A sound that isn't really there
Strike a tuning fork at 440 Hz and a guitar string tuned slightly sharp at 446 Hz at the same time, and you don't hear two separate pitches. You hear one tone, roughly in between, pulsing louder and softer six times a second. Nothing in the room is physically oscillating at that pulse rate — no string, no membrane, no air column. The pulsing is a pattern that emerges purely from two waves overlapping, and it's one of the cleanest examples in all of physics of how addition of two simple signals produces a third phenomenon that looks nothing like either input.
This is the beat phenomenon, and every piano tuner, luthier, and audio engineer uses it constantly, usually without needing to derive the underlying trigonometry. But the trigonometry is short, exact, and worth working through once, because it explains not just that beats happen but exactly how fast they pulse and how loud the peaks get — which is precisely what a tool built around inputs Frequency 1 f1, Frequency 2 f2, Amplitude 1 A1, and Amplitude 2 A2 is designed to show you in real time.
The trig identity that explains everything
Start with two sine waves of equal amplitude for simplicity, both function of time t:
y1(t) = A * sin(2*pi*f1*t)
y2(t) = A * sin(2*pi*f2*t)
Add them together and apply the sum-to-product identity:
y1(t) + y2(t) = 2*A * cos(2*pi * ((f1-f2)/2) * t) * sin(2*pi * ((f1+f2)/2) * t)
Read the right-hand side carefully and it splits into two very different physical roles. The sin term oscillates at the average frequency, (f1+f2)/2 — that's the pitch your ear actually perceives, since human hearing can't resolve individual cycles that close together as separate tones once f1 and f2 are within a few hertz of each other. The cos term oscillates much more slowly, at (f1-f2)/2, and it multiplies the amplitude of the fast oscillation — it's an envelope, not a tone. Because the ear responds to the magnitude of pressure regardless of sign, a full loud-soft-loud cycle of that envelope happens twice per cycle of the cosine term, which is why the perceived beat frequency is the full difference, not half of it:
beat frequency = |f1 - f2|
beat period = 1 / beat frequency
When A1 and A2 differ — which is the general case a real tool has to handle, since two independent oscillators rarely have matched loudness — the envelope no longer dips all the way to silence at its minimum. The composite amplitude swings between the sum and the difference of the two amplitudes:
max composite amplitude = A1 + A2 (constructive peak)
min composite amplitude = |A1 - A2| (destructive trough)
If A1 and A2 are equal, the minimum hits exactly zero and you get full silence at the bottom of each beat cycle — the classic textbook case. If they're unequal, the trough is a quieter version of the tone rather than true silence, which is exactly what you hear when tuning a loud reference tone against a much quieter string.
Working through a tuning example
Suppose you're tuning a guitar's A string against an electronic reference. The reference plays a clean 440 Hz tone at amplitude A1 = 1.0 (normalized), and the string, still slightly sharp, rings out at f2 = 446 Hz with amplitude A2 = 0.8 (strings decay, so it's quieter than the sustained reference tone). Plugging into the formulas:
Beat frequency = |440 - 446| = 6 Hz
Average frequency = (440 + 446) / 2 = 443 Hz
Beat period = 1 / 6 ≈ 0.1667 s (about one pulse every 167 milliseconds)
Max composite amplitude = 1.0 + 0.8 = 1.8
Min composite amplitude = |1.0 - 0.8| = 0.2
So you'd hear a tone centered near 443 Hz, pulsing noticeably louder and softer six times per second, with the quiet moments never quite reaching silence because the two amplitudes don't match. As the guitarist tightens the string and f2 drifts down toward 440 Hz, the beat frequency shrinks continuously, the beat period stretches out, and the pulsing slows until it disappears entirely at f2 = 440 Hz — which is precisely the technique every acoustic tuner-by-ear uses: tune until the beats stop.
It's worth sanity-checking the direction of that relationship, because it trips people up: a slower beat means the two frequencies are closer together, not farther apart. A beginner tuning by ear sometimes assumes fast pulsing means "almost there" when it actually means the opposite — the string is still far from the target pitch.
Where the simple model breaks down
The clean formulas above assume two pure sine tones. Real instruments produce harmonics — a plucked guitar string is a stack of a fundamental plus overtones at roughly integer multiples of that fundamental — so in practice you hear multiple simultaneous beat frequencies between different harmonic pairs, not just one. Piano tuners exploit this deliberately: they don't just listen for beats between fundamentals, they listen for beats between specific harmonics (like the 4th partial of one note against the 5th partial of another) because those beats are often easier to hear and more sensitive to small tuning errors than the fundamental beat alone.
Another subtlety: once |f1 - f2| grows past roughly 15-20 Hz, the ear stops perceiving a slow amplitude pulse and starts perceiving two separate pitches, or in the middle range, a rough, dissonant buzzing sensation called "roughness" rather than a clean beat. The simple beat-frequency formula is still mathematically valid at any frequency difference, but the perceptual experience it predicts (smooth loud-soft pulsing) only matches what you actually hear when f1 and f2 are close — within a few percent of each other. Past that range the two tones just sound like a dissonant chord, and "beat frequency" stops being a useful description of what's audible.
Amplitude asymmetry matters more than people expect too. If one source is much louder than the other (say A1 = 1.0 against A2 = 0.05), the minimum composite amplitude barely dips below the maximum, and the beat becomes nearly inaudible even though the beat frequency and average frequency are exactly the same as in a case with matched amplitudes. This is why a distant, quiet echo rarely produces an audible beat against a loud direct sound, even when the two are close in pitch.
A note on cents and why musicians rarely say "hertz"
Electronic tuners don't usually display a beat frequency in Hz; they display an error in cents, a logarithmic unit where 100 cents equals one semitone. The two descriptions carry the same information but scale very differently across the keyboard, and it's worth knowing why a beat-frequency mental model can mislead you if you're not careful. A 6 Hz beat against a 440 Hz A is a small fractional error — a few cents sharp. The identical 6 Hz beat against a 82 Hz low E on a bass guitar is a much larger fractional error, tens of cents, because 6 Hz is a bigger fraction of 82 Hz than it is of 440 Hz. In other words, the same absolute beat frequency means a wildly different amount of "out of tune" depending on which note you're tuning. This is precisely why tuning by beats works best relative to a nearby reference — piano tuners historically tuned one octave using beats against a fixed reference tone, then propagated outward using fixed frequency-ratio intervals — rather than trying to interpret an absolute beat count as a fixed amount of mistuning across the whole instrument.
It also explains why beats are such a sensitive tuning aid in the first place. As two pitches converge, the beat period stretches toward infinity — a 1 Hz difference produces a full second between pulses, easily audible and easy to count by ear, long after the two pitches are close enough that most people can no longer distinguish them as separate tones by pitch alone. That's the whole reason beat-based tuning outperforms tuning by ear on pitch alone: the beat period keeps giving usable, countable feedback well past the point where raw pitch discrimination runs out.
Try it yourself
Seeing the composite waveform update live as you drag f1, f2, A1, and A2 makes the envelope-versus-carrier distinction obvious in a way that the trig identity alone doesn't. You can experiment with all four parameters and watch the beat frequency, average frequency, beat period, and max composite amplitude update instantly using the Acoustic Beats Simulator here. For a look at how beating behaves inside a resonant structure rather than free air, the acoustic resonance tool is a good next stop.
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