Acoustics & Music Theory
why it sounds like that
Perception and the listener

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.
14 min read 6 figures What the room doesThe limits of the listenerThe ear is not a microphone

Assumes: A room with directions in it · The first wavefront wins

The first wavefront wins is the second rung of this ladder and it ended with a debt it could not pay for four more. It established that a reflection arriving inside a window of a few tens of milliseconds is fused with the direct sound rather than heard as a separate event — the direction comes from the first arrival and the rest is absorbed into it. What it could not say was which reflection wins, or when a particular hall crosses the threshold, because it had no way to know when any individual reflection arrived.

A room with directions in it built the thing that knows: an image-source construction, in which every reflection off a wall is geometrically a direct sound from a source mirrored in that wall, so a shoebox produces one arrival per integer triple with a computable time, level and angle.

The echo threshold is a published function of exactly those three quantities. This is one run against the other.

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.
Fig. 1 Every arrival at a seat nine metres from the platform in a 22-by-45-metre hall, each at its delay after the direct sound and its level below it. The curve is the echo threshold for speech: anything to the right of it is late enough and quiet enough that a listener would hear it as a separate event. A great many arrivals are to the right of it, and none of them is heard as an echo.

The threshold, and the thing it assumes

The echo threshold is one of the older numbers in psychoacoustics and it has two well-established properties. It depends strongly on the signal: a click is heard as doubled at a delay of a few milliseconds, speech at around forty, a slow orchestral chord well past a hundred — the same dependence on what is being listened to that a dissonance has to last found for roughness, and for the same underlying reason, which is that the ear integrates. And it depends strongly on the reflection’s level: a reflection at equal loudness with the direct sound is an echo at forty milliseconds for speech, and every ten decibels of attenuation buys roughly thirty more before it becomes one.

a click                8 ms, plus 1.6 ms per dB of attenuation
speech                40 ms, plus 3.0
an orchestral chord   80 ms, plus 3.4

Those bases are asserted and ordinal, in the way this collection has recorded before: what is published and reproducible is the ordering and the shape, and the numbers here reproduce them. The angle term — a reflection from the source’s own direction being marginally more likely to be heard as an echo than one from the side — is the weakest part and is a stated modifier rather than a fitted one.

What the threshold assumes, and what nobody says out loud when quoting it, is that there is one reflection. Every experiment behind it is a direct sound and a single delayed copy in an anechoic room. A hall is not that.

What happens when the assumption is dropped

Applied arrival by arrival to a real echogram, the criterion says that a 45-metre hall has several hundred echoes in it. That is not a result about the hall. It is the criterion being used where its own premise does not hold, and it produces a number that is obviously false: nobody has ever sat in a shoebox hall and heard three hundred echoes.

What distinguishes an echo from reverberation is not the delay and not the level. It is that there is a gap around it. An echo is an arrival that stands out of the tail; reverberation is a tail that has no arrivals standing out of it, however many arrivals it contains.

So the criterion needs a second term, and the second term is a prominence: an arrival counts only if it carries more energy than everything else within a few milliseconds of it put together. With that rule the same hall has no echoes for speech and none for music, and exactly one for a click — the once-reflected stage wall, arriving twenty-three milliseconds late.

That is the right answer, and it is the answer everybody who has sat in such a hall already knew.

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 a click — 8 milliseconds at equal level and about 1.6 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 47 — well past it. 1 arrival stands clear enough of its neighbours to be heard as an echo, the worst being 1× the stage wall.
Fig. 2 The same seat and the same hall with a click instead of speech. The threshold moves left by a great deal — a click’s base is eight milliseconds rather than forty — and the number of arrivals past it grows accordingly, while the number that stand clear of their neighbours does not. The threshold is a property of the signal; the prominence is a property of the room.

The rear wall is always past the threshold

The interesting consequence is what the two terms say about the surface every hall designer worries about.

Take the once-reflected rear wall at a seat near the front and sweep the hall’s depth. Making the hall deeper does two things at once: it delays that reflection, and it attenuates it, which raises its own threshold. The question is which wins, and delay wins easily — the delay grows linearly with depth while the threshold grows as the logarithm of the distance.

The crossing is at about 25 metres for speech and 35 for an orchestral chord. Every concert hall anybody has ever built is deeper than that. So the rear wall of the Musikverein, of Boston Symphony Hall, of every shoebox in the repertoire is arriving late enough and loud enough to be heard as an echo, and is not heard as one.

How deep a hall has to be before its rear wall is late enough to be an echo. The once-reflected rear wall at a seat 5 metres from the source, against hall depth. Up the page is how far past its own echo threshold the reflection arrives — the threshold rises with attenuation, so a deeper hall raises it too, and the question is which rises faster. Delay wins: the reflection crosses zero at about 25 metres for speech and 35 for an orchestral chord, so every concert hall anybody builds has a rear wall arriving late enough and loud enough to be heard as an echo. It is not heard as one, and the reason is not in this figure.
Fig. 3 The rear wall’s reflection against hall depth, drawn as how far past its own echo threshold it arrives. Zero is the crossing. Both curves cross well inside the range of depths halls are actually built at, and the more forgiving signal — an orchestral chord, with its eighty-millisecond base — buys only ten metres.

What saves it, and how it can be taken away

If the geometry does not save the rear wall, the only thing left is the prominence, and the prominence is a statement about what else is arriving.

Sweep the absorption of every surface with the geometry fixed. The rear wall’s own level relative to the direct sound barely moves — it has reflected once either way — and it stays past its threshold throughout. What moves is everything around it: a hall with five per cent absorption has a dense, loud tail at two hundred milliseconds, and a hall with seventy-five per cent absorption has almost nothing there.

The rear wall’s prominence rises from 0.10 at five per cent absorption to 3.9 at seventy-five, and crosses one at about forty-three per cent. Below that it is buried; above it, it stands alone and is an echo. That crossing is a property of the prominence rule as much as of the hall, and a section below says by how much.

That is the finding, and it is the opposite of the way a hall is usually described. Reverberation is what prevents echoes. A room made drier — more absorptive, more “clear”, with a shorter reverberation time and a better early-to-late ratio by every published measure — is a room in which a reflection that was always past the threshold stops being hidden. The improvement and the defect are the same change.

Absorbing a hall is what turns its rear wall into an echo. The rear wall's reflection carries the same energy relative to the direct sound at every absorption drawn, and it is past its echo threshold at all of them. What changes is how much else arrives beside it. Its prominence — its own energy over everything else within 8 milliseconds of it — rises from 0.39 at 5 per cent absorption to 3.97 at 75, and it crosses one at about 31 per cent, which is where the reflection carries more energy than the whole of the rest of its neighbourhood. An echo is prevented by reverberation and not by geometry, which is the opposite of how a hall is usually described.
Fig. 4 The rear wall’s prominence against how absorbent the hall is: its own energy over the energy of everything within eight milliseconds of it. The reflection does not change. Its neighbours do, and somewhere around a third of the energy being absorbed at each bounce it stops having any.

Where the echoes are, when there are any

Turn the absorption up past the crossing and the model does name surfaces, which is the thing the second rung said it could not do.

At seventy per cent of the energy absorbed at every bounce, exactly one arrival stands clear, and it is the once-reflected rear wall. Not the loudest late arrival, not the latest, and not one of the several hundred others past the threshold — the one whose neighbourhood has emptied.

That the answer is a single named surface is the whole point of the exercise. The second rung of this ladder could say that some reflection wins and could not say which; the fifth built a list with names on it and asked a different question of it. Putting the two together produces a sentence of the form a hall consultant would want: at this seat, in this hall, at this dryness, the surface responsible is the rear wall, arrived at from a geometry and a published threshold with nothing measured.

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 70 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, 29 decibels down, against a threshold of 127 — well past it. 1 arrival stands clear enough of its neighbours to be heard as an echo, the worst being 1× the rear wall.
Fig. 5 The same seat in the same geometry with seventy per cent of the energy absorbed at each reflection. The picture has emptied out and one arrival is left standing alone to the right of the threshold — which is what a very dry hall sounds like, and why the first thing done to one is usually to put some of the reflection back.

The seat matters, and not the way the geometry suggests

The obvious guess is that the seats in trouble are the ones nearest the rear wall, because the reflection is loudest there. The delay is what decides it, and the delay is shortest there.

A listener sitting three rows from the platform hears the rear wall at two hundred milliseconds; a listener at the back hears it at ten, because the path from the source to the wall behind them and back is barely longer than the direct path. So the seat with the echo problem is the front of the hall and the platform itself, which is why the complaint about a rear wall is traditionally made by the performers and not by the audience.

The first eighty milliseconds are a different room drew the same division for a different purpose, and the two are the same fact: what arrives early is fused and counts toward clarity, and what arrives late is either reverberation or an echo depending on whether it has company.

Lateral energy at five seats in a 22 by 45 metre hall. The lateral energy fraction — sideways-weighted energy arriving between 5 and 80 milliseconds, over everything arriving in the first 80 — computed from the image sources of a shoebox with 18 per cent absorption. It is highest at mid side at 0.23 and lowest at front centre at 0.12, and the reason the front seat is worst is not that its reflections are weak: it is that the direct sound is 6 metres away rather than 18, and the direct sound is in the denominator.
Fig. 6 The earlier figure, which is the other reading of the same echogram: the lateral energy fraction at five seats. The front seat is worst for spaciousness for a reason that has nothing to do with its reflections — the direct sound is in the denominator — and it is also the seat where the rear wall arrives latest. The two quantities disagree about which seat is best, and both are computed from one list of arrivals.

Which computation produced the numbers

The arrivals are imageSources, the fifth rung’s own construction, to eighth order — every integer triple whose reflection count is eight or under, with each image’s energy computed as spherical spreading times what survives each reflection. There are about three hundred of them inside a quarter of a second in a hall this size.

A reflection’s level relative to the direct sound is ten times the log of the energy ratio, which is a level in decibels and is the argument the threshold takes. The threshold is the base for the signal plus the per-decibel term times the attenuation plus the angle modifier times the squared cosine of the angle from the lateral axis — the same lateral weight the fifth rung uses for its spaciousness measure, reused here for a different question.

The prominence is the arrival’s energy over the summed energy of every other arrival within eight milliseconds of it. Eight is the window and one is the standing-out ratio, and both are stated rather than fitted: at a ratio of one, half of the local energy is in the single arrival. Nothing about the direction of the result depends on either — the prominence curve is monotone in absorption over the whole range, so a different window moves the crossing and not the finding.

Where the model stops

The hall is a rectangular box with six identical surfaces. Real halls have balconies, coffered ceilings, side galleries and a rear wall that is usually deliberately not flat, and the reason it is not flat is exactly this argument — a diffusing rear wall spreads its reflection over time, which is a way of manufacturing the crowd that reverberation would otherwise have supplied. A room does not decay evenly is the essay about the frequency half of the same design problem.

Absorption is frequency-independent here and is not in any room, which is the simplification how long a room rings spends its whole length on. The echo threshold is also frequency-dependent in ways this treatment ignores; a bass-heavy tail hides an echo less well than a broadband one because the echo’s own spectrum is broad.

And the prominence rule is this essay’s invention. The echo threshold is published and the image-source construction is standard; the requirement that an arrival stand out of its neighbourhood is an obvious thing to add and is not a published criterion with a measured parameter behind it. What is defensible about it is that without some such rule the published criterion returns 188 echoes out of 226 arrivals in this hall. What is not defensible is leaving its two constants unswept, and the section below sweeps them.

What the invented rule is worth, swept

The prominence rule has two numbers in it — a window of eight milliseconds and a standing-out ratio of one — and both were stated rather than derived. Sweeping them says how much of this essay’s finding is the rule’s rather than the room’s.

window ratio 0.5 ratio 1 ratio 2
4 ms 22.4% 36.1% 54.4%
6 25.9 38.8 55.4
8 — as stated 31.5 43.3 57.9
12 54.7 71.3 84.1
16 63.4 77.4 87.6

Each cell is the absorption at which the rear wall’s reflection stops being buried. The direction is the same in every cell — more absorption, more prominence, and the prominence curve is monotone throughout — so the finding this essay is about survives the sweep. Its location does not: the crossing runs from a fifth of the energy absorbed to nine tenths, which is the difference between a hall that is comfortably reverberant and one that does not exist.

Two things follow and both are worth stating plainly.

The thirty per cent quoted earlier is the shipped rule’s number and it is nearer forty-three. At the stated window and ratio the rear wall’s prominence passes one between 0.40 and 0.45, not at 0.30 — the figure crosses where the figure crosses, and the round number in the text was a reading rather than a solve.

And at a window of twelve milliseconds or more the rear wall is never an echo at all, because the crossing lands above any absorption a hall is built with. That is not a moved crossing; it is a qualitative disagreement, and it means the essay’s conclusion depends on the window being eight milliseconds rather than twelve. Eight is defensible — it is the same order as the integration this whole ladder is about — and it is not measured, and nothing here can decide between it and twelve.

What is left is narrower than the section above claimed and is still the useful part: the reflection does not change and its neighbours do, the prominence rises monotonically as a hall is dried out, and somewhere in the ordinary range of absorptions a surface that was always past the threshold stops being hidden. Where that happens is a number this collection cannot supply.

What the picture cannot show

It cannot show a listener changing their mind. The precedence effect builds up and breaks down over the course of several seconds — a listener in a room becomes progressively less likely to hear a reflection separately, and then hears it again when the room changes. That is a time constant this model does not have and the second rung named.

Nor can it show what an echo does to music. The threshold is a criterion for hearing two events rather than one; whether that is a fault depends entirely on the music, and a hall that puts a distinct rear-wall return under a slow chord has a quite different problem from one that does it under a fast passage.

And it cannot show the direction of the echo, which is what makes it intolerable. An arrival from behind, at the level and delay this figure calls an echo, is heard as a separate source in a separate place — the whole content of two ears and one difference is that a listener has an angle for every arrival. The census has each arrival’s direction and does nothing with it beyond the modifier.

Whose halls, and when

The design consequence is a nineteenth-century one turned upside down by a twentieth-century habit. The shoebox halls of the 1870s and 1880s were built with hard plaster surfaces and are, by any modern measure, extremely reverberant; they have no echo problems. The mid-twentieth-century halls that acquired echo problems are the ones built wide, fan-shaped and heavily upholstered, on the reasonable-sounding argument that clarity is good and reverberation is a nuisance.

This figure says what that trade actually was. Every one of those halls still had a rear wall past the echo threshold, because every hall does; what they gave up was the crowd that had been hiding it. The remedies that worked — diffusing the rear wall, breaking it into panels, angling it — are all ways of restoring prominence rather than of removing the reflection, which is what the model says they would have to be.

Where this ladder goes next

Six rungs. Two ears and 655 microseconds; a room full of copies and the first wavefront winning; a periodicity in neither ear’s signal; the frequency above which the two ears stop agreeing; a room with directions in it; and now the threshold that decides which of those directions a listener is allowed to have.

What is owed after this is the term the prominence rule stands in for. Fusion is not a yes or a no: a reflection that is not heard as a separate event still moves the apparent source, widens it, and colours it, and the published apparatus for that — image shift, apparent source width, colouration — takes the same three inputs this figure already has. The census here sorts arrivals into two boxes, and what it should produce is a position and a width for the sound the listener is hearing, computed from a list this ladder has had since the fifth rung.

Part 6 of 12

One essay in the series on localisation. The essays either side of this one:

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 Part 5 — 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. Where the sound is, and how wide. An earlier essay sorted a hall's arrivals into echoes and everything else. Everything else is not nothing: a reflection too early to be heard as a separate event still moves the apparent source, widens it and colours it, and all three come out of the same list of times, levels and angles. At this seat the direct sound arrives from 26.6 degrees off the front and the image is heard 8.5 degrees left of it, pulled by the near side wall. The apparent width is 33 degrees, from a lateral energy fraction of 0.23. And the strongest early reflection arrives 0.6 milliseconds behind off 1× the floor, which is a comb filter with notches every 1608 hertz and 25 decibels deep. The trading ratio and the discount on late arrivals are stipulated rather than measured, so the degrees are ordinal: what the figure claims is the direction and the shape, not the number. A position and a width Part 7 — Sorting a hall's arrivals into echoes and everything else settled that fusion is not a yes or a no. Everything else is not nothing: a reflection too early to be heard separately still moves the apparent source, widens it and colours it. All three come out of the same list of times, levels and angles, and none of them needed a new input.

Where this goes next

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. How late a reflection has to be before it is an echo. What a single reflection does to the sound it follows, against its delay, on a logarithmic axis. Under a millisecond the two combine into one image that is pulled towards the earlier source. From there out to a few tens of milliseconds the reflection is not heard as a separate event at all and does not move the image — it only changes the timbre. Past the echo threshold it becomes a second sound, and the threshold is five times later for speech than for a click. The first wavefront wins A room sends a hundred copies of every note to a listener from a hundred directions, and the listener hears one note in one place. The mechanism that does it is brutal and simple: for the first few tens of milliseconds after a sound arrives, everything that follows is denied a vote on where it came from — even when it is louder than the original. 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.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The objects named here

The third way in, after the field and the series: the things themselves, and every essay that touches each one.

AbsorptionHall designImage-sourceLocalisationPrecedence effectReflectionReverberationRoom acoustics