Low harmonic entropy linear temperaments
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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 a really interesting list of low-complexity yet accurate temperaments. 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. 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]]/[[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]] (all-around) * [[Slendric]] (medium-fine) * [[Liese]]/[[triton]] (fine) Temperaments where 4/3 has higher complexity: * [[Magic]] (5, fine) * [[Hanson]]/[[keemun]] (6, fine) * [[Orwell]] (7, extra fine) * [[Myna]] (10, extra 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)
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<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 "typical" means average, but you throw away the highest and lowest values first), you get a really interesting list of low-complexity yet accurate temperaments. 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.<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="/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">Porcupine</a> (all-around)</li><li><a class="wiki_link" href="/Slendric">Slendric</a> (medium-fine)</li><li><a class="wiki_link" href="/Liese">Liese</a>/<a class="wiki_link" href="/triton">triton</a> (fine)</li></ul>Temperaments where 4/3 has higher complexity:<br /> <ul><li><a class="wiki_link" href="/Magic">Magic</a> (5, fine)</li><li><a class="wiki_link" href="/Hanson">Hanson</a>/<a class="wiki_link" href="/keemun">keemun</a> (6, fine)</li><li><a class="wiki_link" href="/Orwell">Orwell</a> (7, extra fine)</li><li><a class="wiki_link" href="/Myna">Myna</a> (10, extra fine)</li></ul>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>