I know what most of you are saying. But it is from the playing of the instrument viewpoint. There are those of a technical bent who start with the maths and understand from that known position. Mine is steeped in Fourier analysis. What you call octave I know as harmonic or "multiple of".
The first stop for me IS the actual frequencies or combinations thereof. What you call the tone of an instrument's chord I see as non-synchronous frequencies that the western ear has codified as approximately pleasantly related (12 notes does not cover all true sharps and flats). And those chords include a certain non-linear mixing of two or more frequencies - which I know as sum and difference frequencies, and their harmonics also and sum and differences of all above ad nauseum. I hear a chord, but I perceive the gamut technically. It is still as pleasant (or not sometimes) but far more interesting to me.

Now it mayhap I go no further than the spreadsheet - the analysis has a different purpose. But I am still stuck with two melodeons, The toy (in red) in C and an old beasty from Saxony in A (or maybe A# as Pete Grassby put it). Transpositions are a must! And you can see how I will do it.

Anyone remember the Kippers and their "tremolodeon"? it used a wobbly pendulum creating vibrato. Or was it a wheel with an eccentric weight?