Low harmonic entropy linear temperaments: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
If you do a survey of [[MOS|MOS]]es and look for the ones that have the lowest typical [[Harmonic_Entropy|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|cangwu badness]].
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-05-19 04:29:05 UTC</tt>.<br>
: The original revision id was <tt>337248976</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>
<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]].


[[http://xenharmonic.wikispaces.com/file/view/coarse-period1-4-thru-11-trimmed-10-70.pdf|coarse, octave period]]
[[:File: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]]
[[:File:coarse-period2-4-thru-12-trimmed-10-70.pdf|coarse, half-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]]
[[:File:medium-period1-5-thru-12-trimmed-10-70.pdf|medium, 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]]
[[:File:medium-period2-6-thru-14-trimmed-10-70.pdf|medium, half-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]]
[[:File:fine-period1-7-thru-13-trimmed-10-70.pdf|fine, octave period]]
 
[[:File:fine-period2-6-thru-14-trimmed-10-70.pdf|fine, half-octave period]]
 
[[:File:extra-fine-period1-7-thru-13-trimmed-10-70.pdf|extra fine, octave period]]
 
[[:File:extra-fine-period2-8-thru-18-trimmed-10-70.pdf|extra fine, half-octave period]]
 
[[:File: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.


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.
In terms of rankings within this set, [[Meantone|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.
However, for scales with more than 7 notes, meantone chromatic is not the clear winner - it is practically tied with the [[pajara|pajara]] decatonic scale in most categories.


First of all, some small EDOs appear:
First of all, some small EDOs appear:
* [[5edo]] (coarse)
 
* [[7edo]] (coarse)
<ul><li>[[5edo|5edo]] (coarse)</li><li>[[7edo|7edo]] (coarse)</li><li>[[12edo|12edo]] (coarse/medium)</li></ul>Temperaments where 4/3 has complexity 1 all have the same structure:
* [[12edo]] (coarse/medium)
 
Temperaments where 4/3 has complexity 1 all have the same structure:
<ul><li>[[Meantone|Meantone]] (all-around)</li><li>[[Superpyth|Superpyth]] (all-around)</li><li>[[Mavila|Mavila]] (coarse)</li><li>[[Helmholtz|Helmholtz]]/[[Garibaldi|garibaldi]] (fine)</li></ul>Temperaments where 4/3 has complexity 2:
* [[Meantone]] (all-around)
 
* [[Superpyth]] (all-around)
<ul><li>[[Semaphore_and_Godzilla|Semaphore / godzilla]] (all-around)</li><li>[[Neutral_third_scales|Neutral third scales]] (mohajira, maqamic, beatles..., all-around)</li><li>[[Srutal|Srutal]]/[[pajara|pajara]] (all-around)</li></ul>Temperaments where 4/3 has complexity 3:
* [[Mavila]] (coarse)
 
* [[Helmholtz]]/[[garibaldi]] (fine)
<ul><li>[[Porcupine|Porcupine]] (all-around)</li><li>[[Slendric|Slendric]] (all-around; quite accurate)</li><li>[[Liese|Liese]]/[[Triton|triton]] (fine)</li></ul>Temperaments where 4/3 has higher complexity:
Temperaments where 4/3 has complexity 2:
 
* [[Semaphore and Godzilla|Semaphore / godzilla]] (all-around)
<ul><li>[[Magic|Magic]] (5, fine)</li><li>[[Orwell|Orwell]] (7, extra fine)</li><li>[[Myna|Myna]] (10, extra fine)</li><li>[[Hanson|Hanson]]/[[catakleismic|catakleismic]] (6, extra fine)</li><li>[[Miracle|Miracle]] (6, super duper fine)</li><li>[[Valentine|Valentine]] (9, super duper fine)</li></ul>
* [[Neutral third scales]] (mohajira, maqamic, beatles..., all-around)
* [[Srutal]]/[[pajara]] (all-around)
Temperaments where 4/3 has complexity 3:
* [[Porcupine]] (all-around)
* [[Slendric]] (all-around; quite accurate)
* [[Liese]]/[[triton]] (fine)
Temperaments where 4/3 has higher complexity:
* [[Magic]] (5, fine)
* [[Orwell]] (7, extra fine)
* [[Myna]] (10, extra fine)
* [[Hanson]]/[[catakleismic]] (6, extra fine)
* [[Miracle]] (6, super duper fine)
* [[Valentine]] (9, super duper fine)


Finally, a temperament in which 3 has two different mappings:
Finally, a temperament in which 3 has two different mappings:
* [[Pseudo-semaphore]] (medium)
 
<ul><li>[[Pseudo-semaphore|Pseudo-semaphore]] (medium)</li></ul>


All-around runners up:
All-around runners up:
* [[Negri]] (4)
 
* [[Sensi]] (7)
<ul><li>[[Negri|Negri]] (4)</li><li>[[Sensi|Sensi]] (7)</li></ul>


The following temperaments were not included in the list, because they don't stand out as good independent temperaments:
The following temperaments were not included in the list, because they don't stand out as good independent temperaments:
* Augmented (indistinguishable in practice from 12-EDO subsets)
 
* Roulette (index-2 subtemperament of meantone)
<ul><li>Augmented (indistinguishable in practice from 12-EDO subsets)</li><li>Roulette (index-2 subtemperament of meantone)</li><li>Injera (pajara is simply better; injera just appears as a little shoulder on its side in a plot of average HE)</li></ul>     [[Category:consonance]]
* Injera (pajara is simply better; injera just appears as a little shoulder on its side in a plot of average HE)</pre></div>
[[Category:dissonance]]
<h4>Original HTML content:</h4>
[[Category:entropy]]
<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;
[[Category:harmonic_entropy]]
&lt;br /&gt;
[[Category:sonance]]
&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;
[[Category:theory]]
&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;
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;
&lt;br /&gt;
In terms of rankings within this set, &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; 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.&lt;br /&gt;
&lt;br /&gt;
However, for scales with more than 7 notes, meantone chromatic is not the clear winner - it is practically tied with the &lt;a class="wiki_link" href="/pajara"&gt;pajara&lt;/a&gt; decatonic scale in most categories.&lt;br /&gt;
&lt;br /&gt;
First of all, some small EDOs appear:&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link" href="/5edo"&gt;5edo&lt;/a&gt; (coarse)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt; (coarse)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt; (coarse/medium)&lt;/li&gt;&lt;/ul&gt;Temperaments where 4/3 has complexity 1 all have the same structure:&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Meantone"&gt;Meantone&lt;/a&gt; (all-around)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Superpyth"&gt;Superpyth&lt;/a&gt; (all-around)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Mavila"&gt;Mavila&lt;/a&gt; (coarse)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Helmholtz"&gt;Helmholtz&lt;/a&gt;/&lt;a class="wiki_link" href="/garibaldi"&gt;garibaldi&lt;/a&gt; (fine)&lt;/li&gt;&lt;/ul&gt;Temperaments where 4/3 has complexity 2:&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Semaphore%20and%20Godzilla"&gt;Semaphore / godzilla&lt;/a&gt; (all-around)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Neutral%20third%20scales"&gt;Neutral third scales&lt;/a&gt; (mohajira, maqamic, beatles..., all-around)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Srutal"&gt;Srutal&lt;/a&gt;/&lt;a class="wiki_link" href="/pajara"&gt;pajara&lt;/a&gt; (all-around)&lt;/li&gt;&lt;/ul&gt;Temperaments where 4/3 has complexity 3:&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Porcupine"&gt;Porcupine&lt;/a&gt; (all-around)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Slendric"&gt;Slendric&lt;/a&gt; (all-around; quite accurate)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Liese"&gt;Liese&lt;/a&gt;/&lt;a class="wiki_link" href="/triton"&gt;triton&lt;/a&gt; (fine)&lt;/li&gt;&lt;/ul&gt;Temperaments where 4/3 has higher complexity:&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Magic"&gt;Magic&lt;/a&gt; (5, fine)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Orwell"&gt;Orwell&lt;/a&gt; (7, extra fine)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Myna"&gt;Myna&lt;/a&gt; (10, extra fine)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Hanson"&gt;Hanson&lt;/a&gt;/&lt;a class="wiki_link" href="/catakleismic"&gt;catakleismic&lt;/a&gt; (6, extra fine)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Miracle"&gt;Miracle&lt;/a&gt; (6, super duper fine)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Valentine"&gt;Valentine&lt;/a&gt; (9, super duper fine)&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
Finally, a temperament in which 3 has two different mappings:&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Pseudo-semaphore"&gt;Pseudo-semaphore&lt;/a&gt; (medium)&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
All-around runners up:&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Negri"&gt;Negri&lt;/a&gt; (4)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Sensi"&gt;Sensi&lt;/a&gt; (7)&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
The following temperaments were not included in the list, because they don't stand out as good independent temperaments:&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;Augmented (indistinguishable in practice from 12-EDO subsets)&lt;/li&gt;&lt;li&gt;Roulette (index-2 subtemperament of meantone)&lt;/li&gt;&lt;li&gt;Injera (pajara is simply better; injera just appears as a little shoulder on its side in a plot of average HE)&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 00:00, 17 July 2018

If you do a survey of MOSes 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.

coarse, octave period

coarse, half-octave period

medium, octave period

medium, half-octave period

fine, octave period

fine, half-octave period

extra fine, octave period

extra fine, half-octave period

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.

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:

Temperaments where 4/3 has complexity 1 all have the same structure:

Temperaments where 4/3 has complexity 2:

Temperaments where 4/3 has complexity 3:

Temperaments where 4/3 has higher complexity:

Finally, a temperament in which 3 has two different mappings:

All-around runners up:

The following temperaments were not included in the list, because they don't stand out as good independent temperaments:

  • Augmented (indistinguishable in practice from 12-EDO subsets)
  • 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)
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