Acoustics & Music Theory
why it sounds like that
Timbre and acoustics

The sound a listener knows best

A voice is recognisable across every vowel it says, across two octaves of pitch, down a bad telephone line and in a whisper where there is no pitch at all. Nothing that survives all of that can be a frequency. What survives is a ratio: the resonances of a vocal tract are set by its length, so a shorter tract multiplies every formant by the same factor, and identity is a scale on the spectral envelope rather than a position within it. Between an adult man and a child the whole pattern moves by a fifth, and the vowel does not change at all.
20 min read 7 figures The ear is not a microphoneWhat the listener suppliesThe instrument is not the model

Assumes: The other instrument with a reed · A vowel is two resonances

The claim that opens this essay is one everybody has tested thousands of times without noticing.

A familiar voice is identified in about a syllable. It is identified whichever vowel that syllable contains, at whatever pitch it is spoken, over a telephone that has removed everything below 300 hertz and above 3,400, through a wall, over the noise of a crowded room where several other voices are being separated out at the same time, and in a whisper — where the vocal folds are not vibrating at all and there is no pitch to identify anything by — and across the seam where the folds change vibrating mechanism entirely, which alters the source spectrum by six decibels an octave and does not make anybody unrecognisable.

Whatever the ear is using has survived removal of the fundamental, removal of the vowel, and removal of the source. That is a short list of candidates.

What the vowel is, and what it is not

The source–filter model this ladder is built on separates two things completely. The folds chop the airflow into pulses, producing a spectrum that falls smoothly at about twelve decibels per octave and carries no information about vowels at all. The tract above them resonates, and the peaks of those resonances — the formants — are what a vowel is.

The vowel in "heed", sung at 110 HzThe partials of a 110 Hz note, each drawn at the amplitude the vocal tract's resonances give it. The peaks of the curve are the formants — 270 Hz and 2290 Hz — and they stay where they are when the pitch changes, because they are a property of the shape of the mouth and not of the note being sung.F1270 HzF22290 Hz050010001500200025003000hertzamplitudethe filter — the shape of the mouththe partials — the pitch being sungand "hod", for comparisonPeterson & Barney, 1952 — mean adult male values7 partials carry most of the identity
Fig. 1 Two vowels at one pitch. The partials are identical — the same 110-hertz fundamental and the same series above it — and the envelopes are not. Everything that makes one of these “heed” and the other “hod” is in where the peaks are, and nothing whatever is in the source.

So a vowel is a shape of the envelope: the first two formants at 270 and 2,290 hertz make one vowel, and at 730 and 1,090 another. Changing vowel moves the formants relative to each other.

Which immediately rules the formants out as the carrier of identity, because a speaker says every vowel. If identity were a formant position it would change with every syllable.

The length that scales everything at once

The tract’s resonances are those of a tube closed at the larynx and open at the lips, so they sit at odd multiples of the quarter-wavelength: the speed of sound divided by four times the length, and three times that, and five times that. Every one of them is inversely proportional to the same length.

That is the whole mechanism. A shorter tract multiplies every formant by the same factor.

The vowel in "hod", sung at 110 HzThe partials of a 110 Hz note, each drawn at the amplitude the vocal tract's resonances give it. The peaks of the curve are the formants — 730 Hz and 1090 Hz — and they stay where they are when the pitch changes, because they are a property of the shape of the mouth and not of the note being sung.F1730 HzF21090 Hz050010001500200025003000hertzamplitudethe filter — the shape of the mouththe partials — the pitch being sungand "hod, a 11.5 cm tract", for comparisonPeterson & Barney, 1952 — mean adult male values11 partials carry most of the identity
Fig. 2 One vowel, two speakers. The two envelopes are the same shape displaced along a logarithmic frequency axis — not a different shape, a scaled one. The vowel is identical in both; what differs is a single number, and the peaks stay in the same proportion to each other throughout.

The sizes involved are not small. A neutral tract of 17.5 centimetres — an adult male — puts uniform-tube resonances at roughly 490, 1470, 2450 and 3430 hertz. At 14.5 centimetres they are 591, 1774, 2957 and 4140. At 11.5, a child’s, they are 746, 2237, 3728 and 5220.

Tract Resonances, Hz Mean spacing
11.5 cm 746 · 2237 · 3728 1491 Hz
14.5 cm 591 · 1774 · 2957 1183 Hz
17.5 cm 490 · 1470 · 2450 980 Hz

From an adult man to an adult woman the whole pattern moves up by 3.26 semitones; from an adult man to a child, by 7.27 — very nearly a perfect fifth. That is an enormous displacement of the spectral envelope, and it happens with the vowel completely unchanged.

Why the spacing is the invariant

The quantity that identifies the speaker cannot be a formant, because the vowel moves those. It cannot be the fundamental, because a whisper has none. What is left is the relation between formants — how far apart they are on a logarithmic axis — which is what the tract length sets. Whether the vowel leaves it alone is a separate question, it is the one the section after next puts to the data, and the answer for the first three formants is no.

Formant dispersion, the mean spacing between successive formants, is the simplest version of this. It is 980 hertz for the long tract and 1,491 for the short one, a ratio of 1.52 — which is the inverse of the length ratio, exactly, as the tube model requires.

That ratio is exact because it is computed from the tube, whose dispersion is the same for every vowel by construction: the uniform tube has no tongue in it and therefore has no vowels. The section below runs the same measure on nine real vowels of one real speaker, which is the test the model cannot perform on itself.

Every vowel, placed by its first two formants. The nine vowels of the Peterson and Barney survey plotted by their first two resonances, with both axes reversed so the layout matches the position of the tongue. The horizontal lines are sung pitches: above one of them, the first formant is below the fundamental, and no partial of the note can excite it.
Fig. 3 Nine vowels placed by their first two formants. A change of vowel is a move around this plane. A change of speaker is something the plane cannot show at all: it is the whole cloud translating along the diagonal, keeping its internal geometry. Vowel identification and speaker identification are reading orthogonal components of the same two numbers, which is why a listener does both at once and finds neither difficult.

That orthogonality is the reason the system works. A listener hearing an unfamiliar speaker say an unfamiliar word has to solve for two unknowns from one spectrum, and it is soluble because the two unknowns act on the envelope in independent ways — one moves peaks relative to each other, the other scales all of them together.

The name for the second half of that in the speech literature is vocal-tract normalisation, and the reason it needed a name is that it has to happen before vowel identification can work at all. A child’s “heed” has a first formant at 746 hertz, which is higher than an adult man’s “hod” at 730. Without normalising for the speaker, the two would be confused constantly, and they are not.

The invariant, measured on a speaker

Peterson and Barney’s nine vowels are already in this collection, averaged over their adult male speakers, and computing a dispersion for each of them is subtraction.

vowel F1 F2 F3 dispersion tract that spacing implies
heed 270 2,290 3,010 1,370 Hz 12.5 cm
hid 390 1,990 2,550 1,080 15.9
head 530 1,840 2,480 975 17.6
who’d 300 870 2,240 970 17.7
hood 440 1,020 2,240 900 19.1
had 660 1,720 2,410 875 19.6
hod 730 1,090 2,440 855 20.1

One speaker’s dispersion runs from 855 to 1,370 hertz, a ratio of 1.60. An adult man against a child, on the same vowel, is a ratio of 1.52. The quantity proposed as the speaker-invariant varies more across one person’s vowels than it does across the whole range of human tract lengths, and a listener reading tract length off it would place this one man anywhere from 12.5 centimetres to 20.1 — the first of those being a small child’s and the second longer than any adult’s.

So the orthogonality claimed two sections above is false as stated, and it is false in the direction that matters: the vowel does not merely move the formants relative to each other while leaving their mean spacing alone. It moves the mean spacing, hard. Taking only the gap from the second formant to the third makes it worse rather than better — 560 hertz on “hid” against 1,570 on “hawed”, a ratio of 2.8.

The repair is known and this collection’s data cannot carry it. The tongue’s effect on a tract falls away for the shorter wavelengths, so the higher formants — the fourth, the fifth and above — are far less vowel-dependent than the first three, and dispersion measured over those is the quantity the speech literature actually uses. Peterson and Barney published three formants; this site carries three; and three is exactly the range in which the tongue does most of its work.

What the argument of this rung therefore rests on is narrower than it looked. The scaling claim is exact and needs no data: a uniformly scaled resonator of any shape has uniformly scaled resonances, so a change of speaker really is one multiplication. What is not established here is that any measurable function of the first three formants recovers that multiplication, and the table above says one obvious candidate does not. The mechanism is secure and the read-out is owed.

The source is not silent about identity either

Setting the filter aside for a moment is worth doing, because the division of labour is not as clean as the model suggests.

The source is not silent about identity either, and setting the filter aside for a moment is worth doing because the division of labour is not as clean as the model suggests. The shape of the glottal pulse — how long the folds stay open, how abruptly they close — sets the slope of the source spectrum, and it differs between the two laryngeal mechanisms, between speakers, and between one speaker’s moods. So identity has two carriers and they behave differently. The filter carries a scale factor fixed for a given person that survives everything; the source carries pulse shape, breathiness, jitter and shimmer, which vary with effort, health and emotion within one speaker. A listener recognising somebody who has a cold is using the first; a listener noticing that somebody is upset is using the second.

So identity has two carriers and they behave differently. The filter carries a scale factor that is fixed for a given person and survives everything; the source carries pulse shape, breathiness, jitter and shimmer, which vary with effort, health and emotion within one speaker. A listener recognising somebody who has a cold is using the first; a listener noticing that somebody is upset is using the second.

That division explains an otherwise odd asymmetry. Impersonators can do a passable version of somebody’s prosody, pulse quality and articulatory habits and are limited by anatomy in exactly one respect: they cannot change the length of their own vocal tract by more than a small fraction. The imitable part is the source and the manner; the part that gives them away is the scale.

What a whisper proves

The whisper is the cleanest test, and it is available for nothing.

In a whisper the folds do not vibrate. The source is turbulent noise at the glottis — broadband, with no fundamental and no harmonics — and everything above it is unchanged. So a whisper is the tract’s filter with the source’s structure removed.

The vowel in "hod", sung at 523 HzThe partials of a 523 Hz note, each drawn at the amplitude the vocal tract's resonances give it. The peaks of the curve are the formants — 730 Hz and 1090 Hz — and they stay where they are when the pitch changes, because they are a property of the shape of the mouth and not of the note being sung.F1730 HzF21090 Hz050010001500200025003000hertzamplitudethe filter — the shape of the mouththe partials — the pitch being sungand "heed", for comparisonPeterson & Barney, 1952 — mean adult male values8 partials carry most of the identity
Fig. 4 The same two vowels an octave and a half up, which is the case a whisper is the limit of. At 523 hertz the partials are so far apart that only a handful land anywhere near the formants, and the filter’s peaks have to be inferred from three or four samples of a curve rather than read off thirty. A whisper is that argument taken all the way: the folds do not vibrate, the source is broadband turbulence with no fundamental and no harmonics, and the tract’s filter is left standing alone. Vowels remain identifiable in it and so do speakers — identity survives the removal of the entire source, which means identity is in the filter.

Vowels remain identifiable in a whisper, and so do speakers. That is the argument in its strongest form: identity survives the removal of the entire source, which means identity is in the filter. And within the filter it survives the change of vowel, which means it is in the part of the filter the vowel does not move.

The same argument, one instrument along

The structure of this claim is not peculiar to voices, and this collection has already made it about a different object.

A bowed string on 196 Hz, through a violin bodyThe source is a sawtooth at one over n; the filter is the body's measured response, with A0 at 275 Hz, B1− at 460 Hz, B1+ at 540 Hz, bridge hill at 2500 Hz. What is radiated is their product, drawn as the bars. The resonances stay where they are when the note changes, exactly as a vowel's formants do — which is why an instrument has a voice rather than a tone, and why the same argument that identifies a vowel identifies a violin. The body frequencies are measured means over full-size instruments rather than computed from a plate.A0B1−B1+bridge hill05001000150020002500300035004000hertzamplitudethe body — a filter, and it does not movethe string — a sawtooth, and it doesbody response: measured means, full-size instruments
Fig. 5 A violin body’s response: a fixed set of resonances through which every note played on the instrument passes. It is the same architecture as the vocal tract — a source with a wide spectrum, a resonator that shapes it, and a listener who identifies the resonator rather than the source. And it scales the same way: a viola is a larger box and its whole response is lower.

The body is the filter for a string instrument in the way the tract is for a voice, and the consequence is identical. Two violins are told apart by their bodies, not their strings; a violin is told from a viola by a scale factor on the response; and neither identification depends on which note is being played.

Where the two part company is that a violin cannot change its filter and a speaker changes theirs several times a second. That is the entire difference between an instrument and a voice, and it is why the voice needed a two-part model when the string essays did not.

The vowel in "heed", sung at 196 HzThe partials of a 196 Hz note, each drawn at the amplitude the vocal tract's resonances give it. The peaks of the curve are the formants — 326 Hz and 2764 Hz — and they stay where they are when the pitch changes, because they are a property of the shape of the mouth and not of the note being sung.F1326 HzF22764 Hz050010001500200025003000hertzamplitudethe filter — the shape of the mouththe partials — the pitch being sungand "heed, a 17.5 cm tract", for comparisonPeterson & Barney, 1952 — mean adult male values3 partials carry most of the identity
Fig. 6 A close vowel through two tracts, at a pitch where the partials are widely spaced. The two envelopes are the same shape scaled, exactly as with the open vowel — the scaling is a property of the tube and not of the vowel, so it holds for all of them at once. That is the condition for the scale factor to be usable as an identity cue: it has to be the same number whatever is being said.

The part the model gets wrong

The uniform tube is a caricature and it should be said how bad a one.

A real vocal tract is not a tube of constant cross-section; it is a tube with a tongue in it, and the tongue is what makes a vowel. The uniform model’s resonances at 490, 1470 and 2450 hertz correspond to no vowel — they are the neutral schwa, roughly, and every actual vowel departs from them substantially.

What the model gets right is the scaling, and it gets it right for a reason that does not depend on the shape. Any resonator scaled uniformly has its resonances scaled by the inverse of the factor, whatever its shape is. So the claim “a shorter tract multiplies every formant by the same factor” is exact for a uniformly scaled tract of any shape at all, and the uniform tube is doing nothing but supplying illustrative numbers.

That matters because tracts are not uniformly scaled between people. The pharynx grows more than the oral cavity from childhood to adulthood, and men’s pharynges are proportionally longer than women’s — so the real relation between speakers is a scaling plus a distortion, and the distortion is why a synthesised voice made by simply shifting formants sounds like a small adult rather than a child.

What this predicts that is checkable

Two consequences follow directly and both are testable without any equipment beyond a recording.

Resampling a recording should change the speaker and not the words. Playing a voice back at 1.2 times speed multiplies every frequency by 1.2 — fundamental and formants alike — which is the tract-length transformation with a pitch change bundled in. The result sounds like a smaller person saying the same thing, which is what everybody who has sped up a recording has heard, and the striking part is that the vowels are unharmed. A transformation that moves every formant by a fifth ought to destroy vowel identity if vowels were formant positions, and it does not.

Shifting the pitch alone should not change the speaker. Moving the fundamental without moving the formants — which is what a pitch-shifter that preserves formants does, and what a singer does across their range — leaves the speaker recognisable. Both of these are routine in audio production, and the fact that the two operations have such different perceptual consequences is the strongest everyday evidence that the source and the filter are separately read.

The failure case is the one that used to be audible in cheap pitch-shifting, where the formants were dragged along with the pitch and every shifted voice sounded like a cartoon. That artefact was the model being violated, and it is why the fix has a name.

The vowel in "hod", sung at 110 HzThe partials of a 110 Hz note, each drawn at the amplitude the vocal tract's resonances give it. The peaks of the curve are the formants — 881 Hz and 1316 Hz — and they stay where they are when the pitch changes, because they are a property of the shape of the mouth and not of the note being sung.F1881 HzF21316 Hz050010001500200025003000hertzamplitudethe filter — the shape of the mouththe partials — the pitch being sungand "hod, a 17.5 cm tract", for comparisonPeterson & Barney, 1952 — mean adult male values13 partials carry most of the identity
Fig. 7 The resampling prediction, drawn as the two tracts it is equivalent to. Playing a voice back at 1.2 times speed multiplies every frequency by 1.2 — fundamental and formants alike — which is exactly the transformation from a 17.5-centimetre tract to a 14.5-centimetre one with a pitch change bundled in. The result sounds like a smaller person saying the same thing, and the striking part is that the vowels are unharmed. A transformation that moves every formant by a fifth ought to destroy vowel identity if vowels were formant positions, and it does not — which is the whole of the claim that a vowel is a ratio of resonances rather than a pair of frequencies.

Which computation produced the numbers

The resonances are the closed-open tube formula: the nth is (2n−1) times the speed of sound divided by four times the length, with the speed taken as 34,300 centimetres a second. The three tract lengths are the standard ranges quoted for children, adult women and adult men, and are the one input here that is not computed.

The semitone shifts are 12 log₂ of the length ratio, which is exact and needs no acoustics at all: if every frequency scales by the same factor, the shift in semitones is the same for every one of them.

The vowel formants drawn in the figures are the Peterson and Barney survey values, which are adult male, and the scaled versions are those multiplied through by the length ratio.

Whose voices, and where the numbers came from

The vowel data is a 1952 survey of American English, and it is used across this site because it is the measurement everything else was calibrated against. It is not a description of vowels in general — vowel inventories differ enormously between languages, and a formant plot of a language with front rounded vowels or with pharyngeals looks nothing like this one.

The identity argument is not about a language. Recognising a familiar voice is not a linguistic skill and is not confined to speech; it works on a laugh, a cough and a hum. The mechanism proposed here — that a scale factor on the envelope is the carrier — is the standard account in the speech-perception literature and is quoted rather than demonstrated by anything drawn here.

The evolutionary version of the same argument is worth naming because it is where the strongest evidence sits. Formant dispersion tracks body size in a great many mammals, several of which use it in vocal displays, and a few of which have evolved elaborate machinery for lying about it — a descended larynx lengthens the tube and makes the animal sound bigger than it is.

And the long-term average spectra of an orchestra and of a trained solo voice show what the same filter can be made to do when the problem is audibility rather than identity: a lowered larynx clusters the third, fourth and fifth formants near three kilohertz and puts a peak where the orchestra has none. That is the same tract, reshaped for a different job.

What the picture cannot show

It cannot show learning. Everything here is about what information is present in the signal. Recognising a particular familiar voice from that information is a memory task with a large learned component, and nothing in an envelope says how many exposures it took.

It cannot show the source’s own contribution. Two speakers with identical tracts would still be distinguishable, by the shape of their glottal pulse, by jitter and shimmer, by breathiness, and by habits of articulation and prosody that are not acoustic properties at all. The filter carries a great deal of identity; it does not carry all of it.

It cannot show the room. Everything drawn here assumes the envelope reaching the ear is the envelope leaving the mouth, and a room imposes a filter of its own that is often stronger than the differences between two speakers. Voices are still recognisable in rooms, which means the ear is discounting the room’s contribution — a separate problem with its own machinery.

And it stops working at the top of a voice. A soprano above about 700 hertz has a fundamental higher than her own first formant, so the partials are too far apart to sample the envelope. Vowels become hard to identify there, and so do speakers — which is a prediction this model makes and which every listener to a high soprano line has confirmed without meaning to. It is also why the singer’s formant is a low-register phenomenon: a resonance can only be exploited if there are partials near it, and above the first formant there are not enough partials left to exploit anything.

The ladder from here

This rung and the next were both named as owed when this anchor opened, and they are opposite questions about the same thing. This one asks what makes one voice identifiable. The next asks what happens when there are sixteen of them on one note, and the answer is that the identifiability is exactly what goes — a choir is not a loud singer, and the reason is that everything this essay describes as a carrier of identity becomes, at sixteen sources, a spread.

What is still owed after that is the acquisition side: a listener’s ability to normalise for a tract they have never heard before is present in infancy and is not obviously learnable from the data available, and nothing here has anything to say about how it gets there.

Part 5 of 13

One essay in the series on the voice. The essays either side of this one:

A note sung at 440 Hz, drawn in cents. Deviation from the notated pitch against time, for a vibrato of ±71 cents at 6.0 cycles a second — Seashore's and Prame's measured values, which agree. The note is 142 cents wide, and the shaded bands across the middle are the syntonic comma at 21.5 cents, the Pythagorean comma at 23.5 cents, the difference limen at 440 Hz at 4.0 cents. The pitch a listener reports is near the middle of the excursion rather than at either edge — and not exactly at the middle either: because cents are logarithmic and frequency is not, the mean frequency of this trace sits 0.73 cents above the centre line. A note that is never at its pitch Part 4 — Eleven essays in this field argue about differences of one to twenty-four cents. An ordinary operatic vibrato is a hundred and forty cents wide and completes six excursions a second, so every one of those distinctions fits inside a single sung note several times over — and the beat rate a tuner would null passes through zero eleven times a second. The fluctuation stays; the rate goes. A unison of n voices with a spread of 15 cents, averaged over 5 draws. The depth of the amplitude fluctuation does not fall as voices are added — a choir is no steadier than a duet — but the fraction of that fluctuation in any single modulation component falls from 77 per cent at two voices to 24 at 32. Two voices make one beat and it can be counted; 16 make 120 and none of them is a rate. That is why a choir cannot be tuned by nulling anything. What a choir does that a soloist cannot Part 6 — Two singers on one note produce one beat and it can be counted. Sixteen produce a hundred and twenty at once, and the amplitude still fluctuates by as much as it did — a choir is no steadier than a duet. What has gone is not the fluctuation but its rate: the modulation energy that sat in a single line at two voices is spread across a band at sixteen, with no line in it. That is the choral sound, and it is also why the just-intonation drift this site measured describes only ensembles that hold their pitch still.

Where this goes next

The vowel in "hod", sung at 110 Hz. The partials of a 110 Hz note, each drawn at the amplitude the vocal tract's resonances give it. The peaks of the curve are the formants — 730 Hz and 1090 Hz — and they stay where they are when the pitch changes, because they are a property of the shape of the mouth and not of the note being sung. A vowel is two resonances The vowel in "heed" is the same vowel sung high or low, and nothing about it is a property of the note. It is two peaks in the response of the mouth, sitting at fixed frequencies while the partials of the voice slide underneath them. The flow through the larynx, over two periods of a 110 Hz note. Volume flow against time, in Rosenberg's two-half-cosine model of the glottal pulse — a slow opening, a faster closing, and a closed phase during which no air passes at all. M1 — chest is open for 50 per cent of each period and opens 2.4 times as slowly as it closes. Nothing here is a displacement: the folds are a valve on a steady stream of air, and the flat stretches are the moments they are shut. At 110 Hz each period lasts 9.1 milliseconds, of which 4.5 is silence. The other instrument with a reed The folds do not vibrate the way a string does. They open and shut across a steady stream of air, once per period, and what leaves the larynx is a train of flow pulses with a closed phase in it. Everything said about the voice's tone is a statement about the shape of that pulse — and the shape has two numbers in it. A bowed string on 196 Hz, through a violin body. The source is a sawtooth at one over n; the filter is the body's measured response, with A0 at 275 Hz, B1− at 460 Hz, B1+ at 540 Hz, bridge hill at 2500 Hz. What is radiated is their product, drawn as the bars. The resonances stay where they are when the note changes, exactly as a vowel's formants do — which is why an instrument has a voice rather than a tone, and why the same argument that identifies a vowel identifies a violin. The body frequencies are measured means over full-size instruments rather than computed from a plate. The body is the filter A violin string radiates almost nothing. What reaches a room is the string's sawtooth multiplied by the body's response, and that response is a comb of measured resonances that stays put while the note moves. It is the same arithmetic that identifies a vowel, on wood instead of a mouth — which is why an instrument has a voice rather than a tone.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

What this makes readable

Essays that declare this one a prerequisite.

The objects named here

The third way in, after the field and the series: the things themselves, and every essay that touches each one.

FormantResonanceSource-filterSpectral envelopeTimbreVocal tractVoice identity