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The 2.3.7 [[Just_intonation_subgroups|subgroup]] comma for the '''gamelismic clan''' is the gamelisma, [[1029/1024]], with monzo {{monzo|-10 1 0 3}}. For any member of the clan, for the rank three [[Gamelismic family #Gamelan|gamelan temperament]] itself, and for the rank two 2.3.7 temperament [[slendric]], this means three [[8/7]] intervals give a fifth, [[3/2]]. In fact, we find that 3/2 = (8/7)<sup>3</sup> × 1029/1024. From this it follows that gamelismic temperaments tend to flatten both the fifth and the 7/4, or if they do not, the other of the pair must be flattened even more. [[36edo]] is a good tuning for gamelismic itself, though if the full 7-limit is desired, [[72edo]], [[77edo]] or [[118edo]] might be preferred.
The [[2.3.7 subgroup]] comma for the '''gamelismic clan''' is the gamelisma, [[1029/1024]], with [[monzo]] {{monzo| -10 1 0 3 }}. For any member of the clan, for the rank-3 [[Gamelismic family #Gamelan|gamelismic temperament]] itself, and for the rank-2 2.3.7 temperament [[slendric]], this means three [[~]][[8/7]] intervals give a fifth, [[3/2]]. In fact, we find that 3/2 = (8/7)<sup>3</sup> × 1029/1024. From this it follows that gamelismic temperaments tend to flatten both the fifth and the harmonic seventh, or if they do not, the other of the pair must be flattened even more. [[36edo]] is a good tuning for slendric, though if the full 7-limit is desired, [[72edo]], [[77edo]] or [[118edo]] might be preferred.


To the gamelisma itself we need to add the comma which appears next on the modified [[Normal lists #Normal interval list|normal comma list]] for the full 7-limit. The second comma on the list for mothra is 81/80, for rodan 245/243, for guiron 32805/32768, for gorgo 36/35, and for gidorah 256/245. These all use 8/7 as a generator, though in the case of gidorah that's the same as 6/5. Miracle adds 33075/32768 and uses the secor, half an 8/7, as generator. Lemba adds 525/512 to the list, and has a half-octave period. Valentine adds 6144/6125 with a generator of 21/20 and superkleismic adds 875/864 with a generator of 6/5. Unidec adds 4375/4374, and has a generator of 10/9 with a half-octave period. Hemithirds adds 65625/65536 with a generator half of a major third. Finally, tritikleismic adds 15625/15552 and has a generator of 6/5 with a 1/3 octave period.
To the gamelisma itself we need to add the comma which appears next on the modified [[Normal lists #Normal interval list|normal comma list]] for the full 7-limit. The second comma on the list for mothra is [[81/80]], for rodan [[245/243]], for guiron [[32805/32768]], for gorgo [[36/35]], and for gidorah [[256/245]]. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~6/5. Miracle adds [[33075/32768]] and uses the [[secor]], half an 8/7, as generator. Lemba adds [[525/512]] to the list, and has a half-octave [[period]]. Valentine adds [[6144/6125]] with a generator of ~21/20 and superkleismic adds [[875/864]] with a generator of ~6/5. Unidec adds [[4375/4374]], and has a generator of ~10/9 with a half-octave period. Hemithirds adds [[65625/65536]] with a generator half of a classical major third. Finally, tritikleismic adds [[15625/15552]] and has a generator of 6/5 with a 1/3-octave period.


Full 7-limit temperaments discussed elsewhere are:
Full 7-limit temperaments discussed elsewhere are:
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== Slendric ==
== Slendric ==
{{main| Slendric }}
{{Main| Slendric }}
{{See also| No-fives subgroup temperaments #Slendric }}
{{See also| No-fives subgroup temperaments #Slendric }}


Subgroup: 2.3.7
[[Subgroup]]: 2.3.7


[[Comma list]]: 1029/1024
[[Comma list]]: 1029/1024


[[Sval]] [[mapping]]: [{{val| 1 1 3 }}, {{val| 0 3 -1 }}]
{{Mapping|legend=2| 1 1 3 | 0 3 -1 }}


Mapping generators: ~2, ~8/7
: sval mapping generators: ~2, ~8/7


Gencom mapping: [{{val| 1 1 0 3 }}, {{val| 0 3 0 -1 }}]
{{Mapping|legend=3| 1 1 0 3 | 0 3 0 -1 }}


[[Gencom]]: [2 8/7; 1029/1024]  
: [[gencom]]: [2 8/7; 1029/1024]  


[[POTE generator]]: ~8/7 = 233.688
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 233.688


{{Optimal ET sequence|legend=1| 36, 77, 113, 190 }}
{{Optimal ET sequence|legend=1| 36, 77, 113, 190 }}
Scales: [[slendric5]], [[slendric6]], [[slendric11]], [[slendric16]]


=== Baladic ===
=== Baladic ===
Baladic is a 2.3.7.13.17 subgroup temperament that attempts to approximate the Maqam Sikah Baladi scale. 36edo is an excellent baladic tuning.
Baladic is a 2.3.7.13.17 subgroup temperament that attempts to approximate the Maqam Sikah Baladi scale. 36edo is an excellent baladic tuning.


Subgroup: 2.3.7.13.17
[[Subgroup]]: 2.3.7.13.17


[[Comma list]]: 169/168, 273/272, 289/288
[[Comma list]]: 169/168, 273/272, 289/288


[[Sval]] [[mapping]]: [{{val| 2 2 6 7 7 }}, {{val| 0 3 -1 1 3 }}]
{{Mapping|legend=2| 2 2 6 7 7 | 0 3 -1 1 3 }}


Mapping generators: ~17/12, ~8/7
: sval mapping generators: ~17/12, ~8/7


[[POTE generator]]: ~8/7 = 233.6155
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 233.6155


{{Optimal ET sequence|legend=1| 10, 26, 36, 154f, 190ffg }}
{{Optimal ET sequence|legend=1| 10, 26, 36, 154f, 190ffg }}


== Rodan ==
== Rodan ==
{{main| Rodan }}
{{Main| Rodan }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 1029/1024
[[Comma list]]: 245/243, 1029/1024


[[Mapping]]: [{{val| 1 1 -1 3 }}, {{val| 0 3 17 -1 }}]
{{Mapping|legend=1| 1 1 -1 3 | 0 3 17 -1 }}


{{Multival|legend=1| 3 17 -1 20 -10 -50 }}
{{Multival|legend=1| 3 17 -1 20 -10 -50 }}


[[POTE generator]]: ~8/7 = 234.417
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 234.417


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 2/9 0 1/18 -1/18 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 2/9 0 1/18 -1/18 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 }}, {{monzo| 25/9 0 17/18 -17/18 }}, {{monzo| 25/9 0 -1/18 1/18 }}]
: {{monzo list| 1 0 0 0 | 5/3 0 1/6 -1/6 | 25/9 0 17/18 -17/18 | 25/9 0 -1/18 1/18 }}
: [[Eigenmonzo]]s (unchanged-intervals): 2, 7/5
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.7/5


[[Algebraic generator]]: larger root of 20''x''<sup>2</sup> - 36''x'' + 15, or (9 + √6)/10.
[[Algebraic generator]]: larger root of 20''x''<sup>2</sup> - 36''x'' + 15, or (9 + √6)/10.
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[[Badness]]: 0.037112
[[Badness]]: 0.037112
Scales: [[Rodan26opt]], [[Rodan31opt]], [[Rodan41opt]]


=== 11-limit ===
=== 11-limit ===
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Comma list: 245/243, 385/384, 441/440
Comma list: 245/243, 385/384, 441/440


Mapping: [{{val| 1 1 -1 3 6 }}, {{val| 0 3 17 -1 -13 }}]
Mapping: {{mapping| 1 1 -1 3 6 | 0 3 17 -1 -13 }}


POTE generator: ~8/7 = 234.459
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.459


Minimax tuning:  
Minimax tuning:  
* [[11-odd-limit]]: ~8/7 = {{monzo| 4/19 2/19 0 0 -1/19 }}
* [[11-odd-limit]]: ~8/7 = {{monzo| 4/19 2/19 0 0 -1/19 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 31/19 6/19 0 0 -3/19 }}, {{monzo| 49/19 34/19 0 0 -17/19 }}, {{monzo| 53/19 -2/19 0 0 1/19 }}, {{monzo| 62/19 -26/19 0 0 13/19 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 31/19 6/19 0 0 -3/19 }}, {{monzo| 49/19 34/19 0 0 -17/19 }}, {{monzo| 53/19 -2/19 0 0 1/19 }}, {{monzo| 62/19 -26/19 0 0 13/19 }}]
: Eigenmonzos (unchanged-intervals): 2, 11/9
: Eigenmonzo (unchanged-interval) basis: 2.11/9


Algebraic generator: [[Algebraic number|positive root]] of ''x''<sup>2</sup> + 16''x'' - 31, or √95 - 8.
Algebraic generator: [[Algebraic number|positive root]] of ''x''<sup>2</sup> + 16''x'' - 31, or √95 - 8.
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Badness: 0.023093
Badness: 0.023093
Scales: [[Rodan26opt]], [[Rodan31opt]], [[Rodan41opt]]


==== 13-limit ====
==== 13-limit ====
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Comma list: 196/195, 245/243, 352/351, 364/363
Comma list: 196/195, 245/243, 352/351, 364/363


Mapping: [{{val| 1 1 -1 3 6 8 }}, {{val| 0 3 17 -1 -13 -22 }}]
Mapping: {{mapping| 1 1 -1 3 6 8 | 0 3 17 -1 -13 -22 }}


POTE generator: ~8/7 = 234.482
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.482


Minimax tuning:  
Minimax tuning:  
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Badness: 0.018448
Badness: 0.018448
Scales: [[Rodan26opt]], [[Rodan31opt]], [[Rodan41opt]]


===== 17-limit =====
===== 17-limit =====
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Comma list: 154/153, 196/195, 245/243, 256/255, 273/272
Comma list: 154/153, 196/195, 245/243, 256/255, 273/272


Mapping: [{{val| 1 1 -1 3 6 8 8 }}, {{val| 0 3 17 -1 -13 -22 -20 }}]
Mapping: {{mapping| 1 1 -1 3 6 8 8 | 0 3 17 -1 -13 -22 -20 }}


POTE generator: ~8/7 = 234.524
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.524


Minimax tuning:
Minimax tuning:
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Badness: 0.016743
Badness: 0.016743
Scales: [[Rodan26opt]], [[Rodan31opt]], [[Rodan41opt]]


==== Aerodactyl ====
==== Aerodactyl ====
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Comma list: 91/90, 245/243, 385/384, 441/440
Comma list: 91/90, 245/243, 385/384, 441/440


Mapping: [{{val| 1 1 -1 3 6 -1 }}, {{val| 0 3 17 -1 -13 24 }}]
Mapping: {{mapping| 1 1 -1 3 6 -1 | 0 3 17 -1 -13 24 }}


POTE generator: ~8/7 = 234.639
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.639


{{Optimal ET sequence|legend=1| 5, 41f, 46, 133ff }}
{{Optimal ET sequence|legend=1| 5, 41f, 46, 133ff }}
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Comma list: 176/175, 245/243, 1029/1024
Comma list: 176/175, 245/243, 1029/1024


Mapping: [{{val| 1 1 -1 3 -3 }}, {{val| 0 3 17 -1 33 }}]
Mapping: {{mapping| 1 1 -1 3 -3 | 0 3 17 -1 33 }}


POTE generator: ~8/7 = 234.728
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.728


{{Optimal ET sequence|legend=1| 41e, 46 }}
{{Optimal ET sequence|legend=1| 41e, 46 }}
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Comma list: 91/90, 176/175, 245/243, 847/845
Comma list: 91/90, 176/175, 245/243, 847/845


Mapping: [{{val| 1 1 -1 3 -3 -1 }}, {{val| 0 3 17 -1 33 24 }}]
Mapping: {{mapping| 1 1 -1 3 -3 -1 | 0 3 17 -1 33 24 }}


POTE generator: ~8/7 = 234.782
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.782


{{Optimal ET sequence|legend=1| 41ef, 46 }}
{{Optimal ET sequence|legend=1| 41ef, 46 }}
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Comma list: 100/99, 245/243, 1029/1024
Comma list: 100/99, 245/243, 1029/1024


Mapping: [{{val| 1 1 -1 3 -2 }}, {{val| 0 3 17 -1 28 }}]
Mapping: {{mapping| 1 1 -1 3 -2 | 0 3 17 -1 28 }}


POTE generator: ~8/7 = 234.145
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.145


{{Optimal ET sequence|legend=1| 36ce, 41 }}
{{Optimal ET sequence|legend=1| 36ce, 41 }}
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Comma list: 100/99, 105/104, 245/243, 352/351
Comma list: 100/99, 105/104, 245/243, 352/351


Mapping: [{{val| 1 1 -1 3 -2 0 }}, {{val| 0 3 17 -1 28 19 }}]
Mapping: {{mapping| 1 1 -1 3 -2 0 | 0 3 17 -1 28 19 }}


POTE generator: ~8/7 = 234.089
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.089


{{Optimal ET sequence|legend=1| 36ce, 41 }}
{{Optimal ET sequence|legend=1| 36ce, 41 }}
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== Guiron ==
== Guiron ==
{{see also| Schismatic family }}
{{See also| Schismatic family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 10976/10935
[[Comma list]]: 1029/1024, 10976/10935


[[Mapping]]: [{{val| 1 1 7 3 }}, {{val| 0 3 -24 -1 }}]
{{Mapping|legend=1| 1 1 7 3 | 0 3 -24 -1 }}


Mapping generators: ~2, ~8/7
: mapping generators: ~2, ~8/7


{{Multival|legend=1| 3 -24 -1 -45 -10 65 }}
{{Multival|legend=1| 3 -24 -1 -45 -10 65 }}


[[POTE generator]]: ~8/7 = 233.930
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 233.930


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 7/24 0 -1/24 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 7/24 0 -1/24 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 15/8 0 -1/8 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 65/24 0 1/24 0 }}]
: {{monzo list| 1 0 0 0 | 15/8 0 -1/8 0 | 0 0 1 0 | 65/24 0 1/24 0 }}
: [[Eigenmonzo]]s (unchanged-intervals): 2, 5
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5


{{Optimal ET sequence|legend=1| 36, 41, 77, 118, 277d }}
{{Optimal ET sequence|legend=1| 36, 41, 77, 118, 277d }}
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Comma list: 385/384, 441/440, 10976/10935
Comma list: 385/384, 441/440, 10976/10935


Mapping: [{{val| 1 1 7 3 -2 }}, {{val| 0 3 -24 -1 28 }}]
Mapping: {{mapping| 1 1 7 3 -2 | 0 3 -24 -1 28 }}


Mapping generators: ~2, ~8/7
: mapping generators: ~2, ~8/7


POTE generator: ~8/7 = 233.931
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.931


Minimax tuning:
Minimax tuning:
* [[11-odd-limit]]: ~8/7 = {{monzo| 7/24 0 -1/24 }}
* 11-odd-limit: ~8/7 = {{monzo| 7/24 0 -1/24 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 15/8 0 -1/8 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 65/24 0 1/24 0 0 }}, {{monzo| 37/6 0 -7/6 0 0 }}]  
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 15/8 0 -1/8 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 65/24 0 1/24 0 0 }}, {{monzo| 37/6 0 -7/6 0 0 }}]  
: Eigenmonzos (unchanged-intervals): 2, 5
: Eigenmonzo (unchanged-interval) basis: 2.5


{{Optimal ET sequence|legend=1| 36e, 41, 77, 118, 159, 277d }}
{{Optimal ET sequence|legend=1| 36e, 41, 77, 118, 159, 277d }}
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Comma list: 196/195, 352/351, 385/384, 729/728
Comma list: 196/195, 352/351, 385/384, 729/728


Mapping: [{{val| 1 1 7 3 -2 0 }}, {{val| 0 3 -24 -1 28 19 }}]
Mapping: {{mapping| 1 1 7 3 -2 0 | 0 3 -24 -1 28 19 }}


Mapping generators: ~2, ~8/7
: mapping generators: ~2, ~8/7


POTE generator: ~8/7 = 233.890
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.890


{{Optimal ET sequence|legend=1| 36e, 41, 77, 118 }}
{{Optimal ET sequence|legend=1| 36e, 41, 77, 118 }}
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