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:<br>

: This revision was by author [[User:~~genewardsmith~~|~~genewardsmith~~]] and made on <tt>2012-04-~~16 19~~:43:10 UTC</tt>.<br>

: This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2012-06-03 23:24:52 UTC</tt>.<br>

: The original revision id was <tt>~~321254968~~</tt>.<br>

: The original revision id was <tt>342313156</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>

<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 "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, 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.

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

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==Intervals==

|| Degrees of 50-EDO || Cents value ||

|| 0 || 0 ||

Line 69:

Line 68:

|| 48 || 1152 ||

|| 49 || 1176 ||

==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.)

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

|| | -4 4 -1 > ||> 21.51 ||= 81/80 || Syntonic comma || Didymus comma ||

|| | -8 8 -2 > ||> 43.01 ||= 6561/6400 || Mathieu superdiesis || ||

|| | 23 6 -14 > ||> 3.34 ||= 1212717/1210381 || Vishnu comma || ||

|| | 1 2 -3 1 > ||> 13.79 ||= 126/125 || Small septimal comma || ||

|| | -5 2 2 -1 > ||> 7.71 ||= 225/224 || Septimal kleisma || ||

|| | 6 0 -5 2 > ||> 6.08 ||= 3136/3125 || Middle second comma || ||

|| | -6 -8 2 5 > ||> 1.12 ||= 420175/419904 || || ||

|| |-11 2 7 -3 > ||> 1.63 ||= 703125/702464 || || ||

|| | 11 -10 -10 10 > ||> 5.57 ||= 6772805/6751042 || || ||

|| |-13 10 0 -1 > ||> 50.72 ||= 59049/57344 || Harrison's comma || ||

|| | 2 3 1 -2 -1 > ||> 3.21 ||= 540/539 || Swets' comma || ||

|| | -3 4 -2 -2 2 > ||> 0.18 ||= 9801/9800 || Kalisma || Gauss' comma ||

|| | 5 -1 3 0 -3 > ||> 3.03 ||= 4000/3993 || Undecimal schisma || ||

|| | -7 -1 1 1 1 > ||> 4.50 ||= 385/384 || Undecimal kleisma || ||

|| | 2 -1 0 1 -2 1 > ||> 4.76 ||= 364/363 || || ||

|| | 2 3 0 -1 1 -2 > ||> 7.30 ||= 1188/1183 || Kestrel Comma || ||

|| | 3 0 2 0 1 -3 > ||> 2.36 ||= 2200/2197 || Parizek comma || ||

|| | -3 1 1 1 0 -1 > ||> 16.57 ||= 105/104 || Small tridecimal comma || ||

|| | 3 -2 0 1 -1 -1 0 0 1 > ||> 1.34 ||= 1288/1287 || Triaphonisma || ||

[[http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3|Twinkle canon – 50 edo]] by [[http://soonlabel.com/xenharmonic/archives/573|Claudi Meneghin]]

</pre></div>

<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

| -8 8 -2 > 43.01 6561/6400 Mathieu superdiesis

| 23 6 -14 > 3.34 1212717/1210381 Vishnu comma

| 1 2 -3 1 > 13.79 126/125 small septimal comma

| -5 2 2 -1 > 7.71 225/224 septimal kleisma

| 6 0 -5 2 > 6.08 3136/3125 middle second comma

| -6 -8 2 5 > 1.12 420175/419904

|-11 2 7 -3 > 1.63 703125/702464

| 11 -10 -10 10 > 5.57 6772805/6751042

|-13 10 0 -1 > 50.72 59049/57344 Harrison's comma

| 2 3 1 -2 -1 > 3.21 540/539 Swets' comma

| -3 4 -2 -2 2 > 0.18 9801/9800 kalisma, Gauss' comma

| 5 -1 3 0 -3 > 3.03 4000/3993 undecimal schisma

| -7 -1 1 1 1 > 4.50 385/384 undecimal kleisma

| 2 -1 0 1 -2 1 > 4.76 364/363

| 2 3 0 -1 1 -2 > 7.30 1188/1183 Kestrel Comma

| 3 0 2 0 1 -3 > 2.36 2200/2197 Parizek comma

| -3 1 1 1 0 -1 > 16.57 105/104 small tridecimal comma

| 3 -2 0 1 -1 -1 0 0 1 > 1.34 1288/1287 triaphonisma</span></pre></div>

<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 &quot;Harmonics or the Philosophy of Musical Sounds&quot; (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><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 &quot;Harmonics or the Philosophy of Musical Sounds&quot; (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 />

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, 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 />

<br />

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

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<br />

<h2 id="toc1"><a name="x-Intervals"></a>Intervals</h2>

~~<br />~~

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<br />

<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3" rel="nofollow">Twinkle canon – 50 edo</a> by <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/573" rel="nofollow">Claudi Meneghin</a></body></html></pre></div>

<h2 id="toc2"><a name="x-Commas"></a>Commas</h2>

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.)<br />

<tr>

<th><h3 id="toc3"><a name="x-Commas-In bra format"></a>In bra format</h3>

</th>

<th><h3 id="toc4"><a name="x-Commas-In cents"></a>In cents</h3>

</th>

<th><h3 id="toc5"><a name="x-Commas-Ratio"></a>Ratio</h3>

</th>

</th>

</th>

</tr>

<tr>

</td>

</td>

</td>

<td>Syntonic comma<br />

</td>

<td>Didymus comma<br />

</td>

</tr>

<tr>

</td>

</td>

</td>

<td>Mathieu superdiesis<br />

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

<td>Vishnu comma<br />

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

<td>Small septimal comma<br />

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

<td>Septimal kleisma<br />

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

<td>Middle second comma<br />

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

<td>Harrison's comma<br />

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

<td>Swets' comma<br />

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

<td>Kalisma<br />

</td>

<td>Gauss' comma<br />

</td>

</tr>

<tr>

</td>

</td>

</td>

<td>Undecimal schisma<br />

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

<td>Undecimal kleisma<br />

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

<td>Kestrel Comma<br />

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

<td>Parizek comma<br />

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

<td>Small tridecimal comma<br />

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

<td>Triaphonisma<br />

</td>

</td>

</tr>

</table>

<br />

<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3" rel="nofollow">Twinkle canon – 50 edo</a> by <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/573" rel="nofollow">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<br />

| -8 8 -2 &gt; 43.01 6561/6400 Mathieu superdiesis<br />

| 23 6 -14 &gt; 3.34 1212717/1210381 Vishnu comma<br />

| 1 2 -3 1 &gt; 13.79 126/125 small septimal comma<br />

| -5 2 2 -1 &gt; 7.71 225/224 septimal kleisma<br />

| 6 0 -5 2 &gt; 6.08 3136/3125 middle second comma<br />

| -6 -8 2 5 &gt; 1.12 420175/419904<br />

|-11 2 7 -3 &gt; 1.63 703125/702464<br />

| 11 -10 -10 10 &gt; 5.57 6772805/6751042<br />

|-13 10 0 -1 &gt; 50.72 59049/57344 Harrison's comma<br />

| 2 3 1 -2 -1 &gt; 3.21 540/539 Swets' comma<br />

| -3 4 -2 -2 2 &gt; 0.18 9801/9800 kalisma, Gauss' comma<br />

| 5 -1 3 0 -3 &gt; 3.03 4000/3993 undecimal schisma<br />

| -7 -1 1 1 1 &gt; 4.50 385/384 undecimal kleisma<br />

| 2 -1 0 1 -2 1 &gt; 4.76 364/363<br />

| 2 3 0 -1 1 -2 &gt; 7.30 1188/1183 Kestrel Comma<br />

| 3 0 2 0 1 -3 &gt; 2.36 2200/2197 Parizek comma<br />

| -3 1 1 1 0 -1 &gt; 16.57 105/104 small tridecimal comma<br />

| 3 -2 0 1 -1 -1 0 0 1 &gt; 1.34 1288/1287 triaphonisma</span></body></html></pre></div>

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2012-04-16 19:43:10 UTC</tt>.<br>
+: This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2012-06-03 23:24:52 UTC</tt>.<br>
-: The original revision id was <tt>321254968</tt>.<br>
+: The original revision id was <tt>342313156</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>
 <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 "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, 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, 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.
 [[http://www.archive.org/details/harmonicsorphilo00smit|Robert Smith's book online]]
@@ Line 17: / Line 17: @@
 ==Intervals==
 || Degrees of 50-EDO || Cents value ||
 || 0 || 0 ||
@@ Line 69: / Line 68: @@
 || 48 || 1152 ||
 || 49 || 1176 ||
+==Commas==
+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.)
+||~ ===In bra format=== ||~ ===In cents=== ||~ ===Ratio=== ||~ ===Name 1=== ||~ ===Name2=== ||
+|| | -4 4 -1 &gt; ||&gt; 21.51 ||= 81/80 || Syntonic comma || Didymus comma ||
+|| | -8 8 -2 &gt; ||&gt; 43.01 ||= 6561/6400 || Mathieu superdiesis ||   ||
+|| | 23 6 -14 &gt; ||&gt; 3.34 ||= 1212717/1210381 || Vishnu comma ||   ||
+|| | 1 2 -3 1 &gt; ||&gt; 13.79 ||= 126/125 || Small septimal comma ||   ||
+|| | -5 2 2 -1 &gt; ||&gt; 7.71 ||= 225/224 || Septimal kleisma ||   ||
+|| | 6 0 -5 2 &gt; ||&gt; 6.08 ||= 3136/3125 || Middle second comma ||   ||
+|| | -6 -8 2 5 &gt; ||&gt; 1.12 ||= 420175/419904 ||   ||   ||
+|| |-11 2 7 -3 &gt; ||&gt; 1.63 ||= 703125/702464 ||   ||   ||
+|| | 11 -10 -10 10 &gt; ||&gt; 5.57 ||= 6772805/6751042 ||   ||   ||
+|| |-13 10 0 -1 &gt; ||&gt; 50.72 ||= 59049/57344 || Harrison's comma ||   ||
+|| | 2 3 1 -2 -1 &gt; ||&gt; 3.21 ||= 540/539 || Swets' comma ||   ||
+|| | -3 4 -2 -2 2 &gt; ||&gt; 0.18 ||= 9801/9800 || Kalisma || Gauss' comma ||
+|| | 5 -1 3 0 -3 &gt; ||&gt; 3.03 ||= 4000/3993 || Undecimal schisma ||   ||
+|| | -7 -1 1 1 1 &gt; ||&gt; 4.50 ||= 385/384 || Undecimal kleisma ||   ||
+|| | 2 -1 0 1 -2 1 &gt; ||&gt; 4.76 ||= 364/363 ||   ||   ||
+|| | 2 3 0 -1 1 -2 &gt; ||&gt; 7.30 ||= 1188/1183 || Kestrel Comma ||   ||
+|| | 3 0 2 0 1 -3 &gt; ||&gt; 2.36 ||= 2200/2197 || Parizek comma ||   ||
+|| | -3 1 1 1 0 -1 &gt; ||&gt; 16.57 ||= 105/104 || Small tridecimal comma ||   ||
+|| | 3 -2 0 1 -1 -1 0 0 1 &gt; ||&gt; 1.34 ||= 1288/1287 || Triaphonisma ||   ||
 [[http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3|Twinkle canon – 50 edo]] by [[http://soonlabel.com/xenharmonic/archives/573|Claudi Meneghin]]
-</pre></div>
+&lt;span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;"&gt;| -4 4 -1 &gt; 21.51 81/80 syntonic comma, Didymus comma
+| -8 8 -2 &gt; 43.01 6561/6400 Mathieu superdiesis
+| 23 6 -14 &gt; 3.34 1212717/1210381 Vishnu comma
+| 1 2 -3 1 &gt; 13.79 126/125 small septimal comma
+| -5 2 2 -1 &gt; 7.71 225/224 septimal kleisma
+| 6 0 -5 2 &gt; 6.08 3136/3125 middle second comma
+| -6 -8 2 5 &gt; 1.12 420175/419904
+|-11 2 7 -3 &gt; 1.63 703125/702464
+| 11 -10 -10 10 &gt; 5.57 6772805/6751042
+|-13 10 0 -1 &gt; 50.72 59049/57344 Harrison's comma
+| 2 3 1 -2 -1 &gt; 3.21 540/539 Swets' comma
+| -3 4 -2 -2 2 &gt; 0.18 9801/9800 kalisma, Gauss' comma
+| 5 -1 3 0 -3 &gt; 3.03 4000/3993 undecimal schisma
+| -7 -1 1 1 1 &gt; 4.50 385/384 undecimal kleisma
+| 2 -1 0 1 -2 1 &gt; 4.76 364/363
+| 2 3 0 -1 1 -2 &gt; 7.30 1188/1183 Kestrel Comma
+| 3 0 2 0 1 -3 &gt; 2.36 2200/2197 Parizek comma
+| -3 1 1 1 0 -1 &gt; 16.57 105/104 small tridecimal comma
+| 3 -2 0 1 -1 -1 0 0 1 &gt; 1.34 1288/1287 triaphonisma&lt;/span&gt;</pre></div>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;50edo&lt;/title&gt;&lt;/head&gt;&lt;body&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 &amp;quot;Harmonics or the Philosophy of Musical Sounds&amp;quot; (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;
+<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;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 &amp;quot;Harmonics or the Philosophy of Musical Sounds&amp;quot; (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, 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, 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;
 &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;
@@ Line 84: / Line 124: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Intervals&lt;/h2&gt;
-  &lt;br /&gt;
 &lt;table class="wiki_table"&gt;
@@ Line 397: / Line 436: @@
 &lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3" rel="nofollow"&gt;Twinkle canon – 50 edo&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/573" rel="nofollow"&gt;Claudi Meneghin&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Commas&lt;/h2&gt;
+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.)&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-Commas-In bra format"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;In bra format&lt;/h3&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x-Commas-In cents"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;In cents&lt;/h3&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x-Commas-Ratio"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Ratio&lt;/h3&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="x-Commas-Name 1"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Name 1&lt;/h3&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="x-Commas-Name2"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Name2&lt;/h3&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;| -4 4 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;21.51&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;81/80&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Syntonic comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Didymus comma&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;| -8 8 -2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;43.01&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;6561/6400&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Mathieu superdiesis&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;| 23 6 -14 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;3.34&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1212717/1210381&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Vishnu comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;| 1 2 -3 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;13.79&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;126/125&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Small septimal comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;| -5 2 2 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;7.71&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;225/224&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Septimal kleisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;| 6 0 -5 2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;6.08&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;3136/3125&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Middle second comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;| -6 -8 2 5 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;1.12&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;420175/419904&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;|-11 2 7 -3 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;1.63&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;703125/702464&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;| 11 -10 -10 10 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;5.57&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;6772805/6751042&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;|-13 10 0 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;50.72&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;59049/57344&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Harrison's comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;| 2 3 1 -2 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;3.21&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;540/539&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Swets' comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;| -3 4 -2 -2 2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;0.18&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;9801/9800&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Kalisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Gauss' comma&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;| 5 -1 3 0 -3 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;3.03&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;4000/3993&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Undecimal schisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;| -7 -1 1 1 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;4.50&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;385/384&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Undecimal kleisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;| 2 -1 0 1 -2 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;4.76&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;364/363&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;| 2 3 0 -1 1 -2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;7.30&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1188/1183&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Kestrel Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;| 3 0 2 0 1 -3 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;2.36&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;2200/2197&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Parizek comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;| -3 1 1 1 0 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;16.57&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;105/104&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Small tridecimal comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;| 3 -2 0 1 -1 -1 0 0 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;1.34&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1288/1287&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Triaphonisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;br /&gt;
+&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3" rel="nofollow"&gt;Twinkle canon – 50 edo&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/573" rel="nofollow"&gt;Claudi Meneghin&lt;/a&gt;&lt;br /&gt;
+&lt;span style="display: block; height: 1px; left: -40px; overflow: hidden; position: absolute; top: 5675.5px; width: 1px;"&gt;| -4 4 -1 &amp;gt; 21.51 81/80 syntonic comma, Didymus comma&lt;br /&gt;
+| -8 8 -2 &amp;gt; 43.01 6561/6400 Mathieu superdiesis&lt;br /&gt;
+| 23 6 -14 &amp;gt; 3.34 1212717/1210381 Vishnu comma&lt;br /&gt;
+| 1 2 -3 1 &amp;gt; 13.79 126/125 small septimal comma&lt;br /&gt;
+| -5 2 2 -1 &amp;gt; 7.71 225/224 septimal kleisma&lt;br /&gt;
+| 6 0 -5 2 &amp;gt; 6.08 3136/3125 middle second comma&lt;br /&gt;
+| -6 -8 2 5 &amp;gt; 1.12 420175/419904&lt;br /&gt;
+|-11 2 7 -3 &amp;gt; 1.63 703125/702464&lt;br /&gt;
+| 11 -10 -10 10 &amp;gt; 5.57 6772805/6751042&lt;br /&gt;
+|-13 10 0 -1 &amp;gt; 50.72 59049/57344 Harrison's comma&lt;br /&gt;
+| 2 3 1 -2 -1 &amp;gt; 3.21 540/539 Swets' comma&lt;br /&gt;
+| -3 4 -2 -2 2 &amp;gt; 0.18 9801/9800 kalisma, Gauss' comma&lt;br /&gt;
+| 5 -1 3 0 -3 &amp;gt; 3.03 4000/3993 undecimal schisma&lt;br /&gt;
+| -7 -1 1 1 1 &amp;gt; 4.50 385/384 undecimal kleisma&lt;br /&gt;
+| 2 -1 0 1 -2 1 &amp;gt; 4.76 364/363&lt;br /&gt;
+| 2 3 0 -1 1 -2 &amp;gt; 7.30 1188/1183 Kestrel Comma&lt;br /&gt;
+| 3 0 2 0 1 -3 &amp;gt; 2.36 2200/2197 Parizek comma&lt;br /&gt;
+| -3 1 1 1 0 -1 &amp;gt; 16.57 105/104 small tridecimal comma&lt;br /&gt;
+| 3 -2 0 1 -1 -1 0 0 1 &amp;gt; 1.34 1288/1287 triaphonisma&lt;/span&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>