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 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) * [[Archytas clan#Superpyth|Superpyth]] (all-around) * [[Mavila]] (coarse) * [[Helmholtz]]/[[Schismatic family#Garibaldi|garibaldi]] (fine) Temperaments where 4/3 has complexity 2: * [[Semaphore]] (both [[Meantone family#Godzilla|godzilla]] and no-5's semaphore, all-around) * [[Meantone family#Mohajira|Mohajira]] (both mohajira proper and no-5's mohajira, all-around) * [[Srutal]]/[[Diaschismic family#Pajara|pajara]] (all-around) Temperaments where 4/3 has complexity 3: * [[Porcupine family#Porcupine|Porcupine]] (all-around) * 1029/1024 2.3.7 temperament in the [[Gamelismic clan|gamelismic clan]] (medium/fine) * [[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)
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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 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 <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="/Archytas%20clan#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="/Meantone%20family#Godzilla">godzilla</a> and no-5's semaphore, all-around)</li><li><a class="wiki_link" href="/Meantone%20family#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="/Diaschismic%20family#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>1029/1024 2.3.7 temperament in the <a class="wiki_link" href="/Gamelismic%20clan">gamelismic clan</a> (medium/fine)</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>