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Low harmonic entropy linear temperaments: Difference between revisions

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Wikispaces>keenanpepper
**Imported revision 247102809 - Original comment: **
Wikispaces>keenanpepper
**Imported revision 247150291 - 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:keenanpepper|keenanpepper]] and made on <tt>2011-08-19 22:16:40 UTC</tt>.<br>
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-08-20 05:28:47 UTC</tt>.<br>
: The original revision id was <tt>247102809</tt>.<br>
: The original revision id was <tt>247150291</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>
<h4>Original Wikitext content:</h4>
<h4>Original Wikitext content:</h4>
<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">If you do a survey of [[MOS]]es and look for the ones that have the lowest typical [[harmonic entropy]] of an interval (where "typical" means average, but you throw away the highest and lowest values first), you get an interesting list of reasonably low-complexity yet accurate temperaments, which accords well with lists obtained by starting from temperaments rather than MOS. The results are different according to the "sigma" of the harmonic entropy function you use (coarse versus fine), but some temperaments appear for a wide range of sigma values; this might be compared to the situation with [[cangwu badness]].
<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">If you do a survey of [[MOS]]es and look for the ones that have the lowest typical [[harmonic entropy]] of an interval (where "typical" means average, but you throw away the highest and lowest values first), you get an interesting list of reasonably low-complexity yet accurate temperaments, which accords well with lists obtained by starting from temperaments rather than MOS. The results are different according to the "sigma" of the harmonic entropy function you use (coarse versus fine), but some temperaments appear for a wide range of sigma values; this might be compared to the situation with [[cangwu badness]].
[[http://xenharmonic.wikispaces.com/file/view/coarse-period1-4-thru-11-trimmed-10-70.pdf|coarse, octave period]]
[[http://xenharmonic.wikispaces.com/file/view/coarse-period2-4-thru-12-trimmed-10-70.pdf|coarse, half-octave period]]
[[http://xenharmonic.wikispaces.com/file/view/medium-period1-5-thru-12-trimmed-10-70.pdf|medium, octave period]]
[[http://xenharmonic.wikispaces.com/file/view/medium-period2-6-thru-14-trimmed-10-70.pdf|medium, half-octave period]]
[[http://xenharmonic.wikispaces.com/file/view/fine-period1-7-thru-13-trimmed-10-70.pdf|fine, octave period]]
[[http://xenharmonic.wikispaces.com/file/view/fine-period2-6-thru-14-trimmed-10-70.pdf|fine, half-octave period]]
[[http://xenharmonic.wikispaces.com/file/view/extra-fine-period1-7-thru-13-trimmed-10-70.pdf|extra fine, octave period]]
[[http://xenharmonic.wikispaces.com/file/view/extra-fine-period2-8-thru-18-trimmed-10-70.pdf|extra fine, half-octave period]]
[[http://xenharmonic.wikispaces.com/file/view/extra-fine-period3-9-thru-18-trimmed-10-70.pdf|extra fine, third-octave period]]


It makes sense to organize the results by the complexity of 4/3, because 4/3 (or its octave equivalent 3/2) has by far the lowest harmonic entropy of any interval within an octave, and the results are accordingly dominated by temperaments with lots of good 4/3s.
It makes sense to organize the results by the complexity of 4/3, because 4/3 (or its octave equivalent 3/2) has by far the lowest harmonic entropy of any interval within an octave, and the results are accordingly dominated by temperaments with lots of good 4/3s.
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<h4>Original HTML content:</h4>
<h4>Original HTML content:</h4>
<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;Low harmonic entropy linear temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;If you do a survey of &lt;a class="wiki_link" href="/MOS"&gt;MOS&lt;/a&gt;es and look for the ones that have the lowest typical &lt;a class="wiki_link" href="/harmonic%20entropy"&gt;harmonic entropy&lt;/a&gt; of an interval (where &amp;quot;typical&amp;quot; means average, but you throw away the highest and lowest values first), you get an interesting list of reasonably low-complexity yet accurate temperaments, which accords well with lists obtained by starting from temperaments rather than MOS. The results are different according to the &amp;quot;sigma&amp;quot; of the harmonic entropy function you use (coarse versus fine), but some temperaments appear for a wide range of sigma values; this might be compared to the situation with &lt;a class="wiki_link" href="/cangwu%20badness"&gt;cangwu badness&lt;/a&gt;.&lt;br /&gt;
<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;Low harmonic entropy linear temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;If you do a survey of &lt;a class="wiki_link" href="/MOS"&gt;MOS&lt;/a&gt;es and look for the ones that have the lowest typical &lt;a class="wiki_link" href="/harmonic%20entropy"&gt;harmonic entropy&lt;/a&gt; of an interval (where &amp;quot;typical&amp;quot; means average, but you throw away the highest and lowest values first), you get an interesting list of reasonably low-complexity yet accurate temperaments, which accords well with lists obtained by starting from temperaments rather than MOS. The results are different according to the &amp;quot;sigma&amp;quot; of the harmonic entropy function you use (coarse versus fine), but some temperaments appear for a wide range of sigma values; this might be compared to the situation with &lt;a class="wiki_link" href="/cangwu%20badness"&gt;cangwu badness&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;a href="http://xenharmonic.wikispaces.com/file/view/coarse-period1-4-thru-11-trimmed-10-70.pdf"&gt;coarse, octave period&lt;/a&gt;&lt;br /&gt;
&lt;a href="http://xenharmonic.wikispaces.com/file/view/coarse-period2-4-thru-12-trimmed-10-70.pdf"&gt;coarse, half-octave period&lt;/a&gt;&lt;br /&gt;
&lt;a href="http://xenharmonic.wikispaces.com/file/view/medium-period1-5-thru-12-trimmed-10-70.pdf"&gt;medium, octave period&lt;/a&gt;&lt;br /&gt;
&lt;a href="http://xenharmonic.wikispaces.com/file/view/medium-period2-6-thru-14-trimmed-10-70.pdf"&gt;medium, half-octave period&lt;/a&gt;&lt;br /&gt;
&lt;a href="http://xenharmonic.wikispaces.com/file/view/fine-period1-7-thru-13-trimmed-10-70.pdf"&gt;fine, octave period&lt;/a&gt;&lt;br /&gt;
&lt;a href="http://xenharmonic.wikispaces.com/file/view/fine-period2-6-thru-14-trimmed-10-70.pdf"&gt;fine, half-octave period&lt;/a&gt;&lt;br /&gt;
&lt;a href="http://xenharmonic.wikispaces.com/file/view/extra-fine-period1-7-thru-13-trimmed-10-70.pdf"&gt;extra fine, octave period&lt;/a&gt;&lt;br /&gt;
&lt;a href="http://xenharmonic.wikispaces.com/file/view/extra-fine-period2-8-thru-18-trimmed-10-70.pdf"&gt;extra fine, half-octave period&lt;/a&gt;&lt;br /&gt;
&lt;a href="http://xenharmonic.wikispaces.com/file/view/extra-fine-period3-9-thru-18-trimmed-10-70.pdf"&gt;extra fine, third-octave period&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
It makes sense to organize the results by the complexity of 4/3, because 4/3 (or its octave equivalent 3/2) has by far the lowest harmonic entropy of any interval within an octave, and the results are accordingly dominated by temperaments with lots of good 4/3s.&lt;br /&gt;
It makes sense to organize the results by the complexity of 4/3, because 4/3 (or its octave equivalent 3/2) has by far the lowest harmonic entropy of any interval within an octave, and the results are accordingly dominated by temperaments with lots of good 4/3s.&lt;br /&gt;
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