RE: [tuning] Re: octave-equivalent harmonic entropy

John wrote,

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

Nothing. The N-(odd)-limit set has more ratios, since factors of 2 are
"free".

>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?

Yup, that's the "odd limit" of the ratio.

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

>And what is that again?

Have you read Partch's book? Basically, he creates a graph of comparitive
consonance, with ratios of 1 (defined by the same rule as the weighting rule
you mention above) most consonant, ratios of 3 less consonant, ratios of 5
less consonant than that, and so on through ratios of 7, 9, and 11, with
ratios beyond the 11-limit being most dissonant. The increments between the
consonance levels get smaller and smaller.

>[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?

I don't know what you mean "mathematically higher". If you mean higher
numbers are used in the ratio, well you can find examples of that in all the
harmonic entropy graphs, but for smaller s you'd generally need to use
ratios with higher numbers than these -- for example 100/99 has a lower
entropy than 12/11 for reasonable s values. In this case, the s=1.5%
parameter means that 1/3 of the probability comes from more than 25 cents
away from the actual interval. That means that the dips associated with each
of the simple ratios are, in some sense, 50 cents across. So 11/9 is well
within both the 5/4 dip and the 6/5 dip, which overlap, while 9/7 is on the
very outer edge of the 5/4 dip and near no other dip. That's just my
unscientific way of thinking about it, but seems to be the easiest way of
seeing what's going on in these graphs, and fairly accurate too.