Cataharry family

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:

• 7- and 9-odd-limit:

[|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

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:

• 11-odd-limit

[|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

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: Template:Val list

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

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 729/728

Hagrid

Subgroup: 2.3.5.7.11

Comma list: 243/242, 9801/9800

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 729/728

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|]

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: Template:Val list

Badness: 0.999 × 10^-3