Xenharmonic Wiki

Catalog of rank-4 temperaments

Revision as of 16:50, 2 March 2026 by FloraC (talk | contribs) (Units & misc. cleanup)
This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

A rank-4 temperament has a period and three additional independent generators, for a total of four dimensions. 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.

Cake (45/44)

Subgroup: 2.3.5.7.11

Comma list: 45/44

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

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 694.823 ¢, ~5/4 = 378.660 ¢, ~7/4 = 968.826 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 693.197 ¢, ~5/4 = 376.082 ¢, ~7/4 = 965.060 ¢

Optimal ET sequence: 7d, 9, 10, 12, 19, 26, 45

Badness (Smith): 0.0219 × 10-6

Mothwellsmic (99/98)

Subgroup: 2.3.5.7.11

Comma list: 99/98

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 700.306 ¢, ~5/4 = 386.314 ¢, ~7/4 = 974.000 ¢
  • CWE: ~2 = 1200.000 ¢, ~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 (Smith): 0.0299 × 10-6

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]]

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 703.961 ¢, ~5/4 = 382.009 ¢, ~7/4 = 968.826 ¢
  • CWE: ~2 = 1200.000 ¢, ~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 *

* optimal patent val: 104

Badness (Smith): 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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.602 ¢, ~11/10 = 157.450 ¢, ~7/4 = 968.826 ¢
  • CWE: ~2 = 1200.000 ¢, ~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 (Smith): 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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.955 ¢, ~5/4 = 388.882 ¢, ~7/4 = 970.703 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.587 ¢, ~5/4 = 389.393 ¢, ~7/4 = 971.520 ¢

Optimal ET sequence: 22, 31, 46, 53, 58, 80, 111

Badness (Smith): 0.0186 × 10-6

Tridecimal valinorsmic

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.304 ¢, ~5/4 = 388.738 ¢, ~7/4 = 970.598 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.694 ¢, ~5/4 = 389.382 ¢, ~7/4 = 971.530 ¢

Optimal ET sequence: 

Badness (Smith):

  Todo: complete temperament data

Rastmic (243/242)

Rastmic halves 3/2 into two 11/9s, 32/27 into two 12/11s, and 25/24 into two 45/44~55/54s.

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]]

Mapping generators: ~2, ~11/9, ~5, ~7

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 350.572 ¢, ~5/4 = 386.314 ¢, ~7/4 = 968.826 ¢
  • CWE: ~2 = 1200.000 ¢, ~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 (Smith): 0.0509 × 10-6

Frostmic (245/242)

Subgroup: 2.3.5.7.11

Comma list: 245/242

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.955 ¢, ~11/7 = 792.479 ¢, ~7/4 = 964.861 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.753 ¢, ~11/7 = 792.335 ¢, ~7/4 = 964.527 ¢

Optimal ET sequence: 9, 12, 15, 23de, 24, 26, 27e, 38d, 41, 91, 106d

Badness (Smith): 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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.873 ¢, ~5/4 = 386.213 ¢, ~11/8 = 550.306 ¢
  • CWE: ~2 = 1200.000 ¢, ~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 (Smith): 2.550 × 10-6

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.363 ¢, ~5/4 = 385.439 ¢, ~7/4 = 967.547 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.693 ¢, ~5/4 = 385.149 ¢, ~7/4 = 967.300 ¢

Minimax tuning:

  • 11-odd-limit: ~2 = [1 0 0 0 0⟩, ~3 = [0 1 0 0 0⟩, ~5 = [7/3 1/3 2/3 -1/3 -1/3⟩, ~7 = [7/3 1/3 -1/3 2/3 -1/3⟩
Unchanged-interval (eigenmonzo) 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 (Smith): 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]]

Lattice basis:

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

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.629 ¢, ~5/4 = 385.219 ¢, ~7/4 = 967.741 ¢

Minimax tuning: [to be confirmed]

  • 13- and 15-odd-limit: ~2 = ⟨1 0 0 0 0 0], ~3 = ⟨0 1 0 0 0 0], ~5 = ⟨2/3 4/3 1/3 0 0 -1/3], ~7 = ⟨19/6 -1/6 -1/6 1/2 -1/2 1/6]
Unchanged-interval (eigenmonzo) 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 (Smith): 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]]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.358 ¢, ~5/4 = 385.674 ¢, ~7/4 = 967.452 ¢

Optimal ET sequence: 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def

Badness (Smith): 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 tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 701.205 ¢, ~5/4 = 386.715 ¢, ~7/4 = 967.646 ¢

Optimal ET sequence: 19, 22, 26, 31, 46, 53, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 495bdeefff, 545bddeefff, 598bcdeeefff

Badness (Smith): 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.657 ¢, ~5/4 = 385.632 ¢, ~7/4 = 967.829 ¢
  • CWE: ~2 = 1200.000 ¢, ~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 (Smith): 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 tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.761 ¢, ~5/4 = 385.159 ¢, ~7/4 = 966.298 ¢

Optimal ET sequence: 9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159

Badness (Smith): 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 tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.368 ¢, ~5/4 = 385.416 ¢, ~13/11 = 291.066 ¢

Optimal ET sequence: 34, 37, 41, 46, 53, 87, 103, 140, 190, 243e, 330e, 520de, 573dee

Badness (Smith): 4.15 × 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 tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.620 ¢, ~5/4 = 386.673 ¢, ~7/4 = 967.775 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.496 ¢, ~5/4 = 386.432 ¢, ~7/4 = 967.588 ¢

Minimax tuning:

  • 11-odd-limit: ~2 = [1 0 0 0 0⟩, ~3 = [1/2 2/3 1/6 -1/3 1/6⟩, ~5 = [0 0 1 0 0⟩, ~7 = [1 -2/3 1/3 1/3 1/3⟩
unchanged-interval (eigenmonzo) 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

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 701.880 ¢, ~5/4 = 386.523 ¢, ~7/4 = 967.841 ¢

Minimax tuning:

  • 13-odd-limit: ~2 = [1 0 0 0 0 0⟩, ~3 = [5/3 0 1/3 -1/3 -1/3 1/3⟩, ~5 = [1/6 0 5/6 2/3 -5/6 1/3⟩, ~7 = [0 0 0 1 0 0⟩
Eigenmonzos (unchanged-intervals): 2, 11/10, 8/7, 16/13
  • 15-odd-limit: ~2 = [1 0 0 0 0 0⟩, ~3 = [5/4 1/4 1/4 -1/4 -1/4 1/4⟩, ~5 = [5/4 -3/4 5/4 -1/4 -1/4 1/4⟩, ~7 = [17/8 -11/8 5/8 -1/8 3/8 1/8⟩
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 tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 701.175 ¢, ~5/4 = 387.330 ¢, ~7/4 = 968.480 ¢

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 tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.601 ¢, ~5/4 = 387.137 ¢, ~7/4 = 966.418 ¢

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 tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 700.555 ¢, ~5/4 = 387.816 ¢, ~7/4 = 967.984 ¢

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 tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 700.606 ¢, ~5/4 = 387.655 ¢, ~7/4 = 967.876 ¢

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 tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 701.611 ¢, ~5/4 = 386.882 ¢, ~13/11 = 290.287 ¢

Optimal ET sequence: 12f, 16, 17c, 25e, 29, 41, 46, 58, 87, 103, 145, 149, 161, 190, 248, 438d

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 tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.6167 ¢, ~5/4 = 386.0717 ¢, ~7/4 = 969.5334 ¢
  • CWE: ~2 = 1200.000 ¢, ~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 (Smith): 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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.6687 ¢, ~5/4 = 386.0441 ¢, ~7/4 = 969.5668 ¢
  • CWE: ~2 = 1200.000 ¢, ~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 (Smith): 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 tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 701.284 ¢, ~5/4 = 386.211 ¢, ~13/11 = 291.733 ¢

Optimal ET sequence: 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f

Badness (Smith): 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 tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 701.702 ¢, ~5/4 = 385.631 ¢, ~7/4 = 969.467 ¢

Optimal ET sequence: 19, 22, 31, 49f, 50, 53, 72, 103, 121, 152f, 193, 224

Badness (Smith): 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 tuning (CTE): ~2 = 1200.000 ¢, ~26/15 = 950.906 ¢, ~5/4 = 386.474 ¢, ~7/4 = 969.892 ¢

Optimal ET sequence: 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de

Badness (Smith): 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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 703.576 ¢, ~5/4 = 386.314 ¢, ~7/4 = 968.126 ¢
  • CWE: ~2 = 1200.000 ¢, ~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 (Smith): 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 tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 703.786 ¢, ~5/4 = 386.314 ¢, ~7/4 = 967.665 ¢

Optimal ET sequence: 12f, 17c, 22, 29, 34d, 41, 46, 58, 80, 87, 121, 167, 179ef, 208, 266ef, 433bceef, 641bbceeeff, 699bbcdeeeff

Badness (Smith): 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 tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 701.955 ¢, ~5/4 = 386.699 ¢, ~13/10 = 454.558 ¢

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

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 tuning:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.998 ¢, ~5/4 = 386.252 ¢, ~55/36 = 733.436 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.978 ¢, ~5/4 = 386.232 ¢, ~55/36 = 733.430 ¢

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

Commas 2080/2079, 3025/3024

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 3025/3024

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.076 ¢, ~5/4 = 386.208 ¢, ~55/36 = 733.438 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.099 ¢, ~5/4 = 386.248 ¢, ~55/36 = 733.449 ¢

Optimal ET sequence: 15, 26, 41, 46, 72, 87, 111, 183, 224, 270, 311, 494

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 tunings:

  • CTE: ~2 = 1200.0000 ¢, ~3/2 = 701.5449 ¢, ~5/4 = 386.7538 ¢, ~7/4 = 968.8259 ¢
  • CWE: ~2 = 1200.0000 ¢, ~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 (Smith): 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

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~3/2 = 701.5537 ¢, ~5/4 = 386.7680 ¢, ~7/4 = 968.8144 ¢
  • CWE: ~2 = 1200.0000 ¢, ~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 (Smith): 3.33 × 10-6

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 tuning:

  • CTE: ~99/70 = 600.000 ¢, ~3/2 = 701.942 ¢, ~5/4 = 386.327 ¢, ~7/4 = 968.846 ¢
  • CWE: ~99/70 = 600.000 ¢, ~3/2 = 701.940 ¢, ~5/4 = 386.323 ¢, ~7/4 = 968.840 ¢

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

Optimal tuning (CTE): ~99/70 = 600.000 ¢, ~3/2 = 702.024 ¢, ~5/4 = 386.294 ¢, ~7/4 = 969.114 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~99/70 = [1/2 0 0 0 0 0⟩, ~3 = [7/10 4/5 0 -2/5 0 1/5⟩, ~5 = [7/10 -1/5 1 -2/5 0 1/5⟩, ~7 = [7/5 -2/5 0 1/5 0 2/5⟩
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

Comma list: 12005/11979

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

Optimal tunings:

  • CTE and CWE: ~2 = 1200.000 ¢, ~3/2 = 702.031 ¢, ~5/4 = 386.233 ¢, ~11/7 = 783.508 ¢

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.072 ¢, ~5/4 = 386.528 ¢, ~11/7 = 783.823 ¢
  • CWE: ~2 = 1200.000 ¢, ~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

Optimal tunings:

  • CTE and CWE: ~2 = 1200.000 ¢, ~3/2 = 701.952 ¢, ~5/4 = 386.139 ¢, ~11/7 = 783.548 ¢

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

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.334 ¢, ~18/11 = 854.555 ¢, ~7/4 = 968.826 ¢
  • CWE: ~2 = 1200.000 ¢, ~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 (Smith): 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 = 1200.000 ¢, ~3/2 = 702.469 ¢, ~18/11 = 854.644 ¢, ~7/4 = 968.959 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.483 ¢, ~18/11 = 854.642 ¢, ~7/4 = 968.989 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~3/2 = 702.413 ¢, ~18/11 = 854.633 ¢, ~7/4 = 969.032 ¢
  • CWE: ~2 = 1200.000 ¢, ~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 (Smith): 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 = 1200.000 ¢, ~3/2 = 702.368 ¢, ~18/11 = 854.639 ¢, ~7/4 = 969.075 ¢
  • CWE: ~2 = 1200.000 ¢, ~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 (Smith): 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 tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.470 ¢, ~128/99 = 442.106 ¢, ~7/4 = 968.826 ¢

Optimal ET sequence: 19, 22, 41, 65d, 68, 84, 87, 111, 130, 152, 239, 282, 328, 369, 521e, 1370bcdeee, 1411bcdeee, 1609bccdeee

Badness (Smith): 0.219 × 10-6

Symbiotic (19712/19683)

Subgroup: 2.3.5.7.11

Comma list: 19712/19683

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.213 ¢, ~5/4 = 386.314 ¢, ~7/4 = 968.736 ¢
  • CWE: ~2 = 1200.000 ¢, ~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 (Smith): 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.228 ¢, ~5/4 = 386.284 ¢, ~7/4 = 968.789 ¢
  • CWE: ~2 = 1200.000 ¢, ~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 (Smith): 3.31 × 10-6

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 2080/2079

Mapping: [⟨1 0 0 0 -8 -13 -6], ⟨0 1 0 0 9 12 5], ⟨0 0 1 0 0 -1 1], ⟨0 0 0 1 -1 0 0]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.230 ¢, ~5/4 = 386.304 ¢, ~7/4 = 968.797 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.255 ¢, ~5/4 = 386.408 ¢, ~7/4 = 968.902 ¢

Optimal ET sequence: 17ch, 24f, 34dffh, 36ceh, 41, 53, 58h, 94, 111, 152f, 212h, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bfh

Badness (Smith): 4.37 × 10-6

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 tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.103 ¢, ~5/4 = 386.314 ¢, ~7/4 = 969.012 ¢

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 tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.103 ¢, ~5/4 = 386.298 ¢, ~7/4 = 969.007 ¢

Optimal ET sequence: 41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1699, 2280, 2774e *

* optimal patent val: 3044

Seascape (160083/160000)

Subgroup: 2.3.5.7.11

Comma list: 160083/160000

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

Mapping generators: ~2, ~400/231, ~5, ~7

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~400/231 = 950.960 ¢, ~5/4 = 386.417 ¢, ~7/4 = 968.751 ¢

Optimal ET sequence: 15, 19, 29, 43, 53, 58, 68, 72

Fifthchromismic (1610510/1594323)

Subgroup: 2.3.5.7.11

Comma list: 1610510/1594323

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

Mapping generators: ~2, ~3, ~22/9, ~7

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.738 ¢, ~11/9 = 344.406 ¢, ~7/4 = 968.826 ¢

Optimal ET sequence: 77, 80, 87, 94, 167

Fifthchroma

See also: Fifth-chroma temperaments

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 951665/944784

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

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.776 ¢, ~11/9 = 344.410 ¢, ~7/4 = 968.826 ¢

Optimal ET sequence: 77, 80, 87, 94, 167

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 715/714, 155771/155520

Mapping: [⟨1 0 4 0 -1 4 6], ⟨0 1 3 0 2 -1 3], ⟨0 0 -5 0 1 1 -3], ⟨0 0 0 1 0 0 -1]]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.713 ¢, ~11/9 = 344.403 ¢, ~7/4 = 969.193 ¢

Optimal ET sequence: 77, 80, 87, 94, 167

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 352/351, 715/714, 1463/1458, 2057/2052

Mapping: [⟨1 0 4 0 -1 4 6 2], ⟨0 1 3 0 2 -1 3 4], ⟨0 0 -5 0 1 1 -3 -1], ⟨0 0 0 1 0 0 -1 -1]]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 702.736 ¢, ~11/9 = 344.417 ¢, ~7/4 = 969.159 ¢

Optimal ET sequence: 77, 80, 87, 94, 167h

Retrieved from "https://en.xen.wiki/index.php?title=Catalog_of_rank-4_temperaments&oldid=225025"