A room does not decay evenly

Sabine’s reverberation time is one line of arithmetic — volume over absorption — and the whole of concert-hall design followed from it. Every use of it on this site so far has treated the answer as a number: a cathedral is eight seconds, a studio is a third of a second, and the ceiling those put on how fast harmony can change follows.

The absorption in the denominator is a strong function of frequency. Carpet takes the treble and leaves the bass; thin wood panelling does the opposite; an audience takes almost everything above 500 hertz and much less below it. Run the same equation band by band and one room produces six different answers — and the spread between them is larger, in every room examined here, than the differences between rooms that the single number is usually used to draw. The ceiling a long decay puts on harmonic rhythm turns out to be a ceiling on the bass alone.

Why the coefficients pull in different directions #

Absorption is a mechanism rather than a property, and the mechanisms are frequency-selective for reasons that have nothing to do with each other.

A porous absorber works by friction. Air moving through the fibres of a carpet or a curtain loses energy, and the air moves most where the particle velocity is largest — a quarter-wavelength from a hard wall. So a thin porous layer absorbs the frequencies whose quarter-wavelength is about its own thickness and does very little below that. Heavy carpet on concrete takes 65 per cent at 4 kilohertz and 2 per cent at 125.

A panel absorber works by flexing. A thin membrane over an air space is a mass on a spring, it resonates at a low frequency, and it absorbs there. Thin wood panelling takes 30 per cent at 125 hertz and 10 per cent at 4 kilohertz — the opposite slope, from a different physical principle.

And an audience absorbs almost everything. Upholstered seating with people in it runs from 39 per cent at 125 hertz to 94 at 1 kilohertz, which is why a hall’s reverberation time depends on how full it is, and why halls are designed with seats whose absorption when empty resembles a person. It also means that an audience is a treble absorber above all else, so a full hall is darker as well as drier than an empty one — which is a change in the spectrum rather than in the level, and is the difference every performer notices between a rehearsal and a concert.

The number that hall designers specify #

The ratio of the low bands to the middle ones has a name and a target. Beranek’s bass ratio is the sum of the 125 and 250 hertz decay times over the sum of the 500 and 1,000 hertz ones, and concert halls are specified between 1.1 and 1.25 — a modest bass lift, described in the literature as warmth.

The stone church computes to 1.38, which is above the band and is why such rooms are described as boomy rather than warm.

And a plausible concert hall computes to 1.34 as well, on a surface list of plaster, panelling, wood floor and audience, which is above the specification. The two are within four hundredths of each other, which the section on the coefficients below shows is inside the uncertainty of the table they come from — so read them as one verdict on two hard rooms rather than as a comparison. Getting it down to 1.22 needs the panelling area raised by more than half — from 1,400 square metres to 2,200 — and that is not a fudge, it is what the number is telling: hitting the specification takes a great deal of deliberate bass absorption, because a large hard room has none naturally. Panelling and seating in a hall are load-bearing acoustically and not decorative.

The small studio’s 3.5 is off the scale entirely, and it is the ordinary condition of every domestic room. Bass absorption is the expensive kind, because it needs either mass or depth — an absorber that works at 60 hertz has to be a substantial fraction of a 5.7-metre wavelength thick, or heavy enough to resonate down there. Treble absorption is a rug.

What this does to a sustained chord #

A decay that differs by band is not a fade. It is a filter that gets steeper as the sound dies.

That is the argument, and it is worth stating plainly because it changes what a reverberant room is. A hall does not add a quiet copy of the music. It adds a copy that darkens continuously, so a note’s tail is a different timbre from its onset, and the darkening rate is a property of the building.

It also means the room’s contribution to a chord is not the chord. In the church, four seconds after a chord stops, the surviving energy is the bottom two bands — which is a fifth and an octave’s worth of spectrum rather than a triad. The room is a filter, in exactly the sense a violin body is, and unlike a violin body its filter is a function of time.

What music written for such rooms does about it #

The consequence for the repertoire is a constraint that runs alongside the one this ladder has already computed.

A long decay puts a ceiling on how fast harmony can change, because chords pile up. What the band decay adds is that they pile up unequally: the accumulating haze is bass-weighted, so what a fast harmonic rhythm produces in a stone room is not a blur of chords but a blur of their bottom octaves.

That is consistent with what music for those rooms is like. Writing that stays in a narrow high register — chant, and the earliest polyphony above it — puts very little energy in the bands that persist longest, which is a way of using a six-second room without accumulating anything. Writing with a strong independent bass line, which arrives much later and in different buildings, puts its energy precisely where the decay is longest — and it arrives at about the point where the room’s own set of resonances has been pushed below the musical range by the size of the buildings being used.

The room that is hardest to fix #

The studio’s bass ratio of 3.5 is the ordinary state of a small room and it is the hardest number in this essay to move — in both senses. It is hard for an owner to move, for the reasons below, and it is also the number in this essay least sensitive to the published coefficients it is computed from: doubling or halving any single one of them leaves it between 3.04 and 3.87. The church and the hall’s ratios can be argued about at the third significant figure; the studio’s cannot be argued down to anything a concert hall would accept.

Everything a small room contains — carpet, furniture, curtains, acoustic tile, the people in it — is a porous absorber, and a porous absorber works from about its own thickness upward. Five centimetres of anything absorbs above roughly 700 hertz and does almost nothing at 125. So adding treatment to a small room makes the treble decay shorter and leaves the bass where it was, which raises the bass ratio: the more a small room is treated in the ordinary way, the more unequal it becomes.

That is a counterintuitive result and it is the one this essay most needed to be robust, because it runs against what everybody who treats a room believes they are doing. It is: the untreated room’s 0.91 and the treated room’s 3.5 never change places under any twofold error in any coefficient, and the treated figure moves by at most a quarter either way. Ordinary treatment quadruples the bass ratio of a small room, and no plausible revision of the absorption table makes it not do so.

It is the reason bass traps exist as a separate category of object. Absorbing at 60 hertz requires either a depth of half a metre or more, or a resonant device tuned to a band — a membrane or a Helmholtz absorber — and both are large, expensive and specific.

Whose music, and when #

The coefficients are measurements, and they are the largest block of published constants this site has used in one essay. They are the standard tabulated Sabine coefficients used in architectural acoustics, they were measured in reverberation chambers on samples, and they vary between sources by a good deal more than the third decimal place they are usually quoted to. Everything computed from them here is exact and what it rests on is a table.

The site has recorded before that the proportion of measured constants is worth watching, and this essay is the strongest instance of it. The defence offered is that the argument does not depend on any particular value, and it is a defence that can be tested rather than made: multiply each of the nine materials’ coefficients by two and by a half in turn, eighteen perturbations, and see what moves.

Most of it does not move. The ordering of the four rooms by bass ratio — untreated room lowest, treated studio highest — survives fifteen of the eighteen, and the studio’s 3.5 stays between 3.04 and 3.87 under every one of them. The counterintuitive result, that treating a small room roughly quadruples its bass ratio, is not sensitive to any single number in the table.

Two things do move, and both are worth naming rather than leaving inside the defence.

The stone church at 1.38 and the concert hall at 1.34 are four hundredths apart, which is far less than the coefficients support, and three of the eighteen perturbations swap them. Nothing in this essay rests on which of the two is boomier, and nothing should: they are one number to the precision available.

And the claim they are both used for — that both sit above the 1.25 specification — survives fifteen perturbations and fails three. Doubling the panelling coefficient puts the hall at 1.16, halving the audience’s puts it at 1.18 and the church at 1.24, and doubling glass puts the church at 1.24. Over the whole sweep the church runs from 1.24 to 1.52 and the hall from 1.16 to 1.51. So both rooms are above the specification is true at the tabulated values and is within a twofold coefficient error of not being — which is a fair description of how firmly an architectural claim of this kind can be made from a published table, and is one reason a designer measures the finished room.

The repertoire claim is about European sacred music written for stone buildings between roughly the ninth and sixteenth centuries, and about the concert halls of the nineteenth. Both are documented spaces and in many cases still standing, so the surface lists are checkable in a way most repertoire claims on this site are not.

What a reverberation algorithm has to get right #

The same asymmetry decides what a synthetic reverberation has to model, and it is the part that separates a convincing one from an obvious one.

A reverberator that applies one decay time to everything sounds metallic, and the standard fix — a low-pass filter inside the feedback loop, so that each circulation loses more treble than bass — is exactly this essay’s curve implemented as a mechanism rather than as a table. Every algorithmic reverberator has that filter, and its cutoff is the parameter usually labelled damping.

What that makes explicit is that the frequency-dependent decay is not a detail of a room; it is close to being the definition of one. A space that decayed evenly at every frequency would be unlike any room that exists, and would be recognised immediately as artificial.

And it is why an impulse response captures a room and a decay time does not. The site has already used the same distinction about an instrument: a violin’s radiated spectrum is the string’s sawtooth times a comb of measured resonances, and no single number describes the comb. A room is the same object at a larger scale and with a time axis.

Where the model stops #

Sabine’s equation is itself the wrong model in the extreme bands. It predicts that a room whose walls absorb everything still rings, and it is least accurate where absorption is highest — which in a treated small room is the top two bands, exactly where the ratio in this essay is largest. The Eyring correction reduces those numbers and does not change their ordering.

Air absorption dominates the top band in a large room and is included here as the usual term, but it depends strongly on humidity — a hall at 30 per cent relative humidity and the same hall at 70 have measurably different treble decay. That is a real effect that concert halls are known to show seasonally, and it means the top of every curve here is a range rather than a value.

And the modal region does not obey any of this. Below the frequency at which a room stops being a set of resonances, decay is not a single exponential at all — each mode decays at its own rate according to where its pressure maxima sit relative to the absorbing surfaces. The 125-hertz band in a small room is entirely inside that region, so the studio’s 3.39 seconds is a number the model should not really be asked for.

What the picture cannot show #

It cannot show a level. A decay time says how fast energy falls and not how much there was, and how far from the source the reverberant field wins is a separate calculation from the same two numbers.

It cannot show the early part. Everything here is the reverberant tail, and the first eighty milliseconds — the early reflections that fuse with the direct sound — have their own spectrum and their own dependence on which surfaces are near the listener. A hall’s clarity is largely an early-energy quantity and none of it is in these curves.

It cannot show the seats. The audience is the largest absorber in a hall and it is not spread evenly over the surfaces; it is a plane at one height. Sabine’s equation takes total absorption and does not care where it is, which is a good approximation for a diffuse field and a poor one for a room with one very absorbing floor and five hard sides.

It cannot show the direction the bass comes from. Low frequencies arriving from the sides are heard differently from the same energy arriving from the front, and the mechanisms a listener uses to place a sound work quite differently at 125 hertz from the way they do at 4 kilohertz. A band-by-band decay time has no direction in it.

And it cannot show what a listener calls warmth. Bass ratio is a specification with a target and it is a proxy. The perceptual quantity it stands in for involves the early-to-late ratio, the direction the low energy arrives from, and the level — none of which is a decay time.

Where this ladder ends #

Seven rungs. The first four described a room as a set of resonances, as a decay time, as a direction and as a constraint on harmony. These three collapsed those into single numbers: a frequency above which the resonances stop being countable, a distance beyond which the building is louder than the player, and the discovery that the decay time the other two are computed from is not one number but six. The room was the last object on this site to be treated as though one measurement described it, and it is not.