Beats and Tuning by Ear: How to Count the Error
Tuning by ear is not listening for in tune. It is turning an error into a rhythm and taking it to zero.
Short answer: two tones of nearly the same frequency drift in and out of phase, so their combined loudness rises and falls at exactly the difference between them. Play 440 Hz and 442 Hz together and you hear two pulses per second. This is the most useful trick in practical tuning, because it is much more sensitive than comparing two pitches directly: a one-cent error at A4 is only a quarter of a hertz, far below the difference most people can hear as a pitch change, yet it produces one clear beat every four seconds. You are not judging a pitch. You are counting a pulse.
What is actually happening
Add two sine waves of slightly different frequency and the result is a single wave at the average frequency whose amplitude is modulated slowly. Where the two are in phase they reinforce and the sound is loud; where they are 180 degrees out of phase they cancel and it is quiet. The cycle from loud to loud repeats at the difference of the two frequencies. If one tone is 500 Hz and the other 502 Hz, they realign every 250 cycles of the first, which takes half a second, so the beat interval is half a second and the beat rate is 2 Hz.
The rule is that simple: beat rate equals the frequency difference. It is a form of amplitude modulation, and it works while the two components fall inside the same auditory filter. Beyond about 15 Hz of separation the fluctuation is too fast to count and the impression changes to a rough, buzzy quality; beyond a critical band the ear resolves the two tones separately and the effect disappears altogether.
Why this beats listening for the note
Direct pitch comparison is not very precise. Trained listeners can often detect differences of a handful of cents under good conditions, and untrained listeners considerably more, which we cover in how small a difference you can hear. Beating is a different task with a different resolution, because a slow pulse is easy to perceive even when the frequency difference producing it is tiny.
Here is the conversion at A4, 440 Hz. Note that cents are a ratio, so the hertz figures double an octave up and halve an octave down.
| Error in cents | Frequency difference at A4 | Beat rate | One beat every |
|---|---|---|---|
| 1 cent | 0.25 Hz | 0.25 per second | about 4 seconds |
| 2 cents | 0.51 Hz | 0.5 per second | about 2 seconds |
| 5 cents | 1.27 Hz | 1.3 per second | about 0.8 seconds |
| 10 cents | 2.55 Hz | 2.5 per second | about 0.4 seconds |
| 20 cents | 5.11 Hz | 5 per second | about 0.2 seconds |
| 50 cents | 12.9 Hz | 13 per second | too fast to count; roughness |
Read the table the other way round and it becomes a measuring instrument. If you can hear a beat at all, you are detecting an error of a fraction of a cent. If you can count four seconds between beats, you have just measured one cent by ear, which no amount of pitch comparison would give you.
Beating on harmonics, not just on unisons
Unisons are the easy case: the fundamentals themselves beat. But the same trick works on any interval, because complex tones carry a whole series of partials, and any two partials that nearly coincide will beat against each other.
For an octave, compare the second harmonic of the lower note with the fundamental of the upper. If A4 is exactly 440, its second harmonic is 880; tune A5 until the pulsing against that 880 stops. For a fifth, compare the third harmonic of the lower note with the second harmonic of the upper: on A4 that is 1320 Hz against twice the E. For a major third, compare the fifth harmonic of the lower with the fourth of the upper: 2200 Hz against four times the C sharp.
This is where temperament becomes audible. A pure fifth has no beat at all, because 3:2 makes those two harmonics identical. An equal-tempered fifth is about two cents narrow, which puts the tempered E5 at 659.26 Hz, so its second harmonic is 1318.5 Hz against the A's 1320: a beat about one and a half times per second. The equal-tempered major third is worse: C sharp 5 at 554.37 Hz has a fourth harmonic of 2217.5 Hz against 2200, which is 17.5 beats per second, fast enough to be heard as roughness rather than pulsing. Piano tuners do not remove those beats. They set them to specified rates, which is what makes an aurally tuned piano a piece of craft rather than arithmetic.
A working procedure
- Get a reference you trust. A tuning fork, an app, or one string you have decided is correct. Everything else is measured against it, so an error here propagates everywhere.
- Sound both notes together and listen to the combination, not to either note. This is the step people get wrong. You are listening for a pulsing in the loudness, not for whether the second note sounds right.
- Adjust and watch the beat rate. If it speeds up, you are going the wrong way. Reverse. If it slows down, keep going.
- Take it as slow as you can, then stop. The last stretch, from one beat per second to one beat every few seconds, is where most of the accuracy lives.
- Cross-check. Verify the result with a different interval, an octave or a harmonic, so that small errors do not accumulate as you work across an instrument.
Two environmental notes. Beating is a loudness fluctuation, so anything that masks small loudness changes, background noise, heavy reverberation, a compressed monitoring chain, will hide it. And sustained tones work far better than plucked ones, because a decaying note gives you only a second or two of usable beating before it dies.
Where the method runs out
Beating assumes the partials are where the arithmetic says they are. On real strings they are not: stiffness makes upper partials progressively sharp, which is why pianos are tuned with stretched octaves and why the same beat-counting method gives different answers on a short spinet and a concert grand. It also assumes both sounds are steady. Voices with vibrato, wind players with breath variation, and anything with heavy tremolo will produce their own amplitude fluctuations that swamp the beat you are trying to hear.
Finally, beating tells you that two frequencies differ. It does not tell you which one is high. That is what the direction check is for: nudge one note and see whether the beat speeds up or slows down. Beginners lose a lot of time by adjusting confidently in the wrong direction, and the fix is to make the first adjustment deliberately large enough to be unambiguous.
Training the ear for it
Two habits transfer directly. The first is holding attention on a sustained sound for several seconds without your interest wandering, which is harder than it sounds and is what Heardrift is built around: keeping a tone steady over time rather than hitting it once. The second is resolving genuinely small pitch differences, which is what Hearfork narrows down for you, round by round, until you find your current limit in cents. Knowing that number is useful, because it tells you when to trust a direct pitch judgement and when to reach for the beat instead.
If you want the arithmetic behind the conversions on this page, our guide to cents has it, and what frequency is covers the hertz side.
Find the point where beating becomes the better tool. Hearfork narrows the gap between two tones until you stop hearing a difference, and hands you the number in cents. Below that number, stop comparing pitches and start counting beats.
Frequently asked questions
What are beats in acoustics?
When two tones of nearly the same frequency sound together, they drift in and out of phase, so the combined loudness rises and falls. That pulsing is called beating, and it repeats at a rate equal to the difference between the two frequencies. Two tones at 440 and 442 Hz beat twice per second. The effect works up to roughly 15 Hz of separation, above which the pulsing turns into roughness.
Why do tuners listen for beats instead of listening for the note?
Because beating is far more sensitive than pitch comparison. A one-cent error at A4 is only about 0.25 Hz, well below the difference most people can hear when comparing two pitches directly, but it produces one beat every four seconds, which is easy to see and count. Beating converts a tiny frequency error into a slow, obvious rhythm.
How many beats per second is one cent?
It depends on the frequency, because a cent is a ratio, not a fixed number of hertz. At A4, 440 Hz, one cent is about 0.25 Hz, so one beat roughly every four seconds. At A5, 880 Hz, the same one cent is 0.51 Hz, so the beat is twice as fast. Doubling the pitch doubles the beat rate for the same error in cents.
Why do tuned fifths still beat on a piano?
Because equal temperament narrows every fifth by about two cents on purpose. On A4 that leaves the third harmonic of A at 1320 Hz against the second harmonic of the tempered E at about 1318.5 Hz, so the fifth beats about one and a half times a second. Piano tuners set those beat rates deliberately rather than eliminating them.
Can I tune a guitar this way?
Yes, and it is more accurate than matching pitches by ear. Fret the lower string at the fifth fret so it should match the open string above, then listen to the combined sound rather than to either string, and adjust until the pulsing slows to a stop. Check the result against an octave or a harmonic, because small errors accumulate as you work across the strings.
Why do beats disappear when the tones are far apart?
Because beating only happens while the two components fall inside the same auditory filter. Once they are separated by more than a critical band, the ear resolves them as two distinct tones and the amplitude fluctuation stops being audible as pulsing. In between, roughly 15 to 20 pulses per second and above, you hear roughness rather than countable beats.
Keep reading
- What cents are: the unit behind every number on this page.
- Equal temperament versus just intonation: why some beats are set on purpose.
- How small a pitch difference can you hear: where direct comparison gives up.
- Pitch drift and holding a note: the steadiness that beat counting depends on.
Sources: Truax (ed.), Beats, Critical band and Tuning, Handbook for Acoustic Ecology, Simon Fraser University; Nave, Equal temperament, HyperPhysics, Georgia State University; Plomp & Levelt (1965), Tonal consonance and critical bandwidth, Journal of the Acoustical Society of America 38(4); Giordano (2015), Explaining the Railsback stretch in terms of the inharmonicity of piano tones and sensory dissonance, Journal of the Acoustical Society of America 138(4); Jaatinen & Pätynen (2022), Effect of inharmonicity on pitch perception and subjective tuning of piano tones, Journal of the Acoustical Society of America 152(2); Huygens-Fokker Foundation, Logarithmic interval measures; Moore, An Introduction to the Psychology of Hearing, Routledge.
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