Well, fft as such doesn't "track" anything. All you get is energy and phase for a certain band over that frame. From this we can quite easily determine the most prominent frequency, but as you know that need not be the fundamental. It quite likely is the fundamental for many signals but we don't *know* this and without more knowledge about the piece we can't say what the root of a certain chord is, even if we could manage to separate the notes. Pitch trackers often get very confused by chords (and may output all sorts of musically interesting "garbage" in response).

What we can do -given enough frequency resolution- is try to find a series of harmonics and use that to calculate what the fundamental must have been. Harmonics will be integer multiples of the fundamental, after all. This is what our hearing psychology does when listening to something like your oud. It's also why a HPF set loose on a full mix by a DJ in a club will sound less dramatic that a LPF; it's far easier for us to imagine the missing elements in the first case than the second.

I think that might've answered your question but I'm well aware that it only raises more. There are lots and lots of questions here. Nobody knows all the answers, in fact we are sure we will never get perfectly accurate instantaneous pitch-tracking. I'm not even sure that would be as useful as making good guesses based on how sound is experienced by our hearing psychology; as noted the perceived frequency, the most important thing, may not actually be there in the signal at all. In a way it's a bit of a tragic task, but of course lots of fun are still to be had.

Kas.