Beat Frequency

Play two tones that are almost, but not quite, the same, and they drift in and out of step: reinforcing when their crests align, cancelling when they oppose. The result is a single tone that throbs in loudness at exactly the difference of the two frequencies. Strike a 440 Hz tuning fork next to a guitar string vibrating at 437 Hz and you hear three slow pulses per second (by convention we label the higher tone f₁, so the beat is always a positive number).

Musicians tune by driving this number to zero: as the string approaches the fork's pitch the beats slow (three per second, then one, then a long slow swell) vanishing entirely at unison. Piano tuners go further, deliberately setting certain intervals to beat at prescribed rates to lay an equal-tempered scale by ear alone. The ear stops hearing distinct beats past roughly 15 Hz of separation; push further apart and the throb dissolves into the rough dissonance that music theory has been negotiating for centuries.