> In fact, this is part of why a piano sounds like a piano and guitar sounds like a guitar.
My understanding is that it is a bit more complex than that, literally!
The the final waveform is a not just
W = sum(a_i * f_i)
= Psi
where a_i is the amplitude and f_i are the fundamental frequencies.
It is actually
W = sum(a_i * f_i + sqrt(-1) * (b_i * f_i))
= Psi + i * Phi
Loosely, the imaginary part plays a significant role in making an instrument sound like it.
Of course, the brain fills up a lot of stuff that is still a mystery but the elec keyboards can set the a_i, b_i to change from "guitar" to "reed organ".