The ear converts a serious of pressure variations, that is a sound wave into a Fourier-analyzed signal traveling on nerves to the hearing center of the brain. In a highly idealized model of the ear, each frequency of sound wave corresponds to one neuron leading from the ear to the brain. For example, if a sound wave were to enter the ear consisting of two frequencies f 1 and f 2 , then two neurons would be excited, one corresponding to f 1 and the other to f 2 .

A physical ear is more complicated than this model, however, and these differences from ideal can be observed by simple experiment. For instance, if a sound wave of two very similar frequencies enters the ear, the brain hears not two frequencies but one average frequency which slowly turns on and off. The turning on and off is called beats, and the beat frequency is the difference between the two frequencies: f beat = f 1 – f 2 .

Another similar example involves a sound wave of two frequencies, which are not similar but have some harmonic relationship. In this case the brain sometimes hears a third tone, a difference tone, corresponding to the difference of the frequencies of the input: f 3 = f 1 – f 2 .

This seemingly unfortunate phenomenon was a boon to the listeners of early phonographs. The phonographs were not really able to reproduce the lowest frequencies in the music, corresponding to the fundamental of the notes being played, although they would reproduce the harmonics. Often the ear would reconstruct the difference tone which would be the missing fundamental, making it seem as if the phonograph reproduced sound better than it in fact did.

(i) On a piano, someone plays the notes B 0 (30.87 Hz) and C 1 (32.70 Hz) simultaneously. A single note is heard beating. What is the frequency of the note which is heard to beat?