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__FORCETOC__
{{interwiki
<span style="display: block; text-align: right;">[[de:Magische_Temperaturen|Deutsch]]</span>
| de = Magische Temperaturen
| en = Magic family
| es =
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}}
{{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.


A magic temperament is optimal, for some searches, in the 9-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-limit interval better than is possible in [[12edo|12edo]]. Properties may depend on tuning and extension.
== Magic ==
{{Main| Magic }}


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.
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.


=Five limit magic=
[[Subgroup]]: 2.3.5
The 5-limit parent comma for the magic family is 3125/3072, the small diesis or magic comma. Its monzo is |-10 -1 5&gt;, and flipping that yields &lt;&lt;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)^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|19edo]] or [[22edo|22edo]] is always possible.


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


5-limit minimax
{{Mapping|legend=1| 1 0 2 | 0 5 1 }}


[&lt;1 0 0|, &lt;0 1 0|, &lt;2 1/5 0|]
: mapping generators: ~2, ~5/4


[[Eigenmonzo|Eigenmonzos]]: 2, 3
[[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 }}


valid range: [360.000, 400.000] (10 to 3)
[[Minimax tuning]]:  
* [[5-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


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


strict range: [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.


POTE generator: ~5/4 = 380.058
{{Optimal ET sequence|legend=1| 3, 13b, 16, 19, 41, 60, 221cc, 281cc }}


Algebraic generator: Terzbirat, the positive root of 9x^2-8x-4 = (4+2√13)/9; approximately 380.3175 [[cent|cents]].
[[Badness]] (Sintel): 0.919


Map: [&lt;1 0 2|, &lt;0 5 1|]
=== 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.


[[generator|Generators]]: 2, 5/4
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]]


[[EDO|Edos]]: [[6edo|6]], [[16edo|16]], [[19edo|19]], [[22edo|22]], [[41edo|41]], [[60edo|60]], [[221edo|221c]], [[281edo|281c]]
== Septimal magic ==
{{Main| Magic }}


==Seven limit children==
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.
The second comma of the [[Normal_lists|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.


=Magic=
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.  
(See also [[Magic|Magic]].)


Magic tempers out not only 3125/3072 and 875/864, but also 225/224, 245/243, and 10976/10935. [[41edo|41edo]] 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.
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.  


Magic, with its accurate fifths, works well with 9-limit harmony. It's more accurate than [[Meantone_family|meantone]] and simpler than [[Garibaldi_temperament|garibaldi]]. It's a little tricky to work with because in it fifths are a relatively complex interval and it doesn't naturally work with scales of around seven notes to the octave. Its wedgie is &lt;&lt;5 1 12 -10 5 25||.
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.  


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.
[[Subgroup]]: 2.3.5.7


243/242 leads to another essentially tempered 9-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. (This "octarod comma" is shared with [[Sensi|sensi]], [[Semaphore_and_Godzilla|godzilla]], [[Superpyth|superpyth]], [[Tetracot_family|octacot]], [[Gamelismic_clan|rodan]], [[Shrutar|shrutar]], [[Porcupine_family|hedgehog]], [[Clyde_node|clyde]], and [[Sensamagic_clan|bohpier]]. See [http://x31eq.com/cgi-bin/uv.cgi?uvs=245:243 temperament finder].)
[[Comma list]]: 225/224, 245/243


By adding 100/99 to the list of commas, magic can be extended to an 11-limit version, &lt;&lt;5 1 12 -8 ... ||. For this, [[104edo|104edo]] provides an excellent tuning, as it does also for the rank three temperaments tempering out 100/99 with 225/224, 245/243 or 875/864. Septimage (see below) is also an excellent 11-limit magic tuning.
{{Mapping|legend=1| 1 0 2 -1 | 0 5 1 12 }}


Commas: 225/224, 245/243
: mapping generators: ~2, ~5/4


7 and 9 limit minimax
[[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 }}


[|1 0 0 0&gt;, |0 1 0 0&gt;, |2 1/5 0 0&gt;, |-1 12/5 0 0&gt;]
[[Minimax tuning]]:
* 7- and [[9-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


[[Eigenmonzo|Eigenmonzos]]: 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)


valid range: [378.947, 381.818] (19 to 22)
[[Algebraic generator]]: Tirzbirat or Septimage, the real root of 5''x''<sup>5</sup> + 4''x'' - 20, 380.7604 cents.


nice range: [378.910, 386.314]
{{Optimal ET sequence|legend=1| 19, 41, 142cd, 183cd, 224ccd }}


strict range: [378.947, 381.818]
[[Badness]] (Sintel): 0.479


[[POTE_tuning|POTE generator]]: 380.352
=== 11-limit ===
Subgroup: 2.3.5.7.11


Algebraic generators: Tirzbirat or Septimage, the real root of 5x^5+4x-20, 380.7604 cents.
Comma list: 100/99, 225/224, 245/243


Map: [&lt;1 0 2 -1|, &lt;0 5 1 12|]
Mapping: {{mapping| 1 0 2 -1 6 | 0 5 1 12 -8 }}


[[generator|Generators]]: 2, 5/4
Optimal tunings:  
* WE: ~2 = 1200.1372{{c}}, ~5/4 = 380.7399{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7008{{c}}


EDOs: 41, 142cd, 183cd, 224cd
Minimax tuning:  
* 11-odd-limit: ~5/4 = {{monzo| 1/3 1/9 0 0 -1/18 }}
: unchanged-interval (eigenmonzo) basis: 2.11/9


==11-limit==
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)


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)/(26/25)=100/99.)
{{Optimal ET sequence|legend=0| 19, 22, 41, 104 }}


See also [[Chords_of_magic|Chords of magic]]
Badness (Sintel): 0.673


Commas: 225/224, 245/243, 100/99
==== 13-limit ====
A notable [[patent val]] tuning beyond the [[optimal patent val]] of 41edo is [[19edo|19]] + [[41edo|41]] = [[60edo]].


valid range: [378.947, 381.818] (19 to 22)
Subgroup: 2.3.5.7.11.13


nice range: [378.910, 386.314]
Comma list: 100/99, 105/104, 144/143, 196/195


strict range: [378.947, 381.818]
Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 1 12 -8 18 }}


[[POTE_tuning|POTE generator]]: 380.696
Optimal tunings:  
* WE: ~2 = 1200.0331{{c}}, ~5/4 = 380.4377{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4284{{c}}


Map: [&lt;1 0 2 -1 6|, &lt;0 5 1 12 -8|]
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]


EDOs: 19, 22, 41, 104, 145c
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}


Badness: 0.0204
Badness (Sintel): 0.889


==13-limit==
===== Magical =====
Commas: 100/99, 105/104, 144/143, 196/195
Subgroup: 2.3.5.7.11.13.17


valid range: [378.947, 381.818] (19 to 22f)
Comma list: 100/99, 105/104, 120/119, 144/143, 154/153


nice range: [378.617, 386.314]
Mapping: {{mapping| 1 0 2 -1 6 -2 6 | 0 5 1 12 -8 18 -6 }}


strict range: [378.947, 381.818]
Optimal tunings:  
* WE: ~2 = 1199.3584{{c}}, ~5/4 = 380.4006{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5896{{c}}


POTE generator: ~5/4 = 380.427
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}


Map: [&lt;1 0 2 -1 6 -2|, &lt;0 5 1 12 -8 18|]
Badness (Sintel): 1.05


EDOS: 19, 41, 265cdef
====== Magicus ======
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0215
Comma list: 100/99, 105/104, 120/119, 133/132, 144/143, 154/153


===Sorcery===
Mapping: {{mapping| 1 0 2 -1 6 -2 6 9 | 0 5 1 12 -8 18 -6 -15 }}
Commas: 65/64, 78/77, 91/90, 100/99


valid range: 378.947 (19)
Optimal tunings:  
* WE: ~2 = 1199.7173{{c}}, ~5/4 = 380.3808{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4680{{c}}


nice range: [359.472, 386.314]
{{Optimal ET sequence|legend=0| 19, 41 }}


strict range: 378.947
Badness (Sintel): 1.27


POTE generator: ~5/4 = 380.477
====== Magica ======
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 0 2 -1 6 4|, &lt;0 5 1 12 -8 -1|]
Comma list: 100/99, 105/104, 120/119, 144/143, 154/153, 171/169


EDOs: 19, 22, 31f, 41f
Mapping: {{mapping| 1 0 2 -1 6 -2 6 -4 | 0 5 1 12 -8 18 -6 26 }}


Badness: 0.0258
Optimal tunings:  
* WE: ~2 = 1199.3670{{c}}, ~5/4 = 380.4681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6541{{c}}


===Necromancy===
{{Optimal ET sequence|legend=0| 22fh, 41 }}
Commas: 100/99, 225/224, 245/243, 275/273


valid range: [380.488, 380.952] (41 to 63)
Badness (Sintel): 1.21


nice range: [378.910, 386.314]
===== Magia =====
Subgroup: 2.3.5.7.11.13.17


strict range: [380.488, 380.952]
Comma list: 100/99, 105/104, 144/143, 170/169, 196/195


POTE generator: ~5/4 = 380.787
Mapping: {{mapping| 1 0 2 -1 6 -2 -7 | 0 5 1 12 -8 18 35 }}


Map: [&lt;1 0 2 -1 6 11|, &lt;0 5 1 12 -8 -23|]
Optimal tunings:  
* WE: ~2 = 1200.1727{{c}}, ~5/4 = 380.2982{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2483{{c}}


EDOs: 19, 22, 41, 63, 104
{{Optimal ET sequence|legend=0| 19g, 41, 60 }}


Badness: 0.0253
Badness (Sintel): 1.34


==Telepathy==
====== 19-limit ======
Commas: 55/54, 99/98, 176/175
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~5/4 = 381.019
Comma list: 100/99, 105/104, 144/143, 170/169, 171/169, 196/195


Map: [&lt;1 0 2 -1 -1|, &lt;0 5 1 12 14|]
Mapping: {{mapping| 1 0 2 -1 6 -2 -7 -4 | 0 5 1 12 -8 18 35 26 }}


EDOs: 19e, 22, 41e, 63e
Optimal tunings:  
* WE: ~2 = 1200.2179{{c}}, ~5/4 = 380.3942{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.3314{{c}}


Badness: 0.0271
{{Optimal ET sequence|legend=0| 19gh, 41 }}


===13-limit telepathy===
Badness (Sintel): 1.44
Commas: 55/54, 65/64, 91/90, 99/98


POTE generator: ~5/4 = 380.520
===== Evening =====
Evening is a remarkable subgroup temperament of {{nowrap| 19 & 22f }} with prime harmonics of 29 and 31.  


Map: [&lt;1 0 2 -1 -1 4|, &lt;0 5 1 12 14 -1|]
Subgroup: 2.3.5.7.11.13.29.31


EDOs: 19e, 22, 41ef
Comma list: 100/99, 105/104, 144/143, 145/144, 155/154, 196/195


Badness: 0.0255
Subgroup-val mapping: {{mapping| 1 0 2 -1 6 -2 2 4 | 0 5 1 12 -8 18 9 3 }}


==Horcrux==
Optimal tunings:
Commas: 45/44, 56/55, 245/243
* WE: ~2 = 1200.2802{{c}}, ~5/4 = 380.5053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4258{{c}}


POTE generator: ~5/4 = 379.642
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}


Map: [&lt;1 0 2 -1 0|, &lt;0 5 1 12 11|]
Badness (Sintel): 0.807


EDOs: 19, 60e
==== Sorcery ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0393
Comma list: 65/64, 78/77, 91/90, 100/99


=Divination=
Mapping: {{mapping| 1 0 2 -1 6 4 | 0 5 1 12 -8 -1 }}
Commas: 121/120, 225/224, 245/243


POTE generator: ~5/4 = 380.233
Optimal tunings:  
* WE: ~2 = 1201.2397{{c}}, ~5/4 = 380.8698{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5080{{c}}


Map: [&lt;2 0 4 -2 5|, &lt;0 5 1 12 3|]
{{Optimal ET sequence|legend=0| 19, 22, 41f }}


EDOs: 22, 38d, 60e, 142cde
Badness (Sintel): 1.07


Badness: 0.0359
==== Necromancy ====
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 100/99, 225/224, 245/243, 275/273
Commas: 105/104, 121/120, 196/195, 245/243


POTE generator: ~5/4 = 379.920
Mapping: {{mapping| 1 0 2 -1 6 11 | 0 5 1 12 -8 -23 }}


Map: [&lt;2 0 4 -2 5 -4|, &lt;0 5 1 12 3 18|]
Optimal tunings:  
* WE: ~2 = 1199.9675{{c}}, ~5/4 = 380.7770{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7874{{c}}


EDOs: 22f, 60e
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63, 104 }}


Badness: 0.0346
Badness (Sintel): 1.04


=Soothsaying=
===== 17-limit =====
Commas: 100/99, 225/224, 245/243, 1352/1331
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~5/4 = 380.508
Comma list: 100/99, 120/119, 154/153, 225/224, 273/272


Map: [&lt;2 0 4 -2 12 15|, &lt;0 5 1 12 -8 -12|]
Mapping: {{mapping| 1 0 2 -1 6 11 6 | 0 5 1 12 -8 -23 -6 }}


EDOs: 22, 60, 82
Optimal tunings:  
* WE: ~2 = 1199.6176{{c}}, ~5/4 = 380.7053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.8280{{c}}


Badness: 0.0554
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63 }}


=Witchcraft=
Badness (Sintel): 1.12
Commas: 225/224, 245/243, 441/440


POTE generator: ~5/4 = 380.232
==== Soothsaying ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 2 -1 -7|, &lt;0 5 1 12 33|]
Comma list: 100/99, 225/224, 245/243, 1352/1331


EDOs: 41, 60e, 101cd, 243cde
Mapping: {{mapping| 2 0 4 -2 12 15 | 0 5 1 12 -8 -12 }}


Badness: 0.0307
Optimal tunings:  
* WE: ~55/39 = 600.2918{{c}}, ~5/4 = 380.6928{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~5/4 = 380.5121{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 22, 60, 82 }}
Commas: 105/104, 196/195, 245/243, 275/273


POTE generator: ~5/4 = 380.189
Badness (Sintel): 2.29


Map: [&lt;1 0 2 -1 -7 -2|, &lt;0 5 1 12 33 18|]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


EDOs: 41, 60e, 101cd
Comma list: 100/99, 221/220, 225/224, 245/243, 273/272


Badness: 0.0235
Mapping: {{mapping| 2 0 4 -2 12 15 5 | 0 5 1 12 -8 -12 5 }}


=Muggles=
Optimal tunings:
Aside from 3125/3072 and 525/512 muggles also tempers out 126/125 and 1323/1280. A good muggles tuning is [[19edo|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. The muggles wedgie is &lt;&lt;5 1 -7 -10 -25 -19||.
* WE: ~17/12 = 600.2918{{c}}, ~5/4 = 380.6927{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.5135{{c}}


Commas: 126/125, 525/512
{{Optimal ET sequence|legend=0| 22, 60, 82 }}


[[POTE_tuning|POTE generator]]: ~5/4 = 378.479
Badness (Sintel): 1.82


Map: [&lt;1 0 2 5|, &lt;0 5 1 -7|]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


EDOs: 19, 73bcd, 92bcd
Comma list: 100/99, 133/132, 221/220, 225/224, 245/243, 273/272


Badness: 0.0562
Mapping: {{mapping| 2 0 4 -2 12 15 5 18 | 0 5 1 12 -8 -12 5 -15 }}


==11-limit==
Optimal tunings:
Commas: 45/44, 126/125, 385/384
* WE: ~17/12 = 600.3301{{c}}, ~5/4 = 380.6797{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.4704{{c}}


[[POTE_tuning|POTE generator]]: ~5/4 = 377.724
{{Optimal ET sequence|legend=0| 22, 60, 82 }}


Map: [&lt;1 0 2 5 0|, &lt;0 5 1 -7 11|]
Badness (Sintel): 1.90


EDOs: 16, 19, 35, 54bd
=== Telepathy ===
Subgroup: 2.3.5.7.11


Badness: 0.0480
Comma list: 55/54, 99/98, 176/175


==13-limit==
Mapping: {{mapping| 1 0 2 -1 -1 | 0 5 1 12 14 }}
Commas: 45/44, 65/64, 78/77, 126/125


[[POTE_tuning|POTE generator]]: ~5/4 = 377.724
Optimal tunings:  
* WE: ~2 = 1200.7724{{c}}, ~5/4 = 381.2641{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 381.0913{{c}}


Map: [&lt;1 0 2 5 0 4|, &lt;0 5 1 -7 11 -1|]
{{Optimal ET sequence|legend=0| 19e, 22, 41e, 63e }}


EDOs: 16, 19, 35f, 54bdf
Badness (Sintel): 0.896


Badness: 0.0309
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=Astrology=
Comma list: 55/54, 65/64, 91/90, 99/98
Commas: 50/49, 3125/3072


POTE generator: ~5/4 = 380.578
Mapping: {{mapping| 1 0 2 -1 -1 4 | 0 5 1 12 14 -1 }}


Map: [&lt;2 0 4 5|, &lt;0 5 1 1|]
Optimal tunings:  
* WE: ~2 = 1202.5634{{c}}, ~5/4 = 381.3348{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6886{{c}}


Wedgie: &lt;&lt;10 2 2 -20 -25 -1||
{{Optimal ET sequence|legend=0| 19e, 22, 41ef }}


EDOs: 6, 16, 22, 60d, 82d
Badness (Sintel): 1.05


Badness: 0.0827
==== Intuition ====
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 55/54, 66/65, 99/98, 105/104
Commas: 50/49, 121/120, 176/175


POTE generator: ~5/4 = 380.530
Mapping: {{mapping| 1 0 2 -1 -1 -2 | 0 5 1 12 14 18 }}


Map: [&lt;2 0 4 5 5|, &lt;0 5 1 1 3|]
Optimal tunings:  
* WE: ~2 = 1201.3172{{c}}, ~5/4 = 380.9004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5942{{c}}


EDOs: 6, 16, 22, 60de, 82de
{{Optimal ET sequence|legend=0| 19e, 22f }}


Badness: 0.0392
Badness (Sintel): 1.08


==13-limit==
=== Horcrux ===
Commas: 50/49, 65/64, 78/77, 121/120
Subgroup: 2.3.5.7.11


POTE generator: ~5/4 = 379.787
Comma list: 45/44, 56/55, 245/243


Map: [&lt;2 0 4 5 5 8|, &lt;0 5 1 1 3 -1|]
Mapping: {{mapping| 1 0 2 -1 0 | 0 5 1 12 11 }}


EDOs: 6, 16, 22, 38f
Optimal tunings:  
* WE: ~2 = 1200.4670{{c}}, ~5/4 = 379.7895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.6889{{c}}


Badness: 0.0344
{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}


[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]
Badness (Sintel): 1.30


==Horoscope==
==== 13-limit ====
Commas: 50/49, 66/65, 105/104, 121/120
Subgroup: 2.3.5.7.11.13


POTE generator: ~5/4 = 379.837
Comma list: 45/44, 56/55, 78/77, 245/243


Map: [&lt;2 0 4 5 5 3|, &lt;0 5 1 1 3 7|]
Mapping: {{mapping| 1 0 2 -1 0 -2 | 0 5 1 12 11 18 }}


EDOs: 16, 22f, 38
Optimal tunings:  
* WE: ~2 = 1200.2953{{c}}, ~5/4 = 379.8842{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.8165{{c}}


Badness: 0.0353
{{Optimal ET sequence|legend=0| 3def, 16dff, 19 }}


=Spell=
Badness (Sintel): 1.32
Commas: 49/48, 3125/3072


POTE generator: ~28/25 = 189.927
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 0 2 2|, &lt;0 10 2 5|]
Comma list: 45/44, 56/55, 78/77, 85/84, 245/243


Wedgie: &lt;&lt;10 2 5 -20 -20 6||
Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 16 }}


EDOs: 6, 19, 82d
Optimal tunings:  
* WE: ~2 = 1200.2484{{c}}, ~5/4 = 380.2053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1482{{c}}


Badness: 0.0810
{{Optimal ET sequence|legend=0| 3defg, 16dffgg, 19g }}


==11-limit==
Badness (Sintel): 1.43
Commas: 49/48, 56/55, 125/121


POTE generator: ~11/10 = 190.285
===== Horcruxic =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 0 2 2 3|, &lt;0 10 2 5 3|]
Comma list: 35/34, 45/44, 52/51, 56/55, 245/243


EDOs: 6, 19, 44de, 63de, 82de
Mapping: {{mapping| 1 0 2 -1 0 -2 0 | 0 5 1 12 11 18 13 }}


Badness: 0.0598
Optimal tunings:  
* WE: ~2 = 1199.5457{{c}}, ~5/4 = 379.4681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.5713{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 3defg, 16dff, 19 }}
Commas: 49/48, 56/55, 78/77, 125/121


POTE generator: ~11/10 = 189.928
Badness (Sintel): 1.51


Map: [&lt;1 0 2 2 3 4|, &lt;0 10 2 5 3 -2|]
==== Glamour ====
Subgroup: 2.3.5.7.11.13


EDOs: 6, 19, 82def
Comma list: 45/44, 56/55, 65/64, 245/243


Badness: 0.0456
Mapping: {{mapping| 1 0 2 -1 0 4 | 0 5 1 12 11 -1 }}


==Cantrip==
Optimal tunings:
Commas: 49/48, 56/55, 91/90, 125/121
* WE: ~2 = 1202.2187{{c}}, ~5/4 = 379.8171{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.2709{{c}}


POTE generator: ~11/10 = 190.360
{{Optimal ET sequence|legend=0| 3de, 16d, 19 }}


Map: [&lt;1 0 2 2 3 1|, &lt;0 10 2 5 3 17|]
Badness (Sintel): 1.38


EDOs: 19, 44de, 63de, 82de
=== Witchcraft ===
Subgroup: 2.3.5.7.11


Badness: 0.0416
Comma list: 225/224, 245/243, 441/440


=Hocum=
Mapping: {{mapping| 1 0 2 -1 -7 | 0 5 1 12 33 }}
Commas: 3125/3072, 4000/3969


POTE generator: ~63/50 = 400.108
Optimal tunings:  
* WE: ~2 = 1201.2634{{c}}, ~5/4 = 380.6321{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2849{{c}}


Map: [&lt;1 5 3 -3|, &lt;0 -10 -2 17|]
{{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd, 243ccdde }}


Wedgie: &lt;&lt;10 2 -17 -20 -55 -45||
Badness (Sintel): 1.02


EDOs: 38, 41, 161c, 202c, 243c, 284c
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.1071
Comma list: 105/104, 196/195, 245/243, 275/273


=Hocus=
Mapping: {{mapping| 1 0 2 -1 -7 -2 | 0 5 1 12 33 18 }}
Commas: 225/224, 243/242, 245/242


POTE generator: ~14/11 = 409.910
Optimal tunings:  
* WE: ~2 = 1201.0424{{c}}, ~5/4 = 380.5193{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2349{{c}}


Map: [&lt;1 5 3 11 12|, &lt;0 -10 -2 -24 -25|]
{{Optimal ET sequence|legend=0| 19e, 41, 60e, 101cd }}


EDOs: 38d, 41, 120cd, 161cd, 202cd
Badness (Sintel): 0.973


Badness: 0.0385
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


==13-limit==
Comma list: 105/104, 154/153, 170/169, 196/195, 245/243
Commas: 105/104, 196/195, 243/242, 245/242


POTE generator: ~14/11 = 410.004
Mapping: {{mapping| 1 0 2 -1 -7 -2 -7 | 0 5 1 12 33 18 35 }}


Map: [&lt;1 5 3 11 12 16|, &lt;0 -10 -2 -24 -25 -36|]
Optimal tunings:  
* WE: ~2 = 1201.1638{{c}}, ~5/4 = 380.4827{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1599{{c}}


EDOs: 41, 79d, 120cd
{{Optimal ET sequence|legend=0| 19eg, 41, 60e, 101cd }}


Badness: 0.0303
Badness (Sintel): 1.06


=Trismegistus=
=== Divination ===
Commas: 3125/3072, 1029/1024
Subgroup: 2.3.5.7.11


POTE generator: ~147/100 = 673.290
Comma list: 121/120, 225/224, 245/243


Map: [&lt;1 10 4 0|, &lt;0 -15 -3 5|]
Mapping: {{mapping| 2 0 4 -2 5 | 0 5 1 12 3 }}


Wedgie: &lt;&lt;15 3 -5 -30 -50 -20||
Optimal tunings:  
* WE: ~99/70 = 600.8306{{c}}, ~5/4 = 380.7598{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~5/4 = 380.3800{{c}}


EDOs: 16, 25, 41, 139c, 180c, 221c, 262c
{{Optimal ET sequence|legend=0| 22, 38d, 60e, 142cdee, 202ccddeee }}


Badness: 0.0983
Badness (Sintel): 1.19


==11-limit==
==== 13-limit ====
Commas: 385/384, 441/440, 625/616
Subgroup: 2.3.5.7.11.13


POTE generator: ~22/15 = 673.340
Comma list: 105/104, 121/120, 196/195, 245/243


Map: [&lt;1 10 4 0 13|, &lt;0 -15 -3 5 -17|]
Mapping: {{mapping| 2 0 4 -2 5 -4 | 0 5 1 12 3 18 }}


EDOs: 16, 25e, 41, 98c, 139c, 180c
Optimal tunings:  
* WE: ~99/70 = 600.9624{{c}}, ~5/4 = 380.5297{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~5/4 = 380.0614{{c}}


Badness: 0.0456
{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}


==13-limit==
Badness (Sintel): 1.43
Commas: 105/104, 144/143, 275/273, 625/616


POTE generator: ~22/15 = 673.359
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 10 4 0 13 11|, &lt;0 -15 -3 5 -17 -13|]
Comma list: 105/104, 121/120, 154/153, 196/195, 245/243


EDOs: 16, 25e, 41, 98c, 139cf
Mapping: {{mapping| 2 0 4 -2 5 -4 5 | 0 5 1 12 3 18 5 }}


Badness: 0.0331
Optimal tunings:  
* WE: ~17/12 = 600.8921{{c}}, ~5/4 = 380.5094{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.0672{{c}}


=Quadrimage=
{{Optimal ET sequence|legend=0| 22f, 38df, 60e }}
Commas: 2401/2400, 3125/3072


POTE generator: ~28/25 = 204.987
Badness (Sintel): 1.21


Map: [&lt;1 5 3 4|, &lt;0 -20 -4 -7|]
=== Hocus ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;20 4 7 -40 -45 5||
Comma list: 225/224, 243/242, 245/242


EDOs: 6, 35, 41, 158cd, 199cd, 240cd, 281cd
Mapping: {{mapping| 1 -5 1 -13 -13 | 0 10 2 24 25 }}


Badness: 0.1274
Optimal tunings:  
* WE: ~2 = 1201.0749{{c}}, ~11/7 = 790.7980{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.1429{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 38d, 41, 120cd }}
Commas: 245/242, 385/384, 625/616


POTE generator: ~28/25 = 204.956
Badness (Sintel): 1.27


Map: [&lt;1 5 3 4 5|, &lt;0 -20 -4 -7 -9|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 6, 35, 41, 199cde, 240cde, 281cde
Comma list: 105/104, 196/195, 243/242, 245/242


Badness: 0.0616
Mapping: {{mapping| 1 -5 1 -13 -13 -20 | 0 10 2 24 25 36 }}


==13-limit==
Optimal tunings:
Commas: 105/104, 144/143, 245/242, 625/616
* WE: ~2 = 1201.2830{{c}}, ~11/7 = 790.8409{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.0516{{c}}


POTE generator: ~28/25 = 205.028
{{Optimal ET sequence|legend=0| 38df, 41, 79d, 120cd }}


Map: [&lt;1 5 3 4 5 9|, &lt;0 -20 -4 -7 -9 -31|]
Badness (Sintel): 1.25


EDOs: 41, 117c, 158cd, 199cdef
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0440
Comma list: 105/104, 154/153, 196/195, 243/242, 245/242
[[Category:family]]
 
[[Category:magic]]
Mapping: {{mapping| 1 -5 1 -13 -13 -20 -15 | 0 10 2 24 25 36 29 }}
[[Category:overview]]
 
[[Category:theory]]
Optimal tunings:
* WE: ~2 = 1201.1557{{c}}, ~11/7 = 790.7157{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 790.0057{{c}}
 
{{Optimal ET sequence|legend=0| 38df, 41, 79d }}
 
Badness (Sintel): 1.30
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 154/153, 196/195, 243/242, 245/242
 
Mapping: {{mapping| 1 -5 1 -13 -13 -20 -3 | 0 10 2 24 25 36 29 11 }}
 
Optimal tunings:
* WE: ~2 = 1201.3558{{c}}, ~11/7 = 790.8266{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 789.9880{{c}}
 
{{Optimal ET sequence|legend=0| 38df, 41, 79dh }}
 
Badness (Sintel): 1.23
 
== Muggles ==
{{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 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|legend=1| 1 0 2 5 | 0 5 1 -7 }}
 
[[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 }}
 
[[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]
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 0 2 5 0 | 0 5 1 -7 11 }}
 
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]
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 0 2 5 0 4 | 0 5 1 -7 11 -1 }}
 
Optimal tunings:
* WE: ~2 = 1203.4291{{c}}, ~5/4 = 378.7321{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.7336{{c}}
 
{{Optimal ET sequence|legend=0| 16, 19, 35f, 54bdf }}
 
Badness (Sintel): 1.26
 
=== Muggloid ===
Subgroup: 2.3.5.7.11
 
Comma list: 33/32, 126/125, 176/175
 
Mapping: {{mapping| 1 0 2 5 5 | 0 5 1 -7 -5 }}
 
Optimal tunings:
* WE: ~2 = 1205.6044{{c}}, ~5/4 = 379.5966{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8142{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 0 2 5 5 4 | 0 5 1 -7 -5 -1 }}
 
Optimal tunings:
* WE: ~2 = 1205.4897{{c}}, ~5/4 = 379.5667{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 377.8118{{c}}
 
{{Optimal ET sequence|legend=0| 3, 16, 19e, 35eef }}
 
Badness (Sintel): 1.19
 
== Brightstone ==
Brightstone tempers out 64/63 and may be described as {{nowrap| 22 & 25 }}. 22edo itself is a good tuning, in which case it is identical to magic. Brightstone can be extended to the 11- and 13-limit in a similar way to muggles.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 64/63, 3125/3024
 
{{Mapping|legend=1| 1 0 2 6 | 0 5 1 -10 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.1701{{c}}, ~5/4 = 381.3741{{c}}
: error map: {{val| -1.830 +4.915 -8.599 +6.454 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 381.9562{{c}}
: error map: {{val| 0.000 +7.826 -4.358 +11.613 }}
 
{{Optimal ET sequence|legend=1| 3, 19d, 22 }}
 
[[Badness]] (Sintel): 2.23
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 55/54, 64/63, 625/616
 
Mapping: {{mapping| 1 0 2 6 -1 | 0 5 1 -10 14 }}
 
Optimal tunings:
* WE: ~2 = 1198.5372{{c}}, ~5/4 = 381.7556{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.1943{{c}}
 
{{Optimal ET sequence|legend=0| 22, 69b }}
 
Badness (Sintel): 1.85
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 55/54, 64/63, 625/616
 
Mapping: {{mapping| 1 0 2 6 6 4 | 0 5 1 -10 14 -4 }}
 
Optimal tunings:
* WE: ~2 = 1197.2300{{c}}, ~5/4 = 381.6164{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 382.4690{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 47bff }}
 
Badness (Sintel): 1.99
 
== Darkstone ==
Darkstone (16 & 19d) is a low-accuracy temperament which tempers out 36/35 and 1875/1792. It makes the major third and the fifth even flatter than those of muggles.
 
This temperament is known as ''witch'' in [http://www.tonalsoft.com/enc/m/magic.aspx Tonalsoft Encyclopedia].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 36/35, 1875/1792
 
{{Mapping|legend=1| 1 0 2 0 | 0 5 1 9 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.7458{{c}}, ~5/4 = 377.2996{{c}}
: [[error map]]: {{val| +1.746 -15.457 -5.523 +26.870 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 376.9630{{c}}
: error map: {{val| 0.000 -17.140 -9.351 +23.841 }}
 
{{Optimal ET sequence|legend=1| 3d, …, 13b, 16 }}
 
[[Badness]] (Sintel): 2.13
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 45/44, 363/343
 
Mapping: {{mapping| 1 0 2 0 0 | 0 5 1 9 11 }}
 
Optimal tunings:
* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.9735{{c}}
 
{{Optimal ET sequence|legend=0| 3de, 13be, 16 }}
 
Badness (Sintel): 1.55
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 27/26, 36/35, 45/44, 363/343
 
Mapping: {{mapping| 1 0 2 0 0 -1 | 0 5 1 9 11 15 }}
 
Optimal tunings:
* WE: ~2 = 1201.7428{{c}}, ~5/4 = 377.3134{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 376.4221{{c}}
 
{{Optimal ET sequence|legend=0| 3def, 13beff, 16 }}
 
Badness (Sintel): 1.58
 
Scales: [[User:BudjarnLambeth/Volcanic glass]]
 
Music: [https://www.youtube.com/watch?v=oXy_RXrBVWA ''Rain in the crystal mirror caves''] - [[Budjarn Lambeth]] (2026)
 
== Hocum ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 3125/3072, 4000/3969
 
{{Mapping|legend=1| 1 -5 1 14 | 0 10 2 -17 }}
 
: mapping generators: ~2, ~63/40
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8375{{c}}, ~63/40 = 790.7032{{c}}
: [[error map]]: {{val| +0.838 +0.890 -4.070 +0.944 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~63/40 = 790.1542{{c}}
: error map: {{val| 0.000 -0.413 -6.005 -1.447 }}
 
{{Optimal ET sequence|legend=1| 3, 38, 41, 161c }}
 
[[Badness]] (Sintel): 2.71
 
== Trismegistus ==
{{Main| Mabilic and trismegistus }}
{{See also| No-threes subgroup temperaments #Mabilic }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1029/1024, 3125/3072
 
{{Mapping|legend=1| 1 -5 1 5 | 0 15 3 -5 }}
 
: mapping generators: ~2, ~168/125
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.0799{{c}}, ~168/125 = 527.1841{{c}}
: [[error map]]: {{val| +1.080 +0.408 -3.681 +0.653 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~168/125 = 526.7349{{c}}
: error map: {{val| 0.000 -0.932 -6.109 -2.500 }}
 
{{Optimal ET sequence|legend=1| 16, 25, 41, 139c, 180c, 221cc, 262ccd }}
 
[[Badness]] (Sintel): 2.49
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 441/440, 625/616
 
Mapping: {{mapping| 1 -5 1 5 -4 | 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 ET sequence|legend=0| 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: {{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
 
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 ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 3125/3072
 
{{Mapping|legend=1| 1 -15 -1 -3 | 0 20 4 7 }}
 
: mapping generators: ~2, ~25/14
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2708{{c}}, ~25/14 = 996.0669{{c}}
: [[error map]]: {{val| +1.271 +0.322 -3.317 -0.170 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/14 = 995.0515{{c}}
: error map: {{val| 0.000 -0.926 -6.108 -3.466 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 -15 -1 -3 -4 | 0 20 4 7 9 }}
 
Optimal tunings:
* WE: ~2 = 1200.6716{{c}}, ~25/14 = 995.6009{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 995.0633{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{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
 
[[Optimal tuning]]s:
* [[WE]]: ~8/7 = 240.3877{{c}}, ~5/4 = 380.4267{{c}} (~256/245 = 100.3488{{c}})
: error map: {{val| +1.939 +0.178 -2.010 -3.398 }}
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~5/4 = 379.9965{{c}} (~256/245 = 100.0035{{c}})
: error map: {{val| 0.000 -1.972 -6.317 -8.826 }}
 
{{Optimal ET sequence|legend=1| 25, 35, 60 }}
 
[[Badness]] (Sintel): 7.27
 
[[Category:Temperament families]]
[[Category:Magic family| ]] <!-- main article -->
[[Category:Magic| ]] <!-- key article -->
[[Category:Rank 2]]
[[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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