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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


Line 91: Line 88:
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 113:
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


===== Magical =====
===== Magical =====
Line 131: Line 132:
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: {{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


Line 144: Line 147:
Mapping: {{mapping| 1 0 2 -1 6 -2 6 9 | 0 5 1 12 -8 18 -6 -15 }}
Mapping: {{mapping| 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 157: Line 162:
Mapping: {{mapping| 1 0 2 -1 6 -2 6 -4 | 0 5 1 12 -8 18 -6 26 }}
Mapping: {{mapping| 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 170: Line 177:
Mapping: {{mapping| 1 0 2 -1 6 -2 -7 | 0 5 1 12 -8 18 35 }}
Mapping: {{mapping| 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 183: Line 192:
Mapping: {{mapping| 1 0 2 -1 6 -2 -7 -4 | 0 5 1 12 -8 18 35 26 }}
Mapping: {{mapping| 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 tuning (POTE): ~2 = 1\1, ~5/4 = 380.416
Optimal tunings:
* 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 209: Line 224:
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 =====
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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.7839
 
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 ET sequence|legend=0| 19, 22, 41f }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.6248
Badness (Sintel): 1.07
 
Optimal ET sequence: {{Optimal ET sequence| 19, 22, 41f }}
 
Badness: 0.023232


==== Necromancy ====
==== Necromancy ====
Line 248: Line 239:
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 261: Line 254:
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}}
Optimal ET sequence: {{Optimal ET sequence| 19f, 22, 41, 63, 104g }}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.8280{{c}}
 
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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.7411
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63 }}


Optimal ET sequence: {{Optimal ET sequence| 19f, 22, 41, 63h, 104gh }}
Badness (Sintel): 1.12
 
Badness: 0.021101


==== Soothsaying ====
==== Soothsaying ====
Line 287: Line 269:
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 (CTE): ~55/39 = 1\2, ~5/4 = 380.5385
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: {{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 300: Line 284:
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 (CTE): ~17/12 = 1\2, ~5/4 = 380.5533
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: {{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 313: Line 299:
Mapping: {{mapping| 2 0 4 -2 12 15 5 18 | 0 5 1 12 -8 -12 5 -15 }}
Mapping: {{mapping| 2 0 4 -2 12 15 5 18 | 0 5 1 12 -8 -12 5 -15 }}


Optimal tuning (CTE): ~17/12 = 1\2, ~5/4 = 380.4704
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: {{Optimal ET sequence| 22, 60, 82 }}
{{Optimal ET sequence|legend=0| 22, 60, 82 }}


Badness: 0.031291
Badness (Sintel): 1.90


=== Telepathy ===
=== Telepathy ===
Line 326: Line 314:
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 (CTE): ~2 = 1\1, ~5/4 = 381.2311
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: {{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 339: Line 329:
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 (CTE): ~2 = 1\1, ~5/4 = 381.1957
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: {{Optimal ET sequence| 19e, 22, 63eff }}
{{Optimal ET sequence|legend=0| 19e, 22, 41ef }}


Badness: 0.025522
Badness (Sintel): 1.05
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 55/54, 65/64, 85/84, 91/90, 99/98
 
Mapping: {{mapping| 1 0 2 -1 -1 4 -1 | 0 5 1 12 14 -1 16 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 381.2884
 
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 tuning (CTE): ~2 = 1\1, ~5/4 = 381.4641
 
Optimal ET sequence: {{Optimal ET sequence| 19egh, 22 }}
 
Badness: 0.019004


==== Intuition ====
==== Intuition ====
Line 378: Line 344:
Mapping: {{mapping| 1 0 2 -1 -1 -2 | 0 5 1 12 14 18 }}
Mapping: {{mapping| 1 0 2 -1 -1 -2 | 0 5 1 12 14 18 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.8157
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: {{Optimal ET sequence| 19e, 22f, 41e, 63ef }}
{{Optimal ET sequence|legend=0| 19e, 22f }}


Badness: 0.026089
Badness (Sintel): 1.08
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 55/54, 66/65, 85/84, 99/98, 105/104
 
Mapping: {{mapping| 1 0 2 -1 -1 -2 -1 | 0 5 1 12 14 18 16 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.9518
 
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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.8658
 
Optimal ET sequence: {{Optimal ET sequence| 19egh, 22fh, 41eg, 63efg }}
 
Badness: 0.019518


=== Horcrux ===
=== Horcrux ===
Line 417: Line 359:
Mapping: {{mapping| 1 0 2 -1 0 | 0 5 1 12 11 }}
Mapping: {{mapping| 1 0 2 -1 0 | 0 5 1 12 11 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 379.7959
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: {{Optimal ET sequence| 3de, …, 16d, 19, 41ee, 60ee }}
{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}


Badness: 0.039282
Badness (Sintel): 1.30


==== Charisma ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 430: Line 374:
Mapping: {{mapping| 1 0 2 -1 0 -2 | 0 5 1 12 11 18 }}
Mapping: {{mapping| 1 0 2 -1 0 -2 | 0 5 1 12 11 18 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 379.8847
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: {{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 443: Line 389:
Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 16 }}
Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 16 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 380.2159
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: {{Optimal ET sequence| 3defg, …, 19g, 41eeg }}
{{Optimal ET sequence|legend=0| 3defg, 16dffgg, 19g }}


Badness: 0.028074
Badness (Sintel): 1.43


===== Charismic =====
===== Horcruxic =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Line 456: Line 404:
Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 13 }}
Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 13 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 379.5237
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: {{Optimal ET sequence| 3defg, …, 19 }}
{{Optimal ET sequence|legend=0| 3defg, 16dff, 19 }}


Badness: 0.029556
Badness (Sintel): 1.51


==== Glamour ====
==== Glamour ====
Line 469: Line 419:
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 (CTE): ~2 = 1\1, ~5/4 = 379.7574
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: {{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 482: Line 434:
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 (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 495: Line 449:
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 (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 508: Line 464:
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 (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 ====
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 tuning (CTE): ~2 = 1\1, ~5/4 = 380.3389
 
Optimal ET sequence: {{Optimal ET sequence| 19egh, 41, 101cdhh, 142cdfghh }}
 
Badness: 0.018625


=== Divination ===
=== Divination ===
Line 534: Line 479:
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 (CTE): ~99/70 = 1\2, ~5/4 = 380.7317
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| 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 547: Line 494:
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 (CTE): ~99/70 = 1\2, ~5/4 = 380.4171
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: {{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 560: Line 509:
Mapping: {{mapping| 2 0 4 -2 5 -4 5 | 0 5 1 12 3 18 5 }}
Mapping: {{mapping| 2 0 4 -2 5 -4 5 | 0 5 1 12 3 18 5 }}


Optimal tuning (CTE): ~17/12 = 1\2, ~5/4 = 380.4329
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: {{Optimal ET sequence| 22f, 60e, 82ef }}
{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}


Badness: 0.023775
Badness (Sintel): 1.21


=== Hocus ===
=== Hocus ===
Line 571: 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: {{mapping| 1 -5 1 -13 -13 | 0 10 2 24 25 }}


Optimal tuning (CTE): ~2 = 1\1, ~14/11 = 409.7603
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: {{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 584: 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: {{mapping| 1 -5 1 -13 -13 -20 | 0 10 2 24 25 36 }}


Optimal tuning (CTE): ~2 = 1\1, ~14/11 = 409.8421
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: {{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 597: 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: {{mapping| 1 -5 1 -13 -13 -20 -15 | 0 10 2 24 25 36 29 }}


Optimal tuning (CTE): ~2 = 1\1, ~14/11 = 409.8958
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: {{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 610: 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: {{mapping| 1 -5 1 -13 -13 -20 -3 | 0 10 2 24 25 36 29 11 }}


Optimal tuning (CTE): ~2 = 1\1, ~14/11 = 409.8836
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: {{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 629: 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 637: 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 651: Line 610:
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 (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 669: Line 629:
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 (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 682: Line 644:
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 (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 695: Line 659:
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 (CTE): ~2 = 1\1, ~5/4 = 377.652
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 }}


Optimal ET sequence: {{Optimal ET sequence| 3, 10bd, 13bd, 16, 19e, 35eef }}
Badness (Sintel): 1.19


Badness: 0.028732
== 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.  


== Darkstone ==
[[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]] ([[CTE]]): ~2 = 1\1, ~5/4 = 377.385
[[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| 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: {{mapping| 1 0 2 6 -1 | 0 5 1 -10 14 }}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 377.388
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: {{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 6 6 4 | 0 5 1 -10 14 -4 }}


Mapping: {{mapping| 1 0 2 0 0 -1 | 0 5 1 9 11 15 }}
Optimal tunings:  
* WE: ~2 = 1197.2300{{c}}, ~5/4 = 381.6164{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.4690{{c}}


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 376.914
{{Optimal ET sequence|legend=0| 22f, 47bff }}


Optimal ET sequence: {{Optimal ET sequence| 3def, …, 16, 35d }}
Badness (Sintel): 1.99


Badness: 0.038328
== 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].  


== Brightstone ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 3125/3024
[[Comma list]]: 36/35, 1875/1792


{{Mapping|legend=1| 1 0 2 6 | 0 5 1 -10 }}
{{Mapping|legend=1| 1 0 2 0 | 0 5 1 9 }}


{{Multival|legend=1| 5 1 -10 -10 -30 -26 }}
[[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 = 381.955
{{Optimal ET sequence|legend=1| 3d, …, 13b, 16 }}


{{Optimal ET sequence|legend=1| 19d, 22 }}
[[Badness]] (Sintel): 2.13
 
[[Badness]]: 0.088072


=== 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 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


Mapping: {{mapping| 1 0 2 6 6 | 0 5 1 -10 -8 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 381.790
Comma list: 27/26, 36/35, 45/44, 363/343


Optimal ET sequence: {{Optimal ET sequence| 19d, 22 }}
Mapping: {{mapping| 1 0 2 0 0 -1 | 0 5 1 9 11 15 }}


Badness: 0.047379
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 775: 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
: 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 809: 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: {{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 822: 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: {{mapping| 1 -5 1 5 -4 -2 | 0 15 3 -5 17 13 }}


Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 673.359
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=1| 16, 25e, 41, 98c, 139cf }}
{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}


Badness: 0.033081
Badness (Sintel): 1.37
 
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 105/104, 144/143, 441/440, 210/209, 625/616
 
Subgroup-val mapping: {{mapping| 1 -5 1 5 -4 -2 6 | 0 15 3 -5 17 13 -4 }}
 
Optimal tunings:
* WE: ~2 = 1200.5832{{c}}, ~15/11 = 526.8804{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6368{{c}}
 
{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}
 
Badness (Sintel): 1.26


== Quadrimage ==
== Quadrimage ==
Line 835: 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


{{Multival|legend=1| 20 4 7 -40 -45 5 }}
: mapping generators: ~2, ~25/14


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 204.987
[[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 ET sequence|legend=1| 35, 41, 158cd, 199ccd, 240ccd, 281ccd }}
{{Optimal ET sequence|legend=1| 6, …, 35, 41, 158cd, 199ccd, 240ccd, 281ccd }}


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


=== 11-limit ===
=== 11-limit ===
Line 852: 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: {{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 865: 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: {{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 tuning (POTE): ~2 = 1\1, ~28/25 = 205.028
{{Optimal ET sequence|legend=0| 6f, 35f, 41, 117c }}


{{Optimal ET sequence|legend=1| 35f, 41, 117c }}
Badness (Sintel): 1.82


Badness: 0.044047
== 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 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 882: 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"