Didymus rank-3 family

The didymus rank-3 family are rank-3 temperaments tempering out the didymus comma, 81/80. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.

Euterpe

Comma list: 81/80, 99/98

Minimax tuning:

• 11-odd-limit:

[[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [0 0 0 1 0⟩, [-1 0 -1/2 2 0⟩]

Eigenmonzos: 2, 5, 7

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

Badness: 0.536 × 10^-3

Calliope

Comma list: 81/80, 45/44

Minimax tuning:

• 11-odd-limit:

[[1 0 0 0 0⟩, [1 0 0 0 1/6⟩, [0 0 0 0 2/3⟩, [1 -1 0 1 1/6⟩, [0 0 0 0 1⟩]

Eigenmonzos: 2, 7/3, 11

Mapping: [⟨1 0 -4 0 -6], ⟨0 1 4 0 6], ⟨0 0 0 1 0]]

Badness: 0.530 × 10^-3

Erato

Comma list: 81/80, 126/125

Mapping: [⟨1 0 -4 -13 0], ⟨0 1 4 10 0], ⟨0 0 0 0 1]]

Badness: 0.558 × 10^-3

13-limit

Comma list: 81/80, 105/104, 126/125

Mapping: [⟨1 0 -4 -13 0 -20], ⟨0 1 4 10 0 15], ⟨0 0 0 0 1 0]]

Clio

Comma list: 81/80, 176/175

Minimax tuning

• 11-odd-limit:

[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [0 0 0 1 0⟩, [-4 0 2 1 0⟩]

Eigenmonzos: 2, 5, 7

Mapping: [⟨1 0 -4 0 -12], ⟨0 1 4 0 8], ⟨0 0 0 1 1]]

Badness: 0.738 × 10^-3

Polyhymnia

Comma list: 81/80, 385/384

Mapping: [⟨1 0 -4 0 11], ⟨0 1 4 0 -3], ⟨0 0 0 1 -1]]

Thalia

Comma list: 81/80, 55/54

Mapping: [⟨1 0 -4 0 5], ⟨0 1 4 0 -1], ⟨0 0 0 1 0]]

Melpomene

Comma list: 81/80, 56/55

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

Urania

Comma list: 81/80, 121/120

Mapping: [⟨1 1 0 0 2], ⟨0 2 8 0 5], ⟨0 0 0 1 0]]

Lattice basis:

11/9 length = 0.2536, 8/7 length = 2.807

Angle (11/9, 8/7) = 90 degrees

Map to lattice: [⟨0 2 8 0 5], ⟨0 0 0 -1 0]]

Badness: 0.842 × 10^-3

Complexity spectrum: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5

Scale: urania24

Terpsichore

Comma list: 81/80, 540/539

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

Badness: 0.850 × 10^-3

Complexity spectrum: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11