Re: plethora of pentatonics

Paul,

Thanks for posting the list of pentatonics. I'm looking forward to the
results for higher numbers of tones.
I have to look closer at the list. At first glance there are a lot of
intervals between common JI/meantone intervals, but as we go further down,
there are some interesting anamolies. The first scale with disc. 40.01 is
do re mi fa so in meantone, except the third is flat by 3 cents. The minor
version below it also has a flat major third. Looks like the optimizer
prefers a just minor to a either a just major third or a comprimize between
the two in this scale, because this give the widest minor second. From a
musical standpoint that sounds like a bogous reason, and I think it
partially vindicates John's observation about the seconds being
overweighted as harmonic elements, although you can't solve that by leaving
the seconds out of consideration. The same thing happens in both of the
major thirds of first disc. 40.048 scale, and one minor third is made sharp
of just for the same reason. At disc. = 40.398 the optimizer starts to
find the septimal dips in the entropy curve (from 229 to 498 is 8/7) and
some stranger intervals start to creep in. Check out 40.505, there's a
chain of tempered minor thirds and a perfect 5th above the tritone giving a
slightly tempered 27/25.
It looks like the optimizer in the 5-note case will fall into any scale
where every note is related to at least one other by a 7-limit just
consonance with only a very small amount of tempering. If we take that
amount of tempering as a tolerance, then for the 5 note case we'll find
relatively few scales distinct in terms of ratio names which are equivalent
within the tempering tolerance. The exceptions will be scales containing a
series of four fifths, two maj 3rds, three minor 3rds, three 8/7's, et c.
(anything that leaves a comma below a certain minimum size). Because all
the scales of this sort are listable (there are 6^4*4! of them I believe),
we could test this hypothesis by giving the optimizer each possibility,
look at how much it tweaks each one, and see if any of the random imputs
falls into a scale unique from these. Well, 6^4*4! is quite a large number.

jason