COMMENT ⊗   VALID 00004 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	∂15-Aug-79  0135	DON  	rooting around
C00004 00003	∂16-Aug-79  1407	JAM  	rev 
C00006 00004	∂17-Aug-79  1600	JAM  	Re-re-reverb  
C00008 ENDMK
C⊗;
∂15-Aug-79  0135	DON  	rooting around
I think you're out of luck, but I'll let you decide for yourself.  My Complex
Analysis book is in your mailbox (please leave it in mine when you're done with
it).  The relevant theorem is no. 6.1 on pages 216-217 (Rouche's theorem), and
states that a suitable contour integral can be used to find N - P, where N is
the number of zeros (roots) inside the contour and P is the number of poles.
Since no self- respecting polynomial has any poles anywhere except at infinity,
P = 0 in your case.  The contour to integrate along, of course, is the unit
circle; this is usually done by substituting exp(i*theta) for z and integrating
for 0≤theta≤2π.  On the other hand, it's not at all clear to me that you'll be
able to perform the integration.  Even for a simple case like p(z) = (z-a)(z-b),
it turns out that you want to integrate dz/(z-a) + dz/(z-b), and when you
substitute exp(i*theta) I'm not sure it can be done analytically.  (It probably
can't be, since the resulting definite integral would have to be discontinuous
at |a|=1 or |b|=1.)  Example 6.1 on pages 221-222 may help, but I'm dubious.

Good luck!

∂16-Aug-79  1407	JAM  	rev 
Hmmm. Well, if you think about it, you will realize that "the cans
mouths facing each other" only gives you a few more fundamental modes
of vibration. Anyway, echo density is not the problem - it is all plenty
dense enough, much more so than the ear can distinguish. For any case
but pure impulses, I have yet to meet anyone who can distinguish the
"tin can" model from the pure noise model. I don't think we have to
do any more on it for the time being. Anyway, about the stability of
those feedback things, the value of the root with the largest magnitude
can always be determined by running off the impulse response and watching
the decay - the largest magnitude root will always dominate in the long
run, unless there are several of equal magnitude. After the approximate
magnitude of the largest root is determined, you can stabalize the
filter by multiplying the coefficients by (ε/r)↑n, where n is the
coefficient number, r is the magnitude of the largest root, and ε is
any number less than one. (hmm. I think N-n is really the coefficient
number). Indeed, this is roughly the only way to stabalize a high-order
polynomial like that.

∂17-Aug-79  1600	JAM  	Re-re-reverb  
One thing occurred to me: any combination of filters and feedbacks
around a delay does not (necessarily) increase the number of roots,
or natural modes. All it can do is move them around. What we really
need, to be more like real rooms (if that is the goal) is some way
of exploding the number of natural modes. In real rooms, the modes
are also direction dependent, so that you will find different groups
of resonances in different directions. I don't see any way of increasing
the number of modes without increasing the number of multiplications,
except if one just uses longer delays everywhere.