Cataharry family

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

Comma: 19683/19600

7 and 9 limit minimax

[|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

Map: [<1 0 0 -2|, <0 2 0 9|, <0 0 1 -1|]

Generators: 2, 140/81, 5

Edos: 19, 24, 48, 53, 58, 72, 130, 202, 457, 587, 659, 717

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

Albus

Commas: 540/539, 8019/8000

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

Map: [<1 0 0 -2 6|, <0 2 0 9 -12|, <0 0 1 -1 3|]

Generators: 2, 140/81, 5

Edos: 19, 53, 58, 72, 130, 183, 255, 385, 640

Badness: 0.000621

Madagascar

Commas: 351/350, 540/539, 676/675

13 and 15 limit minimax

[|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

Lattice basis: 15/13 0.3019 26/25 0.9477

Angle(15/13, 26/25) = 81.6476

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

Map: [<1 0 0 -2 6 -1|, <0 2 0 9 -12 3|, <0 0 1 -1 3 1|]

Generators: 2, 26/15, 5

Edos: 19, 53, 58, 72, 111, 130, 183, 313

Badness: 0.000560

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

madagascar19

Snape

Commas: 19683/19600, 441/440

13-limit

Commas: 351/350, 441/440, 729/728

Hagrid

Commas: 243/242, 9801/9800

13-limit

Commas: 243/242, 351/350, 729/728

Draco

Commas: 19683/19600, 176/175

Map: [<1 0 0 -2 -6|, <0 2 0 9 9|, <0 0 1 -1 1|]

EDOs: 24, 34d, 53, 58, 111, 357cd, 468cd

Badness: 0.001934

13-limit

Commas: 176/175, 351/350, 676/675

Map: [<1 0 0 -2 -6 -1|, <0 2 0 9 9 3|, <0 0 1 -1 1 1|]

EDOs: 24, 34d, 53, 58, 77, 111, 357cdf, 468cdf

Badness: 0.000999