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:~~MasonGreen1~~|~~MasonGreen1~~]] and made on <tt>2015-11-05 01:52:17 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-05 11:18:21 UTC</tt>.<br>

: The original revision id was <tt>~~565256841~~</tt>.<br>

: The original revision id was <tt>565319065</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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If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44.444... [[cent]]s 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.

Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[harmonic entropy]] possible and thus is, in theory, most dissonant. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison are are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.

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.

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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 [[consistent]]ly and distinctly--that is, everything in the 7-limit [[Diamonds|diamond]] is uniquely represented by a certain number of steps of 27 equal. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament

Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[harmonic entropy]] possible and thus is, in theory, most dissonant. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison are are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.

==Intervals==

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<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="/cent">cent</a>s 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 />~~

Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest <a class="wiki_link" href="/harmonic%20entropy">harmonic entropy</a> possible and thus is, in theory, most dissonant. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison are are more consonant as a result; intervals larger than this have less &quot;tension&quot; and thus are also more consonant.<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 <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 />

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<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 <a class="wiki_link" href="/consistent">consistent</a>ly 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. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament<br />

<br />

Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest <a class="wiki_link" href="/harmonic%20entropy">harmonic entropy</a> possible and thus is, in theory, most dissonant. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison are are more consonant as a result; intervals larger than this have less &quot;tension&quot; and thus are also more consonant.<br />

<br />

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

@@ 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:MasonGreen1|MasonGreen1]] and made on <tt>2015-11-05 01:52:17 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-05 11:18:21 UTC</tt>.<br>
-: The original revision id was <tt>565256841</tt>.<br>
+: The original revision id was <tt>565319065</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 9: / Line 9: @@
 If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44.444... [[cent]]s 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.
-Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[harmonic entropy]] possible and thus is, in theory, most dissonant. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison are are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.
 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.
@@ Line 17: / Line 15: @@
 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 [[consistent]]ly and distinctly--that is, everything in the 7-limit [[Diamonds|diamond]] is uniquely represented by a certain number of steps of 27 equal. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament
+Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[harmonic entropy]] possible and thus is, in theory, most dissonant. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison are are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.
 ==Intervals==
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   &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="/cent"&gt;cent&lt;/a&gt;s 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;
-Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest &lt;a class="wiki_link" href="/harmonic%20entropy"&gt;harmonic entropy&lt;/a&gt; possible and thus is, in theory, most dissonant. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison are are more consonant as a result; intervals larger than this have less &amp;quot;tension&amp;quot; and thus are also more consonant.&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 &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;
@@ Line 144: / Line 142: @@
 &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 &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt;ly 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. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament&lt;br /&gt;
+&lt;br /&gt;
+Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest &lt;a class="wiki_link" href="/harmonic%20entropy"&gt;harmonic entropy&lt;/a&gt; possible and thus is, in theory, most dissonant. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison are are more consonant as a result; intervals larger than this have less &amp;quot;tension&amp;quot; and thus are also more consonant.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:3:&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:3 --&gt;Intervals&lt;/h2&gt;