Didymus rank three 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.

Temperaments discussed elsewhere include:

• Urania (+243/242 or 121/120) → Rastmic rank three clan

Didymus

Subgroup: 2.3.5.7

Comma list: 81/80

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.2387, ~7/4 = 964.9090

Optimal ET sequence: 12, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.1982, ~7/4 = 968.4280

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 (unchanged-intervals): 2, 5, 7

Optimal ET sequence: 12, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.1982, ~7/4 = 968.4280

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⟩]

Eigenmonzo (unchanged-interval) basis: 2.7/3.11

Optimal ET sequence: 7d, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.4949, ~11/8 = 547.0252

Optimal ET sequence: 12, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 695.9883, ~11/8 = 545.6817

Optimal ET sequence: 12f, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 697.2502, ~7/4 = 968.6295

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 (unchanged-intervals): 2, 5, 7

Optimal ET sequence: 7, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.2305, ~7/4 = 964.8695

Optimal ET sequence: 7, 12e, 19, 24, 26, 31

Scales: polyhymnia12

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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 692.0796, ~7/4 = 950.2565

Optimal ET sequence: 5, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.2230, ~7/4 = 964.2363

Optimal ET sequence: 7d, 12, 17c, 19, 24, 31e, 36

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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.2358, ~7/4 = 964.0006

Optimal ET sequence: 14c, 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