Acoustics & Music Theory
why it sounds like that
Concept

Resonance — where it appears

A frequency at which a system responds far more strongly than at its neighbours, because energy put in on one cycle adds to what is already there. Which resonances a body has is fixed by its shape, and they stay put while the notes played through it move.

Named by 33 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

timbre · Spectrum Where each room stops being a set of resonances. The Schroeder frequency of 6 rooms — a carpeted bedroom at 217 Hz, a domestic living room at 183 Hz, a rehearsal room at 120 Hz, a jazz club at 68 Hz, a shoebox concert hall at 21 Hz, a gothic cathedral at 36 Hz — marked on a logarithmic frequency axis with the ranges of 4 instruments underneath. Below the mark a room is a handful of separable modes and a note's loudness depends on where the listener is standing; above it the modes overlap and the room is described by one decay time.

Where a room stops being a room

A small room has frequencies it supports and frequencies it will not, and a hall has a reverberation time. Those are two separate accounts and they are descriptions of the same building at different frequencies. The crossover is one formula, and in a bedroom it lands at about two hundred hertz — in the middle of the bass register, and above nothing at all in a concert hall.

timbre · Room acoustics Long-term average spectra: an orchestra, playing forte against a trained operatic soloist. Each source's mean spectrum over a long passage, in decibels below its own strongest region, on a logarithmic frequency axis. An orchestra, playing forte peaks at 250 Hz and is 30 dB down by 3,150 Hz; a trained operatic soloist peaks at 250 Hz and is 11 dB down by 3,150 Hz. The shapes are the same until about 1 kHz and separate above it: at 3153 Hz the difference is 19.0 decibels, which is the largest anywhere in the range. Nothing here is about level. Both curves are drawn against their own peaks, so what is being compared is shape.

One voice over ninety players

A soloist heard over a full orchestra is not louder than it and could not be. What the trained voice does instead is put a peak of energy at three kilohertz, which is where the orchestra's spectrum has already fallen away and where the ear's own threshold happens to be lowest. Nineteen decibels of advantage, in a place nobody is competing for.

timbre · The voice 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.

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.

timbre · The voice Where A has been. Documented pitch standards and surviving instruments, plotted as cents from A415. The extremes are 392 Hz and 465 Hz, which is 296 cents apart — 3.0 semitones, close enough to a minor third that a piece written at one and played at the other is in a different key. Nothing here is a preference; each is a decision somebody recorded.

The note that moved by a minor third

A has been anywhere between 392 and 465 hertz in the surviving record, which is 296 cents — a minor third short of six. The usual conclusion is that absolute pitch level is a convention and nothing musical depends on it. That is true of everything written on paper and false of everything that happens in a throat or a body: a singer's register break sits at a fixed frequency, so across the historical range it lands three semitones further down the written page. Transposing a piece is not a uniform operation, because the performer does not transpose.

tuning · Pitch standard How near the breaking point each string already is. Frequency times length, as a fraction of what the material allows. The ceiling is half the square root of specific strength — sheep gut 240, music wire 276, nylon 114, brass 127 hertz metres — and it depends on nothing a maker can change: not the gauge, not the tension, not the workmanship. The guitar top E runs at 187 per cent of its own ceiling, which is why it is the string that breaks and why every complaint about rising pitch in the historical record is about that one string.

A standard is a specification

Choosing where to put A looks like a convention and is a mechanical decision. Tension goes as the square of frequency, so a piano built at 440 and tuned to 466 carries twelve per cent more load — a tonne and a half in this model's arithmetic. And there is a hard ceiling nobody can engineer round: frequency times length is capped by half the square root of a material's specific strength, which for gut is 240 hertz-metres. A violin E at A440 runs at 89 per cent of that. At A493 it is at a hundred, and every complaint in the historical record about rising pitch is about that one string.

tuning · Pitch standard One key, nineteen notes, one right answer. A register hole disturbs a mode in proportion to the square of that mode's pressure at the hole, so the place that spoils the fundamental and leaves the third harmonic alone is the third harmonic's own pressure node — a third of the way down whatever length is sounding. The length changes with every fingering and the key does not move, so the two curves cross at one note. At A3 the key sits almost exactly on the node and the twelfth speaks cleanly; at A♭4 it is at 62 per cent of the tube and disturbs the third harmonic by 96 per cent of what an antinode would, which is the throat of the instrument and is exactly where players say the notes are worst.

The hole that spoils a note

A tone hole shortens the tube. A register hole does the opposite job: it is small enough to shorten nothing and is placed where it will wreck the fundamental's resonance and leave the third harmonic's alone, so the note jumps a twelfth instead of retuning. The place that does both is a pressure node of the harmonic being kept — a third of the way along whatever length is sounding — and the length changes with every fingering while the key does not. One key is at the right place for exactly one note, and the note it is worst for is in the throat of the instrument, which is where players say the instrument is worst.

instruments · Tone holes A perfectly tuned unison is the shortest note on the piano. How long a struck unison takes to fall forty decibels, against how far apart its two strings are tuned, for a note whose single string would ring for 20 seconds and whose bridge takes the in-phase mode down in 1.5. At a perfect unison the hammer excites only the mode that drives the bridge, so the note dies in 1.00 seconds. The longest note is at 2.30 cents, at 3.05 seconds — three times as long. Past 4.48 cents the two modes stop sharing a frequency and start sharing a decay, the note beats instead of ringing, and the sustain collapses.

Three strings, and the note that comes back

The usual account of a beat adds two independent sources together. A piano's strings are joined at the bridge and are not independent: the pair has normal modes, one of which drives the bridge and dies in a second and a half while the other cancels at the bridge and rings for twenty. A hammer striking a perfect unison excites only the first — so a perfectly tuned unison is the shortest note on the instrument, dying in one second, and the longest is at a deliberate detuning of about two and a half cents, at three. Past four and a half cents the two modes swap what they share and the sustain collapses.

instruments · Beating One instrument, five spectra. The first 12 partials of a bowed string at 5 pitches, each passed through the same fixed body response and normalised to its own loudest partial. The resonances stay where they are and the partials slide under them, so the pattern is different at every note: the power-weighted centroid runs from 1.02 to 1.71 across the compass and is not monotonic in pitch. A source with no body at all would give 2.35 at every pitch, which is the single number every roughness figure here uses for a string.

An instrument is not one timbre

Every roughness verdict so far uses one partial list per instrument, and a violin does not have one. Its body's resonances stay where they are while the fundamental moves, so the radiated spectrum is different at every pitch: the power-weighted centroid runs from 1.02 to 1.64 across the compass, and it is not monotonic. Which means the result that a spectrum chooses its own scale gives a different scale at every note on the same instrument — three minima in the dissonance curve at the bottom of a violin's range, two in the middle and four at the top.

timbre · Spectrum Blowing harder is playing sharper. How far the played note is pulled from the bore's own resonance, against blowing pressure, for an air jet crossing a 4 mm embouchure. The jet's preferred frequency goes as the square root of the pressure, so it rises by a factor of 3.16 across the tenfold pressure range drawn, and the bore holds it to 272 cents of that — from -174 at the quietest to +98 at the loudest, about the player's own nominal. The rows below give the same pull for a drive one semitone sharp of the bore, for four valves: a clarinet reed is damped by the lip and pulls 4.9 cents, brass lips are not and pull 23.6.

Blowing harder is playing sharper

Every frequency computed so far for an air column is a resonance of the tube, and no wind instrument plays at its bore's resonance. It plays between the bore and whatever is driving it, weighted by how sharply each is tuned — and a flute's driver is an air jet whose own preferred frequency goes as the square root of the blowing pressure. Across a tenfold pressure range the jet's preference rises by a factor of 3.16 and the bore holds it to 112 cents of that. A wind player's dynamics and their intonation are one control, and the size of the coupling between them is the valve's Q.

tuning · Air column Which partials of a natural horn can be lipped into tune. Every partial of the natural series against the nearest note of twelve equal, in cents, with the band the player's lips can actually move it drawn around each one. The band is ±23.1 cents, computed from the Q-weighted mean of a bore at Q 40 and lips at Q 12 rather than chosen. 4 of the first 16 partials fall outside it: 7, 11, 13, 14. The worst is the 11th at -48.7 cents, which would need lips of Q 38 to reach — comparable with the bore's own, at which point the bore has stopped deciding the pitch at all.

The partial the lips cannot reach

A wind instrument plays between its valve's preferred frequency and its bore's, which suggested that the natural trumpet's notorious eleventh partial would therefore turn out to be a statement about the lips. It is not. Run the same Q-weighted mean over the whole series and the lips can move that partial twenty-three cents, which is a quarter of what it needs — and the Q that would fix it is the Q at which the bore stops choosing the note.

tuning · Air column 4 bores of one length, and the series each supports. cylinder, cone, exponential horn, Bessel horn — every one of them 148 cm long, 5.5 mm at the throat and 62 mm at the mouth, so the only thing that differs is the shape between the two. Beside each is the series of resonances Webster's equation gives it, and the number is how far that series is from evenly spaced, in cents. cylinder: 0.0 cents, with each mode sitting at minus 0.50 of a spacing off a whole multiple; cone: 9.6 cents, with each mode sitting at minus 0.12 of a spacing off a whole multiple; exponential horn: 22.8 cents, with each mode sitting at minus 0.16 of a spacing off a whole multiple; Bessel horn: 3.7 cents, with each mode sitting at minus 0.11 of a spacing off a whole multiple.

Only two shapes make a series

A tube's length sets where its modes are and its shape sets whether they are a series at all — and of every shape a maker could choose, exactly two give evenly spaced modes. Neither of them is the shape of a trumpet. Webster's equation says how far the others miss by, and the answer is a quarter of a semitone for the obvious ones and four cents for the family brass bells actually belong to.

instruments · Bore profile What a trumpet mouthpiece does to the series. Each bore's modes before a mouthpiece is fitted and after, on the axis that says which harmonic each mode is: zero means the m-th mode sits at m times the spacing and is the m-th harmonic. Every shape starts near -0.14 and ends near 0.05. The cup holds 3.0 millilitres against a throat 3.7 mm across and pops at 642 hertz. What does not improve is the regularity: the spacing stays as uneven as the flare left it.

What the mouthpiece is actually for

A cup and a throat look like a comfort fitting and are a component. Put a real one on the front of a real bore in the horn equation and the whole mode series slides sideways by about a seventh of its own spacing — which is exactly the distance between a series whose second resonance is 1.85 times the spacing and one whose second resonance is the second harmonic. The flare decides whether the modes are evenly spaced; the mouthpiece decides which harmonic each one is.

timbre · Bore profile 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 hand closing, and the note it is holding. The fourth resonance of a 370 cm horn against how much of the bore the hand occludes, at 97 per cent of the way along where the radius is 36 mm. Nothing much happens for the first ninety per cent. The note then falls to -401 cents — most of a fourth — over the next few, and between 99.5 and 99.90 per cent it jumps to 96 cents SHARP, because the lowest resonance has left the bottom of the range and every mode has taken the place of the one below it. The series' ratios are unchanged across the jump.

The hand goes in, and the note jumps

A horn player's hand closes the bell and the pitch falls — 19 cents, then 55, then 132, then four hundred, accelerating the whole way. Then, in the last half per cent of closure, it stops falling and lands a semitone above where it started. An earlier essay guessed the mechanism was the boundary condition changing kind and the series going odd-only. It is not. The series never changes at all.

tuning · Air column The first five peaks, followed as the hand closes. Each line is one member of the series, tracked by its rank rather than by its frequency, and each dot's size is that peak's height. The lowest peak falls from 38 hertz to 34 as the hand closes and then jumps to 45, which is the renumbering computed earlier: past the wall the series is one member shorter at the bottom and every peak has taken the place of the one below it. The dots shrink through the middle of the travel and grow again at the far end, so the transition costs the player support as well as pitch — and the cost is temporary, which is why a fully stopped horn is a usable instrument and a nearly stopped one is not.

A resonance has a strength as well as a frequency

What eleven earlier essays drew is a row of frequencies, because the solver behind it has no losses and a lossless resonance has no width. Put the losses in and every one of them acquires a height and a Q — and the hand closing a horn's bell turns out to take away nine and a half per cent of the instrument's total support before giving all of it back, in a window a few per cent wide where the horn is genuinely hard to play.

tuning · Air column 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 Three ways a note starts, on one pair of axes. Every instrument in this collection, with how long its note takes to speak measured twice: across, in periods of the note itself; up, in milliseconds. The three clusters are three different pieces of physics. A wind instrument accumulates energy in a resonance, so its wait is the resonance's Q — 27, 26, 26, 19, 37 periods here. A bowed string is at full amplitude the moment Helmholtz motion begins and what takes time is the bow reaching a force inside Schelleng's window, which is 3.0, 3.8, 5.1, 7.6 periods. A plucked or struck string is at full amplitude at once and the only duration in it is the exciter's own contact, under a tenth of a period. The two axes do not agree: the slowest instrument in milliseconds is an alto saxophone at 159, and in periods it is an alto saxophone at 37.

A note takes a number of periods to speak

Two separate accounts stopped at the same object and said so. A wind instrument's note takes as long to arrive as its resonance takes to build, and that time is the resonance's Q divided by its frequency — so the number of periods is the Q and has no pitch in it, while the number of milliseconds does. The two readings order the instruments differently, and both orderings are wanted.

instruments · Onset time Up a trumpet, the two clocks disagree. Every impedance peak of a trumpet, with the settling time each implies. The Q rises up the ladder and so does the frequency, and the settling time is their ratio — so in milliseconds the wait falls from 153 at the C♯2 to 24 at the B5, while in periods it rises from 11 to 24. Both curves are monotone and they point opposite ways, which means a player going up the instrument gets notes that arrive sooner and take longer in their own terms. Nothing here is a measurement of an instrument: it is the transmission-line solve of a trumpet-shaped bore, whose peak Qs are sensitive to how finely the sweep is sampled at about five per cent.

The higher note speaks sooner and takes longer

Up a brass instrument the settling time in milliseconds falls by a factor of seven and the settling time in periods rises by a factor of three. Both curves are read off the same impedance sweep, both are monotone over most of the compass, and they point in opposite directions — so the slowest note of the instrument depends entirely on which clock is used to time it.

tuning · Onset time 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 One vent on a cone, over the notes it has to serve. A cone's modes are a full harmonic series, so the mode a register vent has to keep is the octave rather than the twelfth, and its pressure node sits at half the sounding length measured from the virtual apex. The apex is a fixed point of the instrument and the bell end is not, so every fingering moves the ideal position and there is one hole. The upper line is how much the vent spoils the fundamental, which is what it is for; the lower is how much it damages the octave, which is the cost. Their ratio runs from 451.2 at F♯4 down to 1.9 at B♭3 — a factor of 235 across a single register. The vent is placed at 37 per cent of the longest sounding length, which is where the worst fingering is least badly served.

The vent a cone cannot place

A clarinet's register key has to spoil a fundamental and leave a twelfth. A saxophone's has to spoil a fundamental and leave an octave, whose pressure node sits at half the sounding length from the virtual apex — and the apex is a fixed point while the bell end is not. Over one register the ideal position moves by a factor of two, and one hole is right for one note.

instruments · Tone holes 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 trumpet: what the cup does to every peak. Each impedance peak of a trumpet drawn twice: as the bare bore, and with its own catalogue mouthpiece in front of the throat. The bare heights fall monotonically, from 19.2 at the pedal to 1.96 at the top of the series, which is what a tube does. The fitted heights do not: they fall to 5.4 at E♭6 and rise again to 30.1 at E♭5, giving the instrument a maximum of support in the middle of its compass that the tube alone has nowhere. The dashed line is the mouthpiece's popping frequency, 642 hertz — the note the cup sounds when it is slapped on the palm — and the maximum sits beside it.

What a cup does to the support

The mouthpiece's job was settled four essays ago and settled in cents: it decides which harmonic each mode is. Measured instead in the currency a player buys one in — how hard the note pushes back, and how narrowly it holds its pitch — the cup does something else entirely. It multiplies the support in the written register by about five, and it puts a ceiling on the instrument that the bell had not put there.

instruments · Bore profile The bell and the cup, on one surface. Every combination of a flare exponent and a cup depth, shaded by how far the resonance series is from a harmonic series over partials 2 to 8, in cents. The pale line is the best flare for each cup — the ridge — and it runs diagonally: the exponent that suits the shallowest cup here is 0.93 and the one that suits the deepest is 0.70, a spread of 0.22. The ring marks the bell found earlier by sweeping the flare alone, at an exponent of 1.00 — which reaches 4.6 cents with nothing in front of it, and sits where the ring is, at 24.9 cents, once a catalogue mouthpiece is on it. It is on none of the ridge, and that is the whole finding: the two choices are not separable, and a bell optimised alone is not the bell an instrument wants.

The bell is tuned for the cup

Sweeping a brass bell's two flare parameters found a shape that makes the resonance series harmonic to four and a half cents. Bolt the mouthpiece that instrument is actually played with onto it and the series is twenty-five cents out — worse than a plain cone. The flare and the cup are not two independent choices, and the best bell for a maker to build is one that is deliberately wrong.

timbre · Harmonic series 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 A straight mute where its corks put it, and the annulus left over. The last 220 millimetres of a trumpet's bell, drawn to scale in radius and in length, with a straight mute in it. The mute is a cone 150 mm long running from 35 mm across at the tip to 105 at the base, and it is pushed in until its clearance is the cork's 3.0 mm — which stops it 63 mm inside the rim. The sound goes past it through the annulus between cone and wall, and the narrowest place in that annulus is 393 square millimetres, 59 mm in, which is 27 per cent of the bell's own section there. The lower curve is that annulus turned into the only thing a transmission line cares about: the radius of a round tube of the same area, which falls to 11.2 mm at the throat and recovers to 53 at the rim.

The resonator at the far end

A cup at the throat owns the top of a brass instrument and puts a ceiling on it. A straight mute is a second closure at the other end of the same air column, and the prediction written down before it was computed was that it would push that ceiling further down. It does the opposite: the ceiling goes from C6 to E♭6, and it goes there by reviving exactly the resonances the cup had killed.

instruments · Bore profile 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 bar and its pipe stop being two things. The two normal modes of a bar and a resonator tuned to it, against how strongly they are coupled, at 262 hertz. Below a threshold the pair has one frequency and two different decay rates — the pale curves, which are the damping splitting rather than the pitch — and above it the frequencies separate. The threshold is exact and it is not a matter of degree: it is where the coupling rate equals half the difference between the two damping rates, which for a bar of Q 197 against a tube of Q 80 is a coupling of 0.37 per cent. A marimba's own coupling is 0.62 per cent — 1.67 times the threshold, and not free: it is fixed by how much louder the tube makes the note, since the coupling that splits the pair is the coupling that carries the energy out. So the resonator model's assumption that the tube is a filter downstream of the bar is wrong at middle C, and it is wrong by less than a factor of two.

A bar and its pipe are one object

Three earlier essays treat a marimba's resonator as a filter the bar's output passes through, and both of them said in their own caveats that the coupling was not modelled. It is here, and the debt was right: the coupling is 1.67 times the threshold at which the pair acquires two frequencies instead of two decay rates, so the tube is not downstream of anything. Every consequence of that is smaller than the peaks it would have to be seen between.

instruments · Struck bar An unaccompanied quartet settles where its open strings put it. The average pitch of a quartet correcting toward itself over 480 corrections, in cents from the note it was given, averaged over 24 runs. With no pull from the open strings the ensemble random-walks, and the shaded band is how far: 3.7 cents root-mean-square by the end. With each open string pulling the notes that share its pitch class at a weight of 0.05, the ensemble settles at −0.97 cents in A major, against −1.01 from the open strings' weighted mean; −2.46 cents in C major, against −2.42 from the open strings' weighted mean; −2.53 cents in E♭ major, against −2.54 from the open strings' weighted mean.

An open string pulls the quartet flat

Once the tuning note has stopped, a quartet corrects toward itself and nothing holds its pitch. But four of its pitches do not move: the open strings, on a Pythagorean chain from C 5.9 cents flat to E 2.0 sharp, each ringing when a stopped note shares its pitch class. Give that sympathy a weight of a hundredth of a correction and it beats the random walk within a movement. The quartet settles flat in every major key — by 0.9 cents in E, 2.5 in C and 2.6 in A♭ — and in A♭ major the cellist's tuning scatter moves the whole ensemble by 1.7 cents.

tuning · Open strings Three G strings, and they are not one pitch. Each instrument's four open strings, at the Pythagorean position its own chain of fifths puts them, with the uncertainty its own tuning leaves drawn as a band. The A is given and carries no error; every other string is reached from it one fifth at a time, and a fifth set by ear is set by nulling a beat whose rate falls with frequency — so the error accumulates down the chain and is worst at the bottom. violin: G3 -3.9 ± 1.77, D4 -2.0 ± 0.98, A4 0.0 ± 0.00, E5 2.0 ± 0.44; viola: C3 -5.9 ± 2.83, G3 -3.9 ± 1.77, D4 -2.0 ± 0.98, A4 0.0 ± 0.00; cello: C2 -5.9 ± 2.83, G2 -3.9 ± 1.77, D3 -2.0 ± 0.98, A3 0.0 ± 0.00. The three G strings share a pitch class and are expected to sit 2.5 cents apart; the two C strings 4.0.

The quartet settles at two pitches, not four

The quartet's open strings have been treated as five fixed pitches on one chain, and they are not: the violin, the viola and the cello each tuned a G string by ear and the three are expected to sit two and a half cents apart. Giving each player their own strings, with their own scatter, and pulling each toward only their own, changes the ensemble's settled pitch by a hundredth of a cent. What it does change is systematic rather than random: a violin has an E string and no C, the lower instruments have a C and no E, so the quartet splits by section by a tenth of a cent in every key.

tuning · Open strings

Named alongside it

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

BoreBrassImpedanceCutoffQuality factorRegisterHorn equationDampingIntonationNormal modeStanding waveBoundary condition

All concepts