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Magic family: Difference between revisions

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{{interwiki
| de = Magische Temperaturen
| en = Magic family
| es =
| ja =
}}
{{Technical data page}}
{{Technical data page}}
The '''magic family''' of temperaments tempers out [[3125/3072]], the small diesis or magic comma. The septimal version of magic is locally optimal, for some searches, in the [[9-odd-limit]]. Magic has a slightly higher complexity than [[meantone]] but it is closer to just intonation. It is the simplest rank-2 temperament that tunes every [[9-odd-limit]] interval better than is possible in [[12edo]]. The most prominent deficiency is that it lacks [[Rothenberg propriety|proper]] or nearly-proper [[mos scale]]s in the 5- to 10-note region. Properties may depend on tuning and extension.
The '''magic family''' of temperaments tempers out [[3125/3072]], the small diesis or magic comma. The septimal version of magic is locally optimal, for some searches, in the [[9-odd-limit]]. Magic has a slightly higher complexity than [[meantone]] but it is closer to just intonation. It is the simplest rank-2 temperament that tunes every [[9-odd-limit]] interval better than is possible in [[12edo]]. The most prominent deficiency is that it lacks [[Rothenberg propriety|proper]] or nearly-proper [[mos scale]]s in the 5- to 10-note region. Properties may depend on tuning and extension.
Line 26: Line 20:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.2194{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.2194{{c}}
: error map: {{val| 0.000 -0.858 -6.094 }}
: error map: {{val| 0.000 -0.858 -6.094 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 380.499{{c}}
: [[error map]]: {{val| 0.000 +0.542 -5.814 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 380.058{{c}}
: error map: {{val| 0.000 -1.663 -6.255 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 76: Line 66:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.4576{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.4576{{c}}
: error map: {{val| 0.000 +0.333 -5.856 -3.335 }}
: error map: {{val| 0.000 +0.333 -5.856 -3.335 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 380.651{{c}}
: [[error map]]: {{val| 0.000 +1.301 -5.662 -1.011 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 380.352{{c}}
: error map: {{val| 0.000 -0.195 -5.962 -4.602 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 100: Line 86:
Comma list: 100/99, 225/224, 245/243
Comma list: 100/99, 225/224, 245/243


Mapping: {{mapping| 1 0 2 -1 6 | 0 5 1 12 -8 }}
{{Mapping|legend=0| 1 0 2 -1 6 | 0 5 1 12 -8 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1372{{c}}, ~5/4 = 380.7399{{c}}
* WE: ~2 = 1200.1372{{c}}, ~5/4 = 380.7399{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7008{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7008{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.720{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.696{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 127: Line 111:
Comma list: 100/99, 105/104, 144/143, 196/195
Comma list: 100/99, 105/104, 144/143, 196/195


Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 1 12 -8 18 }}
{{Mapping|legend=0| 1 0 2 -1 6 -2 | 0 5 1 12 -8 18 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0331{{c}}, ~5/4 = 380.4377{{c}}
* WE: ~2 = 1200.0331{{c}}, ~5/4 = 380.4377{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4284{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4284{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.435{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.427{{c}} -->


Tuning ranges:  
Tuning ranges:  
Line 148: Line 130:
Comma list: 100/99, 105/104, 120/119, 144/143, 154/153
Comma list: 100/99, 105/104, 120/119, 144/143, 154/153


Mapping: {{mapping| 1 0 2 -1 6 -2 6 | 0 5 1 12 -8 18 -6 }}
{{Mapping|legend=0| 1 0 2 -1 6 -2 6 | 0 5 1 12 -8 18 -6 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.3584{{c}}, ~5/4 = 380.4006{{c}}
* WE: ~2 = 1199.3584{{c}}, ~5/4 = 380.4006{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5896{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5896{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.510{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.604{{c}} -->


{{Optimal ET sequence|legend=0| 19, 22f, 41 }}
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}
Line 160: Line 140:
Badness (Sintel): 1.05
Badness (Sintel): 1.05


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


Comma list: 100/99, 105/104, 120/119, 133/132, 144/143, 154/153
Comma list: 100/99, 105/104, 120/119, 133/132, 144/143, 154/153


Mapping: {{mapping| 1 0 2 -1 6 -2 6 9 | 0 5 1 12 -8 18 -6 -15 }}
{{Mapping|legend=0| 1 0 2 -1 6 -2 6 9 | 0 5 1 12 -8 18 -6 -15 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.7173{{c}}, ~5/4 = 380.3808{{c}}
* WE: ~2 = 1199.7173{{c}}, ~5/4 = 380.3808{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4680{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4680{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.451{{c}} -->


{{Optimal ET sequence|legend=0| 19, 41 }}
{{Optimal ET sequence|legend=0| 19, 41 }}
Line 181: Line 160:
Comma list: 100/99, 105/104, 120/119, 144/143, 154/153, 171/169
Comma list: 100/99, 105/104, 120/119, 144/143, 154/153, 171/169


Mapping: {{mapping| 1 0 2 -1 6 -2 6 -4 | 0 5 1 12 -8 18 -6 26 }}
{{Mapping|legend=0| 1 0 2 -1 6 -2 6 -4 | 0 5 1 12 -8 18 -6 26 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.3670{{c}}, ~5/4 = 380.4681{{c}}
* WE: ~2 = 1199.3670{{c}}, ~5/4 = 380.4681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6541{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6541{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.573{{c}} -->


{{Optimal ET sequence|legend=0| 22fh, 41 }}
{{Optimal ET sequence|legend=0| 22fh, 41 }}
Line 197: Line 175:
Comma list: 100/99, 105/104, 144/143, 170/169, 196/195
Comma list: 100/99, 105/104, 144/143, 170/169, 196/195


Mapping: {{mapping| 1 0 2 -1 6 -2 -7 | 0 5 1 12 -8 18 35 }}
{{Mapping|legend=0| 1 0 2 -1 6 -2 -7 | 0 5 1 12 -8 18 35 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1727{{c}}, ~5/4 = 380.2982{{c}}
* WE: ~2 = 1200.1727{{c}}, ~5/4 = 380.2982{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2483{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2483{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.271{{c}} -->


{{Optimal ET sequence|legend=0| 19g, 41, 60 }}
{{Optimal ET sequence|legend=0| 19g, 41, 60 }}
Line 213: Line 190:
Comma list: 100/99, 105/104, 144/143, 170/169, 171/169, 196/195
Comma list: 100/99, 105/104, 144/143, 170/169, 171/169, 196/195


Mapping: {{mapping| 1 0 2 -1 6 -2 -7 -4 | 0 5 1 12 -8 18 35 26 }}
{{Mapping|legend=0| 1 0 2 -1 6 -2 -7 -4 | 0 5 1 12 -8 18 35 26 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.2179{{c}}, ~5/4 = 380.3942{{c}}
* WE: ~2 = 1200.2179{{c}}, ~5/4 = 380.3942{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.3314{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.3314{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.360{{c}} -->


{{Optimal ET sequence|legend=0| 19gh, 41 }}
{{Optimal ET sequence|legend=0| 19gh, 41 }}
Line 231: Line 207:
Comma list: 100/99, 105/104, 144/143, 145/144, 155/154, 196/195
Comma list: 100/99, 105/104, 144/143, 145/144, 155/154, 196/195


Subgroup-val mapping: {{mapping| 1 0 2 -1 6 -2 2 4 | 0 5 1 12 -8 18 9 3 }}
{{Mapping|legend=2| 1 0 2 -1 6 -2 2 4 | 0 5 1 12 -8 18 9 3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.2802{{c}}, ~5/4 = 380.5053{{c}}
* WE: ~2 = 1200.2802{{c}}, ~5/4 = 380.5053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4258{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4258{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.478{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.416{{c}} -->


{{Optimal ET sequence|legend=0| 19, 22f, 41 }}
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}
Line 248: Line 222:
Comma list: 65/64, 78/77, 91/90, 100/99
Comma list: 65/64, 78/77, 91/90, 100/99


Mapping: {{mapping| 1 0 2 -1 6 4 | 0 5 1 12 -8 -1 }}
{{Mapping|legend=0| 1 0 2 -1 6 4 | 0 5 1 12 -8 -1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.2397{{c}}, ~5/4 = 380.8698{{c}}
* WE: ~2 = 1201.2397{{c}}, ~5/4 = 380.8698{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5080{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.674{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.477{{c}} -->


{{Optimal ET sequence|legend=0| 19, 22, 41f }}
{{Optimal ET sequence|legend=0| 19, 22, 41f }}


Badness (Sintel): 1.07
Badness (Sintel): 1.07
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 52/51, 65/64, 78/77, 91/90, 100/99
Mapping: {{mapping| 1 0 2 -1 6 4 6 | 0 5 1 12 -8 -1 -6 }}
Optimal tunings:
* WE: ~2 = 1200.4822{{c}}, ~5/4 = 380.8818{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7356{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.784{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.729{{c}} -->
{{Optimal ET sequence|legend=0| 19, 22 }}
Badness (Sintel): 1.21
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 52/51, 65/64, 78/77, 91/90, 100/99, 133/132
Mapping: {{mapping| 1 0 2 -1 6 4 6 9 | 0 5 1 12 -8 -1 -6 -15 }}
Optimal tunings:
* WE: ~2 = 1200.6684{{c}}, ~5/4 = 380.8328{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6212{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.625{{c}} -->
{{Optimal ET sequence|legend=0| 19, 22 }}
Badness (Sintel): 1.41


==== Necromancy ====
==== Necromancy ====
Line 298: Line 237:
Comma list: 100/99, 225/224, 245/243, 275/273
Comma list: 100/99, 225/224, 245/243, 275/273


Mapping: {{mapping| 1 0 2 -1 6 11 | 0 5 1 12 -8 -23 }}
{{Mapping|legend=0| 1 0 2 -1 6 11 | 0 5 1 12 -8 -23 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9675{{c}}, ~5/4 = 380.7770{{c}}
* WE: ~2 = 1199.9675{{c}}, ~5/4 = 380.7770{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7874{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7874{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.788{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.787{{c}} -->


{{Optimal ET sequence|legend=0| 19f, 22, 41, 63, 104 }}
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63, 104 }}
Line 315: Line 252:
Comma list: 100/99, 120/119, 154/153, 225/224, 273/272
Comma list: 100/99, 120/119, 154/153, 225/224, 273/272


Mapping: {{mapping| 1 0 2 -1 6 11 6 | 0 5 1 12 -8 -23 -6 }}
{{Mapping|legend=0| 1 0 2 -1 6 11 6 | 0 5 1 12 -8 -23 -6 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.6176{{c}}, ~5/4 = 380.7053{{c}}
* WE: ~2 = 1199.6176{{c}}, ~5/4 = 380.7053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.8280{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.8280{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.837{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.827{{c}} -->


{{Optimal ET sequence|legend=0| 19f, 22, 41, 63 }}
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63 }}


Badness (Sintel): 1.12
Badness (Sintel): 1.12
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 100/99, 120/119, 133/132, 154/153, 209/208, 225/224
Mapping: {{mapping| 1 0 2 -1 6 11 6 9 | 0 5 1 12 -8 -23 -6 -15 }}
Optimal tunings:
* WE: ~2 = 1199.6176{{c}}, ~5/4 = 380.6981{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7355{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.741{{c}} -->
{{Optimal ET sequence|legend=0| 19f, 22, 41 }}
Badness (Sintel): 1.28


==== Soothsaying ====
==== Soothsaying ====
Line 348: Line 267:
Comma list: 100/99, 225/224, 245/243, 1352/1331
Comma list: 100/99, 225/224, 245/243, 1352/1331


Mapping: {{mapping| 2 0 4 -2 12 15 | 0 5 1 12 -8 -12 }}
{{Mapping|legend=0| 2 0 4 -2 12 15 | 0 5 1 12 -8 -12 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~55/39 = 600.2918{{c}}, ~5/4 = 380.6928{{c}}
* WE: ~55/39 = 600.2918{{c}}, ~5/4 = 380.6928{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~5/4 = 380.5121{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~5/4 = 380.5121{{c}}
<!-- * CTE: ~55/39 = 600.000{{c}}, ~5/4 = 380.539{{c}}
* POTE: ~55/39 = 600.000{{c}}, ~5/4 = 380.508{{c}} -->


{{Optimal ET sequence|legend=0| 22, 60, 82 }}
{{Optimal ET sequence|legend=0| 22, 60, 82 }}
Line 365: Line 282:
Comma list: 100/99, 221/220, 225/224, 245/243, 273/272
Comma list: 100/99, 221/220, 225/224, 245/243, 273/272


Mapping: {{mapping| 2 0 4 -2 12 15 5 | 0 5 1 12 -8 -12 5 }}
{{Mapping|legend=0| 2 0 4 -2 12 15 5 | 0 5 1 12 -8 -12 5 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~17/12 = 600.2918{{c}}, ~5/4 = 380.6927{{c}}
* WE: ~17/12 = 600.2918{{c}}, ~5/4 = 380.6927{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.5135{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.5135{{c}}
<!-- * CTE: ~17/12 = 600.000{{c}}, ~5/4 = 380.553{{c}}
* POTE: ~17/12 = 600.000{{c}}, ~5/4 = 380.508{{c}} -->


{{Optimal ET sequence|legend=0| 22, 60, 82 }}
{{Optimal ET sequence|legend=0| 22, 60, 82 }}
Line 382: Line 297:
Comma list: 100/99, 133/132, 221/220, 225/224, 245/243, 273/272
Comma list: 100/99, 133/132, 221/220, 225/224, 245/243, 273/272


Mapping: {{mapping| 2 0 4 -2 12 15 5 18 | 0 5 1 12 -8 -12 5 -15 }}
{{Mapping|legend=0| 2 0 4 -2 12 15 5 18 | 0 5 1 12 -8 -12 5 -15 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~17/12 = 600.3301{{c}}, ~5/4 = 380.6797{{c}}
* WE: ~17/12 = 600.3301{{c}}, ~5/4 = 380.6797{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.4704{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.4704{{c}}
<!-- * CTE: ~17/12 = 600.000{{c}}, ~5/4 = 380.470{{c}}
* CWE: ~17/12 = 600.000{{c}}, ~5/4 = 380.470{{c}} -->


{{Optimal ET sequence|legend=0| 22, 60, 82 }}
{{Optimal ET sequence|legend=0| 22, 60, 82 }}
Line 399: Line 312:
Comma list: 55/54, 99/98, 176/175
Comma list: 55/54, 99/98, 176/175


Mapping: {{mapping| 1 0 2 -1 -1 | 0 5 1 12 14 }}
{{Mapping|legend=0| 1 0 2 -1 -1 | 0 5 1 12 14 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.7724{{c}}, ~5/4 = 381.2641{{c}}
* WE: ~2 = 1200.7724{{c}}, ~5/4 = 381.2641{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 381.0913{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 381.0913{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 381.231{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 381.019{{c}} -->


{{Optimal ET sequence|legend=0| 19e, 22, 41e, 63e }}
{{Optimal ET sequence|legend=0| 19e, 22, 41e, 63e }}
Line 416: Line 327:
Comma list: 55/54, 65/64, 91/90, 99/98
Comma list: 55/54, 65/64, 91/90, 99/98


Mapping: {{mapping| 1 0 2 -1 -1 4 | 0 5 1 12 14 -1 }}
{{Mapping|legend=0| 1 0 2 -1 -1 4 | 0 5 1 12 14 -1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1202.5634{{c}}, ~5/4 = 381.3348{{c}}
* WE: ~2 = 1202.5634{{c}}, ~5/4 = 381.3348{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6886{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6886{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 381.196{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.520{{c}} -->


{{Optimal ET sequence|legend=0| 19e, 22, 41ef }}
{{Optimal ET sequence|legend=0| 19e, 22, 41ef }}


Badness (Sintel): 1.05
Badness (Sintel): 1.05
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 55/54, 65/64, 85/84, 91/90, 99/98
Mapping: {{mapping| 1 0 2 -1 -1 4 -1 | 0 5 1 12 14 -1 16 }}
Optimal tunings:
* WE: ~2 = 1202.5560{{c}}, ~5/4 = 381.4295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7843{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 381.288{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.619{{c}} -->
{{Optimal ET sequence|legend=0| 19eg, 22, 41efg }}
Badness (Sintel): 1.03
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 55/54, 57/56, 65/64, 76/75, 85/84, 99/98
Mapping: {{mapping| 1 0 2 -1 -1 4 -1 2 | 0 5 1 12 14 -1 16 7 }}
Optimal tunings:
* WE: ~2 = 1203.4043{{c}}, ~5/4 = 381.5989{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7353{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 381.464{{c}} -->
{{Optimal ET sequence|legend=0| 19egh, 22, 41efghh }}
Badness (Sintel): 1.16


==== Intuition ====
==== Intuition ====
Line 466: Line 342:
Comma list: 55/54, 66/65, 99/98, 105/104
Comma list: 55/54, 66/65, 99/98, 105/104


Mapping: {{mapping| 1 0 2 -1 -1 -2 | 0 5 1 12 14 18 }}
{{Mapping|legend=0| 1 0 2 -1 -1 -2 | 0 5 1 12 14 18 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.3172{{c}}, ~5/4 = 380.9004{{c}}
* WE: ~2 = 1201.3172{{c}}, ~5/4 = 380.9004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5942{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5942{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.816{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.483{{c}} -->


{{Optimal ET sequence|legend=0| 19e, 22f }}
{{Optimal ET sequence|legend=0| 19e, 22f }}


Badness (Sintel): 1.08
Badness (Sintel): 1.08
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 55/54, 66/65, 85/84, 99/98, 105/104
Mapping: {{mapping| 1 0 2 -1 -1 -2 -1 | 0 5 1 12 14 18 16 }}
Optimal tunings:
* WE: ~2 = 1201.3593{{c}}, ~5/4 = 381.0356{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7204{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.952{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.604{{c}} -->
{{Optimal ET sequence|legend=0| 19eg, 22f }}
Badness (Sintel): 1.03
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 55/54, 66/65, 77/76, 85/84, 99/98, 105/104
Mapping: {{mapping| 1 0 2 -1 -1 -2 -1 -4 | 0 5 1 12 14 18 16 26 }}
Optimal tunings:
* WE: ~2 = 1201.5266{{c}}, ~5/4 = 381.0019{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6275{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.866{{c}} -->
{{Optimal ET sequence|legend=0| 19egh, 22fh }}
Badness (Sintel): 1.19


=== Horcrux ===
=== Horcrux ===
Line 516: Line 357:
Comma list: 45/44, 56/55, 245/243
Comma list: 45/44, 56/55, 245/243


Mapping: {{mapping| 1 0 2 -1 0 | 0 5 1 12 11 }}
{{Mapping|legend=0| 1 0 2 -1 0 | 0 5 1 12 11 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.4670{{c}}, ~5/4 = 379.7895{{c}}
* WE: ~2 = 1200.4670{{c}}, ~5/4 = 379.7895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.6889{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.6889{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 379.796{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 379.642{{c}} -->


{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}
{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}
Line 533: Line 372:
Comma list: 45/44, 56/55, 78/77, 245/243
Comma list: 45/44, 56/55, 78/77, 245/243


Mapping: {{mapping| 1 0 2 -1 0 -2 | 0 5 1 12 11 18 }}
{{Mapping|legend=0| 1 0 2 -1 0 -2 | 0 5 1 12 11 18 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.2953{{c}}, ~5/4 = 379.8842{{c}}
* WE: ~2 = 1200.2953{{c}}, ~5/4 = 379.8842{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.8165{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.8165{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 379.885{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 379.791{{c}} -->


{{Optimal ET sequence|legend=0| 3def, 16dff, 19 }}
{{Optimal ET sequence|legend=0| 3def, 16dff, 19 }}
Line 550: Line 387:
Comma list: 45/44, 56/55, 78/77, 85/84, 245/243
Comma list: 45/44, 56/55, 78/77, 85/84, 245/243


Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 16 }}
{{Mapping|legend=0| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 16 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.2484{{c}}, ~5/4 = 380.2053{{c}}
* WE: ~2 = 1200.2484{{c}}, ~5/4 = 380.2053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1482{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1482{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.216{{c}} -->


{{Optimal ET sequence|legend=0| 3defg, 16dffgg, 19g }}
{{Optimal ET sequence|legend=0| 3defg, 16dffgg, 19g }}
Line 566: Line 402:
Comma list: 35/34, 45/44, 52/51, 56/55, 245/243
Comma list: 35/34, 45/44, 52/51, 56/55, 245/243


Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 13 }}
{{Mapping|legend=0| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 13 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.5457{{c}}, ~5/4 = 379.4681{{c}}
* WE: ~2 = 1199.5457{{c}}, ~5/4 = 379.4681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.5713{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.5713{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 379.524{{c}} -->


{{Optimal ET sequence|legend=0| 3defg, 16dff, 19 }}
{{Optimal ET sequence|legend=0| 3defg, 16dff, 19 }}
Line 582: Line 417:
Comma list: 45/44, 56/55, 65/64, 245/243
Comma list: 45/44, 56/55, 65/64, 245/243


Mapping: {{mapping| 1 0 2 -1 0 4 | 0 5 1 12 11 -1 }}
{{Mapping|legend=0| 1 0 2 -1 0 4 | 0 5 1 12 11 -1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1202.2187{{c}}, ~5/4 = 379.8171{{c}}
* WE: ~2 = 1202.2187{{c}}, ~5/4 = 379.8171{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.2709{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.2709{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 379.757{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 379.116{{c}} -->


{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}
{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}
Line 599: Line 432:
Comma list: 225/224, 245/243, 441/440
Comma list: 225/224, 245/243, 441/440


Mapping: {{mapping| 1 0 2 -1 -7 | 0 5 1 12 33 }}
{{Mapping|legend=0| 1 0 2 -1 -7 | 0 5 1 12 33 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.2634{{c}}, ~5/4 = 380.6321{{c}}
* WE: ~2 = 1201.2634{{c}}, ~5/4 = 380.6321{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2849{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2849{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.416{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.232{{c}} -->


{{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd, 243ccdde }}
{{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd, 243ccdde }}
Line 616: Line 447:
Comma list: 105/104, 196/195, 245/243, 275/273
Comma list: 105/104, 196/195, 245/243, 275/273


Mapping: {{mapping| 1 0 2 -1 -7 -2 | 0 5 1 12 33 18 }}
{{Mapping|legend=0| 1 0 2 -1 -7 -2 | 0 5 1 12 33 18 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.0424{{c}}, ~5/4 = 380.5193{{c}}
* WE: ~2 = 1201.0424{{c}}, ~5/4 = 380.5193{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2349{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2349{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.352{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.189{{c}} -->


{{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd }}
{{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd }}
Line 633: Line 462:
Comma list: 105/104, 154/153, 170/169, 196/195, 245/243
Comma list: 105/104, 154/153, 170/169, 196/195, 245/243


Mapping: {{mapping| 1 0 2 -1 -7 -2 -7 | 0 5 1 12 33 18 35 }}
{{Mapping|legend=0| 1 0 2 -1 -7 -2 -7 | 0 5 1 12 33 18 35 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.1638{{c}}, ~5/4 = 380.4827{{c}}
* WE: ~2 = 1201.1638{{c}}, ~5/4 = 380.4827{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1599{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1599{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.281{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.114{{c}} -->


{{Optimal ET sequence|legend=0| 19eg, 41, 60e, 101cd }}
{{Optimal ET sequence|legend=0| 19eg, 41, 60e, 101cd }}


Badness (Sintel): 1.06
Badness (Sintel): 1.06
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 105/104, 133/132, 154/153, 170/169, 171/169, 196/195
Mapping: {{mapping| 1 0 2 -1 -7 -2 -7 -4 | 0 5 1 12 33 18 35 26 }}
Optimal tunings:
* WE: ~2 = 1201.2528{{c}}, ~5/4 = 380.5518{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2052{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.339{{c}} -->
{{Optimal ET sequence|legend=0| 19egh, 41, 60eh }}
Badness (Sintel): 1.13


=== Divination ===
=== Divination ===
Line 666: Line 477:
Comma list: 121/120, 225/224, 245/243
Comma list: 121/120, 225/224, 245/243


Mapping: {{mapping| 2 0 4 -2 5 | 0 5 1 12 3 }}
{{Mapping|legend=0| 2 0 4 -2 5 | 0 5 1 12 3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 600.8306{{c}}, ~5/4 = 380.7598{{c}}
* WE: ~99/70 = 600.8306{{c}}, ~5/4 = 380.7598{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~5/4 = 380.3800{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~5/4 = 380.3800{{c}}
<!-- * CTE: ~99/70 = 600.000{{c}}, ~5/4 = 380.732{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~5/4 = 380.223{{c}} -->


{{Optimal ET sequence|legend=0| 22, 38d, 60e, 142cdee, 202ccddeee }}
{{Optimal ET sequence|legend=0| 22, 38d, 60e, 142cdee, 202ccddeee }}
Line 683: Line 492:
Comma list: 105/104, 121/120, 196/195, 245/243
Comma list: 105/104, 121/120, 196/195, 245/243


Mapping: {{mapping| 2 0 4 -2 5 -4 | 0 5 1 12 3 18 }}
{{Mapping|legend=0| 2 0 4 -2 5 -4 | 0 5 1 12 3 18 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 600.9624{{c}}, ~5/4 = 380.5297{{c}}
* WE: ~99/70 = 600.9624{{c}}, ~5/4 = 380.5297{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~5/4 = 380.0614{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~5/4 = 380.0614{{c}}
<!-- * CTE: ~99/70 = 600.000{{c}}, ~5/4 = 380.417{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~5/4 = 379.920{{c}} -->


{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}
{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}
Line 700: Line 507:
Comma list: 105/104, 121/120, 154/153, 196/195, 245/243
Comma list: 105/104, 121/120, 154/153, 196/195, 245/243


Mapping: {{mapping| 2 0 4 -2 5 -4 5 | 0 5 1 12 3 18 5 }}
{{Mapping|legend=0| 2 0 4 -2 5 -4 5 | 0 5 1 12 3 18 5 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~17/12 = 600.8921{{c}}, ~5/4 = 380.5094{{c}}
* WE: ~17/12 = 600.8921{{c}}, ~5/4 = 380.5094{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.0672{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.0672{{c}}
<!-- * CTE: ~17/12 = 600.000{{c}}, ~5/4 = 380.433{{c}} -->


{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}
{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}
Line 716: Line 522:
Comma list: 225/224, 243/242, 245/242
Comma list: 225/224, 243/242, 245/242


Mapping: {{mapping| 1 -5 1 -13 -13 | 0 10 2 24 25 }}
{{Mapping|legend=0| 1 -5 1 -13 -13 | 0 10 2 24 25 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.0749{{c}}, ~11/7 = 790.7980{{c}}
* WE: ~2 = 1201.0749{{c}}, ~11/7 = 790.7980{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.1429{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.1429{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/7 = 790.240{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/7 = 790.090{{c}} -->


{{Optimal ET sequence|legend=0| 38d, 41, 120cd }}
{{Optimal ET sequence|legend=0| 38d, 41, 120cd }}
Line 733: Line 537:
Comma list: 105/104, 196/195, 243/242, 245/242
Comma list: 105/104, 196/195, 243/242, 245/242


Mapping: {{mapping| 1 -5 1 -13 -13 -20 | 0 10 2 24 25 36 }}
{{Mapping|legend=0| 1 -5 1 -13 -13 -20 | 0 10 2 24 25 36 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.2830{{c}}, ~11/7 = 790.8409{{c}}
* WE: ~2 = 1201.2830{{c}}, ~11/7 = 790.8409{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.0516{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.0516{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/7 = 790.158{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/7 = 789.996{{c}} -->


{{Optimal ET sequence|legend=0| 38df, 41, 79d, 120cd }}
{{Optimal ET sequence|legend=0| 38df, 41, 79d, 120cd }}
Line 750: Line 552:
Comma list: 105/104, 154/153, 196/195, 243/242, 245/242
Comma list: 105/104, 154/153, 196/195, 243/242, 245/242


Mapping: {{mapping| 1 -5 1 -13 -13 -20 -15 | 0 10 2 24 25 36 29 }}
{{Mapping|legend=0| 1 -5 1 -13 -13 -20 -15 | 0 10 2 24 25 36 29 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.1557{{c}}, ~11/7 = 790.7157{{c}}
* WE: ~2 = 1201.1557{{c}}, ~11/7 = 790.7157{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.0057{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.0057{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/7 = 790.104{{c}} -->


{{Optimal ET sequence|legend=0| 38df, 41, 79d }}
{{Optimal ET sequence|legend=0| 38df, 41, 79d }}
Line 766: Line 567:
Comma list: 105/104, 154/153, 196/195, 243/242, 245/242
Comma list: 105/104, 154/153, 196/195, 243/242, 245/242


Mapping: {{mapping| 1 -5 1 -13 -13 -20 -3 | 0 10 2 24 25 36 29 11 }}
{{Mapping|legend=0| 1 -5 1 -13 -13 -20 -3 | 0 10 2 24 25 36 29 11 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.3558{{c}}, ~11/7 = 790.8266{{c}}
* WE: ~2 = 1201.3558{{c}}, ~11/7 = 790.8266{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 789.9880{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 789.9880{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/7 = 790.116{{c}} -->


{{Optimal ET sequence|legend=0| 38df, 41, 79dh }}
{{Optimal ET sequence|legend=0| 38df, 41, 79dh }}
Line 793: Line 593:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 378.5328{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 378.5328{{c}}
: error map: {{val| 0.000 -9.291 -7.781 -18.555 }}
: error map: {{val| 0.000 -9.291 -7.781 -18.555 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 378.744{{c}}
: [[error map]]: {{val| 0.000 -8.235 -7.570 -20.034 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 378.479{{c}}
: error map: {{val| 0.000 -9.558 -7.834 -18.181 }} -->


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 812: Line 608:
Comma list: 45/44, 126/125, 385/384
Comma list: 45/44, 126/125, 385/384


Mapping: {{mapping| 1 0 2 5 0 | 0 5 1 -7 11 }}
{{Mapping|legend=0| 1 0 2 5 0 | 0 5 1 -7 11 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1203.0804{{c}}, ~5/4 = 378.6936{{c}}
* WE: ~2 = 1203.0804{{c}}, ~5/4 = 378.6936{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8174{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8174{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 378.228{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 377.724{{c}} -->


Tuning ranges:  
Tuning ranges:  
Line 833: Line 627:
Comma list: 45/44, 65/64, 78/77, 126/125
Comma list: 45/44, 65/64, 78/77, 126/125


Mapping: {{mapping| 1 0 2 5 0 4 | 0 5 1 -7 11 -1 }}
{{Mapping|legend=0| 1 0 2 5 0 4 | 0 5 1 -7 11 -1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1203.4291{{c}}, ~5/4 = 378.7321{{c}}
* WE: ~2 = 1203.4291{{c}}, ~5/4 = 378.7321{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.7336{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.7336{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 378.176{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 377.653{{c}} -->


{{Optimal ET sequence|legend=0| 16, 19, 35f, 54bdf }}
{{Optimal ET sequence|legend=0| 16, 19, 35f, 54bdf }}
Line 850: Line 642:
Comma list: 33/32, 126/125, 176/175
Comma list: 33/32, 126/125, 176/175


Mapping: {{mapping| 1 0 2 5 5 | 0 5 1 -7 -5 }}
{{Mapping|legend=0| 1 0 2 5 5 | 0 5 1 -7 -5 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1205.6044{{c}}, ~5/4 = 379.5966{{c}}
* WE: ~2 = 1205.6044{{c}}, ~5/4 = 379.5966{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8142{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8142{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 377.724{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 377.832{{c}} -->


{{Optimal ET sequence|legend=0| 3, 16, 19e, 35ee, 54bdeee }}
{{Optimal ET sequence|legend=0| 3, 16, 19e, 35ee, 54bdeee }}
Line 867: Line 657:
Comma list: 33/32, 65/64, 105/104, 126/125
Comma list: 33/32, 65/64, 105/104, 126/125


Mapping: {{mapping| 1 0 2 5 5 4 | 0 5 1 -7 -5 -1 }}
{{Mapping|legend=0| 1 0 2 5 5 4 | 0 5 1 -7 -5 -1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1205.4897{{c}}, ~5/4 = 379.5667{{c}}
* WE: ~2 = 1205.4897{{c}}, ~5/4 = 379.5667{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8118{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8118{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 377.652{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 377.838{{c}} -->


{{Optimal ET sequence|legend=0| 3, 16, 19e, 35eef }}
{{Optimal ET sequence|legend=0| 3, 16, 19e, 35eef }}
Line 879: Line 667:
Badness (Sintel): 1.19
Badness (Sintel): 1.19


== Darkstone ==
== Brightstone ==
Darkstone (16 & 19d) is a low-accuacy temperament which tempers out 36/35 and 1875/1792. It makes the major third and the fifth even flatter than those of muggles. In [http://www.tonalsoft.com/enc/m/magic.aspx ''Encyclopedia of Microtonal Music Theory'', Tonalsoft], this temperament is given a name ''witch''.
Brightstone tempers out 64/63 and may be described as {{nowrap| 22 & 25 }}. 22edo itself is a good tuning, in which case it is identical to magic. Brightstone can be extended to the 11- and 13-limit in a similar way to muggles.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 1875/1792
[[Comma list]]: 64/63, 3125/3024


{{Mapping|legend=1| 1 0 2 0 | 0 5 1 9 }}
{{Mapping|legend=1| 1 0 2 6 | 0 5 1 -10 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.7458{{c}}, ~5/4 = 377.2996{{c}}
* [[WE]]: ~2 = 1198.1701{{c}}, ~5/4 = 381.3741{{c}}
: [[error map]]: {{val| +1.746 -15.457 -5.523 +26.870 }}
: error map: {{val| -1.830 +4.915 -8.599 +6.454 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 376.9630{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 381.9562{{c}}
: error map: {{val| 0.000 -17.140 -9.351 +23.841 }}
: error map: {{val| 0.000 +7.826 -4.358 +11.613 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 377.385{{c}}
: error map: {{val| 0.000 -15.028 -8.928 +27.643 }} -->


{{Optimal ET sequence|legend=1| 3d, …, 13b, 16 }}
{{Optimal ET sequence|legend=1| 3, 19d, 22 }}


[[Badness]] (Sintel): 2.13
[[Badness]] (Sintel): 2.23


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 36/35, 45/44, 363/343
Comma list: 55/54, 64/63, 625/616


Mapping: {{mapping| 1 0 2 0 0 | 0 5 1 9 11 }}
{{Mapping|legend=0| 1 0 2 6 -1 | 0 5 1 -10 14 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
* WE: ~2 = 1198.5372{{c}}, ~5/4 = 381.7556{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.9735{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.1943{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 377.388{{c}} -->


{{Optimal ET sequence|legend=0| 3de, 13be, 16 }}
{{Optimal ET sequence|legend=0| 22, 69b }}


Badness (Sintel): 1.55
Badness (Sintel): 1.85


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


Comma list: 27/26, 36/35, 45/44, 363/343
Comma list: 40/39, 55/54, 64/63, 625/616


Mapping: {{mapping| 1 0 2 0 0 -1 | 0 5 1 9 11 15 }}
{{Mapping|legend=0| 1 0 2 6 6 4 | 0 5 1 -10 14 -4 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
* WE: ~2 = 1197.2300{{c}}, ~5/4 = 381.6164{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.4221{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.4690{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 376.914{{c}} -->
 
{{Optimal ET sequence|legend=0| 22f, 47bff }}
 
Badness (Sintel): 1.99


{{Optimal ET sequence|legend=0| 3def, 13beff, 16 }}
== Darkstone ==
Darkstone (16 & 19d) is a low-accuracy temperament which tempers out 36/35 and 1875/1792. It makes the major third and the fifth even flatter than those of muggles.


Badness (Sintel): 1.58
This temperament is known as ''witch'' in [http://www.tonalsoft.com/enc/m/magic.aspx Tonalsoft Encyclopedia].  


== Brightstone ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 3125/3024
[[Comma list]]: 36/35, 1875/1792


{{Mapping|legend=1| 1 0 2 6 | 0 5 1 -10 }}
{{Mapping|legend=1| 1 0 2 0 | 0 5 1 9 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1198.1701{{c}}, ~5/4 = 381.3741{{c}}
* [[WE]]: ~2 = 1201.7458{{c}}, ~5/4 = 377.2996{{c}}
: error map: {{val| -1.830 +4.915 -8.599 +6.454 }}
: [[error map]]: {{val| +1.746 -15.457 -5.523 +26.870 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 381.9562{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 376.9630{{c}}
: error map: {{val| 0.000 +7.826 -4.358 +11.613 }}
: error map: {{val| 0.000 -17.140 -9.351 +23.841 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 381.955{{c}}
: error map: {{val| 0.000 +7.818 -4.359 +11.627 }} -->


{{Optimal ET sequence|legend=1| 3, 19d, 22 }}
{{Optimal ET sequence|legend=1| 3d, …, 13b, 16 }}


[[Badness]] (Sintel): 2.23
[[Badness]] (Sintel): 2.13


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 64/63, 100/99, 605/588
Comma list: 36/35, 45/44, 363/343


Mapping: {{mapping| 1 0 2 6 6 | 0 5 1 -10 -8 }}
{{Mapping|legend=0| 1 0 2 0 0 | 0 5 1 9 11 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1198.8032{{c}}, ~5/4 = 381.3145{{c}}
* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 381.7119{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.9735{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 381.790{{c}} -->


{{Optimal ET sequence|legend=0| 3, 19d, 22 }}
{{Optimal ET sequence|legend=0| 3de, 13be, 16 }}


Badness (Sintel): 1.57
Badness (Sintel): 1.55


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


Comma list: 64/63, 65/63, 100/99, 169/165
Comma list: 27/26, 36/35, 45/44, 363/343


Mapping: {{mapping| 1 0 2 6 6 4 | 0 5 1 -10 -8 -1 }}
{{Mapping|legend=0| 1 0 2 0 0 -1 | 0 5 1 9 11 15 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9726{{c}}, ~5/4 = 381.7271{{c}}
* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 381.7362{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.4221{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 381.732{{c}} -->
 
{{Optimal ET sequence|legend=0| 3def, 13beff, 16 }}
 
Badness (Sintel): 1.58


{{Optimal ET sequence|legend=0| 3, 22 }}
Scales: [[User:BudjarnLambeth/Volcanic glass]]


Badness (Sintel): 1.64
Music: [https://www.youtube.com/watch?v=oXy_RXrBVWA ''Rain in the crystal mirror caves''] - [[Budjarn Lambeth]] (2026)


== Hocum ==
== Hocum ==
Line 997: Line 785:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~63/40 = 790.1542{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~63/40 = 790.1542{{c}}
: error map: {{val| 0.000 -0.413 -6.005 -1.447 }}
: error map: {{val| 0.000 -0.413 -6.005 -1.447 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~63/40 = 790.164{{c}}
: [[error map]]: {{val| 0.000 -0.316 -5.986 -1.612 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~63/40 = 789.892{{c}}
: error map: {{val| 0.000 -0.437 -6.010 -1.406 }} -->


{{Optimal ET sequence|legend=1| 3, 38, 41, 161c }}
{{Optimal ET sequence|legend=1| 3, 38, 41, 161c }}
Line 1,023: Line 807:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~168/125 = 526.7349{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~168/125 = 526.7349{{c}}
: error map: {{val| 0.000 -0.932 -6.109 -2.500 }}
: error map: {{val| 0.000 -0.932 -6.109 -2.500 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~168/125 = 526.813{{c}}
: [[error map]]: {{val| 0.000 +0.240 -5.875 -2.891 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~168/125 = 526.710{{c}}
: error map: {{val| 0.000 -0.932 -6.109 -2.500 }} -->


{{Optimal ET sequence|legend=1| 16, 25, 41, 139c, 180c, 221cc, 262ccd }}
{{Optimal ET sequence|legend=1| 16, 25, 41, 139c, 180c, 221cc, 262ccd }}
Line 1,037: Line 817:
Comma list: 385/384, 441/440, 625/616
Comma list: 385/384, 441/440, 625/616


Mapping: {{mapping| 1 -5 1 5 -4 | 0 15 3 -5 17 }}
{{Mapping|legend=0| 1 -5 1 5 -4 | 0 15 3 -5 17 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.8404{{c}}, ~15/11 = 527.0289{{c}}
* WE: ~2 = 1200.8404{{c}}, ~15/11 = 527.0289{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6826{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6826{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~15/11 = 526.759{{c}}
* POTE: ~2 = 1200.000{{c}}, ~15/11 = 526.660{{c}} -->


{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}
{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}
Line 1,054: Line 832:
Comma list: 105/104, 144/143, 275/273, 625/616
Comma list: 105/104, 144/143, 275/273, 625/616


Mapping: {{mapping| 1 -5 1 5 -4 -2 | 0 15 3 -5 17 13 }}
{{Mapping|legend=0| 1 -5 1 5 -4 -2 | 0 15 3 -5 17 13 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.4759{{c}}, ~15/11 = 526.8502{{c}}
* WE: ~2 = 1200.4759{{c}}, ~15/11 = 526.8502{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6548{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6548{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~15/11 = 526.706{{c}}
* POTE: ~2 = 1200.000{{c}}, ~15/11 = 526.641{{c}} -->


{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}
{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}
Line 1,071: Line 847:
Comma list: 105/104, 144/143, 441/440, 210/209, 625/616
Comma list: 105/104, 144/143, 441/440, 210/209, 625/616


Subgroup-val mapping: {{mapping| 1 -5 1 5 -4 -2 6 | 0 15 3 -5 17 13 -4 }}
{{Mapping|legend=2| 1 -5 1 5 -4 -2 6 | 0 15 3 -5 17 13 -4 }}


Optimal tunings:
Optimal tunings:
Line 1,095: Line 871:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/14 = 995.0515{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/14 = 995.0515{{c}}
: error map: {{val| 0.000 -0.926 -6.108 -3.466 }}
: error map: {{val| 0.000 -0.926 -6.108 -3.466 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~25/14 = 995.140{{c}}
: [[error map]]: {{val| 0.000 +0.853 -5.752 -2.843 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~25/14 = 995.013{{c}}
: error map: {{val| 0.000 -1.691 -6.261 -3.733 }} -->


{{Optimal ET sequence|legend=1| 6, …, 35, 41, 158cd, 199ccd, 240ccd, 281ccd }}
{{Optimal ET sequence|legend=1| 6, …, 35, 41, 158cd, 199ccd, 240ccd, 281ccd }}
Line 1,109: Line 881:
Comma list: 245/242, 385/384, 625/616
Comma list: 245/242, 385/384, 625/616


Mapping: {{mapping| 1 -15 -1 -3 -4 | 0 20 4 7 9 }}
{{Mapping|legend=0| 1 -15 -1 -3 -4 | 0 20 4 7 9 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.6716{{c}}, ~25/14 = 995.6009{{c}}
* WE: ~2 = 1200.6716{{c}}, ~25/14 = 995.6009{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 995.0633{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 995.0633{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~25/14 = 995.119{{c}}
* POTE: ~2 = 1200.000{{c}}, ~25/14 = 995.044{{c}} -->


{{Optimal ET sequence|legend=0| 6, 35, 41 }}
{{Optimal ET sequence|legend=0| 6, 35, 41 }}
Line 1,126: Line 896:
Comma list: 105/104, 144/143, 245/242, 625/616
Comma list: 105/104, 144/143, 245/242, 625/616


Mapping: {{mapping| 1 5 3 4 5 9 | 0 -20 -4 -7 -9 -31 }}
{{Mapping|legend=0| 1 -15 -1 -3 -22 | 0 20 4 7 9 31 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.6276{{c}}, ~25/14 = 995.4920{{c}}
* WE: ~2 = 1200.6276{{c}}, ~25/14 = 995.4920{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 994.9901{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 994.9901{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~25/14 = 995.044{{c}}
* POTE: ~2 = 1200.000{{c}}, ~25/14 = 994.972{{c}} -->


{{Optimal ET sequence|legend=0| 6f, 35f, 41, 117c }}
{{Optimal ET sequence|legend=0| 6f, 35f, 41, 117c }}
Line 1,152: Line 920:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 556.0504{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 556.0504{{c}}
: error map: {{val| 0.000 -0.695 -6.062 -3.868 }}
: error map: {{val| 0.000 -0.695 -6.062 -3.868 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~48/35 = 556.123{{c}}
: [[error map]]: {{val| 0.000 +1.132 -5.696 -2.480 }} -->


{{Optimal ET sequence|legend=1| 13b, 28b, 41, 177bcd, 218bccdd, 259bccdd, 300cccdd }}
{{Optimal ET sequence|legend=1| 13b, 28b, 41, 177bcd, 218bccdd, 259bccdd, 300cccdd }}
Line 1,164: Line 930:
Comma list: 385/384, 625/616, 2401/2376
Comma list: 385/384, 625/616, 2401/2376


Mapping: {{mapping| 1 -10 0 -6 3 | 0 25 5 19 1 }}
{{Mapping|legend=0| 1 -10 0 -6 3 | 0 25 5 19 1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.4252{{c}}, ~11/8 = 556.2831{{c}}
* WE: ~2 = 1200.4252{{c}}, ~11/8 = 556.2831{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 556.0951{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 556.0951{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/8 = 556.122{{c}} -->


{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}
{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}
Line 1,180: Line 945:
Comma list: 196/195, 364/363, 385/384, 625/616
Comma list: 196/195, 364/363, 385/384, 625/616


Mapping: {{mapping| 1 -10 0 -6 3 0 | 0 25 5 19 1 8 }}
{{Mapping|legend=0| 1 -10 0 -6 3 0 | 0 25 5 19 1 8 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.8239{{c}}, ~11/8 = 556.0389{{c}}
* WE: ~2 = 1199.8239{{c}}, ~11/8 = 556.0389{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 556.1171{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 556.1171{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/8 = 556.106{{c}} -->


{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}
{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}
Line 1,205: Line 969:
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~5/4 = 379.9965{{c}} (~256/245 = 100.0035{{c}})
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~5/4 = 379.9965{{c}} (~256/245 = 100.0035{{c}})
: error map: {{val| 0.000 -1.972 -6.317 -8.826 }}
: error map: {{val| 0.000 -1.972 -6.317 -8.826 }}
<!-- * [[CTE]]: ~8/7 = 240.000{{c}}, ~5/4 = 380.499{{c}} (~256/245 = 99.501{{c}})
: [[error map]]: {{val| 0.000 +0.542 -5.814 -8.826 }} -->


{{Optimal ET sequence|legend=1| 25, 35, 60 }}
{{Optimal ET sequence|legend=1| 25, 35, 60 }}
Line 1,212: Line 974:
[[Badness]] (Sintel): 7.27
[[Badness]] (Sintel): 7.27


[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Magic family| ]] <!-- main article -->
[[Category:Magic family| ]] <!-- main article -->
[[Category:Magic| ]] <!-- key article -->
[[Category:Magic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Magic_family"