Makams And Maqamat II

: 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. As we discussed in a previous chapter, the use of an extended Pythagorean analysis for chromatic as well as neutral intervals goes back to Safi al-Din al-Urmawi. Things have changed a lot in middle eastern music theory in the last 700 years since Safi al-Din, but we're still kind of using Pythagorean analysis in modern Turkish makam theory. And that's what we'll focus on in this chapter.

: The Commas

In Turkish music theory, particularly the AEU system laid out by Arel and Ezgi and Uzdilek, the Pythagorean intervals are notated as natural pitches, and there are several accidentals used to describe microtonal deviations from them. 

One of the foundational ideas of the AEU system is 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. And so Turkish music theory is based on 53-EDO as much as Arabic music theory is based on 24-EDO, though Turkish music theory has the benefit that they can use either the Pythagorean comma or the Syntonic comma in determpering analysis, as convenient.

Let's start by looking at how "nine commas to the major second" is equivalent to using 53-EDO. If we want to temper out the difference between nine Pythagorean commas and a Pythagorean major second, we can do an easy tempering analysis in rank-2 interval space.  The difference is

9 * A0 - M2

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

in the (P8, P12) prime harmonic basis. This same interval is much more compact in the Lilley basis (A1, d2),  there having coordinates (-2, -10). The coordinates in either basis for this tiny difference interval 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 coordinates (-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 some of its integer multiples: [53, 106, 159, 212, ...]-EDO. So 53-EDO pops out when we say that there are nine Pythagorean commas to the rank-2 major second. Nice. Easy. Also 53-EDO tunes AcM2 to 9 steps and the Pythagorean comma to 1 step. That's another way of looking at it, if you just want to test every small EDO to see which one/ones have that behavior.

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).  This interval is justly tuned to 16777216000000000/16677181699666569, and is also tempered out by 53-EDO and some of its integer multiples, plus a few extras: (53, 106, 157, 159, 210, 212, ...)-EDO. So  53-EDO also pops out if we say that there are nine syntonic commas to the AcM2. And again, 53-EDO tunes the syntonic comma to 1 step. In summary, it doesn't super matter which of the commas you use. And this will turn out to be highly convenient, in that it lets us e.g. use rank-3 intervals that are in fact 2nds as neutral seconds, instead of using Pythagorean d3 and A1.

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 comma 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 grade school 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} without adornment indicates flattening a tone by 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.

I. Basic Turkish Makams

Çargâh makam: tonic C. [T, T, B, T, T, T, B] # Çargâh pentachord [T, T, B, T] + Çargâh tetrachord [T, T, B]. [C, D, E, F, G, A, B, C].

Bûselik makam (Kürdi ending): tonic A. [T, B, T, T, B, T, T] # Bûselik pentachord [T, B, T, T] + Kürdi tetrachord [B, T, T]. [A, B, C, D, E, F, G, A].

Bûselik makam (Hicaz ending): tonic A. [T, B, T, T, B, A, S] # Bûselik pentachord [T, B, T, T] + Hicaz tetrachord [B, A, S]. [A, B, C, D, E, F, G#, A].

(Basit) Şehnâz Bûselik (descends): tonic A. [-S, -A, -B, -T, -T, -B, -T] # Hicaz tetrachord [-S, -A, -B] + Bûselik pentachord [-T, -T, -B, -T]. [A, B, C, D, E, F, G#, A].

Kürdi makam: tonic A. [B, T, T, T, B, T, T] # Kürdi tetrachord [B, T, T] + Bûselik pentachord [T, B, T, T]. [A, Bb, C, D, E, F, G, A].

Rast makam: tonic G. [T, K, S, T, T, K, S] # Rast pentachord [T, K, S, T] + Rast tetrachord [T, K, S]. [G, A, Bd, C, D, E, F#, G].

Uşşak makam: tonic A. [K, S, T, T, B, T, T] # Uşşak tetrachord [K, S, T] + Bûselik pentachord [T, B, T, T]. [A, Bd, C, D, E, F, G, A]. // Website has a typo in the staff notation. It should be "Bd", not "B", so that Uşşak is the same as Beyâti.

Beyâti makam: tonic A. [K, S, T, T, B, T, T] # Uşşak tetrachord [K, S, T] + Bûselik pentachord [T, B, T, T]. [A, Bd, C, D, E, F, G, A].

(Beste) Isfahân makam (Uşşak ending) (descends): tonic A.  [-T, -T, -B, -T, -T, -S, -K] # Bûselik pentachord [-T, -T, -B, -T] + Uşşak tetrachord [-T, -S, -K].

(Beste) Isfahân makam (Rast ending) (descends): tonic A. [-T, -T, -B, -T, -S, -K, -T] # Bûselik pentachord [-T, -T, -B, -T] + Rast tetrachord [-S, -K, -T].

(Hicaz) Hümâyûn makam: Tonic A. [S, A, S, T, B, T, T] # Hicaz tetrachord [S, A, S] + Bûselik pentachord [T, B, T, T]. [A, B/b, C#, D, E, F, G, A].

Hicaz makam: Tonic A. [S, A, S, T, K, S, T] # Hicaz tetrachord [S, A, S] + Rast pentachord [T, K, S, T]. [A, B/b, C#, D, E, F#, G, A].

Uzzâl makam: Tonic A. [S, A, S, T, K, S, T] # Hicaz pentachord [S, A, S, T] + Uşşak tetrachord [K, S, T]. [A, B/b, C#, D, E, F#, G, A].

Hüseyni makam (ascending): Tonic A. [K, S, T, T, K, S, T] # Hüseyni pentachord [K, S, T, T] + Uşşak tetrachord [K, S, T]. [A, Bd, C, D, E, F#, G, A].

Hüseyni makam (sometimes when descending): Tonic A. [K, S, T, T, T, B, T] # Hüseyni pentachord [K, S, T, T] + Bûselik tetrachord [T, B, T]. [A, Bd, C, D, E, F, G, A]. 

Muhayyer makam (descends): Tonic A. [-T, -S, -K, -T, -T, -S, -K] # Uşşak tetrachord [-T, -S, -K] + Hüseyni pentachord [-T, -T, -S, -K]. [A, Bd, C, D, E, F#, G, A]. // Muhayyer can also be extended upward over the high "A" with a Bûselik pentachord [T, B, T, T].

Gülizâr makam: Just the descending form of Hüseyni makam.

Nevâ makam: Tonic A. [K, S, T, T, K, S, T] # Uşşak tetrachord [K, S, T] + Rast pentachord [T, K, S, T]. [A, Bd, C, D, E, F#, G, A] // Sometimes desends from the high "A" with a Bûselik tetrachord [T, T, B, T].

Tâhir makam (descends): Tonic A. [-T, -S, -K, -T, -T, -S, -K] # Rast pentachord [-T, -S, -K, -T] + Uşşak tetrachord [-T, -S, -K]. [A, G, F#, E, D, C, Bd, A]. // Sometimes expands upward from the high "A" with a Bûselik tetrachord, [T, B, T].

Karcığar makam: Tonic A. [K, S, T, S, A, S, T] # Uşşak tetrachord [K, S, T] + Hicaz pentachord [S, A, S, T]. [A, Bd, C, D, E/b, F, G, A]. // Sometimes extended upward from the "B" below the high "Aw" with a Bûselik tetrachord, [T, B, T, T].

(Basit) Sûzinâk makam: Tonic G. [T, K, S, T, S, A, S] # Rast pentachord [T, K, S, T] + Hicaz tetrachord [S, A, S]. [G, A, Bd, C, D, E/b, F#, G]. // Sometimes extended upward past the high "G" with a Bûselik pentachord [T, B, T, T].

II. Advanced Turkish makams

Mahur makam (descends): Tonic G. [-B, -T, -T, -T, -B, -T, -T] # Çargâh tetrachord [-B, -T, -T] + Çargâh pentachord [-T, -B, -T, -T]. [G, F#, E, D, C, B, A, G].

Acem Aşirân makam: Tonic F. [T, T, B, T, T, T, B] # Çargâh pentachord [T, T, B, T] + Çargâh tetrachord [T, T, B]. [F, G, A, Bb, C, D, E, F].

Nihavend makam: Tonic G. [T, B, T, T, B, T, T] # Bûselik pentachord [T, B, T, T] + Kürdi tetrachord [B, T, T]. [G, A, Bb, C, D, Eb, F, G].

Ruhnüvâz / Ruhnevâz makam (descends): Tonic E. [-T, -T, -B, -T, -T, -B, -T]. Kürdi pentachord [-T, -T, -B, -T] + Bûselik tetrachord [-T, -B, -T]. [E, F#, G, A, B, C, D#, E].

Here's a summary of the basis Turkish ajnas:  

Bûselik tetrachord [T, B, T]  // Bûselik pentachord [T, B, T, T]

Çargâh tetrachord [T, T, B] // Çargâh pentachord [T, T, B, T]

Rast tetrachord [T, K, S] // Rast pentachord [T, K, S, T]

Hicaz tetrachord [S, A, S] // Hicaz pentachord [S, A, S, T]

Uşşak tetrachord [K, S, T] // Uşşak pentachord [K, S, T, T]

Kürdi tetrachord [B, T, T] // Kürdi pentachord [B, T, T, T]

I think all of them have made an appearance but the  Kürdi pentachord. The pentachords here are all formed by adding a Pythagorean major second, with simge "T", to the related tetrachord. There are only two points of deviation from this summary as the ajnas are used in Esendere Kültür Sanat Derneği's makams: 1) the Uşşak pentachord is regularly called the "Hüseyni " pentachord. No biggie. 2) In two places the Hicaz tetrachord is spelled [B, A, S] instead of [S, A, S], namely in Bûselik makam (Hicaz ending) and (descending as [-S, -A, -B]) in (Basit) Şehnâz Bûselik makam. I don't think this is a mistake; jins Hicaz just has a more variable intonation than other ajnas, and maybe this is why we have an the simge "A" with an option of being 12 or 13 commas.

Common Hicaz tetrachord: [S, A, S] → [5, 12, 5] commas.

Buselik Hicaz tetrachord: [B, A, S] → [4, 13, 5] commas.

Nice. This helps us to solve what the 13-step A simge should be called as a rank-3 second interval: it has to be AcAcA2, since that's the interval which, when added to a Grm2 (the "B" simge) and an m2 (the "S" simge), produces a perfect fourth. Here's the interval arithmetic in the rank-3 Lilley basis, (Ac1, A1, d2):

Grm2 + AcAcA2 + m2 = P4

(-1, 1, 1) + (2, 3, 1)  + (0, 1, 1) = (1, 5, 3) 

Lets look at the basic six Turkish tetrachords in terms of their rank-3 intervals and 5-limit just tunings.

We have three Pythagorean tetrachords that are cyclic permutations of each other:

Çargâh tetrachord: [T, T, B] → [AcM2, AcM2, Grm2] # (9/8, 9/8, 256/243)

Bûselik tetrachord: [T, B, T] → [AcM2, Grm2, AcM2] # (9/8, 256/243, 9/8)

Kürdi tetrachord: [B, T, T] → [Grm2, AcM2, AcM2] # (256/243, 9/8, 9/8)

I'd like to note that "Çargâh" as a name for the major tetrachord and major scale are specific to modern Arel-Ezgi-Uzdilek theory. The more traditional name in Turkish music theory is Acem. The name Çargâh is based on the Persian mode Chahargah, with its characteristic repeated tetrachord Chahargah, and these don't sound anything like a major tetrachord /  major scale. I'll keep using the wrong term in this chapter, since this chapter is about AEU theory, but in general, I hope you will do better and use the historical name.

 We also have two simple 5-limit tetrachords that are cyclic permutations of each other:

Rast tetrachord: [T, K, S] → [AcM2, M2, m2] # (9/8, 10/9, 16/15)

Uşşak tetrachord: [K, S, T] → [M2, m2, AcM2] # (10/9, 16/15, 9/8)

And we have have two intonations of the Hicaz tetrachord, but they both have a 3-limit or 5-limit m2, a big jump to the major third, and then finish on P4:

Hicaz tetrachord (Common): [S, A_12, S] → [m2, AcA2, m2] # (16/15, 75/64, 16/15)

Hicaz tetrachord (Buselik intonation): [B, A_13, S] → [Grm2, AcAcA2, m2] # (256/243, 1215/1024, 16/15)

Nice.

:: More Turkish Makams in AEU simgeler

The website for Esendere Kültür Sanat Derneği often doesn't load images. Occasionally I'll come back and see some new images and learn some new makams. Here are a few that I don't think were covered above.

Makam Segah: [S, T, K, T, S, A, B] [Bd, C, D, Ed, F#, G, A#, Bd]

Structure: Segah pentachord [S, T, K, T] on Bd + Hicaz tetrachord [S, A, B] on F#.

Makam Hüzzam: [S, T, S, A, S, A, B] [(A#), Bd, C, D, E\b, F#, G, A#, Bd]

Structure: Huzzam pentachord [S, T, S, A] on Bd + Hicaz tetrachord [S, A, B] on F#.

Makam Aşkefzâ: [B, T, T, T, B, T, T, B, T, T] [E, F, G, A, B, C, D, E, (F, G, A)]

Structure: [B, T, T] + [T, B, T, T] + [B, T, T]. Tonic on E. Kurdi tetrachord + Buselik pentachord on A. Extends past the octave with another Kurdi tetrachord on E.

Makam Zirgüleli Hicaz: [S, A, S, T, S, A, S] [(G#), A, B\b, C#, D, E, Ft, G#, A]

Structure: Hicaz pentachord [S, A, S, T] on A + Hicaz tetrachord [S, A, S] on E.

Makam Sultâni Yegâh: [T, B, T, T, B, A, S] [D, E, F, G, A, Bb, C#, D]

Structure: Buselik pentachord [T, B, T, T] on D + Hicaz tetrachord [B, A, S] on A.

Makam Ferahnümâ: tonic on D. [B, T, T, T, B, T, T, B, T, T]. [D, Eb, F, G, A, Bb, C, D, Eb, F, G]

Structure: Kurdi tetrachord [B, T, T] on D + Buselik pentachord [T, B, T, T] on G + Kurdi tetrachord [B, T, T] on D.

Makam Sûzidil: tonic on E. [S, A, S, T, B, A, S] [E, Ft, G#, A, B, C, D#, E].

Structure: Hicaz pentachord [S, A, S, T] on E + Hicaz tetrachord [B, A, S] on B.

Makam Şedarabân: [S, A, S, T, S, A, S] [D, E\b, F#, G, A, B\b, C#, D]

Structure: Hicaz pentachord [S, A, S, T] on D + Hicaz tetrachord [S, A, S] on A.

Makam Reng-i Dil: [T, S, A, S, S, A, S]. [(Ed), F, G, A\b, Bd, C, D\b, Ed, F]

Structure: Nikriz pentachord [T, S, A, S] on F + Hicaz tetrachord [S, A, S] on A.

:: AEU Makams from Akkoc

The 2015 paper "Experiments on the Relationship between Perde and Seyir in Turkish Makam Music" by Can Akkoc, William A. Sethares, and Mustafa Kemal Karaosmanoğlu has a nice chart of twenty four Turkish makams. The makams don't have simgeler or EDO steps written in, but they do have AEU accidentals on staff notation. Their chart helpfully lists the tuned sized of the accidentals.

The normal flat accidental is tuned to -113.7 cents, like a Pythagorean augmented unison at (2187/2048). The backwards flat accidental is for flattening things by a comma of 23.5 cents. This corresponds to flattening by the Pythagorean comma, known in rank 2 interval space as the augmented zeroth, at (531441/524288). The syntonic comma at 81/80 is perceptually indistinguishable at 21.5 cents. There's also an accidental that looks like a flat symbol with a slash through it. This is a comma sharper than the -113.7 flat by one coma, meaning it has a size of -90.2 cents, which is the size of a pythagorean  minor second, Grm2, just tuned to 256/243, and also indistinguishable from the acute augmented unsison, AcA1, justly tuned to (135/128).

b = -113.7 cents # 1/(2187/2048)

\b = -90.2 cents # 1/(135/128) or 1/(256/243)

d = -23.5 cents # 1/(81/80) or 1/(531441/524288)

The sharpening accidentals are all mirrors of these:

t = +23.5 cents # (81/80) or (531441/524288)

# = +90.2 cents # (135/128) or (256/243)

#t = +113.7 cents # (2187/2048)

You might not have expected that {#} and {b} are differently sized. In AEU, {b} is the opposite of {#t}, while {\b} is the opposite of {#}. What can you do? In the full AEU system, there are five sharpening accidentals and five flattening accidentals. I can't remember all of them and my eyes aren't good enough to distinguish them on staff notation, so I'm quite pleased that Akkoc and company only used six.

In 53-EDO, the accidentals don't just have just tunings, they also have associated steps. The 23.5 cents accidentals are +1 or -1 step. The 90.2 cent accidentals are +4 or -4 steps. And the 113.7 cent accidentals are +5 or -5 steps. The remaining sharpening and flattening accidentals in AEU are associated with plus or minus 8 or 9 commas.

Here are the twenty four Turkish makams of Akkoc at alia:

Acemkürdi: [(G), A, Bb, C, D, E, F, G, A]. // Dominant on F.

Beyati: [(G), A, Bd, C, D, E, F, G, A]. // Dominant on D.

Buselik: [(G#), A, B, C, D, E, F, G, A]. // Dominant on E.

Evic: [(E#), F#, G, A, Bd, C#, D, E, F#]. // Dominant on A.

Hicaz: [(G), A, B\b, C#, D, E, F#, G, A]. // Dominant on D.

Hicazkar: [(F#), G, Ab, Bd, C, D, E\b, F#, G]. // Dominant on D.

Hüseyni: [(G), A, Bd, C, D, E, F#, G, A]. // Dominant on E.

Isfahan: [(G), A, Bd, C, D, E, F, G, A]. // Dominant on D.

Kürdi: [(G), A, Bb, C, D, E, F, G, A]. // Dominant on D.

Kurdilihicazkar: [(F), G, Ab, Bb, C, D, Eb, F, G]. // Dominant on C.

Mahur: [(F#t), G, A, B, C, D, E, F, G]. // Dominant on D.

Muhayyer: [(G), A, Bd, C, D, E, F#, G, A]. // Dominant on E.

Neva: [(G), A, Bd, C, D, E, F#, G, A]. // Dominant on D.

Nihavent: [(F#), G, A, Bb, C, D, Eb, F, G]. // Dominant on D.

Rast: [(F#), G, A, Bd, C, D, E, F#, G]. // Dominant on D.

Rehavi: [(F#), G, A, Bd, C, D, E, F#, G]. // Dominant on D.

Saba: [(G), A, Bd, C, D\b, E, F, G, A]. // Dominant on C.

Sazkâr: [(F#), G, A, Bd, C, D, E, F#, G]. // Dominant on D.

Segah: [(A#), Bd, C, D, Ed, F#, G, A#, B]. // Dominant on D.

Şehnaz: [(G), A, B\b, C#, D, E, Ft, G#, A]. // Dominant on E.

Sultaniyegah: [(C#), D, E, F, G, A, Bb, C#, D]. // Dominant on A.

Suzinak: [(F#), G, A, Bd, C, D, E\b, F#, G]. // Dominant on D.

Uşşak: [(G), A, Bd, C, D, E, F, G, A]. // Dominant on D.

Yegah: [(C#), D, E, F#, G, A, Bd, C, D]. // Dominant on A.

Every makam has a leading tone below the tonic, which I've placed in parentheses. The makams also have an annotated "dominant" tone, which is not always a perfect fifth over the tonic. The dominant makes a difference. For example, makams Muhayyer and Neva have the same key signature, tonic, and leading tone. They only differ on their dominant tone.

When transcribing these, I noticed that often the Makams will have flattening accidentals that proceed along the normal circle of fifth [Bb, Eb, Ab, ...], and sharpening accidentals that proceeda long the normal circle of fifths, [F#, C#, G#, ....], and that these happen independently. For example, makam Evic has both [Bd] and [F#, C#]. Some makam break with this trend, but it's definitely a trend. I wonder if that explains something about the origin of makams.

 It's nice to have data about so many Turkish makams, but how do we actually tune the individual pitches? Here are all the pitch classes that made an appearance: [C, C#, D\b, D, Eb, E\b, Ed, E, E#, F, Ft, F#, F#t, G, G#, Ab, A, A#, Bb, B\b, Bd, B]. We'll have to figure those out. First, what are the tunings of the natural pitches in AEU?

Wikipedia gives a chart with AEU pitch classes and their tunings as steps of 53-EDO over middle C. The table uses a fuller set of accidentals than my eyes are comfortable with, but the natural pitches are clear enough. Below I show the pitch, the step of 53-EDO over C4, and the cute little Turkish name for the pitch. The cute little names do not repeat at the octave, and don't make sense to me in other ways besides that, and I've never had the patience to learn them, but here they are.

C4: 0 _ Kaba Çârgâh

D4: 9 _ Yegâh

E4: 18 _ Hüseynî Aşîrân

F4: 22 _ Acem Aşîrân

G4: 31 _ Rast

A4: 40 _ Dügâh

B4: 49 _ Bûselik

C5: 53 _ Çârgâh

D5: 62 _ Nevâ

E5: 71 _ Hüseynî

F5: 75 _ Acem

G5: 84 _ Gerdâniye

A5: 93 _ Muhayyer

B5: 102 _ Tîz Bûselik

C6: 106 _ Tîz Çârgâh

I strongly support using rank-3 interval names and 5-limit tuning for analyzing the natural pitches of Turkish music. Here are the obvious intervals assocaited with each of those steps of 53-EDO:

C4: 0 _ P1

D4: 9 _ AcM2

E4: 18 _ AcM3

F3: 22 _ P4

G4: 31 _ P5

A4: 40 _ AcM6

B4: 49 _ AcM7

C5: 53 _ P8

We can see that in AEU, the natural pitches are Pythagorean. And we already have tunings for the accidentals, so that should do it, right? We can figure out EDO steps and 5-limit tunings for all twenty four of the makams presented by Akkoc, Sethares, and Karaosmanoğlu.

First, here's a plain-text table with EDO steps for different pitches:

C4: 0

Ct4: 1

C#4: 4

C#t4: 5

Db4: 4

D\b4: 5

Dd4: 8

D4: 9

Dt4: 10

D#4: 13

D#t4: 14

Eb4: 13

E\b4: 14

Ed4: 17

E4: 18

Et4: 19

E#4: 22

E#t4: 23

Fb4: 17

F\b4: 18

Fd4: 21

F4: 22

Ft4: 23

F#4: 26

F#t4: 27

Gb4: 26

G\b4: 27

Gd4: 30

G4: 31

Gt4: 32

G#4: 35

G#t4: 36

Ab4: 35

A\b4: 36

Ad4: 39

A4: 40

At4: 41

A#4: 44

A#t4: 45

Bb4: 44

B\b4: 45

Bd4: 48

B4: 49

Bt4: 50

B#4: 53

B#t4: 54

Cb5: 48

C\b5: 49

Cd5: 52

C5: 53

Ct5: 54

C#5: 57

C#t5: 58

Db5: 57

D\b5: 58

Dd5: 61

D5: 62

Dt5: 63

D#5: 66

D#t5: 67

Eb5: 66

E\b5: 67

Ed5: 70

E5: 71

Et5: 72

E#5: 75

E#t5: 76

Fb5: 70

F\b5: 71

Fd5: 74

F5: 75

Ft5: 76

F#5: 79

F#t5: 80

Gb5: 79

G\b5: 80

Gd5: 83

G5: 84

Gt5: 85

G#5: 88

G#t5: 89

Ab5: 88

A\b5: 89

Ad5: 92

A5: 93

At5: 94

A#5: 97

A#t5: 98

Bb5: 97

B\b5: 98

Bd5: 101

B5: 102

Bt5: 103

B#5: 106

B#t5: 107

Cb6: 101

C\b6: 102

Cd6: 105

C6: 106

Next, here are the makams with each scale degree specifies as a step of 53-EDO over C4: 

Acemkürdi : [(31), 40, 44, 53, 62, 71, 75, 84, 93]

Beyati : [(31), 40, 48, 53, 62, 71, 75, 84, 93]

Buselik : [(35), 40, 49, 53, 62, 71, 75, 84, 93]

Evic : [(22), 22, 31, 40, 48, 4, 62, 71, 79]

Hicaz : [(31), 40, 45, 4, 62, 71, 79, 84, 93]

Hicazkar : [(26), 31, 35, 48, 53, 62, 67, 79, 84]

Hüseyni : [(31), 40, 48, 53, 62, 71, 79, 84, 93]

Isfahan : [(31), 40, 48, 53, 62, 71, 75, 84, 93]

Kürdi : [(31), 40, 44, 53, 62, 71, 75, 84, 93]

Kurdilihicazkar : [(22), 31, 35, 44, 53, 62, 66, 75, 84]

Mahur : [(27), 31, 40, 49, 53, 62, 71, 75, 84]

Muhayyer : [(31), 40, 48, 53, 62, 71, 79, 84, 93]

Neva : [(31), 40, 48, 53, 62, 71, 79, 84, 93]

Nihavent : [(26), 31, 40, 44, 53, 62, 66, 75, 84]

Rast : [(26), 31, 40, 48, 53, 62, 71, 79, 84]

Rehavi : [(26), 31, 40, 48, 53, 62, 71, 79, 84]

Saba : [(31), 40, 48, 53, 58, 71, 75, 84, 93]

Sazkâr : [(26), 31, 40, 48, 53, 62, 71, 79, 84]

Segah : [(44), 48, 53, 62, 70, 79, 84, 97, 102]

Şehnaz : [(31), 40, 45, 4, 62, 71, 76, 88, 93]

Sultaniyegah : [(4), 9, 18, 22, 31, 40, 44, 4, 62]

Suzinak : [(26), 31, 40, 48, 53, 62, 67, 79, 84]

Uşşak : [(31), 40, 48, 53, 62, 71, 75, 84, 93]

Yegah : [(4), 9, 18, 26, 31, 40, 48, 53, 62]

Here's a rank-3 detempering of those edo steps, with all intervals phrased over C4:

Acemkürdi : [(P5), AcM6, Grm7, P8, AcM9, AcM10, P11, P12, AcM13]

Beyati : [(P5), AcM6, M7, P8, AcM9, AcM10, P11, P12, AcM13]

Buselik : [(AcA5), AcM6, AcM7, P8, AcM9, AcM10, P11, P12, AcM13]

Evic : [(AcAcA3), AcA4, P5, AcM6, M7, AcA8, AcM9, AcM10, AcA11]

Hicaz : [(P5), AcM6, m7, AcA8, AcM9, AcM10, AcA11, P12, AcM13]

Hicazkar : [(AcA4), P5, Grm6, M7, P8, AcM9, m10, AcA11, P12]

Hüseyni : [(P5), AcM6, M7, P8, AcM9, AcM10, AcA11, P12, AcM13]

Isfahan : [(P5), AcM6, M7, P8, AcM9, AcM10, P11, P12, AcM13]

Kurdilihicazkar : [(P4), P5, Grm6, Grm7, P8, AcM9, Grm10, P11, P12]

Kürdi : [(P5), AcM6, Grm7, P8, AcM9, AcM10, P11, P12, AcM13]

Mahur : [(AcAcA4), P5, AcM6, AcM7, P8, AcM9, AcM10, P11, P12]

Muhayyer : [(P5), AcM6, M7, P8, AcM9, AcM10, AcA11, P12, AcM13]

Neva : [(P5), AcM6, M7, P8, AcM9, AcM10, AcA11, P12, AcM13]

Nihavent : [(AcA4), P5, AcM6, Grm7, P8, AcM9, Grm10, P11, P12]

Rast : [(AcA4), P5, AcM6, M7, P8, AcM9, AcM10, AcA11, P12]

Rehavi : [(AcA4), P5, AcM6, M7, P8, AcM9, AcM10, AcA11, P12]

Saba : [(P5), AcM6, M7, P8, m9, AcM10, P11, P12, AcM13]

Sazkâr : [(AcA4), P5, AcM6, M7, P8, AcM9, AcM10, AcA11, P12]

Segah : [(AcAcA6), M7, P8, AcM9, M10, AcA11, P12, AcAcA13, AcM14]

Şehnaz : [(P5), AcM6, m7, AcA8, AcM9, AcM10, Ac11, AcA12, AcM13]

Sultaniyegah : [(AcA1), AcM2, AcM3, P4, P5, AcM6, Grm7, AcA8, AcM9]

Suzinak : [(AcA4), P5, AcM6, M7, P8, AcM9, m10, AcA11, P12]

Uşşak : [(P5), AcM6, M7, P8, AcM9, AcM10, P11, P12, AcM13]

Yegah : [(AcA1), AcM2, AcM3, AcA4, P5, AcM6, M7, P8, AcM9]

Here are the 5-limit just tunings for that detempering:

Acemkürdi : [3/2, 27/16, 16/9, 2/1, 9/4, 81/32, 8/3, 3/1, 27/8]

Beyati : [3/2, 27/16, 15/8, 2/1, 9/4, 81/32, 8/3, 3/1, 27/8]

Buselik : [405/256, 27/16, 243/128, 2/1, 9/4, 81/32, 8/3, 3/1, 27/8]

Evic : [10935/8192, 45/32, 3/2, 27/16, 15/8, 135/64, 9/4, 81/32, 45/16]

Hicaz : [3/2, 27/16, 9/5, 135/64, 9/4, 81/32, 45/16, 3/1, 27/8]

Hicazkar : [45/32, 3/2, 128/81, 15/8, 2/1, 9/4, 12/5, 45/16, 3/1]

Hüseyni : [3/2, 27/16, 15/8, 2/1, 9/4, 81/32, 45/16, 3/1, 27/8]

Isfahan : [3/2, 27/16, 15/8, 2/1, 9/4, 81/32, 8/3, 3/1, 27/8]

Kurdilihicazkar : [4/3, 3/2, 128/81, 16/9, 2/1, 9/4, 64/27, 8/3, 3/1]

Kürdi : [3/2, 27/16, 16/9, 2/1, 9/4, 81/32, 8/3, 3/1, 27/8]

Mahur : [729/512, 3/2, 27/16, 243/128, 2/1, 9/4, 81/32, 8/3, 3/1]

Muhayyer : [3/2, 27/16, 15/8, 2/1, 9/4, 81/32, 45/16, 3/1, 27/8]

Neva : [3/2, 27/16, 15/8, 2/1, 9/4, 81/32, 45/16, 3/1, 27/8]

Nihavent : [45/32, 3/2, 27/16, 16/9, 2/1, 9/4, 64/27, 8/3, 3/1]

Rast : [45/32, 3/2, 27/16, 15/8, 2/1, 9/4, 81/32, 45/16, 3/1]

Rehavi : [45/32, 3/2, 27/16, 15/8, 2/1, 9/4, 81/32, 45/16, 3/1]

Saba : [3/2, 27/16, 15/8, 2/1, 20/9, 81/32, 8/3, 3/1, 27/8]

Sazkâr : [45/32, 3/2, 27/16, 15/8, 2/1, 9/4, 81/32, 45/16, 3/1]

Segah : [3645/2048, 15/8, 2/1, 9/4, 5/2, 45/16, 3/1, 3645/1024, 243/64]

Şehnaz : [3/2, 27/16, 9/5, 135/64, 9/4, 81/32, 27/10, 405/128, 27/8]

Sultaniyegah : [135/128, 9/8, 81/64, 4/3, 3/2, 27/16, 16/9, 135/64, 9/4]

Suzinak : [45/32, 3/2, 27/16, 15/8, 2/1, 9/4, 12/5, 45/16, 3/1]

Uşşak : [3/2, 27/16, 15/8, 2/1, 9/4, 81/32, 8/3, 3/1, 27/8]

Yegah : [135/128, 9/8, 81/64, 45/32, 3/2, 27/16, 15/8, 2/1, 9/4]

...

Here are those frequency ratios rooted on the tonic of each makam, so that the first frequency ratios - the leading tone - is less than 1/1:

Acemkürdi : [8/9, 1/1, 256/243, 32/27, 4/3, 3/2, 128/81, 16/9, 2/1]

Beyati : [8/9, 1/1, 10/9, 32/27, 4/3, 3/2, 128/81, 16/9, 2/1]

Buselik : [15/16, 1/1, 9/8, 32/27, 4/3, 3/2, 128/81, 16/9, 2/1]

Evic : [243/256, 1/1, 16/15, 6/5, 4/3, 3/2, 8/5, 9/5, 2/1]

Hicaz : [8/9, 1/1, 16/15, 5/4, 4/3, 3/2, 5/3, 16/9, 2/1]

Hicazkar : [15/16, 1/1, 256/243, 5/4, 4/3, 3/2, 8/5, 15/8, 2/1]

Hüseyni : [8/9, 1/1, 10/9, 32/27, 4/3, 3/2, 5/3, 16/9, 2/1]

Isfahan : [8/9, 1/1, 10/9, 32/27, 4/3, 3/2, 128/81, 16/9, 2/1]

Kurdilihicazkar : [8/9, 1/1, 256/243, 32/27, 4/3, 3/2, 128/81, 16/9, 2/1]

Kürdi : [8/9, 1/1, 256/243, 32/27, 4/3, 3/2, 128/81, 16/9, 2/1]

Mahur : [243/256, 1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 16/9, 2/1]

Muhayyer : [8/9, 1/1, 10/9, 32/27, 4/3, 3/2, 5/3, 16/9, 2/1]

Neva : [8/9, 1/1, 10/9, 32/27, 4/3, 3/2, 5/3, 16/9, 2/1]

Nihavent : [15/16, 1/1, 9/8, 32/27, 4/3, 3/2, 128/81, 16/9, 2/1]

Rast : [15/16, 1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1]

Rehavi : [15/16, 1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1]

Saba : [8/9, 1/1, 10/9, 32/27, 320/243, 3/2, 128/81, 16/9, 2/1]

Sazkâr : [15/16, 1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1]

Segah : [243/256, 1/1, 16/15, 6/5, 4/3, 3/2, 8/5, 243/128, 81/40]

Şehnaz : [8/9, 1/1, 16/15, 5/4, 4/3, 3/2, 8/5, 15/8, 2/1]

Sultaniyegah : [15/16, 1/1, 9/8, 32/27, 4/3, 3/2, 128/81, 15/8, 2/1]

Suzinak : [15/16, 1/1, 9/8, 5/4, 4/3, 3/2, 8/5, 15/8, 2/1]

Uşşak : [8/9, 1/1, 10/9, 32/27, 4/3, 3/2, 128/81, 16/9, 2/1]

Yegah : [15/16, 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 16/9, 2/1]

Those rooted frequency ratios are the just tunings of these intervals:

Acemkürdi : [(Grm0), P1, Grm2, Grm3, P4, P5, Grm6, Grm7, P8]

Beyati : [(Grm0), P1, M2, Grm3, P4, P5, Grm6, Grm7, P8]

Buselik : [(M0), P1, AcM2, Grm3, P4, P5, Grm6, Grm7, P8]

Evic : [(AcM0), P1, m2, m3, P4, P5, m6, m7, P8]

Hicaz : [(Grm0), P1, m2, M3, P4, P5, M6, Grm7, P8]

Hicazkar : [(M0), P1, Grm2, M3, P4, P5, m6, M7, P8]

Hüseyni : [(Grm0), P1, M2, Grm3, P4, P5, M6, Grm7, P8]

Isfahan : [(Grm0), P1, M2, Grm3, P4, P5, Grm6, Grm7, P8]

Kurdilihicazkar : [(Grm0), P1, Grm2, Grm3, P4, P5, Grm6, Grm7, P8]

Kürdi : [(Grm0), P1, Grm2, Grm3, P4, P5, Grm6, Grm7, P8]

Mahur : [(AcM0), P1, AcM2, AcM3, P4, P5, AcM6, Grm7, P8]

Muhayyer : [(Grm0), P1, M2, Grm3, P4, P5, M6, Grm7, P8]

Neva : [(Grm0), P1, M2, Grm3, P4, P5, M6, Grm7, P8]

Nihavent : [(M0), P1, AcM2, Grm3, P4, P5, Grm6, Grm7, P8]

Rast : [(M0), P1, AcM2, M3, P4, P5, AcM6, M7, P8]

Rehavi : [(M0), P1, AcM2, M3, P4, P5, AcM6, M7, P8]

Saba : [(Grm0), P1, M2, Grm3, Gr4, P5, Grm6, Grm7, P8]

Sazkâr : [(M0), P1, AcM2, M3, P4, P5, AcM6, M7, P8]

Segah : [(AcM0), P1, m2, m3, P4, P5, m6, AcM7, Ac8]

Şehnaz : [(Grm0), P1, m2, M3, P4, P5, m6, M7, P8]

Sultaniyegah : [(M0), P1, AcM2, Grm3, P4, P5, Grm6, M7, P8]

Suzinak : [(M0), P1, AcM2, M3, P4, P5, m6, M7, P8]

Uşşak : [(Grm0), P1, M2, Grm3, P4, P5, Grm6, Grm7, P8]

Yegah : [(M0), P1, AcM2, M3, P4, P5, M6, Grm7, P8]

When I look for Turkish makams I might be missing, I see makam Zengüle, but it seems to be a synonym for makam Zirgüleli. And also Zirgüleli has a few named forms differing on the tonic:

 Makam Zirgüleli Hicaz:

Pitch classes: [A, B\b, C#, D, E, Ft, G#, A]

Structure: Hicaz pentachord [S, A, S, T] + Hicaz tetrachord [S, A, S].

Makam Zirgüleli Sûzinâk:

Pitch classes: [G, A\b, Bd, C, D, E\b, F#, G].

Structure: Hicaz pentachord [S, A, S, T] + Hicaz tetrachord [S, A, S].

Makam Şedarabân / Şedd-i Arabân:

Pitch classes: [D, E\b, F#, G, A, B\b, C, D]

Structure: Hicaz pentachord [S, A, S, T] + Hicaz tetrachord [S, A, S].

Makam Sûz-i Dil:

Pitch classes: [E, Ft, G#, A, B, C, D#, E]

Structure: Hicaz pentachord [S, A, S, T] + Hicaz tetrachord [S, A, S].

They have the same structure, just different tonics. But that matters a bit in middle eastern music. There's a makam that's basically identical but with a different intonation of Hicaz called Evcârâ.

Makam Evcârâ:

Pitch classes: [F#, G, A#, Bd, C#, D, E#, F#]

Structure: [S, A_13, B, T] + [S, A_13, B]

I may occasionally fill in other Turkish makams of lesser fame and importance if I learn about them, but for the most part, this analysis is done. This is what Turkish makams sound like at the resolution of 53-EDO, which readily offers both rank2 / 3-limit and rank-3 / 5-limit analyses, although only the 5-limit one is spelled correctly in intervals. In a later post we'll talk about just tunings of some Turkish makams and tetrachords that have 13-limit frequency ratios, based on tetrachords of Zalzal, al Farabi, and Ibn Sina, and some higher prime limit intonations inspired by Ozan Yarman that are less historical, less regular, and less logical than the Zalzalian intonations, but still interesting.