36edo: 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:<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-05-~~19 15~~:15:50 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-05-28 23:05:42 UTC</tt>.<br>

: The original revision id was <tt>~~230097516~~</tt>.<br>

: The original revision id was <tt>232648600</tt>.<br>

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

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>

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=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 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 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). The 2.3.7 subgroup can be extended to 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].

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 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 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). The 2.3.7 subgroup can be extended to the [[k*N subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].

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.

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<h1 id="toc0"><a name="As a harmonic temperament"></a>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 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 <a class="wiki_link" href="/just%20intonation%20subgroup">just intonation subgroup</a>, 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). The 2.3.7 subgroup can be extended to 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as <a class="wiki_link" href="/72edo">72edo</a> does in the full <a class="wiki_link" href="/17-limit">17-limit</a>.<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 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 <a class="wiki_link" href="/just%20intonation%20subgroup">just intonation subgroup</a>, 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). The 2.3.7 subgroup can be extended to the <a class="wiki_link" href="/k%2AN%20subgroups">2*36 subgroup</a> 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as <a class="wiki_link" href="/72edo">72edo</a> does in the full <a class="wiki_link" href="/17-limit">17-limit</a>.<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 />

@@ 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>2011-05-19 15:15:50 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-05-28 23:05:42 UTC</tt>.<br>
-: The original revision id was <tt>230097516</tt>.<br>
+: The original revision id was <tt>232648600</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 16: / Line 16: @@
 =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 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 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). The 2.3.7 subgroup can be extended to 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].
+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 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 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). The 2.3.7 subgroup can be extended to the [[k*N subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].
 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.
@@ Line 76: / Line 76: @@
 &lt;!-- ws:start:WikiTextHeadingRule:1:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="As a harmonic temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:1 --&gt;As a harmonic temperament&lt;/h1&gt;
   &lt;br /&gt;
-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 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 &lt;a class="wiki_link" href="/just%20intonation%20subgroup"&gt;just intonation subgroup&lt;/a&gt;, 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_diesis" rel="nofollow"&gt;Slendro diesis&lt;/a&gt; of around 36 cents, and as 64:63, the so-called &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_comma" rel="nofollow"&gt;septimal comma&lt;/a&gt; of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_third-tone" rel="nofollow"&gt;Septimal third-tone&lt;/a&gt; (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as &lt;a class="wiki_link" href="/72edo"&gt;72edo&lt;/a&gt; does in the full &lt;a class="wiki_link" href="/17-limit"&gt;17-limit&lt;/a&gt;.&lt;br /&gt;
+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 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 &lt;a class="wiki_link" href="/just%20intonation%20subgroup"&gt;just intonation subgroup&lt;/a&gt;, 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_diesis" rel="nofollow"&gt;Slendro diesis&lt;/a&gt; of around 36 cents, and as 64:63, the so-called &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_comma" rel="nofollow"&gt;septimal comma&lt;/a&gt; of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_third-tone" rel="nofollow"&gt;Septimal third-tone&lt;/a&gt; (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the &lt;a class="wiki_link" href="/k%2AN%20subgroups"&gt;2*36 subgroup&lt;/a&gt; 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as &lt;a class="wiki_link" href="/72edo"&gt;72edo&lt;/a&gt; does in the full &lt;a class="wiki_link" href="/17-limit"&gt;17-limit&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 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 &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for the rank four temperament tempering out 56/55, as well as the rank three temperament &lt;a class="wiki_link" href="/Didymus%20rank%20three%20family"&gt;melpomene&lt;/a&gt; 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.&lt;br /&gt;