Acoustics & Music Theory
why it sounds like that
Concept

Absorption — where it appears

The fraction of a wave's energy a surface takes up rather than reflecting, quoted per material and per frequency band. It is the single input a reverberation calculation needs, and because it varies with frequency a room does not decay evenly across the spectrum.

Named by 10 essays across 3 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 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 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 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 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

Named alongside it

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

ReverberationDecaySabine equationPrecedence effectRoom acousticsClarityDiffuse fieldHarmonic rhythmLocalisationOrchestrationSpectrumBass ratio

All concepts