Beat Frequency Calculator

Background.

Play two tones that are almost — but not quite — the same pitch, and you do not hear two notes. You hear one note that pulses, swelling and fading in a slow rhythm. Those pulses are beats, and their rate is the simplest formula in acoustics: the beat frequency is just the difference between the two tones, f_beat = |f₁ − f₂|.

The effect is pure superposition. When the two waves start in step they reinforce; because one completes cycles slightly faster, it steadily pulls ahead until the waves are in opposite phase and cancel; then it laps the slower wave and they reinforce again. Each lap is one beat, and laps happen exactly as often as the frequency gap dictates.

Beats are most useful precisely when they are slow. A gap of 4 Hz between two instruments is hard to name as a pitch difference — about a sixth of a semitone at concert pitch, under 16 cents — yet it produces four unmistakable throbs every second. That inversion, where a tiny frequency error becomes a big audible rhythm, is why piano tuners, string players, and radio engineers have used beats as a precision instrument for centuries: tune until the beating slows to a stop, and the two frequencies match to well under a hertz.

This page computes the beat rate from two steady sine frequencies. It deliberately says nothing about loudness, audibility, or the fused-versus-rough character the ear assigns to faster beats — those live in psychoacoustics, and the scope note beside the result draws that boundary.

What is beat frequency calculator?

Beat frequency is the rate at which the combined loudness of two superposed tones rises and falls, equal to the absolute difference of their frequencies: f_beat = |f₁ − f₂|. Two tuning forks at 440 Hz and 444 Hz beat 4 times per second. The phenomenon requires no special medium or nonlinearity — it falls straight out of adding two sinusoids — and the perceived pitch of the pair sits near the average of the two frequencies while the beat rides on top as a loudness envelope. The beat period, the time from one loudness peak to the next, is the reciprocal: 1 ÷ 4 Hz = 0.25 s. Musicians who think in tempo can multiply by 60: 4 Hz is 240 pulses a minute, a brisk metronome setting. Because the result is a difference, it does not care how high the two frequencies are: two radio carriers in the gigahertz range that differ by 4 Hz beat exactly as slowly as two tuning forks.

How to use this calculator.

1. 01Enter both frequencies in hertz — from a tuner app, signal generator, or instrument specification.
2. 02Order does not matter: the formula takes the absolute difference, so 440 and 444 give the same 4 Hz as 444 and 440.
3. 03Read the beat frequency as pulses per second; its reciprocal is the time between loudness peaks (4 Hz beats repeat every 0.25 s).
4. 04Tuning in practice runs backwards: count the beats you hear against a reference tone over ten seconds and divide by ten, and you have measured your detuning directly — 7 beats in 10 s is 0.7 Hz.
5. 05Expect the clean throb only while the gap is small — past roughly 15–20 Hz the ear stops resolving pulses and hears roughness, then two separate tones.
6. 06For a mistuned octave or fifth, enter the pair of harmonics that nearly coincide rather than the two notes themselves: the second harmonic of 440 Hz is 880 Hz, and that is what beats against a note of 882 Hz.

The formula.

Add two equal-amplitude sinusoids and a trigonometric identity does the rest: sin(2πf₁t) + sin(2πf₂t) = 2 cos(2π·½(f₁−f₂)t)·sin(2π·½(f₁+f₂)t). The second factor is a tone at the average frequency — what your ear assigns as pitch. The first is a slow envelope at half the difference frequency; but loudness peaks every time the cosine touches +1 or −1, twice per envelope cycle, so audible beats arrive at the full difference |f₁ − f₂|. That factor-of-two subtlety is the classic exam trap.

Equal amplitudes are what make the troughs silent. With unequal amplitudes A₁ and A₂ the combined amplitude swings between A₁ + A₂ and |A₁ − A₂| instead of between double and nothing. If one tone is half as strong as the other, the peaks are 1.5 and the troughs 0.5 in the same units — a 3 : 1 swing, about 9.5 dB — so the beating is shallower but its rate is unchanged. That is why the calculator needs no amplitudes: they set how deep the beats are, never how often they come. Unequal amplitudes also pull the heard pitch towards the stronger tone, so the average frequency is only an approximation of what the ear reports.

The formula assumes two steady pure tones. Real instruments carry harmonics, and each pair of nearly coinciding harmonics beats at its own difference frequency, which is how tuners hear intervals other than unisons. With three or more tones, every pair beats at its own difference: 440, 441 and 443 Hz sounding together produce 1, 2 and 3 Hz fluctuations at once, the shimmer of a choir section or a detuned synthesizer patch rather than one clean pulse. A pitch that drifts or wavers — vibrato, a warming string, a sweeping oscillator — makes the beat rate drift with it, and the result then describes only the instant whose frequencies you entered. Slow beats also take time to observe. A difference of 0.1 Hz produces one loudness peak every 10 seconds, so confirming it needs a tone that lasts at least that long; a piano note that dies away in a few seconds cannot show it. This is the practical face of a general limit: two frequencies can be told apart only to about one divided by the time you listen.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, University Physics Volume 1, section 17.6, Beats.
2. [2]Young and Freedman, University Physics with Modern Physics, 15th ed. (PRINT).
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.
4. [4]NIST Special Publication 811, Guide for the Use of the International System of Units, 2008 edition.
5. [5]HyperPhysics, Beats.