27edo: Difference between revisions

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**Imported revision 565256841 - Original comment: **
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**Imported revision 565319065 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2015-11-05 01:52:17 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-05 11:18:21 UTC</tt>.<br>
: The original revision id was <tt>565256841</tt>.<br>
: The original revision id was <tt>565319065</tt>.<br>
: The revision comment was: <tt></tt><br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44.444... [[cent]]s in size. However, 27 is a prime candidate for [[octave shrinking]], and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the [[5_4|third]], [[3_2|fifth]] and [[7_4|7/4]] sharply.
If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44.444... [[cent]]s in size. However, 27 is a prime candidate for [[octave shrinking]], and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the [[5_4|third]], [[3_2|fifth]] and [[7_4|7/4]] sharply.
Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[harmonic entropy]] possible and thus is, in theory, most dissonant. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison are are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.


Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as [[12edo]], sharp 13 2/3 cents. The result is that [[6_5|6/5]], [[7_5|7/5]] and especially [[7_6|7/6]] are all tuned more accurately than this.
Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as [[12edo]], sharp 13 2/3 cents. The result is that [[6_5|6/5]], [[7_5|7/5]] and especially [[7_6|7/6]] are all tuned more accurately than this.
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Though the [[7-limit]] tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both [[consistent]]ly and distinctly--that is, everything in the 7-limit [[Diamonds|diamond]] is uniquely represented by a certain number of steps of 27 equal. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament
Though the [[7-limit]] tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both [[consistent]]ly and distinctly--that is, everything in the 7-limit [[Diamonds|diamond]] is uniquely represented by a certain number of steps of 27 equal. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament
Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[harmonic entropy]] possible and thus is, in theory, most dissonant. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison are are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.


==Intervals==  
==Intervals==  
Line 136: Line 136:
  &lt;br /&gt;
  &lt;br /&gt;
If octaves are kept pure, 27edo divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; in 27 equal parts each exactly 44.444... &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s in size. However, 27 is a prime candidate for &lt;a class="wiki_link" href="/octave%20shrinking"&gt;octave shrinking&lt;/a&gt;, and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the &lt;a class="wiki_link" href="/5_4"&gt;third&lt;/a&gt;, &lt;a class="wiki_link" href="/3_2"&gt;fifth&lt;/a&gt; and &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt; sharply.&lt;br /&gt;
If octaves are kept pure, 27edo divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; in 27 equal parts each exactly 44.444... &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s in size. However, 27 is a prime candidate for &lt;a class="wiki_link" href="/octave%20shrinking"&gt;octave shrinking&lt;/a&gt;, and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the &lt;a class="wiki_link" href="/5_4"&gt;third&lt;/a&gt;, &lt;a class="wiki_link" href="/3_2"&gt;fifth&lt;/a&gt; and &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt; sharply.&lt;br /&gt;
&lt;br /&gt;
Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest &lt;a class="wiki_link" href="/harmonic%20entropy"&gt;harmonic entropy&lt;/a&gt; possible and thus is, in theory, most dissonant. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison are are more consonant as a result; intervals larger than this have less &amp;quot;tension&amp;quot; and thus are also more consonant.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, sharp 13 2/3 cents. The result is that &lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;, &lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt; and especially &lt;a class="wiki_link" href="/7_6"&gt;7/6&lt;/a&gt; are all tuned more accurately than this.&lt;br /&gt;
Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, sharp 13 2/3 cents. The result is that &lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;, &lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt; and especially &lt;a class="wiki_link" href="/7_6"&gt;7/6&lt;/a&gt; are all tuned more accurately than this.&lt;br /&gt;
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&lt;br /&gt;
&lt;br /&gt;
Though the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt;ly and distinctly--that is, everything in the 7-limit &lt;a class="wiki_link" href="/Diamonds"&gt;diamond&lt;/a&gt; is uniquely represented by a certain number of steps of 27 equal. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament&lt;br /&gt;
Though the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt;ly and distinctly--that is, everything in the 7-limit &lt;a class="wiki_link" href="/Diamonds"&gt;diamond&lt;/a&gt; is uniquely represented by a certain number of steps of 27 equal. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament&lt;br /&gt;
&lt;br /&gt;
Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest &lt;a class="wiki_link" href="/harmonic%20entropy"&gt;harmonic entropy&lt;/a&gt; possible and thus is, in theory, most dissonant. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison are are more consonant as a result; intervals larger than this have less &amp;quot;tension&amp;quot; and thus are also more consonant.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x27 tone equal tempertament-Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;Intervals&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x27 tone equal tempertament-Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;Intervals&lt;/h2&gt;

Revision as of 11:18, 5 November 2015

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author genewardsmith and made on 2015-11-05 11:18:21 UTC.
The original revision id was 565319065.
The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

=<span style="color: #0061ff; font-family: 'Times New Roman',Times,serif; font-size: 113%;">27 tone equal tempertament</span>= 

If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44.444... [[cent]]s in size. However, 27 is a prime candidate for [[octave shrinking]], and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the [[5_4|third]], [[3_2|fifth]] and [[7_4|7/4]] sharply.

Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as [[12edo]], sharp 13 2/3 cents. The result is that [[6_5|6/5]], [[7_5|7/5]] and especially [[7_6|7/6]] are all tuned more accurately than this.

27edo, with its 400 cent major third, tempers out the [[diesis]] of 128/125, and also the [[septimal comma]], 64/63 (and hence 126/125 also.) These it shares with 12edo, making some relationships familiar, and as a consequence they both support augene temperament. It shares with [[22edo]] tempering out the allegedly Bohlen-Pierce comma 245/243 as well as 64/63, so that they both support superpyth temperament, with quite sharp "superpythagorean" fifths giving a sharp 9/7 in place of meantone's 5/4.

Though the [[7-limit]] tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both [[consistent]]ly and distinctly--that is, everything in the 7-limit [[Diamonds|diamond]] is uniquely represented by a certain number of steps of 27 equal. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament

Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[harmonic entropy]] possible and thus is, in theory, most dissonant. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison are are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.

==Intervals== 
|| Degrees of 27-EDO || Cents value
DMS value ||= Approximate
Ratios* ||= Solfege ||
|| 0 || 0 ||= 1/1 ||= do ||
|| 1 || 44.44
13°20' ||= 36/35, 49/48, 50/49 ||= di ||
|| 2 || 88.89
26°40' ||= 16/15, 21/20, 25/24 ||= ra ||
|| 3 || 133.33
40° ||= 14/13, 13/12 ||= ru ||
|| 4 || 177.78
53°20' ||= 10/9 ||= reh ||
|| 5 || 222.22
66°40' ||= 8/7, 9/8 ||= re ||
|| 6 || 266.67
80° ||= 7/6 ||= ma ||
|| 7 || 311.11
93°20' ||= 6/5 ||= me ||
|| 8 || 355.56
106°40' ||= 16/13 ||= mu ||
|| 9 || 400
120° ||= 5/4 ||= mi ||
|| 10 || 444.44
133°20' ||= 9/7, 13/10 ||= mo ||
|| 11 || 488.89
146°40' ||= 4/3 ||= fa ||
|| 12 || 533.33
160° ||= 49/36, 48/35 ||= fih ||
|| 13 || 577.78
173°20' ||= 7/5, 18/13 ||= fi ||
|| 14 || 622.22
186°40' ||= 10/7, 13/9 ||= se ||
|| 15 || 666.67
200° ||= 72/49, 35/24 ||= sih ||
|| 16 || 711.11
213°20' ||= 3/2 ||= so/sol ||
|| 17 || 755.56
226°40' ||= 14/9, 20/13 ||= lo ||
|| 18 || 800
240° ||= 8/5 ||= le ||
|| 19 || 844.44
253°20' ||= 13/8 ||= lu ||
|| 20 || 888.89
266°40' ||= 5/3 ||= la ||
|| 21 || 933.33
280° ||= 12/7 ||= li ||
|| 22 || 977.78
293°20' ||= 7/4, 16/9 ||= ta ||
|| 23 || 1022.22
306°40' ||= 9/5 ||= te ||
|| 24 || 1066,67
320° ||= 13/7, 24/13 ||= tu ||
|| 25 || 1111.11
333°20' ||= 40/21 ||= ti ||
|| 26 || 1155.56
346°40' ||= 35/18, 96/49, 49/25 ||= da ||
|| 27 || 1200
360° ||= 2/1 ||= do ||
*based on treating 27-EDO as a 2.3.5.7.13 subgroup temperament; other approaches are possible.
==Rank two temperaments== 
[[List of 27edo rank two temperaments by badness]]
[[List of edo-distinct 27e rank two temperaments]]
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\27 || [[Quartonic]]/Quarto ||
|| 1 || 2\27 || [[Octacot]]/Octocat ||
|| 1 || 4\27 || [[Tetracot]]/Modus/Wollemia ||
|| 1 || 5\27 || [[Machine]]/Kumonga ||
|| 1 || 7\27 || [[Myna]]/Coleto/Minah ||
|| 1 || 8\27 || [[Beatles]]/Ringo ||
|| 1 || 10\27 || [[Sensi]]/Sensis ||
|| 1 || 11\27 || [[Superpyth]] ||
|| 1 || 13\27 || Fervor ||
|| 3 || 1\27 || [[Semiaug]]/Hemiaug ||
|| 3 || 2\27 || [[Augmented]]/[[augene|Augene]]/Ogene ||
|| 3 || 4\27 || Oodako ||
|| 9 || 1\27 || Terrible version of [[Ennealimmal]]
/ Niner ||
==Commas== 
27 EDO tempers out the following commas. (Note: This assumes the val < 27 43 63 76 93 100 |.)
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 128/125 ||< | 7 0 -3 > ||> 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 20000/19683 ||< | 5 -9 4 > ||> 27.66 ||= Minimal Diesis ||= Tetracot Comma ||=   ||
||= 78732/78125 ||< | 2 9 -7 > ||> 13.40 ||= Medium Semicomma ||= Sensipent Comma ||=   ||
||= 4711802/4709457 ||< | 1 -27 18 > ||> 0.86 ||= Ennealimma ||=   ||=   ||
||= 686/675 ||< | 1 -3 -2 3 > ||> 27.99 ||= Senga ||=   ||=   ||
||= 64/63 ||< | 6 -2 0 -1 > ||> 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 50421/50000 ||< | -4 1 -5 5 > ||> 14.52 ||= Trimyna ||=   ||=   ||
||= 245/243 ||< | 0 -5 1 2 > ||> 14.19 ||= Sensamagic ||=   ||=   ||
||= 126/125 ||< | 1 2 -3 1 > ||> 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 ||< | 5 -4 3 -2 > ||> 13.47 ||= Octagar ||=   ||=   ||
||= 1728/1715 ||< | 6 3 -1 -3 > ||> 13.07 ||= Orwellisma ||= Orwell Comma ||=   ||
||= 420175/419904 ||< | -6 -8 2 5 > ||> 1.12 ||= Wizma ||=   ||=   ||
||= 2401/2400 ||< | -5 -1 -2 4 > ||> 0.72 ||= Breedsma ||=   ||=   ||
||= 4375/4374 ||< | -1 -7 4 1 > ||> 0.40 ||= Ragisma ||=   ||=   ||
||= 250047/250000 ||< | -4 6 -6 3 > ||> 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 ||< | -1 2 0 -2 1 > ||> 17.58 ||= Mothwellsma ||=   ||=   ||
||= 896/891 ||< | 7 -4 0 1 -1 > ||> 9.69 ||= Pentacircle ||=   ||=   ||
||= 385/384 ||< | -7 -1 1 1 1 > ||> 4.50 ||= Keenanisma ||=   ||=   ||
||= 91/90 ||< | -1 -2 -1 1 0 1 > ||> 19.13 ||= Superleap ||=   ||=   ||

=Music= 

[[http://www.archive.org/details/MusicForYourEars|Music For Your Ears]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://www.archive.org/download/MusicForYourEars/musicfor.mp3|play]]</span> by [[Gene Ward Smith]] The central portion is in 27edo, the rest in [[46edo]].
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3|Sad Like Winter Leaves]]</span> by Igliashon Jones
//[[file:Superpythagorean Waltz.mp3|Superpythagorean Waltz]]// by Igliashon Jones
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3|Galticeran Sonatina]]</span> by [[http://soundcloud.com/joelgranttaylor/galticeran_sonatina|Joel Taylor]]
<span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link">[[http://www.youtube.com/watch?v=7QcwKlK6z4c|miniature prelude and fugue]]</span> by Kosmorsky[[media type="custom" key="10942764"]]
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://micro.soonlabel.com/27edo/daily20111202-deep-chasm-zeta-cp-1.mp3|Chicago Pile-1]]</span> by [[Chris Vaisvil]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Perc-Sitar.mp3|Tetracot Perc-Sitar]] by [[http://soundcloud.com/dustin-schallert/tetracot-perc-sitar|Dustin Schallert]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Jam.mp3|Tetracot Jam]] by [[http://soundcloud.com/dustin-schallert/tetracot-jam|Dustin Schallert]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Pump.mp3|Tetracot Pump]] by [[http://soundcloud.com/dustin-schallert/tetracot-pump|Dustin Schallert]] all in [[27edo]]
[[https://soundcloud.com/dustin-schallert/27-edo-guitar-1|27-EDO Guitar 1 by Dustin Schallert]]

Original HTML content:

<html><head><title>27edo</title></head><body><!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="x27 tone equal tempertament"></a><!-- ws:end:WikiTextHeadingRule:1 --><span style="color: #0061ff; font-family: 'Times New Roman',Times,serif; font-size: 113%;">27 tone equal tempertament</span></h1>
 <br />
If octaves are kept pure, 27edo divides the <a class="wiki_link" href="/octave">octave</a> in 27 equal parts each exactly 44.444... <a class="wiki_link" href="/cent">cent</a>s in size. However, 27 is a prime candidate for <a class="wiki_link" href="/octave%20shrinking">octave shrinking</a>, and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the <a class="wiki_link" href="/5_4">third</a>, <a class="wiki_link" href="/3_2">fifth</a> and <a class="wiki_link" href="/7_4">7/4</a> sharply.<br />
<br />
Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as <a class="wiki_link" href="/12edo">12edo</a>, sharp 13 2/3 cents. The result is that <a class="wiki_link" href="/6_5">6/5</a>, <a class="wiki_link" href="/7_5">7/5</a> and especially <a class="wiki_link" href="/7_6">7/6</a> are all tuned more accurately than this.<br />
<br />
27edo, with its 400 cent major third, tempers out the <a class="wiki_link" href="/diesis">diesis</a> of 128/125, and also the <a class="wiki_link" href="/septimal%20comma">septimal comma</a>, 64/63 (and hence 126/125 also.) These it shares with 12edo, making some relationships familiar, and as a consequence they both support augene temperament. It shares with <a class="wiki_link" href="/22edo">22edo</a> tempering out the allegedly Bohlen-Pierce comma 245/243 as well as 64/63, so that they both support superpyth temperament, with quite sharp &quot;superpythagorean&quot; fifths giving a sharp 9/7 in place of meantone's 5/4.<br />
<br />
Though the <a class="wiki_link" href="/7-limit">7-limit</a> tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both <a class="wiki_link" href="/consistent">consistent</a>ly and distinctly--that is, everything in the 7-limit <a class="wiki_link" href="/Diamonds">diamond</a> is uniquely represented by a certain number of steps of 27 equal. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament<br />
<br />
Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest <a class="wiki_link" href="/harmonic%20entropy">harmonic entropy</a> possible and thus is, in theory, most dissonant. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison are are more consonant as a result; intervals larger than this have less &quot;tension&quot; and thus are also more consonant.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:3:&lt;h2&gt; --><h2 id="toc1"><a name="x27 tone equal tempertament-Intervals"></a><!-- ws:end:WikiTextHeadingRule:3 -->Intervals</h2>
 

<table class="wiki_table">
    <tr>
        <td>Degrees of 27-EDO<br />
</td>
        <td>Cents value<br />
DMS value<br />
</td>
        <td style="text-align: center;">Approximate<br />
Ratios*<br />
</td>
        <td style="text-align: center;">Solfege<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td style="text-align: center;">1/1<br />
</td>
        <td style="text-align: center;">do<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>44.44<br />
13°20'<br />
</td>
        <td style="text-align: center;">36/35, 49/48, 50/49<br />
</td>
        <td style="text-align: center;">di<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>88.89<br />
26°40'<br />
</td>
        <td style="text-align: center;">16/15, 21/20, 25/24<br />
</td>
        <td style="text-align: center;">ra<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>133.33<br />
40°<br />
</td>
        <td style="text-align: center;">14/13, 13/12<br />
</td>
        <td style="text-align: center;">ru<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>177.78<br />
53°20'<br />
</td>
        <td style="text-align: center;">10/9<br />
</td>
        <td style="text-align: center;">reh<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>222.22<br />
66°40'<br />
</td>
        <td style="text-align: center;">8/7, 9/8<br />
</td>
        <td style="text-align: center;">re<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>266.67<br />
80°<br />
</td>
        <td style="text-align: center;">7/6<br />
</td>
        <td style="text-align: center;">ma<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>311.11<br />
93°20'<br />
</td>
        <td style="text-align: center;">6/5<br />
</td>
        <td style="text-align: center;">me<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>355.56<br />
106°40'<br />
</td>
        <td style="text-align: center;">16/13<br />
</td>
        <td style="text-align: center;">mu<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>400<br />
120°<br />
</td>
        <td style="text-align: center;">5/4<br />
</td>
        <td style="text-align: center;">mi<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>444.44<br />
133°20'<br />
</td>
        <td style="text-align: center;">9/7, 13/10<br />
</td>
        <td style="text-align: center;">mo<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>488.89<br />
146°40'<br />
</td>
        <td style="text-align: center;">4/3<br />
</td>
        <td style="text-align: center;">fa<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>533.33<br />
160°<br />
</td>
        <td style="text-align: center;">49/36, 48/35<br />
</td>
        <td style="text-align: center;">fih<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>577.78<br />
173°20'<br />
</td>
        <td style="text-align: center;">7/5, 18/13<br />
</td>
        <td style="text-align: center;">fi<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>622.22<br />
186°40'<br />
</td>
        <td style="text-align: center;">10/7, 13/9<br />
</td>
        <td style="text-align: center;">se<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>666.67<br />
200°<br />
</td>
        <td style="text-align: center;">72/49, 35/24<br />
</td>
        <td style="text-align: center;">sih<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>711.11<br />
213°20'<br />
</td>
        <td style="text-align: center;">3/2<br />
</td>
        <td style="text-align: center;">so/sol<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>755.56<br />
226°40'<br />
</td>
        <td style="text-align: center;">14/9, 20/13<br />
</td>
        <td style="text-align: center;">lo<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>800<br />
240°<br />
</td>
        <td style="text-align: center;">8/5<br />
</td>
        <td style="text-align: center;">le<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>844.44<br />
253°20'<br />
</td>
        <td style="text-align: center;">13/8<br />
</td>
        <td style="text-align: center;">lu<br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>888.89<br />
266°40'<br />
</td>
        <td style="text-align: center;">5/3<br />
</td>
        <td style="text-align: center;">la<br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>933.33<br />
280°<br />
</td>
        <td style="text-align: center;">12/7<br />
</td>
        <td style="text-align: center;">li<br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>977.78<br />
293°20'<br />
</td>
        <td style="text-align: center;">7/4, 16/9<br />
</td>
        <td style="text-align: center;">ta<br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>1022.22<br />
306°40'<br />
</td>
        <td style="text-align: center;">9/5<br />
</td>
        <td style="text-align: center;">te<br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>1066,67<br />
320°<br />
</td>
        <td style="text-align: center;">13/7, 24/13<br />
</td>
        <td style="text-align: center;">tu<br />
</td>
    </tr>
    <tr>
        <td>25<br />
</td>
        <td>1111.11<br />
333°20'<br />
</td>
        <td style="text-align: center;">40/21<br />
</td>
        <td style="text-align: center;">ti<br />
</td>
    </tr>
    <tr>
        <td>26<br />
</td>
        <td>1155.56<br />
346°40'<br />
</td>
        <td style="text-align: center;">35/18, 96/49, 49/25<br />
</td>
        <td style="text-align: center;">da<br />
</td>
    </tr>
    <tr>
        <td>27<br />
</td>
        <td>1200<br />
360°<br />
</td>
        <td style="text-align: center;">2/1<br />
</td>
        <td style="text-align: center;">do<br />
</td>
    </tr>
</table>

*based on treating 27-EDO as a 2.3.5.7.13 subgroup temperament; other approaches are possible.<br />
<!-- ws:start:WikiTextHeadingRule:5:&lt;h2&gt; --><h2 id="toc2"><a name="x27 tone equal tempertament-Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:5 -->Rank two temperaments</h2>
 <a class="wiki_link" href="/List%20of%2027edo%20rank%20two%20temperaments%20by%20badness">List of 27edo rank two temperaments by badness</a><br />
<a class="wiki_link" href="/List%20of%20edo-distinct%2027e%20rank%20two%20temperaments">List of edo-distinct 27e rank two temperaments</a><br />


<table class="wiki_table">
    <tr>
        <th>Periods<br />
per octave<br />
</th>
        <th>Generator<br />
</th>
        <th>Temperaments<br />
</th>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>1\27<br />
</td>
        <td><a class="wiki_link" href="/Quartonic">Quartonic</a>/Quarto<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>2\27<br />
</td>
        <td><a class="wiki_link" href="/Octacot">Octacot</a>/Octocat<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>4\27<br />
</td>
        <td><a class="wiki_link" href="/Tetracot">Tetracot</a>/Modus/Wollemia<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>5\27<br />
</td>
        <td><a class="wiki_link" href="/Machine">Machine</a>/Kumonga<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>7\27<br />
</td>
        <td><a class="wiki_link" href="/Myna">Myna</a>/Coleto/Minah<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>8\27<br />
</td>
        <td><a class="wiki_link" href="/Beatles">Beatles</a>/Ringo<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>10\27<br />
</td>
        <td><a class="wiki_link" href="/Sensi">Sensi</a>/Sensis<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>11\27<br />
</td>
        <td><a class="wiki_link" href="/Superpyth">Superpyth</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>13\27<br />
</td>
        <td>Fervor<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>1\27<br />
</td>
        <td><a class="wiki_link" href="/Semiaug">Semiaug</a>/Hemiaug<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>2\27<br />
</td>
        <td><a class="wiki_link" href="/Augmented">Augmented</a>/<a class="wiki_link" href="/augene">Augene</a>/Ogene<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>4\27<br />
</td>
        <td>Oodako<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>1\27<br />
</td>
        <td>Terrible version of <a class="wiki_link" href="/Ennealimmal">Ennealimmal</a><br />
/ Niner<br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextHeadingRule:7:&lt;h2&gt; --><h2 id="toc3"><a name="x27 tone equal tempertament-Commas"></a><!-- ws:end:WikiTextHeadingRule:7 -->Commas</h2>
 27 EDO tempers out the following commas. (Note: This assumes the val &lt; 27 43 63 76 93 100 |.)<br />


<table class="wiki_table">
    <tr>
        <th>Comma<br />
</th>
        <th>Monzo<br />
</th>
        <th>Value (Cents)<br />
</th>
        <th>Name 1<br />
</th>
        <th>Name 2<br />
</th>
        <th>Name 3<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">128/125<br />
</td>
        <td style="text-align: left;">| 7 0 -3 &gt;<br />
</td>
        <td style="text-align: right;">41.06<br />
</td>
        <td style="text-align: center;">Diesis<br />
</td>
        <td style="text-align: center;">Augmented Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">20000/19683<br />
</td>
        <td style="text-align: left;">| 5 -9 4 &gt;<br />
</td>
        <td style="text-align: right;">27.66<br />
</td>
        <td style="text-align: center;">Minimal Diesis<br />
</td>
        <td style="text-align: center;">Tetracot Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">78732/78125<br />
</td>
        <td style="text-align: left;">| 2 9 -7 &gt;<br />
</td>
        <td style="text-align: right;">13.40<br />
</td>
        <td style="text-align: center;">Medium Semicomma<br />
</td>
        <td style="text-align: center;">Sensipent Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4711802/4709457<br />
</td>
        <td style="text-align: left;">| 1 -27 18 &gt;<br />
</td>
        <td style="text-align: right;">0.86<br />
</td>
        <td style="text-align: center;">Ennealimma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">686/675<br />
</td>
        <td style="text-align: left;">| 1 -3 -2 3 &gt;<br />
</td>
        <td style="text-align: right;">27.99<br />
</td>
        <td style="text-align: center;">Senga<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">64/63<br />
</td>
        <td style="text-align: left;">| 6 -2 0 -1 &gt;<br />
</td>
        <td style="text-align: right;">27.26<br />
</td>
        <td style="text-align: center;">Septimal Comma<br />
</td>
        <td style="text-align: center;">Archytas' Comma<br />
</td>
        <td style="text-align: center;">Leipziger Komma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">50421/50000<br />
</td>
        <td style="text-align: left;">| -4 1 -5 5 &gt;<br />
</td>
        <td style="text-align: right;">14.52<br />
</td>
        <td style="text-align: center;">Trimyna<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">245/243<br />
</td>
        <td style="text-align: left;">| 0 -5 1 2 &gt;<br />
</td>
        <td style="text-align: right;">14.19<br />
</td>
        <td style="text-align: center;">Sensamagic<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">126/125<br />
</td>
        <td style="text-align: left;">| 1 2 -3 1 &gt;<br />
</td>
        <td style="text-align: right;">13.79<br />
</td>
        <td style="text-align: center;">Septimal Semicomma<br />
</td>
        <td style="text-align: center;">Starling Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4000/3969<br />
</td>
        <td style="text-align: left;">| 5 -4 3 -2 &gt;<br />
</td>
        <td style="text-align: right;">13.47<br />
</td>
        <td style="text-align: center;">Octagar<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1728/1715<br />
</td>
        <td style="text-align: left;">| 6 3 -1 -3 &gt;<br />
</td>
        <td style="text-align: right;">13.07<br />
</td>
        <td style="text-align: center;">Orwellisma<br />
</td>
        <td style="text-align: center;">Orwell Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">420175/419904<br />
</td>
        <td style="text-align: left;">| -6 -8 2 5 &gt;<br />
</td>
        <td style="text-align: right;">1.12<br />
</td>
        <td style="text-align: center;">Wizma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">2401/2400<br />
</td>
        <td style="text-align: left;">| -5 -1 -2 4 &gt;<br />
</td>
        <td style="text-align: right;">0.72<br />
</td>
        <td style="text-align: center;">Breedsma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4375/4374<br />
</td>
        <td style="text-align: left;">| -1 -7 4 1 &gt;<br />
</td>
        <td style="text-align: right;">0.40<br />
</td>
        <td style="text-align: center;">Ragisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">250047/250000<br />
</td>
        <td style="text-align: left;">| -4 6 -6 3 &gt;<br />
</td>
        <td style="text-align: right;">0.33<br />
</td>
        <td style="text-align: center;">Landscape Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">99/98<br />
</td>
        <td style="text-align: left;">| -1 2 0 -2 1 &gt;<br />
</td>
        <td style="text-align: right;">17.58<br />
</td>
        <td style="text-align: center;">Mothwellsma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">896/891<br />
</td>
        <td style="text-align: left;">| 7 -4 0 1 -1 &gt;<br />
</td>
        <td style="text-align: right;">9.69<br />
</td>
        <td style="text-align: center;">Pentacircle<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">385/384<br />
</td>
        <td style="text-align: left;">| -7 -1 1 1 1 &gt;<br />
</td>
        <td style="text-align: right;">4.50<br />
</td>
        <td style="text-align: center;">Keenanisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">91/90<br />
</td>
        <td style="text-align: left;">| -1 -2 -1 1 0 1 &gt;<br />
</td>
        <td style="text-align: right;">19.13<br />
</td>
        <td style="text-align: center;">Superleap<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id="toc4"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:9 -->Music</h1>
 <br />
<a class="wiki_link_ext" href="http://www.archive.org/details/MusicForYourEars" rel="nofollow">Music For Your Ears</a> <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://www.archive.org/download/MusicForYourEars/musicfor.mp3" rel="nofollow">play</a></span> by <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a> The central portion is in 27edo, the rest in <a class="wiki_link" href="/46edo">46edo</a>.<br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3" rel="nofollow">Sad Like Winter Leaves</a></span> by Igliashon Jones<br />
<em><a href="/file/view/Superpythagorean%20Waltz.mp3/392037262/Superpythagorean%20Waltz.mp3" onclick="ws.common.trackFileLink('/file/view/Superpythagorean%20Waltz.mp3/392037262/Superpythagorean%20Waltz.mp3');">Superpythagorean Waltz</a></em> by Igliashon Jones<br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3" rel="nofollow">Galticeran Sonatina</a></span> by <a class="wiki_link_ext" href="http://soundcloud.com/joelgranttaylor/galticeran_sonatina" rel="nofollow">Joel Taylor</a><br />
<span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://www.youtube.com/watch?v=7QcwKlK6z4c" rel="nofollow">miniature prelude and fugue</a></span> by Kosmorsky<!-- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/10942764?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;10942764&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; --><script type="text/javascript" src="http://mediaplayer.yahoo.com/js">
</script><!-- ws:end:WikiTextMediaRule:0 --><br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://micro.soonlabel.com/27edo/daily20111202-deep-chasm-zeta-cp-1.mp3" rel="nofollow">Chicago Pile-1</a></span> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Perc-Sitar.mp3" rel="nofollow">Tetracot Perc-Sitar</a> by <a class="wiki_link_ext" href="http://soundcloud.com/dustin-schallert/tetracot-perc-sitar" rel="nofollow">Dustin Schallert</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Jam.mp3" rel="nofollow">Tetracot Jam</a> by <a class="wiki_link_ext" href="http://soundcloud.com/dustin-schallert/tetracot-jam" rel="nofollow">Dustin Schallert</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Pump.mp3" rel="nofollow">Tetracot Pump</a> by <a class="wiki_link_ext" href="http://soundcloud.com/dustin-schallert/tetracot-pump" rel="nofollow">Dustin Schallert</a> all in <a class="wiki_link" href="/27edo">27edo</a><br />
<a class="wiki_link_ext" href="https://soundcloud.com/dustin-schallert/27-edo-guitar-1" rel="nofollow">27-EDO Guitar 1 by Dustin Schallert</a></body></html>
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