Tetracot family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 204555638 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 213079444 - 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>2011- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-23 00:45:56 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>213079444</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">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. | <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. | |||
[[POTE tuning|POTE generator]]: 176.160 | [[POTE tuning|POTE generator]]: 176.160 | ||
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Since 16/13 is shy of (10/9)^2 by just 325/324, it is likewise natural to extend our winning streak with these temperaments by adding this to the list of commas. This gives us <<4 9 -15 10 -2 ...|| for 13-limit monkey and <<4 9 26 10 -2 ...|| for 13-limit bunya. Once again, 41 is recommended as a tuning for monkey, while banyan can with advantage tune the fifth sharper: 17/116 as a generator with a fifth a cent and a half sharp or 11/75 with a fifth two cents sharp. | Since 16/13 is shy of (10/9)^2 by just 325/324, it is likewise natural to extend our winning streak with these temperaments by adding this to the list of commas. This gives us <<4 9 -15 10 -2 ...|| for 13-limit monkey and <<4 9 26 10 -2 ...|| for 13-limit bunya. Once again, 41 is recommended as a tuning for monkey, while banyan can with advantage tune the fifth sharper: 17/116 as a generator with a fifth a cent and a half sharp or 11/75 with a fifth two cents sharp. | ||
=Monkey= | |||
Commas: 5120/5103, 875/864 | Commas: 5120/5103, 875/864 | ||
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EDOs: 7, 34, 41 | EDOs: 7, 34, 41 | ||
11-limit | ==11-limit== | ||
Commas: 243/242, 385/384, 100/99 | Commas: 243/242, 385/384, 100/99 | ||
| Line 41: | Line 43: | ||
EDOs: 7, 34, 41 | EDOs: 7, 34, 41 | ||
13-limit | ==13-limit== | ||
Commas: 100/99, 105/104, 144/143, 243/242 | Commas: 100/99, 105/104, 144/143, 243/242 | ||
| Line 49: | Line 51: | ||
EDOs: 7, 34, 41 | EDOs: 7, 34, 41 | ||
=Bunya= | |||
Commas: 225/224, 15625/15309 | Commas: 225/224, 15625/15309 | ||
| Line 57: | Line 59: | ||
EDOs: 41, 116 | EDOs: 41, 116 | ||
11-limit | ==11-limit== | ||
Commas: 100/99, 225/224, 1344/1331 | Commas: 100/99, 225/224, 1344/1331 | ||
| Line 65: | Line 67: | ||
EDOs: 41 | EDOs: 41 | ||
13-limit | ==13-limit== | ||
Commas: 100/99, 144/143, 225/224, 243/242 | Commas: 100/99, 144/143, 225/224, 243/242 | ||
| Line 73: | Line 75: | ||
EDOs: 41 | EDOs: 41 | ||
=Octacot= | |||
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 <<8 18 11 10 -5 -25|| and may also be described as 41&68. [[68edo]] or [[109edo]] can be used as tunings, as can (5/2)^(1/18), which gives just major thirds. Another tuning is [[150edo]], which has a generator, 11/150, of exactly 88 cents. This relates octacot to the [[88cET]] non-octave temperament, which like [[Carlos Alpha]] arguably makes more sense viewed as part of a rank two temperament with octaves rather than rank one without them. | 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 <<8 18 11 10 -5 -25|| and may also be described as 41&68. [[68edo]] or [[109edo]] can be used as tunings, as can (5/2)^(1/18), which gives just major thirds. Another tuning is [[150edo]], which has a generator, 11/150, of exactly 88 cents. This relates octacot to the [[88cET]] non-octave temperament, which like [[Carlos Alpha]] arguably makes more sense viewed as part of a rank two temperament with octaves rather than rank one without them. | ||
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EDOs: 27, 41, 68, 109 | EDOs: 27, 41, 68, 109 | ||
11-limit | ==11-limit== | ||
Commas: 100/99, 243/242, 245/242 | Commas: 100/99, 243/242, 245/242 | ||
| Line 93: | Line 95: | ||
EDOs: 41 | EDOs: 41 | ||
13-limit | ==13-limit== | ||
Commas: 100/99, 144/143, 196/195, 243/242 | Commas: 100/99, 144/143, 196/195, 243/242 | ||
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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>The parent of the tetracot family is tetracot, the 5-limit temperament tempering out 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 /> | <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:22:&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:22 --><!-- ws:start:WikiTextTocRule:23: --><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#Monkey">Monkey</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#Bunya">Bunya</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | <a href="#Octacot">Octacot</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | ||
<!-- ws:end:WikiTextTocRule:34 --><br /> | |||
The parent of the tetracot family is tetracot, the 5-limit temperament tempering out 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 /> | ||
| Line 121: | Line 125: | ||
Since 16/13 is shy of (10/9)^2 by just 325/324, it is likewise natural to extend our winning streak with these temperaments by adding this to the list of commas. This gives us &lt;&lt;4 9 -15 10 -2 ...|| for 13-limit monkey and &lt;&lt;4 9 26 10 -2 ...|| for 13-limit bunya. Once again, 41 is recommended as a tuning for monkey, while banyan can with advantage tune the fifth sharper: 17/116 as a generator with a fifth a cent and a half sharp or 11/75 with a fifth two cents sharp. <br /> | Since 16/13 is shy of (10/9)^2 by just 325/324, it is likewise natural to extend our winning streak with these temperaments by adding this to the list of commas. This gives us &lt;&lt;4 9 -15 10 -2 ...|| for 13-limit monkey and &lt;&lt;4 9 26 10 -2 ...|| for 13-limit bunya. Once again, 41 is recommended as a tuning for monkey, while banyan can with advantage tune the fifth sharper: 17/116 as a generator with a fifth a cent and a half sharp or 11/75 with a fifth two cents sharp. <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt; | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Monkey"></a><!-- ws:end:WikiTextHeadingRule:4 -->Monkey</h1> | ||
Commas: 5120/5103, 875/864<br /> | Commas: 5120/5103, 875/864<br /> | ||
<br /> | <br /> | ||
| Line 129: | Line 133: | ||
EDOs: 7, 34, 41<br /> | EDOs: 7, 34, 41<br /> | ||
<br /> | <br /> | ||
11-limit< | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Monkey-11-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit</h2> | ||
Commas: 243/242, 385/384, 100/99<br /> | Commas: 243/242, 385/384, 100/99<br /> | ||
<br /> | <br /> | ||
| Line 137: | Line 141: | ||
EDOs: 7, 34, 41<br /> | EDOs: 7, 34, 41<br /> | ||
<br /> | <br /> | ||
13-limit < | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Monkey-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h2> | ||
Commas: 100/99, 105/104, 144/143, 243/242<br /> | Commas: 100/99, 105/104, 144/143, 243/242<br /> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.622<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 175.622<br /> | ||
| Line 145: | Line 149: | ||
EDOs: 7, 34, 41<br /> | EDOs: 7, 34, 41<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Bunya"></a><!-- ws:end:WikiTextHeadingRule:10 -->Bunya</h1> | ||
Commas: 225/224, 15625/15309<br /> | Commas: 225/224, 15625/15309<br /> | ||
<br /> | <br /> | ||
| Line 153: | Line 157: | ||
EDOs: 41, 116<br /> | EDOs: 41, 116<br /> | ||
<br /> | <br /> | ||
11-limit< | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Bunya-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->11-limit</h2> | ||
Commas: 100/99, 225/224, 1344/1331<br /> | Commas: 100/99, 225/224, 1344/1331<br /> | ||
<br /> | <br /> | ||
| Line 161: | Line 165: | ||
EDOs: 41<br /> | EDOs: 41<br /> | ||
<br /> | <br /> | ||
13-limit< | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Bunya-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h2> | ||
Commas: 100/99, 144/143, 225/224, 243/242<br /> | Commas: 100/99, 144/143, 225/224, 243/242<br /> | ||
<br /> | <br /> | ||
| Line 169: | Line 173: | ||
EDOs: 41<br /> | EDOs: 41<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Octacot"></a><!-- ws:end:WikiTextHeadingRule:16 -->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: 27, 41, 68, 109<br /> | EDOs: 27, 41, 68, 109<br /> | ||
<br /> | <br /> | ||
11-limit< | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Octacot-11-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->11-limit</h2> | ||
Commas: 100/99, 243/242, 245/242<br /> | Commas: 100/99, 243/242, 245/242<br /> | ||
<br /> | <br /> | ||
| Line 189: | Line 193: | ||
EDOs: 41<br /> | EDOs: 41<br /> | ||
<br /> | <br /> | ||
13-limit< | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Octacot-13-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->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 /> | ||