Xenharmonic Wiki

Porwell temperaments

Revision as of 23:55, 12 May 2021 by Xenllium (talk | contribs) (→Countermiracle)

This family of temperaments tempers out the porwell comma, [11 1 -3 -2⟩ = 6144/6125, and includes hendecatonic, hemischis, twothirdtonic, nessafof, septisuperfourth, whoops, and polypyth.

Discussed elsewhere are:

Hendecatonic

The hendecatonic temperament has a period of 1/11 octave, which represents 16/15 and four times of which represents 9/7.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 10976/10935

Mapping: [⟨11 0 43 -4], ⟨0 1 -1 2]]

Wedgie: ⟨⟨ 11 -11 22 -43 4 82 ]]

POTE generator: ~3/2 = 703.054

Template:Val list

Badness: 0.041081

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 10976/10935

Mapping: [⟨11 0 43 -4 38], ⟨0 1 -1 2 0]]

POTE generator: ~3/2 = 702.636

Vals: Template:Val list

Badness: 0.046088

Icosidillic

Subgroup: 2.3.5.7.11

Comma list: 3388/3375, 6144/6125, 9801/9800

Mapping: [⟨22 0 86 -8 111], ⟨0 1 -1 2 -1]]

POTE generator: ~3/2 = 702.914

Vals: Template:Val list

Badness: 0.057725

Hemischis

See also: Schismatic family

Subgroup: 2.3.5.7

Comma list: 6144/6125, 19683/19600

Mapping: [⟨1 0 15 -17], ⟨0 2 -16 25]]

Wedgie: ⟨⟨ 2 -16 25 -30 34 103 ]]

POTE generator: ~81/70 = 249.203

Template:Val list

Badness: 0.045817

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 8019/8000, 5632/5625

Mapping: [⟨1 0 15 -17 51], ⟨0 2 -16 25 -60]]

POTE generator: ~81/70 = 249.199

Vals: Template:Val list

Badness: 0.036289

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675, 4096/4095

Mapping: [⟨1 0 15 -17 51 14], ⟨0 2 -16 25 -60 -13]]

POTE generator: ~15/13 = 249.199

Vals: Template:Val list

Badness: 0.020816

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095

Map: [⟨1 0 15 -17 51 14 -49], ⟨0 2 -16 25 -60 -13 67]]

POTE generator: ~15/13 = 249.190

Vals: Template:Val list

Badness: 0.021073

Twothirdtonic

Subgroup: 2.3.5.7

Comma list: 686/675, 6144/6125

Mapping: [⟨1 3 2 4], ⟨0 -13 3 -11]]

Wedgie: ⟨⟨ 13 -3 11 -35 -19 34 ]]

POTE generator: ~15/14 = 130.401

Template:Val list

Badness: 0.099601

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 686/675

Mapping: [⟨1 3 2 4 4], ⟨0 -13 3 -11 -5]]

POTE generator: ~15/14 = 130.430

Vals: Template:Val list

Badness: 0.040768

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 169/168, 176/175

Mapping: [⟨1 3 2 4 4 5], ⟨0 -13 3 -11 -5 -12]]

POTE generator: ~15/14 = 130.409

Vals: Template:Val list

Badness: 0.025941

Nessafof

Subgroup: 2.3.5.7

Comma list: 6144/6125, 250047/250000

Mapping: [⟨3 2 5 10], ⟨0 7 5 -4]]

Wedgie: ⟨⟨ 21 15 -12 -25 -78 -70 ]]

POTE generator: ~35/32 = 157.480

Template:Val list

Badness: 0.045048

Septisuperfourth

Subgroup: 2.3.5.7

Comma list: 6144/6125, 118098/117649

Mapping: [⟨2 4 4 7], ⟨0 -9 7 -15]]

Wedgie: ⟨⟨ 18 -14 30 -64 -3 109 ]]

POTE generator: ~48/35 = 544.680

Template:Val list

Badness: 0.059241

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4000/3993, 5632/5625

Mapping: [⟨2 4 4 7 6], ⟨0 -9 7 -15 10]]

POTE generator: ~48/35 = 544.696

Vals: Template:Val list

Badness: 0.024619

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4000/3993, 21168/21125

Mapping: [⟨2 4 4 7 6 11], ⟨0 -9 7 -15 10 -39]]

POTE generator: ~48/35 = 544.675

Vals: Template:Val list

Badness: 0.022887

Septisuperquad

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 540/539, 5632/5625

Mapping: [⟨2 4 4 7 6 5], ⟨0 -9 7 -15 10 26]]

POTE generator: ~48/35 = 544.641

Vals: Template:Val list

Badness: 0.033038

Whoops

See also: Very high accuracy temperaments § Whoosh

Subgroup: 2.3.5.7

Comma list: 6144/6125, 244140625/243045684

Mapping: [⟨1 17 14 -7], ⟨0 -33 -25 21]]

Wedgie: ⟨⟨ 33 25 -21 -37 -126 -119 ]]

POTE generator: ~441/320 = 560.519

Template:Val list

Badness: 0.1758

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 6144/6125

Mapping: [⟨1 17 14 -7 10], ⟨0 -33 -25 21 -14]]

POTE generator: ~242/175 = 560.519

Vals: Template:Val list

Badness: 0.0437

Polypyth

Subgroup: 2.3.5.7

Comma list: 6144/6125, 179200/177147

Mapping: [⟨1 0 -31 52], ⟨0 1 21 -31]]

POTE generator: ~3/2 = 704.174

Template:Val list

Badness: 0.137995

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 2200/2187, 6144/6125

Mapping: [⟨1 0 -31 52 59], ⟨0 1 21 -31 -35]]

POTE generator: ~3/2 = 704.177

Vals: Template:Val list

Badness: 0.051131

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363, 1716/1715

Mapping: [⟨1 0 -31 52 59 64], ⟨0 1 21 -31 -35 -38]]

POTE generator: ~3/2 = 704.168

Vals: Template:Val list

Badness: 0.030292

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 325/324, 352/351, 364/363, 1716/1715

Mapping: [⟨1 0 -31 52 59 64 39], ⟨0 1 21 -31 -35 -38 -22]]

POTE generator: ~3/2 = 704.168

Vals: Template:Val list

Badness: 0.019051

Icositritonic

The icositritonic temperament (46&161, named by Xenllium) has a period of 1/23 octave, so six period represents 6/5 and nine period represents 21/16.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 9920232/9765625

Mapping: [⟨23 37 54 64], ⟨0 -1 -1 1]]

Wedgie: ⟨⟨ 23 23 -23 -17 -101 -118 ]]

POTE generator: ~64/63 = 29.3586

Template:Val list

Badness: 0.196622

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 6144/6125, 35937/35840

Mapping: [⟨23 37 54 64 79], ⟨0 -1 -1 1 1]]

POTE generator: ~64/63 = 29.3980

Vals: Template:Val list

Badness: 0.064613

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 847/845, 3584/3575

Mapping: [⟨23 37 54 64 79 84], ⟨0 -1 -1 1 1 2]]

POTE generator: ~64/63 = 29.2830

Vals: Template:Val list

Badness: 0.040484

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 441/440, 561/560, 847/845, 1089/1088

Mapping: [⟨23 37 54 64 79 84 94], ⟨0 -1 -1 1 1 2 0]]

POTE generator: ~64/63 = 29.2800

Vals: Template:Val list

Badness: 0.024676

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 351/350, 441/440, 456/455, 476/475, 513/512, 847/845

Mapping: [⟨23 37 54 64 79 84 94 96], ⟨0 -1 -1 1 1 2 0 3]]

POTE generator: ~64/63 = 29.3760

Vals: Template:Val list

Badness: 0.021579

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 276/275, 351/350, 391/390, 441/440, 456/455, 476/475, 847/845

Mapping: [⟨23 37 54 64 79 84 94 96 104], ⟨0 -1 -1 1 1 2 0 3 0]]

POTE generator: ~64/63 = 29.3471

Vals: Template:Val list

Badness: 0.017745

Countermiracle

The countermiracle temperament (31&145, named by Xenllium) tempers out the trimyna, 50421/50000 and the quince comma, 823543/819200.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 50421/50000

Mapping: [⟨1 4 3 3], ⟨0 -25 -7 -2]]

Wedgie: ⟨⟨ 25 7 2 -47 -67 -15 ]]

POTE generator: ~343/320 = 115.917

Template:Val list

Badness: 0.102326

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3388/3375, 6144/6125

Mapping: [⟨1 4 3 3 8], ⟨0 -25 -7 -2 -47]]

POTE generator: ~77/72 = 115.916

Vals: Template:Val list

Badness: 0.039162

Counterrevelation

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 50421/50000

Mapping: [⟨1 4 3 3 5], ⟨0 -25 -7 -2 -16]]

POTE generator: ~343/320 = 115.919

Vals: Template:Val list

Badness: 0.064070

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