15edt: Difference between revisions
Wikispaces>genewardsmith **Imported revision 250637172 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 314392094 - 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:mbattaglia1|mbattaglia1]] and made on <tt>2012-03-25 15:07:13 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>314392094</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]] | ||
=<span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; line-height: normal;">15 Equal Divisions of the Tritave</span>= | =<span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; line-height: normal;">15 Equal Divisions of the Tritave</span>= | ||
=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. 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= | ||
|| Degrees || Cents || Approximate Ratios || | || Degrees || Cents || Approximate Ratios || | ||
|| 0 || 0 || <span style="color: #660000;">[[1_1|1/1]]</span> || | || 0 || 0 || <span style="color: #660000;">[[1_1|1/1]]</span> || | ||
| Line 23: | Line 22: | ||
|| 6 || 760.782 || <span style="color: #660000;">[[14_9|14/9]]</span> || | || 6 || 760.782 || <span style="color: #660000;">[[14_9|14/9]]</span> || | ||
|| 7 || 887.579 || [[5_3|5/3]] || | || 7 || 887.579 || [[5_3|5/3]] || | ||
|| 8 || 1014.376 || [[9_5|9/5]] | || 8 || 1014.376 || [[9_5|9/5]] || | ||
|| 9 || 1141.173 || <span style="color: #660000;">[[27_14|27/14]]</span> || | || 9 || 1141.173 || <span style="color: #660000;">[[27_14|27/14]]</span> || | ||
|| 10 || 1267.970 || 27/13 || | || 10 || 1267.970 || 27/13 || | ||
| Line 38: | Line 37: | ||
15edt approximates the 5th and 13th harmonics (and 29th) very well. Taking these as consonances one obtains an 3L+3s MOS "augmented scale", in which three 13/9 intervals close to a tritave, and another three are set 5/3 away. | 15edt approximates the 5th and 13th harmonics (and 29th) very well. Taking these as consonances one obtains an 3L+3s MOS "augmented scale", in which three 13/9 intervals close to a tritave, and another three are set 5/3 away. | ||
=Z function= | =Z function= | ||
Below is a plot of the [[The Riemann Zeta Function and Tuning#Removing | 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]]</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: --> | ||
<!-- ws:end:WikiTextTocRule:13 --> | <!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x15 Equal Divisions of the Tritave"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; line-height: normal;">15 Equal Divisions of the Tritave</span></h1> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x15 Equal Divisions of the Tritave"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; line-height: normal;">15 Equal Divisions of the Tritave</span></h1> | |||
<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. 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> | ||
<table class="wiki_table"> | <table class="wiki_table"> | ||
| Line 131: | Line 129: | ||
<td>1014.376<br /> | <td>1014.376<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/9_5">9/5</a> | <td><a class="wiki_link" href="/9_5">9/5</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 200: | Line 198: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Z function"></a><!-- ws:end:WikiTextHeadingRule:6 -->Z function</h1> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Z function"></a><!-- ws:end:WikiTextHeadingRule:6 -->Z function</h1> | ||
Below is a plot of the <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Removing | 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 --></body></html></pre></div> | ||