Robert Linnstaedt wrote: "I hope Professor Liljencrants in his next
discussion can include a word about cylindrical vs. conical resonators.
I have wondered at the 'why' behind their special characteristics and
how one causes the nodes to aggregate at one end only."

I'm not the good professor, but I'll take a crack at it.  The resonator
of a beating reed behaves like a stopped flue pipe, only the reed is
the stopper and the top of the resonator is the mouth.

The air column in a stopped cylindrical pipe will resonate at half the
frequency of the same pipe if it is open.  To achieve the same pitch,
a stopped pipe is approximately half the length of an open pipe.
Furthermore, stopped and open pipes exhibit different harmonic
structure.  An open pipe will produce overtones at 2, 3, 4, 5... times
it's fundamental frequency, until the diameter of the pipe is too large
for higher overtones to be produced.  A stopped cylindrical pipe will
only produce half of the overtones, at 3, 5, 7... times the fundamental
frequency, again, until the diameter suppresses higher overtones.

When a stopped pipe is tapered, something amazing happens.  The
even numbered harmonics are also produced.  A tapered, stopped pipe
will produce overtones 2, 3, 4, 5... times the fundamental frequency.
A tapered and stopped pipe is about 1-1/2 times the length of a
cylindrical stopped pipe, or 3/4 as long as an open pipe.

Thus a cylindrical reed resonator like a clarinet supports only the
odd numbered partial series, while a tapered resonator, like a trumpet,
supports the entire series.

The reed tongue itself does not vibrate audibly.  It is simply a switch
that opens and closes sequentially to modulate a stream of air.  This
causes the compressions and rarefactions that the resonator supports as
standing waves.

John Nolte