50edo: Difference between revisions
Wikispaces>clamengh **Imported revision 491202310 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 491210938 - 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:genewardsmith|genewardsmith]] and made on <tt>2014-02-22 10:58:13 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>491210938</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">//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. | <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. | |||
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. 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. | ||
| Line 13: | Line 15: | ||
[[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= | |||
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= | |||
|| Degrees of 50-EDO || Cents value || | || Degrees of 50-EDO || Cents value || | ||
|| 0 || 0 || | || 0 || 0 || | ||
| Line 69: | Line 71: | ||
|| 49 || 1176 || | || 49 || 1176 || | ||
==Commas | ==Intervals by patent val error== | ||
|| Interval || Error || | |||
|| 16/13 || 0.528 || | |||
|| 15/14 || 0.557 || | |||
|| 11/8 || 0.682 || | |||
|| 13/11 || -1.210 || | |||
|| 13/10 || 1.786 || | |||
|| 5/4 || -2.314 || | |||
|| 7/6 || -2.871 || | |||
|| 11/10 || 2.996 || | |||
|| 9/7 || -3.084 || | |||
|| 6/5 || -3.641 || | |||
|| 13/12 || 5.427 || | |||
|| 4/3 || 5.955 || | |||
|| 7/5 || -6.512 || | |||
|| 12/11 || -6.637 || | |||
|| 15/13 || -7.741 || | |||
|| 16/15 || 8.269 || | |||
|| 14/13 || -8.298 || | |||
|| 8/7 || 8.826 || | |||
|| 15/11 || -8.951 || | |||
|| 14/11 || -9.508 || | |||
|| 10/9 || 9.596 || | |||
|| 18/13 || -11.382 || | |||
|| 9/8 || -11.910 || | |||
|| 11/9 || 12.592 || | |||
=Commas= | |||
50 EDO tempers out the following commas. (Note: This assumes the val < 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. | 50 EDO tempers out the following commas. (Note: This assumes the val < 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=== || | ||~ ===In ket format=== ||~ ===In cents=== ||~ ===Ratio=== ||~ ===Name 1=== ||~ ===Name2=== || | ||
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|| 6115295232.00 ||</pre></div> | || 6115295232.00 ||</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>50edo</title></head><body><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 /> | <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>50edo</title></head><body><!-- ws:start:WikiTextTocRule:18:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><div style="margin-left: 1em;"><a href="#Relations">Relations</a></div> | ||
<!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><div style="margin-left: 1em;"><a href="#Intervals">Intervals</a></div> | |||
<!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><div style="margin-left: 2em;"><a href="#Intervals-Intervals by patent val error">Intervals by patent val error</a></div> | |||
<!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><div style="margin-left: 1em;"><a href="#Commas">Commas</a></div> | |||
<!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><div style="margin-left: 3em;"><a href="#Commas--In ket format">In ket format</a></div> | |||
<!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><div style="margin-left: 3em;"><a href="#Commas--In cents">In cents</a></div> | |||
<!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><div style="margin-left: 3em;"><a href="#Commas--Ratio">Ratio</a></div> | |||
<!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><div style="margin-left: 3em;"><a href="#Commas--Name 1">Name 1</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:28 --><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 /> | ||
| Line 124: | Line 164: | ||
<a class="wiki_link_ext" href="http://www.music.ed.ac.uk/russell/conference/robertsmithkirckman.html" rel="nofollow">More information about Robert Smith's temperament</a><br /> | <a class="wiki_link_ext" href="http://www.music.ed.ac.uk/russell/conference/robertsmithkirckman.html" rel="nofollow">More information about Robert Smith's temperament</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt; | <!-- 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 /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt; | <!-- 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"> | ||
| Line 440: | Line 480: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name=" | <!-- 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"> | |||
<tr> | |||
<td>Interval<br /> | |||
</td> | |||
<td>Error<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>16/13<br /> | |||
</td> | |||
<td>0.528<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>15/14<br /> | |||
</td> | |||
<td>0.557<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>11/8<br /> | |||
</td> | |||
<td>0.682<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>13/11<br /> | |||
</td> | |||
<td>-1.210<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>13/10<br /> | |||
</td> | |||
<td>1.786<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>5/4<br /> | |||
</td> | |||
<td>-2.314<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>7/6<br /> | |||
</td> | |||
<td>-2.871<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>11/10<br /> | |||
</td> | |||
<td>2.996<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>9/7<br /> | |||
</td> | |||
<td>-3.084<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>6/5<br /> | |||
</td> | |||
<td>-3.641<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>13/12<br /> | |||
</td> | |||
<td>5.427<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>4/3<br /> | |||
</td> | |||
<td>5.955<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>7/5<br /> | |||
</td> | |||
<td>-6.512<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>12/11<br /> | |||
</td> | |||
<td>-6.637<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>15/13<br /> | |||
</td> | |||
<td>-7.741<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>16/15<br /> | |||
</td> | |||
<td>8.269<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>14/13<br /> | |||
</td> | |||
<td>-8.298<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>8/7<br /> | |||
</td> | |||
<td>8.826<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>15/11<br /> | |||
</td> | |||
<td>-8.951<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>14/11<br /> | |||
</td> | |||
<td>-9.508<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>10/9<br /> | |||
</td> | |||
<td>9.596<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>18/13<br /> | |||
</td> | |||
<td>-11.382<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>9/8<br /> | |||
</td> | |||
<td>-11.910<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>11/9<br /> | |||
</td> | |||
<td>12.592<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:6 -->Commas</h1> | |||
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.<br /> | 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.<br /> | ||
| Line 446: | Line 643: | ||
<table class="wiki_table"> | <table class="wiki_table"> | ||
<tr> | <tr> | ||
<th><!-- ws:start:WikiTextHeadingRule: | <th><!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Commas--In ket format"></a><!-- ws:end:WikiTextHeadingRule:8 -->In ket format</h3> | ||
</th> | </th> | ||
<th><!-- ws:start:WikiTextHeadingRule: | <th><!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Commas--In cents"></a><!-- ws:end:WikiTextHeadingRule:10 -->In cents</h3> | ||
</th> | </th> | ||
<th><!-- ws:start:WikiTextHeadingRule: | <th><!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Commas--Ratio"></a><!-- ws:end:WikiTextHeadingRule:12 -->Ratio</h3> | ||
</th> | </th> | ||
<th><!-- ws:start:WikiTextHeadingRule: | <th><!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Commas--Name 1"></a><!-- ws:end:WikiTextHeadingRule:14 -->Name 1</h3> | ||
</th> | </th> | ||
<th><!-- ws:start:WikiTextHeadingRule: | <th><!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Commas--Name2"></a><!-- ws:end:WikiTextHeadingRule:16 -->Name2</h3> | ||
</th> | </th> | ||
</tr> | </tr> | ||