Return to my Physics pages
Go to my home page
© Copyright 2002, Jim Loy
Above we have two sine waves and their sum (in blue). When rendered into sound (an electronic instrument may just move the diaphram of a speaker back and forth in time with the oscillations of the above graph), sine waves produce a particularly pure sound, roughly similar to that of a flute. The two sine waves above are an octave apart. Their sum (what we hear when we listen to two tones an octave apart) has the attributes of both tones. Essentially we hear a slightly richer sound with the same frequency as the higher tone.
Above is a fifth, which is a much richer sound, as you can guess by looking at it. A fourth looks very much the same, as it is an inverted fifth.
Above is a minor sixth, which is an even more interesting.
In reality, our musical instruments (including the human voice) do not produce a pure sine wave, nor do we want them to. They are instead much much richer than any of the sounds shown above. Each instrument has harmonics. In essence, this means that each instrument is playing several tones simulataneously (some louder than others), in harmony. Below, we see the sum of two somewhat richer tones, a fifth apart. Compare it to the fifth above. There are the same number of peaks and valleys, but the shapes are all different. And the sound is much richer. Real musical instruments produce even richer sounds than this.
I drew these graphs using Geometer's Sketchpad.