Tetracot family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 337666180 - Original comment: ** |
Wikispaces>phylingual **Imported revision 346577676 - 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:phylingual|phylingual]] and made on <tt>2012-06-19 20:54:32 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>346577676</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|flat]] | <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|flat]] | ||
The parent of the **tetracot family** is **tetracot**, the 5-limit temperament [[tempering out]] 20000/19683 = |5 -9 4>, the minimal diesis or tetracot comma. The dual of this comma is the wedgie <<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 [[34edo]] 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. | The parent of the **tetracot family** is **tetracot**, the 5-limit temperament [[tempering out]] 20000/19683 = |5 -9 4>, the minimal diesis or tetracot comma. The dual of this comma is the wedgie <<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 [[34edo]] 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. | ||
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Badness: 0.0631 | Badness: 0.0631 | ||
==13-limit== | ==13-limit== | ||
Commas: 55/54, 64/63, 66/65, 143/140 | Commas: 55/54, 64/63, 66/65, 143/140 | ||
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Badness: 0.0705 | Badness: 0.0705 | ||
==11-limit== | ==11-limit== | ||
Commas: 56/55, 100/99, 243/242 | Commas: 56/55, 100/99, 243/242 | ||
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Badness: 0.0376 | Badness: 0.0376 | ||
==13-limit== | ==13-limit== | ||
Commas: 56/55, 91/90, 100/99, 352/351 | Commas: 56/55, 91/90, 100/99, 352/351 | ||
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Badness: 0.0233 | Badness: 0.0233 | ||
==Octocat== | ==Octocat== | ||
Commas: 78/77, 91/90, 100/99, 245/242 | Commas: 78/77, 91/90, 100/99, 245/242 | ||
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Badness: 0.0276 | Badness: 0.0276 | ||
==Octopod== | ==Octopod== | ||
Commas: 100/99 105/104 243/242 245/242 | Commas: 100/99 105/104 243/242 245/242 | ||
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Badness: 0.0283 | Badness: 0.0283 | ||
=Dificot= | =Dificot= | ||
Commas: 100/99, 243/242, 245/242, 343/338 | Commas: 100/99, 243/242, 245/242, 343/338 | ||
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Badness: 0.0519 | Badness: 0.0519 | ||
=Duodecicot= | =Duodecicot= | ||
Commas: 3087/3125, 10976/10935 | Commas: 3087/3125, 10976/10935 | ||
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EDOs: 41, 184, 225, 409bcd | EDOs: 41, 184, 225, 409bcd | ||
Badness: 0.1198 | Badness: 0.1198 | ||
</pre></div> | |||
==Musical Examples== | |||
* [[http://soundcloud.com/dustin-schallert/tetracot-perc-sitar|Tetracot Perc-Sitar]] by Dustin Schallert | |||
* [[http://soundcloud.com/dustin-schallert/tetracot-jam|Tetracot Jam]] by Dustin Schallert</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:48:&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:48 --><!-- ws:start:WikiTextTocRule:49: --><!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --> | <a href="#Monkey">Monkey</a><!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --> | <a href="#Bunya">Bunya</a><!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --> | <a href="#Modus">Modus</a><!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --> | <a href="#Ponens">Ponens</a><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --><!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --> | <a href="#Wollemia">Wollemia</a><!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --> | <a href="#Octacot">Octacot</a><!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --><!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --> | <a href="#Dificot">Dificot</a><!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --> | <a href="#Duodecicot">Duodecicot</a><!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:73 -->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 /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 176.160<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 176.160<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Ponens-13-limit"></a><!-- ws:end:WikiTextHeadingRule:24 -->13-limit</h2> | <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Ponens-13-limit"></a><!-- ws:end:WikiTextHeadingRule:24 -->13-limit</h2> | ||
Commas: 55/54, 64/63, 66/65, 143/140<br /> | Commas: 55/54, 64/63, 66/65, 143/140<br /> | ||
<br /> | <br /> | ||
POTE generator: ~10/9 = 177.197<br /> | POTE generator: ~10/9 = 177.197<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="Wollemia-11-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->11-limit</h2> | <!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="Wollemia-11-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->11-limit</h2> | ||
Commas: 56/55, 100/99, 243/242<br /> | Commas: 56/55, 100/99, 243/242<br /> | ||
<br /> | <br /> | ||
POTE generator: ~10/9 = 177.413<br /> | POTE generator: ~10/9 = 177.413<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Wollemia-13-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->13-limit</h2> | <!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Wollemia-13-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->13-limit</h2> | ||
Commas: 56/55, 91/90, 100/99, 352/351<br /> | Commas: 56/55, 91/90, 100/99, 352/351<br /> | ||
<br /> | <br /> | ||
POTE generator: ~10/9 = 177.231<br /> | POTE generator: ~10/9 = 177.231<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><a name="Octacot-Octocat"></a><!-- ws:end:WikiTextHeadingRule:38 -->Octocat</h2> | <!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><a name="Octacot-Octocat"></a><!-- ws:end:WikiTextHeadingRule:38 -->Octocat</h2> | ||
Commas: 78/77, 91/90, 100/99, 245/242<br /> | Commas: 78/77, 91/90, 100/99, 245/242<br /> | ||
<br /> | <br /> | ||
POTE generator: ~22/21 = 88.179<br /> | POTE generator: ~22/21 = 88.179<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Octacot-Octopod"></a><!-- ws:end:WikiTextHeadingRule:40 -->Octopod</h2> | <!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Octacot-Octopod"></a><!-- ws:end:WikiTextHeadingRule:40 -->Octopod</h2> | ||
Commas: 100/99 105/104 243/242 245/242<br /> | Commas: 100/99 105/104 243/242 245/242<br /> | ||
<br /> | <br /> | ||
POTE generator: ~22/21 = 87.697<br /> | POTE generator: ~22/21 = 87.697<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:42:&lt;h1&gt; --><h1 id="toc21"><a name="Dificot"></a><!-- ws:end:WikiTextHeadingRule:42 -->Dificot</h1> | <!-- ws:start:WikiTextHeadingRule:42:&lt;h1&gt; --><h1 id="toc21"><a name="Dificot"></a><!-- ws:end:WikiTextHeadingRule:42 -->Dificot</h1> | ||
Commas: 100/99, 243/242, 245/242, 343/338<br /> | Commas: 100/99, 243/242, 245/242, 343/338<br /> | ||
<br /> | <br /> | ||
POTE generator: ~13/9 = 643.989<br /> | POTE generator: ~13/9 = 643.989<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:44:&lt;h1&gt; --><h1 id="toc22"><a name="Duodecicot"></a><!-- ws:end:WikiTextHeadingRule:44 -->Duodecicot</h1> | <!-- ws:start:WikiTextHeadingRule:44:&lt;h1&gt; --><h1 id="toc22"><a name="Duodecicot"></a><!-- ws:end:WikiTextHeadingRule:44 -->Duodecicot</h1> | ||
Commas: 3087/3125, 10976/10935<br /> | Commas: 3087/3125, 10976/10935<br /> | ||
<br /> | <br /> | ||
POTE generator: ~28/27 = 58.675<br /> | POTE generator: ~28/27 = 58.675<br /> | ||
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Wedgie: &lt;&lt;12 27 37 15 25 10||<br /> | Wedgie: &lt;&lt;12 27 37 15 25 10||<br /> | ||
EDOs: 41, 184, 225, 409bcd<br /> | EDOs: 41, 184, 225, 409bcd<br /> | ||
Badness: 0.1198</body></html></pre></div> | Badness: 0.1198<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:46:&lt;h2&gt; --><h2 id="toc23"><a name="Duodecicot-Musical Examples"></a><!-- ws:end:WikiTextHeadingRule:46 -->Musical Examples</h2> | |||
<ul><li><a class="wiki_link_ext" href="http://soundcloud.com/dustin-schallert/tetracot-perc-sitar" rel="nofollow">Tetracot Perc-Sitar</a> by Dustin Schallert</li><li><a class="wiki_link_ext" href="http://soundcloud.com/dustin-schallert/tetracot-jam" rel="nofollow">Tetracot Jam</a> by Dustin Schallert</li></ul></body></html></pre></div> | |||