Cataharry family

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The cataharry family of rank-3 temperaments tempers out the cataharry comma, 19683/19600.

Cataharry

Subgroup: 2.3.5.7

Comma list: 19683/19600

Mapping: [1 0 0 -2], 0 2 0 9], 0 0 1 -1]]

Mapping generators: ~2, ~140/81, ~5

Minimax tuning:

[[1 0 0 0, [4/9 0 2/9 2/9, [0 0 1 0, [0 0 0 1]
Eigenmonzos: 2, 8/7, 5/4

Vals19, 34d, 39d, 53, 72, 130, 183, 202, 385, 587

Badness: 0.301 × 10-3

Projection pairs: 3 19600/6561 7 1033052339200000000/150094635296999121 to 2.5.81/7

Albus

Subgroup: 2.3.5.7.11

Comma list: 540/539, 8019/8000

Mapping: [1 0 0 -2 6], 0 2 0 9 -12], 0 0 1 -1 3]]

Mapping generators: ~2, ~140/81, ~5

Minimax tuning:

[[1 0 0 0 0, [8/3 0 4/3 -2/3 -2/3, [5 0 7/2 -2 -3/2, [5 0 5/2 -1 -3/2, [5 0 5/2 -2 -1/2]
Eigenmonzos: 2, 11/10, 7/5

Vals19, 33cd, 39d, 53, 58, 72, 130, 183, 255, 385, 640

Badness: 0.621 × 10-3

Madagascar

Subgroup: 2.3.5.7.11.13

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

Mapping: [1 0 0 -2 6 -1], 0 2 0 9 -12 3], 0 0 1 -1 3 1]]

Mapping generators: ~2, ~26/15, ~5

Map to lattice: [0 -2 -3 -6 3 -6], 0 0 -1 1 -3 -1]]

Lattice basis:

15/13 length = 0.3019, 26/25 length = 0.9477
Angle (15/13, 26/25) = 81.6476

Minimax tuning:

  • 13- and 15-odd-limit
[[1 0 0 0 0 0, [8/3 0 4/3 -2/3 -2/3 0, [5 0 7/2 -2 -3/2 0, [5 0 5/2 -1 -3/2 0, [5 0 5/2 -2 -1/2 0, [8 0 11/2 -3 -5/2 0]
Eigenmonzos: 2, 11/10, 7/5

Vals: 19, 33cdff, 39df, 53, 58, 72, 111, 130, 183, 313

Badness: 0.560 × 10-3

Complexity spectrum: 15/13, 4/3, 13/10, 10/9, 6/5, 9/7, 18/13, 9/8, 5/4, 7/6, 13/12, 15/14, 16/15, 14/13, 8/7, 7/5, 16/13, 11/10, 15/11, 11/8, 12/11, 13/11, 11/9, 14/11

Projection pairs: 3 676/225 7 1357375919744/192216796875 11 16218292236328125/1491077447838784 13 8788/675 to 2.5.13/3

Scales: madagascar19

Snape

Subgroup: 2.3.5.7.11

Comma list: 441/440, 19683/19600

Mapping: [1 0 0 -2 -7], 0 2 0 9 22], 0 0 1 -1 -3]]

Mapping generators: ~2, ~140/81, ~5

Vals19e, 33cdee, 39dee, 53e, 58, 72, 130, 202, 351, 423d, 481, 553

Badness: 0.988 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 676/675

Mapping: [1 0 0 -2 -7 -1], 0 2 0 9 22 3], 0 0 1 -1 -3 1]]

Mapping generators: ~2, ~26/15, ~5

Vals: 19e, 33cdeeff, 39deef, 53e, 58, 72, 130, 279, 409

Badness: 0.786 × 10-3

Hagrid

Subgroup: 2.3.5.7.11

Comma list: 243/242, 9801/9800

Mapping: [2 0 0 -4 -1], 0 2 0 9 5], 0 0 1 -1 0]]

Mapping generators: ~99/70, ~140/81, ~5

Vals24, 34d, 48, 58, 72, 130, 202

Badness: 1.116 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 364/363

Mapping: [2 0 0 -4 -1 -2], 0 2 0 9 5 3], 0 0 1 -1 0 1]]

Mapping generators: ~99/70, ~26/15, ~5

Vals: 24, 34d, 58, 72, 130

Badness: 0.876 × 10-3

Draco

Subgroup: 2.3.5.7.11

Comma list: 176/175, 19683/19600

Mapping: [1 0 0 -2 -6], 0 2 0 9 9], 0 0 1 -1 1]]

Vals19e, 24, 34d, 53, 58, 111, 280cd

Badness: 1.934 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 676/675

Mapping: [1 0 0 -2 -6 -1], 0 2 0 9 9 3], 0 0 1 -1 1 1]]

Vals: 19e, 24, 34d, 53, 58, 111, 280cdf

Badness: 0.999 × 10-3