Acoustics & Music Theory
why it sounds like that
Concept

Cutoff — where it appears

The frequency above which a row of open tone holes stops reflecting a wave and lets it out instead. It sets the ceiling of an instrument's normal register and gives a whole family its characteristic timbre above that point.

Named by 15 essays across 4 fields — each of them below, with the objects they name alongside it.

Where each family's tone-hole lattice stops reflecting. The cutoff frequency of an open tone-hole lattice, from Benade's formula, for four woodwind geometries: clarinet 1824 Hz, oboe 2990 Hz, flute 1690 Hz, bassoon 506 Hz. Below its cutoff a note's wave turns round at the first open hole and the instrument is a tube of that length; above it the wave passes through the whole lattice and radiates from the far end, so the upper part of every note's spectrum leaves the instrument from the same place whichever note is fingered. That is what gives a family one recognisable voice across its range.

Above a certain note the holes stop working

A row of open tone holes reflects the wave and makes the tube shorter — up to a frequency. Above it the wave runs straight past the whole lattice and leaves from the bell, so the top of every note's spectrum radiates from the same place whichever note is fingered. That cutoff is computable, it differs by family, and it is most of what makes an oboe sound like an oboe.

instruments · Tone holes What gets out of an opening, for 4 openings. The fraction of the wave's energy radiated at an open end against frequency, in the baffled-piston model — the radiation resistance of a circular piston, normalised to the tube's own impedance. Each curve runs from nothing at the bottom, where the opening is far smaller than a wavelength and the wave simply turns round, to everything above ka ≈ 2. Half the energy leaves at 6364 Hz for a flute's embouchure end (radius 10 mm), 2015 Hz for a clarinet's bell (radius 30 mm), 975 Hz for a trumpet's bell (radius 62 mm), 403 Hz for a horn's bell (radius 150 mm). The crossover goes as one over the radius, so the widest and narrowest here are 15.8 times apart in frequency. The same number decides how strongly the tube resonates and how much sound it makes, which is why a bell cannot brighten an instrument without also weakening its own resonances.

The bell decides what gets out

A tube resonates because the wave turns round at the open end, and it is audible because some of the wave does not. Those are the same number with opposite signs. One quantity — the size of the opening against a wavelength — decides how loud an instrument is, how bright it is and how directional it is, and a bell moves the boundary rather than removing it.

timbre · Air column What a hand in the bell buys, and what it costs. How far the instrument flattens and how much radiation it loses, against the share of the bell's mouth the hand blocks, for an F horn's 15 centimetre mouth on a 3.7 metre acoustic length. The two curves are the same aperture radius read twice: a narrower mouth adds inertance, which lengthens the tube, and is acoustically smaller, which stops radiating. The mark is the 25.6 cents left owed earlier on the eleventh partial — it needs 62 per cent of the mouth blocked and costs 3.8 decibels of radiated power. Fully stopping is a semitone and nine decibels.

The hand that changes the bore

A conjecture refuted here left a question with a number on it: the natural trumpet's eleventh partial is 48.7 cents flat, the lips can move it 23.1, and 25.6 cents are owed by something that is not an embouchure. There is exactly one thing a player can change about the bore while playing, and putting the hand in the bell buys those 25.6 cents at a cost of 3.8 decibels — because the aperture that tunes the instrument is the aperture that radiates it.

tuning · Air column Every modern bell puts the boundary at the same partial. For each brass instrument, the partial at which its bell stops turning the wave round — the frequency at which half the energy escapes at the mouth, divided by the tube's own fundamental. The five modern instruments land between partial 8.4 and partial 10.2 despite tube lengths differing by a factor of four, because a bell is scaled with its instrument. The natural trumpet of the baroque, whose bell is small and whose tube is long, puts it at partial 17.2.

The fourth top is the maker's

The series has three tops and all three are the ear's: where consecutive partials stop being resolved, where their spacing falls below a semitone, and where it falls below what the ear can hear at all. The fourth is how far up a player can go, and it is the only one somebody chose. Every modern brass bell puts the boundary at about the ninth partial across a family whose tubes differ by a factor of four — and the baroque natural trumpet, whose bell is small, puts it at the seventeenth, which is exactly the top of the clarino register.

intervals · Harmonic series cylinder and Bessel flare: the length each mode behaves as though it has. Each mode's own acoustic length, m·c over twice its frequency, for a bore 148 cm long. A cylinder would give one number repeated. This gives 164 cm at the second mode and 154 at the 8th — a spread of 10.1 centimetres, or 110 cents, because a flare's end correction is a length that shrinks as the note rises. The first mode is off the top of this axis and is not a mode a player uses.

A horn has one length per partial

Every tube until now has had an acoustic length: its physical length plus a correction for the wave carrying on past the opening. A flaring bore does not have one. Its second mode behaves as though the tube were 164 centimetres long and its eighth as though it were 154, and the ten centimetres between them are the same physical fact — a fixed correction against a shrinking wavelength — arriving as a hundred cents.

tuning · Bore profile The cutoff computed from the shape, and the cutoff read off the measurement. Two numbers for the same boundary. The hollow point is Webster's local cutoff — the largest value of (c/2π)√Γ along the bore, which is geometry alone. The filled point is where the impedance peaks stop, which is what a maker measures with a loudspeaker and a microphone. cone: no geometric cutoff at all, and peaks stopping at 4731; exponential horn: 89 hertz from the geometry, and peaks stopping at 2614, a factor of 29.26; catenoidal horn: 115 hertz from the geometry, and peaks stopping at 2499, a factor of 21.76; Bessel horn: 1154 hertz from the geometry, and peaks stopping at 1593, a factor of 1.38; cylinder and Bessel flare: 2945 hertz from the geometry, and peaks stopping at 4417, a factor of 1.50. The published figure for a trumpet is about 1500 hertz, marked. The two agree within about half for a bore whose flare is concentrated at the end and disagree by more than an order of magnitude for one whose flare is spread along it, which says what the local formula is and is not a measurement of.

The cutoff a maker can actually measure

A trumpet's bell cutoff computed from the geometry alone comes to 1,150 hertz against a published 1,500, and the calibration was the first thing that left owing. Measuring the model the way a maker measures the instrument — sweep a loudspeaker in, find where the impedance peaks stop — gives 1,593. The discrepancy was almost entirely in the formula, and on a bore whose flare is spread rather than concentrated the same formula is out by a factor of twenty-nine.

instruments · Bore profile A family resemblance, in the heights rather than in the frequencies. The peak heights of trumpet, F horn, tenor trombone, plotted against peak number rather than against frequency. The three differ in length by a factor of 2.4 and their frequency series cannot be made to overlap; their heights agree to 3.2 decibels on average and their Qs to a factor of 1.30. The agreement improves up the series — 6.9 decibels at the first peak and 1.6 at the 8th — which is an earlier claim arriving as a measurement: a family has one voice because it has one filter, and the filter is visible in what the bore pushes back with and not in where its resonances are.

A family resemblance in the heights

Trumpet, horn and trombone differ in length by a factor of two and a half, so their frequency series cannot be laid over one another. Their impedance peaks agree to three decibels in height and to thirty per cent in Q, peak for peak, and the agreement improves with peak number. An earlier essay inferred that a family has one voice because it has one filter; the solver can now be asked directly.

instruments · Bore profile The flare that makes a series harmonic, and how narrow it is. Every combination of a flare exponent and a station at which the flare begins, shaded by how far the bore's resonance series is from a harmonic series over partials 2 to 8, in cents. The best is 4.6 cents at an exponent of 1.00 beginning 43 per cent of the way along, against 127 cents for a plain cylinder, 21 for a plain cone and 26 for a Bessel horn of the exponent used everywhere else. Only 2.1 per cent of the surface is within five cents of the minimum, so the shape is forced rather than chosen — which is what three centuries of empirical brass design were finding.

The flare that makes a series harmonic

Every figure until now treats a partial as n times a fundamental, and the whole of brass instrument design is the business of making that true. A plain cylinder is 127 cents from a harmonic series and a plain cone is 21; the best flare found by sweeping is 4.6, and only two per cent of the swept surface comes within five cents of it. The shape is forced rather than chosen.

timbre · Harmonic series A woodwind cannot be pulled to a new standard. Every earlier figure computes strings, because a string's tension and gauge give a closed-form scaling law. A wind instrument does not have one, and this is why. To move from 440 to 415 hertz the player pulls out 11.7 millimetres at the joint, which lengthens the sounding tube of every fingering by the same absolute amount — so the interval each note drops is a fixed length against a shrinking one, exactly the shape of an end correction. The tuning note lands where it should and nothing else does: D3 is 67 cents sharp of where it belongs and G5 is 75 flat, a spread of 142 cents across the compass. A rebuilt instrument has no such problem — scale every length by one ratio and every mode moves by the same interval, and the tone-hole lattice cutoff moves with it, from 1766 hertz to 1666, which is 101 cents and therefore the same instrument transposed. That is the difference between an afternoon and a year.

A woodwind cannot be pulled to a new standard

Every earlier essay computes strings, because a string has a closed-form scaling law. A wind instrument does not. Pulling out at the joint lengthens every fingering's sounding tube by the same number of millimetres, which is a fixed length against a shrinking one — so the tuning note lands and the compass spreads by 142 cents. The strings could be regauged in an afternoon; the winds had to be rebuilt.

tuning · Pitch standard Seven holes that all sound 196 hertz, and none of them agrees about the twelfth. Each dot is a hole radius, placed at the station that makes the first resonance 196 hertz. The stations run from 411 millimetres for a 7.5-millimetre hole to 307 for a 1.4-millimetre one, which is a fifth of the tube. Up the axis is what the second resonance does: a cylinder's should be three times the first, and it is -2 cents from it for the widest hole and -453 for the narrowest. The hole's inertance rises with frequency, so a narrow hole lengthens the tube more for the twelfth than for the fundamental — and two holes that are interchangeable in the first register are a fourth apart in the second.

A hole is a short tube

Four earlier essays have treated an open tone hole as a point where the pressure is released. It is not: the air in a hole has mass, and a hole with mass does not end the bore, it loads it. Seven holes drilled at seven stations all sound the same G — and their twelfths are spread over a fourth. Cross-fingering falls out of the same arithmetic, and it is not made of what everybody says it is.

instruments · Tone holes Four bells on one tube, and the series each of them supports. The same 1.48-metre trumpet bore, drawn to scale, with its mouth opened from 32 millimetres across to 280 — a factor of 8.8 in radius and everything else held. Beside each is the frequency past which its flare stops reflecting, computed from the geometry: 403 Hz, 1212 Hz, 2945 Hz, 7303 Hz. That boundary moves by a factor of 18. The resonances underneath it barely move at all — the fourth peak sits at 409, 421, 428, 432 hertz across all four.

The mouth that decides nothing

Twelve figures across six earlier essays draw a trumpet with a 124-millimetre bell, and not one of them draws any other. Opened from 32 millimetres across to 280, the bell's own boundary moves by a factor of eighteen and almost nothing else moves at all: the instrument's registration, the sharpness of its notes in the written register, and where it runs out are all within a semitone of themselves. The reason is that the resonances are held by the walls, and a bell cannot reach them.

instruments · Bore profile A woodwind with holes graduated 12 mm to 6 mm, drilled so that every fingering is in tune. A cylindrical bore 567 millimetres of acoustic length and 15 across, with twelve tone holes through a 4-millimetre wall. Opening them one at a time from the far end takes it up a chromatic scale from D3 to D4. The stations are not copied from a maker's drawing: each was solved so that its own fingering sounds its equal-tempered note in this model, one hole at a time down the tube with every hole below it already open, which is what a reamer and a tuning fork do. The worst fingering is 11.9 cents out. The diameters run 12.0 millimetres at the bell end to 6.0 at the top, and the spacings close from 30 millimetres to 19.

The cutoff that is a list

Five earlier essays have quoted one number for a woodwind's cutoff — 1,824 hertz for a clarinet — from a formula written for an infinite lattice of identical holes. Solve a whole twelve-hole chart instead and the number is eleven different numbers, running from 2,193 hertz down to 1,574, which is 574 cents. The lowest fingering has no cutoff at all, and which way the list runs turns out to be a design decision rather than a fact about woodwinds.

instruments · Tone holes 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 Four throats on one instrument, drawn to scale. 4 bores 148 centimetres long, drawn to scale in radius and length, differing only in the radius of the tube itself — 6.0, 9.6, 14.0, 22.0 millimetres across at the throat. The mouth is held at 124 millimetres, which is what a maker changing a leadpipe actually does. A consequence is that the flare ratio falls from 20.7 to 5.6 and the bell's own boundary with it, from 5906 hertz to 1158. Beside each is the fourth resonance and how sharp it is: 427 Hz at Q 23, 428 Hz at Q 34, 427 Hz at Q 44, 421 Hz at Q 51. The notes move by 27 cents across the whole sweep and the sharpness rises by a factor of 2.2.

The throat that decides both

Eight earlier essays draw a tube eleven millimetres across and none of them draws another. Opened from six millimetres to twenty-two, it does exactly what was predicted to the loss in the walls — and it moves the radiation loss three times further, in the opposite direction, and monotonically, which the bell does not. So the two ends of a brass instrument do not divide the two losses between them. One end owns one loss and both ends own the other.

instruments · Bore profile Where a woodwind's A♭3 leaves it, below its corner and above it. The same fingering — 6 holes open on a 15-millimetre bore 567 millimetres long, sounding A♭3 at 207 hertz — drawn twice, with each opening's circle scaled by the share of the radiated power that leaves through it. At 400 hertz 78 per cent of it leaves through the first open hole, the bell takes 0 per cent, and the number of apertures really doing the radiating is 1.6; At 2600 hertz 6 per cent of it leaves through the first open hole, the bell takes 49 per cent, and the number of apertures really doing the radiating is 3.3. The lower frequency is below this fingering's corner and the higher one above it: below the corner the instrument is a short tube with one opening at the end of it, and above the corner it is the whole lattice at once. The power-weighted station — where a listener would say the sound is coming from — moves from 392 millimetres to 515.

Where a woodwind actually sounds from

Every number so far is read at the mouthpiece, and the corner's whole musical meaning is at the other end. Run the same solver forwards and it gives the flow leaving every hole — from which a clarinet's radiating aperture turns out to be a function of fingering and of frequency, but not the way it was predicted to: the fingering sets how far the aperture opens, almost exactly to the number of open holes, and barely moves the frequency at which it does.

instruments · Tone holes

Named alongside it

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

BoreResonanceImpedanceBrassEnd correctionRadiation efficiencyTone holeBrightnessDampingHarmonic seriesHorn equationInstrument design

All concepts