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Keemic temperaments

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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.

This is a collection of linear temperaments that temper out the keema (monzo: [-5 -3 3 1⟩, ratio: 875/864), with S-expression S5/S6. Its fundamental equivalence entails that 6/5 is sharpened so that it stacks three times to reach 7/4, and the interval between 6/5 and 5/4 is compressed so that 7/6–6/5–5/4–9/7 are set equidistant from each other. As the canonical extension of rank-3 keemic to the 11-limit tempers out the commas 100/99 and 385/384 (whereby (6/5)2 is identified with 16/11), this provides a clean way to extend the various keemic temperaments to the 11-limit as well.

Full 7-limit keemic temperaments discussed elsewhere are:

Discussed below are quasitemp, chromo, barbad, hyperkleismic, and sevond, in the order of increasing TE logflat badness.

Quasitemp

For the 5-limit version, see Miscellaneous 5-limit temperaments #Quasitemp.

Quasitemp tempers out 2401/2400 in addition to 875/864 and may be described as the 37 & 41 temperament. It is characterized by equating the interval between the pental and septimal thirds (36/35) with the classical chromatic semitone (25/24), and by tempering together the septimal dieses of 49/48 and 50/49. In that sense, it is opposed to orwellismic temperaments, in particular myna, where the distance between the pental and septimal thirds is the same as the septimal dieses and different from the classical chromatic semitone.

Quasitemp can also be thought of as a strong extension of the 2.5/3.7/3-subgroup temperament called gariberttet, which is defined by tempering out 3125/3087. In gariberttet, three generators reach 5/3 and five reach 7/3, so that the generator itself has the interpretation of 25/21. This implies that 3:5:7 and 5:6:7 chords are reached rather quickly. Quasitemp tempering out 875/864 entails that 8/7 is found after 9 generators, from which the mappings of 3 and 5 follow.

Note that the generator is given as 25/21's octave complement, 42/25, in the data that follow, since a stack of 14 such generators octave-reduced is the perfect fifth, whence the temperament's ploidacot is iota-14-cot. This generator is equated to 22/13 for the 13-limit extension, tempering out 275/273.

Subgroup: 2.3.5.7

Comma list: 875/864, 2401/2400

Mapping: [⟨1 -9 -6 -4], ⟨0 14 11 9]]

mapping generators: ~2, ~42/25

Optimal tunings:

  • WE: ~2 = 1200.9237 ¢, ~42/25 = 907.9887 ¢
error map: ⟨+0.924 +1.573 -3.981 -0.623]
  • CWE: ~2 = 1200.0000 ¢, ~42/25 = 907.3471 ¢
error map: ⟨0.000 +0.905 -5.495 -2.702]

Optimal ET sequence: 4, …, 37, 41

Badness (Sintel): 1.53

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 1375/1372

Mapping: [⟨1 -9 -6 -4 8], ⟨0 14 11 9 -6]]

Optimal tunings:

  • WE: ~2 = 1199.9585 ¢, ~42/25 = 907.4221 ¢
  • CWE: ~2 = 1200.0000 ¢, ~42/25 = 907.4521 ¢

Optimal ET sequence: 4, 37, 41, 119

Badness (Sintel): 1.43

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 196/195, 275/273, 385/384

Mapping: [⟨1 -9 -6 -4 8 9], ⟨0 14 11 9 -6 -7]]

Optimal tunings:

  • WE: ~2 = 1199.4376 ¢, ~22/13 = 907.1175 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/13 = 907.5314 ¢

Optimal ET sequence: 4, 37, 41, 78, 119f

Badness (Sintel): 1.36

Quato

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 625/616

Mapping: [⟨1 -9 -6 -4 -23], ⟨0 14 11 9 35]]

Optimal tunings:

  • WE: ~2 = 1201.2729 ¢, ~42/25 = 908.1116 ¢
  • CWE: ~2 = 1200.0000 ¢, ~42/25 = 907.2109 ¢

Optimal ET sequence: 41, 127cd, 168cd

Badness (Sintel): 1.36

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 243/242, 275/273, 325/324

Mapping: [⟨1 -9 -6 -4 -23 -22], ⟨0 14 11 9 35 34]]

Optimal tunings:

  • WE: ~2 = 1201.4078 ¢, ~42/25 = 908.1362 ¢
  • CWE: ~2 = 1200.0000 ¢, ~42/25 = 907.1370 ¢

Optimal ET sequence: 41, 86ce

Badness (Sintel): 1.24

Chromo

For the 5-limit version, see Miscellaneous 5-limit temperaments #Chromo.

Chromo represents the 13edf chain as a rank-2 temperament, with 6/5 and 5/4 mapped to 6 and 7 steps, respectively. Since the difference of those two intervals is abbreviated considerably from just, keemic provides the most meaningful 7-limit extension (setting 7/6, 6/5, 5/4, 9/7 equidistant) so that the temperament then approximates the 4:5:6:7 tetrad with 0:7:13:18 generator steps.

Note that if one allows a more complex mapping for prime 7 and wants a larger prime limit, one may prefer escapade.

Subgroup: 2.3.5.7

Comma list: 875/864, 2430/2401

Mapping: [⟨1 1 2 2], ⟨0 13 7 18]]

mapping generators: ~2, ~36/35

Optimal tunings:

  • WE: ~2 = 1201.4060 ¢, ~36/35 = 53.8791 ¢
error map: ⟨+1.406 -0.121 -6.348 +3.810]
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 53.9055 ¢
error map: ⟨0.000 -1.183 -8.975 +1.474]

Optimal ET sequence: 22, 45, 67c

Badness (Sintel): 2.30

Barbad

Subgroup: 2.3.5.7

Comma list: 875/864, 16875/16807

Mapping: [⟨1 -10 -5 -10], ⟨0 19 12 21]]

mapping generators: ~2, ~98/75

Optimal tunings:

  • WE: ~2 = 1201.0462 ¢, ~75/49 = 732.3071 ¢
error map: ⟨+1.046 +1.418 -3.859 -0.838]
  • CWE: ~2 = 1200.0000 ¢, ~75/49 = 731.7183 ¢
error map: ⟨0.000 +0.692 -5.694 -2.742]

Optimal ET sequence: 18, 23d, 41

Badness (Sintel): 2.80

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/242, 540/539, 625/616

Mapping: [⟨1 -10 -5 -10 -13], ⟨0 19 12 21 27]]

Optimal tunings:

  • WE: ~2 = 1200.8513 ¢, ~75/49 = 732.1519 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/49 = 731.6740 ¢

Optimal ET sequence: 18e, 23de, 41

Badness (Sintel): 1.66

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 245/242, 275/273

Mapping: [⟨1 -10 -5 -10 -13 -3], ⟨0 19 12 21 27 11]]

Optimal tunings:

  • WE: ~2 = 1199.7960 ¢, ~20/13 = 731.6053 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 731.7208 ¢

Optimal ET sequence: 18e, 23de, 41

Badness (Sintel): 1.62

Hyperkleismic

Subgroup: 2.3.5.7

Comma list: 875/864, 51200/50421

Mapping: [⟨1 -3 -2 2], ⟨0 17 16 3]]

mapping generators: ~2, ~6/5

Optimal tunings:

  • WE: ~2 = 1200.0290 ¢, ~6/5 = 323.7882 ¢
error map: ⟨+0.029 +2.358 -5.759 +2.597]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 323.7816 ¢
error map: ⟨0.000 +2.332 -5.808 +2.519]

Optimal ET sequence: 26, 37, 63

Badness (Sintel): 3.99

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 2420/2401

Mapping: [⟨1 -3 -2 2 4], ⟨0 17 16 3 -2]]

Optimal tunings:

  • WE: ~2 = 1199.9010 ¢, ~6/5 = 323.7691 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 323.7931 ¢

Optimal ET sequence: 26, 37, 63

Badness (Sintel): 2.16

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 169/168, 275/273, 385/384

Mapping: [⟨1 -3 -2 2 4 1], ⟨0 17 16 3 -2 10]]

Optimal tunings:

  • WE: ~2 = 1200.0524 ¢, ~6/5 = 323.8039 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 323.7912 ¢

Optimal ET sequence: 26, 37, 63

Badness (Sintel): 1.48

Sevond

For the 5-limit version, see Syntonic–chromatic equivalence continuum #Sevond (5-limit).

10/9 is tempered to be exactly 1\7. Therefore 3/2 is 1 generator sharp of a 7edo step and 5/4 is 2 generators sharp.

Subgroup: 2.3.5.7

Comma list: 875/864, 327680/321489

Mapping: [⟨7 0 -6 53], ⟨0 1 2 -3]]

mapping generators: ~10/9, ~3

Optimal tunings:

  • WE: ~10/9 = 171.4007 ¢, ~3/2 = 705.4982 ¢
error map: ⟨-0.195 +3.348 -4.112 -0.499]
  • CWE: ~10/9 = 171.4286 ¢, ~3/2 = 705.6057 ¢
error map: ⟨0.000 +3.651 -3.674 +0.071]

Optimal ET sequence: 7, …, 56, 63, 119

Badness (Sintel): 5.23

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 6655/6561

Mapping: [⟨7 0 -6 53 2], ⟨0 1 2 -3 2]]

Optimal tunings:

  • WE: ~11/10 = 171.3859 ¢, ~3/2 = 705.3421 ¢
  • CWE: ~11/10 = 171.4286 ¢, ~3/2 = 705.4973 ¢

Optimal ET sequence: 7, 56, 63, 119

Badness (Sintel): 2.33

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 169/168, 352/351, 385/384

Mapping: [⟨7 0 -6 53 2 37], ⟨0 1 2 -3 2 -1]]

Optimal tunings:

  • WE: ~11/10 = 171.4163 ¢, ~3/2 = 705.2930 ¢
  • CWE: ~11/10 = 171.4286 ¢, ~3/2 = 705.3402 ¢

Optimal ET sequence: 7, 56, 63, 119

Badness (Sintel): 1.70

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