Periodic Sounds Must Have Harmonic Structure

Periodic sounds (sounds with waveforms that have a repeated "motiv", as in the blue trace shown above) will have Fourier spectra which always must consist solely of "harmonics" of the sound. Harmonics are sine waves with periods that are integer multiples of some fundamental period. The red lines above are "cosine phase" harmonics of the blue line. When thinking about Fourier spectra, we want to imagine the blue line being made up of a sum of lots of sine waves like the red and green lines, where we might adjust the phase and amplitude of the sine waves as required. The important thing to note here is that, no matter how we would adjust the phase and amplitude of the green line, it could never be part of the mixture needed to make up the blue line. The reason is this: compare the values that the waveforms have at identical points in the period, for example by comparing the points marked by the stippled gray lines. The red lines will always contribute the same values at each cycle (here for example they are always maximal at the periods marked out by the gray lines). In contrast, the green line does not "fit" an integer number of cycles into the fundamental period, and the contribution it would make to each cycle of the wave would therefore be different, which would destroy the periodicity of the wave. The green line can therefore not be a Fourier component of the periodic blue sound wave. Nor can any other sine wave that has a period which is not a harmonic of the fundamental period of the sound.