Harmonic Ratios - how they arise, some examples

In the case of a tensioned musical string, the vibration of the string is caused by wave motion. Waves travel along the string in both directions and are partially reflected at the boundaries at the end of the speaking length - the part of the length that vibrates. These waves create standing waves on the string, which are wave shapes that do not travel along the string, but do oscillate perpendicularly to the length of the string. Any such standing wave must have two ends, one at each end of the speaking length, and a whole number of half-wave shapes between the two ends. This is just a consequence of simple geometry.

The figure below illustrates a simple wave. The horizontal straight line is the string at rest. The curves are possible shapes of the string, in a snapshot in time, when it is vibrating, the vertical displacement being greatly exaggerated.

Points along the string between the two ends can be displaced, but the two end are fixed (more or less). 

Points between the two ends can be displaced, but the two ends cannot (except for a tiny amount of movement at the soundboard bridge on a musical instrument). The wavelength is the distance from the beginning of the wave on left, up into a crest, down into a trough, and back again to the horizontal line. The only wavelengths that therefore can form on the speaking length of the string, are

1. Twice the speaking length of the string;

2. The speaking length of the string;

3. Half the speaking length of the string;

4. One third the speaking length of the string;

5. One quarter the speaking length of the string

... and so on.

The relationship between wavelength and frequency is given by  

where is frequency, is the velocity of the wave (this is the velocity of the traveling waves that produce the standing wave) and is the wavelength. Because the waves travel at the same speed (approximately), the frequencies of the waves are in the ratio 1:2:3:4 ... and so on. Thus the partials fall approximately in an harmonic series, and have harmonic ratios between them.

Fourier

There is, however, another interesting derivation of the harmonic series at the highest, most generalised level. The discovery of harmonic ratios in strings (as Pythagoras is said to have done) is in fact just a discovery of a specific instance, in one particular phenomenon, of a higher, generalised mathematical principle.

Fourier (1768 - 1830) showed that any bounded, periodic mathematical function, is equivalent to the sum of an infinite series of sine functions whose frequencies are arranged in the harmonic series, as f, 2f, 3f... and so on. This means that any phenomenon involving bounded periodic changes, can also involve harmonics.

To be periodic, a parameter must keep repeating its pattern of change exactly, in relation to another parameter, which may be time. In the case of musical strings, their vibrations are not quite periodic in time. This results in a series of simple vibrations with frequencies that are approximately harmonic. Fourier showed that bounded functions that are not periodic, are still equivalent to an infinite integration over all frequencies, which is equivalent in many cases to a set of simple functions, but not necessarily related as an harmonic series.