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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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| en = Magic family
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<h4>Original Wikitext content:</h4>
{{Technical data page}}
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc]]
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.
&lt;span style="display: block; text-align: right;"&gt;Other languages: [[xenharmonie/Magische Temperaturen|Deutsch]]
&lt;/span&gt;
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]]. Properties may depend on tuning and extension.


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


=Five limit magic=
The [[generator]] of magic is a major third, and to get to the interval class of fifths requires five of these. In fact, (5/4)<sup>5</sup> = 3 × 3125/3072. [[41edo|13\41]] is a highly recommendable generator, though [[60edo|19\60]], the [[optimal patent val]] generator, also makes a lot of sense, and using [[19edo]] or [[22edo]] is always possible.
The 5-limit parent comma for the magic family is 3125/3072, the small diesis or magic comma. Its monzo is |-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]] or [[22edo]] is always possible.


[[Comma]]: [[3125_3072|3125/3072]]
[[Subgroup]]: 2.3.5


5-limit minimax
[[Comma list]]: 3125/3072
[&lt;1 0 0|, &lt;0 1 0|, &lt;2 1/5 0|]
[[Eigenmonzo|Eigenmonzos]]: 2, 3


valid range: [360.000, 400.000] (10 to 3)
{{Mapping|legend=1| 1 0 2 | 0 5 1 }}
nice range: [378.910, 386.314]
strict range: [378.910, 386.314]


POTE generator: ~5/4 = 380.058
: mapping generators: ~2, ~5/4


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


Map: [&lt;1 0 2|, &lt;0 5 1|]
[[Minimax tuning]]:
[[Generator|Generators]]: 2, 5/4
* [[5-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 }}
[[Edo|Edos]]: [[6edo|6]], [[16edo|16]], [[19edo|19]], [[22edo|22]], [[41edo|41]], [[60edo|60]], [[221edo|221c]], [[281edo|281c]]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


==Seven limit children==
[[Tuning ranges]]:
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.
* 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)


=Magic=
[[Algebraic generator]]: Terzbirat, the positive root of 9''x''<sup>2</sup> - 8''x'' - 4 = (4 + 2√13)/9; approximately 380.3175 [[cent]]s.
(See also [[Magic]].)
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 MOS are possible scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1.


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


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.
[[Badness]] (Sintel): 0.919


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]].)
=== 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.  


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]] 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.
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]]


Commas: 225/224, 245/243
== Septimal magic ==
{{Main| Magic }}


7 and 9 limit minimax
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.
[|1 0 0 0&gt;, |0 1 0 0&gt;, |2 1/5 0 0&gt;, |-1 12/5 0 0&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 3


valid range: [378.947, 381.818] (19 to 22)
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.  
nice range: [378.910, 386.314]
strict range: [378.947, 381.818]


[[POTE tuning|POTE generator]]: 380.352
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.  


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


Map: [&lt;1 0 2 -1|, &lt;0 5 1 12|]
[[Subgroup]]: 2.3.5.7
[[Generator|Generators]]: 2, 5/4


EDOs: 41, 142cd, 183cd, 224cd
[[Comma list]]: 225/224, 245/243


==11-limit==
{{Mapping|legend=1| 1 0 2 -1 | 0 5 1 12 }}


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.)
: mapping generators: ~2, ~5/4


See also [[Chords of magic]]
[[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 }}


Commas: 225/224, 245/243, 100/99
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


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


[[POTE tuning|POTE generator]]: 380.696
[[Algebraic generator]]: Tirzbirat or Septimage, the real root of 5''x''<sup>5</sup> + 4''x'' - 20, 380.7604 cents.


Map: [&lt;1 0 2 -1 6|, &lt;0 5 1 12 -8|]
{{Optimal ET sequence|legend=1| 19, 41, 142cd, 183cd, 224ccd }}
EDOs: 19, 22, 41, 104, 145c
Badness: 0.0204


==13-limit==
[[Badness]] (Sintel): 0.479
Commas: 100/99, 105/104, 144/143, 196/195


valid range: [378.947, 381.818] (19 to 22f)
=== 11-limit ===
nice range: [378.617, 386.314]
Subgroup: 2.3.5.7.11
strict range: [378.947, 381.818]


POTE generator: ~5/4 = 380.427
Comma list: 100/99, 225/224, 245/243


Map: [&lt;1 0 2 -1 6 -2|, &lt;0 5 1 12 -8 18|]
Mapping: {{mapping| 1 0 2 -1 6 | 0 5 1 12 -8 }}
EDOS: 19, 41, 265cdef
Badness: 0.0215


===Sorcery===
Optimal tunings:
Commas: 65/64, 78/77, 91/90, 100/99
* WE: ~2 = 1200.1372{{c}}, ~5/4 = 380.7399{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7008{{c}}


valid range: 378.947 (19)
Minimax tuning:  
nice range: [359.472, 386.314]
* 11-odd-limit: ~5/4 = {{monzo| 1/3 1/9 0 0 -1/18 }}
strict range: 378.947
: unchanged-interval (eigenmonzo) basis: 2.11/9


POTE generator: ~5/4 = 380.477
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)


Map: [&lt;1 0 2 -1 6 4|, &lt;0 5 1 12 -8 -1|]
{{Optimal ET sequence|legend=0| 19, 22, 41, 104 }}
EDOs: 19, 22, 31f, 41f
Badness: 0.0258


===Necromancy===
Badness (Sintel): 0.673
Commas: 100/99, 225/224, 245/243, 275/273


POTE generator: ~5/4 = 380.787
==== 13-limit ====
A notable [[patent val]] tuning beyond the [[optimal patent val]] of 41edo is [[19edo|19]] + [[41edo|41]] = [[60edo]].


Map: [&lt;1 0 2 -1 6 11|, &lt;0 5 1 12 -8 -23|]
Subgroup: 2.3.5.7.11.13
EDOs: 19, 22, 41, 63, 104
Badness: 0.0253


==Telepathy==
Comma list: 100/99, 105/104, 144/143, 196/195
Commas: 55/54, 99/98, 176/175


POTE generator: ~5/4 = 381.019
Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 1 12 -8 18 }}


Map: [&lt;1 0 2 -1 -1|, &lt;0 5 1 12 14|]
Optimal tunings:  
EDOs: 19e, 22, 41e, 63e
* WE: ~2 = 1200.0331{{c}}, ~5/4 = 380.4377{{c}}
Badness: 0.0271
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4284{{c}}


===13-limit telepathy===  
Tuning ranges:
Commas: 55/54, 65/64, 91/90, 99/98
* 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]


POTE generator: ~5/4 = 380.520
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}


Map: [&lt;1 0 2 -1 -1 4|, &lt;0 5 1 12 14 -1|]
Badness (Sintel): 0.889
EDOs: 19e, 22, 41ef
Badness: 0.0255


==Horcrux==  
===== Magical =====
Commas: 45/44, 56/55, 245/243
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~5/4 = 379.642
Comma list: 100/99, 105/104, 120/119, 144/143, 154/153


Map: [&lt;1 0 2 -1 0|, &lt;0 5 1 12 11|]
Mapping: {{mapping| 1 0 2 -1 6 -2 6 | 0 5 1 12 -8 18 -6 }}
EDOs: 19, 60e
Badness: 0.0393


=Divination=  
Optimal tunings:
Commas: 121/120, 225/224, 245/243
* WE: ~2 = 1199.3584{{c}}, ~5/4 = 380.4006{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5896{{c}}


POTE generator: ~5/4 = 380.233
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}


Map: [&lt;2 0 4 -2 5|, &lt;0 5 1 12 3|]
Badness (Sintel): 1.05
EDOs: 22, 38d, 60e, 142cde
Badness: 0.0359


==13-limit==  
====== Magicus ======
Commas: 105/104, 121/120, 196/195, 245/243
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~5/4 = 379.920
Comma list: 100/99, 105/104, 120/119, 133/132, 144/143, 154/153


Map: [&lt;2 0 4 -2 5 -4|, &lt;0 5 1 12 3 18|]
Mapping: {{mapping| 1 0 2 -1 6 -2 6 9 | 0 5 1 12 -8 18 -6 -15 }}
EDOs: 22f, 60e
Badness: 0.0346


=Soothsaying=  
Optimal tunings:
Commas: 100/99, 225/224, 245/243, 1352/1331
* WE: ~2 = 1199.7173{{c}}, ~5/4 = 380.3808{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4680{{c}}


POTE generator: ~5/4 = 380.508
{{Optimal ET sequence|legend=0| 19, 41 }}


Map: [&lt;2 0 4 -2 12 15|, &lt;0 5 1 12 -8 -12|]
Badness (Sintel): 1.27
EDOs: 22, 60, 82
Badness: 0.0554


=Witchcraft=  
====== Magica ======
Commas: 225/224, 245/243, 441/440
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~5/4 = 380.232
Comma list: 100/99, 105/104, 120/119, 144/143, 154/153, 171/169


Map: [&lt;1 0 2 -1 -7|, &lt;0 5 1 12 33|]
Mapping: {{mapping| 1 0 2 -1 6 -2 6 -4 | 0 5 1 12 -8 18 -6 26 }}
EDOs: 41, 60e, 101cd, 243cde
Badness: 0.0307


==13-limit==
Optimal tunings:
Commas: 105/104, 196/195, 245/243, 275/273
* WE: ~2 = 1199.3670{{c}}, ~5/4 = 380.4681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6541{{c}}


POTE generator: ~5/4 = 380.189
{{Optimal ET sequence|legend=0| 22fh, 41 }}


Map: [&lt;1 0 2 -1 -7 -2|, &lt;0 5 1 12 33 18|]
Badness (Sintel): 1.21
EDOs: 41, 60e, 101cd
Badness: 0.0235


=Muggles=  
===== Magia =====
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. The muggles wedgie is &lt;&lt;5 1 -7 -10 -25 -19||.
Subgroup: 2.3.5.7.11.13.17


Commas: 126/125, 525/512
Comma list: 100/99, 105/104, 144/143, 170/169, 196/195


[[POTE tuning|POTE generator]]: ~5/4 = 378.479
Mapping: {{mapping| 1 0 2 -1 6 -2 -7 | 0 5 1 12 -8 18 35 }}


Map: [&lt;1 0 2 5|, &lt;0 5 1 -7|]
Optimal tunings:  
EDOs: 19, 73bcd, 92bcd
* WE: ~2 = 1200.1727{{c}}, ~5/4 = 380.2982{{c}}
Badness: 0.0562
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2483{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 19g, 41, 60 }}
Commas: 45/44, 126/125, 385/384


[[POTE tuning|POTE generator]]: ~5/4 = 377.724
Badness (Sintel): 1.34


Map: [&lt;1 0 2 5 0|, &lt;0 5 1 -7 11|]
====== 19-limit ======
EDOs: 16, 19, 35, 54bd
Subgroup: 2.3.5.7.11.13.17.19
Badness: 0.0480


==13-limit==
Comma list: 100/99, 105/104, 144/143, 170/169, 171/169, 196/195
Commas: 45/44, 65/64, 78/77, 126/125


[[POTE tuning|POTE generator]]: ~5/4 = 377.724
Mapping: {{mapping| 1 0 2 -1 6 -2 -7 -4 | 0 5 1 12 -8 18 35 26 }}


Map: [&lt;1 0 2 5 0 4|, &lt;0 5 1 -7 11 -1|]
Optimal tunings:  
EDOs: 16, 19, 35f, 54bdf
* WE: ~2 = 1200.2179{{c}}, ~5/4 = 380.3942{{c}}
Badness: 0.0309
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.3314{{c}}


=Astrology=
{{Optimal ET sequence|legend=0| 19gh, 41 }}
Commas: 50/49, 3125/3072


POTE generator: ~5/4 = 380.578
Badness (Sintel): 1.44


Map: [&lt;2 0 4 5|, &lt;0 5 1 1|]
===== Evening =====
Wedgie: &lt;&lt;10 2 2 -20 -25 -1||
Evening is a remarkable subgroup temperament of {{nowrap| 19 & 22f }} with prime harmonics of 29 and 31.  
EDOs: 6, 16, 22, 60d, 82d
Badness: 0.0827


==11-limit==
Subgroup: 2.3.5.7.11.13.29.31
Commas: 50/49, 121/120, 176/175


POTE generator: ~5/4 = 380.530
Comma list: 100/99, 105/104, 144/143, 145/144, 155/154, 196/195


Map: [&lt;2 0 4 5 5|, &lt;0 5 1 1 3|]
Subgroup-val mapping: {{mapping| 1 0 2 -1 6 -2 2 4 | 0 5 1 12 -8 18 9 3 }}
EDOs: 6, 16, 22, 60de, 82de
Badness: 0.0392


==13-limit==
Optimal tunings:
Commas: 50/49, 65/64, 78/77, 121/120
* WE: ~2 = 1200.2802{{c}}, ~5/4 = 380.5053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4258{{c}}


POTE generator: ~5/4 = 379.787
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}


Map: [&lt;2 0 4 5 5 8|, &lt;0 5 1 1 3 -1|]
Badness (Sintel): 0.807
EDOs: 6, 16, 22, 38f
Badness: 0.0344


[[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]]
==== Sorcery ====
Subgroup: 2.3.5.7.11.13


==Horoscope==
Comma list: 65/64, 78/77, 91/90, 100/99
Commas: 50/49, 66/65, 105/104, 121/120


POTE generator: ~5/4 = 379.837
Mapping: {{mapping| 1 0 2 -1 6 4 | 0 5 1 12 -8 -1 }}


Map: [&lt;2 0 4 5 5 3|, &lt;0 5 1 1 3 7|]
Optimal tunings:  
EDOs: 16, 22f, 38
* WE: ~2 = 1201.2397{{c}}, ~5/4 = 380.8698{{c}}
Badness: 0.0353
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5080{{c}}


=Spell=
{{Optimal ET sequence|legend=0| 19, 22, 41f }}
Commas: 49/48, 3125/3072


POTE generator: ~28/25 = 189.927
Badness (Sintel): 1.07


Map: [&lt;1 0 2 2|, &lt;0 10 2 5|]
==== Necromancy ====
Wedgie: &lt;&lt;10 2 5 -20 -20 6||
Subgroup: 2.3.5.7.11.13
EDOs: 6, 19, 82d
Badness: 0.0810


==11-limit==
Comma list: 100/99, 225/224, 245/243, 275/273
Commas: 49/48, 56/55, 125/121


POTE generator: ~11/10 = 190.285
Mapping: {{mapping| 1 0 2 -1 6 11 | 0 5 1 12 -8 -23 }}


Map: [&lt;1 0 2 2 3|, &lt;0 10 2 5 3|]
Optimal tunings:  
EDOs: 6, 19, 44de, 63de, 82de
* WE: ~2 = 1199.9675{{c}}, ~5/4 = 380.7770{{c}}
Badness: 0.0598
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7874{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63, 104 }}
Commas: 49/48, 56/55, 78/77, 125/121


POTE generator: ~11/10 = 189.928
Badness (Sintel): 1.04


Map: [&lt;1 0 2 2 3 4|, &lt;0 10 2 5 3 -2|]
===== 17-limit =====
EDOs: 6, 19, 82def
Subgroup: 2.3.5.7.11.13.17
Badness: 0.0456


==Cantrip==
Comma list: 100/99, 120/119, 154/153, 225/224, 273/272
Commas: 49/48, 56/55, 91/90, 125/121


POTE generator: ~11/10 = 190.360
Mapping: {{mapping| 1 0 2 -1 6 11 6 | 0 5 1 12 -8 -23 -6 }}


Map: [&lt;1 0 2 2 3 1|, &lt;0 10 2 5 3 17|]
Optimal tunings:  
EDOs: 19, 44de, 63de, 82de
* WE: ~2 = 1199.6176{{c}}, ~5/4 = 380.7053{{c}}
Badness: 0.0416
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.8280{{c}}


=Hocum=
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63 }}
Commas: 3125/3072, 4000/3969


POTE generator: ~63/50 = 400.108
Badness (Sintel): 1.12


Map: [&lt;1 5 3 -3|, &lt;0 -10 -2 17|]
==== Soothsaying ====
Wedgie: &lt;&lt;10 2 -17 -20 -55 -45||
Subgroup: 2.3.5.7.11.13
EDOs: 38, 41, 161c, 202c, 243c, 284c
Badness: 0.1071


=Hocus=
Comma list: 100/99, 225/224, 245/243, 1352/1331
Commas: 225/224, 243/242, 245/242


POTE generator: ~14/11 = 409.910
Mapping: {{mapping| 2 0 4 -2 12 15 | 0 5 1 12 -8 -12 }}


Map: [&lt;1 5 3 11 12|, &lt;0 -10 -2 -24 -25|]
Optimal tunings:  
EDOs: 38d, 41, 120cd, 161cd, 202cd
* WE: ~55/39 = 600.2918{{c}}, ~5/4 = 380.6928{{c}}
Badness: 0.0385
* 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, 243/242, 245/242


POTE generator: ~14/11 = 410.004
Badness (Sintel): 2.29


Map: [&lt;1 5 3 11 12 16|, &lt;0 -10 -2 -24 -25 -36|]
===== 17-limit =====
EDOs: 41, 79d, 120cd
Subgroup: 2.3.5.7.11.13.17
Badness: 0.0303


=Trismegistus=
Comma list: 100/99, 221/220, 225/224, 245/243, 273/272
Commas: 3125/3072, 1029/1024


POTE generator: ~147/100 = 673.290
Mapping: {{mapping| 2 0 4 -2 12 15 5 | 0 5 1 12 -8 -12 5 }}


Map: [&lt;1 10 4 0|, &lt;0 -15 -3 5|]
Optimal tunings:  
Wedgie: &lt;&lt;15 3 -5 -30 -50 -20||
* WE: ~17/12 = 600.2918{{c}}, ~5/4 = 380.6927{{c}}
EDOs: 16, 25, 41, 139c, 180c, 221c, 262c
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.5135{{c}}
Badness: 0.0983


==11-limit==
{{Optimal ET sequence|legend=0| 22, 60, 82 }}
Commas: 385/384, 441/440, 625/616


POTE generator: ~22/15 = 673.340
Badness (Sintel): 1.82


Map: [&lt;1 10 4 0 13|, &lt;0 -15 -3 5 -17|]
===== 19-limit =====
EDOs: 16, 25e, 41, 98c, 139c, 180c
Subgroup: 2.3.5.7.11.13.17.19
Badness: 0.0456


==13-limit==
Comma list: 100/99, 133/132, 221/220, 225/224, 245/243, 273/272
Commas: 105/104, 144/143, 275/273, 625/616


POTE generator: ~22/15 = 673.359
Mapping: {{mapping| 2 0 4 -2 12 15 5 18 | 0 5 1 12 -8 -12 5 -15 }}


Map: [&lt;1 10 4 0 13 11|, &lt;0 -15 -3 5 -17 -13|]
Optimal tunings:  
EDOs: 16, 25e, 41, 98c, 139cf
* WE: ~17/12 = 600.3301{{c}}, ~5/4 = 380.6797{{c}}
Badness: 0.0331
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.4704{{c}}


=Quadrimage=
{{Optimal ET sequence|legend=0| 22, 60, 82 }}
Commas: 2401/2400, 3125/3072


POTE generator: ~28/25 = 204.987
Badness (Sintel): 1.90


Map: [&lt;1 5 3 4|, &lt;0 -20 -4 -7|]
=== Telepathy ===
Wedgie: &lt;&lt;20 4 7 -40 -45 5||
Subgroup: 2.3.5.7.11
EDOs: 6, 35, 41, 158cd, 199cd, 240cd, 281cd
Badness: 0.1274


==11-limit==
Comma list: 55/54, 99/98, 176/175
Commas: 245/242, 385/384, 625/616


POTE generator: ~28/25 = 204.956
Mapping: {{mapping| 1 0 2 -1 -1 | 0 5 1 12 14 }}


Map: [&lt;1 5 3 4 5|, &lt;0 -20 -4 -7 -9|]
Optimal tunings:  
EDOs: 6, 35, 41, 199cde, 240cde, 281cde
* WE: ~2 = 1200.7724{{c}}, ~5/4 = 381.2641{{c}}
Badness: 0.0616
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 381.0913{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 19e, 22, 41e, 63e }}
Commas: 105/104, 144/143, 245/242, 625/616


POTE generator: ~28/25 = 205.028
Badness (Sintel): 0.896


Map: [&lt;1 5 3 4 5 9|, &lt;0 -20 -4 -7 -9 -31|]
==== 13-limit ====
EDOs: 41, 117c, 158cd, 199cdef
Subgroup: 2.3.5.7.11.13
Badness: 0.0440</pre></div>
 
<h4>Original HTML content:</h4>
Comma list: 55/54, 65/64, 91/90, 99/98
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Magic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:70:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:70 --&gt;&lt;!-- ws:start:WikiTextTocRule:71: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Five limit magic"
 
Mapping: {{mapping| 1 0 2 -1 -1 4 | 0 5 1 12 14 -1 }}
 
Optimal tunings:
* WE: ~2 = 1202.5634{{c}}, ~5/4 = 381.3348{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6886{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 0 2 -1 -1 -2 | 0 5 1 12 14 18 }}
 
Optimal tunings:
* WE: ~2 = 1201.3172{{c}}, ~5/4 = 380.9004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5942{{c}}
 
{{Optimal ET sequence|legend=0| 19e, 22f }}
 
Badness (Sintel): 1.08
 
=== Horcrux ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 56/55, 245/243
 
Mapping: {{mapping| 1 0 2 -1 0 | 0 5 1 12 11 }}
 
Optimal tunings:
* WE: ~2 = 1200.4670{{c}}, ~5/4 = 379.7895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.6889{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 0 2 -1 0 -2 | 0 5 1 12 11 18 }}
 
Optimal tunings:
* WE: ~2 = 1200.2953{{c}}, ~5/4 = 379.8842{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 379.8165{{c}}
 
{{Optimal ET sequence|legend=0| 3def, 16dff, 19 }}
 
Badness (Sintel): 1.32
 
===== 17-limit =====
Subgroup:

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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