Acoustics & Music Theory
why it sounds like that
Concept

Damping — where it appears

The loss that turns a resonance from a line into a peak with a width. Without it a mode has a frequency and nothing else, so damping is what gives an instrument's resonances a height and a sharpness as well as a set of places.

Named by 13 essays across 3 fields — each of them below, with the objects they name alongside it.

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 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 What each strike point on a marimba bar excites. The amplitude each of the first 4 partials receives from a strike at each point along the bar, each partial drawn to its own maximum. On a string this comb is |sin nπx| and is periodic, so one strike point silences a whole family; on a bar it is the mode shape itself, whose zeroes are not at rational fractions, so no strike point silences a family. The exception is the middle: every antisymmetric mode is zero there by symmetry, so a centre strike drives partial 1 at 100 per cent of its maximum and partial 2 at nothing at all. The cord's hole at 0.2031 is the mirror case — the fundamental is the partial a strike there cannot reach.

The four ways a marimba loses what its arch placed

The centre of a bar is a node of every even mode by symmetry, so a stroke there gives the second partial exactly nothing and a mallet's width cannot recover it; five millimetres off centre returns 14 per cent. Add the yarn mallet's 3-millisecond contact, the tube's rigid lid and a decay four times faster than the fundamental's, and four independent mechanisms take the tuned partial 80 decibels below it. On a xylophone the same four come to 24.

instruments · Struck bar 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 A bow shows the loss law a blow conceals. The same string on 130.8 hertz under three loss laws, drawn twice each: struck, and held by a continuous drive. The pale marks are the spectrum a blow produces, and they are identical in all three rows — a strike is the source spectrum and has no loss in it yet, which is why both endpoints of a struck note were found to carry nothing about the law. The solid marks are where each partial settles when a drive balances its own loss, at drive over loss, so the steady spectrum rolls off as the source's roll-off plus the exponent. At an exponent of 0.5 the held spectrum's centroid sits at 167 hertz, 5.7 semitones under the strike's 232; At an exponent of 1 the held spectrum's centroid sits at 144 hertz, 8.2 semitones under the strike's 232; At an exponent of 2 the held spectrum's centroid sits at 133 hertz, 9.6 semitones under the strike's 232. The quantity that is invisible at both ends of a struck note is the slope of a bowed one, for as long as the bow moves.

A bow holds the number a blow hides

Two struck notes with different loss laws are identical at the strike and identical at the end, which is why separating them at all meant looking in the middle. Drive the same two strings continuously and the loss law stops being a rate and becomes a slope: each partial settles at its drive over its own loss, so the exponent adds to the source's roll-off and sits in the spectrum for as long as the bow moves. It is 18.7 decibels of separation available from the first instant, against 41.3 that a blow delivers after four tenths of a second and then takes away.

timbre · Envelope The drain does not stop when the key does. The note does.. Semitones of colour gone, against time, for a note on 130.8 hertz left to ring and for the same note released after 0.4 seconds onto a damper of 0.15 seconds. The two curves lie on each other until the key comes up, and the damped one then ends: the note is inaudible at 0.52 seconds with 8.7 of the free note's 9.9 semitones delivered. A damper adds one loss to every partial alike, so it adds the same number to every decay rate and leaves every DIFFERENCE between rates exactly as it was — the spectrum at each instant is the ringing spectrum shifted bodily down by 400 decibels a second. The colour goes on draining at its own rate the whole time. What the damper takes away is not the drain but the seconds.

A damper changes the clock, not the colour

A damper is an extra loss on the string rather than a second decay, so it adds the same number of nepers a second to every partial — and adding a constant to every rate leaves every difference between rates exactly where it was. The damped spectrum at any instant is the ringing spectrum at that instant shifted bodily down, to machine precision. The colour goes on draining at its own rate; the note simply runs out of seconds, and how many it gets is written on the page as a note value and a tempo.

timbre · Envelope A damper is a loss on the string, so a room can overrule it. Decay rate in nepers a second against partial number, for a note on 130.8 hertz in a room of 2 seconds. The rising line is the string's own loss, 1.15 nepers a second at the fundamental and growing as the partial number to the power 1. The line above it is that plus the damper's 46.1, which is what the string does once the key comes up. The flat line is the room. What a listener receives is the SLOWER of the damped string and the room, because a hall goes on radiating what the string has already given it — and here the room is slower on 8 of 8 partials, from the fundamental upward. The composition proposed earlier — take the slower of the string and the room, then add the damper to whichever won — would put the damper outside the minimum, where nothing can overrule it, and would predict a note 2.54 seconds shorter than ringing where the arithmetic here predicts 0.90.

A damper cannot reach into the room

The essay before this one proposed the arithmetic for a damped note in a hall: take the slower of the string's rate and the room's, then add the damper's to whichever won. The composition is wrong, and it is wrong in the one place that decides the answer. A damper is a loss on the string, so it belongs inside the minimum where a room can overrule it — and past about three seconds of reverberation it is overruled on every partial, so the damper removes no audible seconds of note at all.

timbre · Envelope A staccato is the direct sound's, and the room takes it within a fifth of the critical distance. A note on 130.8 Hz held 0.4 s and damped, in a room of 2 s reverberation, heard at distances from 0.02 to 5 times the critical distance: how long after the release the note takes to fall 10 dB and 20 dB. To fall 10 dB: 24 ms at the source, 333 ms far away; 0.02: 24 ms, 0.05: 25 ms, 0.1: 25 ms, 0.15: 26 ms, 0.2: 28 ms, 0.3: 35 ms, 0.5: 103 ms, 0.75: 187 ms, 1: 234 ms, 1.5: 281 ms, 2: 302 ms, 3: 318 ms, 5: 328 ms; doubled by 0.38 of the critical distance. To fall 20 dB: 49 ms at the source, 667 ms far away; 0.02: 49 ms, 0.05: 51 ms, 0.1: 60 ms, 0.15: 117 ms, 0.2: 198 ms, 0.3: 308 ms, 0.5: 436 ms, 0.75: 520 ms, 1: 568 ms, 1.5: 614 ms, 2: 635 ms, 3: 652 ms, 5: 661 ms; doubled by 0.14 of the critical distance. Where the direct sound and the room are equal, the damper's work is already hidden: the room's copy is only 20 dB below the direct sound at a tenth of the critical distance, and a 20 dB fall reaches it there.

Only the player hears a staccato end

A damper stops a string in a seventh of a second, and in a hall the room goes on for two. A listener hears both, mixed in proportion to how close they sit, and the question was at what distance the short part stops mattering. The answer is closer than any seat. A damped note's twenty-decibel fall has doubled in length by a seventh of a hall's critical distance — 77 centimetres in a two-second concert hall — and by a quarter of it in a jazz club. The end of a staccato is something the pianist hears and the front row does not.

timbre · Envelope Pointed, four partials above the fundamental keep their staccato further out than it does, and the highest never catch up. A note on 130.8 Hz held 0.4 s and damped, in a room of 2 s: for each of its first eight partials, the distance along the source's axis, in critical distances of an omnidirectional source, at which that partial's own 20 dB fall after the release has taken twice as long as at the source — with the source radiating equally everywhere, and with it beamed like a piston of 35 cm radius. Partial 1 at 131 Hz: 0.141 omnidirectional, 0.164 beamed; partial 2 at 262 Hz: 0.140 omnidirectional, 0.217 beamed; partial 3 at 392 Hz: 0.139 omnidirectional, 0.283 beamed; partial 4 at 523 Hz: 0.087 omnidirectional, 0.223 beamed; partial 5 at 654 Hz: 0.054 omnidirectional, 0.170 beamed; partial 6 at 785 Hz: 0.034 omnidirectional, 0.126 beamed; partial 7 at 916 Hz: 0.021 omnidirectional, 0.091 beamed; partial 8 at 1047 Hz: 0.013 omnidirectional, 0.065 beamed. Beaming multiplies each partial's reach by about the square root of its directivity factor, but the upper partials start so far behind that only partials 2, 3, 4, 5 pass the fundamental's 0.164.

A pointing source's staccato ends dark, not bright

A damped note's release is heard as the player plays it only close to the instrument: a little way out, the room's copy takes over and the staccato is gone. A source that points, as a grand piano with its lid raised does, beams its upper partials along its axis, so their direct sound reaches further, and the prediction was that the release of a note's colour would survive into the hall where the release of the note does not — heard as a brief brightening before a room-length tail. Beaming does carry the second to fifth partials' articulation beyond the fundamental's. The highest never catch up, the whole note gains a fifth of its reach, and the release is heard not as a brightening but as a darkening.

timbre · Envelope Where in the hall a raised lid makes a C3 staccato darker, and where brighter. A floor in front of a 35 cm piston radiator pointing up the page, in units of the critical distance of a source radiating everywhere, three wide and one and a half deep, in a 2-second room, divided into 160 seats. Each seat is marked by what a damped C3 does to its colour over the 400 ms after its release: darkens (42 seats), brightens but by no more than with no lid (26), or brightens more than with no lid (92). The darkening seats form a narrow wedge on the axis; everywhere else a raised lid makes the end of a note brighter, and over 58 per cent of the floor brighter than it would be with the lid closed.

A lid darkens a staccato only inside a narrowing cone

Along its axis, a raised lid makes a damped note's release darker, because the brightness it beams at the listener is the part the damper stops. Most of an audience does not sit on the axis. Off it, the same beaming does the reverse, and a little way off it does the reverse more strongly than no lid at all would. The region where the release darkens is a cone whose half-width obeys one law — the fundamental times the radiator's radius times the sine of the angle is 23.6 hertz-metres — so it narrows from 28 degrees at C3 to 8 at C5.

timbre · Envelope

Named alongside it

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

DecayEnvelopeResonanceBoreBrassImpedanceBrightnessQuality factorReverberationCritical distanceCutoffIdentification

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