64/63: Difference between revisions
Wikispaces>spt3125 **Imported revision 503937978 - Original comment: ** |
Wikispaces>Stephen_Weigel **Imported revision 609093675 - 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:Stephen_Weigel|Stephen_Weigel]] and made on <tt>2017-03-19 14:04:54 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>609093675</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> | ||
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The interval 64/63, called **septimal comma** or **Archytas' comma** or (in German) **[[@http://de.wikipedia.org/wiki/Komma_%28Musik%29#Leipziger_Komma|Leipziger Komma]]**, 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 interval 64/63, called **septimal comma** or **Archytas' comma** or (in German) **[[@http://de.wikipedia.org/wiki/Komma_%28Musik%29#Leipziger_Komma|Leipziger Komma]]**, 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 >. | The Archytas comma is a 7-limit comma with monzo | 6 -2 0 -1 >. It is | ||
27.264092 cents in size. | |||
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 family|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.) | 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 family|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.) | ||
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The interval 64/63, called <strong>septimal comma</strong> or <strong>Archytas' comma</strong> or (in German) <strong><a class="wiki_link_ext" href="http://de.wikipedia.org/wiki/Komma_%28Musik%29#Leipziger_Komma" rel="nofollow" target="_blank">Leipziger Komma</a></strong>, 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 /> | The interval 64/63, called <strong>septimal comma</strong> or <strong>Archytas' comma</strong> or (in German) <strong><a class="wiki_link_ext" href="http://de.wikipedia.org/wiki/Komma_%28Musik%29#Leipziger_Komma" rel="nofollow" target="_blank">Leipziger Komma</a></strong>, 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 /> | <br /> | ||
The Archytas comma is a 7-limit comma with monzo | 6 -2 0 -1 &gt;.<br /> | The Archytas comma is a 7-limit comma with monzo | 6 -2 0 -1 &gt;. It is <br /> | ||
27.264092 cents in size.<br /> | |||
<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 <a class="wiki_link" href="/Porcupine%20family">Porcupine</a>, 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 /> | 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 <a class="wiki_link" href="/Porcupine%20family">Porcupine</a>, 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 /> | ||