Tetracot family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 289647655 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 309274428 - 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>2012-03-08 21:16:44 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>309274428</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: 7, 27e, 34d, 61de | EDOs: 7, 27e, 34d, 61de | ||
Badness: 0.0238 | Badness: 0.0238 | ||
=Wollemia= | |||
Commas: 126/125, 2240/2187 | |||
POTE generator: ~10/9 = 177.357 | |||
Map: [<1 1 1 0|, <0 4 9 19|] | |||
Wedgie: <<4 9 19 5 19 19|| | |||
EDOs: 27, 61, 88bc, 115bc | |||
Badness: 0.0705 | |||
=Octacot= | =Octacot= | ||
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Badness: 0.0233</pre></div> | Badness: 0.0233</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>Tetracot 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>Tetracot family</title></head><body><!-- ws:start:WikiTextTocRule:30:&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:30 --><!-- ws:start:WikiTextTocRule:31: --><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --> | <a href="#Monkey">Monkey</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --> | <a href="#Bunya">Bunya</a><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --> | <a href="#Modus">Modus</a><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --> | <a href="#Wollemia">Wollemia</a><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --> | <a href="#Octacot">Octacot</a><!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:46 --><br /> | ||
The parent of the <strong>tetracot family</strong> is <strong>tetracot</strong>, the 5-limit temperament <a class="wiki_link" href="/tempering%20out">tempering out</a> 20000/19683 = |5 -9 4&gt;, the minimal diesis or tetracot comma. The dual of this comma is the wedgie &lt;&lt;4 9 5||, which tells us 10/9 is a generator, and that four of them give 3/2. In fact, (10/9)^4 = 20000/19683 * 3/2. We also have (10/9)^9 = (20000/19683)^2 * 5/2. From this it is evident we should flatten the generator a bit, and <a class="wiki_link" href="/34edo">34edo</a> does this and makes for a recommendable tuning. Another possibility is to use (5/2)^(1/9) for a generator. The 13-note MOS gives enough space for eight triads, with the 20-note MOS supplying many more.<br /> | The parent of the <strong>tetracot family</strong> is <strong>tetracot</strong>, the 5-limit temperament <a class="wiki_link" href="/tempering%20out">tempering out</a> 20000/19683 = |5 -9 4&gt;, the minimal diesis or tetracot comma. The dual of this comma is the wedgie &lt;&lt;4 9 5||, which tells us 10/9 is a generator, and that four of them give 3/2. In fact, (10/9)^4 = 20000/19683 * 3/2. We also have (10/9)^9 = (20000/19683)^2 * 5/2. From this it is evident we should flatten the generator a bit, and <a class="wiki_link" href="/34edo">34edo</a> does this and makes for a recommendable tuning. Another possibility is to use (5/2)^(1/9) for a generator. The 13-note MOS gives enough space for eight triads, with the 20-note MOS supplying many more.<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0238<br /> | Badness: 0.0238<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Octacot"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Wollemia"></a><!-- ws:end:WikiTextHeadingRule:22 -->Wollemia</h1> | ||
Commas: 126/125, 2240/2187<br /> | |||
<br /> | |||
POTE generator: ~10/9 = 177.357<br /> | |||
<br /> | |||
Map: [&lt;1 1 1 0|, &lt;0 4 9 19|]<br /> | |||
Wedgie: &lt;&lt;4 9 19 5 19 19||<br /> | |||
EDOs: 27, 61, 88bc, 115bc<br /> | |||
Badness: 0.0705<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Octacot"></a><!-- ws:end:WikiTextHeadingRule:24 -->Octacot</h1> | |||
Octacot cuts the Gordian knot of deciding between the monkey and bunya mappings for 7 by cutting the generator in half and splitting the difference. It adds 245/243 to the normal comma list, and also tempers out 2401/2400. It has wedgie &lt;&lt;8 18 11 10 -5 -25|| and may also be described as 41&amp;68. <a class="wiki_link" href="/68edo">68edo</a> or <a class="wiki_link" href="/109edo">109edo</a> can be used as tunings, as can (5/2)^(1/18), which gives just major thirds. Another tuning is <a class="wiki_link" href="/150edo">150edo</a>, which has a generator, 11/150, of exactly 88 cents. This relates octacot to the <a class="wiki_link" href="/88cET">88cET</a> non-octave temperament, which like <a class="wiki_link" href="/Carlos%20Alpha">Carlos Alpha</a> arguably makes more sense viewed as part of a rank two temperament with octaves rather than rank one without them.<br /> | Octacot cuts the Gordian knot of deciding between the monkey and bunya mappings for 7 by cutting the generator in half and splitting the difference. It adds 245/243 to the normal comma list, and also tempers out 2401/2400. It has wedgie &lt;&lt;8 18 11 10 -5 -25|| and may also be described as 41&amp;68. <a class="wiki_link" href="/68edo">68edo</a> or <a class="wiki_link" href="/109edo">109edo</a> can be used as tunings, as can (5/2)^(1/18), which gives just major thirds. Another tuning is <a class="wiki_link" href="/150edo">150edo</a>, which has a generator, 11/150, of exactly 88 cents. This relates octacot to the <a class="wiki_link" href="/88cET">88cET</a> non-octave temperament, which like <a class="wiki_link" href="/Carlos%20Alpha">Carlos Alpha</a> arguably makes more sense viewed as part of a rank two temperament with octaves rather than rank one without them.<br /> | ||
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EDOs: 14c, 27, 41, 68, 109<br /> | EDOs: 14c, 27, 41, 68, 109<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Octacot-11-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->11-limit</h2> | ||
Commas: 100/99, 243/242, 245/242<br /> | Commas: 100/99, 243/242, 245/242<br /> | ||
<br /> | <br /> | ||
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See also: <a class="wiki_link" href="/Chords%20of%20octacot">Chords of octacot</a><br /> | See also: <a class="wiki_link" href="/Chords%20of%20octacot">Chords of octacot</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="Octacot-13-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->13-limit</h2> | ||
Commas: 100/99, 144/143, 196/195, 243/242<br /> | Commas: 100/99, 144/143, 196/195, 243/242<br /> | ||
<br /> | <br /> | ||