Vishnuzmic family

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The vishnuzmic family of temperaments tempers out the vishnuzma, [23 6 -14⟩ = 6115295232/6103515625, a 5-limit comma of about 3 cents. It is the amount by which seven just chromatic semitones of 25/24 fall short of a just perfect fourth of 4/3, or in other words (4/3)/(25/24)7.

Vishnu

Subgroup: 2.3.5

Comma list: 6115295232/6103515625

Mapping: [⟨2 4 5], ⟨0 -7 -3]]

mapping generators: ~78125/55296, ~25/24

Optimal tunings:

  • CTE: ~78125/55296 = 1\2, ~25/24 = 71.1556
  • POTE: ~78125/55296 = 1\2, ~25/24 = 71.143

Optimal ET sequence: 16, 34, 84, 118, 270, 388, 506, 894, 1400, 3306c, 4706cc

Badness: 0.031181

Septimal vishnu

Subgroup: 2.3.5.7

Comma list: 4375/4374, 29360128/29296875

Mapping: [⟨2 4 5 10], ⟨0 -7 -3 -37]]

Wedgie: ⟨⟨ 14 6 74 -23 78 155 ]]

Optimal tunings:

  • CTE: ~10125/7168 = 1\2, ~25/24 = 71.1175
  • POTE: ~10125/7168 = 1\2, ~25/24 = 71.107

Optimal ET sequence: 34d, 84d, 118, 152, 270, 1232, 1502, 1772c, 2042bc, 2312bc, 2582bc, 4894bbccd

Badness: 0.041912

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 5632/5625

Mapping: [⟨2 4 5 10 10], ⟨0 -7 -3 -37 -26]]

Optimal tunings:

  • CTE: ~99/70 = 1\2, ~25/24 = 71.1142
  • POTE: ~99/70 = 1\2, ~25/24 = 71.105

Optimal ET sequence: 34d, 84de, 118, 152, 270, 962, 1232, 1502, 2734bcd, 4236bbccdd

Badness: 0.014180

Ananta

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 3025/3024, 4096/4095, 4375/4374

Mapping: [⟨2 4 5 10 10 1], ⟨0 -7 -3 -37 -26 54]]

Optimal tuning:

  • CTE: ~99/70 = 1\2, ~25/24 = 71.1173
  • POTE: ~99/70 = 1\2, ~25/24 = 71.117

Optimal ET sequence: 118, 152, 270, 928, 1198, 1468

Badness: 0.023678

Acyuta

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 5632/5625

Mapping: [⟨2 4 5 10 10 17], ⟨0 -7 -3 -37 -26 -81]]

Optimal tunings:

  • CTE: ~99/70 = 1\2, ~25/24 = 71.1079
  • POTE: ~99/70 = 1\2, ~25/24 = 71.101

Optimal ET sequence: 118f, 152f, 270, 692, 962, 1654cd, 2616bcdf

Badness: 0.015805

Vishnean

Subgroup: 2.3.5.7

Comma list: 126/125, 540225/524288

Mapping: [⟨2 4 5 5], ⟨0 -7 -3 5]]

Wedgie: ⟨⟨ 14 6 -10 -23 -55 -40 ]]

Optimal tunings:

  • CTE: ~735/512 = 1\2, ~25/24 = 71.4956
  • POTE: ~735/512 = 1\2, ~25/24 = 71.779

Optimal ET sequence: 16, 34, 50, 134dd

Badness: 0.161398

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 1350/1331

Mapping: [⟨2 4 5 5 8], ⟨0 -7 -3 5 -9]]

Optimal tunings:

  • CTE: ~63/44 = 1\2, ~25/24 = 71.6218
  • POTE: ~63/44 = 1\2, ~25/24 = 71.924

Optimal ET sequence: 16, 34, 50

Badness: 0.071990

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 126/125, 144/143, 847/845

Mapping: [⟨2 4 5 5 8 8], ⟨0 -7 -3 5 -9 -5]]

Optimal tunings:

  • CTE: ~55/39 = 1\2, ~25/24 = 71.6369
  • POTE: ~55/39 = 1\2, ~25/24 = 71.926

Optimal ET sequence: 16, 34, 50

Badness: 0.040739

Narayana

Subgroup: 2.3.5.7

Comma list: 65625/65536, 321489/320000

Mapping: [⟨2 4 5 3], ⟨0 -7 -3 22]]

Wedgie: ⟨⟨ 14 6 -44 -23 -109 -119 ]]

Optimal tunings:

  • CTE: ~567/400 = 1\2, ~25/24 = 71.2706
  • POTE: ~567/400 = 1\2, ~25/24 = 71.265

Optimal ET sequence: 34, 84, 118, 202, 320, 522

Badness: 0.094295

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65625/65536

Mapping: [⟨2 4 5 3 3], ⟨0 -7 -3 22 33]]

Optimal tunings:

  • CTE: ~99/70 = 1\2, ~25/24 = 71.2609
  • POTE: ~99/70 = 1\2, ~25/24 = 71.248

Optimal ET sequence: 34e, 84, 118, 202, 320

Badness: 0.045908

Quarvish

Subgroup: 2.3.5.7

Comma list: 2401/2400, 6115295232/6103515625

Mapping: [⟨2 4 5 6], ⟨0 -28 -12 -13]]

Wedgie: ⟨⟨ 56 24 26 -92 -116 -7 ]]

Optimal tunings:

  • CTE: ~78125/55296 = 1\2, ~7/5 = 582.2115 (~110592/109375 = 17.7885)
  • POTE: ~78125/55296 = 1\2, ~7/5 = 582.213 (~110592/109375 = 17.787)

Optimal ET sequence: 68, 134, 202, 270, 742, 1012

Badness: 0.045029

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 172032/171875

Mapping: [⟨2 4 5 6 6], ⟨0 -28 -12 -13 31]]

Optimal tunings:

  • CTE: ~99/70 = 1\2, ~7/5 = 582.2122 (~99/98 = 17.7878)
  • POTE: ~99/70 = 1\2, ~7/5 = 582.213 (~99/98 = 17.787)

Optimal ET sequence: 68, 134, 202, 270, 742, 1012

Badness: 0.019868

Decavish

The decavish temperament (50 & 270) tempers out the wizma, [-6 -8 2 5⟩ = 420175/419904 and the linus comma, [11 -10 -10 10⟩ = 578509309952/576650390625.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 2202927104/2197265625

Mapping: [⟨10 6 19 14], ⟨0 7 3 10]]

mapping generators: ~15/14, ~54/49

Wedgie: ⟨⟨ 70 30 100 -115 -38 148 ]]

Optimal tunings:

  • CTE: ~15/14 = 1\10, ~54/49 = 168.8587 (~25/24 = 71.1413, or ~36/35 = 48.8587)
  • POTE: ~15/14 = 1\10, ~54/49 = 168.871 (~25/24 = 71.129, or ~36/35 = 48.871)

Optimal ET sequence: 50, …, 220, 270, 1400, 1670, 1940, 2210c, 2480c

Badness: 0.114531

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 41503/41472, 172032/171875

Mapping: [⟨10 6 19 14 36], ⟨0 7 3 10 -1]]

Optimal tunings:

  • CTE: ~15/14 = 1\10, ~54/49 = 168.8583 (~25/24 = 71.1417, or ~36/35 = 48.8583)
  • POTE: ~15/14 = 1\10, ~54/49 = 168.859 (~25/24 = 71.141, or ~36/35 = 48.859)

Optimal ET sequence: 50, …, 220, 270, 860, 1130, 1400

Badness: 0.036938

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 4225/4224, 4459/4455, 6656/6655

Mapping: [⟨10 6 19 14 36 37], ⟨0 7 3 10 -1 0]]

Optimal tunings:

  • CTE: ~15/14 = 1\10, ~54/49 = 168.8583 (~25/24 = 71.1417, or ~36/35 = 48.8583)
  • POTE: ~15/14 = 1\10, ~54/49 = 168.859 (~25/24 = 71.151, or ~36/35 = 48.859)

Optimal ET sequence: 50, 220, 270, 590, 860, 1130

Badness: 0.021906

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