32edo: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 220051368 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 220051876 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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>2011-04-13 18: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-13 18:14:52 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>220051876</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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=Z function= | =Z function= | ||
Below is a plot of the Z function, showing how its peak value is shifted above 32, corresponding to a zeta tuning with octaves flattened to 1197.375 cents. This will improve the fifth, at the expense of the third. | Below is a plot of the Z function, showing how its peak (ie most negative) value is shifted above 32, corresponding to a zeta tuning with octaves flattened to 1197.375 cents. This will improve the fifth, at the expense of the third. | ||
[[image:plot32.png]] | [[image:plot32.png]] | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Z function"></a><!-- ws:end:WikiTextHeadingRule:0 -->Z function</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Z function"></a><!-- ws:end:WikiTextHeadingRule:0 -->Z function</h1> | ||
Below is a plot of the Z function, showing how its peak value is shifted above 32, corresponding to a zeta tuning with octaves flattened to 1197.375 cents. This will improve the fifth, at the expense of the third.<br /> | Below is a plot of the Z function, showing how its peak (ie most negative) value is shifted above 32, corresponding to a zeta tuning with octaves flattened to 1197.375 cents. This will improve the fifth, at the expense of the third.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule:4:&lt;img src=&quot;/file/view/plot32.png/219952208/plot32.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/plot32.png/219952208/plot32.png" alt="plot32.png" title="plot32.png" /><!-- ws:end:WikiTextLocalImageRule:4 --><br /> | <!-- ws:start:WikiTextLocalImageRule:4:&lt;img src=&quot;/file/view/plot32.png/219952208/plot32.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/plot32.png/219952208/plot32.png" alt="plot32.png" title="plot32.png" /><!-- ws:end:WikiTextLocalImageRule:4 --><br /> | ||
Revision as of 18:14, 13 April 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-04-13 18:14:52 UTC.
- The original revision id was 220051876.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
The //32 equal division// divides the octave into 32 equal parts of precisely 37.5 cents each. While even advocates of less-common edos can struggle to find something about it worth noting, it does provide an excellent tuning for [[Petr Parízek]]'s sixix temperament, which tempers out the 5-limit sixix comma, 3125/2916, using its 9\32 generator of size 337.5 cents. Parízek's preferred generator for sixix is (128/15)^(1/11), which is 337.430 cents and which gives equal error to fifths and major thirds, so 32edo does sixix about as well as sixix can be done. It also tempers out 2048/2025 in the 5-limit, and 50/49 with 64/63 in the 7-limit, which means it supports [[Diaschismic family|pajara temperament]], with a very sharp fifth of 712.5 cents which could be experimented with by those with a penchant for fifths even sharper than the fifth of [[27edo]]. =Z function= Below is a plot of the Z function, showing how its peak (ie most negative) value is shifted above 32, corresponding to a zeta tuning with octaves flattened to 1197.375 cents. This will improve the fifth, at the expense of the third. [[image:plot32.png]] =Music= [[http://micro.soonlabel.com/petr_parizek/3125_2916_temp_q32.ogg|Sixix]] by Petr Parízek
Original HTML content:
<html><head><title>32edo</title></head><body>The <em>32 equal division</em> divides the octave into 32 equal parts of precisely 37.5 cents each. While even advocates of less-common edos can struggle to find something about it worth noting, it does provide an excellent tuning for <a class="wiki_link" href="/Petr%20Par%C3%ADzek">Petr Parízek</a>'s sixix temperament, which tempers out the 5-limit sixix comma, 3125/2916, using its 9\32 generator of size 337.5 cents. Parízek's preferred generator for sixix is (128/15)^(1/11), which is 337.430 cents and which gives equal error to fifths and major thirds, so 32edo does sixix about as well as sixix can be done.<br /> <br /> It also tempers out 2048/2025 in the 5-limit, and 50/49 with 64/63 in the 7-limit, which means it supports <a class="wiki_link" href="/Diaschismic%20family">pajara temperament</a>, with a very sharp fifth of 712.5 cents which could be experimented with by those with a penchant for fifths even sharper than the fifth of <a class="wiki_link" href="/27edo">27edo</a>.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Z function"></a><!-- ws:end:WikiTextHeadingRule:0 -->Z function</h1> Below is a plot of the Z function, showing how its peak (ie most negative) value is shifted above 32, corresponding to a zeta tuning with octaves flattened to 1197.375 cents. This will improve the fifth, at the expense of the third.<br /> <br /> <!-- ws:start:WikiTextLocalImageRule:4:<img src="/file/view/plot32.png/219952208/plot32.png" alt="" title="" /> --><img src="/file/view/plot32.png/219952208/plot32.png" alt="plot32.png" title="plot32.png" /><!-- ws:end:WikiTextLocalImageRule:4 --><br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:2 -->Music</h1> <a class="wiki_link_ext" href="http://micro.soonlabel.com/petr_parizek/3125_2916_temp_q32.ogg" rel="nofollow">Sixix</a> by Petr Parízek</body></html>