Catalog of rank-4 temperaments: Difference between revisions

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[[Badness]]: 0.05088 × 10<sup>-6</sup>
[[Badness]]: 0.05088 × 10<sup>-6</sup>
== Akua (352/351, 847/845) ==
Subgroup: 2.3.5.7.11.13
[[Comma list]]: 352/351, 847/845
[[Mapping]]: [{{val| 1 0 0 10 0 5 }}, {{val| 0 1 0 -6 0 -3 }}, {{val| 0 0 1 1 0 0 }}, {{val| 0 0 0 0 1 1 }}]
Mapping generators: ~2, ~3, ~5, ~11
[[POTE generator]]s: ~3/2 = 702.9075, ~5/4 = 387.0723, ~11/8 = 551.4538
{{Val list|legend=1| 12f, 17c, 29, 41, 46, 53, 58, 87, 111, 140, 198, 350f, 437f, 490f }}
[[Badness]]: 2.550 × 10<sup>-6</sup>


== Werckismic (441/440) ==
== Werckismic (441/440) ==
Line 36: Line 51:
[[Mapping]]: [{{val| 1 0 0 0 -3 }}, {{val| 0 1 0 0 2 }}, {{val| 0 0 1 0 -1 }}, {{val| 0 0 0 1 2 }}]
[[Mapping]]: [{{val| 1 0 0 0 -3 }}, {{val| 0 1 0 0 2 }}, {{val| 0 0 1 0 -1 }}, {{val| 0 0 0 1 2 }}]


Generators: ~2, ~3, ~5, ~7
Mapping generators:~2, ~3, ~5, ~7


{{Val list|legend=1| 10, 12, 15, 19e, 26, 27e, 31, 41, 58, 72, 118, 130, 190, 248, 289, 320, 609d }}
{{Val list|legend=1| 10, 12, 15, 19e, 26, 27e, 31, 41, 58, 72, 118, 130, 190, 248, 289, 320, 609d }}
Line 47: Line 62:
[[Mapping]]: [{{val| 1 0 0 0 -3 -8 }}, {{val| 0 1 0 0 2 5 }}, {{val| 0 0 1 0 -1 -2 }}, {{val| 0 0 0 1 2 3 }}]
[[Mapping]]: [{{val| 1 0 0 0 -3 -8 }}, {{val| 0 1 0 0 2 5 }}, {{val| 0 0 1 0 -1 -2 }}, {{val| 0 0 0 1 2 3 }}]


Generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


Map to lattice: [{{val| 0 1 1 -1 -1 0 }}, {{val| 0 0 1 0 -1 -2 }}, {{val| 0 0 1 1 1 1 }}]
Map to lattice: [{{val| 0 1 1 -1 -1 0 }}, {{val| 0 0 1 0 -1 -2 }}, {{val| 0 0 1 1 1 1 }}]
Line 71: Line 86:
[[Mapping]]: [{{val| 1 0 0 0 -3 1 }}, {{val| 0 1 0 0 2 -3 }}, {{val| 0 0 1 0 -1 2 }}, {{val| 0 0 0 1 2 1 }}]
[[Mapping]]: [{{val| 1 0 0 0 -3 1 }}, {{val| 0 1 0 0 2 -3 }}, {{val| 0 0 1 0 -1 2 }}, {{val| 0 0 0 1 2 1 }}]


Generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 12f, 14cf, 19e, 26, 27e, 31, 45ef, 46, 58, 72, 103, 130, 233, 279, 409, 512bf, 642bf }}
{{Val list|legend=1| 12f, 14cf, 19e, 26, 27e, 31, 45ef, 46, 58, 72, 103, 130, 233, 279, 409, 512bf, 642bf }}
Line 82: Line 97:
[[Mapping]]: [{{val| 1 0 0 0 -3 2 }}, {{val| 0 1 0 0 2 -1 }}, {{val| 0 0 1 0 -1 -1 }}, {{val| 0 0 0 1 2 2 }}]
[[Mapping]]: [{{val| 1 0 0 0 -3 2 }}, {{val| 0 1 0 0 2 -1 }}, {{val| 0 0 1 0 -1 -1 }}, {{val| 0 0 0 1 2 2 }}]


Generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 10, 12f, 17c, 19e, 27e, 29, 31, 41, 46, 58, 87, 118, 145, 232 }}
{{Val list|legend=1| 10, 12f, 17c, 19e, 27e, 29, 31, 41, 46, 58, 87, 118, 145, 232 }}
Line 93: Line 108:
[[Mapping]]: [{{val| 1 0 0 0 -3 11 }}, {{val| 0 1 0 0 2 -4 }}, {{val| 0 0 1 0 -1 2 }}, {{val| 0 0 0 1 2 -2 }}]
[[Mapping]]: [{{val| 1 0 0 0 -3 11 }}, {{val| 0 1 0 0 2 -4 }}, {{val| 0 0 1 0 -1 2 }}, {{val| 0 0 0 1 2 -2 }}]


Generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 17c, 26, 29, 31, 43, 46, 60e, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff }}
{{Val list|legend=1| 17c, 26, 29, 31, 43, 46, 60e, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff }}
Line 104: Line 119:
[[Mapping]]: [{{val| 1 0 0 0 -3 11 7 }}, {{val| 0 1 0 0 2 -4 -3 }}, {{val| 0 0 1 0 -1 2 2 }}, {{val| 0 0 0 1 2 -2 -1 }}]
[[Mapping]]: [{{val| 1 0 0 0 -3 11 7 }}, {{val| 0 1 0 0 2 -4 -3 }}, {{val| 0 0 1 0 -1 2 2 }}, {{val| 0 0 0 1 2 -2 -1 }}]


Generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 17cg, 26, 29g, 31, 43, 46, 60e, 72, 103, 149, 221ef }}
{{Val list|legend=1| 17cg, 26, 29g, 31, 43, 46, 60e, 72, 103, 149, 221ef }}
Line 115: Line 130:
[[Mapping]]: [{{val| 1 0 0 0 -3 -3 }}, {{val| 0 1 0 0 2 2 }}, {{val| 0 0 1 1 1 1 }}, {{val| 0 0 0 2 4 5 }}]
[[Mapping]]: [{{val| 1 0 0 0 -3 -3 }}, {{val| 0 1 0 0 2 2 }}, {{val| 0 0 1 1 1 1 }}, {{val| 0 0 0 2 4 5 }}]


Generators: ~2, ~3, ~5, ~13/11
Mapping generators: ~2, ~3, ~5, ~13/11


{{Val list|legend=1| 12f, 16, 17c, 25e, 29, 41, 46, 58, 87, 103, 145, 149, 161, 190, 248, 438d }}
{{Val list|legend=1| 12f, 16, 17c, 25e, 29, 41, 46, 58, 87, 103, 145, 149, 161, 190, 248, 438d }}
Line 128: Line 143:
[[Mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 0 0 1 }}, {{val| 0 0 1 0 -1 }}, {{val| 0 0 0 1 -1 }}]
[[Mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 0 0 1 }}, {{val| 0 0 1 0 -1 }}, {{val| 0 0 0 1 -1 }}]


Generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


[[Transpose]]: [2 3 5 7 385/35]
[[Transpose]]: [2 3 5 7 385/35]
Line 148: Line 163:
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 }}]


Generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


[[Transpose]]: [2 3 5 7 385/35 324/25]
[[Transpose]]: [2 3 5 7 385/35 324/25]
Line 184: Line 199:
[[Mapping]]: [{{val| 1 0 0 0 7 1 }}, {{val| 0 1 0 0 1 -3 }}, {{val| 0 0 1 0 -1 2 }}, {{val| 0 0 0 1 -1 1 }}]
[[Mapping]]: [{{val| 1 0 0 0 7 1 }}, {{val| 0 1 0 0 1 -3 }}, {{val| 0 0 1 0 -1 2 }}, {{val| 0 0 0 1 -1 1 }}]


Generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 19, 22, 26, 31, 46, 53, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 495bdeefff, 545bddeefff, 598bcdeeefff }}
{{Val list|legend=1| 19, 22, 26, 31, 46, 53, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 495bdeefff, 545bddeefff, 598bcdeeefff }}
Line 195: Line 210:
[[Mapping]]: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 0 0 1 -2 }}, {{val| 0 0 1 0 -1 -1 }}, {{val| 0 0 0 1 -1 -1 }}]
[[Mapping]]: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 0 0 1 -2 }}, {{val| 0 0 1 0 -1 -1 }}, {{val| 0 0 0 1 -1 -1 }}]


Generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}
{{Val list|legend=1| 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}
Line 206: Line 221:
[[Mapping]]: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 0 0 1 3 }}, {{val| 0 0 1 0 -1 -2 }}, {{val| 0 0 0 1 -1 -3 }}]
[[Mapping]]: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 0 0 1 3 }}, {{val| 0 0 1 0 -1 -2 }}, {{val| 0 0 0 1 -1 -3 }}]


Generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159 }}
{{Val list|legend=1| 9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159 }}
Line 217: Line 232:
[[Mapping]]: [{{val| 1 0 0 0 7 7 }}, {{val| 0 1 0 0 1 1 }}, {{val| 0 0 1 1 -2 -2 }}, {{val| 0 0 0 2 -2 -1 }}]
[[Mapping]]: [{{val| 1 0 0 0 7 7 }}, {{val| 0 1 0 0 1 1 }}, {{val| 0 0 1 1 -2 -2 }}, {{val| 0 0 0 2 -2 -1 }}]


Generators: ~2, ~3, ~5, ~13/11
Mapping generators: ~2, ~3, ~5, ~13/11


{{Val list|legend=1| 34, 37, 41, 46, 53, 87, 103, 140, 190, 243e, 330e, 520de, 573dee }}
{{Val list|legend=1| 34, 37, 41, 46, 53, 87, 103, 140, 190, 243e, 330e, 520de, 573dee }}
Line 228: Line 243:
[[Mapping]]: [{{val| 1 0 0 0 2 }}, {{val| 0 1 0 0 3 }}, {{val| 0 0 1 0 1 }}, {{val| 0 0 0 1 -2 }}]
[[Mapping]]: [{{val| 1 0 0 0 2 }}, {{val| 0 1 0 0 3 }}, {{val| 0 0 1 0 1 }}, {{val| 0 0 0 1 -2 }}]


Generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 8d, 9, 10, 12e, 14c, 17c, 19, 22, 27e, 31, 41, 53, 58, 72, 130, 152, 224, 354, 506e, 578, 730de, 761d, 985d, 1115de, 1267dde }}
{{Val list|legend=1| 8d, 9, 10, 12e, 14c, 17c, 19, 22, 27e, 31, 41, 53, 58, 72, 130, 152, 224, 354, 506e, 578, 730de, 761d, 985d, 1115de, 1267dde }}
Line 239: Line 254:
[[Mapping]]: [{{val| 1 0 0 0 2 2 }}, {{val| 0 1 0 0 3 3 }}, {{val| 0 0 1 1 -1 -1 }}, {{val| 0 0 0 2 -4 -3 }}]
[[Mapping]]: [{{val| 1 0 0 0 2 2 }}, {{val| 0 1 0 0 3 3 }}, {{val| 0 0 1 1 -1 -1 }}, {{val| 0 0 0 2 -4 -3 }}]


Generators: ~2, ~3, ~5, ~13/11
Mapping generators: ~2, ~3, ~5, ~13/11


{{Val list|legend=1| 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f }}
{{Val list|legend=1| 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f }}
Line 250: Line 265:
[[Mapping]]: [{{val| 1 0 0 0 2 -4 }}, {{val| 0 1 0 0 3 -1 }}, {{val| 0 0 1 0 1 4 }}, {{val| 0 0 0 1 -2 0 }}]
[[Mapping]]: [{{val| 1 0 0 0 2 -4 }}, {{val| 0 1 0 0 3 -1 }}, {{val| 0 0 1 0 1 4 }}, {{val| 0 0 0 1 -2 0 }}]


Generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 19, 22, 31, 49f, 50, 53, 72, 103, 121, 152f, 193, 224 }}
{{Val list|legend=1| 19, 22, 31, 49f, 50, 53, 72, 103, 121, 152f, 193, 224 }}
Line 261: Line 276:
[[Mapping]]: [{{val| 1 0 0 0 2 -1 }}, {{val| 0 2 0 0 6 3 }}, {{val| 0 0 1 0 1 1 }}, {{val| 0 0 0 1 -2 0 }}]
[[Mapping]]: [{{val| 1 0 0 0 2 -1 }}, {{val| 0 2 0 0 6 3 }}, {{val| 0 0 1 0 1 1 }}, {{val| 0 0 0 1 -2 0 }}]


Generators: ~2, ~26/15, ~5, ~7
Mapping generators: ~2, ~26/15, ~5, ~7


{{Val list|legend=1| 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de }}
{{Val list|legend=1| 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de }}
Line 296: Line 311:
[[Mapping]]: [{{val| 1 0 0 2 0 1 }}, {{val| 0 1 0 0 0 0 }}, {{val| 0 0 1 1 1 1 }}, {{val| 0 0 0 4 -3 1 }}]
[[Mapping]]: [{{val| 1 0 0 2 0 1 }}, {{val| 0 1 0 0 0 0 }}, {{val| 0 0 1 1 1 1 }}, {{val| 0 0 0 4 -3 1 }}]


Generators: ~2, ~3, ~5, ~13/10
Mapping generators: ~2, ~3, ~5, ~13/10


{{Val list|legend=1| 16, 21, 24d, 29, 37, 45ef, 50, 53, 58, 87, 103, 111, 140, 190, 198, 301, 388, 689e }}
{{Val list|legend=1| 16, 21, 24d, 29, 37, 45ef, 50, 53, 58, 87, 103, 111, 140, 190, 198, 301, 388, 689e }}
Line 307: Line 322:
[[Mapping]]: [{{val| 1 0 0 0 2 }}, {{val| 0 1 0 1 2 }}, {{val| 0 0 1 0 -1 }}, {{val| 0 0 0 2 1 }}]
[[Mapping]]: [{{val| 1 0 0 0 2 }}, {{val| 0 1 0 1 2 }}, {{val| 0 0 1 0 -1 }}, {{val| 0 0 0 2 1 }}]


Generators: ~2, ~3, ~5, ~55/36
Mapping generators: ~2, ~3, ~5, ~55/36


{{Val list|legend=1| 7d, 8d, 10, 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}
{{Val list|legend=1| 7d, 8d, 10, 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}
Line 318: Line 333:
[[Mapping]]: [{{val| 2 0 0 0 3 }}, {{val| 0 1 0 0 -2 }}, {{val| 0 0 1 0 1 }}, {{val| 0 0 0 1 1 }}]
[[Mapping]]: [{{val| 2 0 0 0 3 }}, {{val| 0 1 0 0 -2 }}, {{val| 0 0 1 0 1 }}, {{val| 0 0 0 1 1 }}]


Generators: ~99/70, ~3, ~5, ~7
Mapping generators: ~99/70, ~3, ~5, ~7


{{Val list|legend=1| 8d, 10, 12, 22, 34d, 46, 58, 72, 118, 130, 152, 224, 270, 342, 612, 836, 1084, 1106, 1236, 1506, 1578, 1848, 2684, 4038, 4190, 4532, 11254, 15786e, 21896e }}
{{Val list|legend=1| 8d, 10, 12, 22, 34d, 46, 58, 72, 118, 130, 152, 224, 270, 342, 612, 836, 1084, 1106, 1236, 1506, 1578, 1848, 2684, 4038, 4190, 4532, 11254, 15786e, 21896e }}
Line 329: Line 344:
[[Mapping]]: [{{val| 2 0 0 0 3 -7 }}, {{val| 0 1 0 0 -2 1 }}, {{val| 0 0 1 0 1 0 }}, {{val| 0 0 0 1 1 2 }}]
[[Mapping]]: [{{val| 2 0 0 0 3 -7 }}, {{val| 0 1 0 0 -2 1 }}, {{val| 0 0 1 0 1 0 }}, {{val| 0 0 0 1 1 2 }}]


Generators: ~99/70, ~3, ~5, ~7
Mapping generators: ~99/70, ~3, ~5, ~7


Lattice basis:  
Lattice basis:  
Line 354: Line 369:
[[Mapping]]: [{{val| 1 0 0 0 17 }}, {{val| 0 1 0 0 -5 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 1 -2 }}]
[[Mapping]]: [{{val| 1 0 0 0 17 }}, {{val| 0 1 0 0 -5 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 1 -2 }}]


Generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 41, 53, 84, 87, 130, 183, 224, 270, 494, 764, 1164, 1205, 1475, 1969, 2239, 3133de, 4608cde, 5102bcde, 10474bbccdddeee }}
{{Val list|legend=1| 41, 53, 84, 87, 130, 183, 224, 270, 494, 764, 1164, 1205, 1475, 1969, 2239, 3133de, 4608cde, 5102bcde, 10474bbccdddeee }}
Line 365: Line 380:
[[Mapping]]: [{{val| 1 0 0 0 17 12 }}, {{val| 0 1 0 0 -5 -2 }}, {{val| 0 0 1 0 0 -1 }}, {{val| 0 0 0 1 -2 -1 }}]
[[Mapping]]: [{{val| 1 0 0 0 17 12 }}, {{val| 0 1 0 0 -5 -2 }}, {{val| 0 0 1 0 0 -1 }}, {{val| 0 0 0 1 -2 -1 }}]


Generators: ~2, ~3, ~5, ~7
Mapping generators: ~2, ~3, ~5, ~7


{{Val list|legend=1| 41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1699, 2280, 2774e }}
{{Val list|legend=1| 41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1699, 2280, 2774e }}
Line 372: Line 387:
[[Category:Rank 4| ]] <!-- main article -->
[[Category:Rank 4| ]] <!-- main article -->
{{IoT}}
{{IoT}}
{{Todo| expand }}

Revision as of 14:24, 25 October 2021

A rank-4 temperament has a period and three additional independent generators. Typical examples include 7-limit JI, full 11-limit temperament with a one-dimensional comma basis, and full 13-limit temperament with a two-dimensional comma basis.

Ptolemismic (100/99)

Subgroup: 2.3.5.7.11

Comma list: 100/99

Mapping: [⟨1 0 0 0 2], ⟨0 1 0 0 -2], ⟨0 0 1 0 2], ⟨0 0 0 1 0]]

POTE generators: ~3/2 = 704.9532, ~5/4 = 384.0675, ~7/4 = 970.8803

Template:Val list *

* optimal patent val: 104

Badness: 0.0225 × 10-6

Rastmic (243/242)

Subgroup: 2.3.5.7.11

Comma list: 243/242

Mapping: [⟨1 1 0 0 2], ⟨0 2 0 0 5], ⟨0 0 1 0 0], ⟨0 0 0 1 0]]

POTE generators: ~11/9 = 350.5254, ~5/4 = 386.1653, ~7/4 = 968.6464

Template:Val list

Badness: 0.05088 × 10-6

Akua (352/351, 847/845)

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845

Mapping: [⟨1 0 0 10 0 5], ⟨0 1 0 -6 0 -3], ⟨0 0 1 1 0 0], ⟨0 0 0 0 1 1]]

Mapping generators: ~2, ~3, ~5, ~11

POTE generators: ~3/2 = 702.9075, ~5/4 = 387.0723, ~11/8 = 551.4538

Template:Val list

Badness: 2.550 × 10-6

Werckismic (441/440)

Subgroup: 2.3.5.7.11

Comma list: 441/440

Mapping: [⟨1 0 0 0 -3], ⟨0 1 0 0 2], ⟨0 0 1 0 -1], ⟨0 0 0 1 2]]

Mapping generators:~2, ~3, ~5, ~7

Template:Val list

Commas 364/363, 441/440

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440

Mapping: [⟨1 0 0 0 -3 -8], ⟨0 1 0 0 2 5], ⟨0 0 1 0 -1 -2], ⟨0 0 0 1 2 3]]

Mapping generators: ~2, ~3, ~5, ~7

Map to lattice: [⟨0 1 1 -1 -1 0], ⟨0 0 1 0 -1 -2], ⟨0 0 1 1 1 1]]

Lattice basis:

3/2 length = 1.2263, 14/11 length = 1.4629, 21/16 length = 1.4657

Minimax tuning:

[[1 0 0 0 0 0⟩, [5/3 0 1/3 -1/3 -1/3 1/3⟩, [1/6 0 5/6 2/3 -5/6 1/3⟩, [0 0 0 1 0 0⟩, [1/6 0 -1/6 2/3 1/6 1/3⟩, [0 0 0 0 0 1⟩]
Eigenmonzos: 2, 11/10, 8/7, 16/13
[[1 0 0 0 0 0⟩, [5/4 1/4 1/4 -1/4 -1/4 1/4⟩, [5/4 -3/4 5/4 -1/4 -1/4 1/4⟩, [17/8 -11/8 5/8 -1/8 3/8 1/8⟩, [5/2 -3/2 1/2 -1/2 1/2 1/2⟩, [17/8 -11/8 5/8 -9/8 3/8 9/8⟩]
Eigenmonzos: 2, 14/13, 6/5, 11/9

Template:Val list

Commas 351/350, 441/440

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440

Mapping: [⟨1 0 0 0 -3 1], ⟨0 1 0 0 2 -3], ⟨0 0 1 0 -1 2], ⟨0 0 0 1 2 1]]

Mapping generators: ~2, ~3, ~5, ~7

Template:Val list

Commas 196/195, 352/351

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351

Mapping: [⟨1 0 0 0 -3 2], ⟨0 1 0 0 2 -1], ⟨0 0 1 0 -1 -1], ⟨0 0 0 1 2 2]]

Mapping generators: ~2, ~3, ~5, ~7

Template:Val list

Tannic

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1287/1280

Mapping: [⟨1 0 0 0 -3 11], ⟨0 1 0 0 2 -4], ⟨0 0 1 0 -1 2], ⟨0 0 0 1 2 -2]]

Mapping generators: ~2, ~3, ~5, ~7

Template:Val list

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 441/440, 561/560

Mapping: [⟨1 0 0 0 -3 11 7], ⟨0 1 0 0 2 -4 -3], ⟨0 0 1 0 -1 2 2], ⟨0 0 0 1 2 -2 -1]]

Mapping generators: ~2, ~3, ~5, ~7

Template:Val list

Commas 441/440, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 847/845

Mapping: [⟨1 0 0 0 -3 -3], ⟨0 1 0 0 2 2], ⟨0 0 1 1 1 1], ⟨0 0 0 2 4 5]]

Mapping generators: ~2, ~3, ~5, ~13/11

Template:Val list

Keenanismic (385/384)

Main article: Keenanismic family

Subgroup: 2.3.5.7.11

Comma list: 385/384

Mapping: [⟨1 0 0 0 7], ⟨0 1 0 0 1], ⟨0 0 1 0 -1], ⟨0 0 0 1 -1]]

Mapping generators: ~2, ~3, ~5, ~7

Transpose: [2 3 5 7 385/35]

Minimax tuning:

[[1 0 0 0 0⟩, [0 1 0 0 0⟩, [7/3 1/3 2/3 -1/3 -1/3⟩, [7/3 1/3 -1/3 2/3 -1/3⟩, [7/3 1/3 -1/3 -1/3 2/3⟩]
Eigenmonzos: 2, 3, 7/5, 11/5

Template:Val list

Badness: 15.159 × 10-9

Martwin

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384

Mapping: [⟨1 0 0 0 7 2], ⟨0 1 0 0 1 4], ⟨0 0 1 0 -1 -2], ⟨0 0 0 1 -1 0]]

Mapping generators: ~2, ~3, ~5, ~7

Transpose: [2 3 5 7 385/35 324/25]

Lattice basis:

4/3 length = 1.0820, 6/5 length = 1.3935, 10/9 length = 1.6247

Minimax tuning: [to be confirmed]

[⟨1 0 0 0 0 0], ⟨0 1 0 0 0 0], ⟨2/3 4/3 1/3 0 0 -1/3], ⟨19/6 -1/6 -1/6 1/2 -1/2 1/6], ⟨19/6 -1/6 -1/6 -1/2 1/2 1/6], ⟨2/3 4/3 -2/3 0 0 2/3]]
Eigenmonzos: 2, 14/11, 13/10, 4/3

Template:Val list

Badness: 2.206 × 10-6

Ancient

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 625/624

Mapping: [⟨1 0 0 0 7 -4], ⟨0 1 0 0 1 -1], ⟨0 0 1 0 -1 4], ⟨0 0 0 1 -1 0]]

Transpose: [2 3 5 7 385/35 625/48]

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Badness: 2.573 × 10-6

Commas 351/350, 385/384

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 385/384

Mapping: [⟨1 0 0 0 7 1], ⟨0 1 0 0 1 -3], ⟨0 0 1 0 -1 2], ⟨0 0 0 1 -1 1]]

Mapping generators: ~2, ~3, ~5, ~7

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Commas 352/351, 385/384

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 0 0 1 -2], ⟨0 0 1 0 -1 -1], ⟨0 0 0 1 -1 -1]]

Mapping generators: ~2, ~3, ~5, ~7

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Commas 364/363, 385/384

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 385/384

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 0 0 1 3], ⟨0 0 1 0 -1 -2], ⟨0 0 0 1 -1 -3]]

Mapping generators: ~2, ~3, ~5, ~7

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Commas 385/384, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 847/845

Mapping: [⟨1 0 0 0 7 7], ⟨0 1 0 0 1 1], ⟨0 0 1 1 -2 -2], ⟨0 0 0 2 -2 -1]]

Mapping generators: ~2, ~3, ~5, ~13/11

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Swetismic (540/539)

Subgroup: 2.3.5.7.11

Comma list: 540/539

Mapping: [⟨1 0 0 0 2], ⟨0 1 0 0 3], ⟨0 0 1 0 1], ⟨0 0 0 1 -2]]

Mapping generators: ~2, ~3, ~5, ~7

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Commas 540/539, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 847/845

Mapping: [⟨1 0 0 0 2 2], ⟨0 1 0 0 3 3], ⟨0 0 1 1 -1 -1], ⟨0 0 0 2 -4 -3]]

Mapping generators: ~2, ~3, ~5, ~13/11

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Commas 540/539, 625/624

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624

Mapping: [⟨1 0 0 0 2 -4], ⟨0 1 0 0 3 -1], ⟨0 0 1 0 1 4], ⟨0 0 0 1 -2 0]]

Mapping generators: ~2, ~3, ~5, ~7

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Commas 540/539, 676/675

Subgroup: 2.3.5.7.11

Comma list: 540/539, 676/675

Mapping: [⟨1 0 0 0 2 -1], ⟨0 2 0 0 6 3], ⟨0 0 1 0 1 1], ⟨0 0 0 1 -2 0]]

Mapping generators: ~2, ~26/15, ~5, ~7

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Pentacircle (896/891)

Subgroup: 2.3.5.7.11

Comma list: 896/891

Mapping: [⟨1 0 0 0 7], ⟨0 1 0 0 -4], ⟨0 0 1 0 0], ⟨0 0 0 1 1]]

POTE generators: ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722

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Badness: 0.0658 × 10-6

Commas 352/351, 364/363

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 0 0 -4 -7], ⟨0 0 1 0 0 0], ⟨0 0 0 1 1 1]]

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Badness: 3.375 × 10-6

Topsy (847/845, 1001/1000)

Subgroup: 2.3.5.7.11.13

Comma list: 847/845, 1001/1000

Mapping: [⟨1 0 0 2 0 1], ⟨0 1 0 0 0 0], ⟨0 0 1 1 1 1], ⟨0 0 0 4 -3 1]]

Mapping generators: ~2, ~3, ~5, ~13/10

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Lehmerismic (3025/3024)

Subgroup: 2.3.5.7.11

Comma list: 3025/3024

Mapping: [⟨1 0 0 0 2], ⟨0 1 0 1 2], ⟨0 0 1 0 -1], ⟨0 0 0 2 1]]

Mapping generators: ~2, ~3, ~5, ~55/36

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Kalismic (9801/9800)

Subgroup: 2.3.5.7.11

Comma list: 9801/9800

Mapping: [⟨2 0 0 0 3], ⟨0 1 0 0 -2], ⟨0 0 1 0 1], ⟨0 0 0 1 1]]

Mapping generators: ~99/70, ~3, ~5, ~7

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Commas 1716/1715, 2080/2079

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079

Mapping: [⟨2 0 0 0 3 -7], ⟨0 1 0 0 -2 1], ⟨0 0 1 0 1 0], ⟨0 0 0 1 1 2]]

Mapping generators: ~99/70, ~3, ~5, ~7

Lattice basis:

3/2 length = 1.1956, 7/4 length = 1.4506, 14/13 length = 1.8299

Minimax tuning:

[[1 0 0 0 0 0⟩, [7/10 4/5 0 -2/5 0 1/5⟩, [7/10 -1/5 1 -2/5 0 1/5⟩, [7/5 -2/5 0 1/5 0 2/5⟩, [11/5 -11/5 1 3/5 0 1/5⟩, [0 0 0 0 0 1⟩]
Eigenmonzos: 2, 6/5, 16/13, 9/7

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Semicanousmic (14641/14580)

See Semicanousmic temperaments #Semicanousmic.

Semiporwellismic (16384/16335)

See Semiporwellismic temperaments #Semiporwellismic.

Olympic (131072/130977)

Subgroup: 2.3.5.7.11

Comma list: 131072/130977

Mapping: [⟨1 0 0 0 17], ⟨0 1 0 0 -5], ⟨0 0 1 0 0], ⟨0 0 0 1 -2]]

Mapping generators: ~2, ~3, ~5, ~7

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Commas 2080/2079, 4096/4095

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4096/4095

Mapping: [⟨1 0 0 0 17 12], ⟨0 1 0 0 -5 -2], ⟨0 0 1 0 0 -1], ⟨0 0 0 1 -2 -1]]

Mapping generators: ~2, ~3, ~5, ~7

Template:Val list Template:IoT

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