This is a fascinating question with a much more complex answer than
anyone would suspect.  The problem is what is actually in tune.  Bernt
Damm stated that all pianos left his shop "in tune", but I'll bet he
has no idea that there are over 150 different tunings for a piano.
Odds are good his tuner used equal-temperament (or at least as close to
that as the tuner was able), and Bernt's ear is good enough to realize
the equal-temperament doesn't sound great -- the chief advantage is it
sounds okay everywhere, but never great.

 [ The pessimist would say that the equal-temperament tuning sounds
 [ equally bad in any key!  -- Robbie

If you just want a simple answer, tell your tuners to use Bill
Bremmer's EBVT-III.  It sounds good, and seems ideal for player pianos.
See http://www.billbremmer.com/  (He has sound samples, including some
from an Ampico player).  If your tuner cannot handle this I'd find one
who can.  I'm sure many tuners are angry at this statement -- but I don't
think they know music well enough to touch my piano.  (If you find the
rare tuner that recommends a different temperament I'd let him try it
once, and let your ears judge.)

For a more complex explanation:  In modern times we have defined A to
be 440 Hz, and that is that, but before 1900 (someone is sure to give
the correct year) a tuner could use whatever note sounded good.  Often
who ever tuned the local church organ choose something easy for that
organ, and everyone else in town used that as standard.  The next town
could be very different.  If you have an older music box (or other
permanently tuned instruments) you are probably well aware that you
can't play along with it and sound good.

The note an octave above A=440 is A=880 Hz, the octave below is 220 Hz
and so on; just multiply or divide by 2 and you get the frequency of
the octave.  If you take a piano and tune the A string, and turn by
4ths and 5ths, and you will eventually tune an A on a different octave,
if you compare that A to the lower one you will discover it is not a
true octave (and doesn't measure 880 Hz)!  If instead you tune by 3rds
you will get still another note.  Tuners need to compromise their
tuning so that all three possible notes can exist one key.

Musicians today are starting to admit that the mathematically pure
equal temperament sounds bad.  However it is very hard to find a piano
tuner who understands this, much less one who can actually tune any of
the 150 different temperaments that have been used.  Electronic tuners
help here, the better ones will let you select a temperament.  For a
more detailed discussion read "How Equal Temperament Ruined Harmony
(and Why You Should Care)"

  http://www.amazon.com/Equal-Temperament-Ruined-Harmony-Should/dp/0393062279 

The above assumes a piano or other full-scale instrument.  Many of our
mechanical instruments play an abbreviated scale with missing notes.
It should be possible to devise a temperament for them that takes
advantage of the useless missing notes to sound better.  I'd be curious
to know of anyone's experiments in this direction.

Henry Miller