Actually, the number of times a piano with new strings needs to be
tuned depends greatly upon the procedures used to bring it up to
pitch to begin with.  I'll try to explain.  Most amateur tuners (and
some pros, for that matter) go about raising the pitch completely
wrong.  Be it an old string or new, there are a couple of things that
should be kept in mind.

I was taught to set the temperament and then tune the piano to itself.
That thinking is wrong for pitch raising, and here's why.  How many
times has a tuner tried that and ended up with the treble progressively
flatter as you go up the scale?  A piano in such a state will have to
be tuned several times to correct that.

I offer this explanation: In the case of a piano with old strings, the
piano may fall a key or more flat during many idle years until someone
comes along and pulls it up, putting increased pressure on practically
everything in the strung back.  But the main reason is that part of the
"dead" section on the lower bridge gets pulled up into the speaking
portion of the string.  The slight kink in the string then slowly
straightens out under the increased tension, and then the pitch falls.

A similar thing happens with a newly strung piano: the beckets will
tighten under pressure, and the strings will adjust to the loops and
bends on the hitch pins and bridges, also causing the pitch to fall.
Either situation calls for a different approach than usual tuning.
I use an electronic pitch generator to set the middle octave, then
I adjust the rest of the octaves to the tone generator, _not_ to the
temperament octave.  While I'm pitch raising I _never_ tune any octave
to the temperament octave, because the temperament octave will have
already fallen in pitch before the next octave can be set.  Always keep
in mind, every string on the piano will fall slightly while tuning, and
almost to the same degree.

New strings will stretch to a certain degree, especially new bass
strings.  Some people think that such a piano needs time to "settle"
first, but that's hogwash.

Once the piano is very close to where it needs to be, then -- and only
then -- will I tune it to itself.  In most cases I can get a piano to
stay to pitch in four tunings, including a newly strung piano.

Andy Taylor