Hemifamity temperaments
The hemifamity temperaments temper out the hemifamity comma, [10 -6 1 -1⟩ = 5120/5103, dividing an exact or approximate septimal diesis, [2 2 -1 -1⟩ = 36/35 into two equal steps.
Belonging to it and considered below are buzzard, undecental, leapday, mystery, quanic and ketchup. Other hemifamity temperaments are:
- Dominant, {36/35, 64/63} → Meantone family
- Garibaldi, {225/224, 3125/3087} → Schismatic family
- Rodan, {245/243, 1029/1024} → Gamelismic clan
- Hemififths, {2401/2400, 5120/5103} → Breedsmic temperaments
- Misty, {3136/3125, 5120/5103} → Misty family
- Amity, {4375/4374, 5120/5103} → Ragismic microtemperaments and Amity family
- Countercata, {5120/5103, 15625/15552} → Kleismic family
- Kwai, {5120/5103, 16875/16807} → Mirkwai clan
- Warrior, {5120/5103, 78732/78125} → Sensipent family
- Supers, {5120/5103, 118098/117649} → Stearnsmic clan
- Undim, {5120/5103, 390625/388962} → Undim family
- Quinticosiennic, {5120/5103, 395136/390625} → Quintaleap family
- Quintakwai, {5120/5103, 9765625/9680832} → Quindromeda family
- Alphaquarter, {5120/5103, 29360128/29296875} → Escapade family
Buzzard
Subgroup: 2.3.5.7
Comma list: 1728/1715, 5120/5103
Mapping: [⟨1 0 -6 4], ⟨0 4 21 -3]]
Wedgie: ⟨⟨ 4 21 -3 24 -16 -66 ]]
POTE generator: ~21/16 = 475.636
Badness: 0.047963
11-limit
Subgroup: 2.3.5.7.11
Comma list: 176/175, 540/539, 5120/5103
Mapping: [⟨1 0 -6 4 -12], ⟨0 4 21 -3 39]]
POTE generator: ~21/16 = 475.700
Optimal GPV sequence: Template:Val list
Badness: 0.034484
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 351/350, 540/539, 676/675
Mapping: [⟨1 0 -6 4 -12 -7], ⟨0 4 21 -3 39 27]]
POTE generator: ~21/16 = 475.697
Optimal GPV sequence: Template:Val list
Badness: 0.018842
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 176/175, 256/255, 351/350, 442/441, 540/539
Mapping: [⟨1 0 -6 4 -12 -7 14], ⟨0 4 21 -3 39 27 -25]]
POTE generator: ~21/16 = 475.692
Optimal GPV sequence: Template:Val list
Badness: 0.018403
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539
Mapping: [⟨1 0 -6 4 -12 -7 14 -12], ⟨0 4 21 -3 39 27 -25 41]]
POTE generator: ~21/16 = 475.679
Optimal GPV sequence: Template:Val list
Badness: 0.015649
Buteo
Subgroup: 2.3.5.7.11
Comma list: 99/98, 385/384, 2200/2187
Mapping: [⟨1 0 -6 4 9], ⟨0 4 21 -3 -14]]
POTE generator: ~21/16 = 475.436
Optimal GPV sequence: Template:Val list
Badness: 0.060238
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 99/98, 275/273, 385/384, 572/567
Mapping: [⟨1 0 -6 4 9 -7], ⟨0 4 21 -3 -14 27]]
POTE generator: ~21/16 = 475.464
Optimal GPV sequence: Template:Val list
Badness: 0.039854
Undecental
Subgroup: 2.3.5.7
Comma list: 5120/5103, 235298/234375
Mapping: [⟨1 0 61 71], ⟨0 1 -37 -43]]
Wedgie: ⟨⟨ 1 -37 -43 -61 -71 4 ]]
POTE generator: ~3/2 = 703.039
Badness: 0.094603
Leapday
- For the 5-limit version of this temperament, see High badness temperaments #Leapday.
- Not to be confused with calendar-based modes such as those in 293edo or Irvian mode.
Leapday tempers out [31 -21 1⟩ (trisayo) in the 5-limit. This temperament can be described as 29&46 temperament, which tempers out the hemifamity and 686/675 (senga). Alternative extension polypyth (46&121) tempers out the same 5-limit comma as the leapday, but with the porwell (6144/6125) rather than the hemifamity tempered out.
Subgroup: 2.3.5.7
Comma list: 686/675, 5120/5103
Mapping: [⟨1 0 -31 -21], ⟨0 1 21 15]]
Wedgie: ⟨⟨ 1 21 15 31 21 -24 ]]
POTE generator: ~3/2 = 704.263
Badness: 0.096123
11-limit
Subgroup: 2.3.5.7.11
Comma list: 121/120, 441/440, 686/675
Mapping: [⟨1 0 -31 -21 -14], ⟨0 1 21 15 11]]
POTE generator: ~3/2 = 704.250
Optimal GPV sequence: Template:Val list
Badness: 0.038624
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 121/120, 169/168, 352/351
Mapping: [⟨1 0 -31 -21 -14 -9], ⟨0 1 21 15 11 8]]
POTE generator: ~3/2 = 704.214
Optimal GPV sequence: Template:Val list
Badness: 0.024732
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 91/90, 121/120, 136/135, 154/153, 169/168
Mapping: [⟨1 0 -31 -21 -14 -9 -34], ⟨0 1 21 15 11 8 24]]
POTE generator: ~3/2 = 704.229
Optimal GPV sequence: Template:Val list
Badness: 0.017863
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168
Mapping: [⟨1 0 -31 -21 -14 -9 -34 9], ⟨0 1 21 15 11 8 24 -3]]
POTE generator: ~3/2 = 704.135
Optimal GPV sequence: Template:Val list
Badness: 0.017356
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 161/160, 169/168
Mapping: [⟨1 0 -31 -21 -14 -9 -34 9 -5], ⟨0 1 21 15 11 8 24 -3 6]]
POTE generator: ~3/2 = 704.141
Optimal GPV sequence: Template:Val list
Badness: 0.014065
Leapling
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168
Mapping: [⟨1 0 -31 -21 -14 -9 -34 -37], ⟨0 1 21 15 11 8 24 26]]
POTE generator: ~3/2 = 704.123
Optimal GPV sequence: Template:Val list
Badness: 0.019065
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 77/76, 91/90, 115/114, 121/120, 136/135, 153/152, 161/160
Mapping: [⟨1 0 -31 -21 -14 -9 -34 -37 -5], ⟨0 1 21 15 11 8 24 26 6]]
POTE generator: ~3/2 = 704.114
Optimal GPV sequence: Template:Val list
Badness: 0.016067
Mystery
- For the 5-limit version of this temperament, see High badness temperaments #Mystery.
Subgroup: 2.3.5.7
Comma list: 5120/5103, 50421/50000
Mapping: [⟨29 46 0 14], ⟨0 0 1 1]]
Wedgie: ⟨⟨ 0 29 29 46 46 -14 ]]
POTE generator: ~5/4 = 388.646
Badness: 0.103734
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 896/891, 3388/3375
Mapping: [⟨29 46 0 14 33], ⟨0 0 1 1 1]]
POTE generator: ~5/4 = 388.460
Optimal GPV sequence: Template:Val list
Badness: 0.034291
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 352/351, 364/363, 676/675
Mapping: [⟨29 46 0 14 33 40], ⟨0 0 1 1 1 1]]
POTE generator: ~5/4 = 388.354
Optimal GPV sequence: Template:Val list
Badness: 0.018591
Quanic
Subgroup: 2.3.5.7
Comma list: 5120/5103, 5832000/5764801
Mapping: [⟨1 1 -4 0], ⟨0 5 54 24]]
POTE generator: ~160/147 = 140.493
Badness: 0.179475
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1331/1323, 5120/5103
Mapping: [⟨1 1 -4 0 1], ⟨0 5 54 24 21]]
POTE generator: ~88/81 = 140.489
Optimal GPV sequence: Template:Val list
Badness: 0.058678
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 540/539, 729/728, 1331/1323
Mapping: [⟨1 1 -4 0 1 3], ⟨0 5 54 24 21 6]]
POTE generator: ~13/12 = 140.496
Optimal GPV sequence: Template:Val list
Badness: 0.032481
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 352/351, 442/441, 540/539, 715/714, 847/845
Mapping: [⟨1 1 -4 0 1 3 -2], ⟨0 5 54 24 21 6 52]]
POTE generator: ~13/12 = 140.497
Optimal GPV sequence: Template:Val list
Badness: 0.021112
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
Mapping: [⟨1 1 -4 0 1 3 -2 -5], ⟨0 5 54 24 21 6 52 79]]
POTE generator: ~13/12 = 140.496
Optimal GPV sequence: Template:Val list
Badness: 0.017273
Septiquarter
Subgroup: 2.3.5.7
Comma list: 5120/5103, 420175/419904
Mapping: [⟨1 3 10 2], ⟨0 -7 -38 4]]
Wedgie: ⟨⟨ 7 38 -4 44 -26 -116 ]]
POTE generator: ~147/128 = 242.453
Badness: 0.053760
Semiseptiquarter
Subgroup: 2.3.5.7.11
Comma list: 5120/5103, 9801/9800, 14641/14580
Mapping: [⟨2 6 20 4 15], ⟨0 -7 -38 4 -20]]
POTE generators: ~121/105 = 242.4511
Optimal GPV sequence: Template:Val list
Badness: 0.064160
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 847/845, 1716/1715, 14641/14580
Mapping: [⟨2 6 20 4 15 7], ⟨0 -7 -38 4 -20 1]]
POTE generators: ~121/105 = 242.4448
Optimal GPV sequence: Template:Val list
Badness: 0.034834
Tricot
The generator for tricot is the real cube root of third harmonic, 31/3, tuned between 63/44 and 13/9. Tricot can be described as 53&70 temperament (also called as "trimot", as confirmed by the data from x31eq), tempering out the tricot comma, [39 -29 3⟩ in the 5-limit, 2430/2401 (nuwell comma) and 5120/5103 in the 7-limit, 99/98 and 121/120 in the 11-limit, 169/168, 352/351, 640/637, and 729/728 in the 13-limit.
Subgroup: 2.3.5.7
Comma list: 2430/2401, 5120/5103
Mapping: [⟨1 0 -13 -3], ⟨0 3 29 11]]
Wedgie: ⟨⟨ 3 29 11 39 9 -56 ]]
POTE generator: ~81/56 = 634.026
Badness: 0.100127
11-limit
Subgroup: 2.3.5.7.11
Comma list: 99/98, 121/120, 5120/5103
Mapping: [⟨1 0 -13 -3 -5], ⟨0 3 29 11 16]]
POTE generator: ~63/44 = 634.027
Optimal GPV sequence: Template:Val list
Badness: 0.056134
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 99/98, 121/120, 169/168, 352/351
Mapping: [⟨1 0 -13 -3 -5 0], ⟨0 3 29 11 16 7]]
POTE generator: ~13/9 = 634.012
Optimal GPV sequence: Template:Val list
Badness: 0.032102
Countriton
- For the 5-limit version of this temperament, see High badness temperaments #Countriton.
Subgroup: 2.3.5.7
Comma list: 5120/5103, 7558272/7503125
Mapping: [⟨1 6 19 -7], ⟨0 -9 -34 20]]
Wedgie: ⟨⟨ 9 34 -20 33 -57 -142 ]]
POTE generator: ~1728/1225 = 588.582
Badness: 0.131191
11-limit
Subgroup: 2.3.5.7.11
Comma list: 176/175, 5120/5103, 41503/41472
Mapping: [⟨1 6 19 -7 27], ⟨0 -9 -34 20 -48]]
POTE generator: ~108/77 = 588.545
Optimal GPV sequence: Template:Val list
Badness: 0.084782
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 351/350, 847/845, 2197/2187
Mapping: [⟨1 6 19 -7 27 14], ⟨0 -9 -34 20 -48 -21]]
POTE generator: ~108/77 = 588.544
Optimal GPV sequence: Template:Val list
Badness: 0.042321
Ketchup
Subgroup: 2.3.5.7
Comma list: 5120/5103, 1071875/1062882
Mapping: [⟨2 3 4 6], ⟨0 4 15 -9]]
Wedgie: ⟨⟨ 8 30 -18 29 -51 -126 ]]
POTE generator: ~64/63 = ~81/80 = 25.719
Badness: 0.084538
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1331/1323, 2200/2187
Mapping: [⟨2 3 4 6 7], ⟨0 4 15 -9 -2]]
POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.693
Optimal GPV sequence: Template:Val list
Badness: 0.039555
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 352/351, 847/845, 1331/1323
Mapping: [⟨2 3 4 6 7 8], ⟨0 4 15 -9 -2 -14]]
POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.697
Optimal GPV sequence: Template:Val list
Badness: 0.024824
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 289/288, 325/324, 352/351, 385/384, 561/560
Mapping: [⟨2 3 4 6 7 8 8], ⟨0 4 15 -9 -2 -14 4]]
POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.701
Optimal GPV sequence: Template:Val list
Badness: 0.016591
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 190/189, 209/208, 289/288, 352/351, 385/384, 561/560
Mapping: [⟨2 3 4 6 7 8 8 9], ⟨0 4 15 -9 -2 -14 4 -12]]
POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.660
Optimal GPV sequence: Template:Val list
Badness: 0.018170
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 190/189, 209/208, 253/252, 289/288, 323/322, 352/351, 385/384
Mapping: [⟨2 3 4 6 7 8 8 9 9], ⟨0 4 15 -9 -2 -14 4 -12 1]]
POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.661
Optimal GPV sequence: Template:Val list
Badness: 0.014033