Horwell family

The horwell family of rank-3 temperaments tempers out the horwell comma, 65625/65536 = [-16 1 5 1⟩.

Horwell

Subgroup: 2.3.5.7

Comma list: 65625/65536

Mapping: [⟨1 0 0 16], ⟨0 1 0 -1], ⟨0 0 1 -5]]

mapping generators: ~2, ~3, ~5

Mapping to lattice: [⟨0 -1 1 -4], ⟨0 -1 0 -1]]

Lattice basis:

5/4 length = 0.6213, 16/15 length = 1.3863

Angle (5/4, 16/15) = 64.6379

Minimax tuning:

• 7-odd-limit

[[1 0 0 0⟩, [16/7 6/7 -5/7 -1/7⟩, [16/7 -1/7 2/7 -1/7⟩, [16/7 -1/7 -5/7 6/7⟩]

eigenmonzo (unchanged-interval) basis: 2.5/3.7/3

• 9-odd-limit

[[1 0 0 0⟩, [16/13 12/13 -5/13 -1/13⟩, [32/13 -2/13 3/13 -2/13⟩, [32/13 -2/13 -10/13 11/13⟩]

eigenmonzo (unchanged-interval) basis: 2.9/5.9/7

Optimal ET sequence: 22, 31, 53, 84, 87, 118, 140, 171, 1592c, 1763c, 1934c, 2074c, 2105c, 2245cd, 2416cd, 2587cd, 2758cd, 2929cd, 3100cd, 3271ccd, 3442ccd, 3613ccd, 5034bcccdd, 5205bcccdd

Badness: 0.173 × 10^-3

Projection pair: 7 65536/9375

Zelda

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3388/3375

Mapping: [⟨1 0 0 16 -9], ⟨0 1 0 -1 2], ⟨0 0 1 -5 4]]

Optimal ET sequence: 22, 31, 53, 84, 87, 118, 258e, 376de, 547de

Badness: 0.642 × 10^-3

Horwellic

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 65625/65536

Mapping: [⟨1 0 0 16 10], ⟨0 1 0 -1 1], ⟨0 0 2 -10 -7]]

Optimal ET sequence: 31, 75e, 87, 118, 193, 224, 311, 342, 1592c, 1934ce, 2276cde, 2587cde, 2929cde, 3271ccde, 3613ccde, 5205bcccddee

Badness: 0.505 × 10^-3