[rspeer]

Close. Each shift of an octave doubles the frequency, so octaves are specifically harmonics that are powers of 2. Other harmonic ratios make other intervals (including intervals that aren't in our scale, when you get to bigger prime numbers).

Most instruments produce harmonic multiples of the base frequency. (Percussion instruments don't.) So if you play a note at 110 Hz, there are also pitches at 220 Hz, 330 Hz, 440 Hz, etc. contained inside it. If you play a note an octave up, at 220 Hz, it contains pitches at 220 Hz, 440 Hz, 660 Hz... which is similar enough that it sounds like "the same note".

(This doesn't explain why 330, 660, 990, ... sounds like a different note, though. Maybe it doesn't to an untrained ear.)

[dharma1]

I haven't tested this, but my intuition is that any musical sound with harmonic (as opposed to inharmonic) partials will sound like a single note, if the first fundamental is clearly the loudest, and the harmonics follow some kind of natural distribution in terms of loudness and temporal decay.

Two notes played in an octave would be quite distinct since you have a clear first fundamental for both notes, and the loudness/decay of the harmonics is slightly different between both notes