Biyatismic clan: Difference between revisions

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The '''biyatismic clan''' of rank-3 temperaments tempers out the [[biyatisma]], 121/120 = {{monzo| -3 -1 -1 0 2 }}.
{{Technical data page}}
The '''biyatismic clan''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the biyatisma, [[121/120]].


Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* ''[[Sonic]]'', {55/54, 100/99} → [[Porcupine rank three family #Sonic|Porcupine rank-3 family]]
* ''[[Sonic]]'' (+55/54 or 100/99) → [[Porcupine rank-3 family #Sonic|Porcupine rank-3 family]]
* ''[[Urania]]'', {81/80, 121/120} → [[Didymus rank three family #Urania|Didymus rank-3 family]]
* ''[[Urania]]'' (+81/80) → [[Rastmic rank-3 clan #Urania|Rastmic rank-3 clan]]
* ''[[Big Brother]]'', {99/98, 121/120} → [[Nuwell family #Big Brother|Nuwell family]]
* ''[[Bisector]]'' (+245/243) → [[Sensamagic family #Bisector|Sensamagic family]]
* ''[[Oxpecker]]'', {121/120, 126/125} → [[Starling family #Oxpecker|Starling family]]
* [[Zeus]], {121/120, 176/175} → [[Porwell family #Zeus|Porwell family]]
* ''[[Artemis]]'', {121/120, 225/224} → [[Marvel family #Artemis|Marvel family]]
* ''[[Bisector]]'', {121/120, 245/243} → [[Sensamagic family #Bisector|Sensamagic family]]


Considered below are aphrodite, and the no-7 subgroup temperament, protomere, but we can begin by looking at the rank-4 temperament.  
Considered below are zeus, artemis, oxpecker, kahoupokane, big brother, aphrodite, and the no-7 subgroup temperament, protomere. For the rank-4 biyatismic temperament, see [[Rank-4 temperament #Biyatismic (121/120)]].  


== Biyatismic ==
== Protomere ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.11
 
[[Comma list]]: 121/120
 
{{Mapping|legend=2| 1 0 1 2 | 0 1 1 1 | 0 0 -2 -1 }}
: mapping generators: ~2, ~3, ~11/10
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6628{{c}}, ~3/2 = 701.8452{{c}}, ~11/10 = 157.8337{{c}}
: [[error map]]: {{val| +0.663 +0.553 +1.190 -5.318 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9848{{c}}, ~12/11 = 157.6099{{c}}
: error map: {{val| 0.000 +0.030 +0.451 -6.943 }}
 
{{Optimal ET sequence|legend=1| 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee }}
 
[[Badness]] (Sintel): 0.245
 
== Zeus ==
{{Main| Zeus }}
{{See also| Porwell family #Zeus }}
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 121/120, 176/175
 
{{Mapping|legend=1| 1 0 1 4 2 | 0 1 1 -1 1 | 0 0 -2 3 1 }}
 
[[Mapping to lattice]]: [{{val| 0 1 -1 2 0 }}, {{val| 0 1 1 -1 1 }}]
 
Lattice basis:
: 11/10, 11/8
: Angle (11/10, 11/8) = 87.464 degrees
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1693{{c}}, ~3/2 = 702.2521{{c}}, ~12/11 = 157.1102{{c}}
: [[error map]]: {{val| +0.169 +0.466 +2.057 +0.761 -5.668 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2478{{c}}, ~12/11 = 157.1265{{c}}
: error map: {{val| 0.000 +0.293 +1.681 +0.306 -6.197 }}
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/9 10/9 -1/3 -2/9 0 }}, {{monzo| 22/9 2/9 1/3 -4/9 0 }}, {{monzo| 22/9 2/9 -2/3 5/9 0 }}, {{monzo| 10/3 2/3 0 -1/3 0 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.9/7
 
{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 68, 77, 99, 130e }}
 
[[Badness]] (Sintel): 0.480
 
[[Projection pair]]s: 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]]
Comma list: 121/120, 176/175, 351/350


[[Mapping]]: [{{val| 1 0 1 0 2 }}, {{val| 0 1 1 0 1 }}, {{val| 0 0 -2 0 -1 }}, {{val| 0 0 0 1 0 }}]
Mapping: {{mapping| 1 0 1 4 2 7 | 0 1 1 -1 1 -2 | 0 0 -2 3 -1 -1 }}


Mapping generators: ~2, ~3, ~11/10, ~7
Mapping to lattice: [{{val| 0 1 -1 2 0 -3 }}, {{val| 0 1 1 -1 1 -2 }}]


[[POTE generator]]s: ~3/2 = 701.4578, ~11/10 = 157.7466, ~7/4 = 966.9589
Lattice basis:  
: 11/10 length = 0.7898, 11/8 length = 1.002
: Angle (11/10, 11/8) = 106.7439 degrees


{{Val list|legend=1| 14c, 15, 22, 31, 46, 53, 60e, 68, 77, 91e, 99, 130e, 159ee, 190ee }}
Optimal tunings:
* WE: ~2 = 1200.2411{{c}}, ~3/2 = 702.0090{{c}}, ~12/11 = 156.9897{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8818{{c}}, ~12/11 = 156.9568{{c}}


[[Badness]]: 0.0345 × 10<sup>-6</sup>
Minimax tuning:
* 13-odd-limit
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 11/9 10/9 -1/3 -2/9 0 0 }}, {{monzo| 22/9 2/9 1/3 -4/9 0 0 }}, {{monzo| 22/9 2/9 -2/3 5/9 0 0 }}, {{monzo| 10/3 2/3 0 -1/3 0 0 }}, {{monzo| 14/3 -8/3 1 1/3 0 0 }}]
: unchanged-interval (eigenmonzo) basis: 2.9/5.9/7
* 15-odd-limit
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 0 1 0 0 0 0 }}, {{monzo| 11/5 1/5 2/5 -2/5 0 0 }}, {{monzo| 11/5 1/5 -3/5 3/5 0 0 }}, {{monzo| 13/5 3/5 1/5 -1/5 0 0 }}, {{monzo| 38/5 -12/5 1/5 -1/5 0 0 }}]
: unchanged-interval (eigenmonzo) basis: 2.3.7/5


== Protomere ==
{{Optimal ET sequence|legend=0| 15f, 22, 31, 46, 53, 77, 99, 130e }}
Subgroup: 2.3.5.11
 
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: {{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{{c}}, ~3/2 = 699.2984{{c}}, ~12/11 = 155.3569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2982{{c}}, ~12/11 = 155.3484{{c}}
 
{{Optimal ET sequence|legend=0| 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]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19673.html Yahoo! Tuning Group | ''Artemis and friends'']</ref>, 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|legend=1| 1 0 1 -3 2 | 0 1 1 4 1 | 0 0 -2 -4 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2783{{c}}, ~3/2 = 700.6174{{c}}, ~11/10 = 158.4919{{c}}
: [[error map]]: {{val| +1.278 -0.059 -0.123 +0.955 -5.357 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.2777{{c}}, ~11/10 = 158.3100{{c}}
: error map: {{val| 0.000 -1.677 -2.656 -0.955 -9.350 }}
 
{{Optimal ET sequence|legend=1| 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: {{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{{c}}, ~3/2 = 699.7509{{c}}, ~11/10 = 158.9484{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.1687{{c}}, ~11/10 = 158.7345{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{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{{c}}, ~3/2 = 701.5110{{c}}, ~11/10 = 159.1256{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.9717{{c}}, ~11/10 = 158.7903{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=1| 1 0 1 2 2 | 0 1 1 1 1 | 0 0 -2 -6 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4124{{c}}, ~3/2 = 701.1291{{c}}, ~12/11 = 155.8292{{c}}
: [[error map]]: {{val| +0.412 -0.414 +3.982 -1.435 -4.781 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2359{{c}}, ~12/11 = 155.7399{{c}}
: error map: {{val| 0.000 -0.719 +3.442 -2.029 -5.822 }}
 
{{Optimal ET sequence|legend=1| 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: {{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{{c}}, ~3/2 = 700.9581{{c}}, ~12/11 = 154.7247{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.6409{{c}}, ~12/11 = 154.9115{{c}}
 
{{Optimal ET sequence|legend=0| 7d, 8d, 15, 23de, 24d, 31 }}
 
Badness (Sintel): 1.02
 
== Kahoupokane ==
Named by [[Tristan Bay]] in 2025, Kahoupokane tempers out [[5120/5103]] and may be described as the {{nowrap| 29 & 46 & 53 }} temperament.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 121/120, 5120/5103
 
{{Mapping|legend=1| 1 0 1 11 2 | 0 1 1 -5 1 | 0 0 -2 -2 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1911{{c}}, ~3/2 = 703.1412{{c}}, ~11/10 = 158.1068{{c}}
: [[error map]]: {{val| +0.191 +1.377 +0.996 +0.401 -5.710 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.0417{{c}}, ~11/10 = 157.9917{{c}}
: error map: {{val| 0.000 +1.087 +0.744 -0.018 -6.268 }}
 
{{Optimal ET sequence|legend=1| 7, 17c, 24d, 29, 46, 53, 82e, 99 }}
 
[[Badness]] (Sintel): 2.73
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 169/168, 352/351
 
Mapping: {{mapping| 1 0 1 11 2 7 | 0 1 1 -5 1 -2 | 0 0 -2 -2 -1 -1 }}
 
Optimal tunings:
* WE: ~2 = 1200.4435{{c}}, ~3/2 = 703.1443{{c}}, ~11/10 = 158.4176{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9013{{c}}, ~11/10 = 158.1657{{c}}
 
{{Optimal ET sequence|legend=0| 7, 17c, 24d, 29, 46, 53, 82e, 99, 181eef }}
 
Badness (Sintel): 1.27
 
== Big brother ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Nuwell]].''


[[Comma list]]: 121/120
[[Subgroup]]: 2.3.5.7.11


[[Sval]] [[mapping]]: [{{val| 1 0 1 2 }}, {{val| 0 1 1 1 }}, {{val| 0 0 -2 -1 }}]
[[Comma list]]: 99/98, 121/120


Sval mapping generators: ~2, ~3, ~11/10
{{Mapping|legend=1| 1 0 -5 -1 -1 | 0 1 3 2 2 | 0 0 4 1 2 }}
: mapping generators: ~2, ~3, ~11/7


[[POTE generator]]s: ~3/2 = 701.4578, ~11/10 = 157.7466
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.6559{{c}}, ~3/2 = 700.2627{{c}}, ~11/7 = 771.8821{{c}}
: [[error map]]: {{val| +0.656 -1.036 +0.691 +4.237 -6.372 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4019{{c}}, ~11/7 = 771.2671{{c}}
: error map: {{val| 0.000 -1.553 -0.039 +3.245 -7.980 }}


{{Val list|legend=1| 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee }}
{{Optimal ET sequence|legend=1| 8d, 9, 14c, 17c, 22, 31, 53, 84e }}


[[Badness]]: 0.0297 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.609


== Aphrodite ==
[[Projection pair]]s: <code>5 2401/486, 11 98/9</code> to 2.3.7
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)<sup>3</sup> with 8/7.


=== 7-limit (squalentine) ===
=== 13-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 64827/64000
Comma list: 66/65, 99/98, 121/120


[[Mapping]]: [{{val| 1 0 1 3 }}, {{val| 0 1 1 0 }}, {{val| 0 0 4 3 }}]
Mapping: {{mapping| 1 0 3 1 3 1 | 0 1 3 2 2 0 | 0 0 -4 -1 -2 2 }}


Mapping generators: ~2, ~3, ~21/20
Optimal tunings:
* WE: ~2 = 1199.0121{{c}}, ~3/2 = 699.9867{{c}}, ~11/7 = 771.9817{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.7360{{c}}, ~11/7 = 773.0154{{c}}


{{Val list|legend=1| 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d }}
{{Optimal ET sequence|legend=0| 8d, 9, 14c, 17c, 22f, 31, 79cf }}


[[Badness]]: 0.943 × 10<sup>-3</sup>
Badness (Sintel): 0.889


[[Projection pair]]s: 5 320000/64827 7 64000/9261 to 2.3.7/5
== Aphrodite ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Squalentine]].''


=== 11-limit ===
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11


[[Comma list]]: 121/120, 441/440
[[Comma list]]: 121/120, 441/440


[[Mapping]]: [{{val| 1 0 1 3 2 }}, {{val| 0 1 1 0 1 }}, {{val| 0 0 4 3 2 }}]
{{Mapping|legend=1| 1 0 1 3 2 | 0 1 1 0 1 | 0 0 -4 -3 -2 }}
: mapping generators: ~2, ~3, ~22/21
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.0691{{c}}, ~3/2 = 700.9439{{c}}, ~22/21 = 78.7122{{c}}
: [[error map]]: {{val| +1.069 +0.058 +1.920 -1.755 -4.591 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.8990{{c}}, ~22/21 = 78.4412{{c}}
: error map: {{val| 0.000 -1.056 +0.820 -4.150 -7.301 }}
 
{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee }}
 
[[Badness]] (Sintel): 0.701


Mapping generators: ~2, ~3, ~22/21
=== 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 has the pleasing property that all the harmonics are on the negative side of the last generator. Specifically, -3 to 2 fifths and -5 to 0 [[~]][[23/22]]'s will provide odd harmonics 1–23 up to octave equivalence; you can think of this as a 6×6 grid, which is a recommendable place to start looking at its structure.


{{Val list|legend=1| 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee }}
Tempering out the less accurate comma 121/120 can be seen as an implication of tempering out [[441/440]] ({{S|21}}), [[484/483]] ({{S|22}}), and [[529/528]] ({{S|23}}). Therefore, characteristic of any good tuning is the prime [[11/1|11]] being the flattest prime, with other primes having strictly less than 5{{cent}} of error.


[[Badness]]: 0.583 × 10<sup>-3</sup>
This temperament was discovered by [[Scott Dakota]]. Note that the 17-limit extension requires the 29g val for 29edo, which has the sizes of 17/16 and 18/17 swapped.


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 351/350, 441/440
Comma list: 121/120, 196/195, 352/351
 
Mapping: {{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{{c}}, ~3/2 = 701.8766{{c}}, ~22/21 = 78.6564{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5612{{c}}, ~22/21 = 78.4778{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }}
 
Badness (Sintel): 1.08
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 121/120, 154/153, 196/195, 352/351
 
Mapping: {{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{{c}}, ~3/2 = 702.1026{{c}}, ~22/21 = 78.7963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7925{{c}}, ~22/21 = 78.6203{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{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{{c}}, ~3/2 = 702.0959{{c}}, ~22/21 = 78.8004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6963{{c}}, ~22/21 = 78.6479{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{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{{c}}, ~3/2 = 702.2463{{c}}, ~22/21 = 78.8824{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8010{{c}}, ~23/22 = 78.7188{{c}}


Mapping: [{{val| 1 0 1 3 2 6 }}, {{val| 0 1 1 0 1 -1 }}, {{val| 0 0 4 3 2 11 }}]
{{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}


Vals: {{Val list| 14cf, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}
Badness (Sintel): 1.08


Badness: 1.456 × 10<sup>-3</sup>
=== Astarte ===
This extension is catalogued as tridecimal aphrodite in [[Graham Breed]]'s temperament finder.  


==== Eros ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 196/195, 352/351
Comma list: 121/120, 351/350, 441/440
 
Mapping: {{mapping| 1 0 1 3 2 6 | 0 1 1 0 1 -1 | 0 0 -4 -3 -2 -11 }}


Mapping: [{{val| 1 0 1 3 2 7 }}, {{val| 0 1 1 0 1 -2 }}, {{val| 0 0 4 3 2 2 }}]
Optimal tunings:  
* WE: ~2 = 1201.0656{{c}}, ~3/2 = 700.7374{{c}}, ~22/21 = 78.5908{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.7005{{c}}, ~22/21 = 78.3253{{c}}


Vals: {{Val list| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }}
{{Optimal ET sequence|legend=0| 14cf, 29ff, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}


Badness: 1.150 × 10<sup>-3</sup>
Badness (Sintel): 1.36


==== Inanna ====
=== Inanna ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 105/104, 121/120, 275/273
Comma list: 105/104, 121/120, 275/273


Mapping: [{{val| 1 0 1 3 2 1 }}, {{val| 0 1 1 0 1 2 }}, {{val| 0 0 4 3 2 7 }}]
Mapping: {{mapping| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }}


Vals: {{Val list| 14cf, 15, 29, 31, 45ef, 60e }}
Optimal tunings:  
* WE: ~2 = 1201.7881{{c}}, ~3/2 = 699.8166{{c}}, ~22/21 = 79.7282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.5926{{c}}, ~22/21 = 79.3822{{c}}


Badness: 1.077 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 14cf, 15, 29, 31, 45ef, 60e }}


==== Ishtar ====
Badness (Sintel): 1.01
 
=== Ishtar ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 91/90, 121/120, 441/440
Comma list: 91/90, 121/120, 441/440


Mapping: [{{val| 1 0 1 3 2 -1 }}, {{val| 0 1 1 0 1 3 }}, {{val| 0 0 4 3 2 1 }}]
Mapping: {{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{{c}}, ~3/2 = 703.8568{{c}}, ~22/21 = 79.0096{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.7640{{c}}, ~22/21 = 78.8025{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}


Vals: {{Val list| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}
Badness (Sintel): 1.08


Badness: 1.151 × 10<sup>-3</sup>
== References ==


[[Category:Regular temperament theory]]
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Biyatismic clan| ]] <!-- main article -->
[[Category:Biyatismic clan| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 03:43, 25 April 2026

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:

Considered below are zeus, artemis, oxpecker, kahoupokane, 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 ¢, ~12/11 = 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 ¢, ~12/11 = 157.1102 ¢
error map: ⟨+0.169 +0.466 +2.057 +0.761 -5.668]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2478 ¢, ~12/11 = 157.1265 ¢
error map: ⟨0.000 +0.293 +1.681 +0.306 -6.197]

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⟩]
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 ¢, ~12/11 = 156.9897 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8818 ¢, ~12/11 = 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 ¢, ~12/11 = 155.3569 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.2982 ¢, ~12/11 = 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[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 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 ¢, ~12/11 = 155.8292 ¢
error map: ⟨+0.412 -0.414 +3.982 -1.435 -4.781]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2359 ¢, ~12/11 = 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 ¢, ~12/11 = 154.7247 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.6409 ¢, ~12/11 = 154.9115 ¢

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

Badness (Sintel): 1.02

Kahoupokane

Named by Tristan Bay in 2025, Kahoupokane tempers out 5120/5103 and may be described as the 29 & 46 & 53 temperament.

Subgroup: 2.3.5.7.11

Comma list: 121/120, 5120/5103

Mapping: [⟨1 0 1 11 2], ⟨0 1 1 -5 1], ⟨0 0 -2 -2 -1]]

Optimal tunings:

  • WE: ~2 = 1200.1911 ¢, ~3/2 = 703.1412 ¢, ~11/10 = 158.1068 ¢
error map: ⟨+0.191 +1.377 +0.996 +0.401 -5.710]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 703.0417 ¢, ~11/10 = 157.9917 ¢
error map: ⟨0.000 +1.087 +0.744 -0.018 -6.268]

Optimal ET sequence: 7, 17c, 24d, 29, 46, 53, 82e, 99

Badness (Sintel): 2.73

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 169/168, 352/351

Mapping: [⟨1 0 1 11 2 7], ⟨0 1 1 -5 1 -2], ⟨0 0 -2 -2 -1 -1]]

Optimal tunings:

  • WE: ~2 = 1200.4435 ¢, ~3/2 = 703.1443 ¢, ~11/10 = 158.4176 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.9013 ¢, ~11/10 = 158.1657 ¢

Optimal ET sequence: 7, 17c, 24d, 29, 46, 53, 82e, 99, 181eef

Badness (Sintel): 1.27

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

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 has the pleasing property that all the harmonics are on the negative side of the last generator. Specifically, -3 to 2 fifths and -5 to 0 ~23/22's will provide odd harmonics 1–23 up to octave equivalence; you can think of this as a 6×6 grid, which is a recommendable place to start looking at its structure.

Tempering out the less accurate comma 121/120 can be seen as an implication of tempering out 441/440 (S21), 484/483 (S22), and 529/528 (S23). Therefore, characteristic of any good tuning is the prime 11 being the flattest prime, with other primes having strictly less than 5 ¢ of error.

This temperament was discovered by Scott Dakota. Note that the 17-limit 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

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

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

Astarte

This extension is catalogued as tridecimal aphrodite in Graham Breed's temperament finder.

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

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

References

  1. ↑ Yahoo! Tuning Group | Artemis and friends
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