Kleismic rank-3 family

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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.

The kleismic rank-3 family of temperaments tempers out the kleisma, 15625/15552. If nothing else is tempered out in the 7-limit, we have a planar temperament. With an 11-limit comma we get an 11-limit temperament, and so forth.

Rank-3 kleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552

Mapping: [⟨1 0 1 0], ⟨0 6 5 0], ⟨0 0 0 1]]

mapping generators: ~2, ~6/5, ~7

Optimal tunings:

  • WE: ~2 = 1200.1659 ¢, ~6/5 = 317.0504 ¢, ~7/4 = 968.4936 ¢
error map: ⟨+0.166 +0.347 -0.896 -0.001]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0308 ¢, ~7/4 = 968.5773 ¢
error map: ⟨0.000 +0.230 -1.160 -0.249]

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [1 5/6 0 0⟩, [0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.3.7

Optimal ET sequence: 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c

Badness (Sintel): 1.64

2.3.5.7.13 subgroup

Subgroup: 2.3.5.7.13

Comma list: 325/324, 625/624

Subgroup-val mapping: [⟨1 0 1 0 0], ⟨0 6 5 0 14], ⟨0 0 0 1 0]]

Optimal tunings:

  • WE: ~2 = 1200.1210 ¢, ~6/5 = 317.1076 ¢, ~7/4 = 968.5830 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0920 ¢, ~7/4 = 968.6426 ¢

Optimal ET sequence: 15, 19, 34d, 49f, 53, 72, 121, 140, 333, 352f, 492f

Badness (Sintel): 0.385

Enlil

Subgroup: 2.3.5.7.11

Comma list: 385/384, 6250/6237

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

Optimal tunings:

  • WE: ~2 = 1200.3806 ¢, ~6/5 = 317.0420 ¢, ~7/4 = 967.7397 ¢
error map: ⟨+0.381 +0.297 -0.723 -0.325 -0.493]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.9734 ¢, ~7/4 = 967.4765 ¢
error map: ⟨0.000 -0.114 -1.447 -1.349 -1.821]

Minimax tuning:

[[1 0 0 0 0⟩, [0 1 0 0 0⟩, [1 5/6 0 0 0⟩, [3 1/12 0 1/2 -1/2⟩, [3 1/12 0 -1/2 1/2⟩]
unchanged-interval (eigenmonzo) basis: 2.3.11/7

Optimal ET sequence: 15, 19, 34, 38, 53, 68, 72, 140, 159, 212, 284, 496ce, 583cde, 655cde, 867ccdee, 939ccddee

Badness (Sintel): 0.959

13-limit (sumatra)

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624

Mapping: [⟨1 0 1 0 6 0], ⟨0 6 5 0 1 14], ⟨0 0 0 1 -1 0]]

Optimal tunings:

  • WE: ~2 = 1200.3557 ¢, ~6/5 = 317.1024 ¢, ~7/4 = 967.7541 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0355 ¢, ~7/4 = 967.5068 ¢

Minimax tuning:

  • 13- and 15-odd-limit
[[1 0 0 0 0 0⟩, [4/3 0 0 -2/9 -2/9 4/9⟩, [19/9 0 0 -5/27 -5/27 10/27⟩, [28/9 0 0 13/27 -14/27 1/27⟩, [28/9 0 0 -14/27 13/27 1/27⟩, [28/9 0 0 -14/27 -14/27 28/27⟩]
unchanged-interval (eigenmonzo) basis: 2.11/7.13/7

Optimal ET sequence: 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 511cdef

Badness (Sintel): 0.636

Complexity spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 8/7, 11/8, 11/10, 7/5, 12/11, 7/6, 16/13, 15/11, 15/14, 11/9, 9/7, 13/11, 14/13, 14/11

Scales: enlil19_13

Isimud

Subgroup: 2.3.5.7.11

Comma list: 4125/4096, 8019/8000

Mapping: [⟨1 0 1 0 9], ⟨0 6 5 0 -21], ⟨0 0 0 1 0]]

Optimal tunings:

  • WE: ~2 = 1200.7273 ¢, ~6/5 = 316.9521 ¢, ~7/4 = 967.3697 ¢
error map: ⟨+0.727 -0.242 -0.826 -0.002 -0.767]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7598 ¢, ~7/4 = 967.1919 ¢
error map: ⟨0.000 -1.396 -2.515 -1.634 -3.273]

Optimal ET sequence: 19, 34e, …, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee

Badness (Sintel): 4.43

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 625/624, 1287/1280

Mapping: [⟨1 0 1 0 9 0], ⟨0 6 5 0 -21 14], ⟨0 0 0 1 0 0]]

Optimal tunings:

  • WE: ~2 = 1200.8711 ¢, ~6/5 = 317.0291 ¢, ~7/4 = 967.0809 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.8070 ¢, ~7/4 = 966.8000 ¢

Optimal ET sequence: 19, 34e, …, 53, 72, 125f, 178def, 197ef, 231eef, 250deff, 303cdeefff

Badness (Sintel): 2.13

Enki

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4356

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

mapping generators: ~2, ~6/5, ~66/25

Optimal tunings:

  • WE: ~2 = 1200.0985 ¢, ~6/5 = 317.0336 ¢, ~33/25 = 484.1262 ¢
error map: ⟨+0.099 +0.246 -1.047 -0.376 +1.140]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0222 ¢, ~33/25 = 484.1546 ¢
error map: ⟨0.000 +0.178 -1.203 -0.517 +0.925]

Minimax tuning:

[[1 0 0 0 0⟩, [-18/13 0 6/13 -6/13 12/13⟩, [-2/13 0 5/13 -5/13 10/13⟩, [-2/13 0 -8/13 8/13 10/13⟩, [0 0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5.11

Optimal ET sequence: 15, 23de, 34d, 49, 72, 159, 193, 265, 352, 424

Badness (Sintel): 1.31

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 625/624

Mapping: [⟨1 0 1 0 1 0], ⟨0 6 5 0 4 14], ⟨0 0 0 2 1 0]]

Optimal tunings:

  • WE: ~2 = 1200.0373 ¢, ~6/5 = 317.0967 ¢, ~33/25 = 484.1601 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 317.0919 ¢, ~33/25 = 484.1707 ¢

Minimax tuning:

  • 13- and 15-odd-limit
[[1 0 0 0 0 0⟩, [2/3 0 -2/3 0 0 2/3⟩, [14/9 0 -5/9 0 0 5/9⟩, [0 0 0 1 0 0⟩, [13/9 0 -4/9 1/2 0 4/9⟩, [14/9 0 -14/9 0 0 14/9⟩]
unchanged-interval (eigenmonzo) basis: 2.7.13/5

Optimal ET sequence: 15, 23deff, 34d, 49f, 72, 87, 121, 159, 193, 280, 352f

Badness (Sintel): 0.817

Catakleismoid

Catakleismoid is the same as catakleismic in the 2.3.5.7.13 subgroup but with an independent generator for prime 11.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 4375/4374

Mapping: [⟨1 0 1 -3 0], ⟨0 6 5 22 0], ⟨0 0 0 0 1]]

mapping generators: ~2, ~6/5, ~11

Optimal tunings:

  • WE: ~2 = 1200.5965 ¢, ~6/5 = 316.8893 ¢, ~11/8 = 549.5258 ¢
error map: ⟨+0.596 -0.619 -1.271 +0.948 -0.003]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7705 ¢, ~11/8 = 550.0343 ¢
error map: ⟨0.000 -1.332 -2.461 +0.126 -1.284]

Optimal ET sequence: 19, 34d, 53, 72, 197e, 269ce, 341ce

Badness (Sintel): 1.53

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 325/324

Mapping: [⟨1 0 1 -3 0 0], ⟨0 6 5 22 0 14], ⟨0 0 0 0 1 0]]

Optimal tunings:

  • WE: ~2 = 1200.8238 ¢, ~6/5 = 316.9478 ¢, ~11/8 = 548.9611 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.7939 ¢, ~11/8 = 549.5903 ¢

Optimal ET sequence: 19, 34d, 53, 72, 125f, 197ef

Badness (Sintel): 0.853

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