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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 380.499
* [[WE]]: ~2 = 1201.2449{{c}}, ~5/4 = 380.4527{{c}}
: [[error map]]: {{val| 0.000 +0.542 -5.814 }}
: [[error map]]: {{val| +1.245 +0.309 -3.371 }}
* [[POTE]]: ~2 = 1200.000, ~5/4 = 380.058
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.2194{{c}}
: error map: {{val| 0.000 -1.663 -6.255 }}
: error map: {{val| 0.000 -0.858 -6.094 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[5-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 }}
* [[5-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 }}
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 39: Line 39:
{{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 57: Line 57:
225/224 is the [[marvel family|marvel]] comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament. 245/243, the [[sensamagic family|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.  
225/224 is the [[marvel family|marvel]] comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament. 245/243, the [[sensamagic family|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]] to the list of commas, magic can be extended to an 11-limit version. 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.
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
[[Subgroup]]: 2.3.5.7
Line 68: Line 68:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 380.651
* [[WE]]: ~2 = 1201.0786{{c}}, ~5/4 = 380.6939{{c}}
: [[error map]]: {{val| 0.000 +1.301 -5.662 -1.011 }}
: [[error map]]: {{val| +1.079 +1.514 -3.463 -1.578 }}
* [[POTE]]: ~2 = 1200.000, ~5/4 = 380.352
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.4576{{c}}
: error map: {{val| 0.000 -0.195 -5.962 -4.602 }}
: error map: {{val| 0.000 +0.333 -5.856 -3.335 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 0 }}
* 7- and [[9-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 0 }}
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 85: Line 85:
{{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 ===
Tempering out 100/99 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. (The tritone of the dominant seventh is (9/5)/(5/4) = 36/25. (16/11)/(36/25) = 100/99.)
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 97: Line 95:


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


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~5/4 = {{monzo| 1/3 1/9 0 0 -1/18 }}
* 11-odd-limit: ~5/4 = {{monzo| 1/3 1/9 0 0 -1/18 }}
: eigenmonzo (unchanged-interval) basis: 2.11/9
: unchanged-interval (eigenmonzo) basis: 2.11/9


Tuning ranges:  
Tuning ranges:  
Line 110: Line 108:
{{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 122: Line 120:


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


Tuning ranges:  
Tuning ranges:  
Line 131: Line 129:
{{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 141: Line 139:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.510
* WE: ~2 = 1199.3584{{c}}, ~5/4 = 380.4006{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 380.604
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5896{{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 ======
====== Magicus ======
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Line 156: Line 154:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.451
* WE: ~2 = 1199.7173{{c}}, ~5/4 = 380.3808{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4680{{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 170: Line 169:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.573
* WE: ~2 = 1199.3670{{c}}, ~5/4 = 380.4681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6541{{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 184: Line 184:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.271
* WE: ~2 = 1200.1727{{c}}, ~5/4 = 380.2982{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2483{{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 198: Line 199:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.360
* WE: ~2 = 1200.2179{{c}}, ~5/4 = 380.3942{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.3314{{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 & 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 211: Line 213:
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, ~5/4 = 380.478
* WE: ~2 = 1200.2802{{c}}, ~5/4 = 380.5053{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 380.416
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4258{{c}}


{{Optimal ET sequence|legend=0| 19, 22f, 41 }}
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}
Badness (Sintel): 0.807


==== Sorcery ====
==== Sorcery ====
Line 227: Line 231:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.674
* WE: ~2 = 1201.2397{{c}}, ~5/4 = 380.8698{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 380.477
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5080{{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 =====
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:
* CTE: ~2 = 1200.000, ~5/4 = 380.784
* POTE: ~2 = 1200.000, ~5/4 = 380.729
 
{{Optimal ET sequence|legend=0| 19, 22 }}
 
Badness (Smith): 0.023768
 
===== 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:
* CTE: ~2 = 1200.000, ~5/4 = 380.625
* CWE: ~2 = 1200.000, ~5/4 = 380.621
 
{{Optimal ET sequence|legend=0| 19, 22 }}
 
Badness (Smith): 0.023232


==== Necromancy ====
==== Necromancy ====
Line 272: Line 246:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.788
* WE: ~2 = 1199.9675{{c}}, ~5/4 = 380.7770{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 380.787
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7874{{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 287: Line 261:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.837
* WE: ~2 = 1199.6176{{c}}, ~5/4 = 380.7053{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 380.827
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.8280{{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 =====
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:
* CTE: ~2 = 1200.000, ~5/4 = 380.741
* CWE: ~2 = 1200.000, ~5/4 = 380.735
 
{{Optimal ET sequence|legend=0| 19f, 22, 41 }}
 
Badness (Smith): 0.021101


==== Soothsaying ====
==== Soothsaying ====
Line 317: Line 276:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 600.000, ~5/4 = 380.539
* WE: ~55/39 = 600.2918{{c}}, ~5/4 = 380.6928{{c}}
* POTE: ~55/39 = 600.000, ~5/4 = 380.508
* CWE: ~55/39 = 600.0000{{c}}, ~5/4 = 380.5121{{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 332: Line 291:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~5/4 = 380.553
* WE: ~17/12 = 600.2918{{c}}, ~5/4 = 380.6927{{c}}
* POTE: ~17/12 = 600.000, ~5/4 = 380.508
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.5135{{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 347: Line 306:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~5/4 = 380.470
* WE: ~17/12 = 600.3301{{c}}, ~5/4 = 380.6797{{c}}
* CWE: ~17/12 = 600.000, ~5/4 = 380.470
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.4704{{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 362: Line 321:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 381.231
* WE: ~2 = 1200.7724{{c}}, ~5/4 = 381.2641{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 381.019
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 381.0913{{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 377: Line 336:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 381.196
* WE: ~2 = 1202.5634{{c}}, ~5/4 = 381.3348{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 380.520
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6886{{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 =====
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:
* CTE: ~2 = 1200.000, ~5/4 = 381.288
* POTE: ~2 = 1200.000, ~5/4 = 380.619
 
{{Optimal ET sequence|legend=0| 19eg, 22, 41efg }}
 
Badness (Smith): 0.020201
 
===== 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:
* CTE: ~2 = 1200.000, ~5/4 = 381.464
* CWE: ~2 = 1200.000, ~5/4 = 380.735
 
{{Optimal ET sequence|legend=0| 19egh, 22, 41efghh }}
 
Badness (Smith): 0.019004


==== Intuition ====
==== Intuition ====
Line 422: Line 351:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.816
* WE: ~2 = 1201.3172{{c}}, ~5/4 = 380.9004{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 380.483
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5942{{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 =====
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:
* CTE: ~2 = 1200.000, ~5/4 = 380.952
* POTE: ~2 = 1200.000, ~5/4 = 380.604
 
{{Optimal ET sequence|legend=0| 19eg, 22f }}
 
Badness (Smith): 0.020274
 
===== 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:
* CTE: ~2 = 1200.000, ~5/4 = 380.866
* CWE: ~2 = 1200.000, ~5/4 = 380.628
 
{{Optimal ET sequence|legend=0| 19egh, 22fh }}
 
Badness (Smith): 0.019518


=== Horcrux ===
=== Horcrux ===
Line 467: Line 366:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 379.796
* WE: ~2 = 1200.4670{{c}}, ~5/4 = 379.7895{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 379.642
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.6889{{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 482: Line 381:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 379.885
* WE: ~2 = 1200.2953{{c}}, ~5/4 = 379.8842{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 379.791
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.8165{{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 497: Line 396:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.216
* WE: ~2 = 1200.2484{{c}}, ~5/4 = 380.2053{{c}}
* CWE: ~2 = 1200.000, ~5/4 = 380.148
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1482{{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 512: Line 411:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 379.524
* WE: ~2 = 1199.5457{{c}}, ~5/4 = 379.4681{{c}}
* CWE: ~2 = 1200.000, ~5/4 = 379.571
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.5713{{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 527: Line 426:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 379.757
* WE: ~2 = 1202.2187{{c}}, ~5/4 = 379.8171{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 379.116
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.2709{{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 542: Line 441:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.416
* WE: ~2 = 1201.2634{{c}}, ~5/4 = 380.6321{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 380.232
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2849{{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 557: Line 456:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.352
* WE: ~2 = 1201.0424{{c}}, ~5/4 = 380.5193{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 380.189
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2349{{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 572: Line 471:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 380.281
* WE: ~2 = 1201.1638{{c}}, ~5/4 = 380.4827{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 380.114
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1599{{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 ====
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:
* CTE: ~2 = 1200.000, ~5/4 = 380.339
* CWE: ~2 = 1200.000, ~5/4 = 380.205
 
{{Optimal ET sequence|legend=0| 19egh, 41, 60eh }}
 
Badness (Smith): 0.018625


=== Divination ===
=== Divination ===
Line 602: Line 486:


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000, ~5/4 = 380.732
* WE: ~99/70 = 600.8306{{c}}, ~5/4 = 380.7598{{c}}
* POTE: ~99/70 = 600.000, ~5/4 = 380.223
* CWE: ~99/70 = 600.0000{{c}}, ~5/4 = 380.3800{{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 617: Line 501:


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000, ~5/4 = 380.417
* WE: ~99/70 = 600.9624{{c}}, ~5/4 = 380.5297{{c}}
* POTE: ~99/70 = 600.000, ~5/4 = 379.920
* CWE: ~99/70 = 600.0000{{c}}, ~5/4 = 380.0614{{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 632: Line 516:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~5/4 = 380.433
* WE: ~17/12 = 600.8921{{c}}, ~5/4 = 380.5094{{c}}
* CWE: ~17/12 = 600.000, ~5/4 = 380.067
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.0672{{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 644: Line 528:
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, ~14/11 = 409.760
* WE: ~2 = 1201.0749{{c}}, ~11/7 = 790.7980{{c}}
* POTE: ~2 = 1200.000, ~14/11 = 409.910
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.1429{{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 659: Line 543:
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, ~14/11 = 409.842
* WE: ~2 = 1201.2830{{c}}, ~11/7 = 790.8409{{c}}
* POTE: ~2 = 1200.000, ~14/11 = 410.004
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.0516{{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 674: Line 558:
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, ~14/11 = 409.896
* WE: ~2 = 1201.1557{{c}}, ~11/7 = 790.7157{{c}}
* CWE: ~2 = 1200.000, ~14/11 = 409.994
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.0057{{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 689: Line 573:
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, ~14/11 = 409.884
* WE: ~2 = 1201.3558{{c}}, ~11/7 = 790.8266{{c}}
* CWE: ~2 = 1200.000, ~14/11 = 410.012
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 789.9880{{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 ==
Line 711: Line 595:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 378.744
* [[WE]]: ~2 = 1203.9554{{c}}, ~5/4 = 379.7269{{c}}
: [[error map]]: {{val| 0.000 -8.235 -7.570 -20.034 }}
: [[error map]]: {{val| +3.955 -3.321 +1.324 -7.137 }}
* [[POTE]]: ~2 = 1200.000, ~5/4 = 378.479
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 378.5328{{c}}
: error map: {{val| 0.000 -9.558 -7.834 -18.181 }}
: error map: {{val| 0.000 -9.291 -7.781 -18.555 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 723: Line 607:
{{Optimal ET sequence|legend=1| 16, 19, 73bcd, 92bcdd, 111bcddd }}
{{Optimal ET sequence|legend=1| 16, 19, 73bcd, 92bcdd, 111bcddd }}


[[Badness]] (Smith): 0.056206
[[Badness]] (Sintel): 1.42


=== 11-limit ===
=== 11-limit ===
Line 733: Line 617:


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


Tuning ranges:  
Tuning ranges:  
Line 742: Line 626:
{{Optimal ET sequence|legend=0| 16, 19, 35, 54bd }}
{{Optimal ET sequence|legend=0| 16, 19, 35, 54bd }}


Badness (Smith): 0.048038
Badness (Sintel): 1.59


==== 13-limit ====
==== 13-limit ====
Line 752: Line 636:


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


{{Optimal ET sequence|legend=0| 16, 19, 35f, 54bdf }}
{{Optimal ET sequence|legend=0| 16, 19, 35f, 54bdf }}


Badness (Smith): 0.030386
Badness (Sintel): 1.26


=== Muggloid ===
=== Muggloid ===
Line 767: Line 651:


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


{{Optimal ET sequence|legend=0| 3, 16, 19e, 35ee, 54bdeee }}
{{Optimal ET sequence|legend=0| 3, 16, 19e, 35ee, 54bdeee }}


Badness (Smith): 0.046970
Badness (Sintel): 1.55


==== 13-limit ====
==== 13-limit ====
Line 782: Line 666:


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


{{Optimal ET sequence|legend=0| 3, 16, 19e, 35eef }}
{{Optimal ET sequence|legend=0| 3, 16, 19e, 35eef }}


Badness (Smith): 0.028732
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:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 377.385
* [[WE]]: ~2 = 1198.1701{{c}}, ~5/4 = 381.3741{{c}}
: [[error map]]: {{val| 0.000 -15.028 -8.928 +27.643 }}
: error map: {{val| -1.830 +4.915 -8.599 +6.454 }}
* [[CWE]]: ~2 = 1200.000, ~5/4 = 376.963
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 381.9562{{c}}
: error map: {{val| 0.000 -18.198 -9.562 +21.937 }}
: error map: {{val| 0.000 +7.826 -4.358 +11.613 }}


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


[[Badness]] (Smith): 0.084213
[[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: {{mapping| 1 0 2 6 -1 | 0 5 1 -10 14 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 377.388
* WE: ~2 = 1198.5372{{c}}, ~5/4 = 381.7556{{c}}
* CWE: ~2 = 1200.000, ~5/4 = 376.973
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.1943{{c}}


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


Badness (Smith): 0.046775
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: {{mapping| 1 0 2 6 6 4 | 0 5 1 -10 14 -4 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 376.914
* WE: ~2 = 1197.2300{{c}}, ~5/4 = 381.6164{{c}}
* CWE: ~2 = 1200.000, ~5/4 = 376.422
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.4690{{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 (Smith): 0.038328
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:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 381.955
* [[WE]]: ~2 = 1201.7458{{c}}, ~5/4 = 377.2996{{c}}
: [[error map]]: {{val| 0.000 +7.818 -4.359 +11.627 }}
: [[error map]]: {{val| +1.746 -15.457 -5.523 +26.870 }}
* [[CWE]]: ~2 = 1200.000, ~5/4 = 381.956
* [[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 }}


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


[[Badness]] (Smith): 0.088072
[[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: {{mapping| 1 0 2 0 0 | 0 5 1 9 11 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 381.790
* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
* CWE: ~2 = 1200.000, ~5/4 = 381.712
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.9735{{c}}


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


Badness (Smith): 0.047379
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: {{mapping| 1 0 2 0 0 -1 | 0 5 1 9 11 15 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 381.732
* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
* CWE: ~2 = 1200.000, ~5/4 = 381.736
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.4221{{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 (Smith): 0.039703
Music: [https://www.youtube.com/watch?v=oXy_RXrBVWA ''Rain in the crystal mirror caves''] - [[Budjarn Lambeth]] (2026)


== Hocum ==
== Hocum ==
Line 890: Line 782:
[[Comma list]]: 3125/3072, 4000/3969
[[Comma list]]: 3125/3072, 4000/3969


{{Mapping|legend=1| 1 5 3 -3 | 0 -10 -2 17 }}
{{Mapping|legend=1| 1 -5 1 14 | 0 10 2 -17 }}


: mapping generators: ~2, ~63/50
: mapping generators: ~2, ~63/40


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~63/50 = 409.836
* [[WE]]: ~2 = 1200.8375{{c}}, ~63/40 = 790.7032{{c}}
: [[error map]]: {{val| 0.000 -0.316 -5.986 -1.612 }}
: [[error map]]: {{val| +0.838 +0.890 -4.070 +0.944 }}
* [[POTE]]: ~2 = 1200.000, ~63/50 = 410.108
* [[CWE]]: ~2 = 1200.0000{{c}}, ~63/40 = 790.1542{{c}}
: error map: {{val| 0.000 -0.437 -6.010 -1.406 }}
: error map: {{val| 0.000 -0.413 -6.005 -1.447 }}


{{Optimal ET sequence|legend=1| 3, 38, 41, 161c }}
{{Optimal ET sequence|legend=1| 3, 38, 41, 161c }}


[[Badness]] (Smith): 0.107115
[[Badness]] (Sintel): 2.71


== Trismegistus ==
== Trismegistus ==
{{Main| Mabilic and trismegistus }}
{{See also| No-threes subgroup temperaments #Mabilic }}
{{See also| No-threes subgroup temperaments #Mabilic }}


Line 911: Line 804:
[[Comma list]]: 1029/1024, 3125/3072
[[Comma list]]: 1029/1024, 3125/3072


{{Mapping|legend=1| 1 10 4 0 | 0 -15 -3 5 }}
{{Mapping|legend=1| 1 -5 1 5 | 0 15 3 -5 }}


: mapping generators: ~2, ~147/100
: mapping generators: ~2, ~168/125


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~147/100 = 673.187
* [[WE]]: ~2 = 1201.0799{{c}}, ~168/125 = 527.1841{{c}}
: [[error map]]: {{val| 0.000 +0.240 -5.875 -2.891 }}
: [[error map]]: {{val| +1.080 +0.408 -3.681 +0.653 }}
* [[POTE]]: ~2 = 1200.000, ~147/100 = 673.290
* [[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 }}


{{Optimal ET sequence|legend=1| 16, 25, 41, 139c, 180c, 221cc, 262ccd }}
{{Optimal ET sequence|legend=1| 16, 25, 41, 139c, 180c, 221cc, 262ccd }}


[[Badness]] (Smith): 0.098334
[[Badness]] (Sintel): 2.49


=== 11-limit ===
=== 11-limit ===
Line 930: Line 823:
Comma list: 385/384, 441/440, 625/616
Comma list: 385/384, 441/440, 625/616


Mapping: {{mapping| 1 10 4 0 13 | 0 -15 -3 5 -17 }}
Mapping: {{mapping| 1 -5 1 5 -4 | 0 15 3 -5 17 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~22/15 = 673.241
* WE: ~2 = 1200.8404{{c}}, ~15/11 = 527.0289{{c}}
* POTE: ~2 = 1200.000, ~22/15 = 673.340
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6826{{c}}


{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}
{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}


Badness (Smith): 0.045623
Badness (Sintel): 1.51


=== 13-limit ===
=== 13-limit ===
Line 945: Line 838:
Comma list: 105/104, 144/143, 275/273, 625/616
Comma list: 105/104, 144/143, 275/273, 625/616


Mapping: {{mapping| 1 10 4 0 13 11 | 0 -15 -3 5 -17 -13 }}
Mapping: {{mapping| 1 -5 1 5 -4 -2 | 0 15 3 -5 17 13 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~22/15 = 673.294
* WE: ~2 = 1200.4759{{c}}, ~15/11 = 526.8502{{c}}
* POTE: ~2 = 1200.000, ~22/15 = 673.359
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6548{{c}}
 
{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}
 
Badness (Sintel): 1.37
 
=== 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: {{mapping| 1 -5 1 5 -4 -2 6 | 0 15 3 -5 17 13 -4 }}
 
Optimal tunings:
* WE: ~2 = 1200.5832{{c}}, ~15/11 = 526.8804{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6368{{c}}


{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}
{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}


Badness (Smith): 0.033081
Badness (Sintel): 1.26


== Quadrimage ==
== Quadrimage ==
Line 960: Line 868:
[[Comma list]]: 2401/2400, 3125/3072
[[Comma list]]: 2401/2400, 3125/3072


{{Mapping|legend=1| 1 5 3 4 | 0 -20 -4 -7 }}
{{Mapping|legend=1| 1 -15 -1 -3 | 0 20 4 7 }}


: mapping generators: ~2, ~28/25
: mapping generators: ~2, ~25/14


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~28/25 = 204.860
* [[WE]]: ~2 = 1201.2708{{c}}, ~25/14 = 996.0669{{c}}
: [[error map]]: {{val| 0.000 +0.853 -5.752 -2.843 }}
: [[error map]]: {{val| +1.271 +0.322 -3.317 -0.170 }}
* [[POTE]]: ~2 = 1200.000, ~28/25 = 204.987
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/14 = 995.0515{{c}}
: error map: {{val| 0.000 -1.691 -6.261 -3.733 }}
: error map: {{val| 0.000 -0.926 -6.108 -3.466 }}


{{Optimal ET sequence|legend=1| 6, 29b, 35, 41, 158cd, 199ccd, 240ccd, 281ccd }}
{{Optimal ET sequence|legend=1| 6, …, 35, 41, 158cd, 199ccd, 240ccd, 281ccd }}


[[Badness]] (Smith): 0.127422
[[Badness]] (Sintel): 3.22


=== 11-limit ===
=== 11-limit ===
Line 979: Line 887:
Comma list: 245/242, 385/384, 625/616
Comma list: 245/242, 385/384, 625/616


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~28/25 = 204.881
* WE: ~2 = 1200.6716{{c}}, ~25/14 = 995.6009{{c}}
* POTE: ~2 = 1200.000, ~28/25 = 204.956
* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 995.0633{{c}}


{{Optimal ET sequence|legend=0| 6, 35, 41 }}
{{Optimal ET sequence|legend=0| 6, 35, 41 }}


Badness (Smith): 0.061572
Badness (Sintel): 2.04


=== 13-limit ===
=== 13-limit ===
Line 994: Line 902:
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: {{mapping| 1 -15 -1 -3 -22 | 0 20 4 7 9 31 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~28/25 = 204.956
* WE: ~2 = 1200.6276{{c}}, ~25/14 = 995.4920{{c}}
* POTE: ~2 = 1200.000, ~28/25 = 205.028
* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 994.9901{{c}}


{{Optimal ET sequence|legend=0| 6f, 35f, 41, 117c }}
{{Optimal ET sequence|legend=0| 6f, 35f, 41, 117c }}


Badness (Smith): 0.044047
Badness (Sintel): 1.82


== Quinmage ==
== Quinmage ==
Line 1,014: Line 922:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~48/35 = 556.123
* [[WE]]: ~2 = 1201.3334{{c}}, ~48/35 = 556.6311{{c}}
: [[error map]]: {{val| 0.000 +1.132 -5.696 -2.480 }}
: error map: {{val| +1.333 +0.489 -3.158 -0.835 }}
* [[CWE]]: ~2 = 1200.000, ~48/35 = 556.050
* [[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 }}


{{Optimal ET sequence|legend=1| 13b, 28b, 41, 177bcd, 218bccdd, 259bccdd, 300cccdd }}
{{Optimal ET sequence|legend=1| 13b, 28b, 41, 177bcd, 218bccdd, 259bccdd, 300cccdd }}


[[Badness]] (Smith): 0.194548
[[Badness]] (Sintel): 4.92


=== 11-limit ===
=== 11-limit ===
Line 1,031: Line 939:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/8 = 556.122
* WE: ~2 = 1200.4252{{c}}, ~11/8 = 556.2831{{c}}
* CWE: ~2 = 1200.000, ~11/8 = 556.095
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 556.0951{{c}}


{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}
{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}


Badness (Smith): 0.101724
Badness (Sintel): 3.36


=== 13-limit ===
=== 13-limit ===
Line 1,046: Line 954:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/8 = 556.106
* WE: ~2 = 1199.8239{{c}}, ~11/8 = 556.0389{{c}}
* CWE: ~2 = 1200.000, ~11/8 = 556.117
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 556.1171{{c}}


{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}
{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}


Badness (Smith): 0.067742
Badness (Sintel): 2.80


== Warlock ==
== Warlock ==
Line 1,063: Line 971:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~8/7 = 240.000, ~5/4 = 380.499 (~256/245 = 99.501)
* [[WE]]: ~8/7 = 240.3877{{c}}, ~5/4 = 380.4267{{c}} (~256/245 = 100.3488{{c}})
: [[error map]]: {{val| 0.000 +0.542 -5.814 -8.826 }}
: error map: {{val| +1.939 +0.178 -2.010 -3.398 }}
* [[CWE]]: ~8/7 = 240.000, ~5/4 = 379.997 (~256/245 = 100.003)
* [[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 }}


{{Optimal ET sequence|legend=1| 25, 35, 60 }}
{{Optimal ET sequence|legend=1| 25, 35, 60 }}


[[Badness]] (Smith): 0.287190
[[Badness]] (Sintel): 7.27


[[Category:Temperament families]]
[[Category:Temperament families]]
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