Beat Frequency: When Two Almost-Matching Waves Pulse

Two waves, one ear

Sound pressure adds linearly, so when two tuning forks near the same pitch ring together, the air pressure at your ear is simply the sum of the two sine waves. If the frequencies are identical the sum is just a louder version of the same tone. If they differ slightly, the two waves drift in and out of phase with each other over time, and the sum's amplitude visibly swells and fades - a beat.

The identity that explains it

A sum-to-product trigonometric identity turns two pure tones into a single carrier wave riding inside a slow amplitude envelope:

sin(2π f1 t) + sin(2π f2 t)
    = 2 · cos(2π · (f1-f2)/2 · t) · sin(2π · (f1+f2)/2 · t)
              [slow envelope]         [audible carrier tone]

The carrier you hear rings at the average frequency, (f1+f2)/2, and its loudness is modulated by the slowly varying cosine envelope, whose frequency is half the difference, (f1-f2)/2.

Why the beat rate equals the difference, not half of it

Here's the subtlety that trips people up: your ear responds to loudness, which follows the absolute value of that envelope, not its raw sign. A cosine envelope crosses zero twice per cycle, so |cos(...)| produces two loud pulses for every one cycle of the underlying envelope frequency. The two halves cancel out, and the perceived pulsing rate you actually count is the full difference, |f1 - f2|.

From slow pulses to roughness to a separate note

Below roughly 15 Hz of difference, the ear resolves distinct, countable pulses - the classic "wow-wow-wow" of two nearly matched guitar strings. Push the gap toward 20 to 30 Hz and the pulsing becomes too fast to count individually; it's perceived instead as roughness or dissonance, because both tones now fall inside the same critical band of the cochlea and interfere at the level of the inner ear's mechanics. Separate the tones further still and they stop interacting altogether, heard as two clearly independent pitches.

Tuning by beats

Because the beat rate is exactly the frequency difference, listening for beats to slow down and stop is a precise way to bring two notes into unison without any electronic tuner - the technique goes back centuries. Piano tuners use a refined version of the same idea to tune stretched octaves and intervals, deliberately counting a target number of beats per second between specific notes rather than aiming for silence, since real piano strings are slightly inharmonic.

Frequently asked questions

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