Biyatismic clan

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The biyatismic clan of rank-3 temperaments tempers out the biyatisma, 121/120 = [-3 -1 -1 0 2⟩.

Temperaments discussed elsewhere are:

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

Minimax tuning:

[[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

Eros fairs impressively into the 23-limit as a rank 3 temperament; not only is it fairly simple (considering this 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

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