Acoustics & Music Theory
why it sounds like that
Intervals and chords

Beats are arithmetic that anybody can hear

Two tones a few hertz apart swell and fade at exactly their difference. It is the most direct evidence available that the ear does sums on what reaches it, and it is how every instrument in the world gets tuned.
17 min read 9 figures The ear is not a microphoneSmall whole numbers

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.

220 Hz against 223 HzTwo tones 3 hertz apart, added. The rapid oscillation is their average; the slow swelling is their difference, heard as 3 beats a second and used by every tuner who has ever worked by ear.00.511.52seconds3 beats per second — the difference, exactlythe carrier is drawn slower than it sounds, or it would be a solid band
Fig. 1 Two tones a few hertz apart, added. The rapid oscillation is their average; the slow swelling is their difference, heard as a pulsing at exactly the number of hertz that separates them. The carrier here is drawn far slower than it sounds, or it would be a solid band — but the envelope is drawn at its true rate, because the beat rate is the entire point.

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⁡(2πf1t)+sin⁡(2πf2t)=2cos⁡ ⁣(π(f1−f2)t)sin⁡ ⁣(π(f1+f2)t).\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: ∣f1−f2∣|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 a fraction of one per second, which at 440 hertz is a precision of about one part in a thousand — four cents, and better than that with practice. Beating converts an unmeasurable quantity into a countable one, 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.

The simple ratios, and the twelve equal stepsOne octave laid out in cents. Above the line, the frequency ratios of small whole numbers, where they actually fall; below it, the twelve equal steps. The two sets almost never coincide.6/5minor third5/4major third4/3fourth3/2fifth8/5minor sixth5/3major sixthCC♯DE♭EFF♯GA♭AB♭BC+16-14-2+2+14-16the ratios of small whole numberstwelve equal steps of exactly 100 cents1200 cents to the octave
Fig. 2 One octave in cents, with the simple ratios where they fall and the twelve equal steps below. Every one of those dashed connectors is a beat rate a tuner can hear: the two-cent gap at the fifth is a slow, patient pulse and the fourteen-cent gap at the major third is a fast one.

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.

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.

Roughness across an octaveSensory dissonance between two complex tones as the upper one is swept through an octave, computed by summing the roughness between every pair of their partials. Nothing here is placed by hand. The deep wells land on the fourth, the fifth and the octave; the thirds sit on shoulders rather than in wells, which is a real feature of this model and not a defect of the drawing.semitones above the lower tone6/55/44/33/25/32/1CC♯DE♭EFF♯GA♭AB♭BCroughest at about a semitonesmooth at the simple ratios
Fig. 3 Roughness across an octave, computed by summing the beating between every pair of partials of two complex tones. The peak just above the unison is the roughness region made visible: a semitone is the roughest interval available, and it is rough because its partials are separated by exactly the wrong amount.

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.

Four spectra of the same noteThe amplitude of each partial for four timbres at the same pitch. These are the exact lists the sound buttons on this site synthesise from, so the picture and the sound are the same data.1pureone partial, nothing else12345678stringall partials, falling12345678clarineteven partials nearly absent123456789bellodd partials onlyamplitude
Fig. 4 The partial content of four timbres. Beating happens between individual partials, so the roughness of an interval depends on which of these lists the two tones are drawn from — a fact about the instruments, not about the interval.

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 first eight partials of a stringA string vibrating in one, two, three and more equal parts, with the frequency ratio and the nearest named note beside each. The seventh partial is a third of a semitone flat of anything on a keyboard, which is a fact about strings rather than about tuning.1130.8 HzC2261.6 HzC3392.4 HzG4523.3 HzC5654.1 HzE -14¢6784.9 HzG7915.7 HzB♭ -31¢81046.5 HzCpartialthe dots are the nodes — the places that do not move
Fig. 5 The partials of a single string, with the frequency of each. Tuning by ear means listening for a specific pair out of two stacks like this and making them agree, which is why a tuner listens to a region of the sound rather than to the note.

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.

Three envelopesHow loudness changes over the life of a note, for a plucked, a bowed and a struck instrument. Remove the attack from a recorded piano and it stops sounding like a piano, which is the shortest demonstration that the envelope carries as much identity as the spectrum.00.511.5200.20.40.60.81secondsamplitudeplucked — no sustainbowedstruck — no sustainkey released
Fig. 6 Three envelopes — how loudness changes over the life of a note. Beating is a modulation of exactly this kind, arriving from a second tone rather than from a player, and the ear treats the two the same way because it has no access to the difference.

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.

The chain of fifths in quarter-comma meantoneThe fifths laid end to end as the chain they are. The bar under each shows how far that fifth departs from a pure three-to-two, and the last one — where the chain is forced to close — is the wolf.E♭B♭FCGDAEBF♯C♯G♯D♯-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4+35.7each fifth's departure from a pure 3:2, in centsone fifth carries the whole error
Fig. 7 The chain of fifths with each fifth’s departure from pure drawn beneath it. Every one of those bars corresponds to a beat rate: a bar of zero height is a beatless interval, and the tall one at the end is a howl.

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.

Notes on the keyboardA piano keyboard with the notes under discussion marked. The keyboard is used throughout this site because it shows distance rather than name, and distance is what the theory is about.CEG3 notes sounding
Fig. 8 Two notes on a keyboard. Everything in this essay concerns what happens when both are held down, which a keyboard makes trivially easy and which is the reason keyboard instruments drove the entire history of tuning theory.

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.

The same partials, drawn as pressureEach spectrum summed into the wave it actually produces, over two cycles. The shapes are strikingly different and the ear has almost no access to that difference — what it hears is the list of partials, not the shape they add up to.pure1 partialstring8 partialsclarinet8 partials2 cycles · each normalised by its own peak
Fig. 9 Three timbres drawn as the waves they produce. Every one of these shapes is a sum of steady components; a beating pair is a sum of two steady components too, and nothing distinguishes the two situations in the air. What distinguishes them is entirely in how the ear parcels the frequencies out.

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.

Where this goes next

one frame = one repeat of the combined wavelower — 2 per frameupper — 3 per frametheir sumthe pattern repeats after 2 cycles of the lower tone Two notes and a ratio, which is the whole of consonance Sound two tones together and the pair either settles or does not. What decides it is the ratio of their frequencies, and the rule is that simpler ratios settle — which is two and a half thousand years old and still not quite an explanation. semitones above the lower tone6/55/44/33/25/32/1CC♯DE♭EFF♯GA♭AB♭BCroughest at about a semitonesmooth at the simple ratios Roughness can be computed, and the answer looks like a scale Feed two spectra into a model of how nearby frequencies interfere, sweep one past the other, and the curve that comes out has dips exactly where the consonant intervals are. Nobody put them there. 024681012-20-1001020semitones above the tonicCC♯DE♭EFF♯GA♭AB♭BCjust intonationPythagoreanquarter-comma meantonesharp of equalflat of equal Where to hide the comma, which is the only real question Four tuning systems, four different places to put an error that cannot be removed. Each one is coherent, each was standard somewhere, and the choice between them decided what music could be written.