As I begin to read this paper I am distressed to see no reference to equitempered scale.
There are two natural sources of harmonics which are small integral multiples of a single fixed underlying frequency: The eigenvalues over a disk in the plane, however, are roots of the Bessel functions Jn(x) and these are not integers, nor are their ratios rational. The ‘musical’ drum has these for harmonics.

Another independent common source of integral overtones results from from non-linear vibrations which are periodic. Fourier showed that periodic functions can be expressed as a sum of sinusoidal functions whose frequencies were integral multiples of the fundamental. The human voice is a non-linear oscillator.

equitempered scale,