Magic family

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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*sqrt(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 1 5 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|, <1 0 5 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|, <1 0 5 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|, <1 0 5 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|, <1 0 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|, <1 0 5 12 14 -1|]
EDOs: 19, 22, 31f, 34ef, 41ef
Badness: 0.0255

=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

Original HTML content:

<html><head><title>Magic family</title></head><body><!-- ws:start:WikiTextTocRule:22:&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:22 --><!-- ws:start:WikiTextTocRule:23: --><div style="margin-left: 1em;"><a href="#Five limit magic">Five limit magic</a></div>
<!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><div style="margin-left: 2em;"><a href="#Five limit magic-Seven limit children">Seven limit children</a></div>
<!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><div style="margin-left: 1em;"><a href="#Magic">Magic</a></div>
<!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><div style="margin-left: 2em;"><a href="#Magic-11-limit">11-limit</a></div>
<!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><div style="margin-left: 2em;"><a href="#Magic-13-limit">13-limit</a></div>
<!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><div style="margin-left: 3em;"><a href="#Magic-13-limit-Sorcery">Sorcery</a></div>
<!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><div style="margin-left: 3em;"><a href="#Magic-13-limit-Necromancy">Necromancy</a></div>
<!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><div style="margin-left: 2em;"><a href="#Magic-Telepathy">Telepathy</a></div>
<!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 3em;"><a href="#Magic-Telepathy-13-limit telepathy">13-limit telepathy</a></div>
<!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><div style="margin-left: 1em;"><a href="#Muggles">Muggles</a></div>
<!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 1em;"><a href="#Astrology">Astrology</a></div>
<!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --></div>
<!-- ws:end:WikiTextTocRule:34 --><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*sqrt(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 1 5 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;1 0 5 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;1 0 5 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;1 0 5 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;1 0 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;1 0 5 12 14 -1|]<br />
EDOs: 19, 22, 31f, 34ef, 41ef<br />
Badness: 0.0255<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc9"><a name="Muggles"></a><!-- ws:end:WikiTextHeadingRule:18 -->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:20:&lt;h1&gt; --><h1 id="toc10"><a name="Astrology"></a><!-- ws:end:WikiTextHeadingRule:20 -->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</body></html>