Low harmonic entropy linear temperaments: Difference between revisions

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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>2011-08-17 04:16:47 UTC</tt>.<br>
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-08-17 16:31:50 UTC</tt>.<br>
: The original revision id was <tt>246434033</tt>.<br>
: The original revision id was <tt>246551953</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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* [[Helmholtz]]/[[Schismatic family#Garibaldi|garibaldi]] (fine)
* [[Helmholtz]]/[[Schismatic family#Garibaldi|garibaldi]] (fine)
Temperaments where 4/3 has complexity 2:
Temperaments where 4/3 has complexity 2:
* [[Semaphore]] (both [[Meantone family#Godzilla|godzilla]] and no-5's semaphore, all-around)
* [[Semaphore]] (both [[godzilla]] and no-5's semaphore, all-around)
* [[Meantone family#Mohajira|Mohajira]] (both mohajira proper and no-5's mohajira, all-around)
* [[Mohajira]] (both mohajira proper and no-5's mohajira, all-around)
* [[Srutal]]/[[Diaschismic family#Pajara|pajara]] (all-around)
* [[Srutal]]/[[pajara]] (all-around)
Temperaments where 4/3 has complexity 3:
Temperaments where 4/3 has complexity 3:
* [[Porcupine family#Porcupine|Porcupine]] (all-around)
* [[Porcupine family#Porcupine|Porcupine]] (all-around)
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&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="/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="/Schismatic%20family#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="/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="/Schismatic%20family#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"&gt;Semaphore&lt;/a&gt; (both &lt;a class="wiki_link" href="/Meantone%20family#Godzilla"&gt;godzilla&lt;/a&gt; and no-5's semaphore, all-around)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Meantone%20family#Mohajira"&gt;Mohajira&lt;/a&gt; (both mohajira proper and no-5's mohajira, 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="/Diaschismic%20family#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="/Semaphore"&gt;Semaphore&lt;/a&gt; (both &lt;a class="wiki_link" href="/godzilla"&gt;godzilla&lt;/a&gt; and no-5's semaphore, all-around)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Mohajira"&gt;Mohajira&lt;/a&gt; (both mohajira proper and no-5's mohajira, 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%20family#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="/Meantone%20family#Liese"&gt;Liese&lt;/a&gt;/&lt;a class="wiki_link" href="/Marvel%20temperaments#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="/Porcupine%20family#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="/Meantone%20family#Liese"&gt;Liese&lt;/a&gt;/&lt;a class="wiki_link" href="/Marvel%20temperaments#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%20family#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="/Starling%20temperaments#Myna temperament"&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="/Kleismic%20family#Catakleismic"&gt;catakleismic&lt;/a&gt; (6, extra fine)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Gamelismic%20clan#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;
&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Magic%20family#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="/Starling%20temperaments#Myna temperament"&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="/Kleismic%20family#Catakleismic"&gt;catakleismic&lt;/a&gt; (6, extra fine)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Gamelismic%20clan#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;

Revision as of 16:31, 17 August 2011

IMPORTED REVISION FROM WIKISPACES

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This revision was by author keenanpepper and made on 2011-08-17 16:31:50 UTC.
The original revision id was 246551953.
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:

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]].

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.

First of all, some small EDOs appear:
* [[5edo]] (coarse)
* [[7edo]] (coarse)
* [[12edo]] (coarse/medium)
Temperaments where 4/3 has complexity 1 all have the same structure:
* [[Meantone]] (all-around)
* [[Superpyth]] (all-around)
* [[Mavila]] (coarse)
* [[Helmholtz]]/[[Schismatic family#Garibaldi|garibaldi]] (fine)
Temperaments where 4/3 has complexity 2:
* [[Semaphore]] (both [[godzilla]] and no-5's semaphore, all-around)
* [[Mohajira]] (both mohajira proper and no-5's mohajira, all-around)
* [[Srutal]]/[[pajara]] (all-around)
Temperaments where 4/3 has complexity 3:
* [[Porcupine family#Porcupine|Porcupine]] (all-around)
* [[Slendric]] (all-around; quite accurate)
* [[Meantone family#Liese|Liese]]/[[Marvel temperaments#Triton|triton]] (fine)
Temperaments where 4/3 has higher complexity:
* [[Magic family#Magic|Magic]] (5, fine)
* [[Orwell]] (7, extra fine)
* [[Starling temperaments#Myna temperament|Myna]] (10, extra fine)
* [[Hanson]]/[[Kleismic family#Catakleismic|catakleismic]] (6, extra fine)
* [[Gamelismic clan#Miracle|Miracle]] (6, super duper fine)
* [[Valentine]] (9, super duper fine)

Finally, a temperament in which 3 has two different mappings:
* [[Pseudo-semaphore]] (medium)

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)

Original HTML content:

<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 />
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 />
First of all, some small EDOs appear:<br />
<ul><li><a class="wiki_link" href="/5edo">5edo</a> (coarse)</li><li><a class="wiki_link" href="/7edo">7edo</a> (coarse)</li><li><a class="wiki_link" href="/12edo">12edo</a> (coarse/medium)</li></ul>Temperaments where 4/3 has complexity 1 all have the same structure:<br />
<ul><li><a class="wiki_link" href="/Meantone">Meantone</a> (all-around)</li><li><a class="wiki_link" href="/Superpyth">Superpyth</a> (all-around)</li><li><a class="wiki_link" href="/Mavila">Mavila</a> (coarse)</li><li><a class="wiki_link" href="/Helmholtz">Helmholtz</a>/<a class="wiki_link" href="/Schismatic%20family#Garibaldi">garibaldi</a> (fine)</li></ul>Temperaments where 4/3 has complexity 2:<br />
<ul><li><a class="wiki_link" href="/Semaphore">Semaphore</a> (both <a class="wiki_link" href="/godzilla">godzilla</a> and no-5's semaphore, all-around)</li><li><a class="wiki_link" href="/Mohajira">Mohajira</a> (both mohajira proper and no-5's mohajira, all-around)</li><li><a class="wiki_link" href="/Srutal">Srutal</a>/<a class="wiki_link" href="/pajara">pajara</a> (all-around)</li></ul>Temperaments where 4/3 has complexity 3:<br />
<ul><li><a class="wiki_link" href="/Porcupine%20family#Porcupine">Porcupine</a> (all-around)</li><li><a class="wiki_link" href="/Slendric">Slendric</a> (all-around; quite accurate)</li><li><a class="wiki_link" href="/Meantone%20family#Liese">Liese</a>/<a class="wiki_link" href="/Marvel%20temperaments#Triton">triton</a> (fine)</li></ul>Temperaments where 4/3 has higher complexity:<br />
<ul><li><a class="wiki_link" href="/Magic%20family#Magic">Magic</a> (5, fine)</li><li><a class="wiki_link" href="/Orwell">Orwell</a> (7, extra fine)</li><li><a class="wiki_link" href="/Starling%20temperaments#Myna temperament">Myna</a> (10, extra fine)</li><li><a class="wiki_link" href="/Hanson">Hanson</a>/<a class="wiki_link" href="/Kleismic%20family#Catakleismic">catakleismic</a> (6, extra fine)</li><li><a class="wiki_link" href="/Gamelismic%20clan#Miracle">Miracle</a> (6, super duper fine)</li><li><a class="wiki_link" href="/Valentine">Valentine</a> (9, super duper fine)</li></ul><br />
Finally, a temperament in which 3 has two different mappings:<br />
<ul><li><a class="wiki_link" href="/Pseudo-semaphore">Pseudo-semaphore</a> (medium)</li></ul><br />
The following temperaments were not included in the list, because they don't stand out as good independent temperaments:<br />
<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></body></html>
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