5edo: Difference between revisions

Wikispaces>guest
**Imported revision 302323736 - Original comment: **
Wikispaces>Cenobyte
**Imported revision 358582621 - 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:guest|guest]] and made on <tt>2012-02-16 07:18:31 UTC</tt>.<br>
: This revision was by author [[User:Cenobyte|Cenobyte]] and made on <tt>2012-08-19 11:26:07 UTC</tt>.<br>
: The original revision id was <tt>302323736</tt>.<br>
: The original revision id was <tt>358582621</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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<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">[[toc|flat]]
<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">[[toc|flat]]
----
----
[[image:5-EDO.png width="240" align="right"]]
 
=5 Equal Divisions of the Octave: Theory=  
=5 Equal Divisions of the Octave: Theory=  
==="Equal Pentatonic"===  
==="Equal Pentatonic"===  


The above is a preliminary visualization of 5EDO courtesy of the primitive TuxPaint program. The colors are arbitrary and no labeling has been done yet, but it may be more than has ever been done before.
5-edo divides the 1200-[[cent]] octave into 5 equal parts, making its smallest interval exactly 240 [[cent|cents]], or the fifth root of 2. 5edo is the 3rd [[prime numbers|prime]] edo, after [[2edo]] and [[3edo]]. Most importantly, 5-edo is the smallest [[edo]] containing xenharmonic intervals! (1edo 2edo 3edo 4edo are all subsets of 12edo.)
 
==Listen to the sound of the 5-edo scale==


5-edo divides the 1200-[[cent]] octave into 5 equal parts, making its smallest interval exactly 240 [[cent|cents]], or the fifth root of 2. 5edo is the 3rd [[prime numbers|prime]] edo, after [[2edo]] and [[3edo]]. Most importantly, 5-edo is the smallest [[edo]] containing xenharmonic intervals! (1edo 2edo 3edo 4edo are all subsets of 12edo.)
For any musician, there is no substitute for the experience of a particular xenharmonic sound. The user going by the name Hyacinth on Wikipedia and Wikimedia Commons has many xenharmonic MIDI's and has graciously copylefted them! This is his 5-edo scale MIDI:
http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid


==Intervals in 5-edo==  
==Intervals in 5-edo==  
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||= 676/675 ||&gt; 2.563 || Parizeksma ||  ||  || | 2 -3 -2 0 0 2 &gt; ||  ||</pre></div>
||= 676/675 ||&gt; 2.563 || Parizeksma ||  ||  || | 2 -3 -2 0 0 2 &gt; ||  ||</pre></div>
<h4>Original HTML content:</h4>
<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;5edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:26:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;a href="#x5 Equal Divisions of the Octave: Theory"&gt;5 Equal Divisions of the Octave: Theory&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt; | &lt;a href="#x5-edo in Musicmaking"&gt;5-edo in Musicmaking&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt;&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&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;5edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:28:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;a href="#x5 Equal Divisions of the Octave: Theory"&gt;5 Equal Divisions of the Octave: Theory&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt; | &lt;a href="#x5-edo in Musicmaking"&gt;5-edo in Musicmaking&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt;&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt;&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;
&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;hr /&gt;
&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;hr /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:468:&amp;lt;img src=&amp;quot;/file/view/5-EDO.png/244670441/5-EDO.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;width: 240px;&amp;quot; align=&amp;quot;right&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/5-EDO.png/244670441/5-EDO.png" alt="5-EDO.png" title="5-EDO.png" style="width: 240px;" align="right" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:468 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;5 Equal Divisions of the Octave: Theory&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;5 Equal Divisions of the Octave: Theory&lt;/h1&gt;
  &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory--&amp;quot;Equal Pentatonic&amp;quot;"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&amp;quot;Equal Pentatonic&amp;quot;&lt;/h3&gt;
  &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory--&amp;quot;Equal Pentatonic&amp;quot;"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&amp;quot;Equal Pentatonic&amp;quot;&lt;/h3&gt;
  &lt;br /&gt;
  &lt;br /&gt;
The above is a preliminary visualization of 5EDO courtesy of the primitive TuxPaint program. The colors are arbitrary and no labeling has been done yet, but it may be more than has ever been done before.&lt;br /&gt;
5-edo divides the 1200-&lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt; octave into 5 equal parts, making its smallest interval exactly 240 &lt;a class="wiki_link" href="/cent"&gt;cents&lt;/a&gt;, or the fifth root of 2. 5edo is the 3rd &lt;a class="wiki_link" href="/prime%20numbers"&gt;prime&lt;/a&gt; edo, after &lt;a class="wiki_link" href="/2edo"&gt;2edo&lt;/a&gt; and &lt;a class="wiki_link" href="/3edo"&gt;3edo&lt;/a&gt;. Most importantly, 5-edo is the smallest &lt;a class="wiki_link" href="/edo"&gt;edo&lt;/a&gt; containing xenharmonic intervals! (1edo 2edo 3edo 4edo are all subsets of 12edo.)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
5-edo divides the 1200-&lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt; octave into 5 equal parts, making its smallest interval exactly 240 &lt;a class="wiki_link" href="/cent"&gt;cents&lt;/a&gt;, or the fifth root of 2. 5edo is the 3rd &lt;a class="wiki_link" href="/prime%20numbers"&gt;prime&lt;/a&gt; edo, after &lt;a class="wiki_link" href="/2edo"&gt;2edo&lt;/a&gt; and &lt;a class="wiki_link" href="/3edo"&gt;3edo&lt;/a&gt;. Most importantly, 5-edo is the smallest &lt;a class="wiki_link" href="/edo"&gt;edo&lt;/a&gt; containing xenharmonic intervals! (1edo 2edo 3edo 4edo are all subsets of 12edo.)&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-Listen to the sound of the 5-edo scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Listen to the sound of the 5-edo scale&lt;/h2&gt;
&lt;br /&gt;
For any musician, there is no substitute for the experience of a particular xenharmonic sound. The user going by the name Hyacinth on Wikipedia and Wikimedia Commons has many xenharmonic MIDI's and has graciously copylefted them! This is his 5-edo scale MIDI:&lt;br /&gt;
&lt;!-- ws:start:WikiTextUrlRule:715:http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid --&gt;&lt;a class="wiki_link_ext" href="http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid" rel="nofollow"&gt;http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:715 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-Intervals in 5-edo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Intervals in 5-edo&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-Intervals in 5-edo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Intervals in 5-edo&lt;/h2&gt;
   
   


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&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-Related scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Related scales&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-Related scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Related scales&lt;/h2&gt;
  &lt;ul&gt;&lt;li&gt;By its cardinality, 5-edo is related to other &lt;a class="wiki_link" href="/pentatonic"&gt;pentatonic&lt;/a&gt; scales, and it is especially close in sound to many Indonesian &lt;a class="wiki_link" href="/slendro"&gt;slendros&lt;/a&gt;.&lt;/li&gt;&lt;li&gt;Due to the interest around the &amp;quot;fifth&amp;quot; interval size, there are many &lt;a class="wiki_link" href="/nonoctave"&gt;nonoctave&lt;/a&gt; &amp;quot;stretch sisters&amp;quot; to 5-edo: square root of 4/3, cube root of 3/2, 8th root of 3, etc.&lt;/li&gt;&lt;li&gt;For the same reason there are many &amp;quot;circle sisters&amp;quot;:&lt;ul&gt;&lt;li&gt;Make a chain of five &amp;quot;bigger fifths&amp;quot; (50/33), which makes three octaves 3.227¢ flat. (50/33)^5=7.985099.&lt;/li&gt;&lt;/ul&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
  &lt;ul&gt;&lt;li&gt;By its cardinality, 5-edo is related to other &lt;a class="wiki_link" href="/pentatonic"&gt;pentatonic&lt;/a&gt; scales, and it is especially close in sound to many Indonesian &lt;a class="wiki_link" href="/slendro"&gt;slendros&lt;/a&gt;.&lt;/li&gt;&lt;li&gt;Due to the interest around the &amp;quot;fifth&amp;quot; interval size, there are many &lt;a class="wiki_link" href="/nonoctave"&gt;nonoctave&lt;/a&gt; &amp;quot;stretch sisters&amp;quot; to 5-edo: square root of 4/3, cube root of 3/2, 8th root of 3, etc.&lt;/li&gt;&lt;li&gt;For the same reason there are many &amp;quot;circle sisters&amp;quot;:&lt;ul&gt;&lt;li&gt;Make a chain of five &amp;quot;bigger fifths&amp;quot; (50/33), which makes three octaves 3.227¢ flat. (50/33)^5=7.985099.&lt;/li&gt;&lt;/ul&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-As a temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;As a temperament&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-As a temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;As a temperament&lt;/h2&gt;
  If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit &lt;a class="wiki_link" href="/Trienstonic%20clan"&gt;father temperament&lt;/a&gt;.&lt;br /&gt;
  If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit &lt;a class="wiki_link" href="/Trienstonic%20clan"&gt;father temperament&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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Despite its lack of accuracy, 5EDO is the second &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta integral edo&lt;/a&gt;, after 2EDO. It also is the smallest equal division representing the &lt;a class="wiki_link" href="/9-limit"&gt;9-limit&lt;/a&gt; &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt;ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how &lt;a class="wiki_link" href="/4edo"&gt;4edo&lt;/a&gt; can be used, and which is discussed in that article, it can be used to represent &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the &lt;a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices"&gt;lattice&lt;/a&gt; of tetrads/pentads together with the number of scale steps in 5EDO. However, while &lt;a class="wiki_link" href="/2edo"&gt;2edo&lt;/a&gt; represents the &lt;a class="wiki_link" href="/3-limit"&gt;3-limit&lt;/a&gt; consistently, &lt;a class="wiki_link" href="/3edo"&gt;3edo&lt;/a&gt; the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;, &lt;a class="wiki_link" href="/4edo"&gt;4edo&lt;/a&gt; the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; and &lt;a class="wiki_link" href="/5edo"&gt;5edo&lt;/a&gt; the &lt;a class="wiki_link" href="/9-limit"&gt;9-limit&lt;/a&gt;, to represent the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; consistently with a &lt;a class="wiki_link" href="/patent%20val"&gt;patent val&lt;/a&gt; requires going all the way to &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;.&lt;br /&gt;
Despite its lack of accuracy, 5EDO is the second &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta integral edo&lt;/a&gt;, after 2EDO. It also is the smallest equal division representing the &lt;a class="wiki_link" href="/9-limit"&gt;9-limit&lt;/a&gt; &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt;ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how &lt;a class="wiki_link" href="/4edo"&gt;4edo&lt;/a&gt; can be used, and which is discussed in that article, it can be used to represent &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the &lt;a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices"&gt;lattice&lt;/a&gt; of tetrads/pentads together with the number of scale steps in 5EDO. However, while &lt;a class="wiki_link" href="/2edo"&gt;2edo&lt;/a&gt; represents the &lt;a class="wiki_link" href="/3-limit"&gt;3-limit&lt;/a&gt; consistently, &lt;a class="wiki_link" href="/3edo"&gt;3edo&lt;/a&gt; the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;, &lt;a class="wiki_link" href="/4edo"&gt;4edo&lt;/a&gt; the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; and &lt;a class="wiki_link" href="/5edo"&gt;5edo&lt;/a&gt; the &lt;a class="wiki_link" href="/9-limit"&gt;9-limit&lt;/a&gt;, to represent the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; consistently with a &lt;a class="wiki_link" href="/patent%20val"&gt;patent val&lt;/a&gt; requires going all the way to &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-Cycles, Divisions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Cycles, Divisions&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-Cycles, Divisions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Cycles, Divisions&lt;/h2&gt;
  5 is a prime number so 5-edo contains no sub-edos. Only simple cycles:&lt;br /&gt;
  5 is a prime number so 5-edo contains no sub-edos. Only simple cycles:&lt;br /&gt;
Cycle of seconds: 0-1-2-3-4-0&lt;br /&gt;
Cycle of seconds: 0-1-2-3-4-0&lt;br /&gt;
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Cycle of sevenths: 0-4-3-2-1-0&lt;br /&gt;
Cycle of sevenths: 0-4-3-2-1-0&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="x5-edo in Musicmaking"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;5-edo in Musicmaking&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="x5-edo in Musicmaking"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;5-edo in Musicmaking&lt;/h1&gt;
  &lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="x5-edo in Musicmaking-Compositions, improvisations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;&lt;strong&gt;Compositions&lt;/strong&gt;, improvisations&lt;/h2&gt;
  &lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="x5-edo in Musicmaking-Compositions, improvisations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;&lt;strong&gt;Compositions&lt;/strong&gt;, improvisations&lt;/h2&gt;
  &lt;ul&gt;&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/" rel="nofollow"&gt;Herman Miller&lt;/a&gt;: &lt;em&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/herman_miller/Daybreak.mp3" rel="nofollow"&gt;Daybreak on Slendro Mountain&lt;/a&gt;&lt;/em&gt; (2000)&lt;/li&gt;&lt;li&gt;Aaron K. Johnson: &lt;em&gt;&lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/5tet_funk.mp3" rel="nofollow"&gt;5tet funk&lt;/a&gt;&lt;/em&gt; (2004)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=1519939" rel="nofollow"&gt;Andrew Heathwaite: //Pinta Penta// (2004)&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+pintapentain5tet.mp3" rel="nofollow"&gt;play&lt;/a&gt; (rendered in 6 alternative pentatonics as well)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Hans%20Straub"&gt;Hans Straub&lt;/a&gt;: &lt;a class="wiki_link_ext" href="http://home.datacomm.ch/straub/mamuth/5tet_e.html#asimchomsaia" rel="nofollow"&gt;Asîmchômsaia&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Straub/asimchomsaia.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Brian%20Wong"&gt;Brian Wong&lt;/a&gt;: &lt;a class="wiki_link_ext" href="http://bwong.ca/template1.php?sub=3" rel="nofollow"&gt;Slendronica#1b&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Wong/Slendronica1b.ogg" rel="nofollow"&gt;play&lt;/a&gt;&lt;/li&gt;&lt;li&gt;Brian McLaren: various and sundry&lt;/li&gt;&lt;li&gt;Paul Rubenstein: various, with electric guitars in 10- and 15-edo&lt;/li&gt;&lt;li&gt;X.J.Scott: &lt;em&gt;Sleeping Through It All&lt;/em&gt; (2004)&lt;/li&gt;&lt;li&gt;Bill Sethares: &lt;em&gt;5-tet funk&lt;/em&gt; (2004), &lt;em&gt;Pentacle&lt;/em&gt; (2004)&lt;/li&gt;&lt;li&gt;&amp;quot;Cenobyte&amp;quot; Ukulele &lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=UKUCRnEJKKU" rel="nofollow"&gt; http://www.youtube.com/watch?v=UKUCRnEJKKU&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/ul&gt;&lt;br /&gt;
  &lt;ul&gt;&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/" rel="nofollow"&gt;Herman Miller&lt;/a&gt;: &lt;em&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/herman_miller/Daybreak.mp3" rel="nofollow"&gt;Daybreak on Slendro Mountain&lt;/a&gt;&lt;/em&gt; (2000)&lt;/li&gt;&lt;li&gt;Aaron K. Johnson: &lt;em&gt;&lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/5tet_funk.mp3" rel="nofollow"&gt;5tet funk&lt;/a&gt;&lt;/em&gt; (2004)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=1519939" rel="nofollow"&gt;Andrew Heathwaite: //Pinta Penta// (2004)&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+pintapentain5tet.mp3" rel="nofollow"&gt;play&lt;/a&gt; (rendered in 6 alternative pentatonics as well)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Hans%20Straub"&gt;Hans Straub&lt;/a&gt;: &lt;a class="wiki_link_ext" href="http://home.datacomm.ch/straub/mamuth/5tet_e.html#asimchomsaia" rel="nofollow"&gt;Asîmchômsaia&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Straub/asimchomsaia.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Brian%20Wong"&gt;Brian Wong&lt;/a&gt;: &lt;a class="wiki_link_ext" href="http://bwong.ca/template1.php?sub=3" rel="nofollow"&gt;Slendronica#1b&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Wong/Slendronica1b.ogg" rel="nofollow"&gt;play&lt;/a&gt;&lt;/li&gt;&lt;li&gt;Brian McLaren: various and sundry&lt;/li&gt;&lt;li&gt;Paul Rubenstein: various, with electric guitars in 10- and 15-edo&lt;/li&gt;&lt;li&gt;X.J.Scott: &lt;em&gt;Sleeping Through It All&lt;/em&gt; (2004)&lt;/li&gt;&lt;li&gt;Bill Sethares: &lt;em&gt;5-tet funk&lt;/em&gt; (2004), &lt;em&gt;Pentacle&lt;/em&gt; (2004)&lt;/li&gt;&lt;li&gt;&amp;quot;Cenobyte&amp;quot; Ukulele &lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=UKUCRnEJKKU" rel="nofollow"&gt; http://www.youtube.com/watch?v=UKUCRnEJKKU&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="x5-edo in Musicmaking-Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Notation&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="x5-edo in Musicmaking-Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Notation&lt;/h2&gt;
  &lt;ul&gt;&lt;ul&gt;&lt;li&gt;via Reinhard's cents notation&lt;/li&gt;&lt;li&gt;Sagittal: naturals on a five-line staff, with enharmonics (used interchangably) E=F and B=C&lt;/li&gt;&lt;li&gt;a four-line hybrid treble/bass staff.&lt;/li&gt;&lt;/ul&gt;&lt;/ul&gt;&lt;br /&gt;
  &lt;ul&gt;&lt;ul&gt;&lt;li&gt;via Reinhard's cents notation&lt;/li&gt;&lt;li&gt;Sagittal: naturals on a five-line staff, with enharmonics (used interchangably) E=F and B=C&lt;/li&gt;&lt;li&gt;a four-line hybrid treble/bass staff.&lt;/li&gt;&lt;/ul&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="x5-edo in Musicmaking-Harmony"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Harmony&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="x5-edo in Musicmaking-Harmony"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Harmony&lt;/h2&gt;
  5edo does not have any strong consonance nor dissonance. The 240 cent interval can serve as either a major second or minor third, and the 960 cent interval as either a major sixth or minor seventh. The fourth is about 18 cents flat of a just fourth, making it rather &amp;quot;dirty&amp;quot; but recognizable. The fifth is likewise about 18 cents sharp of a just fifth, dissonant but still easily recognizable.&lt;br /&gt;
  5edo does not have any strong consonance nor dissonance. The 240 cent interval can serve as either a major second or minor third, and the 960 cent interval as either a major sixth or minor seventh. The fourth is about 18 cents flat of a just fourth, making it rather &amp;quot;dirty&amp;quot; but recognizable. The fifth is likewise about 18 cents sharp of a just fifth, dissonant but still easily recognizable.&lt;br /&gt;
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Important chords:&lt;br /&gt;
Important chords:&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;0+1+3&lt;/li&gt;&lt;li&gt;0+2+3&lt;/li&gt;&lt;li&gt;0+1+3+4&lt;/li&gt;&lt;li&gt;0+2+3+4&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;0+1+3&lt;/li&gt;&lt;li&gt;0+2+3&lt;/li&gt;&lt;li&gt;0+1+3+4&lt;/li&gt;&lt;li&gt;0+2+3+4&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="x5-edo in Musicmaking-Melody"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Melody&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="x5-edo in Musicmaking-Melody"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Melody&lt;/h2&gt;
  First from edos which can be use for melodies in &amp;quot;standard&amp;quot; way. Relatively large step of 240.00 c can be used as major second for the melody construction. The scale has whole-tone as well as pentatonic character.&lt;br /&gt;
  First from edos which can be use for melodies in &amp;quot;standard&amp;quot; way. Relatively large step of 240.00 c can be used as major second for the melody construction. The scale has whole-tone as well as pentatonic character.&lt;br /&gt;
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&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="x5-edo in Musicmaking-Chord or scale?"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Chord or scale?&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="x5-edo in Musicmaking-Chord or scale?"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Chord or scale?&lt;/h2&gt;
  Either way, it is hard to wander very far from where you start. However, it has the scale-like feature that there are (barely) enough notes to create melody, in the form of an equal version of pentatonic.&lt;br /&gt;
  Either way, it is hard to wander very far from where you start. However, it has the scale-like feature that there are (barely) enough notes to create melody, in the form of an equal version of pentatonic.&lt;br /&gt;
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&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="x5-edo in Musicmaking-Commas Tempered"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Commas Tempered&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="x5-edo in Musicmaking-Commas Tempered"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;Commas Tempered&lt;/h2&gt;
  5-EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 5 8 12 14 17 19 |.)&lt;br /&gt;
  5-EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 5 8 12 14 17 19 |.)&lt;br /&gt;
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