Magic family: Difference between revisions

Revision as of 16:50, 14 November 2010

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This revision was by author xenwolf and made on 2010-11-14 16:50:27 UTC.

The original revision id was 179391683.

The revision comment was: links added

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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 also makes 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]]

==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

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

===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||.

Original HTML content:

<html><head><title>Magic family</title></head><body>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 also makes 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><br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->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:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Seven limit children-Magic"></a><!-- ws:end:WikiTextHeadingRule:2 -->Magic</h3>
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 />
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 />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-Seven limit children-Muggles"></a><!-- ws:end:WikiTextHeadingRule:4 -->Muggles</h3>
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||.</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-11-13 14:31:21 UTC</tt>.<br>
+: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2010-11-14 16:50:27 UTC</tt>.<br>
-: The original revision id was <tt>179194027</tt>.<br>
+: The original revision id was <tt>179391683</tt>.<br>
-: The revision comment was: <tt></tt><br>
+: The revision comment was: <tt>links added</tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
 <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">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 also makes sense and using [[19edo]] or [[22edo]] is always possible.
-Comma: 3125/3072
+[[Comma]]: 3125/3072
 -limit minimax
 [&lt;1 0 0|, &lt;0 1 0|, &lt;2 1/5 0|]
-Eigenmonzos: 2, 3
+[[Eigenmonzo|Eigenmonzos]]: 2, 3
-Algebraic generator: Terzbirat, the positive root of 9x^2-8x-4 = (4+2*sqrt(13))/9; approximately 380.3175 cents.
+Algebraic generator: Terzbirat, the positive root of 9x^2-8x-4 = (4+2*sqrt(13))/9; approximately 380.3175 [[Cent|cents]].
 Map: [&lt;1 0 2|, &lt;0 5 1|]
-Generators: 2, 5/4
+[[Generator|Generators]]: 2, 5/4
-Edos: 6, 16, 19, 22, 41, 60
+[[Edo|Edos]]: [[6edo|6]], [[16edo|16]], [[19edo|19]], [[22edo|22]], [[41edo|41]], [[60edo|60]]
 ==Seven limit children==
@@ Line 34: / Line 34: @@
 and 9 limit minimax
 [|1 0 0 0&gt;, |0 1 0 0&gt;, |2 1/5 0 0&gt;, |-1 12/5 0 0&gt;]
-Eigenmonzos: 2, 3
+[[Eigenmonzo|Eigenmonzos]]: 2, 3
 Algebraic generators: Tirzbirat or Septimage, the real root of 5x^5+4x-20, 380.7604 cents.
 Map: [&lt;1 0 2 -1|, &lt;0 5 1 12|]
-Generators: 2, 5/4
+[[Generator|Generators]]: 2, 5/4
 ===Muggles===
@@ Line 46: / Line 46: @@
 <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;The 5-limit parent comma for the magic family is 3125/3072, the small diesis or magic comma. Its monzo is |-10 -1 5&amp;gt;, and flipping that yields &amp;lt;&amp;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 also makes sense and using &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt; or &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; is always possible.&lt;br /&gt;
 &lt;br /&gt;
-Comma: 3125/3072&lt;br /&gt;
+&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;: 3125/3072&lt;br /&gt;
 &lt;br /&gt;
 -limit minimax&lt;br /&gt;
 [&amp;lt;1 0 0|, &amp;lt;0 1 0|, &amp;lt;2 1/5 0|]&lt;br /&gt;
-Eigenmonzos: 2, 3&lt;br /&gt;
+&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 3&lt;br /&gt;
 &lt;br /&gt;
-Algebraic generator: Terzbirat, the positive root of 9x^2-8x-4 = (4+2*sqrt(13))/9; approximately 380.3175 cents.&lt;br /&gt;
+Algebraic generator: Terzbirat, the positive root of 9x^2-8x-4 = (4+2*sqrt(13))/9; approximately 380.3175 &lt;a class="wiki_link" href="/Cent"&gt;cents&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 Map: [&amp;lt;1 0 2|, &amp;lt;0 5 1|]&lt;br /&gt;
-Generators: 2, 5/4&lt;br /&gt;
+&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 5/4&lt;br /&gt;
-Edos: 6, 16, 19, 22, 41, 60&lt;br /&gt;
+&lt;a class="wiki_link" href="/Edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/6edo"&gt;6&lt;/a&gt;, &lt;a class="wiki_link" href="/16edo"&gt;16&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/60edo"&gt;60&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Seven limit children&lt;/h2&gt;
@@ Line 72: / Line 72: @@
 and 9 limit minimax&lt;br /&gt;
 [|1 0 0 0&amp;gt;, |0 1 0 0&amp;gt;, |2 1/5 0 0&amp;gt;, |-1 12/5 0 0&amp;gt;]&lt;br /&gt;
-Eigenmonzos: 2, 3&lt;br /&gt;
+&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 3&lt;br /&gt;
 &lt;br /&gt;
 Algebraic generators: Tirzbirat or Septimage, the real root of 5x^5+4x-20, 380.7604 cents.&lt;br /&gt;
 &lt;br /&gt;
 Map: [&amp;lt;1 0 2 -1|, &amp;lt;0 5 1 12|]&lt;br /&gt;
-Generators: 2, 5/4&lt;br /&gt;
+&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 5/4&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Seven limit children-Muggles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Muggles&lt;/h3&gt;
 Aside from 3125/3072 and 525/512 muggles also tempers out 126/125 and 1323/1280. A good muggles tuning is &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, 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 &amp;lt;&amp;lt;5 1 -7 -10 -25 -19||.&lt;/body&gt;&lt;/html&gt;</pre></div>

Magic family: Difference between revisions

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