Beats are arithmetic that anybody can hear

Sound two tones three hertz apart and the result is not two tones. It is one tone, at the average of the two, swelling and fading three times a second.

Nothing in the air is pulsing. Two steady tones are present, each at constant amplitude, and their sum is a wave whose amplitude varies — because two sinusoids at nearby frequencies drift in and out of phase with each other, reinforcing when they agree and cancelling when they oppose.

$$\sin(2\pi f_1 t) + \sin(2\pi f_2 t) = 2 \cos\!\left(\pi (f_1 - f_2) t\right) \sin\!\left(\pi (f_1 + f_2) t\right).$$

The right-hand side is a tone at the average frequency, multiplied by a slowly varying envelope at half the difference. Because loudness does not distinguish a positive swing from a negative one, the audible pulsing happens at the full difference: $|f_1 - f_2|$ beats a second.

Three hertz apart, three beats a second. Exactly, always, with no adjustable constants.

The most useful measurement in music #

That exactness is why beating is not a curiosity but the primary instrument of practical tuning.

A tuner cannot measure frequency by ear to better than a few per cent. A tuner can count beats to about half of one per second, and what that is worth in cents depends on which interval is being listened to — because the beat happens at a coincidence high up the series, and half a hertz is a smaller fraction of a high frequency than of a low one:

The precision runs from two cents to four tenths of one, and it improves as the interval gets more complex, which is the opposite of the intuition. The third is the interval a listener finds hardest to judge as pure and it is the one that gives the finest reading, because its partials coincide at five times the lower note rather than at two.

So beating converts an unmeasurable quantity into a countable one, at a resolution five to ten times better than the ear’s own pitch discrimination, and it does so with no equipment.

The method: sound two notes that should form a pure interval, listen to the partials that ought to coincide, and adjust until the beating stops. A pure octave, fifth or third is beatless. Any deviation shows up immediately as a pulse, and the size of the deviation is the rate of the pulse.

Setting an equal temperament is therefore not a matter of getting intervals pure. It is a matter of getting each one impure by a prescribed amount, which is the same as making it beat at a prescribed rate. Tuners work from tables of those rates, and the tables are computed from the temperament — which means they can be computed here.

Across the octave below A440, an equal-tempered fifth beats between 0.59 and 0.99 times a second, rising smoothly as the pair rises, and a fourth between about 1.5 and 2.5. Those are the rates a bearing plan is built from, and their smooth rise is the check: a fifth that beats faster than the one below it in the wrong proportion is wrong, and the tuner knows before the octave is finished.

The thirds are a different matter and the difference is worth stating, because it is the one place the method runs out. An equal-tempered major third at A440 beats at 17.5 times a second and a minor third at 23.7. Those are not countable — seventeen a second is inside the roughness region the section below is about, where a listener hears a texture rather than a pulse. So a tuner cannot set an equal-tempered third by counting its beats, and does not: thirds are checked by whether their rates increase smoothly up the scale, which is a comparison between two roughnesses rather than a count of either. The instrument that makes the fifths easy makes the thirds unavailable, and the boundary between the two is the same critical band that decides everything else on this page.

This means the comma is directly audible as a rhythm. A tuner setting a sequence of fifths hears the beat rate climb as the sequence rises, in a pattern the arithmetic dictates, and an error shows up as a rate that does not match. It is one of the very few places where an abstract quantity in music theory is available as a countable physical event.

What happens as the difference grows #

Beating is not one phenomenon but a continuum, and walking up it is the fastest route into how hearing works.

Below about 6 hertz, the swelling is heard as loudness variation. One tone, wavering. Organ builders exploit this deliberately: a voix céleste stop is a rank of pipes tuned slightly sharp of another rank, and the resulting slow beating is the shimmer the stop exists for.

Between about 6 and 25 hertz, the pulsing gets too fast to follow individually and turns into roughness — a buzzing, grating quality. This is the region where mistuning stops being a pitch error and becomes a texture.

Above roughly 25 hertz, the roughness begins to fade, and somewhere above that the two tones separate: the ear stops hearing one wavering thing and starts hearing two distinct pitches.

The roughness curve is the same arithmetic summed rather than counted: every pair of partials close enough to beat contributes, and the wells fall where enough pairs coincide exactly and stop contributing.

The transition points are not fixed frequencies. They depend on where in the register the tones sit, because they are governed by critical bandwidth — the width of the frequency region within which the ear cannot separate two components. Two tones within a critical band interact and produce roughness; two tones further apart than that are resolved separately and do not.

Critical bandwidth is roughly a minor third wide in the middle of the piano’s range and much wider proportionally in the bass. A minor third in the bass is inside one critical band and is genuinely rough; the same interval two octaves up is comfortably outside and is not. This is not a preference or a convention; it is a measurable property of the cochlea, and it is the reason the roughness curve has the shape it has, and it is why every orchestration text says to space chords widely at the bottom.

Beating is a property of hearing, not of air #

It is tempting to describe beating as something the two tones do. They do not.

Two sound waves in air superpose linearly, which is what makes a spectrum additive at all, and pass through one another unchanged. Place a microphone in front of two tuning forks three hertz apart and the pressure signal recorded genuinely does swell and fade — that much is physics. But the sensation of one tone pulsing, rather than two tones present, is a consequence of the ear failing to resolve them, and that failure is the interesting part.

The evidence that resolution is the operative thing: two tones far enough apart do not beat perceptually even though the mathematics of superposition is identical. And two tones presented one to each ear — dichotically — do not produce ordinary beats either, because they never meet on the same part of the basilar membrane. What is heard instead is a much fainter and stranger effect called binaural beating, which is a different mechanism entirely and works only at low frequencies.

That last observation is the cleanest possible demonstration that ordinary beating happens in the ear rather than in the room.

The partials do the beating #

For real instruments, the beating that matters is usually not between the fundamentals at all.

Two notes a fifth apart, at 220 and 330 hertz, have fundamentals 110 hertz apart — far too far to beat. But the third partial of the lower (660) and the second partial of the upper (660) coincide exactly. Detune the fifth slightly and those two partials, not the fundamentals, drift apart and beat.

The partials of one note are what supplies the coincidences: two notes a fifth apart share their third and second partials at the same frequency, so the beat when the fifth is mistuned happens up there rather than between the fundamentals.

This is why a tuner listening to a fifth listens high — the useful information is an octave and a fifth above the notes being played. It is also why beat rate depends on register: the coinciding partials of a fifth in the bass are at a lower absolute frequency than those of the same interval in the treble, so the same cent error produces a slower beat.

And it is why an inharmonic instrument cannot be tuned this way at all. A bell’s partials are not at whole-number multiples, so there is no pair that should coincide, and there is nothing for a beatless interval to mean.

Beating in one instrument #

Everything above concerns two notes. Beating also happens inside a single sustained note, and when it does it is usually a fault being diagnosed.

A piano’s middle and upper notes have two or three strings each, struck together and meant to be in unison. They never are exactly. A pair a tenth of a hertz apart beats once every ten seconds, which is inaudible as pulsing and audible as a very slow bloom in the tone; a pair a full hertz apart produces an obvious wobble that a tuner will fix immediately.

The margin is narrow and the effect is not simply “less error is better”. Perfectly identical strings produce a note that decays faster and sounds duller than very slightly detuned ones, because coupled identical strings transfer energy to the bridge more efficiently. Piano tuners therefore aim for unisons that are as close as they can make them, and the resulting tiny residual mistuning is part of what a piano sounds like.

The same principle explains a guitar’s twelve-string courses, a mandolin’s paired strings, and the deliberate detuning in an accordion’s musette voicing. In every case the beating is present, and in every case somebody chose how much.

What a beat is not #

Two confusions are worth clearing, because both are common and both lead somewhere wrong.

A beat is not a third tone. Nothing at the difference frequency is present in the air. Three hertz is far below the range of hearing in any case, and the pulsing is a variation in the amplitude of the average frequency, not a new pitch. There are real difference tones produced by nonlinearity in the ear at high levels, and they are a separate phenomenon that happens to share the arithmetic.

A beat is not the same as a tremolo. A tremolo is an amplitude modulation applied to one tone by a player or a device. It sounds similar and it is mathematically the same shape, which is why an electronic tremolo effect and a pair of detuned oscillators can be made to sound nearly identical. The difference is where the modulation came from, and the ear cannot tell.

Counting beats to find a comma #

The most satisfying use of beat rates is to hear a piece of arithmetic that would otherwise be abstract.

Twelve fifths overshoot seven octaves by 23.46 cents. That statement can be verified on any piano in about four minutes, with no equipment.

Tune a chain of twelve pure fifths upward from a low A, each one beatless, folding down an octave whenever the note leaves the keyboard. Every fifth in the chain is exact — no pulsing at all, which is a distinctive and slightly eerie stillness. At the end, compare the note reached with the A seven octaves above the start.

It beats. Slowly, unmistakably, at a rate that depends on where the comparison is made — and that beating is the comma, converted from a ratio into a number of pulses per second that can be counted on a watch.

And a chain of twelve fifths in a temperament is laid by exactly this: each fifth is set by counting the beat between the lower note’s third partial and the upper note’s second, and the target rate is a number the tuner has memorised for each note of the bearing octave.

There is no better demonstration in the subject. An arithmetic identity about powers of two and three becomes a pulse an untrained listener can count, and it takes one instrument and no theory.

Whose music, and when #

Beating as an explicit tool is documented wherever fixed-pitch instruments needed tuning, which is nearly everywhere, and it is one of the few pieces of acoustic knowledge that seems to have been arrived at independently many times.

It has also been used as an effect rather than avoided as a fault. The voix céleste and unda maris organ stops are deliberate mistuning. So is the tuning of a mandolin’s or a twelve-string guitar’s paired courses, which are never exactly together. So is the Balinese gamelan’s ombak — the “wave” — where paired instruments are tuned a few hertz apart specifically to produce a shimmering beat, and the beat rate is a tuning parameter that a gamelan’s maker chooses and that distinguishes one ensemble’s sound from another’s.

That last case is worth pausing on. In a European tuning tradition, beating is error. In Balinese practice it is a designed quantity with an aesthetic target. The physics is identical; the value judgement is not, and only one of the two traditions is in the habit of calling the other one out of tune.

The same mathematics, four times over #

Amplitude modulation at a low rate is a single phenomenon that music has found four separate uses for, and noticing that they are one thing is worth more than learning them separately.

Beating is two tones close in frequency, and the modulation is an accident of their proximity. Tremolo is one tone whose amplitude is varied deliberately, by a player’s bow pressure or by a circuit. Vibrato is a frequency modulation, which becomes an amplitude modulation at the listener’s ear because the room and the instrument’s body respond unevenly across frequency. And the celeste and ombak effects are beating chosen and tuned as a design parameter.

All four produce a periodic swelling at a few hertz. The ear treats them as related, which is why an instrument with heavy vibrato blends more easily with a slightly mistuned one — the mistuning is heard as more of the same rather than as an error.

Drawn as waveforms, a pure tone, a string and a clarinet differ in shape and the beating argument does not care: the rate is set by the frequencies and the depth by the amplitudes, and neither is a fact about the shape.

The practical upshot is that “in tune” for an ensemble is not a target but a tolerance, and the width of the tolerance is set by how much modulation is already present. A vibrato-heavy string section can be several cents apart without anybody noticing; an organ’s stopped ranks cannot.

Where the model stops #

Two sinusoids only. The clean formula at the top is exact for two pure tones. Real tones have many partials, each pair of which beats at its own rate, and the resulting sensation is a sum that no single number describes.

Linearity. The superposition argument assumes the ear adds. It does not, quite: the cochlea is mildly nonlinear and generates combination tones — audible pitches at the sum and difference frequencies that are not present in the air at all. At high levels these are easy to hear, and they mean that two tones can produce a third that no instrument played.

The figure slows the carrier. The oscillation drawn in the hero figure is far slower than a real one, because 220 hertz over two seconds is 440 cycles and would be a solid block of ink. The envelope is drawn at its true rate; the carrier is not, and the caption says so because the ratio between them is the only thing in the picture that is wrong.

Roughness is not the whole of dissonance. Beating explains sensory roughness. It does not explain why a chord sounds unresolved, which is a matter of context and expectation rather than of the cochlea.

The ladder from here #

Later rungs: critical bandwidth measured, and where the numbers come from. Combination tones, and the pitches that are not there. Binaural beats, and why they are a different animal. Beat rates as a tuner’s table. The ombak, and beating as an aesthetic target. Roughness as a computed curve. Amplitude modulation as the same mathematics under another name. And the vibrato-versus-beating question, which asks why one periodic amplitude variation is beautiful and another is a fault.

A pair of tuning forks and a hard surface will demonstrate every claim in this essay in about a minute, which is roughly how Helmholtz demonstrated most of them in 1863, using forks, resonators and a great deal of patience.