Biyatismic clan
The biyatismic clan of rank-3 temperaments tempers out the biyatisma, 121/120 = [-3 -1 -1 0 2⟩.
Temperaments discussed elsewhere are:
- Sonic (+55/54 or 100/99) → Porcupine rank-3 family
- Urania (+81/80) → Didymus rank-3 family
- Big brother (+99/98) → Nuwell family
- Bisector (+245/243) → Sensamagic family
Considered below are zeus, artemis, oxpecker, aphrodite, and the no-7 subgroup temperament, protomere. For the rank-4 biyatismic temperament, see Rank-4 temperament #Biyatismic (121/120).
Protomere
Subgroup: 2.3.5.11
Comma list: 121/120
Sval mapping: [⟨1 0 1 2], ⟨0 1 1 1], ⟨0 0 -2 -1]]
- Mapping generators: ~2, ~3, ~11/10
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.4578, ~11/10 = 157.7466
Optimal ET sequence: 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee
Badness: 0.0297 × 10-3
Zeus
- Main article: Zeus
- See also: Porwell family #Zeus
Subgroup: 2.3.5.7.11
Comma list: 121/120, 176/175
Mapping: [⟨1 0 1 4 2], ⟨0 1 1 -1 1], ⟨0 0 -2 3 1]]
Mapping to lattice: [⟨0 1 -1 2 0], ⟨0 1 1 -1 1]]
Lattice basis:
- 11/10, 11/8
- Angle (11/10, 11/8) = 87.464 degrees
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.1530, ~11/10 = 157.0881
- [[1 0 0 0 0⟩, [11/9 10/9 -1/3 -2/9 0⟩, [22/9 2/9 1/3 -4/9 0⟩, [22/9 2/9 -2/3 5/9 0⟩, [10/3 2/3 0 -1/3 0⟩]
- eigenmonzo (unchanged-interval) basis: 2.9/5.9/7
Optimal ET sequence: 15, 22, 31, 46, 53, 68, 77, 99, 130e
Badness: 0.400 × 10-3
Projection pairs: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5
Zeus11[22] hobbit transversal
- 33/32, 16/15, 11/10, 8/7, 64/55, 77/64, 5/4, 14/11, 4/3,
- 11/8, 45/32, 16/11, 3/2, 11/7, 8/5, 5/3, 55/32, 7/4,
- 11/6, 15/8, 64/33, 2
Zeus11[24] hobbit transversal
- 33/32, 16/15, 11/10, 9/8, 8/7, 77/64, 11/9, 5/4, 21/16, 4/3,
- 11/8, 45/32, 16/11, 3/2, 32/21, 8/5, 18/11, 5/3, 7/4, 16/9,
- 11/6, 15/8, 64/33, 2
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 176/175, 351/350
Mapping: [⟨1 0 1 4 2 7], ⟨0 1 1 -1 1 -2], ⟨0 0 -2 3 -1 -1]]
Mapping to lattice: [⟨0 1 -1 2 0 -3], ⟨0 1 1 -1 1 -2]]
Lattice basis:
- 11/10 length = 0.7898, 11/8 length = 1.002
- Angle (11/10, 11/8) = 106.7439 degrees
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.8679, ~11/10 = 156.9582
Minimax tuning:
- 13-odd-limit
- [[1 0 0 0 0 0⟩, [11/9 10/9 -1/3 -2/9 0 0⟩, [22/9 2/9 1/3 -4/9 0 0⟩, [22/9 2/9 -2/3 5/9 0 0⟩, [10/3 2/3 0 -1/3 0 0⟩, [14/3 -8/3 1 1/3 0 0⟩]
- eigenmonzo (unchanged-interval) basis: 2.9/5.9/7
- 15-odd-limit
- [[1 0 0 0 0 0⟩, [0 1 0 0 0 0⟩, [11/5 1/5 2/5 -2/5 0 0⟩, [11/5 1/5 -3/5 3/5 0 0⟩, [13/5 3/5 1/5 -1/5 0 0⟩, [38/5 -12/5 1/5 -1/5 0 0⟩]
- eigenmonzo (unchanged-interval) basis: 2.3.7/5
Optimal ET sequence: 15, 22, 31, 46, 53, 77, 99, 130e
Badness: 0.934 × 10-3
Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5
Zeus13[22] hobbit transversal
- 260/243, 88/81, 11/10, 44/39, 162/143, 11/9, 16/13, 320/243, 4/3, 1040/729, 13/9, 729/520, 3/2, 99/65, 44/27, 18/11, 1280/729, 16/9, 11/6, 24/13, 243/130, 2
Tinia
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 121/120, 176/175
Mapping: [⟨1 0 1 4 2 2], ⟨0 1 1 -1 1 1], ⟨0 0 -2 3 -1 -1]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.3420, ~11/10 = 155.3666
Optimal ET sequence: 7, 9, 15, 22f, 24, 31
Badness: 0.808 × 10-3
Artemis
Subgroup: 2.3.5.7.11
Comma list: 121/120, 225/224
Mapping: [⟨1 0 1 -3 2], ⟨0 1 1 4 1], ⟨0 0 -2 -4 -1]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.8719, ~11/10 = 158.3232
Optimal ET sequence: 9, 15d, 16d, 20, 22, 31, 53, 82e, 84e, 113e, 144ee
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 121/120, 196/195
Mapping: [⟨1 0 1 -3 2 -5], ⟨0 1 1 4 1 6], ⟨0 0 -2 -4 -1 -6]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7090, ~11/10 = 158.7117
Optimal ET sequence: 9, 20, 22f, 29, 31
Diana
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 225/224, 275/273
Mapping: [⟨1 0 1 -3 2 7], ⟨0 1 1 4 1 -2], ⟨0 0 -2 -4 -1 -1]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.9789, ~11/10 = 159.0048
Optimal ET sequence: 22, 29, 31, 53, 82e, 84e, 113e, 166ee
Oxpecker
Subgroup: 2.3.5.7.11
Comma list: 121/120, 126/125
Mapping: [⟨1 0 1 2 2], ⟨0 1 1 1 1], ⟨0 0 -2 -6 -1]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.8882, ~11/10 = 155.7756
Optimal ET sequence: 7d, 8d, 15, 23de, 24d, 31, 46, 77
Badness: 0.699 × 10-3
Woodpecker
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 121/120, 126/125
Mapping: [⟨1 0 1 2 2 2], ⟨0 1 1 1 1 1], ⟨0 0 -2 -6 -1 1]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.5946, ~11/10 = 154.8652
Optimal ET sequence: 7d, 8d, 15, 23de, 24d, 31
Badness: 1.093 × 10-3
Aphrodite
Aphrodite tempers out the squalentine comma, 64827/64000, in the 7-limit. Its generators can be taken to be 2, 3, and 21/20, and it equates (21/20)3 with 8/7.
7-limit (squalentine)
Subgroup: 2.3.5.7
Comma list: 64827/64000
Mapping: [⟨1 0 1 3], ⟨0 1 1 0], ⟨0 0 -4 -3]]
- Mapping generators: ~2, ~3, ~21/20
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.2144, ~21/20 = 78.5694
Optimal ET sequence: 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d
Badness: 0.943 × 10-3
Projection pairs: 5 320000/64827 7 64000/9261 to 2.3.7/5
11-limit
Subgroup: 2.3.5.7.11
Comma list: 121/120, 441/440
Mapping: [⟨1 0 1 3 2], ⟨0 1 1 0 1], ⟨0 0 -4 -3 -2]]
- Mapping generators: ~2, ~3, ~22/21
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.3200, ~21/20 = 78.6421
Optimal ET sequence: 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee
Badness: 0.583 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 351/350, 441/440
Mapping: [⟨1 0 1 3 2 6], ⟨0 1 1 0 1 -1], ⟨0 0 -4 -3 -2 -11]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.1158, ~21/20 = 78.5211
Optimal ET sequence: 14cf, 31, 45ef, 46, 77, 122ee, 137def, 168deef
Badness: 1.456 × 10-3
Eros
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 196/195, 352/351
Mapping: [⟨1 0 1 3 2 7], ⟨0 1 1 0 1 -2], ⟨0 0 -4 -3 -2 -2]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.5014, ~21/20 = 78.6143
Optimal ET sequence: 17c, 29, 31, 46, 60e, 77, 106de, 183dee
Badness: 1.150 × 10-3
Inanna
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 121/120, 275/273
Mapping: [⟨1 0 1 3 2 1], ⟨0 1 1 0 1 2], ⟨0 0 -4 -3 -2 -7]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7754, ~21/20 = 79.6096
Optimal ET sequence: 14cf, 15, 29, 31, 45ef, 60e
Badness: 1.077 × 10-3
Ishtar
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 121/120, 441/440
Mapping: [⟨1 0 1 3 2 -1], ⟨0 1 1 0 1 3], ⟨0 0 -4 -3 -2 -1]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.3952, ~21/20 = 78.9578
Optimal ET sequence: 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def
Badness: 1.151 × 10-3