31edo

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//Thirty-one tone equal temperament//, also called //31-tET//, //31-EDO//, //31-et//, or //tricesimoprimal temperament//, is the scale derived by dividing the octave into 31 [[equal|equally]] large steps. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]].

For more encyclopedic info, see [[http://en.wikipedia.org/wiki/31_equal_temperament|Wikipedia's article]].

=Intervals= 
===Diesis - 1/31 octave=== 
A single step of 31-edo is about 38.7¢. Intervals around this size are called [[diesis|dieses]] (singular 'diesis'). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G#, to B#, but B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). Demonstrated in [[SpiralProgressions]].

===2/31 octave=== 
The difference between a major and minor third. The more 'expressive' of the 'half steps'.

===3/31 octave=== 
The difference between a perfect fourth and a major third. The larger and clunkier of the 'half steps'.

===Neutral second - 4/31 octave=== 
Exactly one half of the minor third.

===5/31 octave=== 
A rather smallish whole tone. Often called melodically dull.

===6/31 octave=== 
A 'supermajor 2nd'.

===7/31 octave=== 
A 'subminor 3rd'.

===8/31 octave=== 
A minor third, closer to the just 6:5 than 12-edo, but not really close enough.

===9/31 octave=== 
A neutral 3rd, practically equivalent to 11:9.

===10/31 octave=== 
A near-just major 3rd. Has led to the characterization of 31-edo as "smooth".

===11/31 octave=== 
A 'supermajor 3rd'.

===12/31 octave=== 
A narrow fourth.

===13/31 octave=== 
Not very satisfying.

===14/31 octave=== 
10¢ off from a just 11:8; barely functional as such.

===15/31 octave=== 
The small tritone.

===16/31 octave=== 
The large tritone.

===17/31 octave=== 
===18/31 octave=== 
===19/31 octave=== 
===20/31 octave=== 
===21/31 octave=== 
===22/31 octave=== 
===23/31 octave=== 
===24/31 octave=== 
===25/31 octave=== 
===26/31 octave=== 
===27/31 octave=== 
===28/31 octave=== 
===29/31 octave=== 
===30/31 octave=== 
===perfect octave - 31/31 octave=== 


=Modes= 
When 31 is too many alternatives to choose from.

A large open list of modes (subsets) from 31-edo that people have named: [[31edo modes]]. Some of the popular ones:

* 31-tone major: 5 5 3 5 5 5 3
* 1/4-comma meantone (Eb-G#): 2 3 3 2 3 2 3 2 3 3 2 3
* Harmonic scale 8: 5 5 4 4 4 4 3 3
* the [[Euler-fokker genera]] (technically [[JI]] but representable in 31) 

=Music in 31-edo= 
[[31-edo compositions|An alphabetical list of Tricesimoprimal Compositions]].

==Thirty-one tone pedagogy== 
The [[MicroPedagogyCollective]] is currently at work producing demonstrative material which will encourage and enable more people to learn this system. There have been two [[ThirtyOneToneSinginCamp]]s as well.


=Other Articles= 
* [<span class="wiki_link_ext">[http://www.xs4all.nl/~huygensf/english/index.html</span>|de Beer, Anton, ''The Development of 31-tone Music'']]
* [[<span class="wiki_link_ext">http://www.xs4all.nl/~huygensf/doc/fokkerorg.html</span>|Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ'']]
* Fokker, A.D., "New Music with 31 Notes" translated by Leigh Gerdine
* [[<span class="wiki_link_ext">http://www.xs4all.nl/~huygensf/doc/rap31.html</span>|Rapoport, Paul, ''About 31-tone Equal Temperament'']
* [[<span class="wiki_link_ext">http://www.xs4all.nl/~huygensf/doc/terp31.html</span>|Terpstra, Siemen, ''Toward a Theory of Meantone (and 31-et) Harmony'']]
* [[<span class="wiki_link_ext">http://tonalsoft.com/enc/number/31edo.aspx</span>|Tonalsoft Encyclopedia article]

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<html><head><title>31edo</title></head><body><!-- ws:start:WikiTextTocRule:72:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --><!-- ws:end:WikiTextTocRule:76 --><!-- ws:start:WikiTextTocRule:77: --><!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --><!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --><!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --><!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --><!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --><!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --><!-- ws:end:WikiTextTocRule:90 --><!-- ws:start:WikiTextTocRule:91: --><!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --><!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --><!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --><!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --><!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --> | <a href="#Modes">Modes</a><!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --> | <a href="#Music in 31-edo">Music in 31-edo</a><!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextTocRule:108: --> | <a href="#Other Articles">Other Articles</a><!-- ws:end:WikiTextTocRule:108 --><!-- ws:start:WikiTextTocRule:109: -->
<!-- ws:end:WikiTextTocRule:109 --><br />
<hr />
<em>Thirty-one tone equal temperament</em>, also called <em>31-tET</em>, <em>31-EDO</em>, <em>31-et</em>, or <em>tricesimoprimal temperament</em>, is the scale derived by dividing the octave into 31 <a class="wiki_link" href="/equal">equally</a> large steps. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 <a class="wiki_link" href="/cents">cents</a>.<br />
<br />
For more encyclopedic info, see <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/31_equal_temperament" rel="nofollow">Wikipedia's article</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:0 -->Intervals</h1>
 <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="Intervals--Diesis - 1/31 octave"></a><!-- ws:end:WikiTextHeadingRule:2 -->Diesis - 1/31 octave</h3>
 A single step of 31-edo is about 38.7¢. Intervals around this size are called <a class="wiki_link" href="/diesis">dieses</a> (singular 'diesis'). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G#, to B#, but B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). Demonstrated in <a class="wiki_link" href="/SpiralProgressions">SpiralProgressions</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="Intervals--2/31 octave"></a><!-- ws:end:WikiTextHeadingRule:4 -->2/31 octave</h3>
 The difference between a major and minor third. The more 'expressive' of the 'half steps'.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Intervals--3/31 octave"></a><!-- ws:end:WikiTextHeadingRule:6 -->3/31 octave</h3>
 The difference between a perfect fourth and a major third. The larger and clunkier of the 'half steps'.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Intervals--Neutral second - 4/31 octave"></a><!-- ws:end:WikiTextHeadingRule:8 -->Neutral second - 4/31 octave</h3>
 Exactly one half of the minor third.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Intervals--5/31 octave"></a><!-- ws:end:WikiTextHeadingRule:10 -->5/31 octave</h3>
 A rather smallish whole tone. Often called melodically dull.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Intervals--6/31 octave"></a><!-- ws:end:WikiTextHeadingRule:12 -->6/31 octave</h3>
 A 'supermajor 2nd'.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Intervals--7/31 octave"></a><!-- ws:end:WikiTextHeadingRule:14 -->7/31 octave</h3>
 A 'subminor 3rd'.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Intervals--8/31 octave"></a><!-- ws:end:WikiTextHeadingRule:16 -->8/31 octave</h3>
 A minor third, closer to the just 6:5 than 12-edo, but not really close enough.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="Intervals--9/31 octave"></a><!-- ws:end:WikiTextHeadingRule:18 -->9/31 octave</h3>
 A neutral 3rd, practically equivalent to 11:9.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Intervals--10/31 octave"></a><!-- ws:end:WikiTextHeadingRule:20 -->10/31 octave</h3>
 A near-just major 3rd. Has led to the characterization of 31-edo as &quot;smooth&quot;.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Intervals--11/31 octave"></a><!-- ws:end:WikiTextHeadingRule:22 -->11/31 octave</h3>
 A 'supermajor 3rd'.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Intervals--12/31 octave"></a><!-- ws:end:WikiTextHeadingRule:24 -->12/31 octave</h3>
 A narrow fourth.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Intervals--13/31 octave"></a><!-- ws:end:WikiTextHeadingRule:26 -->13/31 octave</h3>
 Not very satisfying.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Intervals--14/31 octave"></a><!-- ws:end:WikiTextHeadingRule:28 -->14/31 octave</h3>
 10¢ off from a just 11:8; barely functional as such.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="Intervals--15/31 octave"></a><!-- ws:end:WikiTextHeadingRule:30 -->15/31 octave</h3>
 The small tritone.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="Intervals--16/31 octave"></a><!-- ws:end:WikiTextHeadingRule:32 -->16/31 octave</h3>
 The large tritone.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:34:&lt;h3&gt; --><h3 id="toc17"><a name="Intervals--17/31 octave"></a><!-- ws:end:WikiTextHeadingRule:34 -->17/31 octave</h3>
 <!-- ws:start:WikiTextHeadingRule:36:&lt;h3&gt; --><h3 id="toc18"><a name="Intervals--18/31 octave"></a><!-- ws:end:WikiTextHeadingRule:36 -->18/31 octave</h3>
 <!-- ws:start:WikiTextHeadingRule:38:&lt;h3&gt; --><h3 id="toc19"><a name="Intervals--19/31 octave"></a><!-- ws:end:WikiTextHeadingRule:38 -->19/31 octave</h3>
 <!-- ws:start:WikiTextHeadingRule:40:&lt;h3&gt; --><h3 id="toc20"><a name="Intervals--20/31 octave"></a><!-- ws:end:WikiTextHeadingRule:40 -->20/31 octave</h3>
 <!-- ws:start:WikiTextHeadingRule:42:&lt;h3&gt; --><h3 id="toc21"><a name="Intervals--21/31 octave"></a><!-- ws:end:WikiTextHeadingRule:42 -->21/31 octave</h3>
 <!-- ws:start:WikiTextHeadingRule:44:&lt;h3&gt; --><h3 id="toc22"><a name="Intervals--22/31 octave"></a><!-- ws:end:WikiTextHeadingRule:44 -->22/31 octave</h3>
 <!-- ws:start:WikiTextHeadingRule:46:&lt;h3&gt; --><h3 id="toc23"><a name="Intervals--23/31 octave"></a><!-- ws:end:WikiTextHeadingRule:46 -->23/31 octave</h3>
 <!-- ws:start:WikiTextHeadingRule:48:&lt;h3&gt; --><h3 id="toc24"><a name="Intervals--24/31 octave"></a><!-- ws:end:WikiTextHeadingRule:48 -->24/31 octave</h3>
 <!-- ws:start:WikiTextHeadingRule:50:&lt;h3&gt; --><h3 id="toc25"><a name="Intervals--25/31 octave"></a><!-- ws:end:WikiTextHeadingRule:50 -->25/31 octave</h3>
 <!-- ws:start:WikiTextHeadingRule:52:&lt;h3&gt; --><h3 id="toc26"><a name="Intervals--26/31 octave"></a><!-- ws:end:WikiTextHeadingRule:52 -->26/31 octave</h3>
 <!-- ws:start:WikiTextHeadingRule:54:&lt;h3&gt; --><h3 id="toc27"><a name="Intervals--27/31 octave"></a><!-- ws:end:WikiTextHeadingRule:54 -->27/31 octave</h3>
 <!-- ws:start:WikiTextHeadingRule:56:&lt;h3&gt; --><h3 id="toc28"><a name="Intervals--28/31 octave"></a><!-- ws:end:WikiTextHeadingRule:56 -->28/31 octave</h3>
 <!-- ws:start:WikiTextHeadingRule:58:&lt;h3&gt; --><h3 id="toc29"><a name="Intervals--29/31 octave"></a><!-- ws:end:WikiTextHeadingRule:58 -->29/31 octave</h3>
 <!-- ws:start:WikiTextHeadingRule:60:&lt;h3&gt; --><h3 id="toc30"><a name="Intervals--30/31 octave"></a><!-- ws:end:WikiTextHeadingRule:60 -->30/31 octave</h3>
 <!-- ws:start:WikiTextHeadingRule:62:&lt;h3&gt; --><h3 id="toc31"><a name="Intervals--perfect octave - 31/31 octave"></a><!-- ws:end:WikiTextHeadingRule:62 -->perfect octave - 31/31 octave</h3>
 <br />
<br />
<!-- ws:start:WikiTextHeadingRule:64:&lt;h1&gt; --><h1 id="toc32"><a name="Modes"></a><!-- ws:end:WikiTextHeadingRule:64 -->Modes</h1>
 When 31 is too many alternatives to choose from.<br />
<br />
A large open list of modes (subsets) from 31-edo that people have named: <a class="wiki_link" href="/31edo%20modes">31edo modes</a>. Some of the popular ones:<br />
<br />
<ul><li>31-tone major: 5 5 3 5 5 5 3</li><li>1/4-comma meantone (Eb-G#): 2 3 3 2 3 2 3 2 3 3 2 3</li><li>Harmonic scale 8: 5 5 4 4 4 4 3 3</li><li>the <a class="wiki_link" href="/Euler-fokker%20genera">Euler-fokker genera</a> (technically <a class="wiki_link" href="/JI">JI</a> but representable in 31)</li></ul><br />
<!-- ws:start:WikiTextHeadingRule:66:&lt;h1&gt; --><h1 id="toc33"><a name="Music in 31-edo"></a><!-- ws:end:WikiTextHeadingRule:66 -->Music in 31-edo</h1>
 <a class="wiki_link" href="/31-edo%20compositions">An alphabetical list of Tricesimoprimal Compositions</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:68:&lt;h2&gt; --><h2 id="toc34"><a name="Music in 31-edo-Thirty-one tone pedagogy"></a><!-- ws:end:WikiTextHeadingRule:68 -->Thirty-one tone pedagogy</h2>
 The <a class="wiki_link" href="/MicroPedagogyCollective">MicroPedagogyCollective</a> is currently at work producing demonstrative material which will encourage and enable more people to learn this system. There have been two <a class="wiki_link" href="/ThirtyOneToneSinginCamp">ThirtyOneToneSinginCamp</a>s as well.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:70:&lt;h1&gt; --><h1 id="toc35"><a name="Other Articles"></a><!-- ws:end:WikiTextHeadingRule:70 -->Other Articles</h1>
 <ul><li>[<span class="wiki_link_ext">[<!-- ws:start:WikiTextUrlRule:203:http://www.xs4all.nl/~huygensf/english/index.html --><a class="wiki_link_ext" href="http://www.xs4all.nl/~huygensf/english/index.html" rel="nofollow">http://www.xs4all.nl/~huygensf/english/index.html</a><!-- ws:end:WikiTextUrlRule:203 --></span>|de Beer, Anton, ''The Development of 31-tone Music'']]</li><li>[[<span class="wiki_link_ext"><!-- ws:start:WikiTextUrlRule:204:http://www.xs4all.nl/~huygensf/doc/fokkerorg.html --><a class="wiki_link_ext" href="http://www.xs4all.nl/~huygensf/doc/fokkerorg.html" rel="nofollow">http://www.xs4all.nl/~huygensf/doc/fokkerorg.html</a><!-- ws:end:WikiTextUrlRule:204 --></span>|Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ'']]</li><li>Fokker, A.D., &quot;New Music with 31 Notes&quot; translated by Leigh Gerdine</li><li>[[<span class="wiki_link_ext"><!-- ws:start:WikiTextUrlRule:205:http://www.xs4all.nl/~huygensf/doc/rap31.html --><a class="wiki_link_ext" href="http://www.xs4all.nl/~huygensf/doc/rap31.html" rel="nofollow">http://www.xs4all.nl/~huygensf/doc/rap31.html</a><!-- ws:end:WikiTextUrlRule:205 --></span>|Rapoport, Paul, ''About 31-tone Equal Temperament'']* [[<span class="wiki_link_ext"><!-- ws:start:WikiTextUrlRule:206:http://www.xs4all.nl/~huygensf/doc/terp31.html --><a class="wiki_link_ext" href="http://www.xs4all.nl/~huygensf/doc/terp31.html" rel="nofollow">http://www.xs4all.nl/~huygensf/doc/terp31.html</a><!-- ws:end:WikiTextUrlRule:206 --></span>|Terpstra, Siemen, ''Toward a Theory of Meantone (and 31-et) Harmony'']]</li><li>[[<span class="wiki_link_ext"><!-- ws:start:WikiTextUrlRule:207:http://tonalsoft.com/enc/number/31edo.aspx --><a class="wiki_link_ext" href="http://tonalsoft.com/enc/number/31edo.aspx" rel="nofollow">http://tonalsoft.com/enc/number/31edo.aspx</a><!-- ws:end:WikiTextUrlRule:207 --></span>|Tonalsoft Encyclopedia article]</li></ul></body></html>