I've heard that 53-EDO is used productively to analyze Turkish music. It can be defined by pure octaves and tempering out ddddddd6, which as coordinates (0, -1) in the (A1, d2) basis.
t(P8) = 2/1
t(ddddddd6) = 1/1
The step size of 53-EDO, 2^(1/53), is
1200 * log(2^(1/53)) / log(2) = 1200/53 ~ 22.6 cents
which is really close to the Pythagorean comma, used in analyzing rank-2, 3-limit just intonation music. The Pythagorean comma is the augmented zeroth interval, A0, which Pythagorean tuning tunes to a frequency ratio of 531441/524288 ~= 23.46 cents. Quite close to the step of 53-EDO at 22.6 cents.
The step of 53-EDO is also quite close to the justly tuned syntonic comma, also called the acute unison or Ac1. In normal 5-limit just intonation, this interval is tuned to a frequency ratio of 81/80 ~= 21.5 cents. The acute unison is used in analyzing rank-3, 5-limit just intonation music. Humans can only discriminate about 5-cent differences in frequency, so the justly tuned A0, the 53-EDO tuned A0, and the 5-limit Ac1 are are basically equal as far as we can hear.
I haven't read original sources from Turkish music theorists on 53-EDO, but just the fact that the two commas are mentioned as justifications for 53-EDO in western sources makes me want to give 53-EDO both a rank-2, 3-limit and a rank-3, 5-limit interpretation.
Let's start by finding rank-1 intervals with short names for every 53-EDO step. To do this, I just tune various small intervals, increasing the complexity until I've got one tuned to each of the 53 steps. The results are:
2^(0/53) - P1 : (0, 0)
2^(1/53) - A0 : (0, -1)
2^(2/53) - dddd4 : (1, 3)
2^(3/53) - dd3 : (1, 2)
2^(4/53) - m2 : (1, 1)
2^(5/53) - A1 : (1, 0)
2^(6/53) - AA0 : (1, -1)
2^(7/53) - ddd4 : (2, 3)
2^(8/53) - d3 : (2, 2)
2^(9/53) - M2 : (2, 1)
2^(10/53) - AA1 : (2, 0)
2^(11/53) - dddd5 : (3, 4)
2^(12/53) - dd4 : (3, 3)
2^(13/53) - m3 : (3, 2)
2^(14/53) - A2 : (3, 1)
2^(15/53) - dddd6 : (4, 5)
2^(16/53) - ddd5 : (4, 4)
2^(17/53) - d4 : (4, 3)
2^(18/53) - M3 : (4, 2)
2^(19/53) - AA2 : (4, 1)
2^(20/53) - ddd6 : (5, 5)
2^(21/53) - dd5 : (5, 4)
2^(22/53) - P4 : (5, 3)
2^(23/53) - A3 : (5, 2)
2^(24/53) - dddd7 : (6, 6)
2^(25/53) - dd6 : (6, 5)
2^(26/53) - d5 : (6, 4)
2^(27/53) - A4 : (6, 3)
2^(28/53) - AA3 : (6, 2)
2^(29/53) - ddd7 : (7, 6)
2^(30/53) - d6 : (7, 5)
2^(31/53) - P5 : (7, 4)
2^(32/53) - AA4 : (7, 3)
2^(33/53) - dddd8 : (8, 7)
2^(34/53) - dd7 : (8, 6)
2^(35/53) - m6 : (8, 5)
2^(36/53) - A5 : (8, 4)
2^(37/53) - dddd9 : (9, 8)
2^(38/53) - ddd8 : (9, 7)
2^(39/53) - d7 : (9, 6)
2^(40/53) - M6 : (9, 5)
2^(41/53) - AA5 : (9, 4)
2^(42/53) - ddd9 : (10, 8)
2^(43/53) - dd8 : (10, 7)
2^(44/53) - m7 : (10, 6)
2^(45/53) - A6 : (10, 5)
2^(46/53) - dddd10 : (11, 9)
2^(47/53) - dd9 : (11, 8)
2^(48/53) - d8 : (11, 7)
2^(49/53) - M7 : (11, 6)
2^(50/53) - AA6 : (11, 5)
2^(51/53) - ddd10 : (12, 9)
2^(52/53) - d9 : (12, 8)
2^(53/53) - P8 : (12, 7)
I've also listed the interval coordinates in Lilley's (A1, d2) basis at the end of each line.
Supposedly not all of these steps are used in Turkish music. Unless you're transposing around maybe. But normal practice is to just use intervals with step sized of: [0, 4, 5, 8, 9, 13, 14, 17, 18, 22, 23, 26, 27, 30, 31, 35, 36, 39, 40, 44, 45, 48, 49]. This set makes a lot of sense! If we just remove all the lines that have highly modified intervals - those that are twice or more diminished or twice or more augmented -, then we get the Turkish set at steps [0, 4, 5, ...] plus
2^(1/53) - A0 : (0, -1)
2^(52/53) - d9 : (12, 8)
which are just the 53-EDO step and its octave complement. A0 is also the formal name for the Pythagorean comma.
Here are the rest of the natural and once-modified intervals, the ones corresponding to steps of [0, 4, 5, 8, 9, 13, 14, 17, 18, 22, 23, 26, 27, 30, 31, 35, 36, 39, 40, 44, 45, 48, 49, 52]:
2^(0/53) - P1 : (0, 0)
2^(4/53) - m2 : (1, 1)
2^(5/53) - A1 : (1, 0)
2^(8/53) - d3 : (2, 2)
2^(9/53) - M2 : (2, 1)
2^(13/53) - m3 : (3, 2)
2^(14/53) - A2 : (3, 1)
2^(17/53) - d4 : (4, 3)
2^(18/53) - M3 : (4, 2)
2^(22/53) - P4 : (5, 3)
2^(23/53) - A3 : (5, 2)
2^(26/53) - d5 : (6, 4)
2^(27/53) - A4 : (6, 3)
2^(30/53) - d6 : (7, 5)
2^(31/53) - P5 : (7, 4)
2^(35/53) - m6 : (8, 5)
2^(36/53) - A5 : (8, 4)
2^(39/53) - d7 : (9, 6)
2^(40/53) - M6 : (9, 5)
2^(44/53) - m7 : (10, 6)
2^(45/53) - A6 : (10, 5)
2^(48/53) - d8 : (11, 7)
2^(49/53) - M7 : (11, 6)
Those are normal things! We know how to use them fairly well in western music theory! We probably don't even need a rank-3 analysis with the syntonic comma, but I intend to do that anyway.
This also has intervals between the minor and major versions of (2nds, 3rds, 6ths, 7ths) which is good. Middle eastern music definitely has extra/intermediate/neutral versions of the imperfect ordinal intervals.
If you look up the names for the notes associated with the Turkish subset of 53-EDO on Wikipedia, you get a table with a bunch of Turkish words you can't read and some accidentals you've never seen before (in the Arel-Ezgi-Uzdilek notation column). The analysis above does away with that! We don't need to call 2^(5/53) "the thing that's one comma sharp of a minor second" with a new symbol, it's just an augmented unison! This analysis also reveals what I'm tempted to call a mistake in the Arel-Ezgi-Uzdilek system: F# and Gb aren't the same in 53-EDO: 26 steps is the tuned frequency for the diminished 5th, i.e. Gb over C, and 27 steps is tuned frequency for the augmented 4th, i.e. F# over C. But A-E-U have them both at 26 steps.
I'm going to want to refer to that list of intervals frequently, so I'll condense it here: these are the intervals of Turkish music we arrive at from the rank-2 analysis of 53-EDO with restrictions to steps [0, 4, 5, ...] and so on:
[P1, m2, A1, d3, M2, m3, A2, d4, M3, P4, A3, d5, A4, d6, P5, m6, A5, d7, M6, m7, A6, d8, M7]
These natural and once-modified intervals appear in the same order in 53-EDO as they do in Pythagorean tuning, but in a different order from quarter comma meantone or 5-limit just intonation.
How do we make a rank-3 version of 53-EDO?
I don't really know. I haven't thought a ton about turning 5-limit just intonation into different EDOs. But I think I've got a solution for this specific EDO. Obviously we still want pure octaves. And it would be ideal if the acute unison, Ac1, was tuned to 1^(1/53). That's two parameters fixed. If they're independent, then we still need to fix a third parameter. I think they are independent, and I think the third parameter is naturally fixed by tuning the interval Gr5, which has coordinates (1, 7, 4) in the rank-3 Lilley basis of (Ac1, A1, d2). Normally it's tuned to 40/27 in 5-limit just intonation. We could also use the interval with coordinates (3, 5, 3), which 5-limit just intonation tunes to 2187/1600. Not as good. The (1, 7, 4) interval makes the system of basis vectors of have determinant -1 and the (3, 5, 3) intervals creates a basis matrix with determinant +1. They both make unimodular systems.
Next I want a function to convert interval coordinates from the rank-3 Lilley basis, (Ac1, A1, d2) to this new basis, (Ac1, Gr5, P8). If you've been reading this blog for a while, you probably know the drill: For a change of basis, find the basis vectors of the old system expressed in the new system:
(1, 0, 0) = Ac1
(5, 7, -4) = A1
(-9, -12, 7) = d2
And convert rows to columns:
def convert_rank3_Lilly_basis_to_53edo_basis(interval):
(x, y, z) = interval
a = x * 1 + y * 5 + z * -9
b = x * 0 + y * 7 + z * -12
c = x * 0 + y * -4 + z * 7
return (a, b, c)
From the old coordinates (in the rank-3 Lilley basis for 5-limit just intonation intervals) like
d1 = (0, -1, 0)
P1 = (0, 0, 0)
d2 = (0, 0, 1)
A1 = (0, 1, 0)
m2 = (0, 1, 1)
M2 = (0, 2, 1)
A2 = (0, 3, 1)
d3 = (1, 2, 2)
m3 = (1, 3, 2)
M3 = (1, 4, 2)
d4 = (1, 4, 3)
A3 = (1, 5, 2)
P4 = (1, 5, 3)
A4 = (1, 6, 3)
d5 = (2, 6, 4)
P5 = (2, 7, 4)
d6 = (2, 7, 5)
A5 = (2, 8, 4)
m6 = (2, 8, 5)
M6 = (2, 9, 5)
A6 = (2, 10, 5)
d7 = (3, 9, 6)
m7 = (3, 10, 6)
M7 = (3, 11, 6)
d8 = (3, 11, 7)
A7 = (3, 12, 6)
P8 = (3, 12, 7)
and this change of basis function, we can now get interval coordinates in the (Ac1, Gr5, P8) basis, which we can then tune using the constraints
t(Ac1) = 2^(1/53)
t(Gr5) = ???
t(P8) = 2/1
to investigate whether we can get a 53-EDO with a rank-3 intervallic interpretation, so that we can reverse the interpretation and rational 5-limit just intonation to analyze 53-EDO Turkish music. It's a good plan.
I tried t(Gr5) = 40/27, and it looks bad. And of course it does; there's no place for nice 5-limit fractions in 53-EDO. Let's now try the closest number of steps in 53-EDO to 40/27. We can find this by inverting
2^(x / 53) = 40/27
x = 53 * log(40/27) / log(2) = 30.053...
So let's try t(Gr5) = 2^(30/53). Hopefully once we do this, we'll get a a 53-EDO, and from the intervals that are tempered out, we can find a better way to define 53-EDO.
Woo! It's working really well. There were lots of intervals tuned to the same frequency ratios. The difference between any two intervals tuned to the same frequency ratio will be another interval which the tuning system tempers out. There happened to be two intervals that showed up again and again. I'll just show one example of each. The acute unison and the grave diminished second were tuned to the same 53-EDO frequency ratio. From this, we can say that the difference between them is tempered out. I'll show the difference using coordinates in the rank-3 Lilley basis:
Ac1 - Grd2 :: [1, 0, 0] - [-1, 0, 1] = [2, 0, -1]
Grd2 - Ac1 :: [-1, 0, 1] - [1, 0, 0] = [-2, 0, 1]
The difference in either direction is tempered out, so I've shown both differences. You just negate the coordinates.
The second tempered out interval can be see as the difference between a grave acute unison and a diminished second:
GrA1 - d2 :: [-1, 1, 0] - [0, 0, 1] = [-1, 1, -1]
d2 - GrA1 :: [0, 0, 1] - [-1, 1, 0] = [1, -1, 1]
Are you shaking with excitement yet? These are both forms of famous intervals! The interval [2, 0, -1] is the grave grave diminished zeroth, GrGrd0, better known as the Schisma! In 5-limit just intonation, it's tuned to 32805/32768. The interval [-1, 1, -1] is an Acute diminished diminished zeroth, Acdd0, better known as the Kleisma! In 5-limit just intonation, it's tuned to 15625/15552. So it looks like 53-EDO might be definable by pure octaves and tempering out the Schisma and the Kleisma?
Sadly, this fact is stated fairly openly on the wikipedia page for 53-EDO:
"The 53-TET tuning equates to the unison, or tempers out, the intervals 32805/32768, known as the schisma, and 15625/15552, known as the kleisma. ... The fact that 53 ET tempers out both characterizes it completely as a 5 limit temperament: it is the only regular temperament tempering out both of these intervals, or commas."
But I never knew what it meant before! Because it's mathematical nonsense to equate a positive fraction like 32805/32768 with unison. I still stand by that! The wikipedia page is obviously and pathetically wrong. And it's so sad. Someone clearly did the math right, but then someone wrote about it very wrong. Maybe even the same person. It turns out that the Japanese guy who named the Kleisma also figured out that 53-EDO tempers it out, and he popularized it, and he very well might be the reason why modern Turkish music theorists think 53-EDO is a cool thing to use for analysis. Sorry, Shohe Tanaka. I hope you knew the difference between intervals and frequency ratios. If so, we've done you dirty. If not, I'm still impressed that you got the math right, if not the communication of it.
Okay, rant over and nods to history completed. Now what are the actual rank-3 interval names for the steps of 53-EDO? How are those intervals justly tuned? How do the 5-limit just tunings of the rank-3 intervals compare to the 3-limit/Pythagorean tunings of the rank-2 intervals?
Here are some short rank-3 intervals for each 53-EDO step:
0: P1
1: Ac1, Grd2
2: GrA1, d2
3: A1, Acd2
4: AcA1, Grm2
5: m2
6: Acm2
7: GrM2
8: M2
9: AcM2
10: GrA2, Grd3
11: A2, d3
12: AcA2, Acd3
13: Grm3
14: m3
15: Acm3
16: GrM3
17: M3
18: AcM3, Grd4
19: GrA3, d4
20: A3, Acd4
21: AcA3, Gr4
22: P4
23: Ac4
24: GrA4
25: A4
26: AcA4
27: Grd5
28: d5
29: Acd5
30: Gr5
31: P5
32: Ac5, Grd6
33: GrA5, d6
34: A5, Acd6
35: AcA5, Grm6
36: m6
37: Acm6
38: GrM6
39: M6
40: AcM6
41: GrA6, Grd7
42: A6, d7
43: AcA6, Acd7
44: Grm7
45: m7
46: Acm7
47: GrM7
48: M7
49: AcM7, Grd8
50: GrA7, d8
51: A7, Acd8
52: AcA7, Gr8
53: P8
If we restrict ourselves 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], we get... this set:
0: P1
4: AcA1, Grm2
5: m2
8: M2
9: AcM2
13: Grm3
14: m3
17: M3
18: AcM3, Grd4
22: P4
23: Ac4
26: AcA4
27: Grd5
30: Gr5
31: P5
35: AcA5, Grm6
36: m6
39: M6
40: AcM6
44: Grm7
45: m7
48: M7
49: AcM7, Grd8
Which is okay, I guess? It has the perfect, minor, and major intervals. Then it also has all of [Ac, AcM, AcA, Gr, Grm, Grd] as qualities, which are a little weird, but it's not like we were going to avoid them when we defined the system with Ac1 at step 1 and Gr5 at step 30. This 3-limit interval interpretation of 53-EDO also doesn't have neutral seconds, thirds, sixths, or sevenths, which doesn't look good to me.
Of the rank-2 and rank-3 intervals that have the same names and have that name appearing in both the rank-2 and rank-3 analyses of 53-EDO, there has been some movement. The Pythagorean m2 was at 4 steps, while the 5-limit one is at 5-steps. The Pythagorean M2 narrowed from 9 steps to 8 steps. The m3 widened from 13 to 14 steps. The M3 narrowed from 18 to 17 steps. The perfect intervals all stayed put, and the motions of the 6ths and 7ths can be inferred from the seconds and thirds by octave complementation.
It's kind of interesting that e.g. Turkish music using both steps 4 and 5 means that you have a minor second available to you in either analysis system. The consecutively used steps appear in the right places.
Okay, I'm going to be real with you: I don't know how to use Acute and Grave intervals. "Acute" and "Grave" only exist as modifiers so that we still have names for the 3-limit Pythagorean intervals when we make the imperfect intervals become 5-limit. A tuning system where a quarter of the notes have names like "AcA5" isn't actually that useful to me. The rank-2 analysis is way more compelling. And this saddens me, because Shohe Tanaka thought that 53-EDO was good for representing 5-limit just intonation, and I'm not seeing it.
I will admit that, like, at 4-steps of 53-EDO, the justly tuned AcA1 and Grm2 are basically identical as humans can discern them:
t(AcA1) = 135/128 ~= 92 cents
t(Grm2) = 256/243 ~= 90 cents
2^(4/53) ~= 91 cents
So the system is at least finding 5-limit frequency ratios that are indistinguishable and making them identical. But also, a lot of the good simple intervals with good simple ratios are slipping through the cracks if we only use the Turkish steps, [0, 4, 5, 8, 9, 13, ...]. 53-EDO has lots of divisions and some are of course going to be close to simple 5-limit intervals, but the supposedly-Turkish subset on steps [0, 4, 5, ...] doesn't look nearly as good as the full set.
Let's combine the rank-2 and rank-3 analyses!:
0: P1
4: Pythagorean m2, AcA1, Grm2
5: Pythagorean A1, Just m2
8: Pythagorean d3, Just M2
9: Pythagorean M2, AcM2
13: Pythagorean m3, Grm3
14: Pythagorean A2, Just m3
17: Pythagorean d4, Just M3
18: Pythagorean M3, AcM3, Grd4
22: P4
23: Pythagorean A3, Ac4
26: Pythagorean d5, AcA4
27: Pythagorean A4, Grd5
30: Pythagorean d6, Gr5
31: P5
35: Pythagorean m6, AcA5, Grm6
36: Pythagorean A5, Just m6
39: Pythagorean d7, Just M6
40: Pythagorean M6, AcM6
44: Pythagorean m7, Grm7
45: Pythagorean A6, Just m7
48: Pythagorean d8, Just M7
49: Pythagorean M7, AcM7, Grd8
53: P8
So, like, the popular analysis of Turkish microtones by Arel-Ezgi-Uzdilek uses the rank-2 intervals, and notes that there are some intermediate tones between the major Nths and minor Nths. These happen to closely approximate the 5-limit / justly tuned versions of those intervals. And that's it. Turkish music is Pythagorean with Just microtones. One problem with this combined analysis is that e.g. 2^(5/31) is closer to the Pythagorean A1 than to the Just m2, so in a sense the simple Pythagorean analysis has less error. But it's still pretty close. And a merit of the combines analysis as that musicians use 2^(5/31) as if it were some kind of a second interval, so it's nice that we can explain it's second-ordinal function.
...
I read a cool dissertation of Turkish maqamat by Ozan Yarman. He advocates a few analytic systems based around (53 * 3 =) 159-EDO. He didn't say this, but I'm here to tell you that we can make a 159-EDO by tempering out the the interval dddddddddddddddddddddd5, which has coordinates (-15, 4) in the (A1, d2) basis.
Lol, oh shit, tempering that makes 153 EDO. The actual coordinates of the interavl to temper out are (-21, 1). Which is a dddddddddddddddddddddd2, I think. Because (0, 1) is d2, and then to get -21 on the first coordinated, you just add on 21 more {d}s. Yeah.
Yarman doesn't restrict himself to 24 intervals that are once-modified or natural: he's got names for 79 different intervals within 159-EDO. If you've ever been frustrated that people talking about maqamat are super vague about the true frequency ratios and how they vary by culture and region, look into Ozan Yarman.
I read the paper and I thought it was good enough to share, but I haven't really digested it much. I think I'm going to go through Turkish maqamat first and try to notate them here. I don't know how much they differ in writing from Arabic maqamat that I wrote about in the post on quartertones. Apparently there are subtle differences in performance even between the Arabic versus Turkish maqamat that are written identically. But I'm going to start with the writing and then revisit what Ozan Yarman has to say about the performance after.
...
Okay. I have lots of notes from lots of Turkish PDFs and websites and blurry jpegs. Some of the following might be right. Some of these might be real ascending scales with the correct names:
[K S T S A12 S T]: Karcığar
[K S T T B T T]: Acem or Uşşak or Beyâti
[K S T T K S T]: Nevâ or Hüseynî
[S A12 S T B T T]: Hümâyun
[S A12 S T K S T]: Hicaz or Uzzâl. Might also be called Hicaz Ailesi?
[S A12 S T S A12 S]: I think this one ascending it's called Zirgüleli Hicaz and descending it's called Hicazkâr. It happens to be a palindrome.
[S T K S T T K]: Irak
[S T K T S A13 B]: Segâh
[T B T T B T T]: Bûselik or Nihavend
[T K S T T K S]: Râst
[T K S T S A12 S] : Basit Sûzinâk
[T S A12 S T K S]: Nikriz
[T T B T T T B]: Çârgâh
Those letters in the brackets indicate numbers of steps in 53-EDO, according to the following mapping:
"F": 1,
"E": 3,
"B": 4,
"S": 5,
"K": 8,
"T": 9,
"A12": 12,
"A13": 13,
.
Usually the "A" sized scale step is 12 steps of 53-EDO but sometimes it's 13, so I've indicated "A12" and "A13" throughout the maqamat. My sources mostly didn't notate this, and I just picked the version that made the octave complete and the whole set of scale intervals normal (i.e. natural or once modified). Also, F and E don't really show up. If pretended that E showed up sometimes, then we can call it BASKET notation. I'll just stick with BSKTA in ascending order of size.
By accumulating a running sum of EDO steps as you go through the list of BSKTA letters, you can get EDO steps for every maqam scale degree, which can then be converted to rank-2 intervals through the mapping:
edo_step_to_interval = {
0: "P1",
4: "m2",
5: "A1",
8: "d3",
9: "M2",
13: "m3",
14: "A2",
17: "d4",
18: "M3",
22: "P4",
23: "A3",
26: "d5",
27: "A4",
30: "d6",
31: "P5",
35: "m6",
36: "A5",
39: "d7",
40: "M6",
44: "m7",
45: "A6",
48: "d8",
49: "M7",
53: "P8",
This maqam are also well formed ascending scaled, but I've only see n it referenced once:
[T S A12 S S A12 S]: Yegah'da Nev'eser
I think "Yegah'da" just refers to the first note of the maqam as it was written on the staff. If I only present the intervals, its probably just called Nev'eser.
There's a Saba maqam, but it's a little more than an octave and I'm confused where the tonic is. It's sometimes written like this: [K S S A B T S A S]. If we lop off the first letters under the interpretation that they are an ornament below the tonic, then we get [S A13 B T S A12 S], which adds up to 53 steps and produces the normal intervals of [P1, A1, M3, P4, P5, A5, d8, P8]. That might sound kind of arbitrary, especially with how I snuck in both an A13 and an A12, but I tried a ton of things and that's the only one that works. It at least has an interpretation in terms of tetrachords? I think a Turkish music theorist would recognize [S A13 B] and [S A12 S] as two versions of the Hicaz tetrachord / dörtlüsü.
I think Küçek is also more than an octave. Haven't figured it out yet. I also don't understand the Ferahnâk maqam and I'm done trying. I've seen it presented as two runs of notes, each less than an octave, and the notes aren't sorted, and there are differing accidentals if a note shows up more than once.
Some maqamat have a different form when descending:
[-T -B -T -T -S -K -T]: Râst descending
[-T -S -A12 -S -T -B -T]: Bûselik descending
I was silly and my program analyzed those intervals without the negative signs, i.e. as scales going up. They do work as scales going up, but I haven't checked that they work going down, i.e. that they hit normal intervals . They probably do.
I think you normally go down Çârgâh the way you come up, but there's still a special name for the descending scale? It's Acemasiran or Acem Aşirân: [-B -T -T -T -B -T -T]. The name kind of looks like "the Assyrian form of the Acem makam", right? I don't know. I don't speak Turkish. Also, I think "Mâhûr" might be a synonym for Çârgâh. And I think Kürdî might also be descending Çârgâh? People love to say that although Çârgâh sounds like the western major scale it's not important to Turkish musicians. But they sure seem to have a lot of names for it. Or I'm analyzing things incorrectly.
If you go down Râst the way you came up, it's called Gerdâniye, [-S -K -T -T -S -K -T]. And I think Rast and Gerdaniye are both names for specific pitches in Turkish music, and Gerdaniye is an octave above Rast, so it's kind of a fitting name.
I've seen "Eviç" notated as [S T K S T T K], which is Irak, and as [S T T K S T K]. I've also read the claim that Eviç is the descending form of Irak. I wonder if Irak might refer to Iraq. Stranger things have happened.
Isfahân is a name for descending Beyâti. Which I think is also Acem and Uşşak. Maybe those three aren't all the same and I've made some mistakes. But I still pretty sure that Isfahân is [-T -T -B -T -T -S -K] instead of [K S T T B T T].[0, 5, 14, 19, 31, 36, 49, 53]
I've seen this as a variation on Râst: [T K S T T B T]. It might be called Acem'li or Pençgâh? I've also seen the reverse of that called Acem'li rast descending, [T B T T S K T]. And I've also seen [T S K T S K T] called Acem'li descending. I'm pretty lost.
There are supposedly hundreds of maqamat. I wish someone would make a machine-readable list. I'm doing my best. Hereafter are the ones I want to check my notes and the internet for to see why I don't have them listed above, because I thought I'd taken notes on them: [Büzürk, Hisar, Müsteâr, Nişâbûr, Rehâvi, Zirefkend]. Also, is there a plain Zirgüle besides Zirgüleli Hicaz?
Sometimes you'll see the claim that maqamat are made of a tetrachord (four notes spanning P4, not actually played together harmonically as a chord), and a pentachord (five tones spanning P5, usually just made from a P4 tetrachord plus a major second to reach P5). From what I've seen, only like half of the maqamat are analyzable this way. But maybe I have an impoverished set of tetrachords in my analytical library.
Once I've re-researched the key missing maqamat, I'll compare what's here in 53-EDO to Ozan Yarman's work in 159-EDO.
The Hüzzâm maqam is this: [S T S A12 S A13 B]. And it's frustrating to me. Here it is as steps of 53-EDO from P1 to P8: [0, 5, 14, 19, 31, 36, 49, 53]. The 19 is the frustrating part. That step doesn't correspond to one of our normal intervals. It's not in [0, 4, 5, 8, 9, 13, 14, 17, 18, 22, 23, 26, 27, 30, 31, 35, 36, 39, 40, 44, 45, 48, 49]. No, 19 steps of 53-edo as a rank-2 interval is AA2. Twice modified. I wish this was a typo? But everyone agrees Huzzam is [S T S A S A B], and there isn't an assignment of 12 and 13 steps to the As that makes it any better. I don't get it.
Ali C. Gedik posted some turkish maqamat in terms of 53-EDO steps in their PhD thesis, "Automatic Transcription Of Traditional Turkish Art Music Recordings".
These ones match what I have above:
rast = [0, 9, 17, 22, 31, 40, 48, 53]
ussak = [0, 8, 13, 22, 31, 35, 44, 53]
nihavend = [0, 9, 13, 22, 31, 35, 44, 53]
huzzam = [0, 5, 14, 19, 31, 36, 49, 53]
huseyni = [0, 8, 13, 22, 31, 39, 44, 53]
And these ones do not:
hicaz = [0, 5, 17, 22, 31, 35, 39, 44, 53]
segah = [0, 5, 14, 22, 31, 36, 45, 49, 53]
kurdili_hicazkar = [0, 4, 13, 22, 31, 35, 44, 53]
saba = [0, 8, 13, 18, 31, 35, 44, 49]
.
Hicaz and Segah have one more note here than I expected. Also, Saba doesn't have the octave. I hadn't heard of kurdili hicazkar. The Turkish wikipedia article on it makes it sound like a real banger: popular, uplifting, some kind of super group mecha makam made of three smaller ones but still having the regular number of scale degrees.
I think it's
Kürdi’li Hicazkâr Makamı: [-T -T -B -T -T -T -B]
And sometimes there's an extra [-T -B -T] on top at the start as an ornament.
Here's a classification of maqamat from the Turkish wikipedia:
Simple: Çargah, Bûselik, Rast, Uşşak, Hicaz, Uzzal, Hümâyun, Zirgüleli Hicaz, Neva, Hüseynî, Karacığar, Suzinak, Kürdî
Migrated (sed): Acemaşiran, Mahur, Sultaniyegah, Nihavend, Kürdilihicazkar, Zirgüleli Suzinak, Şedaraban, Evcara, Hicazkar, Suz-i dil, Ruhnevaz, Ferahnüma, Aşkefza, Heftgah
Let's check that we have all of those.
...
Or! Or I could go back to the title of the post. Let's rationalize 53-EDO. Based on the math in the next post ("Tempered Frequency Ratios? Why, I never!"), here are the justly tuned frequency ratios for rank-5 (11-limit) intervals and their number of steps in 53-edo:
0 - 1/1
1 - 50/49, 55/54, 64/63, 81/80, 100/99
2 - 33/32, 36/35, 45/44, 49/48, 56/55, 77/75
3 - 22/21, 25/24, 28/27, 80/77
4 - 21/20, 81/77
5 - 15/14, 16/15, 35/33, 77/72
6 - 27/25, 88/81
7 - 11/10, 12/11, 35/32, 49/45, 54/49
8 - 10/9, 55/49
9 - 9/8, 28/25, 49/44
10 - 8/7, 25/22, 55/48, 112/99
11 - 63/55, 81/70
12 - 7/6, 33/28, 64/55, 75/64, 88/75, 90/77
13 - 25/21, 32/27
14 - 6/5, 77/64, 105/88
15 - 11/9, 40/33, 60/49, 98/81, 121/98
16 - 27/22, 49/40, 99/80, 100/81
17 - 5/4, 44/35, 56/45, 96/77, 121/96
18 - 63/50, 80/63, 81/64, 125/98, 125/99
19 - 9/7, 14/11, 32/25, 77/60
20 - 35/27, 55/42, 64/49, 100/77, 125/96, 128/99
21 - 21/16, 33/25, 72/55, 98/75
22 - 4/3, 66/49, 75/56, 121/90
23 - 27/20, 110/81
24 - 11/8, 15/11, 48/35, 49/36, 135/98
25 - 25/18, 88/63, 112/81
26 - 7/5, 45/32, 99/70, 108/77
27 - 10/7, 64/45, 77/54, 121/84, 125/88, 140/99
28 - 36/25, 63/44, 81/56
29 - 16/11, 22/15, 35/24, 72/49
30 - 40/27, 81/55, 121/81, 125/84
31 - 3/2, 49/33, 112/75, 121/80
32 - 32/21, 50/33, 55/36, 75/49
33 - 49/32, 54/35, 77/50, 84/55, 99/64, 125/81, 135/88
34 - 11/7, 14/9, 25/16, 120/77
35 - 63/40, 100/63, 128/81
36 - 8/5, 35/22, 45/28, 77/48, 121/75
37 - 44/27, 80/49, 81/50, 125/77, 160/99
38 - 18/11, 33/20, 49/30, 81/49, 105/64
39 - 5/3, 121/72, 128/77, 165/98
40 - 27/16, 42/25, 147/88
41 - 12/7, 55/32, 56/33, 75/44, 77/45, 121/70, 128/75
42 - 110/63, 125/72, 140/81
43 - 7/4, 44/25, 96/55, 99/56, 135/77
44 - 16/9, 25/14, 88/49, 175/99
45 - 9/5, 98/55
46 - 11/6, 20/11, 49/27, 64/35, 90/49, 175/96
47 - 50/27, 81/44, 147/80
48 - 15/8, 28/15, 66/35, 121/64, 144/77
49 - 40/21, 121/63, 125/66, 154/81
50 - 21/11, 27/14, 48/25, 77/40
51 - 35/18, 55/28, 64/33, 88/45, 96/49, 125/64, 150/77
52 - 49/25, 63/32, 99/50, 108/55, 125/63, 160/81
53 - 2/1
.
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