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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.
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. Weak extensions include astrology, spell, hocum, trismegistus, quadrimage, and warlock.  


== 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. 13\41 is a highly recommendable generator, though 19\60, the [[optimal patent val]] generator, also makes a lot of sense, and using [[19edo]] or [[22edo]] is always possible.
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


[[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]]: ~5/4 = {{monzo| 0 1/5 0 }}
* [[5-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 }}
: [[Eigenmonzo]]s (unchanged intervals): 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]
* 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.


{{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
 
=== 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, as well as low-accuracy extensions including darkstone and brightstone.
 
Weak extensions considered below are hocum, trismegistus, quadrimage, quinmage and warlock. Discussed elsewhere are
* ''[[Astrology]]'' → [[Jubilismic clan #Astrology|Jubilismic clan]]
* ''[[Spell]]'' → [[Hemimean clan #Spell|Hemimean clan]]


== Septimal magic ==
== Septimal magic ==
{{main| Magic }}
{{Main| Magic }}


Septimal magic tempers out not only 3125/3072 and 875/864, but also [[225/224]], [[245/243]], and [[10976/10935]]. [[41edo]] is a good magic tuning, and 19- or 22-note MOSes 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 in 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.  
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.  


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]] 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.  


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.
[[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
: 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}}
[[POTE generator]]: ~5/4 = 380.352
: [[error map]]: {{val| +1.079 +1.514 -3.463 -1.578 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.4576{{c}}
: error map: {{val| 0.000 +0.333 -5.856 -3.335 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 0 }}
* 7- and [[9-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 0 }}
: [[Eigenmonzo]]s (unchanged intervals): 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]
* 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.


{{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:  
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]]s (unchanged intervals): 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]
* 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 GPV sequence: {{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:  
* 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 GPV sequence: {{Val list| 19, 22f, 41, 265cdef }}
{{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


Comma list: 100/99, 105/104, 120/119, 144/143, 154/153
Comma list: 100/99, 105/104, 120/119, 144/143, 154/153


Mapping: [{{val| 1 0 2 -1 6 -2 6 }}, {{val| 0 5 1 12 -8 18 -6 }}]
Mapping: {{mapping| 1 0 2 -1 6 -2 6 | 0 5 1 12 -8 18 -6 }}


POTE generator: ~5/4 = 380.604
Optimal tunings:  
* WE: ~2 = 1199.3584{{c}}, ~5/4 = 380.4006{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5896{{c}}


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


Badness: 0.020633
Badness (Sintel): 1.05


===== Evening =====
====== Magicus ======
'''Evening''' is a remarkable subgroup temperament of 19&amp;41 with prime harmonics of 29 and 31.
Subgroup: 2.3.5.7.11.13.17.19


Subgroup: 2.3.5.7.11.13.29.31
Comma list: 100/99, 105/104, 120/119, 133/132, 144/143, 154/153


Comma list: 100/99, 105/104, 144/143, 145/144, 155/154, 196/195
Mapping: {{mapping| 1 0 2 -1 6 -2 6 9 | 0 5 1 12 -8 18 -6 -15 }}


Mapping: [{{val| 1 0 2 -1 6 -2 2 4 }}, {{val| 0 5 1 12 -8 18 9 3 }}]
Optimal tunings:  
* WE: ~2 = 1199.7173{{c}}, ~5/4 = 380.3808{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4680{{c}}


POTE generator: ~5/4 = 380.416
{{Optimal ET sequence|legend=0| 19, 41 }}


Optimal GPV sequence: {{Val list| 19, 22f, 41 }}
Badness (Sintel): 1.27


==== Sorcery ====
====== Magica ======
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 65/64, 78/77, 91/90, 100/99
Comma list: 100/99, 105/104, 120/119, 144/143, 154/153, 171/169


Mapping: [{{val| 1 0 2 -1 6 4 }}, {{val| 0 5 1 12 -8 -1 }}]
Mapping: {{mapping| 1 0 2 -1 6 -2 6 -4 | 0 5 1 12 -8 18 -6 26 }}


POTE generator: ~5/4 = 380.477
Optimal tunings:  
* WE: ~2 = 1199.3670{{c}}, ~5/4 = 380.4681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6541{{c}}


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


Badness: 0.025829
Badness (Sintel): 1.21


===== 17-limit =====
===== Magia =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 52/51, 65/64, 78/77, 91/90, 100/99
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}}
 
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}
 
Badness (Sintel): 0.807


Mapping: [{{val| 1 0 2 -1 6 4 6 }}, {{val| 0 5 1 12 -8 -1 -6 }}]
==== Sorcery ====
Subgroup: 2.3.5.7.11.13


POTE generator: ~5/4 = 380.729
Comma list: 65/64, 78/77, 91/90, 100/99


Optimal GPV sequence: {{Val list| 19, 22, 41f }}
Mapping: {{mapping| 1 0 2 -1 6 4 | 0 5 1 12 -8 -1 }}


Badness: 0.023768
Optimal tunings:
* WE: ~2 = 1201.2397{{c}}, ~5/4 = 380.8698{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5080{{c}}
 
{{Optimal ET sequence|legend=0| 19, 22, 41f }}
 
Badness (Sintel): 1.07


==== Necromancy ====
==== Necromancy ====
Line 178: 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 }}


POTE generator: ~5/4 = 380.787
Optimal tunings:  
* WE: ~2 = 1199.9675{{c}}, ~5/4 = 380.7770{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7874{{c}}


Optimal GPV sequence: {{Val list| 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 191: Line 258:
Comma list: 100/99, 120/119, 154/153, 225/224, 273/272
Comma list: 100/99, 120/119, 154/153, 225/224, 273/272


Mapping: [{{val| 1 0 2 -1 6 11 6 }}, {{val| 0 5 1 12 -8 -23 -6 }}]
Mapping: {{mapping| 1 0 2 -1 6 11 6 | 0 5 1 12 -8 -23 -6 }}


POTE generator: ~5/4 = 380.827
Optimal tunings:  
* WE: ~2 = 1199.6176{{c}}, ~5/4 = 380.7053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.8280{{c}}


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


Badness: 0.022032
Badness (Sintel): 1.12


==== Soothsaying ====
==== Soothsaying ====
Line 204: 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 }}


POTE generator: ~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 GPV sequence: {{Val list| 22, 60, 82 }}
{{Optimal ET sequence|legend=0| 22, 60, 82 }}


Badness: 0.055443
Badness (Sintel): 2.29


===== 17-limit =====
===== 17-limit =====
Line 217: Line 288:
Comma list: 100/99, 221/220, 225/224, 245/243, 273/272
Comma list: 100/99, 221/220, 225/224, 245/243, 273/272


Mapping: [{{val| 2 0 4 -2 12 15 5 }}, {{val| 0 5 1 12 -8 -12 5 }}]
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}}
 
{{Optimal ET sequence|legend=0| 22, 60, 82 }}
 
Badness (Sintel): 1.82
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 100/99, 133/132, 221/220, 225/224, 245/243, 273/272


POTE generator: ~5/4 = 380.508
Mapping: {{mapping| 2 0 4 -2 12 15 5 18 | 0 5 1 12 -8 -12 5 -15 }}


Optimal GPV sequence: {{Val list| 22, 60, 82 }}
Optimal tunings:  
* WE: ~17/12 = 600.3301{{c}}, ~5/4 = 380.6797{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.4704{{c}}


Badness: 0.035654
{{Optimal ET sequence|legend=0| 22, 60, 82 }}
 
Badness (Sintel): 1.90


=== Telepathy ===
=== Telepathy ===
Line 230: 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}}


Optimal GPV sequence: {{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 243: 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}}


Optimal GPV sequence: {{Val list| 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: [{{val| 1 0 2 -1 -1 4 -1 }}, {{val| 0 5 1 12 14 -1 16 }}]
Mapping: {{mapping| 1 0 2 -1 -1 -2 | 0 5 1 12 14 18 }}


POTE generator: ~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 GPV sequence: {{Val list| 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: [{{val| 1 0 2 -1 -1 -2 }}, {{val| 0 5 1 12 14 18 }}]
Mapping: {{mapping| 1 0 2 -1 0 | 0 5 1 12 11 }}


POTE generator: ~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 GPV sequence: {{Val list| 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: [{{val| 1 0 2 -1 -1 -2 -1 }}, {{val| 0 5 1 12 14 18 16 }}]
Mapping: {{mapping| 1 0 2 -1 0 -2 | 0 5 1 12 11 18 }}


POTE generator: ~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 GPV sequence: {{Val list| 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: [{{val| 1 0 2 -1 0 }}, {{val| 0 5 1 12 11 }}]
Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 16 }}


POTE generator: ~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 GPV sequence: {{Val list| 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: [{{val| 1 0 2 -1 0 -2 }}, {{val| 0 5 1 12 11 18 }}]
Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 13 }}


POTE generator: ~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 GPV sequence: {{Val list| 3def, 19, 41ee, 60ee }}
{{Optimal ET sequence|legend=0| 3defg, 16dff, 19 }}


Badness: 0.031938
Badness (Sintel): 1.51


==== Glamour ====
==== Glamour ====
Line 321: 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 -1}}]
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}}


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


Badness: 0.033317
Badness (Sintel): 1.38


=== Witchcraft ===
=== Witchcraft ===
Line 334: 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}}


Optimal GPV sequence: {{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 347: 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}}


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


Badness: 0.023547
Badness (Sintel): 0.973


==== 17-limit ====
==== 17-limit ====
Line 360: Line 468:
Comma list: 105/104, 154/153, 170/169, 196/195, 245/243
Comma list: 105/104, 154/153, 170/169, 196/195, 245/243


Mapping: [{{val| 1 0 2 -1 -7 -2 -7 }}, {{val| 0 5 1 12 33 18 35 }}]
Mapping: {{mapping| 1 0 2 -1 -7 -2 -7 | 0 5 1 12 33 18 35 }}


POTE generator: ~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 GPV sequence: {{Val list| 41, 60e, 101cd, 161cde }}
{{Optimal ET sequence|legend=0| 19eg, 41, 60e, 101cd }}


Badness: 0.020756
Badness (Sintel): 1.06


=== Divination ===
=== Divination ===
Line 373: 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}}


Optimal GPV sequence: {{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 386: 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 }}


POTE generator: ~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 GPV sequence: {{Val list| 22f, 60e }}
{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}


Badness: 0.034551
Badness (Sintel): 1.43
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 121/120, 154/153, 196/195, 245/243
 
Mapping: {{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 399: 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}}


Optimal GPV sequence: {{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 412: 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 }}
 
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


POTE generator: ~14/11 = 410.004
Comma list: 105/104, 154/153, 196/195, 243/242, 245/242


Optimal GPV sequence: {{Val list| 41, 79d, 120cd }}
Mapping: {{mapping| 1 -5 1 -13 -13 -20 -3 | 0 10 2 24 25 36 29 11 }}


Badness: 0.030280
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 ==
{{main| Muggles }}
{{Main| Muggles }}


Aside from 3125/3072 and 525/512 muggles also tempers out [[126/125]] and 1323/1280. A good muggles tuning is [[19edo]], in which tuning it'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 and more melodically effective.  
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
[[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}}
[[POTE generator]]: ~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]]:  
* 7-odd-limit [[diamond monotone]]: ~5/4 = [375.000, 378.947] (5\16 to 6\19)
* [[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)
* [[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


{{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 451: 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:  
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 GPV sequence: {{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 ====
Line 469: Line 633:
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 = 377.653
Optimal tunings:  
* WE: ~2 = 1203.4291{{c}}, ~5/4 = 378.7321{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.7336{{c}}


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


Badness: 0.030386
Badness (Sintel): 1.26


=== Muggloid ===
=== Muggloid ===
Line 482: Line 648:
Comma list: 33/32, 126/125, 176/175
Comma list: 33/32, 126/125, 176/175


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


POTE generator: ~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 GPV sequence: {{Val list| 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 495: Line 663:
Comma list: 33/32, 65/64, 105/104, 126/125
Comma list: 33/32, 65/64, 105/104, 126/125


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


POTE generator: ~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 GPV sequence: {{Val list| 3, 10bd, 13bd, 16, 19e }}
{{Optimal ET sequence|legend=0| 3, 16, 19e, 35eef }}


Badness: 0.028732
Badness (Sintel): 1.19


== Astrology ==
== Brightstone ==
{{See also| Jubilismic clan }}
Brightstone tempers out 64/63 and may be described as {{nowrap| 22 & 25 }}. 22edo itself is a good tuning, in which case it is identical to magic. Brightstone can be extended to the 11- and 13-limit in a similar way to muggles.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 50/49, 3125/3072
[[Comma list]]: 64/63, 3125/3024


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


{{Multival|legend=1| 10 2 2 -20 -25 -1 }}
[[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 }}


[[POTE generator]]: ~5/4 = 380.578
{{Optimal ET sequence|legend=1| 3, 19d, 22 }}


{{Val list|legend=1| 6, 16, 22, 60d, 82d }}
[[Badness]] (Sintel): 2.23
 
[[Badness]]: 0.082673


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 50/49, 121/120, 176/175
Comma list: 55/54, 64/63, 625/616


Mapping: [{{val| 2 0 4 5 5 }}, {{val| 0 5 1 1 3 }}]
Mapping: {{mapping| 1 0 2 6 -1 | 0 5 1 -10 14 }}


POTE generator: ~5/4 = 380.530
Optimal tunings:  
* WE: ~2 = 1198.5372{{c}}, ~5/4 = 381.7556{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.1943{{c}}


Optimal GPV sequence: {{Val list| 6, 16, 22, 60de, 82de }}
{{Optimal ET sequence|legend=0| 22, 69b }}


Badness: 0.039151
Badness (Sintel): 1.85


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 65/64, 78/77, 121/120
 
Mapping: [{{val| 2 0 4 5 5 8 }}, {{val| 0 5 1 1 3 -1 }}]
 
POTE generator: ~5/4 = 379.787
 
Optimal GPV sequence: {{Val list| 6, 16, 22, 38f }}
 
Badness: 0.034376
 
; Music
* [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]
 
==== Horoscope ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 50/49, 66/65, 105/104, 121/120
Comma list: 40/39, 55/54, 64/63, 625/616


Mapping: [{{val| 2 0 4 5 5 3 }}, {{val| 0 5 1 1 3 7 }}]
Mapping: {{mapping| 1 0 2 6 6 4 | 0 5 1 -10 14 -4 }}


POTE generator: ~5/4 = 379.837
Optimal tunings:  
* WE: ~2 = 1197.2300{{c}}, ~5/4 = 381.6164{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.4690{{c}}


Optimal GPV sequence: {{Val list| 16, 22f, 38 }}
{{Optimal ET sequence|legend=0| 22f, 47bff }}


Badness: 0.035284
Badness (Sintel): 1.99


== Spell ==
== Darkstone ==
{{See also| Hemimean clan #Spell }}
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.


Subgroup: 2.3.5.7
This temperament is known as ''witch'' in [http://www.tonalsoft.com/enc/m/magic.aspx Tonalsoft Encyclopedia].  


[[Comma list]]: 49/48, 3125/3072
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 0 2 2 }}, {{val| 0 10 2 5 }}]
[[Comma list]]: 36/35, 1875/1792


{{Multival|legend=1| 10 2 5 -20 -20 6 }}
{{Mapping|legend=1| 1 0 2 0 | 0 5 1 9 }}


[[POTE generator]]: ~28/25 = 189.927
[[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 }}


{{Val list|legend=1| 6, 19, 82dd }}
{{Optimal ET sequence|legend=1| 3d, …, 13b, 16 }}


[[Badness]]: 0.080958
[[Badness]] (Sintel): 2.13


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


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


POTE generator: ~11/10 = 190.285
Optimal tunings:  
* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.9735{{c}}


Optimal GPV sequence: {{Val list| 6, 19, 44de, 63dee, 82ddee }}
{{Optimal ET sequence|legend=0| 3de, 13be, 16 }}


Badness: 0.059791
Badness (Sintel): 1.55


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


Comma list: 49/48, 56/55, 78/77, 125/121
Comma list: 27/26, 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 -1 | 0 5 1 9 11 15 }}


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 = 376.4221{{c}}


Optimal GPV sequence: {{Val list| 6, 19, 82ddeeff }}
{{Optimal ET sequence|legend=0| 3def, 13beff, 16 }}


Badness: 0.045591
Badness (Sintel): 1.58


==== Cantrip ====
Scales: [[User:BudjarnLambeth/Volcanic glass]]
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 56/55, 91/90, 125/121
Music: [https://www.youtube.com/watch?v=oXy_RXrBVWA ''Rain in the crystal mirror caves''] - [[Budjarn Lambeth]] (2026)
 
Mapping: [{{val| 1 0 2 2 3 1 }}, {{val| 0 10 2 5 3 17 }}]
 
POTE generator: ~11/10 = 190.360
 
Optimal GPV sequence: {{Val list| 19, 44de, 63dee, 82ddee }}
 
Badness: 0.041603


== 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 = 410.108
[[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 653: 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}}


Optimal GPV sequence: {{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 666: 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 }}
 
Optimal tunings:
* WE: ~2 = 1200.4759{{c}}, ~15/11 = 526.8502{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 526.6548{{c}}
 
{{Optimal ET sequence|legend=0| 16, 25e, 41, 98c }}
 
Badness (Sintel): 1.37
 
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.19


POTE generator: ~22/15 = 673.359
Comma list: 105/104, 144/143, 441/440, 210/209, 625/616


Optimal GPV sequence: {{Val list| 16, 25e, 41, 98c, 139cf }}
Subgroup-val mapping: {{mapping| 1 -5 1 5 -4 -2 6 | 0 15 3 -5 17 13 -4 }}


Badness: 0.033081
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 694: 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}}


Optimal GPV sequence: {{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 707: 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 }}


POTE generator: ~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 GPV sequence: {{Val list| 41, 117c, 158cd, 199cdef }}
{{Optimal ET sequence|legend=0| 13b, 28b, 41 }}


Badness: 0.044047
Badness (Sintel): 2.80


== Warlock ==
== Warlock ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3125/3072, 16807/16384
[[Comma list]]: 3125/3072, 16807/16384


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


Mapping generators: ~8/7, ~5/4
: mapping generators: ~8/7, ~5/4


[[POTE generator]]: ~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 }}


{{Val list|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]]
[[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]]

Latest revision as of 01:34, 12 May 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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

Magic

Main article: Magic

The generator of magic is a major third, and to get to the interval class of fifths requires five of these. In fact, (5/4)5 = 3 × 3125/3072. 13\41 is a highly recommendable generator, though 19\60, the optimal patent val generator, also makes a lot of sense, and using 19edo or 22edo is always possible.

Subgroup: 2.3.5

Comma list: 3125/3072

Mapping: [⟨1 0 2], ⟨0 5 1]]

mapping generators: ~2, ~5/4

Optimal tunings:

  • WE: ~2 = 1201.2449 ¢, ~5/4 = 380.4527 ¢
error map: ⟨+1.245 +0.309 -3.371]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.2194 ¢
error map: ⟨0.000 -0.858 -6.094]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.3

Tuning ranges:

Algebraic generator: Terzbirat, the positive root of 9x2 - 8x - 4 = (4 + 2√13)/9; approximately 380.3175 cents.

Optimal ET sequence: 3, 13b, 16, 19, 41, 60, 221cc, 281cc

Badness (Sintel): 0.919

Overview to extensions

Apart from magic, we also consider other extensions. The second comma of the normal comma list defines which 7-limit family member we are looking at. 875/864, the keemic comma, gives septimal magic, and 525/512, Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator, as well as low-accuracy extensions including darkstone and brightstone.

Weak extensions considered below are hocum, trismegistus, quadrimage, quinmage and warlock. Discussed elsewhere are

Septimal magic

Main article: Magic

Septimal magic tempers out not only 3125/3072 and 875/864, but also 225/224, 245/243, and 10976/10935. 41edo is a good magic tuning, and 19- or 22-note mosses are possible scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1.

This temperament, with its accurate fifths, works well with 9-odd-limit harmony. It is more accurate than meantone and simpler than garibaldi. It is a little tricky to work with because its fifths are a relatively complex interval and it does not naturally work with scales of around seven notes to the octave.

225/224 is the marvel comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament. 245/243, the sensamagic comma, leads to another essentially tempered 9-odd-limit triad with two thirds approximating 9/7 and the other 6/5. It also divides the approximate 3/2 into two steps of 7/6 and one of 10/9.

By adding 100/99 and 105/104 to the list of commas, magic can be extended to the 11-limit and 13-limit. 11-limit magic allows for a tritone substitution where the extended 5-limit tuning of a dominant seventh with a 9/5 above the root shares its tritone with an 8:10:11:12:16 chord rooted on the seventh of the original chord. For this, 104edo provides an excellent tuning, as it does also for the rank-3 temperaments tempering out 100/99 with 225/224, 245/243 or 875/864. Septimage (see below) is also an excellent 11-limit magic tuning. For the 13-limit, 41edo makes for a recommendable tuning.

Subgroup: 2.3.5.7

Comma list: 225/224, 245/243

Mapping: [⟨1 0 2 -1], ⟨0 5 1 12]]

mapping generators: ~2, ~5/4

Optimal tunings:

  • WE: ~2 = 1201.0786 ¢, ~5/4 = 380.6939 ¢
error map: ⟨+1.079 +1.514 -3.463 -1.578]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4576 ¢
error map: ⟨0.000 +0.333 -5.856 -3.335]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.3

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
  • 7- and 9-odd-limit diamond tradeoff: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)

Algebraic generator: Tirzbirat or Septimage, the real root of 5x5 + 4x - 20, 380.7604 cents.

Optimal ET sequence: 19, 41, 142cd, 183cd, 224ccd

Badness (Sintel): 0.479

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 245/243

Mapping: [⟨1 0 2 -1 6], ⟨0 5 1 12 -8]]

Optimal tunings:

  • WE: ~2 = 1200.1372 ¢, ~5/4 = 380.7399 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.7008 ¢

Minimax tuning:

  • 11-odd-limit: ~5/4 = [1/3 1/9 0 0 -1/18⟩
unchanged-interval (eigenmonzo) basis: 2.11/9

Tuning ranges:

  • 11-odd-limit diamond monotone: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
  • 11-odd-limit diamond tradeoff: ~5/4 = [378.910, 386.314] (1/4-comma to untempered)

Optimal ET sequence: 19, 22, 41, 104

Badness (Sintel): 0.673

13-limit

A notable patent val tuning beyond the optimal patent val of 41edo is 19 + 41 = 60edo.

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

Mapping: [⟨1 0 2 -1 6 -2], ⟨0 5 1 12 -8 18]]

Optimal tunings:

  • WE: ~2 = 1200.0331 ¢, ~5/4 = 380.4377 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4284 ¢

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~5/4 = [378.947, 381.818] (6\19 to 7\22)
  • 13- and 15-odd-limit diamond tradeoff: ~5/4 = [378.617, 386.314]

Optimal ET sequence: 19, 22f, 41

Badness (Sintel): 0.889

Magical

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 120/119, 144/143, 154/153

Mapping: [⟨1 0 2 -1 6 -2 6], ⟨0 5 1 12 -8 18 -6]]

Optimal tunings:

  • WE: ~2 = 1199.3584 ¢, ~5/4 = 380.4006 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.5896 ¢

Optimal ET sequence: 19, 22f, 41

Badness (Sintel): 1.05

Magicus

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 120/119, 133/132, 144/143, 154/153

Mapping: [⟨1 0 2 -1 6 -2 6 9], ⟨0 5 1 12 -8 18 -6 -15]]

Optimal tunings:

  • WE: ~2 = 1199.7173 ¢, ~5/4 = 380.3808 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4680 ¢

Optimal ET sequence: 19, 41

Badness (Sintel): 1.27

Magica

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 120/119, 144/143, 154/153, 171/169

Mapping: [⟨1 0 2 -1 6 -2 6 -4], ⟨0 5 1 12 -8 18 -6 26]]

Optimal tunings:

  • WE: ~2 = 1199.3670 ¢, ~5/4 = 380.4681 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.6541 ¢

Optimal ET sequence: 22fh, 41

Badness (Sintel): 1.21

Magia

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 144/143, 170/169, 196/195

Mapping: [⟨1 0 2 -1 6 -2 -7], ⟨0 5 1 12 -8 18 35]]

Optimal tunings:

  • WE: ~2 = 1200.1727 ¢, ~5/4 = 380.2982 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.2483 ¢

Optimal ET sequence: 19g, 41, 60

Badness (Sintel): 1.34

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 144/143, 170/169, 171/169, 196/195

Mapping: [⟨1 0 2 -1 6 -2 -7 -4], ⟨0 5 1 12 -8 18 35 26]]

Optimal tunings:

  • WE: ~2 = 1200.2179 ¢, ~5/4 = 380.3942 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.3314 ¢

Optimal ET sequence: 19gh, 41

Badness (Sintel): 1.44

Evening

Evening is a remarkable subgroup temperament of 19 & 22f with prime harmonics of 29 and 31.

Subgroup: 2.3.5.7.11.13.29.31

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

Subgroup-val mapping: [⟨1 0 2 -1 6 -2 2 4], ⟨0 5 1 12 -8 18 9 3]]

Optimal tunings:

  • WE: ~2 = 1200.2802 ¢, ~5/4 = 380.5053 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.4258 ¢

Optimal ET sequence: 19, 22f, 41

Badness (Sintel): 0.807

Sorcery

Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 78/77, 91/90, 100/99

Mapping: [⟨1 0 2 -1 6 4], ⟨0 5 1 12 -8 -1]]

Optimal tunings:

  • WE: ~2 = 1201.2397 ¢, ~5/4 = 380.8698 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.5080 ¢

Optimal ET sequence: 19, 22, 41f

Badness (Sintel): 1.07

Necromancy

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 225/224, 245/243, 275/273

Mapping: [⟨1 0 2 -1 6 11], ⟨0 5 1 12 -8 -23]]

Optimal tunings:

  • WE: ~2 = 1199.9675 ¢, ~5/4 = 380.7770 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.7874 ¢

Optimal ET sequence: 19f, 22, 41, 63, 104

Badness (Sintel): 1.04

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 120/119, 154/153, 225/224, 273/272

Mapping: [⟨1 0 2 -1 6 11 6], ⟨0 5 1 12 -8 -23 -6]]

Optimal tunings:

  • WE: ~2 = 1199.6176 ¢, ~5/4 = 380.7053 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.8280 ¢

Optimal ET sequence: 19f, 22, 41, 63

Badness (Sintel): 1.12

Soothsaying

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 225/224, 245/243, 1352/1331

Mapping: [⟨2 0 4 -2 12 15], ⟨0 5 1 12 -8 -12]]

Optimal tunings:

  • WE: ~55/39 = 600.2918 ¢, ~5/4 = 380.6928 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~5/4 = 380.5121 ¢

Optimal ET sequence: 22, 60, 82

Badness (Sintel): 2.29

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 221/220, 225/224, 245/243, 273/272

Mapping: [⟨2 0 4 -2 12 15 5], ⟨0 5 1 12 -8 -12 5]]

Optimal tunings:

  • WE: ~17/12 = 600.2918 ¢, ~5/4 = 380.6927 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~5/4 = 380.5135 ¢

Optimal ET sequence: 22, 60, 82

Badness (Sintel): 1.82

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 133/132, 221/220, 225/224, 245/243, 273/272

Mapping: [⟨2 0 4 -2 12 15 5 18], ⟨0 5 1 12 -8 -12 5 -15]]

Optimal tunings:

  • WE: ~17/12 = 600.3301 ¢, ~5/4 = 380.6797 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~5/4 = 380.4704 ¢

Optimal ET sequence: 22, 60, 82

Badness (Sintel): 1.90

Telepathy

Subgroup: 2.3.5.7.11

Comma list: 55/54, 99/98, 176/175

Mapping: [⟨1 0 2 -1 -1], ⟨0 5 1 12 14]]

Optimal tunings:

  • WE: ~2 = 1200.7724 ¢, ~5/4 = 381.2641 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 381.0913 ¢

Optimal ET sequence: 19e, 22, 41e, 63e

Badness (Sintel): 0.896

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 65/64, 91/90, 99/98

Mapping: [⟨1 0 2 -1 -1 4], ⟨0 5 1 12 14 -1]]

Optimal tunings:

  • WE: ~2 = 1202.5634 ¢, ~5/4 = 381.3348 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.6886 ¢

Optimal ET sequence: 19e, 22, 41ef

Badness (Sintel): 1.05

Intuition

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 66/65, 99/98, 105/104

Mapping: [⟨1 0 2 -1 -1 -2], ⟨0 5 1 12 14 18]]

Optimal tunings:

  • WE: ~2 = 1201.3172 ¢, ~5/4 = 380.9004 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.5942 ¢

Optimal ET sequence: 19e, 22f

Badness (Sintel): 1.08

Horcrux

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 2 -1 0], ⟨0 5 1 12 11]]

Optimal tunings:

  • WE: ~2 = 1200.4670 ¢, ~5/4 = 379.7895 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 379.6889 ¢

Optimal ET sequence: 3de, 16d, 19

Badness (Sintel): 1.30

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 2 -1 0 -2], ⟨0 5 1 12 11 18]]

Optimal tunings:

  • WE: ~2 = 1200.2953 ¢, ~5/4 = 379.8842 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 379.8165 ¢

Optimal ET sequence: 3def, 16dff, 19

Badness (Sintel): 1.32

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 0 2 -1 0 -2 0], ⟨0 5 1 12 11 18 16]]

Optimal tunings:

  • WE: ~2 = 1200.2484 ¢, ~5/4 = 380.2053 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.1482 ¢

Optimal ET sequence: 3defg, 16dffgg, 19g

Badness (Sintel): 1.43

Horcruxic

Subgroup: 2.3.5.7.11.13.17

Comma list: 35/34, 45/44, 52/51, 56/55, 245/243

Mapping: [⟨1 0 2 -1 0 -2 0], ⟨0 5 1 12 11 18 13]]

Optimal tunings:

  • WE: ~2 = 1199.5457 ¢, ~5/4 = 379.4681 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 379.5713 ¢

Optimal ET sequence: 3defg, 16dff, 19

Badness (Sintel): 1.51

Glamour

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 65/64, 245/243

Mapping: [⟨1 0 2 -1 0 4], ⟨0 5 1 12 11 -1]]

Optimal tunings:

  • WE: ~2 = 1202.2187 ¢, ~5/4 = 379.8171 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 379.2709 ¢

Optimal ET sequence: 3de, 16d, 19

Badness (Sintel): 1.38

Witchcraft

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/243, 441/440

Mapping: [⟨1 0 2 -1 -7], ⟨0 5 1 12 33]]

Optimal tunings:

  • WE: ~2 = 1201.2634 ¢, ~5/4 = 380.6321 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.2849 ¢

Optimal ET sequence: 19e, 41, 60e, 101cd, 243ccdde

Badness (Sintel): 1.02

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 245/243, 275/273

Mapping: [⟨1 0 2 -1 -7 -2], ⟨0 5 1 12 33 18]]

Optimal tunings:

  • WE: ~2 = 1201.0424 ¢, ~5/4 = 380.5193 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.2349 ¢

Optimal ET sequence: 19e, 41, 60e, 101cd

Badness (Sintel): 0.973

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 154/153, 170/169, 196/195, 245/243

Mapping: [⟨1 0 2 -1 -7 -2 -7], ⟨0 5 1 12 33 18 35]]

Optimal tunings:

  • WE: ~2 = 1201.1638 ¢, ~5/4 = 380.4827 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 380.1599 ¢

Optimal ET sequence: 19eg, 41, 60e, 101cd

Badness (Sintel): 1.06

Divination

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 245/243

Mapping: [⟨2 0 4 -2 5], ⟨0 5 1 12 3]]

Optimal tunings:

  • WE: ~99/70 = 600.8306 ¢, ~5/4 = 380.7598 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~5/4 = 380.3800 ¢

Optimal ET sequence: 22, 38d, 60e, 142cdee, 202ccddeee

Badness (Sintel): 1.19

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 0 4 -2 5 -4], ⟨0 5 1 12 3 18]]

Optimal tunings:

  • WE: ~99/70 = 600.9624 ¢, ~5/4 = 380.5297 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~5/4 = 380.0614 ¢

Optimal ET sequence: 22f, 38df, 60e

Badness (Sintel): 1.43

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 121/120, 154/153, 196/195, 245/243

Mapping: [⟨2 0 4 -2 5 -4 5], ⟨0 5 1 12 3 18 5]]

Optimal tunings:

  • WE: ~17/12 = 600.8921 ¢, ~5/4 = 380.5094 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~5/4 = 380.0672 ¢

Optimal ET sequence: 22f, 38df, 60e

Badness (Sintel): 1.21

Hocus

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 245/242

Mapping: [⟨1 -5 1 -13 -13], ⟨0 10 2 24 25]]

Optimal tunings:

  • WE: ~2 = 1201.0749 ¢, ~11/7 = 790.7980 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 790.1429 ¢

Optimal ET sequence: 38d, 41, 120cd

Badness (Sintel): 1.27

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 243/242, 245/242

Mapping: [⟨1 -5 1 -13 -13 -20], ⟨0 10 2 24 25 36]]

Optimal tunings:

  • WE: ~2 = 1201.2830 ¢, ~11/7 = 790.8409 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 790.0516 ¢

Optimal ET sequence: 38df, 41, 79d, 120cd

Badness (Sintel): 1.25

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 154/153, 196/195, 243/242, 245/242

Mapping: [⟨1 -5 1 -13 -13 -20 -15], ⟨0 10 2 24 25 36 29]]

Optimal tunings:

  • WE: ~2 = 1201.1557 ¢, ~11/7 = 790.7157 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 790.0057 ¢

Optimal ET sequence: 38df, 41, 79d

Badness (Sintel): 1.30

19-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 154/153, 196/195, 243/242, 245/242

Mapping: [⟨1 -5 1 -13 -13 -20 -3], ⟨0 10 2 24 25 36 29 11]]

Optimal tunings:

  • WE: ~2 = 1201.3558 ¢, ~11/7 = 790.8266 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 789.9880 ¢

Optimal ET sequence: 38df, 41, 79dh

Badness (Sintel): 1.23

Muggles

Main article: Muggles

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

Subgroup: 2.3.5.7

Comma list: 126/125, 525/512

Mapping: [⟨1 0 2 5], ⟨0 5 1 -7]]

Optimal tunings:

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

Tuning ranges:

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

Badness (Sintel): 1.42

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 126/125, 385/384

Mapping: [⟨1 0 2 5 0], ⟨0 5 1 -7 11]]

Optimal tunings:

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

Tuning ranges:

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

Optimal ET sequence: 16, 19, 35, 54bd

Badness (Sintel): 1.59

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 2 5 0 4], ⟨0 5 1 -7 11 -1]]

Optimal tunings:

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

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

Badness (Sintel): 1.26

Muggloid

Subgroup: 2.3.5.7.11

Comma list: 33/32, 126/125, 176/175

Mapping: [⟨1 0 2 5 5], ⟨0 5 1 -7 -5]]

Optimal tunings:

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

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

Badness (Sintel): 1.55

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 33/32, 65/64, 105/104, 126/125

Mapping: [⟨1 0 2 5 5 4], ⟨0 5 1 -7 -5 -1]]

Optimal tunings:

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

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

Badness (Sintel): 1.19

Brightstone

Brightstone tempers out 64/63 and may be described as 22 & 25. 22edo itself is a good tuning, in which case it is identical to magic. Brightstone can be extended to the 11- and 13-limit in a similar way to muggles.

Subgroup: 2.3.5.7

Comma list: 64/63, 3125/3024

Mapping: [⟨1 0 2 6], ⟨0 5 1 -10]]

Optimal tunings:

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

Optimal ET sequence: 3, 19d, 22

Badness (Sintel): 2.23

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 22, 69b

Badness (Sintel): 1.85

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 22f, 47bff

Badness (Sintel): 1.99

Darkstone

Darkstone (16 & 19d) is a low-accuracy temperament which tempers out 36/35 and 1875/1792. It makes the major third and the fifth even flatter than those of muggles.

This temperament is known as witch in Tonalsoft Encyclopedia.

Subgroup: 2.3.5.7

Comma list: 36/35, 1875/1792

Mapping: [⟨1 0 2 0], ⟨0 5 1 9]]

Optimal tunings:

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

Optimal ET sequence: 3d, …, 13b, 16

Badness (Sintel): 2.13

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 363/343

Mapping: [⟨1 0 2 0 0], ⟨0 5 1 9 11]]

Optimal tunings:

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

Optimal ET sequence: 3de, 13be, 16

Badness (Sintel): 1.55

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 45/44, 363/343

Mapping: [⟨1 0 2 0 0 -1], ⟨0 5 1 9 11 15]]

Optimal tunings:

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

Optimal ET sequence: 3def, 13beff, 16

Badness (Sintel): 1.58

Scales: User:BudjarnLambeth/Volcanic glass

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

Hocum

Subgroup: 2.3.5.7

Comma list: 3125/3072, 4000/3969

Mapping: [⟨1 -5 1 14], ⟨0 10 2 -17]]

mapping generators: ~2, ~63/40

Optimal tunings:

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

Optimal ET sequence: 3, 38, 41, 161c

Badness (Sintel): 2.71

Trismegistus

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

Subgroup: 2.3.5.7

Comma list: 1029/1024, 3125/3072

Mapping: [⟨1 -5 1 5], ⟨0 15 3 -5]]

mapping generators: ~2, ~168/125

Optimal tunings:

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

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

Badness (Sintel): 2.49

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 16, 25e, 41, 98c

Badness (Sintel): 1.51

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 16, 25e, 41, 98c

Badness (Sintel): 1.37

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 105/104, 144/143, 441/440, 210/209, 625/616

Subgroup-val mapping: [⟨1 -5 1 5 -4 -2 6], ⟨0 15 3 -5 17 13 -4]]

Optimal tunings:

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

Optimal ET sequence: 16, 25e, 41, 98c

Badness (Sintel): 1.26

Quadrimage

Subgroup: 2.3.5.7

Comma list: 2401/2400, 3125/3072

Mapping: [⟨1 -15 -1 -3], ⟨0 20 4 7]]

mapping generators: ~2, ~25/14

Optimal tunings:

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

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

Badness (Sintel): 3.22

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 6, 35, 41

Badness (Sintel): 2.04

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.82

Quinmage

Subgroup: 2.3.5.7

Comma list: 3125/3072, 16875/16807

Mapping: [⟨1 -10 0 -6], ⟨0 25 5 19]]

mapping generators: ~2, ~48/35

Optimal tunings:

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

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

Badness (Sintel): 4.92

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 625/616, 2401/2376

Mapping: [⟨1 -10 0 -6 3], ⟨0 25 5 19 1]]

Optimal tunings:

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

Optimal ET sequence: 13b, 28b, 41

Badness (Sintel): 3.36

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 364/363, 385/384, 625/616

Mapping: [⟨1 -10 0 -6 3 0], ⟨0 25 5 19 1 8]]

Optimal tunings:

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

Optimal ET sequence: 13b, 28b, 41

Badness (Sintel): 2.80

Warlock

Subgroup: 2.3.5.7

Comma list: 3125/3072, 16807/16384

Mapping: [⟨5 0 10 14], ⟨0 5 1 0]]

mapping generators: ~8/7, ~5/4

Optimal tunings:

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

Optimal ET sequence: 25, 35, 60

Badness (Sintel): 7.27

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