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 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.

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 has a strong restriction to the 2.5/3.7/3 subgroup, 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. In quasitemp, tempering out 875/864 entails that 8/7 is found after 9 generators, from which the mappings of 3 and 5 follow.

The generator is equated to 13/11 for the 13-limit extension, tempering out 275/273.

Subgroup: 2.3.5.7

Comma list: 875/864, 2401/2400

Mapping: [⟨1 5 5 5], ⟨0 -14 -11 -9]]

Mapping generators: ~2, ~25/21

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~25/21 = 292.710 ¢

Optimal ET sequence: 4, 37, 41

Badness (Smith): 0.060269

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 5 5 5 2], ⟨0 -14 -11 -9 6]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~25/21 = 292.547 ¢

Optimal ET sequence: 4, 37, 41, 119

Badness (Smith): 0.043209

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 5 5 5 2 2], ⟨0 -14 -11 -9 6 7]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~13/11 = 292.457 ¢

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

Badness (Smith): 0.032913

Quato

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 5 5 5 12], ⟨0 -14 -11 -9 -35]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~25/21 = 292.851 ¢

Optimal ET sequence: 41, 127cd, 168cd

Badness (Smith): 0.041170

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 5 5 5 12 12], ⟨0 -14 -11 -9 -35 -34]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~13/11 = 292.928 ¢

Optimal ET sequence: 41, 86ce, 127cd

Badness (Smith): 0.030081

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, ~25/24

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~25/24 = 53.816 ¢

Optimal ET sequence: 22, 45, 67c

Badness (Smith): 0.090769

Barbad

Subgroup: 2.3.5.7

Comma list: 875/864, 16875/16807

Mapping: [⟨1 9 7 11], ⟨0 -19 -12 -21]]

Mapping generators: ~2, ~98/75

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~98/75 = 468.331 ¢

Optimal ET sequence: 18, 23d, 41

Badness (Smith): 0.110448

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 9 7 11 14], ⟨0 -19 -12 -21 -27]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~98/75 = 468.367 ¢

Optimal ET sequence: 18e, 23de, 41, 228ccdd

Badness (Smith): 0.050105

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 9 7 11 14 8], ⟨0 -19 -12 -21 -27 -11]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~13/10 = 468.270 ¢

Optimal ET sequence: 18e, 23de, 41

Badness (Smith): 0.039183

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 tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 323.780 ¢

Optimal ET sequence: 26, 37, 63

Badness (Smith): 0.157830

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 tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 323.796 ¢

Optimal ET sequence: 26, 37, 63

Badness (Smith): 0.065356

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 tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 323.790 ¢

Optimal ET sequence: 26, 37, 63

Badness (Smith): 0.035724

Sevond

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

10/9 is tempered to be exactly 1\7 of an octave. 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 tuning (POTE): ~10/9 = 171.429 ¢, ~3/2 = 705.613 ¢

Optimal ET sequence: 7, 56, 63, 119

Badness (Smith): 0.206592

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 tuning (POTE): ~10/9 = 171.429 ¢, ~3/2 = 705.518 ¢

Optimal ET sequence: 7, 56, 63, 119

Badness (Smith): 0.070437

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 tuning (POTE): ~10/9 = 171.429 ¢, ~3/2 = 705.344 ¢

Optimal ET sequence: 7, 56, 63, 119

Badness (Smith): 0.041238

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