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The '''magic family''' of temperaments tempers out [[3125/3072]], the small diesis or magic comma. A magic temperament is optimal, for some searches, in the [[9-odd-limit]]. It has slightly higher complexity than [[Meantone family|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 [[12 EDO]]. 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.


The most prominent deficiency of magic temperaments is that they lack [[Rothenberg propriety|proper]] or nearly-proper MOS scales in the 5 to 10 note "diatonic" region.
== Magic ==
{{Main| Magic }}


== Five-limit magic ==
The [[generator]] of magic is a major third, and to get to the interval class of fifths requires five of these. In fact, (5/4)<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 5-limit parent comma for the magic family is [[3125/3072]], the small diesis or magic comma. Its monzo 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. 13\41 is a highly recommendable generator, though 19\60, the [[optimal patent val]] generator, also makes a lot of sense and using [[19 EDO]] or [[22 EDO]] is always possible.


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 3125/3072
[[Comma list]]: 3125/3072


[[Mapping]]: [{{val| 1 0 2 }}, {{val| 0 5 1 }}]
{{Mapping|legend=1| 1 0 2 | 0 5 1 }}


Mapping generators: ~2, ~5/4
: mapping generators: ~2, ~5/4


[[POTE generator]]: ~5/4 = 380.058
[[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-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 }}
: [{{monzo| 1 0 0 }}, {{monzo| 0 1 0 }}, {{monzo| 2 1/5 0 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
: [[Eigenmonzo]]s (unchanged intervals): 2, 3


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[Diamond monotone]] range: ~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)
* [[Diamond tradeoff]] range: ~5/4 = [378.910, 386.314]
* 5-odd-limit [[diamond tradeoff]]: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)
* 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.


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


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


=== Extensions ===
=== Overview to 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 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.  


== Magic ==
Weak extensions considered below are hocum, trismegistus, quadrimage, quinmage and warlock. Discussed elsewhere are
{{main| Magic }}
* ''[[Astrology]]'' → [[Jubilismic clan #Astrology|Jubilismic clan]]
* ''[[Spell]]'' → [[Hemimean clan #Spell|Hemimean clan]]


Magic tempers out not only 3125/3072 and 875/864, but also [[225/224]], [[245/243]], and [[10976/10935]]. [[41 EDO]] is a good magic tuning, and 19 or 22 note MOS are possible scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1.
== Septimal magic ==
{{Main| Magic }}


Magic, 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 in its fifths are a relatively complex interval and it does not naturally work with scales of around seven notes to the octave.  
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.


225/224 is the [[Marvel temperaments|marvel]] comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament.
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.  


245/243, the [[Sensamagic clan|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, [[104 EDO]] 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


[[Comma list]]: 225/224, 245/243
[[Comma list]]: 225/224, 245/243


[[Mapping]]: [{{val| 1 0 2 -1 }}, {{val| 0 5 1 12 }}]
{{Mapping|legend=1| 1 0 2 -1 | 0 5 1 12 }}
 
Mapping generators: ~2, ~5/4


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


[[POTE generator]]: ~5/4 = 380.352
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.0786{{c}}, ~5/4 = 380.6939{{c}}
: [[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]]
* 7- and [[9-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 0 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 2 1/5 0 0 }}, {{monzo| -1 12/5 0 0 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
: [[Eigenmonzo]]s (unchanged intervals): 2, 3


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[Diamond monotone]] range: ~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)
* [[Diamond tradeoff]] range: ~5/4 = [378.910, 386.314]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)
* 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.


{{Val list|legend=1| 19, 41, 142cd, 183cd, 224cd }}
{{Optimal ET sequence|legend=1| 19, 41, 142cd, 183cd, 224ccd }}


[[Badness]]: 0.018918
[[Badness]] (Sintel): 0.479


=== 11-limit ===
=== 11-limit ===
Tempering 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: [{{val| 1 0 2 -1 6 }}, {{val| 0 5 1 12 -8 }}]
Mapping: {{mapping| 1 0 2 -1 6 | 0 5 1 12 -8 }}


POTE generator: ~5/4 = 380.696
Optimal tunings:  
* WE: ~2 = 1200.1372{{c}}, ~5/4 = 380.7399{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7008{{c}}
 
Minimax tuning:
* 11-odd-limit: ~5/4 = {{monzo| 1/3 1/9 0 0 -1/18 }}
: unchanged-interval (eigenmonzo) basis: 2.11/9


Tuning ranges:  
Tuning ranges:  
* Diamond monotone range: ~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)
* Diamond tradeoff range: ~5/4 = [378.910, 386.314]
* 11-odd-limit diamond tradeoff: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)
* Diamond monotone and tradeoff: ~5/4 = [378.947, 381.818]


Vals: {{Val list| 19, 22, 41, 104, 145c }}
{{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: [{{val| 1 0 2 -1 6 -2 }}, {{val| 0 5 1 12 -8 18 }}]
Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 1 12 -8 18 }}


POTE generator: ~5/4 = 380.427
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:  
* Diamond monotone range: ~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)
* Diamond tradeoff range: ~5/4 = [378.617, 386.314]
* 13- and 15-odd-limit diamond tradeoff: ~5/4 = [378.617, 386.314]
* Diamond monotone and tradeoff: ~5/4 = [378.947, 381.818]
 
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}
 
Badness (Sintel): 0.889
 
===== Magical =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 100/99, 105/104, 120/119, 144/143, 154/153
 
Mapping: {{mapping| 1 0 2 -1 6 -2 6 | 0 5 1 12 -8 18 -6 }}
 
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|legend=0| 19gh, 41 }}
 
Badness (Sintel): 1.44
 
===== Evening =====
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
 
Comma list: 100/99, 105/104, 144/143, 145/144, 155/154, 196/195
 
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}}


Vals: {{Val list| 19, 22f, 41, 265cdef }}
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}


Badness: 0.021509
Badness (Sintel): 0.807


==== Sorcery ====
==== Sorcery ====
Line 125: Line 228:
Comma list: 65/64, 78/77, 91/90, 100/99
Comma list: 65/64, 78/77, 91/90, 100/99


Mapping: [{{val| 1 0 2 -1 6 4 }}, {{val| 0 5 1 12 -8 -1 }}]
Mapping: {{mapping| 1 0 2 -1 6 4 | 0 5 1 12 -8 -1 }}
 
POTE generator: ~5/4 = 380.477


Tuning ranges:  
Optimal tunings:  
* Diamond monotone range: ~5/4 = 378.947 (6\19)
* WE: ~2 = 1201.2397{{c}}, ~5/4 = 380.8698{{c}}
* Diamond tradeoff range: ~5/4 = [359.472, 386.314]
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5080{{c}}
* Diamond monotone and tradeoff: ~5/4 = 378.947


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


Badness: 0.025829
Badness (Sintel): 1.07


==== Necromancy ====
==== Necromancy ====
Line 143: Line 243:
Comma list: 100/99, 225/224, 245/243, 275/273
Comma list: 100/99, 225/224, 245/243, 275/273


Mapping: [{{val| 1 0 2 -1 6 11 }}, {{val| 0 5 1 12 -8 -23 }}]
Mapping: {{mapping| 1 0 2 -1 6 11 | 0 5 1 12 -8 -23 }}
 
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|legend=0| 19f, 22, 41, 63, 104 }}


POTE generator: ~5/4 = 380.787
Badness (Sintel): 1.04


Tuning ranges:  
===== 17-limit =====
* Diamond monotone range: ~5/4 = [380.488, 380.952] (13\41 to 20\63)
Subgroup: 2.3.5.7.11.13.17
* Diamond tradeoff range: ~5/4 = [378.910, 386.314]
 
* Diamond monotone and tradeoff: ~5/4 = [380.488, 380.952]
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 }}
 
Optimal tunings:  
* WE: ~2 = 1199.6176{{c}}, ~5/4 = 380.7053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.8280{{c}}


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


Badness: 0.025275
Badness (Sintel): 1.12


==== Soothsaying ====
==== Soothsaying ====
Line 161: Line 273:
Comma list: 100/99, 225/224, 245/243, 1352/1331
Comma list: 100/99, 225/224, 245/243, 1352/1331


Mapping: [{{val| 2 0 4 -2 12 15 }}, {{val| 0 5 1 12 -8 -12 }}]
Mapping: {{mapping| 2 0 4 -2 12 15 | 0 5 1 12 -8 -12 }}
 
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=0| 22, 60, 82 }}
 
Badness (Sintel): 2.29
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 100/99, 221/220, 225/224, 245/243, 273/272
 
Mapping: {{mapping| 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}}


POTE generator: ~5/4 = 380.508
{{Optimal ET sequence|legend=0| 22, 60, 82 }}


Vals: {{Val list| 22, 60, 82 }}
Badness (Sintel): 1.82


Badness: 0.055443
===== 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 174: Line 318:
Comma list: 55/54, 99/98, 176/175
Comma list: 55/54, 99/98, 176/175


Mapping: [{{val| 1 0 2 -1 -1 }}, {{val| 0 5 1 12 14 }}]
Mapping: {{mapping| 1 0 2 -1 -1 | 0 5 1 12 14 }}


POTE generator: ~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}}


Vals: {{Val list| 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 187: Line 333:
Comma list: 55/54, 65/64, 91/90, 99/98
Comma list: 55/54, 65/64, 91/90, 99/98


Mapping: [{{val| 1 0 2 -1 -1 4 }}, {{val| 0 5 1 12 14 -1 }}]
Mapping: {{mapping| 1 0 2 -1 -1 4 | 0 5 1 12 14 -1 }}


POTE generator: ~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}}


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


Badness: 0.025522
Badness (Sintel): 1.05
 
==== Intuition ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 55/54, 66/65, 99/98, 105/104
 
Mapping: {{mapping| 1 0 2 -1 -1 -2 | 0 5 1 12 14 18 }}
 
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=0| 19e, 22f }}
 
Badness (Sintel): 1.08


=== Horcrux ===
=== Horcrux ===
Line 200: Line 363:
Comma list: 45/44, 56/55, 245/243
Comma list: 45/44, 56/55, 245/243


Mapping: [{{val| 1 0 2 -1 0 }}, {{val| 0 5 1 12 11 }}]
Mapping: {{mapping| 1 0 2 -1 0 | 0 5 1 12 11 }}


POTE generator: ~5/4 = 379.642
Optimal tunings:  
* WE: ~2 = 1200.4670{{c}}, ~5/4 = 379.7895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.6889{{c}}


Vals: {{Val list| 3de, 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


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


Mapping: [{{val| 1 0 2 -1 0 -2 }}, {{val| 0 5 1 12 11 18 }}]
Mapping: {{mapping| 1 0 2 -1 0 -2 | 0 5 1 12 11 18 }}


POTE generator: ~5/4 = 379.791
Optimal tunings:  
* WE: ~2 = 1200.2953{{c}}, ~5/4 = 379.8842{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.8165{{c}}


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


Badness: 0.031938
Badness (Sintel): 1.32
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 45/44, 56/55, 78/77, 85/84, 245/243
 
Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 16 }}
 
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=0| 3defg, 16dffgg, 19g }}
 
Badness (Sintel): 1.43
 
===== Horcruxic =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 35/34, 45/44, 52/51, 56/55, 245/243
 
Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 13 }}
 
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=0| 3defg, 16dff, 19 }}
 
Badness (Sintel): 1.51


==== Glamour ====
==== Glamour ====
Line 226: Line 423:
Comma list: 45/44, 56/55, 65/64, 245/243
Comma list: 45/44, 56/55, 65/64, 245/243


Mapping: [{{val| 1 0 2 -1 0 4 }}, {{val| 0 5 1 12 11 -2 }}]
Mapping: {{mapping| 1 0 2 -1 0 4 | 0 5 1 12 11 -1 }}


POTE generator: ~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}}


Vals: {{Val list| 3de, 19 }}
{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}


Badness: 0.033317
Badness (Sintel): 1.38


=== Witchcraft ===
=== Witchcraft ===
Line 239: Line 438:
Comma list: 225/224, 245/243, 441/440
Comma list: 225/224, 245/243, 441/440


Mapping: [{{val| 1 0 2 -1 -7 }}, {{val| 0 5 1 12 33 }}]
Mapping: {{mapping| 1 0 2 -1 -7 | 0 5 1 12 33 }}


POTE generator: ~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}}


Vals: {{Val list| 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 252: Line 453:
Comma list: 105/104, 196/195, 245/243, 275/273
Comma list: 105/104, 196/195, 245/243, 275/273


Mapping: [{{val| 1 0 2 -1 -7 -2 }}, {{val| 0 5 1 12 33 18 }}]
Mapping: {{mapping| 1 0 2 -1 -7 -2 | 0 5 1 12 33 18 }}


POTE generator: ~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}}


Vals: {{Val list| 41, 60e, 101cd }}
{{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd }}


Badness: 0.023547
Badness (Sintel): 0.973
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 154/153, 170/169, 196/195, 245/243
 
Mapping: {{mapping| 1 0 2 -1 -7 -2 -7 | 0 5 1 12 33 18 35 }}
 
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=0| 19eg, 41, 60e, 101cd }}
 
Badness (Sintel): 1.06


=== Divination ===
=== Divination ===
Line 265: Line 483:
Comma list: 121/120, 225/224, 245/243
Comma list: 121/120, 225/224, 245/243


Mapping: [{{val| 2 0 4 -2 5 }}, {{val| 0 5 1 12 3 }}]
Mapping: {{mapping| 2 0 4 -2 5 | 0 5 1 12 3 }}


POTE generator: ~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}}


Vals: {{Val list| 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 278: Line 498:
Comma list: 105/104, 121/120, 196/195, 245/243
Comma list: 105/104, 121/120, 196/195, 245/243


Mapping: [{{val| 2 0 4 -2 5 -4 }}, {{val| 0 5 1 12 3 18 }}]
Mapping: {{mapping| 2 0 4 -2 5 -4 | 0 5 1 12 3 18 }}
 
Optimal tunings:
* WE: ~99/70 = 600.9624{{c}}, ~5/4 = 380.5297{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~5/4 = 380.0614{{c}}


POTE generator: ~5/4 = 379.920
{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}


Vals: {{Val list| 22f, 60e }}
Badness (Sintel): 1.43


Badness: 0.034551
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 121/120, 154/153, 196/195, 245/243
 
Mapping: {{mapping| 2 0 4 -2 5 -4 5 | 0 5 1 12 3 18 5 }}
 
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 291: Line 528:
Comma list: 225/224, 243/242, 245/242
Comma list: 225/224, 243/242, 245/242


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


POTE generator: ~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}}


Vals: {{Val list| 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 304: Line 543:
Comma list: 105/104, 196/195, 243/242, 245/242
Comma list: 105/104, 196/195, 243/242, 245/242


Mapping: [{{val| 1 5 3 11 12 16 }}, {{val| 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 }}
 
Badness (Sintel): 1.25
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 154/153, 196/195, 243/242, 245/242
 
Mapping: {{mapping| 1 -5 1 -13 -13 -20 -15 | 0 10 2 24 25 36 29 }}


POTE generator: ~14/11 = 410.004
Optimal tunings:  
* WE: ~2 = 1201.1557{{c}}, ~11/7 = 790.7157{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.0057{{c}}


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


Badness: 0.030280
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 ==
Aside from 3125/3072 and 525/512 muggles also tempers out [[126/125]] and 1323/1280. A good muggles tuning is [[19 EDO]], in which tuning it's the same thing as magic. Muggles works better for small scales than magic in the sense that 7 or 10 note MOS are reasonable choices, as while the flatter generator compromises the accuracy of the 5-limit intervals, it grants simpler access to some higher limit ones, and makes the small steps larger & more melodically effective.
{{Main| Muggles }}


Subgroup: 2.3.5.7
Aside from 3125/3072 and 525/512 muggles also tempers out [[126/125]] and 1323/1280. A good muggles tuning is [[19edo]], in which tuning it is the same thing as magic. Muggles works better for small scales than magic in the sense that 7- or 10-note mosses are reasonable choices, as while the flatter generator compromises the accuracy of the 5-limit intervals, it grants simpler access to some higher-limit ones, and makes the small steps larger and more melodically effective.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 126/125, 525/512
[[Comma list]]: 126/125, 525/512


[[Mapping]]: [{{val| 1 0 2 5 }}, {{val| 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}}
: [[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 }}


[[POTE generator]]: ~5/4 = 378.479
[[Tuning ranges]]:
* [[7-odd-limit]] [[diamond monotone]]: ~5/4 = [375.000, 378.947] (5\16 to 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]


{{Val list|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 334: Line 614:
Comma list: 45/44, 126/125, 385/384
Comma list: 45/44, 126/125, 385/384


Mapping: [{{val| 1 0 2 5 0 }}, {{val| 0 5 1 -7 11 }}]
Mapping: {{mapping| 1 0 2 5 0 | 0 5 1 -7 11 }}


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


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


Badness: 0.048038
Badness (Sintel): 1.59


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


Comma list: 45/44, 65/64, 78/77, 126/125
Comma list: 45/44, 65/64, 78/77, 126/125


Mapping: [{{val| 1 0 2 5 0 4 }}, {{val| 0 5 1 -7 11 -1 }}]
Mapping: {{mapping| 1 0 2 5 0 4 | 0 5 1 -7 11 -1 }}


POTE generator: ~5/4 = 377.724
Optimal tunings:  
* WE: ~2 = 1203.4291{{c}}, ~5/4 = 378.7321{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.7336{{c}}


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


Badness: 0.030386
Badness (Sintel): 1.26


== Astrology ==
=== Muggloid ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11


[[Comma list]]: 50/49, 3125/3072
Comma list: 33/32, 126/125, 176/175


[[Mapping]]: [{{val| 2 0 4 5 }}, {{val| 0 5 1 1 }}]
Mapping: {{mapping| 1 0 2 5 5 | 0 5 1 -7 -5 }}


{{Multival|legend=1| 10 2 2 -20 -25 -1 }}
Optimal tunings:
* WE: ~2 = 1205.6044{{c}}, ~5/4 = 379.5966{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8142{{c}}


[[POTE generator]]: ~5/4 = 380.578
{{Optimal ET sequence|legend=0| 3, 16, 19e, 35ee, 54bdeee }}


{{Val list|legend=1| 6, 16, 22, 60d, 82d }}
Badness (Sintel): 1.55


[[Badness]]: 0.082673
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 11-limit ===
Comma list: 33/32, 65/64, 105/104, 126/125
Subgroup: 2.3.5.7.11


Comma list: 50/49, 121/120, 176/175
Mapping: {{mapping| 1 0 2 5 5 4 | 0 5 1 -7 -5 -1 }}


Mapping: [{{val| 2 0 4 5 5 }}, {{val| 0 5 1 1 3 }}]
Optimal tunings:  
* WE: ~2 = 1205.4897{{c}}, ~5/4 = 379.5667{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8118{{c}}


POTE generator: ~5/4 = 380.530
{{Optimal ET sequence|legend=0| 3, 16, 19e, 35eef }}


Vals: {{Val list| 6, 16, 22, 60de, 82de }}
Badness (Sintel): 1.19


Badness: 0.039151
== 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.  


==== 13-limit ====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13


Comma list: 50/49, 65/64, 78/77, 121/120
[[Comma list]]: 64/63, 3125/3024


Mapping: [{{val| 2 0 4 5 5 8 }}, {{val| 0 5 1 1 3 -1 }}]
{{Mapping|legend=1| 1 0 2 6 | 0 5 1 -10 }}


POTE generator: ~5/4 = 379.787
[[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 }}


Vals: {{Val list| 6, 16, 22, 38f }}
{{Optimal ET sequence|legend=1| 3, 19d, 22 }}


Badness: 0.034376
[[Badness]] (Sintel): 2.23


; Music
=== 11-limit ===
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/AstrologyPercQuintet1_c.mp3 Astrology Percussion Quintet No 1] by [https://soundcloud.com/joelgranttaylor/astrology-percussion-quintet Joel Taylor]
Subgroup: 2.3.5.7.11


==== Horoscope ====
Comma list: 55/54, 64/63, 625/616
Subgroup: 2.3.5.7.11.13


Comma list: 50/49, 66/65, 105/104, 121/120
Mapping: {{mapping| 1 0 2 6 -1 | 0 5 1 -10 14 }}


Mapping: [{{val| 2 0 4 5 5 3 }}, {{val| 0 5 1 1 3 7 }}]
Optimal tunings:  
* WE: ~2 = 1198.5372{{c}}, ~5/4 = 381.7556{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.1943{{c}}


POTE generator: ~5/4 = 379.837
{{Optimal ET sequence|legend=0| 22, 69b }}


Vals: {{Val list| 16, 22f, 38 }}
Badness (Sintel): 1.85


Badness: 0.035284
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== Spell ==
Comma list: 40/39, 55/54, 64/63, 625/616
Subgroup: 2.3.5.7


[[Comma list]]: 49/48, 3125/3072
Mapping: {{mapping| 1 0 2 6 6 4 | 0 5 1 -10 14 -4 }}


[[Mapping]]: [{{val| 1 0 2 2 }}, {{val| 0 10 2 5 }}]
Optimal tunings:  
* WE: ~2 = 1197.2300{{c}}, ~5/4 = 381.6164{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.4690{{c}}


{{Multival|legend=1| 10 2 5 -20 -20 6 }}
{{Optimal ET sequence|legend=0| 22f, 47bff }}


[[POTE generator]]: ~28/25 = 189.927
Badness (Sintel): 1.99


{{Val list|legend=1| 6, 19, 82dd }}
== Darkstone ==
Darkstone (16 & 19d) is a low-accuracy temperament which tempers out 36/35 and 1875/1792. It makes the major third and the fifth even flatter than those of muggles.


[[Badness]]: 0.080958
This temperament is known as ''witch'' in [http://www.tonalsoft.com/enc/m/magic.aspx Tonalsoft Encyclopedia].  


=== 11-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11


Comma list: 49/48, 56/55, 125/121
[[Comma list]]: 36/35, 1875/1792


Mapping: [{{val| 1 0 2 2 3 }}, {{val| 0 10 2 5 3 }}]
{{Mapping|legend=1| 1 0 2 0 | 0 5 1 9 }}


POTE generator: ~11/10 = 190.285
[[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 }}


Vals: {{Val list| 6, 19, 44de, 63dee, 82ddee }}
{{Optimal ET sequence|legend=1| 3d, …, 13b, 16 }}


Badness: 0.059791
[[Badness]] (Sintel): 2.13


==== 13-limit ====
=== 11-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11


Comma list: 49/48, 56/55, 78/77, 125/121
Comma list: 36/35, 45/44, 363/343


Mapping: [{{val| 1 0 2 2 3 4 }}, {{val| 0 10 2 5 3 -2 }}]
Mapping: {{mapping| 1 0 2 0 0 | 0 5 1 9 11 }}


POTE generator: ~11/10 = 189.928
Optimal tunings:  
* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.9735{{c}}


Vals: {{Val list| 6, 19, 82ddeeff }}
{{Optimal ET sequence|legend=0| 3de, 13be, 16 }}


Badness: 0.045591
Badness (Sintel): 1.55


==== Cantrip ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 56/55, 91/90, 125/121
Comma list: 27/26, 36/35, 45/44, 363/343
 
Mapping: {{mapping| 1 0 2 0 0 -1 | 0 5 1 9 11 15 }}


Mapping: [{{val| 1 0 2 2 3 1 }}, {{val| 0 10 2 5 3 17 }}]
Optimal tunings:  
* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.4221{{c}}


POTE generator: ~11/10 = 190.360
{{Optimal ET sequence|legend=0| 3def, 13beff, 16 }}


Vals: {{Val list| 19, 44de, 63dee, 82ddee }}
Badness (Sintel): 1.58


Badness: 0.041603
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 ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3125/3072, 4000/3969
[[Comma list]]: 3125/3072, 4000/3969


[[Mapping]]: [{{val| 1 5 3 -3 }}, {{val| 0 -10 -2 17 }}]
{{Mapping|legend=1| 1 -5 1 14 | 0 10 2 -17 }}


{{Multival|legend=1| 10 2 -17 -20 -55 -45 }}
: mapping generators: ~2, ~63/40


[[POTE generator]]: ~63/50 = 400.108
[[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 }}


{{Val list|legend=1| 38, 41, 161c, 202c, 243c, 284c }}
{{Optimal ET sequence|legend=1| 3, 38, 41, 161c }}


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


== Trismegistus ==
== Trismegistus ==
Subgroup: 2.3.5.7
{{Main| Mabilic and trismegistus }}
{{See also| No-threes subgroup temperaments #Mabilic }}
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 3125/3072
[[Comma list]]: 1029/1024, 3125/3072


[[Mapping]]: [{{val| 1 10 4 0 }}, {{val| 0 -15 -3 5 }}]
{{Mapping|legend=1| 1 -5 1 5 | 0 15 3 -5 }}


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


[[POTE generator]]: ~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 }}


{{Val list|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 501: Line 823:
Comma list: 385/384, 441/440, 625/616
Comma list: 385/384, 441/440, 625/616


POTE generator: ~22/15 = 673.340
Mapping: {{mapping| 1 -5 1 5 -4 | 0 15 3 -5 17 }}


Mapping: [{{val| 1 10 4 0 13 }}, {{val| 0 -15 -3 5 -17 }}]
Optimal tunings:  
* WE: ~2 = 1200.8404{{c}}, ~15/11 = 527.0289{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6826{{c}}


Vals: {{Val list| 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 514: Line 838:
Comma list: 105/104, 144/143, 275/273, 625/616
Comma list: 105/104, 144/143, 275/273, 625/616


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


POTE generator: ~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}}


Vals: {{Val list| 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 ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 3125/3072
[[Comma list]]: 2401/2400, 3125/3072


[[Mapping]]: [{{val| 1 5 3 4 }}, {{val| 0 -20 -4 -7 }}]
{{Mapping|legend=1| 1 -15 -1 -3 | 0 20 4 7 }}


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


[[POTE generator]]: ~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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 542: Line 887:
Comma list: 245/242, 385/384, 625/616
Comma list: 245/242, 385/384, 625/616


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


POTE generator: ~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}}


Vals: {{Val list| 6, 35, 41, 199cde, 240cde, 281cde }}
{{Optimal ET sequence|legend=0| 6, 35, 41 }}


Badness: 0.061572
Badness (Sintel): 2.04


=== 13-limit ===
=== 13-limit ===
Line 555: Line 902:
Comma list: 105/104, 144/143, 245/242, 625/616
Comma list: 105/104, 144/143, 245/242, 625/616


Mapping: [{{val| 1 5 3 4 5 9 }}, {{val| 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 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 ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 3125/3072, 16807/16384
 
{{Mapping|legend=1| 5 0 10 14 | 0 5 1 0 }}
 
: mapping generators: ~8/7, ~5/4


POTE generator: ~28/25 = 205.028
[[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 }}


Vals: {{Val list| 41, 117c, 158cd, 199cdef }}
{{Optimal ET sequence|legend=1| 25, 35, 60 }}


Badness: 0.044047
[[Badness]] (Sintel): 7.27


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Magic family| ]] <!-- main article -->
[[Category:Magic family| ]] <!-- main article -->
[[Category:Magic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Magic]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Magic_family"