48edo: Difference between revisions

Revision as of 08:31, 16 July 2014

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[[toc|flat]]

=Basics= 
The //48 equal division// divides the [[octave]] into 48 1/8th tones of precisely 25 [[cent]]s each. Since 48 is a multiple of 12, it has attracted a small amount of interest. However, its best major third, of 375 cents, is over 11 cents flat. An alternative third is the familiar 400 cent major third. Using this third, 48 tunes to the same values as 12 in the [[5-limit]], but [[tempering out|tempers out]] [[2401_2400|2401/2400]] in the [[7-limit]], making it a tuning for [[Meantone family|squares temperament]]. In the [[11-limit]] we can add [[99_98|99/98]] and [[121_120|121/120]] to the list, and in the [[13-limit]], [[66_65|66/65]]. While [[31edo]] can also do 13-limit squares, 48 might be preferred for some purposes.

=Temperaments= 
Using its best major third, 48 tempers out 20000/19683, but [[34edo]] does a much better job for this temperament, known as [[Tetracot family|tetracot]]. However in the 7-limit it can be used for [[Jubilismic clan|doublewide temperament]], the 1/2 octave period temperament with minor third generator tempering out 50/49 and 875/864, for which it is the [[optimal patent val]]. In the 11-limit, we may add 99/98, leading to 11-limit doublewide for which 48 again gives the optimal patent val. It is also the optimal patent val for the rank three temperament [[Jubilismic family|jubilee]], which tempers out 50/49 and 99/98.

If 48 is treated as a no-fives system, it still tempers out 99/98 and 243/242 in the 11-limit, leading to a no-fives version of squares for which it does well as a tuning. In the 13 no-fives limit, we can add 144/143 to the list of commas, and we get the no-fives version of 13-limit squares, for which 48 actually defines the [[optimal patent val]]. No-fives squares should probably be considered by anyone interested in 48edo; the generator is 17\48, a 425 cent interval serving as both [[9_7|9/7]] and [[14_11|14/11]].

Something close to 48edo is what you get if you cross 16edo with pure fifths, for instance, on a 16-tone guitar.

=Compositions= 
[[http://archive.org/download/Quincunx/Quincunx.mp3|Quincunx]] by Jon Lyle Smith
[[http://www.seraph.it/blog_files/a0aafbec9519cfe9600e7a82118da2ee-26.html|Two At Once]] [[http://www.seraph.it/dep/det/twoatonce.mp3|play]] by [[Carlo Serafini]]
[[http://www.seraph.it/blog_files/a8435fb03236157e8a60b047e9892594-27.html|Tim's Flutes]] [[http://www.seraph.it/dep/det/Tim%27sFlutes.mp3|play]] by [[Carlo Serafini]]

Original HTML content:

<html><head><title>48edo</title></head><body><!-- ws:start:WikiTextTocRule:6:&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:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Basics">Basics</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Temperaments">Temperaments</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#Compositions">Compositions</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: -->
<!-- ws:end:WikiTextTocRule:10 --><br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:0 -->Basics</h1>
 The <em>48 equal division</em> divides the <a class="wiki_link" href="/octave">octave</a> into 48 1/8th tones of precisely 25 <a class="wiki_link" href="/cent">cent</a>s each. Since 48 is a multiple of 12, it has attracted a small amount of interest. However, its best major third, of 375 cents, is over 11 cents flat. An alternative third is the familiar 400 cent major third. Using this third, 48 tunes to the same values as 12 in the <a class="wiki_link" href="/5-limit">5-limit</a>, but <a class="wiki_link" href="/tempering%20out">tempers out</a> <a class="wiki_link" href="/2401_2400">2401/2400</a> in the <a class="wiki_link" href="/7-limit">7-limit</a>, making it a tuning for <a class="wiki_link" href="/Meantone%20family">squares temperament</a>. In the <a class="wiki_link" href="/11-limit">11-limit</a> we can add <a class="wiki_link" href="/99_98">99/98</a> and <a class="wiki_link" href="/121_120">121/120</a> to the list, and in the <a class="wiki_link" href="/13-limit">13-limit</a>, <a class="wiki_link" href="/66_65">66/65</a>. While <a class="wiki_link" href="/31edo">31edo</a> can also do 13-limit squares, 48 might be preferred for some purposes.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Temperaments"></a><!-- ws:end:WikiTextHeadingRule:2 -->Temperaments</h1>
 Using its best major third, 48 tempers out 20000/19683, but <a class="wiki_link" href="/34edo">34edo</a> does a much better job for this temperament, known as <a class="wiki_link" href="/Tetracot%20family">tetracot</a>. However in the 7-limit it can be used for <a class="wiki_link" href="/Jubilismic%20clan">doublewide temperament</a>, the 1/2 octave period temperament with minor third generator tempering out 50/49 and 875/864, for which it is the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a>. In the 11-limit, we may add 99/98, leading to 11-limit doublewide for which 48 again gives the optimal patent val. It is also the optimal patent val for the rank three temperament <a class="wiki_link" href="/Jubilismic%20family">jubilee</a>, which tempers out 50/49 and 99/98.<br />
<br />
If 48 is treated as a no-fives system, it still tempers out 99/98 and 243/242 in the 11-limit, leading to a no-fives version of squares for which it does well as a tuning. In the 13 no-fives limit, we can add 144/143 to the list of commas, and we get the no-fives version of 13-limit squares, for which 48 actually defines the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a>. No-fives squares should probably be considered by anyone interested in 48edo; the generator is 17\48, a 425 cent interval serving as both <a class="wiki_link" href="/9_7">9/7</a> and <a class="wiki_link" href="/14_11">14/11</a>.<br />
<br />
Something close to 48edo is what you get if you cross 16edo with pure fifths, for instance, on a 16-tone guitar.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:4 -->Compositions</h1>
 <a class="wiki_link_ext" href="http://archive.org/download/Quincunx/Quincunx.mp3" rel="nofollow">Quincunx</a> by Jon Lyle Smith<br />
<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/a0aafbec9519cfe9600e7a82118da2ee-26.html" rel="nofollow">Two At Once</a> <a class="wiki_link_ext" href="http://www.seraph.it/dep/det/twoatonce.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a><br />
<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/a8435fb03236157e8a60b047e9892594-27.html" rel="nofollow">Tim's Flutes</a> <a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Tim%27sFlutes.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a></body></html>

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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>2012-05-29 16:34:59 UTC</tt>.<br>
+: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2014-07-16 08:31:51 UTC</tt>.<br>
-: The original revision id was <tt>340664616</tt>.<br>
+: The original revision id was <tt>516359276</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 8: / Line 8: @@
 <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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]
+=Basics=
-=Basics=
 The //48 equal division// divides the [[octave]] into 48 1/8th tones of precisely 25 [[cent]]s each. Since 48 is a multiple of 12, it has attracted a small amount of interest. However, its best major third, of 375 cents, is over 11 cents flat. An alternative third is the familiar 400 cent major third. Using this third, 48 tunes to the same values as 12 in the [[5-limit]], but [[tempering out|tempers out]] [[2401_2400|2401/2400]] in the [[7-limit]], making it a tuning for [[Meantone family|squares temperament]]. In the [[11-limit]] we can add [[99_98|99/98]] and [[121_120|121/120]] to the list, and in the [[13-limit]], [[66_65|66/65]]. While [[31edo]] can also do 13-limit squares, 48 might be preferred for some purposes.
 =Temperaments=
 Using its best major third, 48 tempers out 20000/19683, but [[34edo]] does a much better job for this temperament, known as [[Tetracot family|tetracot]]. However in the 7-limit it can be used for [[Jubilismic clan|doublewide temperament]], the 1/2 octave period temperament with minor third generator tempering out 50/49 and 875/864, for which it is the [[optimal patent val]]. In the 11-limit, we may add 99/98, leading to 11-limit doublewide for which 48 again gives the optimal patent val. It is also the optimal patent val for the rank three temperament [[Jubilismic family|jubilee]], which tempers out 50/49 and 99/98.
 If 48 is treated as a no-fives system, it still tempers out 99/98 and 243/242 in the 11-limit, leading to a no-fives version of squares for which it does well as a tuning. In the 13 no-fives limit, we can add 144/143 to the list of commas, and we get the no-fives version of 13-limit squares, for which 48 actually defines the [[optimal patent val]]. No-fives squares should probably be considered by anyone interested in 48edo; the generator is 17\48, a 425 cent interval serving as both [[9_7|9/7]] and [[14_11|14/11]].
+Something close to 48edo is what you get if you cross 16edo with pure fifths, for instance, on a 16-tone guitar.
 =Compositions=
@@ Line 24: / Line 25: @@
 <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;48edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:6:&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:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;&lt;a href="#Basics"&gt;Basics&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;!-- ws:start:WikiTextTocRule:8: --&gt; | &lt;a href="#Temperaments"&gt;Temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt; | &lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt;
 &lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;br /&gt;
-&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Basics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Basics&lt;/h1&gt;
-The &lt;em&gt;48 equal division&lt;/em&gt; divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; into 48 1/8th tones of precisely 25 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s each. Since 48 is a multiple of 12, it has attracted a small amount of interest. However, its best major third, of 375 cents, is over 11 cents flat. An alternative third is the familiar 400 cent major third. Using this third, 48 tunes to the same values as 12 in the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;, but &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; &lt;a class="wiki_link" href="/2401_2400"&gt;2401/2400&lt;/a&gt; in the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt;, making it a tuning for &lt;a class="wiki_link" href="/Meantone%20family"&gt;squares temperament&lt;/a&gt;. In the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; we can add &lt;a class="wiki_link" href="/99_98"&gt;99/98&lt;/a&gt; and &lt;a class="wiki_link" href="/121_120"&gt;121/120&lt;/a&gt; to the list, and in the &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt;, &lt;a class="wiki_link" href="/66_65"&gt;66/65&lt;/a&gt;. While &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; can also do 13-limit squares, 48 might be preferred for some purposes.&lt;br /&gt;
+ The &lt;em&gt;48 equal division&lt;/em&gt; divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; into 48 1/8th tones of precisely 25 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s each. Since 48 is a multiple of 12, it has attracted a small amount of interest. However, its best major third, of 375 cents, is over 11 cents flat. An alternative third is the familiar 400 cent major third. Using this third, 48 tunes to the same values as 12 in the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;, but &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; &lt;a class="wiki_link" href="/2401_2400"&gt;2401/2400&lt;/a&gt; in the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt;, making it a tuning for &lt;a class="wiki_link" href="/Meantone%20family"&gt;squares temperament&lt;/a&gt;. In the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; we can add &lt;a class="wiki_link" href="/99_98"&gt;99/98&lt;/a&gt; and &lt;a class="wiki_link" href="/121_120"&gt;121/120&lt;/a&gt; to the list, and in the &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt;, &lt;a class="wiki_link" href="/66_65"&gt;66/65&lt;/a&gt;. While &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; can also do 13-limit squares, 48 might be preferred for some purposes.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Temperaments&lt;/h1&gt;
-Using its best major third, 48 tempers out 20000/19683, but &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt; does a much better job for this temperament, known as &lt;a class="wiki_link" href="/Tetracot%20family"&gt;tetracot&lt;/a&gt;. However in the 7-limit it can be used for &lt;a class="wiki_link" href="/Jubilismic%20clan"&gt;doublewide temperament&lt;/a&gt;, the 1/2 octave period temperament with minor third generator tempering out 50/49 and 875/864, for which it is the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt;. In the 11-limit, we may add 99/98, leading to 11-limit doublewide for which 48 again gives the optimal patent val. It is also the optimal patent val for the rank three temperament &lt;a class="wiki_link" href="/Jubilismic%20family"&gt;jubilee&lt;/a&gt;, which tempers out 50/49 and 99/98.&lt;br /&gt;
+ Using its best major third, 48 tempers out 20000/19683, but &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt; does a much better job for this temperament, known as &lt;a class="wiki_link" href="/Tetracot%20family"&gt;tetracot&lt;/a&gt;. However in the 7-limit it can be used for &lt;a class="wiki_link" href="/Jubilismic%20clan"&gt;doublewide temperament&lt;/a&gt;, the 1/2 octave period temperament with minor third generator tempering out 50/49 and 875/864, for which it is the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt;. In the 11-limit, we may add 99/98, leading to 11-limit doublewide for which 48 again gives the optimal patent val. It is also the optimal patent val for the rank three temperament &lt;a class="wiki_link" href="/Jubilismic%20family"&gt;jubilee&lt;/a&gt;, which tempers out 50/49 and 99/98.&lt;br /&gt;
 &lt;br /&gt;
 If 48 is treated as a no-fives system, it still tempers out 99/98 and 243/242 in the 11-limit, leading to a no-fives version of squares for which it does well as a tuning. In the 13 no-fives limit, we can add 144/143 to the list of commas, and we get the no-fives version of 13-limit squares, for which 48 actually defines the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt;. No-fives squares should probably be considered by anyone interested in 48edo; the generator is 17\48, a 425 cent interval serving as both &lt;a class="wiki_link" href="/9_7"&gt;9/7&lt;/a&gt; and &lt;a class="wiki_link" href="/14_11"&gt;14/11&lt;/a&gt;.&lt;br /&gt;
+&lt;br /&gt;
+Something close to 48edo is what you get if you cross 16edo with pure fifths, for instance, on a 16-tone guitar.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Compositions&lt;/h1&gt;

48edo: Difference between revisions

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