Augmented family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 326260872 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 332171402 - 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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-05-08 21:16:32 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>332171402</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: 6, 21, 27e | EDOs: 6, 21, 27e | ||
Badness: 0.0423 | Badness: 0.0423 | ||
=Hemisemiaug= | |||
Commas: 128/125, 12005/11664 | |||
POTE generator: ~15/14 = 144.515 | |||
Map: [<6 1 14 4|, <0 2 0 3|] | |||
Wedgie: <<12 0 18 -28 -5 42|| | |||
Edos: 18, 24, 42, 66c, 108bc | |||
Badness: 0.2110 | |||
==11-limit== | |||
Commas: 56/55, 128/125, 3773/3645 | |||
POTE generator: ~15/14 = 144.780 | |||
Map: [<6 1 14 4 8|, <0 2 0 3 3|] | |||
EDOs: 24, 42e, 66ce, 108bce | |||
Badness: 0.0807 | |||
=Niner= | =Niner= | ||
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Map: [<9 0 21 11 17 19|, <0 1 0 1 1 1|] | Map: [<9 0 21 11 17 19|, <0 1 0 1 1 1|] | ||
EDOs: 9, 27e, 36, 63ce | EDOs: 9, 27e, 36, 63ce | ||
Badness: 0.0241 | Badness: 0.0241</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:56:&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:56 --><!-- ws:start:WikiTextTocRule:57: --><div style="margin-left: 1em;"><a href="#Seven limit children">Seven limit children</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><div style="margin-left: 1em;"><a href="#August">August</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><div style="margin-left: 2em;"><a href="#August-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><div style="margin-left: 2em;"><a href="#August-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --><div style="margin-left: 2em;"><a href="#August-Augustus">Augustus</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --><div style="margin-left: 1em;"><a href="#Augene">Augene</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><div style="margin-left: 2em;"><a href="#Augene-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><div style="margin-left: 3em;"><a href="#Augene-11-limit-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><div style="margin-left: 3em;"><a href="#Augene-11-limit-Ogene">Ogene</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><div style="margin-left: 2em;"><a href="#Augene-Eugene">Eugene</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><div style="margin-left: 1em;"><a href="#Inflated">Inflated</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --><div style="margin-left: 2em;"><a href="#Inflated-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><div style="margin-left: 1em;"><a href="#Deflated">Deflated</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --><div style="margin-left: 1em;"><a href="#Hexe">Hexe</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><div style="margin-left: 1em;"><a href="#Hemiaug">Hemiaug</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><div style="margin-left: 2em;"><a href="#Hemiaug-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><div style="margin-left: 2em;"><a href="#Hemiaug-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><div style="margin-left: 1em;"><a href="#Triforce">Triforce</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><div style="margin-left: 2em;"><a href="#Triforce-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --><div style="margin-left: 2em;"><a href="#Triforce-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:76 --><!-- ws:start:WikiTextTocRule:77: --><div style="margin-left: 1em;"><a href="#Oodako">Oodako</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --><div style="margin-left: 2em;"><a href="#Oodako-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --><div style="margin-left: 2em;"><a href="#Oodako-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --><div style="margin-left: 1em;"><a href="#Hemisemiaug">Hemisemiaug</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><div style="margin-left: 2em;"><a href="#Hemisemiaug-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><div style="margin-left: 1em;"><a href="#Niner">Niner</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><div style="margin-left: 2em;"><a href="#Niner-11-limit">11-limit</a></div> | ||
< | <!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><div style="margin-left: 2em;"><a href="#Niner-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --></div> | |||
<!-- ws:end:WikiTextTocRule:85 --><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 /> | ||
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Badness: 0.0202<br /> | Badness: 0.0202<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Triforce-11-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->11-limit</h2> | <!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Triforce-11-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->11-limit</h2> | ||
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Badness: 0.0423<br /> | Badness: 0.0423<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:46:&lt;h1&gt; --><h1 id="toc23"><a name="Niner"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:46:&lt;h1&gt; --><h1 id="toc23"><a name="Hemisemiaug"></a><!-- ws:end:WikiTextHeadingRule:46 -->Hemisemiaug</h1> | ||
Commas: 128/125, 12005/11664<br /> | |||
<br /> | |||
POTE generator: ~15/14 = 144.515<br /> | |||
<br /> | |||
Map: [&lt;6 1 14 4|, &lt;0 2 0 3|]<br /> | |||
Wedgie: &lt;&lt;12 0 18 -28 -5 42||<br /> | |||
Edos: 18, 24, 42, 66c, 108bc<br /> | |||
Badness: 0.2110<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:48:&lt;h2&gt; --><h2 id="toc24"><a name="Hemisemiaug-11-limit"></a><!-- ws:end:WikiTextHeadingRule:48 -->11-limit</h2> | |||
Commas: 56/55, 128/125, 3773/3645<br /> | |||
<br /> | |||
POTE generator: ~15/14 = 144.780<br /> | |||
<br /> | |||
Map: [&lt;6 1 14 4 8|, &lt;0 2 0 3 3|]<br /> | |||
EDOs: 24, 42e, 66ce, 108bce<br /> | |||
Badness: 0.0807<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:50:&lt;h1&gt; --><h1 id="toc25"><a name="Niner"></a><!-- ws:end:WikiTextHeadingRule:50 -->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:52:&lt;h2&gt; --><h2 id="toc26"><a name="Niner-11-limit"></a><!-- ws:end:WikiTextHeadingRule:52 -->11-limit</h2> | ||
Commas: 56/55, 128/125, 540/539<br /> | Commas: 56/55, 128/125, 540/539<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0349<br /> | Badness: 0.0349<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:54:&lt;h2&gt; --><h2 id="toc27"><a name="Niner-13-limit"></a><!-- ws:end:WikiTextHeadingRule:54 -->13-limit</h2> | ||
Commas: 56/55, 78/77, 91/90, 128/125<br /> | Commas: 56/55, 78/77, 91/90, 128/125<br /> | ||
<br /> | <br /> | ||