Has anybody any numerical data on how the force of a simple "board" 
pneumatic varies with the opening?  I've derived an equation, but 
it's fairly complex and I'd sure like to check it with actual 
measured numbers.  

This data is needed to describe the effect of the hammer-lift 
"Soft Pedal" in a grand piano reproducer.  When the hammer rest 
rail is lifted it cuts the hammer travel distance to one-half.  
If the pneumatic force were constant that would result in 3 
decibels of sound volume reduction.  But my ear and microphone 
indicates its actually more, and the reason is because the 
available pneumatic force is much less as the device closes.

-- Robbie Rhodes