Another thing to consider is that each instrument produces a (different) variety of 'overtones' which are multiples of the fundamental pitch. (If they didn't, they would all be producing pure sine waves, and all sound EXACTLY the same!)

The first overtone is double, and is an octave above the fundamental, so no problem there. The second is three times the fundamental, and is equal to an octave and a fifth (Natural, not tempered!). This may therefore, if loud enough, clash with a tempered fifth being played.
The third overtone is four times, and is another octave. The fourth is five times the fundamental, and APPROXIMATES to a third, if I remember my calculations correctly. But this and higher overtones are usually faint enough not to clash.

This also provides part of the explanation for the old 'valve versus transistor amp' argument. Overloaded transistor amps used to introduce more third harmonic (=second overtone) distortion, which is quite noticeable.