A fundamental-less E0 "note" (as in, a bundle of specific partials, namely E1-B1-E2-G#2-B2-D3-E3 ...) does sound different from an E1 "note" precisely because the latter's harmonic makeup is different (as evident from the first two-octave chunk, i.e. E1-E2-B2-E3 and...that's it).This was discussed before (a missing fundamental harmonic), and I think it’s not so much that it sounds the same, or that your brain fills in the missing fundamental. It’s more like the note is recognizable even if the fundamental is missing.
So, a low E0 sounds like the same note even if the fundamental is missing, and not like the note E1 an octave up, due to the intervals on the harmonics. However it’s timbre is different, thinner or more tinny, when the fundamental is missing. It is surprising though, since you might think that an E0 with the fundamental missing would sound the same as the E1.
I listened to a bunch of YouTube “missing fundamental” demonstrations to convince myself……
Specifically, the former sounds an octave below: it gets its pitch assigned by the brain to "E0", and the brain does fill in a missing fundamental, triangulating where to place it from the pitch intervals between the partials it actually hears.
Our noggin is capable of playing us this trick at any register, of course. Ask someone to whistle a held note (better if in the lowest register of whistling this person is capable of, with the tongue in the "oo" position - it is a tone reasonably close to a sine wave), then add a whistled note a perfect fifth above: in my experience, a virtual fundamental an octave below the other person's note immediately appears.
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