Biyatismic clan: Difference between revisions

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* ''[[Bisector]]'' (+245/243) → [[Sensamagic family #Bisector|Sensamagic family]]
* ''[[Bisector]]'' (+245/243) → [[Sensamagic family #Bisector|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)]].  
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 20: Line 20:
* [[WE]]: ~2 = 1200.6628{{c}}, ~3/2 = 701.8452{{c}}, ~11/10 = 157.8337{{c}}
* [[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 }}
: [[error map]]: {{val| +0.663 +0.553 +1.190 -5.318 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9848{{c}}, ~11/10 = 157.6099{{c}}
* [[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 }}
: error map: {{val| 0.000 +0.030 +0.451 -6.943 }}
<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4578{{c}}, ~11/10 = 157.7466{{c}} -->


{{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 }}
Line 45: Line 44:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1693{{c}}, ~3/2 = 702.2521{{c}}, ~11/10 = 157.1102{{c}}
* [[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 }}
: [[error map]]: {{val| +0.169 +0.466 +2.057 +0.761 -5.668 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2478{{c}}, ~11/10 = 157.1265{{c}}
* [[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 }}
: error map: {{val| 0.000 +0.293 +1.681 +0.306 -6.197 }}
<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1530{{c}}, ~11/10 = 157.0881{{c}} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 86: Line 84:


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.2411{{c}}, ~3/2 = 702.0090{{c}}, ~11/10 = 156.9897{{c}}
* 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}}, ~11/10 = 156.9568{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8818{{c}}, ~12/11 = 156.9568{{c}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8679{{c}}, ~11/10 = 156.9582{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 115: Line 112:


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9251{{c}}, ~3/2 = 699.2984{{c}}, ~11/10 = 155.3569{{c}}
* 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}}, ~11/10 = 155.3484{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2982{{c}}, ~12/11 = 155.3484{{c}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 699.3420{{c}}, ~11/10 = 155.3666{{c}} -->


{{Optimal ET sequence|legend=0| 7, 9, 15, 22f, 24, 31 }}
{{Optimal ET sequence|legend=0| 7, 9, 15, 22f, 24, 31 }}
Line 137: Line 133:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.2777{{c}}, ~11/10 = 158.3100{{c}}
* [[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 }}
: error map: {{val| 0.000 -1.677 -2.656 -0.955 -9.350 }}
<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 699.8719{{c}}, ~11/10 = 158.3232{{c}} -->


{{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, 60e, 84e, 91e, 113e, 144ee }}
Line 153: Line 148:
* WE: ~2 = 1201.7896{{c}}, ~3/2 = 699.7509{{c}}, ~11/10 = 158.9484{{c}}
* 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}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.1687{{c}}, ~11/10 = 158.7345{{c}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 698.7090{{c}}, ~11/10 = 158.7117{{c}} -->


{{Optimal ET sequence|legend=0| 9, 20, 22f, 29, 31, 60e, 129cddee }}
{{Optimal ET sequence|legend=0| 9, 20, 22f, 29, 31, 60e, 129cddee }}
Line 169: Line 163:
* WE: ~2 = 1200.9110{{c}}, ~3/2 = 701.5110{{c}}, ~11/10 = 159.1256{{c}}
* 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}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.9717{{c}}, ~11/10 = 158.7903{{c}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 700.9789{{c}}, ~11/10 = 159.0048{{c}} -->


{{Optimal ET sequence|legend=0| 22, 29, 31, 53, 82e, 84e, 113e }}
{{Optimal ET sequence|legend=0| 22, 29, 31, 53, 82e, 84e, 113e }}
Line 183: Line 176:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.4124{{c}}, ~3/2 = 701.1291{{c}}, ~11/10 = 155.8292{{c}}
* [[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 }}
: [[error map]]: {{val| +0.412 -0.414 +3.982 -1.435 -4.781 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2359{{c}}, ~11/10 = 155.7399{{c}}
* [[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 }}
: error map: {{val| 0.000 -0.719 +3.442 -2.029 -5.822 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.8882{{c}}, ~11/10 = 155.7756{{c}} -->


{{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 }}
Line 201: Line 193:


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1198.9113{{c}}, ~3/2 = 700.9581{{c}}, ~11/10 = 154.7247{{c}}
* 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}}, ~11/10 = 154.9115{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.6409{{c}}, ~12/11 = 154.9115{{c}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5946{{c}}, ~11/10 = 154.8652{{c}} -->


{{Optimal ET sequence|legend=0| 7d, 8d, 15, 23de, 24d, 31 }}
{{Optimal ET sequence|legend=0| 7d, 8d, 15, 23de, 24d, 31 }}


Badness (Sintel): 1.02
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 261: Line 286:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.8990{{c}}, ~22/21 = 78.4412{{c}}
* [[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 }}
: error map: {{val| 0.000 -1.056 +0.820 -4.150 -7.301 }}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 700.3200{{c}}, ~22/21 = 78.6421{{c}} -->


{{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 }}
Line 267: Line 291:
[[Badness]] (Sintel): 0.701
[[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
 
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}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 700.1158{{c}}, ~21/20 = 78.5211{{c}} -->
 
{{Optimal ET sequence|legend=0| 14cf, 29ff, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}


Badness (Sintel): 1.36
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.  


==== Eros ====
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.
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 295: Line 307:
* WE: ~2 = 1200.6419{{c}}, ~3/2 = 701.8766{{c}}, ~22/21 = 78.6564{{c}}
* 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}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5612{{c}}, ~22/21 = 78.4778{{c}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5014{{c}}, ~21/20 = 78.6143{{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 }}
Line 301: Line 312:
Badness (Sintel): 1.08
Badness (Sintel): 1.08


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


Line 313: Line 322:
* WE: ~2 = 1200.6172{{c}}, ~3/2 = 702.1026{{c}}, ~22/21 = 78.7963{{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}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9299{{c}}, ~22/21 = 78.2539{{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 }}
Line 319: Line 327:
Badness (Sintel): 0.931
Badness (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 329: Line 337:
* WE: ~2 = 1200.6224{{c}}, ~3/2 = 702.0959{{c}}, ~22/21 = 78.8004{{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}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5642{{c}}, ~22/21 = 78.2353{{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 }}
Line 335: Line 342:
Badness (Sintel): 1.16
Badness (Sintel): 1.16


===== 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 345: Line 352:
* WE: ~2 = 1200.7268{{c}}, ~3/2 = 702.2463{{c}}, ~22/21 = 78.8824{{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}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7115{{c}}, ~23/22 = 78.2054{{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 }}
Line 351: Line 357:
Badness (Sintel): 1.08
Badness (Sintel): 1.08


==== Inanna ====
=== 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: {{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 361: Line 384:
* WE: ~2 = 1201.7881{{c}}, ~3/2 = 699.8166{{c}}, ~22/21 = 79.7282{{c}}
* 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}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.5926{{c}}, ~22/21 = 79.3822{{c}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 698.7754{{c}}, ~21/20 = 79.6096{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 15, 29, 31, 45ef, 60e }}
{{Optimal ET sequence|legend=0| 14cf, 15, 29, 31, 45ef, 60e }}
Line 367: Line 389:
Badness (Sintel): 1.01
Badness (Sintel): 1.01


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


Line 377: Line 399:
* WE: ~2 = 1200.7875{{c}}, ~3/2 = 703.8568{{c}}, ~22/21 = 79.0096{{c}}
* 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}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.7640{{c}}, ~22/21 = 78.8025{{c}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 703.3952{{c}}, ~21/20 = 78.9578{{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 }}
Line 387: Line 408:
[[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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