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Catalog of rank-4 temperaments: Difference between revisions

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== Ptolemismic (100/99) ==
== Ptolemismic (100/99) ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[100/99]]
[[Comma list]]: [[100/99]]
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[[Mapping]]: [{{val| 1 0 0 0 2 }}, {{val| 0 1 0 0 -2 }}, {{val| 0 0 1 0 2 }}, {{val| 0 0 0 1 0 }}]
[[Mapping]]: [{{val| 1 0 0 0 2 }}, {{val| 0 1 0 0 -2 }}, {{val| 0 0 1 0 2 }}, {{val| 0 0 0 1 0 }}]


[[POTE generator]]s: ~3/2 = 704.9532, ~5/4 = 384.0675, ~7/4 = 970.8803
[[Optimal tuning]] ([[POTE]]): ~3/2 = 704.9532, ~5/4 = 384.0675, ~7/4 = 970.8803


{{Val list|legend=1| 7d, 8d, 10e, 12, 15, 19, 22, 27e, 34d, 41, 90e, 131e }} <nowiki>*</nowiki>
{{Val list|legend=1| 7d, 8d, 10e, 12, 15, 19, 22, 27e, 34d, 41, 90e, 131e }} <nowiki>*</nowiki>
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== Biyatismic (121/120) ==
== Biyatismic (121/120) ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[121/120]]
[[Comma list]]: [[121/120]]
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Mapping generators: ~2, ~3, ~11/10, ~7
Mapping generators: ~2, ~3, ~11/10, ~7


[[POTE generator]]s: ~3/2 = 701.4578, ~11/10 = 157.7466, ~7/4 = 966.9589
[[Optimal tuning]] ([[POTE]]): ~3/2 = 701.4578, ~11/10 = 157.7466, ~7/4 = 966.9589


{{Val list|legend=1| 14c, 15, 22, 31, 46, 53, 60e, 68, 77, 91e, 99, 130e, 159ee, 190ee }}
{{Val list|legend=1| 14c, 15, 22, 31, 46, 53, 60e, 68, 77, 91e, 99, 130e, 159ee, 190ee }}
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== Valinorsmic (176/175) ==
== Valinorsmic (176/175) ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[176/175]]
[[Comma list]]: [[176/175]]
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Mapping generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


[[POTE generator]]s: ~3/2 = 703.0449, ~5/4 = 389.7641, ~7/4 = 972.1113
[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.0449, ~5/4 = 389.7641, ~7/4 = 972.1113


{{Val list|legend=1| 22, 31, 46, 53, 58, 80, 111 }}
{{Val list|legend=1| 22, 31, 46, 53, 58, 80, 111 }}
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== Rastmic (243/242) ==
== Rastmic (243/242) ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[243/242]]
[[Comma list]]: [[243/242]]
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[[Mapping]]: [{{val| 1 1 0 0 2 }}, {{val| 0 2 0 0 5 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 1 0}}]
[[Mapping]]: [{{val| 1 1 0 0 2 }}, {{val| 0 2 0 0 5 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 1 0}}]


[[POTE generator]]s: ~11/9 = 350.5254, ~5/4 = 386.1653, ~7/4 = 968.6464
[[Optimal tuning]] ([[POTE]]): ~11/9 = 350.5254, ~5/4 = 386.1653, ~7/4 = 968.6464


{{Val list|legend=1| 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202 }}
{{Val list|legend=1| 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202 }}
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== Akua (352/351, 847/845) ==
== Akua (352/351, 847/845) ==
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 352/351, 847/845
[[Comma list]]: 352/351, 847/845
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Mapping generators: ~2, ~3, ~5, ~11
Mapping generators: ~2, ~3, ~5, ~11


[[POTE generator]]s: ~3/2 = 702.9075, ~5/4 = 387.0723, ~11/8 = 551.4538
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.9075, ~5/4 = 387.0723, ~11/8 = 551.4538


{{Val list|legend=1| 12f, 17c, 24d, 29, 41, 46, 53, 58, 87, 111, 140, 152f, 198, 350f, 437f, 490f }}
{{Val list|legend=1| 12f, 17c, 24d, 29, 41, 46, 53, 58, 87, 111, 140, 152f, 198, 350f, 437f, 490f }}
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== Werckismic (441/440) ==
== Werckismic (441/440) ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[441/440]]
[[Comma list]]: [[441/440]]
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=== Commas 364/363, 441/440 ===
=== Commas 364/363, 441/440 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 364/363, 441/440
[[Comma list]]: 364/363, 441/440
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=== Commas 351/350, 441/440 ===
=== Commas 351/350, 441/440 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 351/350, 441/440
[[Comma list]]: 351/350, 441/440
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=== Commas 196/195, 352/351 ===
=== Commas 196/195, 352/351 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 196/195, 352/351
[[Comma list]]: 196/195, 352/351
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=== Tannic ===
=== Tannic ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 441/440, 1287/1280
[[Comma list]]: 441/440, 1287/1280
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==== 17-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
[[Subgroup]]: 2.3.5.7.11.13.17


[[Comma list]]: 273/272, 441/440, 561/560
[[Comma list]]: 273/272, 441/440, 561/560
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=== Commas 441/440, 847/845 ===
=== Commas 441/440, 847/845 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 441/440, 847/845
[[Comma list]]: 441/440, 847/845
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{{Main| Keenanismic family }}
{{Main| Keenanismic family }}


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[385/384]]
[[Comma list]]: [[385/384]]
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=== Martwin ===
=== Martwin ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 325/324, 385/384
[[Comma list]]: 325/324, 385/384


Mapping: [{{val| 1 0 0 0 7 2 }}, {{val| 0 1 0 0 1 4 }}, {{val| 0 0 1 0 -1 -2 }}, {{val| 0 0 0 1 -1 0 }}]
[[Mapping]]: [{{val| 1 0 0 0 7 2 }}, {{val| 0 1 0 0 1 4 }}, {{val| 0 0 1 0 -1 -2 }}, {{val| 0 0 0 1 -1 0 }}]


Mapping generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7
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=== Ancient ===
=== Ancient ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 385/384, 625/624
[[Comma list]]: 385/384, 625/624
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=== Commas 351/350, 385/384 ===
=== Commas 351/350, 385/384 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 351/350, 385/384
[[Comma list]]: 351/350, 385/384
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=== Commas 352/351, 385/384 ===
=== Commas 352/351, 385/384 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 352/351, 385/384
[[Comma list]]: 352/351, 385/384
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=== Commas 364/363, 385/384 ===
=== Commas 364/363, 385/384 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 364/363, 385/384
[[Comma list]]: 364/363, 385/384
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=== Commas 385/384, 847/845 ===
=== Commas 385/384, 847/845 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 385/384, 847/845
[[Comma list]]: 385/384, 847/845
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== Swetismic (540/539) ==
== Swetismic (540/539) ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[540/539]]
[[Comma list]]: [[540/539]]
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=== Commas 540/539, 847/845 ===
=== Commas 540/539, 847/845 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 540/539, 847/845
[[Comma list]]: 540/539, 847/845
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=== Commas 540/539, 625/624 ===
=== Commas 540/539, 625/624 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 540/539, 625/624
[[Comma list]]: 540/539, 625/624
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=== Commas 540/539, 676/675 ===
=== Commas 540/539, 676/675 ===
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 540/539, 676/675
[[Comma list]]: 540/539, 676/675
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== Pentacircle (896/891) ==
== Pentacircle (896/891) ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 896/891
[[Comma list]]: 896/891
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[[Mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 0 0 -4 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 1 1 }}]
[[Mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 0 0 -4 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 1 1 }}]


[[POTE generator]]s: ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722
[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722


{{Val list|legend=1| 12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee }}
{{Val list|legend=1| 12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee }}
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=== Commas 352/351, 364/363 ===
=== Commas 352/351, 364/363 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 352/351, 364/363
[[Comma list]]: 352/351, 364/363
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== Topsy (847/845, 1001/1000) ==
== Topsy (847/845, 1001/1000) ==
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 847/845, 1001/1000
[[Comma list]]: 847/845, 1001/1000
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== Lehmerismic (3025/3024) ==
== Lehmerismic (3025/3024) ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[3025/3024]]
[[Comma list]]: [[3025/3024]]
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== Trimitone (8019/8000) ==
== Trimitone (8019/8000) ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[8019/8000]]
[[Comma list]]: [[8019/8000]]
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== Kalismic (9801/9800) ==
== Kalismic (9801/9800) ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[9801/9800]]
[[Comma list]]: [[9801/9800]]
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=== Commas 1716/1715, 2080/2079 ===
=== Commas 1716/1715, 2080/2079 ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 1716/1715, 2080/2079
[[Comma list]]: 1716/1715, 2080/2079
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== Semicanousmic (14641/14580) ==
== Semicanousmic (14641/14580) ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[14641/14580]]
[[Comma list]]: [[14641/14580]]
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Mapping generators: ~2, ~3, ~18/11, ~7
Mapping generators: ~2, ~3, ~18/11, ~7


[[POTE generator]]s: ~3/2 = 702.2503, ~18/11 = 854.5421, ~7/4 = 968.6866
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.2503, ~18/11 = 854.5421, ~7/4 = 968.6866


{{Val list|legend=1| 14c, 17c, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e }}
{{Val list|legend=1| 14c, 17c, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e }}
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Mapping: [{{val| 1 0 2 0 1 -6 }}, {{val| 0 1 2 0 2 3 }}, {{val| 0 0 -4 0 -1 3 }}, {{val| 0 0 0 1 0 1 }}]
Mapping: [{{val| 1 0 2 0 1 -6 }}, {{val| 0 1 2 0 2 3 }}, {{val| 0 0 -4 0 -1 3 }}, {{val| 0 0 0 1 0 1 }}]


POTE generators: ~3/2 = 702.4931, ~18/11 = 854.6400, ~7/4 = 969.0099
Optimal tuning (POTE): ~3/2 = 702.4931, ~18/11 = 854.6400, ~7/4 = 969.0099


Optimal GPV sequence: {{val list| 87, 111, 181, 198, 323, 410 }}
Optimal GPV sequence: {{val list| 87, 111, 181, 198, 323, 410 }}
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Mapping: [{{val| 1 0 2 0 1 -6 -4 }}, {{val| 0 1 2 0 2 3 6 }}, {{val| 0 0 -4 0 -1 3 -2 }}, {{val| 0 0 0 1 0 1 0 }}]
Mapping: [{{val| 1 0 2 0 1 -6 -4 }}, {{val| 0 1 2 0 2 3 6 }}, {{val| 0 0 -4 0 -1 3 -2 }}, {{val| 0 0 0 1 0 1 0 }}]


POTE generators: ~3/2 = 702.4099, ~18/11 = 854.6338, ~7/4 = 969.0228
Optimal tuning (POTE): ~3/2 = 702.4099, ~18/11 = 854.6338, ~7/4 = 969.0228


Optimal GPV sequence: {{val list| 87, 94, 111, 181, 198g, 212g, 292, 323, 410 }}
Optimal GPV sequence: {{val list| 87, 94, 111, 181, 198g, 212g, 292, 323, 410 }}
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Mapping: [{{val| 1 0 2 0 1 -6 -4 -4 }}, {{val| 0 1 2 0 2 3 6 7 }}, {{val| 0 0 -4 0 -1 3 -2 -4 }}, {{val| 0 0 0 1 0 1 0 0 }}]
Mapping: [{{val| 1 0 2 0 1 -6 -4 -4 }}, {{val| 0 1 2 0 2 3 6 7 }}, {{val| 0 0 -4 0 -1 3 -2 -4 }}, {{val| 0 0 0 1 0 1 0 0 }}]


POTE generators: ~3/2 = 702.3413, ~18/11 = 854.6472, ~7/4 = 968.9734
Optimal tuning (POTE): ~3/2 = 702.3413, ~18/11 = 854.6472, ~7/4 = 968.9734


Optimal GPV sequence: {{val list| 87, 94, 111, 181, 205, 212gh, 292h, 299, 323, 410, 622ef }}
Optimal GPV sequence: {{val list| 87, 94, 111, 181, 205, 212gh, 292h, 299, 323, 410, 622ef }}
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== Olympic (131072/130977) ==
== Olympic (131072/130977) ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 131072/130977
[[Comma list]]: 131072/130977
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=== Tridecimal olympic ===
=== Tridecimal olympic ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 2080/2079, 4096/4095
[[Comma list]]: 2080/2079, 4096/4095
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