Beats

Play two notes whose frequencies are close but not equal, say 256 Hz and 260 Hz. You hear one note at about the average frequency, 258 Hz, but its loudness swells and fades, over and over. Each swell-and-fade is one beat.

Why? The two waves keep drifting in and out of step. When they are in step they add up to a loud sound; a little later one has gained half a cycle on the other and they nearly cancel. Adding them gives y = 2a cos(π(ν₁ – ν₂)t) sin(π(ν₁ + ν₂)t): a fast note whose amplitude rises and falls slowly with time.

The loud moments come once every 1/(ν₁ – ν₂) seconds, so the beat frequency is ν_beat = |ν₁ – ν₂|. For 256 Hz and 260 Hz, that is 4 beats per second.

Beats are heard clearly only when the difference is small, a few per second. If the frequencies are far apart, the beats come too fast and you just hear two separate notes.

Finding an unknown frequency. The beat count tells you the gap, but not which note is higher. To decide, change one source a little and watch the beats:

• Tightening a string raises its frequency; loosening it lowers it. Sticking a little wax on a tuning fork's prong lowers the fork's frequency.
• If the change brings the two notes closer, the beats slow down; if it pushes them apart, they speed up.

Example: a 384 Hz tuning fork and a sitar string give 4 beats per second, so the string is at 380 or 388 Hz. Loosening the string (lowering it) makes 6 beats per second, so it moved away from 384. It must have been below the fork: the string was at 380 Hz.

This is how musicians tune: play the instrument with a reference note and adjust until the beats slow down and disappear. Then the two frequencies are equal.