A buzz and a tube.

Strike it, and listen.

Close your eyes. Someone knocks on a wooden table, then a steel pipe, then a glass window. You name each one instantly, even though the knock was the same each time: the same knuckle, the same force, the same brief instant of contact. The knock only lasted a few milliseconds. Everything you heard afterwards, the dull thud or the long bright ring, came from the object answering the knock in its own way.

Try it. Click a slab to strike it, then change what you strike it with.

What makes them sound different?

A struck object vibrates in a handful of natural modes, each a pattern of motion with its own frequency, and each dying away at its own rate. The spectra above show them as peaks. Two physical properties set almost everything:

And the striker? The hard hammer touches for a fraction of a millisecond, delivering an almost perfect instantaneous kick that excites every mode, low and high. The soft mallet stays in contact for a few milliseconds, which is too slow to shake the high modes, so you hear the same object, darker. Notice what did not change when you switched strikers: the positions of the peaks. They belong to the object. The striker only decides how strongly each one gets excited.

That last formula already contains the whole of this page. The striker is \(X(f)\), the source. The slab is \(H(f)\), the filter. What you hear is their product, and your ear is remarkably good at telling the two factors apart.

Impulse responses are everywhere

Once you look for it, “excite it briefly, then listen to how it answers” is one of the most common ways people and machines probe the world.

Now tap faster.

One strike gives one ring. What if you strike the same slab again and again, at a steady rate? At two taps a second you hear a rhythm: tap, tap, tap. Speed up. Somewhere around 20 taps per second something remarkable happens: your ear stops hearing separate events and hears a single continuous buzz with a pitch. The pitch is the tapping rate.

Look at the spectrum as you raise the rate. This picture is the heart of the source–filter model, so it deserves a slow read:

• The taps set the pitch. A tap train repeating every \(T\) seconds is periodic, so its sound can only contain frequencies at whole-number multiples of \(1/T\): the harmonics. The spectrum turns into a comb of lines, spaced exactly at the tap rate. Double the rate and the lines spread twice as far apart; the pitch goes up an octave.
• The slab sets the colour. The heights of those lines follow the slab's resonance curve (red). Lines that fall near a resonance are loud; lines between resonances are weak. Swap wood for steel at the same rate and the pitch stays the same while the timbre changes completely.
• The two are independent. The tapper knows nothing about the slab, and the slab doesn't care how often it is tapped. What you hear is simply comb × resonance curve.

Your voice does exactly this.

Map the pieces across, one to one:

Musicians know this arrangement too. In a trumpet, the player's lips buzz (source) and the brass tube resonates (filter). The player changes the note by changing the buzz (with the tube's strong resonances helping to lock the lips onto one of its notes); the instrument's character comes from the tube. In speech we mostly do the opposite: we keep the buzz roughly steady and change the tube, and that is what spells out vowels.

The grid below does the same experiment with a computer. Each row is a source, each column a filter (a vowel-shaped tube, or none at all). Click any cell.

Listen to the first column: the raw source on its own is a harsh, featureless buzz (or a plain hiss). It isn't any vowel. All the “vowel-ness” is added by the filter. Along a column the vowel stays the same while the voice changes; along a row the voice stays the same while the vowel changes. Two knobs, independent, just like the tapper and the slab.

The source–filter model in one picture.

Formalised by Gunnar Fant in 1960 (Acoustic Theory of Speech Production), the model says that a speech signal can be treated as the output of a filter (the vocal tract) driven by a source (the vocal folds, or turbulent noise), with the two varying independently:

In symbols, this is the same statement as the slab: the output is the source convolved with the tract's impulse response, or equivalently the source spectrum multiplied by the tract's frequency response.

The last line is why the model is so useful: on a log (dB) scale, the source and the filter add. The source contributes fast ripples across frequency (the harmonics, one every \(F_0\) Hz). The filter contributes a slow, smooth envelope (the formant peaks). Different speeds of variation means they can be pulled apart again, and much of speech analysis is ways of doing exactly that.

Does real speech really look like this?

Here is one speaker, ksp, an Indian-English male speaker from the CMU ARCTIC corpus, saying the vowels /i/ and /ɑ/ in different sentences. Like everyone, he does not speak on a monotone: his pitch drifts up and down, so we can pick each vowel once where his pitch was lower and once where it was higher. For each we show a 40 ms spectrum from the middle of the vowel, with its smooth envelope drawn on top.

Compare the two /i/ spectra: with harmonics every 173 Hz instead of every 111 Hz, the same envelope is sampled more sparsely, so its shape is harder to see. Most women's voices (roughly 165–255 Hz) and children's voices (higher still) make this worse. This is a real, practical difficulty (formant estimation is harder for high-pitched voices), and a direct consequence of “comb × envelope”; Figure 8 below shows it for a 220 Hz voice.

Now let's open up the two boxes, starting with the source.

The vocal folds: a tapper made of muscle.

Inside your larynx (the “Adam's apple”), two small folds of tissue stretch across the airway from front to back: about 1.5–2 cm long in adult men, 1.2–1.5 cm in women. The gap between them is the glottis. When you breathe, the folds are held wide apart. When you speak a voiced sound, muscles bring them together, and the air from your lungs does the rest.

One cycle, four moments

Nothing in your brain fires once per cycle. Muscles set the folds' tension and position, and the oscillation then runs itself on air pressure and elasticity, the same way a flag flaps in the wind or your lips buzz when you blow a raspberry (or play a trumpet). This is the myoelastic–aerodynamic theory of voicing. Tighter, thinner folds vibrate faster, giving a higher pitch.

The animation is slowed down more than a hundred times. Watch the three traces on the right. The airflow rises gently as the folds open but falls off a cliff when they close. Its rate of change is therefore small during opening and a big, sharp negative spike at closure. That spike is the kick that excites the tract, and the impulse train keeps just that: one impulse per closure.

Can we see the taps in real speech?

Yes, if we zoom in far enough. Here are 50 ms from the middle of the vowel in “dark” (from “Ah, it is growing dark and darker”), spoken by ksp.

An impulse train in time is a comb in frequency

Why does a periodic source produce harmonics? Any signal that repeats every \(T_0\) seconds can be written as a Fourier series, a sum of sinusoids at \(F_0, 2F_0, 3F_0, \dots\) and nothing in between. For an impulse train the answer is especially clean: every harmonic has the same strength,

An impulse train in time is an impulse train in frequency, with reciprocal spacing: pulses far apart in time (low voice) give harmonics close together in frequency, and vice versa. A flat comb is the ideal source for a filter to act on. Every frequency the tract might want to emphasise is available, in equal measure, on the harmonic grid.

Refinement: real glottal pulses are smooth

Real folds don't produce infinitely sharp impulses. The airflow through the glottis is a smooth, lopsided hump each cycle (switch the explorer to realistic glottal pulses). Its harmonics fall off at about −12 dB per octave: still a comb at exactly the same frequencies, just tilted. We don't need a new model for this. A smooth pulse train is an impulse train passed through a smoothing filter \(G(f)\):

Since \(G\), \(V\) and \(R\) are all filters, they can be lumped together into one overall filter driven by a flat impulse train. That keeps the source as simple as possible (just where the impulses are), and it is exactly what the AR model later on will do. The leftover net tilt of about −6 dB/octave is why speech is usually pre-emphasised (high frequencies boosted slightly) before AR analysis.

Pulse shape also carries voice quality. Folds that close gently and incompletely give smooth pulses and a breathy voice. Folds that slam shut give sharp closures, stronger high harmonics and a pressed, tense voice.

And the tapping rate is not constant. \(F_0\) rises and falls through every sentence: that is intonation (a question rises at the end, a statement falls). The model handles this simply by letting \(T_0\) change from one short frame (~20–30 ms) to the next.

Not every sound is tapped: noise sources

If the folds are held apart and air is forced through a narrow constriction instead, the flow becomes turbulent: a random, hissing pressure fluctuation with no period at all. That is the second kind of source in the model: noise. It can arise at the glottis itself (/h/, whispering) or further forward, where the tongue or lips narrow the tract (/s/, /ʃ/, /f/). Voiced fricatives such as /z/ and /v/ run both sources at once.

Isolated phone segments cut straight from ksp's sentences at the (automatic) phone boundaries: they sound clipped out of context, which is part of the point. Real speech is continuous.

We know what kicks the tract. Next: why it rings where it does.

Why things ring where they ring.

Every slab in section 01 rang at its own set of frequencies, and the vocal tract does too. To see where those frequencies come from, start with the simplest thing that can ring: a weight hanging on a spring.

A mass on a spring

Pull the mass down and let go: it bobs up and down at one particular rate, its natural frequency. The spring pulls it back, its inertia carries it past the middle, and so on. A stiffer spring pulls harder, so it bobs faster. A heavier mass is more sluggish, so it bobs slower:

Friction (damping) makes the bobbing die away. Now do what a child's swing teaches everyone: instead of one pull, keep pushing it rhythmically, by moving the top of the spring up and down. Try it below. Change how fast you shake it and watch how far the mass swings.

What you should find:

• Shake slowly and the mass just follows your hand. Shake fast and it barely moves: it can't keep up.
• Shake at the natural frequency and each push arrives exactly in step with the motion, adding to it. The swing grows many times larger than your hand's movement. This is resonance. (It is how a singer can shatter a wine glass, and why soldiers break step on bridges.)
• With light damping the resonance peak is tall and narrow: you have to hit the rate precisely, but then the response is huge. With heavy damping it is low and broad.
• Kick it once (with the shaking stopped) and it rings at its natural frequency, dying away at a rate set by the damping. That decaying ring is its impulse response.

Keep that formula in mind for a moment. A typical vocal-tract resonance has a bandwidth of 50–150 Hz, which means it rings for only \(\tau \approx 2\text{--}6\) ms. That is exactly what Figure 4 showed: each glottal kick rings and has largely died away before the next one, about 8 ms later.

Air is a spring, too

A column of air has both ingredients of a resonator. It has mass (air is light, but not weightless) and it is springy (squeeze it and it pushes back). So an air-filled tube resonates, and that is how every wind instrument works. Blow across the top of a bottle and you hear its resonance. Pour in some water to shorten the air column and the note goes up. Your vocal tract is a bent tube of air about 17 cm long, so it has resonances too, and those resonances are the formants.

The vocal tract as a tube, one step at a time

Why should a 17 cm tube ring at 500, 1500 and 2500 Hz, and not at any other frequencies? We'll build the answer in small steps, starting from bathtime, and let the formula fall out of a picture at the end.

Step 1 · Air in a tube sloshes, like water in a bathtub

Sit in a bathtub and rock back and forth. If you rock at just the right speed, the water sloshes higher and higher, and it slops over the edge. Rock at any other speed and the waves stay small and messy. The tub has its own favourite sloshing rhythm, set by its size: a small basin sloshes quickly, a big tub slowly.

The air inside a tube behaves the same way. It can slosh back and forth along the tube, and it has its own favourite rhythms, again set by the tube's size. For a tube the size of your vocal tract, those rhythms are hundreds of times a second: they are audible tones. Blow across a bottle and you hear one of them. Those favourite rhythms are the resonances, the formants. The only question is: which rhythms does a tube like best?

Step 2 · The two ends set the rules

The answer depends entirely on what happens at the two ends of the tube. Our tube has two very different ends:

Step 3 · Which wave shapes obey both rules?

When the air sloshes at a steady rhythm, the amount it swings changes smoothly along the tube in the shape of a wave. So we ask: what piece of a wave starts at zero (still, at the glottis) and ends at its biggest swing (at the lips)? Try the simplest pieces one by one. The curve in each tube below shows how far the air swings at each point.

So the tube's favourite waves are the ones that fit \(\tfrac14, \tfrac34, 1\tfrac14, \dots\) of a wavelength into its length \(L\):

Step 4 · From wavelength to frequency

One more fact: sound travels through warm air at \(c \approx 350\) m/s, and for any wave, \(\text{speed} = \text{wavelength} \times \text{frequency}\). So a frequency is just the speed divided by the wavelength, \(F = c/\lambda\). Turn each fitting wave into a frequency:

Every row has the same shape, an odd number times \(c/4L\), so we can write them all at once:

Step 5 · Watch it, and hear it

Below, each dot is a little parcel of air. Pick a resonance and watch: the parcels by the glottis barely move, and those at the lips swing the most. For \(F_2\) and \(F_3\) you'll also see still points inside the tube, where the wave passes through zero. Change the tube length and the formants follow the formula.

Does the uniform tube match real speech? A uniform tube is the average tract shape. Every real vowel bends it one way or another, so its formants should sit near the middle of the vowel space. They do:

Real tracts aren't uniform: glue tubes together

Your vocal tract is not a straight pipe of constant width. The throat is wide in some places and narrow in others, the tongue bulges into it, and the lips can nearly close. The standard trick for modelling that shape is beautifully simple: approximate it by a chain of short tubes, each with its own width, like a pipe built from pieces of different diameters.

What happens at a joint between a wide piece and a narrow one? It is a partial wall. Some of the pulse passes through and some echoes back. The bigger the change in width, the stronger the echo:

In a uniform tube, sound only bounces off the two ends. Now it also bounces, a little, off every joint, so the sloshing patterns that fit are different. Each vowel's pattern of wide and narrow pieces creates its own resonances. Build some vowels yourself below: drag each piece up or down to change its width. The formants are recomputed from the echoes at every joint.

Notice how little it takes. Two pieces, a narrow one behind a wide one, already sound like “ah”; flip them and you are heading towards “ee”. Linguists and engineers have used these two- and three-tube models since the 1960s (Fant's work) to reason about vowels with pencil and paper. The more pieces you use, the closer you get to a real tract shape. The virtual vocal tract below simply uses 34 thinner slices and lets a “tongue” shape them for you.

Shaping the tube: how vowels are made

When you raise your tongue, drop your jaw or round your lips, the tube is no longer uniform: it gets narrow in some places and wide in others. Its resonances move, and different vowels come out. Three rules of thumb cover most of it:

(The general principle, from acoustic perturbation theory: narrowing the tube where a resonance's air velocity is greatest lowers that resonance, and narrowing it where the pressure swing is greatest raises it. Each formant has its own pattern of velocity and pressure maxima, which is why one tongue movement can push \(F_1\) and \(F_2\) in opposite directions.)

Four real vowels: mouth shape, waveform, spectrogram

Here are the rules of thumb in action on four vowels spoken by one speaker (ksp, CMU ARCTIC). Each column is one vowel. Read it top to bottom: how the mouth is shaped, what the waveform looks like (40 ms from the middle of the vowel), and where the energy sits in the spectrogram of the whole word, with \(F_1\) and \(F_2\) measured and marked. The four vowels sit at the corners of the vowel space, so their shapes and formants differ as much as vowels can.

How to read the sketches: the head faces left. Blue is the air inside the vocal tract, the “tube” from the vocal folds to the lips; the pink shape is the tongue; the red circle marks where the tube is narrowest. The small oval at the top left is the lips seen from the front. The nasal cavity (dashed) is shut off by the raised velum for all four vowels. The sketches are schematic: they show the tongue, jaw and lip positions textbooks give for these vowels, not measurements of this speaker.

/i/
“seat”

/æ/
“sad”

/ɑ/
“dark”

/u/
“two”

Mouth shape
(side view, sketch)

tongue high & front, lips spread

tongue low & front, jaw open

tongue low & back, jaw wide open

tongue high & back, lips rounded

Waveform
(40 ms)

Spectrogram
(word, 0–4 kHz)

Formants
& audio

\(F_1 \approx 310\) · \(F_2 \approx 2290\)

\(F_1\) low, \(F_2\) high: far apart

\(F_1 \approx 650\) · \(F_2 \approx 1650\)

\(F_1\) high, \(F_2\) fairly high

\(F_1 \approx 690\) · \(F_2 \approx 1330\)

\(F_1\) high, \(F_2\) low: closer together

\(F_1 \approx 320\) · \(F_2 \approx 810\)

\(F_1\) low, \(F_2\) low

Sketches use the same articulatory presets as the vocal-tract demo. Spectrograms use a short (6 ms) window so formants show up as bright bands; thin white vertical lines mark where the vowel starts and ends, the dashed lines show \(F_1\) and \(F_2\), and their values are measured automatically (with the AR method of the next section) at the vowel's middle.

What to look for:

• Down each column, one tract shape gives one waveform shape and one pair of formant bands. The shape of the tube is written into the sound.
• Tongue height ↔ \(F_1\). The high-tongue vowels /i/ and /u/ both have a low \(F_1\) (about 310–320 Hz). The low-tongue, open-jaw vowels /æ/ and /ɑ/ have a high \(F_1\) (650–690 Hz). In the waveforms, /ɑ/'s strong, fast \(F_1\) ringing is obvious, while /i/ is a slow wave with fine high-frequency ripples on top (its high \(F_2\)).
• Tongue front/back ↔ \(F_2\). The front vowels /i/ and /æ/ have a high \(F_2\) (about 1650–2290 Hz), and the back vowels /ɑ/ and /u/ a low one (810–1330 Hz). In the spectrograms you can see \(F_1\) and \(F_2\) far apart for /i/ and much closer together for /ɑ/ and /u/.
• Formants move. Look at the edges of the vowels in “sad” and “dark”: the bands bend into and out of the vowel as the tongue travels from and to the consonants. Those glides (formant transitions) are how listeners hear which consonant is next to the vowel.

A note on accents: this speaker's /u/ in “two” has \(F_2 \approx 810\) Hz, right at the textbook back-vowel value (~870 Hz for Peterson & Barney's men). Many American English speakers pronounce the same vowel with the tongue further forward after /t/ and /s/ (“/u/-fronting”), with \(F_2\) well above 1,000 Hz. Formant values are properties of a speaker and an accent, not of the vowel symbol, and the phone labels used here come from an automatic aligner with an American-English pronunciation dictionary.

Try it below. It is the same chain-of-tubes model, now with 34 thin slices shaped for you by a “tongue” and the lips. The formants are computed from the shape, not looked up.

Model details, for the curious: a chain of 34 lossless cylindrical sections (each with its own area) with the glottis end closed and the lip end open, solved exactly in the frequency domain with a small uniform loss so that the peaks have realistic bandwidths (~70 Hz). Moving the tongue also bunches it, widening the cavity on the opposite side, and pulling it back also opens the mouth (as the jaw drops for “ah”). The sound is synthesised by weighting each harmonic of a 120 Hz source by the tube's response at that frequency. For the articulatory side of the same story, see the vocal-tract demo.

Formants are not harmonics

A very common confusion, worth heading off explicitly. Harmonics belong to the source: they sit at exact multiples of \(F_0\) and move whenever the pitch changes. Formants belong to the filter: they are the peaks of the smooth envelope and move only when the tract changes shape. The harmonics are the only frequencies actually present in the sound, so they sample the envelope. A formant can sit between two harmonics and still be heard in their relative loudness.