intonation-subset with factors of (2, 3, 11, and 17) that I'll introduce at some point for analyzing 24-EDO scales intervallically.

Almost all makams and maqamat can be characterized using just the normal western pitches plus Ed and Bd. In principle, we can also stack the quartertone accidentals with the half tone accidentals. For example, in 24-EDO, a C half flat, Ct, is equivalent to D three-quarters flat, Dbd. 

Now that we have provisionally established some notation and hinted at an interpretation, here's our first bit of data. These are my transcriptions of some Arabic maqam scales based on staff notation on Wikipedia.

Hijaz (Nahawand ending) [0, 2, 8, 10, 14, 16, 20, 24] [D, Eb, F#, G, A, Bb, C, D] # Tonal. 

Nawa Athar [0, 4, 6, 12, 14, 16, 22, 24] [C, D, Eb, F#, G, Ab, B, C] # Tonal.

Shad 'Araban [0, 2, 8, 10, 14, 16, 22, 24] [G, Ab, B, C, D, Eb, F#, G] # Tonal.

Bayati [0, 3, 6, 10, 14, 16, 20, 24] [D, Ed, F, G, A, Bb, C, D] # Has Ed.

Jiharkah [0, 4, 8, 10, 14, 18, 21, 24] [F, G, A, Bb, C, D, Ed, F] # Has Ed.

Huzam [0, 3, 7, 9, 15, 17, 21, 24] [Ed, F, G, Ab, B, C, D, Ed]. # Has Ed.

Rahat al Arwah [0, 3, 7, 9, 15, 17, 21, 24] [Bd, C, D, Eb, F#, G, A, Bd] # Has Bd.

Saba [0, 3, 6, 8, 14, 16, 20, 24] [D, Ed, F, Gb, A, Bb, C, D] # Has Ed.

Rast [0, 4, 7, 10, 14, 18, 21, 24] [C, D, Ed, F, G, A, Bd, C] # Has Ed and Bd.

Husayni 'Ushayran [0, 3, 6, 10, 13, 16, 20, 24] [A, Bd, C, D, Ed, F, G, A] # Has Bd and Ed.

On each line, I show the name of the maqam, the number of steps of 24-EDO of each scale degree relative to the tonic in the nominal tuning, the pitch classes of the maqam using "d"s as a half-flat accidental, and finally a summary comment on the presence or absence of any notes with half flat accidentals. Wikipedia also tells us that the maqamat 'Ajam, Nahawand, and Kurd are tonal, and correspond to western Major, Minor, and Phrygian modes. Wikipedia does not tell us the tonic pitches though. Maybe we'll figure them out from another data source.

So those are the scales in terms of pitch classes and tuned steps of 24-EDO. But what are the scales in terms of intervals? There's a pretty easy association between pitch classes and intervals over C natural. If there's a "d" on the pitch class, then the associated interval over C is a Sub-(whatever the interval over C would have been without the "d" accidental). And if there's a "t" on the pitch class, then the associated interval is a Super-(whatever it would have been otherwise). Here are three example pitch classes with their intervals over C:

Dbt <> Spm2 over C

C#t <> SpA1 over C

Dd <> SbM2 over C

24-EDO tunes all of them to 3 steps.

Here are some simple intervals that 24-EDO tunes to each of its steps:

^0: P1

^1: Sbm2 = Sp1

^2: m2

^3: SbM2 = Spm2

^4: M2

^5: Sbm3 = SpM2

^6: m3

^7: SbM3 = Spm3

^8: M3

^9: Sb4 = SpM3

^10: P4

^11: Sbd5, Sp4

^12: A4, d5

^13: Sb5, SpA4

^14: P5

^15: Sbm6 = Sp5

^16: m6

^17: SbM6 = Spm6

^18: M6

^19: Sbm7 = SpM6

^20: m7

^21: SbM7 = Spm7

^22: M7

^23: Sb8 = SpM7

^24: P8

For every odd step, i.e. for every quarter tone, we have two options for naming. For example, step 1 of 24-EDO could be called a half-sharp unison or a half-flat minor second. From the pitch classes listed above for each maqam, we can see that Arabic maqamat use mostly half-flats and few half-sharps (only half-flats in these ones, but that patterns won't continue to hold absolutely).At first I tried figuring out maqam intervals based on the 24-EDO steps (rather than the pitch classes), and I regret it as a failure that could have been avoided with a forethought. But I think the diagram above still nicely shows off how rank-4 septimal interval fill in the gaps between familiar intervals of 12-EDO.

Curiously, none of the maqamat use 19 steps of 24-EDO. This is weird because the septimal sub-minor seventh, Sbm7, is justly tuned to the reduced seventh harmonic, 7/4, and 24-EDO tunes Sbm7 to 19 steps. I think this is a little bit of an argument against 24-EDO maqam music being secretly septimal under the hood. If maqamat were septimal (with 24-EDO being a convenient shorthand for collapsing the septimal half-sharp/half-flat distinction), then I'd expect that Arabic music would be using that 7th harmonic to good effect. But even if the maqamat didn't come from 7-limit just intonation, still rank-4 intervals are still the obvious simplest way to analyze 24-EDO intervallically. 

If we infer intervals for maqam steps by looking at the pitch classes, then we get some acute and grave intervals, which feels a little silly to me but I don't know a better way. I definitely don't want to just look at the EDO steps for every scale degree and pick one without reference to the pitch classes. Partly this is because it didn't work well for me. Partly, I just don't believe that maqams really exist in 24-EDO frequency space. They exist in interval space and can be tuned to 24-EDO. So we'll use the rank-4 names, I suppose. Here are intervals for each maqam scale degree:

Hijaz (Nahawand ending): P1, Acm2, M3, Ac4, P5, Acm6, m7, P8.

Nawa Athar: P1, M2, m3, A4, P5, m6, M7, P8.

Shad 'Araban: P1, m2, M3, P4, Gr5, m6, GrM7, P8.

Bayati: P1, SbAcM2, m3, Ac4, P5, Acm6, m7, P8.

Jiharkah: P1, AcM2, M3, Ac4, P5, M6, SbM7, P8.

Huzam: P1, Spm2, Spm3, Spd4, Sp5, Spm6, SpGrm7, P8.

Rahat al Arwah: P1, Spm2, SpGrm3, Spd4, SpGr5, Spm6, SpGrm7, P8.

Saba: P1, SbAcM2, m3, Acd4, P5, Acm6, m7, P8.

Rast: P1, M2, SbM3, P4, P5, M6, SbM7, P8.

Husayni 'Ushayran: P1, SbAcM2, m3, P4, Sb5, m6, m7, P8.

and here are the relative intervals between steps:

Hijaz (Nahawand ending): Acm2, A2, Acm2, M2, Acm2, M2, M2.

Nawa Athar: M2, Acm2, A2, Acm2, m2, AcA2, m2.

Shad 'Araban: m2, AcA2, m2, M2, Acm2, A2, Acm2.

Bayati: SbAcM2, Spm2, AcM2, M2, Acm2, M2, M2.

Jiharkah: AcM2, M2, Acm2, M2, M2, SbAcM2, Spm2.

Huzam: Spm2, AcM2, m2, AcA2, m2, M2, SbAcM2.

Rahat al Arwah: Spm2, M2, Acm2, A2, Acm2, M2, SbAcM2.

Saba: SbAcM2, Spm2, Acm2, A2, Acm2, M2, M2.

Rast: M2, SbAcM2, Spm2, AcM2, M2, SbAcM2, Spm2.

Husayni 'Ushayran: SbAcM2, Spm2, M2, SbAcM2, Spm2, AcM2, M2.

If we lop off the "acute" prefixes, then all of the relative intervals come from [m2, Spm2, SbM2, M2, A2].

This intervallic description allows you to freely transpose maqamat to other tonics, however in Arabic music, the scales are not actually transposed, e.g., part of the definition of Rast is that it starts on C. So far as maqam tones can be quantized down to 24-EDO, Arabic music uses a very limited pallet of notes, e.g. the only quarter tones we see above are B half-flat and E half-flat. Here's the full set of pitch classes from the Wikipedia transcriptions:

[C, D, Ed, Eb, F, F#, Gb, G, Ab, A, Bb, Bd, B]

although we'll expand this in time. You might notice that there's no E natural. A very limited pallet!

In practice, 24-EDO started out as more a notational convenience than a tuning system for actual use, and the tuning/intonation of things are instead hugely dependent on the region, the time period, the musician, and the maqam. Like, the D of Maqam Huzam might not be exactly the D in Maqam Rahat al Arwah, for all I know, and middle eastern musicians know - by feel or vibes or muscle memory - the tiny variations frequency between them and can have discussions about which should be flatter or sharper by amounts that are basically imperceptible to untrained ears.

But "it's contextual and sometimes imperceptible to untrained ears" is not useful data that can be transmitted for teaching, so Arabic musicians notate things in 24-EDO as a start, and we'll do our best to learn more about the regional variation as we go.

In addition to this rank-4/septimal analysis, I have an interesting 17-limit interpretation of 24-EDO maqamat that we'll get to eventually. But this is a good start for now.

: Turkish Makams: Arel-Ezgi-Uzdilek, Simge, 53-EDO Analysis And Beyond

Turkish makams have some names in common with Arabic ones; sometimes with the same intervals and tonics and sometimes varied. Turkish tradition also has some scales in common Arabic tradition while giving them different names.

The most widespread analysis of Turkish makams is based on a Pythagorean spiral of fifths. 53-EDO provides an excellent finite representation of Pythagorean tuning and is also used extensively in this tradition. This Pythagorean spiral analysis is mostly due to the 13th century musician Safi al-Din al-Urmawi (sometimes spelled Safiaddin Ormavi) who extended the spiral of fifths from a 12-tone chromatic scale to a 17-tone system. Even in the time of Safi al-Din, it was recognized that this 17-tone Pythagorean scale was too simple of a model to accurately describe the frequency ratios used in practice in middle eastern music. We some have knowledge about what these frequency ratios really were from texts by lutenists describing the geometry of fretting and fingering, and from measurements of old instruments, but we'll start here with the Pythagorean analysis, much as we started with 24-EDO for modern Arabic practice. Pythagorean analyses provide a different, historically important, and finer grained perspective on makam/maqam music than 24-EDO, and I'm taking any data that I can get. Also, we'll use this Pythagorean analyses to learn the names for the modern Turkish makams, at least the simpler ones. We'll also explore rank-3 respellings of the Pythagorean intervals.

: The Spiral

Safi al-Din al-Urmawi wrote a book called Kitab al-Adwār that outlined a system with 17 tones per octave for analyzing Arabic modal music (which we must imagine was closer to its parent system, Persian Dastgāh, at the time). Here are his 17 tones:

Rank 2 interval name = Coordinates in (P5, P8) basis # Just tuning

-

d9 = (-12, 8) # 1048576/531441

d6 = (-11, 7) # 262144/177147

d3 = (-10, 6) # 65536/59049

d7 = (-9, 6) # 32768/19683

d4 = (-8, 5) # 8192/6561

d8 = (-7, 5) # 4096/2187

d5 = (-6, 4) # 1024/729

m2 = (-5, 3) # 256/243

m6 = (-4, 3) # 128/81

m3 = (-3, 2) # 32/27

m7 = (-2, 2) # 16/9

P4 = (-1, 1) # 4/3

P1 = (0, 0) # 1/1

P5 = (1, 0) # 3/2

M2 = (2, -1) # 9/8

M6 = (3, -1) # 27/16

M3 = (4, -2) # 81/64

Here they are in ascending order, sorted by just/Pythagorean tuning, with the octave added in as a closer: 

P1 = (0, 0) # 1/1

m2 = (-5, 3) # 256/243

d3 = (-10, 6) # 65536/59049

M2 = (2, -1) # 9/8

m3 = (-3, 2) # 32/27

d4 = (-8, 5) # 8192/6561

M3 = (4, -2) # 81/64

P4 = (-1, 1) # 4/3

d5 = (-6, 4) # 1024/729

d6 = (-11, 7) # 262144/177147

P5 = (1, 0) # 3/2

m6 = (-4, 3) # 128/81

d7 = (-9, 6) # 32768/19683

M6 = (3, -1) # 27/16

m7 = (-2, 2) # 16/9

d8 = (-7, 5) # 4096/2187

d9 = (-12, 8) # 1048576/531441

P8 = (0, 1) # 2/1

You can see that he extended the chromatic scale farther in the direction of minor and diminished intervals and didn't extend along the direction into augmented intervals. When you sort these by increasing frequency ratio, you can see that successive tones are separated by either the "Pythagorean limma", i.e. the minor second with a tuned value of 256/243, or "Pythagorean comma", i.e. the augmented zeroth with a tuned value of 531441/524288.

You might have noticed that this 17-tone scale has no major seventh, M7. If we perform a cyclic permutation to start on the P4 as our tonic, then we get our usual chromatic scale (including a M7) and the non-chromatic diminished intervals basically stay the same: the only difference is that the d9 becomes the Pythagorean M7.

M7 = (5, -2) # 243/128

If we don't perform the cyclic permutation, then Safi al-Din's scale has more of mixolydian feel.

Intervals and pitches live in 1-to-1 correspondence, so let's find pitches over C that correspond to the 17-tone system. If we do the cyclic permutation, then we have the usual chromatic pitch classes,

(C, Db, D, Eb, E, F, Gb, G, Ab, A, Bb, B)

and the five additional notes of 

(Ebb, Fb, Abb, Bbb, Cb).

The full set is ordered like this if we assume just / Pythgorean tuning for the rank-2 intervals:

[C, Db, Ebb, D, Eb, Fb, E, F, Gb, Abb, G, Ab, Bbb, A, Bb, Cb, B, C]

Now, I think we must say that al-Urmawi's contribution to music theory is an early description of makams/maqamat using Pythagorean tuning, and not the development of this 17-tone system in particular, because this is still just a Pythagorean spiral, taken verbatim from Pythagoras who lived 1700 years before, and cutting the spiral off at 17 tones doesn't really make al-Urmawi an innovator. There's no new math here. But there is data. Although I'm not great with medieval Arabic, so we're going to come back to that data a little later and progress right on to modern Turkish Pythagorean analyses.

: The Commas

In Turkish music theory, there is a claim that a major second (by which they mean the Pythagorean major second, justly tuned to 9/8), is made up of 9 commas - nine small distinguishable intervals, of about 22 or 23 cents. Whether we use the Pythagorean comma or the syntonic comma here, which are similarly sized, this leads us pretty obviously to 53-EDO.

For example, if we want to temper out the difference between nine Pythagorean commas and and one Pythagorean major second, that interval is 

9 * A0 - M2

9 * (-19, 12) - (-3, 2) = (-171, 108) - (-3, 2) = (-168, 106)

in the (P8, P12) prime harmonic basis. This happens to be the Pythagorean AAAAAAAAAAAAAAA-9, with coordinates (-2, -10) in the (A1, d2) basis. The coordinates in either basis have a common factor of two, and EDOs that temper out (-168, 106) will also temper out the related interval with coprime coordinates (-84, 53). This has coordiantes (-1, -5) in the (A1, d2) basis, and variously goes by the name AAAAAAA-4 or "Mercator's comma". The EDOs that temper out the rank-2 AAAAAAA-4 are 53-EDO and its integer multiples: (53, 106, 159, 212, ...)-EDO.

If instead we want to temper out the difference between 9 syntonic commas and the Pythagorean major second, that interval is a GrGrGrGrGrGrGrGrM2, with coordinates (33, -34, 9) in the rank-3 prime harmonic basis, (P8, P12, M17), or coordinates (-8, 2, 1) in the rank-3 Lilley-Johnston basis, (Ac1, A1, d2).  It's justly tuned to 16777216000000000/16677181699666569, and the is also tempered out by 53-EDO and its integer multiples, plus a few extras: (53, 106, 157, 159, 210, 212, ...)-EDO. 

You might know that 53-EDO can be defined over rank-3 intervals by its having pure octaves  and tempering out the "schisma", or AcAcA0, justly tuned to 32805/32768 (definable as the difference between the Pythagorean and Syntonic commas) and the "kleisma", or Acdd0, justly tuned to 15625/15552 (which also exists). The difference between 9 syntonic commas and the Pythagorean major second can therefore be constructed from the schisma and kleisma. It happens to be the difference between two kleismas and three schismas. We can verify this with the just tunings:

AcM2 - 9 * Ac1 = 2 * Acdd0 - 3 * AcAcA0

(9/8) / (81/80)^9 = (15625/15552)^2 / (32805/32768)^3

Whichever we use Pythagorean or Syntonic, it's pretty obvious that if you're good at recognizing a 22 or 23 cent comma, and you want to simplify your life by tuning intervals so that there are exactly 9 commas in a Pythagorean major second, then you should be using 53-EDO as a tuning system. And that's what most gradeschool Turkish music theory does. Even better, 53-EDO is basically Pythagorean, in the sense of having a very pure perfect fifth: the tuned P5 of 53-EDO, at 2^(31/53),  is flat of the pure value by less than a tenth of a cent. 

Another useful feature of this tuning system: since 53-EDO also tempers out the schisma (and tunes the syntonic comma and the Pythagorean comma to one step of 53-EDO), then any time we flatten a tone by a step, we have some choice of interpretation as to which of the commas we're using under the hood in interval space.

: AEU and The Simgeler

The system of accidentals used in most Turkish sheet music is called Arel-Ezgi-Uzdilek notation. It has like 10 accidentals and I hate it. I don't even particularly like reading western sheet music with just sharps and flats, so AEU really does not fit in my head. Fortunately, in educational diagrams describing makams, if not in sheet music, there's a second system for notating numbers of steps of 53-EDO. The system uses letters. One letter is called a sign or "simge". Multiple letters are "simgeler". These are my main source of knowledge about Turkish makams.

I can never remember it when looking at staff notation, but the main things you need to read AEU are that :

backwards b is flat a comma: "Ad"

b with a slash is flat four commas: "A\b"

b is flat 5 commas: "Ab"

So the backwards flat, "d", isn't at all like a half-flat, as it is in Arabic music.

In addition to not liking AEU in practice, I'm a little opposed to AEU notation even in theory. Most Turkish makams restrict themselves to steps [0, 4, 5, 8, 9, 13, 14, 17, 18, 22, 23, 26, 27, 30, 31, 35, 36, 39, 40, 44, 45, 48, 49, 53] of 53-EDO, and these are all interpretable as natural intervals or once modified intervals (i.e. once diminished or once augmented). We already have a great system for notating those. It's sharps and flats, as used in Pythagorean tuning and quarter comma meantone and western grade school music education. Now, there are definitely many worse notation systems than AEU used for middle eastern music - I'm not saying AEU is without merit. But I propose throwing them all out.

The simgeler are ok though. They show relative intervals between makam steps. Here's a guide:

F: 1 comma

E: 3 commas

B: 4 commas = m2

S: 5 commas = A1

K: 8 commas = d3

T: 9 commas = M2

A: 12 or 13 commas

That's how they're usually presented. Simge F and Simge E aren't really used, but we could call them A0 and dd3. I don't why A is ambiguous in its size. But 12 and 13 steps of 53-EDO could be tuned versions of dd4 and m3. More often it's the 12-step dd4 that's used, e.g. here's a common tetrachord spanning a perfect 4th:

P4 = [S, A_12, S] = 5 + 12 + 5 steps = 22 steps

The A1 and d3 are something like neutral seconds - one of them is about 23 cents sharp of m2 and one is about 23 cents flat of M2 - whereas the 24-EDO neutral second is right smack in the middle. If you've ever heard the claim that Turkish music has "eighth tones" in contrast to quarter tones, this is why. 

1200 * log_2((9/8)^(1/8)) = 25.5 cents ~ 1 comma

It's just Pythagorean commas all over the place. That's the great advanced secret of Turkish microtonalism. The Pythagorean A1 is about the flattest you can go while still calling something a neutral second, and the d3 is about as sharp as you can go while still calling something a neutral second. Most neutral seconds played across the middle east today, and throughout history so far as we can tell,  fall somewhere in the middle of these, which is part of why al-Urmawi's system was insufficient from the start. Still, it's data, and I'm going to use any data that I can in characterizing this long-lived, widespread, beloved microtonal tradition that is middle eastern modal music. I think maybe we should call the Turkish A1 and d3 "middle seconds" rather than "neutral seconds", because they're somwehre in the middle, but not very close to neutral 24-EDO value.

Honestly, I suspect that some Arabic musicians must be influenced to play their music more like 24-EDO than the practiced and personally transmitted tradition would dictate, just as a "spelling pronunciation", let's call it. And similarly, some Turkish musicians must be influenced to play their music more like Pythagorean / 53-EDO than person-to-person instruction would dictate. And so this data might do a decent job of describing some performed middle eastern music, if not the elusive traditional tunings of trained experts.

Since 53-EDO treats the Pythagorean comma and the syntonic comma the same, the Turkish middle seconds can also be interpreted as 1) raising the Pythagorean m2 by a syntonic comma and 2) lowering the Pythagorean M2 by a syntonic comma, which of course produce the rank-3, 5-limit m2 and M2. So perhaps instead of saying that Turkish music, as notated, has neutral intervals, we could say that Turkish music has options for 5-limit just intonation intervals. This interpretation has the added benefit that scales can be spelled correctly: the things functioning as seconds will actually be called second intervals, instead of A1 and d3, and consequently the scales in terms of pitch classes will also be spelled correctly / alphabetically.

Turkish staff notation doesn't really support this rank-3 interpretation, but I don't particularly care for Turkish staff notation and maybe we can do better. With rank-3 intervals, we have these interpretations for the simgeler:

B: 4 steps - Grm2

S: 5 steps - m2

K: 8 steps - M2

T: 9 steps - AcM2

A_12: 12 steps - AcA2

A_13: 13 steps - ?

A rank-3 interpretation of the "A" simge at 13-steps as a rank-3 2nd interval could be AcAcA2 or GrAA2. I'll argue later that simge A_13 should be associated with the rank-3 second interval AcAcA2. 

: The Makams

I'll start with transcriptions of Turkish makams from "Esendere Kültür Sanat Derneği", https://www.eksd.org.tr/. These transcriptions feature comments about shorter scale fragments from which they're composed. The scale fragments are called "ajnas" (singular "jins"), which is derived from the greek word "genus". The ajneas generally span a perfect fourth (a tetrachord) or a perfect fifth (a pentachord), and usually the pentachords are just a tetrachord + a pythagorean major second.

Each jins provides a temporary tonic center for melodic exploration. You noodle around on one jins and then move over to another one. So the root of each jins is marked for special attention. We'll use the makams to introduce the ajnas and then talk about them more in depth later.

:: 24-EDO Ajnas

Now that we're familiar with ajnas, let's learn some Arabic ones in 24-EDO:

Sikah trichord: [3, 4]

'Ajam trichord: [4, 4]

Kurd tetrachord: [2, 4, 4]

Hijaz tetrachord: [2, 6, 2]

Saba tetrachord: [3, 3, 2]

Bayati tetrachord: [3, 3, 4]

Nahawand tetrachord: [4, 2, 4]

Rast tetrachord: [4, 3, 3]

Nikriz pentachord: [4, 2, 6, 2]

Those are all taken from Wikipedia, although Wiki uses "1" for a M2 whole tone, so I multiplied through by a factor of 4 to get 24-EDO commas. The Saba tetrachord falls flat of P4 at 10 steps of 24-EDO, but the other tetrachords reach it. The ones with odd-valued commas are microtonal and the ones with all even-valued commas are representable in 12-EDO.

: MaqamWorld

: MaqamWorld Maqamt

MaqamWorld.com has tons of Arabic maqams, and almost all of them have the easily read staff notation of flats, "b", and half-flats, "d". They also have notes about arabic ajnas, but I don't have them all written down yet.

'Ajam Family:

'Ajam (Upper Ajam Ending): [C, D, E, F, G, A, B, C] # 'Ajam pentachord + upper 'Ajam tetrachord. Major scale.

'Ajam (Nahawand Ending): [C, D, E, F, G, A, Bb, C] # 'Ajam pentachord + Nahawand tetrachord.

'Ajam 'Ushayran (descends): [Bb, A, G, F, Eb, D, C, Bb] # Nahawand trichord down to Kurd tetrachord down to 'Ajam tetrachord.

Shawq Afza: [C, D, E, F, G, Ab, B, C] # 'Ajam pentachord + Hijaz tetrachord

.

Bayati Family:

Bayati (Nahawand Ending): [D, Ed, F, G, A, Bb, C, D]

Bayati (Rast Ending): [D, Ed, F, G, A, Bd, C, D]

Bayati Shuri: [D, Ed, F, G, Ab, B, C, D]

Husayni: [D, Ed, F, G, A, Bd or Bb, C, D] # This has both descending and ascending parts, as written on MaqamWorld, and it has multiple sixth scale degrees. I don't get it. I've just written it ascending.

Muhayyar: Bayati (Rast Ending) but then you emphasize Jins Bayati on the octave?

Hijaz Family:

Hijaz (Nahawand Ending): [D, Eb, F#, G, A, Bb, C, D]

Hijaz (Rast Ending): [D, Eb, F#, G, A, Bd, C, D]

Hijazkar (descends) : [E, Db, C, B, Ab, G, F, E, Db, C] # Also called Shadd 'Araban or Suzidil or Shahnaz

Zanjaran (descends): [C, Bb, A, G, F, E, Db, C]

Kurd Family:

Kurd: [D, Eb, F, G, A, Bb, C, D]

Hijazkar Kurd (descends): [E, Db, C, B or Bb, Ab, G, F, Eb, Db, C]

Nahawand Family:

Nahawand (Hijaz Ending): [C, D, Eb, F, G, Ab, B, C] # Nahawand pentachord + Hijaz tetrachord. Harmonic minor scale.

Nahawand (Kurd Ending): [C, D, Eb, F, G, Ab, Bb, C] # Nahawand pentachord + Kurd tetrachord. Natural minor scale.

Farahfaza: (Nahawand transposed to start on G.)

Nahawand Murassa': [C, D, Eb, F, Gb, A, Bb, C] # # Nahawand Murassa pentachord on the tonic, overlapped by Hijaz tetrachord on the 4th degree. And then add on the octave.

'Ushaq Masri: [D, E, F, G, A, Bd, C, D] # Nahawand pentachord + Bayati tetrachord

Nikriz Family:

Nikriz (descends from 9 rather than octave): [D, C, Bb, A, G, F#, Eb, D, C]. Ascending, Nikriz pentachord + Nahawand pentachord.

Nawa Athar: [C, D, Eb, F, G, Ab, B, C] # Nikriz pentachord on tonic, overlapping with a Hijazkar Hexachord starting on third degree (centered on fifth degree)

Athar Kurd: [C, Db, Eb, F#, G, Ab, B, C]

Rast Family:

Rast (Upper Rast ending): [C, D, Ed, F, G, A, Bd, C]

Rast (Nahawand ending): [C, D, Ed, F, G, A, Bb, C]

Kirdan: (descending Rast with upper Rast ending)

Sazkar (descends): [C, Bd, A, G, F, Ed, D#, C]

Suznak: [C, D, Ed, F, G, Ab, B, C] # Rast pentachord + Hijaz tetrachord.

Nairuz: [C, D, Ed, F, G, Ad, Bb, C] # Modern transposed maqam Yakah. Weird in that it has A half flat, which none of the other scales do.

Yakah: [G, A, Bd, C, D, Ed, F, G] # Older less common version of maqam Nairuz. Normal in that it has Bd and Ed, which many of the other scales do. Go team Yakah.

Dalanshin (descends): [E, Db, C, Bd, A, G, F, Ed, D, C]

Suzdalara (descends): [C, Bb, A, G, F, Ed, D, C] # Just the descending form of Rast with Nahawand ending.

Mahur: [C, D, Ed, F, G, A, B, C]

Sikah family:

Sikah: [Ed, F, G, A, Bd, C, D, Ed] # Sikah trichord + Upper Rast tetrachord + Rast trichord

Huzam: [Ed, F, G, Ab, B, C, D, Ed] # Sikah trichord + Hijaz tetrachord + Rast trichord

Maqam Rahat al-Arwah: (Huzzam rooted on Bd)

'Iraq: [Bd, C, D, Ed, F, G, A, Bd] # Sika trichord + Bayati tetrachord + Rast trichord

Awj ‘Iraq (descends): [D, C, Bd, A#, G, F#, Eb, D, C, Bd] 

Bastanikar: [Bd, C, D, Ed, F, Gb, A, Bb, C, Db, E, F] # "Maqam Bastanikar is effectively Jins Sikah followed by Maqam Saba. Its scale starts with the root Jins Sikah on the tonic, then Jins Saba on the 3rd degree, an overlapping Jins Hijaz on the 5th degree, and finally Jins Nikriz on the octave."

Musta'ar: [Ed, F#, G, A, Bb, C, D, Ed]

No family:

Jiharkah: [Ed, F, G, A, Bd, C, D, Ed, F] # F is the tonic, I think, not Ed. As much as Maqams have tonics, i.e. final notes. Jiharkah hexachord + Upper Rast tetrachord.

Lami: [D, Eb, F, G, Ab, Bb, C, D] # Lami pentachord, overlapped with Kurd tetrachord on the fourth, then add on the octave.

Saba ('Ajam ending): [D, Ed, F, Gb, A, Bb, C, D]

Saba (Nikriz ending): [D, Ed, F, Gb, A, Bb, C, Db, E, F]

Saba Zamzam ('Ajam ending): [D, Eb, F, Gb, A, Bb, C, D]

Saba Zamzam (Nikriz ending): [D, Eb, F, Gb, A, Bb, C, Db, E, F]

The one maqam I didn't transcribe, Sikah Baladi, has weird accidentals that I don't know how to interpret. I think its the only one with weird intervals? There's a small chance I saw other weird flats and rounded them off in my head to "the half flat accidental".

Here's the comment on Sikah Baladi from maqam world: 

    "Maqam Sikah Baladi is arguably the most challenging Arabic maqam. Its scale (and sayr) is something of a hybrid between a transposition of Maqam Huzam to an ordinary non-Sikah note, and Maqam Hijazkar – the intervals are not quite the same as either, but it sounds a bit like both." They also have a silly comment about how the pitches of Sikah Baladi are neither just nor equally tempered - even through just intonation and high division EDOs can approxiamte or exactly represent any frequency ratio. I think we'll just have to learn about Sikah Baladi from another source that isn't salivating with excitement about the magic and mystery of its forbidden unrepresentable tones.

Let's try transcribing it all the same. Maqam Sikah Baladi descends. The accidentals are modified by arrows up and arrows down, which I'm going to ignore at first. After that we have 

Sikah Baladi (descends): [C, B/b, A/b, G, Ft, E/b, D, C#, C, B/b, A/b, G]

You might notice that there's both a C and a C# right next to each other. I don't know what to do about that. The notes that are highlighted as tonics or jins roots are [G, B/b, D], but that doesn't line up with the annotated ajnas. The annotated ajnas are a Jins Sikah Baladi hexacord down from the top, [C, B/b, A/b, G, Ft, E/b], overlapping with Pseudo-Hijazkar pentachord [G, Ft, E/b, D, C#] down to C#, and then we pretend we're at C instead of C#, and go down with a tetrachord fragment of the previous Jin Sikah Baladi hexachord, [C, B/b, A/b, G]. So where do B/b and D come from as temporary tonics? I think the answer is just that Jins Sikah Baladi and Jins Psuedo-Hijazkar both have their tonics in the middle of their note sets. When I write "Ft", I'm not positive that their accidenal is supposed to be half sharp. It looks weird.

Anyway. On to the arrow accidentals: All of the "A/b"s have an arrow down to indicate a little flatter than half flat, and all of the "B/b"s have an arrow up to indicate a little sharper than half-flat, and the C naturals have an arrow down, and also the F half sharp has an arrow up.

After a little reflection, here's what I think is really going on. Suppose we start with Maqam Hijaz Kar, which is made of a Hijaz tetrachord + 9/8 + a Hijaz tetrachord. In 24-EDO we can write this as [2, 6, 2] + [4] + [2, 6, 2], specified with relative intervals between steps. Now we slightly alter the Hijaz tetrachord to make the Sikah tetrachord: [3, 4, 3], again stated relatively in 24-EDO. This Sikah tetrachord isn't a jins of Arabic music theory so far as I know, but it's the obvious extension to the Sikah trichord, which is given as [3, 4] in relative steps of 24-EDO.

If we do this, then Hijaz Kar becomes becomes an new maqam with relative commas [3, 4, 3] + [4] + [3, 4, 3]. The septimal spelling of this in absolute intervals is:

[P1, SbM2, SbM3, P4, P5, SbM6, SbM7, P8]

which, when rooted on G, becomes

[G, Ad, Bd, C, D, Ed, F#d, G]

modulo a few acute unisons, which don't show up tuned in 24-EDO anyway.

Simple enough. I don't know if it's right, but it's simple and regular and probably right enough. I still don't know why the original had both C natural and C# though.

And really I don't know how any of the maqamworld stuff is tuned. We could pretend that it's all 24-EDO, except for when there are arrows up and down, I suppose? I'll do a rank-4 / 24-EDO analysis at some point.

: MaqamWorld Ajnas:

...

: Alsiadi

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

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

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

: Combined 24-EDO and 53-EDO analysis

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

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

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

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

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

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

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

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

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

I know it's only two data points, but I enjoy the tight grouping. Turkish Rast third: 377.5 cents.

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

You can see that the interval up from the minor third is much simpler than the interval down from the major third for both for both of these Turkish Rast thirds. I should probably learn something from that.

: The  Kitāb al-Adwār

...

: Makamlar.net

...

: Yalcin Tura's Baglama And 24-EDO

A baglama is a long-necked Turkish lute. Wikipedia lists a tuning for the frets of a baglama due to modern Turkish composer Yalçın Tura. It has weird 17-limit frequency ratios that I found really interesting, so I played around with the math until I figured out how Tura must have constructed his tuning, which is like a western chromatic scale that has a few additional middle eastern microtones. This will be one more source of data on the intervals and frequencies that are used in middle eastern music.

We'll start with a rank-2 major scale, specified in terms of relative intervals between steps:

[M2, M2, m2, M2, M2 M2 m2]

These steps can be accumulated into our old friend

[P1, M2, M3, P4, P5, M6, M7, P8]

which we also could have constructed (without the closing octave) by a spiral of fifth, starting on P4 and cycling upward to M7.

In Pythagorean tuning, the M2 is tuned to 9/8 and the m2 is tuned to 256/243. So, we have 

    [9/8, 9/8, 256/243, 9/8, 9/8, 9/8, 256/243]

as the tuned intervals between steps of a Pythagorean major scale. We can accumulate these frequency ratios (multiplicatively) to get a the frequency ratios for each step of the major scale in Pythagorean tuning:

P1: 1/1 - 0c

M2: 9/8 - 203c

M3: 81/64 - 407c

P4: 4/3 - 498c

P5: 3/2 - 701c

M6: 27/16 - 905c

M7: 16/9 - 996c

P8: 2/1 - 1200c

Nothing new yet. But now Tura does thing inspired by a different ancient Greek music theorist. To get a minor second-like scale degree of ~100 cents, Tura splits the tuned M2 by taking the arithmetic mean of 9/8 with 1/1. 

((9/8) + (1/1)) / 2 = 17/16 at 105c.

We already were using a Pythagorean minor second as a relative-step in the construction of the Major scale, so coming up with a second m2 in a surprising turn. However, for super-particular ratios like 9/8, the arithmetic mean with 1/1 provides a good approximation for the square root. Consequently, 9/8 divided by 17/16 will also be a good approximation for the square root of 9/8:

(9/8) / (17/16) = 18/17 at 99c.

You can see that both frequency ratios are about 100c, which is half of the ~200c for 9/8 and also a good 12-EDO-ish minor second. We also could have constructed the 18/17 fraction as the harmonic mean of 9/8 with 1/1. This procedure of splitting a super-particular frequency ratio into parts using the arithmetic and harmonic means with 1/1 was used extensively by the ancient Greek music theorist Archytas.

If we expand all of the tuned M2s of the relative-step Pythagorean major scale into these parts, we get a chromatic scale specified by relative-steps:

[(18/17, 17/16), (18/17, 17/16), 256/243, (18/17, 17/16), (18/17, 17/16), (18/17, 17/16), 256/243]

I've included parentheses just to help show the grouping of what used to be M2 intervals. This splitting gives us intervals that are like minor Nths below the major Nths of the major scale. Also, we happen to introduce below the P5 an interval that's like a diminished fifth, since there was a gap of a M2 between P4 and P5 and this also became more fine-grained through division.

Why Tura chose to put the 18/17 before the 17/16 in each expansion of the tuned M2, I don't know, but it will turn out to not matter much in the end of my analysis.

To get middle-eastern microtones, we're just going to break up one of the intervals of our (now chromatic) scale one more time. All of the 17/16 ratios that take us from a minor Nth to a major Nth are going to get split into an arithmetic mean and a harmonic mean. 

17/16 → 33/32 and 34/33

When we accumulate all of the relative steps, this will give us neutral 2nds, 3rds, 6ths, and 7ths. It will also add a half-flat fifth  between the diminished fifth and the perfect fifth.

Here are the expanded relative steps:

[1/1, 18/17, (34/33, 33/32), 18/17, (34/33, 33/32), 256/243, 18/17, (34/33, 33/32), 18/17, (34/33, 33/32), 18/17, (34/33, 33/32), 256/243]

If we accumulate the frequency ratios between scale degrees multiplicatively, we get this for the tuned steps of each scale degree relative to the tonic:

     P1: 1/1 - 0c

m2: 18/17 - 98c

n2: 12/11 - 150c

M2: 9/8 - 203c

m3: 81/68 - 302c

n3: 27/22 - 354c

M3: 81/64 - 407c

P4: 4/3 - 498c

d5: 24/17 - 596c

n5: 16/11 - 648c

P5: 3/2 - 701c

m6: 27/17 - 800c

n6: 18/11 - 852c

M6: 27/16 - 905c

m7: 243/136 - 1004c

n7: 81/44 - 1056c

M7: 243/128 - 1109c

P8: 2/1 - 1200c

All of these frequency ratios are made of factors of 2, 3, 11, and 17. We could say that they lie on a four dimensional subspace of 17-limit just intonation. In the language of the Xenharmonic community, this is called the "2.3.11.17" just intonation subspace. Fine.

These just ratios are very close to 24-EDO frequency ratios. The largest deviation is the tuned M7 at 9 cents sharp of 24-EDO, and that's just because it's the unmodified Pythagorean frequency ratio that we started with. All of Tura's modifications to the Pythagorean major scale got us closer to 24-EDO.

This scale wasn't constructed with regular interval arithmetic - it was constructed in the manner of Archytas's rational approximations to square roots - and so it doesn't obey octave complementation. For example, the Archytas m2 + Pythagorean M7 != P8.

But what if we try to relate Tura's scale to its octave complement? With octave complementation, we find that e.g. the Pythagorean m2 and the Archytas m2 are separated by a small 9 cent frequency ratio

(18/17) / (256/243) = 2187/2176 at 9 cents.

that we might want to temper out so that we have this cool 17-limit scale from Tura in which regular interval arithmetic still works.

Another comma that shows up when we do octave complementation is the Archytas neutral third divided by the octave complement of the Archytas neutral sixth:

(27/22) / (11/9) = 243/242 at 7 cents.

We might want to temper that one out as well.

There's one more comma that shows up. If we take the octave complement of the Archytas d5 (24/17), we get 17/12, which is also at 12-steps of 24-EDO. The ratio is 

(17/12)/(24/17)  = 289/288

These three commas are not quite independent: we can simplify the first comma, 2187/2176, a little bit by dividing it by the second comma,  243/242:

 (2187/2176)  / (243/242) = 1089/1088

And now we've got three beautiful independent super-particular commas.

I don't know if you'll find it a surprise at this point, but if we start with the four dimensional 2.3.11.17 just intonation space, and then  we tune our octaves purely and temper out these three commas, we get 24-EDO.

Here's how that works.

Start with the intervals justly associated with the three Tura commas and the octave, all expressed in the rank-7 prime harmonic basis:

[-5, -2, 0, 0, 0, 0, 1] # 289/288 = (17/16) / (18/17)

[-1, 5, 0, 0, -2, 0, 0] # 243/242 = (27/22) / (11/9)

[-6, 2, 0, 0, 2, 0, -1] # 1089/1088 = (33/32) / (34/33)

[1, 0, 0, 0, 0, 0, 0] # 2/1

Since we're only working in the 2.3.11.17 subgroup, just remove all of the coordinates associated with 5, 7, and 13. They were all zero anyway:

[-5, -2, 0, 2] # 289/288 = (17/16) / (18/17)

[-1, 5, -2, 0] # 243/242 = (27/22) / (11/9)

[-6, 2, 2, -1] # 1089/1088 = (33/32) / (34/33)

[1, 0, 0, 0] # 2/1

This matrix has an absolute determinant of 24. You can swap round some rows to get +24 instead of -24 if you care about that, but it's still 24-EDO.

But maybe the determinant isn't convincing to you. How about we tune some intervals using the comma+octave matrix?

First find the inverse of the matrix. Wolfram Alpha gives

> inverse of [[-5, -2, 0, 2], [-1, 5, -2, 0], [-6, 2, 2, -1], [1, 0, 0, 0]]

= 1/24 * [[0, 0, 0, 24], [2, 4, 4, 38], [5, -2, 10, 83], [14, 4, 4, 98]]

Now let's look at two Tura intervals that were justly tuned to nearly the same 24-EDO step, say

~350c: DeAcM3 = [-1, 3, -1, 0] # 27/22

~350c: AsGrm3 = [0, -2, 1, 0] # 11/9

both at around (350c/50c = ) 7 steps of 24-EDO. Again, I've removed the coordinates above that were associated with prime harmonic 5, 7, and 13. Now we just multiply these by the inverse of the comma+octave matrix to see how they're tuned:

[-1, 3, -1, 0] * (1/24 * [[0, 0, 0, 24], [4, 4, 2, 38], [10, -2, 5, 83], [4, 4, 14, 98]]) = (1/12, 7/12, 1/24, 7/24)

[0, -2, 1, 0] * (1/24 * [[0, 0, 0, 24], [4, 4, 2, 38], [10, -2, 5, 83], [4, 4, 14, 98]]) = (1/12, -5/12, 1/24, 7/24)

Since we're tempering out the commas (i.e. tuning them to a frequency ratio of 1/1) it doesn't matter what coordinates show up in the first three slots: one raised to a real power is still one. The only coordinate that maters is the last one, associated with the octave, which shows that [-1, 3, -1, 0] and [0, -2, 1, 0] are both tuned to 2^(7/24), i.e. 7 steps of 24-EDO.

What does all of this show? It shows that 24-EDO is not such a bad scale for describing middle eastern music, if Tura's baglama tuning is actually used anywhere. It also shows that if you want a just analysis of music written in 24-EDO, you can do worse than using fractions in the 2.3.11.17 J.I. subspace. It also just shows how you can do cool temperament things in high dimensions. Like now you could try tuning one of the other commas justly to see what happens.

I had a dream of relating the half-flat and half-sharp interval names more directly to the 2.3.7.13 intervals, like so that I could say whether

~350c: DeAcM3 = [-1, 3, 0, 0, -1, 0, 0] # 27/22

~350c: AsGrm3 = [0, -2, 0, 0, 1, 0, 0] # 11/9

were each SbM3 or Spm3 or anything else. I think I'm going to give up on that though. I've got more advanced and fined grained analyses I want to do of middle eastern music than focus on the stuff related to 24-EDO.

: Zalzal, al Farabi, 87-EDO, Margo Schulter

We've seen 24-EDO neutral seconds and we've seen 7-limit neutral seconds . We've seen 53-EDO neutral seconds and neutral second made by an extended Pythagorean spiral and neutral seconds made by 5-limit adjustments with syntonic commas. We've even seen 17-limit neutral second inspired by Archytas. What else could there be? (We've seen neutral thirds and sixths and sevenths in all those varieties.)

A medieval Iranian musician and music theorist named Manṣūr Zalzal, who lived around 800 CE, is credited with introducing specific neutral seconds and neutral thirds as intervals in Arabic lute music. Credit comes from almost-as-old-Philosopher and music theorist al-Farabi, who will be even more important in this section.

If a minor second has a cent value around 100 cents and a major second has a cent value around 200 cents, then a neutral second is something like 150 cents.

Zalzal analyzed neutral seconds with four small super-particular ratios:

11/10 ~ 165 cents

12/11 ~ 155 cents

13/12 ~ 140 cents

14/13 ~ 130 cents

Following Zalzal, al-Farabi invented some tetrachords using these neutral seconds - tetrachords being little scales of four tones extending from P1 to P4. All of al-Farabi's tetrachords hit P1, Pythagorean M2, some intermediate neutral third, and P4.

[P1, M2, ?, P4]

Here are the tetrachords, given with relative intervals on the left and absolute on the right:

[9/8, 11/10, 320/297] : [1/1, 9/8, 99/80, 4/3]

[9/8, 12/11, 88/81] : [1/1, 9/8, 27/22, 4/3]

[9/8, 13/12, 128/117] : [1/1, 9/8, 39/32, 4/3]

[9/8, 14/13, 208/189] : [1/1, 9/8, 63/52, 4/3]

Al-Farabi's tetrachords look a little funny at first, but they're kind of beautiful if you squint. In the first tetrachord based on 11/10, the wonky guy at the end of the relative intervals, 320/297, is really close to 14/13. The difference is like 1 cent, which is too small for humans to hear.

Likewise in the last tetrachord based on 14/13, the wonky guy at the end of the relative intervals, 208/189, is really close to 11/10. Only off by one cent. The same one cent, actually:

(320/297) / (14/13) = 2080/2079

(208/189) / (11/10) = 2080/2079

The other two tetrachords have a similar relationship: The wonky 88/81 in the second tetrachord based on 12/11 is basically the same as 13/12. And the wonky 128/117 in the third tetrachord based on 13/12 is basically the same as 12/11. The difference is larger here, at 5 cents, but that's still basically the smallest thing that humans can hear:

(128/117) / (12/11) = 352/351

(88/81) / (13/12) = 352/351

It would be nice if we could just use the simple fractions all throughout, right? We could say things like:

(9/8) * (11/10) * (14/13) ~= (4/3)

It's not like there aren't error bars on these things. There are limits to the fidelity of our recognition and our performance. So what system will let us do impressionistic arithmetic that expresses al-Farabi's tetrachords in terms of the Zalzalian neutral seconds, without any wonky bits? One solution is to find a new fine-grained EDO in which e.g.  (320/297) and (14/13)  are tuned to the same step. Then we can describe the tetrachord in terms of EDO steps instead of ratios.

The simplest EDO that has distinct steps for all of the Zalzalian neutral seconds is 45-EDO, but it doesn't put them in ascending order and doesn't .  After that we have 71-EDO, 78-EDO, 85-EDO, all well formed. 

But here's the list of EDOs that tune the Zalzalian seconds to distinct ascending step while also tempering out the intervals that are justly associated with (2080/2079) and (352/351): 

(87, 94, 99, 111, 128, 133, 135, 140, 145, 157, 174, 181, 186, 198, 205, 210, 227, 232, 244, 251, 268, 269, ...]-EDO.

In 87-EDO, 

{9/8 * 11/10 * 14/13 ~= 4/3} 

{9/8 * 12/11 * 13/12 ~= 4/3} 

are explained as 

(15 + 12 + 9 = 36) steps

(15 + 11 + 10 = 36) steps

respectively. In 94-EDO, the explanations are (16 + 13 + 10 = 39) and (16 + 12 + 11 = 39), respectively.

The people on the Xenharmonic Alliance discord point out that 87-EDO is a "Parapyth" EDO, identified by Margo Schulter for use in analyzing middle eastern music just like this. I got scooped. Many of the other higher EDOs mentioned above are Parapyth EDOs as well.

Ozan Yarman is a Turkish music theorist who also participates in the Xenharmonic community a little. He often gives just analyses of tetrachords which are literally false but perhaps still useful in this program of impressionistic arithmetic. I don't think he ever actually drawn attention to the fact that most of his work contains errors of normal arithmetic, but I'm pretty sure he knows and just doesn't care. But we don't have to pretend that {9/8 * 11/10 * 14/13 ~= 4/3}! We can just say that (15 + 12 + 9 = 36) steps of 87-EDO. I think that's a good way to elevate his work to mathematical validity.

And 87-EDO is great for this; it also tempers out the intervals justly associated with 256/255, 406/405, and 154/153, which all show up as deviations from truth when Yarman does arithmetic. Although 94-EDO tempers out the interval justly associated with 225/224, which Yarman also ignores. I think I still prefer 87-EDO, partly because it's the simplest one that works with Al-Farabi, but I admdit that I haven't investigated the merits of the higher EDOs in detail.

When we did a combined 24-EDO + 53-EDO analysis to just neutral thirds for Arabic and Turkish tradition, three of the four Arabic frequency ratios (16/13, 21/17, 27/22) that we found are justly associated with intervals that are tuned to 26 steps of 87-EDO. Both of the Turkish frequency ratios (56/45, 68/55) and one of the Arabic ones (99/80) get tuned one step higher to 27 steps of 87-EDO. So in today's  music, a tetrachord phrased with absolute intervals of the form [P1, M2, n3, P4] will look like

[0, 15, 26 or 27, 36]

when tuned in 87-EDO. This is the modern baseline that we'll use for looking at al-Farabi's medieval tetrachords.

The 11/10 tetrachord of al-Farabi is actually modern, since  (9/8) * (11/10) = (99/80) was one of our modern Arabic ratios in the combined 24 & 53 analysis. Here's the tetrachord, relative on the left, absolute on the right:

[15, 12, 9] : [0, 15, 27, 36]

A fine modern Turkish Rast. 

The 12/11 tetrachord of al-Farabi is also modern, since (9/8) * (12/11) = (27/22), a ratios we've seen which leads us to

[15, 11, 10] : [0, 15, 26, 36]

A fine modern Arabic Rast.

The 13/12 tetrachord of al-Farabi is flatter on the third than a modern Rast, as (9/8) * (13/12) = (39/32), which gives us

[15, 10, 11] : [0, 15, 25, 36]

The 14/13 of al-Farabi is flatter still than that on the third, as (9/8) * (14/13) = (63/52), which gives us

[15, 9, 12] : [0, 15, 24, 36]

as a tetrachord. 

I'm of the opinion that these last two are not Rast tetrachords at all, since Rast, in relative intervals, has a larger neutral second followed by a smaller neutral second (or at least two equal neutral seconds in 24-EDO), whereas these last two tetrachords of al-Farabi have a smaller neutral second followed by a larger one. That is to say, [T, K, S] is Rast, while these last two tetrachords of al-Farabi are [T, S, K]. Margo Schulter sometimes calls this Rast Jadid ("New Rast") or Mustaqim.

...

: Ozan Yarman on Just Tunings of Turkish Makams

Ozan Yarman is a Turkish music theorist who has written at length about the just tuning of Turkish makams. That sounds amazing, right? It's definitely cool, but sometimes his work seems a little sloppy to me. I'll start with some of his data from his doctorate thesis, "79-Tone Tuning & Theory For Turkish Maqam Music", and then I'll pick it apart a bit.

Rast (ascends and descends the same way, tonic on C): [1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1].

Acemli Rast (rises the same as Rast but descends as follows, tonic on C): [2/1, 16/9, 5/3, 3/2, 4/3, 5/4, 9/8, 1/1].

Mahur (ascending), tonic on C: [1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2/1].

Mahur (descending): [2/1, 15/8, 27/16, 3/2, 4/3, 5/4, 9/8, 1/1]. # Descends just like Rast.

Pencgah (ascends and descends the same way), tonic on C: [1/1, 9/8, 5/4, 7/5, 3/2, 27/16, 15/8, 2/1].

Nihavend (ascending), tonic on C: [1/1, 9/8, 6/5, 4/3, 3/2, 8/5, 32/17, 2/1].

Nihavend (descending): [2/1, 9/5, 8/5, 3/2, 4/3, 6/5, 9/8, 1/1].

All of those had C as a tonic. When we get makams with different tonics, we see that Ozan Yarman still notates the frequency ratios relative to C, so that the first note isn't tuned to 1/1.

Hicaz (ascending), tonic on D: [9/8, 6/5, 7/5, 3/2, 27/16, 50/27, 2/1, 9/4]. If we divide through by 9/8, we get [1/1, 16/15, 56/45, 4/3, 3/2, 400/243, 16/9, 2/1].

Hicaz (descending), tonic on D: [9/4, 2/1, 9/5, 27/16, 3/2, 7/5, 6/5, 9/8]. Dividing through by 9/8 to re-root gives [2/1, 16/9, 8/5, 3/2, 4/3, 56/45, 16/15, 1/1]. 

Huseyni (ascends and descends the same way), tonic on D: [9/8, 21/17, 4/3, 3/2, 27/16, 63/34, 2/1, 9/4]. Dividing through by 9/8 gives this rooted makam: [1/1, 56/51, 32/27, 4/3, 3/2, 28/17, 16/9, 2/1].

Segah (ascending), tonic on E: [(20/17), 5/4, 4/3, 3/2, 5/3, 15/8, 2/1, 40/17, 5/2] # The 20/17 note in parentheses is a leading tone below the tonic. Dividing through by 5/4 gives [(16/17), 1/1, 16/15, 6/5, 4/3, 3/2, 8/5, 32/17, 2/1].

Segah (descending): [40/17, 9/4, 2/1, 30/17, 5/3, 3/2, 4/3, 5/4]. Dividing through by 5/4 gives [32/17, 9/5, 8/5, 24/17, 4/3, 6/5, 16/15, 1/1]. # This doesn't descend from the octave and that's how Yarman has it written.

Huzzam (ascending), tonic on a slightly flat E: [(7/6), 36/29, 4/3, 3/2, 48/29, 54/29, 2/1, 7/3, 72/29] # The (7/6) note in parentheses is a leading tone below the tonic. Dividing through by 36/29 gives [(203/216), 1/1, 29/27, 29/24, 4/3, 3/2, 29/18, 203/108, 2/1]

Huzzam (descending): [40/17, 9/4, 2/1, 30/17, 48/29, 3/2, 4/3, 36/29]. Dividing through by 36/29 gives [290/153, 29/16, 29/18, 145/102, 4/3, 29/24, 29/27, 1/1]. # Like Segah makam (descending), this doesn't descend from the octave and that's how Yarman has it written.

Saba (ascending), tonic on D: [9/8, 21/17, 27/20, 16/11, 27/16, 11/6, 2/1, 9/4]. Dividing through by 9/8 gives [1/1, 56/51, 6/5, 128/99, 3/2, 44/27, 16/9, 2/1].

Saba (descending): [8/3, 5/2, 32/15, 2/1, 25/14, 27/16, 10/7, 4/3, 11/9, 9/8]. Dividing through by 9/8 gives [64/27, 20/9, 256/135, 16/9, 100/63, 3/2, 80/63, 32/27, 88/81, 1/1].

Now these are mostly fine. It's a little weird that there are a lot of fractions with factors of 17 (all in the denominators): [16/17, 20/17, 21/17, 24/17, 28/17, 30/17, 32/17, 40/17, 56/51, 63/34, 145/102] and a bunch of fractions with factors of 29 (mostly in the numerators): [29/16, 29/18, 29/24, 29/27, 36/29, 48/29, 54/29, 72/29, 203/108, 203/216, 290/153], but maybe that regularity is a consequence of some regular construction. I'm open to that. Like if you want 5-limit major intervas to be 60 cents sharper, you multiply them all by (30/29). Whatever.

What's weird is that Yarman gives tetrachord glosses on top of the makams, and the math doesn't work out. A bunch of the tetrachords don't exactly form a perfect fourth, but that might be excusable since Yarman's dissertation is about constructing Turkish microtones using weird irregular chains of tempered perfect fifths; if your P5s are all messed up, then your P4s will be messed up too. The weirder part to me is that he gives tetrachord glosses on top of frequency ratios that are obviously different from the tetrachords.

For example, Rast starts out with a dead simple 5-limit scale: [1/1, 9/8, 5/4, 4/3], which has relative intervals of 

[9/8 * 10/9 * 16/15].

But Yarman's notes on top say that the makam starts out with the "tempered Rast" tetrachord:

[28/25 * 28/25 * 17/16]

which would actually form 

[1/1, 28/25, 784/625, 833/625]

This is flat of 4/3 by a tiny 2500/2499 at 0.7 cents.  So maybe Ozan Yaramn uses 2500/2499 as part of his tempering scheme: I don't know. But if he does, that means that he thinks Rast makam starts 

[1/1, 28/25, 784/625, 833/625]

and he should notate it that way. He definitely shouldn't have it notated such that the first two relative steps differ by 22 cents and then in a parenthetical remark say "Never mind, they're actually equal to each other at 28/25".

Okay, so: weird frequency ratios, don't bother me, but they befuddle me. Tetrachords that don't form P4, bother me a little, but there's a chance that's on purpose. The inconsistent descriptions between the scale steps and the tetrachords bother me a lot. There are many more of those which I haven't mention. Let's do one more. If we divide his Huseyni through by 9/8 so that it's rooted on unison, the opening tetrachord is:

[1/1, 56/51, 32/27, 4/3]

with relative intervals of

[(56/51) * (68/63) * (9/8)]

However the gloss above the scale says the tetrachord is

[(11/10) * (13/12) * (9/8)]

which in absolute intervals is

[1/1, 11/10, 143/120,  429/320]

and it can't be both. The number 56/51 does not equal 11/10. He doesn't comment on this anywhere in the section. He just makes errors of arithemtic all over the place. Maybe he explains it in the section with the irregular tempering. I wouldn't know. Anyway,  the 429/320 is sharp of 4/3 by a factor of 1287/1280 at 9 cents. So if he is mistuning his tetrachords on purpose, then it looks like he sometimes goes flat and sometimes goes sharp.

There other inconsistencies not related to the tetrachords. For example, the Segah genus on page 137 has frequency ratio (40/7) indicated as a D# in the ascending form, and then it's notated as an Eb with some Saggital Notation accidentals in the descending form. Am I supposed to believe that his tuning system tunes both of those intervals over C to the same just ratio, and that one of the pitches happens to occur in ascending form of Segah and the other pitch occurs in the descending form?

It would make more sense if Yarman was just thinking in terms of some high-division EDO and then finding simple frequency ratios that were associated with each step, but he's clearly not doing that or he wouldn't have factors of 17 in only the numerators and it wouldn't be the case that he had factors of 17 and 29 but not factors of 13, 19 or 23.

Anyway, as much as his inconsistencies and math errors annoy me, it's still useful data for characterizing Turkish tuning. So let's go over his tetrachords. If he has a pentachord ending in (9/8), I just lop that off and call it a tetrachord:

Pure Rast tetrachord: (9/8) * (10/9) * (16/15)

Tempered Rast tetrachord: (28/25) * (28/25) * (17/16)

Mahur tetrachord: (9/8) * (9/8) * (256/243)

Pencgah tetrachord: (9/8) * (10/9) * (28/25)

Nihavend tetrachord: (9/8) * (16/15) * (10/9)

Segah tetrachord: (16/15) * (9/8) * (10/9)

Hicaz tetrachord: (16/15) * (7/6) * (15/14)

Wide Hicaz tetrachord: (16/15) * (20/17) * (17/16)

Huzzam tetrachord: (15/14) * (9/8) * (32/29)

Huseyni tetrachord: (11/10) * (13/12) * (9/8)

Ussak tetrachord: (12/11) * (12/11) * (9/8)

Çargah tetrachord: (15/14) * (13/11) * (55/52)

Saba pentachord: (11/10) * (12/11) * (13/12) * (15/13)

Here's a summary of their factor structure:

3-limit:

(9/8) * (9/8) * (256/243): Mahur tetrachord

5-limit:

(9/8) * (10/9) * (16/15): Pure Rast tetrachord

(9/8) * (16/15) * (10/9): Nihavend tetrachord

(16/15) * (9/8) * (10/9): Segah tetrachord

7-limit:

(9/8) * (10/9) * (28/25): Pencgah tetrachord

(16/15) * (7/6) * (15/14): Hicaz tetrachord

11 and 13 limit:

(11/10) * (13/12) * (9/8): Huseyni tetrachord

(12/11) * (12/11) * (9/8): Ussak tetrachord

(15/14) * (13/11) * (55/52): Çargah tetrachord

(11/10) * (12/11) * (13/12) * (15/13): Saba pentachord

17-limit:

(16/15) * (20/17) * (17/16): Wide Hicaz tetrachord

(28/25) * (28/25) * (17/16): Tempered Rast tetrachord

29-limit:

(15/14) * (9/8) * (32/29): Huzzam tetrachord

Remember the good old days when we though Turkish music theory was 53-EDO and it only had [Rast, Çârgâh, Bûselik, Kürdî, Uşşâk, Hicaz] as tetrachords? 

Actually, let's tune Yarman's tetrachrods to 53-EDO and see how they compare to the standard Turkish ajnas  and the ajnas of Alsiadi. I'll phrase them all in relative steps.

Yarman's Hicaz tetrachord and his Wide Hicaz tetrachord are both tuned to [5, 12, 5] , which is a normal Turkish Hicaz intonation.  An intonation at [4, 13, 5] would also have been standard. 

What he calls "Mahur" is tuned to [9, 9, 4] steps of 53-EDO. This is normally called Çargâh in Turkish theory and Ajam in Arabic theory. "Mahur" is the Persian name for this tetrachord, so that's not crazy.

In contrast, Ozan Yarman's Çargâh tetrachord is tuned to [5, 13, 4] steps of 53-EDO, which is an Arabic intonation for Hijaz. Weird.

Yarman's Pure Rast is [9, 8, 5] and this is the standard 53-EDO Turkish tuning for Rast. His Tempered Rast tetrachord is [9, 9, 5], which adds up to 23 steps, not 53-EDO's tuned P4 at 22 steps.

Yarman's Nihavend is tuned to [9, 5, 8] steps of 53-EDO, which is like a permutation of traditional Turkish Rast at [9, 8, 5] or Turkish Uşşak at [8, 5, 9]. I'd like to point out that Nihavend is an alternative spelling of Nahawand, which is a city in present day Iran. In Arabic music theory, the Nahawand or Busalik tetrachord is [9, 4, 9]. In standard Turkish music theory based on 53-EDO, the [9, 4, 9] tetrachord is called Bûselik. This [9, 4, 9] intonation is just a Pythagorean [M2, m2, M2], like the first four notes of a rank-2 minor scale. Ozan Yarman's Nihavend is justly tuned to [(9/8) * (16/15) * (10/9)], which are the just tunings for the rank-3 intervals [AcM2, m2, M2], i.e. the start of the rank-3 minor scale. So I think of Yarman's tetrachord as a refreshing confirmation that some of Turkish music's tetrachordal structure is pushing toward scales of 5-limit just intonation in a way that Arabic music isn't.

Both Yarman's Huzzam and Segah tetrachords are tuned to [5, 9, 8] which, again, is like a permutation of traditional Turkish Rast at [9, 8, 5] or Uşşak at [8, 5, 9].  Yarman's Segah is justly tuned in 5-limit while his Huzzam had a factors of 7 and 29. The Arabic Maqam Huzam starts on an E half flat. Yarman's Huzzam starts on an E somewhat flat, at (1200 * log_2(36/29) = ) 374 cents relative to C. I think this is fantastic. Ozan Yarman really wants you to know that Huzzam should be played with an Ed at 374 cents over C, and he introduced a factor of 29 into this and only this tetrachord to make it happen. Want some weirdly specific insight into the regional intonation of microtonal middle eastern scales? Ozan Yarman has got you covered, and he's packing a factor of 29 for coverage.

Ozan Yarman's Uşşak is tuned to [7, 7, 9], which overshoots P4 by a step like his Tempered Rast genus did. Perhaps he could tune the second intervals a little flatter.

Yarman's Huseyni tetrachord is tuned to [7, 6, 9]. This is like a permutation of Arabic Rast at [9, 7, 6].

His Pencgah tetrachord is [9, 8, 9]. This is hugely sharp of P4 at 26 steps of 53-EDO instead of 22. I don't know anything about Pencgah.

His Saba pentachord is [7, 7, 6, 11] at 31-steps of 53-EDO. This is actually a perfect fifth! Well done, Ozan.

So that's a 53-EDO analysis of the tetrachord glosses that Ozan Yarman puts on diagrams of his makams. Next let's look at the actual frequency ratios between steps of his makams, which should be equivalent to the notated tetrachords, or at least equivalent up to tempering, but rarely are.

First let's write all of Ozan Yarman's makams in terms of relative frequency ratios between steps. I'll ignore leading tones and write desending makam forms as though they were ascending, so that all of the fequency ratios will be larger than 1/1.

Rast: [9/8, 10/9, 16/15] + [9/8] + [9/8, 10/9, 16/15] # Rast + T + Rast

Acemli Rast (descending): [9/8, 10/9, 16/15] + [9/8, 10/9, 16/15] + [9/8] # Rast + Rast + T

Mahur (ascending): [9/8, 9/8, 256/243] + [9/8] + [9/8, 9/8, 256/243] # Mahur + T + Mahur

Mahur (descending) [9/8, 10/9, 16/15] + [9/8] +  [9/8, 10/9, 16/15] # Rast + T + Rast

Pencgah: [9/8, 10/9, 28/25, 15/14] + [9/8, 10/9, 16/15] # Pencgah pentachord + Rast

Nihavend (ascending): [9/8, 16/15, 10/9, 9/8] + [16/15, 20/17, 17/16] # Nihavend pentachord + Wide Hicaz

Nihavend (descending): [9/8, 16/15, 10/9, 9/8] + [16/15, 9/8, 10/9] #  Nihavend pentachord + Segah

Hicaz (ascending): [16/15, 7/6, 15/14] + [9/8] + [800/729, 27/25, 9/8] # Hicaz + T + Huseyni

Hicaz (descending): [16/15, 7/6, 15/14] +  [9/8, 16/15, 10/9, 9/8] # Hicaz + Nihavend pentachord

Huseyni: [56/51, 68/63, 9/8, 9/8] + [56/51, 68/63, 9/8] # Huseyni pentachord + Huseyni

Segah (ascending): [16/15, 9/8, 10/9] + [9/8] + [16/15, 20/17, 17/16] # Segah + T + Wide Hicaz

Segah (descending): [16/15, 9/8, 10/9] + [18/17] + [17/15, 9/8, 160/153] # Segah + 18/17 + "Mahur" + 17/16

Huzzam (ascending): [29/27, 9/8, 32/29] + [9/8] + [29/27, 7/6, 216/203] # Huzzam + T + "Hicaz"

Huzzam (descending): [29/27, 9/8, 32/29] + [145/136] + [17/15, 9/8, 160/153] # Huzzam + "16/15" + "Mahur"

Saba (ascending): [56/51, 153/140, 320/297, 297/256] + [88/81, 12/11, 9/8] # Saba pentachord + Ussak

Saba (descending): [88/81, 12/11] + [15/14, 189/160, 200/189] + [28/25] + [16/15, 75/64, 16/15]  # The first trichord isn't given a gloss, but it looks a lot like the start of Ussak from Saba ascending. Above that we have Çargah + "9/8" + "Hicaz".

There's this crazy thing with Ozan Yarman's Rast, where the given frequency ratios for each scale degree are the same ascending and descending, but the notated tetrachords and the EDO-like steps of his tempered 79-tone tuning system both differ in the ascending and descending forms. It's on page 134 of his doctoral thesis.

I looked up Pencgah a little. It seems to be cognate with the Persian dastgah scale "Rast Panjgah". The scale is a lot like Rast, with the main difference being that the fourth scale degree is sharpened by a factor of 21/20, which, at 84 cents, is a little larger than the 5-limit augmented unison, 25/24 at 71 cents. So, rooted on C, it's like we take the F of Rast up to an F# plus a little more. Although it's still less than a 24-EDO augmented unison. In so much as Turkish Rast looks like a push from Pythagorean major toward 5-limit major, Pencgah looks a lot like a 5-limit Locrian scale.

The Hicaz makam (ascending) ends with a Huseyni tetrachord, (800/729 * 27/25 * 9/8). But later we see the Huseyni tetrachord given as (56/51 * 68/63 * 9/8) in the Huseyni makam, and the Huseyni pentachord given as (56/51 * 68/63 * 9/8 * 9/8)  within the same makam. Both intonation of the Huseyni tetrachord are tuned to [7, 6, 9] steps of 53-EDO. The 56/51 is sharp of 800/729 by a factor of 1701/1700 at 1 cent. The 68/63 is flat of 27/25 by the saem 1 cent. So they're perceptually indistinguishable. The version from the Hicaz makam has shorter fractions, but a factor of 17. The version from the Huseyni makam has more complicated fractions, but it's 5-limit.

Segah (descending) has the tetrachord [17/15, 9/8, 160/153], which Ozan Yarman glosses as Mahur, instead of his previous Pythagorean version [9/8, 9/8, 256/243].  The first ratio, (17/15), is sharp of the usual 9/8 by 136/135 at 13 cents. And the last ratio, (160/153), is flat of the usual (256/243) by the same amount. I think 13 cents is enough of a difference that he might have given this tetrachord intonation a different name, like "17-limit mahur" or something. Segah (descending) also has some weird stuff going on where there's an 18/17 in the middle and a 17/16 up top to reach the octave. I don't feel qualified enough on Turkish makam intonation to comment on this.

Huzzam (descending) has the same 17-limit intonation of Mahur that we saw in Segah (descending). It also has a step of 145/136 between the tetrachords, which is 77 cents, but it's glossed as 16/15, which is also 77 cents. If you forget that and just put in the regular Pythagorean major second, (9/8), between the tetrachords, then Huzzam reaches the octave. I don't feel qualified enough on Turkish makam intonation to comment on this.

Huzzam (ascending) has a third intonation for the Hicaz tetrachord,  [29/27, 7/6, 216/203]. We'd previously been given Wide Hicaz as [16/15, 20/17, 17/16] and normal Hicaz as [16/15, 7/6, 15/14]. All three of these are just tuned to [5, 12, 5] in 53-EDO. In the Huzzam intonation, the first ratio, (29/27), is sharp of the usual 16/15 by a factor of 145/144 at 12 cents. The last ratio in the Huzzam intonation, (216/203) is flat of (17/16) by this same 12-cents. In the glosses, Yarman gave a version of a Huzzam tetrachord that didn't equal a justly tuned P4: 

(15/14 * 9/8 * 32/29) != 4/3

This is sharp of 4/3 by a factor of 406/405 at 4 cents. But if you look at the just tunings of the scale degrees, the tetrachord works out fine:

(29/27 * 9/8 * 32/29) = 4/3

In Saba (descending) ozan Yarman has a gap of 9/8 notated, but the actual relative step ratio is 28/25. This is flat of 9/8 by 225/224 at 8 cents. The Çargah tetrachord he gave right before that was sharp of a just P4 by that same 8 cents.

(15/14 * 189/160 *  200/189) / (225/224) = 4/3

I really don't know with Ozan Yarman whether these things that seem to be error of arithmetic are intentional or not. It you want a 7-limit Çargah tetrachord that equals 4/3, I'd reccomend flattening the last ratio, giving

(15/14 * 189/160 *  256/243) 

Saba (descending) has one more intonation for Hicaz.  This one, [16/15, 75/64, 16/15], is also tuned to [5, 12, 5] in 53-EDO.

So these are the tetrachords and pentachords we have infer from the justly tuned scale degrees:

Rast tetrachord: (9/8 * 10/9 * 16/15)

Mahur tetrachord: (9/8 * 9/8 * 256/243)

Mahur tetrachord from Segah:  (17/15 * 9/8 * 160/153)

Pencgah pentachord: (9/8 * 10/9 * 28/25 * 15/14)

Nihavend pentachord: (9/8 * 16/15 * 10/9 * 9/8)

Wide Hicaz tetrachord: (16/15 * 20/17 * 17/16)

Normal Hicaz tetrachord: (16/15 * 7/6 * 15/14)

Hicaz tetrachord from Huzzam: (29/27 * 7/6 * 216/203)

Hicaz tetrachord from Saba: (16/15 * 75/64 * 16/15)

Segah tetrachord: (16/15 * 9/8 * 10/9)

Huseyni tetrachord: (56/51 * 68/63 * 9/8) or (800/729 * 27/25 * 9/8)

Huzzam tetrachord: (29/27 * 9/8 * 32/29)

Saba pentachord: (56/51 * 153/140 * 320/297 * 297/256)

Ussak tetrachord: (88/81 * 12/11 * 9/8) 

Çargah tetrachord: (15/14 * 189/160 * 200/189) or maybe (15/14 * 189/160 *  256/243) if you want it to hit P4

In contrast to the glosses

Pure Rast tetrachord: (9/8) * (10/9) * (16/15)

Tempered Rast tetrachord: (28/25) * (28/25) * (17/16)

Mahur tetrachord: (9/8) * (9/8) * (256/243)

Pencgah tetrachord: (9/8) * (10/9) * (28/25)

Nihavend tetrachord: (9/8) * (16/15) * (10/9)

Segah tetrachord: (16/15) * (9/8) * (10/9)

Hicaz tetrachord: (16/15) * (7/6) * (15/14)

Wide Hicaz tetrachord: (16/15) * (20/17) * (17/16)

Huzzam tetrachord: (15/14) * (9/8) * (32/29)

Huseyni tetrachord: (11/10) * (13/12) * (9/8)

Ussak tetrachord: (12/11) * (12/11) * (9/8)

Çargah tetrachord: (15/14) * (13/11) * (55/52)

Saba pentachord: (11/10) * (12/11) * (13/12) * (15/13)

It's kind of crazy how often he has two expressions that are almost equal, but differ by a complex super particular ratio. Ozan Yarman's math would make a lot more sense in a system that tempered out the intervals justly associated with the frequency ratios [145/144, 225/224, 243/242, 406/405, 1701/1700, 2500/2499] . No EDO tempers out all of these. The only EDOs that temper out all but 1 are:

12-EDO, which tunes  243/242 to -1 step.

41-EDO, which tunes 2500/2499 to -1 step.

60-EDO, which tuned 243/242 to -1 step.

The members of the set [10, 19, 29, 53, 72, 82, 96]-EDO also do a decent job, tempering out all but 2 of Ozan Yarman's problematic commas.

: 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.

...

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.

:: The Analysis

...