5edo
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[[toc|flat]] ---- [[image:5-EDO.png width="240" align="right"]] "C<span style="background-color: #ffffff;">annot possibly sound wrong no matter what notes you press." (Sevish on 5EDO)</span> =5 Equal Divisions of the Octave: Theory= ==="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.) ==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 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 1-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]] ==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== 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== 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. ==Commas Tempered== 5-EDO tempers out the following commas. (Note: This assumes the val < 5 8 12 14 17 19 |.) ||~ Comma ||~ Value (cents) ||~ Name ||~ Second Name ||~ Third Name ||~ Val || ||= 256/243 ||> 90.225 || Limma || Pythagorean Minor 2nd || || | 8 -5 > || ||= 81/80 ||> 21.506 || Syntonic Comma || Didymos Comma || Meantone Comma || | -4 4 -1 > || ||= 2889416/2882415 ||> 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 > || ||= 1065875/1063543 ||> 3.792 || Wadisma || || || | -26 -1 1 9 > || ||= 420175/419904 ||> 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 > || ||
Original HTML content:
<html><head><title>5edo</title></head><body><!-- ws:start:WikiTextTocRule:26:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><a href="#x5 Equal Divisions of the Octave: Theory">5 Equal Divisions of the Octave: Theory</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --> | <a href="#x5-edo in Musicmaking">5-edo in Musicmaking</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: -->
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<!-- ws:start:WikiTextLocalImageRule:468:<img src="/file/view/5-EDO.png/244670441/5-EDO.png" alt="" title="" style="width: 240px;" align="right" /> --><img src="/file/view/5-EDO.png/244670441/5-EDO.png" alt="5-EDO.png" title="5-EDO.png" style="width: 240px;" align="right" /><!-- ws:end:WikiTextLocalImageRule:468 --><br />
<br />
"C<span style="background-color: #ffffff;">annot possibly sound wrong no matter what notes you press." (Sevish on 5EDO)</span><br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="x5 Equal Divisions of the Octave: Theory"></a><!-- ws:end:WikiTextHeadingRule:0 -->5 Equal Divisions of the Octave: Theory</h1>
<!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x5 Equal Divisions of the Octave: Theory--"Equal Pentatonic""></a><!-- ws:end:WikiTextHeadingRule:2 -->"Equal Pentatonic"</h3>
<br />
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.<br />
<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 2. 5edo 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 />
<!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x5 Equal Divisions of the Octave: Theory-Intervals in 5-edo"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals in 5-edo</h2>
<table class="wiki_table">
<tr>
<td><strong>Interval,</strong><br />
<strong>in fifths of</strong><br />
<strong>an octave</strong><br />
</td>
<td><strong>Interval</strong><br />
<strong>in ¢</strong><br />
</td>
<td><strong>Closest</strong><br />
<strong>diatonic</strong><br />
<strong>interval name</strong><br />
</td>
<td><strong>The "neighborhood" 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:<h2> --><h2 id="toc3"><a name="x5 Equal Divisions of the Octave: Theory-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 "fifth" interval size, there are many <a class="wiki_link" href="/nonoctave">nonoctave</a> "stretch sisters" 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 "circle sisters":<ul><li>Make a chain of five "bigger fifths" (50/33), which makes three octaves 3.227¢ flat. (50/33)^5=7.985099.</li></ul></li></ul><br />
<!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="x5 Equal Divisions of the Octave: Theory-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 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 1-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:10:<h2> --><h2 id="toc5"><a name="x5 Equal Divisions of the Octave: Theory-Cycles, Divisions"></a><!-- ws:end:WikiTextHeadingRule:10 -->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:12:<h1> --><h1 id="toc6"><a name="x5-edo in Musicmaking"></a><!-- ws:end:WikiTextHeadingRule:12 -->5-edo in Musicmaking</h1>
<!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="x5-edo in Musicmaking-Compositions, improvisations"></a><!-- ws:end:WikiTextHeadingRule:14 --><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&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>"Cenobyte" 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></ul></ul><br />
<!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc8"><a name="x5-edo in Musicmaking-Notation"></a><!-- ws:end:WikiTextHeadingRule:16 -->Notation</h2>
<ul><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></ul><br />
<!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="x5-edo in Musicmaking-Harmony"></a><!-- ws:end:WikiTextHeadingRule:18 -->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 "dirty" 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:20:<h2> --><h2 id="toc10"><a name="x5-edo in Musicmaking-Melody"></a><!-- ws:end:WikiTextHeadingRule:20 -->Melody</h2>
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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:<h2> --><h2 id="toc11"><a name="x5-edo in Musicmaking-Chord or scale?"></a><!-- ws:end:WikiTextHeadingRule:22 -->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:24:<h2> --><h2 id="toc12"><a name="x5-edo in Musicmaking-Commas Tempered"></a><!-- ws:end:WikiTextHeadingRule:24 -->Commas Tempered</h2>
5-EDO tempers out the following commas. (Note: This assumes the val < 5 8 12 14 17 19 |.)<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>Val<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 ><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 ><br />
</td>
</tr>
<tr>
<td style="text-align: center;">2889416/2882415<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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><br />
</td>
</tr>
<tr>
<td style="text-align: center;">1065875/1063543<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 ><br />
</td>
</tr>
<tr>
<td style="text-align: center;">420175/419904<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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><br />
</td>
<td><br />
</td>
</tr>
</table>
</body></html>