Where a room stops being a room

This ladder has described a room twice and the two descriptions have nothing in common.

The first says a room is a set of resonances: frequencies it supports and frequencies it does not, so a bass note is loud in one corner and absent in another, and no equipment fixes it. The second says a room is a decay time: volume over absorption, one number, from which the whole of concert-hall design follows.

Both are right. They are right about the same building at different frequencies, and the frequency at which one hands over to the other is computable from two numbers a room already has.

Counting the modes #

A rectangular room of 4.2 by 3.4 by 2.5 metres has its first mode at 40.8 hertz, and then 50.4, 64.9, 68.6, 79.8, 81.7. Nine modes below 100 hertz, forty-six below 200, and three hundred and twelve below 400.

That acceleration is not an accident of these dimensions. The number of modes below a frequency grows as the cube of the frequency, because the modes are lattice points in a three-dimensional space and the count is a volume. So modal density — how many modes there are per hertz — grows as the square of the frequency and in proportion to the volume of the room.

At 100 hertz this room has about one mode every nine hertz. At 500 hertz it has nearly three per hertz.

Why counting them stops being the point #

Each mode is not a spike. It has a width, and the width comes from the decay: a resonance that dies away in time T has a bandwidth of roughly 2.2 divided by T hertz. In a room with a half-second decay that is 4.4 hertz.

Now compare. At 100 hertz the modes are nine hertz apart and 4.4 wide, so they stand separately. At 250 hertz they are 1.4 hertz apart and still 4.4 wide, so each one overlaps three of its neighbours and there is no longer any such thing as “the mode at 250 hertz” — what exists is a dense thicket in which the response at any frequency is a sum of several overlapping resonances with random phases.

The spacing follows from the density and the density has a closed form: the number of modes per hertz is 4π times the volume times the frequency squared, over the cube of the speed of sound. For this room that is 0.111 modes per hertz at 100 and 0.695 at 250, which are spacings of 9.0 and 1.4 — the two numbers above, and neither of them measured. The whole of Schroeder’s criterion is that expression set equal to three over the bandwidth, and solving it for frequency is where the 2,000 comes from.

Which is why the crossover depends on the volume and not on the shape. Two rooms of the same volume and the same decay time have completely different mode frequencies and identical mode densities, so they hand over from one description to the other at the same place. A long thin room and a cubic one of equal volume are different instruments below the crossover and the same statistics above it.

Schroeder’s criterion is that the transition happens when about three modes overlap within one bandwidth, and working that through gives a single formula: the crossover is 2,000 times the square root of the decay time over the volume, with the decay time in seconds and the volume in cubic metres.

For the room above, with a half-second decay, it is 237 hertz — and at that frequency the count says there are seventy-seven modes below it.

The two numbers in that criterion are worth keeping apart, because they are easy to run together. Counting the modes in a forty-hertz band around the crossover gives thirty-one of them, so the mean spacing is 1.3 hertz, not three. What is three is the overlap: a 4.4-hertz bandwidth divided by a 1.3-hertz spacing is 3.4 modes per bandwidth, which is Schroeder’s criterion satisfied to a tenth. The spacing is the smaller number and the overlap is the criterion, and the formula returns the frequency at which the second reaches three rather than the first.

What the formula says about rooms people actually use #

The two variables pull in opposite directions and volume wins, because it is under a square root but ranges over four orders of magnitude while decay times range over one.

A carpeted bedroom at 34 cubic metres and a 0.4-second decay: 217 hertz. That is between A3 and A♭3 — the middle of a cello’s range, the bottom of a voice’s, and everything a bass guitar plays.

A rehearsal room at 250 cubic metres and 0.9 seconds: 120 hertz, which is around B2.

A jazz club at 700 cubic metres and a 0.8-second decay: 68 hertz, which is the bottom C of a cello and below almost everything a club actually amplifies.

A concert hall at 18,700 cubic metres and a two-second decay: 21 hertz, which is below the lowest note of an orchestra.

And a cathedral at 25,000 cubic metres and eight seconds: 36 hertz — higher than the hall’s, because the enormous decay time narrows every mode and pushes the transition up, and still below almost everything. That inversion is worth pausing on: the largest room on the list does not have the lowest crossover, because the crossover is not a measure of size.

So the halls that music is written for are statistical everywhere it matters, and the rooms most music is now made in are modal across their bass registers. Those are two different acoustic objects and the difference is not a matter of quality.

The gap between the two ends of that list is worth reading as a ratio rather than as five numbers. From a bedroom to a concert hall the volume rises by a factor of 550 and the decay time by five, so the crossover falls by the square root of 110 — a factor of ten and a half, which is three and a half octaves. Three and a half octaves is the entire difference between a room whose modal region covers a bass guitar and one whose modal region is below the audible range, and it is bought almost entirely with volume, since the decay time enters under the same square root and moves the wrong way.

What a listener actually notices #

The consequence in a small room is that the bass is a function of position and of note, and both dependencies are strong.

Two adjacent semitones can differ by fifteen decibels at a given seat, because one falls on a mode and the other between two. Move two metres and the two swap. A bass line played in such a room is therefore not a bass line with a level; it is a sequence of notes whose levels are set by the room, and no amount of equalisation fixes it because the fault is a function of position as well as frequency.

Above the crossover none of that happens. The response is still not flat — it is a random-looking thicket with peaks and dips of several decibels — but the peaks are dense, they are different at every position, and the ear integrates over them. That is why the same room can be dreadful in its bass and unremarkable above it.

What each régime does to a note #

The two descriptions imply two different fates for a sustained note, and it is worth stating both because they sound nothing alike.

Below the crossover, the room selects. The note excites whichever modes lie near its partials, and those modes ring on at their own frequencies rather than at the note’s. A bass note whose fundamental falls between two modes gets almost no support at the fundamental and a good deal at whichever of its upper partials happens to land on one — so the room changes the spectrum, which is the same thing an instrument body does and by exactly the same mechanism.

Above the crossover, the room copies. Every partial finds dozens of overlapping modes, none of them favoured, so what returns is a delayed and decaying version of the whole sound with its spectrum roughly intact. That is what “reverberation” describes and it is why it can be modelled as a filter that does not depend on what is played.

So the room is a resonator down low and a repeater up high, and a listener who says the bass of a room is bad and its top is fine is reporting the crossover.

The rule that follows for anybody recording anything #

The crossover explains a piece of studio practice that is usually given as a list of tips.

A small room’s bass is its own and belongs to the room rather than to the instrument, so a recording made in one carries the room’s modal pattern in its bottom two octaves whatever microphone is used. This is why bass is so often recorded without a microphone at all — taken directly from the instrument, bypassing the room entirely — and why bass traps, which are absorbers wide enough to work at long wavelengths, are the one acoustic treatment that changes a small room’s character rather than merely its liveliness.

And above the crossover the room can be treated statistically, which is what every reverberation algorithm assumes. A digital reverberator models a dense diffuse field; it is a good model of a hall and a poor model of a small room’s bass, and its failures are exactly where the formula says they should be.

Whose music, and when #

The formula is Manfred Schroeder’s, published in 1954 and 1962, and the criterion of three overlapping modes per bandwidth is a convention of that literature rather than a measured threshold. Different sources use a factor of 2,000 or 4,000 depending on whether the criterion is three modes or ten. That is a factor of two in frequency, which is a whole octave rather than the third of one previously recorded here — so the crossover in the bedroom is somewhere between 217 and 434 hertz depending on which convention is used, and the figure’s marks should be read as the bottom of a band an octave deep rather than as lines.

The consequence for music is a fact about the last seventy years rather than about the repertoire. Music written before recording was performed in rooms large enough to be statistical everywhere; music made since about 1960 is very often made in rooms that are modal across their bass. That is a genuine change in the medium and it is the reason a whole industry of small-room acoustic treatment exists, none of which is needed in a hall.

The instruments the crossover cuts through are the low ones, and they are also the ones whose own behaviour is least forgiving. A double bass string’s partials are already stretched by stiffness and its fundamental radiates poorly, so its bottom octave arrives at a listener as a set of upper partials — which the modal region of a small room then treats one at a time, individually, according to whether each lands on a resonance.

And it interacts with what the room was chosen for. A cathedral’s eight-second decay puts a ceiling on how fast harmony can change, which is a constraint from the statistical régime; a bedroom’s modal bass puts no such ceiling on anything and instead makes individual notes unreliable. Two rooms, two completely different constraints, from the same physics read at different frequencies.

The same shape, three fields apart #

A boundary between a régime in which individual objects can be counted and one in which only statistics survive is not peculiar to rooms, and this site has met it twice before under other names.

A tone-hole lattice reflects the wave below a computable cutoff and lets it straight past above, so a woodwind’s low notes are shaped by which holes are open and its high ones all radiate from the same place. A series of events is a rhythm below about twenty a second and a pitch above it, and the same events change category without changing.

In each case a single frequency separates two descriptions that share no vocabulary, in each case the number is computable from the object’s dimensions, and in each case an argument made on one side of it is simply not about the other side. That is worth carrying as a habit rather than as three facts: when two accounts of the same object disagree completely, the first thing to look for is the frequency at which they hand over.

The room is the clearest instance because both accounts were already on this site, each with its own essay, and neither said where it stopped.

Where the model stops #

The mode count assumes a rectangular box with hard walls. Real rooms have furniture, doors, non-parallel surfaces and absorption that varies over each wall, all of which shift individual modes and none of which changes the density, because the density comes from the volume. So the crossover is robust and the individual mode frequencies are not.

That robustness has a limit worth stating, because the density formula is asymptotic. Counting the modes of this room directly gives nine below 100 hertz where the volume term alone predicts three; the difference is made up by surface and edge terms that matter enormously at low frequency and hardly at all high up. By 400 hertz the direct count is 312 against a volume term of 237, so the correction is still a quarter of the total. The formula is exact about the density and approximate about the count, and the crossover depends on the first, which is why it survives being applied to a room whose lowest modes the same arithmetic gets wrong by a factor of three.

And nothing here is measured in a room. Every number is a rectangular box with hard walls and a single decay time. What a real small room does below its crossover is measured with a microphone at several positions, and the measurement is the only way to find out which of its modes are actually excited by a source in a particular corner — a question the mode list cannot answer, because a mode with a node where the source is does not sound at all.

Sabine’s decay time is itself an approximation, and the site has already recorded where it fails: it predicts that a room whose walls absorb everything still rings, and it is least accurate in exactly the small, heavily damped rooms this essay is about. The crossover inherits that error under a square root, which halves it.

And the transition is not a line. Nothing happens at 217 hertz. Modal behaviour fades out over about an octave, and describing the boundary as a frequency is a convenience — a useful one, because the two régimes are genuinely different a factor of two either side of it.

What the picture cannot show #

It cannot show the position dependence. Everything below the crossover depends on where the listener and the source are, and the figure has one axis and it is frequency. The same room measured at two seats produces two different pictures below the mark and nearly the same picture above it.

It cannot show absorption’s frequency dependence. The formula takes one decay time, and a real room has a different one in every octave band — which is the last rung of this ladder and means the crossover itself is computed from a number that is not single.

It cannot show the first mode. The lowest mode of a room is set by its longest dimension — 40.8 hertz for a 4.2-metre length — and below that frequency a room supports nothing at all and the pressure simply rises and falls together everywhere. That third régime is below the crossover and below the modal region, and it is where the very bottom of an organ’s pedal range lives.

Where the ladder goes next #

This rung collapsed two earlier ones into a frequency. The next collapses two others into a distance — the reverberation time from the second rung and the directivity from the third — and produces the radius beyond which a listener stops hearing the instrument and starts hearing the building.