You have an awesome intuitive understanding of how this works physically! Joseph Fourier is officially my favorite genius from history.

I think what you are suggesting would work perfectly but our formulation of a Fourier analysis with Real numbers would prevent us from seeing the pattern even if we were close. Part of the reason for this is because we use e and pi in most of our mathematics. Both are irrational and on top of that, the natural log just isn't a number born of any geometry.

Our Real numbers are all approximations. As a result, our mathematics has trouble representing things that are physically real. From what I have read, this includes music. Perhaps you already know this but, Real numbers can't represent musical scales exactly. I am still learning quite a bit but I've turned to music theory to look for physical inspiration on what our mathematics "should" look like. I very much look forward to learning more.

Thanks for your comment!