27edo: 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>2011-06-~~07 17~~:25:51 UTC</tt>.<br>

: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-06-11 12:14:33 UTC</tt>.<br>

: The original revision id was <tt>~~234971772~~</tt>.<br>

: The original revision id was <tt>235939348</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>

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<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">=<span style="color: #0061ff; font-size: 103%;">27 tone equal tempertament</span>=

If octaves are kept pure, 27edo divides the octave in 27 equal parts each exactly 44.444... cents in size. However, 27 is a prime candidate for octave shrinking, and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the third, fifth and 7/4 sharply.

If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44.444... [[cents]] in size. However, 27 is a prime candidate for [[octave shrinking]], and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the [[5_4|third]], [[3_2|fifth]] and [[7_4|7/4]] sharply.

Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as 12edo, sharp 13 2/3 cents. The result is that 6/5, 7/5 and especially 7/6 are all tuned more accurately than this.

Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as [[12edo]], sharp 13 2/3 cents. The result is that [[6_5|6/5]], [[7_5|7/5]] and especially [[7_6|7/6]] are all tuned more accurately than this.

27edo, with its 400 cent major third, tempers out the diesis of 128/125, and also the septimal comma, 64/63 (and hence 126/125 also.) These it shares with 12edo, making some relationships familiar, and as a consequence they both support augene temperament. It shares with [[22edo]] tempering out the allegedly Bohlen-Pierce comma 245/243 as well as 64/63, so that they both support superpyth temperament, with quite sharp "superpythagorean" fifths giving a sharp 9/7 in place of meantone's 5/4.

27edo, with its 400 cent major third, tempers out the [[diesis]] of 128/125, and also the [[septimal comma]], 64/63 (and hence 126/125 also.) These it shares with 12edo, making some relationships familiar, and as a consequence they both support augene temperament. It shares with [[22edo]] tempering out the allegedly Bohlen-Pierce comma 245/243 as well as 64/63, so that they both support superpyth temperament, with quite sharp "superpythagorean" fifths giving a sharp 9/7 in place of meantone's 5/4.

Though the 7-limit tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both consistently and distinctly--that is, everything in the 7-limit [[Diamonds|diamond]] is uniquely represented by a certain number of steps of 27 equal.

Though the [[7-limit]] tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both consistently and distinctly--that is, everything in the 7-limit [[Diamonds|diamond]] is uniquely represented by a certain number of steps of 27 equal.

==Intervals==

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=Music=

[[http://www.archive.org/details/MusicForYourEars|Music For Your Ears]] [[http://www.archive.org/download/MusicForYourEars/musicfor.mp3|play]] by [[Gene Ward Smith]] The central portion is in 27edo, the rest in 46edo.</pre></div>

[[http://www.archive.org/details/MusicForYourEars|Music For Your Ears]] [[http://www.archive.org/download/MusicForYourEars/musicfor.mp3|play]] by [[Gene Ward Smith]] The central portion is in 27edo, the rest in [[46edo]].</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>27edo</title></head><body><h1 id="toc0"><a name="x27 tone equal tempertament"></a><span style="color: #0061ff; font-size: 103%;">27 tone equal tempertament</span></h1>

<br />

If octaves are kept pure, 27edo divides the octave in 27 equal parts each exactly 44.444... cents in size. However, 27 is a prime candidate for octave shrinking, and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the third, fifth and 7/4 sharply.<br />

If octaves are kept pure, 27edo divides the <a class="wiki_link" href="/octave">octave</a> in 27 equal parts each exactly 44.444... <a class="wiki_link" href="/cents">cents</a> in size. However, 27 is a prime candidate for <a class="wiki_link" href="/octave%20shrinking">octave shrinking</a>, and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the <a class="wiki_link" href="/5_4">third</a>, <a class="wiki_link" href="/3_2">fifth</a> and <a class="wiki_link" href="/7_4">7/4</a> sharply.<br />

<br />

Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as 12edo, sharp 13 2/3 cents. The result is that 6/5, 7/5 and especially 7/6 are all tuned more accurately than this.<br />

Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as <a class="wiki_link" href="/12edo">12edo</a>, sharp 13 2/3 cents. The result is that <a class="wiki_link" href="/6_5">6/5</a>, <a class="wiki_link" href="/7_5">7/5</a> and especially <a class="wiki_link" href="/7_6">7/6</a> are all tuned more accurately than this.<br />

<br />

27edo, with its 400 cent major third, tempers out the diesis of 128/125, and also the septimal comma, 64/63 (and hence 126/125 also.) These it shares with 12edo, making some relationships familiar, and as a consequence they both support augene temperament. It shares with <a class="wiki_link" href="/22edo">22edo</a> tempering out the allegedly Bohlen-Pierce comma 245/243 as well as 64/63, so that they both support superpyth temperament, with quite sharp &quot;superpythagorean&quot; fifths giving a sharp 9/7 in place of meantone's 5/4. <br />

27edo, with its 400 cent major third, tempers out the <a class="wiki_link" href="/diesis">diesis</a> of 128/125, and also the <a class="wiki_link" href="/septimal%20comma">septimal comma</a>, 64/63 (and hence 126/125 also.) These it shares with 12edo, making some relationships familiar, and as a consequence they both support augene temperament. It shares with <a class="wiki_link" href="/22edo">22edo</a> tempering out the allegedly Bohlen-Pierce comma 245/243 as well as 64/63, so that they both support superpyth temperament, with quite sharp &quot;superpythagorean&quot; fifths giving a sharp 9/7 in place of meantone's 5/4. <br />

<br />

Though the 7-limit tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both consistently and distinctly--that is, everything in the 7-limit <a class="wiki_link" href="/Diamonds">diamond</a> is uniquely represented by a certain number of steps of 27 equal.<br />

Though the <a class="wiki_link" href="/7-limit">7-limit</a> tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both consistently and distinctly--that is, everything in the 7-limit <a class="wiki_link" href="/Diamonds">diamond</a> is uniquely represented by a certain number of steps of 27 equal.<br />

<br />

<h2 id="toc1"><a name="x27 tone equal tempertament-Intervals"></a>Intervals</h2>

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<br />

<h1 id="toc3"><a name="Music"></a>Music</h1>

<a class="wiki_link_ext" href="http://www.archive.org/details/MusicForYourEars" rel="nofollow">Music For Your Ears</a> <a class="wiki_link_ext" href="http://www.archive.org/download/MusicForYourEars/musicfor.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a> The central portion is in 27edo, the rest in 46edo.</body></html></pre></div>

<a class="wiki_link_ext" href="http://www.archive.org/details/MusicForYourEars" rel="nofollow">Music For Your Ears</a> <a class="wiki_link_ext" href="http://www.archive.org/download/MusicForYourEars/musicfor.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a> The central portion is in 27edo, the rest in <a class="wiki_link" href="/46edo">46edo</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:genewardsmith|genewardsmith]] and made on <tt>2011-06-07 17:25:51 UTC</tt>.<br>
+: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-06-11 12:14:33 UTC</tt>.<br>
-: The original revision id was <tt>234971772</tt>.<br>
+: The original revision id was <tt>235939348</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>
@@ Line 8: / Line 8: @@
 <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">=&lt;span style="color: #0061ff; font-size: 103%;"&gt;27 tone equal tempertament&lt;/span&gt;=
-If octaves are kept pure, 27edo divides the octave in 27 equal parts each exactly 44.444... cents in size. However, 27 is a prime candidate for octave shrinking, and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the third, fifth and 7/4 sharply.
+If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44.444... [[cents]] in size. However, 27 is a prime candidate for [[octave shrinking]], and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the [[5_4|third]], [[3_2|fifth]] and [[7_4|7/4]] sharply.
-Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as 12edo, sharp 13 2/3 cents. The result is that 6/5, 7/5 and especially 7/6 are all tuned more accurately than this.
+Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as [[12edo]], sharp 13 2/3 cents. The result is that [[6_5|6/5]], [[7_5|7/5]] and especially [[7_6|7/6]] are all tuned more accurately than this.
-edo, with its 400 cent major third, tempers out the diesis of 128/125, and also the septimal comma, 64/63 (and hence 126/125 also.) These it shares with 12edo, making some relationships familiar, and as a consequence they both support augene temperament. It shares with [[22edo]] tempering out the allegedly Bohlen-Pierce comma 245/243 as well as 64/63, so that they both support superpyth temperament, with quite sharp "superpythagorean" fifths giving a sharp 9/7 in place of meantone's 5/4.
+edo, with its 400 cent major third, tempers out the [[diesis]] of 128/125, and also the [[septimal comma]], 64/63 (and hence 126/125 also.) These it shares with 12edo, making some relationships familiar, and as a consequence they both support augene temperament. It shares with [[22edo]] tempering out the allegedly Bohlen-Pierce comma 245/243 as well as 64/63, so that they both support superpyth temperament, with quite sharp "superpythagorean" fifths giving a sharp 9/7 in place of meantone's 5/4.
-Though the 7-limit tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both consistently and distinctly--that is, everything in the 7-limit [[Diamonds|diamond]] is uniquely represented by a certain number of steps of 27 equal.
+Though the [[7-limit]] tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both consistently and distinctly--that is, everything in the 7-limit [[Diamonds|diamond]] is uniquely represented by a certain number of steps of 27 equal.
 ==Intervals==
@@ Line 69: / Line 69: @@
 =Music=
-[[http://www.archive.org/details/MusicForYourEars|Music For Your Ears]] [[http://www.archive.org/download/MusicForYourEars/musicfor.mp3|play]] by [[Gene Ward Smith]] The central portion is in 27edo, the rest in 46edo.</pre></div>
+[[http://www.archive.org/details/MusicForYourEars|Music For Your Ears]] [[http://www.archive.org/download/MusicForYourEars/musicfor.mp3|play]] by [[Gene Ward Smith]] The central portion is in 27edo, the rest in [[46edo]].</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;27edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x27 tone equal tempertament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="color: #0061ff; font-size: 103%;"&gt;27 tone equal tempertament&lt;/span&gt;&lt;/h1&gt;
   &lt;br /&gt;
-If octaves are kept pure, 27edo divides the octave in 27 equal parts each exactly 44.444... cents in size. However, 27 is a prime candidate for octave shrinking, and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the third, fifth and 7/4 sharply.&lt;br /&gt;
+If octaves are kept pure, 27edo divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; in 27 equal parts each exactly 44.444... &lt;a class="wiki_link" href="/cents"&gt;cents&lt;/a&gt; in size. However, 27 is a prime candidate for &lt;a class="wiki_link" href="/octave%20shrinking"&gt;octave shrinking&lt;/a&gt;, and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the &lt;a class="wiki_link" href="/5_4"&gt;third&lt;/a&gt;, &lt;a class="wiki_link" href="/3_2"&gt;fifth&lt;/a&gt; and &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt; sharply.&lt;br /&gt;
 &lt;br /&gt;
-Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as 12edo, sharp 13 2/3 cents. The result is that 6/5, 7/5 and especially 7/6 are all tuned more accurately than this.&lt;br /&gt;
+Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, sharp 13 2/3 cents. The result is that &lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;, &lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt; and especially &lt;a class="wiki_link" href="/7_6"&gt;7/6&lt;/a&gt; are all tuned more accurately than this.&lt;br /&gt;
 &lt;br /&gt;
-edo, with its 400 cent major third, tempers out the diesis of 128/125, and also the septimal comma, 64/63 (and hence 126/125 also.) These it shares with 12edo, making some relationships familiar, and as a consequence they both support augene temperament. It shares with &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; tempering out the allegedly Bohlen-Pierce comma 245/243 as well as 64/63, so that they both support superpyth temperament, with quite sharp &amp;quot;superpythagorean&amp;quot; fifths giving a sharp 9/7 in place of meantone's 5/4. &lt;br /&gt;
+edo, with its 400 cent major third, tempers out the &lt;a class="wiki_link" href="/diesis"&gt;diesis&lt;/a&gt; of 128/125, and also the &lt;a class="wiki_link" href="/septimal%20comma"&gt;septimal comma&lt;/a&gt;, 64/63 (and hence 126/125 also.) These it shares with 12edo, making some relationships familiar, and as a consequence they both support augene temperament. It shares with &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; tempering out the allegedly Bohlen-Pierce comma 245/243 as well as 64/63, so that they both support superpyth temperament, with quite sharp &amp;quot;superpythagorean&amp;quot; fifths giving a sharp 9/7 in place of meantone's 5/4. &lt;br /&gt;
 &lt;br /&gt;
-Though the 7-limit tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both consistently and distinctly--that is, everything in the 7-limit &lt;a class="wiki_link" href="/Diamonds"&gt;diamond&lt;/a&gt; is uniquely represented by a certain number of steps of 27 equal.&lt;br /&gt;
+Though the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both consistently and distinctly--that is, everything in the 7-limit &lt;a class="wiki_link" href="/Diamonds"&gt;diamond&lt;/a&gt; is uniquely represented by a certain number of steps of 27 equal.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x27 tone equal tempertament-Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Intervals&lt;/h2&gt;
@@ Line 544: / Line 544: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Music&lt;/h1&gt;
-&lt;a class="wiki_link_ext" href="http://www.archive.org/details/MusicForYourEars" rel="nofollow"&gt;Music For Your Ears&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://www.archive.org/download/MusicForYourEars/musicfor.mp3" rel="nofollow"&gt;play&lt;/a&gt; by &lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&gt;Gene Ward Smith&lt;/a&gt; The central portion is in 27edo, the rest in 46edo.&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;a class="wiki_link_ext" href="http://www.archive.org/details/MusicForYourEars" rel="nofollow"&gt;Music For Your Ears&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://www.archive.org/download/MusicForYourEars/musicfor.mp3" rel="nofollow"&gt;play&lt;/a&gt; by &lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&gt;Gene Ward Smith&lt;/a&gt; The central portion is in 27edo, the rest in &lt;a class="wiki_link" href="/46edo"&gt;46edo&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>