Every time I try to analyze some bit of Turkish music theory from Ozan Yarman, it doesn't work out, but I'm not ready to give up. Let's look at his just tunings for some makams. These start on page 134 of his doctoral thesis.
Here are some makams with 5-limit tunings:
Rast (ascends and descends the same way):
[P1, AcM2, M3, P4, P5, AcM6, M7, P8] [1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1]
Acemli Rast (rises the same as Rast but descends as follows):
[P8, Grm7, M6, P5, P4, M3, AcM2, P1] [2/1, 16/9, 5/3, 3/2, 4/3, 5/4, 9/8, 1/1]
Mahur (ascending):
[P1, AcM2, AcM3, P4, P5, AcM6, AcM7, P8] [1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2/1]
Mahur (descending):
[P8, M7, AcM6, P5, P4, M3, AcM2, P1] [2/1, 15/8, 27/16, 3/2, 4/3, 5/4, 9/8, 1/1]
Nihavend (descending):
[P8, m7, m6, P5, P4, m3, AcM2, P1] [2/1, 9/5, 8/5, 3/2, 4/3, 6/5, 9/8, 1/1]
These intervals all come from a very small list: [P1, AcM2, m3, M3, AcM3, P4, P5, m6, M6, AcM6, Grm7, m7, M7, AcM7, P8]
Yarman specifies other makams relative to the same P1 as these makams above, even when they have a different tonic.
If we add in 7-limit ratios, we get these makams:
Pencgah (ascends and descends the same way):
[P1, AcM2, M3, Sbd5, P5, AcM6, M7, P8] [1/1, 9/8, 5/4, 7/5, 3/2, 27/16, 15/8, 2/1]
Hicaz (ascending):
[AcM2, m3, Sbd5, P5, AcM6, GrM7, P8, AcM9] [9/8, 6/5, 7/5, 3/2, 27/16, 50/27, 2/1, 9/4]
Hicaz (descending):
[AcM9, P8, m7, AcM6, P5, Sbd5, m3, AcM2] [9/4, 2/1, 9/5, 27/16, 3/2, 7/5, 6/5, 9/8]
To the previous tone collection, these makams add
* Sbd5, justly tuned to 7/5. Yarman uses it as a fourth interval in every case (i.e. between m3 and P5 or between M3 and P5).
* GrM7, a neutral tone, justly tuned to 50/27.
* AcM9, but of course that's just AcM2 an octave up.
In the 11-limit, we get one more makam:
Saba (descending):
[P11, M10, m9, P8, SpA6, AcM6, SpA4, P4, AsGrm3, AcM2] [8/3, 5/2, 32/15, 2/1, 25/14, 27/16, 10/7, 4/3, 11/9, 9/8]
The P11 is an octave displaced P4 and the M10 is an octave displaced M3, both of which we've seen. The m9 is new: it's of course an octave displaced just m2. The SpA6 is also new, justly tuned to 25/14, and Yarman uses it like a 7th interval. The SpA4 is new, justly tuned to 10/7, and Yarman uses it like a 5th interval. Finally the AsGrm3 is new, a neutral tone justly tuned to 11/9. We got a lot of new tones in this one. Saba is pretty crazy.
Yarman's just tunings of the makams don't use factors of 13. He skips straight to 17. I don't know why. He glosses the makams impressionistically with tetrachords that include factors of 13, but the actual notes don't include them.
In the 17-limit, we have these makams:
Huseyni (ascends and descends the same way):
[AcM2, HbSbAcd4, P4, P5, AcM6, HbSbAcd8, P8, AcM9] [9/8, 21/17, 4/3, 3/2, 27/16, 63/34, 2, 9/4]
Nihavend (ascending):
[P1, AcM2, m3, P4, P5, m6, Hbd8, P8] [1/1, 9/8, 6/5, 4/3, 3/2, 8/5, 32/17, 2/1]
Saba (ascending):
[AcM2, HbSbAcd4, Ac4, De5, AcM6, AsGrm7, P8, AcM9] [9/8, 21/17, 27/20, 16/11, 27/16, 11/6, 2, 9/4]
Segah (ascending):
[(Hbm3), M3, P4, P5, M6, M7, P8, Hbm10, M10] [(20/17), 5/4, 4/3, 3/2, 5/3, 15/8, 2, 40/17, 5/2]
Segah (descending):
[Hbm10, AcM9, P8, Hbm7, M6, P5, P4, M3] [40/17, 9/4, 2/1, 30/17, 5/3, 3/2, 4/3, 5/4]
The Hbm3 of Segah ascending, justly tuned to 20/17, is a leading tone - the actual tonic is a M3 over the tonic of Rast. Yarman uses it like a 2nd interval.
In this set of makams we also get
* the HbSbAcd4, justly tuned to 21/17, which Yarman treats as a 3rd interval.
* the Ac4 at 27/20
* the De5 at 16/11
* the AsGrm7 at 11/6
* Hbm7 at 30/17
* HbSbAcd8 at 63/34, which Yarman treats as a 7th interval
* Hbd8 at 32/17 which Yarman treats as a 7th interval
* Hbm10 at 40/17, which is just an octave displaced version of the Hbm3 leading tone we saw when we ascending in Segah.
Yarman gives two more just tunings, for makam Huzzam ascending and descending. They include factors of 29, which I don't have interval names for, but I'll show you what I can:
Huzzam (ascending):
[(Sbm3), ?, P4, P5, ?, ?, P8, Sbm10, ?] [(7/6), 36/29, 4/3, 3/2, 48/29, 54/29, 2/1, 7/3, 72/29]
Huzzam (descending):
[Hbm10, AcM9, P8, Hbm7, ?, P5, P4, ?] [40/17, 9/4, 2/1, 30/17, 48/29, 3/2, 4/3, 36/29]
Yarman uses
* Sbm3 is a 2nd interval
* 36/29 like a 3rd interval
* 48/29 like a 6th interval
* 54/29 like a 7th interval
* 63/34 like a 7th interval
* Sbm10 at 7/3 like a 9th interval
* 72/29 like a 10th interval
It seems worth pointing out that every time Yarman uses a factor of 17 or 29, it's in the denominator.
Let's look at relative intervals. For descending forms of scales, I'll reverse them to ascending, so that the lowest intervals come first and al the relative frequency ratios between consecutive scale degrees are greater than 1. These makams have fairly clear tetrachord/pentachord structure:
Rast (ascends and descends the same way) [9/8, 10/9, 16/15] * 9/8 * [9/8, 10/9, 16/15].
Mahur (ascending) [9/8, 9/8, 256/243] * 9/8 * [9/8, 9/8, 256/243].
Acemli Rast (rises the same as Rast but descends as follows): [9/8, 10/9, 16/15] * [9/8, 10/9, 16/15] * 9/8.
Pencgah (ascends and descends the same way): [9/8, 10/9, 28/25, 15/14] * [9/8, 10/9, 16/15].
Nihavend (ascending) [9/8, 16/15, 10/9] * 9/8 * [16/15, 20/17, 17/16].
Nihavend (descending) [9/8, 16/15, 10/9] * 9/8 * [16/15, 9/8, 10/9].
Huseyni (ascends and descends the same way) [56/51, 68/63, 9/8] * 9/8 * [56/51, 68/63, 9/8].
Hicaz (descending) [16/15, 7/6, 15/14] * 9/8 * [16/15, 10/9, 9/8].
Hicaz (ascending) [16/15, 7/6, 15/14] * 9/8 * [800/729, 27/25, 9/8].
I don't know where Yarman got the weird repeated tetrachord in Huseyni:
[56/51, 68/63, 9/8] @ [162c, 132c, 204c]
but it looks to me like he wanted something that would sound like the Zalzalian tetrachords
[11/10, 320/297, 9/8] @ [165c, 129c, 204c]
[208/189, 14/13, 9/8] @ [166c, 128c, 204c]
but with smaller numerators, and decided to achieve this by going up to 31-limit. They're all separated by 5 cents or less.
When I go back and look how Yarman glossed the Huseyni scale with tetrachords, he uses [11/10, 13/12, 9/8]. I bet this is a typo for [11/10, 14/13, 9/8], which also doesn't reach P4, but at least it's perceptually indistinguishable from one of those two Zalzalian tetrachords that does. I'm generally in favor of using factors of {5 and 11} over using {7 and 13}, so unless you've got a simpler perceptually indistinguishable tetrachord with similar historical basis, I recommend [11/10, 320/297, 9/8] for the intonation of Huseyni:
[11/10, 320/297, 9/8] * [9/8] * [11/10, 320/297, 9/8]
We can accumulate this multiplicatively to get:
[P1, Asm2, Grm3, P4, P5, Asm6, Grm7, P8] # [1/1, 11/10, 32/27, 4/3, 3/2, 33/20, 16/9, 2/1]
This also has the perk of being spelled correctly in intervals in the Lilley-Johnston interval naming system, i.e. we don't have to treat an 8th interval, HbSbAcd8, as a 7th.
For makams with less clear structure, Segah looks alright ascending but falls apart in its descent:
Segah (ascending): (17/16) * [16/15, 9/8, 10/9] * 9/8 * [16/15, 20/17, 17/16].
Segah (descending): [16/15, 9/8, 10/9] * [18/17, 17/15, 9/8, 160/153]
And I'm pretty confused by both ascending and descending forms of Saba and Huzzam:
Saba (ascending): [56/51, 153/140, 320/297, 297/256] * [88/81, 12/11, 9/8].
Saba (descending) [88/81, 12/11] * [15/14, 189/160, 200/189] * [28/25] * [16/15, 75/64, 16/15].
Huzzam (ascending): (216/203) * [29/27, 9/8, 32/29] * 9/8 * [29/27, 7/6, 216/203].
Huzzam (descending) [29/27, 9/8, 32/29] * [145/136, 17/15, 9/8, 160/153].
Yarman glosses the lower tones of Saba (ascending) with the pentachord
[11/10 * 12/11 * 13/12 * 15/13]
from D4 to A4 and then rises to D5 with a nonsensical Ussak tetrachord, [12/11 * 12/11 * 9/8] that obviously doesn't reach P4, and there's no reason for that because the actual relative ratios between ratios for the makam scale degrees,
[88/81 * 12/11 * 9/8],
are perfectly simple reasonable ratios on their own without any arithmetic errors.
Let's compare the two ascending Saba pentachords by cents. The calculated pentachord:
[56/51, 153/140, 320/297, 297/256] @ [162c, 154c, 129c, 257c]
versus the glossed pentachord:
[11/10 * 12/11 * 13/12 * 15/13] @ [165c, 151c, 139c, 248c]
So the first two ratios of each tetrachord are perceptually indistinguishable (and each form 6/5 when multiplied within the tetrachord), but the next two ratios of the tetrachord differ by 10 cents from each other positionally (and both form 5/4 when multiplied within the tetrachord). To me this means that there's no reason to use the crazy {56/51 and 153/140} ratios, and the complicated {320/297 and 297/256} ratios are only worth using in so far as that 10 cent difference is both necessary and can't be achieved by simpler ratios.
If we use the pentachord with nice Zalzalian super-particular ratios and the Ussak tetrachord that actually hits P4, we get this for Saba (ascending):
[11/10 * 12/11 * 13/12 * 15/13] * [88/81 * 12/11 * 9/8]
Accumulating frequency ratios multiplicatively, we have these scale degrees:
[P1, Asm2, m3, Prd4, P5, AsGrm6, Grm7, P8] # [1/1, 11/10, 6/5, 13/10, 3/2, 44/27, 16/9, 2/1]
And since Yarman roots Saba on D4 a Pythagorean 9/8 over the C of Rast, lets' do the same:
[AcM2, Asm3, Ac4, Prd5, AcM6, AsGrm7, P8, AcM9] # [9/8, 99/80, 27/20, 117/80, 27/16, 11/6, 2/1, 9/4]
I think this is fairly nice. And if it were too nice, it wouldn't be Saba. We still have to address the descending form:
Saba (descending): [88/81, 12/11] * [15/14, 189/160, 200/189] * 28/25 * [16/15, 75/64, 16/15]
So 28/25 is smaller than 9/8 by a factor of 225/224 (at 8 cents), and the pseudo-tetrachord [15/14 * 189/160 * 200/189] is larger than 4/3 by 8 cents. Clearly we need to move a factor around.
One option is
[SpA1, SbAcm3, Grm2] # [15/14 * 189/160 * 256/243] at [119c, 288c, 90c]
which not only reaches P4 but also gets us a nice Pythagorean minor second, Grm2, at the end. Another option is
[m2, SbAcm3, SpGrA1] # [16/15 * 189/160 * 200/189] at [112c, 288c, 98c]
Either way it looks a little weird. We could shuffle around an acute unison to clean things up nicely:
[m2, Sbm3, SpA1] # [16/15, 7/6, 15/14] at [112c, 267c, 119c]
But I don't think that's allowed. Yarman's tetrachords might not always add up to P4, but they're usually only off by like 7 cents, not 22. His intonation is more precise than that, even if you have to squint to see his equalities. If you wanted to simplify the numerators of the ratios, 189/160 at 288 cents is perceptually indistinguishable from 13/11 at 289 cents. And then you could do
[m2, PrDem3, ReAsA1] # [16/15, 13/11, 55/52] at [112c, 289c, 97c]
But I don't know anything that's recommending those intervals other than small numerators in the tunings.
Lets look at Yarman's glosses for Saba descending. The lowest two notes don't have any description. Over the next four notes, we have a Cargah tetrachord, then an acute major second, then a Hicaz tetrachord. Here's Yarman's intonation from the gloss:
[15/14, 13/11, 55/52] * 9/8 * [16/15, 7/6, 15/14]
The lower Cargah tetrachord is off from P4 by a SpA0, justly tuned to 225/224, at 8 cents. This is weird. Both the glossed and the calculated Cargah tetrachords were off by 225/224. Maybe Saba has impure tetrachords when it descends? I don't think so. Yarman's upper tetrachord, calculated, was
[16/15, 75/64, 16/15] at [112c, 275c, 112c]
whereas the gloss gives
[16/15, 7/6, 15/14] at [112c, 267c, 119c]
which we've used before for Hicaz tetrachord. We actually just saw this Hicaz intonation as a simplification of the calculated Cargah tetrachord if we were willing to move a whole syntonic comma around. If these two tetrachords only differ by moving around a syntonic comma, should we notate them as differing by the placement of a syntonic comma?
[m2, SbAcm3, SpGrA1] # [16/15 * 189/160 * 200/189] at [112c, 288c, 98c] # Possible Cargah tetrachord intonation
[16/15, 7/6, 15/14] at [112c, 267c, 119c] # Hicaz tetrachord
Let's try it and see how it looks. If that 28/25 tone between the tetrachords ends up being important, we can put it back in, but for now, here's my guess as to Saba's descending intonation:
[88/81, 12/11] * [16/15 * 189/160 * 200/189] * 9/8 * [16/15, 7/6, 15/14]
If we accumulate multiplicatively, we get this scale:
[88/81, 32/27, 512/405, 112/75, 128/81, 16/9, 256/135, 896/405, 64/27]
and this should be rooted on 9/8:
[9/8, 11/9, 4/3, 64/45, 42/25, 16/9, 2/1, 32/15, 112/45, 8/3]
...
The lowest two notes of Saba descending outline a Pythagorean minor third, Grm3, justly tuned to 32/27, so that looks like the upper fragment of a
[9/8, 88/81, 12/11]
tetrachord, which I'm not familiar with, but it's interesting because Saba (ascending) had an Ussak tetrachord like
[88/81, 12/11, 9/8]
at the top of its range.
...
You know, I hardly made any changes to Saba (descending) and none of them were good. Let's come back to this in a bit.
...
Let's see if we can clear up Huzzam. Here are the two directions in relative frequency ratios again:
Huzzam (ascending) [(216/203), 29/27, 9/8, 32/29, 9/8, 29/27, 7/6, 216/203].
Huzzam (descending) [29/27, 9/8, 32/29, 145/136, 17/15, 9/8, 160/153].
Despite these ratios, Yarman glosses the ascending form as
[15/14 * 9/8 * 32/29] * 9/8 * [16/15 * 7/6 * 15/14]
The second tetrachord reaches a just P4 at 4/3, but the first one which (which Yarman calls a Huzzam tetrachord) doesn't. Since 32/29 is the obvious outlier, we should see what ratio could stand in its place to reach 4/3. This gives us:
[15/14 * 9/8 * 448/405] @ [119c, 204c, 175c]
The 32/29 is flat of 448/405 by about 5 cents. If you wanted a flatter interval there, you could use 11/10 or even 21/19. No idea why he thought a factor of 29 was logical.
If instead you replace 15/14 to get a pure tetrachord, you get
[29/27 * 9/8 * 32/29] @ [124c, 204c, 170c]
which is what Yarman's scale actually has, despite the glossed Huzzam tetrachord with the 15/14. If the 15/14 isn't definite, then I think I'd rather spell the Huzzam tetrachord as
[14/13 * 9/8 * 208/189] @ [128c, 204c, 166c]
since it has history via Zalzal. It's a good tetrachord.
I don't yet have strong opinions about the upper tetrachord (Hicaz) of Huzzam. It seems like the main options are
[16/15 * 7/6 * 15/14] // Repeated in multiple glosses
[29/27 * 7/6 * 216/203] // Calculated from Huzzam
[16/15 * 75/64 * 16/15] // Calculated from Saba
...
Okay, back to Saba. Most other sources will tell you that Saba has two forms: one that reaches the octave and one that goes past the octave.
Form 1 has pitch classes [D4, E/b4, F4, Gb4, A4, Bb4, C5, D5]
Form 2 has pitch classes: [D4, E/b4, F4, Gb4, A4, Bb4, C5, Db5, E5, F5]
Also form 1 sometimes has lower ornamental pitches of [B/b3, C4].
Form 1 is:
* jins saba on D4 up to Gb4: [D4, E/b4, F4, Gb4],
* then an overlapping jins hijaz starting on F4: [F4, Gb4, A4, Bb4].
* then the start of jins ajam on Bb: [Bb4, C5, D5], i.e. two major seconds.
Form 2 could be described as:
* jins saba on D4 up to Gb4: [D4, E/b4, F4, Gb4],
* the same overlapping jins hijaz on F4: [F4, Gb4, A4, Bb4].
* jins Nikriz on Bb4: [Bb4, C5, Db5, E5, F5]
This second form seems to be the scale that Yarman's descending Saba. Let's see if this knowledge helps us to make better sense of Yarman's intonation.
The most boring Arabic intonation we could choose for these pitch classes is Pythagorean on the chromatic pitches and 11-limit on the neutral E\b:
[D4, E/b4, F4, Gb4, A4, Bb4, C5, Db5, E5, F5] : [9/8, 27/22, 4/3, 1024/729, 27/16, 16/9, 2/1, 512/243, 81/32, 8/3]
Here's Yarman's descending Saba scale in frequency ratios again for comparison:
[AcM2, AsGrm3, P4, SpA4, AcM6, SpA6, P8, m9, M10, P11] # [9/8, 11/9, 4/3, 10/7, 27/16, 25/14, 2/1, 32/15, 5/2, 8/3]
Differences:
Yarman's neutral E\b4 at 11/9 is fine: 11/9 and 27/22 are the two most obvious 11-limit neutral thirds, and they only differ by 243/242 at 7 cents.
Yarman uses a 7-limit ratio for Gb4, 10/7, instead of the Pythagorean one, 1024/729. These differ by 225/224 at 8 cents.
Yarman uses a 7-limit ratio for Bb4, 25/14, instead the the Pythagorean one, 16/9. These differ by 225/224.
Yarman uses 5-limit ratios for Db5 and E5, namely 32/15 and 5/2, instead of the Pythagorean 512/243 and 81/32. These seem fine. Turkish music uses 5-limit intonation a lot more than Arabic and Persian.
Now let's compare the relative intervals and tetrachord intonation.
Oh! We should fix the intonation of the hijaz/hicaz tetrachord before we go any further. We're going to need it. Yarman's septimal intonation:
[m2, Sbm3, SpA1] # [16/15, 7/6, 15/14] _ [112c, 267c, 119c]
doesn't make sense intervallically. Those should all be second intervals, not [2nd, 3rd, 1st]. What tetrachords are spelled correctly and similarly sized? Here are my first few ideas:
[m2, SpM2, SbAcM2] # [16/15, 8/7, 35/32] _ [112c, 231c, 155c]
[m2, A2, Acm2] # [16/15, 125/108, 27/25] _ [112c, 253c, 133c]
[m2, AcA2, m2] # [16/15, 75/64, 16/15] _ [112c, 275c, 112c]
[m2, AsSpM2, DeSbAcM2] # [16/15, 33/28, 35/33] _ [112c, 284c, 102c]
[m2, SpA2, SbAcm2] # [16/15, 25/21, 21/20] _ [112c, 302c, 84c]
Yarman uses the [16/15, 75/64, 16/15] tetrachord in the scale itself (but not in the gloss), and that one's closest to the malformed one in the gloss with the Sbm3 at 7/6 (only 225/224 at 8 cents away).
I'd like to propose this as another possible Hicaz intonation:
[Prm2, ReAcA2, m2] # [13/12, 15/13, 16/15] _ [139c, 248c, 112c
which is somewhat close to other intonations and more importantly can overlap with jins Saba that ends in [... 13/12, 15/13].
This in hand, I've found my best Saba (descending) intonation yet:
Absolute scale degrees over D (which is 9/8 over C of Rast):
[AcM2, Asm3, Ac4, Prd5, AcM6, m7, P8, m9, M10, P11] [9/8, 99/80, 27/20, 117/80, 27/16, 9/5, 2, 32/15, 5/2, 8/3]
Relative scale degrees:
[Asm2, DeAcM2, Prm2, ReAcA2, m2, M2, m2, AcA2, m2] [11/10, 12/11, 13/12, 15/13, 16/15, 10/9, 16/15, 75/64, 16/15]
And here it is with pitches separated by relative frequency ratios for easy reading:
[D, (11/10), E/b, (12/11), F, (13/12), Gb, (15/13), A, (16/15), Bb, (10/9) C, (16/15), Db,(75/64), E, (16/15), F]
In this intonation, we've got the Saba pentachord verbatim from the ascending gloss,
[11/10, 12/11, 13/12, 15/13]
We've got two different intonations of the Hicaz tetrachord,
[13/12, 15/13, 16/15]
[16/15, 75/64, 16/15]
But Hicaz is a tetrachord with a famously changeable intonation, and at at least these two intonations are both spelled in second intervals, and indeed have the same rough form of [m2, A2, m2], with some microtonal accidentals. is a 5-limit intonation, normal for Turkish Makam, and one is a mix of 5-limit and Zalzalian intonation. They're good tetrachords.
Like Yarman, I use a small ratio than 9/8 to span the gap between Bb and C. He uses Sbd3 justly tuned to 28/25 at 196c and I use M2 justly tuned to 10/9 at 182c. These differ by a SbAcd2, justly tuned to 126/125 at 14c. I could probably fiddle around with 10/9 or one of the adjacent 16/15 relative ratios to get a closer intonation to Yarman's, but I'm pretty happy with this already.
Yarman's descending form isn't glossed with multiple Hicaz tetrachords and a Saba pentachord, but with one Hicaz tetrachord and a Cargah tetrachord. So we should still look at that a little.
...
Let's do the same treatment of Cargah that we did of Hicaz/Hijaz: make the relative intervals all seconds. Here are the two intonations of Cargah that Yarman gives us as hints:
[SpA1, SbAcm3, SpGrA1] # [15/14, 189/160, 200/189] _ [119c, 288c, 98c]
[SpA1, PrDem3, ReAsA1] # [15/14, 13/11, 55/52] _ [119c, 289c, 97c]
The first is calculated from scale degrees of Saba, the second is from the annotation of Saba. My first thought of how to construct similarly sized tetrachords consisting of 2nd intervals is
[m2, AsSpM2, DeSbAcM2] # [16/15, 33/28, 35/33] _ [112c, 284c, 102c]
And these options might also work:
[m2, AcAcA2, Grm2] # [16/15, 1215/1024, 256/243] _ [112c, 296c, 90C]
[m2, AcA2, m2] # [16/15, 75/64, 16/15] _ [112c, 275c, 112c]
In Yarman's gloss of Saba descending,
* The lowest tone is D4 that's 9/8 over C4 of Rast
* The next tone up is a neutral E at 11/8 over C4 (or 88/81 over D4)
* then up by 12/11 to get to an F natural that's 4/3 over C4.
* then the Cargah genus starts on F4: [F4, G?4, A4, Bb4], with some weird accidentals on G that I'm not going to try figuring out or notating because his accidentals are just EDO steps and not strong clues to just tunings,
* then there's a 9/8 tone from Bb4 to C5
* and finally a Hicaz tetrachord from C5 to F5
When I try both the 5-limit and Zalzalian intonations of the Hicaz genus:
[16/15, 75/64, 16/15] # 5-limit Hicaz
[13/12, 15/13, 16/15] # Zalzalian Hicaz
I looked for a weird intonation on the high Db, since that's what Yarman has written in staff notation. The 5-limit version of Hicaz just tunes this to 32/15, i.e. a just m9 over C4. That's a Db, but not weird enough. The Zalzalian Hicaz gives us a Db at 13/6, a justly tuned Prm9, and an octave over the Zalzalian 13/12. This is nice to me.
Let's see the whole thing all together. An intonation for Saba descending that we're investigating, first in relative intervals and ratios:
[AsGrm2, DeAcM2] + [m2, AsSpM2, DeSbAcM2] + AcM2 + [Prm2, ReAcA2, m2]
[88/81, 12/11] * [16/15, 33/28, 35/33] * 9/8 * [13/12, 15/13, 16/15]
And now in absolute intervals and ratios, over a D4 that's 9/8 over C4.
[AcM2, AsGrm3, P4, Grd5, AsSpGrm6, Grm7, P8, Prm9, M10, P11]
[9/8, 11/9, 4/3, 64/45, 176/105, 16/9, 2/1, 13/6, 5/2, 8/3]
In Yarman's staff notation, D4, F4, A4, Bb4, C5, E5 and F5 have no microtonal accidentals, so it would be nice if these were tuned to 5-limit or Pythagorean ratios. In contrast, [E?4, G?4, D?5] all have weird accidentals, and so we should expect non-chromatic tunings for these. In the proposed intonation, the first four scale degrees look fine, but then A4 is associated with AsSpGrm6 over C, and tuned to a non-chromatic 176/105. In Yarman's intonation, this pitch was assocaited with a Pythagorean M6 over C4 tuned to 27/16. The other pitches all look good though in being chromatic or non-chromatic as expected. So let's work on that A4 intonation.
If we just change the scale to have A4 on (AcM6 -> 27/16) or (M6 -> 5/3), what does that Cargah tetrachord look like?
Saba descending with a Pythagorean M6 looks like this in terms of relative intervals and ratios:
[AsGrm2, DeAcM2] + [m2, AcAcA2, Grm2] + AcM2 + [Prm2, ReAcA2, m2]
[88/81, 12/11] * [16/15, 1215/1024, 256/243] * 9/8 * [13/12, 15/13, 16/15]
And Saba descending with a just M6 looks like this for relative intervals and ratios:
[AsGrm2, DeAcM2] + [m2, AcA2, m2] + AcM2 + [Prm2, ReAcA2, m2]
[88/81, 12/11] * [16/15, 75/64, 16/15] * 9/8] * [13/12, 15/13, 16/15]
The Cargah intonation in this first one looks pretty dumb, but it gives us the AcM6 that Yarman had. The second Cargah intonation looks reasonable, but it's identical to my recommendation for a 5-limit intonation of the Hicaz tetrachord, and why have a name for a Cargah tetrachord if it's going to be the same as Hicaz?
Oh, I've got it! The G isn't chromatic so the transition from F to G should be weirder than 16/15. That part of the tetrachord is wrong. I did a lot of math to figure out what tetrachord might work and...It's just the
[16/15, 33/28, 35/33]
intonation of the Cargah tetrachord but backwards:
[35/33, 33/28, 16/15]
I shouldn't have had to do any math to see that as a good option. Anyway, new description of Saba descending in relative terms:
[AsGrm2, DeAcM2] + [DeSbAcM2, AsSpM2, m2] + AcM2 + [Prm2, ReAcA2, m2]
[88/81, 12/11] + [35/33, 33/28, 16/15] + 9/8 + [13/12, 15/13, 16/15]
And in absolute terms over P1 but with a tonic on AcM2:
[AcM2, AsGrm3, P4, DeSb5, M6, Grm7, P8, Prm9, M10, P11] [9/8, 11/9, 4/3, 140/99, 5/3, 16/9, 2, 13/6, 5/2, 8/3]
Nice. If we root this on P1 instead of AcM2, that fourth scale degree looks even crazier: DeSb5 becomes DeSb4, justly tuned to 1120/891 at 396 cents, in contrast to Yarman's (10/7) / (9/8) = 80/63 at 414 cents. Not amazing agreement, but I've done worse than 18 cents before.
If you still want A4 to be an AcM6 over C4, you could also use this intonation of Cargah:
[DeSbAcAcM2, AsSpM2, Grm2] : [189/176, 33/28, 256/243]
instead of this one:
[m2, AcAcA2, Grm2] : [16/15, 1215/1024, 256/243]
If you use that mess then you get this in relative terms:
[AsGrm2, DeAcM2] + [DeSbAcAcM2, AsSpM2, Grm2] + AcM2 + [Prm2, ReAcA2, m2]
[88/81, 12/11] * [189/176, 33/28, 256/243] * 9/8 * [13/12, 15/13, 16/15]
Which looks like this in absolute terms:
[AsGrm2, DeAcM2, DeSbAcAcM2, AsSpM2, Grm2, AcM2, Prm2, ReAcA2, m2]
[88/81, 12/11] * [189/176, 33/28, 256/243] * 9/8 * [13/12, 15/13, 16/15]
...
Let's take stock of what's done and what's left to do. These makams were already spelled alphabetically as presented by Yarman:
* Rast (ascending & descending): [P1, AcM2, M3, P4, P5, AcM6, M7, P8] [1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1]
* Acemli Rast (ascends as Rast, descends as follows): [P8, Grm7, M6, P5, P4, M3, AcM2, P1] [2/1, 16/9, 5/3, 3/2, 4/3, 5/4, 9/8, 1/1]
* Mahur (ascending): [P1, AcM2, AcM3, P4, P5, AcM6, AcM7, P8] [1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2/1]
* Mahur (descending): [P8, M7, AcM6, P5, P4, M3, AcM2, P1] [2/1, 15/8, 27/16, 3/2, 4/3, 5/4, 9/8, 1/1]
* Nihavend (descending): [P8, m7, m6, P5, P4, m3, AcM2, P1] [2/1, 9/5, 8/5, 3/2, 4/3, 6/5, 9/8, 1/1]
* Segah (descending): [Hbm10, AcM9, P8, Hbm7, M6, P5, P4, M3] [40/17, 9/4, 2/1, 30/17, 5/3, 3/2, 4/3, 5/4]
This one was not spelled right:
* Saba (ascending, broken): [AcM2, HbSbAcd4, Ac4, De5, AcM6, AsGrm7, P8, AcM9] [9/8, 21/17, 27/20, 16/11, 27/16, 11/6, 2/1, 9/4]
But we fixed that:
* Saba (ascending, fixed): [AcM2, Asm3, Ac4, Prd5, AcM6, AsGrm7, P8, AcM9] # [9/8, 99/80, 27/20, 117/80, 27/16, 11/6, 2/1, 9/4]
This one was not spelled right:
* Saba (descending, broken): [AcM2, AsGrm3, P4, SpA4, AcM6, SpA6, P8, m9, M10, P11] [9/8, 11/9, 4/3, 10/7, 27/16, 25/14, 2/1, 32/15, 5/2, 8/3]
But we fixed that:
* Saba (descending, fixed) [AcM2, AsGrm3, P4, DeSb5, M6, Grm7, P8, Prm9, M10, P11] [9/8, 11/9, 4/3, 140/99, 5/3, 16/9, 2/1, 13/6, 5/2, 8/3]
This one was not spelled right:
* Huseyni (ascending and descending, broken): [AcM2, HbSbAcd4, P4, P5, AcM6, HbSbAcd8, P8, AcM9] [9/8, 21/17, 4/3, 3/2, 27/16, 63/34, 2/1, 9/4]
But we fixed that:
* Huseyni (ascending and descendnig, fixed): [P1, Asm2, Grm3, P4, P5, Asm6, Grm7, P8] # [1/1, 11/10, 32/27, 4/3, 3/2, 33/20, 16/9, 2/1]
These are spelled wrong and we haven't addressed it:
?! Pencgah (ascends and descends the same way): [P1, AcM2, M3, Sbd5, P5, AcM6, M7, P8] [1/1, 9/8, 5/4, 7/5, 3/2, 27/16, 15/8, 2/1]
?! Hicaz (ascending): [AcM2, m3, Sbd5, P5, AcM6, GrM7, P8, AcM9] [9/8, 6/5, 7/5, 3/2, 27/16, 50/27, 2/1, 9/4]
?! Hicaz (descending): [AcM9, P8, m7, AcM6, P5, Sbd5, m3, AcM2] [9/4, 2/1, 9/5, 27/16, 3/2, 7/5, 6/5, 9/8]
?! Nihavend (ascending): [P1, AcM2, m3, P4, P5, m6, Hbd8, P8] [1/1, 9/8, 6/5, 4/3, 3/2, 8/5, 32/17, 2/1]
?! Segah (ascending): [(Hbm3), M3, P4, P5, M6, M7, P8, Hbm10, M10] [(20/17), 5/4, 4/3, 3/2, 5/3, 15/8, 2/1, 40/17, 5/2]
?! Huzzam (ascending): [(Sbm3), ?, P4, P5, ?, ?, P8, Sbm10, ?] [(7/6), 36/29, 4/3, 3/2, 48/29, 54/29, 2/1, 7/3, 72/29]
?! Huzzam (descending): [Hbm10, AcM9, P8, Hbm7, ?, P5, P4, ?] [40/17, 9/4, 2/1, 30/17, 48/29, 3/2, 4/3, 36/29]
Well, we looked at Huzzam a little, but I don't think we solved it.
Here's a first attempt at Huzzam (ascending):
Relative intervals:
[ReSbAcM2, AcM2, PrSpGrm2] + AcM2 + [ReSbAcM2, PrSpM2, m2]
[14/13, 9/8, 208/189] * 9/8 * [14/13, 65/56, 16/15]
Yarman roots Huzzam 36/29 (at 259 cents) over C4, which happens to be 12 cents away from 5/4 or 5 cents away from 26/21, or 15 cents away from 16/13. Margo Schulter relates that 16/13 is an Arabic value for the tonic of Sikah.
...
I feel a little lost. Let's look up some other sources.
Here's a Turkish description of Huzzam from Kazim Yigiter:
Pitch classes: [(A#4), Bd4, C5, D5, E\b5, F#5, G5, A#5, Bd5]
Tetrachords in terms of simgeler:
Huzzam pentachord: [S, T, S, A] -> [Bd4, C5, D5, E\b5, F#5]
Hicaz tetrachord: [S, A, B] -> [F#5, G5, A#5, Bd5]
The "A" in the simgeler is ambiguous between 12 and 13 steps of 53-EDO but we can figure out which is the case in each instance, since the pentachord has to sum to 31 steps and the tetrachord has to sum to 22 steps.
Here's the intonation in 53-EDO steps:
Huzzam pentachord: [5, 9, 5, 12]
Hicaz tetrachord: [5, 13, 4]
...
In Yarman's staff notation, [E4, F4, and C5] have no microtonal accidentals and are tuned to [4/3, 3/2, 2/1] respectively. These are our most definite landmarks that we will try to hit. The rest is changeable.
...
I hear that Segâh, Müstear, and Hüzzam are closely related. Maybe we can transfer knowledge between them as we solve them.
...
Alsiadi gives an Arabic intonation in 53-EDO steps for Huzam as:
[6, 9, 4, 14, 4, 9, 7]
...
Maqam world gives a 24-EDO intonation:
[3, 4, 2, 6, 2, 4, 3]
...
I need to take a break from Huzam. Let's look at Pencgah. It only has one misspelled interval. Yarman uses Sbd5 tuned to 7/5 at 583c as a fourth interval. What fourth intervals have just tunings around there? Here are some options:
ReAcA4 # 18/13 _ 563c
ExAsGr4 # 187/135 _ 564c
HbAsSp4 # 165/119 _ 566c
SpAc4 # 243/175 _ 568c
A4 # 25/18 _ 569c
ExDeAcA4 # 153/110 _ 571c
PrSp4 # 39/28 _ 574c
ExReA4 # 272/195 _ 576c
AsSpGr4 # 88/63 _ 579c
ReDeAcAA4 # 200/143 _ 581c
AsAsGr4 # 605/432 _ 583c
DeSpAcA4 # 108/77 _ 586c
AcA4 # 45/32 _ 590c
ReSpA4 # 128/91 _ 591c
ReAsA4 # 55/39 _ 595c
AsSp4 # 99/70 _ 600c
PrSpSpGr4 # 208/147 _ 601c
ExA4 # 17/12 _ 603c
I think our strongest candidates are probably
PrSp4 # 39/28 _ 574c
AcA4 # 45/32 _ 590c
Even though those don't have the best fit. Let's just call it AcA4. It's only off by 7 cents and it's a nice 9/8 over the scale degree below it, M3 at 5/4.
Here's our fixed Pencgah in relative intervals:
[AcM2, M2, AcM2, m2] + [AcM2, M2, m2]
[9/8, 10/9, 9/8, 16/15] + [9/8, 10/9, 16/15]
And in absolute intervals:
[P1, AcM2, M3, AcA4, P5, AcM6, M7, P8]
[1, 9/8, 5/4, 45/32, 3/2, 27/16, 15/8, 2/1]
Fixed! And it was easy. And I liked it. Hicaz ascending and descending were only misspelled in using Sbd5 as 4th interval also, so maybe we've fixed those too? Let's check.
Hicaz (ascending) in relative terms:
[m2, AcA2, m2, AcM2, GrM2, Acm2, AcM2]
[16/15, 75/64, 16/15] + 9/8 + [800/729, 27/25, 9/8]
and absolute terms:
[AcM2, m3, AcA4, P5, AcM6, GrM7, P8, AcM9]
[9/8, 6/5, 45/32, 3/2, 27/16, 50/27, 2/1, 9/4]
Let's look at the intonation of the upper tetrachord. Yarman calls it a Huseyni tetrachord and it looks like this (as calculated between scale degrees):
[GrM2, Acm2, AcM9] # [800/729, 27/25, 9/8] _ [161c, 133c, 204c]
This is a fine 5-limit intonation of a tetrachord with [large neutral second, small neutral second, Pythagorean major second]. Yarman has an annotation which instead describes the tetrachord in this intonation:
[Asm2, Prm2, AcM2] # [11/10 * 13/10 * 9/8] _ [165c, 139c, 204c]
which you can see, from the accidentals or from the factor structure, doesn't equal a just P4 at 4/3.
If we want to keep two of these three ratios and adjust the third to reach P4, we have these options for tetrachords:
[11/10 * 320/297 * 9/8] _ [165c, 129c, 204c]
[128/117 * 13/12 * 9/8] _ [156c, 139c, 204c]
[11/10, 13/12, 160/143] _ [165c, 139c, 194c]
which each differs by 10 cents from Yarman's annotated tetrachord (differs on the substituted ratio). The 5-limit intonation splits that 10-cent deviation into two 5-cent deviations, across the neutral seconds. It's a very good compromise, and since the lower tetrachord was also 5-limit, the entire scale ends up being 5-limit. Pretty slick.
Here's our fixed Hicaz (descending), but written ascending because I prefer my scales that way. First in relative terms:
[m2, AcA2, m2] + [AcM2, m2, M2, AcM2]
[16/15, 75/64, 16/15] * [9/8, 16/15, 10/9, 9/8]
and now in absolute terms:
[AcM2, m3, AcA4, P5, AcM6, m7, P8, AcM9]
[9/8, 6/5, 45/32, 3/2, 27/16, 9/5, 2/1, 9/4]
Looks good to me. Yarman calls the high pentachord is "Nihavend". Speaking of which, let's fix the spelling of makam Nihavend. The descending form was fine. The ascending form looked like this:
[P1, AcM2, m3, P4, P5, m6, Hbd8, P8] [1/1, 9/8, 6/5, 4/3, 3/2, 8/5, 32/17, 2/1]
you can see that there's a low 8th interval standing in for a 7th interval. Here it is in relative intervals and ratios:
[AcM2, m2, M2, AcM2] + [m2, Hbm3, ExA1]
[9/8, 16/15, 10/9, 9/8] * [16/15, 20/17, 17/16] _ [204c, 112c, 182c, 204c] + [112c, 281c, 105c]
With the low pentachord being Nihavend and the high tetrachord being "Wide Hicaz". My first thought of how to spell Wide Hicaz by 2nd intervals is:
[m2, AsSpM2, DeSbAcM2] # [16/15, 33/28, 35/33] _ [112c, 284c, 102c]
which only differs from Yarman's tetrachord by 3 cents. Let's see how it looks in the makam. New version of Nihavend in relative degrees:
[AcM2, m2, M2, AcM2, m2, AsSpM2, DeSbAcM2] # [9/8, 16/15, 10/9, 9/8, 16/15, 33/28, 35/33]
And in absolute degrees:
[P1, AcM2, m3, P4, P5, m6, AsSpGrm7, P8] [1/1, 9/8, 6/5, 4/3, 3/2, 8/5, 66/35, 2/1]
Seems fine to me. Fixed. Great success.
Okay, let's try Segah (ascending).Yarman's intonation:
[(Hbm3), M3, P4, P5, M6, M7, P8, Hbm10, M10] [(20/17), 5/4, 4/3, 3/2, 5/3, 15/8, 2/1, 40/17, 5/2]
First step, the ornamental Hbm3 is lower than M3 by an ExA1 tuned to 17/16. We saw this in the last makam actually, in the Wide Hicaz genus. My solution is the same: use DeSbAcM2 tuned to 35/33 instead of ExA1. This gives us AsSpM2 over C4 tuned to 33/28. Higher up in the makam, the Hbm10 tuned to 40/17 is also lower than a just m10 by ExA1. To get a ninth interval where we want it, we just lower m10 by DeSbAcM2 instead, as before, giving AsSpM9 tuned to 33/14. Here's our fixed Segah in relative terms:
(DeSbAcM2) + [m2, AcM2, M2] + AcM2 + [m2, AsSpM2, DeSbAcM2]
(35/33) * [16/15, 9/8, 10/9] * 9/8 * [16/15, 33/28, 35/33]
And here it is in absolute terms:
[(AsSpM2), M3, P4, P5, M6, M7, P8, AsSpM9, M10]
[(33/28), 5/4, 4/3, 3/2, 5/3, 15/8, 2, 33/14, 5/2]
Nice. All that's left is Huzzam ascending and descending. Do you think we can do it? I'm going to try solving it without looking at my previous notes, and then I'll go back and compare.
Here's Yarman's makam Huzzam (ascending):
[(Sbm3), ?, P4, P5, ?, ?, P8, Sbm10, ?] [(7/6), 36/29, 4/3, 3/2, 48/29, 54/29, 2/1, 7/3, 72/29]
We want a 3rd interval that sounds like 36/29 @ 374 cents. The ReAsAsGrm3, tuned to 242/195 is 374c, but that's nonsense. Let's find a normal interval to stand in.
I think these are pretty good for simple intervals nearby:
Asm3 # 99/80 _ 369c
ReAcM3 # 81/65 _ 381c
The tonic in Segah, which is related to Huzzam, is a nearby M3 at 5/4. So using a typ eof minor third, Asm3, seems a little wrong, even though that interval looks a little better in terms of closeness of tuning. I guess we should use ReAcM3 for the tonic.
Oh, no, wait! I want 99/80 for tetrachord structure. You'll see in a minute.
The Sbm3 should be replaced with some kind of second around
1200 * log_2(7/6) = 267c
Here are some options:
SpAcM2 # 81/70 _ 253c
ExDeAcA2 # 51/44 _ 256c
PrSpM2 # 65/56 _ 258c
DeSpAcA2 # 90/77 _ 270c
AcA2 # 75/64 _ 275c
AsSpM2 # 33/28 _ 284c
Nothing great. I guess I pick AcA2 tuned to 75/64. It's only 8 cents sharp of Yarman's thing.
Next we need a 6th interval around
1200 * log_2(48/29) = 872 cents
Here are some options:
Asm6 # 33/20 _ 867c
PrSpGrm6 # 104/63 _ 868c
ExSbM6 # 119/72 _ 870c
ReAcM6 # 108/65 _ 879c
DeSpM6 # 128/77 _ 880c
M6 # 5/3 _ 884c
PrSpm6 # 117/70 _ 889c
ExGrM6 # 136/81 _ 897c
I think I like the Asm6 at 33/20 best. AND 33/20 gives us a tetrachord where we want it! Here are the scale degrees over C4:
[99/80, 4/3, 3/2, 33/20]
which have these relative intervals:
[DeM2, AcM2, Asm2] # [320/297 * 9/8 * 11/10]
Tight. That's the huzzam tetrachord. It's also zalzalian and perceptually indistinguishable from [14/13, 9/8, 208/189]. So good.
Now we need a 7th interval around
1200 * log_2(54/29) = 1076 cents
Here are some options:
GrM7 # 50/27 _ 1067c
ExDeM7 # 102/55 _ 1069c
PrSpGrm7 # 13/7 _ 1072c
ExSbM7 # 119/64 _ 1074c
ReAsAsGrm7 # 121/65 _ 1076c
DeSpM7 # 144/77 _ 1084c
M7 # 15/8 _ 1088c
...
Looking at all those options for sixths and all those options for sevenths, here are some simple tetrachords we can make (with a fifth of 3/2 and an octave at 2/1:
[sixth, seventh] :: [tetrachord]
[33/20, 15/8] :: [11/10, 25/22, 16/15] [165c, 221c, 112c]
[5/3, 50/27] :: [10/9, 10/9, 27/25] [182c, 182c, 133c]
[5/3, 13/7] :: [10/9, 39/35, 14/13] [182c, 187c, 128c]
[5/3, 15/8] :: [10/9, 9/8, 16/15] [182c, 204c, 112c]
[117/70, 13/7] :: [39/35, 10/9, 14/13] [187c, 182c, 128c]
I think the first one is going to be our best bet. Yarman's annotations say that the 7th should be 16/15 under 2/1, ie.e 15/8, and I already liked 33/20 for the 6th. The 16/15 is also the start of a Hicaz tetrachord. We've found multiple intonations for that, but this one starts with 16/15:
[m2, AcA2, m2] # [16/15, 75/64, 16/15] _ [112c, 275c, 112c]
and I'm keen to try it. This transforms Yarman's pseudo-ninth interval, the Sbm10 at 7/3, into 75/32, which sure looks nice to me. See how they're basically the same fraction? It's cute. Then we end the ascending form 16/15 higher on 5/2, which is 12 cents off from Yarman's high pitch at 72/29. Actually this is a little bit of a problem since it doesn't form an octave with our tonic, unless we accept that Huzzam starts on 5/4 like Segah instead of 12 cents below 5/4. I think we lost 12 cents somewhere in there. Probably we don't have AcM2 separating the tetrachords. Please hold.
Right, I don't want the 7th and the 6th to form a tetrachord, I want the 7th to be 9/8 over the 6th, which means it has to be 297/160. And then we need 320/297 to reach 2/1, and that has to be the first interval of a Hicaz tetrachord? Let's see how that could work. Here are three options:
[320/297, 81/70, 77/72] _ [129c, 253c, 116c]
[320/297, 297/256, 16/15] _ [129c, 257c, 112c]
...
Here's huzzam ascending with the first one of those weird Hicaz intonations (without the low ornament):
Relative:
[DeM2, AcM2, Asm2] + AcM2 + [DeM2, SpAcM2, AsSbm2]
[320/297, 9/8, 11/10] * 9/8 * [320/297, 81/70, 77/72]
Absolute:
[Asm3, P4, P5, Asm6, Asm7, P8, SpAcM9, Asm10]
[99/80, 4/3, 3/2, 33/20, 297/160, 2/1, 81/35, 99/40]
Here's Huzzam ascending with the second of those weird Hicaz intonations (without the low ornament).
Relative:
[DeM2, AcM2, Asm2, AcM2, DeM2, AsAcM2, m2]
[320/297, 9/8, 11/10, 9/8, 320/297, 297/256, 16/15]
Absolute:
[Asm3, P4, P5, Asm6, Asm7, P8, AsAcM9, Asm10]
[99/80, 4/3, 3/2, 33/20, 297/160, 2/1, 297/128, 99/40]
They both seem fine. The first one has simpler frequency ratios, but I like the factor structure better in the second one. They only differ by 4 cents, so that isn't much help.
...
Oh, cute. I looked back a little bit about what I'd already written concerning Huzzam. On my first analysis, I I used
[14/13 * 9/8 * 208/189]
instead of the perceptually indistinguishable
[320/297, 9/8, 11/10]
for the lower tetrachord. We could try that a little more.
The tonic will now be
(4/3) / (14/13) = 26/21.
Our sixth interval will now be
(3/2) * (208/189) = 104/63
Our seventh interval will now be
(104/63) * (9/8) = 13/7
I had two hicaz intonations that I hadn't chosen between. The analogue of the one that ends in a just m2 at16/15 is
[14/13, 65/56, 16/15]
And the other one doesn't translate as obviously. I think it the analogue would be
[14/13, 624/539, 77/72]
which is pretty ugly, so let's just use the first one. Here it is all together, without the low ornament:
Relative:
[ReSbAcM2, AcM2, PrSpGrm2, AcM2, ReSbAcM2, PrSpM2, m2]
[14/13, 9/8, 208/189, 9/8, 14/13, 65/56, 16/15]
Absolute:
[PrSpGrm3, P4, P5, PrSpGrm6, PrSpGrm7, P8, PrSpM9, PrSpGrm10]
[26/21, 4/3, 3/2, 104/63, 13/7, 2/1, 65/28, 52/21]
I don't know if the low ornament needs to change. I think we should just keep it where it is.
I don't know if this is solved. Maybe? Let's look at Huzzam descending now. Here's Yarman's intonation (written descending):
[Hbm10, AcM9, P8, Hbm7, ?, P5, P4, ?] [40/17, 9/4, 2/1, 30/17, 48/29, 3/2, 4/3, 36/29]
There is no annotation of tetrachords to help us along, except that the weird A and weird Bb are connected by just m2 at 16/15, i.e. between the 6th interval (48/29) and the 7th interval (30/17). The actual ratio here is
(30/17) / (48/29) = 145/136 at 111 cents,
and a m2 at 16/15 is 112 cents. So that's something.
For the scale to be alphabetical, we need a 6th interval around 48/29 at 872 cents, and a 3rd interval around 36/29 at 374 cents. And we already fond candidates for those on the ascent. The 36/29 is well represented by 99/80 or 26/21 (which are perceptually indistinguishable). The 48/29 is well represented by 33/20 or 104/63. Let's see what it looks like with the 11-limit options first. I'll write the scale ascending from here on.
Relative:
[DeM2, AcM2, Asm2, HbDeAcM2, ExM2, AcM2, Hbm2] [320/297, 9/8, 11/10, 200/187, 17/15, 9/8, 160/153]
Absolute:
[Asm3, P4, P5, Asm6, Hbm7, P8, AcM9, Hbm10] [99/80, 4/3, 3/2, 33/20, 30/17, 2/1, 9/4, 40/17]
And now the 13-limit version.
Relative:
[ReSbAcM2, AcM2, PrSpGrm2, HbReSbAcAcM2, ExM2, AcM2, Hbm2] [14/13, 9/8, 208/189, 945/884, 17/15, 9/8, 160/153]
Absolute:
[PrSpGrm3, P4, P5, PrSpGrm6, Hbm7, P8, AcM9, Hbm10] [26/21, 4/3, 3/2, 104/63, 30/17, 2/1, 9/4, 40/17]
You may notice some garbage. The 11-limit scale has HbDeAcM2 in its relative intervals and the 13-limit scale has HbReSbAcAcM2. These are not real things and these are not acceptable. Also we don't have 16/15 anywhere.
If we replace our 6th intervals with the ratio that's actually 16/15 below our 7th interval at 30/17, i.e. 225/136, then things look a lot better.
Here's Huzzam descending with the 11-limit tonic (99/80):
Relative:
[DeM2, AcM2, HbAcM2, m2, ExM2, AcM2, Hbm2] [320/297, 9/8, 75/68, 16/15, 17/15, 9/8, 160/153]
Absolute:
[Asm3, P4, P5, HbAcM6, Hbm7, P8, AcM9, Hbm10] [99/80, 4/3, 3/2, 225/136, 30/17, 2/1, 9/4, 40/17]
Here's Huzzam descending with the 13-limit tonic (26/21):
Relative:
[ReSbAcM2, AcM2, HbAcM2, m2, ExM2, AcM2, Hbm2] [14/13, 9/8, 75/68, 16/15, 17/15, 9/8, 160/153]
Absolute:
[PrSpGrm3, P4, P5, HbAcM6, Hbm7, P8, AcM9, Hbm10] [26/21, 4/3, 3/2, 225/136, 30/17, 2/1, 9/4, 40/17]
One problem with this is that we no longer have a perfect fourth in our bottom tetrachord (since we adjusted the 6th interval but not the 3rd interval, which is our tonic written relative to Rast). We could change our tonic the same way we changed our sixth to re-establish a tetrachord, or, since we changed our sixth to get 16/15 up to the 7th, we could change the 7th (and the 10th interval that we want to be P4 over that) and leave our 3rd and 6th how they were. I don't think I want to change the tonic. It's pretty good and we'd have to monkey with the ascending form also. And what's so good about 30/17 and 40/17? Nothing worth keeping there. Let's try this:
Relative:
[DeM2, AcM2, Asm2, m2, DeAcA2, AcM2, AsGrd2]
[320/297, 9/8, 11/10, 16/15, 25/22, 9/8, 704/675]
Absolute:
[Asm3, P4, P5, Asm6, AsGrd7, P8, AcM9, AsGrd10]
[99/80, 4/3, 3/2, 33/20, 44/25, 2/1, 9/4, 176/75]
It's not beautiful, is it? But if the high note is
P4 + m2 + P4 = Grd8
4/3 * 16/15 * 4/3 = 256/135
over the tonic, then
Asm3 + Grd8 = AsGrd10
99/80 * 256/135 = 176/75
our hands are tied, up to our choice of tonic. If we use Yarman's tuning for the higher note, 40/17, that give us a tonic at 675/544, which is 374c in contrast to 99/80 at 369c. There's only 5 cents of difference, and my way we get the use the nice Zalzalian ratios of 320/297 and 11/10.
If you think (99/80 * 256/135 = 175) looks bad, check out what happens when we use the perceptually indistinguishable 13-limit tonic:
26/21 * 256/135 = 6656/2835
So I think we're doing fine with the 11-limt intonation. Solved.
