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.
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
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
- [[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
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
- [[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
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
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
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
- [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
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
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
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
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
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
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