Tetracot family: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings, per community consensus (3/3)
- CTE & POTE tunings
Line 20: Line 20:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 176.0965{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 176.0965{{c}}
: error map: {{val| 0.000 +2.431 -1.445 }}
: error map: {{val| 0.000 +2.431 -1.445 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 176.028{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 176.160{{c}} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 55: Line 53:
Comma list: 100/99, 243/242
Comma list: 100/99, 243/242


Sval mapping: {{mapping| 1 1 1 2 | 0 4 9 10 }}
Subgroup-val mapping: {{mapping| 1 1 1 2 | 0 4 9 10 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.3274{{c}}, ~10/9 = 175.8862{{c}}
* WE: ~2 = 1199.3274{{c}}, ~10/9 = 175.8862{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.8847{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.8847{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~10/9 = 175.7765{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/9 = 175.985{{c}} -->


{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e }}
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e }}
Line 72: Line 68:
Comma list: 100/99, 144/143, 243/242
Comma list: 100/99, 144/143, 243/242


Sval mapping: {{mapping| 1 1 1 2 4 | 0 4 9 10 -2 }}
Subgroup-val mapping: {{mapping| 1 1 1 2 4 | 0 4 9 10 -2 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1198.6852{{c}}, ~10/9 = 176.0034{{c}}
* WE: ~2 = 1198.6852{{c}}, ~10/9 = 176.0034{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.0854{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.0854{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~10/9 = 175.8150{{c}}
* POTE ~2 = 1200.000{{c}}, ~10/9 = 176.196{{c}} -->


{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e, 109ef }}
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e, 109ef }}
Line 89: Line 83:
Comma list: 325/324, 512/507
Comma list: 325/324, 512/507


Sval mapping: {{mapping| 1 1 1 4 | 0 4 9 -2 }}
Subgroup-val mapping: {{mapping| 1 1 1 4 | 0 4 9 -2 }}


Optimal tunings:
Optimal tunings:
* WE: ~2 = 1198.8502{{c}}, ~10/9 = 176.2195{{c}}
* WE: ~2 = 1198.8502{{c}}, ~10/9 = 176.2195{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.2975{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.2975{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~10/9 = 176.0783{{c}} -->


{{Optimal ET sequence|legend=0| 7, 20c, 27, 34, 245bff, 279bfff }}
{{Optimal ET sequence|legend=0| 7, 20c, 27, 34, 245bff, 279bfff }}
Line 114: Line 107:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 175.6622{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 175.6622{{c}}
: error map: {{val| 0.000 +0.694 -5.354 -3.759 }}
: error map: {{val| 0.000 +0.694 -5.354 -3.759 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 175.676{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 175.659{{c}} -->


{{Optimal ET sequence|legend=1| 7, 34, 41 }}
{{Optimal ET sequence|legend=1| 7, 34, 41 }}
Line 131: Line 122:
* WE: ~2 = 1200.3988{{c}}, ~10/9 = 175.6287{{c}}
* WE: ~2 = 1200.3988{{c}}, ~10/9 = 175.6287{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.5750{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.5750{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 175.598{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/9 = 175.570{{c}} -->


{{Optimal ET sequence|legend=0| 7, 34, 41 }}
{{Optimal ET sequence|legend=0| 7, 34, 41 }}
Line 148: Line 137:
* WE: ~2 = 1199.9206{{c}}, ~10/9 = 175.6108{{c}}
* WE: ~2 = 1199.9206{{c}}, ~10/9 = 175.6108{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.6217{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.6217{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 175.618{{c}}
* POTE ~2 = 1200.000{{c}}, ~10/9 = 175.622{{c}} -->


{{Optimal ET sequence|legend=0| 7, 34, 41 }}
{{Optimal ET sequence|legend=0| 7, 34, 41 }}
Line 165: Line 152:
* WE: ~2 = 1199.5029{{c}}, ~10/9 = 175.6832{{c}}
* WE: ~2 = 1199.5029{{c}}, ~10/9 = 175.6832{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.7558{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.7558{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 175.754{{c}} -->


{{Optimal ET sequence|legend=0| 7, 34, 41 }}
{{Optimal ET sequence|legend=0| 7, 34, 41 }}
Line 181: Line 167:
* WE: ~2 = 1199.7318{{c}}, ~10/9 = 175.6498{{c}}
* WE: ~2 = 1199.7318{{c}}, ~10/9 = 175.6498{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.6901{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.6901{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 175.697{{c}} -->


{{Optimal ET sequence|legend=0| 7, 34, 41 }}
{{Optimal ET sequence|legend=0| 7, 34, 41 }}
Line 201: Line 186:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 175.7567{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 175.7567{{c}}
: error map: {{val| 0.000 +1.072 -4.503 +0.849 }}
: error map: {{val| 0.000 +1.072 -4.503 +0.849 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 175.785{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 175.741{{c}} -->


{{Optimal ET sequence|legend=1| 7d, …, 34d, 41, 116, 157c, 198c }}
{{Optimal ET sequence|legend=1| 7d, …, 34d, 41, 116, 157c, 198c }}
Line 218: Line 201:
* WE: ~2 = 1199.7481{{c}}, ~10/9 = 175.7401{{c}}
* WE: ~2 = 1199.7481{{c}}, ~10/9 = 175.7401{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.7637{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.7637{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 175.738{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/9 = 175.777{{c}} -->


{{Optimal ET sequence|legend=0| 7d, …, 34d, 41, 116e }}
{{Optimal ET sequence|legend=0| 7d, …, 34d, 41, 116e }}
Line 235: Line 216:
* WE: ~2 = 1199.1044{{c}}, ~10/9 = 175.7545{{c}}
* WE: ~2 = 1199.1044{{c}}, ~10/9 = 175.7545{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.8526{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.8526{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 175.748{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/9 = 175.886{{c}} -->


{{Optimal ET sequence|legend=0| 7d, 34d, 41, 116ef }}
{{Optimal ET sequence|legend=0| 7d, 34d, 41, 116ef }}
Line 252: Line 231:
* WE: ~2 = 1198.7905{{c}}, ~10/9 = 175.7757{{c}}
* WE: ~2 = 1198.7905{{c}}, ~10/9 = 175.7757{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.9302{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.9302{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 175.811{{c}} -->


{{Optimal ET sequence|legend=0| 34d, 41, 75e }}
{{Optimal ET sequence|legend=0| 34d, 41, 75e }}
Line 268: Line 246:
* WE: ~2 = 1198.7904{{c}}, ~10/9 = 175.7755{{c}}
* WE: ~2 = 1198.7904{{c}}, ~10/9 = 175.7755{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.9287{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.9287{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 175.802{{c}} -->


{{Optimal ET sequence|legend=0| 34dh, 41, 75e }}
{{Optimal ET sequence|legend=0| 34dh, 41, 75e }}
Line 290: Line 267:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 177.1188{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 177.1188{{c}}
: error map: {{val| 0.000 +6.520 +7.755 +14.224 }}
: error map: {{val| 0.000 +6.520 +7.755 +14.224 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 176.818{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 177.203{{c}} -->


{{Optimal ET sequence|legend=1| 7, 20c, 27, 61d, 88bcd, 149bccddd }}
{{Optimal ET sequence|legend=1| 7, 20c, 27, 61d, 88bcd, 149bccddd }}
Line 307: Line 282:
* WE: ~2 = 1196.4227{{c}}, ~10/9 = 176.5252{{c}}
* WE: ~2 = 1196.4227{{c}}, ~10/9 = 176.5252{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.9286{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.9286{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 176.446{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/9 = 177.053{{c}} -->


{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34d, 61de }}
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34d, 61de }}
Line 324: Line 297:
* WE: ~2 = 1196.8686{{c}}, ~10/9 = 176.4915{{c}}
* WE: ~2 = 1196.8686{{c}}, ~10/9 = 176.4915{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.8735{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.8735{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 176.471{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/9 = 176.953{{c}} -->


{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34d, 61de }}
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34d, 61de }}
Line 341: Line 312:
* WE: ~2 = 1196.8783{{c}}, ~10/9 = 176.5241{{c}}
* WE: ~2 = 1196.8783{{c}}, ~10/9 = 176.5241{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.8969{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.8969{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 176.453{{c}} -->


{{Optimal ET sequence|legend=0| 7g, …, 27eg, 34d }}
{{Optimal ET sequence|legend=0| 7g, …, 27eg, 34d }}
Line 357: Line 327:
* WE: ~2 = 1196.6939{{c}}, ~10/9 = 176.5426{{c}}
* WE: ~2 = 1196.6939{{c}}, ~10/9 = 176.5426{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.9645{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.9645{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 176.538{{c}} -->


{{Optimal ET sequence|legend=0| 7g, …, 27eg, 34dh, 61degh }}
{{Optimal ET sequence|legend=0| 7g, …, 27eg, 34dh, 61degh }}
Line 380: Line 349:
* WE: ~2 = 1198.5026{{c}}, ~10/9 = 176.9786{{c}}
* WE: ~2 = 1198.5026{{c}}, ~10/9 = 176.9786{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1589{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 176.990{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/9 = 177.200{{c}} -->


{{Optimal ET sequence|legend=0| 7, 20c, 27 }}
{{Optimal ET sequence|legend=0| 7, 20c, 27 }}
Line 397: Line 364:
* WE: ~2 = 1198.5149{{c}}, ~10/9 = 176.9778{{c}}
* WE: ~2 = 1198.5149{{c}}, ~10/9 = 176.9778{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1681{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 177.017{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/9 = 177.197{{c}} -->


{{Optimal ET sequence|legend=0| 7, 20c, 27 }}
{{Optimal ET sequence|legend=0| 7, 20c, 27 }}
Line 414: Line 379:
* WE: ~2 = 1197.4542{{c}}, ~10/9 = 177.1828{{c}}
* WE: ~2 = 1197.4542{{c}}, ~10/9 = 177.1828{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.5355{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.5355{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 177.378{{c}} -->


{{Optimal ET sequence|legend=0| 7, 20c }}
{{Optimal ET sequence|legend=0| 7, 20c }}
Line 430: Line 394:
* WE: ~2 = 1197.3233{{c}}, ~10/9 = 177.2025{{c}}
* WE: ~2 = 1197.3233{{c}}, ~10/9 = 177.2025{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.5878{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.5878{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 177.505{{c}} -->


{{Optimal ET sequence|legend=0| 7, 20c }}
{{Optimal ET sequence|legend=0| 7, 20c }}
Line 450: Line 413:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 177.1667{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 177.1667{{c}}
: error map: {{val| 0.000 +6.712 +8.186 -2.659 }}
: error map: {{val| 0.000 +6.712 +8.186 -2.659 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 176.900{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 177.357{{c}} -->


{{Optimal ET sequence|legend=1| 7d, 20cd, 27, 61, 88bc, 115bc }}
{{Optimal ET sequence|legend=1| 7d, 20cd, 27, 61, 88bc, 115bc }}
Line 467: Line 428:
* WE: ~2 = 1196.6462{{c}}, ~10/9 = 176.9174{{c}}
* WE: ~2 = 1196.6462{{c}}, ~10/9 = 176.9174{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1370{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1370{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 176.704{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/9 = 177.413{{c}} -->


{{Optimal ET sequence|legend=0| 7d, 20cde, 27e }}
{{Optimal ET sequence|legend=0| 7d, 20cde, 27e }}
Line 484: Line 443:
* WE: ~2 = 1197.4576{{c}}, ~10/9 = 176.8557{{c}}
* WE: ~2 = 1197.4576{{c}}, ~10/9 = 176.8557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.0949{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.0949{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 176.716{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/9 = 177.231{{c}} -->


{{Optimal ET sequence|legend=0| 7d, 20cde, 27e }}
{{Optimal ET sequence|legend=0| 7d, 20cde, 27e }}
Line 501: Line 458:
* WE: ~2 = 1197.4770{{c}}, ~10/9 = 176.7733{{c}}
* WE: ~2 = 1197.4770{{c}}, ~10/9 = 176.7733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.0123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.0123{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 176.641{{c}} -->


{{Optimal ET sequence|legend=0| 7dg, 27eg }}
{{Optimal ET sequence|legend=0| 7dg, 27eg }}
Line 517: Line 473:
* WE: ~2 = 1197.4380{{c}}, ~10/9 = 176.8774{{c}}
* WE: ~2 = 1197.4380{{c}}, ~10/9 = 176.8774{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1216{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1216{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/9 = 176.749{{c}} -->


{{Optimal ET sequence|legend=0| 7dgh, 27eg }}
{{Optimal ET sequence|legend=0| 7dgh, 27eg }}
Line 541: Line 496:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 88.0525{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 88.0525{{c}}
: error map: {{val| 0.000 +2.465 -1.369 -0.248 }}
: error map: {{val| 0.000 +2.465 -1.369 -0.248 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~21/20 = 88.023{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~21/20 = 88.076{{c}} -->


{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109 }}
{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109 }}
Line 558: Line 511:
* WE: ~2 = 1199.6025{{c}}, ~21/20 = 87.9460{{c}}
* WE: ~2 = 1199.6025{{c}}, ~21/20 = 87.9460{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.9453{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.9453{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 87.910{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 87.975{{c}} -->


{{Optimal ET sequence|legend=0| 14c, 27e, 41, 109e }}
{{Optimal ET sequence|legend=0| 14c, 27e, 41, 109e }}
Line 575: Line 526:
* WE: ~2 = 1198.8609{{c}}, ~21/20 = 87.0219{{c}}
* WE: ~2 = 1198.8609{{c}}, ~21/20 = 87.0219{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0557{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 87.926{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.106{{c}} -->


{{Optimal ET sequence|legend=0| 14c, 27e, 41, 68e, 109ef }}
{{Optimal ET sequence|legend=0| 14c, 27e, 41, 68e, 109ef }}
Line 592: Line 541:
* WE: ~2 = 1198.4494{{c}}, ~21/20 = 87.9878{{c}}
* WE: ~2 = 1198.4494{{c}}, ~21/20 = 87.9878{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0324{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0324{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 87.842{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.102{{c}} -->


{{Optimal ET sequence|legend=0| 14c, 27eg, 41, 68egg }}
{{Optimal ET sequence|legend=0| 14c, 27eg, 41, 68egg }}
Line 609: Line 556:
* WE: ~2 = 1198.5995{{c}}, ~20/19 = 88.0081{{c}}
* WE: ~2 = 1198.5995{{c}}, ~20/19 = 88.0081{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.0471{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.0471{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~20/19 = 87.866{{c}}
* POTE: ~2 = 1200.000{{c}}, ~20/19 = 88.111{{c}} -->


{{Optimal ET sequence|legend=0| 14c, 27eg, 41, 68egg }}
{{Optimal ET sequence|legend=0| 14c, 27eg, 41, 68egg }}
Line 626: Line 571:
* WE: ~2 = 1199.4441{{c}}, ~21/20 = 88.1380{{c}}
* WE: ~2 = 1199.4441{{c}}, ~21/20 = 88.1380{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1375{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1375{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.090{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.179{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 27e, 41f }}
{{Optimal ET sequence|legend=0| 14cf, 27e, 41f }}
Line 643: Line 586:
* WE: ~2 = 1198.4257{{c}}, ~21/20 = 88.1636{{c}}
* WE: ~2 = 1198.4257{{c}}, ~21/20 = 88.1636{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1642{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1642{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.011{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 27eg }}
{{Optimal ET sequence|legend=0| 14cf, 27eg }}
Line 659: Line 601:
* WE: ~2 = 1198.5748{{c}}, ~20/19 = 88.1631{{c}}
* WE: ~2 = 1198.5748{{c}}, ~20/19 = 88.1631{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.1637{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.1637{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~20/19 = 88.017{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 27eg }}
{{Optimal ET sequence|legend=0| 14cf, 27eg }}
Line 675: Line 616:
* WE: ~2 = 1200.5116{{c}}, ~21/20 = 87.7346{{c}}
* WE: ~2 = 1200.5116{{c}}, ~21/20 = 87.7346{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.7257{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.7257{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 87.770{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 87.697{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 27eff, 41 }}
{{Optimal ET sequence|legend=0| 14cf, 27eff, 41 }}
Line 692: Line 631:
* WE: ~2 = 1199.6667{{c}}, ~21/20 = 87.7494{{c}}
* WE: ~2 = 1199.6667{{c}}, ~21/20 = 87.7494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.7559{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.7559{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 87.728{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 27effg, 41 }}
{{Optimal ET sequence|legend=0| 14cf, 27effg, 41 }}
Line 708: Line 646:
* WE: ~2 = 1199.9909{{c}}, ~20/19 = 87.7474{{c}}
* WE: ~2 = 1199.9909{{c}}, ~20/19 = 87.7474{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 87.7476{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 87.7476{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~20/19 = 87.750{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 27effg, 41 }}
{{Optimal ET sequence|legend=0| 14cf, 27effg, 41 }}
Line 725: Line 662:
* WE: ~2 = 1199.1496{{c}}, ~13/9 = 643.5328{{c}}
* WE: ~2 = 1199.1496{{c}}, ~13/9 = 643.5328{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 643.9567{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 643.9567{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~13/9 = 643.916{{c}}
* POTE: ~2 = 1200.000{{c}}, ~13/9 = 643.989{{c}} -->


{{Optimal ET sequence|legend=0| 13cdeef, 28ccdef, 41 }}
{{Optimal ET sequence|legend=0| 13cdeef, 28ccdef, 41 }}
Line 742: Line 677:
* WE: ~2 = 1199.8843{{c}}, ~21/20 = 88.0261{{c}}
* WE: ~2 = 1199.8843{{c}}, ~21/20 = 88.0261{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0329{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0329{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.026{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.035{{c}} -->


{{Optimal ET sequence|legend=0| 27, 41, 68, 109, 150, 259 }}
{{Optimal ET sequence|legend=0| 27, 41, 68, 109, 150, 259 }}
Line 759: Line 692:
* WE: ~2 = 1199.5060{{c}}, ~21/20 = 88.0388{{c}}
* WE: ~2 = 1199.5060{{c}}, ~21/20 = 88.0388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0697{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0697{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.041{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.075{{c}} -->


{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}
{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}
Line 776: Line 707:
* WE: ~2 = 1199.3845{{c}}, ~21/20 = 88.0589{{c}}
* WE: ~2 = 1199.3845{{c}}, ~21/20 = 88.0589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1027{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1027{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.093{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.104{{c}} -->


{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}
{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}
Line 793: Line 722:
* WE: ~2 = 1199.4449{{c}}, ~20/19 = 88.0723{{c}}
* WE: ~2 = 1199.4449{{c}}, ~20/19 = 88.0723{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.1107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.1107{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~20/19 = 88.093{{c}}
* POTE: ~2 = 1200.000{{c}}, ~20/19 = 88.113{{c}} -->


{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}
{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}
Line 813: Line 740:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 58.6624{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 58.6624{{c}}
: error map: {{val| 0.000 +1.993 -2.430 +1.681 }}
: error map: {{val| 0.000 +1.993 -2.430 +1.681 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~28/27 = 58.648{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~28/27 = 58.675{{c}} -->


{{Optimal ET sequence|legend=1| 20cd, 41, 143d, 184, 225 }}
{{Optimal ET sequence|legend=1| 20cd, 41, 143d, 184, 225 }}
Line 830: Line 755:
* WE: ~2 = 1199.3125{{c}}, ~28/27 = 58.6317{{c}}
* WE: ~2 = 1199.3125{{c}}, ~28/27 = 58.6317{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 58.6360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 58.6360{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~28/27 = 58.602{{c}}
* POTE: ~2 = 1200.000{{c}}, ~28/27 = 58.665{{c}} -->


{{Optimal ET sequence|legend=0| 20cde, 41 }}
{{Optimal ET sequence|legend=0| 20cde, 41 }}
Line 847: Line 770:
* WE: ~2 = 1199.0713{{c}}, ~28/27 = 58.5932{{c}}
* WE: ~2 = 1199.0713{{c}}, ~28/27 = 58.5932{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 58.5982{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 58.5982{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~28/27 = 58.551{{c}}
* POTE: ~2 = 1200.000{{c}}, ~28/27 = 58.639{{c}} -->


{{Optimal ET sequence|legend=0| 20cdef, 41 }}
{{Optimal ET sequence|legend=0| 20cdef, 41 }}
Line 869: Line 790:
* [[CWE]]: ~7/5 = 1200.0000{{c}}, ~10/9 = 175.5632{{c}}
* [[CWE]]: ~7/5 = 1200.0000{{c}}, ~10/9 = 175.5632{{c}}
: error map: {{val| 0.000 +0.298 -6.245 +11.243 }}
: error map: {{val| 0.000 +0.298 -6.245 +11.243 }}
<!-- * [[CTE]]: ~7/5 = 600.000{{c}}, ~10/9 = 175.472{{c}} -->


{{Optimal ET sequence|legend=1| 14c, 34d, 48 }}
{{Optimal ET sequence|legend=1| 14c, 34d, 48 }}
Line 885: Line 805:
* WE: ~7/5 = 599.6525{{c}}, ~10/9 = 175.5103{{c}}
* WE: ~7/5 = 599.6525{{c}}, ~10/9 = 175.5103{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.5086{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.5086{{c}}
<!-- * CTE: ~7/5 = 600.000{{c}}, ~10/9 = 175.401{{c}} -->


{{Optimal ET sequence|legend=0| 14c, 34d, 48 }}
{{Optimal ET sequence|legend=0| 14c, 34d, 48 }}
Line 901: Line 820:
* WE: ~7/5 = 599.4539{{c}}, ~10/9 = 175.7393{{c}}
* WE: ~7/5 = 599.4539{{c}}, ~10/9 = 175.7393{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.7502{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.7502{{c}}
<!-- * CTE: ~7/5 = 600.000{{c}}, ~10/9 = 175.586{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 20cdef, 34d }}
{{Optimal ET sequence|legend=0| 14cf, 20cdef, 34d }}
Line 917: Line 835:
* WE: ~7/5 = 599.7509{{c}}, ~10/9 = 175.6684{{c}}
* WE: ~7/5 = 599.7509{{c}}, ~10/9 = 175.6684{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.6839{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.6839{{c}}
<!-- * CTE: ~7/5 = 600.000{{c}}, ~10/9 = 175.596{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 20cdef, 34d }}
{{Optimal ET sequence|legend=0| 14cf, 20cdef, 34d }}
Line 933: Line 850:
* WE: ~7/5 = 599.6682{{c}}, ~10/9 = 175.5994{{c}}
* WE: ~7/5 = 599.6682{{c}}, ~10/9 = 175.5994{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.6190{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.6190{{c}}
<!-- * CTE: ~7/5 = 600.000{{c}}, ~10/9 = 175.498{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 20cdefhh, 34dh, 48f }}
{{Optimal ET sequence|legend=0| 14cf, 20cdefhh, 34dh, 48f }}

Revision as of 16:25, 24 November 2025

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 parent of the tetracot family is tetracot, the 5-limit temperament tempering out the tetracot comma (ratio: 20000/19683, monzo: [5 -9 4⟩).

Tetracot

Main article: Tetracot

The generator of tetracot is ~10/9, and that four of these give ~3/2. In fact, (10/9)4 = (20000/19683)⋅(3/2). We also have (10/9)9 = (20000/19683)2⋅(5/2). From this it is evident we should flatten the generator a bit, and 34edo does this and makes for a recommendable tuning. Another possibility is to use (5/2)1/9 for a generator. The 13-note mos gives enough space for eight triads, with the 20-note mos supplying many more.

The name comes from members of the Araucaria family of conifers, which have four cotyledons (though sometimes these are fused).

Subgroup: 2.3.5

Comma list: 20000/19683

Mapping: [⟨1 1 1], ⟨0 4 9]]

Optimal tunings:

  • WE: ~2 = 1199.5586 ¢, ~10/9 = 176.0950 ¢
error map: ⟨-0.441 +1.984 -1.900]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 176.0965 ¢
error map: ⟨0.000 +2.431 -1.445]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.5

Optimal ET sequence: 7, 20c, 27, 34, 75, 109

Badness (Sintel): 1.14

Overview to extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at.

  • 875/864, the keema, gives monkey;
  • 179200/177147 (or equivalently 225/224) gives bunya;
  • 245/243 gives octacot, which splits the generator in half.

Monkey and bunya

Monkey tempers out the keema. The keema, 875/864, is the amount by which three just minor thirds fall short of 7/4, and tells us the ~7/4 of monkey is reached by three minor thirds in succession. It can be described as the 34 & 41 temperament. 41edo is an excellent tuning for monkey, and has the effect of making monkey identical to bunya with the same tuning.

Bunya adds 225/224 to the list of commas and may be described as the 34d & 41 temperament. 41edo can again be used as a tuning, in which case it is the same as monkey. However an excellent alternative is 141/26 as a generator, giving just ~7's and an improved value for ~5, at the cost of a slightly sharper, but still less than a cent sharp, fifth. Octave stretching, if employed, also serves to distinguish bunya from monkey, as its octaves should be stretched considerably less.

Since the generator in all cases is between 10/9 and 11/10, it is natural to extend these temperaments to the 11-limit by tempering out (10/9)/(11/10) = 100/99. This gives 11-limit monkey and 11-limit bunya. Again, 41edo can be used as a tuning, making the two identical, which is also the case if we turn to the 2.3.5.11 subgroup temperament, dispensing with 7. However, 11-limit bunya, like 7-limit bunya, profits a little from a slightly sharper fifth, such as the 141/26 generator supplies, or even sharper yet, as for instance by the val ⟨355 563 823 997 1230], with a 52/355 generator.

Since 16/13 is shy of (10/9)2 by just 325/324, it is likewise natural to extend our winning streak with these temperaments by adding this to the list of commas. This gives us 13-limit monkey and 13-limit bunya. Once again, 41edo is recommended as a tuning for monkey, while bunya can with advantage tune the fifth sharper: 17\116 as a generator with a fifth a cent and a half sharp or 11\75 with a fifth two cents sharp.

2.3.5.11 subgroup

As discussed above, tetracot works well for the 2.3.5.11.13 subgroup, in which it tempers out 100/99, 144/143 and 243/242.

The S-expression-based comma list of this temperament is {S9/S11, S10}.

Subgroup: 2.3.5.11

Comma list: 100/99, 243/242

Subgroup-val mapping: [⟨1 1 1 2], ⟨0 4 9 10]]

Optimal tunings:

  • WE: ~2 = 1199.3274 ¢, ~10/9 = 175.8862 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.8847 ¢

Optimal ET sequence: 7, 20ce, 27e, 34, 41, 75e

Badness (Sintel): 0.459

2.3.5.11.13 subgroup

Subgroup: 2.3.5.11.13

Comma list: 100/99, 144/143, 243/242

Subgroup-val mapping: [⟨1 1 1 2 4], ⟨0 4 9 10 -2]]

Optimal tunings:

  • WE: ~2 = 1198.6852 ¢, ~10/9 = 176.0034 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 176.0854 ¢

Optimal ET sequence: 7, 20ce, 27e, 34, 41, 75e, 109ef

Badness (Sintel): 0.489

2.3.5.13 subgroup

Subgroup: 2.3.5.13

Comma list: 325/324, 512/507

Subgroup-val mapping: [⟨1 1 1 4], ⟨0 4 9 -2]]

Optimal tunings:

  • WE: ~2 = 1198.8502 ¢, ~10/9 = 176.2195 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 176.2975 ¢

Optimal ET sequence: 7, 20c, 27, 34, 245bff, 279bfff

Badness (Sintel): 0.551

Monkey

See also: Tetracot

Subgroup: 2.3.5.7

Comma list: 875/864, 5120/5103

Mapping: [⟨1 1 1 5], ⟨0 4 9 -15]]

Optimal tunings:

  • WE: ~2 = 1200.7982 ¢, ~10/9 = 175.7757 ¢
error map: ⟨+0.798 +1.946 -3.534 -1.470]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.6622 ¢
error map: ⟨0.000 +0.694 -5.354 -3.759]

Optimal ET sequence: 7, 34, 41

Badness (Sintel): 1.86

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 243/242, 385/384

Mapping: [⟨1 1 1 5 2], ⟨0 4 9 -15 10]]

Optimal tunings:

  • WE: ~2 = 1200.3988 ¢, ~10/9 = 175.6287 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.5750 ¢

Optimal ET sequence: 7, 34, 41

Badness (Sintel): 1.28

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 243/242

Mapping: [⟨1 1 1 5 2 4], ⟨0 4 9 -15 10 -2]]

Optimal tunings:

  • WE: ~2 = 1199.9206 ¢, ~10/9 = 175.6108 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.6217 ¢

Optimal ET sequence: 7, 34, 41

Badness (Sintel): 1.17

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 144/143, 154/153, 170/169

Mapping: [⟨1 1 1 5 2 4 6], ⟨0 4 9 -15 10 -2 -13]]

Optimal tunings:

  • WE: ~2 = 1199.5029 ¢, ~10/9 = 175.6832 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.7558 ¢

Optimal ET sequence: 7, 34, 41

Badness (Sintel): 1.32

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 144/143, 154/153, 170/169, 171/169

Mapping: [⟨1 1 1 5 2 4 6 6], ⟨0 4 9 -15 10 -2 -13 -12]]

Optimal tunings:

  • WE: ~2 = 1199.7318 ¢, ~10/9 = 175.6498 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.6901 ¢

Optimal ET sequence: 7, 34, 41

Badness (Sintel): 1.35

Bunya

See also: Tetracot

Subgroup: 2.3.5.7

Comma list: 225/224, 15625/15309

Mapping: [⟨1 1 1 -1], ⟨0 4 9 26]]

Optimal tunings:

  • WE: ~2 = 1200.2991 ¢, ~10/9 = 175.7844 ¢
error map: ⟨+0.299 +1.482 -3.955 +1.270]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.7567 ¢
error map: ⟨0.000 +1.072 -4.503 +0.849]

Optimal ET sequence: 7d, …, 34d, 41, 116, 157c, 198c

Badness (Sintel): 1.59

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 243/242

Mapping: [⟨1 1 1 -1 2], ⟨0 4 9 26 10]]

Optimal tunings:

  • WE: ~2 = 1199.7481 ¢, ~10/9 = 175.7401 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.7637 ¢

Optimal ET sequence: 7d, …, 34d, 41, 116e

Badness (Sintel): 1.04

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 144/143, 225/224, 243/242

Mapping: [⟨1 1 1 -1 2 4], ⟨0 4 9 26 10 -2]]

Optimal tunings:

  • WE: ~2 = 1199.1044 ¢, ~10/9 = 175.7545 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.8526 ¢

Optimal ET sequence: 7d, 34d, 41, 116ef

Badness (Sintel): 1.03

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 120/119, 144/143, 170/169, 225/224

Mapping: [⟨1 1 1 -1 2 4 6], ⟨0 4 9 26 10 -2 -13]]

Optimal tunings:

  • WE: ~2 = 1198.7905 ¢, ~10/9 = 175.7757 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.9302 ¢

Optimal ET sequence: 34d, 41, 75e

Badness (Sintel): 1.19

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 120/119, 144/143, 170/169, 190/189, 225/224

Mapping: [⟨1 1 1 -1 2 4 6 0], ⟨0 4 9 26 10 -2 -13 29]]

Optimal tunings:

  • WE: ~2 = 1198.7904 ¢, ~10/9 = 175.7755 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.9287 ¢

Optimal ET sequence: 34dh, 41, 75e

Badness (Sintel): 1.18

Modus

See also: Tetracot

Modus was named by Mike Battaglia in 2012 for its fantastic modmos structures[1].

Subgroup: 2.3.5.7

Comma list: 64/63, 4375/4374

Mapping: [⟨1 1 1 4], ⟨0 4 9 -8]]

Optimal tunings:

  • WE: ~2 = 1196.7884 ¢, ~10/9 = 176.7292 ¢
error map: ⟨-3.212 +1.750 +1.038 +4.494]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.1188 ¢
error map: ⟨0.000 +6.520 +7.755 +14.224]

Optimal ET sequence: 7, 20c, 27, 61d, 88bcd, 149bccddd

Badness (Sintel): 1.73

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 243/242

Mapping: [⟨1 1 1 4 2], ⟨0 4 9 -8 10]]

Optimal tunings:

  • WE: ~2 = 1196.4227 ¢, ~10/9 = 176.5252 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 176.9286 ¢

Optimal ET sequence: 7, 20ce, 27e, 34d, 61de

Badness (Sintel): 1.16

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 78/77, 100/99, 144/143

Mapping: [⟨1 1 1 4 2 4], ⟨0 4 9 -8 10 -2]]

Optimal tunings:

  • WE: ~2 = 1196.8686 ¢, ~10/9 = 176.4915 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 176.8735 ¢

Optimal ET sequence: 7, 20ce, 27e, 34d, 61de

Badness (Sintel): 0.984

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 64/63, 78/77, 100/99, 120/119, 144/143

Mapping: [⟨1 1 1 4 2 4 1], ⟨0 4 9 -8 10 -2 21]]

Optimal tunings:

  • WE: ~2 = 1196.8783 ¢, ~10/9 = 176.5241 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 176.8969 ¢

Optimal ET sequence: 7g, …, 27eg, 34d

Badness (Sintel): 1.10

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 64/63, 78/77, 96/95, 100/99, 120/119, 144/143

Mapping: [⟨1 1 1 4 2 4 1 5], ⟨0 4 9 -8 10 -2 21 -5]]

Optimal tunings:

  • WE: ~2 = 1196.6939 ¢, ~10/9 = 176.5426 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 176.9645 ¢

Optimal ET sequence: 7g, …, 27eg, 34dh, 61degh

Badness (Sintel): 1.09

Music

Ponens

The error of 11 is about the same as that of modus, but flat instead of sharp, and much more abundant. Since the other primes are all sharp, however, this leads to a much larger error for other intervals involving 11.

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 363/350

Mapping: [⟨1 1 1 4 3], ⟨0 4 9 -8 3]]

Optimal tunings:

  • WE: ~2 = 1198.5026 ¢, ~10/9 = 176.9786 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.1589 ¢

Optimal ET sequence: 7, 20c, 27

Badness (Sintel): 2.09

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 66/65, 143/140

Mapping: [⟨1 1 1 4 3 4], ⟨0 4 9 -8 3 -2]]

Optimal tunings:

  • WE: ~2 = 1198.5149 ¢, ~10/9 = 176.9778 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.1681 ¢

Optimal ET sequence: 7, 20c, 27

Badness (Sintel): 1.61

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 52/51, 55/54, 64/63, 66/65, 143/140

Mapping: [⟨1 1 1 4 3 4 5], ⟨0 4 9 -8 3 -2 -6]]

Optimal tunings:

  • WE: ~2 = 1197.4542 ¢, ~10/9 = 177.1828 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.5355 ¢

Optimal ET sequence: 7, 20c

Badness (Sintel): 1.79

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 52/51, 55/54, 64/63, 66/65, 77/76, 143/140

Mapping: [⟨1 1 1 4 3 4 5 5], ⟨0 4 9 -8 3 -2 -6 -5]]

Optimal tunings:

  • WE: ~2 = 1197.3233 ¢, ~10/9 = 177.2025 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.5878 ¢

Optimal ET sequence: 7, 20c

Badness (Sintel): 1.70

Wollemia

See also: Tetracot

Subgroup: 2.3.5.7

Comma list: 126/125, 2240/2187

Mapping: [⟨1 1 1 0], ⟨0 4 9 19]]

Optimal tunings:

  • WE: ~2 = 1197.6555 ¢, ~10/9 = 177.0104 ¢
error map: ⟨-2.345 +3.742 +4.435 -5.628]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.1667 ¢
error map: ⟨0.000 +6.712 +8.186 -2.659]

Optimal ET sequence: 7d, 20cd, 27, 61, 88bc, 115bc

Badness (Sintel): 1.78

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99, 243/242

Mapping: [⟨1 1 1 0 2], ⟨0 4 9 19 10]]

Optimal tunings:

  • WE: ~2 = 1196.6462 ¢, ~10/9 = 176.9174 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.1370 ¢

Optimal ET sequence: 7d, 20cde, 27e

Badness (Sintel): 1.24

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 91/90, 100/99, 243/242

Mapping: [⟨1 1 1 0 2 4], ⟨0 4 9 19 10 -2]]

Optimal tunings:

  • WE: ~2 = 1197.4576 ¢, ~10/9 = 176.8557 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.0949 ¢

Optimal ET sequence: 7d, 20cde, 27e

Badness (Sintel): 1.29

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 56/55, 91/90, 100/99, 136/135, 154/153

Mapping: [⟨1 1 1 0 2 4 1], ⟨0 4 9 19 10 -2 21]]

Optimal tunings:

  • WE: ~2 = 1197.4770 ¢, ~10/9 = 176.7733 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.0123 ¢

Optimal ET sequence: 7dg, 27eg

Badness (Sintel): 1.25

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 56/55, 76/75, 91/90, 100/99, 136/135, 154/153

Mapping: [⟨1 1 1 0 2 4 1 1], ⟨0 4 9 19 10 -2 21 22]]

Optimal tunings:

  • WE: ~2 = 1197.4380 ¢, ~10/9 = 176.8774 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.1216 ¢

Optimal ET sequence: 7dgh, 27eg

Badness (Sintel): 1.28

Octacot

See also: Chords of octacot

Octacot cuts the Gordian knot of deciding between the monkey and bunya mappings for 7 by cutting the generator in half and splitting the difference. It adds 245/243 to the normal comma list, and also tempers out 2401/2400. It may also be described as 41 & 68. 68edo or 109edo can be used as tunings, as can (5/2)1/18, which gives just major thirds. Another tuning is 150edo, which has a generator, 11\150, of exactly 88 cents. This relates octacot to the 88cET non-octave temperament, which like Carlos Alpha arguably makes more sense viewed as part of a rank-2 temperament with octaves rather than rank-1 without them.

Once again and for the same reasons, it is natural to add 100/99 and 325/324 to the list of commas. Generators of 3\41, 8\109 and 11\150 (88 cents) are all good choices for the 7, 11 and 13 limits.

Subgroup: 2.3.5.7

Comma list: 245/243, 2401/2400

Mapping: [⟨1 1 1 2], ⟨0 8 18 11]]

Optimal tunings:

  • WE: ~2 = 1199.6782 ¢, ~21/20 = 88.0528 ¢
error map: ⟨-0.322 +2.145 -1.686 -0.889]
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 88.0525 ¢
error map: ⟨0.000 +2.465 -1.369 -0.248]

Optimal ET sequence: 14c, 27, 41, 68, 109

Badness (Sintel): 0.857

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 243/242, 245/242

Mapping: [⟨1 1 1 2 2], ⟨0 8 18 11 20]]

Optimal tunings:

  • WE: ~2 = 1199.6025 ¢, ~21/20 = 87.9460 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 87.9453 ¢

Optimal ET sequence: 14c, 27e, 41, 109e

Badness (Sintel): 0.796

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 144/143, 196/195, 243/242

Mapping: [⟨1 1 1 2 2 4], ⟨0 8 18 11 20 -4]]

Optimal tunings:

  • WE: ~2 = 1198.8609 ¢, ~21/20 = 87.0219 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 88.0557 ¢

Optimal ET sequence: 14c, 27e, 41, 68e, 109ef

Badness (Sintel): 0.962

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 120/119, 144/143, 154/153, 189/187

Mapping: [⟨1 1 1 2 2 4 3], ⟨0 8 18 11 20 -4 15]]

Optimal tunings:

  • WE: ~2 = 1198.4494 ¢, ~21/20 = 87.9878 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 88.0324 ¢

Optimal ET sequence: 14c, 27eg, 41, 68egg

Badness (Sintel): 1.07

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 120/119, 133/132, 144/143, 154/153, 189/187

Mapping: [⟨1 1 1 2 2 4 3 3], ⟨0 8 18 11 20 -4 15 17]]

Optimal tunings:

  • WE: ~2 = 1198.5995 ¢, ~20/19 = 88.0081 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/19 = 88.0471 ¢

Optimal ET sequence: 14c, 27eg, 41, 68egg

Badness (Sintel): 1.01

Octocat

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 91/90, 100/99, 245/242

Mapping: [⟨1 1 1 2 2 2], ⟨0 8 18 11 20 23]]

Optimal tunings:

  • WE: ~2 = 1199.4441 ¢, ~21/20 = 88.1380 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 88.1375 ¢

Optimal ET sequence: 14cf, 27e, 41f

Badness (Sintel): 1.14

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 52/51, 78/77, 91/90, 100/99, 189/187

Mapping: [⟨1 1 1 2 2 2 3], ⟨0 8 18 11 20 23 15]]

Optimal tunings:

  • WE: ~2 = 1198.4257 ¢, ~21/20 = 88.1636 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 88.1642 ¢

Optimal ET sequence: 14cf, 27eg

Badness (Sintel): 1.19

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 52/51, 78/77, 91/90, 100/99, 133/132, 189/187

Mapping: [⟨1 1 1 2 2 2 3 3], ⟨0 8 18 11 20 23 15 17]]

Optimal tunings:

  • WE: ~2 = 1198.5748 ¢, ~20/19 = 88.1631 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/19 = 88.1637 ¢

Optimal ET sequence: 14cf, 27eg

Badness (Sintel): 1.09

Octopod

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 243/242, 245/242

Mapping: [⟨1 1 1 2 2 1], ⟨0 8 18 11 20 37]]

Optimal tunings:

  • WE: ~2 = 1200.5116 ¢, ~21/20 = 87.7346 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 87.7257 ¢

Optimal ET sequence: 14cf, 27eff, 41

Badness (Sintel): 1.17

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 120/119, 154/153, 243/242

Mapping: [⟨1 1 1 2 2 1 3], ⟨0 8 18 11 20 37 15]]

Optimal tunings:

  • WE: ~2 = 1199.6667 ¢, ~21/20 = 87.7494 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 87.7559 ¢

Optimal ET sequence: 14cf, 27effg, 41

Badness (Sintel): 1.26

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 120/119, 133/132, 154/153, 209/208

Mapping: [⟨1 1 1 2 2 1 3 3], ⟨0 8 18 11 20 37 15 17]]

Optimal tunings:

  • WE: ~2 = 1199.9909 ¢, ~20/19 = 87.7474 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/19 = 87.7476 ¢

Optimal ET sequence: 14cf, 27effg, 41

Badness (Sintel): 1.19

Dificot

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 243/242, 245/242, 343/338

Mapping: [⟨1 -7 -17 -9 -18 -14], ⟨0 16 36 22 40 33]]

mapping generators: ~2, ~13/9

Optimal tunings:

  • WE: ~2 = 1199.1496 ¢, ~13/9 = 643.5328 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/9 = 643.9567 ¢

Optimal ET sequence: 13cdeef, 28ccdef, 41

Badness (Sintel): 2.14

October

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 1375/1372

Mapping: [⟨1 1 1 2 5], ⟨0 8 18 11 -21]]

Optimal tunings:

  • WE: ~2 = 1199.8843 ¢, ~21/20 = 88.0261 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 88.0329 ¢

Optimal ET sequence: 27, 41, 68, 109, 150, 259

Badness (Sintel): 1.31

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 275/273, 385/384

Mapping: [⟨1 1 1 2 5 4], ⟨0 8 18 11 -21 -4]]

Optimal tunings:

  • WE: ~2 = 1199.5060 ¢, ~21/20 = 88.0388 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 88.0697 ¢

Optimal ET sequence: 27, 41, 68, 109f

Badness (Sintel): 1.29

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 154/153, 170/169, 196/195, 245/243, 256/255

Mapping: [⟨1 1 1 2 5 4 6], ⟨0 8 18 11 -21 -4 -26]]

Optimal tunings:

  • WE: ~2 = 1199.3845 ¢, ~21/20 = 88.0589 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 88.1027 ¢

Optimal ET sequence: 27, 41, 68, 109f

Badness (Sintel): 1.37

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 154/153, 170/169, 190/189, 196/195, 209/208, 245/243

Mapping: [⟨1 1 1 2 5 4 6 3], ⟨0 8 18 11 -21 -4 -26 17]]

Optimal tunings:

  • WE: ~2 = 1199.4449 ¢, ~20/19 = 88.0723 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/19 = 88.1107 ¢

Optimal ET sequence: 27, 41, 68, 109f

Badness (Sintel): 1.25

Dodecacot

Subgroup: 2.3.5.7

Comma list: 3125/3087, 10976/10935

Mapping: [⟨1 1 1 1], ⟨0 12 27 37]]

mapping generators: ~2, ~28/27

Optimal tunings:

  • WE: ~2 = 1199.6912 ¢, ~28/27 = 58.6600 ¢
error map: ⟨-0.309 +1.657 -2.802 +1.287]
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 58.6624 ¢
error map: ⟨0.000 +1.993 -2.430 +1.681]

Optimal ET sequence: 20cd, 41, 143d, 184, 225

Badness (Sintel): 3.03

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 243/242, 1375/1372

Mapping: [⟨1 1 1 1 2], ⟨0 12 27 37 30]]

Optimal tunings:

  • WE: ~2 = 1199.3125 ¢, ~28/27 = 58.6317 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 58.6360 ¢

Optimal ET sequence: 20cde, 41

Badness (Sintel): 1.97

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 196/195, 243/242, 275/273

Mapping: [⟨1 1 1 1 2 2], ⟨0 12 27 37 30 35]]

Optimal tunings:

  • WE: ~2 = 1199.0713 ¢, ~28/27 = 58.5932 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 58.5982 ¢

Optimal ET sequence: 20cdef, 41

Badness (Sintel): 1.80

Byhearted

See also: No-fives subgroup temperaments § Byhearted

Subgroup: 2.3.5.7

Comma list: 50/49, 19683/19208

Mapping: [⟨2 2 2 3], ⟨0 4 9 9]]

mapping generators: ~7/5, ~10/9

Optimal tunings:

  • WE: ~7/5 = 599.6934 ¢, ~10/9 = 175.5626 ¢
error map: ⟨-0.613 -0.318 -6.864 +10.318]
  • CWE: ~7/5 = 1200.0000 ¢, ~10/9 = 175.5632 ¢
error map: ⟨0.000 +0.298 -6.245 +11.243]

Optimal ET sequence: 14c, 34d, 48

Badness (Sintel): 2.82

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 99/98, 243/242

Mapping: [⟨2 2 2 3 4], ⟨0 4 9 9 10]]

Optimal tunings:

  • WE: ~7/5 = 599.6525 ¢, ~10/9 = 175.5103 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~10/9 = 175.5086 ¢

Optimal ET sequence: 14c, 34d, 48

Badness (Sintel): 1.45

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 78/77, 99/98, 243/242

Mapping: [⟨2 2 2 3 4 3], ⟨0 4 9 9 10 15]]

Optimal tunings:

  • WE: ~7/5 = 599.4539 ¢, ~10/9 = 175.7393 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~10/9 = 175.7502 ¢

Optimal ET sequence: 14cf, 20cdef, 34d

Badness (Sintel): 1.32

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 78/77, 85/84, 99/98, 243/242

Mapping: [⟨2 2 2 3 4 3 7], ⟨0 4 9 9 10 15 4]]

Optimal tunings:

  • WE: ~7/5 = 599.7509 ¢, ~10/9 = 175.6684 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~10/9 = 175.6839 ¢

Optimal ET sequence: 14cf, 20cdef, 34d

Badness (Sintel): 1.33

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 50/49, 78/77, 85/84, 99/98, 135/133, 243/242

Mapping: [⟨2 2 2 3 4 3 7 5], ⟨0 4 9 9 10 15 4 12]]

Optimal tunings:

  • WE: ~7/5 = 599.6682 ¢, ~10/9 = 175.5994 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~10/9 = 175.6190 ¢

Optimal ET sequence: 14cf, 20cdefhh, 34dh, 48f

Badness (Sintel): 1.28

Other subgroup extensions

Devisemi (2.3.5.19)

Subgroup: 2.3.5.19

Comma list: 361/360, 20000/19683

Subgroup-val mapping: [⟨1 1 1 3], ⟨0 8 18 17]]

Gencom mapping: [⟨1 1 1 0 0 0 0 3], ⟨0 8 18 0 0 0 0 17]]

mapping generators: ~2, ~20/19

Optimal tunings:

  • WE: ~2 = 1199.6900 ¢, ~20/19 = 88.0541 ¢
error map: ⟨-0.310 +2.168 -1.649 -1.523]
  • CWE: ~2 = 1200.0000 ¢, ~20/19 = 88.0538 ¢
error map: ⟨0.000 +2.475 -1.345 -0.598]

Optimal ET sequence: 14c, 27, 41, 68, 109

Badness (Sintel): 1.30

2.3.5.7.19 subgroup

Subgroup: 2.3.5.7.19

Comma list: 190/189, 245/243, 361/360

Subgroup-val mapping: [⟨1 1 1 2 3], ⟨0 8 18 11 17]]

Gencom mapping: [⟨1 1 1 2 0 0 0 3], ⟨0 8 18 11 0 0 0 17]]

Optimal tunings:

  • WE: ~2 = 1199.7591 ¢, ~20/19 = 88.0570 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/19 = 88.0564 ¢

Optimal ET sequence: 14c, 27, 41, 68, 109

Badness (Sintel): 0.508

References

  1. ↑ Yahoo! Tuning Group | Guaranteed meantone successor
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