Chords

Two pure tones played together exhibit beats at the difference of their frequencies—a consequence of the fact that

Sin[ω_{1} t]+Sin[ω_{2} t]==2 Sin[(ω_{1}+ω_{2}) t/2] Cos[(ω_{1}-ω_{2}) t/2]

With ω≃500 Hz, one can explicitly hear the time variation of the beats if their frequency is below about 15 Hz, and the result is quite pleasant. But between 15 Hz and about 60 Hz, the sound tends to be rather grating—possibly because this frequency range conflicts with that used for signals in the auditory nerve.

In music it is usually thought that chords consisting of tones with frequencies whose ratios have small denominators (such as 3/2, corresponding to a perfect fifth) yield the most pleasing sounds. The mechanics of the ear imply that if two tones of reasonable amplitude are played together, progressively smaller additional signals will effectively be generated at frequencies Abs[n_{1} ω_{1} ± n_{2} ω_{2}]. The picture below shows the extent to which such frequencies tend to be in the range that yield grating effects. The minima at values of ω_{2}/ω_{1} corresponding to rationals with small denominators may explain why such chords seem more pleasing. (See also page 917.)