Catalog of rank-4 temperaments
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.
Mothwellsmic (99/98)
Subgroup: 2.3.5.7.11
Mapping: [⟨1 0 0 0 1], ⟨0 1 0 0 -2], ⟨0 0 1 0 0], ⟨0 0 0 1 2]]
- mapping generators: ~2, ~3, ~5, ~7
- CTE: ~2 = 1\1, ~3/2 = 700.306, ~5/4 = 386.314, ~7/4 = 974.000
- CWE: ~2 = 1\1, ~3/2 = 700.060, ~5/4 = 385.931, ~7/4 = 973.388
Optimal ET sequence: 5, 8d, 9, 12, 17c, 19e, 22, 31, 53, 84e, 96, 127
Badness: 0.0299 × 10-6
Ptolemismic (100/99)
Subgroup: 2.3.5.7.11
Mapping: [⟨1 0 0 0 2], ⟨0 1 0 0 -2], ⟨0 0 1 0 2], ⟨0 0 0 1 0]]
- mapping generators: ~2, ~3, ~5, ~7
- CTE: ~2 = 1\1, ~3/2 = 703.961, ~5/4 = 382.009, ~7/4 = 968.826
- CWE: ~2 = 1\1, ~3/2 = 704.475, ~5/4 = 383.076, ~7/4 = 969.890
Optimal ET sequence: 7d, 8d, 10e, 12, 15, 19, 22, 27e, 34d, 41, 90e, 131e *
Badness: 0.0225 × 10-6
Biyatismic (121/120)
Subgroup: 2.3.5.7.11
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
- CTE: ~2 = 1\1, ~3/2 = 702.602, ~11/10 = 157.450, ~7/4 = 968.826
- CWE: ~2 = 1\1, ~3/2 = 701.985, ~11/10 = 157.610, ~7/4 = 967.819
Optimal ET sequence: 14c, 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
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
- CTE: ~2 = 1\1, ~3/2 = 701.955, ~5/4 = 388.882, ~7/4 = 970.703
- CWE: ~2 = 1\1, ~3/2 = 702.587, ~5/4 = 389.393, ~7/4 = 971.520
Optimal ET sequence: 22, 31, 46, 53, 58, 80, 111
Badness: 0.0186 × 10-6
Rastmic (243/242)
Subgroup: 2.3.5.7.11
Mapping: [⟨1 1 0 0 2], ⟨0 2 0 0 5], ⟨0 0 1 0 0], ⟨0 0 0 1 0]]
- mapping generators: ~2, ~11/9, ~5, ~7
- CTE: ~2 = 1\1, ~11/9 = 350.572, ~5/4 = 386.314, ~7/4 = 968.826
- CWE: ~2 = 1\1, ~11/9 = 350.549, ~5/4 = 386.239, ~7/4 = 968.735
Optimal ET sequence: 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202
Badness: 0.0509 × 10-6
Frostmic (245/242)
Subgroup: 2.3.5.7.11
Mapping: [⟨1 0 1 0 0], ⟨0 1 0 0 0], ⟨0 0 2 0 1], ⟨0 0 0 1 1]]
- mapping generators: ~2, ~3, ~11/7, ~7
- CTE: ~2 = 1\1, ~3/2 = 701.955, ~11/7 = 792.479, ~7/4 = 964.861
- CWE: ~2 = 1\1, ~3/2 = 701.753, ~5/4 = 792.335, ~7/4 = 964.527
Optimal ET sequence: 9, 12, 15, 23de, 24, 26, 27e, 38d, 41, 91, 106d
Badness: 0.0897 × 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
- CTE: ~2 = 1\1, ~3/2 = 702.873, ~5/4 = 386.213, ~11/8 = 550.306
- CWE: ~2 = 1\1, ~3/2 = 702.896, ~5/4 = 386.784, ~11/8 = 551.068
Optimal ET sequence: 12f, 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
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
- [[1 0 0 0 0⟩, [1/2 2/3 1/6 -1/3 1/6⟩, [0 0 1 0 0⟩, [1 -2/3 1/3 1/3 1/3⟩, [0 0 0 0 1⟩]
- eigenmonzo (unchanged-interval) basis: 2.5.9/7.11
Optimal ET sequence: 10, 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 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
- [[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 (unchanged-intervals): 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 (unchanged-intervals): 2, 14/13, 6/5, 11/9
Optimal ET sequence: 12f, 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]]
Optimal ET sequence: 12f, 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]]
Optimal ET sequence: 10, 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]]
Optimal ET sequence: 17c, 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]]
Optimal ET sequence: 17cg, 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 ET sequence: 12f, 16, 17c, 25e, 29, 41, 46, 58, 87, 103, 145, 149, 161, 190, 248, 438d
Keenanismic (385/384)
Subgroup: 2.3.5.7.11
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]
- [[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⟩]
- Eigenmonzo (unchanged-interval) basis: 2.3.7/5.11/5
Optimal ET sequence: 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de
Badness: 15.2 × 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]]
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]
- 13- and 15-odd-limit
- [⟨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]]
- Eigenmonzo (unchanged-interval) basis: 2.3.11/7.13/5
Optimal ET sequence: 12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff
Badness: 2.21 × 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 ET sequence: 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def
Badness: 2.57 × 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]]
Optimal ET sequence: 19, 22, 26, 31, 46, 53, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 495bdeefff, 545bddeefff, 598bcdeeefff
Badness: 2.98 × 10-6
Zaxa
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]]
- CTE: ~2 = 1\1, ~3/2 = 702.657, ~5/4 = 385.632, ~7/4 = 967.829
- CWE: ~2 = 1\1, ~3/2 = 702.603, ~5/4 = 385.644, ~7/4 = 967.903
Optimal ET sequence: 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e
Badness: 3.35 × 10-6
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]]
Optimal ET sequence: 9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159
Badness: 3.32 × 10-6
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 ET sequence: 34, 37, 41, 46, 53, 87, 103, 140, 190, 243e, 330e, 520de, 573dee
Badness: 4.15 × 10-6
Swetismic (540/539)
Subgroup: 2.3.5.7.11
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
- CTE: ~2 = 1\1, ~3/2 = 701.6167, ~5/4 = 386.0717, ~7/4 = 969.5334
- CWE: ~2 = 1\1, ~3/2 = 701.6950, ~5/4 = 386.1796, ~7/4 = 969.6366
Optimal ET sequence: 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
Badness: 0.0105 × 10-6
Commas 540/539, 729/728
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728
Mapping: [⟨1 0 0 0 2 -3], ⟨0 1 0 0 3 6], ⟨0 0 1 0 1 0], ⟨0 0 0 1 -2 -1]]
- mapping generators: ~2, ~3, ~5, ~7
- CTE: ~2 = 1\1, ~3/2 = 701.6687, ~5/4 = 386.0441, ~7/4 = 969.5668
- CWE: ~2 = 1\1, ~3/2 = 701.7230, ~5/4 = 386.1818, ~7/4 = 969.6607
Optimal ET sequence: 12e, 14cf, 17c, 19, 22f, 31f, 39df, 41, 53, 58, 72, 111, 130, 183, 224, 354, 578, 985d, 1267ddef
Badness: 1.73 × 10-6
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 ET sequence: 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f
Badness: 3.97 × 10-6
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]]
Optimal ET sequence: 19, 22, 31, 49f, 50, 53, 72, 103, 121, 152f, 193, 224
Badness: 3.59 × 10-6
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 ET sequence: 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de
Badness: 3.06 × 10-6
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]]
- mapping generators: ~2, ~3, ~5, ~7
- CTE: ~2 = 1\1, ~3/2 = 703.576, ~5/4 = 386.314, ~7/4 = 968.126
- CWE: ~2 = 1\1, ~3/2 = 703.743, ~5/4 = 387.245, ~7/4 = 969.048
Optimal ET sequence: 12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee
Badness: 0.0658 × 10-6
Tridecimal pentacircle a.k.a. gentle
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 ET sequence: 12f, 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 ET sequence: 16, 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
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 ET sequence: 7d, 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
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
- CTE: ~2 = 1\1, ~3/2 = 701.5449, ~5/4 = 386.7538, ~7/4 = 968.8259
- CWE: ~2 = 1\1, ~3/2 = 701.4729, ~5/4 = 386.5374, ~7/4 = 968.6210
Optimal ET sequence: 12, 19, 26, 39d, 46, 53, 58, 72, 118, 130, 183, 190, 248, 255, 301, 373, 804, 876, 1177be
Badness: 0.0820 × 10-6
Commas 729/728, 1001/1000
Subgroup: 2.3.5.7.11.13
Comma list: 729/728, 1001/1000
Mapping: [⟨1 0 0 0 6 -3], ⟨0 1 0 0 -6 6], ⟨0 0 1 0 3 0], ⟨0 0 0 1 0 -1]]
- mapping generators: ~2, ~3, ~5, ~7
- CTE: ~2 = 1\1, ~3/2 = 701.5537, ~5/4 = 386.7680, ~7/4 = 968.8144
- CWE: ~2 = 1\1, ~3/2 = 701.4770, ~5/4 = 386.5437, ~7/4 = 968.6150
Optimal ET sequence: 53, 58, 72, 111, 130, 183, 190, 243e, 248, 301, 373, 804, 1177be
Badness: 3.33 × 10-6
Kalismic (9801/9800)
Subgroup: 2.3.5.7.11
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 ET sequence: 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
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]]
Lattice basis:
- 3/2 length = 1.1956, 7/4 length = 1.4506, 14/13 length = 1.8299
- 13- and 15-odd-limit
- [[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⟩]
- Eigenmonzo (unchanged-intervals) basis: 2, 6/5, 16/13, 9/7
Optimal ET sequence: 10e, 12f, 14cf, 22f, 26, 36ce, 46, 58, 72, 130, 198, 224, 270, 494, 764, 1258, 1810d, 2304d, 2574d
Unisquary (12005/11979)
Subgroup: 2.3.5.7.11
Mapping: [⟨1 0 0 0 0], ⟨0 1 0 2 2], ⟨0 0 1 -1 -1], ⟨0 0 0 3 4]]
- mapping generators: ~2, ~3, ~5, ~11/7
- CTE: ~2 = 1\1, ~3/2 = 702.031, ~5/4 = 386.233, ~11/7 = 783.508
- CWE: ~2 = 1\1, ~3/2 = 702.031, ~5/4 = 386.232, ~11/7 = 783.507
Optimal ET sequence: 12, 21, 23de, 26, 37, 46, 58, 72, 118, 130, 190, 239, 311, 441, 559, 752e, 870
Hensquary
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 1716/1715
Mapping: [⟨1 0 0 0 0 -2], ⟨0 1 0 2 2 3], ⟨0 0 1 -1 -1 -1], ⟨0 0 0 3 4 5]]
- Mapping generators: ~2, ~3, ~5, ~11/7
- CTE: ~2 = 1\1, ~3/2 = 702.072, ~5/4 = 386.528, ~11/7 = 783.823
- CWE: ~2 = 1\1, ~3/2 = 702.253, ~5/4 = 386.776, ~11/7 = 783.875
Optimal ET sequence: 26, 37, 46, 58, 72, 121, 130, 193, 239, 251e, 323e, 369
Ekasquary
Subgroup: 2.3.5.7.11.13
Comma list: 1575/1573, 4459/4455
Mapping: [⟨1 0 0 0 0 0], ⟨0 1 0 2 2 0], ⟨0 0 1 -1 -1 3], ⟨0 0 0 3 4 -5]]
- Mapping generators: ~2, ~3, ~5, ~11/7
- CTE: ~2 = 1\1, ~3/2 = 701.952, ~5/4 = 386.139, ~11/7 = 783.188
- CWE: ~2 = 1\1, ~3/2 = 701.952, ~5/4 = 386.139, ~11/7 = 783.188
Optimal ET sequence: 46f, 49f, 58, 72, 118, 121, 130, 190, 193, 248, 311, 441, 752e, 1072
Semicanousmic (14641/14580)
Subgroup: 2.3.5.7.11
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
- CTE: ~2 = 1\1, ~3/2 = 702.334, ~18/11 = 854.555, ~7/4 = 968.826
- CWE: ~2 = 1\1, ~3/2 = 702.292, ~18/11 = 854.548, ~7/4 = 968.756
Optimal ET sequence: 14c, 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]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 702.469, ~18/11 = 854.644, ~7/4 = 968.959
- CWE: ~2 = 1\1, ~3/2 = 702.483, ~18/11 = 854.642, ~7/4 = 968.989
Optimal ET 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]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 702.413, ~18/11 = 854.633, ~7/4 = 969.032
- CWE: ~2 = 1\1, ~3/2 = 702.411, ~18/11 = 854.634, ~7/4 = 969.026
Optimal ET 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]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 702.368, ~18/11 = 854.639, ~7/4 = 969.075
- CWE: ~2 = 1\1, ~3/2 = 702.351, ~18/11 = 854.644, ~7/4 = 969.010
Optimal ET sequence: 87, 94, 111, 181, 205, 212gh, 292h, 299, 323, 410, 622ef
Badness: 41.9 × 10-6
Semiporwellismic (16384/16335)
Subgroup: 2.3.5.7.11
Comma list: 16384/16335
Mapping: [⟨1 0 0 0 7], ⟨0 1 1 0 -2], ⟨0 0 2 0 -1], ⟨0 0 0 1 0]]
- mapping generators: ~2, ~3, ~128/99, ~7
Optimal ET sequence: 19, 22, 41, 65d, 68, 84, 87, 111, 130, 152, 239, 282, 328, 369, 521e, 1370bcdeee, 1411bcdeee, 1609bccdeee
Badness: 0.219 × 10-6
Symbiotic (19712/19683)
Subgroup: 2.3.5.7.11
Mapping: [⟨1 0 0 0 -8], ⟨0 1 0 0 9], ⟨0 0 1 0 0], ⟨0 0 0 1 -1]]
- mapping generators: ~2, ~3, ~5, ~7
- CTE: ~2 = 1\1, ~3/2 = 702.213, ~5/4 = 386.314, ~7/4 = 968.736
- CWE: ~2 = 1\1, ~3/2 = 702.244, ~5/4 = 386.406, ~7/4 = 968.859
Optimal ET sequence: 17c, 19e, 24, 34d, 41, 53, 58, 94, 99e, 118, 152, 270, 581, 733, 851, 1003, 1273, 1854, 2124b
Badness: 0.120 × 10-6
Tridecimal symbiotic
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 19712/19683
Mapping: [⟨1 0 0 0 -8 -13], ⟨0 1 0 0 9 12], ⟨0 0 1 0 0 -1], ⟨0 0 0 1 -1 0]]
- CTE: ~2 = 1\1, ~3/2 = 702.228, ~5/4 = 386.284, ~7/4 = 968.789
- CWE: ~2 = 1\1, ~3/2 = 702.254, ~5/4 = 386.401, ~7/4 = 968.898
Optimal ET sequence: 17c, 34dff, 36ce, 41, 53, 58, 94, 111, 152f, 212, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bf
Badness: 3.31 × 10-6
Olympic (131072/130977)
Subgroup: 2.3.5.7.11
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 ET sequence: 41, 53, 84, 87, 130, 183, 224, 270, 494, 764, 1164, 1205, 1475, 1969, 2239, 3133de, 4608cde, 5102bcde, 10474bbccdddeee
Tridecimal olympic
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]]
Optimal ET sequence: 41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1699, 2280, 2774e *