I was experimenting with Octave and NumPy/SciPy by synthesizing (reverse) sawtooth waves and decided to find out what a sawtooth comprised of harmonics with arbitrary phases would sound like. Note that all waveforms shown here were normalized before being previewed in Audacity (max/min sample value being 1/-1).
I began with two five-second clips of 1000 Hz sawtooth made up of 22 harmonics, one with all initial zero phases and the other with random phases. They look like this (zoomed in):
The most immediate observation is the erratic nature of the random phase waveform; how it appears to have a DC-offset. More surprising to me is that they sound, to the best of my ability to compare them, identical except for a slight difference in volume.
I decided to go all out and render two new PCMs, this time being 1 Hz with 22050 harmonics:
I listened to both of these, and predictably the zero phase PCM sounded silent except for the jump across the zero boundary once every second. But the random phase waveform sounded very noisey in addition to having some semblance to a one-beat-per-second sound.
My theory on this is that the all-zero phase sawtooth is an example of the least "information" loss due to destructive interference, hence the noisey and offset nature of the random phase sawtooth. Perhaps the normalization has hidden the extent of this information loss. Assuming that is true, I suspect that the mere 22 partials in my 1000 Hz wasn't enough to produce any audible noise in that case.
Is this theory about "information loss" correct?