Ozan Yarman's doctoral thesis, "79-tone Tuning & Theory for Turkish Maqam Music", is a lot to take in, and I've been looking at it on and off for a while. Today I'm going to talk about the Turkish names for different pitches and the intonation than Yarman gives them, i.e. their suggested just frequency ratios or ranges of cent values. He also gives descriptions of pitches as numbers of steps in 159 edo (which is like 53 edo but divide every step in three), but his step assignments are pretty bad - at the very least not consistent with mapping prime harmonics to their nearest steps - and I don't care about EDO analysis that isn't intervallically consistent, so we're going to ignore that.
We'll start with the easy ones. For any of these, if he suggests some crazy ratios with 6 digits in the numerator and a factor of 17, I just throw it away.
In Turkish music, a "perde" is a pitch name, sometimes corresponding to a frequency ratio and sometimes corresponding to a range of frequency ratios. Here are the ones that only have one option after we through away the crazy just ratio suggestions:
Rast: 1/1 # 0c
Zengule: 10/9 # 182
Nihavend: 6/5 # 316c
Çargah: 4/3 # 498c
Saba: 10/7 # 617c
Neva: 3/2 # 702c
Hisar(ek): 5/3 # 884c
Gerdaniye: 2/1 # 1200c
So far we've got a 5-limit just scale and a 7-limit tritone. Nice. Things quickly get less nice. Some of the perdes have turkish adjectives in front like "Nim", "Nerm", "Dik", and "Sarp". If we get rid of those, and also any perdes that Yarman puts in parentheses for some reason like they're perhaps theoretical rather than attested in music, then we we have these perdes in addition to the previous perdes:
Şuri: 25/24, 22/21 # 71c to 81c
Dügah: 28/25, 9/8 # 196c to 204c
Kürdi: 33/28, 13/11, 32/27 # 284c to 294c
Hicazi Segah: 63/52, 40/33, 17/14 # 332c to 336c
Uşşaki Segah: 39/32, 11/9, 27/22 # 342c to 355c
Sabai Segah: 16/13, 100/81, 21/17 # 359c to 366c
Segahçe: 31/25, 41/33, 46/37, 5/4 # 372c to 386c
Segah: 5/4, 64/51, 59/47 # 386c to 394c
Buselik: 81/64, 19/15, 33/26 # 408c to 413c
Nişabür: 14/11, 23/18, 32/25 # 418c to 427c
Hicaz: 7/5, 1024/729, 45/32 # 583c to 590c
Uzzal: 24/17, 17/12 # 597c to 603c
Bayati: 25/16, 47/30, 11/7 # 773c to 782c
Hüseyni: 5/3, 42/25, 27/16 # 884c to 906c
Acem: 23/13, 16/9 # 988c to 996c
Evc: 15/8, 32/17, 17/9 # 1088c to 1101c
Mahur: 256/135, 243/128, 40/21 # 1108c to 1116c
I don't have any quick explanations for these or how to choose among them, but I do appreciate that Yarman gives precise intonation and suggestions of just ratios. This is, from surface plausibility, the best data I know of about makam/maqam music intonation. It could all be wrong as far as I know, but it looks pretty good.
Of the suggestions of just ratios above, some of them have fairly high primes in their factors. I'm good with 7-limit just intonation and I'm getting better at 11-limit just intonation, but I have very little idea how to use factors 13 and higher, so I'm going to hide those high-prime-limit ratios for a second. For the perdes above and their 11-limit frequency ratios suggestions, here are the interval names:
Şuri: A1, AsSpGr1
Dügah: Sbd3, AcM2
Kürdi: Grm3
Hicazi Segah: DeM3
Uşşaki Segah: AsGrm3, DeAcM3
Sabai Segah: GrM3
Segahçe: M3
Segah: M3
Buselik: AcM3
Nişabür: DeSbAc4, d4
Hicaz: Sbd5, GrGrd5, AcA4
Bayati: A5, AsSpGr5
Hüseyni: M6, Sbd7, AcM6
Acem: Grm7
Evc: M7
Mahur: Grd8, AcM7
I think this is cool. Also, the interval names give me some hope that we can narrow down the ratios: presumably perde Huseyni is treated like a 6th or like a 7th in Turkish music, but not both, and so we can throw something out if we learn more about how perde Huseyni is used, written, annotated, things like that.
Next there are a bunch of perdes that have turkish adjectives as prefixes, something like "sharp" and "flat" in meaning but with more nuance. I'm hopeful that we can narrow down how much of a change, in cents, each adjective is associated with in frequency space, and less hopeful that there will be a single just ratios associated with each one, but that would be amazingly useful and we're going to check.
...
Dik {perde} is a little sharp of {perde}. It's not sharp by a consistent amount. It is variously [8, 8, 13, 15, 17, 18] cents sharp for perdes [Dugah, Acem, Mahur, Rast, Huseyni, Cargah] respectively. That's 13 cents sharp on average.
Nerm {perde} is a little flat of {perde}. Nerm {perde} is variously flat of {perde} by [14, 15, 16, 16, 22, 23] cents for perdes [Evc, Hicaz, Cargah, Gerdaniye, Kurdi, Acem] respectively. That's 18 cents flat on average.
Nim {perde} is flatter still compared to Nerm {perde}. Nim {perde} is variously [40, 44, 46, 91, 92] cents flat for perdes [Hicaz, Kurdi, Acem, Hisar, Zengule]. This is an average of 63 cent flat.
Let's also look if the difference between Nim and Nerm is consistent, since it sure isn't consistent just compared to its modified perde.
...
Actually, let's pause on that. There's also an accidental/adjective "Sarp". But we're going to change tactics.
Yarman gives perde Rast" as 1/1 and perde Dik Rast as [126/125, 100/99, or 81/80]. This is great! If we had three options for Rast, and Dik came in three varieties, we'd potentially have 3x3 = 9 possible frequency ratios to reason about when we see Dik Rast in order to figure out what Dik is, but since there's only one Rast tuning option, then we're pretty close to figuring out Dik. Hopefully it's just one of those three!
Perde Çargah is only tuned to 4/3, and perde Dik Çargah is tuned to [39/29, 35/26, or 27/20]. Dividing through by 4/3, we get suggestions that the accidental Dik might be associated with frequency ratios [117/116, 105/104, 81/80]. This is less great! Very little overlap. But they at least agree that 81/80 is an option. Maybe Dik is just the syntonic comma. Of these frequency ratios, 117/116 is 29-limit, 105/104 is 13-limit, and 81/80 is 5-limit, if you're counting primes.
Perde Gerdaniye is only associated with 2/1, so when we see that perde Nerm Geraniye is [2025/1024 or 105/53], we can say that Nerm probably flattens things by 2/1 divided by those things, i.e. 2048/2025 and 106/105. The first of those is the grave diminished second, also called the diaschisma, and the second one I'm not familiar with, and it has a factor of 53 which is crazy, but I mean, there are crazier things than associating an accidental with a super-particular ratio, and for all I know this one has some historic use in makam music. I doubt it, but maybe. So we have one tuned value for Nerm that's somewhat plausible in so far as Turkish uses 5-limit just intonation ratios, which we've already seen, and another tuned value that looks pretty crazy, but maybe some scholarship would justify its use, who knows.
Nerm is a flattening accidental. Yarman gives just tuning options for Nerm Çargah [33/25, 37/28], and since the plain perde Çargah is only tuned to [4/3], you'd think it would be pretty cut and dry: Nerm can be flattening by 100/99 or 112/111. Unfortunately nothing's ever easy with Yarman's arithmetic. If you divide every {perde} tuning by its Nerm {perde} tunings, when Yarman gives both, then the only ratios we see repeated are 99/98 (three times) and 572/567 (twice). These are respectively 11-limit and 13-limit commas. Among all the Nerm options generated this way, most of which don't repeat, we find ratios of sizes [5, 8, 8, 9, 10, 10, 11, 11, 12, 12, 13, 14, 15, 15, 15, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 20, 20, 21, 22, 22, 22, 27] cents, with an average size of 15 cents.
I think we have to talk a little bit about Yarman's 159 edo subset at this point. He selects a subset of all 159 tones in which almost every pair of adjacent scale tones is separated by 2 steps of 159 edo. Since 159 = 3 * 53, and 53-EDO is very common for analyzing turkish music, this is like separating tones at a resolution of 2/3 tone of a 53-EDO tone. Steps of 159 edo is tuned to 15.1 cents, and I'm increasingly convinced that Yarman just uses this as the basic building block of his theory and all of the frequency ratios are put on decoratively as an afterthought.
I guess we need to familiarize ourselves with 159-EDO if we're going to provide an intervallically consistent detempering that can analyze Turkish music at fine resolution. Let's look at some simple 17-limit ratios that are justly associated with steps [0, 1, 2, 3].
Here are just frequency ratios for some intervals are tuned to 3 steps of 159 edo along with cent values:
65/64 : 26.8c
78/77 : 22.3c
81/80 : 21.5c
85/84 : 20.5c
99/98 : 17.6c
224/221 : 23.3c
289/286 : 18.1c
294/289 : 29.7c
I had suspected that 81/80 would show up here instead of at 2 steps, which is one reason why I was a little suspicious of using it as the tuning for a 2-step accidental. On the other hand, Turkish music definitely uses the syntonic comma, so if Yarman doesn't use 3-step commas, then so much the worse for his theory.
Here are some simple 2-step associated ratios:
91/90 : 19.1c
100/99 : 17.4c
105/104 : 16.6c
120/119 : 14.5c
121/120 : 14.4c
126/125 : 13.8c
136/135 : 12.8c
144/143 : 12.1c
170/169 : 10.2c
176/175 : 9.9c
189/187 : 18.4c
245/242 : 21.3c
275/272 : 19.0c
275/273 : 12.6c
I don't see the appeal for using most any of these as accidentals, but we'll look into them in a little more detail shortly.
Here are just ratios for some intervals associated with 1 step of 159 edo:
154/153 : 11.3c
169/168 : 10.3c
196/195 : 8.9c
221/220 : 7.9c
225/224 : 7.7c
243/242 : 7.1c
245/243 : 14.2c
289/288 : 6.0c
If you wanted to generate Yarman's interval space by a cycle of pure and tempered fifths, as he claims you can, you could do much worse than tempering your fifths by variously flattening them by some of these ratios.
And, uh, if you're wondering about tempered commas of 159-EDO, here are a few:
273/272 : 6.4c
325/324 : 5.3c
364/363 : 4.8c
375/374 : 4.6c
385/384 : 4.5c
.
Okay, done with the EDO stuff for a bit. Let's look at the just tunings for Sarp.
Sarp is even sharper than Dik. For perde Rast Sarp, Yarman gives these just tuning options [64/63, 3125/3072, 55/54]. And since Rast is just tuned to 1/1, we'd like to think that in so far as Sarp has a tuning, it's one of these. Across all the perdes where Yarman gives both a natural and a Sarp version, we see a few frequency ratios repeated as possible ratios for the accidentals: [64/63, 50/49, 2048/2025], respectively at [27c, 35c, 20c]. The first of these occurs 6 times, the second occurs 3 times, and the last occurs 2 times. That 6 looks pretty good, but it's 6 out of 25 calculated values for Sarp, so it's not overwhelming identified.
If we do a similar analysis of Nim, we get a surprising number of repeats: [19/18, 135/128, 250/243, 256/243, 311296/295245]. These occur, respectively, 2 times, 2 times, 2 times, 3 times, and 2 times. The regular old Pythagorean minor second, 256/243, comes most often. Weird, right? That shouldn't be an accidental, that should separate natural perdes. It's almost as weird as 311296/295245, which is 19-limit and has a ton of digits. I don't think Turkish lutenists are raising their pitches by that frequency ratios. You would have to work very hard to convince me that 311296/295245 isn't nonsense. Other than Grm2, which of these is the least nonsensical? I'm not sure. 135/128 and 250/243 are respectively AcA1 and GrA1, and it makes sense for accidentals to be unisons, so I don't hate those, although they differ from each other by syntonic commas, so we're really not narrowing Nim down very much - there's a 43 cent gap just between these two. I wonder why plain A1, 25/24, doesn't show up among any of the 38 calculated commas for Nim. The reduced 19th harmonic, 19/18, is ... I don't know anything about 19-limit just intonation, but if you're going to do something with it, you might as well put the reduced 19th harmonic to good use. If I restrict the natural perdes to being 5-limit, then the only repeats are [19/18, 135/128, 250/243], with two copies of each. These are 94c, 92c, and 49 cents respectively.
I confess, I expected that the two flat accidentals and the two sharp accidentals would be mirror images. It doesn't particularly look that way on further inspection though.
Let's do the full factorization based analysis of Dik. When we just looked at Dik Rast and Dik Çargah, the only ratio shared was 81/80. Let's look at every perde with a Dik accidental, because there are a lot of them - ten in total. Across the full set of calculated ratios for Dik, we have all [81/80, 100/99, 126/125, 208/207, 225/224, 275/273, 325/322, 896/891, 2048/2025] all repeated, among many others including 1/1 since Yarman lists 9/8 as a just tuning for both perde Dügah and perde Dik Dügah, and 32/27 for Kürdi and Dik Kürdi, and 16/9 for Acem and Dik Acem, and so on! Among all those repeated commas we have:
81/80 four times,
100/99 three times,
126/125 twice,
208/207 three times,
225/224 five times,
275/273 twice,
325/322 three times,
896/891 twice, and
2048/2025 twice.
So there's a mystery for you. Also 225/224 is most common, but only 7.7 cents, while 81/80 is second common at a stately 21.5 cents. So we haven't narrowed anything down The average of all the calculated Dik ratios is 14 cents, so maybe that makes some of the middle-sized ratios more plausible, e.g.
275/273 : 12.6c, 13-limit
126/125 : 13.8c, 7-limit
325/322 : 16.1c, 23-limit
100/99 : 17.4c, 11-limit
.
Okay, back to EDO stuff. Yarman gives steps of 159 edo for every perde so we can see how many steps of difference there are between {accidental} {perde} and natural {perde}.
...
Next we'll look at Turkish notation for the various perdes.
...
New plan: let's look at Amin Ad-Dik's 24-tone Egyptian Tuning (also presented in Yarman's thesis), which approximates 24 EDO:
0: Yakgāh: 1/1
1: Nīm Qarār Ḥiṣār: 1053/1024
2: Qarār Ḥiṣār: 256/243
3: Tik Qarār Ḥiṣār: 12/11
4: ‘Ušayrān: 9/8
5: Nīm ‘Ajam-‘Ušayrān: 147/128
6: ‘Ajam-‘Ušayrān: 32/27
7: ‘Irāq: 27/22
8: Gavašt: 5/4
9: Tik Gavašt: 9/7
10: Rāst: 4/3
11: Nīm Zīrgūlah: 48/35
12: Zīrgūlah: 1024/729
13: Tik Zīrgūlah: 81/56
14: Dūgāh: 3/2
15: Nīm Kurdī: 49/32
16: Kurdī: 128/81
17: Sahgāh: 18/11
18: Būsalīk: 27/16
19: Tik Būsalīk: 26/15
20: Tšahārgāh: 9/5
21: Nīm Ḥijāz: 11/6
22: Ḥijāz: 15/8
23: Tik Ḥijāz: 35/18
24: Nawā: 2/1
In this system, all the even-numbered tones are 3-limit or 5-limit, and the odd-numbered tones have factors up to, oh, it looks like 13 to me, and also they all have Tik or Nim in their names, except for
7: ‘Irāq: 27/22
17: Sahgāh: 18/11
Let's see what 13-limit ratios are associated with the Tik and Nim accidentals in Amin Ad-Dik's system.
Right off the bat it doesn't look good. The Nīm that modifies Qarār Ḥiṣār is
(256/243) / (1053/1024) = 262144/255879
which is 13-limit and sporting 6 digits in the numerator. Pretty nonsensical. But that's how much you have to flatten Grm2 to get the 1053/1024 thing.
The Tik associated with Qarār Ḥiṣār is only a little better:
(12/11) / (256/243) = 729/704
This is 11 limit. If we'd used the just minor second, 6/5, for Qarār Ḥiṣār instead of Grm2, then the Tik we'd need to get to 12/11 would be a very simple 11/10. Too bad.
The Nim associated with ‘Ajam-‘Ušayrān is
(32/27) / (147/128) = 4096/3969
which is 7-limit. And also garbage.
You know what, I'm skipping ahead to the good ones.
If we look at Gavašt, we see a very sensible value for Tik:
(9/7) / (5/4) = 36/35.
Beautiful. The accidental around Zīrgūlah are very bad. Those of Kurdī are no prize either. Būsalīk's accidentals are bad. Hijaz might be okay!
21: Nīm Ḥijāz: 11/6
22: Ḥijāz: 15/8
23: Tik Ḥijāz: 35/18
These gives us a Nim of 45/44 and a Tik of 28/27. I'll take it! Let's look at the scale in total some more instead of the accidentals.
If we root the tones on Rast (i.e divide through by 4/3), the upper perdes become:
1/1 : Rāst
36/35 : Nīm Zīrgūlah
256/243 : Zīrgūlah
243/224 : Tik Zīrgūlah
9/8 : Dūgāh
147/128 : Nīm Kurdī
32/27 : Kurdī
27/22 : Sahgāh
81/64 : Būsalīk
13/10 : Tik Būsalīk
27/20 : Tšahārgāh
11/8 : Nīm Ḥijāz
45/32 : Ḥijāz
35/24 : Tik Ḥijāz
3/2 : Nawā
and the lower perdes are less than 1/1, which I think will make comparison harder, so let's root on Rast and then raise an octave for easy comparison:
3/2 - P8 : Yakgāh
3159/2048 - P8 : Nīm Qarār Ḥiṣār
128/81 - P8 : Qarār Ḥiṣār
18/11 - P8 : Tik Qarār Ḥiṣār
27/16 - P8 : ‘Ušayrān
441/256 - P8 : Nīm ‘Ajam-‘Ušayrān
16/9 - P8 : ‘Ajam-‘Ušayrān
81/44 - P8 : ‘Irāq
15/8 - P8 : Gavašt
27/14 - P8 : Tik Gavašt
Fantastic. We picked up right where we left off at 3/2.
You might wonder if we have a repeated structure where the tones between 1/1 and 4/3 repeat raised by a factor of 3/2 across the span from 3/2 to 2/1. And the answer is not really, but there are some repetitions. You can see them between the list rooted on Yakgāh the list rooted on Rāst, but I'll mention them in a little detail.
Qarār Ḥiṣār is 256/243 over Yakgāh, just like Zīrgūlah is 256/243 over Rāst. A quarter tone below that, we have Nīm Zīrgūlah which is 36/35 over Rāst whereas Nīm Qarār Ḥiṣār is a 48 cent 1053/1024 over Yakgāh. We have both [243/224 and 12/11] as options for the ratio between 256/243 and 9/8. The two tetrachords agree on 147/128, 32/27, 27/22. Then we have a disagreement of [81/64 versus 5/4], and after that a disagreement between [13/10 and 9/7], and finally a disagreement where Rāst is 4/3 over Yakgāh, but the analogous perde Tšahārgāh is 27/20 over Rāst, i.e. Tšahārgāh is an acute 4th over Rāst, whereas Rāst is perfect 4th over Yakgāh.
This looks to me like an Arabic intonation, rather than a Turkish one, but it's still a little enlightening: it suggests at least that Nim and Tik should be less than a 100 cent minor second.
...
I'm getting a little sick of Yarman. Let's see if we can do some good looking at standard AEU analysis of Turkish perdes in 53-EDO.
Here are steps of 53-EDO over Kaba Çârgâh.
0: "Kaba Çârgâh",
4: "Kaba Nim Hicâz",
5: "Kaba Hicâz",
8: "Kaba Dik Hicâz",
9: "Yegâh",
13: "Kaba Nim Hisâr",
14: "Kaba Hisâr",
17: "Kaba Dik Hisâr",
18: "Hüseynî Aşîrân",
22: "Acem Aşîrân",
23: "Dik Acem Aşîrân",
26: "Irak",
27: "Gevest",
30: "Dik Gevest",
31: "Rast",
35: "Nim Zirgüle",
36: "Zirgüle",
39: "Dik Zirgüle",
40: "Dügâh",
44: "Kürdi",
45: "Dik Kürdi",
48: "Segâh",
49: "Bûselik",
52: "Dik Bûselik",
53: "Çârgâh",
57: "Nim Hicâz",
58: "Hicâz",
61: "Dik Hicâz",
62: "Nevâ",
66: "Nim Hisâr",
67: "Hisâr",
70: "Dik Hisâr",
71: "Hüseynî",
75: "Acem",
76: "Dik Acem",
79: "Eviç",
80: "Mâhûr",
83: "Dik Mâhûr",
84: "Gerdâniye",
88: "Nim Şehnâz",
89: "Şehnâz",
92: "Dik Şehnâz",
93: "Muhayyer",
97: "Sünbüle",
98: "Dik Sünbüle",
101: "Tîz Segâh",
102: "Tîz Bûselik",
105: "Tîz Dik Bûselik",
106: "Tîz Çârgâh",
In this data, Nim {x} is always 1 step flatter than {x}. Dik is inconsistent between being 1 step sharp and 3 steps sharp.
Dik is 1 step sharp for [Acem Aşîrân, Kürdi, Acem, Sünbüle]. Sünbüle is an octave over Kürdi, and Acem is an octave over Acem Aşîrân, so those make some sense.
Dik is 3 steps sharp for [Kaba Hicâz, Kaba Hisâr, Gevest, Zirgüle, Bûselik, Hicâz, Hisâr, Mâhûr, Şehnâz, Tîz Bûselik]
Tîz Bûselik is an octave over Bûselik, Hisâr is an octave over Kaba Hisâr, Şehnâz is an octave over Zirgüle, Mâhûr is an octave over Gevest, and Hicâz is an octave over Kaba Hicâz.
So while Dik might come in two sizes, at least those sizes are applied regularly across octaves.
The AEU steps of 53-EDO happen to be assocaited with simple 3-limit and 5-limit intervals:
0 : 1/1 _ P1
4 : 256/243 _ Grm2
5 : 16/15 _ m2
8 : 10/9 _ M2
9 : 9/8 _ AcM2
13 : 32/27 _ Grm3
14 : 6/5 _ m3
17 : 5/4 _ M3
18 : 81/64 _ AcM3
22 : 4/3 _ P4
23 : 27/20 _ Ac4
26 : 1024/729, 45/32 _ GrGrd5, AcA4
27 : 729/512, 64/45 _ AcAcA4, Grd5
30 : 40/27 _ Gr5
31 : 3/2 _ P5
35 : 128/81 _ Grm6
36 : 8/5 _ m6
39 : 5/3 _ M6
40 : 27/16 _ AcM6
44 : 16/9 _ Grm7
45 : 9/5 _ m7
48 : 15/8 _ M7
49 : 243/128 _ AcM7
52 : 160/81 _ Gr8
53 : 2/1 _ P8
If we pair everything up in the obvious way,
0 : 1/1 _ P1 # "Kaba Çârgâh",
4 : 256/243 _ Grm2 # "Kaba Nim Hicâz",
5 : 16/15 _ m2 # "Kaba Hicâz",
8 : 10/9 _ M2 # "Kaba Dik Hicâz",
9 : 9/8 _ AcM2 # "Yegâh",
13 : 32/27 _ Grm3 # "Kaba Nim Hisâr",
14 : 6/5 _ m3 # "Kaba Hisâr",
17 : 5/4 _ M3 # "Kaba Dik Hisâr",
18 : 81/64 _ AcM3 # "Hüseynî Aşîrân",
22 : 4/3 _ P4 # "Acem Aşîrân",
23 : 27/20 _ Ac4 # "Dik Acem Aşîrân",
26 : 1024/729, 45/32 _ GrGrd5, AcA4 # "Irak",
27 : 729/512, 64/45 _ AcAcA4, Grd5 # "Gevest",
30 : 40/27 _ Gr5 # "Dik Gevest",
31 : 3/2 _ P5 # "Rast",
35 : 128/81 _ Grm6 # "Nim Zirgüle",
36 : 8/5 _ m6 # "Zirgüle",
39 : 5/3 _ M6 # "Dik Zirgüle",
40 : 27/16 _ AcM6 # "Dügâh",
44 : 16/9 _ Grm7 # "Kürdi",
45 : 9/5 _ m7 # "Dik Kürdi",
48 : 15/8 _ M7 # "Segâh",
49 : 243/128 _ AcM7 # "Bûselik",
52 : 160/81 _ Gr8 # "Dik Bûselik",
53 : 2/1 _ P8 # "Çârgâh",
we get a pretty weird system, but there are some patterns. Nim lowers a 5-limit minor interval by Ac1 to give a 3-limit Grm interval. Dik raises a 5-limit minor interval by A1 to give a 5-limit major interval. So {Grm, m, M} all have the same base perde for a given ordinal. But then the Pythagorean major interval, i.e. the acute major, gets a new base perde name entirely.
In this perspective, we can say that
Yegâh is acute major Hicâz
Hüseynî Aşîrân is acute major Hisâr.
Dügâh is acute major Zirgüle.
For some reason this sequence breaks for Kürdi, and the sequence of [Grm7, m7, M7, AcM7] goes [Kürdi, Dik Kürdi, Segâh, Bûselik], i.e. Dik functions as Acute unison instead of its usual augmented unison. The tone over Bûselik is also a little odd: instead of being called Nim Çârgâh, which we'd expect to be Gr8 at 160/81, instead the perde for 52 steps of 53 EDO is called Dik Bûselik, which would be AcA7 tuned to 2025/1024. It's fine to do this: AcA7 is also tuned to 52 steps of 53-EDO, it just seems unusual to me.
The peculiarity with the tuning of Dik happens again around P4, where the tones [P4, Ac4] are named [Acem Aşîrân, Dik Acem Aşîrân], i.e. Dik is acting as acute unison instead of acting, as it usually does, as an augmented unison.
I'd argue that [Irak, Gevest, Dik Gevest, Rast] should be considered [GrGrd5, Grd5, Gr5, P5]. I think you'll agree if you have a close look.
This characterization of Turkish music as being 5-limit just intonation with some unusual names seems fine to me, perhaps because I don't know better, but Yarman thought it wasn't fine-grained enough.
I think it gives us a good start though. Maybe we can regularize/systemize Yarman's messy thing by seeing how it deviates from AEU. I really hope so. I'm getting sick of this thesis.
...
Okay! If the simple diatonic 5-limit just intonation intervals aren't precise enough to represent Turkish makam intonation, we can use more convoluted 5-limit ratios and/or we can introduce some higher prime-limit ratios. I'm ready to try the latter. Here are some simple 13-limit ratios:
SpGr1 : 64/63 at 27c
Sp1 : 36/35 at 49c
As1 : 33/32 at 53c
Sbm2 : 28/27 at 63c
ReAcA1 : 27/26 at 65c
SbAcm2 : 21/20 at 84c
SpA1 : 15/14 at 119c
ReSbAcM2 : 14/13 at 128c
Prm2 : 13/12 at 139c
AsGrm2 : 88/81 at 143c
DeAcM2 : 12/11 at 151c
Asm2 : 11/10 at 165c
Sbd3 : 28/25 at 196c
DeAcA2 : 25/22 at 221c
SpM2 : 8/7 at 231c
ReAcA2 : 15/13 at 248c
Sbm3 : 7/6 at 267c
AsGrm3 : 11/9 at 347c
DeAcM3 : 27/22 at 355c
ReM3 : 16/13 at 359c
PrSpGrm3 : 26/21 at 370c
SpGrM3 : 80/63 at 414c
SpM3 : 9/7 at 435c
Sb4 : 35/27 at 449c
Prd4 : 13/10 at 454c
SbAc4 : 21/16 at 471c
Asd4 : 33/25 at 481c
SpA3 : 75/56 at 506c
DeAcA4 : 15/11 at 537c
Sp4 : 48/35 at 547c
As4 : 11/8 at 551c
ReAcA4 : 18/13 at 563c
Sbd5 : 7/5 at 583c
SpA4 : 10/7 at 617c
PrGrd5 : 13/9 at 637c
De5 : 16/11 at 649c
Sb5 : 35/24 at 653c
AsGrd5 : 22/15 at 663c
SpGr5 : 32/21 at 729c
ReA5 : 20/13 at 746c
Sp5 : 54/35 at 751c
Sbm6 : 14/9 at 765c
SbAcm6 : 63/40 at 786c
Prm6 : 13/8 at 841c
AsGrm6 : 44/27 at 845c
DeAcM6 : 18/11 at 853c
Asm6 : 33/20 at 867c
Sbd7 : 42/25 at 898c
SpM6 : 12/7 at 933c
PrGrd7 : 26/15 at 952c
Sbm7 : 7/4 at 969c
SpA6 : 25/14 at 1004c
DeM7 : 20/11 at 1035c
AsGrm7 : 11/6 at 1049c
DeAcM7 : 81/44 at 1057c
ReM7 : 24/13 at 1061c
PrSpGrm7 : 13/7 at 1072c
Sbd8 : 28/15 at 1081c
SpGrM7 : 40/21 at 1116c
SpM7 : 27/14 at 1137c
De8 : 64/33 at 1147c
Sb8 : 35/18 at 1151c
As8 : 33/16 at 1253c
ReSbAcM9 : 28/13 at 1328c
Prm9 : 13/6 at 1339c
DeAcM9 : 24/11 at 1351c
Asm9 : 11/5 at 1365c
Perhaps Turkish makams make use of those. Let's see where they fall on 53-EDO.
0\53 : 1/1 _ P1
1\53 : 64/63 _ SpGr1
2\53 : 33/32, 36/35 _ As1, Sp1
3\53 : 27/26, 28/27 _ ReAcA1, Sbm2
4\53 : 256/243, 21/20 _ Grm2, SbAcm2
5\53 : 15/14, 16/15 _ SpA1, m2
6\53 : 88/81, 13/12, 14/13 _ AsGrm2, Prm2, ReSbAcM2
7\53 : 11/10, 12/11 _ Asm2, DeAcM2
8\53 : 10/9 _ M2
9\53 : 9/8, 28/25 _ AcM2, Sbd3
10\53 : 25/22, 8/7 _ DeAcA2, SpM2
11\53 : 15/13 _ ReAcA2
12\53 : 7/6 _ Sbm3
13\53 : 32/27 _ Grm3
14\53 : 6/5 _ m3
15\53 : 11/9 _ AsGrm3
16\53 : 27/22, 26/21, 16/13 _ DeAcM3, PrSpGrm3, ReM3
17\53 : 5/4 _ M3
18\53 : 81/64, 80/63 _ AcM3, SpGrM3
19\53 : 9/7 _ SpM3
20\53 : 13/10, 35/27 _ Prd4, Sb4
21\53 : 33/25, 21/16 _ Asd4, SbAc4
22\53 : 4/3, 75/56 _ P4, SpA3
24\53 : 11/8, 15/11, 48/35 _ As4, DeAcA4, Sp4
25\53 : 18/13 _ ReAcA4
26\53 : 1024/729, 7/5 _ GrGrd5, Sbd5
27\53 : 729/512, 10/7 _ AcAcA4, SpA4
28\53 : 13/9 _ PrGrd5
29\53 : 22/15, 16/11, 35/24 _ AsGrd5, De5, Sb5
31\53 : 3/2 _ P5
32\53 : 32/21 _ SpGr5
33\53 : 20/13, 54/35 _ ReA5, Sp5
34\53 : 14/9 _ Sbm6
35\53 : 128/81, 63/40 _ Grm6, SbAcm6
36\53 : 8/5 _ m6
37\53 : 44/27, 13/8, 21/13 _ AsGrm6, Prm6, ReSbAcM6
38\53 : 33/20, 18/11 _ Asm6, DeAcM6
39\53 : 5/3 _ M6
40\53 : 27/16, 42/25 _ AcM6, Sbd7
41\53 : 12/7 _ SpM6
42\53 : 26/15 _ PrGrd7
43\53 : 7/4 _ Sbm7
44\53 : 16/9, 25/14 _ Grm7, SpA6
45\53 : 9/5 _ m7
46\53 : 11/6, 20/11 _ AsGrm7, DeM7
47\53 : 81/44, 13/7, 24/13 _ DeAcM7, PrSpGrm7, ReM7
48\53 : 15/8, 28/15 _ M7, Sbd8
49\53 : 243/128, 40/21 _ AcM7, SpGrM7
50\53 : 27/14 _ SpM7
51\53 : 64/33, 35/18 _ De8, Sb8
53\53 : 2/1 _ P8
There are a few intervals that share a step of 53-EDO. In so far as Turkish music uses higher prime ratios and 53-EDO is inadequate at distinguishing them from lower prime ratios, the steps where we have multiple options are likely places that a Turkish musician with a precise intonation would want to make finer distinctions, particularly when one of the shared intervals is 3-limit or 5-limit.
I myself don't see too many collisions like that. Most of the good 5-limit, 7-limit, and 11-limit ratios are on their own rows. And I don't know what 13-limit ratios are good.
Possible collisions that would make someone want more than 53-divisions:
2\53 : 33/32, 36/35 _ As1, Sp1
The septimal and the undecimal comma are tempered together. You'd think this would mean that there were lots of places where As{X} was tuned the same as Sp{X}, but for the most part, that doesn't happen. Like, the normal places to put {Sp} are on {M} and {GrM} qualities, whereas the normal places to put {As} are on {m} and {Grm} intervals. So there aren't that many collisions - we have widened septimal majors versus undecimal minors narrowed toward neutral.
I think it's possible but not likely that a musician would be dismayed that 53-EDO fails to distinguish
4\53 : 256/243, 21/20 _ Grm2, SbAcm2
Finally a real one: the medieval music theorist and lutenist Mansur Zalzal distinguished between [14/13, 13/12, 12/11, 11/10] in the mathematical descriptions of his tetrachords. Commonly, modern theorists will settle for tone collections that only have two such Zalzalian intervals, and indeed 53-EDO only gives us two:
6\53 : 88/81, 13/12, 14/13 _ AsGrm2, Prm2, ReSbAcM2
7\53 : 11/10, 12/11 _ Asm2, DeAcM2
If you wanted to distinguish among these, that's one reason to use finer divisions that 53-EDO. Although I partly generated my list of simple 13-limit intervals by looking at 3-limit and 5-limit diatonic intervals and looking whether raising or lowering them by a Zalzalian interval produced a simple ratio, so we'll have to keep an eye out as we progress just how much disambiguating the tuning of these intervals would disambiguate the tunings of others.
Let's organize the remaining steps where simple intervals are tempered together by the factors of their just tunings:
3-limit conflated with 7-limit:
18\53 : 81/64, 80/63 _ AcM3, SpGrM3
26\53 : 1024/729, 7/5 _ GrGrd5, Sbd5
27\53 : 729/512, 10/7 _ AcAcA4, SpA4
35\53 : 128/81, 63/40 _ Grm6, SbAcm6
40\53 : 27/16, 42/25 _ AcM6, Sbd7
49\53 : 243/128, 40/21 _ AcM7, SpGrM7
7-limit conflated with 11-limit:
10\53 : 25/22, 8/7 _ DeAcA2, SpM2
24\53 : 11/8, 15/11, 48/35 _ As4, DeAcA4, Sp4
29\53 : 22/15, 16/11, 35/24 _ AsGrd5, De5, Sb5
11-limit conflated with itself:
38\53 : 33/20, 18/11 _ Asm6, DeAcM6
46\53 : 11/6, 20/11 _ AsGrm7, DeM7
11-limit conflated with 13-limit:
37\53 : 44/27, 13/8, 21/13 _ AsGrm6, Prm6, ReSbAcM6
If Yarman was disturbed by the lack of 11-limit and 13-limit ratio approximations in AEU steps, he could just use steps associated with 11-limit and 13-limit ratios. A priori, without looking at which steps of 159-EDO he uses, my guess is that he wants fined grained representation of 7-limit ratios, distinguished from 3-limit ratios, or possibly more Zalzalian intervals.
Let's look at how 159-EDO tunes various second intervals:
08\159 ; Sbm2 _ 28/27 __ 63c
11\159 ; SbAcm2 _ 21/20 __ 84c
12\159 ; Grm2 _ 256/243 __ 90c
15\159 ; m2 _ 16/15 __ 112c
18\159 ; Prm2 _ 13/12 __ 139c
19\159 ; AsGrm2 _ 88/81 __ 143c
20\159 ; DeAcM2 _ 12/11 __ 151c
22\159 ; Asm2 _ 11/10 __ 165c
24\159 ; M2 _ 10/9 __ 182c
27\159 ; AcM2 _ 9/8 __ 204c
31\159 ; SpM2 _ 8/7 __ 231c
All of these differ from m2 or M2 by a comma justly tuned to one of these super particular ratios, [33/32, 36/35, 55/54, 64/63, 65/64, 81/80], which are respectively [11, 7, 11, 7, 13, and 5]-limit. You might notice a conspicuous absence: one of the Zalzalian neutral thirds,
17\159 ; ReSbAcM2 _ 14/13 __ 128c
differs from M2 by 65/63. All the other second intervals had simple enough names, but ReSbAcM2, the "recessed sub acute major second", sure is ugly. We can pair up second intervals that sum to Grm3, justly tuned to 32/27:
(SpM2, 8/7) _ (Sbm2, 28/27)
(AcM2, 9/8) _ (Grm2, 256/243)
(M2, 10/9) _ (m2, 16/15)
(Asm2, 11/10) _ (DeM2, 320/297)
(DeAcM2, 12/11) _ (AsGrm2, 88/81)
(Prm2, 13/12) _ (ReM2, 128/117)
(ReSbAcM2, 14/13) _ (PrSpGrm2, 208/189)
I've seen 320/297 in the writings of Margo Schulter on Ibn Sina, so that's familiar, and I just found a reference to 128/117 in the works of Julien Jalâl Ed-Dine Weiss, so everything is coming up roses. No search results for 208/189 though.
Let's see what the analogous versions of these look like for thirds, sixths, and sevenths. Here are intervals with the same qualitie but thirds instead of seconds:
(SpM3, 9/7) _ (Sbm3, 7/6)
(AcM3, 81/64) _ (Grm3, 32/27)
(M3, 5/4) _ (m3, 6/5)
(Asm3, 99/80) _ (DeM3, 40/33)
(DeAcM3, 27/22) _ (AsGrm3, 11/9)
(Prm3, 39/32) _ (ReM3, 16/13)
(ReSbAcM3, 63/52) _ (PrSpGrm3, 26/21)
The intervals on each line sum to P5 instead of Grm3 now.
Here are the analogous sixths:
(SpM6, 12/7) _ (Sbm6, 14/9)
(AcM6, 27/16) _ (Grm6, 128/81)
(M6, 5/3) _ (m6, 8/5)
(Asm6, 33/20) _ (DeM6, 160/99)
(DeAcM6, 18/11) _ (AsGrm6, 44/27)
(Prm6, 13/8) _ (ReM6, 64/39)
(ReSbAcM6, 21/13) _ (PrSpGrm6, 104/63)
They sum to P11.
Here are the analogous sevenths:
(SpM7, 27/14) _ (Sbm7, 7/4)
(AcM7, 243/128) _ (Grm7, 16/9)
(M7, 15/8) _ (m7, 9/5)
(Asm7, 297/160) _ (DeM7, 20/11)
(DeAcM7, 81/44) _ (AsGrm7, 11/6)
(Prm7, 117/64) _ (ReM7, 24/13)
(ReSbAcM7, 189/104) _ (PrSpGrm7, 13/7)
Each line sums to AcM13.
I also considered adding these in to the mix:
(SbAcm2, 21/20) _ (SpGrM2, 640/567)
(SbAcm3, 189/160) _ (SpGrM3, 80/63)
(SbAcm6, 63/40) _ (SpGrM6, 320/189)
(SbAcm7, 567/320) _ (SpGrM7, 40/21)
since one of every pair has a fairly low complexity just tuning.
I claim that these are the intervals that you want in order to play 13-limit just intonation music with neutral-sounding imperfect tones.
...
Whoa! Weiss 2004 relates that Necdet Yaşar’s gives a tone collection spanning a perfect fourth for playing makams on a tanbūr, and it's a nice subset of the justly tuned interval that I highlighted above. In particular, it's 1/1 and and 4/3, and between those it's
(AcM2, 9/8) _ (Grm2, 256/243)
(M2, 10/9) _ (m2, 16/15)
(Prm2, 13/12) _ (ReM2, 128/117)
(AcM3, 81/64) _ (Grm3, 32/27)
(M3, 5/4) _ (m3, 6/5)
(Prm3, 39/32) _ (ReM3, 16/13)
but sorted by frequency ratio instead of sorted into pairs of complementary interval qualities. All together thats:
[1/1, 256/243, 16/15, 13/12, 128/117, 10/9, 9/8, 32/27, 6/5, 39/32, 16/13, 5/4, 81/64, 4/3]
I'm pretty impressed with this, since it matches what I was doing. It's nice to find agreement. The short summary of the tone collection is that we have 3-limit and 5-limit chromatic intervals, and then we also have 13-limit neutral tones, namely:
(Prm2, 13/12) _ (ReM2, 128/117)
(Prm3, 39/32) _ (ReM3, 16/13)
Good shit. The neutral seconds are at (139c, 156c) and the neutral thirds are at (342c, 359c). These ratios are the just tunings of the intervals separating the pitches over the G below middle C, going up to middle C. I don't know how to write all the accidentals in a way that you can understand them, but 1/1 is associated with a G natural, 256/243 through 9/8 are associated with various A pitches, 32/27 through 81/64 are associated with various B pitches, and 4/3 is associated to middle C, since that's P4 over G.
For the next tone collection I look at, I need adjectives/interval qualities for intervals with just tunings that are 19-limit ratios. I can't decide on the adjectives though. I had decided once and I don't like them any more. And since I haven't used them anywhere, I have one chance to get it right before my choice ends up in far flung corners of my blogs. Well, I'll just call them 19# for the 19-limit sharpening quality and 19b for the 19-limit flattening quality. Here are even more imperfect intervals with simple ratios up to 19-limit:
(19#)m2 _ 19/18 __ 94c
HbAcm2 _ 18/17 __ 99c
DeSbAcM2 _ 35/33 __ 102c
ReSbAcM2 _ 14/13 __ 128c
Prm2 _ 13/12 __ 139c
(19#)Spm2 _ 38/35 __ 142c
DeAcM2 _ 12/11 __ 151c
Asm2 _ 11/10 __ 165c
(19b)SbAcM2 _ 21/19 __ 173c
PrSpm2 _ 39/35 __ 187c
ExM2 _ 17/15 __ 217c
(19#)Dem3 _ 38/33 __ 244c
(19b)AsM2 _ 22/19 __ 254c
Hbm3 _ 20/17 __ 281c
AsSpM2 _ 33/28 __ 284c
PrDem3 _ 13/11 __ 289c
(19#)m3 _ 19/16 __ 298c
Prm3 _ 39/32 __ 342c
AsGrm3 _ 11/9 __ 347c
DeAcM3 _ 27/22 __ 355c
ReM3 _ 16/13 __ 359c
PrSpGrm3 _ 26/21 __ 370c
ExGrM3 _ 34/27 __ 399c
(19b)M3 _ 24/19 __ 404c
ReAsM3 _ 33/26 __ 413c
(19#)m6 _ 19/12 __ 796c
HbAcm6 _ 27/17 __ 801c
DeSbAcM6 _ 35/22 __ 804c
ReSbAcM6 _ 21/13 __ 830c
Prm6 _ 13/8 __ 841c
(19#)Spm6 _ 57/35 __ 844c
DeAcM6 _ 18/11 __ 853c
Asm6 _ 33/20 __ 867c
(19b)M6 _ 32/19 __ 902c
ReAsM6 _ 22/13 __ 911c
ExM6 _ 17/10 __ 919c
(19#)Dem7 _ 19/11 __ 946c
(19b)AsM6 _ 33/19 __ 956c
Hbm7 _ 30/17 __ 983c
PrDem7 _ 39/22 __ 991c
(19#)m7 _ 57/32 __ 999c
(19#)SpGrm7 _ 38/21 __ 1027c
DeM7 _ 20/11 __ 1035c
AsGrm7 _ 11/6 __ 1049c
(19b)SbM7 _ 35/19 __ 1058c
ReM7 _ 24/13 __ 1061c
PrSpGrm7 _ 13/7 __ 1072c
ExGrM7 _ 17/9 __ 1101c
(19b)M7 _ 36/19 __ 1106c
DeSbAc8 _ 21/11 __ 1119c
...
Maybe I need to think more about 159-EDO.
159-EDO is minimally definable in rank-4 space, roughly for the same reason that its tuning of prime harmonics [2, 3, 5, 7, ...], namely:
[159, 252, 369, 446, ...]-steps
first has GCD = 1 when we combine the first four elements.
Here's a rank-4 basis for 159-EDO in prime harmonic coordinates, which has a 7-limit just tuning:
[[1, 0, 0, 0], [5, -7, -1, 3], [-4, 6, -6, 3], [-10, 1, 0, 3]] :: [2/1, 10976/10935, 250047/250000, 1029/1024]
Here's a rank-5 basis for 159-EDO with an 11-limit just tuning:
[[1, 0, 0, 0, 0], [1, -10, 1, 2, 2], [2, 2, -4, -1, 2], [3, -2, 1, -2, 1], [-7, -1, 1, 1, 1]] :: [2/1, 59290/59049, 4356/4375, 440/441, 385/384]
Here's a rank-6 basis and with a 13-limit just tuning:
[[1, 0, 0, 0, 0, 0] [4, 1, -4, 0, 0, 1], [3, -2, 1, -2, 1, 0], [3, -6, -1, 2, 2, -1], [5, -3, 1, -1, -1, 1], [-7, -1, 1, 1, 1, 0]] :: [2/1, 624/625, 440/441, 47432/47385, 2080/2079, 385/384]
All of these consist of an octave that you can tune purely and then a bunch of little intervals that you can temper out to get 159-EDO over that interval space.
159 ; P8 _ 2/1 __ 1200c
0 ; SbSbSbGrdd3 _ 10976/10935 __ 6c
0 ; SbSbSbAcAcddd3 _ 250047/250000 __ 0c
0 ; SbSbSbAcAcm2 _ 1029/1024 __ 8c
0 ; AsAsSbSbGrGrd2 _ 59290/59049 __ 7c
0 ; DeDeSbAcAA1 _ 4375/4356 __ 8c
0 ; DeSbSbAcAcm2 _ 441/440 __ 4c
0 ; AsSb1 _ 385/384 __ 5c
0 ; ReAA0 _ 625/624 __ 3c
0 ; DeSbSbAcAcm2 _ 441/440 __ 4c
0 ; ReAsAsSbSbGrd2 _ 47432/47385 __ 2c
0 ; PrDeSpGr1 _ 2080/2079 __ 1c
0 ; AsSb1 _ 385/384 __ 5c
If we replace the octave in any of these bases with an interval that 159-EDO tunes to 1 step, e.g. SpA0 (which has prime harmonic coordinates [-5, 2, 2, -1] and just tuning 225/224), then we get a unimodular basis.
...
Yeah, I don't know.
...
I think we pretty much have to ignore all of his frequency ratios and just look at his EDO steps. He uses an 80 tone subset of 159 edo to describe perdes. It looks like this:
"Rast": 0,
"Dik Rast": 2,
"(Sarp Rast)": 4,
"(Rast+irha)": 6,
"(nerm Şuri)": 8,
"Şuri": 10,
"Nim Zengule": 12,
"Zengule cluster": 14, 16, 18, 20, 22,
"Zengule": 24,
"Dügah": 26,
"Dik Dügah": 28,
"(Sarp Dügah)": 30,
"Nim Kürdi": 32,
"(Nim Nihavend)": 34,
"Nerm Kürdi": 36,
"Kürdi": 38,
"Dik Kürdi": 40,
"Nihavend": 42,
"Hicazi Segah": 44,
"Uşşaki Segah": 46,
"Sabai Segah": 48,
"Segahçe": 50,
"Segah": 52,
"Buselik": 54,
"Nişabür": 56,
"(Dik Nişabür)": 58,
"(Buselik+irha)": 60,
"(Nişabür+irha)": 62,
"Nerm Çargah": 64,
"Çargah": 66,
"Dik Çargah": 68,
"(Sarp Çargah)": 70,
"Nim Hicaz": 72,
"(Nim Saba)": 74,
"Nerm Hicaz": 76,
"Hicaz": 78,
"Uzzal": 80,
"Saba": 82,
"Saba cluster": 84, 86, 88, 90,
"Neva": 92, 93,
"Dik Neva": 95,
"(Sarp Neva)": 97,
"(Neva+irha)": 99,
"(Nerm Bayati)": 101,
"Bayati": 103,
"Nim Hisar": 105,
"Hisar/Hüzzam cluster": 107, 109, 111, 113, 115,
"Hisar(ek)": 117,
"Hüseyni": 119,
"Dik Hüseyni": 121,
"(Sarp Hüseyni)": 123,
"Nim Acem": 125,
"(Nim Dik Acem)": 127,
"Nerm Acem": 129,
"Acem": 131,
"Dik Acem": 133,
"Sarp Acem": 135,
"Evc cluster": 137, 139, 141,
"Nerm Evc": 143,
"Evc": 145,
"Mahur": 147,
"Dik Mahur": 149,
"(Mahurek)": 151,
"(Mahur+irha)": 153,
"(Dik Mahur+irha)": 155,
"Nerm Gerdaniye": 157,
"Gerdaniye": 159,
...
I started to look at which simple frequency ratios are closest to these steps. My first surprise was that none of them are close to 9/8. The step 27\159 is very close to 9/8, and we might expect this for perde Dugah, but Yarman uses 26\159 for Dugah, which is about 8 cents flatter than 9/8 and 27\159. His Dugah is closer to 19/17.
A lot of the perdes that you'd expect to be associated with simple frequency ratios are instead associated with weird ones. Not all of the following are weird, but almost all of them are weird:
0 _ Rast # 1/1
10 _ Şuri # 23/22
24 _ Zengule # 10/9
26 _ Dügah # 19/17
38 _ Kürdi # 13/11
42 _ Nihavend # 6/5
44 _ Hicazi Segah # 23/19
46 _ Uşşaki Segah # 11/9
48 _ Sabai Segah # 16/13
50 _ Segahçe # 36/29
52 _ Segah # 44/35
54 _ Buselik # 19/15
56 _ Nişabür # 23/18
66 _ Çargah # 4/3
78 _ Hicaz # 38/27
80 _ Uzzal # 17/12
82 _ Saba # 10/7
93 _ Neva # 3/2
103 _ Bayati # 36/23
117 _ Hisar(ek) # 5/3
119 _ Hüseyni # 37/22
131 _ Acem # 23/13
145 _ Evc # 32/17
147 _ Mahur # 19/10
159 _ Gerdaniye # 2/1
I do appreciate that he gives five flavors of perde Segah though. You might wonder what "Segahçe" is. Yarman made it up - "little Segah". I don't yet know in what makams he uses it.
...
If we allow Yarman's perdes to be a full step of 159 edo off from the nearest step of their just tunings in either direction, and use a frequency ratio complexity norm that favors pythagorean ratios somewhat more than simple numerator magnitute would, then we get a basic gamut of the most basic perdes that looks something like this:
0 _ Rast # 1/1
10 _ Şuri # 21/20, 25/24, 256/245
24 _ Zengule # 10/9
26 _ Dügah # 9/8, 28/25
38 _ Kürdi # 32/27
42 _ Nihavend # 6/5
44 _ Hicazi Segah # 98/81, 135/112, 243/200
46 _ Uşşaki Segah # 11/9, 49/40, 243/200
48 _ Sabai Segah # 49/40, 100/81, 243/196
52 _ Segah # 5/4
54 _ Buselik # 63/50, 80/63, 81/64
56 _ Nişabür # 32/25, 80/63, 245/192
66 _ Çargah # 4/3
78 _ Hicaz # 7/5
80 _ Uzzal # 64/45
82 _ Saba # 10/7, 64/45
93 _ Neva # 3/2
103 _ Bayati # 25/16
119 _ Hüseyni # 27/16
131 _ Acem # 16/9
145 _ Evc # 15/8
147 _ Mahur # 243/128
159 _ Gerdaniye # 2/1
I think this has strayed from Yarman's presentation enough that we can't call it his, but perhaps we can still call it an essentially Turkish intonation. If so, it's exciting to have a just Turkish intonation which is not 53-EDO.
It's somewhat common above that when one perde is associated with multiple frequency ratios, two of them are separated by 225/224. This is about 8 cents.
Since I allow perdes to take ratios that are more close in their tuning two a step up or a step down from what Yarman gives, and since the perdes are almost all separated by 2 steps, we get a few cases where two perdes with multiple options for ratios end up sharing a ratio more closely tuned to the step between them. I'll have to figure out what to do with those.
Allowing us to go a step up or down from Yarman's steps is basically reducing our precision by 3, so that we're working in 53-EDO instead of 159. And that's fine. I don't trust Yarman to get anything precisely correct.
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