Low harmonic entropy linear temperaments

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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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