Recently I saw a book in which two researchers had made spectographs of every sound in various compositions ranging from full orchestra to solo cello. The interesting thing in these scientifically produced pictures is that they reveal a frequency range that tests the limits of human hearing.
I learned about overtones from having spent many years accompanying singers. Every day in the life of a singer releases a new mixture of overtones; the voice is alive and constantly changing as if daring the pianist to adapt to these shifting timbres. Listening for pitch is far from simple.
Too many pianists play as if overtones did not matter. Rubinstein, the master of overtones, could build a crescendo that felt like infinitely increasing intensity because he knew how to manipulate resonance.
For me the basis of a solid piano technique is touch sensitive enough to touch overtones without turning them into real tones. This requires strength, agility, infinite adaptability, and total familiarity with the instrument one is playing. The result is tone that sings the way singers sing, i.e., with the variability intrinsic to a living creature.