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Ragismic microtemperaments: Difference between revisions

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Ennealimmal extensions discussed elsewhere include [[Compton family #Omicronbeta|omicronbeta]], [[Tritrizo clan #Undecentic|undecentic]], [[Tritrizo clan #Schisennealimmal|schisennealimmal]], and [[Tritrizo clan #Lunennealimmal|lunennealimmal]].
Ennealimmal extensions discussed elsewhere include [[Compton family #Omicronbeta|omicronbeta]], [[Tritrizo clan #Undecentic|undecentic]], [[Tritrizo clan #Schisennealimmal|schisennealimmal]], and [[Tritrizo clan #Lunennealimmal|lunennealimmal]].


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 4375/4374
[[Comma list]]: 2401/2400, 4375/4374
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Mapping generators: ~27/25, ~5/3
Mapping generators: ~27/25, ~5/3


[[POTE generator]]s: ~5/3 = 884.3129 or ~36/35 = 49.0205
[[Optimal tuning]] ([[POTE]]): ~5/3 = 884.3129 (~36/35 = 49.0205)


[[Tuning ranges]]:  
[[Tuning ranges]]:  
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=== 11-limit ===
=== 11-limit ===
The ennealimmal temperament can be described as 99e&270 temperament, which tempers out 5632/5625 (vishdel comma) and 19712/19683 (symbiotic comma).
The ennealimmal temperament can be described as 99e & 171e, which tempers out [[5632/5625]] (vishdel comma) and [[19712/19683]] (symbiotic comma).


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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Mapping: [{{val| 9 1 1 12 -75 }}, {{val| 0 2 3 2 16 }}]
Mapping: [{{val| 9 1 1 12 -75 }}, {{val| 0 2 3 2 16 }}]


POTE generator: ~5/3 = 884.4679 or ~36/35 = 48.8654
Optimal tuning (POTE): ~5/3 = 884.4679 (~36/35 = 48.8654)


Optimal GPV sequence: {{Val list| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}
Optimal GPV sequence: {{Val list| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}
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Mapping: [{{val| 9 1 1 12 -75 93 }}, {{val| 0 2 3 2 16 -9 }}]
Mapping: [{{val| 9 1 1 12 -75 93 }}, {{val| 0 2 3 2 16 -9 }}]


POTE generator: ~5/3 = 884.4304 or ~36/35 = 48.9030
Optimal tuning (POTE): ~5/3 = 884.4304 (~36/35 = 48.9030)


Optimal GPV sequence: {{Val list| 99e, 171e, 270 }}
Optimal GPV sequence: {{Val list| 99e, 171e, 270 }}
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Mapping: [{{val| 9 1 1 12 -75 93 -3 }}, {{val| 0 2 3 2 16 -9 6 }}]
Mapping: [{{val| 9 1 1 12 -75 93 -3 }}, {{val| 0 2 3 2 16 -9 6 }}]


POTE generator: ~5/3 = 884.4304 or ~36/35 = 48.9030
Optimal tuning (POTE): ~5/3 = 884.4304 (~36/35 = 48.9030)


Optimal GPV sequence: {{Val list| 99e, 171e, 270 }}
Optimal GPV sequence: {{Val list| 99e, 171e, 270 }}
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Mapping: [{{val| 9 1 1 12 -75 93 -3 -48 }}, {{val| 0 2 3 2 16 -9 6 13 }}]
Mapping: [{{val| 9 1 1 12 -75 93 -3 -48 }}, {{val| 0 2 3 2 16 -9 6 13 }}]


POTE generator: ~5/3 = 884.4304 or ~36/35 = 48.9030
Optimal tuning (POTE): ~5/3 = 884.4304 (~36/35 = 48.9030)


Optimal GPV sequence: {{Val list| 99e, 171e, 270 }}
Optimal GPV sequence: {{Val list| 99e, 171e, 270 }}
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Mapping: [{{val| 9 1 1 12 -75 -106 }}, {{val| 0 2 3 2 16 21 }}]
Mapping: [{{val| 9 1 1 12 -75 -106 }}, {{val| 0 2 3 2 16 21 }}]


CTE generator: ~5/3 = 884.4560 (~36/35 = 48.8773)
Optimal tuning (CTE): ~5/3 = 884.4560 (~36/35 = 48.8773)


Optimal GPV sequence: {{Val list| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}
Optimal GPV sequence: {{Val list| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}
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=== Ennealimmia ===
=== Ennealimmia ===
Ennealimmal temperament has various extensions to the 11-limit. Tempering out 131072/130977 (salururu comma) leads to the ''ennealimmia'' temperament (171&270).
The ennealimmia temperament is an alternative extension and can be described as 99 & 171, which tempers out [[131072/130977]] (olympia).  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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Mapping: [{{val| 9 1 1 12 124 }}, {{val| 0 2 3 2 -14 }}]
Mapping: [{{val| 9 1 1 12 124 }}, {{val| 0 2 3 2 -14 }}]


POTE generator: ~5/3 = 884.4089 or ~36/35 = 48.9244
Optimal tuning (POTE): ~5/3 = 884.4089 (~36/35 = 48.9244)


Optimal GPV sequence: {{Val list| 99, 171, 270, 711, 981, 1251, 2232e }}
Optimal GPV sequence: {{Val list| 99, 171, 270, 711, 981, 1251, 2232e }}
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Mapping: [{{val| 9 1 1 12 124 93 }}, {{val| 0 2 3 2 -14 -9 }}]
Mapping: [{{val| 9 1 1 12 124 93 }}, {{val| 0 2 3 2 -14 -9 }}]


POTE generator: ~5/3 = 884.3997 or ~36/35 = 48.9336
Optimal tuning (POTE): ~5/3 = 884.3997 (~36/35 = 48.9336)


Optimal GPV sequence: {{Val list| 99, 171, 270, 711, 981, 1692e, 2673e }}
Optimal GPV sequence: {{Val list| 99, 171, 270, 711, 981, 1692e, 2673e }}
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Mapping: [{{val| 9 1 1 12 124 93 -3 }}, {{val| 0 2 3 2 -14 -9 6 }}]
Mapping: [{{val| 9 1 1 12 124 93 -3 }}, {{val| 0 2 3 2 -14 -9 6 }}]


POTE generator: ~5/3 = 884.3997 or ~36/35 = 48.9336
Optimal tuning (POTE): ~5/3 = 884.3997 (~36/35 = 48.9336)


Optimal GPV sequence: {{Val list| 99, 171, 270 }}
Optimal GPV sequence: {{Val list| 99, 171, 270 }}
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Mapping: [{{val| 9 1 1 12 124 93 -3 -48 }}, {{val| 0 2 3 2 -14 -9 6 13 }}]
Mapping: [{{val| 9 1 1 12 124 93 -3 -48 }}, {{val| 0 2 3 2 -14 -9 6 13 }}]


POTE generator: ~5/3 = 884.3997 or ~36/35 = 48.9336
Optimal tuning (POTE): ~5/3 = 884.3997 (~36/35 = 48.9336)


Optimal GPV sequence: {{Val list| 99, 171, 270 }}
Optimal GPV sequence: {{Val list| 99, 171, 270 }}


=== Ennealimnic ===
=== Ennealimnic ===
Ennealimnic temperament (72&171) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.
Ennealimnic (72&171) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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Mapping: [{{val| 9 1 1 12 -2 }}, {{val| 0 2 3 2 5 }}]
Mapping: [{{val| 9 1 1 12 -2 }}, {{val| 0 2 3 2 5 }}]


POTE generator: ~5/3 = 883.9386 or ~36/35 = 49.3948
Optimal tuning (POTE): ~5/3 = 883.9386 (~36/35 = 49.3948)


Tuning ranges:  
Tuning ranges:  
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Mapping: [{{val| 9 1 1 12 -2 -33 }}, {{val| 0 2 3 2 5 10 }}]
Mapping: [{{val| 9 1 1 12 -2 -33 }}, {{val| 0 2 3 2 5 10 }}]


POTE generator: ~5/3 = 883.9920 or ~36/35 = 49.3414
Optimal tuning (POTE): ~5/3 = 883.9920 (~36/35 = 49.3414)


Tuning ranges:  
Tuning ranges:  
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Mapping: [{{val| 9 1 1 12 -2 -33 -3 }}, {{val| 0 2 3 2 5 10 6 }}]
Mapping: [{{val| 9 1 1 12 -2 -33 -3 }}, {{val| 0 2 3 2 5 10 6 }}]


POTE generator: ~5/3 = 883.9981 or ~36/35 = 49.3353
Optimal tuning (POTE): ~5/3 = 883.9981 (~36/35 = 49.3353)


Tuning ranges:  
Tuning ranges:  
Line 243: Line 243:
Mapping: [{{val| 9 1 1 12 -2 20 }}, {{val| 0 2 3 2 5 2 }}]
Mapping: [{{val| 9 1 1 12 -2 20 }}, {{val| 0 2 3 2 5 2 }}]


POTE generator: ~5/3 = 883.6257 or ~36/35 = 49.7076
Optimal tuning (POTE): ~5/3 = 883.6257 (~36/35 = 49.7076)


Optimal GPV sequence: {{Val list| 27e, 45ef, 72 }}
Optimal GPV sequence: {{Val list| 27e, 45ef, 72 }}
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Mapping: [{{val| 9 1 1 12 -2 20 -3 }}, {{val| 0 2 3 2 5 2 6 }}]
Mapping: [{{val| 9 1 1 12 -2 20 -3 }}, {{val| 0 2 3 2 5 2 6 }}]


POTE generator: ~5/3 = 883.6257 or ~36/35 = 49.7076
Optimal tuning (POTE): ~5/3 = 883.6257 (~36/35 = 49.7076)


Optimal GPV sequence: {{Val list| 27eg, 45efg, 72 }}
Optimal GPV sequence: {{Val list| 27eg, 45efg, 72 }}
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Mapping: [{{val| 9 1 1 12 -2 20 -3 25 }}, {{val| 0 2 3 2 5 2 6 2 }}]
Mapping: [{{val| 9 1 1 12 -2 20 -3 25 }}, {{val| 0 2 3 2 5 2 6 2 }}]


POTE generator: ~5/3 = 883.6257 or ~36/35 = 49.7076
Optimal tuning (POTE): ~5/3 = 883.6257 (~36/35 = 49.7076)


Optimal GPV sequence: {{Val list| 27eg, 45efg, 72 }}
Optimal GPV sequence: {{Val list| 27eg, 45efg, 72 }}
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Mapping: [{{val| 9 1 1 12 51 }}, {{val| 0 2 3 2 -3 }}]
Mapping: [{{val| 9 1 1 12 51 }}, {{val| 0 2 3 2 -3 }}]


POTE generator: ~5/3 = 883.8298 or ~36/35 = 49.5036
Optimal tuning (POTE): ~5/3 = 883.8298 (~36/35 = 49.5036)


Optimal GPV sequence: {{Val list| 27, 45, 72, 171e, 243e, 315e }}
Optimal GPV sequence: {{Val list| 27, 45, 72, 171e, 243e, 315e }}
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Mapping: [{{val| 9 1 1 12 51 20 }}, {{val| 0 2 3 2 -3 2 }}]
Mapping: [{{val| 9 1 1 12 51 20 }}, {{val| 0 2 3 2 -3 2 }}]


POTE generator: ~5/3 = 883.8476 or ~36/35 = 49.4857
Optimal tuning (POTE): ~5/3 = 883.8476 (~36/35 = 49.4857)


Optimal GPV sequence: {{Val list| 27, 45f, 72, 171ef, 243ef }}
Optimal GPV sequence: {{Val list| 27, 45f, 72, 171ef, 243ef }}
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Mapping: [{{val| 9 1 1 12 51 20 50 }}, {{val| 0 2 3 2 -3 2 -2 }}]
Mapping: [{{val| 9 1 1 12 51 20 50 }}, {{val| 0 2 3 2 -3 2 -2 }}]


POTE generator: ~5/3 = 883.8476 or ~36/35 = 49.4857
Optimal tuning (POTE): ~5/3 = 883.8476 (~36/35 = 49.4857)


Optimal GPV sequence: {{Val list| 27, 45f, 72 }}
Optimal GPV sequence: {{Val list| 27, 45f, 72 }}
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Mapping: [{{val| 9 1 1 12 51 20 50 25 }}, {{val| 0 2 3 2 -3 2 -2 2 }}]
Mapping: [{{val| 9 1 1 12 51 20 50 25 }}, {{val| 0 2 3 2 -3 2 -2 2 }}]


POTE generator: ~5/3 = 883.8476 or ~36/35 = 49.4857
Optimal tuning (POTE): ~5/3 = 883.8476 (~36/35 = 49.4857)


Optimal GPV sequence: {{Val list| 27, 45f, 72 }}
Optimal GPV sequence: {{Val list| 27, 45f, 72 }}
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Mapping generators: ~80/77, ~400/231
Mapping generators: ~80/77, ~400/231


POTE generator: ~400/231 = 950.9553
Optimal tuning (POTE): ~400/231 = 950.9553


Tuning ranges:  
Tuning ranges:  
Line 348: Line 348:
Mapping: [{{val| 18 0 -1 22 48 -19 }}, {{val| 0 2 3 2 1 6 }}]
Mapping: [{{val| 18 0 -1 22 48 -19 }}, {{val| 0 2 3 2 1 6 }}]


POTE generator ~26/15 = 951.0837
Optimal tuning (POTE): ~26/15 = 951.0837


Tuning ranges:  
Tuning ranges:  
Line 369: Line 369:
Mapping: [{{val| 18 0 -1 22 48 -19 -12 }}, {{val| 0 2 3 2 1 6 6 }}]
Mapping: [{{val| 18 0 -1 22 48 -19 -12 }}, {{val| 0 2 3 2 1 6 6 }}]


POTE generator ~26/15 = 951.0837
Optimal tuning (POTE): ~26/15 = 951.0837


Optimal GPV sequence: {{Val list| 72, 198g, 270 }}
Optimal GPV sequence: {{Val list| 72, 198g, 270 }}
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Mapping: [{{val| 18 0 -1 22 48 -19 -12 48 105 }}, {{val| 0 2 3 2 1 6 6 -2 }}]
Mapping: [{{val| 18 0 -1 22 48 -19 -12 48 105 }}, {{val| 0 2 3 2 1 6 6 -2 }}]


POTE generator ~26/15 = 951.0837
Optimal tuning (POTE): ~26/15 = 951.0837


Optimal GPV sequence: {{Val list| 72, 198g, 270 }}
Optimal GPV sequence: {{Val list| 72, 198g, 270 }}
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Mapping generators: ~80/77, ~1053/800
Mapping generators: ~80/77, ~1053/800


POTE generator: ~1053/800 = 475.4727
Optimal tuning (POTE): ~1053/800 = 475.4727


Optimal GPV sequence: {{Val list| 126, 144, 270, 684, 954 }}
Optimal GPV sequence: {{Val list| 126, 144, 270, 684, 954 }}
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Mapping generators: ~27/25, ~140/121
Mapping generators: ~27/25, ~140/121


POTE generator: ~140/121 = 250.3367
Optimal tuning (POTE): ~140/121 = 250.3367


Optimal GPV sequence: {{Val list| 72, 369, 441 }}
Optimal GPV sequence: {{Val list| 72, 369, 441 }}
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Mapping: [{{val| 9 3 4 14 18 -8 }}, {{val| 0 6 9 6 7 22 }}]
Mapping: [{{val| 9 3 4 14 18 -8 }}, {{val| 0 6 9 6 7 22 }}]


POTE generator: ~140/121 = 250.3375
Optimal tuning (POTE): ~140/121 = 250.3375


Optimal GPV sequence: {{Val list| 72, 297ef, 369f, 441 }}
Optimal GPV sequence: {{Val list| 72, 297ef, 369f, 441 }}
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Mapping generators: ~27/25, ~25/22
Mapping generators: ~27/25, ~25/22


POTE generator: ~25/22 = 221.0717
Optimal tuning (POTE): ~25/22 = 221.0717


Optimal GPV sequence: {{Val list| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}
Optimal GPV sequence: {{Val list| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}
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Mapping generators: ~2744/2673, ~2352/1375
Mapping generators: ~2744/2673, ~2352/1375


POTE generator: ~2352/1375 = 928.8000
Optimal tuning (POTE): ~2352/1375 = 928.8000


Optimal GPV sequence: {{Val list| 27, 243, 270, 783, 1053, 1323 }}
Optimal GPV sequence: {{Val list| 27, 243, 270, 783, 1053, 1323 }}
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