Frequency Ratios

When you hear a pleasing musical chord, you are hearing simple frequency ratios. A perfect fifth — like C and G played together — has a frequency ratio of 3:2. An octave is 2:1. The simpler the ratio, the more consonant the combination sounds to most human ears.

According to tradition, Pythagoras and his followers are credited with noticing this link more than 2,500 years ago, supposedly while experimenting with strings of different lengths. Pythagoras left no surviving writings, so the story comes to us only through much later accounts — and its most famous version is physically wrong. The acoustics, though, hold up: a string half as long vibrates at twice the frequency, producing a note one octave higher. Music's beauty, at its root, is the beauty of small whole-number ratios. Helmholtz later showed in 1863 that dissonance arises from clashing overtones between complex ratios.