64/63: Difference between revisions

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:~~kalleaho~~|~~kalleaho~~]] and made on <tt>2012-01-~~11 13~~:14:19 UTC</tt>.<br>

: This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2012-10-07 19:22:45 UTC</tt>.<br>

: The original revision id was <tt>~~291373213~~</tt>.<br>

: The original revision id was <tt>370951818</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 **septimal** or **Archytas comma**, 64/63, is a [[xenharmonic/superparticular|superparticular ratio]] which equates [[xenharmonic/9_8|9/8]] and [[xenharmonic/8_7|8/7]] if tempered out and has the eighth square number as a numerator. It also equates [[xenharmonic/7_4|7/4]] with [[xenharmonic/16_9|16/9]], so that the just dominant seventh chord, 1-5/4-3/2-16/9, and the otonal tetrad, 1-5/4-3/2-7/4, are equated to the same chord when 64/63 is tempered out. Equal divisions of the octave tempering out 64/63 include 12, 15, 22, 27, 37, 49 and 59.

The Archytas comma is a 7-limit comma with monzo | 6 -2 0 -1 >.

It is similar to the Didymus or syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths equal a major third (octave equivalent). In the case of 81/80, however, the major third is 5/4, while with the Archytas comma, the major third is 9/7. (Note that Porcupine, which tempers out 64/63, uses a minor tone as a generator and generally is considered to have 5/4 major thirds, so it doesn't depend on this equivalency.)

If you are using 9/7 major thirds, this also implies that the major third is split into two equal steps that represent both 9/8 and 8/7: If a stack of four fifths gets you to (octave-equivalent) 9/7, and a stack of two fifths gets you to 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equal, however, as a result of the generation process.

[[http://en.wikipedia.org/wiki/Septimal_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>64_63</title></head><body>The <strong>septimal</strong> or <strong>Archytas comma</strong>, 64/63, is a <a class="wiki_link" href="http://xenharmonic.wikispaces.com/superparticular">superparticular ratio</a> which equates <a class="wiki_link" href="http://xenharmonic.wikispaces.com/9_8">9/8</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/8_7">8/7</a> if tempered out and has the eighth square number as a numerator. It also equates <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7_4">7/4</a> with <a class="wiki_link" href="http://xenharmonic.wikispaces.com/16_9">16/9</a>, so that the just dominant seventh chord, 1-5/4-3/2-16/9, and the otonal tetrad, 1-5/4-3/2-7/4, are equated to the same chord when 64/63 is tempered out. Equal divisions of the octave tempering out 64/63 include 12, 15, 22, 27, 37, 49 and 59.<br />

<br />

The Archytas comma is a 7-limit comma with monzo | 6 -2 0 -1 &gt;.<br />

<br />

It is similar to the Didymus or syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths equal a major third (octave equivalent). In the case of 81/80, however, the major third is 5/4, while with the Archytas comma, the major third is 9/7. (Note that Porcupine, which tempers out 64/63, uses a minor tone as a generator and generally is considered to have 5/4 major thirds, so it doesn't depend on this equivalency.)<br />

<br />

If you are using 9/7 major thirds, this also implies that the major third is split into two equal steps that represent both 9/8 and 8/7: If a stack of four fifths gets you to (octave-equivalent) 9/7, and a stack of two fifths gets you to 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equal, however, as a result of the generation process.<br />

<br />

<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_comma" rel="nofollow">http://en.wikipedia.org/wiki/Septimal_comma</a></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:kalleaho|kalleaho]] and made on <tt>2012-01-11 13:14:19 UTC</tt>.<br>
+: This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2012-10-07 19:22:45 UTC</tt>.<br>
-: The original revision id was <tt>291373213</tt>.<br>
+: The original revision id was <tt>370951818</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 **septimal** or **Archytas comma**, 64/63, is a [[xenharmonic/superparticular|superparticular ratio]] which equates [[xenharmonic/9_8|9/8]] and [[xenharmonic/8_7|8/7]] if tempered out and has the eighth square number as a numerator. It also equates [[xenharmonic/7_4|7/4]] with [[xenharmonic/16_9|16/9]], so that the just dominant seventh chord, 1-5/4-3/2-16/9, and the otonal tetrad, 1-5/4-3/2-7/4, are equated to the same chord when 64/63 is tempered out. Equal divisions of the octave tempering out 64/63 include 12, 15, 22, 27, 37, 49 and 59.
+The Archytas comma is a 7-limit comma with monzo | 6 -2 0 -1 &gt;.
+It is similar to the Didymus or syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths equal a major third (octave equivalent). In the case of 81/80, however, the major third is 5/4, while with the Archytas comma, the major third is 9/7. (Note that Porcupine, which tempers out 64/63, uses a minor tone as a generator and generally is considered to have 5/4 major thirds, so it doesn't depend on this equivalency.)
+If you are using 9/7 major thirds, this also implies that the major third is split into two equal steps that represent both 9/8 and 8/7: If a stack of four fifths gets you to (octave-equivalent) 9/7, and a stack of two fifths gets you to 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equal, however, as a result of the generation process.
 [[http://en.wikipedia.org/wiki/Septimal_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;64_63&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;strong&gt;septimal&lt;/strong&gt; or &lt;strong&gt;Archytas comma&lt;/strong&gt;, 64/63, is a &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/superparticular"&gt;superparticular ratio&lt;/a&gt; which equates &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/9_8"&gt;9/8&lt;/a&gt; and &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/8_7"&gt;8/7&lt;/a&gt; if tempered out and has the eighth square number as a numerator. It also equates &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/7_4"&gt;7/4&lt;/a&gt; with &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/16_9"&gt;16/9&lt;/a&gt;, so that the just dominant seventh chord, 1-5/4-3/2-16/9, and the otonal tetrad, 1-5/4-3/2-7/4, are equated to the same chord when 64/63 is tempered out. Equal divisions of the octave tempering out 64/63 include 12, 15, 22, 27, 37, 49 and 59.&lt;br /&gt;
+&lt;br /&gt;
+The Archytas comma is a 7-limit comma with monzo | 6 -2 0 -1 &amp;gt;.&lt;br /&gt;
+&lt;br /&gt;
+It is similar to the Didymus or syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths equal a major third (octave equivalent). In the case of 81/80, however, the major third is 5/4, while with the Archytas comma, the major third is 9/7. (Note that Porcupine, which tempers out 64/63, uses a minor tone as a generator and generally is considered to have 5/4 major thirds, so it doesn't depend on this equivalency.)&lt;br /&gt;
+&lt;br /&gt;
+If you are using 9/7 major thirds, this also implies that the major third is split into two equal steps that represent both 9/8 and 8/7: If a stack of four fifths gets you to (octave-equivalent) 9/7, and a stack of two fifths gets you to 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equal, however, as a result of the generation process.&lt;br /&gt;
+&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_comma" rel="nofollow"&gt;http://en.wikipedia.org/wiki/Septimal_comma&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>