Mirkwai family

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The mirkwai family of planar temperaments tempers out the mirkwai comma, 16875/16807.

Mirkwai

Subgroup: 2.3.5.7

Comma list: 16875/16807 (mirkwai)

Mapping: [1 0 0 0], 0 1 3 3], 0 0 -5 -4]]

Mapping generators: ~2, ~3, ~7/5

Minimax tuning:

[[1 0 0 0, [0 4/7 -4/7 5/7, [0 -3/7 3/7 5/7, [0 0 0 1]
Eigenmonzos (unchanged intervals): 2, 8/7, 6/5
[[1 0 0 0, [0 8/11 -4/11 5/11, [0 -6/11 3/11 10/11, [0 0 0 1]
Eigenmonzos (unchanged intervals): 2, 10/9, 8/7

Vals31, 41, 72, 152, 224

Badness: 0.342 × 10-3

Projection pairs: 5 84375/16807 7 16875/2401 to 2.3.7/5

Indra

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372

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

Mapping generators: ~2, ~3, ~7/5

Map to lattice: [0 2 1 2 3], 0 -1 2 1 -3]]

Lattice basis:

15/14 length = 0.6184, 55/36 length = 0.7585
Angle (15/14, 55/36) = 68.1048 deg

Minimax tuning:

[[1 0 0 0 0, [10 10 -3 0 -5, [0 0 1 0 0, [6 6 -1 0 -3, [20 18 -6 0 -9]
Eigenmonzos (unchanged intervals): 2, 11/9, 5/4

Vals31, 41, 72, 152, 224, 631d, 855d

Badness: 0.266 × 10-3

Projection pairs: 5 84375/16807 7 16875/2401 11 1372/125 to 2.3.7/5

Scales: indra31

Shibi

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1375/1372

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

Vals: 31f, 39df, 41, 72, 111, 152f, 183, 224, 631d, 855d

Badness: 0.721 × 10-3

Yajna

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 1375/1372

Mapping: [1 0 0 0 2 -4], 0 1 3 3 0 11], 0 0 -5 -4 3 -20]]

Mapping generators: ~2, ~3, ~7/5

Minimax tuning:

  • 13-odd-limit
[[1 0 0 0 0 0, [4/5 6/5 -4/5 0 0 1/5, [8/5 2/5 -3/5 0 0 2/5, [44/25 26/25 -24/25 0 0 11/25, [62/25 48/25 -27/25 0 0 3/25, [8/5 2/5 -8/5 0 0 7/5]
Eigenmonzos (unchanged intervals): 2, 10/9, 13/10
  • 15-odd-limit minimax
[[1 0 0 0 0 0, [0 1 0 0 0 0, [5/4 1/2 0 -5/16 0 5/16, [1 1 0 -1/4 0 1/4, [5/4 3/2 0 3/16 0 -3/16, [1 1 0 -5/4 0 5/4]
Eigenmonzos (unchanged intervals): 2, 14/13, 4/3

Vals: 31, 49f, 72, 103, 121, 152f, 193, 224

Badness: 0.700 × 10-3

Vibhu

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 1375/1372

Mapping: [1 0 0 0 2 1], 0 1 3 3 0 6], 0 0 -5 -4 3 -14]]

Vals: 31, 64cdf, 72, 103, 111, 152, 175f, 183, 469f

Badness: 0.826 × 10-3

Vish

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 540/539

Mapping: [1 0 0 0 2 2], 0 1 3 3 0 -2], 0 0 -5 -4 3 10]]

Vals: 31f, 41, 72, 121, 193

Badness: 0.930 × 10-3