Acoustics & Music Theory
why it sounds like that
Concept

Nonlinearity — where it appears

A response that is not proportional to what caused it, which is what allows a steady supply of energy to be converted into an oscillation. Every self-sustaining instrument has one — a reed, a bow, a pair of vocal folds — and no struck one needs it.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

The same note, hit at a middling dynamic. The spectrum of a struck string with the hammer's own contact time applied as a low-pass. Contact lasts 1.60 ms at this force, against 2.26 ms at the softest and 0.95 ms at the loudest drawn — felt is a nonlinear spring, so a harder blow is a shorter contact and a brighter note. The spectral centroid moves from partial 1.5 to partial 2.2, which is a change of timbre and not of loudness.

A hammer is not an impulse

Contact lasts a couple of milliseconds, which low-passes the note — any partial whose half-period is shorter than the contact is barely excited. Piano felt is a spring that stiffens as it compresses, so a harder blow makes the contact shorter, the corner higher and the note brighter. A loud note is not a scaled-up quiet one, and no linear model gives that.

instruments · Excitation point Helmholtz motion, bowed at 9% of the way from the bridge. Above: the string at 5 instants of one period. It is two straight lines meeting at a corner, and the corner travels round the string rather than the string swinging. Below: the resulting force on the bridge, a sawtooth whose two segments are in the ratio 0.09 to 0.91 — the bow's own position. A sawtooth contains every harmonic at exactly one over n, so the spectrum barely changes with bow position even though the waveform plainly does.

The bow makes a corner

A bowed string is not a string swinging. It is two straight lines meeting at a single sharp corner that travels round the string once per period, triggering the slip that keeps it going. The force on the bridge is therefore a sawtooth, and a sawtooth is every harmonic at exactly one over n — which is why a bowed string is the most nearly perfect harmonic series in the orchestra.

instruments · Bowed string How much bow force is allowed, and where. Schelleng's diagram. The lower bound is the least force that will trigger a slip on every pass of the corner and goes as one over beta squared; the upper bound is the most the string will take before it sticks for more than a period and goes as one over beta. At beta = 0.09 the usable range spans a factor of 9.0; at 0.03, near the bridge, it is 3.0, and at 0.2, over the fingerboard, 20.0. The window closes in proportion to beta, so the difficulty of playing near the bridge is a slope on this picture rather than a matter of opinion.

How much bow is allowed

Too little force and the corner fails to trigger a slip on every pass; too much and the string sticks for more than a period. Both bounds depend on where the bow is, and they depend on it differently — one as the square of the distance from the bridge and one linearly — so the window between them closes in proportion as the bow approaches the bridge. Sul ponticello is difficult by a power law.

instruments · Bowed string Two mechanisms, the notes both of them make, and the seam. The frequency range of each laryngeal mechanism for an adult male voice, on a logarithmic axis, with the band both can produce shaded. M1 — chest runs 82–349 Hz and M2 — falsetto runs 220–698 Hz, so 799 cents of the range — 8.0 semitones — can be sung either way. The two dots inside that band are the measured signature that this is a bifurcation rather than a threshold: the change upward happens at 330 Hz and the change downward at 294 Hz, 200 cents lower. A threshold is crossed at the same place in both directions and this is not.

Two mechanisms, and the seam between them

Every singer has a place in the range where the voice changes character, and eight semitones of it can be produced either way. The measurement that settles what kind of a place it is takes ten seconds: the change upward happens two hundred cents higher than the change downward, and a threshold cannot do that.

instruments · The voice A reed that shuts at 5000 pascals, and the air it lets through. Volume flow through the reed channel against the pressure across it, in the quasi-static model: Bernoulli flow through an opening that closes linearly with pressure. The flow peaks at 1667 pascals — exactly a third of the closing pressure, for any reed, because that is where the two effects balance — and it is 0.18 litres a second there. Everything to the right of that peak is the argument: the flow falls as the player blows harder, from 0.18 to 0.05 litres a second by 4500 pascals, and a resistance that behaves that way supplies energy instead of taking it. There is no reed inertia in this model, so it cannot squeak.

The reed is a valve, not a vibrator

Three essays here have said that a clarinet's reed does not choose the note, and none of them said what it does instead. It chops a steady stream of air, and past a third of the pressure that closes it the flow falls as the player blows harder — a resistance with the wrong sign, which is the only thing in the instrument capable of putting energy into an oscillation that is otherwise losing it.

instruments · Air column C4: the pulse computed and the pulse assumed. Above, the force the hammer delivers to the string at C4, integrated forward against the felt's nonlinear force and the string's returning corner, drawn against the half-sine of 1.60 milliseconds that every earlier figure assumed. The computed contact lasts 2.21 milliseconds and the corner comes home 4.6 times inside it. Below, the excitation each pulse gives to each partial. They agree at the bottom and part company higher up — worst at partial 6, by 34 decibels — because the assumed pulse has nulls the computed one does not.

The pulse that was assumed

Every figure until now low-passes the string's excitation with the spectrum of a half-sine, which is what a hammer would deliver against a rigid wall. An earlier essay said so and declined to do better. Doing better takes forty lines and refuses the prediction that came with it: the corner's round trips govern the spectrum as expected, and the contact time is governed by something else entirely — the mass ratio discovered one essay earlier.

timbre · Excitation point How much of each instrument is narrow enough to steepen a wave. For each bore, the quantity that decides how nonlinear it is: the narrowest radius divided by the radius at each station, integrated along the tube. The shading is that integrand, so a bar that stays dark is a tube still doing damage to the wave and a bar that fades is a flare that has thinned it out. Divided by the instrument's own length the integral is a pure number: a plain cylinder 1.00, a tenor trombone 0.88, a trumpet 0.86, an F horn 0.59, a cone of a trumpet's length 0.24. A plain cylinder is 1 by construction, a cone of the same length and mouth is 0.24, and the ordering across the brass family is the one players give when asked which of them can be made to blare.

The partials the tube makes itself

Eleven earlier essays compute a passive linear resonator, and none of them ever says so. At a real fortissimo the air in a brass instrument is not linear: a compression outruns a rarefaction, the wave leans forward as it travels, and the fourth partial of a loud trumpet note is seventy decibels louder than a scaled-up quiet one — generated in the tube rather than at the lips. How much of it happens is an integral over the bore, and it is why a flugelhorn cannot be blown into being a trumpet.

instruments · Air column The partials the air makes, on the series the bore actually has. The input impedance of a trumpet, with the partials wave steepening manufactures from its A4 at 428 hertz drawn on it as vertical marks. The steepening is a distortion of one periodic waveform, so it makes its partials at exact integer multiples — 856, 1285, 1713, 2141, 2569 hertz. The bore's own resonances are not at integer multiples of anything: the peaks the manufactured partials aim at sit at 886, 1346, 1812, 2278, 2741. So every manufactured partial lands flat of the peak it might have used, by 59, 81, 98, 108, 112 cents — 2.6, 2.7, 3.0, 4.0, 4.9 half-widths of the peaks in question, which is outside the half-power point of every one of them. The dashed line is where the bell stops reflecting, at 2945 hertz; above it there is no peak to land on or miss.

Which notes go brassy first

Wave steepening manufactures partials at exact integer multiples of the note being played. A brass instrument's resonances are not at integer multiples of anything, and twelve earlier essays have measured how far off they are without ever putting the two on one axis. Put them there and a manufactured partial never lands on a peak — never once, at any note, by a miss that is the same number of hertz every time. So the alignment cannot be what decides which notes blare, and the thing that does turns out to be the bell.

instruments · Air column

Named alongside it

The objects these essays reach for when they reach for this one.

SpectrumEnvelopeBoreExcitation pointPartialAttack transientBrassBrightnessImpedanceOverblowingStanding waveBoundary condition

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