Magic family: Difference between revisions

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{{interwiki
{{Technical data page}}
| de = Magische Temperaturen
| en = Magic family
| es =
| ja =
}}
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 21: Line 16:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~5/4 = 380.4994
* [[WE]]: ~2 = 1201.2449{{c}}, ~5/4 = 380.4527{{c}}
* [[POTE]]: ~2 = 1\1, ~5/4 = 380.058
: [[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 }}


[[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 34: Line 31:
[[Algebraic generator]]: Terzbirat, the positive root of 9''x''<sup>2</sup> - 8''x'' - 4 = (4 + 2√13)/9; approximately 380.3175 [[cent]]s.
[[Algebraic generator]]: Terzbirat, the positive root of 9''x''<sup>2</sup> - 8''x'' - 4 = (4 + 2√13)/9; approximately 380.3175 [[cent]]s.


{{Optimal ET sequence|legend=1| 3, 13b, 16, 19, 22, 41 }}
{{Optimal ET sequence|legend=1| 3, 13b, 16, 19, 41, 60, 221cc, 281cc }}


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


=== Overview to extensions ===
=== Overview to extensions ===
Line 50: Line 47:
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.
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 temperament|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.  
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 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, {{multival| 5 1 12 -8 … }}. 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 62: Line 59:
{{Mapping|legend=1| 1 0 2 -1 | 0 5 1 12 }}
{{Mapping|legend=1| 1 0 2 -1 | 0 5 1 12 }}


: Mapping generators: ~2, ~5/4
: mapping generators: ~2, ~5/4
 
{{Multival|legend=1| 5 1 12 -10 5 25 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~5/4 = 380.6512
* [[WE]]: ~2 = 1201.0786{{c}}, ~5/4 = 380.6939{{c}}
* [[POTE]]: ~2 = 1\1, ~5/4 = 380.352
: [[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 }}


[[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 80: Line 77:
[[Algebraic generator]]: Tirzbirat or Septimage, the real root of 5''x''<sup>5</sup> + 4''x'' - 20, 380.7604 cents.
[[Algebraic generator]]: Tirzbirat or Septimage, the real root of 5''x''<sup>5</sup> + 4''x'' - 20, 380.7604 cents.


{{Optimal ET sequence|legend=1| 19, 22, 41, 104, 145c, 186c }}
{{Optimal ET sequence|legend=1| 19, 41, 142cd, 183cd, 224ccd }}


[[Badness]]: 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


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 }}
 
{{Multival|legend=1| 5 1 12 -8 -10 5 -30 25 -22 -64 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 380.7200
* WE: ~2 = 1200.1372{{c}}, ~5/4 = 380.7399{{c}}
* POTE: ~2 = 1\1, ~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 107: Line 100:
* 11-odd-limit diamond tradeoff: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)
* 11-odd-limit diamond tradeoff: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)


Optimal ET sequence: {{Optimal ET sequence| 19, 22, 41, 104, 145c, 249cce }}
{{Optimal ET sequence|legend=0| 19, 22, 41, 104 }}


Badness: 0.020352
Badness (Sintel): 0.673


==== 13-limit ====
==== 13-limit ====
Line 118: 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:  
* CTE: ~2 = 1\1, ~5/4 = 380.4354
* WE: ~2 = 1200.0331{{c}}, ~5/4 = 380.4377{{c}}
* POTE: ~2 = 1\1, ~5/4 = 380.427
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4284{{c}}


Tuning ranges:  
Tuning ranges:  
Line 128: Line 121:
* 13- and 15-odd-limit diamond tradeoff: ~5/4 = [378.617, 386.314]
* 13- and 15-odd-limit diamond tradeoff: ~5/4 = [378.617, 386.314]


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


Badness: 0.021509
Badness (Sintel): 0.889


===== Magical =====
===== Magical =====
Line 137: 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:  
* CTE: ~2 = 1\1, ~5/4 = 380.5103
* WE: ~2 = 1199.3584{{c}}, ~5/4 = 380.4006{{c}}
* POTE: ~2 = 1\1, ~5/4 = 380.604
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5896{{c}}


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


Badness: 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


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:  
* CTE: ~2 = 1\1, ~5/4 = 380.4507
* WE: ~2 = 1199.7173{{c}}, ~5/4 = 380.3808{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4680{{c}}


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


Badness: 0.020881
Badness (Sintel): 1.27


====== Magica ======
====== Magica ======
Line 166: 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:  
* CTE: ~2 = 1\1, ~5/4 = 380.5733
* WE: ~2 = 1199.3670{{c}}, ~5/4 = 380.4681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6541{{c}}


Optimal ET sequence: {{Optimal ET sequence| 19h, 22fh, 41 }}
{{Optimal ET sequence|legend=0| 22fh, 41 }}


Badness: 0.019945
Badness (Sintel): 1.21


===== Magia =====
===== Magia =====
Line 180: 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:  
* CTE: ~2 = 1\1, ~5/4 = 380.2706
* WE: ~2 = 1200.1727{{c}}, ~5/4 = 380.2982{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2483{{c}}


Optimal ET sequence: {{Optimal ET sequence| 19g, 41, 101cde, 142cdefg }}
{{Optimal ET sequence|legend=0| 19g, 41, 60 }}


Badness: 0.026232
Badness (Sintel): 1.34


====== 19-limit ======
====== 19-limit ======
Line 194: 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:  
* CTE: ~2 = 1\1, ~5/4 = 380.3604
* WE: ~2 = 1200.2179{{c}}, ~5/4 = 380.3942{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.3314{{c}}


Optimal ET sequence: {{Optimal ET sequence| 19g, 41 }}
{{Optimal ET sequence|legend=0| 19gh, 41 }}


Badness: 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 210: 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


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:  
* POTE: ~2 = 1\1, ~5/4 = 380.416
* WE: ~2 = 1200.2802{{c}}, ~5/4 = 380.5053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4258{{c}}


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


==== Sorcery ====
==== Sorcery ====
Line 222: 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:
* CTE: ~2 = 1\1, ~5/4 = 380.6741
* POTE: ~2 = 1\1, ~5/4 = 380.477
 
Optimal ET sequence: {{Optimal ET sequence| 19, 22, 41f }}
 
Badness: 0.025829
 
===== 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 = 1\1, ~5/4 = 380.7839
* POTE: ~2 = 1\1, ~5/4 = 380.729
 
Optimal ET sequence: {{Optimal ET sequence| 19, 22, 41f }}
 
Badness: 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:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 380.6248
* WE: ~2 = 1201.2397{{c}}, ~5/4 = 380.8698{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5080{{c}}


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


Badness: 0.023232
Badness (Sintel): 1.07


==== Necromancy ====
==== Necromancy ====
Line 266: 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:  
* CTE: ~2 = 1\1, ~5/4 = 380.7876
* WE: ~2 = 1199.9675{{c}}, ~5/4 = 380.7770{{c}}
* POTE: ~2 = 1\1, ~5/4 = 380.787
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7874{{c}}


Optimal ET sequence: {{Optimal ET sequence| 19f, 22, 41, 63, 104 }}
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63, 104 }}


Badness: 0.025275
Badness (Sintel): 1.04


===== 17-limit =====
===== 17-limit =====
Line 281: 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:
* CTE: ~2 = 1\1, ~5/4 = 380.8373
* POTE: ~2 = 1\1, ~5/4 = 380.827
 
Optimal ET sequence: {{Optimal ET sequence| 19f, 22, 41, 63, 104g }}
 
Badness: 0.022032
 
===== 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:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 380.7411
* WE: ~2 = 1199.6176{{c}}, ~5/4 = 380.7053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.8280{{c}}


Optimal ET sequence: {{Optimal ET sequence| 19f, 22, 41, 63h, 104gh }}
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63 }}


Badness: 0.021101
Badness (Sintel): 1.12


==== Soothsaying ====
==== Soothsaying ====
Line 310: 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:  
* CTE: ~55/39 = 1\2, ~5/4 = 380.5385
* WE: ~55/39 = 600.2918{{c}}, ~5/4 = 380.6928{{c}}
* POTE: ~55/39 = 1\2, ~5/4 = 380.508
* CWE: ~55/39 = 600.0000{{c}}, ~5/4 = 380.5121{{c}}


Optimal ET sequence: {{Optimal ET sequence| 22, 60, 82 }}
{{Optimal ET sequence|legend=0| 22, 60, 82 }}


Badness: 0.055443
Badness (Sintel): 2.29


===== 17-limit =====
===== 17-limit =====
Line 325: 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:  
* CTE: ~17/12 = 1\2, ~5/4 = 380.5533
* WE: ~17/12 = 600.2918{{c}}, ~5/4 = 380.6927{{c}}
* POTE: ~17/12 = 1\2, ~5/4 = 380.508
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.5135{{c}}


Optimal ET sequence: {{Optimal ET sequence| 22, 60, 82 }}
{{Optimal ET sequence|legend=0| 22, 60, 82 }}


Badness: 0.035654
Badness (Sintel): 1.82


===== 19-limit =====
===== 19-limit =====
Line 340: 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:  
* CTE: ~17/12 = 1\2, ~5/4 = 380.4704
* WE: ~17/12 = 600.3301{{c}}, ~5/4 = 380.6797{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.4704{{c}}


Optimal ET sequence: {{Optimal ET sequence| 22, 60, 82 }}
{{Optimal ET sequence|legend=0| 22, 60, 82 }}


Badness: 0.031291
Badness (Sintel): 1.90


=== Telepathy ===
=== Telepathy ===
Line 354: 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 }}
 
{{Multival|legend=1| 5 1 12 14 -10 5 5 25 29 -2 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 381.2311
* WE: ~2 = 1200.7724{{c}}, ~5/4 = 381.2641{{c}}
* POTE: ~2 = 1\1, ~5/4 = 381.019
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 381.0913{{c}}


Optimal ET sequence: {{Optimal ET sequence| 19e, 22, 63e, 85ee }}
{{Optimal ET sequence|legend=0| 19e, 22, 41e, 63e }}


Badness: 0.027109
Badness (Sintel): 0.896


==== 13-limit ====
==== 13-limit ====
Line 371: 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:
* CTE: ~2 = 1\1, ~5/4 = 381.1957
* POTE: ~2 = 1\1, ~5/4 = 380.520
 
Optimal ET sequence: {{Optimal ET sequence| 19e, 22, 63eff }}
 
Badness: 0.025522
 
===== 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 = 1\1, ~5/4 = 381.2884
* POTE: ~2 = 1\1, ~5/4 = 380.619
 
Optimal ET sequence: {{Optimal ET sequence| 19eg, 22, 63effg }}
 
Badness: 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:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 381.4641
* WE: ~2 = 1202.5634{{c}}, ~5/4 = 381.3348{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6886{{c}}


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


Badness: 0.019004
Badness (Sintel): 1.05


==== Intuition ====
==== Intuition ====
Line 415: 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:
* CTE: ~2 = 1\1, ~5/4 = 380.8157
* POTE: ~2 = 1\1, ~5/4 = 380.483
 
Optimal ET sequence: {{Optimal ET sequence| 19e, 22f, 41e, 63ef }}
 
Badness: 0.026089
 
===== 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 = 1\1, ~5/4 = 380.9518
* POTE: ~2 = 1\1, ~5/4 = 380.604
 
Optimal ET sequence: {{Optimal ET sequence| 19eg, 22f, 41eg, 63efg }}
 
Badness: 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:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 380.8658
* WE: ~2 = 1201.3172{{c}}, ~5/4 = 380.9004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5942{{c}}


Optimal ET sequence: {{Optimal ET sequence| 19egh, 22fh, 41eg, 63efg }}
{{Optimal ET sequence|legend=0| 19e, 22f }}


Badness: 0.019518
Badness (Sintel): 1.08


=== Horcrux ===
=== Horcrux ===
Line 459: 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:  
* CTE: ~2 = 1\1, ~5/4 = 379.7959
* WE: ~2 = 1200.4670{{c}}, ~5/4 = 379.7895{{c}}
* POTE: ~2 = 1\1, ~5/4 = 379.642
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.6889{{c}}


Optimal ET sequence: {{Optimal ET sequence| 3de, …, 16d, 19, 41ee, 60ee }}
{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}


Badness: 0.039282
Badness (Sintel): 1.30


==== 13-limit ====
==== 13-limit ====
Line 474: 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:  
* CTE: ~2 = 1\1, ~5/4 = 379.8847
* WE: ~2 = 1200.2953{{c}}, ~5/4 = 379.8842{{c}}
* POTE: ~2 = 1\1, ~5/4 = 379.791
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.8165{{c}}


Optimal ET sequence: {{Optimal ET sequence| 3def, …, 19, 41ee, 60ee }}
{{Optimal ET sequence|legend=0| 3def, 16dff, 19 }}


Badness: 0.031938
Badness (Sintel): 1.32


===== 17-limit =====
===== 17-limit =====
Line 489: 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:  
* CTE: ~2 = 1\1, ~5/4 = 380.2159
* WE: ~2 = 1200.2484{{c}}, ~5/4 = 380.2053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1482{{c}}


Optimal ET sequence: {{Optimal ET sequence| 3defg, …, 19g, 41eeg }}
{{Optimal ET sequence|legend=0| 3defg, 16dffgg, 19g }}


Badness: 0.028074
Badness (Sintel): 1.43


===== Horcruxic =====
===== Horcruxic =====
Line 503: 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:  
* CTE: ~2 = 1\1, ~5/4 = 379.5237
* WE: ~2 = 1199.5457{{c}}, ~5/4 = 379.4681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.5713{{c}}


Optimal ET sequence: {{Optimal ET sequence| 3defg, …, 19 }}
{{Optimal ET sequence|legend=0| 3defg, 16dff, 19 }}


Badness: 0.029556
Badness (Sintel): 1.51


==== Glamour ====
==== Glamour ====
Line 517: 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:  
* CTE: ~2 = 1\1, ~5/4 = 379.7574
* WE: ~2 = 1202.2187{{c}}, ~5/4 = 379.8171{{c}}
* POTE: ~2 = 1\1, ~5/4 = 379.116
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.2709{{c}}


Optimal ET sequence: {{Optimal ET sequence| 3de, …, 16d, 19, 41eef, 60eeff }}
{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}


Badness: 0.033317
Badness (Sintel): 1.38


=== Witchcraft ===
=== Witchcraft ===
Line 532: 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:  
* CTE: ~2 = 1\1, ~5/4 = 380.4163
* WE: ~2 = 1201.2634{{c}}, ~5/4 = 380.6321{{c}}
* POTE: ~2 = 1\1, ~5/4 = 380.232
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2849{{c}}


Optimal ET sequence: {{Optimal ET sequence| 19e, 22e, 41 }}
{{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd, 243ccdde }}


Badness: 0.030706
Badness (Sintel): 1.02


==== 13-limit ====
==== 13-limit ====
Line 547: 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:  
* CTE: ~2 = 1\1, ~5/4 = 380.3523
* WE: ~2 = 1201.0424{{c}}, ~5/4 = 380.5193{{c}}
* POTE: ~2 = 1\1, ~5/4 = 380.189
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2349{{c}}


Optimal ET sequence: {{Optimal ET sequence| 19e, 41, 142cdf }}
{{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd }}


Badness: 0.023547
Badness (Sintel): 0.973


==== 17-limit ====
==== 17-limit ====
Line 562: 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:
* CTE: ~2 = 1\1, ~5/4 = 380.2809
* POTE: ~2 = 1\1, ~5/4 = 380.114
 
Optimal ET sequence: {{Optimal ET sequence| 19eg, 41, 101cd, 142cdfg }}
 
Badness: 0.020756
 
==== 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:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 380.3389
* WE: ~2 = 1201.1638{{c}}, ~5/4 = 380.4827{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1599{{c}}


Optimal ET sequence: {{Optimal ET sequence| 19egh, 41, 101cdhh, 142cdfghh }}
{{Optimal ET sequence|legend=0| 19eg, 41, 60e, 101cd }}


Badness: 0.018625
Badness (Sintel): 1.06


=== Divination ===
=== Divination ===
Line 591: 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:  
* CTE: ~99/70 = 1\2, ~5/4 = 380.7317
* WE: ~99/70 = 600.8306{{c}}, ~5/4 = 380.7598{{c}}
* POTE: ~99/70 = 1\2, ~5/4 = 380.223
* CWE: ~99/70 = 600.0000{{c}}, ~5/4 = 380.3800{{c}}


Optimal ET sequence: {{Optimal ET sequence| 22, 60e, 82e, 104e }}
{{Optimal ET sequence|legend=0| 22, 38d, 60e, 142cdee, 202ccddeee }}


Badness: 0.035864
Badness (Sintel): 1.19


==== 13-limit ====
==== 13-limit ====
Line 606: 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:  
* CTE: ~99/70 = 1\2, ~5/4 = 380.4171
* WE: ~99/70 = 600.9624{{c}}, ~5/4 = 380.5297{{c}}
* POTE: ~99/70 = 1\2, ~5/4 = 379.920
* CWE: ~99/70 = 600.0000{{c}}, ~5/4 = 380.0614{{c}}


Optimal ET sequence: {{Optimal ET sequence| 22f, 60e, 82ef }}
{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}


Badness: 0.034551
Badness (Sintel): 1.43


==== 17-limit ====
==== 17-limit ====
Line 621: 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:  
* CTE: ~17/12 = 1\2, ~5/4 = 380.4329
* WE: ~17/12 = 600.8921{{c}}, ~5/4 = 380.5094{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.0672{{c}}


Optimal ET sequence: {{Optimal ET sequence| 22f, 60e, 82ef }}
{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}


Badness: 0.023775
Badness (Sintel): 1.21


=== Hocus ===
=== Hocus ===
Line 635: Line 522:
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|legend=0| 1 -5 1 -13 -13 | 0 10 2 24 25 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~14/11 = 409.7603
* WE: ~2 = 1201.0749{{c}}, ~11/7 = 790.7980{{c}}
* POTE: ~2 = 1\1, ~14/11 = 409.910
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.1429{{c}}


Optimal ET sequence: {{Optimal ET sequence| 38d, 41 }}
{{Optimal ET sequence|legend=0| 38d, 41, 120cd }}


Badness: 0.038519
Badness (Sintel): 1.27


==== 13-limit ====
==== 13-limit ====
Line 650: 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 3 11 12 16 | 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:  
* CTE: ~2 = 1\1, ~14/11 = 409.8421
* WE: ~2 = 1201.2830{{c}}, ~11/7 = 790.8409{{c}}
* POTE: ~2 = 1\1, ~14/11 = 410.004
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.0516{{c}}


Optimal ET sequence: {{Optimal ET sequence| 38df, 41 }}
{{Optimal ET sequence|legend=0| 38df, 41, 79d, 120cd }}


Badness: 0.030280
Badness (Sintel): 1.25


==== 17-limit ====
==== 17-limit ====
Line 665: 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 3 11 12 16 14 | 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:  
* CTE: ~2 = 1\1, ~14/11 = 409.8958
* WE: ~2 = 1201.1557{{c}}, ~11/7 = 790.7157{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.0057{{c}}


Optimal ET sequence: {{Optimal ET sequence| 38df, 41, 120cdg, 161cdg, 202ccddfgg }}
{{Optimal ET sequence|legend=0| 38df, 41, 79d }}


Badness: 0.025491
Badness (Sintel): 1.30


==== 19-limit ====
==== 19-limit ====
Line 679: 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 3 11 12 16 14 8 | 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:  
* CTE: ~2 = 1\1, ~14/11 = 409.8836
* WE: ~2 = 1201.3558{{c}}, ~11/7 = 790.8266{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 789.9880{{c}}


Optimal ET sequence: {{Optimal ET sequence| 38df, 41, 120cdgh, 161cdgh, 202ccddfgghh }}
{{Optimal ET sequence|legend=0| 38df, 41, 79dh }}


Badness: 0.020277
Badness (Sintel): 1.23


== Muggles ==
== Muggles ==
Line 698: Line 587:


{{Mapping|legend=1| 1 0 2 5 | 0 5 1 -7 }}
{{Mapping|legend=1| 1 0 2 5 | 0 5 1 -7 }}
{{Multival|legend=1| 5 1 -7 -10 -25 -19 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~5/4 = 378.744
* [[WE]]: ~2 = 1203.9554{{c}}, ~5/4 = 379.7269{{c}}
* [[POTE]]: ~2 = 1\1, ~5/4 = 378.479
: [[error map]]: {{val| +3.955 -3.321 +1.324 -7.137 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 378.5328{{c}}
: error map: {{val| 0.000 -9.291 -7.781 -18.555 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 712: Line 601:
{{Optimal ET sequence|legend=1| 16, 19, 73bcd, 92bcdd, 111bcddd }}
{{Optimal ET sequence|legend=1| 16, 19, 73bcd, 92bcdd, 111bcddd }}


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


=== 11-limit ===
=== 11-limit ===
Line 719: 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:  
* CTE: ~2 = 1\1, ~5/4 = 378.228
* WE: ~2 = 1203.0804{{c}}, ~5/4 = 378.6936{{c}}
* POTE: ~2 = 1\1, ~5/4 = 377.724
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8174{{c}}


Tuning ranges:  
Tuning ranges:  
Line 729: Line 618:
* 11-odd-limit diamond tradeoff: ~5/4 = [347.408, 386.314]
* 11-odd-limit diamond tradeoff: ~5/4 = [347.408, 386.314]


Optimal ET sequence: {{Optimal ET sequence| 16, 19, 54bd }}
{{Optimal ET sequence|legend=0| 16, 19, 35, 54bd }}


Badness: 0.048038
Badness (Sintel): 1.59


==== 13-limit ====
==== 13-limit ====
Line 738: 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:  
* CTE: ~2 = 1\1, ~5/4 = 378.1761
* WE: ~2 = 1203.4291{{c}}, ~5/4 = 378.7321{{c}}
* POTE: ~2 = 1\1, ~5/4 = 377.653
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.7336{{c}}


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


Badness: 0.030386
Badness (Sintel): 1.26


=== Muggloid ===
=== Muggloid ===
Line 753: 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:  
* CTE: ~2 = 1\1, ~5/4 = 377.724
* WE: ~2 = 1205.6044{{c}}, ~5/4 = 379.5966{{c}}
* POTE: ~2 = 1\1, ~5/4 = 377.832
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8142{{c}}


Optimal ET sequence: {{Optimal ET sequence| 3, 10bd, 13bd, 16, 19e, 35ee }}
{{Optimal ET sequence|legend=0| 3, 16, 19e, 35ee, 54bdeee }}


Badness: 0.046970
Badness (Sintel): 1.55


==== 13-limit ====
==== 13-limit ====
Line 768: 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:  
* CTE: ~2 = 1\1, ~5/4 = 377.652
* WE: ~2 = 1205.4897{{c}}, ~5/4 = 379.5667{{c}}
* POTE: ~2 = 1\1, ~5/4 = 377.838
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8118{{c}}


Optimal ET sequence: {{Optimal ET sequence| 3, 10bd, 13bd, 16, 19e, 35eef }}
{{Optimal ET sequence|legend=0| 3, 16, 19e, 35eef }}


Badness: 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 }}
 
{{Multival|legend=1| 5 1 9 -10 0 18 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~5/4 = 377.385
* [[WE]]: ~2 = 1198.1701{{c}}, ~5/4 = 381.3741{{c}}
: error map: {{val| -1.830 +4.915 -8.599 +6.454 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 381.9562{{c}}
: error map: {{val| 0.000 +7.826 -4.358 +11.613 }}


{{Optimal ET sequence|legend=1| 3d, …, 13b, 16, 19d, 35d }}
{{Optimal ET sequence|legend=1| 3, 19d, 22 }}


[[Badness]]: 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|legend=0| 1 0 2 6 -1 | 0 5 1 -10 14 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 377.388
* WE: ~2 = 1198.5372{{c}}, ~5/4 = 381.7556{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.1943{{c}}


Optimal ET sequence: {{Optimal ET sequence| 3de, …, 13be, 16, 19d, 35d }}
{{Optimal ET sequence|legend=0| 22, 69b }}


Badness: 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|legend=0| 1 0 2 6 6 4 | 0 5 1 -10 14 -4 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 376.914
* WE: ~2 = 1197.2300{{c}}, ~5/4 = 381.6164{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.4690{{c}}


Optimal ET sequence: {{Optimal ET sequence| 3def, …, 16, 35d }}
{{Optimal ET sequence|legend=0| 22f, 47bff }}


Badness: 0.038328
Badness (Sintel): 1.99
 
== 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.  


== Brightstone ==
This temperament is known as ''witch'' in [http://www.tonalsoft.com/enc/m/magic.aspx Tonalsoft Encyclopedia].
{{See also| Archytas clan }}


[[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 }}
 
{{Multival|legend=1| 5 1 -10 -10 -30 -26 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~5/4 = 381.955
* [[WE]]: ~2 = 1201.7458{{c}}, ~5/4 = 377.2996{{c}}
: [[error map]]: {{val| +1.746 -15.457 -5.523 +26.870 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 376.9630{{c}}
: 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]]: 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|legend=0| 1 0 2 0 0 | 0 5 1 9 11 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 381.790
* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.9735{{c}}


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


Badness: 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|legend=0| 1 0 2 0 0 -1 | 0 5 1 9 11 15 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/4 = 381.732
* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
* 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: {{Optimal ET sequence| 3, …, 19d, 22 }}
Scales: [[User:BudjarnLambeth/Volcanic glass]]


Badness: 0.039703
Music: [https://www.youtube.com/watch?v=oXy_RXrBVWA ''Rain in the crystal mirror caves''] - [[Budjarn Lambeth]] (2026)


== Hocum ==
== Hocum ==
Line 875: Line 776:
[[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
 
{{Multival|legend=1| 10 2 -17 -20 -55 -45 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~63/50 = 409.836
* [[WE]]: ~2 = 1200.8375{{c}}, ~63/40 = 790.7032{{c}}
* [[POTE]]: ~2 = 1\1, ~63/50 = 410.108
: [[error map]]: {{val| +0.838 +0.890 -4.070 +0.944 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~63/40 = 790.1542{{c}}
: error map: {{val| 0.000 -0.413 -6.005 -1.447 }}


{{Optimal ET sequence|legend=1| 3, …, 35d, 38, 41, 202cc, 243cc, 284cc }}
{{Optimal ET sequence|legend=1| 3, 38, 41, 161c }}


[[Badness]]: 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 896: Line 798:
[[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
 
{{Multival|legend=1| 15 3 -5 -30 -50 -20 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~147/100 = 673.187
* [[WE]]: ~2 = 1201.0799{{c}}, ~168/125 = 527.1841{{c}}
* [[POTE]]: ~2 = 1\1, ~147/100 = 673.290
: [[error map]]: {{val| +1.080 +0.408 -3.681 +0.653 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~168/125 = 526.7349{{c}}
: error map: {{val| 0.000 -0.932 -6.109 -2.500 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 915: Line 817:
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|legend=0| 1 -5 1 5 -4 | 0 15 3 -5 17 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~22/15 = 673.241
* WE: ~2 = 1200.8404{{c}}, ~15/11 = 527.0289{{c}}
* POTE: ~2 = 1\1, ~22/15 = 673.340
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6826{{c}}


Optimal ET sequence: {{Optimal ET sequence| 16, 25e, 41, 221cc, 262ccde }}
{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}


Badness: 0.045623
Badness (Sintel): 1.51


=== 13-limit ===
=== 13-limit ===
Line 930: 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 10 4 0 13 11 | 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:  
* CTE: ~2 = 1\1, ~22/15 = 673.294
* WE: ~2 = 1200.4759{{c}}, ~15/11 = 526.8502{{c}}
* POTE: ~2 = 1\1, ~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 ET sequence: {{Optimal ET sequence| 16, 25e, 41, 139cf, 180cf, 221ccf }}
Optimal tunings:
* WE: ~2 = 1200.5832{{c}}, ~15/11 = 526.8804{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6368{{c}}


Badness: 0.033081
{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}
 
Badness (Sintel): 1.26


== Quadrimage ==
== Quadrimage ==
Line 945: Line 862:
[[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
 
{{Multival|legend=1| 20 4 7 -40 -45 5 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~28/25 = 204.860
* [[WE]]: ~2 = 1201.2708{{c}}, ~25/14 = 996.0669{{c}}
* [[POTE]]: ~2 = 1\1, ~28/25 = 204.987
: [[error map]]: {{val| +1.271 +0.322 -3.317 -0.170 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/14 = 995.0515{{c}}
: error map: {{val| 0.000 -0.926 -6.108 -3.466 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 964: Line 881:
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|legend=0| 1 -15 -1 -3 -4 | 0 20 4 7 9 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~28/25 = 204.881
* WE: ~2 = 1200.6716{{c}}, ~25/14 = 995.6009{{c}}
* POTE: ~2 = 1\1, ~28/25 = 204.956
* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 995.0633{{c}}


Optimal ET sequence: {{Optimal ET sequence| 6, …, 29b, 35, 41 }}
{{Optimal ET sequence|legend=0| 6, 35, 41 }}


Badness: 0.061572
Badness (Sintel): 2.04


=== 13-limit ===
=== 13-limit ===
Line 979: 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:  
* CTE: ~2 = 1\1, ~28/25 = 204.956
* WE: ~2 = 1200.6276{{c}}, ~25/14 = 995.4920{{c}}
* POTE: ~2 = 1\1, ~28/25 = 205.028
* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 994.9901{{c}}


Optimal ET sequence: {{Optimal ET sequence| 35f, 41, 199ccdef, 240ccdef, 281ccdeff }}
{{Optimal ET sequence|legend=0| 6f, 35f, 41, 117c }}


Badness: 0.044047
Badness (Sintel): 1.82


== Quinmage ==
== Quinmage ==
Line 996: Line 913:
{{Mapping|legend=1| 1 -10 0 -6 | 0 25 5 19 }}
{{Mapping|legend=1| 1 -10 0 -6 | 0 25 5 19 }}


{{Multival|legend=1| 25 5 19 -50 -40 30 }}
: mapping generators: ~2, ~48/35


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~48/35 = 556.123
* [[WE]]: ~2 = 1201.3334{{c}}, ~48/35 = 556.6311{{c}}
: error map: {{val| +1.333 +0.489 -3.158 -0.835 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 556.0504{{c}}
: error map: {{val| 0.000 -0.695 -6.062 -3.868 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 1,010: 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:  
* CTE: ~2 = 1\1, ~11/8 = 556.122
* WE: ~2 = 1200.4252{{c}}, ~11/8 = 556.2831{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 556.0951{{c}}


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


Badness: 0.101724
Badness (Sintel): 3.36


=== 13-limit ===
=== 13-limit ===
Line 1,024: 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:  
* CTE: ~2 = 1\1, ~11/8 = 556.106
* WE: ~2 = 1199.8239{{c}}, ~11/8 = 556.0389{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 556.1171{{c}}


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


Badness: 0.067742
Badness (Sintel): 2.80


== Warlock ==
== Warlock ==
Line 1,043: Line 965:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~8/7 = 1\5, ~5/4 = 380.499 (~256/245 = 99.501)
* [[WE]]: ~8/7 = 240.3877{{c}}, ~5/4 = 380.4267{{c}} (~256/245 = 100.3488{{c}})
* [[POTE]]: ~8/7 = 1\5, ~5/4 = 379.7131 (~256/245 = 100.2869)
: error map: {{val| +1.939 +0.178 -2.010 -3.398 }}
* [[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 }}


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


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


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

Latest revision as of 12:33, 24 September 2026

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

Magicus

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

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

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

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

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

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:

  • WE: ~2 = 1203.9554 ¢, ~5/4 = 379.7269 ¢
error map: ⟨+3.955 -3.321 +1.324 -7.137]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 378.5328 ¢
error map: ⟨0.000 -9.291 -7.781 -18.555]

Tuning ranges:

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

Badness (Sintel): 1.42

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:

  • WE: ~2 = 1203.0804 ¢, ~5/4 = 378.6936 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 377.8174 ¢

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 (Sintel): 1.59

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:

  • WE: ~2 = 1203.4291 ¢, ~5/4 = 378.7321 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 377.7336 ¢

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

Badness (Sintel): 1.26

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:

  • WE: ~2 = 1205.6044 ¢, ~5/4 = 379.5966 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 377.8142 ¢

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

Badness (Sintel): 1.55

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:

  • WE: ~2 = 1205.4897 ¢, ~5/4 = 379.5667 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 377.8118 ¢

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

Badness (Sintel): 1.19

Brightstone

Brightstone tempers out 64/63 and may be described as 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

Comma list: 64/63, 3125/3024

Mapping: [⟨1 0 2 6], ⟨0 5 1 -10]]

Optimal tunings:

  • WE: ~2 = 1198.1701 ¢, ~5/4 = 381.3741 ¢
error map: ⟨-1.830 +4.915 -8.599 +6.454]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 381.9562 ¢
error map: ⟨0.000 +7.826 -4.358 +11.613]

Optimal ET sequence: 3, 19d, 22

Badness (Sintel): 2.23

11-limit

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 625/616

Mapping: [⟨1 0 2 6 -1], ⟨0 5 1 -10 14]]

Optimal tunings:

  • WE: ~2 = 1198.5372 ¢, ~5/4 = 381.7556 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 382.1943 ¢

Optimal ET sequence: 22, 69b

Badness (Sintel): 1.85

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 55/54, 64/63, 625/616

Mapping: [⟨1 0 2 6 6 4], ⟨0 5 1 -10 14 -4]]

Optimal tunings:

  • WE: ~2 = 1197.2300 ¢, ~5/4 = 381.6164 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 382.4690 ¢

Optimal ET sequence: 22f, 47bff

Badness (Sintel): 1.99

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.

This temperament is known as witch in Tonalsoft Encyclopedia.

Subgroup: 2.3.5.7

Comma list: 36/35, 1875/1792

Mapping: [⟨1 0 2 0], ⟨0 5 1 9]]

Optimal tunings:

  • WE: ~2 = 1201.7458 ¢, ~5/4 = 377.2996 ¢
error map: ⟨+1.746 -15.457 -5.523 +26.870]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 376.9630 ¢
error map: ⟨0.000 -17.140 -9.351 +23.841]

Optimal ET sequence: 3d, …, 13b, 16

Badness (Sintel): 2.13

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:

  • WE: ~2 = 1201.7428 ¢, ~5/4 = 377.3134 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 376.9735 ¢

Optimal ET sequence: 3de, 13be, 16

Badness (Sintel): 1.55

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:

  • WE: ~2 = 1201.7428 ¢, ~5/4 = 377.3134 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 376.4221 ¢

Optimal ET sequence: 3def, 13beff, 16

Badness (Sintel): 1.58

Scales: User:BudjarnLambeth/Volcanic glass

Music: Rain in the crystal mirror caves - Budjarn Lambeth (2026)

Hocum

Subgroup: 2.3.5.7

Comma list: 3125/3072, 4000/3969

Mapping: [⟨1 -5 1 14], ⟨0 10 2 -17]]

mapping generators: ~2, ~63/40

Optimal tunings:

  • WE: ~2 = 1200.8375 ¢, ~63/40 = 790.7032 ¢
error map: ⟨+0.838 +0.890 -4.070 +0.944]
  • CWE: ~2 = 1200.0000 ¢, ~63/40 = 790.1542 ¢
error map: ⟨0.000 -0.413 -6.005 -1.447]

Optimal ET sequence: 3, 38, 41, 161c

Badness (Sintel): 2.71

Trismegistus

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

Subgroup: 2.3.5.7

Comma list: 1029/1024, 3125/3072

Mapping: [⟨1 -5 1 5], ⟨0 15 3 -5]]

mapping generators: ~2, ~168/125

Optimal tunings:

  • WE: ~2 = 1201.0799 ¢, ~168/125 = 527.1841 ¢
error map: ⟨+1.080 +0.408 -3.681 +0.653]
  • CWE: ~2 = 1200.0000 ¢, ~168/125 = 526.7349 ¢
error map: ⟨0.000 -0.932 -6.109 -2.500]

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

Badness (Sintel): 2.49

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~2 = 1200.8404 ¢, ~15/11 = 527.0289 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/11 = 526.6826 ¢

Optimal ET sequence: 16, 25e, 41, 98c

Badness (Sintel): 1.51

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -5 1 5 -4 -2], ⟨0 15 3 -5 17 13]]

Optimal tunings:

  • WE: ~2 = 1200.4759 ¢, ~15/11 = 526.8502 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/11 = 526.6548 ¢

Optimal ET sequence: 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: [⟨1 -5 1 5 -4 -2 6], ⟨0 15 3 -5 17 13 -4]]

Optimal tunings:

  • WE: ~2 = 1200.5832 ¢, ~15/11 = 526.8804 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/11 = 526.6368 ¢

Optimal ET sequence: 16, 25e, 41, 98c

Badness (Sintel): 1.26

Quadrimage

Subgroup: 2.3.5.7

Comma list: 2401/2400, 3125/3072

Mapping: [⟨1 -15 -1 -3], ⟨0 20 4 7]]

mapping generators: ~2, ~25/14

Optimal tunings:

  • WE: ~2 = 1201.2708 ¢, ~25/14 = 996.0669 ¢
error map: ⟨+1.271 +0.322 -3.317 -0.170]
  • CWE: ~2 = 1200.0000 ¢, ~25/14 = 995.0515 ¢
error map: ⟨0.000 -0.926 -6.108 -3.466]

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

Badness (Sintel): 3.22

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -15 -1 -3 -4], ⟨0 20 4 7 9]]

Optimal tunings:

  • WE: ~2 = 1200.6716 ¢, ~25/14 = 995.6009 ¢
  • CWE: ~2 = 1200.0000 ¢, ~25/14 = 995.0633 ¢

Optimal ET sequence: 6, 35, 41

Badness (Sintel): 2.04

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -15 -1 -3 -22], ⟨0 20 4 7 9 31]]

Optimal tunings:

  • WE: ~2 = 1200.6276 ¢, ~25/14 = 995.4920 ¢
  • CWE: ~2 = 1200.0000 ¢, ~25/14 = 994.9901 ¢

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

Badness (Sintel): 1.82

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:

  • WE: ~2 = 1201.3334 ¢, ~48/35 = 556.6311 ¢
error map: ⟨+1.333 +0.489 -3.158 -0.835]
  • CWE: ~2 = 1200.0000 ¢, ~48/35 = 556.0504 ¢
error map: ⟨0.000 -0.695 -6.062 -3.868]

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

Badness (Sintel): 4.92

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:

  • WE: ~2 = 1200.4252 ¢, ~11/8 = 556.2831 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 556.0951 ¢

Optimal ET sequence: 13b, 28b, 41

Badness (Sintel): 3.36

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:

  • WE: ~2 = 1199.8239 ¢, ~11/8 = 556.0389 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 556.1171 ¢

Optimal ET sequence: 13b, 28b, 41

Badness (Sintel): 2.80

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:

  • WE: ~8/7 = 240.3877 ¢, ~5/4 = 380.4267 ¢ (~256/245 = 100.3488 ¢)
error map: ⟨+1.939 +0.178 -2.010 -3.398]
  • CWE: ~8/7 = 240.0000 ¢, ~5/4 = 379.9965 ¢ (~256/245 = 100.0035 ¢)
error map: ⟨0.000 -1.972 -6.317 -8.826]

Optimal ET sequence: 25, 35, 60

Badness (Sintel): 7.27

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