Biyatismic clan: Difference between revisions

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* ''[[Sonic]]'' (+55/54 or 100/99) → [[Porcupine rank-3 family #Sonic|Porcupine rank-3 family]]
* ''[[Sonic]]'' (+55/54 or 100/99) → [[Porcupine rank-3 family #Sonic|Porcupine rank-3 family]]
* ''[[Urania]]'' (+81/80) → [[Rastmic rank-3 clan #Urania|Rastmic rank-3 clan]]
* ''[[Urania]]'' (+81/80) → [[Rastmic rank-3 clan #Urania|Rastmic rank-3 clan]]
* ''[[Big brother]]'' (+99/98) → [[Nuwell family #big Brother|Nuwell family]]
* ''[[Bisector]]'' (+245/243) → [[Sensamagic family #Bisector|Sensamagic family]]
* ''[[Bisector]]'' (+245/243) → [[Sensamagic family #Bisector|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)]].  
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 ==
== Protomere ==
Line 18: Line 17:
: mapping generators: ~2, ~3, ~11/10
: mapping generators: ~2, ~3, ~11/10


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.4578{{c}}, ~11/10 = 157.7466{{c}}
[[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 }}
{{Optimal ET sequence|legend=1| 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee }}


[[Badness]] (Smith): 0.0297 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.245


== Zeus ==
== Zeus ==
Line 40: Line 43:
: Angle (11/10, 11/8) = 87.464 degrees
: Angle (11/10, 11/8) = 87.464 degrees


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.1530{{c}}, ~11/10 = 157.0881{{c}}
[[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]]:  
[[Minimax tuning]]:  
Line 49: Line 56:
{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 68, 77, 99, 130e }}
{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 68, 77, 99, 130e }}


[[Badness]] (Smith): 0.400 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.480


[[Projection pair]]s: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5
[[Projection pair]]s: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5
Line 76: Line 83:
: Angle (11/10, 11/8) = 106.7439 degrees
: Angle (11/10, 11/8) = 106.7439 degrees


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.8679{{c}}, ~11/10 = 156.9582{{c}}
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}}


Minimax tuning:  
Minimax tuning:  
Line 86: Line 95:
: unchanged-interval (eigenmonzo) basis: 2.3.7/5
: unchanged-interval (eigenmonzo) basis: 2.3.7/5


{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 77, 99, 130e }}
{{Optimal ET sequence|legend=0| 15f, 22, 31, 46, 53, 77, 99, 130e }}


Badness (Smith): 0.934 × 10<sup>-3</sup>
Badness (Sintel): 0.873


Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5
Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5
Line 102: Line 111:
Mapping: {{mapping| 1 0 1 4 2 2 | 0 1 1 -1 1 1 | 0 0 -2 3 -1 -1 }}
Mapping: {{mapping| 1 0 1 4 2 2 | 0 1 1 -1 1 1 | 0 0 -2 3 -1 -1 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 699.3420{{c}}, ~11/10 = 155.3666{{c}}
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 }}
{{Optimal ET sequence|legend=0| 7, 9, 15, 22f, 24, 31 }}


Badness (Smith): 0.808 × 10<sup>-3</sup>
Badness (Sintel): 0.756


== Artemis ==
== Artemis ==
Line 117: Line 128:
{{Mapping|legend=1| 1 0 1 -3 2 | 0 1 1 4 1 | 0 0 -2 -4 -1 }}
{{Mapping|legend=1| 1 0 1 -3 2 | 0 1 1 4 1 | 0 0 -2 -4 -1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 699.8719{{c}}, ~11/10 = 158.3232{{c}}
[[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 }}


{{Optimal ET sequence|legend=1| 9, 15d, 16d, 20, 22, 31, 53, 82e, 84e, 113e, 144ee }}
[[Badness]] (Sintel): 0.713


=== 13-limit ===
=== 13-limit ===
Line 128: Line 145:
Mapping: {{mapping| 1 0 1 -3 2 -5 | 0 1 1 4 1 6 | 0 0 -2 -4 -1 -6 }}
Mapping: {{mapping| 1 0 1 -3 2 -5 | 0 1 1 4 1 6 | 0 0 -2 -4 -1 -6 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 698.7090{{c}}, ~11/10 = 158.7117{{c}}
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 }}
{{Optimal ET sequence|legend=0| 9, 20, 22f, 29, 31, 60e, 129cddee }}
 
Badness (Sintel): 1.04


=== Diana ===
=== Diana ===
Line 139: Line 160:
Mapping: {{mapping| 1 0 1 -3 2 7 | 0 1 1 4 1 -2 | 0 0 -2 -4 -1 -1 }}
Mapping: {{mapping| 1 0 1 -3 2 7 | 0 1 1 4 1 -2 | 0 0 -2 -4 -1 -1 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 700.9789{{c}}, ~11/10 = 159.0048{{c}}
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, 166ee }}
{{Optimal ET sequence|legend=0| 22, 29, 31, 53, 82e, 84e, 113e }}
 
Badness (Sintel): 1.07


== Oxpecker ==
== Oxpecker ==
Line 150: Line 175:
{{Mapping|legend=1| 1 0 1 2 2 | 0 1 1 1 1 | 0 0 -2 -6 -1 }}
{{Mapping|legend=1| 1 0 1 2 2 | 0 1 1 1 1 | 0 0 -2 -6 -1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 700.8882{{c}}, ~11/10 = 155.7756{{c}}
[[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 }}
{{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31, 46, 77 }}


[[Badness]] (Smith): 0.699 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.840


=== Woodpecker ===
=== Woodpecker ===
Line 163: Line 192:
Mapping: {{mapping| 1 0 1 2 2 2 | 0 1 1 1 1 1 | 0 0 -2 -6 -1 1 }}
Mapping: {{mapping| 1 0 1 2 2 2 | 0 1 1 1 1 1 | 0 0 -2 -6 -1 1 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.5946{{c}}, ~11/10 = 154.8652{{c}}
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 }}
{{Optimal ET sequence|legend=0| 7d, 8d, 15, 23de, 24d, 31 }}


Badness (Smith): 1.093 × 10<sup>-3</sup>
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 ==
== Big brother ==
Line 176: Line 241:
[[Comma list]]: 99/98, 121/120
[[Comma list]]: 99/98, 121/120


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


{{Optimal ET sequence|legend=1| 8d, 9, 22, 31, 53 }}
[[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 }}


[[Badness]]: 0.507 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=1| 8d, 9, 14c, 17c, 22, 31, 53, 84e }}


[[Projection pair]]s: 5 2401/486 11 98/9 to 2.3.7
[[Badness]] (Sintel): 0.609
 
[[Projection pair]]s: <code>5 2401/486, 11 98/9</code> to 2.3.7


=== 13-limit ===
=== 13-limit ===
Line 191: Line 263:
Mapping: {{mapping| 1 0 3 1 3 1 | 0 1 3 2 2 0 | 0 0 -4 -1 -2 2 }}
Mapping: {{mapping| 1 0 3 1 3 1 | 0 1 3 2 2 0 | 0 0 -4 -1 -2 2 }}


{{Optimal ET sequence|legend=0| 8d, 9, 22f, 31 }}
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}}
 
{{Optimal ET sequence|legend=0| 8d, 9, 14c, 17c, 22f, 31, 79cf }}
 
Badness (Sintel): 0.889


== Aphrodite ==
== Aphrodite ==
Line 203: Line 281:
: mapping generators: ~2, ~3, ~22/21
: mapping generators: ~2, ~3, ~22/21


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 700.3200{{c}}, ~21/20 = 78.6421{{c}}
[[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 }}
{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee }}


[[Badness]] (Smith): 0.583 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.701


==== 13-limit ====
=== Eros ===
Subgroup: 2.3.5.7.11.13
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.  


Comma list: 121/120, 351/350, 441/440
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.


Mapping: {{mapping| 1 0 1 3 2 6 | 0 1 1 0 1 -1 | 0 0 -4 -3 -2 -11 }}
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.
 
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 700.1158{{c}}, ~21/20 = 78.5211{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}
 
Badness (Smith): 1.456 × 10<sup>-3</sup>
 
==== 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 [[121/120|S11]] can be seen as a consequence of tempering {[[441/440|S21]], [[484/483|S22]], [[529/528|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{{cent}} of error. This temperament was first logged on x31eq by [[Scott Dakota]].


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 231: Line 304:
Mapping: {{mapping| 1 0 1 3 2 7 | 0 1 1 0 1 -2 | 0 0 -4 -3 -2 -2 }}
Mapping: {{mapping| 1 0 1 3 2 7 | 0 1 1 0 1 -2 | 0 0 -4 -3 -2 -2 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.5014{{c}}, ~21/20 = 78.6143{{c}}
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 }}
{{Optimal ET sequence|legend=0| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }}


Badness (Smith): 1.150 × 10<sup>-3</sup>
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.


==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Line 247: Line 320:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9299{{c}}, ~22/21 = 78.2539{{c}}
* 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}}
* 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 }}
{{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 77, 106de }}


Badness:
Badness (Sintel): 0.931
* Smith: 0.979 × 10<sup>-3</sup>
* Sintel: 0.931


===== 19-limit =====
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Line 264: Line 335:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5642{{c}}, ~22/21 = 78.2353{{c}}
* 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}}
* 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 }}
{{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}


Badness:
Badness (Sintel): 1.16
* Smith: 1.13 × 10<sup>-3</sup>
* Sintel: 1.159


===== 23-limit =====
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.19.23


Line 281: Line 350:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7115{{c}}, ~23/22 = 78.2054{{c}}
* 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}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8010{{c}}, ~23/22 = 78.7188{{c}}


{{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}
{{Optimal ET sequence|legend=0| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}


Badness:  
Badness (Sintel): 1.08
* Smith: 0.939 × 10<sup>-3</sup>
 
* Sintel: 1.084
=== Astarte ===
This extension is catalogued as tridecimal aphrodite in [[Graham Breed]]'s temperament finder.  


==== Inanna ====
Subgroup: 2.3.5.7.11.13
 
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 }}
 
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}}
 
{{Optimal ET sequence|legend=0| 14cf, 29ff, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}
 
Badness (Sintel): 1.36
 
=== Inanna ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 297: Line 381:
Mapping: {{mapping| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 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 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 698.7754{{c}}, ~21/20 = 79.6096{{c}}
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}}


{{Optimal ET sequence|legend=0| 14cf, 15, 29, 31, 45ef, 60e }}
{{Optimal ET sequence|legend=0| 14cf, 15, 29, 31, 45ef, 60e }}


Badness (Sintel): 1.077 × 10<sup>-3</sup>
Badness (Sintel): 1.01


==== Ishtar ====
=== Ishtar ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 310: Line 396:
Mapping: {{mapping| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 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 tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 703.3952{{c}}, ~21/20 = 78.9578{{c}}
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 }}
{{Optimal ET sequence|legend=0| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}


Badness (Sintel): 1.151 × 10<sup>-3</sup>
Badness (Sintel): 1.08


== Notes ==
== References ==


[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Biyatismic clan| ]] <!-- main article -->
[[Category:Biyatismic clan| ]] <!-- main article -->
[[Category:Biyatismic| ]] <!-- key 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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