31edo

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The revision comment was: 31EDO interval naming sheme added. Broken link removed.

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In music, '''31 equal temperament''', called 31-tet, 31-edo, 31-et, or tricesimoprimal temperament, is the scale derived by dividing the octave into 31 equally large steps. Each step represents a frequency ratio of the 31st root of 2, or 38.71 cents.

Nicola Vicentino had very nearly produced 31-equal temperament on his extended keyboard instrument, the Archicembalo, in 1555, but it was not until 1666 that Lemme Rossi first proposed it theoretically. Shortly thereafter, having discovered it independently, famed scientist Christiaan Huygens wrote about it also. Since the standard system of tuning at that time was quarter-comma meantone, in which the fifth is tuned to the fourth root of 5, the appeal of this method is immediate, as the fifth of 31-et, at 696.77 cents, is only a fifth of a cent sharper than the fifth of quarter-comma meantone. Huygens not only realized that, he went farther and noted that 31-et provides an excellent approximation of septimal, or 7-limit harmony, which was a very advanced insight for the time. In the twentieth century, physicist, music theorist and composer Adriaan Fokker, after reading Huygens's work, led a revival of interest in this system of tuning which led to a number of compositions, particularly by Dutch composers.

=Theoretical properties= 

The single most important fact about 31-et is that it equates to the unison, or ''tempers out'', the syntonic comma of 81/80. It is therefore a meantone temperament. It also tempers the 5-limit intervals 393216/390625, known as the Würschmidt comma after music theorist José Würschmidt, and 2109375/2097152, known as the semicomma.

More significantly, perhaps, it tempers out 126/125, the septimal semicomma, or starling comma. Because it tempers out both 81/80 and 126/125, it supports septimal meantone temperament. It also tempers out 1029/1024, the gamelan residue, and 1728/1715, the orwell comma. Consequently it supports a wide variety of [[http://xenharmonic.wikispaces.com/Regular+Temperaments#lineartemperaments|linear temperaments]].

==Chords of 31 equal temperament== 

Many of the most interesting chords of 31-et are those of septimal meantone temperament. Others are the neutral thirds triad, which might be written either C-Dx-G or C-Fbb-G, and the orwell tetrad, which is C-E-Fx-Bbb.

=Scales= 

[[shortspeak|31EDO interval naming scheme]]

==Septatonic tetrachordal scales== 
[[31edo Tetra 5-3-5]] [[31edo Tetra 5-4-4]] [[31edo Tetra 5-5-3]] [[31edo Tetra 5-6-2]] [[31edo Tetra 5-7-1]]
[[31edo Tetra 6-1-6]] [[31edo Tetra 6-2-5]] [[31edo Tetra 6-3-4]] [[31edo Tetra 6-4-3]] [[31edo Tetra 6-5-2]] [[31edo Tetra 6-6-1]]
[[31edo Tetra 7-1-5]] [[31edo Tetra 7-2-4]] [[31edo Tetra 7-3-3]] [[31edo Tetra 7-4-2]] [[31edo Tetra 7-5-1]]
[[31edo Tetra 8-1-4]] [[31edo Tetra 8-2-3]] [[31edo Tetra 8-4-1]]
[[31edo Tetra 9-1-3]] [[31edo Tetra 9-2-2]] [[31edo Tetra 9-3-1]]
[[31edo Tetra 10-1-2]] [[31edo Tetra 10-2-1]] [[31edo Tetra 11-1-1]]

=External links= 
* [[[http://www.xs4all.nl/%7Ehuygensf/english/index.html|http://www.xs4all.nl/~huygensf/english/index.html]] de Beer, Anton, ''The Development of 31-tone Music'']
* [[[http://www.xs4all.nl/%7Ehuygensf/doc/fokkerorg.html|http://www.xs4all.nl/~huygensf/doc/fokkerorg.html]] Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ'']
* [[[http://www.xs4all.nl/%7Ehuygensf/doc/rap31.html|http://www.xs4all.nl/~huygensf/doc/rap31.html]] Rapoport, Paul, ''About 31-tone Equal Temperament'']
* [[[http://www.xs4all.nl/%7Ehuygensf/doc/terp31.html|http://www.xs4all.nl/~huygensf/doc/terp31.html]] Terpstra, Siemen, ''Toward a Theory of Meantone (and 31-et) Harmony'']
* [http://tonalsoft.com/enc/number/31edo.aspx Tonalsoft Encyclopedia of Microtonal Music-theory]

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<html><head><title>31edo</title></head><body><!-- ws:start:WikiTextTocRule:10:&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:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Theoretical properties">Theoretical properties</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#Scales">Scales</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#External links">External links</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: -->
<!-- ws:end:WikiTextTocRule:16 --><br />
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In music, '''31 equal temperament''', called 31-tet, 31-edo, 31-et, or tricesimoprimal temperament, is the scale derived by dividing the octave into 31 equally large steps. Each step represents a frequency ratio of the 31st root of 2, or 38.71 cents.<br />
<br />
Nicola Vicentino had very nearly produced 31-equal temperament on his extended keyboard instrument, the Archicembalo, in 1555, but it was not until 1666 that Lemme Rossi first proposed it theoretically. Shortly thereafter, having discovered it independently, famed scientist Christiaan Huygens wrote about it also. Since the standard system of tuning at that time was quarter-comma meantone, in which the fifth is tuned to the fourth root of 5, the appeal of this method is immediate, as the fifth of 31-et, at 696.77 cents, is only a fifth of a cent sharper than the fifth of quarter-comma meantone. Huygens not only realized that, he went farther and noted that 31-et provides an excellent approximation of septimal, or 7-limit harmony, which was a very advanced insight for the time. In the twentieth century, physicist, music theorist and composer Adriaan Fokker, after reading Huygens's work, led a revival of interest in this system of tuning which led to a number of compositions, particularly by Dutch composers.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Theoretical properties"></a><!-- ws:end:WikiTextHeadingRule:0 -->Theoretical properties</h1>
 <br />
The single most important fact about 31-et is that it equates to the unison, or ''tempers out'', the syntonic comma of 81/80. It is therefore a meantone temperament. It also tempers the 5-limit intervals 393216/390625, known as the Würschmidt comma after music theorist José Würschmidt, and 2109375/2097152, known as the semicomma.<br />
<br />
More significantly, perhaps, it tempers out 126/125, the septimal semicomma, or starling comma. Because it tempers out both 81/80 and 126/125, it supports septimal meantone temperament. It also tempers out 1029/1024, the gamelan residue, and 1728/1715, the orwell comma. Consequently it supports a wide variety of <a href="http://xenharmonic.wikispaces.com/Regular+Temperaments#lineartemperaments">linear temperaments</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Theoretical properties-Chords of 31 equal temperament"></a><!-- ws:end:WikiTextHeadingRule:2 -->Chords of 31 equal temperament</h2>
 <br />
Many of the most interesting chords of 31-et are those of septimal meantone temperament. Others are the neutral thirds triad, which might be written either C-Dx-G or C-Fbb-G, and the orwell tetrad, which is C-E-Fx-Bbb.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:4 -->Scales</h1>
 <br />
<a class="wiki_link" href="/shortspeak">31EDO interval naming scheme</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Scales-Septatonic tetrachordal scales"></a><!-- ws:end:WikiTextHeadingRule:6 -->Septatonic tetrachordal scales</h2>
 <a class="wiki_link" href="/31edo%20Tetra%205-3-5">31edo Tetra 5-3-5</a> <a class="wiki_link" href="/31edo%20Tetra%205-4-4">31edo Tetra 5-4-4</a> <a class="wiki_link" href="/31edo%20Tetra%205-5-3">31edo Tetra 5-5-3</a> <a class="wiki_link" href="/31edo%20Tetra%205-6-2">31edo Tetra 5-6-2</a> <a class="wiki_link" href="/31edo%20Tetra%205-7-1">31edo Tetra 5-7-1</a><br />
<a class="wiki_link" href="/31edo%20Tetra%206-1-6">31edo Tetra 6-1-6</a> <a class="wiki_link" href="/31edo%20Tetra%206-2-5">31edo Tetra 6-2-5</a> <a class="wiki_link" href="/31edo%20Tetra%206-3-4">31edo Tetra 6-3-4</a> <a class="wiki_link" href="/31edo%20Tetra%206-4-3">31edo Tetra 6-4-3</a> <a class="wiki_link" href="/31edo%20Tetra%206-5-2">31edo Tetra 6-5-2</a> <a class="wiki_link" href="/31edo%20Tetra%206-6-1">31edo Tetra 6-6-1</a><br />
<a class="wiki_link" href="/31edo%20Tetra%207-1-5">31edo Tetra 7-1-5</a> <a class="wiki_link" href="/31edo%20Tetra%207-2-4">31edo Tetra 7-2-4</a> <a class="wiki_link" href="/31edo%20Tetra%207-3-3">31edo Tetra 7-3-3</a> <a class="wiki_link" href="/31edo%20Tetra%207-4-2">31edo Tetra 7-4-2</a> <a class="wiki_link" href="/31edo%20Tetra%207-5-1">31edo Tetra 7-5-1</a><br />
<a class="wiki_link" href="/31edo%20Tetra%208-1-4">31edo Tetra 8-1-4</a> <a class="wiki_link" href="/31edo%20Tetra%208-2-3">31edo Tetra 8-2-3</a> <a class="wiki_link" href="/31edo%20Tetra%208-4-1">31edo Tetra 8-4-1</a><br />
<a class="wiki_link" href="/31edo%20Tetra%209-1-3">31edo Tetra 9-1-3</a> <a class="wiki_link" href="/31edo%20Tetra%209-2-2">31edo Tetra 9-2-2</a> <a class="wiki_link" href="/31edo%20Tetra%209-3-1">31edo Tetra 9-3-1</a><br />
<a class="wiki_link" href="/31edo%20Tetra%2010-1-2">31edo Tetra 10-1-2</a> <a class="wiki_link" href="/31edo%20Tetra%2010-2-1">31edo Tetra 10-2-1</a> <a class="wiki_link" href="/31edo%20Tetra%2011-1-1">31edo Tetra 11-1-1</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="External links"></a><!-- ws:end:WikiTextHeadingRule:8 -->External links</h1>
 <ul><li>[<a class="wiki_link_ext" href="http://www.xs4all.nl/%7Ehuygensf/english/index.html" rel="nofollow">http://www.xs4all.nl/~huygensf/english/index.html</a> de Beer, Anton, ''The Development of 31-tone Music'']</li><li>[<a class="wiki_link_ext" href="http://www.xs4all.nl/%7Ehuygensf/doc/fokkerorg.html" rel="nofollow">http://www.xs4all.nl/~huygensf/doc/fokkerorg.html</a> Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ'']</li><li>[<a class="wiki_link_ext" href="http://www.xs4all.nl/%7Ehuygensf/doc/rap31.html" rel="nofollow">http://www.xs4all.nl/~huygensf/doc/rap31.html</a> Rapoport, Paul, ''About 31-tone Equal Temperament'']</li><li>[<a class="wiki_link_ext" href="http://www.xs4all.nl/%7Ehuygensf/doc/terp31.html" rel="nofollow">http://www.xs4all.nl/~huygensf/doc/terp31.html</a> Terpstra, Siemen, ''Toward a Theory of Meantone (and 31-et) Harmony'']</li><li>[<!-- ws:start:WikiTextUrlRule:85: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:85 --> Tonalsoft Encyclopedia of Microtonal Music-theory]</li></ul></body></html>