31edo

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[[toc|flat]]
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//Thirty-one tone equal temperament//, also called //31-tET//, //31-EDO//, //31-et//, or //tricesimoprimal meantone temperament//, is the scale derived by dividing the octave into 31 [[equal|equally]] large steps. The term 'Tricesimoprimal' was first used by Adriaan Fokker. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. The fifth is flat (over five cents) as befits a tuning supporting meantone, but the major third is less than a cent sharp. The 7/4, a cent flat, is also very good. Because of these good values 31-et is relatively quite accurate and is in fact the sixth Zeta function integral tuning, http://www.research.att.com/~njas/sequences/A117538.

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= 

|||| ====31 tone equal modes:==== ||
|| {{**2 3 3 2 3 2 3 2 3 3 2 3**}} || Meantone Chromatic (53/220-comma) ||
|| {{**5 5 3 5 5 5 3**}} || Thirty-one tone Major, Intense Diatonic Lydian, M.Ionian ||
|| {{**5 3 5 5 3 5 5**}} || Thirty-one tone Natural Minor, Intense Diatonic Hypodorian, Aeolian ||
|| {{**5 3 5 5 5 5 3**}} || Thirty-one tone Melodic Minor ||
|| {{**5 3 5 5 3 7 3**}} || Thirty-one tone Harmonic Minor ||
|| {{**5 5 3 5 3 7 3**}} || Thirty-one tone Harmonic Major ||
|| {{**5 5 3 5 3 5 5**}} || Thirty-one tone Major-Minor ||
|| {{**5 8 5 13**}} || Genus primum ||
|| {{**10 3 5 5 5 3**}} || Genus secundum ||
|| {{**8 2 8 3 7 3**}} || Genus tertium ||
|| {{**10 10 10 1**}} || Genus quartum ||
|| {{**5 7 6 7 5 1**}} || Genus quintum ||
|| {{**4 6 2 6 4 3 3 3**}} || Genus sextum ||
|| {{**4 6 5 6 4 6**}} || Genus septimum ||
|| {{**6 6 6 1 6 6**}} || Genus octavum ||
|| {{**4 6 9 6 4 2**}} || Genus nonum ||
|| {{**13 6 6 6**}} || Genus decimum ||
|| {{**5 5 3 5 5 3 2 3**}} || Genus diatonicum ||
|| {{**3 5 2 3 5 3 2 5 3**}} || Genus chromaticum ||
|| {{**5 5 2 1 5 5 2 3 3**}} || Genus diatonicum cum septimis ||
|| {{**3 4 3 3 2 1 4 1 4 1 2 3**}} || Genus enharmonicum vocale ||
|| {{**2 2 4 2 2 3 3 3 1 3 3 3**}} || Genus enharmonicum instrumentale ||
|| {{**3 2 3 2 3 2 3 3 2 3 2 3**}} || Genus diatonico-chromaticum ||
|| {{**5 2 1 2 5 3 2 1 4 1 2 3**}} || Genus bichromaticum ||
|| {{**4 4 5 4 4 5 5**}} || Neutral Diatonic Mixolydian ||
|| {{**4 5 4 4 5 5 4**}} || Neutral Diatonic Lydian ||
|| {{**5 4 4 5 5 4 4**}} || Neutral Diatonic Phrygian ||
|| {{**4 4 5 5 4 4 5**}} || Neutral Diatonic Dorian ||
|| {{**4 5 5 4 4 5 4**}} || Neutral Diatonic Hypolydian ||
|| {{**5 5 4 4 5 4 4**}} || Neutral Diatonic Hypophrygian ||
|| {{**5 4 4 5 4 4 5**}} || Neutral Diatonic Hypodorian ||
|| {{**4 5 4 4 5 4 5**}} || Neutral Mixolydian ||
|| {{**5 4 4 5 4 5 4**}} || Neutral Lydian ||
|| {{**4 4 5 4 5 4 5**}} || Neutral Phrygian ||
|| {{**4 5 4 5 4 5 4**}} || Neutral Dorian ||
|| {{**5 4 5 4 5 4 4**}} || Neutral Hypolydian ||
|| {{**4 5 4 5 4 4 5**}} || Neutral Hypophrygian ||
|| {{**5 4 5 4 4 5 4**}} || Neutral Hypodorian ||
|| {{**2 2 9 2 2 9 5**}} || Hemiolic Chromatic Mixolydian ||
|| {{**2 9 2 2 9 5 2**}} || Hemiolic Chromatic Lydian ||
|| {{**9 2 2 9 5 2 2**}} || Hemiolic Chromatic Phrygian ||
|| {{**2 2 9 5 2 2 9**}} || Hemiolic Chromatic Dorian ||
|| {{**2 9 5 2 2 9 2**}} || Hemiolic Chromatic Hypolydian ||
|| {{**9 5 2 2 9 2 2**}} || Hemiolic Chromatic Hypophrygian ||
|| {{**5 2 2 9 2 2 9**}} || Hemiolic Chromatic Hypodorian ||
|| {{**2 3 8 2 3 8 5**}} || Ratio 2:3 Chromatic Mixolydian ||
|| {{**3 8 2 3 8 5 2**}} || Ratio 2:3 Chromatic Lydian ||
|| {{**8 2 3 8 5 2 3**}} || Ratio 2:3 Chromatic Phrygian ||
|| {{**2 3 8 5 2 3 8**}} || Ratio 2:3 Chromatic Dorian ||
|| {{**3 8 5 2 3 8 2**}} || Ratio 2:3 Chromatic Hypolydian ||
|| {{**8 5 2 3 8 2 3**}} || Ratio 2:3 Chromatic Hypophrygian ||
|| {{**5 2 3 8 2 3 8**}} || Ratio 2:3 Chromatic Hypodorian ||
|| {{**3 5 5 3 5 5 5**}} || Intense Diatonic Mixolydian, M.Locrian ||
|| {{**5 3 5 5 5 3 5**}} || Intense Diatonic Phrygian, M.Dorian ||
|| {{**3 5 5 5 3 5 5**}} || Intense Diatonic Dorian, M.Phrygian ||
|| {{**5 5 5 3 5 5 3**}} || Intense Diatonic Hypolydian, M.Lydian ||
|| {{**5 5 3 5 5 3 5**}} || Intense Diatonic Hypophrygian, M.Mixolydian ||
|| {{**2 5 6 2 5 6 5**}} || Soft Diatonic Mixolydian ||
|| {{**5 6 2 5 6 5 2**}} || Soft Diatonic Lydian ||
|| {{**6 2 5 6 5 2 5**}} || Soft Diatonic Phrygian ||
|| {{**2 5 6 5 2 5 6**}} || Soft Diatonic Dorian ||
|| {{**5 6 5 2 5 6 2**}} || Soft Diatonic Hypolydian ||
|| {{**6 5 2 5 6 2 5**}} || Soft Diatonic Hypophrygian ||
|| {{**5 2 5 6 2 5 6**}} || Soft Diatonic Hypodorian ||
|| {{**1 2 10 1 2 10 5**}} || Enharmonic Mixolydian ||
|| {{**2 10 1 2 10 5 1**}} || Enharmonic Lydian ||
|| {{**10 1 2 10 5 1 2**}} || Enharmonic Phrygian ||
|| {{**1 2 10 5 1 2 10**}} || Enharmonic Dorian ||
|| {{**2 10 5 1 2 10 1**}} || Enharmonic Hypolydian ||
|| {{**10 5 1 2 10 1 2**}} || Enharmonic Hypophrygian ||
|| {{**5 1 2 10 1 2 10**}} || Enharmonic Hypodorian ||
|| {{**6 6 7 6 6**}} || Quasi-equal Pentatonic ||
|| {{**3 2 2 3 3 2 3 3 2 2 3 3**}} || Fokker 12-tone ||
|| {{**5 3 5 3 5 2 5 3**}} || Modus conjunctus ||
|| {{**3 5 2 5 3 5 3 5**}} || Octatonic ||
|| {{**3 3 4 3 5 3 4 3 3**}} || Hahn symmetric pentachordal ||
|| {{**3 4 3 3 5 3 4 3 3**}} || Hahn pentachordal ||
|| {{**4 4 2 5 3 3 4 3 3**}} || Hahn Nonatonic ||
|| {{**5 1 5 1 5 1 5 1 5 1 1**}} || de Vries 11-tone ||
|| {{**4 1 4 4 4 1 4 4 1 4**}} || Breed 10-tone ||
|| {{**4 2 4 2 4 2 4 3 3 3**}} || Lumma Decatonic ||
|| {{**5 3 3 3 3 5 3 3 3**}} || Rothenberg Generalized Diatonic ||
|| {{**5 2 6 5 2 5 6**}} || "Septimal" Natural Minor ||
|| {{**4 3 4 3 4 3 4 3 3**}} || Thirty-one tone Orwell ||
|| {{**2 5 2 2 5 2 2 2 5 2 2**}} || Secor Sentinel ||

A large open list of modes (subsets) from 31-edo that people have named: [[31edo modes]]. See also [[31edo MOS scales]]. 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.

=Practical Theory / Books= 

[[image:http://ronsword.com/images/TSG_sm.jpg width="87" height="116" link="@http://www.ronsword.com/books.html"]][[@http://www.ronsword.com/books.html|Sword, Ronald. "Tricesimoprimal Scales for Guitar." IAAA Press, UK-USA. First Ed: March 2009.]] - A comprehensive approach to 31-EDO and all the families associated for Guitar. Features over 300 scale charts / scale examples.

=Other Articles= 
* <span class="wiki_link_ext">[[http://www.xs4all.nl/%7Ehuygensf/english/index.html|de Beer, Anton, ''The Development of 31-tone Music]]</span>
* <span class="wiki_link_ext">[[http://www.xs4all.nl/%7Ehuygensf/doc/fokkerorg.html|Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ]]</span>
* Fokker, A.D., "New Music with 31 Notes" translated by Leigh Gerdine
* <span class="wiki_link_ext">[[http://www.xs4all.nl/%7Ehuygensf/doc/rap31.html|Rapoport, Paul, ''About 31-tone Equal Temperament]]</span>
* <span class="wiki_link_ext">[[http://www.xs4all.nl/%7Ehuygensf/doc/terp31.html|Terpstra, Siemen, ''Toward a Theory of Meantone (and 31-et) Harmony'']]</span>
* <span class="wiki_link_ext">[[http://tonalsoft.com/enc/number/31edo.aspx|Tonalsoft Encyclopedia article]]</span>

Original HTML content:

<html><head><title>31edo</title></head><body><!-- ws:start:WikiTextTocRule:50:&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:50 --><!-- ws:start:WikiTextTocRule:51: --><a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --><!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --><!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --><!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --> | <a href="#Modes">Modes</a><!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --> | <a href="#Music in 31-edo">Music in 31-edo</a><!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --> | <a href="#Practical Theory / Books">Practical Theory / Books</a><!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --> | <a href="#Other Articles">Other Articles</a><!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: -->
<!-- ws:end:WikiTextTocRule:76 --><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 meantone temperament</em>, is the scale derived by dividing the octave into 31 <a class="wiki_link" href="/equal">equally</a> large steps. The term 'Tricesimoprimal' was first used by Adriaan Fokker. 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>. The fifth is flat (over five cents) as befits a tuning supporting meantone, but the major third is less than a cent sharp. The 7/4, a cent flat, is also very good. Because of these good values 31-et is relatively quite accurate and is in fact the sixth Zeta function integral tuning, <!-- ws:start:WikiTextUrlRule:868:http://www.research.att.com/~njas/sequences/A117538 --><a class="wiki_link_ext" href="http://www.research.att.com/~njas/sequences/A117538" rel="nofollow">http://www.research.att.com/~njas/sequences/A117538</a><!-- ws:end:WikiTextUrlRule:868 -->.<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 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"></a><!-- ws:end:WikiTextHeadingRule:34 -->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</h3>
 <!-- ws:start:WikiTextHeadingRule:36:&lt;h3&gt; --><h3 id="toc18"><a name="Intervals--29/31 octave 30/31 octave perfect octave - 31/31 octave"></a><!-- ws:end:WikiTextHeadingRule:36 -->29/31 octave 30/31 octave perfect octave - 31/31 octave</h3>
 <br />
<br />
<!-- ws:start:WikiTextHeadingRule:38:&lt;h1&gt; --><h1 id="toc19"><a name="Modes"></a><!-- ws:end:WikiTextHeadingRule:38 -->Modes</h1>
 <br />


<table class="wiki_table">
    <tr>
        <td colspan="2"><!-- ws:start:WikiTextHeadingRule:40:&lt;h4&gt; --><h4 id="toc20"><a name="Modes---31 tone equal modes:"></a><!-- ws:end:WikiTextHeadingRule:40 -->31 tone equal modes:</h4>
</td>
    </tr>
    <tr>
        <td><tt><strong>2 3 3 2 3 2 3 2 3 3 2 3</strong></tt><br />
</td>
        <td>Meantone Chromatic (53/220-comma)<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 5 3 5 5 5 3</strong></tt><br />
</td>
        <td>Thirty-one tone Major, Intense Diatonic Lydian, M.Ionian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 3 5 5 3 5 5</strong></tt><br />
</td>
        <td>Thirty-one tone Natural Minor, Intense Diatonic Hypodorian, Aeolian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 3 5 5 5 5 3</strong></tt><br />
</td>
        <td>Thirty-one tone Melodic Minor<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 3 5 5 3 7 3</strong></tt><br />
</td>
        <td>Thirty-one tone Harmonic Minor<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 5 3 5 3 7 3</strong></tt><br />
</td>
        <td>Thirty-one tone Harmonic Major<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 5 3 5 3 5 5</strong></tt><br />
</td>
        <td>Thirty-one tone Major-Minor<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 8 5 13</strong></tt><br />
</td>
        <td>Genus primum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>10 3 5 5 5 3</strong></tt><br />
</td>
        <td>Genus secundum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>8 2 8 3 7 3</strong></tt><br />
</td>
        <td>Genus tertium<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>10 10 10 1</strong></tt><br />
</td>
        <td>Genus quartum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 7 6 7 5 1</strong></tt><br />
</td>
        <td>Genus quintum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 6 2 6 4 3 3 3</strong></tt><br />
</td>
        <td>Genus sextum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 6 5 6 4 6</strong></tt><br />
</td>
        <td>Genus septimum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>6 6 6 1 6 6</strong></tt><br />
</td>
        <td>Genus octavum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 6 9 6 4 2</strong></tt><br />
</td>
        <td>Genus nonum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>13 6 6 6</strong></tt><br />
</td>
        <td>Genus decimum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 5 3 5 5 3 2 3</strong></tt><br />
</td>
        <td>Genus diatonicum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 5 2 3 5 3 2 5 3</strong></tt><br />
</td>
        <td>Genus chromaticum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 5 2 1 5 5 2 3 3</strong></tt><br />
</td>
        <td>Genus diatonicum cum septimis<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 4 3 3 2 1 4 1 4 1 2 3</strong></tt><br />
</td>
        <td>Genus enharmonicum vocale<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 2 4 2 2 3 3 3 1 3 3 3</strong></tt><br />
</td>
        <td>Genus enharmonicum instrumentale<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 2 3 2 3 2 3 3 2 3 2 3</strong></tt><br />
</td>
        <td>Genus diatonico-chromaticum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 2 1 2 5 3 2 1 4 1 2 3</strong></tt><br />
</td>
        <td>Genus bichromaticum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 4 5 4 4 5 5</strong></tt><br />
</td>
        <td>Neutral Diatonic Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 5 4 4 5 5 4</strong></tt><br />
</td>
        <td>Neutral Diatonic Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 4 4 5 5 4 4</strong></tt><br />
</td>
        <td>Neutral Diatonic Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 4 5 5 4 4 5</strong></tt><br />
</td>
        <td>Neutral Diatonic Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 5 5 4 4 5 4</strong></tt><br />
</td>
        <td>Neutral Diatonic Hypolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 5 4 4 5 4 4</strong></tt><br />
</td>
        <td>Neutral Diatonic Hypophrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 4 4 5 4 4 5</strong></tt><br />
</td>
        <td>Neutral Diatonic Hypodorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 5 4 4 5 4 5</strong></tt><br />
</td>
        <td>Neutral Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 4 4 5 4 5 4</strong></tt><br />
</td>
        <td>Neutral Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 4 5 4 5 4 5</strong></tt><br />
</td>
        <td>Neutral Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 5 4 5 4 5 4</strong></tt><br />
</td>
        <td>Neutral Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 4 5 4 5 4 4</strong></tt><br />
</td>
        <td>Neutral Hypolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 5 4 5 4 4 5</strong></tt><br />
</td>
        <td>Neutral Hypophrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 4 5 4 4 5 4</strong></tt><br />
</td>
        <td>Neutral Hypodorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 2 9 2 2 9 5</strong></tt><br />
</td>
        <td>Hemiolic Chromatic Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 9 2 2 9 5 2</strong></tt><br />
</td>
        <td>Hemiolic Chromatic Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>9 2 2 9 5 2 2</strong></tt><br />
</td>
        <td>Hemiolic Chromatic Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 2 9 5 2 2 9</strong></tt><br />
</td>
        <td>Hemiolic Chromatic Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 9 5 2 2 9 2</strong></tt><br />
</td>
        <td>Hemiolic Chromatic Hypolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>9 5 2 2 9 2 2</strong></tt><br />
</td>
        <td>Hemiolic Chromatic Hypophrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 2 2 9 2 2 9</strong></tt><br />
</td>
        <td>Hemiolic Chromatic Hypodorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 3 8 2 3 8 5</strong></tt><br />
</td>
        <td>Ratio 2:3 Chromatic Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 8 2 3 8 5 2</strong></tt><br />
</td>
        <td>Ratio 2:3 Chromatic Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>8 2 3 8 5 2 3</strong></tt><br />
</td>
        <td>Ratio 2:3 Chromatic Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 3 8 5 2 3 8</strong></tt><br />
</td>
        <td>Ratio 2:3 Chromatic Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 8 5 2 3 8 2</strong></tt><br />
</td>
        <td>Ratio 2:3 Chromatic Hypolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>8 5 2 3 8 2 3</strong></tt><br />
</td>
        <td>Ratio 2:3 Chromatic Hypophrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 2 3 8 2 3 8</strong></tt><br />
</td>
        <td>Ratio 2:3 Chromatic Hypodorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 5 5 3 5 5 5</strong></tt><br />
</td>
        <td>Intense Diatonic Mixolydian, M.Locrian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 3 5 5 5 3 5</strong></tt><br />
</td>
        <td>Intense Diatonic Phrygian, M.Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 5 5 5 3 5 5</strong></tt><br />
</td>
        <td>Intense Diatonic Dorian, M.Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 5 5 3 5 5 3</strong></tt><br />
</td>
        <td>Intense Diatonic Hypolydian, M.Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 5 3 5 5 3 5</strong></tt><br />
</td>
        <td>Intense Diatonic Hypophrygian, M.Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 5 6 2 5 6 5</strong></tt><br />
</td>
        <td>Soft Diatonic Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 6 2 5 6 5 2</strong></tt><br />
</td>
        <td>Soft Diatonic Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>6 2 5 6 5 2 5</strong></tt><br />
</td>
        <td>Soft Diatonic Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 5 6 5 2 5 6</strong></tt><br />
</td>
        <td>Soft Diatonic Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 6 5 2 5 6 2</strong></tt><br />
</td>
        <td>Soft Diatonic Hypolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>6 5 2 5 6 2 5</strong></tt><br />
</td>
        <td>Soft Diatonic Hypophrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 2 5 6 2 5 6</strong></tt><br />
</td>
        <td>Soft Diatonic Hypodorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>1 2 10 1 2 10 5</strong></tt><br />
</td>
        <td>Enharmonic Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 10 1 2 10 5 1</strong></tt><br />
</td>
        <td>Enharmonic Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>10 1 2 10 5 1 2</strong></tt><br />
</td>
        <td>Enharmonic Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>1 2 10 5 1 2 10</strong></tt><br />
</td>
        <td>Enharmonic Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 10 5 1 2 10 1</strong></tt><br />
</td>
        <td>Enharmonic Hypolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>10 5 1 2 10 1 2</strong></tt><br />
</td>
        <td>Enharmonic Hypophrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 1 2 10 1 2 10</strong></tt><br />
</td>
        <td>Enharmonic Hypodorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>6 6 7 6 6</strong></tt><br />
</td>
        <td>Quasi-equal Pentatonic<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 2 2 3 3 2 3 3 2 2 3 3</strong></tt><br />
</td>
        <td>Fokker 12-tone<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 3 5 3 5 2 5 3</strong></tt><br />
</td>
        <td>Modus conjunctus<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 5 2 5 3 5 3 5</strong></tt><br />
</td>
        <td>Octatonic<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 3 4 3 5 3 4 3 3</strong></tt><br />
</td>
        <td>Hahn symmetric pentachordal<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 4 3 3 5 3 4 3 3</strong></tt><br />
</td>
        <td>Hahn pentachordal<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 4 2 5 3 3 4 3 3</strong></tt><br />
</td>
        <td>Hahn Nonatonic<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 1 5 1 5 1 5 1 5 1 1</strong></tt><br />
</td>
        <td>de Vries 11-tone<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 1 4 4 4 1 4 4 1 4</strong></tt><br />
</td>
        <td>Breed 10-tone<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 2 4 2 4 2 4 3 3 3</strong></tt><br />
</td>
        <td>Lumma Decatonic<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 3 3 3 3 5 3 3 3</strong></tt><br />
</td>
        <td>Rothenberg Generalized Diatonic<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 2 6 5 2 5 6</strong></tt><br />
</td>
        <td>&quot;Septimal&quot; Natural Minor<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 3 4 3 4 3 4 3 3</strong></tt><br />
</td>
        <td>Thirty-one tone Orwell<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 5 2 2 5 2 2 2 5 2 2</strong></tt><br />
</td>
        <td>Secor Sentinel<br />
</td>
    </tr>
</table>

<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>. See also <a class="wiki_link" href="/31edo%20MOS%20scales">31edo MOS scales</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:42:&lt;h1&gt; --><h1 id="toc21"><a name="Music in 31-edo"></a><!-- ws:end:WikiTextHeadingRule:42 -->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:44:&lt;h2&gt; --><h2 id="toc22"><a name="Music in 31-edo-Thirty-one tone pedagogy"></a><!-- ws:end:WikiTextHeadingRule:44 -->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 />
<!-- ws:start:WikiTextHeadingRule:46:&lt;h1&gt; --><h1 id="toc23"><a name="Practical Theory / Books"></a><!-- ws:end:WikiTextHeadingRule:46 -->Practical Theory / Books</h1>
 <br />
<!-- ws:start:WikiTextRemoteImageRule:619:&lt;a href=&quot;http://www.ronsword.com/books.html&quot; target=&quot;_blank&quot; rel=&quot;nofollow&quot;&gt;&lt;img src=&quot;http://ronsword.com/images/TSG_sm.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 116px; width: 87px;&quot; /&gt;&lt;/a&gt; --><a href="http://www.ronsword.com/books.html" target="_blank" rel="nofollow"><img src="http://ronsword.com/images/TSG_sm.jpg" alt="external image TSG_sm.jpg" title="external image TSG_sm.jpg" style="height: 116px; width: 87px;" /></a><!-- ws:end:WikiTextRemoteImageRule:619 --><a class="wiki_link_ext" href="http://www.ronsword.com/books.html" rel="nofollow" target="_blank">Sword, Ronald. &quot;Tricesimoprimal Scales for Guitar.&quot; IAAA Press, UK-USA. First Ed: March 2009.</a> - A comprehensive approach to 31-EDO and all the families associated for Guitar. Features over 300 scale charts / scale examples.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:48:&lt;h1&gt; --><h1 id="toc24"><a name="Other Articles"></a><!-- ws:end:WikiTextHeadingRule:48 -->Other Articles</h1>
 <ul><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.xs4all.nl/%7Ehuygensf/english/index.html" rel="nofollow">de Beer, Anton, ''The Development of 31-tone Music</a></span></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.xs4all.nl/%7Ehuygensf/doc/fokkerorg.html" rel="nofollow">Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ</a></span></li><li>Fokker, A.D., &quot;New Music with 31 Notes&quot; translated by Leigh Gerdine</li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.xs4all.nl/%7Ehuygensf/doc/rap31.html" rel="nofollow">Rapoport, Paul, ''About 31-tone Equal Temperament</a></span></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.xs4all.nl/%7Ehuygensf/doc/terp31.html" rel="nofollow">Terpstra, Siemen, ''Toward a Theory of Meantone (and 31-et) Harmony''</a></span></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://tonalsoft.com/enc/number/31edo.aspx" rel="nofollow">Tonalsoft Encyclopedia article</a></span></li></ul></body></html>