The model has nobody in it

Eight rungs of this ladder have described a room by its volume, its surfaces and their absorption. Every one of those is a property of a building with nobody in it, and the last item on rung seven’s list of things it could not show was 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.”

It is the largest absorber. In the hall computed below it is 45 per cent of the total absorption when full — more than the walls, the ceiling and the floor together.

Sabine has exactly one term for a person #

The equation takes a volume and a total absorption, and absorption is a sum of coefficients times areas. An audience enters it as a coefficient — about 0.85 per square metre of audience area at mid frequencies — and nothing else about it can be expressed.

That the coefficient is per square metre rather than per person is itself worth noticing. A row of people absorbs as a surface, not as a set of objects, which is a hint that the model is a model of geometry: what matters is how much area is covered and by what, not who is there.

So the arithmetic is immediate. Nine hundred square metres of audience at 0.85 is 765 square metres of absorption, and a hall whose total is about 1,590 when full has more than two-fifths of it in the seats.

The seat decides how much the hall changes #

The interesting quantity is not how absorbing an audience is. It is how much more absorbing it is than the seats it sits in, because that difference is the whole change between a rehearsal and a concert.

Hard wooden seats absorb around 0.1 to 0.25 per square metre. The difference between that and an audience’s 0.85 is nearly the whole of the audience’s contribution, so a hall with wooden seats is a substantially different room when full — 2.76 seconds against 1.90, a change of 31 per cent.

Heavily upholstered seats absorb around 0.75. The difference is then small, and the hall changes by 3 per cent.

That is not an accident of taste. Upholstered seats were adopted because they make the empty hall and the full hall the same room, which is the single most useful thing a hall can do for the people who have to rehearse in it. The acoustic requirement produced the furniture.

The three per cent is a mid-band figure #

That three per cent is measured at 500 hertz, and the section below establishes that an audience does not absorb evenly. Taking the same hall band by band, with a seat whose coefficient is flat:

and the full hall, for comparison, is 2.48, 2.19, 1.90, 1.76, 1.78, 1.83 seconds from 125 hertz upward.

No flat seat can make the two halls the same room. The best available leaves a seventeen per cent error in some band, and the heavily upholstered seat that gives three per cent at 500 hertz is twenty-one per cent out at its worst — the empty hall a fifth short in the bass and a tenth long at the top, which is the tilt this essay is about, arriving as the residual of the very fix that was supposed to remove it.

The reason is a mismatch of shapes rather than of sizes. A person absorbs 0.39 in the bass and 0.94 at a kilohertz; a cushion absorbs much the same at both, because a porous absorber of a few centimetres is thin against a three-metre wavelength and thick against a thirty-centimetre one. So upholstery can be made to match a person at any one frequency and cannot be made to match one across the range with a single number.

Which turns the design problem into a stated one. The seat that makes a hall invariant is not the most absorbing seat but the one whose absorption curve rises the way an audience’s does — less in the bass than heavy upholstery and more in the treble — and the best flat approximation is 0.665 rather than the 0.75 the trade settled on. A hall specified at 500 hertz will always be specified with a seat that is too absorbing in the bass, because 500 hertz is where the audience’s own curve is steepest and a single number chosen there overshoots everywhere below it.

It tilts as well as shortens #

The audience’s absorption is not flat. Measured band by band it runs 0.39 at 125 hertz, 0.57 at 250, 0.80 at 500 and 0.94 at a kilohertz — more than doubling across the range that matters most.

So an audience does not merely shorten a hall’s decay. It takes far more out of the middle and top than out of the bass.

The bass ratio — the low bands’ decay against the middle ones, which is the standard specification for warmth — goes from 0.86 empty to 1.26 full. An empty hall is bright and a full one is warm, and the number a hall is specified at is the full one.

That connects directly to the rung about uneven decay, and it inverts one of its examples. A hall does not have a decay spectrum; it has one for each state of occupation, and the difference is larger than the difference between many pairs of halls.

Which band the coefficient is quoted at #

One number has been doing a lot of work above — 0.85 for an audience — and it is a mid-frequency value.

Quoted at 125 hertz it is 0.39, and at a kilohertz 0.94. So the audience’s share of the total absorption is itself a function of frequency: in the hall drawn below it is 37 per cent at 125 hertz, 63 at 500 and 64 at a kilohertz, falling back to 44 at 4 kilohertz as the air itself starts absorbing.

That is the same fact as the tilt, seen from the accounting side, and it has a practical consequence for the arithmetic in every handbook: a hall specified by one absorption figure has been specified in one band, and the band is almost always 500 or 1,000 hertz because that is where a room’s specification and a listener’s sensitivity are both most interesting.

And the players are inside the loop #

The audience can at least be entered as a coefficient. The performers cannot be entered at all.

A source in Sabine’s equation radiates and the room responds. A performer radiates, hears the room’s response, and changes what they radiate — which is a feedback path, and there is no term in the equation for a source that depends on its own output.

What players do is measured and consistent. In a long room they play more slowly, more detached, and with fewer simultaneous changes.

What players do in a long room is measured and consistent: they play more slowly, more detached, and with fewer simultaneous changes. Rung four argued that the room’s decay sets a ceiling on chords per second and that the repertoires sit under it, and gave the room the causal role. The loop says the mechanism is not the room acting on the music; it is the players hearing the room and adjusting, bar by bar, in a way that would happen to a competent musician on their first evening in an unfamiliar building. Which is testable, and has been in part: put the same performers in rooms of different reverberation times and their tempo and articulation move, without instruction and often without their noticing. A correlation with a mechanism behind it is a different object from a constraint, and only the second is a property of the building.

That distinction is the whole of this section. Rung four established a correlation between the room’s decay and the repertoire’s harmonic rhythm and gave the room the causal role. The loop says the mechanism is not the room acting on the music; it is the players hearing the room and adjusting, bar by bar, in a way that would happen to a competent musician on their first evening in an unfamiliar building.

Which is testable, and has been in part: put the same performers in rooms of different reverberation times and their tempo and articulation move, without instruction and often without their noticing.

What the loop does to every earlier rung #

Once the source is inside the loop, several of this ladder’s earlier results change status from facts about a building to facts about a building and the people using it.

Clarity is measured with an omnidirectional source at a fixed level. A player who articulates more sharply in a reverberant room is raising the early energy relative to the late by changing the source, which improves clarity at every seat without touching the building.

Directivity is a property of an instrument at a stated frequency, and a player who turns, or lifts the bell, or moves upstage, is changing where the direct energy goes.

The critical distance depends on the source’s directivity factor, which is therefore partly under the player’s control.

And directivity is the same problem one level down. An instrument’s radiation pattern is a property of the source at a stated frequency — narrow at five kilohertz, nearly spherical at two hundred — and a player who turns, or lifts the bell, or moves upstage is choosing which of those patterns faces the audience. The critical distance is computed from a directivity factor, so it too is partly under the control of the person the model is a model of. None of that makes the earlier rungs wrong. It makes them statements about a fixed source, which is what they said they were, and it says the fixed source is a modelling assumption rather than a description of a concert.

None of that makes the earlier rungs wrong. It makes them statements about a fixed source, which is what they said they were, and it says the fixed source is a modelling assumption rather than a description of a concert.

The empty hall is the one the music is made in #

There is a consequence of the seat arithmetic that is easy to state and easy to miss.

An orchestra rehearses in the empty hall and performs in the full one. With hard seats those are 2.76 seconds and 1.90 — rooms as different as a large church and a concert hall — and every decision made in the rehearsal about tempo, articulation and balance was made in the wrong room.

That is the practical reason for the upholstery and it is a strong one. It is also the reason for the older practice, still followed, of hanging the hall with drapes for a rehearsal or seating a paid audience in a dress rehearsal.

Reading that as a design problem is the ordinary way round. Reading it the other way is more interesting: the audience is a component of the instrument, in exactly the sense rung one gave the room, and a hall with nobody in it is an incomplete apparatus rather than a quiet one.

The ladder closes here #

Nine rungs, and the case for stopping is the same shape as the one made for the consonance ladder: not that no further rung could be written, but that the model this ladder built has been bounded in every dimension it has.

Sabine’s model takes a volume, a set of surface areas, their absorption coefficients as a function of frequency, a source with a directivity factor, and a listener at a distance. Those are its inputs. The ladder has bounded it in each:

Frequency, twice — rung one on the modal region where the statistical model does not apply, and rung five on the crossover between them, at 2,000√(T/V).

Time, twice — rung two on the decay itself, and rung eight on where a line drawn across that decay divides early energy from late.

Direction — rung three, radiation omnidirectional below ka = 1 and beamed above it.

Musical rate — rung four, the ceiling on chords per second that a decay time implies.

Distance — rung six, direct against reverberant and the 0.057√(QV/T) at which they cross.

Spectrum — rung seven, absorption as a strong function of frequency and a room with six decay times rather than one.

Occupation — this rung, the audience as the largest single term and the tilt it produces.

The test is the one the consonance closure used and it is checkable rather than rhetorical: name a variable Sabine’s answer moves with, and it is already on that list. Volume and surface area are the two that have no rung of their own, and they enter the model only through the ratio that gives the decay time, which rung two is about.

Where the remaining questions have gone #

Closing a ladder is only honest if the questions it did not answer have somewhere to be, and each of these does.

How a listener discounts the room — the fact that a violin in a cathedral is heard as a violin rather than as a cathedral — belongs to the precedence effect and to the ear’s construction of objects. It is a large and interesting problem and it is a problem about hearing, not about rooms.

Where a sound seems to be belongs to localisation, whose two rungs are about a head rather than a hall.

What a room does to two sounds at once — whether reverberation makes one instrument mask another — belongs to masking, and the mechanism there is cochlear rather than architectural.

What a room does inside an instrument — a pipe, a body, a bore — belongs to the air column and to the body as a filter, which are the same physics at a scale where the wavelength is comparable to the object and the statistical model does not apply at all.

And the feedback path — the performer adjusting to what they hear — is the one this rung has named and not solved. It does not belong to room acoustics, because it is a model of a musician, and this site does not have an anchor for it. That is the honest place for it to sit: a stated gap rather than a filed one.

The one number a hall is remembered by #

It is worth ending the arithmetic on how thin the usual summary of all this is.

A hall is quoted as a reverberation time: a single number, at one band, in one state of occupation, averaged over positions. This ladder has now produced, from the same underlying measurement, six decay times, six clarities, a bass ratio, a critical distance, a modal crossover, a directivity-dependent direct field, and two states of occupation that differ by 31 per cent.

Every one of those is derived rather than separately measured, which is the pleasing part: one decay curve per band, honestly obtained, and the rest is arithmetic. It is also the reason the single number persists — it does carry most of the information, and everything else is a function of it and of something about the listener.

What the picture cannot show #

It cannot show where the audience is. Sabine’s equation takes total absorption and does not care where it is, which is a fair approximation for a diffuse field and a poor one for a room with one very absorbing plane and five hard sides. Every result in this rung inherits that.

It cannot show a partly full hall. Occupancy is drawn as empty or full. A hall at two-thirds is a real and common case and its absorption is not two-thirds of the way between, because the empty seats are not distributed like the full ones.

It cannot show what the empty seat’s spectrum is. The band figure enters the empty seats as one frequency-flat number, which is fair for hard wooden seats and poor for upholstered ones — whose absorption rises with frequency much as an audience’s does, which is exactly why they work. So the tilt drawn is an upper bound for a modern hall.

It cannot show the players’ room. Everything here is about the audience’s side of the stage. The stage has its own acoustics, its own reflectors, and its own long-standing problem — musicians needing to hear each other — which is a different room from the one the audience is in and is where the feedback path actually lives.

And it cannot show the closure being wrong. The list above is a claim that the model has seven dimensions and that each has a rung. If somebody names an eighth, the ladder reopens, and that is the form the claim is deliberately made in.

What closing means, and what it does not #

consonance closed because Plomp–Levelt roughness had been bounded in every dimension it has, and the closure was explicitly a claim about that model rather than about consonance. The same qualification applies here and is worth stating in the same words.

This ladder closes on Sabine’s statistical model of a room. A wave-based model of the same room has different variables — modal density, boundary conditions, phase — and rung one is where this ladder touched it and declined to go further. A model of a room as a source of early reflection patterns rather than of a decay has different variables again, and rung eight touched that one.

Closing the anchor is a claim that this site has finished arguing about the room as a decaying reservoir of energy, which is what the ladder was. It is not a claim that rooms are exhausted.