Beat frequency
Two sine waves with slightly different frequencies produce a pulsing amplitude pattern — beats. The ear hears a single tone whose loudness waxes and wanes at the difference frequency. Piano tuners have used this phenomenon for centuries.
fbeat = |f1 − f2|
Interference in time
When two waves of slightly different frequency overlap, they periodically reinforce and cancel each other. At any given moment, the two waves are either in phase (crests aligned, producing a loud sum) or out of phase (one crest meets the other’s trough, producing silence). The result is an amplitude envelope that pulses at the beat frequency — the absolute difference between the two original frequencies.
Mathematically, the sum of two cosines gives: cos(2πf1t) + cos(2πf2t) = 2 cos(2π·½(f1−f2)·t) · cos(2π·½(f1+f2)·t). The first cosine is the slowly-varying envelope (the beat); the second is the rapidly-varying carrier at the average frequency. Your ear hears a single pitch at (f1+f2)/2 whose volume pulses |f1−f2| times per second.
The piano tuner’s ear
Piano tuners rely on beats to achieve precise tuning. When tuning an octave, the tuner strikes both notes simultaneously and listens for the wah-wah-wah of beating between the fundamental of the upper note and the second harmonic of the lower note. As the upper string is tightened or loosened, the beat frequency changes: faster beats mean the notes are further apart; slower beats mean they’re converging. When the beating disappears entirely, the interval is pure.
In equal temperament (the standard tuning of modern pianos), most intervals are intentionally impure — only octaves are beatless. A skilled tuner learns the correct beat rates for every interval. For example, the equal-tempered fifth A4–E5 should beat at about 1.5 Hz, while a pure fifth would be beatless. This deliberate slight mistuning is the price paid for being able to play in all keys.
The critical band
As the frequency difference increases beyond about 15–20 Hz, the pulsing quality of beats gives way to a sensation of roughness or dissonance. This transition happens when the two frequencies fall within the same critical band of the cochlea — a frequency range over which the basilar membrane cannot resolve individual tones. The width of a critical band is roughly 1/3 octave at mid-frequencies (about 100 Hz around 440 Hz).
When the frequency difference exceeds the critical bandwidth, the ear resolves the two tones as separate pitches and the roughness disappears. This perceptual transition from beats to roughness to separate tones is central to theories of musical consonance. Helmholtz proposed in 1863 that dissonance arises precisely from rapid beating between partials — a theory that, with modern refinements, remains influential.
Heterodyne detection and quantum beats
The beat phenomenon appears whenever two oscillations of slightly different frequency are superposed, not just in acoustics. In radio engineering, heterodyne receivers mix an incoming signal with a local oscillator to produce a beat at an intermediate frequency that is easier to amplify and filter — the principle behind virtually all modern radios. In optics, optical heterodyne detection (or homodyne when the frequencies are equal) achieves extraordinary sensitivity, enabling the detection of gravitational waves at LIGO.
At the quantum level, quantum beats occur when an atom is excited into a superposition of two energy levels with slightly different energies. The fluorescence emitted by the atom oscillates at the beat frequency (E1−E2)/h, providing a direct measurement of energy-level splittings. The mathematics is identical to the classical case: superposition of two oscillations yields a modulated signal.