The room is the slower of the two

Every figure on the account here has been drawn in a room with no walls. A partial is given a level, it falls at its own rate, and the moment it goes under the threshold of hearing it stops existing. That is a complete account of a note in an anechoic chamber and nowhere else, and the difference is not a refinement — it decides whether the thing the account here has spent three essays building is available to a listener at all.

The essay that found the colour draining found that a struck note’s upper partials leave faster than its lower ones, so the sound moves toward its own fundamental while it dies, and made an identity cue out of the rate at which that happens. The fourth found the whole of the rate in the middle of the note. Both are statements about how fast the colour goes, and a reverberant room is a second thing that decides how fast a sound goes.

Two decays in series do not add #

The natural guess is that a room is a second loss and that losses add. It is the wrong guess, and getting it wrong inverts the finding.

A partial leaving a string is a source. What reaches a listener in a room is that source convolved with the room’s impulse response, and the convolution of two decaying exponentials decays at the slower of the two rates, not at their sum. The reason is easy to state without the algebra: the room is still returning energy that the string emitted a second ago. Once the string has fallen silent, the room goes on delivering what it has stored, and the tail a listener hears is the room emptying rather than the string ringing.

So for each partial there are two candidate rates and a listener gets the smaller:

$$\gamma_n^{\text{heard}} = \min\!\left(\gamma_1 n^{\,p}, \; \frac{\ln 10^3}{T_{60}(n f_0)}\right)$$

A string’s rate rises steeply with partial number — that is what the exponent $p$ means and it is the whole of the account’s third essay. A room’s rate rises gently with frequency, because absorption rises gently with frequency. Two curves with very different slopes cross once, and above the crossing the partials the string was losing fastest are the ones the room holds up.

In a room with no tilt the crossing is a partial number #

Take the room’s decay time as one number for every frequency, which is how a reverberation time is usually quoted. Then the crossing condition is

$$\gamma_1 n^{\,p} = \frac{\ln 10^3}{T_{60}} \qquad\Longrightarrow\qquad n = \left(\frac{T_{60}^{\text{string}}}{T_{60}^{\text{room}}}\right)^{1/p}$$

and there is no $f_0$ in it. A flat room takes over at the same partial number on every note of the instrument. A note on C2 and a note on C6 hand the same fraction of their spectrum to the room, even though the frequencies involved differ by sixteen. At the numbers used here — a string whose fundamental rings for six seconds, a loss rising in proportion to partial number — the crossing is at partial four in a two-second room and at partial two in a four-second one.

That is a genuinely odd result and it is worth being explicit that it is a property of the idealisation. It says a room does the same damage to the top of every note’s spectrum regardless of register, which is not what any room does, because no room has one reverberation time.

A real room’s decay time falls with frequency — 3.07 seconds at 125 hertz and 1.53 at four kilohertz for the hall above, a factor of two across the audible span. A room does not decay evenly is the essay that established it, and here it has a consequence that essay had no occasion to compute: it makes the crossing a frequency rather than a partial number, because the room’s rate is now climbing while the string’s climbs faster.

In that hall the crossing sits at partial two on C2, partial three on C3 and C4, and partial four on C5 and C6 — which is to say at roughly 130, 390, 785, 2,090 and 4,190 hertz. The room hands back more of a bass note’s spectrum than of a treble note’s, because the room is slowest where the bass note’s partials are.

What a room changes, and the one thing it cannot #

Held against the drain itself, the effect is large and it is entirely in the rate.

Half of a C3’s colour drain is done after 0.163 seconds with no room at all. In a two-second room it takes 0.270, in the hall above 0.313, and in a stone church whose bass band rings for 6.3 seconds it takes 1.441 — nine times as long. Read at the instant the fourth essay identified as the most informative moment of the note, 0.384 seconds in, a dry note has given up 7.61 semitones of centroid, the same note in the hall has given up 5.79, and in the church 1.37.

What no room changes is where the note ends up, and the reason is one sentence: a room cannot hold a partial up above the fundamental it is also holding. Whatever the reverberation time, the fundamental is the loudest partial and the slowest-decaying one, so the last thing audible is the fundamental alone, and the terminal centroid is the fundamental’s own frequency in every room. Every curve in the figure above climbs to 9.9 semitones.

That is a sharper statement than it looks. It means the total colour drain is a room-independent quantity, and every figure the account here has drawn of that total — the register sweep, the compass dependence, the factor of 3.8 between the bottom and top of a keyboard — survives being moved into a hall untouched. It is only the rate that moves, and the rate is what the cue was.

The church is the limiting case and it is the one worth carrying. When the room’s decay time exceeds the string’s own at every frequency, every partial is held by the room, every partial therefore decays at nearly the same rate, and the spectrum keeps its shape while the level goes. A note in a cathedral does not get duller as it dies. The cathedral does the dying, and the cathedral has no colour drain of its own beyond its own gentle tilt.

What that does to the cue #

The fourth essay’s measurement was the difference between two notes whose loss exponents differ — a half against two — and it came to 41.3 decibels at 0.384 seconds. Run in rooms, the same measurement falls off a cliff.

Dry, the two notes reach half their drain after 0.49 and 0.04 seconds respectively — a factor of twelve, which is an enormous perceptual difference and is why the cue was trusted. In a four-second room the same two figures are 0.99 and 0.97, a factor of 1.02. Two instruments that are a factor of twelve apart in an anechoic chamber are a factor of a fiftieth of that apart in a large church, and the reason is that the room is setting the rate for precisely the partials whose rates carried the difference.

The fall is not gradual. A recording studio at 0.35 seconds leaves the cue entirely intact, because its rate is faster than every partial’s and the minimum never selects it. A jazz club takes 57 per cent of it. From there the loss is steep, and by the time a room is large enough for an orchestra there is almost nothing left.

The room where the cue was measured #

The fall is not gradual in the other direction either, and the dry end of it is where the evidence for it comes from.

A studio at a third of a second leaves the whole of the cue intact, because its rate is faster than the fastest partial’s and the minimum is always the string’s. That is not a coincidence of the numbers; it is what a studio is built to do, and it is the condition under which every measurement of a decay exponent has been made. It is also why the room is part of the instrument is a statement about the performance and not about the workshop.

the third essay argued the drain rate is an identity cue by comparing a piano with a harpsichord. Any such comparison made from recordings is a comparison of two instruments in two rooms, and the numbers above say the room’s contribution to the measured rate is not small: the same instrument in a club and in a hall differs in half-drain time by more than two instruments differ in a studio. So a measured exponent is a property of a recording, and separating the instrument’s exponent from the room’s requires either an anechoic measurement or a room whose decay is known and divided out.

That cuts both ways and the second way is the more useful one. Because the heard rate is a minimum rather than a sum, a measurement made in any sufficiently dry room returns the string’s own rate exactly rather than approximately. There is no correction to apply and no room term to estimate: below the crossing, the room contributes nothing at all. The condition for “sufficiently dry” is stated by the crossing formula and it is easy to meet — a room whose reverberation time is under the string’s own sixty-decibel time divided by the number of partials to the power $p$, which for a six-second string and eight partials at $p = 1$ is three quarters of a second.

So the instrument’s exponent is measurable in a modest room and unmeasurable in a good one, and the rooms music is performed in are the second kind. A listener in a concert hall is not receiving a degraded version of the cue; they are receiving the room’s rate instead of the instrument’s, and the substitution is exact rather than noisy.

The register the room takes it from #

The two halves of this essay point in opposite directions across the compass, and where they meet is the finding.

The essay that put a pitch into the decay model established that the amount of colour available to drain falls monotonically with pitch: 25.2 semitones at the bottom of a keyboard and 6.6 at the top, because a partial has to fit under twenty kilohertz to exist and the top note has four of them. The cue is a bass phenomenon.

A real room’s absorption rises with frequency, so its decay time falls with frequency, so the room is slowest — and therefore takes over most — exactly where the bass note’s partials are. In the hall above, a C2 has half its drain delayed to 0.390 seconds against a dry 0.161, and a C6 to 0.217 against a dry 0.161. The bass note’s rate is pushed out by a factor of 2.4 and the treble note’s by 1.35.

So the register with the most colour to lose is the register a hall stops from losing it in any particular way, and the register with almost none is the one a hall leaves alone. On the numbers here the two effects come close to cancelling: in a hall, what survives of the cue is roughly flat across the compass, where dry it was strongly weighted to the bass. That is a prediction about halls rather than about instruments, and it is the kind of statement nothing here could make before.

The room takes one cue and leaves the other #

The essay that drove the note instead of striking it found that a continuously driven note carries its loss law as a slope rather than as a rate, because each partial settles at its drive divided by its own loss. A rate is exactly what a room overwrites. A slope is not.

For a steady source the reverberant energy density in a room is proportional to that room’s decay time at the frequency concerned — a long decay stores more of what is being supplied. So a held note’s spectrum in a room is its own steady spectrum multiplied, band by band, by $T_{60}(f)$. In the hall above that multiplier falls from 3.07 to 1.53 across five octaves, which is three decibels of extra tilt, or about 0.6 decibels per octave of darkening added to whatever the instrument supplies.

That multiplier is the same for both instruments. It darkens a note with a loss exponent of a half and a note with an exponent of two by identical amounts at identical frequencies, so the difference between them — the 18.7 decibels the previous essay measured — passes through the room unchanged. What the room adds is common to both, and what distinguishes them is not touched.

So the two cues the account here has found behave in opposite ways in the one place music is actually heard. The struck note’s cue is a rate, the room replaces the rate, and a large hall takes essentially all of it. The held note’s cue is a ratio between partials of one steady spectrum, the room applies a common factor to that spectrum, and the ratio survives. A hall does not merely reduce the information available about which instrument is playing; it reallocates it, from the struck instruments to the sustaining ones.

That is a claim about rooms and it is worth naming what would falsify it. The multiplier is common-mode only if the two sources are at the same place in the same room radiating in the same pattern. Two instruments with different directivities excite the reverberant field differently, and a source with directions in it is not equivalent to a point one, so the factor is common to two spectra and not necessarily to two instruments.

Which computation produced the numbers #

The string is this collection’s eight-partial model at a stated pitch, with partial $n$ decaying at $\gamma_1 n^{\,p}$ and $\gamma_1$ set by a six-second sixty-decibel time on the fundamental. Rates are in nepers per second throughout, so a sixty-decibel time of $T$ is a rate of $\ln(10^3)/T$.

The room’s decay time at a partial’s frequency comes from Sabine’s equation applied band by band to a stated surface list, with the standard air-absorption term above one kilohertz, and interpolated in log frequency between the octave-band centres. The named rooms are the two used throughout: a hall of 18,700 cubic metres with an audience in it, and a stone church of 7,000.

The heard rate is the minimum of the two, which is the convolution result stated as a rate. The colour gone by time $t$ is the fall of the power-weighted centroid of the audible partials, in semitones, from its value at the strike — the same measure the third and fifth essay use, so that the numbers here and there are the same quantity.

The single-number rooms are drawn flat deliberately, because the flat case is what makes the crossing-is-a-partial-number result visible, and the banded rooms are drawn beside them because that result is exactly what a real room’s tilt destroys.

Where the model stops #

The minimum is an asymptotic answer. The convolution of two exponentials is a difference of exponentials, and near the strike both terms matter — the sound builds toward its reverberant level over the first part of the note rather than starting there. Taking the minimum describes the tail correctly and the first tenth of a second approximately, and the first tenth of a second is where the room behaves as a set of discrete arrivals rather than as a decaying field in any case.

The listener has no position here. How much of what arrives is reverberant depends on where the listener is: close to the source the direct sound dominates and the string’s own rates survive, and past the critical distance the reverberant field is most of what arrives. Every number above is the fully reverberant case, which is where most of an audience sits and is not where a player sits.

And the levels are not in it. The reverberant field raises the level of everything by a factor that depends on the room and the distance, which lengthens the audible life of every partial and is the main reason a note sounds longer in a hall. Holding the level fixed isolates the rate, which is what the essay is about, and it means the lives quoted here understate what a hall does.

What the picture cannot show #

It cannot show the direct sound and the tail as separate objects. A listener does not receive one decaying spectrum; they receive the string’s own sound on time and the room’s copy of it spread out behind. Whether the auditory system separates those — and there is good reason to think it does, since it separates a source from its reflections for the purposes of localisation — would change the answer entirely, because the direct sound carries the string’s own rates undisturbed.

Nor a room with a floor to its decay. Sabine’s model has one exponential per band. Real decays have a knee: an early part governed by the first strong reflections and a late part by the diffuse field, and the two are different rooms. Which of them the colour drain is competing against depends on when the drain happens, and most of it happens early.

It cannot show what a player does about it. A performer in a live room does not play as they would in a dry one; tempo, articulation and dynamic all change, and the essay that found a rest needs a dry room is one measurement of that adaptation. If the adaptation includes holding notes differently, the drain a listener receives is not the drain computed here.

And it cannot show two notes at once. A struck chord in a hall is three or four collapsing spectra plus a reverberant field containing all of them, and the field does not know which note each of its partials came from. Everything on this page is one note in an empty room.

Still open: whether the direct sound rescues the cue #

The whole of the loss above depends on the listener receiving the reverberant field rather than the string. Every figure here is drawn at a distance where that is true, and it is true for most of an audience — but the direct sound is always there, arriving first and carrying the string’s own rates exactly.

What would settle it is the same computation with the two summed rather than the reverberant one taken alone: the direct sound at its own level, falling at $\gamma_1 n^{\,p}$, plus the room’s copy at the level the critical-distance ratio gives it, falling at the minimum rate. The cue would then be a function of where the listener sits, and the quantity to read off it is the distance at which the separation between two loss laws falls below whatever a listener can use. That distance would say something the essays on rooms have not yet said: not how reverberant a seat is, but how far from a stage an instrument stops sounding like itself.