--- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:

> I have recently had an extensive discussion with Margo Schulter
(for
> whom I would presume margo2.gif was made)

yes, but i specifically offered it to her not because of the position
of the global maximum of discordance (which i wasn't even looking at
at the time), but because it shows ratios such as 17:14 as local
minima of discordance -- margo likes to target such ratios in her
tunings and the fact that the original, smooth formulations of
harmonic entropy did not show local minima at such ratios led her
to "declare her independence" from harmonic entropy -- so i offered
this graph to her as a reconciliatory gesture. the newer, "pointy"
formulation of harmonic entropy did not originate from margo's
objections, though, but rather from the research of joos vos -- you
should search for "vos" on the tuning list and see my 4-part summary
of the beginning of one of his articles -- i'm sure you'll be
fascinated . . .

> > > Perhaps there is not enough resolution in the vertical
direction
> in
> > > your graph -- 7:6 and 9:5 barely show up as consonances -- and
a
> > > higher resolution might make a difference for 8:7 and 9:7. (A
> bump
> > > in the road isn't going to show up on a topographical map, but
I
> sure
> > > feel it when I drive my car over it.)

given the topographical analogy, i guess you'd be ok with a "fractal"
curve, which has a local minumum of discordance at *every* rational
number, but most are invisibly small?

> > it sounds like you'd probably prefer the "pointy" formulation as
in
> > margo2.gif above -- let me know.
>
> Yes, this is very much what I was looking for.
>
> > there's also the s (resolution)
> > parameter which can always be tweaked to give more or less
> importance
> > to more complex ratios (the better your hearing resolution, the
> more
> > easily you can identify the complex ratios "as such", because the
> > complex ratios are in more "crowded" areas amongst all the
ratios).
>
> For that we can refer to my graph:
>
> http://groups.yahoo.com/group/tuning-math/files/secor/consonce.gif
>
> It appears that the point at which I no longer heard n:d as a local
> consonance is when n*d reached a value around 150 (with 16:9 and
> 17:10). So margo2.gif would have a little more sensitivity than
what
> I observed.

i think you're misunderstanding how s works here in the vos-based
curve. also, my question above would seem to be relevant again --
could it be that you're simply not noticing tinier and tinier bumps
in the road, corresponding to more and more complex ratios?

now, notice that in margo2.gif, the most discordant interval not near
80 cents is between 1100 and 1200 cents, and the most discordant
interval not near either of these is between 700 and 800 cents. this
seems very promising for being able to attain the specifications you
requested. however, please note that "near" in this context doesn't
mean quite what it meant in the original, "rounder" harmonic entropy
formulation -- even very near to the extreme peaks of discordance,
there can be significant interruptions in the "plateau", 14:9 being a
perfect example for margo2.gif.

anyway, i'm now computing a version of margo2.gif where an s of 1%
will be assumed -- should be hot off the grill within the hour . . .