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<h4>Original Wikitext content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]

----

[[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]], 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 5-limit [[Trienstonic clan|father temperament]]. 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, 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 EDO lists|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 > || ||

=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]], or the fifth root of 2.

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||= 676/675 ||> 2.563 || Parizeksma || || || | 2 -3 -2 0 0 2 > || ||</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>5edo</title></head><body><a href="#x5 Equal Divisions of the Octave: Theory">5 Equal Divisions of the Octave: Theory</a> | <a href="#x5-edo in Musicmaking">5-edo in Musicmaking</a>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>5edo</title></head><body><a href="#x5 Equal Divisions of the Octave: Theory">5 Equal Divisions of the Octave: Theory</a> | <a href="#x5-edo in Musicmaking">5-edo in Musicmaking</a> | <a href="#x5 Equal Divisions of the Octave: Theory">5 Equal Divisions of the Octave: Theory</a> | <a href="#x5-edo in Musicmaking">5-edo in Musicmaking</a>

<hr />

<hr />

<a href="#x5 Equal Divisions of the Octave: Theory">5 Equal Divisions of the Octave: Theory</a> | <a href="#x5-edo in Musicmaking">5-edo in Musicmaking</a> | <a href="#x5 Equal Divisions of the Octave: Theory">5 Equal Divisions of the Octave: Theory</a> | <a href="#x5-edo in Musicmaking">5-edo in Musicmaking</a>

<hr />

<h1 id="toc0"><a name="x5 Equal Divisions of the Octave: Theory"></a>5 Equal Divisions of the Octave: Theory</h1>

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Line 369:

<h2 id="toc12"><a name="x5-edo in Musicmaking-Commas Tempered"></a>Commas Tempered</h2>

5-EDO tempers out the following commas. (Note: This assumes the val &lt; 5 8 12 14 17 19 |.)

<tr>

<th>Comma

</th>

<th>Value (cents)

</th>

<th>Name

</th>

<th>Second Name

</th>

<th>Third Name

</th>

<th>Val

</th>

</tr>

<tr>

<td style="text-align: center;">256/243

</td>

<td style="text-align: right;">90.225

</td>

<td>Limma

</td>

<td>Pythagorean Minor 2nd

</td>

<td>

</td>

<td>| 8 -5 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">81/80

</td>

<td style="text-align: right;">21.506

</td>

<td>Syntonic Comma

</td>

<td>Didymos Comma

</td>

<td>Meantone Comma

</td>

<td>| -4 4 -1 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">2889416/2882415

</td>

<td style="text-align: right;">4.200

</td>

<td>Vulture

</td>

<td>

</td>

<td>

</td>

<td>| 24 -21 4 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">36/35

</td>

<td style="text-align: right;">48.770

</td>

<td>Septimal Quarter Tone

</td>

<td>

</td>

<td>

</td>

<td>| 2 2 -1 -1 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">49/48

</td>

<td style="text-align: right;">35.697

</td>

<td>Slendro Diesis

</td>

<td>

</td>

<td>

</td>

<td>| -4 -1 0 2 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">64/63

</td>

<td style="text-align: right;">27.264

</td>

<td>Septimal Comma

</td>

<td>Archytas' Comma

</td>

<td>Leipziger Komma

</td>

<td>| 6 -2 0 -1 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">245/243

</td>

<td style="text-align: right;">14.191

</td>

<td>Sensamagic

</td>

<td>

</td>

<td>

</td>

<td>| 0 -5 1 2 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">1728/1715

</td>

<td style="text-align: right;">13.074

</td>

<td>Orwellisma

</td>

<td>Orwell Comma

</td>

<td>

</td>

<td>| 6 3 -1 -3 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">1029/1024

</td>

<td style="text-align: right;">8.433

</td>

<td>Gamelisma

</td>

<td>

</td>

<td>

</td>

<td>| -10 1 0 3 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">19683/19600

</td>

<td style="text-align: right;">7.316

</td>

<td>Cataharry

</td>

<td>

</td>

<td>

</td>

<td>| -4 9 -2 -2 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">5120/5103

</td>

<td style="text-align: right;">5.758

</td>

<td>Hemifamity

</td>

<td>

</td>

<td>

</td>

<td>| 10 -6 1 -1 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">1065875/1063543

</td>

<td style="text-align: right;">3.792

</td>

<td>Wadisma

</td>

<td>

</td>

<td>

</td>

<td>| -26 -1 1 9 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">420175/419904

</td>

<td style="text-align: right;">1.117

</td>

<td>Wizma

</td>

<td>

</td>

<td>

</td>

<td>| -6 -8 2 5 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">99/98

</td>

<td style="text-align: right;">17.576

</td>

<td>Mothwellsma

</td>

<td>

</td>

<td>

</td>

<td>| -1 2 0 -2 1 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">896/891

</td>

<td style="text-align: right;">9.688

</td>

<td>Pentacircle

</td>

<td>

</td>

<td>

</td>

<td>| 7 -4 0 1 -1 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">385/384

</td>

<td style="text-align: right;">4.503

</td>

<td>Keenanisma

</td>

<td>

</td>

<td>

</td>

<td>| -7 -1 1 1 1 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">441/440

</td>

<td style="text-align: right;">3.930

</td>

<td>Werckisma

</td>

<td>

</td>

<td>

</td>

<td>| -3 2 -1 2 -1 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">3025/3024

</td>

<td style="text-align: right;">0.572

</td>

<td>Lehmerisma

</td>

<td>

</td>

<td>

</td>

<td>| -4 -3 2 -1 2 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">91/90

</td>

<td style="text-align: right;">19.130

</td>

<td>Superleap

</td>

<td>

</td>

<td>

</td>

<td>| -1 -2 -1 1 0 1 &gt;

</td>

</tr>

<tr>

<td style="text-align: center;">676/675

</td>

<td style="text-align: right;">2.563

</td>

<td>Parizeksma

</td>

<td>

</td>

<td>

</td>

<td>| 2 -3 -2 0 0 2 &gt;

</td>

<td>

</td>

</tr>

</table>

<h1 id="toc13"><a name="x5 Equal Divisions of the Octave: Theory"></a>5 Equal Divisions of the Octave: Theory</h1>

<h3 id="toc14"><a name="x5 Equal Divisions of the Octave: Theory--&quot;Equal Pentatonic&quot;"></a>&quot;Equal Pentatonic&quot;</h3>

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">cent</a>, or the fifth root of 2.

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)

<h2 id="toc15"><a name="x5 Equal Divisions of the Octave: Theory-Intervals in 5-edo"></a>Intervals in 5-edo</h2>

<tr>

<td>Interval,

in fifths of

an octave

</td>

<td>Interval

in ¢

</td>

<td>Closest

diatonic

interval name

</td>

<td>The &quot;neighborhood&quot; of just intervals

</td>

</tr>

<tr>

<td>0

</td>

<td>0.0

</td>

<td>unison / prime

</td>

<td>exactly 1/1

</td>

</tr>

<tr>

<td>1

</td>

<td>240.0

</td>

<td>second / third

</td>

<td>+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

</td>

</tr>

<tr>

<td>2

</td>

<td>480.0

</td>

<td>fourth

</td>

<td>+9.219 c from narrow fourth 21/16

-0.686 c from smaller fourth 33/25

-18.045 c from just fourth 4/3

</td>

</tr>

<tr>

<td>3

</td>

<td>720.0

</td>

<td>fifth

</td>

<td>+18.045 c from just fifth 3/2

+0.686 c from bigger fifth 50/33

-9.219 c from wide fifth 32/21

</td>

</tr>

<tr>

<td>4

</td>

<td>960.0

</td>

<td>sixth, seventh

</td>

<td>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

</td>

</tr>

<tr>

<td>5

</td>

<td>1200.0

</td>

<td>eighth

</td>

<td>exactly 2/1

</td>

</tr>

</table>

<h2 id="toc16"><a name="x5 Equal Divisions of the Octave: Theory-Related scales"></a>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>

<h2 id="toc17"><a name="x5 Equal Divisions of the Octave: Theory-As a temperament"></a>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>. 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, 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.

Despite its lack of accuracy, 5EDO is the second <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists">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>.

<h2 id="toc18"><a name="x5 Equal Divisions of the Octave: Theory-Cycles, Divisions"></a>Cycles, Divisions</h2>

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

<h1 id="toc19"><a name="x5-edo in Musicmaking"></a>5-edo in Musicmaking</h1>

<h2 id="toc20"><a name="x5-edo in Musicmaking-Compositions, improvisations"></a>Compositions, improvisations</h2>

<ul><ul><li><a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/" rel="nofollow">Herman Miller</a>: <a class="wiki_link_ext" href="http://micro.soonlabel.com/herman_miller/Daybreak.mp3" rel="nofollow">Daybreak on Slendro Mountain</a> (2000)</li><li>Aaron K. Johnson: <a class="wiki_link_ext" href="http://www.akjmusic.com/audio/5tet_funk.mp3" rel="nofollow">5tet funk</a> (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: Sleeping Through It All (2004)</li><li>Bill Sethares: 5-tet funk (2004), Pentacle (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></ul></ul>

<h2 id="toc21"><a name="x5-edo in Musicmaking-Notation"></a>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>

<h2 id="toc22"><a name="x5-edo in Musicmaking-Harmony"></a>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.

Important chords:

<h2 id="toc23"><a name="x5-edo in Musicmaking-Melody"></a>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.

<h2 id="toc24"><a name="x5-edo in Musicmaking-Chord or scale?"></a>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.

<h2 id="toc25"><a name="x5-edo in Musicmaking-Commas Tempered"></a>Commas Tempered</h2>

5-EDO tempers out the following commas. (Note: This assumes the val &lt; 5 8 12 14 17 19 |.)

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-: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-06-28 03:00:10 UTC</tt>.<br>
+: This revision was by author [[User:Cenobyte|Cenobyte]] and made on <tt>2011-08-07 01:00:19 UTC</tt>.<br>
-: The original revision id was <tt>239087229</tt>.<br>
+: The original revision id was <tt>244670371</tt>.<br>
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 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]
+----
+[[toc|flat]]
 ----
 =5 Equal Divisions of the Octave: Theory=
 ==="equal pentatonic"===
+-edo divides the 1200-[[cent]] octave into 5 equal parts, making its smallest interval exactly 240 [[cent]], or the fifth root of 2.
+-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
+.076 c from diminished seventh 216/125
+.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]]. 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, 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 EDO lists|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==
+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&amp;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==
+edo 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==
+-EDO tempers out the following commas. (Note: This assumes the val &lt; 5 8 12 14 17 19 |.)
+||~ Comma ||~ Value (cents) ||~ Name ||~ Second Name ||~ Third Name ||~ Val ||
+||= 256/243 ||&gt; 90.225 || Limma || Pythagorean Minor 2nd ||   || | 8 -5 &gt; ||
+||= 81/80 ||&gt; 21.506 || Syntonic Comma || Didymos Comma || Meantone Comma || | -4 4 -1 &gt; ||
+||= 2889416/2882415 ||&gt; 4.200 || Vulture ||   ||   || | 24 -21 4 &gt; ||
+||= 36/35 ||&gt; 48.770 || Septimal Quarter Tone ||   ||   || | 2 2 -1 -1 &gt; ||
+||= 49/48 ||&gt; 35.697 || Slendro Diesis ||   ||   || | -4 -1 0 2 &gt; ||
+||= 64/63 ||&gt; 27.264 || Septimal Comma || Archytas' Comma || Leipziger Komma || | 6 -2 0 -1 &gt; ||
+||= 245/243 ||&gt; 14.191 || Sensamagic ||   ||   || | 0 -5 1 2 &gt; ||
+||= 1728/1715 ||&gt; 13.074 || Orwellisma || Orwell Comma ||   || | 6 3 -1 -3 &gt; ||
+||= 1029/1024 ||&gt; 8.433 || Gamelisma ||   ||   || | -10 1 0 3 &gt; ||
+||= 19683/19600 ||&gt; 7.316 || Cataharry ||   ||   || | -4 9 -2 -2 &gt; ||
+||= 5120/5103 ||&gt; 5.758 || Hemifamity ||   ||   || | 10 -6 1 -1 &gt; ||
+||= 1065875/1063543 ||&gt; 3.792 || Wadisma ||   ||   || | -26 -1 1 9 &gt; ||
+||= 420175/419904 ||&gt; 1.117 || Wizma ||   ||   || | -6 -8 2 5 &gt; ||
+||= 99/98 ||&gt; 17.576 || Mothwellsma ||   ||   || | -1 2 0 -2 1 &gt; ||
+||= 896/891 ||&gt; 9.688 || Pentacircle ||   ||   || | 7 -4 0 1 -1 &gt; ||
+||= 385/384 ||&gt; 4.503 || Keenanisma ||   ||   || | -7 -1 1 1 1 &gt; ||
+||= 441/440 ||&gt; 3.930 || Werckisma ||   ||   || | -3 2 -1 2 -1 &gt; ||
+||= 3025/3024 ||&gt; 0.572 || Lehmerisma ||   ||   || | -4 -3 2 -1 2 &gt; ||
+||= 91/90 ||&gt; 19.130 || Superleap ||   ||   || | -1 -2 -1 1 0 1 &gt; ||
+||= 676/675 ||&gt; 2.563 || Parizeksma ||   ||   || | 2 -3 -2 0 0 2 &gt; ||   ||
+=5 Equal Divisions of the Octave: Theory=
+==="Equal Pentatonic"===
 -edo divides the 1200-[[cent]] octave into 5 equal parts, making its smallest interval exactly 240 [[cent]], or the fifth root of 2.
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+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextTocRule:80:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:80 --&gt;&lt;!-- ws:start:WikiTextTocRule:81: --&gt;&lt;a href="#x5 Equal Divisions of the Octave: Theory"&gt;5 Equal Divisions of the Octave: Theory&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:81 --&gt;&lt;!-- ws:start:WikiTextTocRule:82: --&gt;&lt;!-- ws:end:WikiTextTocRule:82 --&gt;&lt;!-- ws:start:WikiTextTocRule:83: --&gt;&lt;!-- ws:end:WikiTextTocRule:83 --&gt;&lt;!-- ws:start:WikiTextTocRule:84: --&gt;&lt;!-- ws:end:WikiTextTocRule:84 --&gt;&lt;!-- ws:start:WikiTextTocRule:85: --&gt;&lt;!-- ws:end:WikiTextTocRule:85 --&gt;&lt;!-- ws:start:WikiTextTocRule:86: --&gt;&lt;!-- ws:end:WikiTextTocRule:86 --&gt;&lt;!-- ws:start:WikiTextTocRule:87: --&gt; | &lt;a href="#x5-edo in Musicmaking"&gt;5-edo in Musicmaking&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:87 --&gt;&lt;!-- ws:start:WikiTextTocRule:88: --&gt;&lt;!-- ws:end:WikiTextTocRule:88 --&gt;&lt;!-- ws:start:WikiTextTocRule:89: --&gt;&lt;!-- ws:end:WikiTextTocRule:89 --&gt;&lt;!-- ws:start:WikiTextTocRule:90: --&gt;&lt;!-- ws:end:WikiTextTocRule:90 --&gt;&lt;!-- ws:start:WikiTextTocRule:91: --&gt;&lt;!-- ws:end:WikiTextTocRule:91 --&gt;&lt;!-- ws:start:WikiTextTocRule:92: --&gt;&lt;!-- ws:end:WikiTextTocRule:92 --&gt;&lt;!-- ws:start:WikiTextTocRule:93: --&gt;&lt;!-- ws:end:WikiTextTocRule:93 --&gt;&lt;!-- ws:start:WikiTextTocRule:94: --&gt; | &lt;a href="#x5 Equal Divisions of the Octave: Theory"&gt;5 Equal Divisions of the Octave: Theory&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:94 --&gt;&lt;!-- ws:start:WikiTextTocRule:95: --&gt;&lt;!-- ws:end:WikiTextTocRule:95 --&gt;&lt;!-- ws:start:WikiTextTocRule:96: --&gt;&lt;!-- ws:end:WikiTextTocRule:96 --&gt;&lt;!-- ws:start:WikiTextTocRule:97: --&gt;&lt;!-- ws:end:WikiTextTocRule:97 --&gt;&lt;!-- ws:start:WikiTextTocRule:98: --&gt;&lt;!-- ws:end:WikiTextTocRule:98 --&gt;&lt;!-- ws:start:WikiTextTocRule:99: --&gt;&lt;!-- ws:end:WikiTextTocRule:99 --&gt;&lt;!-- ws:start:WikiTextTocRule:100: --&gt; | &lt;a href="#x5-edo in Musicmaking"&gt;5-edo in Musicmaking&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:100 --&gt;&lt;!-- ws:start:WikiTextTocRule:101: --&gt;&lt;!-- ws:end:WikiTextTocRule:101 --&gt;&lt;!-- ws:start:WikiTextTocRule:102: --&gt;&lt;!-- ws:end:WikiTextTocRule:102 --&gt;&lt;!-- ws:start:WikiTextTocRule:103: --&gt;&lt;!-- ws:end:WikiTextTocRule:103 --&gt;&lt;!-- ws:start:WikiTextTocRule:104: --&gt;&lt;!-- ws:end:WikiTextTocRule:104 --&gt;&lt;!-- ws:start:WikiTextTocRule:105: --&gt;&lt;!-- ws:end:WikiTextTocRule:105 --&gt;&lt;!-- ws:start:WikiTextTocRule:106: --&gt;&lt;!-- ws:end:WikiTextTocRule:106 --&gt;&lt;!-- ws:start:WikiTextTocRule:107: --&gt;
+&lt;!-- ws:end:WikiTextTocRule:107 --&gt;&lt;hr /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;5 Equal Divisions of the Octave: Theory&lt;/h1&gt;
@@ Line 253: / Line 369: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="x5-edo in Musicmaking-Commas Tempered"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Commas Tempered&lt;/h2&gt;
+-EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 5 8 12 14 17 19 |.)&lt;br /&gt;
+&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;Comma&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Value (cents)&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Name&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Second Name&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Third Name&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Val&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;256/243&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;90.225&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Limma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Pythagorean Minor 2nd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 8 -5 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;81/80&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;21.506&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Syntonic Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Didymos Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Meantone Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -4 4 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;2889416/2882415&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;4.200&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Vulture&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 24 -21 4 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;36/35&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;48.770&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Septimal Quarter Tone&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 2 2 -1 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;49/48&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;35.697&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Slendro Diesis&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -4 -1 0 2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;64/63&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;27.264&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Septimal Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Archytas' Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Leipziger Komma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 6 -2 0 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;245/243&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;14.191&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Sensamagic&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 0 -5 1 2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;1728/1715&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;13.074&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Orwellisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Orwell Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 6 3 -1 -3 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;1029/1024&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;8.433&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Gamelisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -10 1 0 3 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;19683/19600&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;7.316&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Cataharry&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -4 9 -2 -2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;5120/5103&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;5.758&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Hemifamity&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 10 -6 1 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;1065875/1063543&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;3.792&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Wadisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -26 -1 1 9 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;420175/419904&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;1.117&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Wizma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -6 -8 2 5 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;99/98&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;17.576&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Mothwellsma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -1 2 0 -2 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;896/891&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;9.688&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Pentacircle&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 7 -4 0 1 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;385/384&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;4.503&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Keenanisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -7 -1 1 1 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;441/440&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;3.930&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Werckisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -3 2 -1 2 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;3025/3024&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;0.572&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Lehmerisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -4 -3 2 -1 2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;91/90&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;19.130&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Superleap&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -1 -2 -1 1 0 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;676/675&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;2.563&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Parizeksma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 2 -3 -2 0 0 2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc13"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;5 Equal Divisions of the Octave: Theory&lt;/h1&gt;
+ &lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory--&amp;quot;Equal Pentatonic&amp;quot;"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;&amp;quot;Equal Pentatonic&amp;quot;&lt;/h3&gt;
+ &lt;br /&gt;
+&lt;br /&gt;
+-edo divides the 1200-&lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt; octave into 5 equal parts, making its smallest interval exactly 240 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;, or the fifth root of 2.&lt;br /&gt;
+&lt;br /&gt;
+-edo is the smallest &lt;a class="wiki_link" href="/edo"&gt;edo&lt;/a&gt; containing xenharmonic intervals! (1edo 2edo 3edo 4edo are all subsets of 12edo)&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-Intervals in 5-edo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;Intervals in 5-edo&lt;/h2&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;strong&gt;Interval,&lt;/strong&gt;&lt;br /&gt;
+&lt;strong&gt;in fifths of&lt;/strong&gt;&lt;br /&gt;
+&lt;strong&gt;an octave&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;Interval&lt;/strong&gt;&lt;br /&gt;
+&lt;strong&gt;in ¢&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;Closest&lt;/strong&gt;&lt;br /&gt;
+&lt;strong&gt;diatonic&lt;/strong&gt;&lt;br /&gt;
+&lt;strong&gt;interval name&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;The &amp;quot;neighborhood&amp;quot; of just intervals&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0.0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;unison / prime&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;exactly 1/1&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;240.0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;second / third&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;+8.826 c from septimal second 8/7&lt;br /&gt;
+-4.969 c from diminished third 144/125&lt;br /&gt;
+-13.076 c from augmented second 125/108&lt;br /&gt;
+-26.871 c from septimal minor third 7/6&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;480.0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;fourth&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;+9.219 c from narrow fourth 21/16&lt;br /&gt;
+-0.686 c from smaller fourth 33/25&lt;br /&gt;
+-18.045 c from just fourth 4/3&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;720.0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;fifth&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;+18.045 c from just fifth 3/2&lt;br /&gt;
++0.686 c from bigger fifth 50/33&lt;br /&gt;
+-9.219 c from wide fifth 32/21&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;960.0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;sixth, seventh&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;26.871 c from septimal major sixth 12/7&lt;br /&gt;
+.076 c from diminished seventh 216/125&lt;br /&gt;
+.969 c from augmented sixth 125/72&lt;br /&gt;
+-8.826 c from septimal seventh 7/4&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1200.0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;eighth&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;exactly 2/1&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-Related scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;Related scales&lt;/h2&gt;
+ &lt;ul&gt;&lt;li&gt;By its cardinality, 5-edo is related to other &lt;a class="wiki_link" href="/pentatonic"&gt;pentatonic&lt;/a&gt; scales, and it is especially close in sound to many Indonesian &lt;a class="wiki_link" href="/slendro"&gt;slendros&lt;/a&gt;.&lt;/li&gt;&lt;li&gt;Due to the interest around the &amp;quot;fifth&amp;quot; interval size, there are many &lt;a class="wiki_link" href="/nonoctave"&gt;nonoctave&lt;/a&gt; &amp;quot;stretch sisters&amp;quot; to 5-edo: square root of 4/3, cube root of 3/2, 8th root of 3, etc.&lt;/li&gt;&lt;li&gt;For the same reason there are many &amp;quot;circle sisters&amp;quot;:&lt;ul&gt;&lt;li&gt;Make a chain of five &amp;quot;bigger fifths&amp;quot; (50/33), which makes three octaves 3.227¢ flat. (50/33)^5=7.985099.&lt;/li&gt;&lt;/ul&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-As a temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;As a temperament&lt;/h2&gt;
+ If 5-edo is regarded as a temperament, which is to say as 5-et, then the most salient fact is that 16/15 is tempered out. This means in 5-et the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit &lt;a class="wiki_link" href="/Trienstonic%20clan"&gt;father temperament&lt;/a&gt;. This is at the very edge what can sensibly be called temperament, but it does make sense and can be used.&lt;br /&gt;
+&lt;br /&gt;
+Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain &lt;a class="wiki_link" href="/Bug%20family"&gt;bug temperament&lt;/a&gt;, 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.&lt;br /&gt;
+&lt;br /&gt;
+Despite its lack of accuracy, 5EDO is the second &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&gt;zeta integral edo&lt;/a&gt;, after 2EDO. It also is the smallest equal division representing the &lt;a class="wiki_link" href="/9-limit"&gt;9-limit&lt;/a&gt; &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt;ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how &lt;a class="wiki_link" href="/4edo"&gt;4edo&lt;/a&gt; can be used, and which is discussed in that article, it can be used to represent &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the &lt;a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices"&gt;lattice&lt;/a&gt; of tetrads/pentads together with the number of scale steps in 5EDO. However, while &lt;a class="wiki_link" href="/2edo"&gt;2edo&lt;/a&gt; represents the &lt;a class="wiki_link" href="/3-limit"&gt;3-limit&lt;/a&gt; consistently, &lt;a class="wiki_link" href="/3edo"&gt;3edo&lt;/a&gt; the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;, &lt;a class="wiki_link" href="/4edo"&gt;4edo&lt;/a&gt; the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; and &lt;a class="wiki_link" href="/5edo"&gt;5edo&lt;/a&gt; the &lt;a class="wiki_link" href="/9-limit"&gt;9-limit&lt;/a&gt;, to represent the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; consistently with a &lt;a class="wiki_link" href="/patent%20val"&gt;patent val&lt;/a&gt; requires going all the way to &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="x5 Equal Divisions of the Octave: Theory-Cycles, Divisions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;Cycles, Divisions&lt;/h2&gt;
+is a prime number so 5-edo contains no sub-edos. Only simple cycles:&lt;br /&gt;
+Cycle of seconds: 0-1-2-3-4-0&lt;br /&gt;
+Cycle of fourths: 0-2-4-1-3-0&lt;br /&gt;
+Cycle of fifths: 0-3-1-4-2-0&lt;br /&gt;
+Cycle of sevenths: 0-4-3-2-1-0&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc19"&gt;&lt;a name="x5-edo in Musicmaking"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;5-edo in Musicmaking&lt;/h1&gt;
+ &lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="x5-edo in Musicmaking-Compositions, improvisations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;&lt;strong&gt;Compositions&lt;/strong&gt;, improvisations&lt;/h2&gt;
+ &lt;ul&gt;&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/" rel="nofollow"&gt;Herman Miller&lt;/a&gt;: &lt;em&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/herman_miller/Daybreak.mp3" rel="nofollow"&gt;Daybreak on Slendro Mountain&lt;/a&gt;&lt;/em&gt; (2000)&lt;/li&gt;&lt;li&gt;Aaron K. Johnson: &lt;em&gt;&lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/5tet_funk.mp3" rel="nofollow"&gt;5tet funk&lt;/a&gt;&lt;/em&gt; (2004)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=1519939" rel="nofollow"&gt;Andrew Heathwaite: //Pinta Penta// (2004)&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+pintapentain5tet.mp3" rel="nofollow"&gt;play&lt;/a&gt; (rendered in 6 alternative pentatonics as well)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Hans%20Straub"&gt;Hans Straub&lt;/a&gt;: &lt;a class="wiki_link_ext" href="http://home.datacomm.ch/straub/mamuth/5tet_e.html#asimchomsaia" rel="nofollow"&gt;Asîmchômsaia&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Straub/asimchomsaia.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Brian%20Wong"&gt;Brian Wong&lt;/a&gt;: &lt;a class="wiki_link_ext" href="http://bwong.ca/template1.php?sub=3" rel="nofollow"&gt;Slendronica#1b&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Wong/Slendronica1b.ogg" rel="nofollow"&gt;play&lt;/a&gt;&lt;/li&gt;&lt;li&gt;Brian McLaren: various and sundry&lt;/li&gt;&lt;li&gt;Paul Rubenstein: various, with electric guitars in 10- and 15-edo&lt;/li&gt;&lt;li&gt;X.J.Scott: &lt;em&gt;Sleeping Through It All&lt;/em&gt; (2004)&lt;/li&gt;&lt;li&gt;Bill Sethares: &lt;em&gt;5-tet funk&lt;/em&gt; (2004), &lt;em&gt;Pentacle&lt;/em&gt; (2004)&lt;/li&gt;&lt;li&gt;&amp;quot;Cenobyte&amp;quot; Ukulele &lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=UKUCRnEJKKU" rel="nofollow"&gt; http://www.youtube.com/watch?v=UKUCRnEJKKU&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/ul&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="x5-edo in Musicmaking-Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;Notation&lt;/h2&gt;
+ &lt;ul&gt;&lt;ul&gt;&lt;li&gt;via Reinhard's cents notation&lt;/li&gt;&lt;li&gt;Sagittal: naturals on a five-line staff, with enharmonics (used interchangably) E=F and B=C&lt;/li&gt;&lt;li&gt;a four-line hybrid treble/bass staff.&lt;/li&gt;&lt;/ul&gt;&lt;/ul&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;a name="x5-edo in Musicmaking-Harmony"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;Harmony&lt;/h2&gt;
+edo does not have any strong consonance nor dissonance. The 240 cent interval can serve as either a major second or minor third, and the 960 cent interval as either a major sixth or minor seventh. The fourth is about 18 cents flat of a just fourth, making it rather &amp;quot;dirty&amp;quot; but recognizable. The fifth is likewise about 18 cents sharp of a just fifth, dissonant but still easily recognizable.&lt;br /&gt;
+&lt;br /&gt;
+Important chords:&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;0+1+3&lt;/li&gt;&lt;li&gt;0+2+3&lt;/li&gt;&lt;li&gt;0+1+3+4&lt;/li&gt;&lt;li&gt;0+2+3+4&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc23"&gt;&lt;a name="x5-edo in Musicmaking-Melody"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;Melody&lt;/h2&gt;
+ First from edos which can be use for melodies in &amp;quot;standard&amp;quot; way. Relatively large step of 240.00 c can be used as major second for the melody construction. The scale has whole-tone as well as pentatonic character.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc24"&gt;&lt;a name="x5-edo in Musicmaking-Chord or scale?"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;Chord or scale?&lt;/h2&gt;
+ Either way, it is hard to wander very far from where you start. However, it has the scale-like feature that there are (barely) enough notes to create melody, in the form of an equal version of pentatonic.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:50:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc25"&gt;&lt;a name="x5-edo in Musicmaking-Commas Tempered"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;Commas Tempered&lt;/h2&gt;
 -EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 5 8 12 14 17 19 |.)&lt;br /&gt;
 &lt;br /&gt;