Tetracot family: Difference between revisions
Wikispaces>Kosmorsky **Imported revision 319222690 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 319241638 - 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:genewardsmith|genewardsmith]] and made on <tt>2012-04-10 18:12:03 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>319241638</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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=Ponens= | =Ponens= | ||
The error of 11 is about the same as that of Modus, but flat instead of sharp, and much more abundant. Since the other primes are all sharp, however, this leads to a much larger error for other intervals involving 11. | |||
POTE generator: ~ | Commas: 55/54, 64/63, 363/350 | ||
POTE generator: ~10/9 = 177.200 | |||
Map: [<1 1 1 4 3|, <0 4 9 -8 3|] | |||
EDOs: 7, 20c, 27, 61de, 88bcde | |||
Badness: 0.0631 | |||
==13-limit== | |||
Commas: 55/54, 64/63, 66/65, 143/140 | |||
POTE generator: ~10/9 = 177.197 | |||
Map: [<1 1 1 4 3 4|, <0 4 9 -8 3 -2|] | Map: [<1 1 1 4 3 4|, <0 4 9 -8 3 -2|] | ||
EDOs: 7, 20c, 27, 61de, 88bcde | |||
Badness: 0.039 | |||
=Wollemia= | =Wollemia= | ||
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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:34:&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:34 --><!-- ws:start:WikiTextTocRule:35: --><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --> | <a href="#Monkey">Monkey</a><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | <a href="#Bunya">Bunya</a><!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --> | <a href="#Modus">Modus</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: --> | <a href="#Ponens">Ponens</a><!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --> | <a href="#Wollemia">Wollemia</a><!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --> | <a href="#Octacot">Octacot</a><!-- 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: | <!-- ws:end:WikiTextTocRule:52 -->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:22:&lt;h1&gt; --><h1 id="toc11"><a name="Ponens"></a><!-- ws:end:WikiTextHeadingRule:22 -->Ponens</h1> | <!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Ponens"></a><!-- ws:end:WikiTextHeadingRule:22 -->Ponens</h1> | ||
Commas:<br /> | The error of 11 is about the same as that of Modus, but flat instead of sharp, and much more abundant. Since the other primes are all sharp, however, this leads to a much larger error for other intervals involving 11.<br /> | ||
<br /> | |||
Commas: 55/54, 64/63, 363/350<br /> | |||
<br /> | |||
POTE generator: ~10/9 = 177.200<br /> | |||
<br /> | |||
Map: [&lt;1 1 1 4 3|, &lt;0 4 9 -8 3|]<br /> | |||
EDOs: 7, 20c, 27, 61de, 88bcde<br /> | |||
Badness: 0.0631<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> | |||
Commas: 55/54, 64/63, 66/65, 143/140<br /> | |||
<br /> | <br /> | ||
POTE generator: ~ | POTE generator: ~10/9 = 177.197<br /> | ||
<br /> | <br /> | ||
Map: [&lt;1 1 1 4 3 4|, &lt;0 4 9 -8 3 -2|]<br /> | Map: [&lt;1 1 1 4 3 4|, &lt;0 4 9 -8 3 -2|]<br /> | ||
EDOs: 7, 20c, 27, 61de, 88bcde<br /> | |||
Badness: 0.039<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc13"><a name="Wollemia"></a><!-- ws:end:WikiTextHeadingRule:26 -->Wollemia</h1> | ||
Commas: 126/125, 2240/2187<br /> | Commas: 126/125, 2240/2187<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0705<br /> | Badness: 0.0705<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="Octacot"></a><!-- ws:end:WikiTextHeadingRule:28 -->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 /> | ||
<br /> | <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:30:&lt;h2&gt; --><h2 id="toc15"><a name="Octacot-11-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->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:32:&lt;h2&gt; --><h2 id="toc16"><a name="Octacot-13-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->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 /> | ||