Hello all.  Thank you for so many on-line and off-line responces!
Okay, I understand what you mean by the Celeste and Musette effect.
I have recently tuned an accordion.  There, I think the detuning is
not constant in cents for each reed?

I don't think that the Celeste effect is the "magic" answer, though,
as it sounds exactly as it is equal to somewhat artificial.

Equal temperament is the problem, you say?  I tend to agree a bit.
At our shop we had three tuners and they all tuned pianos somewhat
differently.  I am talking about that first octave.  One tuner managed
to make that sound absolutely fantastic to my ears, compared to the
other two.  I was always able to tell just by playing some notes and
chords, who tuned the piano.  None of them understood what I meant with
the "magic", though.

I bought an electronic tuning machine from Renner, Germany, and at some
stage, when one of "those" pianos was available, I tried to measure
each strings pitch and write it down.  I gave up after some hours,
though, as it was too tedious and because everything was off by heaps.
It was not like the Celeste, with a string being a few cents high or
low; it was all off by quite a bit.  I concluded that, because the
whole pitch of the piano was off, it was near impossible to simply
measure a "detune pattern".

It is not necessarily the hammers, either, as I have personally tried
facing them and making them mellow soft and the "magic" was still
there.  I have also tried putting drawing pins into some hammers,
mandolin rails, etc., without any success of duplicating the "magic".

Octave spread [stretch]?  I don't know, but I have heard pianos where
the spread was either dead (not enough) or glassy and terrible (too
much).  None ever sounded like the "magic".

In short, I personally, together with the tuners in our shop, never
came to a duplicable conclusion.  And then there is still that Fats
Domino piano.  Hmmm...

Best regards,
Bernt Damm
Sydney