Hemifamity temperaments: Difference between revisions

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{{See also| 28ed5 #28ed5 as a generator }}
{{See also| 28ed5 #28ed5 as a generator }}


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 (the difference between 16/15 and 17/16), 289/288 (the difference between 17/16 and 18/17), 324/323 (the difference between 18/17 and 19/18), and 361/360 (the difference between 19/18 and 20/19) are equated together, and 400/399 (the difference between 20/19 and 21/20) is tempered out. The name ''quintakwai'' is so named because the generator is 1/5 of the [[Mirkwai clan #Kwai|kwai]] fourth (~4/3, about 497.4 cents).
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 (the difference between 20/19 and 21/20) is tempered out. The name ''quintakwai'' is so named because the generator is 1/5 of the [[Mirkwai clan #Kwai|kwai]] fourth (~4/3, about 497.4 cents).


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7

Revision as of 13:23, 28 June 2021

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

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

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

Comma list: 10737418240/10460353203

Mapping: [⟨1 0 -31], ⟨0 1 21]]

POTE generator: ~3/2 = 704.179

Template:Val list

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

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

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

Template:Val list

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

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

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

Template:Val list

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

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

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

Template:Val list

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

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 § 28ed5 as a generator

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 (the difference 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

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