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Pentacircle clan: Difference between revisions

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"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
2.3.7.11 parapyth -> parapythic, by Margo's request; update keys
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For the rank-4 pentacircle temperament, see [[Rank-4 temperament #Pentacircle (896/891)]].  
For the rank-4 pentacircle temperament, see [[Rank-4 temperament #Pentacircle (896/891)]].  


== Parapyth ==
== Parapythic ==
{{Main| Parapyth }}
{{Main| Parapyth }}


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[[Comma list]]: 896/891
[[Comma list]]: 896/891


[[Sval]] [[mapping]]: [{{val| 1 0 0 7 }}, {{val| 0 1 0 -4 }}, {{val| 0 0 1 1 }}]
{{mapping|legend=1| 1 0 0 7 | 0 1 0 -4 | 0 0 1 1 }}


: sval mapping generators: ~2, ~3, ~7
: sval mapping generators: ~2, ~3, ~7
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Considered below are tolerant, kujuku, and terrapyth.  
Considered below are tolerant, kujuku, and terrapyth.  


=== 2.3.7.11.13 subgroup ===
=== Parapyth ===
{{Main| Parapyth }}
 
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


Comma list: 352/351, 364/363
Comma list: 352/351, 364/363


Sval mapping: [{{val| 1 0 0 7 12 }}, {{val| 0 1 0 -4 -7 }}, {{val| 0 0 1 1 1 }}]
Sval mapping: {{mapping| 1 0 0 7 12 | 0 1 0 -4 -7 | 0 0 1 1 1 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.8563, ~7/4 = 969.9074
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.8563, ~7/4 = 969.9074
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Comma list: 352/351, 364/363, 442/441
Comma list: 352/351, 364/363, 442/441


Sval mapping: [{{val| 1 0 0 7 12 -13 }}, {{val| 0 1 0 -4 -7 9 }}, {{val| 0 0 1 1 1 1 }}]
Sval mapping: {{mapping| 1 0 0 7 12 -13 | 0 1 0 -4 -7 9 | 0 0 1 1 1 1 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.0315, ~7/4 = 970.6051
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.0315, ~7/4 = 970.6051
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[[Comma list]]: 896/891, 585640/583443
[[Comma list]]: 896/891, 585640/583443


[[Mapping]]: [{{val| 1 0 -31 0 7 }}, {{val| 0 1 21 0 -4 }}, {{val| 0 0 0 1 1 }}]
[[Mapping]]: {{mapping| 1 0 -31 0 7 | 0 1 21 0 -4 | 0 0 0 1 1 }}


: mapping generators: ~2, ~3, ~7
: mapping generators: ~2, ~3, ~7
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Comma list: 352/351, 364/363, 9295/9261
Comma list: 352/351, 364/363, 9295/9261


Mapping: [{{val| 1 0 -31 0 7 12 }}, {{val| 0 1 0 21 0 4 -7 }}, {{val| 0 0 0 1 1 1 }}]
Mapping: {{mapping| 1 0 -31 0 7 12 | 0 1 0 21 0 4 -7 | 0 0 0 1 1 1 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1691, ~7/4 = 970.8432
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1691, ~7/4 = 970.8432
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Comma list: 352/351, 364/363, 442/441, 715/714
Comma list: 352/351, 364/363, 442/441, 715/714


Mapping: [{{val| 1 0 -31 0 7 12 -13 }}, {{val| 0 1 0 21 0 4 -7 9 }}, {{val| 0 0 0 1 1 1 1 1 }}]
Mapping: {{mapping| 1 0 -31 0 7 12 -13 | 0 1 0 21 0 4 -7 9 | 0 0 0 1 1 1 1 1 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1628, ~7/4 = 970.6620
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1628, ~7/4 = 970.6620
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[[Comma list]]: 179200/177147
[[Comma list]]: 179200/177147


[[Mapping]]: [{{val| 1 0 0 -10 }}, {{val| 0 1 0 11 }}, {{val| 0 0 1 -2 }}]
{{Mapping|legend=1| 1 0 0 -10 | 0 1 0 11 | 0 0 1 -2 }}


: mapping generators: ~2, ~3, ~5
: mapping generators: ~2, ~3, ~5
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[[Comma list]]: 896/891, 2200/2187
[[Comma list]]: 896/891, 2200/2187


[[Mapping]]: [{{val| 1 0 0 -10 -3 }}, {{val| 0 1 0 11 7 }}, {{val| 0 0 1 -2 -2 }}]
{{Mapping|legend=1| 1 0 0 -10 -3 | 0 1 0 11 7 | 0 0 1 -2 -2 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.0412, ~5/4 = 387.2927
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.0412, ~5/4 = 387.2927
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Comma list: 325/324, 352/351, 364/363
Comma list: 325/324, 352/351, 364/363


Mapping: [{{val| 1 0 0 -10 -3 2 }}, {{val| 0 1 0 11 7 4 }}, {{val| 0 0 1 -2 -2 -2 }}]
Mapping: {{mapping| 1 0 0 -10 -3 2 | 0 1 0 11 7 4 | 0 0 1 -2 -2 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.9605, ~5/4 = 386.9831
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.9605, ~5/4 = 386.9831
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Comma list: 256/255, 325/324, 352/351, 364/363
Comma list: 256/255, 325/324, 352/351, 364/363


Mapping: [{{val| 1 0 0 -10 -3 2 8 }}, {{val| 0 1 0 11 7 4 -1 }}, {{val| 0 0 1 -2 -2 -2 -1 }}]
Mapping: {{mapping| 1 0 0 -10 -3 2 8 | 0 1 0 11 7 4 -1 | 0 0 1 -2 -2 -2 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.0831, ~5/4 = 387.3269
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.0831, ~5/4 = 387.3269
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[[Comma list]]: 896/891, 14700/14641
[[Comma list]]: 896/891, 14700/14641


[[Mapping]]: [{{val| 1 0 0 -13 -6 }}, {{val| 0 2 0 17 9 }}, {{val| 0 0 1 1 1 }}]
{{Mapping|legend=1| 1 0 0 -13 -6 | 0 2 0 17 9 | 0 0 1 1 1 }}


: mapping generators: ~2, ~121/70, ~5
: mapping generators: ~2, ~121/70, ~5
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Comma list: 352/351, 364/363, 676/675
Comma list: 352/351, 364/363, 676/675


Mapping: [{{val| 1 0 0 -13 -6 -1 }}, {{val| 0 2 0 17 9 3 }}, {{val| 0 0 1 1 1 1 }}]
Mapping: {{mapping| 1 0 0 -13 -6 -1 | 0 2 0 17 9 3 | 0 0 1 1 1 1 }}


Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.8367, ~5/4 = 386.4048
Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.8367, ~5/4 = 386.4048
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Comma list: 256/255, 352/351, 364/363, 676/675
Comma list: 256/255, 352/351, 364/363, 676/675


Mapping: [{{val| 1 0 0 -13 -6 -1 8 }}, {{val| 0 2 0 17 9 3 -2 }}, {{val| 0 0 1 1 1 1 -1 }}]
Mapping: {{mapping| 1 0 0 -13 -6 -1 8 | 0 2 0 17 9 3 -2 | 0 0 1 1 1 1 -1 }}


Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.8015, ~5/4 = 386.9912
Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.8015, ~5/4 = 386.9912
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