Acoustics & Music Theory
why it sounds like that
Concept

Room acoustics — where it appears

What an enclosure does to a sound made inside it, treated as a reservoir of energy that fills and drains. Below a certain frequency the room is a set of separable modes; above it, a statistical decay described by one time constant.

Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.

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 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 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 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.

ReverberationCritical distanceAbsorptionClarityDampingDecayEnvelopeLocalisationPrecedence effectDiffuse fieldImage-sourceInteraural time difference

All concepts