36edo

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36edo, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33.333... cents.

36 is a highly composite number, factoring into 2x2x3x3. Since 36 is divisible by 12, it contains the overly-familiar [[12edo]] as a subset. It divides 12edo's 100-cent half step into three microtonal step of approximately 33 cents, which could be called "sixth tones." 36edo also contains [[18edo]] ("third tones") and [[9edo]] ("two-thirds tones") as subsets, not to mention the [[6edo]] whole tone scale, [[4edo]] full-diminished seventh chord, and the [[3edo]] augmented triad, all of which are present in 12edo.

That 36edo contains 12edo as a subset makes in compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33 cents, one can arrive at a 24-tone subset of 36 edo (see, for instance, Jacob Barton's piece for two clarinets, [[http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/|De-quinin']]). Three 12edo instruments could play the entire gamut.

=As a harmonic temperament= 

For those interested in approximations to just intonation, 36edo offers no improvement over 12edo in the 5-limit, since its nearest approximation to 5:4 is the overly-familiar 400-cent sharp third. However, it excels at approximations involving 3 & 7. As a 3 & 7 tuning, 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [[http://en.wikipedia.org/wiki/Septimal_diesis|Slendro diesis]] of around 36 cents, and as 64:63, the so-called [[http://en.wikipedia.org/wiki/Septimal_comma|septimal comma]] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [[http://en.wikipedia.org/wiki/Septimal_third-tone|Septimal third-tone]] (which = 49:48 x 64:63).

36 tempers out, like 12, 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. In the 11-limit, it tempers out 56/55, 245/242 and 540/539, and is the [[optimal patent val]] for the rank four temperament tempering out 56/55, as well as the rank three temperament [[Didymus rank three family|melpomene]] tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95.

As a 5-limit temperament, the patent val for 36edo is [[Wedgies and Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is <36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" |29 0 -9> is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val. 

Heinz Bohlen proposed it as a suitable temperament for approximating his 833-cents scale.

=Approximations=
==3-limit (Pythagorean) approximations (same as 12edo):== 

3/2 = 701.955... cents; 21 degrees of 36edo = 700 cents.
4/3 = 498.045... cents; 15 degrees of 36edo = 500 cents.
9/8 = 203.910... cents; 6 degrees of 36edo = 200 cents.
16/9 = 996.090... cents; 30 degrees of 36edo = 1000 cents.
27/16 = 905.865... cents; 27 degrees of 36edo = 900 cents.
32/27 = 294.135... cents; 9 degrees of 36edo = 300 cents.
81/64 = 407.820... cents; 12 degrees of 36edo = 400 cents.
128/81 = 792.180... cents; 24 degrees of 36edo = 800 cents.


==7-limit approximations:== 

===7 only:=== 
7/4 = 968.826... cents; 29 degrees of 36edo = 966.666... cents.
8/7 = 231.174... cents; 7 degrees of 36edo = 233.333... cents.
49/32 = 737.652... cents; 22 degrees of 36edo = 733.333... cents.
64/49 = 462.348... cents; 14 degrees of 36edo = 466.666... cents.

===7 & 3:=== 
7/6 = 266.871... cents; 8 degrees of 36edo = 266.666... cents.
12/7 = 933.129... cents; 28 degrees of 36 = 933.333... cents.
9/7 = 435.084... cents; 13 degrees of 36edo = 433.333... cents.
14/9 = 764.916... cents; 23 degrees of 36edo = 766.666... cents.
28/27 = 62.961... cents; 2 degrees of 36edo = 66.666... cents.
27/14 = 1137.039... cents; 34 degrees of 36edo = 1133.333... cents.
21/16 = 470.781... cents; 14 degrees of 36edo = 466.666... cents.
32/21 = 729.219... cents; 22 degrees of 36edo = 733.333... cents.
49/48 = 35.697... cents; 1 degree of 36edo = 33.333... cents.
96/49 = 1164.303... cents; 35 degrees of 36edo = 1166.666... cents.
49/36 = 533.742... cents; 16 degrees of 36edo = 533.333... cents.
72/49 = 666.258... cents; 20 degrees of 36edo = 666.666... cents.
64/63 = 27.264... cents; 1 degree of 36edo = 33.333... cents.
63/32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents.

=Music=
* [[http://micro.soonlabel.com/gene_ward_smith/36edo/something.mp3|Something]] by Herman Klein
* [[http://micro.soonlabel.com/gene_ward_smith/36edo/hay.mp3|Hay]] by Joe Hayseed

Original HTML content:

<html><head><title>36edo</title></head><body><!-- ws:start:WikiTextTocRule:14:&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:14 --><!-- ws:start:WikiTextTocRule:15: --><a href="#As a harmonic temperament">As a harmonic temperament</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#Approximations">Approximations</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: -->
<!-- ws:end:WikiTextTocRule:22 --><br />
<br />
36edo, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33.333... cents.<br />
<br />
36 is a highly composite number, factoring into 2x2x3x3. Since 36 is divisible by 12, it contains the overly-familiar <a class="wiki_link" href="/12edo">12edo</a> as a subset. It divides 12edo's 100-cent half step into three microtonal step of approximately 33 cents, which could be called &quot;sixth tones.&quot; 36edo also contains <a class="wiki_link" href="/18edo">18edo</a> (&quot;third tones&quot;) and <a class="wiki_link" href="/9edo">9edo</a> (&quot;two-thirds tones&quot;) as subsets, not to mention the <a class="wiki_link" href="/6edo">6edo</a> whole tone scale, <a class="wiki_link" href="/4edo">4edo</a> full-diminished seventh chord, and the <a class="wiki_link" href="/3edo">3edo</a> augmented triad, all of which are present in 12edo.<br />
<br />
That 36edo contains 12edo as a subset makes in compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33 cents, one can arrive at a 24-tone subset of 36 edo (see, for instance, Jacob Barton's piece for two clarinets, <a class="wiki_link_ext" href="http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/" rel="nofollow">De-quinin'</a>). Three 12edo instruments could play the entire gamut.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="As a harmonic temperament"></a><!-- ws:end:WikiTextHeadingRule:0 -->As a harmonic temperament</h1>
 <br />
For those interested in approximations to just intonation, 36edo offers no improvement over 12edo in the 5-limit, since its nearest approximation to 5:4 is the overly-familiar 400-cent sharp third. However, it excels at approximations involving 3 &amp; 7. As a 3 &amp; 7 tuning, 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_diesis" rel="nofollow">Slendro diesis</a> of around 36 cents, and as 64:63, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_comma" rel="nofollow">septimal comma</a> of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_third-tone" rel="nofollow">Septimal third-tone</a> (which = 49:48 x 64:63).<br />
<br />
36 tempers out, like 12, 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. In the 11-limit, it tempers out 56/55, 245/242 and 540/539, and is the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for the rank four temperament tempering out 56/55, as well as the rank three temperament <a class="wiki_link" href="/Didymus%20rank%20three%20family">melpomene</a> tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95.<br />
<br />
As a 5-limit temperament, the patent val for 36edo is <a class="wiki_link" href="/Wedgies%20and%20Multivals">contorted</a>, meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is &lt;36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the &quot;comma&quot; |29 0 -9&gt; is also tempered out, and the &quot;fifth&quot;, 29\36, is actually approximately 7/4, whereas the &quot;major third&quot;, 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a <a class="wiki_link" href="/transversal">transversal</a> for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val. <br />
<br />
Heinz Bohlen proposed it as a suitable temperament for approximating his 833-cents scale.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Approximations"></a><!-- ws:end:WikiTextHeadingRule:2 -->Approximations</h1>
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Approximations-3-limit (Pythagorean) approximations (same as 12edo):"></a><!-- ws:end:WikiTextHeadingRule:4 -->3-limit (Pythagorean) approximations (same as 12edo):</h2>
 <br />
3/2 = 701.955... cents; 21 degrees of 36edo = 700 cents.<br />
4/3 = 498.045... cents; 15 degrees of 36edo = 500 cents.<br />
9/8 = 203.910... cents; 6 degrees of 36edo = 200 cents.<br />
16/9 = 996.090... cents; 30 degrees of 36edo = 1000 cents.<br />
27/16 = 905.865... cents; 27 degrees of 36edo = 900 cents.<br />
32/27 = 294.135... cents; 9 degrees of 36edo = 300 cents.<br />
81/64 = 407.820... cents; 12 degrees of 36edo = 400 cents.<br />
128/81 = 792.180... cents; 24 degrees of 36edo = 800 cents.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Approximations-7-limit approximations:"></a><!-- ws:end:WikiTextHeadingRule:6 -->7-limit approximations:</h2>
 <br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Approximations-7-limit approximations:-7 only:"></a><!-- ws:end:WikiTextHeadingRule:8 -->7 only:</h3>
 7/4 = 968.826... cents; 29 degrees of 36edo = 966.666... cents.<br />
8/7 = 231.174... cents; 7 degrees of 36edo = 233.333... cents.<br />
49/32 = 737.652... cents; 22 degrees of 36edo = 733.333... cents.<br />
64/49 = 462.348... cents; 14 degrees of 36edo = 466.666... cents.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Approximations-7-limit approximations:-7 &amp; 3:"></a><!-- ws:end:WikiTextHeadingRule:10 -->7 &amp; 3:</h3>
 7/6 = 266.871... cents; 8 degrees of 36edo = 266.666... cents.<br />
12/7 = 933.129... cents; 28 degrees of 36 = 933.333... cents.<br />
9/7 = 435.084... cents; 13 degrees of 36edo = 433.333... cents.<br />
14/9 = 764.916... cents; 23 degrees of 36edo = 766.666... cents.<br />
28/27 = 62.961... cents; 2 degrees of 36edo = 66.666... cents.<br />
27/14 = 1137.039... cents; 34 degrees of 36edo = 1133.333... cents.<br />
21/16 = 470.781... cents; 14 degrees of 36edo = 466.666... cents.<br />
32/21 = 729.219... cents; 22 degrees of 36edo = 733.333... cents.<br />
49/48 = 35.697... cents; 1 degree of 36edo = 33.333... cents.<br />
96/49 = 1164.303... cents; 35 degrees of 36edo = 1166.666... cents.<br />
49/36 = 533.742... cents; 16 degrees of 36edo = 533.333... cents.<br />
72/49 = 666.258... cents; 20 degrees of 36edo = 666.666... cents.<br />
64/63 = 27.264... cents; 1 degree of 36edo = 33.333... cents.<br />
63/32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:12 -->Music</h1>
<ul><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/36edo/something.mp3" rel="nofollow">Something</a> by Herman Klein</li><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/36edo/hay.mp3" rel="nofollow">Hay</a> by Joe Hayseed</li></ul></body></html>