Keemic temperaments

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.

These temper out the keema, [-5 -3 3 1⟩ = 875/864 = S5/S6, whose 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:

• Keemun (+49/48) → Kleismic family
• Doublewide (+50/49) → Jubilismic clan
• Porcupine (+64/63) → Porcupine family
• Flattone (+81/80) → Meantone family
• Magic (+225/224) → Magic family
• Sycamore (+686/675) → Sycamore family
• Superkleismic (+1029/1024) → Gamelismic clan
• Undeka (+3200/3087) → 11th-octave temperaments

Discussed below are quasitemp, chromo, barbad, hyperkleismic, and sevond.

Quasitemp

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

Quasitemp is a full 7-limit strong extension of gariberttet, the 2.5/3.7/3 subgroup temperament 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 (which is equated to 13/11 in the 13-limit extension). 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.

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 = 1\1, ~25/21 = 292.710

Optimal ET sequence: 4, 37, 41

Badness: 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 = 1\1, ~25/21 = 292.547

Optimal ET sequence: 4, 37, 41, 119

Badness: 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 = 1\1, ~13/11 = 292.457

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

Badness: 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 = 1\1, ~25/21 = 292.851

Optimal ET sequence: 41, 127cd, 168cd

Badness: 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 = 1\1, ~13/11 = 292.928

Optimal ET sequence: 41, 86ce, 127cd

Badness: 0.030081

Chromo

For the 5-limit version of this temperament, 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 = 1\1, ~25/24 = 53.816

Optimal ET sequence: 22, 45, 67c

Badness: 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 = 1\1, ~98/75 = 468.331

Optimal ET sequence: 18, 23d, 41

Badness: 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 = 1\1, ~98/75 = 468.367

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

Badness: 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 = 1\1, ~13/10 = 468.270

Optimal ET sequence: 18e, 23de, 41

Badness: 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 = 1\1, ~6/5 = 323.780

Optimal ET sequence: 26, 37, 63

Badness: 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 = 1\1, ~6/5 = 323.796

Optimal ET sequence: 26, 37, 63

Badness: 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 = 1\1, ~6/5 = 323.790

Optimal ET sequence: 26, 37, 63

Badness: 0.035724

Sevond

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 = 1\7, ~3/2 = 705.613

Optimal ET sequence: 7, 56, 63, 119

Badness: 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 = 1\7, ~3/2 = 705.518

Optimal ET sequence: 7, 56, 63, 119

Badness: 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 = 1\7, ~3/2 = 705.344

Optimal ET sequence: 7, 56, 63, 119

Badness: 0.041238