Biyatismic clan

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The biyatismic clan of rank-3 temperaments tempers out the biyatisma, 121/120.

Temperaments discussed elsewhere are:

• Sonic (+55/54 or 100/99) → Porcupine rank-3 family
• Urania (+81/80) → Rastmic rank-3 clan
• Bisector (+245/243) → Sensamagic family

Considered below are zeus, artemis, oxpecker, big brother, 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

Subgroup-val mapping: [⟨1 0 1 2], ⟨0 1 1 1], ⟨0 0 -2 -1]]

mapping generators: ~2, ~3, ~11/10

Optimal tunings:

• WE: ~2 = 1200.6628 ¢, ~3/2 = 701.8452 ¢, ~11/10 = 157.8337 ¢

error map: ⟨+0.663 +0.553 +1.190 -5.318]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9848 ¢, ~11/10 = 157.6099 ¢

error map: ⟨0.000 +0.030 +0.451 -6.943]

Optimal ET sequence: 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee

Badness (Sintel): 0.245

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 tunings:

• WE: ~2 = 1200.1693 ¢, ~3/2 = 702.2521 ¢, ~11/10 = 157.1102 ¢

error map: ⟨+0.169 +0.466 +2.057 +0.761 -5.668]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2478 ¢, ~11/10 = 157.1265 ¢

error map: ⟨0.000 +0.293 +1.681 +0.306 -6.197]

Minimax tuning:

• 11-odd-limit

[[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⟩]

unchanged-interval (eigenmonzo) basis: 2.9/5.9/7

Optimal ET sequence: 15, 22, 31, 46, 53, 68, 77, 99, 130e

Badness (Sintel): 0.480

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 tunings:

• WE: ~2 = 1200.2411 ¢, ~3/2 = 702.0090 ¢, ~11/10 = 156.9897 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8818 ¢, ~11/10 = 156.9568 ¢

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⟩]

unchanged-interval (eigenmonzo) 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⟩]

unchanged-interval (eigenmonzo) basis: 2.3.7/5

Optimal ET sequence: 15f, 22, 31, 46, 53, 77, 99, 130e

Badness (Sintel): 0.873

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 tunings:

• WE: ~2 = 1199.9251 ¢, ~3/2 = 699.2984 ¢, ~11/10 = 155.3569 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.2982 ¢, ~11/10 = 155.3484 ¢

Optimal ET sequence: 7, 9, 15, 22f, 24, 31

Badness (Sintel): 0.756

Artemis

Named by Graham Breed in 2011, artemis was found to be locally efficient in the higher limits among rank-3 extensions of marvel, 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 tunings:

• WE: ~2 = 1201.2783 ¢, ~3/2 = 700.6174 ¢, ~11/10 = 158.4919 ¢

error map: ⟨+1.278 -0.059 -0.123 +0.955 -5.357]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.2777 ¢, ~11/10 = 158.3100 ¢

error map: ⟨0.000 -1.677 -2.656 -0.955 -9.350]

Optimal ET sequence: 9, 15d, 16d, 20, 22, 31, 53, 60e, 84e, 91e, 113e, 144ee

Badness (Sintel): 0.713

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 tunings:

• WE: ~2 = 1201.7896 ¢, ~3/2 = 699.7509 ¢, ~11/10 = 158.9484 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.1687 ¢, ~11/10 = 158.7345 ¢

Optimal ET sequence: 9, 20, 22f, 29, 31, 60e, 129cddee

Badness (Sintel): 1.04

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 tunings:

• WE: ~2 = 1200.9110 ¢, ~3/2 = 701.5110 ¢, ~11/10 = 159.1256 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.9717 ¢, ~11/10 = 158.7903 ¢

Optimal ET sequence: 22, 29, 31, 53, 82e, 84e, 113e

Badness (Sintel): 1.07

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 tunings:

• WE: ~2 = 1200.4124 ¢, ~3/2 = 701.1291 ¢, ~11/10 = 155.8292 ¢

error map: ⟨+0.412 -0.414 +3.982 -1.435 -4.781]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2359 ¢, ~11/10 = 155.7399 ¢

error map: ⟨0.000 -0.719 +3.442 -2.029 -5.822]

Optimal ET sequence: 7d, 8d, 15, 23de, 24d, 31, 46, 77

Badness (Sintel): 0.840

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 tunings:

• WE: ~2 = 1198.9113 ¢, ~3/2 = 700.9581 ¢, ~11/10 = 154.7247 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.6409 ¢, ~11/10 = 154.9115 ¢

Optimal ET sequence: 7d, 8d, 15, 23de, 24d, 31

Badness (Sintel): 1.02

Big brother

For the 7-limit version, see Miscellaneous 7-limit temperaments #Nuwell.

Subgroup: 2.3.5.7.11

Comma list: 99/98, 121/120

Mapping: [⟨1 0 -5 -1 -1], ⟨0 1 3 2 2], ⟨0 0 4 1 2]]

mapping generators: ~2, ~3, ~11/7

Optimal tunings:

• WE: ~2 = 1200.6559 ¢, ~3/2 = 700.2627 ¢, ~11/7 = 771.8821 ¢

error map: ⟨+0.656 -1.036 +0.691 +4.237 -6.372]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.4019 ¢, ~11/7 = 771.2671 ¢

error map: ⟨0.000 -1.553 -0.039 +3.245 -7.980]

Optimal ET sequence: 8d, 9, 14c, 17c, 22, 31, 53, 84e

Badness (Sintel): 0.609

Projection pairs: 5 2401/486, 11 98/9 to 2.3.7

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 121/120

Mapping: [⟨1 0 3 1 3 1], ⟨0 1 3 2 2 0], ⟨0 0 -4 -1 -2 2]]

Optimal tunings:

• WE: ~2 = 1199.0121 ¢, ~3/2 = 699.9867 ¢, ~11/7 = 771.9817 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.7360 ¢, ~11/7 = 773.0154 ¢

Optimal ET sequence: 8d, 9, 14c, 17c, 22f, 31, 79cf

Badness (Sintel): 0.889

Aphrodite

For the 7-limit version, see Miscellaneous 7-limit temperaments #Squalentine.

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 tunings:

• WE: ~2 = 1201.0691 ¢, ~3/2 = 700.9439 ¢, ~22/21 = 78.7122 ¢

error map: ⟨+1.069 +0.058 +1.920 -1.755 -4.591]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8990 ¢, ~22/21 = 78.4412 ¢

error map: ⟨0.000 -1.056 +0.820 -4.150 -7.301]

Optimal ET sequence: 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee

Badness (Sintel): 0.701

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 tunings:

• WE: ~2 = 1201.0656 ¢, ~3/2 = 700.7374 ¢, ~22/21 = 78.5908 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.7005 ¢, ~22/21 = 78.3253 ¢

Optimal ET sequence: 14cf, 29ff, 31, 45ef, 46, 77, 122ee, 137def, 168deef

Badness (Sintel): 1.36

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 tunings:

• WE: ~2 = 1200.6419 ¢, ~3/2 = 701.8766 ¢, ~22/21 = 78.6564 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.5612 ¢, ~22/21 = 78.4778 ¢

Optimal ET sequence: 17c, 29, 31, 46, 60e, 77, 106de, 183dee

Badness (Sintel): 1.08

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:

• WE: ~2 = 1200.6172 ¢, ~3/2 = 702.1026 ¢, ~22/21 = 78.7963 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7925 ¢, ~22/21 = 78.6203 ¢

Optimal ET sequence: 17cg, 29g, 31, 46, 60e, 77, 106de

Badness (Sintel): 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:

• WE: ~2 = 1200.6224 ¢, ~3/2 = 702.0959 ¢, ~22/21 = 78.8004 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6963 ¢, ~22/21 = 78.6479 ¢

Optimal ET sequence: 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de

Badness (Sintel): 1.16

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:

• WE: ~2 = 1200.7268 ¢, ~3/2 = 702.2463 ¢, ~22/21 = 78.8824 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8010 ¢, ~23/22 = 78.7188 ¢

Optimal ET sequence: 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de

Badness (Sintel): 1.08

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 tunings:

• WE: ~2 = 1201.7881 ¢, ~3/2 = 699.8166 ¢, ~22/21 = 79.7282 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.5926 ¢, ~22/21 = 79.3822 ¢

Optimal ET sequence: 14cf, 15, 29, 31, 45ef, 60e

Badness (Sintel): 1.01

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 tunings:

• WE: ~2 = 1200.7875 ¢, ~3/2 = 703.8568 ¢, ~22/21 = 79.0096 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.7640 ¢, ~22/21 = 78.8025 ¢

Optimal ET sequence: 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def

Badness (Sintel): 1.08