5edo

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=5 Equal Divisions of the Octave= 
==="equal pentatonic"=== 

5-edo divides the 1200-[[cents|cent]] octave into 5 equal parts, making its smallest interval exactly [[240¢]], or the fifth root of 2.

5-edo is the smallest [[edo]] containing xenharmonic intervals! (1edo 2edo 3edo 4edo are all subsets of 12edo)

==Intervals in 5-edo== 
|| **Interval,
in fifths of
an octave** || **Interval
in ¢** || **Closest
diatonic
interval name** || **The "neighborhood" of just intervals** ||
|| 0 || 0.0 || unison / prime || exactly 1/1 ||
|| 1 || 240.0 || second / third || +8.826 c from septimal second 8/7
-4.969 c from diminished third 144/125
-13.076 c from augmented second 125/108
-26.871 c from septimal minor third 7/6 ||
|| 2 || 480.0 || fourth || +9.219 c from narrow fourth 21/16
-0.686 c from smaller fourth 33/25
-18.045 c from just fourth 4/3 ||
|| 3 || 720.0 || fifth || +18.045 c from just fifth 3/2
+0.686 c from bigger fifth 50/33
-9.219 c from wide fifth 32/21 ||
|| 4 || 960.0 || sixth, seventh || 26.871 c from septimal major sixth 12/7
13.076 c from diminished seventh 216/125
4.969 c from augmented sixth 125/72
-8.826 c from septimal seventh 7/4 ||
|| 5 || 1200.0 || eighth || exactly 2/1 ||

==Related scales== 
* By its cardinality, 5-edo is related to other [[pentatonic]] scales, and it is especially close in sound to many Indonesian [[slendro|slendros]].
* Due to the interest around the "fifth" interval size, there are many [[nonoctave]] "stretch sisters" to 5-edo: square root of 4/3, cube root of 3/2, 8th root of 3, etc.
* For the same reason there are many "circle sisters":
** Make a chain of five "bigger fifths" (50/33), which makes three octaves 3.227¢ flat. (50/33)^5=7.985099.

==As a temperament== 
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 at the very edge what can sensibly be called temperament, but it does make sense and can be used.

Also tempered out is 27/25, leading to [[beep temperament]], which equates 10/9 with 6/5: it is a little more perverse even than [[father temperament|father]]. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role.

//How? Show me!//

==Cycles, Divisions== 
5 is a prime number -> 5-edo contains no sub-edos. Only simple cycles:
Cycle of seconds: 0-1-2-3-4-0
Cycle of fourths: 0-2-4-1-3-0
Cycle of fifths: 0-3-1-4-2-0
Cycle of sevenths: 0-4-3-2-1-0


=5-edo in Musicmaking= 
== == 
==**Compositions**, improvisations== 
* Brian McLaren: various and sundry
* [[http://home.comcast.net/%7Eteamouse/daybreak-vsc.mp3|Herman Miller]]: //[[http://home.comcast.net/%7Eteamouse/daybreak-vsc.mp3|Daybreak on Slendro Mountain]]// (2000)
* Paul Rubenstein: various, with electric guitars in 10- and 15-edo
* Aaron K. Johnson: //5tet funk// (2004)
* Bill Sethares: //5-tet funk// (2004), //Pentacle// (2004)
* X.J.Scott: //Sleeping Through It All// (2004) 
* Andrew Heathwaite: //Pinta Penta// (2004) (rendered in 6 alternative pentatonics as well)
* Hans Straub:  //[[http://home.datacomm.ch/straub/mamuth/5tet_e.html#asimchomsaia|Asîmchômsaia]]//[[#asimchomsaia]]
* [[#asimchomsaia]][[#asimchomsaia]][[#asimchomsaia]][[#asimchomsaia]]

==Notation== 
* via Reinhard's cents notation
* Sagittal: naturals on a five-line staff, with enharmonics (used interchangably) E=F and B=C
* a four-line hybrid treble/bass staff.

==Harmony== 
Scale does not have any strong consonance nor dissonance. Interval 240,000 c can serve as major second or minor third. Interval 960,000 c can serve as major sixth or minor seventh. Fourth is about 18 c flat than just fourth, it is rather "dirty"but jednoznačně identifikovatelná. Fifth is about 18 c sharp than just fifth, it is more dissonant than the fourth but still jednoznačně identifikovatelná.

Important chords:
0+1+3
0+2+3
0+1+3+4
0+2+3+4

==Melody== 
First from edos which can be use for melodies in "standard" 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.

==Chord or scale?== 
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.

Original HTML content:

<html><head><title>5edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x5 Equal Divisions of the Octave"></a><!-- ws:end:WikiTextHeadingRule:0 -->5 Equal Divisions of the Octave</h1>
 <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x5 Equal Divisions of the Octave--&quot;equal pentatonic&quot;"></a><!-- ws:end:WikiTextHeadingRule:2 -->&quot;equal pentatonic&quot;</h3>
 <br />
5-edo divides the 1200-<a class="wiki_link" href="/cents">cent</a> octave into 5 equal parts, making its smallest interval exactly <a class="wiki_link" href="/240%C2%A2">240¢</a>, or the fifth root of 2.<br />
<br />
5-edo is the smallest <a class="wiki_link" href="/edo">edo</a> containing xenharmonic intervals! (1edo 2edo 3edo 4edo are all subsets of 12edo)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x5 Equal Divisions of the Octave-Intervals in 5-edo"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals in 5-edo</h2>
 

<table class="wiki_table">
    <tr>
        <td><strong>Interval,<br />
in fifths of<br />
an octave</strong><br />
</td>
        <td><strong>Interval<br />
in ¢</strong><br />
</td>
        <td><strong>Closest<br />
diatonic<br />
interval name</strong><br />
</td>
        <td><strong>The &quot;neighborhood&quot; of just intervals</strong><br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0.0<br />
</td>
        <td>unison / prime<br />
</td>
        <td>exactly 1/1<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>240.0<br />
</td>
        <td>second / third<br />
</td>
        <td>+8.826 c from septimal second 8/7<br />
-4.969 c from diminished third 144/125<br />
-13.076 c from augmented second 125/108<br />
-26.871 c from septimal minor third 7/6<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>480.0<br />
</td>
        <td>fourth<br />
</td>
        <td>+9.219 c from narrow fourth 21/16<br />
-0.686 c from smaller fourth 33/25<br />
-18.045 c from just fourth 4/3<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>720.0<br />
</td>
        <td>fifth<br />
</td>
        <td>+18.045 c from just fifth 3/2<br />
+0.686 c from bigger fifth 50/33<br />
-9.219 c from wide fifth 32/21<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>960.0<br />
</td>
        <td>sixth, seventh<br />
</td>
        <td>26.871 c from septimal major sixth 12/7<br />
13.076 c from diminished seventh 216/125<br />
4.969 c from augmented sixth 125/72<br />
-8.826 c from septimal seventh 7/4<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>1200.0<br />
</td>
        <td>eighth<br />
</td>
        <td>exactly 2/1<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x5 Equal Divisions of the Octave-Related scales"></a><!-- ws:end:WikiTextHeadingRule:6 -->Related scales</h2>
 <ul><li>By its cardinality, 5-edo is related to other <a class="wiki_link" href="/pentatonic">pentatonic</a> scales, and it is especially close in sound to many Indonesian <a class="wiki_link" href="/slendro">slendros</a>.</li><li>Due to the interest around the &quot;fifth&quot; interval size, there are many <a class="wiki_link" href="/nonoctave">nonoctave</a> &quot;stretch sisters&quot; to 5-edo: square root of 4/3, cube root of 3/2, 8th root of 3, etc.</li><li>For the same reason there are many &quot;circle sisters&quot;:<ul><li>Make a chain of five &quot;bigger fifths&quot; (50/33), which makes three octaves 3.227¢ flat. (50/33)^5=7.985099.</li></ul></li></ul><br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x5 Equal Divisions of the Octave-As a temperament"></a><!-- ws:end:WikiTextHeadingRule:8 -->As a temperament</h2>
 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 at the very edge what can sensibly be called temperament, but it does make sense and can be used.<br />
<br />
Also tempered out is 27/25, leading to <a class="wiki_link" href="/beep%20temperament">beep temperament</a>, which equates 10/9 with 6/5: it is a little more perverse even than <a class="wiki_link" href="/father%20temperament">father</a>. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role.<br />
<br />
<em>How? Show me!</em><br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="x5 Equal Divisions of the Octave-Cycles, Divisions"></a><!-- ws:end:WikiTextHeadingRule:10 -->Cycles, Divisions</h2>
 5 is a prime number -&gt; 5-edo contains no sub-edos. Only simple cycles:<br />
Cycle of seconds: 0-1-2-3-4-0<br />
Cycle of fourths: 0-2-4-1-3-0<br />
Cycle of fifths: 0-3-1-4-2-0<br />
Cycle of sevenths: 0-4-3-2-1-0<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="x5-edo in Musicmaking"></a><!-- ws:end:WikiTextHeadingRule:12 -->5-edo in Musicmaking</h1>
 <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><!-- ws:end:WikiTextHeadingRule:14 --> </h2>
 <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="x5-edo in Musicmaking-Compositions, improvisations"></a><!-- ws:end:WikiTextHeadingRule:16 --><strong>Compositions</strong>, improvisations</h2>
 <ul><li>Brian McLaren: various and sundry</li><li><a class="wiki_link_ext" href="http://home.comcast.net/%7Eteamouse/daybreak-vsc.mp3" rel="nofollow">Herman Miller</a>: <em><a class="wiki_link_ext" href="http://home.comcast.net/%7Eteamouse/daybreak-vsc.mp3" rel="nofollow">Daybreak on Slendro Mountain</a></em> (2000)</li><li>Paul Rubenstein: various, with electric guitars in 10- and 15-edo</li><li>Aaron K. Johnson: <em>5tet funk</em> (2004)</li><li>Bill Sethares: <em>5-tet funk</em> (2004), <em>Pentacle</em> (2004)</li><li>X.J.Scott: <em>Sleeping Through It All</em> (2004)</li><li>Andrew Heathwaite: <em>Pinta Penta</em> (2004) (rendered in 6 alternative pentatonics as well)</li><li>Hans Straub:  <em><a class="wiki_link_ext" href="http://home.datacomm.ch/straub/mamuth/5tet_e.html#asimchomsaia" rel="nofollow">Asîmchômsaia</a></em><!-- ws:start:WikiTextAnchorRule:26:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@asimchomsaia&quot; title=&quot;Anchor: asimchomsaia&quot;/&gt; --><a name="asimchomsaia"></a><!-- ws:end:WikiTextAnchorRule:26 --></li><li><!-- ws:start:WikiTextAnchorRule:27:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@asimchomsaia&quot; title=&quot;Anchor: asimchomsaia&quot;/&gt; --><a name="asimchomsaia"></a><!-- ws:end:WikiTextAnchorRule:27 --><!-- ws:start:WikiTextAnchorRule:28:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@asimchomsaia&quot; title=&quot;Anchor: asimchomsaia&quot;/&gt; --><a name="asimchomsaia"></a><!-- ws:end:WikiTextAnchorRule:28 --><!-- ws:start:WikiTextAnchorRule:29:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@asimchomsaia&quot; title=&quot;Anchor: asimchomsaia&quot;/&gt; --><a name="asimchomsaia"></a><!-- ws:end:WikiTextAnchorRule:29 --><!-- ws:start:WikiTextAnchorRule:30:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@asimchomsaia&quot; title=&quot;Anchor: asimchomsaia&quot;/&gt; --><a name="asimchomsaia"></a><!-- ws:end:WikiTextAnchorRule:30 --></li></ul><br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="x5-edo in Musicmaking-Notation"></a><!-- ws:end:WikiTextHeadingRule:18 -->Notation</h2>
 <ul><li>via Reinhard's cents notation</li><li>Sagittal: naturals on a five-line staff, with enharmonics (used interchangably) E=F and B=C</li><li>a four-line hybrid treble/bass staff.</li></ul><br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="x5-edo in Musicmaking-Harmony"></a><!-- ws:end:WikiTextHeadingRule:20 -->Harmony</h2>
 Scale does not have any strong consonance nor dissonance. Interval 240,000 c can serve as major second or minor third. Interval 960,000 c can serve as major sixth or minor seventh. Fourth is about 18 c flat than just fourth, it is rather &quot;dirty&quot;but jednoznačně identifikovatelná. Fifth is about 18 c sharp than just fifth, it is more dissonant than the fourth but still jednoznačně identifikovatelná.<br />
<br />
Important chords:<br />
0+1+3<br />
0+2+3<br />
0+1+3+4<br />
0+2+3+4<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="x5-edo in Musicmaking-Melody"></a><!-- ws:end:WikiTextHeadingRule:22 -->Melody</h2>
 First from edos which can be use for melodies in &quot;standard&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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="x5-edo in Musicmaking-Chord or scale?"></a><!-- ws:end:WikiTextHeadingRule:24 -->Chord or scale?</h2>
 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.</body></html>