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. Keemic temperaments include doublewide, flattone, porcupine, superkleismic, magic, keemun, undeka, and sycamore. Discussed below are quasitemp, chromo, barbad, hyperkleismic, and sevond.
Tempering out the keema is one of the two main ways septimal harmony is organized in EDOs of medium size, alongside myna. While myna makes the distance between 5/4 and 6/5 twice the distance between 9/7 and 5/4, keemic makes the two distances equal, resulting in the distance between the classical major and minor thirds being narrowed, or in other words 7/6 - 6/5 - 5/4 - 9/7 being made equidistant. EDOs with this structure include 15, 19, and 22.
Quasitemp
- For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Quasitemp.
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
Wedgie: ⟨⟨ 14 11 9 -15 -25 -10 ]]
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
Wedgie: ⟨⟨ 19 12 21 -25 -20 15 ]]
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
Wedgie: ⟨⟨ 17 16 3 -14 -43 -38 ]]
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