Keemic temperaments

These temper out the keema, [-5 -3 3 1⟩ = 875/864. Keemic temperaments include doublewide, flattone, porcupine, superkleismic, magic, keemun, and sycamore. Discussed below are quasitemp, chromo, undeka, barbad, hyperkleismic, and sevond.

Quasitemp

For the 5-limit version of this temperament, see High badness 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 High badness temperaments #Chromo.

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

Undeka

For the 5-limit version of this temperament, see High badness temperaments #Undeka.

Subgroup: 2.3.5.7

Comma list: 875/864, 3200/3087

Mapping: [⟨11 0 8 31], ⟨0 1 1 0]]

mapping generators: ~21/20, ~3

Wedgie: ⟨⟨ 11 11 0 -8 -31 -31 ]]

Optimal tuning (POTE): ~21/20 = 1\11, ~3/2 = 708.792

Optimal ET sequence: 11c, 22

Badness: 0.141782

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 352/343, 385/384

Mapping: [⟨11 0 8 31 38], ⟨0 1 1 0 0]]

Optimal tuning (POTE): ~21/20 = 1\11, ~3/2 = 706.768

Optimal ET sequence: 11c, 22

Badness: 0.068672

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 65/63, 100/99, 169/165, 352/343

Mapping: [⟨11 0 8 31 38 23], ⟨0 1 1 0 0 1]]

Optimal tuning (POTE): ~13/12 = 1\11, ~3/2 = 707.764

Optimal ET sequence: 11cf, 22

Badness: 0.056528

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