50edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:hearneg|hearneg]] and made on <tt>2014-04-19 12:21:28 UTC</tt>.

: This revision was by author [[User:hearneg|hearneg]] and made on <tt>2014-04-19 12:31:02 UTC</tt>.

: The original revision id was <tt>~~503313280~~</tt>.

: The original revision id was <tt>503313912</tt>.

: The revision comment was: <tt></tt>

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//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 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&50 temperament. It is also the unique equal temperament tempering out both 81/80 and the [[vishnuzma]], 6115295232/6103515625 = |23 6 -14>, 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.

50 tempers out 126/125 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&50 temperament ([[http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&limit=11|Coblack]]). It is also the unique equal temperament tempering out both 81/80 and the [[vishnuzma]], 6115295232/6103515625 = |23 6 -14>, 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.

[[http://www.archive.org/details/harmonicsorphilo00smit|Robert Smith's book online]]

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50edo divides the <a class="wiki_link" href="/octave">octave</a> into 50 equal parts of precisely 24 <a class="wiki_link" href="/cent">cent</a>s each. In the <a class="wiki_link" href="/5-limit">5-limit</a>, it tempers out 81/80, making it a <a class="wiki_link" href="/meantone">meantone</a> system, and in that capacity has historically has drawn some notice. In <a class="wiki_link_ext" href="http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf" rel="nofollow">&quot;Harmonics or the Philosophy of Musical Sounds&quot;</a> (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 <a class="wiki_link" href="/Target%20tunings">least squares</a> tuning for 5-limit meantone. 50, however, is especially interesting from a higher limit point of view. While <a class="wiki_link" href="/31edo">31edo</a> extends meantone with a <a class="wiki_link" href="/7_4">7/4</a> which is nearly pure, 50 has a flat 7/4 but both <a class="wiki_link" href="/11_8">11/8</a> and <a class="wiki_link" href="/13_8">13/8</a> are nearly pure.

50 tempers out 126/125 in the <a class="wiki_link" href="/7-limit">7-limit</a>, indicating it supports septimal meantone; 245/242, 385/384 and 540/539 in the <a class="wiki_link" href="/11-limit">11-limit</a> and 105/104, 144/143 and 196/195 in the <a class="wiki_link" href="/13-limit">13-limit</a>, 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. It is also the unique equal temperament tempering out both 81/80 and the <a class="wiki_link" href="/vishnuzma">vishnuzma</a>, 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.

50 tempers out 126/125 in the <a class="wiki_link" href="/7-limit">7-limit</a>, indicating it supports septimal meantone; 245/242, 385/384 and 540/539 in the <a class="wiki_link" href="/11-limit">11-limit</a> and 105/104, 144/143 and 196/195 in the <a class="wiki_link" href="/13-limit">13-limit</a>, 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 (<a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&amp;limit=11" rel="nofollow">Coblack</a>). It is also the unique equal temperament tempering out both 81/80 and the <a class="wiki_link" href="/vishnuzma">vishnuzma</a>, 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.

<a class="wiki_link_ext" href="http://www.archive.org/details/harmonicsorphilo00smit" rel="nofollow">Robert Smith's book online</a>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:hearneg|hearneg]] and made on <tt>2014-04-19 12:21:28 UTC</tt>.<br>
+: This revision was by author [[User:hearneg|hearneg]] and made on <tt>2014-04-19 12:31:02 UTC</tt>.<br>
-: The original revision id was <tt>503313280</tt>.<br>
+: The original revision id was <tt>503313912</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>
@@ Line 9: / Line 9: @@
 //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.
-tempers out 126/125 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. 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.
+tempers out 126/125 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.
 [[http://www.archive.org/details/harmonicsorphilo00smit|Robert Smith's book online]]
@@ Line 159: / Line 159: @@
 &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;
 &lt;br /&gt;
-tempers out 126/125 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. 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;
+tempers out 126/125 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;br /&gt;
 &lt;a class="wiki_link_ext" href="http://www.archive.org/details/harmonicsorphilo00smit" rel="nofollow"&gt;Robert Smith's book online&lt;/a&gt;&lt;br /&gt;