50edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 491210938 - Original comment: ** |
Wikispaces>clamengh **Imported revision 492994218 - 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:clamengh|clamengh]] and made on <tt>2014-03-02 10:34:56 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>492994218</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]] | <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]] | ||
//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. | //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. | ||
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[[http://www.music.ed.ac.uk/russell/conference/robertsmithkirckman.html|More information about Robert Smith's temperament]] | [[http://www.music.ed.ac.uk/russell/conference/robertsmithkirckman.html|More information about Robert Smith's temperament]] | ||
=Relations= | =Relations= | ||
The 50-edo system is related to [[7edo]], [[12edo]], [[19edo]], [[31edo]] as the next approximation to the "Golden Tone System" ([[Das Goldene Tonsystem]]) of Thorvald Kornerup. | The 50-edo system is related to [[7edo]], [[12edo]], [[19edo]], [[31edo]] as the next approximation to the "Golden Tone System" ([[Das Goldene Tonsystem]]) of Thorvald Kornerup. | ||
=Intervals= | =Intervals= | ||
|| Degrees of 50-EDO || Cents value || | || Degrees of 50-EDO || Cents value || | ||
|| 0 || 0 || | || 0 || 0 || | ||
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|| 49 || 1176 || | || 49 || 1176 || | ||
==Intervals by patent val error== | ==Intervals by patent val error== | ||
|| Interval || Error || | || Interval || Error || | ||
|| 16/13 || 0.528 || | || 16/13 || 0.528 || | ||
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[[@http://soonlabel.com/xenharmonic/archives/1118|Fantasia Catalana by Claudi Meneghin]] | [[@http://soonlabel.com/xenharmonic/archives/1118|Fantasia Catalana by Claudi Meneghin]] | ||
<span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;">| -4 4 -1 > 21.51 81/80 syntonic comma, Didymus comma</span> | <span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;">| -4 4 -1 > 21.51 81/80 syntonic comma, Didymus comma</span> | ||
[[http://soonlabel.com/xenharmonic/archives/1929|Fugue on the Dragnet theme by Claudi Meneghin]] | |||
<span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;">| -8 8 -2 > 43.01 6561/6400 Mathieu superdiesis</span> | <span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;">| -8 8 -2 > 43.01 6561/6400 Mathieu superdiesis</span> | ||
<span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;">| 23 6 -14 > 3.34 1212717/1210381 Vishnu comma</span> | <span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;">| 23 6 -14 > 3.34 1212717/1210381 Vishnu comma</span> | ||
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<!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><div style="margin-left: 3em;"><a href="#Commas--Name2">Name2</a></div> | <!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><div style="margin-left: 3em;"><a href="#Commas--Name2">Name2</a></div> | ||
<!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --></div> | <!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --></div> | ||
<!-- ws:end:WikiTextTocRule:28 --> | <!-- ws:end:WikiTextTocRule:28 --><em>50edo</em> 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.<br /> | ||
<em>50edo</em> 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.<br /> | |||
<br /> | <br /> | ||
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.<br /> | 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.<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Relations"></a><!-- ws:end:WikiTextHeadingRule:0 -->Relations</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Relations"></a><!-- ws:end:WikiTextHeadingRule:0 -->Relations</h1> | ||
The 50-edo system is related to <a class="wiki_link" href="/7edo">7edo</a>, <a class="wiki_link" href="/12edo">12edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/31edo">31edo</a> as the next approximation to the &quot;Golden Tone System&quot; (<a class="wiki_link" href="/Das%20Goldene%20Tonsystem">Das Goldene Tonsystem</a>) of Thorvald Kornerup.<br /> | The 50-edo system is related to <a class="wiki_link" href="/7edo">7edo</a>, <a class="wiki_link" href="/12edo">12edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/31edo">31edo</a> as the next approximation to the &quot;Golden Tone System&quot; (<a class="wiki_link" href="/Das%20Goldene%20Tonsystem">Das Goldene Tonsystem</a>) of Thorvald Kornerup.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h1> | ||
<table class="wiki_table"> | <table class="wiki_table"> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Intervals-Intervals by patent val error"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals by patent val error</h2> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Intervals-Intervals by patent val error"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals by patent val error</h2> | ||
<table class="wiki_table"> | <table class="wiki_table"> | ||
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<a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/1118" rel="nofollow" target="_blank">Fantasia Catalana by Claudi Meneghin</a><br /> | <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/1118" rel="nofollow" target="_blank">Fantasia Catalana by Claudi Meneghin</a><br /> | ||
<span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;">| -4 4 -1 &gt; 21.51 81/80 syntonic comma, Didymus comma</span><br /> | <span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;">| -4 4 -1 &gt; 21.51 81/80 syntonic comma, Didymus comma</span><br /> | ||
<a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/1929" rel="nofollow">Fugue on the Dragnet theme by Claudi Meneghin</a><br /> | |||
<span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;">| -8 8 -2 &gt; 43.01 6561/6400 Mathieu superdiesis</span><br /> | <span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;">| -8 8 -2 &gt; 43.01 6561/6400 Mathieu superdiesis</span><br /> | ||
<span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;">| 23 6 -14 &gt; 3.34 1212717/1210381 Vishnu comma</span><br /> | <span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;">| 23 6 -14 &gt; 3.34 1212717/1210381 Vishnu comma</span><br /> | ||