Magic family: Difference between revisions

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Consolidate existing info on 11-limit magic. Address 13-limit magic.
Switch to Sintel's badness, WE & CWE tunings, per community consensus (1/)
Line 22: Line 22:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 380.499{{c}}
* [[WE]]: ~2 = 1201.2449{{c}}, ~5/4 = 380.4527{{c}}
: [[error map]]: {{val| +1.245 +0.309 -3.371 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.2194{{c}}
: 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 }}
: [[error map]]: {{val| 0.000 +0.542 -5.814 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 380.058{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 380.058{{c}}
: error map: {{val| 0.000 -1.663 -6.255 }}
: error map: {{val| 0.000 -1.663 -6.255 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 39: Line 43:
{{Optimal ET sequence|legend=1| 3, 13b, 16, 19, 41, 60, 221cc, 281cc }}
{{Optimal ET sequence|legend=1| 3, 13b, 16, 19, 41, 60, 221cc, 281cc }}


[[Badness]] (Smith): 0.039163
[[Badness]] (Sintel): 0.919


=== Overview to extensions ===
=== Overview to extensions ===
Line 68: Line 72:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 380.651{{c}}
* [[WE]]: ~2 = 1201.0786{{c}}, ~5/4 = 380.6939{{c}}
: [[error map]]: {{val| +1.079 +1.514 -3.463 -1.578 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.4576{{c}}
: 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 }}
: [[error map]]: {{val| 0.000 +1.301 -5.662 -1.011 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 380.352{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 380.352{{c}}
: error map: {{val| 0.000 -0.195 -5.962 -4.602 }}
: error map: {{val| 0.000 -0.195 -5.962 -4.602 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 85: Line 93:
{{Optimal ET sequence|legend=1| 19, 41, 142cd, 183cd, 224ccd }}
{{Optimal ET sequence|legend=1| 19, 41, 142cd, 183cd, 224ccd }}


[[Badness]] (Smith): 0.018918
[[Badness]] (Sintel): 0.479


=== 11-limit ===
=== 11-limit ===
Line 95: Line 103:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.720{{c}}
* WE: ~2 = 1200.1372{{c}}, ~5/4 = 380.7399{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.696{{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 108: Line 118:
{{Optimal ET sequence|legend=0| 19, 22, 41, 104 }}
{{Optimal ET sequence|legend=0| 19, 22, 41, 104 }}


Badness (Smith): 0.020352
Badness (Sintel): 0.673


==== 13-limit ====
==== 13-limit ====
Line 120: Line 130:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.435{{c}}
* WE: ~2 = 1200.0331{{c}}, ~5/4 = 380.4377{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.427{{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 129: Line 141:
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}


Badness (Smith): 0.021509
Badness (Sintel): 0.889


===== Magical =====
===== Magical =====
Line 139: Line 151:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.510{{c}}
* WE: ~2 = 1199.3584{{c}}, ~5/4 = 380.4006{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.604{{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 }}


Badness (Smith): 0.020633
Badness (Sintel): 1.05


====== 19-limit ======
====== 19-limit ======
Line 154: Line 168:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.451{{c}}
* WE: ~2 = 1199.7173{{c}}, ~5/4 = 380.3808{{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 }}


Badness (Smith): 0.020881
Badness (Sintel): 1.27


====== Magica ======
====== Magica ======
Line 168: Line 184:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.573{{c}}
* WE: ~2 = 1199.3670{{c}}, ~5/4 = 380.4681{{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 }}


Badness (Smith): 0.019945
Badness (Sintel): 1.21


===== Magia =====
===== Magia =====
Line 182: Line 200:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.271{{c}}
* WE: ~2 = 1200.1727{{c}}, ~5/4 = 380.2982{{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 }}


Badness (Smith): 0.026232
Badness (Sintel): 1.34


====== 19-limit ======
====== 19-limit ======
Line 196: Line 216:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.360{{c}}
* WE: ~2 = 1200.2179{{c}}, ~5/4 = 380.3942{{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 }}


Badness (Smith): 0.023709
Badness (Sintel): 1.44


===== Evening =====
===== Evening =====
Evening is a remarkable subgroup temperament of 19 &amp; 41 with prime harmonics of 29 and 31.  
Evening is a remarkable subgroup temperament of {{nowrap| 19 & 22f }} with prime harmonics of 29 and 31.  


Subgroup: 2.3.5.7.11.13.29.31
Subgroup: 2.3.5.7.11.13.29.31
Line 209: Line 231:
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


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.478{{c}}
* WE: ~2 = 1200.2802{{c}}, ~5/4 = 380.5053{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.416{{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 }}
Badness (Sintel): 0.807


==== Sorcery ====
==== Sorcery ====
Line 225: Line 251:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.674{{c}}
* WE: ~2 = 1201.2397{{c}}, ~5/4 = 380.8698{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.477{{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 (Smith): 0.025829
Badness (Sintel): 1.07


===== 17-limit =====
===== 17-limit =====
Line 240: Line 268:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.784{{c}}
* WE: ~2 = 1200.4822{{c}}, ~5/4 = 380.8818{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.729{{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 }}
{{Optimal ET sequence|legend=0| 19, 22 }}


Badness (Smith): 0.023768
Badness (Sintel): 1.21


===== 19-limit =====
===== 19-limit =====
Line 255: Line 285:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.625{{c}}
* WE: ~2 = 1200.6684{{c}}, ~5/4 = 380.8328{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 380.621{{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 }}
{{Optimal ET sequence|legend=0| 19, 22 }}


Badness (Smith): 0.023232
Badness (Sintel): 1.41


==== Necromancy ====
==== Necromancy ====
Line 270: Line 301:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.788{{c}}
* WE: ~2 = 1199.9675{{c}}, ~5/4 = 380.7770{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.787{{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 }}


Badness (Smith): 0.025275
Badness (Sintel): 1.04


===== 17-limit =====
===== 17-limit =====
Line 285: Line 318:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.837{{c}}
* WE: ~2 = 1199.6176{{c}}, ~5/4 = 380.7053{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.827{{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 (Smith): 0.022032
Badness (Sintel): 1.12


===== 19-limit =====
===== 19-limit =====
Line 300: Line 335:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.741{{c}}
* WE: ~2 = 1199.6176{{c}}, ~5/4 = 380.6981{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 380.735{{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 }}
{{Optimal ET sequence|legend=0| 19f, 22, 41 }}


Badness (Smith): 0.021101
Badness (Sintel): 1.28


==== Soothsaying ====
==== Soothsaying ====
Line 315: Line 351:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 600.000{{c}}, ~5/4 = 380.539{{c}}
* WE: ~55/39 = 600.2918{{c}}, ~5/4 = 380.6928{{c}}
* POTE: ~55/39 = 600.000{{c}}, ~5/4 = 380.508{{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 }}


Badness (Smith): 0.055443
Badness (Sintel): 2.29


===== 17-limit =====
===== 17-limit =====
Line 330: Line 368:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~5/4 = 380.553{{c}}
* WE: ~17/12 = 600.2918{{c}}, ~5/4 = 380.6927{{c}}
* POTE: ~17/12 = 600.000{{c}}, ~5/4 = 380.508{{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 }}


Badness (Smith): 0.035654
Badness (Sintel): 1.82


===== 19-limit =====
===== 19-limit =====
Line 345: Line 385:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~5/4 = 380.470{{c}}
* WE: ~17/12 = 600.3301{{c}}, ~5/4 = 380.6797{{c}}
* CWE: ~17/12 = 600.000{{c}}, ~5/4 = 380.470{{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 }}


Badness (Smith): 0.031291
Badness (Sintel): 1.90


=== Telepathy ===
=== Telepathy ===
Line 360: Line 402:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 381.231{{c}}
* WE: ~2 = 1200.7724{{c}}, ~5/4 = 381.2641{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 381.019{{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 }}


Badness (Smith): 0.027109
Badness (Sintel): 0.896


==== 13-limit ====
==== 13-limit ====
Line 375: Line 419:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 381.196{{c}}
* WE: ~2 = 1202.5634{{c}}, ~5/4 = 381.3348{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.520{{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 (Smith): 0.025522
Badness (Sintel): 1.05


===== 17-limit =====
===== 17-limit =====
Line 390: Line 436:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 381.288{{c}}
* WE: ~2 = 1202.5560{{c}}, ~5/4 = 381.4295{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.619{{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 }}
{{Optimal ET sequence|legend=0| 19eg, 22, 41efg }}


Badness (Smith): 0.020201
Badness (Sintel): 1.03


===== 19-limit =====
===== 19-limit =====
Line 405: Line 453:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 381.464{{c}}
* WE: ~2 = 1203.4043{{c}}, ~5/4 = 381.5989{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 380.735{{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 }}
{{Optimal ET sequence|legend=0| 19egh, 22, 41efghh }}


Badness (Smith): 0.019004
Badness (Sintel): 1.16


==== Intuition ====
==== Intuition ====
Line 420: Line 469:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.816{{c}}
* WE: ~2 = 1201.3172{{c}}, ~5/4 = 380.9004{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.483{{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 (Smith): 0.026089
Badness (Sintel): 1.08


===== 17-limit =====
===== 17-limit =====
Line 435: Line 486:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.952{{c}}
* WE: ~2 = 1201.3593{{c}}, ~5/4 = 381.0356{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.604{{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 }}
{{Optimal ET sequence|legend=0| 19eg, 22f }}


Badness (Smith): 0.020274
Badness (Sintel): 1.03


===== 19-limit =====
===== 19-limit =====
Line 450: Line 503:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.866{{c}}
* WE: ~2 = 1201.5266{{c}}, ~5/4 = 381.0019{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 380.628{{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 }}
{{Optimal ET sequence|legend=0| 19egh, 22fh }}


Badness (Smith): 0.019518
Badness (Sintel): 1.19


=== Horcrux ===
=== Horcrux ===
Line 465: Line 519:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 379.796{{c}}
* WE: ~2 = 1200.4670{{c}}, ~5/4 = 379.7895{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 379.642{{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 }}


Badness (Smith): 0.039282
Badness (Sintel): 1.30


==== 13-limit ====
==== 13-limit ====
Line 480: Line 536:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 379.885{{c}}
* WE: ~2 = 1200.2953{{c}}, ~5/4 = 379.8842{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 379.791{{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 }}


Badness (Smith): 0.031938
Badness (Sintel): 1.32


===== 17-limit =====
===== 17-limit =====
Line 495: Line 553:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.216{{c}}
* WE: ~2 = 1200.2484{{c}}, ~5/4 = 380.2053{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 380.148{{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 }}


Badness (Smith): 0.028074
Badness (Sintel): 1.43


===== Horcruxic =====
===== Horcruxic =====
Line 510: Line 569:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 379.524{{c}}
* WE: ~2 = 1199.5457{{c}}, ~5/4 = 379.4681{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 379.571{{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 }}


Badness (Smith): 0.029556
Badness (Sintel): 1.51


==== Glamour ====
==== Glamour ====
Line 525: Line 585:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 379.757{{c}}
* WE: ~2 = 1202.2187{{c}}, ~5/4 = 379.8171{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 379.116{{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 }}


Badness (Smith): 0.033317
Badness (Sintel): 1.38


=== Witchcraft ===
=== Witchcraft ===
Line 540: Line 602:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.416{{c}}
* WE: ~2 = 1201.2634{{c}}, ~5/4 = 380.6321{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.232{{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 }}


Badness (Smith): 0.030706
Badness (Sintel): 1.02


==== 13-limit ====
==== 13-limit ====
Line 555: Line 619:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.352{{c}}
* WE: ~2 = 1201.0424{{c}}, ~5/4 = 380.5193{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.189{{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 }}


Badness (Smith): 0.023547
Badness (Sintel): 0.973


==== 17-limit ====
==== 17-limit ====
Line 570: Line 636:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.281{{c}}
* WE: ~2 = 1201.1638{{c}}, ~5/4 = 380.4827{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 380.114{{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 (Smith): 0.020756
Badness (Sintel): 1.06


==== 19-limit ====
==== 19-limit ====
Line 585: Line 653:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 380.339{{c}}
* WE: ~2 = 1201.2528{{c}}, ~5/4 = 380.5518{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 380.205{{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 }}
{{Optimal ET sequence|legend=0| 19egh, 41, 60eh }}


Badness (Smith): 0.018625
Badness (Sintel): 1.13


=== Divination ===
=== Divination ===
Line 600: Line 669:


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000{{c}}, ~5/4 = 380.732{{c}}
* WE: ~99/70 = 600.8306{{c}}, ~5/4 = 380.7598{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~5/4 = 380.223{{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 }}


Badness (Smith): 0.035864
Badness (Sintel): 1.19


==== 13-limit ====
==== 13-limit ====
Line 615: Line 686:


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000{{c}}, ~5/4 = 380.417{{c}}
* WE: ~99/70 = 600.9624{{c}}, ~5/4 = 380.5297{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~5/4 = 379.920{{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 }}


Badness (Smith): 0.034551
Badness (Sintel): 1.43


==== 17-limit ====
==== 17-limit ====
Line 630: Line 703:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~5/4 = 380.433{{c}}
* WE: ~17/12 = 600.8921{{c}}, ~5/4 = 380.5094{{c}}
* CWE: ~17/12 = 600.000{{c}}, ~5/4 = 380.067{{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 }}


Badness (Smith): 0.023775
Badness (Sintel): 1.21


=== Hocus ===
=== Hocus ===
Line 642: Line 716:
Comma list: 225/224, 243/242, 245/242
Comma list: 225/224, 243/242, 245/242


Mapping: {{mapping| 1 5 3 11 12 | 0 -10 -2 -24 -25 }}
Mapping: {{mapping| 1 -5 1 -13 -13 | 0 10 2 24 25 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~14/11 = 409.760{{c}}
* WE: ~2 = 1201.0749{{c}}, ~11/7 = 790.7980{{c}}
* POTE: ~2 = 1200.000{{c}}, ~14/11 = 409.910{{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 }}


Badness (Smith): 0.038519
Badness (Sintel): 1.27


==== 13-limit ====
==== 13-limit ====
Line 657: Line 733:
Comma list: 105/104, 196/195, 243/242, 245/242
Comma list: 105/104, 196/195, 243/242, 245/242


Mapping: {{mapping| 1 5 3 11 12 16 | 0 -10 -2 -24 -25 -36 }}
Mapping: {{mapping| 1 -5 1 -13 -13 -20 | 0 10 2 24 25 36 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~14/11 = 409.842{{c}}
* WE: ~2 = 1201.2830{{c}}, ~11/7 = 790.8409{{c}}
* POTE: ~2 = 1200.000{{c}}, ~14/11 = 410.004{{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 }}


Badness (Smith): 0.030280
Badness (Sintel): 1.25


==== 17-limit ====
==== 17-limit ====
Line 672: Line 750:
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 3 11 12 16 14 | 0 -10 -2 -24 -25 -36 -29 }}
Mapping: {{mapping| 1 -5 1 -13 -13 -20 -15 | 0 10 2 24 25 36 29 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~14/11 = 409.896{{c}}
* WE: ~2 = 1201.1557{{c}}, ~11/7 = 790.7157{{c}}
* CWE: ~2 = 1200.000{{c}}, ~14/11 = 409.994{{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 }}


Badness (Smith): 0.025491
Badness (Sintel): 1.30


==== 19-limit ====
==== 19-limit ====
Line 687: Line 766:
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 3 11 12 16 14 8 | 0 -10 -2 -24 -25 -36 -29 -11 }}
Mapping: {{mapping| 1 -5 1 -13 -13 -20 -3 | 0 10 2 24 25 36 29 11 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~14/11 = 409.884{{c}}
* WE: ~2 = 1201.3558{{c}}, ~11/7 = 790.8266{{c}}
* CWE: ~2 = 1200.000{{c}}, ~14/11 = 410.012{{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 }}


Badness (Smith): 0.020277
Badness (Sintel): 1.23


== Muggles ==
== Muggles ==

Revision as of 13:02, 25 September 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 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 proper or nearly-proper mos scales in the 5- to 10-note region. Properties may depend on tuning and extension.

Magic

Main article: Magic

The generator of magic is a major third, and to get to the interval class of fifths requires five of these. In fact, (5/4)5 = 3 × 3125/3072. 13\41 is a highly recommendable generator, though 19\60, the optimal patent val generator, also makes a lot of sense, and using 19edo or 22edo is always possible.

Subgroup: 2.3.5

Comma list: 3125/3072

Mapping: [⟨1 0 2], ⟨0 5 1]]

mapping generators: ~2, ~5/4

Optimal tunings:

  • WE: ~2 = 1201.2449 ¢, ~5/4 = 380.4527 ¢
error map: ⟨+1.245 +0.309 -3.371]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.2194 ¢
error map: ⟨0.000 -0.858 -6.094]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.3

Tuning ranges:

Algebraic generator: Terzbirat, the positive root of 9x2 - 8x - 4 = (4 + 2√13)/9; approximately 380.3175 cents.

Optimal ET sequence: 3, 13b, 16, 19, 41, 60, 221cc, 281cc

Badness (Sintel): 0.919

Overview to extensions

Apart from magic, we also consider other extensions. The second comma of the normal comma list defines which 7-limit family member we are looking at. 875/864, the keemic comma, gives septimal magic, and 525/512, Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator, as well as low-accuracy extensions including darkstone and brightstone.

Weak extensions considered below are hocum, trismegistus, quadrimage, quinmage and warlock. Discussed elsewhere are

Septimal magic

Main article: Magic

Septimal magic tempers out not only 3125/3072 and 875/864, but also 225/224, 245/243, and 10976/10935. 41edo is a good magic tuning, and 19- or 22-note mosses are possible scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1.

This temperament, with its accurate fifths, works well with 9-odd-limit harmony. It is more accurate than meantone and simpler than garibaldi. It is a little tricky to work with because its fifths are a relatively complex interval and it does not naturally work with scales of around seven notes to the octave.

225/224 is the marvel comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament. 245/243, the sensamagic comma, leads to another essentially tempered 9-odd-limit triad with two thirds approximating 9/7 and the other 6/5. It also divides the approximate 3/2 into two steps of 7/6 and one of 10/9.

By adding 100/99 and 105/104 to the list of commas, magic can be extended to the 11-limit and 13-limit. 11-limit magic allows for a tritone substitution where the extended 5-limit tuning of a dominant seventh with a 9/5 above the root shares its tritone with an 8:10:11:12:16 chord rooted on the seventh of the original chord. For this, 104edo provides an excellent tuning, as it does also for the rank-3 temperaments tempering out 100/99 with 225/224, 245/243 or 875/864. Septimage (see below) is also an excellent 11-limit magic tuning. For the 13-limit, 41edo makes for a recommendable tuning.

Subgroup: 2.3.5.7

Comma list: 225/224, 245/243

Mapping: [⟨1 0 2 -1], ⟨0 5 1 12]]

mapping generators: ~2, ~5/4

Optimal tunings:

  • WE: ~2 = 1201.0786 ¢, ~5/4 = 380.6939 ¢
error map: ⟨+1.079 +1.514 -3.463 -1.578]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4576 ¢
error map: ⟨0.000 +0.333 -5.856 -3.335]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.3

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
  • 7- and 9-odd-limit diamond tradeoff: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)

Algebraic generator: Tirzbirat or Septimage, the real root of 5x5 + 4x - 20, 380.7604 cents.

Optimal ET sequence: 19, 41, 142cd, 183cd, 224ccd

Badness (Sintel): 0.479

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 245/243

Mapping: [⟨1 0 2 -1 6], ⟨0 5 1 12 -8]]

Optimal tunings:

  • WE: ~2 = 1200.1372 ¢, ~5/4 = 380.7399 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.7008 ¢

Minimax tuning:

  • 11-odd-limit: ~5/4 = [1/3 1/9 0 0 -1/18⟩
unchanged-interval (eigenmonzo) basis: 2.11/9

Tuning ranges:

  • 11-odd-limit diamond monotone: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
  • 11-odd-limit diamond tradeoff: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)

Optimal ET sequence: 19, 22, 41, 104

Badness (Sintel): 0.673

13-limit

A notable patent val tuning beyond the optimal patent val of 41edo is 19 + 41 = 60edo.

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

Mapping: [⟨1 0 2 -1 6 -2], ⟨0 5 1 12 -8 18]]

Optimal tunings:

  • WE: ~2 = 1200.0331 ¢, ~5/4 = 380.4377 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4284 ¢

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
  • 13- and 15-odd-limit diamond tradeoff: ~5/4 = [378.617, 386.314]

Optimal ET sequence: 19, 22f, 41

Badness (Sintel): 0.889

Magical

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 0 2 -1 6 -2 6], ⟨0 5 1 12 -8 18 -6]]

Optimal tunings:

  • WE: ~2 = 1199.3584 ¢, ~5/4 = 380.4006 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.5896 ¢

Optimal ET sequence: 19, 22f, 41

Badness (Sintel): 1.05

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 0 2 -1 6 -2 6 9], ⟨0 5 1 12 -8 18 -6 -15]]

Optimal tunings:

  • WE: ~2 = 1199.7173 ¢, ~5/4 = 380.3808 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4680 ¢

Optimal ET sequence: 19, 41

Badness (Sintel): 1.27

Magica

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 120/119, 144/143, 154/153, 171/169

Mapping: [⟨1 0 2 -1 6 -2 6 -4], ⟨0 5 1 12 -8 18 -6 26]]

Optimal tunings:

  • WE: ~2 = 1199.3670 ¢, ~5/4 = 380.4681 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.6541 ¢

Optimal ET sequence: 22fh, 41

Badness (Sintel): 1.21

Magia

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 144/143, 170/169, 196/195

Mapping: [⟨1 0 2 -1 6 -2 -7], ⟨0 5 1 12 -8 18 35]]

Optimal tunings:

  • WE: ~2 = 1200.1727 ¢, ~5/4 = 380.2982 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.2483 ¢

Optimal ET sequence: 19g, 41, 60

Badness (Sintel): 1.34

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 144/143, 170/169, 171/169, 196/195

Mapping: [⟨1 0 2 -1 6 -2 -7 -4], ⟨0 5 1 12 -8 18 35 26]]

Optimal tunings:

  • WE: ~2 = 1200.2179 ¢, ~5/4 = 380.3942 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.3314 ¢

Optimal ET sequence: 19gh, 41

Badness (Sintel): 1.44

Evening

Evening is a remarkable subgroup temperament of 19 & 22f with prime harmonics of 29 and 31.

Subgroup: 2.3.5.7.11.13.29.31

Comma list: 100/99, 105/104, 144/143, 145/144, 155/154, 196/195

Subgroup-val mapping: [⟨1 0 2 -1 6 -2 2 4], ⟨0 5 1 12 -8 18 9 3]]

Optimal tunings:

  • WE: ~2 = 1200.2802 ¢, ~5/4 = 380.5053 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4258 ¢

Optimal ET sequence: 19, 22f, 41

Badness (Sintel): 0.807

Sorcery

Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 78/77, 91/90, 100/99

Mapping: [⟨1 0 2 -1 6 4], ⟨0 5 1 12 -8 -1]]

Optimal tunings:

  • WE: ~2 = 1201.2397 ¢, ~5/4 = 380.8698 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.5080 ¢

Optimal ET sequence: 19, 22, 41f

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: [⟨1 0 2 -1 6 4 6], ⟨0 5 1 12 -8 -1 -6]]

Optimal tunings:

  • WE: ~2 = 1200.4822 ¢, ~5/4 = 380.8818 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.7356 ¢

Optimal ET sequence: 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: [⟨1 0 2 -1 6 4 6 9], ⟨0 5 1 12 -8 -1 -6 -15]]

Optimal tunings:

  • WE: ~2 = 1200.6684 ¢, ~5/4 = 380.8328 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.6212 ¢

Optimal ET sequence: 19, 22

Badness (Sintel): 1.41

Necromancy

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 225/224, 245/243, 275/273

Mapping: [⟨1 0 2 -1 6 11], ⟨0 5 1 12 -8 -23]]

Optimal tunings:

  • WE: ~2 = 1199.9675 ¢, ~5/4 = 380.7770 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.7874 ¢

Optimal ET sequence: 19f, 22, 41, 63, 104

Badness (Sintel): 1.04

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 120/119, 154/153, 225/224, 273/272

Mapping: [⟨1 0 2 -1 6 11 6], ⟨0 5 1 12 -8 -23 -6]]

Optimal tunings:

  • WE: ~2 = 1199.6176 ¢, ~5/4 = 380.7053 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.8280 ¢

Optimal ET sequence: 19f, 22, 41, 63

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: [⟨1 0 2 -1 6 11 6 9], ⟨0 5 1 12 -8 -23 -6 -15]]

Optimal tunings:

  • WE: ~2 = 1199.6176 ¢, ~5/4 = 380.6981 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.7355 ¢

Optimal ET sequence: 19f, 22, 41

Badness (Sintel): 1.28

Soothsaying

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 225/224, 245/243, 1352/1331

Mapping: [⟨2 0 4 -2 12 15], ⟨0 5 1 12 -8 -12]]

Optimal tunings:

  • WE: ~55/39 = 600.2918 ¢, ~5/4 = 380.6928 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~5/4 = 380.5121 ¢

Optimal ET sequence: 22, 60, 82

Badness (Sintel): 2.29

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 221/220, 225/224, 245/243, 273/272

Mapping: [⟨2 0 4 -2 12 15 5], ⟨0 5 1 12 -8 -12 5]]

Optimal tunings:

  • WE: ~17/12 = 600.2918 ¢, ~5/4 = 380.6927 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~5/4 = 380.5135 ¢

Optimal ET sequence: 22, 60, 82

Badness (Sintel): 1.82

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 133/132, 221/220, 225/224, 245/243, 273/272

Mapping: [⟨2 0 4 -2 12 15 5 18], ⟨0 5 1 12 -8 -12 5 -15]]

Optimal tunings:

  • WE: ~17/12 = 600.3301 ¢, ~5/4 = 380.6797 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~5/4 = 380.4704 ¢

Optimal ET sequence: 22, 60, 82

Badness (Sintel): 1.90

Telepathy

Subgroup: 2.3.5.7.11

Comma list: 55/54, 99/98, 176/175

Mapping: [⟨1 0 2 -1 -1], ⟨0 5 1 12 14]]

Optimal tunings:

  • WE: ~2 = 1200.7724 ¢, ~5/4 = 381.2641 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 381.0913 ¢

Optimal ET sequence: 19e, 22, 41e, 63e

Badness (Sintel): 0.896

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 65/64, 91/90, 99/98

Mapping: [⟨1 0 2 -1 -1 4], ⟨0 5 1 12 14 -1]]

Optimal tunings:

  • WE: ~2 = 1202.5634 ¢, ~5/4 = 381.3348 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.6886 ¢

Optimal ET sequence: 19e, 22, 41ef

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: [⟨1 0 2 -1 -1 4 -1], ⟨0 5 1 12 14 -1 16]]

Optimal tunings:

  • WE: ~2 = 1202.5560 ¢, ~5/4 = 381.4295 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.7843 ¢

Optimal ET sequence: 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: [⟨1 0 2 -1 -1 4 -1 2], ⟨0 5 1 12 14 -1 16 7]]

Optimal tunings:

  • WE: ~2 = 1203.4043 ¢, ~5/4 = 381.5989 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.7353 ¢

Optimal ET sequence: 19egh, 22, 41efghh

Badness (Sintel): 1.16

Intuition

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 66/65, 99/98, 105/104

Mapping: [⟨1 0 2 -1 -1 -2], ⟨0 5 1 12 14 18]]

Optimal tunings:

  • WE: ~2 = 1201.3172 ¢, ~5/4 = 380.9004 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.5942 ¢

Optimal ET sequence: 19e, 22f

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: [⟨1 0 2 -1 -1 -2 -1], ⟨0 5 1 12 14 18 16]]

Optimal tunings:

  • WE: ~2 = 1201.3593 ¢, ~5/4 = 381.0356 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.7204 ¢

Optimal ET sequence: 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: [⟨1 0 2 -1 -1 -2 -1 -4], ⟨0 5 1 12 14 18 16 26]]

Optimal tunings:

  • WE: ~2 = 1201.5266 ¢, ~5/4 = 381.0019 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.6275 ¢

Optimal ET sequence: 19egh, 22fh

Badness (Sintel): 1.19

Horcrux

Subgroup: 2.3.5.7.11

Comma list: 45/44, 56/55, 245/243

Mapping: [⟨1 0 2 -1 0], ⟨0 5 1 12 11]]

Optimal tunings:

  • WE: ~2 = 1200.4670 ¢, ~5/4 = 379.7895 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 379.6889 ¢

Optimal ET sequence: 3de, 16d, 19

Badness (Sintel): 1.30

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 78/77, 245/243

Mapping: [⟨1 0 2 -1 0 -2], ⟨0 5 1 12 11 18]]

Optimal tunings:

  • WE: ~2 = 1200.2953 ¢, ~5/4 = 379.8842 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 379.8165 ¢

Optimal ET sequence: 3def, 16dff, 19

Badness (Sintel): 1.32

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 56/55, 78/77, 85/84, 245/243

Mapping: [⟨1 0 2 -1 0 -2 0], ⟨0 5 1 12 11 18 16]]

Optimal tunings:

  • WE: ~2 = 1200.2484 ¢, ~5/4 = 380.2053 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.1482 ¢

Optimal ET sequence: 3defg, 16dffgg, 19g

Badness (Sintel): 1.43

Horcruxic

Subgroup: 2.3.5.7.11.13.17

Comma list: 35/34, 45/44, 52/51, 56/55, 245/243

Mapping: [⟨1 0 2 -1 0 -2 0], ⟨0 5 1 12 11 18 13]]

Optimal tunings:

  • WE: ~2 = 1199.5457 ¢, ~5/4 = 379.4681 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 379.5713 ¢

Optimal ET sequence: 3defg, 16dff, 19

Badness (Sintel): 1.51

Glamour

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 65/64, 245/243

Mapping: [⟨1 0 2 -1 0 4], ⟨0 5 1 12 11 -1]]

Optimal tunings:

  • WE: ~2 = 1202.2187 ¢, ~5/4 = 379.8171 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 379.2709 ¢

Optimal ET sequence: 3de, 16d, 19

Badness (Sintel): 1.38

Witchcraft

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/243, 441/440

Mapping: [⟨1 0 2 -1 -7], ⟨0 5 1 12 33]]

Optimal tunings:

  • WE: ~2 = 1201.2634 ¢, ~5/4 = 380.6321 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.2849 ¢

Optimal ET sequence: 19e, 41, 60e, 101cd, 243ccdde

Badness (Sintel): 1.02

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 245/243, 275/273

Mapping: [⟨1 0 2 -1 -7 -2], ⟨0 5 1 12 33 18]]

Optimal tunings:

  • WE: ~2 = 1201.0424 ¢, ~5/4 = 380.5193 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.2349 ¢

Optimal ET sequence: 19e, 41, 60e, 101cd

Badness (Sintel): 0.973

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 154/153, 170/169, 196/195, 245/243

Mapping: [⟨1 0 2 -1 -7 -2 -7], ⟨0 5 1 12 33 18 35]]

Optimal tunings:

  • WE: ~2 = 1201.1638 ¢, ~5/4 = 380.4827 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.1599 ¢

Optimal ET sequence: 19eg, 41, 60e, 101cd

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: [⟨1 0 2 -1 -7 -2 -7 -4], ⟨0 5 1 12 33 18 35 26]]

Optimal tunings:

  • WE: ~2 = 1201.2528 ¢, ~5/4 = 380.5518 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.2052 ¢

Optimal ET sequence: 19egh, 41, 60eh

Badness (Sintel): 1.13

Divination

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 245/243

Mapping: [⟨2 0 4 -2 5], ⟨0 5 1 12 3]]

Optimal tunings:

  • WE: ~99/70 = 600.8306 ¢, ~5/4 = 380.7598 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~5/4 = 380.3800 ¢

Optimal ET sequence: 22, 38d, 60e, 142cdee, 202ccddeee

Badness (Sintel): 1.19

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 196/195, 245/243

Mapping: [⟨2 0 4 -2 5 -4], ⟨0 5 1 12 3 18]]

Optimal tunings:

  • WE: ~99/70 = 600.9624 ¢, ~5/4 = 380.5297 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~5/4 = 380.0614 ¢

Optimal ET sequence: 22f, 38df, 60e

Badness (Sintel): 1.43

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 121/120, 154/153, 196/195, 245/243

Mapping: [⟨2 0 4 -2 5 -4 5], ⟨0 5 1 12 3 18 5]]

Optimal tunings:

  • WE: ~17/12 = 600.8921 ¢, ~5/4 = 380.5094 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~5/4 = 380.0672 ¢

Optimal ET sequence: 22f, 38df, 60e

Badness (Sintel): 1.21

Hocus

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 245/242

Mapping: [⟨1 -5 1 -13 -13], ⟨0 10 2 24 25]]

Optimal tunings:

  • WE: ~2 = 1201.0749 ¢, ~11/7 = 790.7980 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 790.1429 ¢

Optimal ET sequence: 38d, 41, 120cd

Badness (Sintel): 1.27

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 243/242, 245/242

Mapping: [⟨1 -5 1 -13 -13 -20], ⟨0 10 2 24 25 36]]

Optimal tunings:

  • WE: ~2 = 1201.2830 ¢, ~11/7 = 790.8409 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 790.0516 ¢

Optimal ET sequence: 38df, 41, 79d, 120cd

Badness (Sintel): 1.25

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 154/153, 196/195, 243/242, 245/242

Mapping: [⟨1 -5 1 -13 -13 -20 -15], ⟨0 10 2 24 25 36 29]]

Optimal tunings:

  • WE: ~2 = 1201.1557 ¢, ~11/7 = 790.7157 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 790.0057 ¢

Optimal ET sequence: 38df, 41, 79d

Badness (Sintel): 1.30

19-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 154/153, 196/195, 243/242, 245/242

Mapping: [⟨1 -5 1 -13 -13 -20 -3], ⟨0 10 2 24 25 36 29 11]]

Optimal tunings:

  • WE: ~2 = 1201.3558 ¢, ~11/7 = 790.8266 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 789.9880 ¢

Optimal ET sequence: 38df, 41, 79dh

Badness (Sintel): 1.23

Muggles

Main article: Muggles

Aside from 3125/3072 and 525/512 muggles also tempers out 126/125 and 1323/1280. A good muggles tuning is 19edo, in which tuning it is the same thing as magic. Muggles works better for small scales than magic in the sense that 7- or 10-note mosses are reasonable choices, as while the flatter generator compromises the accuracy of the 5-limit intervals, it grants simpler access to some higher-limit ones, and makes the small steps larger and more melodically effective.

Subgroup: 2.3.5.7

Comma list: 126/125, 525/512

Mapping: [⟨1 0 2 5], ⟨0 5 1 -7]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~5/4 = 378.744 ¢
error map: ⟨0.000 -8.235 -7.570 -20.034]
  • POTE: ~2 = 1200.000 ¢, ~5/4 = 378.479 ¢
error map: ⟨0.000 -9.558 -7.834 -18.181]

Tuning ranges:

Optimal ET sequence: 16, 19, 73bcd, 92bcdd, 111bcddd

Badness (Smith): 0.056206

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 126/125, 385/384

Mapping: [⟨1 0 2 5 0], ⟨0 5 1 -7 11]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~5/4 = 378.228 ¢
  • POTE: ~2 = 1200.000 ¢, ~5/4 = 377.724 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~5/4 = 378.947 (6\19)
  • 11-odd-limit diamond tradeoff: ~5/4 = [347.408, 386.314]

Optimal ET sequence: 16, 19, 35, 54bd

Badness (Smith): 0.048038

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 65/64, 78/77, 126/125

Mapping: [⟨1 0 2 5 0 4], ⟨0 5 1 -7 11 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~5/4 = 378.176 ¢
  • POTE: ~2 = 1200.000 ¢, ~5/4 = 377.653 ¢

Optimal ET sequence: 16, 19, 35f, 54bdf

Badness (Smith): 0.030386

Muggloid

Subgroup: 2.3.5.7.11

Comma list: 33/32, 126/125, 176/175

Mapping: [⟨1 0 2 5 5], ⟨0 5 1 -7 -5]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~5/4 = 377.724 ¢
  • POTE: ~2 = 1200.000 ¢, ~5/4 = 377.832 ¢

Optimal ET sequence: 3, 16, 19e, 35ee, 54bdeee

Badness (Smith): 0.046970

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 33/32, 65/64, 105/104, 126/125

Mapping: [⟨1 0 2 5 5 4], ⟨0 5 1 -7 -5 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~5/4 = 377.652 ¢
  • POTE: ~2 = 1200.000 ¢, ~5/4 = 377.838 ¢

Optimal ET sequence: 3, 16, 19e, 35eef

Badness (Smith): 0.028732

Darkstone

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 Encyclopedia of Microtonal Music Theory, Tonalsoft, this temperament is given a name witch.

Subgroup: 2.3.5.7

Comma list: 36/35, 1875/1792

Mapping: [⟨1 0 2 0], ⟨0 5 1 9]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~5/4 = 377.385 ¢
error map: ⟨0.000 -15.028 -8.928 +27.643]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 376.963 ¢
error map: ⟨0.000 -18.198 -9.562 +21.937]

Optimal ET sequence: 3d, …, 13b, 16

Badness (Smith): 0.084213

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 363/343

Mapping: [⟨1 0 2 0 0], ⟨0 5 1 9 11]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~5/4 = 377.388 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 376.973 ¢

Optimal ET sequence: 3de, 13be, 16

Badness (Smith): 0.046775

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 45/44, 363/343

Mapping: [⟨1 0 2 0 0 -1], ⟨0 5 1 9 11 15]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~5/4 = 376.914 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 376.422 ¢

Optimal ET sequence: 3def, 13beff, 16

Badness (Smith): 0.038328

Brightstone

Subgroup: 2.3.5.7

Comma list: 64/63, 3125/3024

Mapping: [⟨1 0 2 6], ⟨0 5 1 -10]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~5/4 = 381.955 ¢
error map: ⟨0.000 +7.818 -4.359 +11.627]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 381.956 ¢
error map: ⟨0.000 +7.826 -4.358 +11.613]

Optimal ET sequence: 3, 19d, 22

Badness (Smith): 0.088072

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 605/588

Mapping: [⟨1 0 2 6 6], ⟨0 5 1 -10 -8]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~5/4 = 381.790 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 381.712 ¢

Optimal ET sequence: 3, 19d, 22

Badness (Smith): 0.047379

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 65/63, 100/99, 169/165

Mapping: [⟨1 0 2 6 6 4], ⟨0 5 1 -10 -8 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~5/4 = 381.732 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 381.736 ¢

Optimal ET sequence: 3, 22

Badness (Smith): 0.039703

Hocum

Subgroup: 2.3.5.7

Comma list: 3125/3072, 4000/3969

Mapping: [⟨1 5 3 -3], ⟨0 -10 -2 17]]

mapping generators: ~2, ~63/50

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~63/50 = 409.836 ¢
error map: ⟨0.000 -0.316 -5.986 -1.612]
  • POTE: ~2 = 1200.000 ¢, ~63/50 = 410.108 ¢
error map: ⟨0.000 -0.437 -6.010 -1.406]

Optimal ET sequence: 3, 38, 41, 161c

Badness (Smith): 0.107115

Trismegistus

See also: No-threes subgroup temperaments § Mabilic
Main article: Mabilic and trismegistus

Subgroup: 2.3.5.7

Comma list: 1029/1024, 3125/3072

Mapping: [⟨1 10 4 0], ⟨0 -15 -3 5]]

mapping generators: ~2, ~147/100

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~147/100 = 673.187 ¢
error map: ⟨0.000 +0.240 -5.875 -2.891]
  • POTE: ~2 = 1200.000 ¢, ~147/100 = 673.290 ¢
error map: ⟨0.000 -0.932 -6.109 -2.500]

Optimal ET sequence: 16, 25, 41, 139c, 180c, 221cc, 262ccd

Badness (Smith): 0.098334

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 625/616

Mapping: [⟨1 10 4 0 13], ⟨0 -15 -3 5 -17]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~22/15 = 673.241 ¢
  • POTE: ~2 = 1200.000 ¢, ~22/15 = 673.340 ¢

Optimal ET sequence: 16, 25e, 41, 98c

Badness (Smith): 0.045623

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 275/273, 625/616

Mapping: [⟨1 10 4 0 13 11], ⟨0 -15 -3 5 -17 -13]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~22/15 = 673.294 ¢
  • POTE: ~2 = 1200.000 ¢, ~22/15 = 673.359 ¢

Optimal ET sequence: 16, 25e, 41, 98c

Badness (Smith): 0.033081

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 105/104, 144/143, 441/440, 210/209, 625/616

Subgroup-val mapping: [⟨1 10 4 0 13 11 2], ⟨0 -15 -3 5 -17 -13 4]]

Optimal tunings:

  • CWE: ~2 = 1200.000 ¢, ~22/15 = 673.363 ¢

Optimal ET sequence: 16, 25e, 41, 98c

Quadrimage

Subgroup: 2.3.5.7

Comma list: 2401/2400, 3125/3072

Mapping: [⟨1 5 3 4], ⟨0 -20 -4 -7]]

mapping generators: ~2, ~28/25

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~28/25 = 204.860 ¢
error map: ⟨0.000 +0.853 -5.752 -2.843]
  • POTE: ~2 = 1200.000 ¢, ~28/25 = 204.987 ¢
error map: ⟨0.000 -1.691 -6.261 -3.733]

Optimal ET sequence: 6, 29b, 35, 41, 158cd, 199ccd, 240ccd, 281ccd

Badness (Smith): 0.127422

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/242, 385/384, 625/616

Mapping: [⟨1 5 3 4 5], ⟨0 -20 -4 -7 -9]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~28/25 = 204.881 ¢
  • POTE: ~2 = 1200.000 ¢, ~28/25 = 204.956 ¢

Optimal ET sequence: 6, 35, 41

Badness (Smith): 0.061572

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 245/242, 625/616

Mapping: [⟨1 5 3 4 5 9], ⟨0 -20 -4 -7 -9 -31]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~28/25 = 204.956 ¢
  • POTE: ~2 = 1200.000 ¢, ~28/25 = 205.028 ¢

Optimal ET sequence: 6f, 35f, 41, 117c

Badness (Smith): 0.044047

Quinmage

Subgroup: 2.3.5.7

Comma list: 3125/3072, 16875/16807

Mapping: [⟨1 -10 0 -6], ⟨0 25 5 19]]

mapping generators: ~2, ~48/35

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~48/35 = 556.123 ¢
error map: ⟨0.000 +1.132 -5.696 -2.480]
  • CWE: ~2 = 1200.000 ¢, ~48/35 = 556.050 ¢
error map: ⟨0.000 -0.695 -6.062 -3.868]

Optimal ET sequence: 13b, 28b, 41, 177bcd, 218bccdd, 259bccdd, 300cccdd

Badness (Smith): 0.194548

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 625/616, 2401/2376

Mapping: [⟨1 -10 0 -6 3], ⟨0 25 5 19 1]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/8 = 556.122 ¢
  • CWE: ~2 = 1200.000 ¢, ~11/8 = 556.095 ¢

Optimal ET sequence: 13b, 28b, 41

Badness (Smith): 0.101724

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 364/363, 385/384, 625/616

Mapping: [⟨1 -10 0 -6 3 0], ⟨0 25 5 19 1 8]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/8 = 556.106 ¢
  • CWE: ~2 = 1200.000 ¢, ~11/8 = 556.117 ¢

Optimal ET sequence: 13b, 28b, 41

Badness (Smith): 0.067742

Warlock

Subgroup: 2.3.5.7

Comma list: 3125/3072, 16807/16384

Mapping: [⟨5 0 10 14], ⟨0 5 1 0]]

mapping generators: ~8/7, ~5/4

Optimal tunings:

  • CTE: ~8/7 = 240.000 ¢, ~5/4 = 380.499 ¢ (~256/245 = 99.501 ¢)
error map: ⟨0.000 +0.542 -5.814 -8.826]
  • CWE: ~8/7 = 240.000 ¢, ~5/4 = 379.997 ¢ (~256/245 = 100.003 ¢)
error map: ⟨0.000 -1.972 -6.317 -8.826]

Optimal ET sequence: 25, 35, 60

Badness (Smith): 0.287190

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