Acoustics & Music Theory
why it sounds like that
Concept

Reverberation — where it appears

The decaying tail of reflections a room adds to a sound once the direct path has arrived. Its length follows from the room's volume and its absorption, and it is not the same length in every frequency band.

Named by 31 essays across 6 fields — each of them below, with the objects they name alongside it.

A small room's lowest modes. The first few axial standing waves of a room, drawn in plan, with the frequency of every mode below 160 hertz listed underneath. The low modes are far apart in frequency, so some bass notes are loud in one corner and absent in another. The sound buttons play these two octaves above their real pitch, because a room's lowest modes are below what most speakers reproduce.

The room is part of the instrument

A room has frequencies it supports and frequencies it will not. In a small one those frequencies are far apart, so some bass notes are loud in one corner and absent in another — and no equipment fixes it.

timbre · Room acoustics Reverberation time, by two formulas. Sixty-decibel decay time against average absorption, divided by the room's volume-to-surface ratio so that every room sits on the same pair of curves. Sabine's equation, which is the one every textbook gives, and Eyring's correction to it. They agree in the reflective rooms Sabine measured and separate above ᾱ ≈ 0.18: at 0.6 Sabine reads 53% high, and at ᾱ = 1 — a room whose walls absorb everything, which is the outdoors — it still returns a positive time for a space with no reverberation at all. Marked: a concert hall 1.59 s, a stone church 3.37 s, a studio live room 0.45 s, a carpeted bedroom 0.13 s, an anechoic chamber 0.03 s.

How long a room rings, and where the formula stops

Sabine's reverberation time is one line of arithmetic — volume over absorption — and it built the modern concert hall. It also predicts that a room whose walls absorb everything still rings, which is a room with no reverberation at all, and the error is largest in exactly the rooms most music is now made in.

timbre · Room acoustics Where each frequency goes, from a source 18 cm across. Polar response of a circular radiator of radius 9 cm at 200 Hz (ka = 0.3), 800 Hz (ka = 1.3), 2000 Hz (ka = 3.3), 5000 Hz (ka = 8.2). Zero degrees is straight ahead. The low frequency is a circle — it goes everywhere — and the high one is a narrow lobe with nulls either side of it, so a listener off to the side hears the same note with its top missing.

An instrument points

A source radiates evenly while it is small compared with the wavelength and beams once it is not, and the crossover is one number. So the same instrument is omnidirectional in its bottom octave and a searchlight in its top one — which means its spectrum depends on where the listener is standing, and a microphone position is a choice about what the instrument sounds like.

timbre · Room acoustics How often the chord changes, and what a room allows. Chord changes a second implied by each style's stated rate and tempo, on a logarithmic axis, with the rate above which a room leaves more than one earlier chord above 20 dB marked for six rooms. The style rates are conventions rather than corpus measurements and the figure says so; the room rates are arithmetic from the reverberation time.

How often the chord changes

Two pieces can use the same chords in the same order and be nothing alike, because a progression says which chords and not how fast. Harmonic rhythm is the second variable, it runs over a factor of thirty between the styles that use it, and both of its limits are set by things that are not harmony — a listener's memory at the slow end and a building at the fast one.

harmony · Progression 6 chords in a gothic cathedral. Each chord's reverberant decay in a room with a 8 second reverberation time, at 1 chord a second. Decay is linear in decibels, so each line is straight with a slope of -7.5 dB a second. When a chord arrives, 2 earlier ones are still above 20 dB down.

The room chooses the harmonic rhythm

A chord in a cathedral is still sounding, seven decibels down, when the next one arrives — and the one after that, and the one after that. Reverberation is linear in decibels, so the number of chords audible at once is one number divided by another, and it puts a hard ceiling on how fast a composer writing for that building can change harmony. The ceiling is computable, and the music written for those rooms sits under it.

timbre · Room acoustics 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 Direct and reverberant sound in a shoebox concert hall. The direct sound falls six decibels for every doubling of distance and the reverberant field does not fall at all, so they cross once — at 5.5 metres in a room of 18700 cubic metres with a 2-second decay. Both are drawn relative to their level at that crossing. Everything past the crossing is a seat at which the room is louder than the instrument.

How far away the room takes over

Direct sound falls six decibels every time the distance doubles and the reverberant field does not fall at all, so the two cross once. In a concert hall the crossing is at about five and a half metres, which is nearer than nearly every seat — so almost everybody in almost every hall is hearing the building more than the players, and the number that says so is built from two quantities already computed — a room's reverberation time and an instrument's directivity.

perception · Room acoustics Six reverberation times for one room. Sabine's arithmetic evaluated in each octave band from the published absorption coefficients of the surfaces. a large stone church runs from 6.3 seconds at 125 Hz to 2.4 at 4 kHz — a bass ratio of 1.38, where concert halls are specified between 1.1 and 1.25.

A room does not decay evenly

Sabine's arithmetic gives one number and absorption is a strong function of frequency, so a room has six reverberation times rather than one. A stone church rings for 6.3 seconds at 125 hertz and 2.4 at 4 kilohertz, which means a chord left in it does not fade — it changes shape, losing its top before it loses its bottom, and arriving at the listener as a different sonority from the one played.

timbre · Room acoustics One decay, two verdicts, and the line is the listener's. The early-to-late energy ratio against reverberation time, for 50 millisecond and 80 millisecond windows. Nothing about the room differs between the curves; only where the line is drawn across its decay. Zero comes at 1.00 seconds for the 50 ms window and 1.59 seconds for the 80 ms window — which are, to two figures, the published rules of thumb for a room for speech and a room for music. The design targets were not put in; they came out.

The first eighty milliseconds are a different room

Draw a line across a room's decay and the energy on either side is two opposite verdicts about one building — clarity before it, reverberation after. The line is a property of the ear, not of the room, and putting it at 50 milliseconds and at 80 makes the two published design targets fall out — a room for speech at one second, a room for music at 1.6.

timbre · Room acoustics The same hall, empty and full. A hall of 18700 cubic metres with 900 square metres of audience, designed to 1.9 seconds occupied, with three kinds of seat under the audience. It is 2.76 s empty and 1.90 s full with hard wooden seats, a change of 31 per cent; 2.29 s empty and 1.90 s full with lightly padded, a change of 17 per cent; 1.96 s empty and 1.90 s full with heavily upholstered, a change of 3 per cent. The audience is 45 per cent of the total absorption when the hall is full, which is the largest single term in the equation — and how much the hall changes is decided entirely by what the seats were doing before anybody sat on them.

The model has nobody in it

Sabine's room is an empty box. The audience is 45 per cent of a full hall's absorption, a hall with hard seats goes from 2.76 seconds empty to 1.90 full, and because an audience absorbs far more treble than bass it does not shorten the decay so much as tilt it. And the players are inside the loop the model has no term for at all.

instruments · Room acoustics A silence measured in two ways that were not chosen to agree. How far a carried beat drifts during a silence, in fractions of a beat, for beat periods of 200 ms, 550 ms, 2000 ms and a tempo estimate 5 per cent wrong — which is the published discrimination limen rather than a figure chosen here. Half a beat of drift is where the metre coming out of the silence is no longer the one that went in, and it is reached after 10 beats whatever the tempo, which is 2.0 seconds at 200 ms, 5.5 seconds at 550 ms, 20.0 seconds at 2000 ms. The shaded band is the psychological present, 2 to 8 seconds, measured by a completely different literature and used in this collection to bound a phrase. The two answers overlap: a silence under about 2 seconds is a rest inside something and one over about 5.5 is after it.

A silence long enough to be an ending

Four of the five closure components are present or absent. Silence is the one with a continuous scale, so it is the one that can be given a threshold — and the threshold arrives from two literatures that were not chosen to agree, landing between three and a half and five and a half seconds. In a large hall, the room's own decay uses up most of it.

form · Closure Where a room stops sending the two ears the same sound. The correlation between the two ears' signals against frequency, for a seat 15 metres from the source in a 15,000 cubic metre room with a 2 second reverberation time. The pale curve is the diffuse field alone — sin(kd)/(kd) for an ear separation of 17.5 centimetres, which first crosses zero at 980 hertz. The heavy curve adds the direct sound, which is coherent and lifts the whole thing by an amount the direct-to-reverberant ratio sets. At 125 hertz the coherence is 0.98 and at 1000 it is 0.16.

Where the two ears stop agreeing

A room sends both ears versions of the same sound, alike at low frequencies and increasingly unlike at high ones. Where they stop resembling each other is 980 hertz, and it is set by the 17.5 centimetres between the ears rather than by anything about the room — which is within a quarter of a frequency found earlier for a completely different reason. One minus that correlation is spaciousness, and it is computable from a room's own reverberation.

perception · Localisation Why a concert hall is narrow. The lateral energy fraction at the middle seat as the same hall is widened, everything else held. It peaks at 12 metres across at 0.235 and falls to 0.000 at 44. A wide hall's side walls are further away, so their reflections arrive later, weaker and — this is the part Sabine's model cannot say — from nearer the front, where the sideways weighting discounts them. The shoebox halls the orchestral repertoire was written for are all between about eighteen and twenty-five metres wide, and this is the arithmetic they are the answer to.

A room with directions in it

Every room until now has been a reservoir of energy that drains at a rate. That model has no directions in it at all, so it cannot say the one thing every published measure of spaciousness is about: how much of what arrives comes from the side. Mirror the source in six walls and every reflection acquires an angle and a time — and the answer to why a concert hall is narrow falls out at eighteen metres.

perception · Localisation Every arrival at one seat, against the delay at which it would be an echo. The echogram: each reflection at its delay after the direct sound and its level relative to it, in a 22 by 45 by 15 metre hall with 18 per cent absorption. The line is the published echo threshold for speech — 40 milliseconds at equal level and about 3.0 more for each decibel of attenuation — so anything to the RIGHT of it is late enough and loud enough to be heard separately. The once-reflected rear wall arrives at 210 milliseconds, 25 decibels down, against a threshold of 114 — well past it. Nothing here stands clear enough of its neighbours to be heard as an echo, and 181 arrivals are fused with the direct sound instead.

An echo is prevented by the crowd around it

The echogram says when every reflection arrives and how loud it is; the published echo threshold says when a reflection that late and that quiet is heard separately. Put one against the other and the rear wall of every hall anybody builds is past the threshold — a 45-metre hall puts it 210 milliseconds late and 25 decibels down against a threshold of 114. It is not heard as an echo, and what saves it is not the geometry. It is everything else arriving at the same time.

perception · Localisation The hall, as two numbers a listener has. Every reflection at this seat, placed by the interaural delay it produces rather than by the direction it came from. Time runs down; the dot's size is its energy. The direct sound is at 0 microseconds and the reverberation spreads over the whole available range, with a root-mean-square width of 322 against a geometric maximum of 656. That is the position and width computed earlier, in the units a listener has instead of the vectors used until now. 10 of the 56 reflections arrive from behind and carry 9 per cent of the energy — and they are drawn where they are because the interaural delay of a reflection from 120 degrees is identical to one from 60.

The hall through a head

Every direction computed so far is a vector from a seat to an image source, and a listener has no vectors. Run the echogram through the head computed six essays ago and two things happen: the position and width become microseconds, and a third of the room disappears — because both cues fold at ninety degrees and a reflection from behind is identical to one in front.

perception · Localisation A turn of 0.99 degrees tells front from back. A source 45 degrees off centre and its mirror image 135 degrees off, which produce the same interaural delay and are therefore the same signal to a listener who does not move. As the head turns the two predictions separate: the front source's delay falls and the rear source's rises, because the fold at ninety degrees puts them on opposite branches of the same curve. They differ by the 15-microsecond threshold after 0.99 degrees of turn — which is exactly half the 1.97 degrees a source would have to move for the same listener to notice it moving, and it is half for a reason: a turn displaces the two hypotheses from each other by twice what it displaces either of them from where it started.

The turn is half the angle

A stationary head cannot tell a sound in front from the same sound behind, and an earlier essay said so at length. The turn that breaks the confusion is 0.84 degrees — exactly half the angle a source would have to move for the same listener to notice it moving, and half for a reason. In a hall the same turn does something else: the source swings at 8.9 microseconds a degree and the room swings at 2.7, so a listener who moves is separating the soloist from the reverberation as well as the front from the back.

perception · Localisation A general pause of a bar is worth 4.9 decibels in a hall and 8.1 in silence. What a rest is worth as a diminuendo, in rooms with different reverberation times. The upper curve is the earlier figure, which assumed the sound stops when the players do; each curve below it lets the hall go on sounding, falling sixty decibels in its own reverberation time until it reaches the background. At the bar of silence a general pause usually is — about 1.2 seconds — the dry value is 8.1 decibels, shoebox concert hall keeps 60 per cent of it and gothic cathedral keeps 22. At 3.5 seconds, where a gap stops being a pause and becomes an ending, the same hall keeps 86 per cent. A long silence outlives any hall's tail and a short one does not, which is why the room costs the device most at exactly the length a composer writes it.

A rest needs a dry room

The twenty-four decibels a general pause is worth assume the sound stops when the players do. Put a hall under it and a bar of silence keeps 60 per cent of its value in a shoebox concert hall and 22 per cent in a cathedral, while a three-and-a-half-second one keeps 86 and 45 — because a hall's tail has a length and a rest either outlives it or does not. A written bar of silence is worth half its dry value at 2.76 seconds of reverberation, which falls between the concert hall and the stone church.

form · Closure Weighted by the ring of the note before, a note every 0.6 seconds, Raga Deshkar moves least. Each scale placed by how smooth it is against 2000 random scales of its size under 4 instruments, each with its own spectrum and its own ring, with every pair of notes its plainest melody sounds weighted by how much of the earlier note is still sounding when the later one begins, a note every 0.6 seconds. Low is smooth. Raga Bhupali: plucked string, 6 s 53.6, long-ringing string, 12 s 35.8, blown note, 2 s hall 41.3, free bar, 4 s 58.5; spread 22.8; Raga Deshkar: plucked string, 6 s 39.0, long-ringing string, 12 s 25.9, blown note, 2 s hall 26.1, free bar, 4 s 42.0; spread 16.1; one measured slendro: plucked string, 6 s 42.9, long-ringing string, 12 s 27.8, blown note, 2 s hall 32.6, free bar, 4 s 46.4; spread 18.6; Rast, Arabic theory: plucked string, 6 s 92.0, long-ringing string, 12 s 65.2, blown note, 2 s hall 92.3, free bar, 4 s 96.1; spread 30.9; Rast, Turkish theory: plucked string, 6 s 88.0, long-ringing string, 12 s 58.1, blown note, 2 s hall 90.8, free bar, 4 s 93.6; spread 35.5; the diatonic major, tempered: plucked string, 6 s 84.8, long-ringing string, 12 s 54.2, blown note, 2 s hall 89.5, free bar, 4 s 91.3; spread 37.0; the diatonic major, five-limit just: plucked string, 6 s 87.7, long-ringing string, 12 s 56.7, blown note, 2 s hall 90.3, free bar, 4 s 92.9; spread 36.2.

A scale is committed to how long its instrument rings

A traditional scale's standing on the roughness model moves when the instrument changes, and that movement was read as a commitment to the instrument's spectrum. Weight every pair of notes a melody sounds by how much of the earlier note is still ringing when the later one begins, and the movement grows — the tempered diatonic's by 37 percentile points at a note every 0.6 seconds — but almost none of it is the spectrum. Put four instruments' rings on one spectrum and the scale moves 38.6 points; put four spectra under one ring and it moves 9.4. And the partials that make a fifth smooth are the first to stop sounding.

scales · Beyond twelve How long a doubled pizzicato keeps its note, seat by seat, in two rooms. How long a doubled violin pizzicato on 392 hertz keeps its note against the metres from the players, the pluck starting 12 dB up and decaying over 1 s with a loss exponent of 1. a concert hall, a flue pipe: 1 → 86 ms, 1.5 → 123 ms, 2 → 226 ms, 3 → 359 ms, 5 → 445 ms, 7 → 481 ms, 10 → 506 ms, 15 → 522 ms, 20 → 528 ms, 30 → 532 ms; a concert hall, an oboe: 1 → 52 ms, 1.5 → 55 ms, 2 → 59 ms, 3 → 77 ms, 5 → 149 ms, 7 → 195 ms, 10 → 224 ms, 15 → 242 ms, 20 → 248 ms, 30 → 254 ms; a concert hall, a clarinet: 1 → 44 ms, 1.5 → 45 ms, 2 → 45 ms, 3 → 47 ms, 5 → 53 ms, 7 → 65 ms, 10 → 86 ms, 15 → 105 ms, 20 → 112 ms, 30 → 118 ms; a large stone church, a flue pipe: 1 → 440 ms, 1.5 → 578 ms, 2 → 651 ms, 3 → 728 ms, 5 → 784 ms, 7 → 803 ms, 10 → 814 ms, 15 → 820 ms, 20 → 822 ms, 30 → 824 ms; a large stone church, an oboe: 1 → 56 ms, 1.5 → 65 ms, 2 → 89 ms, 3 → 207 ms, 5 → 281 ms, 7 → 303 ms, 10 → 315 ms, 15 → 322 ms, 20 → 325 ms, 30 → 326 ms; a large stone church, a clarinet: 1 → 44 ms, 1.5 → 43 ms, 2 → 43 ms, 3 → 44 ms, 5 → 53 ms, 7 → 67 ms, 10 → 80 ms, 15 → 89 ms, 20 → 92 ms, 30 → 94 ms. The mid-band critical distance is 5.3 m in a concert hall and 2.3 m in a large stone church. In none of the 60 cases does the note return to the pluck once it has left.

A room keeps a pizzicato from giving its note away

Doubled by a flute, a one-second pizzicato loses its note in 70 milliseconds dry, because its upper partials go first. The question left open was whether a room, whose reverberation keeps those partials alive, gives the note back afterwards. It does not give it back. It stops the note going: ten metres into a concert hall the pluck keeps it for 506 milliseconds, in a stone church for 814, and the room's own uneven decay takes back between a quarter and two fifths of that. In a room the loss law that decided everything dry matters a tenth as much, because the room's decay has become the clock.

timbre · Spectrum From partial 3 the room is the slower of the two. Decay rates in nepers a second for each partial of a note on 130.8 hertz, in a concert hall. The rising curve is the string's own loss, which grows as the partial number to the power 1. The flat-ish curve is the room's, from its reverberation time at that partial's frequency. A reverberant field is the source convolved with the room, so a partial's tail falls at the SLOWER of the two — the heavy line — and the room keeps returning energy the string has stopped making. From partial 3, at 392 hertz, the room is in charge: 6 of the note's 8 partials are held up by the room rather than let go by the string. Those are exactly the partials the string was losing fastest, which is why the room does not merely lengthen the note.

The room is the slower of the two

A reverberant field is the source convolved with the room, so a partial's tail falls at the slower of the two rates rather than at their sum — and the room is slower for exactly the partials the string is losing fastest. Half a note's colour is gone in 0.163 seconds in no room at all, 0.313 in a concert hall and 1.441 in a stone church. The destination is identical in all three, because a room cannot hold a partial up above the fundamental it is also holding. What a hall takes away is the rate, and the rate was the whole of the identity cue.

timbre · Envelope A room pulls the compass apart rather than evening it out. How long a pizzicato entering 6 decibels above a held note keeps the composite spectrum, at eight pitches across two and a half octaves, heard 15 metres from the stage. no room: 0.07, 0.05, 0.06, 0.06, 0.02, 0.05, 0.05, 0.04 seconds; a concert hall: 0.61, 0.27, 0.50, 0.38, 0.01, 0.25, 0.27, 0.19 seconds; a large stone church: 1.03, 0.39, 0.79, 0.56, never, 0.33, 0.35, 0.23 seconds. Dry the figures barely move — a spread of 3.0 across the whole compass — because a room is the thing that varies with frequency and there is none. In a hall the spread is 44. The room does not scale the dry answer by a constant: it multiplies it by between four and nine times depending on the note, and at C6 it makes the pluck's position worse rather than better, because the two instruments' spectra nearly coincide there and the pluck starts only 4.0 decibels ahead instead of twelve.

One note in the compass loses its pizzicato

Dry, how long a pluck keeps the composite spectrum barely depends on which note it plays: three-hundredths of a second at the worst pitch and seven at the best, a spread of three. In a concert hall the same eight notes spread by a factor of forty-three, and in a stone church one of them never gets the note at all. The room does not scale the dry answer by a constant — it multiplies it by between four and nine times depending on the pitch, and at the one note where the two instruments' spectra nearly coincide it makes the pluck's position worse instead of better.

timbre · Spectrum 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 In a hall the balanced ending is a louder chord, and the hall sets how much louder. The gain over the tutti, in phons, that a final chord held after no rest needs for the listener's running impression to finish exactly at the tutti's weight when the chord's tail has died into the silence, against the hall's reverberation time, for chords held half a second, one second and two. 0.5 s: 0.2 s 0.7, 0.3 s 1.2, 0.45 s 1.9, 0.6 s 2.7, 0.8 s 3.9, 1 s 5.0, 1.25 s 6.6, 1.5 s 8.2, 1.75 s 9.9, 2 s 11.7, 2.25 s 13.7, 2.5 s 15.8, 3 s 20.4. 1 s: 0.2 s 0.7, 0.3 s 1.1, 0.45 s 1.9, 0.6 s 2.7, 0.8 s 3.8, 1 s 5.0, 1.25 s 6.5, 1.5 s 8.1, 1.75 s 9.9, 2 s 11.7, 2.25 s 13.6, 2.5 s 15.7, 3 s 20.4. 2 s: 0.2 s 0.7, 0.3 s 1.1, 0.45 s 1.9, 0.6 s 2.7, 0.8 s 3.8, 1 s 5.0, 1.25 s 6.5, 1.5 s 8.1, 1.75 s 9.9, 2 s 11.7, 2.25 s 13.6, 2.5 s 15.7, 3 s 20.4. The dashed lines are one dynamic step, 7.12 phons, and two; a one-second chord a step louder balances in a hall of about 1.34 s and two steps louder in one of about 2.32 s.

In a hall the balanced ending is a louder chord

In a dry room a rest before a final stamp a step louder can leave a passage finishing exactly at its tutti's weight, but only for a stamp shorter than a ninth of a second. The prediction was that a hall would make it impossible: its tail fills the rest and lengthens the stamp. That is right for a small room and wrong for a hall. Read when the chord's tail has died, a live hall pulls every ending below its tutti, and the ending that finishes at its own weight there is a held chord marked louder than the passage — five phons louder at a second of reverberation, a whole step at about 1.3 seconds, two at 2.3.

form · Closure In a hall a section's shallow shelf goes under all at once. The margin at the strike for a pizzicato over a held clarinet, note by note from C4 to F♯5, dry (thin) and 15 metres into a hall (thick): the modelled violin, and the median of six sections of twelve with corpus modes spread by a semitone and air modes by half. C4 2.1/2.1 dry, -1.9/-1.9 in the hall; C♯4 -0.4/1.2 dry, -4.3/-2.8 in the hall; D4 2.6/3.3 dry, -1.6/-0.8 in the hall; E♭4 4.2/4.5 dry, 0.0/0.3 in the hall; E4 4.9/5.2 dry, 0.7/0.9 in the hall; F4 5.1/5.3 dry, 1.0/1.1 in the hall; F♯4 5.0/5.1 dry, 0.9/1.0 in the hall; G4 4.4/4.6 dry, 0.4/0.6 in the hall; A♭4 3.3/3.7 dry, -0.7/-0.3 in the hall; A4 0.5/2.2 dry, -3.3/-1.7 in the hall; B♭4 -2.2/0.4 dry, -5.9/-3.4 in the hall; B4 1.2/0.2 dry, -2.6/-3.5 in the hall; C5 -0.1/0.5 dry, -3.9/-3.3 in the hall; C♯5 0.3/2.1 dry, -3.5/-1.7 in the hall; D5 5.0/5.0 dry, 1.1/1.2 in the hall; E♭5 11.1/11.0 dry, 7.3/7.2 in the hall; E5 12.2/12.1 dry, 8.5/8.4 in the hall; F5 12.7/12.6 dry, 9.1/9.0 in the hall; F♯5 13.0/12.9 dry, 9.5/9.4 in the hall. Dry, the section fills the violin's notch at B♭4 into a shelf a few tenths above zero. In the hall the room lowers every note by the same few decibels for both, and the shelf goes under with the notch: the section owns 0 of the six notes from A♭4 to C♯5, the violin 0.

In a hall a section of violins loses the notes it saved

Dry, twelve violins with different bodies fill in the note one violin's pizzicato gives away to a held clarinet, leaving a shallow shelf a few tenths of a decibel above zero. In a hall that shelf goes under all at once. The room takes the same few decibels from each note whether it comes from one violin or twelve, so a section's smoothing survives only to the front of the hall — about two metres over a clarinet, six over an oboe — and from ten metres back a section loses exactly as many notes as one violin does.

timbre · Spectrum A hall fills the gaps a thinning ending leaves only slowly, so the written diminuendo reaches the seats nearly whole. Short-term loudness through the closing bars of a thirty-two-bar scheme of six parts, computed from each bar's part count on the excitation pattern, with a hall of 2 seconds' reverberation under it, its reverberant field in power six decibels over the direct sound in the steady state; the dashed line is the thinning ending with no room. thins to one: arriving -3.83 phons against the running impression; full final chord: arriving +0.13 phons against the running impression; unchanged: arriving +0.00 phons against the running impression; thins then full: arriving +0.43 phons against the running impression. With no room the thinning arrives at -4.46.

A hall leaves a thinning ending most of its diminuendo

A written diminuendo made by dropping parts arrives about four and a half phons under the listener's impression in a dry room. A hall's tail sounds under every bar and holds the level up as parts leave, and the obvious worry is that it fills the gaps a thinning leaves, as it fills a rest. It barely does. A concert hall leaves the thinning 86 to 100 per cent of its contrast and a stone church about three quarters, where the same rooms leave a two-second general pause three quarters and a half. A thinning steps down a decibel or two a bar, and a hall's tail falls nine decibels in the time a listener takes to register the arrival; only a cathedral is slow enough to blunt the last step, and it takes half.

perception · Loudness A section whose upper partials beam keeps its pizzicato deep into a hall on its axis, and loses it faster than an unpointing one a little way off. The margin at the strike for a pizzicato section of twelve over a held clarinet, 15 m into a hall, from C4 to F♯5, with each partial's direct sound scaled as from a piston of 10 cm radius. Above zero the pluck owns the note. no beam: A♭4 to C♯5 -0.3, -1.7, -3.4, -3.5, -3.3, -1.7; on the axis: A♭4 to C♯5 5.2, 4.1, 2.8, 3.0, 3.5, 5.3; 15° off: A♭4 to C♯5 4.4, 3.2, 1.8, 1.8, 2.2, 3.9; 30° off: A♭4 to C♯5 1.9, 0.3, -1.6, -2.2, -5.1, -2.1.

A section keeps its pizzicato only inside a cone

In a hall a section's pizzicato loses its corpus notes to a held clarinet within two metres, on the assumption that a violin radiates equally in every direction. It does not, and twelve players sharing a beam cannot average it away. Directly on the players' axis, the same section keeps every note for twenty-five metres. Thirty degrees off, with the beam's upper partials thrown elsewhere, it is worse than if it had never pointed. A question about how far back a pizzicato carries turns out to be a question about which way the listener's seat faces.

timbre · Spectrum Three orders of leaving, one passage: the part left sounding sets the drop, and the room lifts it behind them all. Short-term loudness on the excitation pattern of six string parts, G3 C4 E4 G4 C5 E5 at 62 dB each, dropping one a bar in three orders, with a hall of 2 s in the middle bands, longer in the bass and shorter in the treble. top down: -9.39 phons from the steady texture to the last bar; bass up: -7.40 phons from the steady texture to the last bar; inner first: -8.05 phons from the steady texture to the last bar.

A hall forgets almost everything but the last part to leave

A hall's bass outlasts its treble, so a thinning ending ought to care which parts it drops first: top down should leave the tail's darkness to fill the gap, and bass up should remove what the tail holds longest. Searched over all 720 orders, it barely does. Give the tail a spectrum and the drop moves by under half a phon, in the direction that helps the order expected to suffer. What sets the ending is two notes: the one left sounding, and the one that went just before it. Everything earlier has gone.

perception · Loudness Antiphonal second violins own their worst pizzicato note almost nowhere, unless a hard wall stands close behind them. A plan of a 24 by 36 m hall with the second violins at the right of the stage, plates turned away from the audience, and a perfectly reflecting wall 2 m behind them. A seat is shaded if the section's worst note from A♭4 to C♯5 stays a pizzicato rather than a clarinet with a click. The seconds keep 30 per cent of seats, the firsts on the other side 30.

A wall puts antiphonal seconds back beside the firsts

Second violins seated opposite the firsts have their plates turned away from the audience, and in a hall a section's pizzicato depends on its beam. With nothing behind them they keep their worst note in about one seat in a hundred. A perfectly hard wall two metres behind gives them back a cone as large as the firsts' own, but a wall that loses three decibels takes most of it away again. And the seats that cone serves turn out to be, within a seat or two, the seats the same section would serve sitting beside the firsts.

timbre · Spectrum

Named alongside it

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

DecayAbsorptionRoom acousticsLocalisationPrecedence effectTimbreCritical distanceDiffuse fieldEnvelopeSabine equationDampingDynamics

All concepts