Magic family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 396378808 - Original comment: ** |
Wikispaces>x31eq **Imported revision 401588902 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:x31eq|x31eq]] and made on <tt>2013-01-26 06:54:49 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>401588902</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<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> | <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">[[toc]] | <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]] | ||
A magic temperament is optimal, for some searches, in the 9-limit. It has slightly higher complexity than 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. | |||
=Five limit magic= | =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 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>, 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 and using [[19edo]] or [[22edo]] is always possible. | ||
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Badness: 0.0562 | Badness: 0.0562 | ||
==11-limit== | ==11-limit== | ||
Commas: 45/44, 126/125, 385/384 | Commas: 45/44, 126/125, 385/384 | ||
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Badness: 0.0480 | Badness: 0.0480 | ||
==13-limit== | ==13-limit== | ||
Commas: 45/44, 65/64, 78/77, 126/125 | Commas: 45/44, 65/64, 78/77, 126/125 | ||
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<!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><div style="margin-left: 2em;"><a href="#Quadrimage-13-limit">13-limit</a></div> | <!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><div style="margin-left: 2em;"><a href="#Quadrimage-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --></div> | <!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --></div> | ||
<!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Five limit magic"></a><!-- ws:end:WikiTextHeadingRule:0 -->Five limit magic</h1> | <!-- ws:end:WikiTextTocRule:106 --><br /> | ||
A magic temperament is optimal, for some searches, in the 9-limit. It has slightly higher complexity than 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 <a class="wiki_link" href="/12edo">12edo</a>. Properties may depend on tuning and extension.<br /> | |||
<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 and using <a class="wiki_link" href="/19edo">19edo</a> or <a class="wiki_link" href="/22edo">22edo</a> is always possible.<br /> | 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 <a class="wiki_link" href="/19edo">19edo</a> or <a class="wiki_link" href="/22edo">22edo</a> is always possible.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Muggles-11-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->11-limit</h2> | <!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Muggles-11-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->11-limit</h2> | ||
Commas: 45/44, 126/125, 385/384<br /> | Commas: 45/44, 126/125, 385/384<br /> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~5/4 = 377.724<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~5/4 = 377.724<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Muggles-13-limit"></a><!-- ws:end:WikiTextHeadingRule:34 -->13-limit</h2> | <!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Muggles-13-limit"></a><!-- ws:end:WikiTextHeadingRule:34 -->13-limit</h2> | ||
Commas: 45/44, 65/64, 78/77, 126/125<br /> | Commas: 45/44, 65/64, 78/77, 126/125<br /> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~5/4 = 377.724<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~5/4 = 377.724<br /> | ||