Keenanismic family

The keenanismic family of rank-4 temperaments tempers out the keenanisma, the 11-limit comma 385/384 = [-7 -1 1 1 1⟩ of 4.503 cents.

Keenanismic

Subgroup: 2.3.5.7.11

Comma list: 385/384

Mapping: [⟨1 0 0 0 7], ⟨0 1 0 0 1], ⟨0 0 1 0 -1], ⟨0 0 0 1 -1]]

Generators: ~2, ~3, ~5, ~7

Transpose: [2 3 5 7 385/35]

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [0 1 0 0 0⟩, [7/3 1/3 2/3 -1/3 -1/3⟩, [7/3 1/3 -1/3 2/3 -1/3⟩, [7/3 1/3 -1/3 -1/3 2/3⟩]

Eigenmonzos: 2, 3, 7/5, 11/5

Optimal GPV sequence: 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de

Badness: 15.159 × 10^-9

Martwin

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384

Mapping: [⟨1 0 0 0 7 2], ⟨0 1 0 0 1 4], ⟨0 0 1 0 -1 -2], ⟨0 0 0 1 -1 0]]

Generators: ~2, ~3, ~5, ~7

Transpose: [2 3 5 7 385/35 324/25]

Minimax tuning: ^{[to be confirmed]}

[[1 0 0 0 0 0⟩, [0 1 0 0 0 0⟩, [2/3 4/3 1/3 0 0 -1/3⟩, [17/3 -5/3 1/3 0 -1 2/3⟩, [2/3 4/3 -2/3 0 1 -1/3⟩, [2/3 4/3 -2/3 0 0 2/3⟩]

Eigenmonzos: 2, 3, 11/5, 13/5

Optimal GPV sequence: 12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff

Badness: 2.206 × 10^-6

Ancient

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 625/624

Mapping: [⟨1 0 0 0 7 -4], ⟨0 1 0 0 1 -1], ⟨0 0 1 0 -1 4], ⟨0 0 0 1 -1 0]]

Generators: ~2, ~3, ~5, ~7

Transpose: [2 3 5 7 385/35 625/48]

Optimal GPV sequence: 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def

Badness: 2.573 × 10^-6