31edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

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: This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2011-05-31 22:41:37 UTC</tt>.

: The original revision id was <tt>~~216456556~~</tt>.

: The original revision id was <tt>233352940</tt>.

: The revision comment was: <tt></tt>

: The revision comment was: <tt>Added comma table.</tt>

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<h4>Original Wikitext content:</h4>

Line 180:

The large tritone.

=Commas=

31 EDO tempers out the following commas. (Note: This assumes the val < 31 49 72 87 107 115 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||

|| 9931568/9752117 || | -25 7 6 > || 31.57 || Ampersand's Comma || || ||

|| 81/80 || | -4 4 -1 > || 21.51 || Syntonic Comma || Didymos Comma || Meantone Comma ||

|| 393216/390625 || | 17 1 -8 > || 11.45 || Wuerschmidt Comma || || ||

|| 2109375/2097152 || | -21 3 7 > || 10.06 || Semicomma || Fokker Comma || ||

|| 6719816/6714445 || | 38 -2 -15 > || 1.38 || Hemithirds Comma || || ||

|| 9859966/9733137 || | -10 7 8 -7 > || 22.41 || Blackjackisma || || ||

|| 64827/64000 || | -9 3 -3 4 > || 22.23 || Squalentine || || ||

|| 2430/2401 || | 1 5 1 -4 > || 20.79 || Nuwell || || ||

|| 50421/50000 || | -4 1 -5 5 > || 14.52 || Trimyna || || ||

|| 126/125 || | 1 2 -3 1 > || 13.79 || Septimal Semicomma || Starling Comma || ||

|| 1728/1715 || | 6 3 -1 -3 > || 13.07 || Orwellisma || Orwell Comma || ||

|| 1029/1024 || | -10 1 0 3 > || 8.43 || Gamelisma || || ||

|| 225/224 || | -5 2 2 -1 > || 7.71 || Septimal Kleisma || Marvel Comma || ||

|| 16875/16807 || | 0 3 4 -5 > || 6.99 || Mirkwai || || ||

|| 3136/3125 || | 6 0 -5 2 > || 6.08 || Hemimean || || ||

|| 6144/6125 || | 11 1 -3 -2 > || 5.36 || Porwell || || ||

|| 1065875/1063543 || | -26 -1 1 9 > || 3.79 || Wadisma || || ||

|| 65625/65536 || | -16 1 5 1 > || 2.35 || Horwell || || ||

|| 703125/702464 || | -11 2 7 -3 > || 1.63 || Meter || || ||

|| 2401/2400 || | -5 -1 -2 4 > || 0.72 || Breedsma || || ||

|| 99/98 || | -1 2 0 -2 1 > || 17.58 || Mothwellsma || || ||

|| 121/120 || | -3 -1 -1 0 2 > || 14.37 || Biyatisma || || ||

|| 176/175 || | 4 0 -2 -1 1 > || 9.86 || Valinorsma || || ||

|| 243/242 || | -1 5 0 0 -2 > || 7.14 || Rastma || || ||

|| 385/384 || | -7 -1 1 1 1 > || 4.50 || Keenanisma || || ||

|| 441/440 || | -3 2 -1 2 -1 > || 3.93 || Werckisma || || ||

|| 540/539 || | 2 3 1 -2 -1 > || 3.21 || Swetisma || || ||

|| 3025/3024 || | -4 -3 2 -1 2 > || 0.57 || Lehmerisma || || ||

=Modes=

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* [[http://tonalsoft.com/enc/number/31edo.aspx|Tonalsoft Encyclopedia article]] [[http://www.webcitation.org/5xeGYj7QU|Permalink]]</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>31edo</title></head><body><a href="#Intervals">Intervals</a> | <a href="#~~Modes~~">~~Modes~~</a> | <a href="#Music in 31-edo">Music in 31-edo</a> | <a href="#Practical Theory / Books">Practical Theory / Books</a> | <a href="#Other Articles">Other Articles</a>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>31edo</title></head><body><a href="#Intervals">Intervals</a> | <a href="#Commas">Commas</a> | <a href="#Modes">Modes</a> | <a href="#Music in 31-edo">Music in 31-edo</a> | <a href="#Practical Theory / Books">Practical Theory / Books</a> | <a href="#Other Articles">Other Articles</a>

<hr />

<hr />

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 <a class="wiki_link" href="/equal">equally</a> large steps. The term 'Tricesimoprimal' was first used by <a class="wiki_link" href="/Adriaan%20Fokker">Adriaan Fokker</a>. 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>. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth Zeta function integral tuning, <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>. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). For JI that uses primes 3 and 7, but no 5, try <a class="wiki_link" href="/36edo">36edo</a>.

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 <a class="wiki_link" href="/equal">equally</a> large steps. The term 'Tricesimoprimal' was first used by <a class="wiki_link" href="/Adriaan%20Fokker">Adriaan Fokker</a>. 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>. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth Zeta function integral tuning, <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>. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). For JI that uses primes 3 and 7, but no 5, try <a class="wiki_link" href="/36edo">36edo</a>.

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>.

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The large tritone.

<h1 id="toc31"><a name="Modes"></a>Modes</h1>

<h1 id="toc31"><a name="Commas"></a>Commas</h1>

31 EDO tempers out the following commas. (Note: This assumes the val &lt; 31 49 72 87 107 115 |.)

<tr>

<th>Comma

</th>

<th>Monzo

</th>

<th>Value (Cents)

</th>

<th>Name 1

</th>

<th>Name 2

</th>

<th>Name 3

</th>

</tr>

<tr>

<td>9931568/9752117

</td>

<td>| -25 7 6 &gt;

</td>

<td>31.57

</td>

<td>Ampersand's Comma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>81/80

</td>

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

</td>

<td>21.51

</td>

<td>Syntonic Comma

</td>

<td>Didymos Comma

</td>

<td>Meantone Comma

</td>

</tr>

<tr>

<td>393216/390625

</td>

<td>| 17 1 -8 &gt;

</td>

<td>11.45

</td>

<td>Wuerschmidt Comma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>2109375/2097152

</td>

<td>| -21 3 7 &gt;

</td>

<td>10.06

</td>

<td>Semicomma

</td>

<td>Fokker Comma

</td>

<td>

</td>

</tr>

<tr>

<td>6719816/6714445

</td>

<td>| 38 -2 -15 &gt;

</td>

<td>1.38

</td>

<td>Hemithirds Comma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>9859966/9733137

</td>

<td>| -10 7 8 -7 &gt;

</td>

<td>22.41

</td>

<td>Blackjackisma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>64827/64000

</td>

<td>| -9 3 -3 4 &gt;

</td>

<td>22.23

</td>

<td>Squalentine

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>2430/2401

</td>

<td>| 1 5 1 -4 &gt;

</td>

<td>20.79

</td>

<td>Nuwell

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>50421/50000

</td>

<td>| -4 1 -5 5 &gt;

</td>

<td>14.52

</td>

<td>Trimyna

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>126/125

</td>

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

</td>

<td>13.79

</td>

<td>Septimal Semicomma

</td>

<td>Starling Comma

</td>

<td>

</td>

</tr>

<tr>

<td>1728/1715

</td>

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

</td>

<td>13.07

</td>

<td>Orwellisma

</td>

<td>Orwell Comma

</td>

<td>

</td>

</tr>

<tr>

<td>1029/1024

</td>

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

</td>

<td>8.43

</td>

<td>Gamelisma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>225/224

</td>

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

</td>

<td>7.71

</td>

<td>Septimal Kleisma

</td>

<td>Marvel Comma

</td>

<td>

</td>

</tr>

<tr>

<td>16875/16807

</td>

<td>| 0 3 4 -5 &gt;

</td>

<td>6.99

</td>

<td>Mirkwai

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>3136/3125

</td>

<td>| 6 0 -5 2 &gt;

</td>

<td>6.08

</td>

<td>Hemimean

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>6144/6125

</td>

<td>| 11 1 -3 -2 &gt;

</td>

<td>5.36

</td>

<td>Porwell

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>1065875/1063543

</td>

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

</td>

<td>3.79

</td>

<td>Wadisma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>65625/65536

</td>

<td>| -16 1 5 1 &gt;

</td>

<td>2.35

</td>

<td>Horwell

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>703125/702464

</td>

<td>| -11 2 7 -3 &gt;

</td>

<td>1.63

</td>

<td>Meter

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>2401/2400

</td>

<td>| -5 -1 -2 4 &gt;

</td>

<td>0.72

</td>

<td>Breedsma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>99/98

</td>

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

</td>

<td>17.58

</td>

<td>Mothwellsma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>121/120

</td>

<td>| -3 -1 -1 0 2 &gt;

</td>

<td>14.37

</td>

<td>Biyatisma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>176/175

</td>

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

</td>

<td>9.86

</td>

<td>Valinorsma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>243/242

</td>

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

</td>

<td>7.14

</td>

<td>Rastma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>385/384

</td>

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

</td>

<td>4.50

</td>

<td>Keenanisma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>441/440

</td>

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

</td>

<td>3.93

</td>

<td>Werckisma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>540/539

</td>

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

</td>

<td>3.21

</td>

<td>Swetisma

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>3025/3024

</td>

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

</td>

<td>0.57

</td>

<td>Lehmerisma

</td>

<td>

</td>

<td>

</td>

</tr>

</table>

<h1 id="toc32"><a name="Modes"></a>Modes</h1>

A large open list of modes (subsets) from 31edo that people have named: <a class="wiki_link" href="/31edo%20modes">31edo modes</a>. <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rothenberg_propriety" rel="nofollow">Strictly proper</a> <a class="wiki_link" href="/Strictly%20proper%207-note%2031edo%20scales">7-note 31edo scales</a> in the sense of <a class="wiki_link" href="/David%20Rothenberg">David Rothenberg</a>. See also <a class="wiki_link" href="/31edo%20MOS%20scales">31edo MOS scales</a>. Some of the popular ones:

Line 7,046:

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<tr>

<td colspan="2"><h4 id="~~toc32~~"><a name="Modes---Some 31 tone equal modes:"></a>Some 31 tone equal modes:</h4>

<td colspan="2"><h4 id="toc33"><a name="Modes---Some 31 tone equal modes:"></a>Some 31 tone equal modes:</h4>

</td>

</tr>

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<h1 id="~~toc33~~"><a name="Music in 31-edo"></a>Music in 31-edo</h1>

<h1 id="toc34"><a name="Music in 31-edo"></a>Music in 31-edo</h1>

<a class="wiki_link" href="/31-edo%20compositions">An alphabetical list of Tricesimoprimal Compositions</a>.

<h2 id="~~toc34~~"><a name="Music in 31-edo-Thirty-one tone pedagogy"></a>Thirty-one tone pedagogy</h2>

<h2 id="toc35"><a name="Music in 31-edo-Thirty-one tone pedagogy"></a>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.

<h1 id="~~toc35~~"><a name="Practical Theory / Books"></a>Practical Theory / Books</h1>

<h1 id="toc36"><a name="Practical Theory / Books"></a>Practical Theory / Books</h1>

<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><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.

<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><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.

<h1 id="~~toc36~~"><a name="Other Articles"></a>Other Articles</h1>

<h1 id="toc37"><a name="Other Articles"></a>Other Articles</h1>

<ul><li><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/beerart.html" rel="nofollow">de Beer, Anton, ''The Development of 31-tone Music</a> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeFzBM9b" rel="nofollow">Permalink</a></li><li><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/fokkerorg.html" rel="nofollow">Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ</a> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeG6Tmli" rel="nofollow">Permalink</a></li><li>Fokker, A.D., &quot;New Music with 31 Notes&quot; translated by Leigh Gerdine</li><li><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/rap31.html" rel="nofollow">Rapoport, Paul, ''About 31-tone Equal Temperament</a> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeGH4uBH" rel="nofollow">Permalink</a></li><li><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/terp31.html" rel="nofollow">Terpstra, Siemen, ''Toward a Theory of Meantone (and 31-et) Harmony''</a> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeGMeCMd" rel="nofollow">Permalink</a></li><li><a class="wiki_link_ext" href="http://tonalsoft.com/enc/number/31edo.aspx" rel="nofollow">Tonalsoft Encyclopedia article</a> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeGYj7QU" rel="nofollow">Permalink</a></li></ul></body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-02 12:51:06 UTC</tt>.<br>
+: This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2011-05-31 22:41:37 UTC</tt>.<br>
-: The original revision id was <tt>216456556</tt>.<br>
+: The original revision id was <tt>233352940</tt>.<br>
-: The revision comment was: <tt></tt><br>
+: The revision comment was: <tt>Added comma table.</tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
@@ Line 180: / Line 180: @@
 The large tritone.
+=Commas=
+EDO tempers out the following commas. (Note: This assumes the val &lt; 31 49 72 87 107 115 |.)
+||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
+|| 9931568/9752117 || | -25 7 6 &gt; || 31.57 || Ampersand's Comma ||   ||   ||
+|| 81/80 || | -4 4 -1 &gt; || 21.51 || Syntonic Comma || Didymos Comma || Meantone Comma ||
+|| 393216/390625 || | 17 1 -8 &gt; || 11.45 || Wuerschmidt Comma ||   ||   ||
+|| 2109375/2097152 || | -21 3 7 &gt; || 10.06 || Semicomma || Fokker Comma ||   ||
+|| 6719816/6714445 || | 38 -2 -15 &gt; || 1.38 || Hemithirds Comma ||   ||   ||
+|| 9859966/9733137 || | -10 7 8 -7 &gt; || 22.41 || Blackjackisma ||   ||   ||
+|| 64827/64000 || | -9 3 -3 4 &gt; || 22.23 || Squalentine ||   ||   ||
+|| 2430/2401 || | 1 5 1 -4 &gt; || 20.79 || Nuwell ||   ||   ||
+|| 50421/50000 || | -4 1 -5 5 &gt; || 14.52 || Trimyna ||   ||   ||
+|| 126/125 || | 1 2 -3 1 &gt; || 13.79 || Septimal Semicomma || Starling Comma ||   ||
+|| 1728/1715 || | 6 3 -1 -3 &gt; || 13.07 || Orwellisma || Orwell Comma ||   ||
+|| 1029/1024 || | -10 1 0 3 &gt; || 8.43 || Gamelisma ||   ||   ||
+|| 225/224 || | -5 2 2 -1 &gt; || 7.71 || Septimal Kleisma || Marvel Comma ||   ||
+|| 16875/16807 || | 0 3 4 -5 &gt; || 6.99 || Mirkwai ||   ||   ||
+|| 3136/3125 || | 6 0 -5 2 &gt; || 6.08 || Hemimean ||   ||   ||
+|| 6144/6125 || | 11 1 -3 -2 &gt; || 5.36 || Porwell ||   ||   ||
+|| 1065875/1063543 || | -26 -1 1 9 &gt; || 3.79 || Wadisma ||   ||   ||
+|| 65625/65536 || | -16 1 5 1 &gt; || 2.35 || Horwell ||   ||   ||
+|| 703125/702464 || | -11 2 7 -3 &gt; || 1.63 || Meter ||   ||   ||
+|| 2401/2400 || | -5 -1 -2 4 &gt; || 0.72 || Breedsma ||   ||   ||
+|| 99/98 || | -1 2 0 -2 1 &gt; || 17.58 || Mothwellsma ||   ||   ||
+|| 121/120 || | -3 -1 -1 0 2 &gt; || 14.37 || Biyatisma ||   ||   ||
+|| 176/175 || | 4 0 -2 -1 1 &gt; || 9.86 || Valinorsma ||   ||   ||
+|| 243/242 || | -1 5 0 0 -2 &gt; || 7.14 || Rastma ||   ||   ||
+|| 385/384 || | -7 -1 1 1 1 &gt; || 4.50 || Keenanisma ||   ||   ||
+|| 441/440 || | -3 2 -1 2 -1 &gt; || 3.93 || Werckisma ||   ||   ||
+|| 540/539 || | 2 3 1 -2 -1 &gt; || 3.21 || Swetisma ||   ||   ||
+|| 3025/3024 || | -4 -3 2 -1 2 &gt; || 0.57 || Lehmerisma ||   ||   ||
 =Modes=
@@ Line 294: / Line 325: @@
 * &lt;span class="wiki_link_ext"&gt;[[http://tonalsoft.com/enc/number/31edo.aspx|Tonalsoft Encyclopedia article]]&lt;/span&gt; [[http://www.webcitation.org/5xeGYj7QU|Permalink]]</pre></div>
 <h4>Original HTML content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;31edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:74:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:74 --&gt;&lt;!-- ws:start:WikiTextTocRule:75: --&gt;&lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:75 --&gt;&lt;!-- ws:start:WikiTextTocRule:76: --&gt;&lt;!-- ws:end:WikiTextTocRule:76 --&gt;&lt;!-- ws:start:WikiTextTocRule:77: --&gt;&lt;!-- ws:end:WikiTextTocRule:77 --&gt;&lt;!-- ws:start:WikiTextTocRule:78: --&gt;&lt;!-- ws:end:WikiTextTocRule:78 --&gt;&lt;!-- ws:start:WikiTextTocRule:79: --&gt;&lt;!-- ws:end:WikiTextTocRule:79 --&gt;&lt;!-- ws:start:WikiTextTocRule:80: --&gt;&lt;!-- ws:end:WikiTextTocRule:80 --&gt;&lt;!-- ws:start:WikiTextTocRule:81: --&gt;&lt;!-- ws:end:WikiTextTocRule:81 --&gt;&lt;!-- ws:start:WikiTextTocRule:82: --&gt;&lt;!-- ws:end:WikiTextTocRule:82 --&gt;&lt;!-- ws:start:WikiTextTocRule:83: --&gt;&lt;!-- ws:end:WikiTextTocRule:83 --&gt;&lt;!-- ws:start:WikiTextTocRule:84: --&gt;&lt;!-- ws:end:WikiTextTocRule:84 --&gt;&lt;!-- ws:start:WikiTextTocRule:85: --&gt;&lt;!-- ws:end:WikiTextTocRule:85 --&gt;&lt;!-- ws:start:WikiTextTocRule:86: --&gt;&lt;!-- ws:end:WikiTextTocRule:86 --&gt;&lt;!-- ws:start:WikiTextTocRule:87: --&gt;&lt;!-- ws:end:WikiTextTocRule:87 --&gt;&lt;!-- ws:start:WikiTextTocRule:88: --&gt;&lt;!-- ws:end:WikiTextTocRule:88 --&gt;&lt;!-- ws:start:WikiTextTocRule:89: --&gt;&lt;!-- ws:end:WikiTextTocRule:89 --&gt;&lt;!-- ws:start:WikiTextTocRule:90: --&gt;&lt;!-- ws:end:WikiTextTocRule:90 --&gt;&lt;!-- ws:start:WikiTextTocRule:91: --&gt;&lt;!-- ws:end:WikiTextTocRule:91 --&gt;&lt;!-- ws:start:WikiTextTocRule:92: --&gt;&lt;!-- ws:end:WikiTextTocRule:92 --&gt;&lt;!-- ws:start:WikiTextTocRule:93: --&gt;&lt;!-- ws:end:WikiTextTocRule:93 --&gt;&lt;!-- ws:start:WikiTextTocRule:94: --&gt;&lt;!-- ws:end:WikiTextTocRule:94 --&gt;&lt;!-- ws:start:WikiTextTocRule:95: --&gt;&lt;!-- ws:end:WikiTextTocRule:95 --&gt;&lt;!-- ws:start:WikiTextTocRule:96: --&gt;&lt;!-- ws:end:WikiTextTocRule:96 --&gt;&lt;!-- ws:start:WikiTextTocRule:97: --&gt;&lt;!-- ws:end:WikiTextTocRule:97 --&gt;&lt;!-- ws:start:WikiTextTocRule:98: --&gt;&lt;!-- ws:end:WikiTextTocRule:98 --&gt;&lt;!-- ws:start:WikiTextTocRule:99: --&gt;&lt;!-- ws:end:WikiTextTocRule:99 --&gt;&lt;!-- ws:start:WikiTextTocRule:100: --&gt;&lt;!-- ws:end:WikiTextTocRule:100 --&gt;&lt;!-- ws:start:WikiTextTocRule:101: --&gt;&lt;!-- ws:end:WikiTextTocRule:101 --&gt;&lt;!-- ws:start:WikiTextTocRule:102: --&gt;&lt;!-- ws:end:WikiTextTocRule:102 --&gt;&lt;!-- ws:start:WikiTextTocRule:103: --&gt;&lt;!-- ws:end:WikiTextTocRule:103 --&gt;&lt;!-- ws:start:WikiTextTocRule:104: --&gt;&lt;!-- ws:end:WikiTextTocRule:104 --&gt;&lt;!-- ws:start:WikiTextTocRule:105: --&gt;&lt;!-- ws:end:WikiTextTocRule:105 --&gt;&lt;!-- ws:start:WikiTextTocRule:106: --&gt; | &lt;a href="#Modes"&gt;Modes&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:106 --&gt;&lt;!-- ws:start:WikiTextTocRule:107: --&gt;&lt;!-- ws:end:WikiTextTocRule:107 --&gt;&lt;!-- ws:start:WikiTextTocRule:108: --&gt; | &lt;a href="#Music in 31-edo"&gt;Music in 31-edo&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:108 --&gt;&lt;!-- ws:start:WikiTextTocRule:109: --&gt;&lt;!-- ws:end:WikiTextTocRule:109 --&gt;&lt;!-- ws:start:WikiTextTocRule:110: --&gt; | &lt;a href="#Practical Theory / Books"&gt;Practical Theory / Books&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:110 --&gt;&lt;!-- ws:start:WikiTextTocRule:111: --&gt; | &lt;a href="#Other Articles"&gt;Other Articles&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:111 --&gt;&lt;!-- ws:start:WikiTextTocRule:112: --&gt;
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;31edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:76:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:76 --&gt;&lt;!-- ws:start:WikiTextTocRule:77: --&gt;&lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:77 --&gt;&lt;!-- ws:start:WikiTextTocRule:78: --&gt;&lt;!-- ws:end:WikiTextTocRule:78 --&gt;&lt;!-- ws:start:WikiTextTocRule:79: --&gt;&lt;!-- ws:end:WikiTextTocRule:79 --&gt;&lt;!-- ws:start:WikiTextTocRule:80: --&gt;&lt;!-- ws:end:WikiTextTocRule:80 --&gt;&lt;!-- ws:start:WikiTextTocRule:81: --&gt;&lt;!-- ws:end:WikiTextTocRule:81 --&gt;&lt;!-- ws:start:WikiTextTocRule:82: --&gt;&lt;!-- ws:end:WikiTextTocRule:82 --&gt;&lt;!-- ws:start:WikiTextTocRule:83: --&gt;&lt;!-- ws:end:WikiTextTocRule:83 --&gt;&lt;!-- ws:start:WikiTextTocRule:84: --&gt;&lt;!-- ws:end:WikiTextTocRule:84 --&gt;&lt;!-- ws:start:WikiTextTocRule:85: --&gt;&lt;!-- ws:end:WikiTextTocRule:85 --&gt;&lt;!-- ws:start:WikiTextTocRule:86: --&gt;&lt;!-- ws:end:WikiTextTocRule:86 --&gt;&lt;!-- ws:start:WikiTextTocRule:87: --&gt;&lt;!-- ws:end:WikiTextTocRule:87 --&gt;&lt;!-- ws:start:WikiTextTocRule:88: --&gt;&lt;!-- ws:end:WikiTextTocRule:88 --&gt;&lt;!-- ws:start:WikiTextTocRule:89: --&gt;&lt;!-- ws:end:WikiTextTocRule:89 --&gt;&lt;!-- ws:start:WikiTextTocRule:90: --&gt;&lt;!-- ws:end:WikiTextTocRule:90 --&gt;&lt;!-- ws:start:WikiTextTocRule:91: --&gt;&lt;!-- ws:end:WikiTextTocRule:91 --&gt;&lt;!-- ws:start:WikiTextTocRule:92: --&gt;&lt;!-- ws:end:WikiTextTocRule:92 --&gt;&lt;!-- ws:start:WikiTextTocRule:93: --&gt;&lt;!-- ws:end:WikiTextTocRule:93 --&gt;&lt;!-- ws:start:WikiTextTocRule:94: --&gt;&lt;!-- ws:end:WikiTextTocRule:94 --&gt;&lt;!-- ws:start:WikiTextTocRule:95: --&gt;&lt;!-- ws:end:WikiTextTocRule:95 --&gt;&lt;!-- ws:start:WikiTextTocRule:96: --&gt;&lt;!-- ws:end:WikiTextTocRule:96 --&gt;&lt;!-- ws:start:WikiTextTocRule:97: --&gt;&lt;!-- ws:end:WikiTextTocRule:97 --&gt;&lt;!-- ws:start:WikiTextTocRule:98: --&gt;&lt;!-- ws:end:WikiTextTocRule:98 --&gt;&lt;!-- ws:start:WikiTextTocRule:99: --&gt;&lt;!-- ws:end:WikiTextTocRule:99 --&gt;&lt;!-- ws:start:WikiTextTocRule:100: --&gt;&lt;!-- ws:end:WikiTextTocRule:100 --&gt;&lt;!-- ws:start:WikiTextTocRule:101: --&gt;&lt;!-- ws:end:WikiTextTocRule:101 --&gt;&lt;!-- ws:start:WikiTextTocRule:102: --&gt;&lt;!-- ws:end:WikiTextTocRule:102 --&gt;&lt;!-- ws:start:WikiTextTocRule:103: --&gt;&lt;!-- ws:end:WikiTextTocRule:103 --&gt;&lt;!-- ws:start:WikiTextTocRule:104: --&gt;&lt;!-- ws:end:WikiTextTocRule:104 --&gt;&lt;!-- ws:start:WikiTextTocRule:105: --&gt;&lt;!-- ws:end:WikiTextTocRule:105 --&gt;&lt;!-- ws:start:WikiTextTocRule:106: --&gt;&lt;!-- ws:end:WikiTextTocRule:106 --&gt;&lt;!-- ws:start:WikiTextTocRule:107: --&gt;&lt;!-- ws:end:WikiTextTocRule:107 --&gt;&lt;!-- ws:start:WikiTextTocRule:108: --&gt; | &lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:108 --&gt;&lt;!-- ws:start:WikiTextTocRule:109: --&gt; | &lt;a href="#Modes"&gt;Modes&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:109 --&gt;&lt;!-- ws:start:WikiTextTocRule:110: --&gt;&lt;!-- ws:end:WikiTextTocRule:110 --&gt;&lt;!-- ws:start:WikiTextTocRule:111: --&gt; | &lt;a href="#Music in 31-edo"&gt;Music in 31-edo&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:111 --&gt;&lt;!-- ws:start:WikiTextTocRule:112: --&gt;&lt;!-- ws:end:WikiTextTocRule:112 --&gt;&lt;!-- ws:start:WikiTextTocRule:113: --&gt; | &lt;a href="#Practical Theory / Books"&gt;Practical Theory / Books&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:113 --&gt;&lt;!-- ws:start:WikiTextTocRule:114: --&gt; | &lt;a href="#Other Articles"&gt;Other Articles&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:114 --&gt;&lt;!-- ws:start:WikiTextTocRule:115: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:112 --&gt;&lt;hr /&gt;
+&lt;!-- ws:end:WikiTextTocRule:115 --&gt;&lt;hr /&gt;
-&lt;em&gt;Thirty-one tone equal temperament&lt;/em&gt;, also called &lt;em&gt;31-tET&lt;/em&gt;, &lt;em&gt;31-EDO&lt;/em&gt;, &lt;em&gt;31-et&lt;/em&gt;, or &lt;em&gt;tricesimoprimal meantone temperament&lt;/em&gt;, is the scale derived by dividing the octave into 31 &lt;a class="wiki_link" href="/equal"&gt;equally&lt;/a&gt; large steps. The term 'Tricesimoprimal' was first used by &lt;a class="wiki_link" href="/Adriaan%20Fokker"&gt;Adriaan Fokker&lt;/a&gt;. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 &lt;a class="wiki_link" href="/cents"&gt;cents&lt;/a&gt;. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth Zeta function integral tuning, &lt;!-- ws:start:WikiTextUrlRule:10842:http://www.research.att.com/~njas/sequences/A117538 --&gt;&lt;a class="wiki_link_ext" href="http://www.research.att.com/~njas/sequences/A117538" rel="nofollow"&gt;http://www.research.att.com/~njas/sequences/A117538&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:10842 --&gt;. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). For JI that uses primes 3 and 7, but no 5, try &lt;a class="wiki_link" href="/36edo"&gt;36edo&lt;/a&gt;.&lt;br /&gt;
+&lt;em&gt;Thirty-one tone equal temperament&lt;/em&gt;, also called &lt;em&gt;31-tET&lt;/em&gt;, &lt;em&gt;31-EDO&lt;/em&gt;, &lt;em&gt;31-et&lt;/em&gt;, or &lt;em&gt;tricesimoprimal meantone temperament&lt;/em&gt;, is the scale derived by dividing the octave into 31 &lt;a class="wiki_link" href="/equal"&gt;equally&lt;/a&gt; large steps. The term 'Tricesimoprimal' was first used by &lt;a class="wiki_link" href="/Adriaan%20Fokker"&gt;Adriaan Fokker&lt;/a&gt;. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 &lt;a class="wiki_link" href="/cents"&gt;cents&lt;/a&gt;. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth Zeta function integral tuning, &lt;!-- ws:start:WikiTextUrlRule:11428:http://www.research.att.com/~njas/sequences/A117538 --&gt;&lt;a class="wiki_link_ext" href="http://www.research.att.com/~njas/sequences/A117538" rel="nofollow"&gt;http://www.research.att.com/~njas/sequences/A117538&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:11428 --&gt;. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). For JI that uses primes 3 and 7, but no 5, try &lt;a class="wiki_link" href="/36edo"&gt;36edo&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 For more encyclopedic info, see &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/31_equal_temperament" rel="nofollow"&gt;Wikipedia's article&lt;/a&gt;.&lt;br /&gt;
@@ Line 7,037: / Line 7,068: @@
   The large tritone.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc31"&gt;&lt;a name="Modes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;Modes&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc31"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;Commas&lt;/h1&gt;
+EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 31 49 72 87 107 115 |.)&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;Comma&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Monzo&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Value (Cents)&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Name 1&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Name 2&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Name 3&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;9931568/9752117&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -25 7 6 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;31.57&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Ampersand's Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;81/80&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -4 4 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;21.51&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Syntonic Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Didymos Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Meantone Comma&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;393216/390625&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 17 1 -8 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;11.45&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Wuerschmidt Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;2109375/2097152&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -21 3 7 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;10.06&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Semicomma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Fokker Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;6719816/6714445&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 38 -2 -15 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1.38&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Hemithirds Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;9859966/9733137&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -10 7 8 -7 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;22.41&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Blackjackisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;64827/64000&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -9 3 -3 4 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;22.23&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Squalentine&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;2430/2401&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 1 5 1 -4 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;20.79&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Nuwell&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;50421/50000&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -4 1 -5 5 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;14.52&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Trimyna&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;126/125&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 1 2 -3 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;13.79&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Septimal Semicomma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Starling Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;1728/1715&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 6 3 -1 -3 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;13.07&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Orwellisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Orwell Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;1029/1024&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -10 1 0 3 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;8.43&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Gamelisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;225/224&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -5 2 2 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7.71&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Septimal Kleisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Marvel Comma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;16875/16807&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 0 3 4 -5 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;6.99&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Mirkwai&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;3136/3125&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 6 0 -5 2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;6.08&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Hemimean&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;6144/6125&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 11 1 -3 -2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;5.36&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Porwell&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;1065875/1063543&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -26 -1 1 9 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;3.79&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Wadisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;65625/65536&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -16 1 5 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;2.35&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Horwell&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;703125/702464&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -11 2 7 -3 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1.63&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Meter&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;2401/2400&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -5 -1 -2 4 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0.72&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Breedsma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;99/98&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -1 2 0 -2 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;17.58&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Mothwellsma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;121/120&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -3 -1 -1 0 2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;14.37&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Biyatisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;176/175&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 4 0 -2 -1 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;9.86&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Valinorsma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;243/242&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -1 5 0 0 -2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7.14&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Rastma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;385/384&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -7 -1 1 1 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;4.50&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Keenanisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;441/440&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -3 2 -1 2 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;3.93&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Werckisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;540/539&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 2 3 1 -2 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;3.21&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Swetisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;3025/3024&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -4 -3 2 -1 2 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0.57&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Lehmerisma&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:64:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc32"&gt;&lt;a name="Modes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:64 --&gt;Modes&lt;/h1&gt;
   &lt;br /&gt;
 A large open list of modes (subsets) from 31edo that people have named: &lt;a class="wiki_link" href="/31edo%20modes"&gt;31edo modes&lt;/a&gt;. &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rothenberg_propriety" rel="nofollow"&gt;Strictly proper&lt;/a&gt; &lt;a class="wiki_link" href="/Strictly%20proper%207-note%2031edo%20scales"&gt;7-note 31edo scales&lt;/a&gt; in the sense of &lt;a class="wiki_link" href="/David%20Rothenberg"&gt;David Rothenberg&lt;/a&gt;. See also &lt;a class="wiki_link" href="/31edo%20MOS%20scales"&gt;31edo MOS scales&lt;/a&gt;. Some of the popular ones:&lt;br /&gt;
@@ Line 7,046: / Line 7,490: @@
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;td colspan="2"&gt;&lt;!-- ws:start:WikiTextHeadingRule:64:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc32"&gt;&lt;a name="Modes---Some 31 tone equal modes:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:64 --&gt;Some 31 tone equal modes:&lt;/h4&gt;
+         &lt;td colspan="2"&gt;&lt;!-- ws:start:WikiTextHeadingRule:66:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc33"&gt;&lt;a name="Modes---Some 31 tone equal modes:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:66 --&gt;Some 31 tone equal modes:&lt;/h4&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 7,562: / Line 8,006: @@
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:66:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc33"&gt;&lt;a name="Music in 31-edo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:66 --&gt;Music in 31-edo&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:68:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc34"&gt;&lt;a name="Music in 31-edo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:68 --&gt;Music in 31-edo&lt;/h1&gt;
   &lt;a class="wiki_link" href="/31-edo%20compositions"&gt;An alphabetical list of Tricesimoprimal Compositions&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:68:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc34"&gt;&lt;a name="Music in 31-edo-Thirty-one tone pedagogy"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:68 --&gt;Thirty-one tone pedagogy&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:70:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc35"&gt;&lt;a name="Music in 31-edo-Thirty-one tone pedagogy"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:70 --&gt;Thirty-one tone pedagogy&lt;/h2&gt;
   The &lt;a class="wiki_link" href="/MicroPedagogyCollective"&gt;MicroPedagogyCollective&lt;/a&gt; is currently at work producing demonstrative material which will encourage and enable more people to learn this system. There have been two &lt;a class="wiki_link" href="/ThirtyOneToneSinginCamp"&gt;ThirtyOneToneSinginCamp&lt;/a&gt;s as well.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:70:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc35"&gt;&lt;a name="Practical Theory / Books"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:70 --&gt;Practical Theory / Books&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:72:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc36"&gt;&lt;a name="Practical Theory / Books"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:72 --&gt;Practical Theory / Books&lt;/h1&gt;
   &lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:7279:&amp;lt;a href=&amp;quot;http://www.ronsword.com/books.html&amp;quot; target=&amp;quot;_blank&amp;quot; rel=&amp;quot;nofollow&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;http://ronsword.com/images/TSG_sm.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 116px; width: 87px;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="http://www.ronsword.com/books.html" target="_blank" rel="nofollow"&gt;&lt;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;" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:7279 --&gt;&lt;a class="wiki_link_ext" href="http://www.ronsword.com/books.html" rel="nofollow" target="_blank"&gt;Sword, Ronald. &amp;quot;Tricesimoprimal Scales for Guitar.&amp;quot; IAAA Press, UK-USA. First Ed: March 2009.&lt;/a&gt; - A comprehensive approach to 31-EDO and all the families associated for Guitar. Features over 300 scale charts / scale examples.&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:7690:&amp;lt;a href=&amp;quot;http://www.ronsword.com/books.html&amp;quot; target=&amp;quot;_blank&amp;quot; rel=&amp;quot;nofollow&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;http://ronsword.com/images/TSG_sm.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 116px; width: 87px;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="http://www.ronsword.com/books.html" target="_blank" rel="nofollow"&gt;&lt;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;" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:7690 --&gt;&lt;a class="wiki_link_ext" href="http://www.ronsword.com/books.html" rel="nofollow" target="_blank"&gt;Sword, Ronald. &amp;quot;Tricesimoprimal Scales for Guitar.&amp;quot; IAAA Press, UK-USA. First Ed: March 2009.&lt;/a&gt; - A comprehensive approach to 31-EDO and all the families associated for Guitar. Features over 300 scale charts / scale examples.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:72:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc36"&gt;&lt;a name="Other Articles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:72 --&gt;Other Articles&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:74:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc37"&gt;&lt;a name="Other Articles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:74 --&gt;Other Articles&lt;/h1&gt;
   &lt;ul&gt;&lt;li&gt;&lt;span class="wiki_link_ext"&gt;&lt;a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/beerart.html" rel="nofollow"&gt;de Beer, Anton, ''The Development of 31-tone Music&lt;/a&gt;&lt;/span&gt; &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xeFzBM9b" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;span class="wiki_link_ext"&gt;&lt;a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/fokkerorg.html" rel="nofollow"&gt;Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ&lt;/a&gt;&lt;/span&gt; &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xeG6Tmli" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;Fokker, A.D., &amp;quot;New Music with 31 Notes&amp;quot; translated by Leigh Gerdine&lt;/li&gt;&lt;li&gt;&lt;span class="wiki_link_ext"&gt;&lt;a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/rap31.html" rel="nofollow"&gt;Rapoport, Paul, ''About 31-tone Equal Temperament&lt;/a&gt;&lt;/span&gt; &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xeGH4uBH" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;span class="wiki_link_ext"&gt;&lt;a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/terp31.html" rel="nofollow"&gt;Terpstra, Siemen, ''Toward a Theory of Meantone (and 31-et) Harmony''&lt;/a&gt;&lt;/span&gt; &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xeGMeCMd" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;span class="wiki_link_ext"&gt;&lt;a class="wiki_link_ext" href="http://tonalsoft.com/enc/number/31edo.aspx" rel="nofollow"&gt;Tonalsoft Encyclopedia article&lt;/a&gt;&lt;/span&gt; &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xeGYj7QU" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>