Didymus rank three family

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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.

Didymus

Subgroup: 2.3.5.7

Comma list: 81/80

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

POTE generators: ~3/2 = 696.2387, ~7/4 = 964.9090

Vals12, 19, 31, 81

Badness: 0.095 × 10-3

Euterpe

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98

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

POTE generators: ~3/2 = 696.1982, ~7/4 = 968.4280

Minimax tuning:

[[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 (unchanged intervals): 2, 5, 7

Vals12, 17c, 19e, 26, 31, 88

Badness: 0.536 × 10-3

Calliope

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80

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

POTE generators: ~3/2 = 696.1982, ~7/4 = 968.4280

Minimax tuning:

[[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 (unchanged intervals): 2, 7/3, 11

Vals7d, 12, 19, 26, 45

Badness: 0.530 × 10-3

Erato

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125

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

POTE generators: ~3/2 = 696.4949, ~11/8 = 547.0252

Vals12, 19, 31, 50, 81

Badness: 0.558 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

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]]

POTE generators: ~3/2 = 695.9883, ~11/8 = 545.6817

Vals12f, 19, 31, 50, 81

Clio

Subgroup: 2.3.5.7.11

Comma list: 81/80, 176/175

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

POTE generators: ~3/2 = 697.2502, ~7/4 = 968.6295

Minimax tuning:

[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 (unchanged intervals): 2, 5, 7

Vals7, 12, 19e, 24, 31, 105, 129

Badness: 0.738 × 10-3

Polyhymnia

Subgroup: 2.3.5.7.11

Comma list: 81/80, 385/384

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

POTE generators: ~3/2 = 696.2305, ~7/4 = 964.8695

Vals7, 12e, 19, 24, 26, 31

Thalia

Subgroup: 2.3.5.7.11

Comma list: 33/32, 55/54

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

POTE generators: ~3/2 = 692.0796, ~7/4 = 950.2565

Vals5, 7, 12e, 14c, 19e

Melpomene

Subgroup: 2.3.5.7.11

Comma list: 81/80, 56/55

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

POTE generators: ~3/2 = 699.2230, ~7/4 = 964.2363

Vals7d, 12, 17c, 19, 24, 31e, 36

Urania

Subgroup: 2.3.5.7.11

Comma list: 81/80, 121/120

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

Map to lattice: [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

POTE generators: ~11/9 = 348.0938, ~7/4 = 963.6042

Vals7, 14c, 17c, 24, 31, 100de, 131bde, 162bde

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

Scales: urania24

Terpsichore

Subgroup: 2.3.5.7.11

Comma list: 81/80, 540/539

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

POTE generators: ~3/2 = 696.2358, ~7/4 = 964.0006

Vals14c, 17c, 19, 31, 81, 112b

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