5edo

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<span style="display: block; text-align: right;">[[５平均律|日本語]]
</span>
[[toc|flat]]
----

=5 Equal Divisions of the Octave: Theory= 
==="Equal Pentatonic"=== 

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

There is a lot of near-equipentatonic world music, just google "gyil" or "amadinda" or "slendro".

==Listen to the sound of the 5-edo scale== 

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== 
||~ degrees ||~ size
in [[cent|cents]] ||~ Closest diatonic
interval name ||~ The "neighborhood" of just intervals ||
||= 0 ||= 0 ||= unison / prime || exactly 1/1 ||
||= 1 ||= 240 ||= second, third || +8.826¢ from septimal second [[8_7|8/7]]
-4.969¢ from diminished third [[144_125|144/125]]
-13.076¢ from augmented second 125/108
-26.871¢ from septimal minor third [[7_6|7/6]] ||
||= 2 ||= 480 ||= fourth || +9.219¢ from narrow fourth [[21_16|21/16]]
-0.686¢ from smaller fourth [[33_25|33/25]]
-18.045¢ from just fourth [[4_3|4/3]] ||
||= 3 ||= 720 ||= fifth || +18.045¢ from just fifth [[3_2|3/2]]
+0.686¢ from bigger fifth [[50_33|50/33]]
-9.219¢ from wide fifth [[32_21|32/21]] ||
||= 4 ||= 960 ||= sixth, seventh || 26.871¢ from septimal major sixth [[12_7|12/7]]
13.076¢ from diminished seventh 216/125
4.969¢ from augmented sixth [[125_72|125/72]]
-8.826¢ from septimal seventh [[7_4|7/4]] ||
||= 5 ||= 1200 ||= octave / eighth || exactly 2/1 ||

[[media type="custom" key="24802268"]]
[[file:5ed2-001.svg]]

==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 5-limit [[Trienstonic clan|father temperament]].

Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain [[Bug family|bug temperament]], which equates 10/9 with 6/5: it is a little more perverse even than 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. For example, I-IV-V-I is the same as I-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.

Despite its lack of accuracy, 5EDO is the second [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]], after 2EDO. It also is the smallest equal division representing the [[9-limit]] [[consistent]]ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[The Seven Limit Symmetrical Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5EDO. However, while [[2edo]] represents the [[3-limit]] consistently, [[3edo]] the [[5-limit]], [[4edo]] the [[7-limit]] and [[5edo]] the [[9-limit]], to represent the [[11-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].

==Cycles, Divisions== 
5 is a prime number so 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== 
** [[http://www.io.com/%7Ehmiller/|Herman Miller]]: //[[http://micro.soonlabel.com/herman_miller/Daybreak.mp3|Daybreak on Slendro Mountain]]// (2000)
** Aaron K. Johnson: //[[http://www.akjmusic.com/audio/5tet_funk.mp3|5tet funk]]// (2004)
** [[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=1519939|Andrew Heathwaite: //Pinta Penta// (2004)]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+pintapentain5tet.mp3|play]] (rendered in 6 alternative pentatonics as well)
** [[Hans Straub]]: [[http://home.datacomm.ch/straub/mamuth/5tet_e.html#asimchomsaia|Asîmchômsaia]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Straub/asimchomsaia.mp3|play]]
** [[Brian Wong]]: [[http://bwong.ca/template1.php?sub=3|Slendronica#1b]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Wong/Slendronica1b.ogg|play]]
** Brian McLaren: various and sundry
** Paul Rubenstein: various, with electric guitars in 10- and 15-edo
** X.J.Scott: //Sleeping Through It All// (2004)
** Bill Sethares: //5-tet funk// (2004), //Pentacle// (2004)
** "Cenobyte" Ukulele [[http://www.youtube.com/watch?v=UKUCRnEJKKU| http://www.youtube.com/watch?v=UKUCRnEJKKU]]
** "[[@http://www.jamendo.com/en/list/a104474/true-island-5-equal-divisions-of-the-octave-ukulele|True Island]]" (album) by Small Scale Revolution (2011)
** Ralph Jarzombek: [[http://webzoom.freewebs.com/ralphjarzombek/micro12.mp3|Micro12]]

There is a lot of 5edo world music, search for "gyil" or "amadinda" or "slendro".

==Ear Training== 
5edo ear-training exercises by Alex Ness available [[@https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web|here]].

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

==Harmony== 
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 "dirty" but recognizable. The fifth is likewise about 18 cents sharp of a just fifth, dissonant but still easily recognizable.

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

==Melody== 
Smallest edo which can be used for melodies in a "standard" way. Relatively large step of 240 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.

==Commas Tempered== 
5-EDO tempers out the following commas. (Note: This assumes the val < 5 8 12 14 17 19/1 |.)

||~ Comma ||~ Value (cents) ||~ Name ||~ Second Name ||~ Third Name ||~ Monzo ||
||= 256/243 ||> 90.225 || Limma || Pythagorean Minor 2nd ||   || | 8 -5 > ||
||= 81/80 ||> 21.506 || Syntonic Comma || Didymos Comma || Meantone Comma || | -4 4 -1 > ||
||=   ||> 4.200 || Vulture ||   ||   || | 24 -21 4 > ||
||= 36/35 ||> 48.770 || Septimal Quarter Tone ||   ||   || | 2 2 -1 -1 > ||
||= 49/48 ||> 35.697 || Slendro Diesis ||   ||   || | -4 -1 0 2 > ||
||= 64/63 ||> 27.264 || Septimal Comma || Archytas' Comma || Leipziger Komma || | 6 -2 0 -1 > ||
||= 245/243 ||> 14.191 || Sensamagic ||   ||   || | 0 -5 1 2 > ||
||= 1728/1715 ||> 13.074 || Orwellisma || Orwell Comma ||   || | 6 3 -1 -3 > ||
||= 1029/1024 ||> 8.433 || Gamelisma ||   ||   || | -10 1 0 3 > ||
||= 19683/19600 ||> 7.316 || Cataharry ||   ||   || | -4 9 -2 -2 > ||
||= 5120/5103 ||> 5.758 || Hemifamity ||   ||   || | 10 -6 1 -1 > ||
||=   ||> 3.792 || Wadisma ||   ||   || | -26 -1 1 9 > ||
||=   ||> 1.117 || Wizma ||   ||   || | -6 -8 2 5 > ||
||= 99/98 ||> 17.576 || Mothwellsma ||   ||   || | -1 2 0 -2 1 > ||
||= 896/891 ||> 9.688 || Pentacircle ||   ||   || | 7 -4 0 1 -1 > ||
||= 385/384 ||> 4.503 || Keenanisma ||   ||   || | -7 -1 1 1 1 > ||
||= 441/440 ||> 3.930 || Werckisma ||   ||   || | -3 2 -1 2 -1 > ||
||= 3025/3024 ||> 0.572 || Lehmerisma ||   ||   || | -4 -3 2 -1 2 > ||
||= 91/90 ||> 19.130 || Superleap ||   ||   || | -1 -2 -1 1 0 1 > ||
||= 676/675 ||> 2.563 || Parizeksma ||   ||   || | 2 -3 -2 0 0 2 > ||
||= 16/15 ||> 111.731 || Diatonic semitone ||   ||   || | 4 -1 -1 > ||
||= 14/13 ||> 128.298 ||   ||   ||   || | 1 0 0 1 0 -1 > ||
||= 27/25 ||> 133.238 || Large diatonic semit. ||   ||   || | 0 3 -2 > ||
||= 11/10 ||> 165.004 || Large neutral second ||   ||   || | -1 0 -1 0 1 > ||

Original HTML content:

<html><head><title>5edo</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="/%EF%BC%95%E5%B9%B3%E5%9D%87%E5%BE%8B">日本語</a><br />
</span><br />
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<!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="x5 Equal Divisions of the Octave: Theory"></a><!-- ws:end:WikiTextHeadingRule:1 -->5 Equal Divisions of the Octave: Theory</h1>
 <!-- ws:start:WikiTextHeadingRule:3:&lt;h3&gt; --><h3 id="toc1"><a name="x5 Equal Divisions of the Octave: Theory--&quot;Equal Pentatonic&quot;"></a><!-- ws:end:WikiTextHeadingRule:3 -->&quot;Equal Pentatonic&quot;</h3>
 <br />
5-edo divides the 1200-<a class="wiki_link" href="/cent">cent</a> octave into 5 equal parts, making its smallest interval exactly 240 <a class="wiki_link" href="/cent">cents</a>, or the fifth root of two. 5-edo is the 3rd <a class="wiki_link" href="/prime%20numbers">prime</a> edo, after <a class="wiki_link" href="/2edo">2edo</a> and <a class="wiki_link" href="/3edo">3edo</a>. Most importantly, 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 />
There is a lot of near-equipentatonic world music, just google &quot;gyil&quot; or &quot;amadinda&quot; or &quot;slendro&quot;.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:5:&lt;h2&gt; --><h2 id="toc2"><a name="x5 Equal Divisions of the Octave: Theory-Listen to the sound of the 5-edo scale"></a><!-- ws:end:WikiTextHeadingRule:5 -->Listen to the sound of the 5-edo scale</h2>
 <br />
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:<br />
<a class="wiki_link_ext" href="http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid" rel="nofollow" target="_blank">http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:7:&lt;h2&gt; --><h2 id="toc3"><a name="x5 Equal Divisions of the Octave: Theory-Intervals in 5-edo"></a><!-- ws:end:WikiTextHeadingRule:7 -->Intervals in 5-edo</h2>
 

<table class="wiki_table">
    <tr>
        <th>degrees<br />
</th>
        <th>size<br />
in <a class="wiki_link" href="/cent">cents</a><br />
</th>
        <th>Closest diatonic<br />
interval name<br />
</th>
        <th>The &quot;neighborhood&quot; of just intervals<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: center;">unison / prime<br />
</td>
        <td>exactly 1/1<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1<br />
</td>
        <td style="text-align: center;">240<br />
</td>
        <td style="text-align: center;">second, third<br />
</td>
        <td>+8.826¢ from septimal second <a class="wiki_link" href="/8_7">8/7</a><br />
-4.969¢ from diminished third <a class="wiki_link" href="/144_125">144/125</a><br />
-13.076¢ from augmented second 125/108<br />
-26.871¢ from septimal minor third <a class="wiki_link" href="/7_6">7/6</a><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">2<br />
</td>
        <td style="text-align: center;">480<br />
</td>
        <td style="text-align: center;">fourth<br />
</td>
        <td>+9.219¢ from narrow fourth <a class="wiki_link" href="/21_16">21/16</a><br />
-0.686¢ from smaller fourth <a class="wiki_link" href="/33_25">33/25</a><br />
-18.045¢ from just fourth <a class="wiki_link" href="/4_3">4/3</a><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3<br />
</td>
        <td style="text-align: center;">720<br />
</td>
        <td style="text-align: center;">fifth<br />
</td>
        <td>+18.045¢ from just fifth <a class="wiki_link" href="/3_2">3/2</a><br />
+0.686¢ from bigger fifth <a class="wiki_link" href="/50_33">50/33</a><br />
-9.219¢ from wide fifth <a class="wiki_link" href="/32_21">32/21</a><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4<br />
</td>
        <td style="text-align: center;">960<br />
</td>
        <td style="text-align: center;">sixth, seventh<br />
</td>
        <td>26.871¢ from septimal major sixth <a class="wiki_link" href="/12_7">12/7</a><br />
13.076¢ from diminished seventh 216/125<br />
4.969¢ from augmented sixth <a class="wiki_link" href="/125_72">125/72</a><br />
-8.826¢ from septimal seventh <a class="wiki_link" href="/7_4">7/4</a><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">5<br />
</td>
        <td style="text-align: center;">1200<br />
</td>
        <td style="text-align: center;">octave / eighth<br />
</td>
        <td>exactly 2/1<br />
</td>
    </tr>
</table>

<br />
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<!-- ws:start:WikiTextHeadingRule:9:&lt;h2&gt; --><h2 id="toc4"><a name="x5 Equal Divisions of the Octave: Theory-Related scales"></a><!-- ws:end:WikiTextHeadingRule:9 -->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:11:&lt;h2&gt; --><h2 id="toc5"><a name="x5 Equal Divisions of the Octave: Theory-As a temperament"></a><!-- ws:end:WikiTextHeadingRule:11 -->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 5-limit <a class="wiki_link" href="/Trienstonic%20clan">father temperament</a>.<br />
<br />
Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain <a class="wiki_link" href="/Bug%20family">bug temperament</a>, which equates 10/9 with 6/5: it is a little more perverse even than 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. For example, I-IV-V-I is the same as I-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.<br />
<br />
Despite its lack of accuracy, 5EDO is the second <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>, after 2EDO. It also is the smallest equal division representing the <a class="wiki_link" href="/9-limit">9-limit</a> <a class="wiki_link" href="/consistent">consistent</a>ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how <a class="wiki_link" href="/4edo">4edo</a> can be used, and which is discussed in that article, it can be used to represent <a class="wiki_link" href="/7-limit">7-limit</a> intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the <a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices">lattice</a> of tetrads/pentads together with the number of scale steps in 5EDO. However, while <a class="wiki_link" href="/2edo">2edo</a> represents the <a class="wiki_link" href="/3-limit">3-limit</a> consistently, <a class="wiki_link" href="/3edo">3edo</a> the <a class="wiki_link" href="/5-limit">5-limit</a>, <a class="wiki_link" href="/4edo">4edo</a> the <a class="wiki_link" href="/7-limit">7-limit</a> and <a class="wiki_link" href="/5edo">5edo</a> the <a class="wiki_link" href="/9-limit">9-limit</a>, to represent the <a class="wiki_link" href="/11-limit">11-limit</a> consistently with a <a class="wiki_link" href="/patent%20val">patent val</a> requires going all the way to <a class="wiki_link" href="/22edo">22edo</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:13:&lt;h2&gt; --><h2 id="toc6"><a name="x5 Equal Divisions of the Octave: Theory-Cycles, Divisions"></a><!-- ws:end:WikiTextHeadingRule:13 -->Cycles, Divisions</h2>
 5 is a prime number so 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 />
<!-- ws:start:WikiTextHeadingRule:15:&lt;h1&gt; --><h1 id="toc7"><a name="x5-edo in Musicmaking"></a><!-- ws:end:WikiTextHeadingRule:15 -->5-edo in Musicmaking</h1>
 <!-- ws:start:WikiTextHeadingRule:17:&lt;h2&gt; --><h2 id="toc8"><a name="x5-edo in Musicmaking-Compositions, improvisations"></a><!-- ws:end:WikiTextHeadingRule:17 --><strong>Compositions</strong>, improvisations</h2>
 <ul><ul><li><a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/" rel="nofollow">Herman Miller</a>: <em><a class="wiki_link_ext" href="http://micro.soonlabel.com/herman_miller/Daybreak.mp3" rel="nofollow">Daybreak on Slendro Mountain</a></em> (2000)</li><li>Aaron K. Johnson: <em><a class="wiki_link_ext" href="http://www.akjmusic.com/audio/5tet_funk.mp3" rel="nofollow">5tet funk</a></em> (2004)</li><li><a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=1519939" rel="nofollow">Andrew Heathwaite: //Pinta Penta// (2004)</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+pintapentain5tet.mp3" rel="nofollow">play</a> (rendered in 6 alternative pentatonics as well)</li><li><a class="wiki_link" href="/Hans%20Straub">Hans Straub</a>: <a class="wiki_link_ext" href="http://home.datacomm.ch/straub/mamuth/5tet_e.html#asimchomsaia" rel="nofollow">Asîmchômsaia</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Straub/asimchomsaia.mp3" rel="nofollow">play</a></li><li><a class="wiki_link" href="/Brian%20Wong">Brian Wong</a>: <a class="wiki_link_ext" href="http://bwong.ca/template1.php?sub=3" rel="nofollow">Slendronica#1b</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Wong/Slendronica1b.ogg" rel="nofollow">play</a></li><li>Brian McLaren: various and sundry</li><li>Paul Rubenstein: various, with electric guitars in 10- and 15-edo</li><li>X.J.Scott: <em>Sleeping Through It All</em> (2004)</li><li>Bill Sethares: <em>5-tet funk</em> (2004), <em>Pentacle</em> (2004)</li><li>&quot;Cenobyte&quot; Ukulele <a class="wiki_link_ext" href="http://www.youtube.com/watch?v=UKUCRnEJKKU" rel="nofollow"> http://www.youtube.com/watch?v=UKUCRnEJKKU</a></li><li>&quot;<a class="wiki_link_ext" href="http://www.jamendo.com/en/list/a104474/true-island-5-equal-divisions-of-the-octave-ukulele" rel="nofollow" target="_blank">True Island</a>&quot; (album) by Small Scale Revolution (2011)</li><li>Ralph Jarzombek: <a class="wiki_link_ext" href="http://webzoom.freewebs.com/ralphjarzombek/micro12.mp3" rel="nofollow">Micro12</a></li></ul></ul><br />
There is a lot of 5edo world music, search for &quot;gyil&quot; or &quot;amadinda&quot; or &quot;slendro&quot;.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:19:&lt;h2&gt; --><h2 id="toc9"><a name="x5-edo in Musicmaking-Ear Training"></a><!-- ws:end:WikiTextHeadingRule:19 -->Ear Training</h2>
 5edo ear-training exercises by Alex Ness available <a class="wiki_link_ext" href="https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&amp;usp=drive_web" rel="nofollow" target="_blank">here</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:21:&lt;h2&gt; --><h2 id="toc10"><a name="x5-edo in Musicmaking-Notation"></a><!-- ws:end:WikiTextHeadingRule:21 -->Notation</h2>
 <ul><ul><li>via Reinhard's cents notation</li><li>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></ul><br />
<!-- ws:start:WikiTextHeadingRule:23:&lt;h2&gt; --><h2 id="toc11"><a name="x5-edo in Musicmaking-Harmony"></a><!-- ws:end:WikiTextHeadingRule:23 -->Harmony</h2>
 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 &quot;dirty&quot; but recognizable. The fifth is likewise about 18 cents sharp of a just fifth, dissonant but still easily recognizable.<br />
<br />
Important chords:<br />
<ul><li>0+1+3</li><li>0+2+3</li><li>0+1+3+4</li><li>0+2+3+4</li></ul><br />
<!-- ws:start:WikiTextHeadingRule:25:&lt;h2&gt; --><h2 id="toc12"><a name="x5-edo in Musicmaking-Melody"></a><!-- ws:end:WikiTextHeadingRule:25 -->Melody</h2>
 Smallest edo which can be used for melodies in a &quot;standard&quot; way. Relatively large step of 240 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:27:&lt;h2&gt; --><h2 id="toc13"><a name="x5-edo in Musicmaking-Chord or scale?"></a><!-- ws:end:WikiTextHeadingRule:27 -->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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:29:&lt;h2&gt; --><h2 id="toc14"><a name="x5-edo in Musicmaking-Commas Tempered"></a><!-- ws:end:WikiTextHeadingRule:29 -->Commas Tempered</h2>
 5-EDO tempers out the following commas. (Note: This assumes the val &lt; 5 8 12 14 17 19/1 |.)<br />
<br />


<table class="wiki_table">
    <tr>
        <th>Comma<br />
</th>
        <th>Value (cents)<br />
</th>
        <th>Name<br />
</th>
        <th>Second Name<br />
</th>
        <th>Third Name<br />
</th>
        <th>Monzo<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">256/243<br />
</td>
        <td style="text-align: right;">90.225<br />
</td>
        <td>Limma<br />
</td>
        <td>Pythagorean Minor 2nd<br />
</td>
        <td><br />
</td>
        <td>| 8 -5 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">81/80<br />
</td>
        <td style="text-align: right;">21.506<br />
</td>
        <td>Syntonic Comma<br />
</td>
        <td>Didymos Comma<br />
</td>
        <td>Meantone Comma<br />
</td>
        <td>| -4 4 -1 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: right;">4.200<br />
</td>
        <td>Vulture<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| 24 -21 4 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">36/35<br />
</td>
        <td style="text-align: right;">48.770<br />
</td>
        <td>Septimal Quarter Tone<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| 2 2 -1 -1 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">49/48<br />
</td>
        <td style="text-align: right;">35.697<br />
</td>
        <td>Slendro Diesis<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| -4 -1 0 2 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">64/63<br />
</td>
        <td style="text-align: right;">27.264<br />
</td>
        <td>Septimal Comma<br />
</td>
        <td>Archytas' Comma<br />
</td>
        <td>Leipziger Komma<br />
</td>
        <td>| 6 -2 0 -1 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">245/243<br />
</td>
        <td style="text-align: right;">14.191<br />
</td>
        <td>Sensamagic<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| 0 -5 1 2 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1728/1715<br />
</td>
        <td style="text-align: right;">13.074<br />
</td>
        <td>Orwellisma<br />
</td>
        <td>Orwell Comma<br />
</td>
        <td><br />
</td>
        <td>| 6 3 -1 -3 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1029/1024<br />
</td>
        <td style="text-align: right;">8.433<br />
</td>
        <td>Gamelisma<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| -10 1 0 3 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">19683/19600<br />
</td>
        <td style="text-align: right;">7.316<br />
</td>
        <td>Cataharry<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| -4 9 -2 -2 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">5120/5103<br />
</td>
        <td style="text-align: right;">5.758<br />
</td>
        <td>Hemifamity<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| 10 -6 1 -1 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: right;">3.792<br />
</td>
        <td>Wadisma<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| -26 -1 1 9 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: right;">1.117<br />
</td>
        <td>Wizma<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| -6 -8 2 5 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">99/98<br />
</td>
        <td style="text-align: right;">17.576<br />
</td>
        <td>Mothwellsma<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| -1 2 0 -2 1 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">896/891<br />
</td>
        <td style="text-align: right;">9.688<br />
</td>
        <td>Pentacircle<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| 7 -4 0 1 -1 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">385/384<br />
</td>
        <td style="text-align: right;">4.503<br />
</td>
        <td>Keenanisma<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| -7 -1 1 1 1 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">441/440<br />
</td>
        <td style="text-align: right;">3.930<br />
</td>
        <td>Werckisma<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| -3 2 -1 2 -1 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3025/3024<br />
</td>
        <td style="text-align: right;">0.572<br />
</td>
        <td>Lehmerisma<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| -4 -3 2 -1 2 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">91/90<br />
</td>
        <td style="text-align: right;">19.130<br />
</td>
        <td>Superleap<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| -1 -2 -1 1 0 1 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">676/675<br />
</td>
        <td style="text-align: right;">2.563<br />
</td>
        <td>Parizeksma<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| 2 -3 -2 0 0 2 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">16/15<br />
</td>
        <td style="text-align: right;">111.731<br />
</td>
        <td>Diatonic semitone<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| 4 -1 -1 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">14/13<br />
</td>
        <td style="text-align: right;">128.298<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| 1 0 0 1 0 -1 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">27/25<br />
</td>
        <td style="text-align: right;">133.238<br />
</td>
        <td>Large diatonic semit.<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| 0 3 -2 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">11/10<br />
</td>
        <td style="text-align: right;">165.004<br />
</td>
        <td>Large neutral second<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>| -1 0 -1 0 1 &gt;<br />
</td>
    </tr>
</table>

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