Musical sound exists simultaneously in several dimensions: in addition to time there are verticality and horizontality. Verticality is a function of overtones. These depend on timbre, among other things. A cappella voices, string quartets and certain well-tuned wind ensembles not in equal temperament produce clear vertical blends. The horizontality expresses tension against the vertical, an example of which is the so-called blue note of jazz, produced by the clarinet playing just slightly under pitch.
Some people believe that the piano furnishes a model of neutral pitch: I heard one conservatory graduate student say that the piano has "no overtones." How did he come to that conclusion? Or was he merely mimicking a non-fact he had been force-fed in an effort to avoid confusion?
I hear many pianists attempt to block overtones by humming loudly while playing or by playing always with the soft pedal. Not hearing the overtones is like being told to sit in the corner and not think about a big brown bear.