Greetings from soggy, sunny Florida.  The small 1" pneumatics to which
you refer are connected to the sustain valve via the tracker size "T".
Their purpose is to slightly reduce the value of the spring pneumatic
while the dampers are lifted.

The idea is that when the dampers lift there is less weight for the
finger pneumatic to lift in order to get the hammer to the string.
This causes damper lift to increase the volume of playing.  We already
know that the damper is 'not' a loud pedal, so these little pneumatics
decrease the vacuum needed to maintain the given volume at the time of
damper use.  Do not confuse this with the control size "T" found on
some Ampicos on the soft side of the pedal valve block.  That is tubed
to the wippin compensation pneumatics which remove lost motion from the
action when the hammers lift for soft play.

If this sounds confusing, I'm sorry because it really is not.  Over the
last few years I have rebuilt eight Ampicos and not one has been alike.
Each has had something different in the ever evolving player action.
All of these have been model A's, some very early and as late as 1929
with a 'B' pump and pedal valves.  Once you get the concept well in the
mind, you can figure out where tubing goes by what it does.

Some of these pianos have come in the basket form, that is, they were
taken apart by the owner and then the owner gave up.  However, most are
like they were when they last played and I always take pictures for
future assembly.

Ferdinand Pointer
Clearwater, FL