Beat Frequency Calculator
When two sound waves of slightly different frequencies overlap, their amplitudes alternately reinforce and cancel, producing a slow throb called a beat. Find the beat frequency and period from two tones, reverse solve a missing frequency from a measured beat, read the average pitch and amplitude swing, or analyse how far apart two notes are in cents for tuning.
Formula
Worked example
A 440 Hz tuning fork sounded with a 443 Hz string: f_beat = |440 - 443| = 3 Hz. You hear the combined sound swell and fade 3 times per second, each pulse lasting 1 / 3 = 0.333 s, around a perceived pitch of (440 + 443) / 2 = 441.5 Hz. That 443 Hz string is 1200 x log2(443 / 440) = 11.8 cents sharp.
What a beat frequency is
Two tones close in pitch do not simply add to a louder steady sound. Because their crests gradually drift in and out of alignment, the combined amplitude rises and falls in a slow envelope. Each time the waves line up you hear a swell of loudness, and each time they oppose you hear a dip. The number of these swells per second is the beat frequency, and it equals the absolute difference of the two source frequencies, |f1 - f2|, regardless of which tone is higher. The beat period is simply one divided by that rate, the seconds between swells.
Why the formula works
Adding two sinusoids of frequencies f1 and f2 can be rewritten as a single tone at the average frequency, (f1 + f2) / 2, multiplied by a slow envelope that oscillates at half the difference frequency. The envelope reaches a loudness peak twice per envelope cycle, once on its positive swing and once on its negative swing, so the audible beat rate is the full difference |f1 - f2|, not half of it. The average term sets the pitch you actually perceive, while the difference term sets how fast that pitch throbs. If the two tones have amplitudes A1 and A2, the loudness swings between A1 + A2 when they are in phase and |A1 - A2| when they are exactly out of phase, which is why beats are deepest when the two tones are equally loud.
Reverse solving and tuning in cents
Tuners often work the formula backward. If you know one reference frequency and you can count the beats against an unknown tone, the unknown must be the reference plus or minus the beat rate. The beat alone cannot say which side the unknown is on, so this calculator lets you pick the expected direction, and in practice you confirm it by nudging the reference and watching whether the beats speed up or slow down. The tuning mode goes a step further and reports the error in cents, the musician unit where one semitone is 100 cents and an octave is 1200. Cents come from the ratio of the two frequencies, 1200 times the base-2 logarithm of played divided by reference, so equal cent errors sound equally out of tune at any pitch, which raw hertz cannot do.
Hearing beats and their limits
The ear resolves beats clearly when the difference is small, roughly up to 15 to 20 Hz. Within that range you can literally count the throbs, which is why piano tuners and string players listen for beats slowing to zero as two notes are brought into unison. As the separation grows past about 20 Hz the individual swells smear together into a sensation of roughness, and at larger separations the two tones are simply heard as distinct pitches with no beating at all. This calculator reports the raw beat frequency for any inputs; whether it is audible as beats depends on the size of the difference.
Beat rate, perception and tuning
| Beat frequency | How it sounds | Cents at 440 Hz | Typical use |
|---|---|---|---|
| 0 Hz | Steady, in tune | 0 | Unison locked in |
| 0.5-2 Hz | Slow, easy to count | ~2-8 | Final fine tuning |
| 2-7 Hz | Clear pulsing | ~8-27 | Coarse tuning by ear |
| 7-20 Hz | Fast flutter, roughness | ~27-79 | Audibly out of tune |
| Over 20 Hz | Two separate pitches | Over ~79 | No beating heard |
How the beat frequency between two tones is heard, with the rough cent error at A4 = 440 Hz.
Frequently asked questions
Is the beat frequency the difference or half the difference of the two frequencies?
It is the full difference, |f1 - f2|. Although the amplitude envelope oscillates at half the difference frequency, the loudness peaks twice in each envelope cycle, at both the positive and negative crests, so you hear |f1 - f2| beats per second. For 440 Hz and 443 Hz that gives 3 beats per second.
Can I find an unknown frequency from the beats I hear?
Yes. Switch to the missing-frequency mode, enter the frequency you know and the number of beats per second, and the unknown is the known frequency plus or minus the beat rate. The beat by itself cannot tell which side the unknown is on, so pick the expected direction. You can confirm it by nudging the reference: if the beats speed up, the unknown is on the opposite side.
What does the detuning in cents tell me?
Cents are a logarithmic pitch unit where one semitone is 100 cents and an octave is 1200. The tuning mode reports how many cents sharp or flat the played pitch is relative to the target, which is more meaningful than raw hertz because equal cent errors sound equally out of tune at any pitch. Most listeners hear a note as in tune within about 5 cents.
Why do the beats disappear when I tune two strings together?
As the two frequencies converge, their difference shrinks, so the beat frequency drops and the swells come further apart. When the pitches match exactly the difference is zero, the waves stay permanently in step, and the loudness becomes steady with no throbbing. Listening for that vanishing beat is the classic way to tune by ear.
Does it matter which frequency I enter first?
No. The formula takes the absolute value of the difference, so 440 and 443 give the same 3 Hz beat as 443 and 440. The beat frequency only depends on how far apart the two tones are, not on which one is higher in pitch.