RE: [tuning] tetradic model [harmonic entropy]

I wrote,

> > A true triadic or tetradic
> >harmonic entropy model (rather than a sum of dyadic harmonic
entropy
> >models, which is what this is) would take this into account."

Joseph Pehrson wrote,

>So, would this be possible to do??... or is it too difficult to model
>mathematically?

A few months ago I posted some ideas and graphs involving Voronoi cells (see
http://www.egroups.com/files/tuning/triads.jpg) which would be an approach
to solving this problem. Unfortunately, I can't do the calculations at the
moment, because, though Matlab can give you the set of line segments making
up the full spread of Voronoi cells for a set of points, it provides no way
to then find just the edges of the Voronoi cell corresponding to a single,
selected point. Perhaps I just need the help of a clever programmer on this
list . . .

>And, how different would it be than the
>diadically-inspired model that Paul just did??

Quite different -- most notably, observe how the model I just presented
gives any chord the same rating as its mirror "inversion". So utonal and
otonal chords of the same identities come out with the same rating. The true
chordal harmonic entropy model would find the otonal chord to be more
concordant (lower entropy) than the utonal chord. Look at
http://www.egroups.com/files/tuning/triads.jpg. The three red dots are the
major triad, in root position, first inversion, and second inversion. The
three blue dots are the minor triad, in root position, first inversion, and
second inversion. Notice how the red dots have more space around them than
the blue dots. Just as in the diadic case, where simple ratios have more
space around them than complex ratios, here the major triads have more space
around them than the minor triads. This has got to be related (though I
don't have a proof) to the fact that the major triads (4:5:6, 5:6:8, 3:4:5)
use simpler numbers than the minor triads (10:12:15, 12:15:20, 15:20:24)
when expressed otonally, and remember, it is the otonal representation that
is relevant for harmonic entropy -- since harmonic entropy means "how hard
it is for the brain's central pitch processor to pin down the fundamental of
the harmonic series that the chord comes from".