27edo: Difference between revisions

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Wikispaces>MasonGreen1
**Imported revision 583137091 - Original comment: **
Wikispaces>JosephRuhf
**Imported revision 597593470 - 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>2016-05-15 02:06:38 UTC</tt>.<br>
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-31 16:51:10 UTC</tt>.<br>
: The original revision id was <tt>583137091</tt>.<br>
: The original revision id was <tt>597593470</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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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, assuming the relatively common values of a=2 and s=1%. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.
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, assuming the relatively common values of a=2 and s=1%. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.


The 27 note system ore one similar like a well temperament can be notated very easily, by a variation on the quartertone accidentals. In this case a sharp raises a note just a hair beneath the following nominal (for example C to C# describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. D flat, C half-sharp, D half flat, and C sharp are all different. The composer can decide for himself which tertiary accidental is necessary if he will need redundancy to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.) otherwise is simple enough and the same tendency for A# to be higher than Bb is not only familiar, though here very exaggerated, to those working with pythagorean scale, but also to many classically trained violinists. et voila
The 27 note system or one similar like a well temperament can be notated very easily, by a variation on the quartertone accidentals. In this case a sharp raises a note just a hair beneath the following nominal (for example C to C# describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. D flat, C half-sharp, D half flat, and C sharp are all different. The composer can decide for himself which tertiary accidental is necessary if he will need redundancy to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.) otherwise is simple enough and the same tendency for A# to be higher than Bb is not only familiar, though here very exaggerated, to those working with pythagorean scale, but also to many classically trained violinists. et voila


==Intervals==  
==Intervals==  
|| Degrees of 27-EDO || Cents value
|| Degrees of 27-EDO || Cents value coarse/fine
DMS value ||= Approximate
DMS value ||= Approximate
Ratios* ||= Solfege ||
Ratios* ||= Solfege ||
|| 0 || 0 ||= 1/1 ||= do ||
|| 0 || 0 ||= 1/1 ||= do ||
|| 1 || 44.44
|| 1 || 44.44, 53.33
13°20' ||= 36/35, 49/48, 50/49 ||= di ||
13°20' ||= 36/35, 49/48, 50/49 ||= di ||
|| 2 || 88.89
|| 2 || 88.89, 106.67
26°40' ||= 16/15, 21/20, 25/24 ||= ra ||
26°40' ||= 16/15, 21/20, 25/24 ||= ra ||
|| 3 || 133.33
|| 3 || 133.33, 160
40° ||= 14/13, 13/12 ||= ru ||
40° ||= 14/13, 13/12 ||= ru ||
|| 4 || 177.78
|| 4 || 177.78, 213.33
53°20' ||= 10/9 ||= reh ||
53°20' ||= 10/9 ||= reh ||
|| 5 || 222.22
|| 5 || 222.22, 266.67
66°40' ||= 8/7, 9/8 ||= re ||
66°40' ||= 8/7, 9/8 ||= re ||
|| 6 || 266.67
|| 6 || 266.67, 320
80° ||= 7/6 ||= ma ||
80° ||= 7/6 ||= ma ||
|| 7 || 311.11
|| 7 || 311.11, 373.33
93°20' ||= 6/5 ||= me ||
93°20' ||= 6/5 ||= me ||
|| 8 || 355.56
|| 8 || 355.56, 426.67
106°40' ||= 16/13 ||= mu ||
106°40' ||= 16/13 ||= mu ||
|| 9 || 400
|| 9 || 400, 480
120° ||= 5/4 ||= mi ||
120° ||= 5/4 ||= mi ||
|| 10 || 444.44
|| 10 || 444.44, 513.33
133°20' ||= 9/7, 13/10 ||= mo ||
133°20' ||= 9/7, 13/10 ||= mo ||
|| 11 || 488.89
|| 11 || 488.89, 566.67
146°40' ||= 4/3 ||= fa ||
146°40' ||= 4/3 ||= fa ||
|| 12 || 533.33
|| 12 || 533.33, 640
160° ||= 49/36, 48/35 ||= fih ||
160° ||= 49/36, 48/35 ||= fih ||
|| 13 || 577.78
|| 13 || 577.78, 693.33
173°20' ||= 7/5, 18/13 ||= fi ||
173°20' ||= 7/5, 18/13 ||= fi ||
|| 14 || 622.22
|| 14 || 622.22, 746.67
186°40' ||= 10/7, 13/9 ||= se ||
186°40' ||= 10/7, 13/9 ||= se ||
|| 15 || 666.67
|| 15 || 666.67, 800
200° ||= 72/49, 35/24 ||= sih ||
200° ||= 72/49, 35/24 ||= sih ||
|| 16 || 711.11
|| 16 || 711.11, 853.33
213°20' ||= 3/2 ||= so/sol ||
213°20' ||= 3/2 ||= so/sol ||
|| 17 || 755.56
|| 17 || 755.56, 906.67
226°40' ||= 14/9, 20/13 ||= lo ||
226°40' ||= 14/9, 20/13 ||= lo ||
|| 18 || 800
|| 18 || 800, 960
240° ||= 8/5 ||= le ||
240° ||= 8/5 ||= le ||
|| 19 || 844.44
|| 19 || 844.44, 1013.33
253°20' ||= 13/8 ||= lu ||
253°20' ||= 13/8 ||= lu ||
|| 20 || 888.89
|| 20 || 888.89, 1066.67
266°40' ||= 5/3 ||= la ||
266°40' ||= 5/3 ||= la ||
|| 21 || 933.33
|| 21 || 933.33, 1120
280° ||= 12/7 ||= li ||
280° ||= 12/7 ||= li ||
|| 22 || 977.78
|| 22 || 977.78, 1173.33
293°20' ||= 7/4, 16/9 ||= ta ||
293°20' ||= 7/4, 16/9 ||= ta ||
|| 23 || 1022.22
|| 23 || 1022.22, 1226.67
306°40' ||= 9/5 ||= te ||
306°40' ||= 9/5 ||= te ||
|| 24 || 1066,67
|| 24 || 1066,67, 1280
320° ||= 13/7, 24/13 ||= tu ||
320° ||= 13/7, 24/13 ||= tu ||
|| 25 || 1111.11
|| 25 || 1111.11, 1333.33
333°20' ||= 40/21 ||= ti ||
333°20' ||= 40/21 ||= ti ||
|| 26 || 1155.56
|| 26 || 1155.56, 1386.67
346°40' ||= 35/18, 96/49, 49/25 ||= da ||
346°40' ||= 35/18, 96/49, 49/25 ||= da ||
|| 27 || 1200
|| 27 || 1200, 1440
360° ||= 2/1 ||= do ||
360° ||= 2/1 ||= do ||
*based on treating 27-EDO as a 2.3.5.7.13 subgroup temperament; other approaches are possible.
*based on treating 27-EDO as a 2.3.5.7.13 subgroup temperament; other approaches are possible.
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/ Niner ||
/ Niner ||
==Commas==  
==Commas==  
27 EDO tempers out the following commas. (Note: This assumes the val &lt; 27 43 63 76 93 100 |.)
27 EDO tempers out the following commas. (Note: This assumes the val &lt; [[tel:27 43 63 76 93 100|27 43 63 76 93 100]] |.)
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 128/125 ||&lt; | 7 0 -3 &gt; ||&gt; 41.06 ||= Diesis ||= Augmented Comma ||=  ||
||= 128/125 ||&lt; | 7 0 -3 &gt; ||&gt; 41.06 ||= Diesis ||= Augmented Comma ||=  ||
||= 20000/19683 ||&lt; | 5 -9 4 &gt; ||&gt; 27.66 ||= Minimal Diesis ||= Tetracot Comma ||=  ||
||= 20000/19683 ||&lt; | 5 -9 4 &gt; ||&gt; 27.66 ||= Minimal Diesis ||= Tetracot Comma ||=  ||
||= 78732/78125 ||&lt; | 2 9 -7 &gt; ||&gt; 13.40 ||= Medium Semicomma ||= Sensipent Comma ||=  ||
||= 78732/78125 ||&lt; | 2 9 -7 &gt; ||&gt; 13.40 ||= Medium Semicomma ||= Sensipent Comma ||=  ||
||= 4711802/4709457 ||&lt; | 1 -27 18 &gt; ||&gt; 0.86 ||= Ennealimma ||=  ||=  ||
||= [[tel:4711802/4709457|4711802/4709457]] ||&lt; | 1 -27 18 &gt; ||&gt; 0.86 ||= Ennealimma ||=  ||=  ||
||= 686/675 ||&lt; | 1 -3 -2 3 &gt; ||&gt; 27.99 ||= Senga ||=  ||=  ||
||= 686/675 ||&lt; | 1 -3 -2 3 &gt; ||&gt; 27.99 ||= Senga ||=  ||=  ||
||= 64/63 ||&lt; | 6 -2 0 -1 &gt; ||&gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 64/63 ||&lt; | 6 -2 0 -1 &gt; ||&gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
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||= 4000/3969 ||&lt; | 5 -4 3 -2 &gt; ||&gt; 13.47 ||= Octagar ||=  ||=  ||
||= 4000/3969 ||&lt; | 5 -4 3 -2 &gt; ||&gt; 13.47 ||= Octagar ||=  ||=  ||
||= 1728/1715 ||&lt; | 6 3 -1 -3 &gt; ||&gt; 13.07 ||= Orwellisma ||= Orwell Comma ||=  ||
||= 1728/1715 ||&lt; | 6 3 -1 -3 &gt; ||&gt; 13.07 ||= Orwellisma ||= Orwell Comma ||=  ||
||= 420175/419904 ||&lt; | -6 -8 2 5 &gt; ||&gt; 1.12 ||= Wizma ||=  ||=  ||
||= [[tel:420175/419904|420175/419904]] ||&lt; | -6 -8 2 5 &gt; ||&gt; 1.12 ||= Wizma ||=  ||=  ||
||= 2401/2400 ||&lt; | -5 -1 -2 4 &gt; ||&gt; 0.72 ||= Breedsma ||=  ||=  ||
||= 2401/2400 ||&lt; | -5 -1 -2 4 &gt; ||&gt; 0.72 ||= Breedsma ||=  ||=  ||
||= 4375/4374 ||&lt; | -1 -7 4 1 &gt; ||&gt; 0.40 ||= Ragisma ||=  ||=  ||
||= 4375/4374 ||&lt; | -1 -7 4 1 &gt; ||&gt; 0.40 ||= Ragisma ||=  ||=  ||
||= 250047/250000 ||&lt; | -4 6 -6 3 &gt; ||&gt; 0.33 ||= Landscape Comma ||=  ||=  ||
||= [[tel:250047/250000|250047/250000]] ||&lt; | -4 6 -6 3 &gt; ||&gt; 0.33 ||= Landscape Comma ||=  ||=  ||
||= 99/98 ||&lt; | -1 2 0 -2 1 &gt; ||&gt; 17.58 ||= Mothwellsma ||=  ||=  ||
||= 99/98 ||&lt; | -1 2 0 -2 1 &gt; ||&gt; 17.58 ||= Mothwellsma ||=  ||=  ||
||= 896/891 ||&lt; | 7 -4 0 1 -1 &gt; ||&gt; 9.69 ||= Pentacircle ||=  ||=  ||
||= 896/891 ||&lt; | 7 -4 0 1 -1 &gt; ||&gt; 9.69 ||= Pentacircle ||=  ||=  ||
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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, assuming the relatively common values of a=2 and s=1%. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison and 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;
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, assuming the relatively common values of a=2 and s=1%. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison and 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;
The 27 note system ore one similar like a well temperament can be notated very easily, by a variation on the quartertone accidentals. In this case a sharp raises a note just a hair beneath the following nominal (for example C to C# describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. D flat, C half-sharp, D half flat, and C sharp are all different. The composer can decide for himself which tertiary accidental is necessary if he will need redundancy to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.) otherwise is simple enough and the same tendency for A# to be higher than Bb is not only familiar, though here very exaggerated, to those working with pythagorean scale, but also to many classically trained violinists. et voila&lt;br /&gt;
The 27 note system or one similar like a well temperament can be notated very easily, by a variation on the quartertone accidentals. In this case a sharp raises a note just a hair beneath the following nominal (for example C to C# describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. D flat, C half-sharp, D half flat, and C sharp are all different. The composer can decide for himself which tertiary accidental is necessary if he will need redundancy to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.) otherwise is simple enough and the same tendency for A# to be higher than Bb is not only familiar, though here very exaggerated, to those working with pythagorean scale, but also to many classically trained violinists. et voila&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;
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         &lt;td&gt;Degrees of 27-EDO&lt;br /&gt;
         &lt;td&gt;Degrees of 27-EDO&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Cents value&lt;br /&gt;
         &lt;td&gt;Cents value coarse/fine&lt;br /&gt;
DMS value&lt;br /&gt;
DMS value&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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         &lt;td&gt;1&lt;br /&gt;
         &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;44.44&lt;br /&gt;
         &lt;td&gt;44.44, 53.33&lt;br /&gt;
13°20'&lt;br /&gt;
13°20'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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         &lt;td&gt;2&lt;br /&gt;
         &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;88.89&lt;br /&gt;
         &lt;td&gt;88.89, 106.67&lt;br /&gt;
26°40'&lt;br /&gt;
26°40'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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         &lt;td&gt;3&lt;br /&gt;
         &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;133.33&lt;br /&gt;
         &lt;td&gt;133.33, 160&lt;br /&gt;
40°&lt;br /&gt;
40°&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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         &lt;td&gt;4&lt;br /&gt;
         &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;177.78&lt;br /&gt;
         &lt;td&gt;177.78, 213.33&lt;br /&gt;
53°20'&lt;br /&gt;
53°20'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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         &lt;td&gt;5&lt;br /&gt;
         &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;222.22&lt;br /&gt;
         &lt;td&gt;222.22, 266.67&lt;br /&gt;
66°40'&lt;br /&gt;
66°40'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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         &lt;td&gt;6&lt;br /&gt;
         &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;266.67&lt;br /&gt;
         &lt;td&gt;266.67, 320&lt;br /&gt;
80°&lt;br /&gt;
80°&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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         &lt;td&gt;7&lt;br /&gt;
         &lt;td&gt;7&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;311.11&lt;br /&gt;
         &lt;td&gt;311.11, 373.33&lt;br /&gt;
93°20'&lt;br /&gt;
93°20'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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         &lt;td&gt;8&lt;br /&gt;
         &lt;td&gt;8&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;355.56&lt;br /&gt;
         &lt;td&gt;355.56, 426.67&lt;br /&gt;
106°40'&lt;br /&gt;
106°40'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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         &lt;td&gt;9&lt;br /&gt;
         &lt;td&gt;9&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;400&lt;br /&gt;
         &lt;td&gt;400, 480&lt;br /&gt;
120°&lt;br /&gt;
120°&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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         &lt;td&gt;10&lt;br /&gt;
         &lt;td&gt;10&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;444.44&lt;br /&gt;
         &lt;td&gt;444.44, 513.33&lt;br /&gt;
133°20'&lt;br /&gt;
133°20'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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         &lt;td&gt;11&lt;br /&gt;
         &lt;td&gt;11&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;488.89&lt;br /&gt;
         &lt;td&gt;488.89, 566.67&lt;br /&gt;
146°40'&lt;br /&gt;
146°40'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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         &lt;td&gt;12&lt;br /&gt;
         &lt;td&gt;12&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;533.33&lt;br /&gt;
         &lt;td&gt;533.33, 640&lt;br /&gt;
160°&lt;br /&gt;
160°&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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         &lt;td&gt;13&lt;br /&gt;
         &lt;td&gt;13&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;577.78&lt;br /&gt;
         &lt;td&gt;577.78, 693.33&lt;br /&gt;
173°20'&lt;br /&gt;
173°20'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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         &lt;td&gt;14&lt;br /&gt;
         &lt;td&gt;14&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;622.22&lt;br /&gt;
         &lt;td&gt;622.22, 746.67&lt;br /&gt;
186°40'&lt;br /&gt;
186°40'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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         &lt;td&gt;15&lt;br /&gt;
         &lt;td&gt;15&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;666.67&lt;br /&gt;
         &lt;td&gt;666.67, 800&lt;br /&gt;
200°&lt;br /&gt;
200°&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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         &lt;td&gt;16&lt;br /&gt;
         &lt;td&gt;16&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;711.11&lt;br /&gt;
         &lt;td&gt;711.11, 853.33&lt;br /&gt;
213°20'&lt;br /&gt;
213°20'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 354: Line 354:
         &lt;td&gt;17&lt;br /&gt;
         &lt;td&gt;17&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;755.56&lt;br /&gt;
         &lt;td&gt;755.56, 906.67&lt;br /&gt;
226°40'&lt;br /&gt;
226°40'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 365: Line 365:
         &lt;td&gt;18&lt;br /&gt;
         &lt;td&gt;18&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;800&lt;br /&gt;
         &lt;td&gt;800, 960&lt;br /&gt;
240°&lt;br /&gt;
240°&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 376: Line 376:
         &lt;td&gt;19&lt;br /&gt;
         &lt;td&gt;19&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;844.44&lt;br /&gt;
         &lt;td&gt;844.44, 1013.33&lt;br /&gt;
253°20'&lt;br /&gt;
253°20'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 387: Line 387:
         &lt;td&gt;20&lt;br /&gt;
         &lt;td&gt;20&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;888.89&lt;br /&gt;
         &lt;td&gt;888.89, 1066.67&lt;br /&gt;
266°40'&lt;br /&gt;
266°40'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 398: Line 398:
         &lt;td&gt;21&lt;br /&gt;
         &lt;td&gt;21&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;933.33&lt;br /&gt;
         &lt;td&gt;933.33, 1120&lt;br /&gt;
280°&lt;br /&gt;
280°&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 409: Line 409:
         &lt;td&gt;22&lt;br /&gt;
         &lt;td&gt;22&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;977.78&lt;br /&gt;
         &lt;td&gt;977.78, 1173.33&lt;br /&gt;
293°20'&lt;br /&gt;
293°20'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 420: Line 420:
         &lt;td&gt;23&lt;br /&gt;
         &lt;td&gt;23&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;1022.22&lt;br /&gt;
         &lt;td&gt;1022.22, 1226.67&lt;br /&gt;
306°40'&lt;br /&gt;
306°40'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 431: Line 431:
         &lt;td&gt;24&lt;br /&gt;
         &lt;td&gt;24&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;1066,67&lt;br /&gt;
         &lt;td&gt;1066,67, 1280&lt;br /&gt;
320°&lt;br /&gt;
320°&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 442: Line 442:
         &lt;td&gt;25&lt;br /&gt;
         &lt;td&gt;25&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;1111.11&lt;br /&gt;
         &lt;td&gt;1111.11, 1333.33&lt;br /&gt;
333°20'&lt;br /&gt;
333°20'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 453: Line 453:
         &lt;td&gt;26&lt;br /&gt;
         &lt;td&gt;26&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;1155.56&lt;br /&gt;
         &lt;td&gt;1155.56, 1386.67&lt;br /&gt;
346°40'&lt;br /&gt;
346°40'&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 464: Line 464:
         &lt;td&gt;27&lt;br /&gt;
         &lt;td&gt;27&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;1200&lt;br /&gt;
         &lt;td&gt;1200, 1440&lt;br /&gt;
360°&lt;br /&gt;
360°&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 598: Line 598:


&lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x27 tone equal tempertament-Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Commas&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x27 tone equal tempertament-Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Commas&lt;/h2&gt;
  27 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 27 43 63 76 93 100 |.)&lt;br /&gt;
  27 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; &lt;a class="wiki_link" href="http://tel.wikispaces.com/27%2043%2063%2076%2093%20100"&gt;27 43 63 76 93 100&lt;/a&gt; |.)&lt;br /&gt;




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     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;4711802/4709457&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;[[tel:4711802/4709457|4711802/4709457]]&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: left;"&gt;| 1 -27 18 &amp;gt;&lt;br /&gt;
         &lt;td style="text-align: left;"&gt;| 1 -27 18 &amp;gt;&lt;br /&gt;
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     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;420175/419904&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;[[tel:420175/419904|420175/419904]]&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: left;"&gt;| -6 -8 2 5 &amp;gt;&lt;br /&gt;
         &lt;td style="text-align: left;"&gt;| -6 -8 2 5 &amp;gt;&lt;br /&gt;
Line 813: Line 813:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;250047/250000&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;[[tel:250047/250000|250047/250000]]&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: left;"&gt;| -4 6 -6 3 &amp;gt;&lt;br /&gt;
         &lt;td style="text-align: left;"&gt;| -4 6 -6 3 &amp;gt;&lt;br /&gt;

Revision as of 16:51, 31 October 2016

IMPORTED REVISION FROM WIKISPACES

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

This revision was by author JosephRuhf and made on 2016-10-31 16:51:10 UTC.
The original revision id was 597593470.
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, assuming the relatively common values of a=2 and s=1%. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.

The 27 note system or one similar like a well temperament can be notated very easily, by a variation on the quartertone accidentals. In this case a sharp raises a note just a hair beneath the following nominal (for example C to C# describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. D flat, C half-sharp, D half flat, and C sharp are all different. The composer can decide for himself which tertiary accidental is necessary if he will need redundancy to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.) otherwise is simple enough and the same tendency for A# to be higher than Bb is not only familiar, though here very exaggerated, to those working with pythagorean scale, but also to many classically trained violinists. et voila

==Intervals== 
|| Degrees of 27-EDO || Cents value coarse/fine
DMS value ||= Approximate
Ratios* ||= Solfege ||
|| 0 || 0 ||= 1/1 ||= do ||
|| 1 || 44.44, 53.33
13°20' ||= 36/35, 49/48, 50/49 ||= di ||
|| 2 || 88.89, 106.67
26°40' ||= 16/15, 21/20, 25/24 ||= ra ||
|| 3 || 133.33, 160
40° ||= 14/13, 13/12 ||= ru ||
|| 4 || 177.78, 213.33
53°20' ||= 10/9 ||= reh ||
|| 5 || 222.22, 266.67
66°40' ||= 8/7, 9/8 ||= re ||
|| 6 || 266.67, 320
80° ||= 7/6 ||= ma ||
|| 7 || 311.11, 373.33
93°20' ||= 6/5 ||= me ||
|| 8 || 355.56, 426.67
106°40' ||= 16/13 ||= mu ||
|| 9 || 400, 480
120° ||= 5/4 ||= mi ||
|| 10 || 444.44, 513.33
133°20' ||= 9/7, 13/10 ||= mo ||
|| 11 || 488.89, 566.67
146°40' ||= 4/3 ||= fa ||
|| 12 || 533.33, 640
160° ||= 49/36, 48/35 ||= fih ||
|| 13 || 577.78, 693.33
173°20' ||= 7/5, 18/13 ||= fi ||
|| 14 || 622.22, 746.67
186°40' ||= 10/7, 13/9 ||= se ||
|| 15 || 666.67, 800
200° ||= 72/49, 35/24 ||= sih ||
|| 16 || 711.11, 853.33
213°20' ||= 3/2 ||= so/sol ||
|| 17 || 755.56, 906.67
226°40' ||= 14/9, 20/13 ||= lo ||
|| 18 || 800, 960
240° ||= 8/5 ||= le ||
|| 19 || 844.44, 1013.33
253°20' ||= 13/8 ||= lu ||
|| 20 || 888.89, 1066.67
266°40' ||= 5/3 ||= la ||
|| 21 || 933.33, 1120
280° ||= 12/7 ||= li ||
|| 22 || 977.78, 1173.33
293°20' ||= 7/4, 16/9 ||= ta ||
|| 23 || 1022.22, 1226.67
306°40' ||= 9/5 ||= te ||
|| 24 || 1066,67, 1280
320° ||= 13/7, 24/13 ||= tu ||
|| 25 || 1111.11, 1333.33
333°20' ||= 40/21 ||= ti ||
|| 26 || 1155.56, 1386.67
346°40' ||= 35/18, 96/49, 49/25 ||= da ||
|| 27 || 1200, 1440
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 < [[tel:27 43 63 76 93 100|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 ||=   ||
||= [[tel:4711802/4709457|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 ||=   ||
||= [[tel:420175/419904|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 ||=   ||=   ||
||= [[tel:250047/250000|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, assuming the relatively common values of a=2 and s=1%. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less &quot;tension&quot; and thus are also more consonant.<br />
<br />
The 27 note system or one similar like a well temperament can be notated very easily, by a variation on the quartertone accidentals. In this case a sharp raises a note just a hair beneath the following nominal (for example C to C# describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. D flat, C half-sharp, D half flat, and C sharp are all different. The composer can decide for himself which tertiary accidental is necessary if he will need redundancy to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.) otherwise is simple enough and the same tendency for A# to be higher than Bb is not only familiar, though here very exaggerated, to those working with pythagorean scale, but also to many classically trained violinists. et voila<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 coarse/fine<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, 53.33<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, 106.67<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, 160<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, 213.33<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, 266.67<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, 320<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, 373.33<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, 426.67<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, 480<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, 513.33<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, 566.67<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, 640<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, 693.33<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, 746.67<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, 800<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, 853.33<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, 906.67<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, 960<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, 1013.33<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, 1066.67<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, 1120<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, 1173.33<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, 1226.67<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, 1280<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, 1333.33<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, 1386.67<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, 1440<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; <a class="wiki_link" href="http://tel.wikispaces.com/27%2043%2063%2076%2093%20100">27 43 63 76 93 100</a> |.)<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;">[[tel:4711802/4709457|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;">[[tel:420175/419904|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;">[[tel:250047/250000|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">
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<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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