Augmented family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 321203394 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 326105814 - 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:genewardsmith|genewardsmith]] and made on <tt>2012-04- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-04-27 13:41:40 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>326105814</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> | ||
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EDOs: 12, 15, 27e | EDOs: 12, 15, 27e | ||
Badness: 0.0196 | Badness: 0.0196 | ||
===13-limit=== | |||
Commas: 40/39, 56/55, 64/63, 66/65 | |||
POTE generator: ~3/2 = 712.013 | |||
Map: [<3 0 7 18 20 16|, <0 1 0 -2 -2 -1|] | |||
EDOs: 12f, 15, 27ef | |||
Badness: 0.0208 | |||
===Ogene=== | |||
Commas: 56/55, 64/63, 91/90, 100/99 | |||
POTE generator: ~3/2 = 712.609 | |||
Map: [<3 0 7 18 20 -8|, <0 1 0 -2 -2 4|] | |||
EDOs: 12, 15, 27e, 69bcef | |||
Badness: 0.0229 | |||
==Eugene== | ==Eugene== | ||
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Map: [<9 0 21 11 17|, <0 1 0 1 1|] | Map: [<9 0 21 11 17|, <0 1 0 1 1|] | ||
EDOs: 9, 27e, 36, 63ce | EDOs: 9, 27e, 36, 63ce | ||
Badness: 0.0349 | Badness: 0.0349</pre></div> | ||
</pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Augmented family</title></head><body><!-- ws:start:WikiTextTocRule: | <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"><html><head><title>Augmented family</title></head><body><!-- ws:start:WikiTextTocRule:42:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><a href="#Seven limit children">Seven limit children</a><!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --> | <a href="#August">August</a><!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --> | <a href="#Augene">Augene</a><!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --> | <a href="#Inflated">Inflated</a><!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --> | <a href="#Deflated">Deflated</a><!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --> | <a href="#Hexe">Hexe</a><!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --> | <a href="#Hemiaug">Hemiaug</a><!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --> | <a href="#Triforce">Triforce</a><!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --><!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --> | <a href="#Niner">Niner</a><!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:64 --><br /> | ||
The 5-limit parent comma for the <strong>augmented family</strong> is 128/125, the <a class="wiki_link" href="/128_125">diesis</a>. Its <a class="wiki_link" href="/monzo">monzo</a> is |7 0 -3&gt;, and flipping that yields &lt;&lt;3 0 -7|| for the <a class="wiki_link" href="/wedgie">wedgie</a>. Hence the period is 1/3 octave, and this is what is used for a major third. The <a class="wiki_link" href="/generator">generator</a> can be taken as a fifth or a semitone, and <a class="wiki_link" href="/12edo">12edo</a>, with its excellent fifth, is an obvious tuning for <a class="wiki_link" href="/5-limit">5-limit</a> augmented, though a sharper fifth might be preferred to go with the sharp third. <br /> | The 5-limit parent comma for the <strong>augmented family</strong> is 128/125, the <a class="wiki_link" href="/128_125">diesis</a>. Its <a class="wiki_link" href="/monzo">monzo</a> is |7 0 -3&gt;, and flipping that yields &lt;&lt;3 0 -7|| for the <a class="wiki_link" href="/wedgie">wedgie</a>. Hence the period is 1/3 octave, and this is what is used for a major third. The <a class="wiki_link" href="/generator">generator</a> can be taken as a fifth or a semitone, and <a class="wiki_link" href="/12edo">12edo</a>, with its excellent fifth, is an obvious tuning for <a class="wiki_link" href="/5-limit">5-limit</a> augmented, though a sharper fifth might be preferred to go with the sharp third. <br /> | ||
<br /> | <br /> | ||
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Badness: 0.0196<br /> | Badness: 0.0196<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id=" | <!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Augene-11-limit-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h3> | ||
Commas: 40/39, 56/55, 64/63, 66/65<br /> | |||
<br /> | |||
POTE generator: ~3/2 = 712.013<br /> | |||
<br /> | |||
Map: [&lt;3 0 7 18 20 16|, &lt;0 1 0 -2 -2 -1|]<br /> | |||
EDOs: 12f, 15, 27ef<br /> | |||
Badness: 0.0208<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Augene-11-limit-Ogene"></a><!-- ws:end:WikiTextHeadingRule:16 -->Ogene</h3> | |||
Commas: 56/55, 64/63, 91/90, 100/99<br /> | |||
<br /> | |||
POTE generator: ~3/2 = 712.609<br /> | |||
<br /> | |||
Map: [&lt;3 0 7 18 20 -8|, &lt;0 1 0 -2 -2 4|]<br /> | |||
EDOs: 12, 15, 27e, 69bcef<br /> | |||
Badness: 0.0229<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Augene-Eugene"></a><!-- ws:end:WikiTextHeadingRule:18 -->Eugene</h2> | |||
Commas: 55/54, 64/63, 77/75 <br /> | Commas: 55/54, 64/63, 77/75 <br /> | ||
<br /> | <br /> | ||
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<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3" rel="nofollow">Galticeran Sonatina</a> by <a class="wiki_link_ext" href="http://soundcloud.com/joelgranttaylor/galticeran_sonatina" rel="nofollow">Joel Taylor</a><br /> | <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3" rel="nofollow">Galticeran Sonatina</a> by <a class="wiki_link_ext" href="http://soundcloud.com/joelgranttaylor/galticeran_sonatina" rel="nofollow">Joel Taylor</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Inflated"></a><!-- ws:end:WikiTextHeadingRule:20 -->Inflated</h1> | ||
Commas: 28/27, 128/125<br /> | Commas: 28/27, 128/125<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0547<br /> | Badness: 0.0547<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Inflated-11-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->11-limit</h2> | ||
Commas: 28/27, 55/54, 128/125<br /> | Commas: 28/27, 55/54, 128/125<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0312<br /> | Badness: 0.0312<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Deflated"></a><!-- ws:end:WikiTextHeadingRule:24 -->Deflated</h1> | ||
Commas: 21/20, 128/125<br /> | Commas: 21/20, 128/125<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0591<br /> | Badness: 0.0591<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc13"><a name="Hexe"></a><!-- ws:end:WikiTextHeadingRule:26 -->Hexe</h1> | ||
Commas: 50/49, 128/125<br /> | Commas: 50/49, 128/125<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0577<br /> | Badness: 0.0577<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="Hemiaug"></a><!-- ws:end:WikiTextHeadingRule:28 -->Hemiaug</h1> | ||
Commas: 128/125, 245/243<br /> | Commas: 128/125, 245/243<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0705<br /> | Badness: 0.0705<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Hemiaug-11-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->11-limit</h2> | ||
Commas: 56/55, 128/125, 245/243<br /> | Commas: 56/55, 128/125, 245/243<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0382<br /> | Badness: 0.0382<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc16"><a name="Triforce"></a><!-- ws:end:WikiTextHeadingRule:32 -->Triforce</h1> | ||
Commas: 49/48, 128/125<br /> | Commas: 49/48, 128/125<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0202<br /> | Badness: 0.0202<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:65:&lt;img src=&quot;/file/view/triforce9.jpg/271381934/triforce9.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/triforce9.jpg/271381934/triforce9.jpg" alt="triforce9.jpg" title="triforce9.jpg" /><!-- ws:end:WikiTextLocalImageRule:65 --><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Triforce-11-limit"></a><!-- ws:end:WikiTextHeadingRule:34 -->11-limit</h2> | ||
Commas: 56/55, 77/75, 128/125<br /> | Commas: 56/55, 77/75, 128/125<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0262<br /> | Badness: 0.0262<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Triforce-13-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->13-limit</h2> | ||
Commas: 49/48, 56/55, 66/65, 77/75<br /> | Commas: 49/48, 56/55, 66/65, 77/75<br /> | ||
<br /> | <br /> | ||
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<a class="wiki_link" href="/triphi">Triforce(9) with L:s = phi</a><br /> | <a class="wiki_link" href="/triphi">Triforce(9) with L:s = phi</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:38:&lt;h1&gt; --><h1 id="toc19"><a name="Niner"></a><!-- ws:end:WikiTextHeadingRule:38 -->Niner</h1> | ||
Niner gives 9 as the complexity of an otonal tetrad, tying it with augene as a temperament supported by 27edo. Niner[18], therefore, has nine such tetrads.<br /> | Niner gives 9 as the complexity of an otonal tetrad, tying it with augene as a temperament supported by 27edo. Niner[18], therefore, has nine such tetrads.<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0672<br /> | Badness: 0.0672<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Niner-11-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->11-limit</h2> | ||
Commas: 56/55, 128/125, 540/539<br /> | Commas: 56/55, 128/125, 540/539<br /> | ||
<br /> | <br /> | ||