Beat Frequency of Two Close Tones

Beat Frequency of Two Close Tones

When two tones very close in pitch play simultaneously, the ear hears a periodic swelling and fading of loudness called beating, at a rate equal to the difference between the two frequencies. Musicians use this effect directly — piano tuners and guitarists listen for beats to slow down and stop as two strings or notes are brought into tune.The same beating phenomenon shows up outside music too: it explains the throbbing sound of two out-of-sync engines or propellers, and is used deliberately in some measurement instruments (like beat-frequency oscillators) to detect very small frequency differences that would otherwise be imperceptible.

The beat frequency is f_beat = |f_1 − f_2|. where f_1t and f_2t are the two tone frequencies and f_beat is the resulting audible beat frequency.

Take the absolute difference between the two frequencies — the sign does not matter, only the size of the mismatch, which sets how fast the loudness pulses.

Results

A 6 Hz beat frequency is easily heard as a slow, distinct throbbing — roughly six pulses of loudness per second — which is exactly the kind of mismatch a tuner would still want to correct. As the two tones are brought closer together the beats slow down further, disappearing entirely (zero beat frequency) once the tones match exactly.

Assumptions & limitations

• Both tones have similar amplitude (beats are most audible when the two loudnesses are comparable).
• Frequency difference is small compared to either tone (a few Hz to a few tens of Hz, within audible beating range).
• Steady, continuous tones (not short transient sounds).

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