RE: [tuning] RE: Plant cells under a microscope?

David Canright wrote,

>It's a beautiful and fascinating figure. (I've always been partial to the
>look of Voronoi diagrams.) So, the largish cell near the southwest end of
>the main vein would be 5:6:7 ...?

Yup!

>Your limit of 64 seems a reasonable stopping point, but do you really think
>all those triads are really distinctly recognizable to listeners, even
>well-tranined ones? I would guess not.

You're right, I wouldn't. That wasn't the point though -- I gave my reasons
for doing this graph a few days ago. If you've been following the harmonic
entropy discussion, the point is to let the limit go to infinity and then
use the harmonic entropy function to give a dissonance measure over the
plane. You end up with lower dissonance where the cells are bigger and
higher dissonance where they are more crowded. You only get a few local
minima at the simplest ratios -- how simple is a function of the value of
sigma you assume for the central pitch processor of the listener (I usually
use 1%).

>If you agree, how would you reduce
>the number of triads to more distinguisable ones?

Again, that isn't the point, but I've discussed (on this list with Dan
Stearns) that in the dyad case, a nice answer to this problem comes out of
just looking at the dyad that is given the largest probability at each point
on the x-axis. So in the triad case, a nice answer to this problem comes out
of just looking at the dyad that is given the largest probability at each
point on the x-axis. You'll get the simple ones that correspond to local
minima of the harmonic entropy, and you'll also get a similar number of
slightly more complex ones, whose borders are a highly sensitive function of
the input parameters.

I wrote,

>So in the triad case, a nice answer to this problem comes out
>of just looking at the dyad that is given the largest probability at each
>point on the x-axis.

I meant, "at each point on the x-y plane".