Magic family: Difference between revisions

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[[toc]]

=Five limit magic=
The 5-limit parent comma for the magic family is 3125/3072, the small diesis or magic comma. Its monzo is |-10 -1 5>, and flipping that yields <<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 sense and using [[19edo]] or [[22edo]] is always possible.

[[Comma]]: 3125/3072

5-limit minimax
[<1 0 0|, <0 1 0|, <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]].

Map: [<1 0 2|, <0 5 1|]
[[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==
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=
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 and simpler than 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 <<5 1 12 -10 5 25||.

By adding 100/99 to the list of commas, magic can be extended to an 11-limit version, <<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.

Commas: 225/224, 245/243

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

[[POTE tuning|POTE generator]]: 380.352

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

Map: [<1 0 2 -1|, <0 5 1 12|]
[[Generator|Generators]]: 2, 5/4

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

==11-limit==
Commas: 225/224, 245/243, 100/99

[[POTE tuning|POTE generator]]: 380.696

Map: [<1 0 2 -1 6|, <0 5 1 12 -8|]
EDOs: 19, 22, 41, 104, 145c
Badness: 0.0204

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

POTE generator: ~5/4 = 380.427

Map: [<1 0 2 -1 6 -2|, <0 5 1 12 -8 18|]
EDOS: 19, 41, 265cdef
Badness: 0.0215

===Sorcery===
Commas: 65/64, 78/77, 91/90, 100/99

POTE generator: ~5/4 = 380.477

Map: [<1 0 2 -1 6 4|, <0 5 1 12 -8 -1|]
EDOs: 19, 22, 31f, 41f
Badness: 0.0258

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

POTE generator: ~5/4 = 380.787

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

==Telepathy==
Commas: 55/54, 99/98, 176/175

POTE generator: ~5/4 = 381.019

Map: [<1 0 2 -1 -1|, <0 1 5 12 14|]
EDOs: 19, 22, 41e, 63e
Badness: 0.0271

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

POTE generator: ~5/4 = 380.520

Map: [<1 0 2 -1 -1 4|, <0 5 1 12 14 -1|]
EDOs: 19, 22, 31f, 34ef, 41ef
Badness: 0.0255

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

POTE generator: ~5/4 = 379.642

Map: [<1 0 2 -1 0|, <0 5 1 12 11|]
EDOs: 19, 60e
Badness: 0.0393

=Hocus=
Commas: 225/224, 243/242, 245/242

POTE generator: ~14/11 = 409.910

Map: [<1 5 3 11 12|, <0 -10 -2 -24 -25|]
EDOs: 38d, 41, 120cd, 161cd, 202cd
Badness: 0.0385

==13-limit==
Commas: 105/104, 196/195, 243/242, 245/242

POTE generator: ~14/11 = 410.004

Map: [<1 5 3 11 12 16|, <0 -10 -2 -24 -25 -36|]
EDOs: 41, 79d, 120cd
Badness: 0.0303

=Divination=
Commas: 121/120, 225/224, 245/243

POTE generator: ~5/4 = 380.233

Map: [<2 0 4 -2 5|, <0 5 1 12 3|]
EDOs: 22, 38d, 60e, 142cde
Badness: 0.0359

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

POTE generator: ~5/4 = 379.920

Map: [<2 0 4 -2 5 -4|, <0 5 1 12 3 18|]
EDOs: 22f, 60e
Badness: 0.0346

=Soothsaying=
Commas: 100/99, 225/224, 245/243, 1352/1331

POTE generator: ~5/4 = 380.508

Map: [<2 0 4 -2 12 15|, <0 5 1 12 -8 -12|]
EDOs: 22, 60, 82
Badness: 0.0554

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

POTE generator: ~5/4 = 380.232

Map: [<1 0 2 -1 -7|,  <0 5 1 12 33|]
EDOs: 41, 60e, 101cd, 243cde
Badness: 0.0307

==13-limit==
Commas: 105/104, 196/195, 245/243, 275/273

POTE generator: ~5/4 =  380.189

Map: [<1 0 2 -1 -7 -2|,  <0 5 1 12 33 18|]
EDOs: 41, 60e, 101cd
Badness: 0.0235

=Muggles=
Aside from 3125/3072 and 525/512 muggles also tempers out 126/125 and 1323/1280. A good muggles tuning is [[19edo]], in which tuning it's the same thing as magic. Muggles works better for small scales than magic in the sense that 7 or 10 note MOS are reasonable choices. The muggles wedgie is <<5 1 -7 -10 -25 -19||.

Commas: 126/125, 525/512

[[POTE tuning|POTE generator]]: 378.479

Map: [<1 0 2 5|, <0 5 1 -7|]

EDOs: 19, 73bcd, 92bcd

=Astrology=
Commas: 50/49, 3125/3072

POTE generator: ~5/4 = 380.578

Map: [<2 0 4 5|, <0 5 1 1|]
Wedgie: <<10 2 2 -20 -25 -1||
EDOs: 6, 16, 22, 60d, 82d
Badness: 0.0827

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

POTE generator: ~5/4 = 380.530

Map: [<2 0 4 5 5|, <0 5 1 1 3|]
EDOs: 6, 16, 22, 60de, 82de
Badness: 0.0392

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

POTE generator: ~5/4 = 379.787

Map: [<2 0 4 5 5 8|, <0 5 1 1 3 -1|]
EDOs: 6, 16, 22, 38f
Badness: 0.0344

==Horoscope==
Commas: 50/49, 66/65, 105/104, 121/120

POTE generator: ~5/4 = 379.837

Map: [<2 0 4 5 5 3|, <0 5 1 1 3 7|]
EDOs: 16, 22f, 38
Badness: 0.0353

=Spell=
Commas: 49/48, 3125/3072

POTE generator: ~28/25 = 189.927

Map: [<1 0 2 2|, <0 10 2 5|]
Wedgie: <<10 2 5 -20 -20 6||
EDOs: 6, 19, 82d
Badness: 0.0810

==11-limit==
Commas: 49/48, 56/55, 125/121

POTE generator: ~11/10 = 190.285

Map: [<1 0 2 2 3|, <0 10 2 5 3|]
EDOs: 6, 19, 44de, 63de, 82de
Badness: 0.0598

==13-limit==
Commas: 49/48, 56/55, 78/77, 125/121

POTE generator: ~11/10 = 189.928

Map: [<1 0 2 2 3 4|, <0 10 2 5 3 -2|]
EDOs: 6, 19, 82def
Badness: 0.0456

==Cantrip==
Commas: 49/48, 56/55, 91/90, 125/121

POTE generator: ~11/10 = 190.360

Map: [<1 0 2 2 3 1|, <0 10 2 5 3 17|]
EDOs: 19, 44de, 63de, 82de
Badness: 0.0416

=Trismegistus=
Commas: 3125/3072, 1029/1024

POTE generator: ~147/100 = 673.290

Map: [<1 10 4 0|,  <0 -15 -3 5|]
Wedgie: <<15 3 -5 -30 -50 -20||
EDOs: 16, 25, 41, 139c, 180c, 221c, 262c
Badness: 0.0983

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

POTE generator: ~22/15 = 673.340

Map: [<1 10 4 0 13|,  <0 -15 -3 5 -17|]
EDOs: 16, 25e, 41, 98c, 139c, 180c
Badness: 0.0456

==13-limit==
Commas: 105/104, 144/143, 275/273, 625/616

POTE generator: ~22/15 = 673.359

Map: [<1 10 4 0 13 11|,  <0 -15 -3 5 -17 -13|]
EDOs: 16, 25e, 41, 98c, 139cf
Badness: 0.0331

=Quadrimage=
Commas: 2401/2400, 3125/3072

POTE generator: ~28/25 = 204.987

Map: [<1 5 3 4|, <0 -20 -4 -7|]
Wedgie: <<20 4 7 -40 -45 5||
EDOs: 6, 35, 41, 158cd, 199cd, 240cd, 281cd
Badness: 0.1274

==11-limit==
Commas: 245/242, 385/384, 625/616

POTE generator: ~28/25 = 204.956

Map: [<1 5 3 4 5|, <0 -20 -4 -7 -9|]
EDOs: 6, 35, 41, 199cde, 240cde, 281cde
Badness: 0.0616

==13-limit==
Commas: 105/104, 144/143, 245/242, 625/616

POTE generator: ~28/25 = 205.028

Map: [<1 5 3 4 5 9|, <0 -20 -4 -7 -9 -31|]
EDOs: 41, 117c, 158cd, 199cdef
Badness: 0.0440

Original HTML content:

<html><head><title>Magic family</title></head><body><!-- ws:start:WikiTextTocRule:64:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><div style="margin-left: 1em;"><a href="#Five limit magic">Five limit magic</a></div>
<!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><div style="margin-left: 2em;"><a href="#Five limit magic-Seven limit children">Seven limit children</a></div>
<!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><div style="margin-left: 1em;"><a href="#Magic">Magic</a></div>
<!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --><div style="margin-left: 2em;"><a href="#Magic-11-limit">11-limit</a></div>
<!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><div style="margin-left: 2em;"><a href="#Magic-13-limit">13-limit</a></div>
<!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --><div style="margin-left: 3em;"><a href="#Magic-13-limit-Sorcery">Sorcery</a></div>
<!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><div style="margin-left: 3em;"><a href="#Magic-13-limit-Necromancy">Necromancy</a></div>
<!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><div style="margin-left: 2em;"><a href="#Magic-Telepathy">Telepathy</a></div>
<!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><div style="margin-left: 3em;"><a href="#Magic-Telepathy-13-limit telepathy">13-limit telepathy</a></div>
<!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><div style="margin-left: 2em;"><a href="#Magic-Horcrux">Horcrux</a></div>
<!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><div style="margin-left: 1em;"><a href="#Hocus">Hocus</a></div>
<!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --><div style="margin-left: 2em;"><a href="#Hocus-13-limit">13-limit</a></div>
<!-- ws:end:WikiTextTocRule:76 --><!-- ws:start:WikiTextTocRule:77: --><div style="margin-left: 1em;"><a href="#Divination">Divination</a></div>
<!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --><div style="margin-left: 2em;"><a href="#Divination-13-limit">13-limit</a></div>
<!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --><div style="margin-left: 1em;"><a href="#Soothsaying">Soothsaying</a></div>
<!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --><div style="margin-left: 1em;"><a href="#Witchcraft">Witchcraft</a></div>
<!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><div style="margin-left: 2em;"><a href="#Witchcraft-13-limit">13-limit</a></div>
<!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><div style="margin-left: 1em;"><a href="#Muggles">Muggles</a></div>
<!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><div style="margin-left: 1em;"><a href="#Astrology">Astrology</a></div>
<!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><div style="margin-left: 2em;"><a href="#Astrology-11-limit">11-limit</a></div>
<!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><div style="margin-left: 2em;"><a href="#Astrology-13-limit">13-limit</a></div>
<!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><div style="margin-left: 2em;"><a href="#Astrology-Horoscope">Horoscope</a></div>
<!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><div style="margin-left: 1em;"><a href="#Spell">Spell</a></div>
<!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --><div style="margin-left: 2em;"><a href="#Spell-11-limit">11-limit</a></div>
<!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --><div style="margin-left: 2em;"><a href="#Spell-13-limit">13-limit</a></div>
<!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --><div style="margin-left: 2em;"><a href="#Spell-Cantrip">Cantrip</a></div>
<!-- ws:end:WikiTextTocRule:90 --><!-- ws:start:WikiTextTocRule:91: --><div style="margin-left: 1em;"><a href="#Trismegistus">Trismegistus</a></div>
<!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><div style="margin-left: 2em;"><a href="#Trismegistus-11-limit">11-limit</a></div>
<!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><div style="margin-left: 2em;"><a href="#Trismegistus-13-limit">13-limit</a></div>
<!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --><div style="margin-left: 1em;"><a href="#Quadrimage">Quadrimage</a></div>
<!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><div style="margin-left: 2em;"><a href="#Quadrimage-11-limit">11-limit</a></div>
<!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><div style="margin-left: 2em;"><a href="#Quadrimage-13-limit">13-limit</a></div>
<!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --></div>
<!-- ws:end:WikiTextTocRule:97 --><br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Five limit magic"></a><!-- ws:end:WikiTextHeadingRule:0 -->Five limit magic</h1>
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 sense and using <a class="wiki_link" href="/19edo">19edo</a> or <a class="wiki_link" href="/22edo">22edo</a> is always possible.<br />
<br />
<a class="wiki_link" href="/Comma">Comma</a>: 3125/3072<br />
<br />
5-limit minimax<br />
[&lt;1 0 0|, &lt;0 1 0|, &lt;2 1/5 0|]<br />
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 3<br />
<br />
Algebraic generator: Terzbirat, the positive root of 9x^2-8x-4 = (4+2√13)/9; approximately 380.3175 <a class="wiki_link" href="/Cent">cents</a>.<br />
<br />
Map: [&lt;1 0 2|, &lt;0 5 1|]<br />
<a class="wiki_link" href="/Generator">Generators</a>: 2, 5/4<br />
<a class="wiki_link" href="/Edo">Edos</a>: <a class="wiki_link" href="/6edo">6</a>, <a class="wiki_link" href="/16edo">16</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/22edo">22</a>, <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/60edo">60</a>, <a class="wiki_link" href="/221edo">221c</a>, <a class="wiki_link" href="/281edo">281c</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Five limit magic-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:2 -->Seven limit children</h2>
The second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Magic"></a><!-- ws:end:WikiTextHeadingRule:4 -->Magic</h1>
Magic tempers out not only 3125/3072 and 875/864, but also 225/224, 245/243, and 10976/10935. <a class="wiki_link" href="/41edo">41edo</a> 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.<br />
<br />
Magic, with its accurate fifths, works well with 9-limit harmony. It's more accurate than meantone and simpler than 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||.<br />
<br />
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, <a class="wiki_link" href="/104edo">104edo</a> 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.<br />
<br />
Commas: 225/224, 245/243<br />
<br />
7 and 9 limit minimax<br />
[|1 0 0 0&gt;, |0 1 0 0&gt;, |2 1/5 0 0&gt;, |-1 12/5 0 0&gt;]<br />
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 3<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 380.352<br />
<br />
Algebraic generators: Tirzbirat or Septimage, the real root of 5x^5+4x-20, 380.7604 cents.<br />
<br />
Map: [&lt;1 0 2 -1|, &lt;0 5 1 12|]<br />
<a class="wiki_link" href="/Generator">Generators</a>: 2, 5/4<br />
<br />
EDOs: 41, 142cd, 183cd, 224cd<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Magic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit</h2>
Commas: 225/224, 245/243, 100/99<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 380.696<br />
<br />
Map: [&lt;1 0 2 -1 6|, &lt;0 5 1 12 -8|]<br />
EDOs: 19, 22, 41, 104, 145c<br />
Badness: 0.0204<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Magic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h2>
Commas: 100/99, 105/104, 144/143, 196/195<br />
<br />
POTE generator: ~5/4 = 380.427<br />
<br />
Map: [&lt;1 0 2 -1 6 -2|, &lt;0 5 1 12 -8 18|]<br />
EDOS: 19, 41, 265cdef<br />
Badness: 0.0215<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Magic-13-limit-Sorcery"></a><!-- ws:end:WikiTextHeadingRule:10 -->Sorcery</h3>
Commas: 65/64, 78/77, 91/90, 100/99<br />
<br />
POTE generator: ~5/4 = 380.477<br />
<br />
Map: [&lt;1 0 2 -1 6 4|, &lt;0 5 1 12 -8 -1|]<br />
EDOs: 19, 22, 31f, 41f<br />
Badness: 0.0258<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Magic-13-limit-Necromancy"></a><!-- ws:end:WikiTextHeadingRule:12 -->Necromancy</h3>
Commas: 100/99, 225/224, 245/243, 275/273<br />
<br />
POTE generator: ~5/4 = 380.787<br />
<br />
Map: [&lt;1 0 2 -1 6 11|, &lt;0 5 1 12 -8 -23|]<br />
EDOs: 19, 22, 41, 63, 104<br />
Badness: 0.0253<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Magic-Telepathy"></a><!-- ws:end:WikiTextHeadingRule:14 -->Telepathy</h2>
Commas: 55/54, 99/98, 176/175<br />
<br />
POTE generator: ~5/4 = 381.019<br />
<br />
Map: [&lt;1 0 2 -1 -1|, &lt;0 1 5 12 14|]<br />
EDOs: 19, 22, 41e, 63e<br />
Badness: 0.0271<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Magic-Telepathy-13-limit telepathy"></a><!-- ws:end:WikiTextHeadingRule:16 -->13-limit telepathy</h3>
Commas: 55/54, 65/64, 91/90, 99/98<br />
<br />
POTE generator: ~5/4 = 380.520<br />
<br />
Map: [&lt;1 0 2 -1 -1 4|, &lt;0 5 1 12 14 -1|]<br />
EDOs: 19, 22, 31f, 34ef, 41ef<br />
Badness: 0.0255<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Magic-Horcrux"></a><!-- ws:end:WikiTextHeadingRule:18 -->Horcrux</h2>
Commas: 45/44, 56/55, 245/243<br />
<br />
POTE generator: ~5/4 = 379.642<br />
<br />
Map: [&lt;1 0 2 -1 0|, &lt;0 5 1 12 11|]<br />
EDOs: 19, 60e<br />
Badness: 0.0393<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Hocus"></a><!-- ws:end:WikiTextHeadingRule:20 -->Hocus</h1>
Commas: 225/224, 243/242, 245/242<br />
<br />
POTE generator: ~14/11 = 409.910<br />
<br />
Map: [&lt;1 5 3 11 12|, &lt;0 -10 -2 -24 -25|]<br />
EDOs: 38d, 41, 120cd, 161cd, 202cd<br />
Badness: 0.0385<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Hocus-13-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->13-limit</h2>
Commas: 105/104, 196/195, 243/242, 245/242<br />
<br />
POTE generator: ~14/11 = 410.004<br />
<br />
Map: [&lt;1 5 3 11 12 16|, &lt;0 -10 -2 -24 -25 -36|]<br />
EDOs: 41, 79d, 120cd<br />
Badness: 0.0303<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Divination"></a><!-- ws:end:WikiTextHeadingRule:24 -->Divination</h1>
Commas: 121/120, 225/224, 245/243<br />
<br />
POTE generator: ~5/4 = 380.233<br />
<br />
Map: [&lt;2 0 4 -2 5|, &lt;0 5 1 12 3|]<br />
EDOs: 22, 38d, 60e, 142cde<br />
Badness: 0.0359<br />
<br />
<!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Divination-13-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->13-limit</h2>
Commas: 105/104, 121/120, 196/195, 245/243<br />
<br />
POTE generator: ~5/4 = 379.920<br />
<br />
Map: [&lt;2 0 4 -2 5 -4|, &lt;0 5 1 12 3 18|]<br />
EDOs: 22f, 60e<br />
Badness: 0.0346<br />
<br />
<!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="Soothsaying"></a><!-- ws:end:WikiTextHeadingRule:28 -->Soothsaying</h1>
Commas: 100/99, 225/224, 245/243, 1352/1331<br />
<br />
POTE generator: ~5/4 = 380.508<br />
<br />
Map: [&lt;2 0 4 -2 12 15|, &lt;0 5 1 12 -8 -12|]<br />
EDOs: 22, 60, 82<br />
Badness: 0.0554<br />
<br />
<!-- ws:start:WikiTextHeadingRule:30:&lt;h1&gt; --><h1 id="toc15"><a name="Witchcraft"></a><!-- ws:end:WikiTextHeadingRule:30 -->Witchcraft</h1>
Commas: 225/224, 245/243, 441/440<br />
<br />
POTE generator: ~5/4 = 380.232<br />
<br />
Map: [&lt;1 0 2 -1 -7|,  &lt;0 5 1 12 33|]<br />
EDOs: 41, 60e, 101cd, 243cde<br />
Badness: 0.0307<br />
<br />
<!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Witchcraft-13-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->13-limit</h2>
Commas: 105/104, 196/195, 245/243, 275/273<br />
<br />
POTE generator: ~5/4 =  380.189<br />
<br />
Map: [&lt;1 0 2 -1 -7 -2|,  &lt;0 5 1 12 33 18|]<br />
EDOs: 41, 60e, 101cd<br />
Badness: 0.0235<br />
<br />
<!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="Muggles"></a><!-- ws:end:WikiTextHeadingRule:34 -->Muggles</h1>
Aside from 3125/3072 and 525/512 muggles also tempers out 126/125 and 1323/1280. A good muggles tuning is <a class="wiki_link" href="/19edo">19edo</a>, 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||.<br />
<br />
Commas: 126/125, 525/512<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 378.479<br />
<br />
Map: [&lt;1 0 2 5|, &lt;0 5 1 -7|]<br />
<br />
EDOs: 19, 73bcd, 92bcd<br />
<br />
<!-- ws:start:WikiTextHeadingRule:36:&lt;h1&gt; --><h1 id="toc18"><a name="Astrology"></a><!-- ws:end:WikiTextHeadingRule:36 -->Astrology</h1>
Commas: 50/49, 3125/3072<br />
<br />
POTE generator: ~5/4 = 380.578<br />
<br />
Map: [&lt;2 0 4 5|, &lt;0 5 1 1|]<br />
Wedgie: &lt;&lt;10 2 2 -20 -25 -1||<br />
EDOs: 6, 16, 22, 60d, 82d<br />
Badness: 0.0827<br />
<br />
<!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><a name="Astrology-11-limit"></a><!-- ws:end:WikiTextHeadingRule:38 -->11-limit</h2>
Commas: 50/49, 121/120, 176/175<br />
<br />
POTE generator: ~5/4 = 380.530<br />
<br />
Map: [&lt;2 0 4 5 5|, &lt;0 5 1 1 3|]<br />
EDOs: 6, 16, 22, 60de, 82de<br />
Badness: 0.0392<br />
<br />
<!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Astrology-13-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->13-limit</h2>
Commas: 50/49, 65/64, 78/77, 121/120<br />
<br />
POTE generator: ~5/4 = 379.787<br />
<br />
Map: [&lt;2 0 4 5 5 8|, &lt;0 5 1 1 3 -1|]<br />
EDOs: 6, 16, 22, 38f<br />
Badness: 0.0344<br />
<br />
<!-- ws:start:WikiTextHeadingRule:42:&lt;h2&gt; --><h2 id="toc21"><a name="Astrology-Horoscope"></a><!-- ws:end:WikiTextHeadingRule:42 -->Horoscope</h2>
Commas: 50/49, 66/65, 105/104, 121/120<br />
<br />
POTE generator: ~5/4 = 379.837<br />
<br />
Map: [&lt;2 0 4 5 5 3|, &lt;0 5 1 1 3 7|]<br />
EDOs: 16, 22f, 38<br />
Badness: 0.0353<br />
<br />
<!-- ws:start:WikiTextHeadingRule:44:&lt;h1&gt; --><h1 id="toc22"><a name="Spell"></a><!-- ws:end:WikiTextHeadingRule:44 -->Spell</h1>
Commas: 49/48, 3125/3072<br />
<br />
POTE generator: ~28/25 = 189.927<br />
<br />
Map: [&lt;1 0 2 2|, &lt;0 10 2 5|]<br />
Wedgie: &lt;&lt;10 2 5 -20 -20 6||<br />
EDOs: 6, 19, 82d<br />
Badness: 0.0810<br />
<br />
<!-- ws:start:WikiTextHeadingRule:46:&lt;h2&gt; --><h2 id="toc23"><a name="Spell-11-limit"></a><!-- ws:end:WikiTextHeadingRule:46 -->11-limit</h2>
Commas: 49/48, 56/55, 125/121<br />
<br />
POTE generator: ~11/10 = 190.285<br />
<br />
Map: [&lt;1 0 2 2 3|, &lt;0 10 2 5 3|]<br />
EDOs: 6, 19, 44de, 63de, 82de<br />
Badness: 0.0598<br />
<br />
<!-- ws:start:WikiTextHeadingRule:48:&lt;h2&gt; --><h2 id="toc24"><a name="Spell-13-limit"></a><!-- ws:end:WikiTextHeadingRule:48 -->13-limit</h2>
Commas: 49/48, 56/55, 78/77, 125/121<br />
<br />
POTE generator: ~11/10 = 189.928<br />
<br />
Map: [&lt;1 0 2 2 3 4|, &lt;0 10 2 5 3 -2|]<br />
EDOs: 6, 19, 82def<br />
Badness: 0.0456<br />
<br />
<!-- ws:start:WikiTextHeadingRule:50:&lt;h2&gt; --><h2 id="toc25"><a name="Spell-Cantrip"></a><!-- ws:end:WikiTextHeadingRule:50 -->Cantrip</h2>
Commas: 49/48, 56/55, 91/90, 125/121<br />
<br />
POTE generator: ~11/10 = 190.360<br />
<br />
Map: [&lt;1 0 2 2 3 1|, &lt;0 10 2 5 3 17|]<br />
EDOs: 19, 44de, 63de, 82de<br />
Badness: 0.0416<br />
<br />
<!-- ws:start:WikiTextHeadingRule:52:&lt;h1&gt; --><h1 id="toc26"><a name="Trismegistus"></a><!-- ws:end:WikiTextHeadingRule:52 -->Trismegistus</h1>
Commas: 3125/3072, 1029/1024<br />
<br />
POTE generator: ~147/100 = 673.290<br />
<br />
Map: [&lt;1 10 4 0|,  &lt;0 -15 -3 5|]<br />
Wedgie: &lt;&lt;15 3 -5 -30 -50 -20||<br />
EDOs: 16, 25, 41, 139c, 180c, 221c, 262c<br />
Badness: 0.0983<br />
<br />
<!-- ws:start:WikiTextHeadingRule:54:&lt;h2&gt; --><h2 id="toc27"><a name="Trismegistus-11-limit"></a><!-- ws:end:WikiTextHeadingRule:54 -->11-limit</h2>
Commas: 385/384, 441/440, 625/616<br />
<br />
POTE generator: ~22/15 = 673.340<br />
<br />
Map: [&lt;1 10 4 0 13|,  &lt;0 -15 -3 5 -17|]<br />
EDOs: 16, 25e, 41, 98c, 139c, 180c<br />
Badness: 0.0456<br />
<br />
<!-- ws:start:WikiTextHeadingRule:56:&lt;h2&gt; --><h2 id="toc28"><a name="Trismegistus-13-limit"></a><!-- ws:end:WikiTextHeadingRule:56 -->13-limit</h2>
Commas: 105/104, 144/143, 275/273, 625/616<br />
<br />
POTE generator: ~22/15 = 673.359<br />
<br />
Map: [&lt;1 10 4 0 13 11|,  &lt;0 -15 -3 5 -17 -13|]<br />
EDOs: 16, 25e, 41, 98c, 139cf<br />
Badness: 0.0331<br />
<br />
<!-- ws:start:WikiTextHeadingRule:58:&lt;h1&gt; --><h1 id="toc29"><a name="Quadrimage"></a><!-- ws:end:WikiTextHeadingRule:58 -->Quadrimage</h1>
Commas: 2401/2400, 3125/3072<br />
<br />
POTE generator: ~28/25 = 204.987<br />
<br />
Map: [&lt;1 5 3 4|, &lt;0 -20 -4 -7|]<br />
Wedgie: &lt;&lt;20 4 7 -40 -45 5||<br />
EDOs: 6, 35, 41, 158cd, 199cd, 240cd, 281cd<br />
Badness: 0.1274<br />
<br />
<!-- ws:start:WikiTextHeadingRule:60:&lt;h2&gt; --><h2 id="toc30"><a name="Quadrimage-11-limit"></a><!-- ws:end:WikiTextHeadingRule:60 -->11-limit</h2>
Commas: 245/242, 385/384, 625/616<br />
<br />
POTE generator: ~28/25 = 204.956<br />
<br />
Map: [&lt;1 5 3 4 5|, &lt;0 -20 -4 -7 -9|]<br />
EDOs: 6, 35, 41, 199cde, 240cde, 281cde<br />
Badness: 0.0616<br />
<br />
<!-- ws:start:WikiTextHeadingRule:62:&lt;h2&gt; --><h2 id="toc31"><a name="Quadrimage-13-limit"></a><!-- ws:end:WikiTextHeadingRule:62 -->13-limit</h2>
Commas: 105/104, 144/143, 245/242, 625/616<br />
<br />
POTE generator: ~28/25 = 205.028<br />
<br />
Map: [&lt;1 5 3 4 5 9|, &lt;0 -20 -4 -7 -9 -31|]<br />
EDOs: 41, 117c, 158cd, 199cdef<br />
Badness: 0.0440</body></html>

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