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 — it is under a fiftieth of a semitone at concert pitch — 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.

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, and you have measured your detuning directly.
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.

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. The calculator subtracts and takes the absolute value in Decimal arithmetic, rounding once at the output — trivial mathematics, but the point of the page is the physics of why a difference is what you hear.

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.