Sympathetic resonance in a piano is the phenomenon of one string being excited, transmitting its sound to other strings that will then start vibrating if they have common frequencies.
For example C2 strings will have a fundamental frequency F and harmonics n times F, n being an integer. And C3 strings will also resonate after C2 is pressed, when sustain pedal is down (i.e. when the strings are free to ring) because C3 fundamental is C2 first harmonic.
My question is this. The strings that will resonate in sympathy have common frequencies with the strings that were hammered. But they also have frequencies that are not harmonics of the initial strings excited.
So will these other frequencies also happen or will the strings only resonate in a limited number of modes? Is that even possible? Can modes of a string be excited separately?
Beware. I am not asking about a simplistic mathematical model of the strings. I am asking about the real piano.
Not a physicist, but familiar with acoustics.