RE: RE: Consistency generalized

>the average error of random intervals is
>1/4 step [given] level 2 consistency . . .

Paul H., surely the average error is less than 1/4 step, since that is the
maximum error. The mean error is actually zero, since errors can be positive
or negative, but the standard deviation (square root of mean of squared
errors) is a better measure of the overall size of the errors. Graphing the
standard deviation with respect to all the JI intervals up to a given limit
for all ETs starting with 1-tET gives a bumpy graph (with dips at famous
ETs) which for the most part closely follows an overall curve. This curve is
given by 1/sqrt(12) times the step size of the ET. In other words, without
assuming any consistency at all, the standard error of random intervals is
.2887 step. Of course, the mean absolute error of random intervals, without
assuming any consistency at all, is .25 step, since all values from 0 to .5
step are taken on with equal probability.

I have not yet figured out how to calculate the expected errors given
consistency, but a good measure of how "special" an ET is is given by
assuming "in-tune-ness" is given by a normal curve centered around the just
interval with some appropriately psychoacoustically researched standard
deviation (in this context, standard deviation refers to the "width" of the
curve). Then the difference between the "in-tune-ness" of the standard
deviation of the ET and the "in-tune-ness" of the nominal value of
1/sqrt(12) times the step size gives you the "specialness" of the ET. My
first post to this digest was a table of the most "special" tunings, for
various harmonic limits and various reasonable values of the standard
deviation, and I later reposted it with all the inconsistent tunings
deleted. Needless to say, most everyone's favorite ET was to be found
somewhere on the table -- they're all the most special!

-Paul E.


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