Pluck two guitar strings that are almost, but not quite, the same note, and something odd happens. You do not hear two pitches. You hear one pitch that swells and fades, swells and fades, like a slow pulse. Musicians call that pulse a beat, and it is one of the oldest tools for tuning an instrument by ear.
A pure tone is a wave: air pressure rising and falling at a steady rate. Concert A, the note orchestras tune to, is 440 of those cycles every second, or 440 hertz. Now add a second wave at 442 Hz. Most of the time the two are out of step by a little, but every so often they line up crest to crest and add into a loud swell. A moment later they line up crest to trough and cancel into near silence.
How often does that happen? The waves drift apart by two cycles every second, so they fall back into step exactly twice a second. The rule is short: the number of beats per second equals the difference between the two frequencies. 440 Hz against 442 Hz gives 2 beats per second. 440 against 441 gives 1.
Below, the top wave is fixed and the middle wave is the one you change. The bottom wave is what reaches your ear: the two added together. Drag the slider and watch the swells. The lamp flashes once per beat. Tap Play to hear the pair, and turn your volume down first.
Try 0 first: identical waves, no swelling, one steady tone. Then move up. At 1 or 2 beats per second the pulse is slow and easy to count. By around 6 to 8 it flutters. Push past roughly 15 or 20 and the individual swells blur into a rough, buzzing sound, and eventually your ear starts to hear two separate pitches.
Most notes in the middle of a piano have three strings struck by one hammer. If those three strings disagree even slightly, the note wobbles. A tuner mutes two strings, tunes one against a reference, then brings the second in until the beating slows and stops. Zero beats means the strings match. The ear is far better at hearing a slow pulse than at judging whether two pitches are 0.5 Hz apart.
The same trick works between different notes. Take A (440 Hz) and the E above it. In equal temperament, that E is 659.26 Hz. The third harmonic of the A is 3 × 440 = 1320 Hz, and the second harmonic of the E is 2 × 659.26 = 1318.5 Hz. The two overtones are about 1.5 Hz apart, so a correctly tuned fifth beats about once every two thirds of a second. Tuners count that slow beat to check their work.
The maths is one line of trigonometry. Adding two cosines gives cos a + cos b = 2 cos((a − b)/2) × cos((a + b)/2). The second factor is a tone at the average of the two frequencies, which is the pitch you hear. The first factor is a slow envelope at half the difference. Because loudness depends on the size of the swing, not its sign, the envelope peaks twice per cycle, so you hear a full beat at the whole difference.
This string is slightly out of tune with a reference at 440 Hz, and the offset is hidden. Drag the slider to change its pitch. Watch the lamp: when it stops flashing, you are in tune. Then press Check.
You can feel the lesson: the closer you get, the slower the lamp, until a single long beat stretches out and disappears. That slowing is the whole method. It turns a tiny pitch difference, too small to judge directly, into a slow rhythm you can count.