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Biyatismic clan: Difference between revisions

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The '''biyatismic clan''' of [[Rank-3 temperament|rank-3]] [[Temperament|temperaments]] [[Tempering out|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 or 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) → [[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) → [[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 15: Line 15:


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


: Mapping generators: ~2, ~3, ~11/10
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.6628{{c}}, ~3/2 = 701.8452{{c}}, ~11/10 = 157.8337{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.4578, ~11/10 = 157.7466
: [[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]]: 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 = 1\1, ~3/2 = 702.1530, ~11/10 = 157.0881
[[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]]:  
* [[11-odd-limit]]
* [[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 }}]
: [{{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|eigenmonzo (unchanged-interval) basis]]: 2.9/5.9/7
: [[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 }}
{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 68, 77, 99, 130e }}


[[Badness]]: 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 = 1\1, ~3/2 = 701.8679, ~11/10 = 156.9582
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:  
* 13-odd-limit  
* 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 }}]
: [{{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 }}]
: eigenmonzo (unchanged-interval) basis: 2.9/5.9/7
: unchanged-interval (eigenmonzo) basis: 2.9/5.9/7
* 15-odd-limit
* 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 }}]
: [{{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 }}]
: eigenmonzo (unchanged-interval) 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: 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 = 1\1, ~3/2 = 699.3420, ~11/10 = 155.3666
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=1| 7, 9, 15, 22f, 24, 31 }}
{{Optimal ET sequence|legend=0| 7, 9, 15, 22f, 24, 31 }}


Badness: 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 = 1\1, ~3/2 = 699.8719, ~11/10 = 158.3232
[[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, 82e, 84e, 113e, 144ee }}
{{Optimal ET sequence|legend=1| 9, 15d, 16d, 20, 22, 31, 53, 60e, 84e, 91e, 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 = 1\1, ~3/2 = 698.7090, ~11/10 = 158.7117
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=1| 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 = 1\1, ~3/2 = 700.9789, ~11/10 = 159.0048
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 }}


{{Optimal ET sequence|legend=1| 22, 29, 31, 53, 82e, 84e, 113e, 166ee }}
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 = 1\1, ~3/2 = 700.8882, ~11/10 = 155.7756
[[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]]: 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 = 1\1, ~3/2 = 701.5946, ~11/10 = 154.8652
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


{{Optimal ET sequence|legend=1| 7d, 8d, 15, 23de, 24d, 31 }}
{{Mapping|legend=1| 1 0 1 11 2 | 0 1 1 -5 1 | 0 0 -2 -2 -1 }}


Badness: 1.093 × 10<sup>-3</sup>
[[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 }}


== Aphrodite ==
{{Optimal ET sequence|legend=1| 7, 17c, 24d, 29, 46, 53, 82e, 99 }}
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) ===
[[Badness]] (Sintel): 2.73
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64827/64000
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Mapping|legend=1| 1 0 1 3 | 0 1 1 0 | 0 0 -4 -3 }}
Comma list: 121/120, 169/168, 352/351


: Mapping generators: ~2, ~3, ~21/20
Mapping: {{mapping| 1 0 1 11 2 7 | 0 1 1 -5 1 -2 | 0 0 -2 -2 -1 -1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.2144, ~21/20 = 78.5694
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=1| 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d }}
{{Optimal ET sequence|legend=0| 7, 17c, 24d, 29, 46, 53, 82e, 99, 181eef }}


[[Badness]]: 0.943 × 10<sup>-3</sup>
Badness (Sintel): 1.27


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


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


[[Comma list]]: 121/120, 441/440
[[Comma list]]: 99/98, 121/120


{{Mapping|legend=1| 1 0 1 3 2 | 0 1 1 0 1 | 0 0 -4 -3 -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


: Mapping generators: ~2, ~3, ~22/21
[[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 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.3200, ~21/20 = 78.6421
{{Optimal ET sequence|legend=1| 8d, 9, 14c, 17c, 22, 31, 53, 84e }}


{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee }}
[[Badness]] (Sintel): 0.609


[[Badness]]: 0.583 × 10<sup>-3</sup>
[[Projection pair]]s: <code>5 2401/486, 11 98/9</code> to 2.3.7


==== 13-limit ====
=== 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: 66/65, 99/98, 121/120
 
Mapping: {{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{{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 ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Squalentine]].''
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 121/120, 441/440
 
{{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 }}


Mapping: {{mapping| 1 0 1 3 2 6 | 0 1 1 0 1 -1 | 0 0 -4 -3 -2 -11 }}
{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.1158, ~21/20 = 78.5211
[[Badness]] (Sintel): 0.701


{{Optimal ET sequence|legend=1| 14cf, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}
=== 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.


Badness: 1.456 × 10<sup>-3</sup>
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 226: 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 = 1\1, ~3/2 = 701.5014, ~21/20 = 78.6143
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=1| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }}
{{Optimal ET sequence|legend=0| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }}


Badness: 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 242: Line 320:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.9299, ~22/21 = 78.2539
* WE: ~2 = 1200.6172{{c}}, ~3/2 = 702.1026{{c}}, ~22/21 = 78.7963{{c}}
* CWE: ~2 = 1\1, ~3/2 = 701.7925, ~22/21 = 78.6203
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7925{{c}}, ~22/21 = 78.6203{{c}}


Optimal ET sequence: {{Optimal ET sequence| 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>
* Dirichlet: 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 259: Line 335:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.5642, ~22/21 = 78.2353
* WE: ~2 = 1200.6224{{c}}, ~3/2 = 702.0959{{c}}, ~22/21 = 78.8004{{c}}
* CWE: ~2 = 1\1, ~3/2 = 701.6963, ~22/21 = 78.6479
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6963{{c}}, ~22/21 = 78.6479{{c}}


Optimal ET sequence: {{Optimal ET sequence| 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>
* Dirichlet: 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 276: Line 350:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3 = 1901.7115, ~23/22 = 78.2054
* WE: ~2 = 1200.7268{{c}}, ~3/2 = 702.2463{{c}}, ~22/21 = 78.8824{{c}}
* CWE: ~2 = 1\1, ~3 = 1901.8010, ~23/22 = 78.7188
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8010{{c}}, ~23/22 = 78.7188{{c}}


Optimal ET sequence: {{Optimal ET sequence| 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>
 
* Dirichlet: 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 292: 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 = 1\1, ~3/2 = 698.7754, ~21/20 = 79.6096
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=1| 14cf, 15, 29, 31, 45ef, 60e }}
{{Optimal ET sequence|legend=0| 14cf, 15, 29, 31, 45ef, 60e }}


Badness: 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 305: 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 = 1\1, ~3/2 = 703.3952, ~21/20 = 78.9578
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=1| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}
{{Optimal ET sequence|legend=0| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}


Badness: 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]]
Retrieved from "https://en.xen.wiki/w/Biyatismic_clan"