When they vibrate, objects do not simply move backwards and forwards in one plane, nor in one simple pattern. When you pluck a stretched string the visible vibration shows an arc centred on the centre of the string. But a secondary hidden vibration produces two arcs, centred one quarter and three quarters along the length of the string.
Depending upon the characteristics of the string itself, and on how hard you pluck it, a complex of many different simultaneous patterns of vibration is set up. Each different segment of movement vibrates at its own frequency. The string may vibrate in 1, 2, 3, 4, etc. segments, and the respective frequencies of these modes of vibration are in the ratio 1, 2, 3, 4, etc. This complex of vibrating patterns makes up the characteristic sound of an object, which is defined as its quality.
Musical sounds are created by those vibrating objects which can produce a clearly dominating vibration (an identifiable note) and whose sub-vibrations are in mathematical, aesthetic sympathy with that note. Furthermore, the quality, or colour, of a particular musical sound is defined by its unique combination of sub-vibrations. Importantly, each sub-vibration is a note in itself, which relates harmonically to the main one.
All of the lesser notes in a musical sound make up what is known as its harmonic structure, and are individually known as partials or harmonics. The mathematical relationship between all these notes is constant in all musical sounds.
For example, the first partial has double the frequency of the main note. This particular relationship is known in music as an octave.
