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.) ==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== || **Interval,** **in fifths of** **an octave** || **Interval** **in ¢, DMS** || **Closest** **diatonic** **interval name** || **The "neighborhood" of just intervals** || || 0 || 0.0, 0° || unison / prime || exactly 1/1 || || 1 || 240.0. 72° || second / third || +8.826 c, +2°38'52" from septimal second 8/7 -4.969 c, -1°29'26" from diminished third 144/125 -13.076 c, -3<span style="line-height: 1.5;">°55'22"</span> from augmented second 125/108 -26.871 c, -8<span style="line-height: 1.5;">°3'41"</span> from septimal minor third 7/6 || || 2 || 480.0, 144° || fourth || +9.219 c, +2°45'57" from narrow fourth 21/16 -0.686 c, -11'37" from smaller fourth 33/25 -18.045 c, -5°24'49" from just fourth 4/3 || || 3 || 720.0, 216° || fifth || +18.045 c, +5°24'49" from just fifth 3/2 +0.686 c<span style="line-height: 1.5;">, +11'37" </span>from bigger fifth 50/33 -9.219 c<span style="line-height: 1.5;">, +2°45'57" </span>from wide fifth 32/21 || || 4 || 960.0, 288° || sixth, seventh || 26.871 c, 8°3'41" from septimal major sixth 12/7 13.076 c, 3°55'22" from diminished seventh 216/125 4.969 c<span style="line-height: 1.5;">, 1°29'26" </span>from augmented sixth 125/72 -8.826 c, -2°38'52" from septimal seventh 7/4 || || 5 || 1200.0, 360° || 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]] ==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 ** 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== 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 |.) ||~ 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 > || ||= 16/15 ||> 111.731 || Diatonic semitone || || || || ||= 27/25 ||> 133.238 || Large diatonic semit. || || || ||
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<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 />
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<!-- ws:start:WikiTextTocRule:31:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><a href="#x5 Equal Divisions of the Octave: Theory">5 Equal Divisions of the Octave: Theory</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- 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: --> | <a href="#x5-edo in Musicmaking">5-edo in Musicmaking</a><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: -->
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<!-- ws:start:WikiTextHeadingRule:1:<h1> --><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:<h3> --><h3 id="toc1"><a name="x5 Equal Divisions of the Octave: Theory--"Equal Pentatonic""></a><!-- ws:end:WikiTextHeadingRule:3 -->"Equal Pentatonic"</h3>
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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 />
<!-- ws:start:WikiTextHeadingRule:5:<h2> --><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:<h2> --><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>
<td><strong>Interval,</strong><br />
<strong>in fifths of</strong><br />
<strong>an octave</strong><br />
</td>
<td><strong>Interval</strong><br />
<strong>in ¢, DMS</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, 0°<br />
</td>
<td>unison / prime<br />
</td>
<td>exactly 1/1<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>240.0.<br />
72°<br />
</td>
<td>second / third<br />
</td>
<td>+8.826 c, +2°38'52" from septimal second 8/7<br />
-4.969 c, -1°29'26" from diminished third 144/125<br />
-13.076 c, -3<span style="line-height: 1.5;">°55'22"</span> from augmented second 125/108<br />
-26.871 c, -8<span style="line-height: 1.5;">°3'41"</span> from septimal minor third 7/6<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>480.0,<br />
144°<br />
</td>
<td>fourth<br />
</td>
<td>+9.219 c, +2°45'57" from narrow fourth 21/16<br />
-0.686 c, -11'37" from smaller fourth 33/25<br />
-18.045 c, -5°24'49" from just fourth 4/3<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>720.0,<br />
216°<br />
</td>
<td>fifth<br />
</td>
<td>+18.045 c, +5°24'49" from just fifth 3/2<br />
+0.686 c<span style="line-height: 1.5;">, +11'37" </span>from bigger fifth 50/33<br />
-9.219 c<span style="line-height: 1.5;">, +2°45'57" </span>from wide fifth 32/21<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>960.0,<br />
288°<br />
</td>
<td>sixth, seventh<br />
</td>
<td>26.871 c, 8°3'41" from septimal major sixth 12/7<br />
13.076 c, 3°55'22" from diminished seventh 216/125<br />
4.969 c<span style="line-height: 1.5;">, 1°29'26" </span>from augmented sixth 125/72<br />
-8.826 c, -2°38'52" from septimal seventh 7/4<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>1200.0,<br />
360°<br />
</td>
<td>eighth<br />
</td>
<td>exactly 2/1<br />
</td>
</tr>
</table>
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<!-- ws:start:WikiTextHeadingRule:9:<h2> --><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 "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:11:<h2> --><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:<h2> --><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:<h1> --><h1 id="toc7"><a name="x5-edo in Musicmaking"></a><!-- ws:end:WikiTextHeadingRule:15 -->5-edo in Musicmaking</h1>
<!-- ws:start:WikiTextHeadingRule:17:<h2> --><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&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><li>"<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>" (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 />
<!-- ws:start:WikiTextHeadingRule:19:<h2> --><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&usp=drive_web" rel="nofollow" target="_blank">here</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:21:<h2> --><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>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:23:<h2> --><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 "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:25:<h2> --><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 "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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:27:<h2> --><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:<h2> --><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 < 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>
</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><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><br />
</td>
</tr>
</table>
</body></html>