Gordon Stelter has responded to my comments and tactfully posed
some good questions that I may not be able to answer except in
a speculative manner.

I think that most of us are fascinated by the designs of these
mechanical machines.  Indeed, they did represent some of the best
engineering of their period.  In a delightful mix of science and
experimentation, these machines present a very broad approach to
accomplish the goal of producing music.

The common denominators all of them must adhere to are the laws of
physics.  Specifically, Force is equal to the product of Mass times
Acceleration: F = M x A.  When applied to closing stack pneumatics to
play a note, the force needed to move (accelerate) the pneumatic comes
from the pump and atmospheric pressure.  The pump removes the air
within the pneumatic, and the greater atmospheric pressure squeezes
the pneumatic closed.  Calculating the strength of the "squeeze" is
dependent on two factors: The differential pressure just mentioned,
and the surface area (size) of the working area of the pneumatic.

The differential pressure isn't applied instantaneously.  It requires
time to exhaust the pneumatic.  This brings mass flow rates into the
discussion.  The valve gap and valve size become important in
determining flow rates.  Most Duo-Art pedal pneumatics have two valves
working together.  Some even have three valves giving a very rapid
exhausting action.

Typically, as the size of the pneumatic is increased, working area
increases.  That is why larger pneumatics are stronger than little ones
at the same operating pressures.  At this point my head is spinning
with thoughts of dynamics and statics, both needed to fully analyze the
player systems.

So, ending the discussion before it gets too boring, player pianos
don't work on the moon.  Coming back to Earth, size is important.
Smaller stacks are more attractive and may have smaller pumps.  The
6-inch Ampico pneumatic works just fine if the flow through the valve
is not restricted and the pump is not "maxed out".  I think the switch
to smaller pneumatics was a business decision to make the player look
better.  I no longer own any Ampico with the larger pneumatics,
whatever that means.

I suppose someone could plot the curves using the variables above,
finding where everything crosses, and then declare the optimum size.
Right now, I don't want to spend my time doing that, but it could be
interesting.

Bob Taylor
Missouri