Beat Frequency: How Far Out of Tune, and Whether You Can Hear It

Q: Is the beat frequency the difference or the average of the two frequencies?

The difference, and its size: fbeat = |f1 − f2|. The average is a different and also real quantity — it is the PITCH you hear while the beating happens. For 440 Hz against 443 Hz the beat rate is 3 Hz and the pitch is 441.5 Hz. Both appear on this page as separate rows because reporting one for the other is the standard mistake.

Two close notes do not average into one. They rise and fall at the DIFFERENCE of their frequencies while sounding at the MEAN of them, and below about seven beats a second you can count the throb. This page gives the beat rate, the beat period, the pitch you actually hear and the interval in cents, and says which side of countable you are on.

Last updated 11 Nov 2026

Beat rate, beat period and the interval in cents

Two frequencies, one judgement

First frequencyIn hertz. 440 Hz is the orchestral A. Order does not matter: the beat rate is the size of the difference, so swapping the two fields changes nothing except the sign of the interval in cents.

Second frequencyIn hertz. 443 Hz is where several European orchestras actually tune, which is why an instrument brought from one country to another can beat audibly against the local A even though both are nominally in tune. Three hertz out at 440 is three beats a second and about twelve cents, which is clearly audible to a trained ear and nearly invisible on a cheap tuner.

3.0000HzExample

An A at 440 Hz sounded against an A at 443 Hz

A sum of two sines, and the two different frequencies hiding in it

f_beat = |f₁ − f₂| · T_beat = 1/f_beat · pitch heard = (f₁ + f₂)/2 · cents = 1200 log₂(f₂/f₁)

|f₁ − f₂|: the beat frequency: how many times a second the sound swells. It is the DIFFERENCE and the modulus matters — 440 against 443 and 443 against 440 both beat three times a second. It is not the average, not the sum and not half the difference, and all three of those appear in wrong answers
(f₁ + f₂)/2: the pitch you hear while the beating happens. The identity behind the whole topic is sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2): a fast tone at the MEAN frequency, multiplied by a slow envelope at HALF the difference. The envelope goes through two loud moments per cycle, which is why the audible beat rate is the full difference and not half of it — the single most common slip in deriving this
T_beat = 1/f_beat: the time from one swell to the next. At an exact match it is infinite, which is not a failure of the formula but the definition of being in tune, and this page leaves the row blank rather than printing a number for it
cents: the musician’s logarithmic measure: 1200 cents to the octave, 100 to an equal-tempered semitone. Worth having beside the hertz because the ear is logarithmic and hertz are not. Three hertz out is 12 cents at 440 Hz and 47 cents at 110 Hz — the same absolute error, four times the musical offence. About 5 cents is roughly where a trained listener starts to object
what beats do NOT tell you: the direction. Three beats a second says the two notes are 3 Hz apart and says nothing about which is higher. The fix is procedural rather than arithmetical: move one note deliberately and watch whether the beating slows or quickens

Worked example

An A at 440 Hz sounded against an A at 443 Hz
THE BEAT RATE. |443 − 440| = 3 Hz. Three swells a second, so a beat period of 0.333 s and fifteen beats in five seconds — which is the number you would actually count, because counting over five seconds divides your timing error by five.
THE PITCH, which is not 3 Hz. You hear a tone at (440 + 443)/2 = 441.5 Hz, swelling and fading three times a second. A 3 Hz tone is inaudible and no such tone is present; the 3 Hz is in the ENVELOPE. Students who answer “you hear a 3 Hz note” have confused the two, and the page prints them as separate rows for that reason.
WHERE THE FACTOR OF TWO HIDES. sin(2π·440t) + sin(2π·443t) = 2 sin(2π·441.5t) cos(2π·1.5t). The envelope term is at 1.5 Hz, half the difference — and yet the beat rate is 3 Hz, because loudness follows the SIZE of the envelope and a cosine passes through maximum size twice per cycle. Two loud moments per 1.5 Hz cycle is 3 per second. This is the step worth writing out once by hand.
THE SAME ERROR IN CENTS. 1200 log₂(443/440) = 11.76 cents, about an eighth of a semitone. Small, and audible: the usual rule of thumb is that a trained ear notices around 5 cents on a sustained note. The same 3 Hz at a low A of 110 Hz would be 47 cents, nearly half a semitone, which is why orchestras tune the basses with more care than the number of hertz suggests.
AND WHICH ONE IS SHARP? The beats cannot tell you. Slacken the string slightly: if the beating slows, you were above and are coming down towards the target; if it quickens, you were below and have just gone further. That procedure is the whole of practical tuning, and no amount of arithmetic replaces it.

The same 3 Hz error at six different pitches

Five of these six rows have an identical beat rate and nothing else in common. That is the case for printing cents beside hertz: the ear judges ratios, so the same 3 Hz is a quarter of a semitone at the bottom of the bass stave and inconsequential two octaves above the treble. It also explains a practical asymmetry — beats are a wonderfully sensitive tuning tool at low pitches and a blunt one at high pitches, which is why piano tuners work upwards from a tuned middle octave rather than trying to zero-beat the top notes directly. The last row is the sensitivity of the method stated as a number: one beat in five seconds is 0.2 Hz at 440, which is 0.8 cents, or one part in 2,200.
Reference, Hz	Sounded, Hz	Beats per second	In cents	Beats in 5 s	Musician’s verdict
110.00	113.00	3	46.6	15	Badly out, nearly a quarter tone
220.00	223.00	3	23.5	15	Clearly out
440.00	443.00	3	11.8	15	Out, and audible on a long note
880.00	883.00	3	5.9	15	Marginal, about the threshold
1760.00	1763.00	3	2.9	15	Inaudible as a mistuning
440.00	440.20	0.2	0.8	1	A zero-beat; this is tuned

The difference is what you count; the mean is what you hear

The beat frequency is the difference of the two frequencies, and that is one subtraction. What makes the topic worth a page is that two different frequencies come out of the same sum and students routinely report the wrong one. Add two sines of nearly equal frequency and the trigonometric identity gives you a product: a fast tone at the MEAN of the two, multiplied by a slow envelope at HALF the difference. The ear hears the fast tone as the pitch and the envelope as a throb. So 440 against 443 is a 441.5 Hz note swelling three times a second, and anyone who answers that you hear 3 Hz has answered with the envelope.

The factor of two in that sentence is the part worth deriving once. The envelope term is cos(2π·1.5t) — half the 3 Hz difference. But loudness does not care about the SIGN of the envelope, only its size, and a cosine reaches maximum size twice per cycle. Two loud moments per 1.5 Hz cycle gives three per second. If you ever find yourself with an answer half the expected beat rate, this is where it went: the envelope frequency and the beat frequency differ by exactly two, and only one of them is audible as a beat.

Beats are the most sensitive tuning instrument most people own. A listener can reliably detect one beat in five seconds on a sustained pair of notes. At 440 Hz that is 0.2 Hz, which is 0.8 cents, or one part in 2,200 — better than a cheap electronic tuner and achieved with no equipment at all. The method is called zero-beating, and it is why organ builders, piano tuners and orchestral wind players all still use it. Its single limitation is that it is blind to direction: a 3 Hz beat rate tells you the notes are 3 Hz apart and nothing about which is sharp. The remedy is to move one deliberately and watch which way the rate goes.

Where it stops working is the other end of the range, and the edges are conventions rather than measurements. Below about half a beat a second there is nothing to count within a reasonable listening time. Above about seven a second the throb blurs into a sensation usually described as roughness. Past about twenty a second you simply hear two notes, plus — if both are loud — a low difference tone that is a nonlinearity in the ear rather than anything present in the air. The bar on this page puts its boundaries at 0.5, 7 and 20 Hz because those are the round numbers the literature uses; they are not thresholds anybody measured, the transitions are gradual, and they depend on pitch and loudness.

Where this goes wrong, in order of frequency. Reporting the mean as the beat rate, or the beat rate as the pitch. Forgetting the modulus and reporting a negative beat frequency, which is not a thing. Halving the answer because the envelope term is at half the difference. And using beats between fundamentals when the measurement you want is between upper partials — tuning an octave by beats works on the lower note’s second harmonic against the upper note’s fundamental, and if you type the two fundamentals into this page you will get a difference of several hundred hertz that nobody can hear as a beat.

Frequently asked questions

Is the beat frequency the difference or the average of the two frequencies?

The difference, and its size: f_beat = |f₁ − f₂|. The average is a different and also real quantity — it is the PITCH you hear while the beating happens. For 440 Hz against 443 Hz the beat rate is 3 Hz and the pitch is 441.5 Hz. Both appear on this page as separate rows because reporting one for the other is the standard mistake.

Why is the beat frequency the full difference when the envelope is at half of it?

Because loudness follows the size of the envelope, not its sign. Writing sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2) gives an envelope at half the difference, but |cos| reaches a maximum twice per cycle, so there are two loud moments per envelope cycle. Two per half-difference cycle is one per difference cycle. If your answer is half the expected beat rate, this is almost certainly the reason.

How accurately can you tune by beats?

To about one beat in five seconds on a sustained note, which is 0.2 Hz. At 440 Hz that is roughly 0.8 cents, or one part in 2,200 — finer than most inexpensive electronic tuners. The sensitivity is in absolute hertz rather than in cents, so the method is at its best on low notes and progressively blunter on high ones: the same 0.2 Hz is 0.2 cents at 1760 Hz, below anything that matters, and 3 cents at 110 Hz.

Can the beat rate tell me which note is sharp?

No, and that is a property of the physics rather than a limitation of this page. The rate depends on the size of the difference only. The standard procedure is to perturb one note deliberately: slacken the string or pull the tuning slide out a little, and if the beating slows you were on the sharp side, if it quickens you were on the flat side. Then reverse and converge.

What happens when the two frequencies are exactly equal?

The beat rate is zero and the beat period is infinite, so this page leaves the beat-period row blank rather than printing a number for it. That is the target state, not an error: tuners call it a zero-beat and it is what they are listening for. The sum of the two sines becomes a single steady tone of twice the amplitude — or of zero amplitude, if the two happen to be exactly out of phase, which is the same arithmetic and the reason two identical loudspeakers wired backwards cancel.

Why do two loud notes a long way apart produce a low tone that is not in either?

That is a difference tone, and it is generated in the ear rather than present in the air. A nonlinear system fed two frequencies produces sums and differences of them, and the ear is mildly nonlinear at high levels. It is a real perception and a real measurement, but it is not the beat phenomenon this page computes: beats are an interference effect in the sound itself and need no nonlinearity at all.

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References

NCERT, Physics Textbook for Class XI, Part II, Chapter 14, Waves. The chapter number was checked against NCERT’s own answers appendix for Part II (keph2an.pdf, read 11 November 2026), which runs Chapters 8 to 14 and puts Oscillations at 13 and Waves at 14; older printings numbered Waves as 15 and much of the web still says 15. Cited for the syllabus treatment of wave speed, standing waves, beats and the Doppler effect. Citing the chapter is not a claim that any board has approved this page.
The identity that the whole topic rests on — sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2) — is standard trigonometry and is where both the mean-frequency tone and the half-difference envelope come from. The factor of two between the envelope frequency and the audible beat rate follows from |cos| having twice the period of cos, and is derived in full in the worked example rather than asserted, because it is the step that goes wrong.
The cent is defined as one twelve-hundredth of an octave, so an interval of ratio r is 1200 log₂(r) cents. That is a definition and needs no citation beyond itself. The perceptual figures on this page are of a different and weaker kind and are flagged as such: that a trained listener notices around 5 cents on a sustained note, that beats blur into roughness around 7 Hz, and that they separate into two tones around 20 Hz are rules of thumb from the psychoacoustics literature which this batch did not read in the original. They are printed as round conventional numbers and the band edges on this page are drawn at them for want of anything better; they are not thresholds anybody measured, they vary with pitch and loudness, and they should not be quoted as if they were data.
The orchestral tuning pitches used in the examples — A = 440 Hz as the international standard and A = 443 Hz as a pitch used by several European orchestras — are conventions rather than physical constants, and they have drifted upwards by well over a semitone since the eighteenth century. They are used here only because they give a realistic 3 Hz beat; nothing on the page depends on either figure.