Beatrix

Meet Beatrix, two tones a little out of tune whose sum swells and fades.

Play two sounds with slightly different frequencies f₁ and f₂ together. Sometimes their crests line up and the sound is loud; a little later a crest meets a trough and it goes quiet. The loudness rises and falls at the difference frequency, |f₁ − f₂|. These pulses are called beats. Mathematically, the sum is a tone at the average frequency inside a slowly varying envelope.

Example: tuning forks at 440 Hz and 443 Hz played together give 3 beats per second. As one is adjusted toward the other, the beats slow down and stop when they match.

Musicians and piano tuners listen for beats to bring strings into tune. The same principle of mixing two frequencies to get their difference is used in radio receivers and in laser measurements.

Beats are easy to hear only when the difference is small, below roughly 10 to 20 Hz. Larger differences sound rough, then as two separate notes.