[birdbrain]

See my reply to your sibling comment - I do in fact mean spectral complexity at any given moment, not just over the course of the song. Many species of birds can and do vocalize with complicated timbre, including harmonic stacks, buzzes, clicks, and others harder to characterize.

And it's great that you bring up the possibility that two modes might be engaged. The avian vocal organ, the syrinx, has two sets of membranes which can vibrate as air is passed over them. Many species (particularly, as you'd expect, the ones best at imitation, like corvids, parrots, lyrebirds, etc) are able to control these two sources independently (but even those which generally don't control them separately can produce syllables with rich timbre), layering a harmonic stack with a click or a buzz.

I feel I should reiterate: nothing here is meant to detract from TFA's demonstration of what looks like a nice acoustic analysis tool. But TFA is, unfortunately, just plain wrong in its conclusion that birdsong is mostly pure tones.