Turkish Makams

: Intro

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.

In 1910, Rauf Yekta published a popular account of Turkish makam intonation using an unequal 24-tone scale that was just a Pythagorean spiral. Over the next four decades, Hüseyin Saadettin Arel and Suphi Ezgi and Murat Uzdilek developed a closely related theory of Turkish music based on 53-EDO, which provides an excellent finite representation of Pythagorean tuning and is also used extensively in this tradition. I think AEU theory has some problems, but it's detailed, concrete, and widely taught, so it will be our starting point.

: 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. And just as Arabic music theory nerds like to talk about the inadequacy of 24-EDO, Turkish music theory nerds like to talk about the inadequacy of 53-EDO. It's great!

Let's start by looking at how "nine commas in a 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 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, i.e. (P8, P12, M17), or the interval can be written with 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 

1. Pure octaves 

2. Tempering out the "schisma", or AcAcA0, justly tuned to 32805/32768 (definable as the difference between the Pythagorean and Syntonic commas).

3. Tempering out 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 aurally, 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 your tuning system. And that's what modern 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 Accidentals and 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. I can never remember it when looking at staff notation, but the main things you need to read AEU accidentals are that :

A backwards flat symbol indicated flattening by one comma relative to natural: "Ad"

A flat symbol with a slash through the stem indicates that the pitch is flattened by  four commas: "A\b"

A fllat symbol 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. There are many more accidentals, but these three are fairly easy to write out and they mostly get the job done.

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. The simgeler show steps between scale degrees of a makam. 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:

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

The Pythagorean A1 and d3 are something like neutral seconds in this system - one of them is about 23 cents sharp of Pyrthagorean m2 and one is about 23 cents flat of a Pythagorean M2. Clearly these are not as neutral as 24-EDO's neutral second. Since 53-EDO treats the Pythagorean comma and the syntonic comma the same, the Turkish middle seconds can also be interpreted as being alterations of Pythagorean tuning by syntonic commas, i.e. they're simply the rank-3 or 5-limit m2 and M2. This interpretation has the added benefit that scales can be spelled correctly: the steps 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. 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 - AcAcA2

The 13\53 steps could be interpreted as a few differankt rank-3 second intervals, like AcAcA2 or GrAA2. I'll argue later that simge A_13 should be associated with AcAcA2. 

: The Makams

I'll start with transcriptions of Turkish makams from "Esendere Kültür Sanat Derneği", https://www.eksd.org.tr/.  Turkish makams have ajnas (or the etymologically related "genera") just as Arabic maqamat do. Each jins provides a temporary tonic center for melodic exploration. A player or compose will noodle around on one jins and then move over to another one.

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].

: Turkish Ajnas

Here's a summary of the basic 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 3-limit and 5-limit analyses, although only the 5-limit one is spelled correctly in pitches. 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.