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

Template:Val list

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

Template:Val list

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

Template:Val list

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

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

Template:Val list

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

Template:Val list

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

Alphaquarter

See also: Escapade family

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

Template:Val list

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

Template:Val list

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 (aka trimot)

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 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: 2.3.5.7

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

Template:Val list

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

Template:Val list

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

Template:Val list

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

Template:Val list

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

Quintakwai

See also: 28ed5 § Regular temperaments

The quintakwai temperament (12&193) tempers out the hemifamity comma (5120/5103) and the compass comma (9765625/9680832, quinruyoyo) in the 7-limit; 1375/1372 and 4375/4356 in the 11-limit. In the 2.3.5.7.17.19 subgroup, 225/224 (the difference between 15/14 and 16/15), 256/255 (between 16/15 and 17/16), 289/288 (between 17/16 and 18/17), 324/323 (between 18/17 and 19/18), and 361/360 (between 19/18 and 20/19) are equated together, and 400/399 (between 20/19 and 21/20) is tempered out. The name quintakwai is so named because the generator is 1/5 of the kwai fourth (~4/3, about 497.4 cents).


Subgroup: 2.3.5.7

Comma list: 5120/5103, 9765625/9680832

Mapping: [⟨1 2 0 -2], ⟨0 -5 28 58]]

Wedgie: ⟨⟨ 5 -28 -58 -56 -106 -56 ]]

POTE generator: ~625/588 = 99.483

Template:Val list

Badness: 0.155536

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4375/4356, 5120/5103

Mapping: [⟨1 2 0 -2 -4], ⟨0 -5 28 58 90]]

POTE generator: ~35/33 = 99.472

Vals: Template:Val list

Badness: 0.073158

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 1375/1372, 1575/1573, 4096/4095

Mapping: [⟨1 2 0 -2 -4 10], ⟨0 -5 28 58 90 -76]]

POTE generator: ~35/33 = 99.468

Vals: Template:Val list

Badness: 0.062737

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 375/374, 595/594, 1275/1274, 4096/4095

Mapping: [⟨1 2 0 -2 -4 10 5], ⟨0 -5 28 58 90 -76 -11]]

POTE generator: ~18/17 = 99.469

Vals: Template:Val list

Badness: 0.037855

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 375/374, 400/399, 595/594, 1216/1215, 1275/1274

Mapping: [⟨1 2 0 -2 -4 10 5 4], ⟨0 -5 28 58 90 -76 -11 3]]

POTE generator: ~18/17 = 99.469

Vals: Template:Val list

Badness: 0.025861

Quinkwai

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 1375/1372, 4375/4356

Mapping: [⟨1 2 0 -2 -4 -5], ⟨0 -5 28 58 90 105]]

POTE generator: ~35/33 = 99.456

Vals: Template:Val list

Badness: 0.061873

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 375/374, 595/594, 833/832, 1375/1372

Mapping: [⟨1 2 0 -2 -4 -5 5], ⟨0 -5 28 58 90 105 -11]]

POTE generator: ~18/17 = 99.458

Vals: Template:Val list

Badness: 0.043506

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 352/351, 375/374, 400/399, 495/494, 595/594, 1375/1372

Mapping: [⟨1 2 0 -2 -4 -5 5 4], ⟨0 -5 28 58 90 105 -11 3]]

POTE generator: ~18/17 = 99.459

Vals: Template:Val list

Badness: 0.031313

Quintakwoid

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 625/624, 1375/1372, 5120/5103

Mapping: [⟨1 2 0 -2 -4 -6], ⟨0 -5 28 58 90 117]]

POTE generator: ~35/33 = 99.484

Vals: Template:Val list

Badness: 0.057357

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 375/374, 442/441, 595/594, 5120/5103

Mapping: [⟨1 2 0 -2 -4 -6 5], ⟨0 -5 28 58 90 117 -11]]

POTE generator: ~18/17 = 99.484

Vals: Template:Val list

Badness: 0.039289

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215

Mapping: [⟨1 2 0 -2 -4 -6 5 4], ⟨0 -5 28 58 90 117 -11 3]]

POTE generator: ~18/17 = 99.485

Vals: Template:Val list

Badness: 0.028101

Quinticosiennic

See also: 16ed5/2 § Regular temperaments

The quinticosiennic temperament (12&145) tempers out the hemifamity comma (5120/5103) and 395136/390625 (trizo-aquadbigu) in the 7-limit; 441/440 (werckisma), 896/891 (pentacircle), and 78408/78125 (lolosepgu) in the 11-limit. The word "quinticosiennic" means 5 (quintuple) × 29 (είκοσι εννέα) = 145, and so named because 1/5 of 29EDO fourth, i.e. 12\145, is a possible generator.


Subgroup: 2.3.5.7

Comma list: 5120/5103, 395136/390625

Mapping: [⟨1 2 1 -1], ⟨0 -5 16 46]]

Wedgie: ⟨⟨ 5 -16 -46 -37 -87 -62 ]]

POTE generator: ~135/128 = 99.345

Template:Val list

Badness: 0.158041

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891, 78408/78125

Mapping: [⟨1 2 1 -1 -2], ⟨0 -5 16 46 66]]

POTE generator: ~35/33 = 99.318

Vals: Template:Val list

Badness: 0.080674

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363, 78408/78125

Mapping: [⟨1 2 1 -1 -2 -3], ⟨0 -5 16 46 66 81]]

POTE generator: ~35/33 = 99.307

Vals: Template:Val list

Badness: 0.052464

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 256/255, 352/351, 364/363, 3757/3750

Mapping: [⟨1 2 1 -1 -2 -3 5], ⟨0 -5 16 46 66 81 -11]]

POTE generator: ~18/17 = 99.308

Vals: Template:Val list

Badness: 0.037108

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 196/195, 256/255, 352/351, 361/360, 364/363, 476/475

Mapping: [⟨1 2 1 -1 -2 -3 5 4], ⟨0 -5 16 46 66 81 -11 3]]

POTE generator: ~18/17 = 99.303

Vals: Template:Val list

Badness: 0.028440

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