81/80: 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>2014-06-20 01:02:42 UTC</tt>.<br>

: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-06-20 21:10:12 UTC</tt>.<br>

: The original revision id was <tt>~~514493824~~</tt>.<br>

: The original revision id was <tt>514562090</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">The **syntonic** or **Didymus comma** (frequency ratio **81/80**) is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit]]. Like [[16_15|16/15]], [[625_624|625/624]], [[2401_2400|2401/2400]] and [[4096_4095|4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between [[10_9|10/9]] and [[9_8|9/8]], the product of which is the just major third, [[5_4|5/4]]. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2. Tempering it out gives a tuning for the whole tone which is intermediate between 10/9 and 9/8, and leads to [[Meantone family|meantone temperament]].

<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">**81/80**

|-4 4 -1>

21.506290 cents

The **syntonic** or **Didymus comma** (frequency ratio **81/80**) is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit]]. Like [[16_15|16/15]], [[625_624|625/624]], [[2401_2400|2401/2400]] and [[4096_4095|4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between [[10_9|10/9]] and [[9_8|9/8]], the product of which is the just major third, [[5_4|5/4]]. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2.

Tempering out 81/80 gives a tuning for the [[tone|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[Meantone family|meantone temperament]].

Youtube video of "[[http://www.youtube.com/watch?v=IpWiEWFRGAY|Five senses of 81/80]]", demonstratory video by Jacob Barton.

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[[http://en.wikipedia.org/wiki/Syntonic_comma]]</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>81_80</title></head><body>The <strong>syntonic</strong> or <strong>Didymus comma</strong> (frequency ratio <strong>81/80</strong>) is the smallest <a class="wiki_link" href="/superparticular">superparticular interval</a> which belongs to the <a class="wiki_link" href="/5-limit">5-limit</a>. Like <a class="wiki_link" href="/16_15">16/15</a>, <a class="wiki_link" href="/625_624">625/624</a>, <a class="wiki_link" href="/2401_2400">2401/2400</a> and <a class="wiki_link" href="/4096_4095">4096/4095</a> it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between <a class="wiki_link" href="/10_9">10/9</a> and <a class="wiki_link" href="/9_8">9/8</a>, the product of which is the just major third, <a class="wiki_link" href="/5_4">5/4</a>. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2. Tempering it out gives a tuning for the whole tone which is intermediate between 10/9 and 9/8, and leads to <a class="wiki_link" href="/Meantone%20family">meantone temperament</a>.<br />

|-4 4 -1&gt;<br />

21.506290 cents<br />

<br />

The <strong>syntonic</strong> or <strong>Didymus comma</strong> (frequency ratio <strong>81/80</strong>) is the smallest <a class="wiki_link" href="/superparticular">superparticular interval</a> which belongs to the <a class="wiki_link" href="/5-limit">5-limit</a>. Like <a class="wiki_link" href="/16_15">16/15</a>, <a class="wiki_link" href="/625_624">625/624</a>, <a class="wiki_link" href="/2401_2400">2401/2400</a> and <a class="wiki_link" href="/4096_4095">4096/4095</a> it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between <a class="wiki_link" href="/10_9">10/9</a> and <a class="wiki_link" href="/9_8">9/8</a>, the product of which is the just major third, <a class="wiki_link" href="/5_4">5/4</a>. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2.<br />

<br />

Tempering out 81/80 gives a tuning for the <a class="wiki_link" href="/tone">whole tone</a> which is intermediate between 10/9 and 9/8, and leads to <a class="wiki_link" href="/Meantone%20family">meantone temperament</a>.<br />

<br />

Youtube video of &quot;<a class="wiki_link_ext" href="http://www.youtube.com/watch?v=IpWiEWFRGAY" rel="nofollow">Five senses of 81/80</a>&quot;, demonstratory video by Jacob Barton.<br />

@@ 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>2014-06-20 01:02:42 UTC</tt>.<br>
+: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-06-20 21:10:12 UTC</tt>.<br>
-: The original revision id was <tt>514493824</tt>.<br>
+: The original revision id was <tt>514562090</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">The **syntonic** or **Didymus comma** (frequency ratio **81/80**) is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit]]. Like [[16_15|16/15]], [[625_624|625/624]], [[2401_2400|2401/2400]] and [[4096_4095|4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between [[10_9|10/9]] and [[9_8|9/8]], the product of which is the just major third, [[5_4|5/4]]. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2. Tempering it out gives a tuning for the whole tone which is intermediate between 10/9 and 9/8, and leads to [[Meantone family|meantone temperament]].
+<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">**81/80**
+|-4 4 -1&gt;
+.506290 cents
+The **syntonic** or **Didymus comma** (frequency ratio **81/80**) is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit]]. Like [[16_15|16/15]], [[625_624|625/624]], [[2401_2400|2401/2400]] and [[4096_4095|4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between [[10_9|10/9]] and [[9_8|9/8]], the product of which is the just major third, [[5_4|5/4]]. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2.
+Tempering out 81/80 gives a tuning for the [[tone|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[Meantone family|meantone temperament]].
 Youtube video of "[[http://www.youtube.com/watch?v=IpWiEWFRGAY|Five senses of 81/80]]", demonstratory video by Jacob Barton.
@@ Line 14: / Line 20: @@
 [[http://en.wikipedia.org/wiki/Syntonic_comma]]</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;81_80&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;strong&gt;syntonic&lt;/strong&gt; or &lt;strong&gt;Didymus comma&lt;/strong&gt; (frequency ratio &lt;strong&gt;81/80&lt;/strong&gt;) is the smallest &lt;a class="wiki_link" href="/superparticular"&gt;superparticular interval&lt;/a&gt; which belongs to the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;. Like &lt;a class="wiki_link" href="/16_15"&gt;16/15&lt;/a&gt;, &lt;a class="wiki_link" href="/625_624"&gt;625/624&lt;/a&gt;, &lt;a class="wiki_link" href="/2401_2400"&gt;2401/2400&lt;/a&gt; and &lt;a class="wiki_link" href="/4096_4095"&gt;4096/4095&lt;/a&gt; it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between &lt;a class="wiki_link" href="/10_9"&gt;10/9&lt;/a&gt; and &lt;a class="wiki_link" href="/9_8"&gt;9/8&lt;/a&gt;, the product of which is the just major third, &lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2. Tempering it out gives a tuning for the whole tone which is intermediate between 10/9 and 9/8, and leads to &lt;a class="wiki_link" href="/Meantone%20family"&gt;meantone temperament&lt;/a&gt;.&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;81_80&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;strong&gt;81/80&lt;/strong&gt;&lt;br /&gt;
+|-4 4 -1&amp;gt;&lt;br /&gt;
+.506290 cents&lt;br /&gt;
+&lt;br /&gt;
+The &lt;strong&gt;syntonic&lt;/strong&gt; or &lt;strong&gt;Didymus comma&lt;/strong&gt; (frequency ratio &lt;strong&gt;81/80&lt;/strong&gt;) is the smallest &lt;a class="wiki_link" href="/superparticular"&gt;superparticular interval&lt;/a&gt; which belongs to the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;. Like &lt;a class="wiki_link" href="/16_15"&gt;16/15&lt;/a&gt;, &lt;a class="wiki_link" href="/625_624"&gt;625/624&lt;/a&gt;, &lt;a class="wiki_link" href="/2401_2400"&gt;2401/2400&lt;/a&gt; and &lt;a class="wiki_link" href="/4096_4095"&gt;4096/4095&lt;/a&gt; it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between &lt;a class="wiki_link" href="/10_9"&gt;10/9&lt;/a&gt; and &lt;a class="wiki_link" href="/9_8"&gt;9/8&lt;/a&gt;, the product of which is the just major third, &lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2.&lt;br /&gt;
+&lt;br /&gt;
+Tempering out 81/80 gives a tuning for the &lt;a class="wiki_link" href="/tone"&gt;whole tone&lt;/a&gt; which is intermediate between 10/9 and 9/8, and leads to &lt;a class="wiki_link" href="/Meantone%20family"&gt;meantone temperament&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 Youtube video of &amp;quot;&lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=IpWiEWFRGAY" rel="nofollow"&gt;Five senses of 81/80&lt;/a&gt;&amp;quot;, demonstratory video by Jacob Barton.&lt;br /&gt;