Play the A above middle C on a flute, then on an oboe, a trumpet and a violin. Each time the air in front of the instrument vibrates 440 times a second, the pitch is the same, and still you could tell the four apart with your eyes shut. What differs is the tone colour, or timbre. Most of it lives in the overtones: fainter vibrations at higher frequencies that sound along with every note. They fuse so completely with it that you hear one tone, not a chord.
This page is about those overtones. It asks where they come from, why for most instruments they sit at exact whole-number multiples of the note, what happens when they donβt, and how much of an instrumentβs identity they carry. The mathematics starts from a single vibrating string. The page ends with real orchestral instruments: notes recorded in an anechoic chamber, taken apart into their overtones, and rebuilt from nothing but those numbers until you can no longer tell the copy from the original.
Every figure makes sound, and each takes over the screen as you reach it; keep scrolling to move on.
π§ Headphones or decent speakers help. Phone speakers cannot reproduce the lowest notes, which turns out to be a demonstration in itself. On an iPhone, if nothing plays, switch off silent mode.
Here are nine instruments playing that same A. These are recordings, from the University of Iowaβs collection of single notes, played (all but the piano) in an anechoic chamber, a room built to have no echo at all. Tap an instrument to hear it.
The upper panel shows three cycles of the sound wave: air pressure at the microphone, against time. Every one of these waves repeats about every 2.27 milliseconds, which is 1/440 of a second. What differs is the shape of the cycle. The lower panel breaks the same sound down into its ingredients. It has one bar at 440 Hz, one at 880, one at 1,320, and so on, and each barβs height is that ingredientβs loudness in decibels, relative to the loudest.
The bars are the partials of the sound. The first, at 440 Hz, is the fundamental; it sets the pitch. Partial number n vibrates exactly n times as fast. Every instrument here has partials at the same frequencies. The heights are what differ, and the pattern of heights is, to a first approximation, the tone colour. The fluteβs energy sits in its fundamental and the next few partials and then falls away, so its wave is smooth. The oboe puts more into its third and fourth partials than into the fundamental, which is 18 dB down. The trumpetβs stack reaches high: its tenth partial is still within 30 dB of the loudest, and its wave is spiky. The violinβs heights jump up and down from one partial to the next, a fingerprint of its wooden body.
A partial is any sinusoidal component of a sound. When partials sit at whole-number multiples f, 2f, 3f, β¦ of a fundamental f, they are called harmonics, and the nth harmonic has frequency nf. Overtones are the partials above the fundamental, so the second harmonic is the first overtone. The off-by-one is a historical accident; this page counts harmonics.
Three questions follow, and the rest of the page answers them. Why should the partials of a note be whole-number multiples of anything? What sets their heights? And is the bar chart really all there is to an instrumentβs voice?
Start with the simplest instrument: a string stretched between two fixed points. Pluck it and it vibrates. Some shapes of vibration, though, are special: the string keeps its shape and only its size swings back and forth. These shapes are the stringβs modes. In the first mode the whole string swings as one arc. In the second, the two halves swing in opposite directions, with a still point, a node, in the middle. The nth mode has n arcs and n β 1 nodes.
Each mode vibrates at its own frequency, and the frequencies follow from a single fact: a wave travels along a given string at a fixed speed. The nth mode is a wave of length 2L/n. A wave n times shorter, travelling at the same speed, passes any point n times as often. Mode n therefore vibrates at n times the frequency of mode 1. That is the whole reason for the whole numbers.
A real pluck pulls the string into a triangle and lets go. A triangle is not any single mode, but it is a mixture of all of them, each with its own amount, and each mode then swings at its own frequency. The stringβs motion is the sum of those swings. In pluck mode, drag the string anywhere to pull it and release. The motion is slowed down a few hundred times. It is a surprising sight: two kinks race apart and reflect off the ends, and the sum of smooth arcs has sharp corners.
The pluck point decides the mixture. Any mode with a node where you pluck cannot be excited at all, because the string is pulled exactly where that mode never moves. Pluck at the middle and every even harmonic is missing; the sound turns hollow. Pluck at one fifth and the 5th, 10th, 15th β¦ harmonics vanish. Pluck close to the end and the high harmonics come in strongly, which is why guitarists play near the bridge for a bright, nasal sound and over the sound hole for a round one.
A string of length L, tension T and mass per length ΞΌ with displacement y(x, t) obeys
$$\frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2}, \qquad c = \sqrt{T/\mu}, \qquad y(0,t) = y(L,t) = 0.$$Trying a shape that only changes size, y = X(x) cos(2Οft), turns this into Xβ³ = β(2Οf/c)Β²X with X(0) = X(L) = 0. The only solutions are sines that fit a whole number of half-waves between the ends:
$$X_n(x) = \sin\frac{n\pi x}{L}, \qquad f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}} = n f_1 .$$The formula for f1 is Mersenneβs law of 1636: halve the length and the pitch goes up an octave; quadruple the tension and so does it. Every motion of the ideal string is a sum of the modes, y(x, t) = Ξ£ bn sin(nΟx/L) cos(2Οfnt), and so every sound it makes contains only harmonics of f1.
Pull the string to height h at x = pL and let go. The initial shape is a triangle, and its amount of mode n is its Fourier sine coefficient:
$$b_n = \frac{2}{L}\int_0^L y_0(x)\,\sin\frac{n\pi x}{L}\,dx = \frac{2h}{\pi^2 p(1-p)}\cdot\frac{\sin(n\pi p)}{n^2}.$$The factor sin(nΟp) is zero exactly when np is a whole number, which is when mode n has a node at the pluck point. What you hear is not the stringβs displacement but the force it exerts on the bridge, which follows the stringβs slope at the end. That slope is proportional to nΒ·bn, so the soundβs harmonics fall off as sin(nΟp)/n; the bars in the figure show these amplitudes. The sound also lets high modes die away faster than low ones, as real strings do, so a pluck darkens as it rings.
The string shows that harmonics can add up to a sound. Joseph Fourier claimed in 1807 that the reverse is also true, and Peter Gustav Lejeune Dirichlet proved it in 1829 for every well-behaved shape: anything that repeats can be built from harmonics. Choose the amplitude of each harmonic and you get a wave; choose a wave and its harmonics are fixed. The figure is a drawbar organ with sixteen drawbars. Drag the bars to set the harmonics and watch the wave change. Or switch to draw and sketch any cycle you like with your finger or the mouse; the page finds its harmonics.
A few classic shapes are worth trying. The sawtooth contains every harmonic, at amplitude 1/n; it is the bright, buzzy wave of a bowed string and of the classic synthesizer. The square keeps only the odd harmonics, at 1/n, and sounds hollow, like a clarinet, for reasons the section on air columns explains. The triangle also uses only odd harmonics, but falls off as 1/nΒ², so it is soft, almost a sine. Sixteen harmonics cannot build a perfectly sharp corner. Near every jump the sum overshoots by about 9% of the jump and ripples. This is the Gibbs phenomenon, and adding more harmonics narrows the ripples but never lowers them.
Now press scramble phases while the sound plays. Each harmonic keeps its amplitude, but it is shifted in time by a random fraction of its own cycle. The wave on the screen changes beyond recognition. The sound, to most ears, does not change at all. Georg Ohm proposed in 1843 that the ear hears a sound as the sum of its sine components. Twenty years later Hermann von Helmholtz pictured the inner ear as a bank of tuned resonators that registers how much of each frequency there is, and found that shifting the phases of a toneβs harmonics leaves its colour essentially unchanged. This is why the bar chart, and not the wave shape, is the right picture of tone colour. The rule has exceptions (a very low note with many harmonics crowded into one band of hearing can reveal its phases), but it holds well enough that the rest of this page mostly ignores phase.
A wave with period T, fundamental frequency f = 1/T, is a sum of harmonics, each with an amplitude An and a phase Οn:
$$x(t) = \sum_{n=1}^{\infty} A_n \sin(2\pi n f t + \varphi_n).$$The harmonics are orthogonal over one period, so each amount is found on its own by correlating the wave with that harmonic: AneiΟn = (2/T)β«0T x(t) ieβ2Οinft dt. Draw mode does this on the drawn cycle. Parsevalβs theorem says the power of the wave is the sum of the harmonicsβ powers, AnΒ²/2, which is why amplitudes, and not phases, carry the loudness of each ingredient.
The presets are the textbook series:
$$\begin{aligned} \text{sawtooth} &: \textstyle\sum_{n} \frac{(-1)^{n+1}}{n}\sin n\omega t,\\ \text{square} &: \textstyle\sum_{n\ \text{odd}} \frac{1}{n}\sin n\omega t,\\ \text{triangle} &: \textstyle\sum_{n\ \text{odd}} \frac{(-1)^{(n-1)/2}}{n^2}\sin n\omega t . \end{aligned}$$The string gave one reason for harmonic partials. Fourier gives a deeper one, and it runs in both directions. If a sound repeats every T seconds, it can contain only frequencies that fit a whole number of cycles into T: 1/T, 2/T, 3/T, β¦ Any other frequency would be out of step by the next repetition. Conversely, a sum of harmonics of f repeats every 1/f, since each harmonic completes whole cycles in that time. Periodic sound and harmonic partials are two faces of one thing, and periodicity is what the ear hears as a clear pitch.
That has a strange consequence. Take away the fundamental and the sound still repeats every 1/f: the harmonics 2f, 3f, 4f all fall back into step after exactly that long. The ear goes by the repetition, not by the lowest ingredient. Remove the fundamental, then the second harmonic, then the third, and the pitch stays put. Only the colour thins.
This is the missing fundamental, also called residue pitch or virtual pitch. August Seebeck heard it with a siren in 1841. The telephone depends on it: classic phone lines carried only frequencies between 300 and 3,400 Hz, which leaves out the fundamental of most voices, and yet nobody on the line sounded higher-pitched. Small loudspeakers depend on it too: a phone cannot reproduce a celloβs low C at 65 Hz, but it plays the harmonics, and your brain supplies the note. The limit is reached when the remaining partials no longer pin down the repetition. Keep only the even harmonics, 2, 4, 6, 8 and 10, and the wave repeats twice as often; the pitch jumps up an octave. The pitch the ear assigns is, to a good approximation, the greatest common divisor of the frequencies present.
Suppose x(t + T) = x(t) and x contains a component e2ΟiΞ½t. Shifting by T multiplies that component by e2ΟiΞ½T, and periodicity demands the factor be 1. So Ξ½T is a whole number: Ξ½ = k/T. A set of partials f1, β¦, fm repeats if and only if all ratios fi/fj are rational, and then the period is 1/gcd(f1, β¦, fm). For 600, 800 and 1,000 Hz that is 1/200 s, even though nothing in the sound happens at 200 Hz.
Lay the harmonics of a low note out on a keyboard and they spell out music. The second harmonic is an octave above the first. The third is a fifth above that, the fourth another octave, the fifth a major third above it, and the sixth completes a major chord. The gaps keep shrinking: whole tones, then semitones, then intervals the piano has no keys for. Tap a harmonic to hear it, or play the series as a bugler would: the bugle has no valves, and every note it can play is a harmonic of its tube.
The numbers under each harmonic are its distance from the nearest piano key, in cents, hundredths of an equal-tempered semitone. The fifths of the series are almost exactly the pianoβs fifths, 2 cents wide. Its major thirds are 14 cents flat of the pianoβs, which is the price equal temperament pays for letting every key sound equally in tune. The 7th harmonic is 31 cents flat of Bβ, the 11th falls almost exactly between F and Fβ―, and the 13th between Aβ and A. Brass players live with these notes. Natural horns and alphorns play the series as it is; the alphornβs 11th harmonic is famous enough to have a name, the βalphorn faβ. Benjamin Britten opens and closes his Serenade for tenor, horn and strings with a solo to be played on the natural harmonics alone, odd tunings and all.
Pitch is heard on a logarithmic scale: an octave is a doubling of frequency wherever it starts. The size of the interval between two frequencies, in cents, is
$$\text{cents}(f_2 : f_1) = 1200 \log_2 \frac{f_2}{f_1},$$and equal temperament puts its keys 100 cents apart. Harmonic n sits 1200 logβ n cents above the fundamental. The fifth 3/2 is 701.96 cents against the pianoβs 700. The major third 5/4 is 386.31 against 400. The harmonic seventh 7/4 is 968.83 against 1,000. Since no power of 3/2 is ever a power of 2, no scale of equal steps can contain the pure fifth; twelve steps come closest among small scales, which is part of why the piano has twelve keys to the octave.
Why do some intervals sound smooth and others rough? Helmholtzβs answer, refined a century later by Reinier Plomp and Willem Levelt, is about overtones. Two pure tones close in frequency beat against each other. Slow beats sound like a wobble, but faster ones, about a quarter of a band of hearing apart, sound rough, and the roughness fades again once the tones are far enough apart to be heard separately. Play two notes with rich spectra and every partial of one rubs against every partial of the other. Add up all that rubbing and you get a dissonance curve.
The figure plots it for two notes as the upper one slides from a unison to just past an octave. Below, the partials of the lower note are drawn in blue, those of the upper note in orange, and every pair close enough to rub is shaded. Drag along the curve, or use the arrow keys, and press play to hear the interval.
With pure tones the curve has one hump just above the unison and is flat afterwards: nothing makes a fifth special. With harmonic timbres the curve grows dips at exactly the simple ratios, 2/1, 3/2, 5/3, 4/3, 5/4, 6/5. At those ratios many partials of the two notes coincide and the rest stay apart. In a fifth, every second harmonic of the upper note lands on every third of the lower. The consonant intervals of Western music are where harmonic spectra mesh.
Change the spectrum and the consonances move. The pianoβs partials run slightly sharp, as the next section shows, so its octave is smoothest a little wider than 2:1, and piano tuners stretch their octaves to match. The bell-like spectrum has dips at ratios no scale uses. William Sethares turned this around: given any spectrum, compute its dissonance curve and you have the scale that suits it. He showed that the tunings of Javanese gamelan, whose gongs and metal bars have strongly inharmonic spectra, sit close to the dips of their own instrumentsβ dissonance curves.
For two partials with frequencies f1 < f2 and amplitudes a1, a2, the roughness is modelled as
$$d = \min(a_1, a_2)\left(e^{-b_1 s \Delta f} - e^{-b_2 s \Delta f}\right), \qquad s = \frac{0.24}{0.0207\,f_1 + 18.96},$$with Ξf = f2 β f1, b1 = 3.51 and b2 = 5.75. It is zero at Ξf = 0, peaks near a quarter of a critical band, and decays beyond. The curve plotted is the sum over all pairs of partials of both notes, lower note at middle C. It is a model of sensory roughness, not of everything that makes harmony work, but it gets the main consonances of harmonic sounds right from nothing but their spectra.
Whole-number overtones are a special property of an idealised string, and of the air column, which the next section treats. Everything else that vibrates has modes too, but their frequencies follow other rules. A drumhead is the simplest case. It is a two-dimensional string: a membrane stretched over a circle. Its modes have nodal lines, circles and diameters, and their frequencies come out as zeros of Bessel functions, not as whole numbers.
Strike the ideal membrane and its partials pile up at 1, 1.59, 2.14, 2.30, 2.65, β¦ times the lowest. No repetition, no clear pitch: a drum, not a note. The modes with a nodal line through the centre are barely excited by a strike in the middle, which is why drummers hit off-centre to get more tone. The orchestral kettledrum cheats. The air sealed in its bowl loads the membrane and drags the frequencies of the modes with one, two, three and four nodal diameters to nearly 2 : 3 : 4 : 5. These are harmonics 2 to 5 of a missing fundamental, so the kettledrum has a definite pitch. The timpanist strikes about a quarter of the way in from the rim, where the lowest mode, the one that fits no series, is barely excited, and it dies away quickly anyway.
The wave equation in two dimensions, βΒ²z/βtΒ² = cΒ²βΒ²z, with z = 0 on the rim r = a, separates in polar coordinates into
$$z_{mn}(r,\theta,t) = J_m\!\left(\frac{j_{mn}\, r}{a}\right)\cos(m\theta)\,\cos(2\pi f_{mn} t), \qquad f_{mn} = \frac{j_{mn}}{2\pi a}\, c ,$$where Jm is the Bessel function of order m and jmn its nth zero. Mode (m, n) has m nodal diameters and n β 1 nodal circles inside the rim. The zeros of Bessel functions are not in whole-number ratios: j11/j01 = 3.832/2.405 = 1.593, j21/j01 = 2.136, and so on. The string is the one-dimensional case, where the corresponding functions are sines and the zeros are evenly spaced.
A bar of wood or metal, free at both ends, also has modes. It resists bending rather than being held taut, so its equation of motion has a fourth derivative in space instead of a second, and its mode frequencies grow roughly like the square of n + Β½ instead of like n: 1, 2.76, 5.40, 8.93. That is roughly the glockenspiel, whose bars are plain rectangles of steel; real bars are wide enough to twist as well as bend, which adds modes of their own, and the measured glockenspiel below has strong partials at 3.3, 4.8 and 7.2 times its lowest. Marimba and xylophone makers carve an arch under the middle of each bar, which lowers the first mode more than the others, until the partials land on musical ratios: 1 : 4 : 10 for the marimba, two octaves and then three octaves plus a major third, and 1 : 3 for the xylophone, an octave plus a fifth. A church bell is a shell, and generations of founders learned to tune five of its partials, the hum, prime, tierce, quint and nominal, to 0.5, 1, 1.2, 1.5 and 2 times the strike note: an octave below it, the note itself, a minor third and a fifth above, and an octave above. That minor third, sounding inside every bell, is part of why bells can sound faintly mournful.
Even a string is not ideal. A piano string is steel wire under great tension, and steel is stiff. Stiffness adds a little of the barβs behaviour to the stringβs, so each partial runs slightly sharp of its harmonic, more so the higher the partial. The figure shows each sound as a comb: one tooth per partial at its frequency relative to the lowest, tooth height for loudness. The faint grid marks the whole numbers. The measured rows come from the Iowa recordings, analysed for this page; tap any row to hear it.
The pianoβs stretching is small but audible, and it shapes how a piano is tuned. The next figure plots, for four notes of the Iowa piano, how far each partial sits above the exact harmonic, in cents. The curves bend upward as nΒ², as the theory says, and the fitted stiffness B grows from the low tenor upward as the strings get shorter. The bass strings are wound with copper to add mass without stiffness, but their great length lets them bend, and they still stretch more than the strings of the tenor.
Because the upper partials of a low note run sharp, an octave tuned by ear, with no beating between the lower noteβs second partial and the upper noteβs first, comes out slightly wider than 2:1. Piano tuners stretch every octave a little, and over seven octaves the top of a well-tuned piano ends up some 30 cents sharp of equal temperament and the bottom as far flat. O. L. Railsback measured this stretch on pianos tuned by ear in 1938, and the plot is still called the Railsback curve. The Iowa piano shows it too: measured by their first partials, its B0 sits 10 cents flat of equal temperament and its C7 14 cents sharp. The second button plays the same note with every partial moved onto its exact harmonic: a little smoother, and a little less like a piano.
Adding bending stiffness to the string gives ΞΌβΒ²y/βtΒ² = TβΒ²y/βxΒ² β EIββ΄y/βxβ΄, where E is Youngβs modulus and I the area moment of the wireβs cross-section. With pinned ends the modes stay sines, and
$$f_n = n f_0 \sqrt{1 + B n^2}, \qquad B = \frac{\pi^3 E d^4}{64\, T L^2}$$for a wire of diameter d (Fletcher, 1964). The fitted values of B are shown in the readout. Drop the tension and only the stiffness term is left: the free bar, whose modes satisfy cos(kL) cosh(kL) = 1, so kL β 4.730, 7.853, 10.996, 14.137, and f β kΒ², giving 1 : 2.757 : 5.404 : 8.933.
Wind instruments vibrate a column of air instead of a string. Sound waves travel along the tube and reflect at both ends. At an open end the pressure cannot build up, so the pressure has a node there. At a closed end the air cannot move, so the pressure swings hardest there. The reed of a clarinet or an oboe, and the lips of a brass player, close their end of the tube. The flute is open at both ends: the player blows across a hole.
The boundary conditions decide which modes fit. A tube open at both ends holds whole numbers of half-waves, like a string, so it has every harmonic. A cylinder closed at one end holds odd numbers of quarter-waves, so only the odd harmonics fit, and its lowest note is an octave lower than that of an open tube of the same length. A cone, closed at its tip, holds every harmonic again, because sound spreads in it as spherical waves.
The measured spectra bear this out where they should. The clarinetβs low notes really do lean on the odd harmonics: on its D3 the 2nd and 4th harmonics are about 20 dB below the fundamental, the 3rd and 5th only about 10. That is the hollow, woody chalumeau sound. The saxophone is also played with a single reed and mouthpiece, but its tube is a cone, and all its harmonics are there. It also explains a fact of fingering. A flute or an oboe, blown harder, jumps up an octave to its second mode. A clarinet jumps an octave and a fifth, to its third mode, because it has no second mode, so it needs more keys to fill the gap.
The rule is not absolute. Real bores have tone holes, a mouthpiece and a bell, and the pattern holds only for the lowest few harmonics: on the same D3, the 8th harmonic is louder than both its odd neighbours. Higher up the range it fades altogether; the clarinetβs A4 in the first figure has no odd-harmonic signature left.
For a tube of length L (plus a small end correction) and speed of sound c, the openβopen and closedβopen cylinders have
$$f_n = \frac{n c}{2L} \quad\text{and}\quad f_n = \frac{(2n-1)\,c}{4L},$$and a complete cone has the open-pipe series. One further fact makes wind and bowed instruments harmonic even when their resonators are not quite. The player excites a self-sustained oscillation: the reed, the lips, the air jet or the stick-and-slip of the bow is a strongly nonlinear valve that feeds energy in once per cycle. The result is a strictly periodic wave, and so, by the theorem of the previous section, exact harmonics. The resonatorβs slightly mistuned modes are pulled into line; this is called mode locking (Fletcher, 1978). A plucked string, once released, has no such master clock, and its stiffness shows.
So far each instrument has been heard on one or two notes. But its spectrum changes from note to note, and how it changes says a lot about the instrument. The figure lays out the measured spectrum of every note in an instrumentβs range, from the Iowa recordings. Each row is one note, lowest at the bottom. Each dot is a harmonic at its frequency, darker for louder. The curve on top is the average level of the harmonics that fall at each frequency, across all notes. Tap a row to hear that note rebuilt from its row of dots.
If an instrument simply scaled its spectrum with the note, its dots would form diagonal stripes: the 3rd harmonic always loudest, wherever it falls. Instead, dark vertical bands appear: frequency regions that are loud whatever note is played. These are formants, fixed resonances of the body, the bore or the reed. Look at the bassoon: for half of its notes the strongest harmonic lies between 470 and 560 Hz. On its lowest note that is the 9th harmonic; on its highest, the fundamental. The tuba is even more single-minded: on every note of its range the strongest harmonic falls between about 330 and 420 Hz, the 10th harmonic of its lowest note and the 2nd of its highest. The oboeβs and the trumpetβs bands both sit around 1.2 to 1.5 kHz. The violin is a box of wooden resonances, and its rows are patchy, loud here and weak there, from note to note.
Speech works the same way. Your vocal cords produce a harmonic spectrum at whatever pitch you speak, and the shape of your mouth and throat imposes formants that make an βahβ an βahβ at any pitch. Formants are also why a recording played back faster sounds like chipmunks rather than like the same voice higher up: speeding it up moves the formants along with the pitch. Sampled instruments need a separate recording every few notes for the same reason.
Is the bar chart all there is? Play a steady spectrum and the answer comes quickly: no. A frozen spectrum with the right heights sounds like the instrument played by an electric organ. Jean-Claude Risset found this in the 1960s at Bell Labs, trying to synthesise a trumpet. The steady state is only part of the identity. The rest is in how the partials change over time.
This figure takes real notes apart. For each recording, the pageβs build script followed every partial through the note, 200 times a second in the attack and 50 times a second after. It measured the partialβs amplitude, the noteβs exact pitch from moment to moment, and the energy of everything left over, such as breath, bow noise, the thump of a hammer. The upper panel plots the partialsβ amplitudes over time, the lowest ones darkest. The middle panel shows the pitch, and the lower one the leftover noise from low to high frequencies. Press resynthesis to hear the note rebuilt from these numbers alone, then switch layers off to hear what each one contributes.
Listen for these differences:
Each note is represented as
$$x(t) = \sum_{p=1}^{P} A_p(t)\cos\phi_p(t) + e(t), \qquad \phi_p'(t) = 2\pi\, r_p\, f_0\, 2^{c(t)/1200},$$where Ap is the measured amplitude envelope of partial p, rp its frequency ratio (exactly p for blown and bowed notes, measured for struck and plucked ones), c(t) the pitch track in cents and e(t) noise whose energy in each of the 25 critical bands of hearing follows the measured residual. This is the βdeterministic plus stochasticβ model of Xavier Serra and Julius Smith (1990). Amplitudes and frequencies are read off short-time Fourier transforms with a BlackmanβHarris window of about 4.6 periods. The residual is what remains after the partials are resynthesised with their measured phases and subtracted from the recording; for blown and bowed notes it is typically 30 to 45 dB below the original, for struck ones, whose attacks are largely noise, much less. Resynthesised from the stored numbers, the notes on this page match their recordings to within half a decibel on average, in level per critical band and in loudness over time; the worst is 0.6 dB off. A 2.5-second note becomes about 10,000 numbers, against 110,000 samples in the recording.
Spectra are not fixed even for a single note. Blow harder into a trumpet and it doesnβt just get louder: the pressure waves in the tube steepen as they travel, and the steeper wave carries more high harmonics. Strike a piano key harder and the felt hammer, which stiffens as it is compressed, makes a shorter, sharper impact that excites more high modes. The Iowa recordings include the same notes at pianissimo, mezzo forte and fortissimo. Here are their spectra, each relative to its own fundamental, so only the shape is compared.
In these recordings the trumpetβs big change comes between pianissimo and mezzo forte: its 10th harmonic rises from 47 to 17 dB below the fundamental, and fortissimo adds almost nothing more. The piano keeps brightening all the way: its 10th partial climbs from 60 to 52 to 26 dB below.
This is why turning down a recording of a loud trumpet does not make it sound like a soft trumpet, and why a sampled instrument needs several recordings per note at different strengths. Our ears read loudness off the spectrum as much as off the level. A trumpet heard from far away is still a loud trumpet.
Time for a test. Each round picks a note at random from the recordings on this page. One of the two buttons plays the recording; the other plays its resynthesis from the measured partials, pitch track and noise. Listen to both, as often as you like, and pick the one you think is real.
If you score near 50%, you are guessing, and the model has captured everything your ears use. John Grey and James Moorer ran this experiment in 1977 with simplified resyntheses of orchestral tones, and their listeners found many of the pairs hard to tell apart. Where a copy does give itself away, the likeliest culprit is the noise. The model keeps only the noiseβs loudness in 25 frequency bands, so a fine-grained rasp such as the catch of the bow hair or the click of a key comes back as a generic hiss.
An anechoic chamber removes the room, and a single microphone a few feet away records one direction. Instruments radiate different partials in different directions. A violinβs upper partials beam from the top plate in narrow lobes that change from note to note, and in a hall they reach you via thousands of reflections. Much of the warmth of a live instrument is the room. And a single note is not music. Phrasing, the way one note leads into the next, and the playerβs choices of attack and vibrato carry as much of an instrumentβs character as its spectrum.
Still, the core of it is in the partials. Their frequencies are set by the geometry of whatever vibrates. They are whole numbers for strings and air columns because those are one-dimensional and, when sustained, strictly periodic. Their strengths and their life over time are set by how the vibration is excited and filtered on its way out. Tone colour, the property that tells a flute from a violin, turns out to be a list of numbers you can measure, plot, change and play back.
Constantin Lisson Β· 2026