Low harmonic entropy linear temperaments: Difference between revisions
Wikispaces>keenanpepper **Imported revision 246794913 - Original comment: ** |
Wikispaces>keenanpepper **Imported revision 247102809 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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- | : This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-08-19 22:16:40 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>247102809</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> | ||
| Line 9: | Line 9: | ||
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. | ||
In terms of rankings within this set, [[meantone]] is the clear winner, with the pentatonic and diatonic scales having the lowest average HE in all four fineness categories from course to extra fine. This provides a theoretical explanation for the fact that these are the most popular scales in the world, by any reasonable metric. | |||
However, for scales with more than 7 notes, meantone chromatic is not the clear winner - it is practically tied with the [[pajara]] decatonic scale in most categories. | |||
First of all, some small EDOs appear: | First of all, some small EDOs appear: | ||
| Line 41: | Line 45: | ||
* Augmented (indistinguishable in practice from 12-EDO subsets) | * Augmented (indistinguishable in practice from 12-EDO subsets) | ||
* Roulette (index-2 subtemperament of meantone) | * Roulette (index-2 subtemperament of meantone) | ||
* Injera (pajara is simply better; injera just appears as a little shoulder on its side in a plot of average HE) | * Injera (pajara is simply better; injera just appears as a little shoulder on its side in a plot of average HE)</pre></div> | ||
</pre></div> | |||
<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"><html><head><title>Low harmonic entropy linear temperaments</title></head><body>If you do a survey of <a class="wiki_link" href="/MOS">MOS</a>es and look for the ones that have the lowest typical <a class="wiki_link" href="/harmonic%20entropy">harmonic entropy</a> of an interval (where &quot;typical&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 &quot;sigma&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 <a class="wiki_link" href="/cangwu%20badness">cangwu badness</a>.<br /> | <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"><html><head><title>Low harmonic entropy linear temperaments</title></head><body>If you do a survey of <a class="wiki_link" href="/MOS">MOS</a>es and look for the ones that have the lowest typical <a class="wiki_link" href="/harmonic%20entropy">harmonic entropy</a> of an interval (where &quot;typical&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 &quot;sigma&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 <a class="wiki_link" href="/cangwu%20badness">cangwu badness</a>.<br /> | ||
<br /> | <br /> | ||
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.<br /> | 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.<br /> | ||
<br /> | |||
In terms of rankings within this set, <a class="wiki_link" href="/meantone">meantone</a> is the clear winner, with the pentatonic and diatonic scales having the lowest average HE in all four fineness categories from course to extra fine. This provides a theoretical explanation for the fact that these are the most popular scales in the world, by any reasonable metric.<br /> | |||
<br /> | |||
However, for scales with more than 7 notes, meantone chromatic is not the clear winner - it is practically tied with the <a class="wiki_link" href="/pajara">pajara</a> decatonic scale in most categories.<br /> | |||
<br /> | <br /> | ||
First of all, some small EDOs appear:<br /> | First of all, some small EDOs appear:<br /> | ||