I have learned a great deal about pitch approximation from Beethoven's Sonata for (Natural) Horn and Piano, Op. 17, which I performed the other evening for the 5th time with my son on horn. There would be no reason to compose such a duo if there were not the possibility of revealing something about the mutual resonance of both instruments.
Using that as the starting point, and reading Beethoven's phrasing and dynamic indications with insight, I find my instincts about the piano once more affirmed: i.e., that, contrary to what we are told, the piano is not a pitch-specific instrument. Rather, it is an instrument capable of taking into its resonating chamber, via the damper pedal, vibrations from any source and allowing them to clarify by a physical process over which we have little control, but which is ours to savor and respond to.
With this as the definition of a sonata all playing is transformed. This piece in particular, with all its danger and unpredictability, becomes highly engaging. On a modern horn it would be simplistic, a piece of cake; it would have no inherent interest.
Many years ago I began thinking of the piano as a "translating" instrument, one that speaks with and for other instruments, that takes them into itself and transforms them. That doesn't include the piano that Chopin had in mind, of course; his music is the piano.