Cataharry family
The cataharry family of rank-3 temperaments tempers out the cataharry comma, 19683/19600.
Cataharry
Subgroup: 2.3.5.7
Mapping: [⟨1 0 0 -2], ⟨0 2 0 9], ⟨0 0 1 -1]]
- mapping generators: ~2, ~140/81, ~5
- 7- and 9-odd-limit:
- [[1 0 0 0⟩, [4/9 0 2/9 2/9⟩, [0 0 1 0⟩, [0 0 0 1⟩]
- eigenmonzo (unchanged-interval) basis: 2.5.7
Optimal ET sequence: 19, 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
- [[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⟩]
- eigenmonzo (unchanged-interval) basis: 2.7/5.11/5
Optimal ET sequence: 19, 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 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⟩]
- eigenmonzo (unchanged-interval) basis: 2.7/5.11/5
Optimal ET sequence: 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]]
Optimal ET sequence: 19e, 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]]
Optimal ET sequence: 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
Optimal ET sequence: 24, 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
Optimal ET sequence: 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]]
Optimal ET sequence: 19e, 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]]
Optimal ET sequence: 19e, 24, 34d, 53, 58, 111, 280cdf
Badness: 0.999 × 10-3