50edo: Difference between revisions

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-04-19 16:33:23 UTC</tt>.<br>

: The original revision id was <tt>503334194</tt>.<br>

//50edo// divides the [[octave]] into 50 equal parts of precisely 24 [[cent]]s each. In the [[5-limit]], it tempers out 81/80, making it a [[meantone]] system, and in that capacity has historically has drawn some notice. In [[http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf|"Harmonics or the Philosophy of Musical Sounds"]] (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts - 50edo, in one word. Later W. S. B. Woolhouse noted it was fairly close to the [[Target tunings|least squares]] tuning for 5-limit meantone. 50, however, is especially interesting from a higher limit point of view. While [[31edo]] extends meantone with a [[7_4|7/4]] which is nearly pure, 50 has a flat 7/4 but both [[11_8|11/8]] and [[13_8|13/8]] are nearly pure.

50 tempers out 126/125, 225/224 and 3136/3125 in the [[7-limit]], indicating it supports septimal meantone; 245/242, 385/384 and 540/539 in the [[11-limit]] and 105/104, 144/143 and 196/195 in the [[13-limit]], and can be used for even higher limits. Aside from meantone and its extension meanpop, it can be used to advantage for the 15&amp;50 temperament ([[http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&amp;limit=11|Coblack]]). It is also the unique equal temperament tempering out both 81/80 and the [[vishnuzma]], 6115295232/6103515625 = |23 6 -14&gt;, so that in 50et seven chromatic semitones are a perfect fourth. In 12et by comparison this gives a fifth, in 31et a doubly diminished fifth, and in 19et a diminished fourth.

|| 11 || 264 || 7/6 ||   ||

50 EDO tempers out the following commas. (Note: This assumes the val &lt; 50 79 116 140 173 185 204 212 226 |, comma values rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2.

||~ ===In ket format=== ||~ ===In cents=== ||~ ===Ratio=== ||~ ===Name 1=== ||~ ===Name2=== ||

&lt;span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;"&gt;| 1 2 -3 1 &gt; 13.79 126/125 small septimal comma&lt;/span&gt;

&lt;span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;"&gt;| -5 2 2 -1 &gt; 7.71 225/224 septimal kleisma&lt;/span&gt;

<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;50edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:18:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Relations"&gt;Relations&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Intervals-Intervals by patent val error"&gt;Intervals by patent val error&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Commas--In ket format"&gt;In ket format&lt;/a&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;&lt;/div&gt;

&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;em&gt;50edo&lt;/em&gt; divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; into 50 equal parts of precisely 24 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s each. In the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;, it tempers out 81/80, making it a &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; system, and in that capacity has historically has drawn some notice. In &lt;a class="wiki_link_ext" href="http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf" rel="nofollow"&gt;&amp;quot;Harmonics or the Philosophy of Musical Sounds&amp;quot;&lt;/a&gt; (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts - 50edo, in one word. Later W. S. B. Woolhouse noted it was fairly close to the &lt;a class="wiki_link" href="/Target%20tunings"&gt;least squares&lt;/a&gt; tuning for 5-limit meantone. 50, however, is especially interesting from a higher limit point of view. While &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; extends meantone with a &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt; which is nearly pure, 50 has a flat 7/4 but both &lt;a class="wiki_link" href="/11_8"&gt;11/8&lt;/a&gt; and &lt;a class="wiki_link" href="/13_8"&gt;13/8&lt;/a&gt; are nearly pure.&lt;br /&gt;

50 tempers out 126/125, 225/224 and 3136/3125 in the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt;, indicating it supports septimal meantone; 245/242, 385/384 and 540/539 in the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; and 105/104, 144/143 and 196/195 in the &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt;, and can be used for even higher limits. Aside from meantone and its extension meanpop, it can be used to advantage for the 15&amp;amp;50 temperament (&lt;a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&amp;amp;limit=11" rel="nofollow"&gt;Coblack&lt;/a&gt;). It is also the unique equal temperament tempering out both 81/80 and the &lt;a class="wiki_link" href="/vishnuzma"&gt;vishnuzma&lt;/a&gt;, 6115295232/6103515625 = |23 6 -14&amp;gt;, so that in 50et seven chromatic semitones are a perfect fourth. In 12et by comparison this gives a fifth, in 31et a doubly diminished fifth, and in 19et a diminished fourth.&lt;br /&gt;

         &lt;td&gt;&lt;br /&gt;

  50 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 50 79 116 140 173 185 204 212 226 |, comma values rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2.&lt;br /&gt;

         &lt;th&gt;&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="Commas--In ket format"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;In ket format&lt;/h3&gt;

         &lt;th&gt;&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Commas--In cents"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;In cents&lt;/h3&gt;

         &lt;th&gt;&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="Commas--Ratio"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Ratio&lt;/h3&gt;

         &lt;th&gt;&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="Commas--Name 1"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Name 1&lt;/h3&gt;

         &lt;th&gt;&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="Commas--Name2"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Name2&lt;/h3&gt;

&lt;span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;"&gt;| 1 2 -3 1 &amp;gt; 13.79 126/125 small septimal comma&lt;/span&gt;&lt;br /&gt;

&lt;span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;"&gt;| -5 2 2 -1 &amp;gt; 7.71 225/224 septimal kleisma&lt;/span&gt;&lt;br /&gt;