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
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
Named by Graham Breed in 2011, artemis was found to be locally efficient in the higher limits among rank-3 extensions of marvel[1], although it is a weak extension. However, the alternative 13-limit extension called diana is more accurate.
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
Eros fairs impressively into the 23-limit as a rank 3 temperament; not only is it fairly simple (considering this is a subgroup as complex as the full 23-limit, with many challenges) but all the generators are positive (or only 1 into the negatives in the case of the fifth) meaning it's even simpler than it might appear and has the pleasing property of all harmonics and subharmonics being "on the same side"; specifically: -3 to 1 fifths (2L 3s) and -5 to 0 ~23/22's will get you every prime, up to octave equivalence; you can think of this as a 5 by 6 grid if you like and is a recommendable place to start looking at its structure. Tempering the less accurate comma S11 can be seen as a consequence of tempering {S21, S22, S23} so is very natural and given its properties certainly excusable. Therefore characteristic of any good tuning is the ~11 being the most flat prime, with other primes having strictly less than 5 ¢ of error. This temperament was first logged on x31eq by Scott Dakota.
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
17-limit
Note that this extension requires the 29g val for 29edo, which has the sizes of 17/16 and 18/17 swapped.
Subgroup: 2.3.5.7.11.13.17
Comma list: 121/120, 154/153, 196/195, 352/351
Mapping: [⟨1 0 1 3 2 7 6], ⟨0 1 1 0 1 -2 -1], ⟨0 0 -4 -3 -2 -2 -5]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 701.9299, ~22/21 = 78.2539
- CWE: ~2 = 1\1, ~3/2 = 701.7925, ~22/21 = 78.6203
Optimal ET sequence: 17cg, 29g, 31, 46, 60e, 77, 106de
Badness:
- Smith: 0.979 × 10-3
- Dirichlet: 0.931
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 121/120, 154/153, 196/195, 286/285, 352/351
Mapping: [⟨1 0 1 3 2 7 6 9], ⟨0 1 1 0 1 -2 -1 -3], ⟨0 0 -4 -3 -2 -2 -5 0]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 701.5642, ~22/21 = 78.2353
- CWE: ~2 = 1\1, ~3/2 = 701.6963, ~22/21 = 78.6479
Optimal ET sequence: 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de
Badness:
- Smith: 1.13 × 10-3
- Dirichlet: 1.159
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 121/120, 154/153, 161/160, 196/195, 286/285, 352/351
Mapping: [⟨1 0 1 3 2 7 6 9 3], ⟨0 1 1 0 1 -2 -1 -3 1], ⟨0 0 -4 -3 -2 -2 -5 0 -1]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3 = 1901.7115, ~23/22 = 78.2054
- CWE: ~2 = 1\1, ~3 = 1901.8010, ~23/22 = 78.7188
Optimal ET sequence: 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de
Badness:
- Smith: 0.939 × 10-3
- Dirichlet: 1.084
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