Harmonic Entropy for the Everyman's Ear

John deLaubenfels wrote:

>Here's another concern I have: 9/7 vs. 11/9. The graph shows 9/7 as
>the more concordant of the two intervals, with a pronounced dip of its
>own; 11/9 is perched at the top of an almost-flat mesa between 6/5
>and 5/4.

>But 11/9 is the "neutral third", which several people have described
>as relatively pleasant, whereas 9/7 is the "car horn third", quite
>irritating-sounding on its own, though part of a well-tuned 7-limit
>dominant 9th chord.

>The graph "makes sense" just looking at the larger numbers in 11/9 vs.
>9/7, yet for some reason the impression of the intervals does not
>match, to my ears.

Remember, John, the s=0.6% curve corresponds to an ideal listener listening
in the ideal frequency range, around 3000Hz, pretty damn high. Try listening
to 11:9 and 9:7 (and, of course, out-of-tune versions of those intervals) up
there and see what you think. More appropriate for typical musical purposes
might be s=1.5%. I graphed that -- see
http://www.egroups.com/files/tuning/perlich/ent_015.jpg. Although 11:9 is
still higher than 9:7, they are both local maxima and 9:7 sits atop a much
taller hump. As I've said before, the overall downward slope of the curve is
an artifact of the Farey series, and, controlling for that, 9:7 may be
higher in this graph. I would need to make some graphs using a Tenney
(numerator times denominator) limit, or other series, to make it easier to
make direct comparisons of intervals that are not roughly the same size. I
anticipate that virtually the same local minima will come up (given an s
value) regardless of the type of series I start with, but it would be good
to be sure . . .