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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.
{{Technical data page}}
The '''magic family''' of temperaments tempers out [[3125/3072]], the small diesis or magic comma. The septimal version of magic is locally optimal, for some searches, in the [[9-odd-limit]]. Magic has a slightly higher complexity than [[meantone]] but it is closer to just intonation. It is the simplest rank-2 temperament that tunes every [[9-odd-limit]] interval better than is possible in [[12edo]]. The most prominent deficiency is that it lacks [[Rothenberg propriety|proper]] or nearly-proper [[mos scale]]s in the 5- to 10-note region. Properties may depend on tuning and extension.


== 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 21:
: 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 53:
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 67:
: 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


Line 91: Line 94:
Mapping: {{mapping| 1 0 2 -1 6 | 0 5 1 12 -8 }}
Mapping: {{mapping| 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


Line 113: Line 119:
Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 1 12 -8 18 }}
Mapping: {{mapping| 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


===== 17-limit =====
===== Magical =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Line 131: Line 138:
Mapping: {{mapping| 1 0 2 -1 6 -2 6 | 0 5 1 12 -8 18 -6 }}
Mapping: {{mapping| 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|legend=0| 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: {{mapping| 1 0 2 -1 6 -2 6 9 | 0 5 1 12 -8 18 -6 -15 }}
 
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|legend=0| 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: {{mapping| 1 0 2 -1 6 -2 6 -4 | 0 5 1 12 -8 18 -6 26 }}
 
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|legend=0| 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: {{mapping| 1 0 2 -1 6 -2 -7 | 0 5 1 12 -8 18 35 }}
 
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|legend=0| 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: {{mapping| 1 0 2 -1 6 -2 -7 -4 | 0 5 1 12 -8 18 35 26 }}
 
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| 19, 22f, 41 }}
{{Optimal ET sequence|legend=0| 19gh, 41 }}


Badness: 0.020633
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 144: Line 213:
Comma list: 100/99, 105/104, 144/143, 145/144, 155/154, 196/195
Comma list: 100/99, 105/104, 144/143, 145/144, 155/154, 196/195


Sval mapping: {{mapping| 1 0 2 -1 6 -2 2 4 | 0 5 1 12 -8 18 9 3 }}
Subgroup-val mapping: {{mapping| 1 0 2 -1 6 -2 2 4 | 0 5 1 12 -8 18 9 3 }}
 
Optimal tunings:
* 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 157: Line 230:
Mapping: {{mapping| 1 0 2 -1 6 4 | 0 5 1 12 -8 -1 }}
Mapping: {{mapping| 1 0 2 -1 6 4 | 0 5 1 12 -8 -1 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.6741
Optimal tunings:
 
* WE: ~2 = 1201.2397{{c}}, ~5/4 = 380.8698{{c}}
Optimal ET sequence: {{Optimal ET sequence| 19, 22, 41f }}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5080{{c}}
 
Badness: 0.025829


===== 17-limit =====
{{Optimal ET sequence|legend=0| 19, 22, 41f }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 52/51, 65/64, 78/77, 91/90, 100/99
Badness (Sintel): 1.07
 
Mapping: {{mapping| 1 0 2 -1 6 4 6 | 0 5 1 12 -8 -1 -6 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.7839
 
Optimal ET sequence: {{Optimal ET sequence| 19, 22, 41f }}
 
Badness: 0.023768


==== Necromancy ====
==== Necromancy ====
Line 183: Line 245:
Mapping: {{mapping| 1 0 2 -1 6 11 | 0 5 1 12 -8 -23 }}
Mapping: {{mapping| 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 196: Line 260:
Mapping: {{mapping| 1 0 2 -1 6 11 6 | 0 5 1 12 -8 -23 -6 }}
Mapping: {{mapping| 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 209: Line 275:
Mapping: {{mapping| 2 0 4 -2 12 15 | 0 5 1 12 -8 -12 }}
Mapping: {{mapping| 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 222: Line 290:
Mapping: {{mapping| 2 0 4 -2 12 15 5 | 0 5 1 12 -8 -12 5 }}
Mapping: {{mapping| 2 0 4 -2 12 15 5 | 0 5 1 12 -8 -12 5 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 380.508
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 }}


{{Optimal ET sequence|legend=1| 22, 60, 82 }}
Badness (Sintel): 1.82


Badness: 0.035654
===== 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: {{mapping| 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 235: Line 320:
Mapping: {{mapping| 1 0 2 -1 -1 | 0 5 1 12 14 }}
Mapping: {{mapping| 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 248: Line 335:
Mapping: {{mapping| 1 0 2 -1 -1 4 | 0 5 1 12 14 -1 }}
Mapping: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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 324: Line 423:
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: {{mapping| 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 339: Line 440:
Mapping: {{mapping| 1 0 2 -1 -7 | 0 5 1 12 33 }}
Mapping: {{mapping| 1 0 2 -1 -7 | 0 5 1 12 33 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 380.232
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|legend=1| 41, 60e, 101cd, 243cde }}
{{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd, 243ccdde }}


Badness: 0.030706
Badness (Sintel): 1.02


==== 13-limit ====
==== 13-limit ====
Line 352: Line 455:
Mapping: {{mapping| 1 0 2 -1 -7 -2 | 0 5 1 12 33 18 }}
Mapping: {{mapping| 1 0 2 -1 -7 -2 | 0 5 1 12 33 18 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 380.189
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|legend=1| 41, 60e, 101cd }}
{{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd }}


Badness: 0.023547
Badness (Sintel): 0.973


==== 17-limit ====
==== 17-limit ====
Line 365: Line 470:
Mapping: {{mapping| 1 0 2 -1 -7 -2 -7 | 0 5 1 12 33 18 35 }}
Mapping: {{mapping| 1 0 2 -1 -7 -2 -7 | 0 5 1 12 33 18 35 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 380.114
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|legend=1| 41, 60e, 101cd, 161cde }}
{{Optimal ET sequence|legend=0| 19eg, 41, 60e, 101cd }}


Badness: 0.020756
Badness (Sintel): 1.06


=== Divination ===
=== Divination ===
Line 378: Line 485:
Mapping: {{mapping| 2 0 4 -2 5 | 0 5 1 12 3 }}
Mapping: {{mapping| 2 0 4 -2 5 | 0 5 1 12 3 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~5/4 = 380.233
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|legend=1| 22, 38d, 60e, 142cde }}
{{Optimal ET sequence|legend=0| 22, 38d, 60e, 142cdee, 202ccddeee }}


Badness: 0.035864
Badness (Sintel): 1.19


==== 13-limit ====
==== 13-limit ====
Line 391: Line 500:
Mapping: {{mapping| 2 0 4 -2 5 -4 | 0 5 1 12 3 18 }}
Mapping: {{mapping| 2 0 4 -2 5 -4 | 0 5 1 12 3 18 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~5/4 = 379.920
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=0| 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


{{Optimal ET sequence|legend=1| 22f, 60e }}
Mapping: {{mapping| 2 0 4 -2 5 -4 5 | 0 5 1 12 3 18 5 }}


Badness: 0.034551
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=0| 22f, 38df, 60e }}
 
Badness (Sintel): 1.21


=== Hocus ===
=== Hocus ===
Line 402: Line 528:
Comma list: 225/224, 243/242, 245/242
Comma list: 225/224, 243/242, 245/242


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


Optimal 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 415: Line 543:
Comma list: 105/104, 196/195, 243/242, 245/242
Comma list: 105/104, 196/195, 243/242, 245/242


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


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 410.004
Badness (Sintel): 1.25


{{Optimal ET sequence|legend=1| 41, 79d, 120cd }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.030280
Comma list: 105/104, 154/153, 196/195, 243/242, 245/242
 
Mapping: {{mapping| 1 -5 1 -13 -13 -20 -15 | 0 10 2 24 25 36 29 }}
 
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: {{mapping| 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 434: Line 594:
{{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]] ([[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 442: Line 604:
* [[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, 73bcd, 92bcd }}
{{Optimal ET sequence|legend=1| 16, 19, 73bcd, 92bcdd, 111bcddd }}


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


=== 11-limit ===
=== 11-limit ===
Line 456: Line 616:
Mapping: {{mapping| 1 0 2 5 0 | 0 5 1 -7 11 }}
Mapping: {{mapping| 1 0 2 5 0 | 0 5 1 -7 11 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 377.724
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|legend=1| 16, 19, 35, 54bd }}
{{Optimal ET sequence|legend=0| 16, 19, 35, 54bd }}


Badness: 0.048038
Badness (Sintel): 1.59


==== 13-limit ====
==== 13-limit ====
Line 474: Line 635:
Mapping: {{mapping| 1 0 2 5 0 4 | 0 5 1 -7 11 -1 }}
Mapping: {{mapping| 1 0 2 5 0 4 | 0 5 1 -7 11 -1 }}


Optimal tuning (POTE): ~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|legend=1| 16, 19, 35f, 54bdf }}
{{Optimal ET sequence|legend=0| 16, 19, 35f, 54bdf }}


Badness: 0.030386
Badness (Sintel): 1.26


=== Muggloid ===
=== Muggloid ===
Line 487: Line 650:
Mapping: {{mapping| 1 0 2 5 5 | 0 5 1 -7 -5 }}
Mapping: {{mapping| 1 0 2 5 5 | 0 5 1 -7 -5 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 377.832
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|legend=1| 3, 10bd, 13bd, 16, 19e }}
{{Optimal ET sequence|legend=0| 3, 16, 19e, 35ee, 54bdeee }}


Badness: 0.046970
Badness (Sintel): 1.55


==== 13-limit ====
==== 13-limit ====
Line 500: Line 665:
Mapping: {{mapping| 1 0 2 5 5 4 | 0 5 1 -7 -5 -1 }}
Mapping: {{mapping| 1 0 2 5 5 4 | 0 5 1 -7 -5 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 377.838
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
 
Mapping: {{mapping| 1 0 2 6 -1 | 0 5 1 -10 14 }}
 
Optimal tunings:
* WE: ~2 = 1198.5372{{c}}, ~5/4 = 381.7556{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.1943{{c}}
 
{{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: {{mapping| 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 (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.  


{{Optimal ET sequence|legend=1| 3, 10bd, 13bd, 16, 19e }}
This temperament is known as ''witch'' in [http://www.tonalsoft.com/enc/m/magic.aspx Tonalsoft Encyclopedia].


Badness: 0.028732
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 36/35, 1875/1792
 
{{Mapping|legend=1| 1 0 2 0 | 0 5 1 9 }}
 
[[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 ET sequence|legend=1| 3d, …, 13b, 16 }}
 
[[Badness]] (Sintel): 2.13
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 45/44, 363/343
 
Mapping: {{mapping| 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
 
Mapping: {{mapping| 1 0 2 0 0 -1 | 0 5 1 9 11 15 }}
 
Optimal tunings:
* 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
 
Scales: [[User:BudjarnLambeth/Volcanic glass]]
 
Music: [https://www.youtube.com/watch?v=oXy_RXrBVWA ''Rain in the crystal mirror caves''] - [[Budjarn Lambeth]] (2026)


== Hocum ==
== Hocum ==
Line 511: Line 782:
[[Comma list]]: 3125/3072, 4000/3969
[[Comma list]]: 3125/3072, 4000/3969


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


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


{{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
: mapping generators: ~2, ~168/125


{{Multival|legend=1| 15 3 -5 -30 -50 -20 }}
[[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 tuning]] ([[POTE]]): ~2 = 1\1, ~147/100 = 673.290
{{Optimal ET sequence|legend=1| 16, 25, 41, 139c, 180c, 221cc, 262ccd }}


{{Optimal ET sequence|legend=1| 16, 25, 41, 139c, 180c, 221c, 262c }}
[[Badness]] (Sintel): 2.49
 
[[Badness]]: 0.098334


=== 11-limit ===
=== 11-limit ===
Line 545: Line 823:
Comma list: 385/384, 441/440, 625/616
Comma list: 385/384, 441/440, 625/616


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


Optimal 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 558: Line 838:
Comma list: 105/104, 144/143, 275/273, 625/616
Comma list: 105/104, 144/143, 275/273, 625/616


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


Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 673.359
Badness (Sintel): 1.37


{{Optimal ET sequence|legend=1| 16, 25e, 41, 98c, 139cf }}
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.19


Badness: 0.033081
Comma list: 105/104, 144/143, 441/440, 210/209, 625/616
 
Subgroup-val mapping: {{mapping| 1 -5 1 5 -4 -2 6 | 0 15 3 -5 17 13 -4 }}
 
Optimal tunings:
* WE: ~2 = 1200.5832{{c}}, ~15/11 = 526.8804{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6368{{c}}
 
{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}
 
Badness (Sintel): 1.26


== Quadrimage ==
== Quadrimage ==
Line 571: Line 868:
[[Comma list]]: 2401/2400, 3125/3072
[[Comma list]]: 2401/2400, 3125/3072


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


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


{{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 588: Line 887:
Comma list: 245/242, 385/384, 625/616
Comma list: 245/242, 385/384, 625/616


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


Optimal 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 601: Line 902:
Comma list: 105/104, 144/143, 245/242, 625/616
Comma list: 105/104, 144/143, 245/242, 625/616


Mapping: {{mapping| 1 5 3 4 5 9 | 0 -20 -4 -7 -9 -31 }}
Mapping: {{mapping| 1 -15 -1 -3 -22 | 0 20 4 7 9 31 }}
 
Optimal tunings:
* 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
 
== Quinmage ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 3125/3072, 16875/16807
 
{{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: {{mapping| 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: {{mapping| 1 -10 0 -6 3 0 | 0 25 5 19 1 8 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 205.028
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=1| 35f, 41, 117c }}
{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}


Badness: 0.044047
Badness (Sintel): 2.80


== Warlock ==
== Warlock ==
Line 618: Line 970:
: 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:Temperament families]]
Retrieved from "https://en.xen.wiki/w/Magic_family"