Hemifamity temperaments

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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:

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

Optimal GPV sequence5, 43c, 48, 53, 111, 164d, 275d

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: 53, 58, 111, 280cd, 391cd

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: 53, 58, 111, 280cdf, 391cdf

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: 53, 58, 111, 321cdfg

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: 53, 58h, 111

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: 5, 48, 53

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: 5, 48f, 53

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

Optimal GPV sequence29, 70, 99, 722bc, 821bc, 920bc, 1019bc, 1118bbcc, 1217bbcc, 1316bbccd

Badness: 0.094603

Leapday

For the 5-limit version of this temperament, see High badness temperaments #Leapday.
Not to be confused with calendar-based temperaments modes such as those in 293edo, 400edo, 353edo 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

Optimal GPV sequence17c, 29, 46, 167d, 213d, 259cdd, 305bcdd

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: 17c, 29, 46, 167de, 213de, 259cdde

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: 17c, 29, 46, 121def, 167def, 213deff

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: 17cg, 29g, 46, 121defg, 167defg, 213deffg

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: 17cg, 29g, 46, 75dfgh, 121defgh

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: 17cg, 29g, 46, 75dfgh, 121defgh

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: 17cgh, 29g, 46h, 75dfg, 121defghh

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: 17cgh, 29g, 46h, 75dfg, 121defghh

Badness: 0.016067

Mystery

Main article: 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

Optimal GPV sequence29, 58, 87, 145

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: 29, 58, 87, 145

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

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

Optimal GPV sequence94, 111, 205

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: 94, 111, 205

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: 94, 111, 205

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: 94, 111, 205

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: 94, 111, 205

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

Optimal GPV sequence94, 99, 292, 391, 881bd, 1272bcd

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: 94, 198, 292, 490

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: 94, 198, 490f

Badness: 0.034834

Tricot

See also: Tricot family

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

Optimal GPV sequence17c, 36c, 53, 229dd, 282dd

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: 17c, 36ce, 53, 70, 123de

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: 17c, 36ce, 53, 70, 123de

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

Optimal GPV sequence53, 157, 210

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: 53, 104c, 157

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: 53, 104c, 157

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

Optimal GPV sequence46, 94, 140

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: 46, 94, 140

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: 46, 94, 140

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: 46, 94, 140

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: 46, 94, 140h, 234eh

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: 46, 94, 140h, 234ehi

Badness: 0.014033