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 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

Vals5, 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals29, 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 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

Vals17c, 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

Vals: 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

Vals: 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

Vals: 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

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

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

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

Vals29, 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

Vals: 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

Vals: 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

Vals94, 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

Vals: 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

Vals: 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

Vals: 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

Vals: 94, 111, 205

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

Vals58, 94, 152

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: 58, 94, 152

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: 58, 94, 152f

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: 58, 94, 152f

Badness: 0.021316

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

Vals87, 152, 239, 391

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: 87, 152, 239, 391

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

Vals94, 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

Vals: 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

Vals: 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 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

Vals53, 229, 282, 335, 388, 441, 1376, 1817, 2258

Badness: 0.046093

7-limit (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

Vals17c, 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

Vals: 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

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

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

Vals46, 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: 46, 94, 140h, 234ehi

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

Vals12, 104, 116, 128, 140, 152, 620, 772, 924c, 1076bc, 1228bc

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

Vals12, 128, 140, 152, 292

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: 12, 128e, 140, 152, 292, 444d, 596d

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

Vals12, 169, 181, 193

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: 12, 181, 193, 374, 567ce

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: 12, 181, 193, 374, 567ce, 941bce

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: 12, 181, 193, 374, 567ce, 941bceg

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: 12, 181, 193, 374, 567ce, 941bcegh, 1508bccdeegghh

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: 12f, 169e, 181

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: 12f, 169e, 181, 374ff, 555cff

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: 12f, 169e, 181, 374ff, 555cff

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: 12f, 181f, 193

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: 12f, 181f, 193

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: 12f, 181f, 193

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

Vals12, 133, 145, 157, 302c, 459bcc

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: 12, 133, 145

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: 12f, 133, 145

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: 12f, 133, 145

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: 12f, 133, 145

Badness: 0.028440