Counting & naming the step sizes in Meantone Maquams

Hi,

I sent this first to Ozan, but I think he's busy elsewhere
for now. Anyway, FWIW, here are my reflections on the
subject. Your C&C are welcome.
Regards,
Yahya

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Based on Ozan's "hints", I've calculated the steps in the
four different scales.

Please don't take the _names_ I've given the steps seriously
- they're just so I could readily tell the difference between
the step sizes. A long search through Scala didn't help much
with more "standard" names.

Rast has just two step sizes, with one size of tone, and one
size of semitone. It is the most regular of these four scales.

Suz-i-Dilara (Oh, Susie!) has three step sizes, with one size
of tone, and one of semitone.

Altered Rast has four step sizes, with three sizes of tone,
and one of semitone.

Altered Suz-i-Dilara has four step sizes, with two sizes of
tone, and two of semitone.

Rast:
C 0 197 MeantoneD 197 197 MeantoneE 394 94 Minor semitoneF
498 197 MeantoneG 695 197 MeantoneA 892 197 MeantoneB 1089 111
Minor semitone (111.731 minor diatonic semitone)C' 1200
Altered Rast:
C 0 197 MeantoneD 197 212 Large MeantoneE 409 89 Sub-Minor
semitone (90.225 limma, Pythagorean minor second)F 498 197 MeantoneG
695 212 Large MeantoneA 907 89 Sub-Minor semitone (90.225 limma,
Pythagorean minor second)Bb 996 204 Major whole tone (203.910 major whole
tone)C' 1200
Suz-i Dilara:
C 0 212 Large MeantoneD 212 212 Large MeantoneE 424 74
TriquitoneF 498 212 Large MeantoneG 710 212 Large MeantoneA 922
212 Large MeantoneB 1134 66 Tristone (66.765 Pythagorean double
diminished third)C' 1200
Altered Suz-i Dilara:
C 0 212 Large MeantoneD 212 207 Large Major toneE 409 89
Sub-Minor semitone (90.225 limma, Pythagorean minor second)F 498 212
Large MeantoneG 710 197 MeantoneA 907 212 Large MeantoneB 1119 81
Small Sub-Minor semitoneC' 1200
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There are NINE different step sizes in just these four scales.
I was surprised; the object of temperaments such as meantone
is surely to reduce the number of steps to a manageable handful.
How many step sizes are there, then, in all of the maqam repertoire
if expressed in these meantones? It would probably be just as
easy to stick with strict ratios - assuming this is how the maqam
tunings originated.
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I looked in Scala for names for these intervals.

First, I input the raw scale, Rast.
Note that Scala does not name anything much -
0: 1/1 0.000 unison, perfect prime 1:
197.000 cents 197.000 2: 394.000 cents 394.000 3:
498.000 cents 498.000 4: 695.000 cents 695.000 5:
892.000 cents 892.000 6: 1089.000 cents 1089.000 7:
2/1 1200.000 octave
These four scales require six small step sizes and four large step
sizes, as follows -
66 74 81 89 94 111 197 204 207 212
Do they have accepted names?

I input these pitches into Scala, then tried approximating them in
the 7, 5 and 3 limits. It gave these names -

Raw -
0: 1/1 0.000 unison, perfect prime 1:
66.000 cents 66.000 2: 74.000 cents 74.000 3:
81.000 cents 81.000 4: 89.000 cents 89.000 5:
94.000 cents 94.000 6: 111.000 cents 111.000 7:
197.000 cents 197.000 8: 204.000 cents 204.000 9:
207.000 cents 207.000 10: 212.000 cents 212.000
7-limit approximation -
0: 1/1 0.000 unison, perfect prime 1:
536870912/516796875 65.973 2: 254803968/244140625 74.010 3:
281302875/268435456 81.059 4: 629407744/597871125 88.992 5:
201768035/191102976 94.017 6: 2560/2401 111.010 7:
5359375/4782969 196.990 8: 9765625/8680203 203.980 9:
18907875/16777216 206.981 10: 15625/13824 212.017
5-limit approximation -
0: 1/1 0.000 unison, perfect prime 1:
134217728/129140163 66.765 Pythagorean double diminished third 2:
254803968/244140625 74.010 3: 17578125/16777216 80.733 4:
8388608/7971615 88.271 5: 4428675/4194304 94.132 6:
16/15 111.731 minor diatonic semitone 7: 14348907/12800000
197.756 8: 9/8 203.910 major whole tone 9:
1310720000/1162261467 208.110 10: 15625/13824 212.017
3-limit approximation -
0: 1/1 0.000 unison, perfect prime 1:
134217728/129140163 66.765 Pythagorean double diminished third 2:
134217728/129140163 66.765 Pythagorean double diminished third 3:
256/243 90.225 limma, Pythagorean minor second 4: 256/243
90.225 limma, Pythagorean minor second 5: 256/243 90.225
limma, Pythagorean minor second 6: 2187/2048 113.685 apotome
7: 9/8 203.910 major whole tone 8: 9/8
203.910 major whole tone 9: 9/8 203.910 major whole
tone 10: 9/8 203.910 major whole tone
The last (3-limit approximation) is clearly useless for distinguishing
these different steps.

Only the 111 and 204 have (approximate) names in the 5-limit, namely
minor (diatonic) semitone and major whole tone, as they are close to
16/15 and 9/8.

Why didn't the minor whole tone 10/9 show up? It's 182.404 cents,
that's why, and a long way from the nearest step in Ozan's 4 scales,
namely 197 cents.

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What names does Scala give for pitches within 3 cents of these nine
intervals?

The 5-limit gives these distinct ratios & note names -
for ~66 (63 to 69) cents -
1: 648/625 62.565 major diesis 2: 531441/512000
64.519 3: 134217728/129140163 66.765 Pythagorean double diminished
third 4: 20480/19683 68.719 grave minor second
for ~74 (71 to 77) cents -
1: 25/24 70.672 classic chromatic semitone, minor
chroma 2: 273375/262144 72.626 3: 254803968/244140625
74.010 4: 51018336/48828125 75.964 5: 5000000/4782969
76.826
for ~81 (78 to 84) cents -
1: 390625/373248 78.780 2: 536870912/512578125 80.164
3: 17578125/16777216 80.733 4: 16384/15625 82.118 two
(minor diesis, diesis) 5: 6561/6250 84.071
for ~89 (86 to 92) cents -
1: 43046721/40960000 86.025 two (Mathieu superdiesis) 2:
8388608/7971615 88.271 3: 256/243 90.225 limma,
Pythagorean minor second 4: 135/128 92.179 major chroma,
major limma
for ~94 (91 to 97) cents -
1: 256/243 90.225 limma, Pythagorean minor second 2:
135/128 92.179 major chroma, major limma 3: 4428675/4194304
94.132 4: 1289945088/1220703125 95.517 5: 409600000/387420489
96.379 6: 258280326/244140625 97.470
for ~111 (108 to 114) cents -
1: 1220703125/1146617856 108.393 2: 524288/492075 109.778
3: 16/15 111.731 minor diatonic semitone 4:
2187/2048 113.685 apotome
(Note that the 7-limit also gives these smallish-integer ratios -
1: 2560/2401 111.010 2: 83349/78125 112.057
BP minor link
for ~197 (194 to 200) cents -
1: 262144/234375 193.849 2: 17496/15625 195.803
3: 14348907/12800000 197.756 4: 1220703125/1088391168 198.618 5:
134217728/119574225 200.003
for ~204 (201 to 207) cents -
1: 134217728/119574225 200.003 2: 4096/3645 201.956
3: 9/8 203.910 major whole tone 4: 295245/262144
205.864 for ~207 (204 to 210) cents -
1: 9/8 203.910 major whole tone 2: 295245/262144
205.864 3: 1310720000/1162261467 208.110 4: 1377495072/1220703125
209.202 5: 200000/177147 210.064
for ~212 (209 to 215) cents -
1: 1377495072/1220703125 209.202 2: 200000/177147 210.064
3: 15625/13824 212.017 4: 536870912/474609375 213.402 5:
18984375/16777216 213.971 6: 442368/390625 215.355
As you will see, there's some overlap between these ranges.

Still, the only named 5-limit intervals approximating the nine steps
of Ozan's four meantone maqam scales are the following (numbers
refer to the fuller lists above) -

for ~66 (63 to 69) cents -
1: 648/625 62.565 major diesis 3: 134217728/129140163
66.765 Pythagorean double diminished third 4: 20480/19683
68.719 grave minor second
for ~74 (71 to 77) cents -
1: 25/24 70.672 classic chromatic semitone, minor
chroma for ~81 (78 to 84) cents -
4: 16384/15625 82.118 two (minor diesis, diesis) for ~89 (86
to 92) cents -
1: 43046721/40960000 86.025 two (Mathieu superdiesis) 3:
256/243 90.225 limma, Pythagorean minor second 4: 135/128
92.179 major chroma, major limmafor ~94 (91 to 97) cents -
1: 256/243 90.225 limma, Pythagorean minor second 2:
135/128 92.179 major chroma, major limma
for ~111 (108 to 114) cents -
3: 16/15 111.731 minor diatonic semitone 4:
2187/2048 113.685 apotome
for ~197 (194 to 200) cents -

NONE

for ~204 (201 to 207) cents -
3: 9/8 203.910 major whole tone for ~207 (204 to 210)
cents -
1: 9/8 203.910 major whole tone
for ~212 (209 to 215) cents -

NONE

Allowing for -
(a) an inaccuracy of up to half a cent in Ozan's interval
specifications, and
(b) a 3-cent region of "indifference" either side of a perfectly
tuned interval;

I'm tempted to make the following identifications -

66 cents -
4: 20480/19683 68.719 grave minor second
74 cents -
1: 25/24 70.672 classic chromatic semitone, minor
chroma 81 cents -
4: 16384/15625 82.118 two (minor diesis, diesis) 89 cents -
3: 256/243 90.225 limma, Pythagorean minor second
94 cents -
2: 135/128 92.179 major chroma, major limma
111 cents -
3: 16/15 111.731 minor diatonic semitone
197 cents -

NONE

204 cents -
3: 9/8 203.910 major whole tone 207 cents -
1: 9/8 203.910 major whole tone
212 cents -

NONE

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To me, there's something unsatisfactory about a temperament system
(even a meantone system), as an approximation to a rationally based
tuning system, that -

1. produces so many different step sizes; and
2. doesn't relate at least two of them to any smallish-integer ratio
in the five-limit.

I can't help but think that this would be a very awkward system to
tune on the 'ud ... How does it work in practice, I wonder? For
example, can you tune the three different sizes of fifths (695, 702,
and 710 cents) by counting beats?

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