I did not respond to Michael Concordia's request for advice in quieting
a piano, as I felt that there would be a flood of answers that would
make my response unnecessary; however, I have not yet seen my answer
published, so here goes.

The most common way to quiet an upright piano is to simply prop up the
hammer rail so it is on continuously.  If more quieting is desired,
simply hang a sheet of cloth or felt across the strings where the
hammers hit, which is the same principal as a rinky-tink or honky-tonk
rail, except that they use metal clips hanging on the cloth.

Different cloths will give different muting, so you can experiment.
Grand pianos are more difficult to mute, but you can try to simply lay
pieces of felt on the strings.  Too much felt will mute the strings so
much that each note thunks instead of plays, but a little
experimentation can produce acceptable results.

Another technique might be to wedge foam in between the soundboard and
the beams.  Unlike the other suggestions regarding foam, this suggestion
has the foam actually preventing the soundboard from vibrating as much
as previously, instead of simply absorbing sound waves from the back of
the piano, and not doing anything about the sound waves coming off the
front of the board.

The best thing about these suggestions is that they are easy to
install, or modify or remove without permanent changes to the piano.

Randolph Herr