The harmonic product spectrum (or harmonic spectrum for short) is the
geometric mean of amplitudes of the harmonics associated with a particular
frequency in the spectrum:

where HPS is the harmonic product spectrum, *k* is the *k*-th
indexed frequency bin in the harmonic spectrum, *Y* is the magnitude
spectrum of the positive frequencies, and *N* is the number of
harmonics to consider.

The following schematic demonstrates how the harmonic spectrum
is calculates. The original spectrum is squeezed so that each successive
harmonic of the original signal is aligned with the fundamental
harmonic. To do this, the spectrum is squeezed by 1/2 to align the
first overtone with the fundamental (it doesn't matter what
frequency the fundamental is at because the first overtone is always
twice the frequency of the fundamental). Then the spectrum is
squeezed to 1/3 to align the second overtone with the fundamental, and
so on (typically for about 5 harmonics).

Here is a comparison of the plain magnitude spectrum
and the harmonic product spectrum:

The nice property of the harmonic spectrum is that it gives
a simple estimate of the pitch in the audio signal.