It happens because our ears and our brain are not doing the Fourier transform, even the fast one. We hear the shape of a wave in its entirety, not as a spectrum of harmonics.
This is actually not exactly an accurate premise. In fact our inner ears ARE doing something very much like a Fourier transform. When a soundwave reaches the fluid in the cochlea through the ossicular bones rapping on the cochlea's oval window, the basilar membrane of the cochlea responds according to place. A complex wave has multiple frequency components and each of those components most excites a particular point on the basilar membrane where the frequency displacement is greatest and different neurons are fired at different locations along the basilar membrane corresponding to these separate peaks. It's modelled as a series of auditory filters. In fact it's very similar to a Fourier transform. Our hearing does in fact work by breaking down a complex sound into component frequency elements that each fire different sets of neurons.
I don't know what's happening with the square wave example in the video, I haven't watched it. But any explanation of the effect based on the assumption that our ears and brain are not doing anything like a Fourier transform or are not separating the wave form into component frequencies and responding with different nerve impulses corresponding individually to the separate frequency components, is incorrect. A sine tone at a single frequency activates one place on the basilar membrane but a complex tone activates multiple places that drive separate neurons to fire in correspondence to each separate frequency component.
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