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, garibaldi, hemififths, amity, misty, rodan, countercata and kwai.
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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Subgroup: 2.3.5
Comma list: 10737418240/10460353203
Mapping: [⟨1 0 -31], ⟨0 1 21]]
POTE generator: ~3/2 = 704.179
Badness: 0.523182
7-limit
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
Vals: 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
Vals: 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
Vals: 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
Vals: Template:Val list
Badness: 0.017356
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
Vals: Template:Val list
Badness: 0.019065
Mystery
Subgroup: 2.3.5
Comma list: [46 -29⟩
Mapping: [⟨29 46 0], ⟨0 0 1]]
Mapping generators: ~531441/524288, ~5
POTE generator: ~5/4 = 387.408
Badness: 1.020556
7-limit
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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: Template:Val list
Badness: 0.017273
Supers
Subgroup: 2.3.5.7
Comma list: 5120/5103, 118098/117649
Mapping: [⟨2 1 -12 2], ⟨0 3 23 5]]
Wedgie: ⟨⟨ 6 46 10 59 -1 -106 ]]
POTE generator: ~9/7 = 434.218
Badness: 0.092748
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4000/3993, 5120/5103
Mapping: [⟨2 1 -12 2 -9], ⟨0 3 23 5 22]]
POTE generator: ~9/7 = 434.217
Vals: Template:Val list
Badness: 0.028240
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 540/539, 729/728, 1575/1573
Mapping: [⟨2 1 -12 2 -9 -2], ⟨0 3 23 5 22 13]]
POTE generator: ~9/7 = 434.221
Vals: Template:Val list
Badness: 0.021645
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 170/169, 289/288, 352/351, 442/441, 561/560
Mapping: [⟨2 1 -12 2 -9 -2 6], ⟨0 3 23 5 22 13 3]]
POTE generator: ~9/7 = 434.181
Vals: Template:Val list
Badness: 0.021316
Alphaquarter
Subgroup: 2.3.5.7
Comma list: 5120/5103, 29360128/29296875
Mapping: [⟨1 2 2 0], ⟨0 -9 7 61]]
Wedgie: ⟨⟨ 9 -7 -61 -32 -122 -122 ]]
POTE generator: ~16128/15625 = 55.243
Badness: 0.116594
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4000/3993, 5120/5103
Mapping: [⟨1 2 2 0 3], ⟨0 -9 7 61 10]]
POTE generator: ~33/32 = 55.243
Vals: Template:Val list
Badness: 0.029638
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
Vals: 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
Vals: 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 known as trimot), 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
Comma: [39 -29 3⟩ = 68719476736000/68630377364883
Mapping: [<1 0 -13|, <0 3 29|]
POTE generator: ~59049/40960 = 634.012
Badness: 0.046093
7-limit (aka Trimot)
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
Vals: 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
Vals: Template:Val list
Badness: 0.032102
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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: Template:Val list
Badness: 0.014033
Undim
Subgroup: 2.3.5
Comma list: [41 -20 -4⟩ = 2199023255552/2179240250625
Mapping: [⟨4 0 41], ⟨0 1 -5]]
Mapping generators: ~1215/1024, ~3
POTE generator: ~3/2 = 702.736
Badness: 0.241703
7-limit
Subgroup: 2.3.5.7
Comma list: 5120/5103, 390625/388962
Mapping: [⟨4 0 41 81], ⟨0 1 -5 -11]]
Wedgie: ⟨⟨ 4 -20 -44 -41 -81 -46 ]]
POTE generator: ~3/2 = 702.736
Badness: 0.062754
11-limit
Subgroup: 2.3.5.7.11
Comma list: 1375/1372, 5120/5103, 5632/5625
Mapping: [⟨4 0 41 81 128], ⟨0 1 -5 -11 -18]]
POTE generator: ~3/2 = 702.689
Vals: Template:Val list
Badness: 0.034837