: 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, but Safi al-Din wrote more than his critics did and is better remembered. We have some have knowledge about what the practical 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. The Pythagorean analysis provides a different, historically important, and finer grained perspective on makam/maqam music than does 24-EDO, and I'm taking any data that I can get. Also, we'll use this Pythagorean analysis 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.
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].
Zirgüleli Hicaz makam: Tonic A. [S, A, S, T, S, A, S] # Hicaz pentachord [S, A, S, T] + Hicaz tetrachord [S, A, S]. [A, B/b, C#, D, E, Ft, 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 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)
We 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.