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Low harmonic entropy linear temperaments: Difference between revisions

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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]].
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]].


[[:File: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]]


[[:File:coarse-period2-4-thru-12-trimmed-10-70.pdf|coarse, half-octave period]]
[[:File:coarse-period2-4-thru-12-trimmed-10-70.pdf|Coarse, half-octave period]]


[[:File:medium-period1-5-thru-12-trimmed-10-70.pdf|medium, octave period]]
[[:File:medium-period1-5-thru-12-trimmed-10-70.pdf|Medium, octave period]]


[[:File:medium-period2-6-thru-14-trimmed-10-70.pdf|medium, half-octave period]]
[[:File:medium-period2-6-thru-14-trimmed-10-70.pdf|Medium, half-octave period]]


[[:File:fine-period1-7-thru-13-trimmed-10-70.pdf|fine, 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: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-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-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]]
[[: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.
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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:


<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 and dissonance]]
<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 and dissonance]]
[[Category:entropy]]
[[Category:entropy]]
[[Category:harmonic_entropy]]
[[Category:harmonic_entropy]]
[[Category:Lists of temperaments]]
[[Category:Lists of temperaments]]
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