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{{interwiki
{{Technical data page}}
| de = Magische Temperaturen
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.
| 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 optimal, for some searches, in the [[9-odd-limit]]. It has slightly higher complexity than [[meantone]] and is also 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.


== Magic ==
== Magic ==
{{Main| Magic }}
{{Main| Magic }}


The [[monzo]] of the magic comma is {{monzo| -10 -1 5 }}, and flipping that yields {{multival| 5 1 -10 }} for the [[wedgie]]. This tells us the [[generator]] is a major third, and that to get to the interval class of fifths will require five of these. In fact, (5/4)<sup>5</sup> = 3 × 3125/3072. [[41edo|13\41]] is a highly recommendable generator, though [[60edo|19\60]], the [[optimal patent val]] generator, also makes a lot of sense, and using [[19edo]] or [[22edo]] is always possible.
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)<sup>5</sup> = 3 × 3125/3072. [[41edo|13\41]] is a highly recommendable generator, though [[60edo|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
[[Subgroup]]: 2.3.5
Line 20: Line 15:
: mapping generators: ~2, ~5/4
: mapping generators: ~2, ~5/4


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 380.4994
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2449{{c}}, ~5/4 = 380.4527{{c}}
: [[error map]]: {{val| +1.245 +0.309 -3.371 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.2194{{c}}
: error map: {{val| 0.000 -0.858 -6.094 }}


[[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]]:  
* 5-odd-limit [[diamond monotone]]: ~5/4 = [360.000, 400.000] (3\10 to 1\3)
* 5-odd-limit [[diamond monotone]]: ~5/4 = [360.000, 400.000] (3\10 to 1\3)
* 5-odd-limit [[diamond tradeoff]]: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)
* 5-odd-limit [[diamond tradeoff]]: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)
* 5-odd-limit diamond monotone and tradeoff: ~5/4 = [378.910, 386.314]


[[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 ===
Apart from magic, we also consider other extensions. The second comma of the [[Normal lists #Normal interval list|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.  
Apart from magic, we also consider other extensions. The second comma of the [[Normal lists #Normal interval list|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, and warlock. Discussed elsewhere are
Weak extensions considered below are hocum, trismegistus, quadrimage, quinmage and warlock. Discussed elsewhere are
* ''[[Astrology]]'' → [[Jubilismic clan #Astrology|Jubilismic clan]]
* ''[[Astrology]]'' → [[Jubilismic clan #Astrology|Jubilismic clan]]
* ''[[Spell]]'' → [[Hemimean clan #Spell|Hemimean clan]]
* ''[[Spell]]'' → [[Hemimean clan #Spell|Hemimean clan]]
Line 49: 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 63: Line 61:
: mapping generators: ~2, ~5/4
: mapping generators: ~2, ~5/4


{{Multival|legend=1| 5 1 12 -10 5 25 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1201.0786{{c}}, ~5/4 = 380.6939{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 380.6512
: [[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]]:  
* 7- and 9-odd-limit [[diamond monotone]]: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
* 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)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)
* 7- and 9-odd-limit diamond monotone and tradeoff: ~5/4 = [378.947, 381.818]


[[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 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.7200
Optimal tunings:
* WE: ~2 = 1200.1372{{c}}, ~5/4 = 380.7399{{c}}
* 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:  
* 11-odd-limit diamond monotone: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
* 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)
* 11-odd-limit diamond tradeoff: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)
* 11-odd-limit diamond monotone and tradeoff: ~5/4 = [378.947, 381.818]


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 ====
A notable [[patent val]] tuning beyond the [[optimal patent val]] of 41edo is [[19edo|19]] + [[41edo|41]] = [[60edo]].
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.4354
Optimal tunings:
* WE: ~2 = 1200.0331{{c}}, ~5/4 = 380.4377{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4284{{c}}


Tuning ranges:  
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 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]
* 13- and 15-odd-limit diamond tradeoff: ~5/4 = [378.617, 386.314]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~5/4 = [378.947, 381.818]


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 129: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.5103
Optimal tunings:
* WE: ~2 = 1199.3584{{c}}, ~5/4 = 380.4006{{c}}
* 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.4507
Optimal tunings:
* 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 155: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.5733
Optimal tunings:
* 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 168: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.2706
Optimal tunings:
* 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 181: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.3604
Optimal tunings:
* 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 196: 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:
* WE: ~2 = 1200.2802{{c}}, ~5/4 = 380.5053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4258{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 380.416
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}


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


==== Sorcery ====
==== Sorcery ====
Line 207: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.6741
 
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:  
* WE: ~2 = 1201.2397{{c}}, ~5/4 = 380.8698{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5080{{c}}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.7839
{{Optimal ET sequence|legend=0| 19, 22, 41f }}


Optimal ET sequence: {{Optimal ET sequence| 19, 22, 41f }}
Badness (Sintel): 1.07
 
Badness: 0.023768


==== Necromancy ====
==== Necromancy ====
Line 233: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.7876
Optimal tunings:
* WE: ~2 = 1199.9675{{c}}, ~5/4 = 380.7770{{c}}
* 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 246: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.8373
Optimal tunings:
* 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, 63, 104g }}
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63 }}


Badness: 0.022032
Badness (Sintel): 1.12


==== Soothsaying ====
==== Soothsaying ====
Line 259: 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 tuning (POTE): ~55/39 = 1\2, ~5/4 = 380.508
Optimal tunings:
* WE: ~55/39 = 600.2918{{c}}, ~5/4 = 380.6928{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~5/4 = 380.5121{{c}}


{{Optimal ET sequence|legend=1| 22, 60, 82 }}
{{Optimal ET sequence|legend=0| 22, 60, 82 }}


Badness: 0.055443
Badness (Sintel): 2.29


===== 17-limit =====
===== 17-limit =====
Line 272: 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:
* WE: ~17/12 = 600.2918{{c}}, ~5/4 = 380.6927{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.5135{{c}}
 
{{Optimal ET sequence|legend=0| 22, 60, 82 }}
 
Badness (Sintel): 1.82


Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 380.508
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


{{Optimal ET sequence|legend=1| 22, 60, 82 }}
Comma list: 100/99, 133/132, 221/220, 225/224, 245/243, 273/272


Badness: 0.035654
{{Mapping|legend=0| 2 0 4 -2 12 15 5 18 | 0 5 1 12 -8 -12 5 -15 }}
 
Optimal tunings:
* 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|legend=0| 22, 60, 82 }}
 
Badness (Sintel): 1.90


=== Telepathy ===
=== Telepathy ===
Line 285: Line 312:
Comma list: 55/54, 99/98, 176/175
Comma list: 55/54, 99/98, 176/175


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


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


{{Optimal ET sequence|legend=1| 19e, 22, 41e, 63e }}
{{Optimal ET sequence|legend=0| 19e, 22, 41e, 63e }}


Badness: 0.027109
Badness (Sintel): 0.896


==== 13-limit ====
==== 13-limit ====
Line 298: 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 tuning (POTE): ~2 = 1\1, ~5/4 = 380.520
Optimal tunings:
* WE: ~2 = 1202.5634{{c}}, ~5/4 = 381.3348{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6886{{c}}


{{Optimal ET sequence|legend=1| 19e, 22, 41ef }}
{{Optimal ET sequence|legend=0| 19e, 22, 41ef }}


Badness: 0.025522
Badness (Sintel): 1.05


===== 17-limit =====
==== Intuition ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13


Comma list: 55/54, 65/64, 85/84, 91/90, 99/98
Comma list: 55/54, 66/65, 99/98, 105/104


Mapping: {{mapping| 1 0 2 -1 -1 4 -1 | 0 5 1 12 14 -1 16 }}
{{Mapping|legend=0| 1 0 2 -1 -1 -2 | 0 5 1 12 14 18 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 380.619
Optimal tunings:
* WE: ~2 = 1201.3172{{c}}, ~5/4 = 380.9004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5942{{c}}


{{Optimal ET sequence|legend=1| 19eg, 22, 41efg }}
{{Optimal ET sequence|legend=0| 19e, 22f }}


Badness: 0.020201
Badness (Sintel): 1.08


==== Intuition ====
=== Horcrux ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11


Comma list: 55/54, 66/65, 99/98, 105/104
Comma list: 45/44, 56/55, 245/243


Mapping: {{mapping| 1 0 2 -1 -1 -2 | 0 5 1 12 14 18 }}
{{Mapping|legend=0| 1 0 2 -1 0 | 0 5 1 12 11 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 380.483
Optimal tunings:
* WE: ~2 = 1200.4670{{c}}, ~5/4 = 379.7895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.6889{{c}}


{{Optimal ET sequence|legend=1| 19e, 22f, 41e }}
{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}


Badness: 0.026089
Badness (Sintel): 1.30


===== 17-limit =====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13


Comma list: 55/54, 66/65, 85/84, 99/98, 105/104
Comma list: 45/44, 56/55, 78/77, 245/243


Mapping: {{mapping| 1 0 2 -1 -1 -2 -1 | 0 5 1 12 14 18 16 }}
{{Mapping|legend=0| 1 0 2 -1 0 -2 | 0 5 1 12 11 18 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 380.604
Optimal tunings:
* WE: ~2 = 1200.2953{{c}}, ~5/4 = 379.8842{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.8165{{c}}


{{Optimal ET sequence|legend=1| 19eg, 22f, 41eg }}
{{Optimal ET sequence|legend=0| 3def, 16dff, 19 }}


Badness: 0.020274
Badness (Sintel): 1.32


=== Horcrux ===
===== 17-limit =====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17


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


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


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 379.642
Optimal tunings:
* WE: ~2 = 1200.2484{{c}}, ~5/4 = 380.2053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1482{{c}}


{{Optimal ET sequence|legend=1| 3de, 19, 41ee, 60ee }}
{{Optimal ET sequence|legend=0| 3defg, 16dffgg, 19g }}


Badness: 0.039282
Badness (Sintel): 1.43


==== Charisma ====
===== Horcruxic =====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17


Comma list: 45/44, 56/55, 78/77, 245/243
Comma list: 35/34, 45/44, 52/51, 56/55, 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 | 0 5 1 12 11 18 13 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 379.791
Optimal tunings:
* WE: ~2 = 1199.5457{{c}}, ~5/4 = 379.4681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.5713{{c}}


{{Optimal ET sequence|legend=1| 3def, 19, 41ee, 60ee }}
{{Optimal ET sequence|legend=0| 3defg, 16dff, 19 }}


Badness: 0.031938
Badness (Sintel): 1.51


==== Glamour ====
==== Glamour ====
Line 376: 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 tuning (POTE): ~2 = 1\1, ~5/4 = 379.116
Optimal tunings:
* WE: ~2 = 1202.2187{{c}}, ~5/4 = 379.8171{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.2709{{c}}


{{Optimal ET sequence|legend=1| 3de, 19 }}
{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}


Badness: 0.033317
Badness (Sintel): 1.38


=== Witchcraft ===
=== Witchcraft ===
Line 389: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.4163
Optimal tunings:
* WE: ~2 = 1201.2634{{c}}, ~5/4 = 380.6321{{c}}
* 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 402: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.3523
Optimal tunings:
* WE: ~2 = 1201.0424{{c}}, ~5/4 = 380.5193{{c}}
* 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 415: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.2809
Optimal tunings:
* 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| 19eg, 41, 101cd, 142cdfg }}
{{Optimal ET sequence|legend=0| 19eg, 41, 60e, 101cd }}


Badness: 0.020756
Badness (Sintel): 1.06


==== 19-limit ====
=== Divination ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11


Comma list: 105/104, 133/132, 154/153, 170/169, 171/169, 196/195
Comma list: 121/120, 225/224, 245/243


Mapping: {{mapping| 1 0 2 -1 -7 -2 -7 -4 | 0 5 1 12 33 18 35 26 }}
{{Mapping|legend=0| 2 0 4 -2 5 | 0 5 1 12 3 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.3389
Optimal tunings:
* WE: ~99/70 = 600.8306{{c}}, ~5/4 = 380.7598{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~5/4 = 380.3800{{c}}


Optimal ET sequence: {{Optimal ET sequence| 19egh, 41, 101cdhh, 142cdfghh }}
{{Optimal ET sequence|legend=0| 22, 38d, 60e, 142cdee, 202ccddeee }}


Badness: 0.018625
Badness (Sintel): 1.19


=== Divination ===
==== 13-limit ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 225/224, 245/243
Comma list: 105/104, 121/120, 196/195, 245/243


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


Optimal tuning (POTE): ~99/70 = 1\2, ~5/4 = 380.233
Optimal tunings:
* WE: ~99/70 = 600.9624{{c}}, ~5/4 = 380.5297{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~5/4 = 380.0614{{c}}


{{Optimal ET sequence|legend=1| 22, 38d, 60e, 142cde }}
{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}


Badness: 0.035864
Badness (Sintel): 1.43


==== 13-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17


Comma list: 105/104, 121/120, 196/195, 245/243
Comma list: 105/104, 121/120, 154/153, 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 5 | 0 5 1 12 3 18 5 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~5/4 = 379.920
Optimal tunings:
* 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|legend=1| 22f, 60e }}
{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}


Badness: 0.034551
Badness (Sintel): 1.21


=== Hocus ===
=== Hocus ===
Line 467: 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 tuning (POTE): ~2 = 1\1, ~14/11 = 409.910
Optimal tunings:
* WE: ~2 = 1201.0749{{c}}, ~11/7 = 790.7980{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.1429{{c}}


{{Optimal ET sequence|legend=1| 38d, 41, 120cd, 161cd, 202cd }}
{{Optimal ET sequence|legend=0| 38d, 41, 120cd }}


Badness: 0.038519
Badness (Sintel): 1.27


==== 13-limit ====
==== 13-limit ====
Line 480: 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:
* WE: ~2 = 1201.2830{{c}}, ~11/7 = 790.8409{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.0516{{c}}
 
{{Optimal ET sequence|legend=0| 38df, 41, 79d, 120cd }}
 
Badness (Sintel): 1.25
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 410.004
Comma list: 105/104, 154/153, 196/195, 243/242, 245/242


{{Optimal ET sequence|legend=1| 41, 79d, 120cd }}
{{Mapping|legend=0| 1 -5 1 -13 -13 -20 -15 | 0 10 2 24 25 36 29 }}


Badness: 0.030280
Optimal tunings:
* WE: ~2 = 1201.1557{{c}}, ~11/7 = 790.7157{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.0057{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=0| 1 -5 1 -13 -13 -20 -3 | 0 10 2 24 25 36 29 11 }}
 
Optimal tunings:
* WE: ~2 = 1201.3558{{c}}, ~11/7 = 790.8266{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 789.9880{{c}}
 
{{Optimal ET sequence|legend=0| 38df, 41, 79dh }}
 
Badness (Sintel): 1.23


== Muggles ==
== Muggles ==
Line 499: Line 588:
{{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:
 
* [[WE]]: ~2 = 1203.9554{{c}}, ~5/4 = 379.7269{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 378.744
: [[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 507: Line 598:
* [[9-odd-limit]] diamond monotone: ~5/4 = 378.947 (6\19)
* [[9-odd-limit]] diamond monotone: ~5/4 = 378.947 (6\19)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~5/4 = [375.882, 386.314]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~5/4 = [375.882, 386.314]
* 7-odd-limit diamond monotone and tradeoff: ~5/4 = [375.882, 378.947]
* 9-odd-limit diamond monotone and tradeoff: ~5/4 = 378.947


{{Optimal ET sequence|legend=1| 16, 19 }}
{{Optimal ET sequence|legend=1| 16, 19, 73bcd, 92bcdd, 111bcddd }}


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


=== 11-limit ===
=== 11-limit ===
Line 519: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 378.228
Optimal tunings:
* WE: ~2 = 1203.0804{{c}}, ~5/4 = 378.6936{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8174{{c}}


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


Optimal ET sequence: {{Optimal ET sequence| 16, 19, 54bd, 73bcd, 92bcdde }}
{{Optimal ET sequence|legend=0| 16, 19, 35, 54bd }}


Badness: 0.048038
Badness (Sintel): 1.59


==== 13-limit ====
==== 13-limit ====
Line 537: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 377.653
Optimal tunings:
* WE: ~2 = 1203.4291{{c}}, ~5/4 = 378.7321{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.7336{{c}}


Optimal ET sequence: {{Optimal ET sequence| 16, 19, 54bdf, 73bcdff, 92bcddeff }}
{{Optimal ET sequence|legend=0| 16, 19, 35f, 54bdf }}


Badness: 0.030386
Badness (Sintel): 1.26


=== Muggloid ===
=== Muggloid ===
Line 550: 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 tuning (CTE): ~2 = 1\1, ~5/4 = 377.724
Optimal tunings:
* WE: ~2 = 1205.6044{{c}}, ~5/4 = 379.5966{{c}}
* 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 563: 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:
* WE: ~2 = 1205.4897{{c}}, ~5/4 = 379.5667{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8118{{c}}
 
{{Optimal ET sequence|legend=0| 3, 16, 19e, 35eef }}
 
Badness (Sintel): 1.19
 
== Brightstone ==
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
 
[[Comma list]]: 64/63, 3125/3024
 
{{Mapping|legend=1| 1 0 2 6 | 0 5 1 -10 }}
 
[[Optimal tuning]]s:
* [[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| 3, 19d, 22 }}
 
[[Badness]] (Sintel): 2.23
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 55/54, 64/63, 625/616


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 377.652
{{Mapping|legend=0| 1 0 2 6 -1 | 0 5 1 -10 14 }}


Optimal ET sequence: {{Optimal ET sequence| 3, 10bd, 13bd, 16, 19e, 35eef }}
Optimal tunings:  
* WE: ~2 = 1198.5372{{c}}, ~5/4 = 381.7556{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.1943{{c}}


Badness: 0.028732
{{Optimal ET sequence|legend=0| 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|legend=0| 1 0 2 6 6 4 | 0 5 1 -10 14 -4 }}
 
Optimal tunings:
* WE: ~2 = 1197.2300{{c}}, ~5/4 = 381.6164{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.4690{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 47bff }}
 
Badness (Sintel): 1.99


== Darkstone ==
== 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 [http://www.tonalsoft.com/enc/m/magic.aspx Tonalsoft Encyclopedia].
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 578: Line 727:
{{Mapping|legend=1| 1 0 2 0 | 0 5 1 9 }}
{{Mapping|legend=1| 1 0 2 0 | 0 5 1 9 }}


{{Multival|legend=1| 5 1 9 -10 0 18 }}
[[Optimal tuning]]s:
* [[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 tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 377.385
{{Optimal ET sequence|legend=1| 3d, …, 13b, 16 }}


{{Optimal ET sequence|legend=1| 13b, 16, 19d, 35d }}
[[Badness]] (Sintel): 2.13


[[Badness]]: 0.084213
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Brightstone ==
Comma list: 36/35, 45/44, 363/343
[[Subgroup]]: 2.3.5.7
 
{{Mapping|legend=0| 1 0 2 0 0 | 0 5 1 9 11 }}
 
Optimal tunings:
* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.9735{{c}}
 
{{Optimal ET sequence|legend=0| 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


[[Comma list]]: 64/63, 3125/3024
{{Mapping|legend=0| 1 0 2 0 0 -1 | 0 5 1 9 11 15 }}


{{Mapping|legend=1| 1 0 2 6 | 0 5 1 -10 }}
Optimal tunings:
* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.4221{{c}}


{{Multival|legend=1| 5 1 -10 -10 -30 -26 }}
{{Optimal ET sequence|legend=0| 3def, 13beff, 16 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~5/4 = 381.955
Badness (Sintel): 1.58


{{Optimal ET sequence|legend=1| 19d, 22 }}
Scales: [[User:BudjarnLambeth/Volcanic glass]]


[[Badness]]: 0.088072
Music: [https://www.youtube.com/watch?v=oXy_RXrBVWA ''Rain in the crystal mirror caves''] - [[Budjarn Lambeth]] (2026)


== Hocum ==
== Hocum ==
Line 606: 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:
* [[WE]]: ~2 = 1200.8375{{c}}, ~63/40 = 790.7032{{c}}
: [[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 tuning]] ([[POTE]]): ~2 = 1\1, ~63/50 = 410.108
{{Optimal ET sequence|legend=1| 3, 38, 41, 161c }}


{{Optimal ET sequence|legend=1| 38, 41, 161c, 202c, 243c, 284c }}
[[Badness]] (Sintel): 2.71


[[Badness]]: 0.107115
== Trismegistus ==
{{Main| Mabilic and trismegistus }}
{{See also| No-threes subgroup temperaments #Mabilic }}


== Trismegistus ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[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


{{Multival|legend=1| 15 3 -5 -30 -50 -20 }}
: mapping generators: ~2, ~168/125


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~147/100 = 673.290
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.0799{{c}}, ~168/125 = 527.1841{{c}}
: [[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, 139c, 180c, 221c, 262c }}
{{Optimal ET sequence|legend=1| 16, 25, 41, 139c, 180c, 221cc, 262ccd }}


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


=== 11-limit ===
=== 11-limit ===
Line 640: 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 tuning (POTE): ~2 = 1\1, ~22/15 = 673.340
Optimal tunings:
* WE: ~2 = 1200.8404{{c}}, ~15/11 = 527.0289{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6826{{c}}


{{Optimal ET sequence|legend=1| 16, 25e, 41, 98c, 139c, 180c }}
{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}


Badness: 0.045623
Badness (Sintel): 1.51


=== 13-limit ===
=== 13-limit ===
Line 653: 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:
* WE: ~2 = 1200.4759{{c}}, ~15/11 = 526.8502{{c}}
* 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


Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 673.359
Subgroup-val mapping: {{mapping| 1 -5 1 5 -4 -2 6 | 0 15 3 -5 17 13 -4 }}


{{Optimal ET sequence|legend=1| 16, 25e, 41, 98c, 139cf }}
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 666: 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:
* [[WE]]: ~2 = 1201.2708{{c}}, ~25/14 = 996.0669{{c}}
: [[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 tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 204.987
{{Optimal ET sequence|legend=1| 6, …, 35, 41, 158cd, 199ccd, 240ccd, 281ccd }}


{{Optimal ET sequence|legend=1| 35, 41, 158cd, 199ccd, 240ccd, 281ccd }}
[[Badness]] (Sintel): 3.22
 
[[Badness]]: 0.127422


=== 11-limit ===
=== 11-limit ===
Line 683: 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 tuning (POTE): ~2 = 1\1, ~28/25 = 204.956
Optimal tunings:
* WE: ~2 = 1200.6716{{c}}, ~25/14 = 995.6009{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 995.0633{{c}}


{{Optimal ET sequence|legend=1| 35, 41 }}
{{Optimal ET sequence|legend=0| 6, 35, 41 }}


Badness: 0.061572
Badness (Sintel): 2.04


=== 13-limit ===
=== 13-limit ===
Line 696: 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:
* WE: ~2 = 1200.6276{{c}}, ~25/14 = 995.4920{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 994.9901{{c}}
 
{{Optimal ET sequence|legend=0| 6f, 35f, 41, 117c }}
 
Badness (Sintel): 1.82


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 205.028
== Quinmage ==
[[Subgroup]]: 2.3.5.7


{{Optimal ET sequence|legend=1| 35f, 41, 117c }}
[[Comma list]]: 3125/3072, 16875/16807


Badness: 0.044047
{{Mapping|legend=1| 1 -10 0 -6 | 0 25 5 19 }}
 
: mapping generators: ~2, ~48/35
 
[[Optimal tuning]]s:
* [[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, 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|legend=0| 1 -10 0 -6 3 | 0 25 5 19 1 }}
 
Optimal tunings:
* WE: ~2 = 1200.4252{{c}}, ~11/8 = 556.2831{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 556.0951{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=0| 1 -10 0 -6 3 0 | 0 25 5 19 1 8 }}
 
Optimal tunings:
* WE: ~2 = 1199.8239{{c}}, ~11/8 = 556.0389{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 556.1171{{c}}
 
{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}
 
Badness (Sintel): 2.80


== Warlock ==
== Warlock ==
Line 713: Line 964:
: mapping generators: ~8/7, ~5/4
: mapping generators: ~8/7, ~5/4


[[Optimal tuning]] ([[POTE]]): ~8/7 = 1\5, ~5/4 = 379.7131
[[Optimal tuning]]s:
* [[WE]]: ~8/7 = 240.3877{{c}}, ~5/4 = 380.4267{{c}} (~256/245 = 100.3488{{c}})
: 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]]
Retrieved from "https://en.xen.wiki/w/Magic_family"