octave-equivalent harmonic entropy

I finally got around to coding this up. Instead of a Farey series, the set
of ratios is just those within the N-(odd)-limit. Mediants do not obey
octave-equivalence (e.g.: the mediant of 4/3 and 7/5 is 11/8; the mediant of
3/2 and 10/7 is 13/9; but the inversion of 13/9 is 18/13, not 11/8). So I
thought of an octave-equivalent analogue of mediant -- a limit-weighted
midpoint. For example, the limit-weighted midpoint of 4/3 and 7/5 would be

(3*498.04 + 7*582.51)/(3+7) = 557.17

which, as expected, is between 11/8 (551.32) and 18/13 (563.38).

I tried s=0.6% (the "best case" scenario) and N=167. You can see the graph
at http://www.egroups.com/files/tuning/perlich/o006_167.jpg. The local
minima were:

cents entropy ratio limit
0 2.3993 1/1 1
115 5.0136 ? -
174 4.9665 11/10-10/9 -
204 4.9792 9/8 9
231 4.9259 8/7 7
267 4.9218 7/6 7
316 4.8134 6/5 5
349 4.995 11/9-16/13 -
386 4.8171 5/4 5
433 4.9626 9/7 9
498 4.4748 4/3 3
557 4.9724 11/8 11
581 4.9215 7/5 7
619 4.9215 10/7 7
643 4.9724 16/11 11
702 4.4748 3/2 3
767 4.9626 14/9 7
814 4.8171 8/5 5
851 4.995 13/8-18/11 -
884 4.8134 5/3 5
933 4.9218 12/7 7
969 4.9259 7/4 7
996 4.9792 16/9 9
1026 4.9665 9/5-20/11 -
1085 5.0136 ? -

A very clear pattern is seen -- for ratios within the 7-(odd)-limit, the
entropy is determined almost exactly by the limit: 2.39 for 1-limit, 4.47
for 3-limit, 4.81 for 5-limit, and 4.92 for 7-limit. The other local minima
are at considerably higher entropy values, though the pattern begins to fall
apart, due to the finite auditory resolution represented by the s=0.6%
parameter. The consistency through the 7-limit is remarkable. Qualitatively,
this is an impressive validation of Partch's one-footed bride concept,
though the details are tweakable in various ways. I believe that the minima
marked "?" (15/8 and 16/15 in the graph) are artifacts of the choice N=167
and should disappear when I try N=223 (next post).

[Paul E:]
>I finally got around to coding this up. Instead of a Farey series, the
>set of ratios is just those within the N-(odd)-limit.

Cool, Paul! So what, exactly, is left out of an "N-(odd)-limit" set
that is included in a full Farey series?

[Paul:]
>So I thought of an octave-equivalent analogue of mediant -- a
>limit-weighted midpoint. For example, the limit-weighted midpoint of
>4/3 and 7/5 would be

>(3*498.04 + 7*582.51)/(3+7) = 557.17

So... is the rule for finding the weighting as follows? Divide the
numerator and denominator, independently, by as many factors of 2 as
possible, then take the larger number?

[Paul:]
>Qualitatively, this is an impressive validation of Partch's one-footed
>bride concept,

And what is that again?

[Paul ("Everyman's octave-equivalent harmonic entropy curve (for John deL .)":]
>I repeated the process for s=1.5% and N=67 (see
>http://www.egroups.com/files/tuning/perlich/o015_67.jpg). John
>deLaubenfels would like this curve since it repeats exactly at the
>octave (not shown) and the 9/7 is more dissonant than the ET major
>second or tritone, while the 11/9 is less dissonant than the ET major
>second or tritone.

Yes, that DOES match better with what my ears would consider a good
ranking. How does it manage to get away with that, though, when 11/9
is mathematically higher than 9/7?

JdL