[wwer]

> 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".

[AstralStorm]

That is way too simplistic as well, as this model is time invariant. Most instruments, due to physical nature of excitation, are variable in both frequencies and phases over time.

It is the main reason why modelling physically is the best way for realistic results right now - lossy lumped finite element models typically - digital waveguides are one of such models.

In such a model you can incorporate nonlinear damping and resonance functions over time at desired accuracy.

[djaychela]

I have a terrible maths background, so that's -way- above me.

However, I have a good grasp of why instruments sound like they do, so I'm hoping that your statement is a complex way (no pun intended) of showing that the waveform has many harmonics, and that those harmonics vary over time? Not looking for any kind of argument, just hoping for a bit of explanation of the above; From what I've learned over the years it's the balance of harmonics and the way that they change over time that gives an instrument its timbre and explains the difference in tone between instruments despite them playing nominally the same note (i.e. fundamental at the same frequency).