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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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| de = Magische Temperaturen
: This revision was by author [[User:x31eq|x31eq]] and made on <tt>2013-01-26 10:35:07 UTC</tt>.<br>
| 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.
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
[[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


Algebraic generator: Terzbirat, the positive root of 9x^2-8x-4 = (4+2√13)/9; approximately 380.3175 [[Cent|cents]].
{{Mapping|legend=1| 1 0 2 | 0 5 1 }}


Map: [&lt;1 0 2|, &lt;0 5 1|]
: mapping generators: ~2, ~5/4
[[Generator|Generators]]: 2, 5/4
[[Edo|Edos]]: [[6edo|6]], [[16edo|16]], [[19edo|19]], [[22edo|22]], [[41edo|41]], [[60edo|60]], [[221edo|221c]], [[281edo|281c]]


==Seven limit children==
[[Optimal tuning]]s:
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.
* [[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 }}


=Magic=
[[Minimax tuning]]:
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.
* [[5-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


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||.
[[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)


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.
[[Algebraic generator]]: Terzbirat, the positive root of 9''x''<sup>2</sup> - 8''x'' - 4 = (4 + 2√13)/9; approximately 380.3175 [[cent]]s.


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


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.
[[Badness]] (Sintel): 0.919


Commas: 225/224, 245/243
=== 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.


7 and 9 limit minimax
Weak extensions considered below are hocum, trismegistus, quadrimage, quinmage and warlock. Discussed elsewhere are
[|1 0 0 0&gt;, |0 1 0 0&gt;, |2 1/5 0 0&gt;, |-1 12/5 0 0&gt;]
* ''[[Astrology]]'' → [[Jubilismic clan #Astrology|Jubilismic clan]]
[[Eigenmonzo|Eigenmonzos]]: 2, 3
* ''[[Spell]]'' → [[Hemimean clan #Spell|Hemimean clan]]


[[POTE tuning|POTE generator]]: 380.352
== Septimal magic ==
{{Main| Magic }}


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


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


EDOs: 41, 142cd, 183cd, 224cd
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.


==11-limit==
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.


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.)
[[Subgroup]]: 2.3.5.7


See also [[Chords of magic]]
[[Comma list]]: 225/224, 245/243


Commas: 225/224, 245/243, 100/99
{{Mapping|legend=1| 1 0 2 -1 | 0 5 1 12 }}


[[POTE tuning|POTE generator]]: 380.696
: mapping generators: ~2, ~5/4


Map: [&lt;1 0 2 -1 6|, &lt;0 5 1 12 -8|]
[[Optimal tuning]]s:  
EDOs: 19, 22, 41, 104, 145c
* [[WE]]: ~2 = 1201.0786{{c}}, ~5/4 = 380.6939{{c}}
Badness: 0.0204
: [[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 }}


==13-limit==
[[Minimax tuning]]:
Commas: 100/99, 105/104, 144/143, 196/195
* 7- and [[9-odd-limit]]: ~5/4 = {{monzo| 0 1/5 0 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


POTE generator: ~5/4 = 380.427
[[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)


Map: [&lt;1 0 2 -1 6 -2|, &lt;0 5 1 12 -8 18|]
[[Algebraic generator]]: Tirzbirat or Septimage, the real root of 5''x''<sup>5</sup> + 4''x'' - 20, 380.7604 cents.
EDOS: 19, 41, 265cdef
Badness: 0.0215


===Sorcery===
{{Optimal ET sequence|legend=1| 19, 41, 142cd, 183cd, 224ccd }}
Commas: 65/64, 78/77, 91/90, 100/99


POTE generator: ~5/4 = 380.477
[[Badness]] (Sintel): 0.479


Map: [&lt;1 0 2 -1 6 4|, &lt;0 5 1 12 -8 -1|]
=== 11-limit ===
EDOs: 19, 22, 31f, 41f
Subgroup: 2.3.5.7.11
Badness: 0.0258


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


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


Map: [&lt;1 0 2 -1 6 11|, &lt;0 5 1 12 -8 -23|]
Optimal tunings:  
EDOs: 19, 22, 41, 63, 104
* WE: ~2 = 1200.1372{{c}}, ~5/4 = 380.7399{{c}}
Badness: 0.0253
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7008{{c}}


==Telepathy==
Minimax tuning:
Commas: 55/54, 99/98, 176/175
* 11-odd-limit: ~5/4 = {{monzo| 1/3 1/9 0 0 -1/18 }}
: unchanged-interval (eigenmonzo) basis: 2.11/9


POTE generator: ~5/4 = 381.019
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 -1|, &lt;0 5 1 12 14|]
{{Optimal ET sequence|legend=0| 19, 22, 41, 104 }}
EDOs: 19, 22, 41e, 63e
Badness: 0.0271


===13-limit telepathy===
Badness (Sintel): 0.673
Commas: 55/54, 65/64, 91/90, 99/98


POTE generator: ~5/4 = 380.520
==== 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 -1 4|, &lt;0 5 1 12 14 -1|]
Subgroup: 2.3.5.7.11.13
EDOs: 19, 22, 31f, 34ef, 41ef
Badness: 0.0255


==Horcrux==
Comma list: 100/99, 105/104, 144/143, 196/195
Commas: 45/44, 56/55, 245/243


POTE generator: ~5/4 = 379.642
Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 1 12 -8 18 }}


Map: [&lt;1 0 2 -1 0|, &lt;0 5 1 12 11|]
Optimal tunings:  
EDOs: 19, 60e
* WE: ~2 = 1200.0331{{c}}, ~5/4 = 380.4377{{c}}
Badness: 0.0393
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4284{{c}}


=Divination=
Tuning ranges:
Commas: 121/120, 225/224, 245/243
* 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 = 380.233
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}


Map: [&lt;2 0 4 -2 5|, &lt;0 5 1 12 3|]
Badness (Sintel): 0.889
EDOs: 22, 38d, 60e, 142cde
Badness: 0.0359


==13-limit==  
===== Magical =====
Commas: 105/104, 121/120, 196/195, 245/243
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~5/4 = 379.920
Comma list: 100/99, 105/104, 120/119, 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 | 0 5 1 12 -8 18 -6 }}
EDOs: 22f, 60e
Badness: 0.0346


=Soothsaying=  
Optimal tunings:
Commas: 100/99, 225/224, 245/243, 1352/1331
* WE: ~2 = 1199.3584{{c}}, ~5/4 = 380.4006{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5896{{c}}


POTE generator: ~5/4 = 380.508
{{Optimal ET sequence|legend=0| 19, 22f, 41 }}


Map: [&lt;2 0 4 -2 12 15|, &lt;0 5 1 12 -8 -12|]
Badness (Sintel): 1.05
EDOs: 22, 60, 82
Badness: 0.0554


=Witchcraft=  
====== Magicus ======
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, 133/132, 144/143, 154/153


Map: [&lt;1 0 2 -1 -7|, &lt;0 5 1 12 33|]
Mapping: {{mapping| 1 0 2 -1 6 -2 6 9 | 0 5 1 12 -8 18 -6 -15 }}
EDOs: 41, 60e, 101cd, 243cde
Badness: 0.0307


==13-limit==
Optimal tunings:
Commas: 105/104, 196/195, 245/243, 275/273
* WE: ~2 = 1199.7173{{c}}, ~5/4 = 380.3808{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4680{{c}}


POTE generator: ~5/4 = 380.189
{{Optimal ET sequence|legend=0| 19, 41 }}


Map: [&lt;1 0 2 -1 -7 -2|, &lt;0 5 1 12 33 18|]
Badness (Sintel): 1.27
EDOs: 41, 60e, 101cd
Badness: 0.0235


=Muggles=  
====== Magica ======
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: 2.3.5.7.11.13.17.19


Commas: 126/125, 525/512
Comma list: 100/99, 105/104, 120/119, 144/143, 154/153, 171/169


[[POTE tuning|POTE generator]]: ~5/4 = 378.479
Mapping: {{mapping| 1 0 2 -1 6 -2 6 -4 | 0 5 1 12 -8 18 -6 26 }}


Map: [&lt;1 0 2 5|, &lt;0 5 1 -7|]
Optimal tunings:  
EDOs: 19, 73bcd, 92bcd
* WE: ~2 = 1199.3670{{c}}, ~5/4 = 380.4681{{c}}
Badness: 0.0562
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.6541{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 22fh, 41 }}
Commas: 45/44, 126/125, 385/384


[[POTE tuning|POTE generator]]: ~5/4 = 377.724
Badness (Sintel): 1.21


Map: [&lt;1 0 2 5 0|, &lt;0 5 1 -7 11|]
===== Magia =====
EDOs: 16, 19, 35, 54bd
Subgroup: 2.3.5.7.11.13.17
Badness: 0.0480


==13-limit==
Comma list: 100/99, 105/104, 144/143, 170/169, 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 | 0 5 1 12 -8 18 35 }}


Map: [&lt;1 0 2 5 0 4|, &lt;0 5 1 -7 11 -1|]
Optimal tunings:  
EDOs: 16, 19, 35f, 54bdf
* WE: ~2 = 1200.1727{{c}}, ~5/4 = 380.2982{{c}}
Badness: 0.0309
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.2483{{c}}


=Astrology=
{{Optimal ET sequence|legend=0| 19g, 41, 60 }}
Commas: 50/49, 3125/3072


POTE generator: ~5/4 = 380.578
Badness (Sintel): 1.34


Map: [&lt;2 0 4 5|, &lt;0 5 1 1|]
====== 19-limit ======
Wedgie: &lt;&lt;10 2 2 -20 -25 -1||
Subgroup: 2.3.5.7.11.13.17.19
EDOs: 6, 16, 22, 60d, 82d
Badness: 0.0827


==11-limit==
Comma list: 100/99, 105/104, 144/143, 170/169, 171/169, 196/195
Commas: 50/49, 121/120, 176/175


POTE generator: ~5/4 = 380.530
Mapping: {{mapping| 1 0 2 -1 6 -2 -7 -4 | 0 5 1 12 -8 18 35 26 }}


Map: [&lt;2 0 4 5 5|, &lt;0 5 1 1 3|]
Optimal tunings:  
EDOs: 6, 16, 22, 60de, 82de
* WE: ~2 = 1200.2179{{c}}, ~5/4 = 380.3942{{c}}
Badness: 0.0392
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.3314{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 19gh, 41 }}
Commas: 50/49, 65/64, 78/77, 121/120


POTE generator: ~5/4 = 379.787
Badness (Sintel): 1.44


Map: [&lt;2 0 4 5 5 8|, &lt;0 5 1 1 3 -1|]
===== Evening =====
EDOs: 6, 16, 22, 38f
Evening is a remarkable subgroup temperament of {{nowrap| 19 & 22f }} with prime harmonics of 29 and 31.  
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]]
Subgroup: 2.3.5.7.11.13.29.31


==Horoscope==
Comma list: 100/99, 105/104, 144/143, 145/144, 155/154, 196/195
Commas: 50/49, 66/65, 105/104, 121/120


POTE generator: ~5/4 = 379.837
Subgroup-val mapping: {{mapping| 1 0 2 -1 6 -2 2 4 | 0 5 1 12 -8 18 9 3 }}


Map: [&lt;2 0 4 5 5 3|, &lt;0 5 1 1 3 7|]
Optimal tunings:  
EDOs: 16, 22f, 38
* WE: ~2 = 1200.2802{{c}}, ~5/4 = 380.5053{{c}}
Badness: 0.0353
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.4258{{c}}


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


POTE generator: ~28/25 = 189.927
Badness (Sintel): 0.807


Map: [&lt;1 0 2 2|, &lt;0 10 2 5|]
==== Sorcery ====
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: 65/64, 78/77, 91/90, 100/99
Commas: 49/48, 56/55, 125/121


POTE generator: ~11/10 = 190.285
Mapping: {{mapping| 1 0 2 -1 6 4 | 0 5 1 12 -8 -1 }}


Map: [&lt;1 0 2 2 3|, &lt;0 10 2 5 3|]
Optimal tunings:  
EDOs: 6, 19, 44de, 63de, 82de
* WE: ~2 = 1201.2397{{c}}, ~5/4 = 380.8698{{c}}
Badness: 0.0598
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.5080{{c}}


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


POTE generator: ~11/10 = 189.928
Badness (Sintel): 1.07


Map: [&lt;1 0 2 2 3 4|, &lt;0 10 2 5 3 -2|]
==== Necromancy ====
EDOs: 6, 19, 82def
Subgroup: 2.3.5.7.11.13
Badness: 0.0456


==Cantrip==
Comma list: 100/99, 225/224, 245/243, 275/273
Commas: 49/48, 56/55, 91/90, 125/121


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


Map: [&lt;1 0 2 2 3 1|, &lt;0 10 2 5 3 17|]
Optimal tunings:  
EDOs: 19, 44de, 63de, 82de
* WE: ~2 = 1199.9675{{c}}, ~5/4 = 380.7770{{c}}
Badness: 0.0416
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.7874{{c}}


=Hocum=
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63, 104 }}
Commas: 3125/3072, 4000/3969


POTE generator: ~63/50 = 400.108
Badness (Sintel): 1.04


Map: [&lt;1 5 3 -3|, &lt;0 -10 -2 17|]
===== 17-limit =====
Wedgie: &lt;&lt;10 2 -17 -20 -55 -45||
Subgroup: 2.3.5.7.11.13.17
EDOs: 38, 41, 161c, 202c, 243c, 284c
Badness: 0.1071


=Hocus=
Comma list: 100/99, 120/119, 154/153, 225/224, 273/272
Commas: 225/224, 243/242, 245/242


POTE generator: ~14/11 = 409.910
Mapping: {{mapping| 1 0 2 -1 6 11 6 | 0 5 1 12 -8 -23 -6 }}


Map: [&lt;1 5 3 11 12|, &lt;0 -10 -2 -24 -25|]
Optimal tunings:  
EDOs: 38d, 41, 120cd, 161cd, 202cd
* WE: ~2 = 1199.6176{{c}}, ~5/4 = 380.7053{{c}}
Badness: 0.0385
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.8280{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 19f, 22, 41, 63 }}
Commas: 105/104, 196/195, 243/242, 245/242


POTE generator: ~14/11 = 410.004
Badness (Sintel): 1.12


Map: [&lt;1 5 3 11 12 16|, &lt;0 -10 -2 -24 -25 -36|]
==== Soothsaying ====
EDOs: 41, 79d, 120cd
Subgroup: 2.3.5.7.11.13
Badness: 0.0303


=Trismegistus=
Comma list: 100/99, 225/224, 245/243, 1352/1331
Commas: 3125/3072, 1029/1024


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


Map: [&lt;1 10 4 0|, &lt;0 -15 -3 5|]
Optimal tunings:  
Wedgie: &lt;&lt;15 3 -5 -30 -50 -20||
* WE: ~55/39 = 600.2918{{c}}, ~5/4 = 380.6928{{c}}
EDOs: 16, 25, 41, 139c, 180c, 221c, 262c
* CWE: ~55/39 = 600.0000{{c}}, ~5/4 = 380.5121{{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): 2.29


Map: [&lt;1 10 4 0 13|, &lt;0 -15 -3 5 -17|]
===== 17-limit =====
EDOs: 16, 25e, 41, 98c, 139c, 180c
Subgroup: 2.3.5.7.11.13.17
Badness: 0.0456


==13-limit==
Comma list: 100/99, 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 | 0 5 1 12 -8 -12 5 }}


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 = 600.2918{{c}}, ~5/4 = 380.6927{{c}}
Badness: 0.0331
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.5135{{c}}


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


POTE generator: ~28/25 = 204.987
Badness (Sintel): 1.82


Map: [&lt;1 5 3 4|, &lt;0 -20 -4 -7|]
===== 19-limit =====
Wedgie: &lt;&lt;20 4 7 -40 -45 5||
Subgroup: 2.3.5.7.11.13.17.19
EDOs: 6, 35, 41, 158cd, 199cd, 240cd, 281cd
Badness: 0.1274


==11-limit==
Comma list: 100/99, 133/132, 221/220, 225/224, 245/243, 273/272
Commas: 245/242, 385/384, 625/616


POTE generator: ~28/25 = 204.956
Mapping: {{mapping| 2 0 4 -2 12 15 5 18 | 0 5 1 12 -8 -12 5 -15 }}


Map: [&lt;1 5 3 4 5|, &lt;0 -20 -4 -7 -9|]
Optimal tunings:  
EDOs: 6, 35, 41, 199cde, 240cde, 281cde
* WE: ~17/12 = 600.3301{{c}}, ~5/4 = 380.6797{{c}}
Badness: 0.0616
* CWE: ~17/12 = 600.0000{{c}}, ~5/4 = 380.4704{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 22, 60, 82 }}
Commas: 105/104, 144/143, 245/242, 625/616


POTE generator: ~28/25 = 205.028
Badness (Sintel): 1.90


Map: [&lt;1 5 3 4 5 9|, &lt;0 -20 -4 -7 -9 -31|]
=== Telepathy ===
EDOs: 41, 117c, 158cd, 199cdef
Subgroup: 2.3.5.7.11
Badness: 0.0440</pre></div>
 
<h4>Original HTML content:</h4>
Comma list: 55/54, 99/98, 176/175
<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
 
Mapping: {{mapping| 1 0 2 -1 -1 | 0 5 1 12 14 }}
 
Optimal tunings:
* WE: ~2 = 1200.7724{{c}}, ~5/4 = 381.2641{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 381.0913{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{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
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