Rank-4 temperament

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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

Optimal GPV sequence7d, 8d, 10e, 12, 15, 19, 22, 27e, 34d, 41, 90e, 131e *

* optimal patent val: 104

Badness: 0.0225 × 10-6

Biyatismic (121/120)

Subgroup: 2.3.5.7.11

Comma list: 121/120

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

Mapping generators: ~2, ~3, ~11/10, ~7

POTE generators: ~3/2 = 701.4578, ~11/10 = 157.7466, ~7/4 = 966.9589

Optimal GPV sequence14c, 15, 22, 31, 46, 53, 60e, 68, 77, 91e, 99, 130e, 159ee, 190ee

Badness: 0.0345 × 10-6

Valinorsmic (176/175)

Subgroup: 2.3.5.7.11

Comma list: 176/175

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

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

POTE generators: ~3/2 = 703.0449, ~5/4 = 389.7641, ~7/4 = 972.1113

Optimal GPV sequence22, 31, 46, 53, 58, 80, 111

Badness: 0.0186 × 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

Optimal GPV sequence7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202

Badness: 0.0509 × 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

Optimal GPV sequence12f, 17c, 24d, 29, 41, 46, 53, 58, 87, 111, 140, 152f, 198, 350f, 437f, 490f

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

Optimal GPV sequence10, 12, 15, 19e, 26, 27e, 31, 41, 58, 72, 118, 130, 190, 248, 289, 320, 609d

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

Mapping 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

Optimal GPV sequence12f, 14cf, 15, 17c, 26, 29, 31f, 41, 46, 58, 72, 87, 130, 217, 289

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

Optimal GPV sequence12f, 14cf, 19e, 26, 27e, 31, 45ef, 46, 58, 72, 103, 130, 233, 279, 409, 512bf, 642bf

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

Optimal GPV sequence10, 12f, 17c, 19e, 27e, 29, 31, 41, 46, 58, 87, 118, 145, 232

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

Optimal GPV sequence17c, 26, 29, 31, 43, 46, 60e, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff

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

Optimal GPV sequence17cg, 26, 29g, 31, 43, 46, 60e, 72, 103, 149, 221ef

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

Optimal GPV sequence12f, 16, 17c, 25e, 29, 41, 46, 58, 87, 103, 145, 149, 161, 190, 248, 438d

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

Optimal GPV sequence9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de

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

Optimal GPV sequence12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff

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]

Optimal GPV sequence15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def

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

Optimal GPV sequence19, 22, 26, 31, 46, 53, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 495bdeefff, 545bddeefff, 598bcdeeefff

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

Optimal GPV sequence22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e

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

Optimal GPV sequence9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159

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

Optimal GPV sequence34, 37, 41, 46, 53, 87, 103, 140, 190, 243e, 330e, 520de, 573dee

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

Optimal GPV sequence8d, 9, 10, 12e, 14c, 17c, 19, 22, 27e, 31, 41, 53, 58, 72, 130, 152, 224, 354, 506e, 578, 730de, 761d, 985d, 1115de, 1267dde

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

Optimal GPV sequence8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f

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

Optimal GPV sequence19, 22, 31, 49f, 50, 53, 72, 103, 121, 152f, 193, 224

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

Optimal GPV sequence9, 10, 14cf, 19, 33cdff, 39df, 48c, 49, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de

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

Optimal GPV sequence12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee

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]]

Optimal GPV sequence12f, 17c, 22, 29, 34d, 41, 46, 58, 80, 87, 121, 167, 179ef, 208, 266ef, 433bceef, 641bbceeeff, 699bbcdeeeff

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

Optimal GPV sequence16, 21, 24d, 29, 37, 45ef, 50, 53, 58, 87, 103, 111, 140, 190, 198, 301, 388, 689e

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

Optimal GPV sequence7d, 8d, 10, 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee

Trimitone (8019/8000)

Subgroup: 2.3.5.7.11

Comma list: 8019/8000

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

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

Optimal GPV sequence12, 19, 26, 39d, 46, 53, 58, 72, 118, 130, 183, 190, 248, 255, 301, 373, 804, 876, 1177be

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

Optimal GPV sequence8d, 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

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

Optimal GPV sequence10e, 12f, 14cf, 22f, 26, 36ce, 46, 58, 72, 130, 198, 224, 270, 494, 764, 1258, 1810d, 2304d, 2574d

Semicanousmic (14641/14580)

Subgroup: 2.3.5.7.11

Comma list: 14641/14580

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

Mapping generators: ~2, ~3, ~18/11, ~7

POTE generators: ~3/2 = 702.2503, ~18/11 = 854.5421, ~7/4 = 968.6866

Optimal GPV sequence14c, 17c, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e

Badness: 0.351 × 10-6

Tridecimal semicanousmic

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 14641/14580

Mapping: [1 0 2 0 1 -6], 0 1 2 0 2 3], 0 0 -4 0 -1 3], 0 0 0 1 0 1]]

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

Optimal GPV sequence: 87, 111, 181, 198, 323, 410

Badness: 17.1 × 10-6

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 1089/1088, 14641/14580

Mapping: [1 0 2 0 1 -6 -4], 0 1 2 0 2 3 6], 0 0 -4 0 -1 3 -2], 0 0 0 1 0 1 0]]

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

Optimal GPV sequence: 87, 94, 111, 181, 198g, 212g, 292, 323, 410

Badness: 34.0 × 10-6

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 715/714, 1089/1088, 1216/1215, 1445/1444

Mapping: [1 0 2 0 1 -6 -4 -4], 0 1 2 0 2 3 6 7], 0 0 -4 0 -1 3 -2 -4], 0 0 0 1 0 1 0 0]]

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

Optimal GPV sequence: 87, 94, 111, 181, 205, 212gh, 292h, 299, 323, 410, 622ef

Badness: 41.9 × 10-6

Semiporwellismic (16384/16335)

See Semiporwellismic clan #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

Optimal GPV sequence41, 53, 84, 87, 130, 183, 224, 270, 494, 764, 1164, 1205, 1475, 1969, 2239, 3133de, 4608cde, 5102bcde, 10474bbccdddeee

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

Optimal GPV sequence41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1699, 2280, 2774e *

* optimal patent val: 3044