15edt: Difference between revisions
Wikispaces>xenwolf **Imported revision 314408666 - Original comment: a few links added, most of them are "red" (it seems to me that the restriction to octave-normalized intervals can also be contra-productive)** |
Wikispaces>guest **Imported revision 342518720 - 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:guest|guest]] and made on <tt>2012-06-04 14:05:06 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>342518720</tt>.<br> | ||
: The revision comment was: <tt> | : 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> | ||
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=Properties= | =Properties= | ||
The 15 equal division of 3, the tritave, divides it into 15 equal parts of 126.797 cents each, corresponding to 9.464 edo. It has 5 and 13 closely in tune, but does not do so well for 7 and 11, which are quite sharp. It tempers out the comma |0 22 -15> in the 5-limit, which is tempered out by [[19edo]] but has an [[optimal patent val]] of [[303edo]]. As a 3.5.13 subgroup system, it tempers out 2197/2187 and 3159/3125. In the 7-limit it tempers out 375/343 and 6561/6125, and in the 11-limit, 81/77, 125/121 and 363/343. 15edt is related to the 2.3.5.13 subgroup temperament 19&123, which has a mapping [<1 0 0 0|, <0 15 22 35|], where the generator, an approximate 27/25, has a POTE tuning of 126.773, very close to 15edt. | The 15 equal division of 3, the tritave, divides it into 15 equal parts of 126.797 cents each, corresponding to 9.464 edo, or 18.928 ed4. It has 5 and 13 closely in tune, but does not do so well for 7 and 11, which are quite sharp. It tempers out the comma |0 22 -15> in the 5-limit, which is tempered out by [[19edo]] but has an [[optimal patent val]] of [[303edo]]. As a 3.5.13 subgroup system, it tempers out 2197/2187 and 3159/3125. In the 7-limit it tempers out 375/343 and 6561/6125, and in the 11-limit, 81/77, 125/121 and 363/343. 15edt is related to the 2.3.5.13 subgroup temperament 19&123, which has a mapping [<1 0 0 0|, <0 15 22 35|], where the generator, an approximate 27/25, has a POTE tuning of 126.773, very close to 15edt. | ||
=Intervals of 15edt= | =Intervals of 15edt= | ||
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Below is a plot of the [[The Riemann Zeta Function and Tuning#Removing%20primes|no-twos Z function]] in the vicinity of 15edt: | Below is a plot of the [[The Riemann Zeta Function and Tuning#Removing%20primes|no-twos Z function]] in the vicinity of 15edt: | ||
[[image:15edt.png]]</pre></div> | [[image:15edt.png]] | ||
Music: | |||
http://www.youtube.com/watch?v=bC_Pc4jKm2k</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>15edt</title></head><body><!-- ws:start:WikiTextTocRule:8:&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:8 --><!-- ws:start:WikiTextTocRule:9: --><a href="#x15 Equal Divisions of the Tritave">15 Equal Divisions of the Tritave</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <a href="#Properties">Properties</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | <a href="#Intervals of 15edt">Intervals of 15edt</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Z function">Z function</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <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>15edt</title></head><body><!-- ws:start:WikiTextTocRule:8:&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:8 --><!-- ws:start:WikiTextTocRule:9: --><a href="#x15 Equal Divisions of the Tritave">15 Equal Divisions of the Tritave</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <a href="#Properties">Properties</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | <a href="#Intervals of 15edt">Intervals of 15edt</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Z function">Z function</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Properties"></a><!-- ws:end:WikiTextHeadingRule:2 -->Properties</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Properties"></a><!-- ws:end:WikiTextHeadingRule:2 -->Properties</h1> | ||
The 15 equal division of 3, the tritave, divides it into 15 equal parts of 126.797 cents each, corresponding to 9.464 edo. It has 5 and 13 closely in tune, but does not do so well for 7 and 11, which are quite sharp. It tempers out the comma |0 22 -15&gt; in the 5-limit, which is tempered out by <a class="wiki_link" href="/19edo">19edo</a> but has an <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> of <a class="wiki_link" href="/303edo">303edo</a>. As a 3.5.13 subgroup system, it tempers out 2197/2187 and 3159/3125. In the 7-limit it tempers out 375/343 and 6561/6125, and in the 11-limit, 81/77, 125/121 and 363/343. 15edt is related to the 2.3.5.13 subgroup temperament 19&amp;123, which has a mapping [&lt;1 0 0 0|, &lt;0 15 22 35|], where the generator, an approximate 27/25, has a POTE tuning of 126.773, very close to 15edt.<br /> | The 15 equal division of 3, the tritave, divides it into 15 equal parts of 126.797 cents each, corresponding to 9.464 edo, or 18.928 ed4. It has 5 and 13 closely in tune, but does not do so well for 7 and 11, which are quite sharp. It tempers out the comma |0 22 -15&gt; in the 5-limit, which is tempered out by <a class="wiki_link" href="/19edo">19edo</a> but has an <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> of <a class="wiki_link" href="/303edo">303edo</a>. As a 3.5.13 subgroup system, it tempers out 2197/2187 and 3159/3125. In the 7-limit it tempers out 375/343 and 6561/6125, and in the 11-limit, 81/77, 125/121 and 363/343. 15edt is related to the 2.3.5.13 subgroup temperament 19&amp;123, which has a mapping [&lt;1 0 0 0|, &lt;0 15 22 35|], where the generator, an approximate 27/25, has a POTE tuning of 126.773, very close to 15edt.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Intervals of 15edt"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals of 15edt</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Intervals of 15edt"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals of 15edt</h1> | ||
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Below is a plot of the <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Removing%20primes">no-twos Z function</a> in the vicinity of 15edt:<br /> | Below is a plot of the <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Removing%20primes">no-twos Z function</a> in the vicinity of 15edt:<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule:152:&lt;img src=&quot;/file/view/15edt.png/250617832/15edt.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/15edt.png/250617832/15edt.png" alt="15edt.png" title="15edt.png" /><!-- ws:end:WikiTextLocalImageRule:152 --></body></html></pre></div> | <!-- ws:start:WikiTextLocalImageRule:152:&lt;img src=&quot;/file/view/15edt.png/250617832/15edt.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/15edt.png/250617832/15edt.png" alt="15edt.png" title="15edt.png" /><!-- ws:end:WikiTextLocalImageRule:152 --><br /> | ||
<br /> | |||
Music:<br /> | |||
<!-- ws:start:WikiTextUrlRule:259:http://www.youtube.com/watch?v=bC_Pc4jKm2k --><a class="wiki_link_ext" href="http://www.youtube.com/watch?v=bC_Pc4jKm2k" rel="nofollow">http://www.youtube.com/watch?v=bC_Pc4jKm2k</a><!-- ws:end:WikiTextUrlRule:259 --></body></html></pre></div> | |||