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

See also: Vulture family

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

Commas: 5120/5103, 50421/50000

POTE generator: ~5/4 = 388.646

Map: [<29 46 0 14|, <0 0 1 1|]

Wedgie: <<0 29 29 46 46 -14||

EDOs: 29, 58, 87, 145

Badness: 0.1037

11-limit

Commas: 441/440, 896/891, 3388/3375

POTE generator: ~5/4 = 388.460

Map: [<29 46 0 14 33|, <0 0 1 1 1|]

EDOs: 29, 58, 87, 145

Badness: 0.0343

13-limit

Commas: 196/195, 352/351, 364/363, 676/675

POTE generator: ~5/4 = 388.354

Map: [<29 46 0 14 33 40|, <0 0 1 1 1 1|]

EDOs: 29, 58, 87, 145, 232, 377

Badness: 0.0186

Quanic

Commas: 5120/5103, 5832000/5764801

POTE generator: ~160/147 = 140.493

Map: [<1 1 -4 0|, <0 5 54 24|]

EDOs: 94, 111, 205

Badness: 0.1795

11-limit

Commas: 540/539, 1331/1323, 5120/5103

POTE generator: ~88/81 = 140.489

Map: [<1 1 -4 0 1|, <0 5 54 24 21|]

EDOs: 94, 111, 205

Badness: 0.0587

13-limit

Commas: 352/351, 540/539, 729/728, 1331/1323

POTE generator: ~13/12 = 140.496

Map: [<1 1 -4 0 1 3|, <0 5 54 24 21 6|]

EDOs: 94, 111, 205

Badness: 0.0325

17-limit

Commas: 352/351, 442/441, 540/539, 715/714, 847/845

POTE generator: ~13/12 = 140.497

Map: [<1 1 -4 0 1 3 -2|, <0 5 54 24 21 6 52|]

EDOs: 94, 111, 205

Badness: 0.0211

19-limit

Commas: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714

POTE generator: ~13/12 = 140.496

Map: [<1 1 -4 0 1 3 -2 -5|, <0 5 54 24 21 6 52 79|]

EDOs: 94, 111, 205

Badness: 0.0173

Supers

Commas: 5120/5103, 118098/117649

POTE generator: ~9/7 = 434.218

Map: [<2 1 -12 2|, <0 3 23 5|]

Wedgie: <<6 46 10 59 -1 -106||

EDOs: 58, 94, 152

Badness: 0.092748

11-limit

Commas: 540/539, 4000/3993, 5120/5103

POTE generator: ~9/7 = 434.217

Map: [<2 1 -12 2 -9|, <0 3 23 5 22|]

EDOs: 58, 94, 152

Badness: 0.028240

13-limit

Commas: 352/351, 540/539, 729/728, 1575/1573

POTE generator: ~9/7 = 434.221

Map: [<2 1 -12 2 -9 -2|, <0 3 23 5 22 13|]

EDOs: 58, 94, 152f

Badness: 0.021645

17-limit

Commas: 170/169, 289/288, 352/351, 442/441, 561/560

POTE generator: ~9/7 = 434.181

Map: [<2 1 -12 2 -9 -2 6|, <0 3 23 5 22 13 3|]

EDOs: 58, 94, 152f

Badness: 0.021316

Alphaquarter

Commas: 5120/5103, 29360128/29296875

POTE generator: ~16128/15625 = 55.243

Map: [<1 2 2 0|, <0 -9 7 61|]

Wedgie: <<9 -7 -61 -32 -122 -122||

EDOs: 87, 152, 239, 391

Badness: 0.1166

11-limit

Commas: 3025/3024, 4000/3993, 5120/5103

POTE generator: ~33/32 = 55.243

Map: [<1 2 2 0 3|, <0 -9 7 61 10|]

EDOs: 87, 152, 239, 391

Badness: 0.0296

Septiquarter

Commas: 5120/5103, 420175/419904

POTE generator: ~147/128 = 242.453

Map: [<1 3 10 2|, <0 -7 -38 4|]

Wedgie: <<7 38 -4 44 -26 -116||

EDOs: 94, 99, 292, 391, 881bd, 1272bcd

Badness: 0.0538

Semiseptiquarter

Commas: 5120/5103, 9801/9800, 14641/14580

POTE generators: ~121/105 = 242.4511

Map: [<2 6 20 4 15|, <0 -7 -38 4 -20|]

EDOs: 94, 198, 292, 490

Badness: 0.0642

13-limit

Commas: 352/351, 847/845, 1716/1715, 14641/14580

POTE generators: ~121/105 = 242.4448

Map: [<2 6 20 4 15 7|, <0 -7 -38 4 -20 1|]

EDOs: 94, 198, 490f

Badness: 0.0348

Tricot (aka Trimot)

See also: Tricot family

The generator for tricot is the real cube root of third harmonic, 3^1/3, 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.

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

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

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

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

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

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

Comma list: 289/288, 325/324, 352/351, 385/384, 561/560

Map: [<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

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

Commas: 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