A rest needs a dry room

A rest is a diminuendo found the parameter three earlier sweeps had missed. A listener’s running impression of loudness falls at a two-second release, so a gap in the sound is a diminuendo nobody wrote: a bar of silence is worth eight decibels, and at the three and a half seconds where a gap stops being a pause and becomes an ending it is worth twenty-four — more than the whole range from a fortissimo to a pianissimo.

Its last paragraph names what every one of those numbers assumes:

Every number on this page assumes the sound stops when the players do, and it does not: a hall’s reverberation is the thing that fills a general pause.

That is not a caveat about realism. It is a claim that the input to the whole calculation is wrong, and the collection has the correction: a room ladder that computes exactly how a level decays after a source stops.

The only thing that changes is the input #

The correction is one line and it is worth being clear how small it is, because the size of the result is out of proportion to it.

The ninth rung builds a level track — a steady passage, a stated number of seconds at the hall’s background, then the passage again — and runs it through the loudness ladder’s pair of smoothers. During the gap the level is the background and nothing else.

Here the gap is not the background. It starts at whatever the passage was sounding at and falls in a straight line in decibels, at sixty decibels per reverberation time, until it reaches the background and stays there. That is the definition of a reverberation time and nothing more has been added. Everything downstream — the loudness curve, the two smoothers, the reading at the instant the sound returns, the conversion into an equivalent diminuendo — is unchanged, which is why setting the reverberation to nothing reproduces the ninth rung’s table to every figure it printed.

The result is decided by two durations, and they are two orders of magnitude apart.

The smoother’s long release is two seconds. That is a property of a listener and it does not change with the room.

The hall’s tail has a length of its own, and it is how long the reverberation takes to fall from the passage’s level to the background: about a third of a second in a recording studio, one and three quarters in a shoebox hall, three and a half in a stone church and seven in a cathedral.

A rest shorter than the tail never reaches the smoother at all, because the hall is still filling it. A rest much longer than the tail costs the room almost nothing, because the reverberation has run out and the remaining silence is real silence. The room is not a general reduction. It is a threshold on the length of a silence, and the threshold is the tail.

Which costs the device most where composers actually write it #

That threshold falls in an unfortunate place.

A general pause in the repertoire is a bar, or two — a second or so at most tempi, and rarely more than three. A hall’s tail is between one and four seconds. So the device sits squarely inside the range where the room decides the answer, and it is the short pause, the ordinary one, that is destroyed.

The half-value reverberation time is the number to carry. A rest of a bar — 1.2 seconds — is worth half its dry value at 2.76 seconds of reverberation. A rest of three and a half is worth half its dry value at 7.01.

2.76 seconds is a specific place. It is above every concert hall built for orchestral music, which run from about 1.4 in an opera house to 2.2 in the largest shoebox halls, and it is below every large stone church, which run from 4 to 8. The threshold falls in the gap between the two kinds of room, and it is not near either edge of it.

So the answer to the debt is yes, with a number: the general pause is a device that needs a dry room, and “dry” means under about two and three quarter seconds for a pause of a bar. That is a claim about a building, and it is falsifiable by anybody with a stopwatch in the back of a hall.

The asymmetry between the two lengths is worth naming on its own, because it is not a detail of the curves but the whole of what a room does. In a concert hall a bar of silence loses 40 per cent of its value and a three-and-a-half-second one loses 14. The room takes most from the ordinary pause and least from the definitive one — from the rhetorical gap in the middle of a phrase rather than from the long stop before the last chord — so a wet hall does not weaken endings so much as flatten the gradations between them. A composer who has an ordinal scale of about eight dynamic markings and a continuous control made of rests finds the continuous one losing its bottom end first.

The other end of the same curve is the opening of a piece. The ninth rung noticed that a first entry lands against a reference sitting at the hall’s noise floor, so the first chord of a piece is louder than the same chord anywhere in it, and it costs the players nothing. That argument is untouched here and it is the only one that is: an audience settling for thirty seconds is far past any tail, so the running impression has come all the way down in every room. A cathedral takes almost everything from a bar of silence and nothing at all from an opening.

What that says about the repertoire, and how far it can be pushed #

The written general pause — everybody stops for a bar, then the final gesture — is a classical and romantic orchestral commonplace. It is all over Haydn, it is a standard rhetorical move by the middle of the nineteenth century, and it is rare in the music written before, which is thin-textured music written for stone rooms.

The correlation is real and the mechanism is computed, and it is still not evidence. A great many things changed between a stone church and a shoebox hall — the size of the orchestra, the dynamic range available, the whole idea of a concert — and each of them is a candidate explanation for a device appearing.

What the arithmetic adds is a prediction that is not about history at all, and it is the honest version of the claim. The same piece, in two rooms, should have a general pause worth three times as much in one of them. A conductor moving a programme from a hall to a cathedral would have to lengthen the pauses to keep the effect, and by a computable amount: to recover the 4.9 decibels a bar gives in a shoebox hall, a cathedral needs a little over two seconds rather than one and a bit. That is a rehearsal decision, it is made constantly by people who have never seen this curve, and if they are not lengthening pauses in wet rooms then something here is wrong.

The other silence, which the room removes entirely #

There is a second device made of gaps in the sound, and putting it through the same arithmetic separates the two cleanly.

A staccato is a dynamic mark established that the sounding fraction of each beat is worth real loudness at a fixed written dynamic: 1.5 phons at a slow tempo and 8.1 phons on semiquavers at 230 a minute, because at speed the notes stop reaching their own short-term loudness and the two effects add. Its finding was that articulation is a dynamic control the notation records exactly and nobody counts.

Those gaps are the same object as a rest, two orders of magnitude smaller. A staccato semiquaver at 230 leaves 49 milliseconds of silence; a bar of general pause leaves 1,200.

A concert hall erases the articulation entirely. The tail falls only 1.5 decibels inside a 49-millisecond gap, so the gap is filled before the smoother can see it, and the eight phons collapse to a quarter of one.

And the ordering inverts. Dry, a staccato is worth most at speed. Wet, it is worth least at speed, because a faster passage has shorter gaps and short gaps are exactly the ones a tail fills. That reversal is the clearest signature the mechanism has, and it is a thing an instrumentalist would recognise: short articulation is a device for dry rooms and small ensembles, and in a cathedral the difference between a staccato and a legato line is a difference in attack rather than in level.

So the two devices, made of the same material, are separated by the room. A rest is robust and an articulation is not, and the dividing length is the hall’s tail. That also sorts the closure ladder’s own five components by how much of each a room leaves alone: a cadence is harmony and the room does not touch it, a deceleration was worth nothing to begin with, and of the two that are made of level the larger one survives and the smaller one does not.

Which finishes the eighth rung’s account of its own nulls #

There is a tidy consequence for the rung that started all of this.

The reading was a step response swept the tempo over a factor of ten and the deceleration over a factor of three and got the same reading to five significant figures from both; the closing gesture, swept over a factor of forty-eight, moved it by seven per cent and did so by bar count rather than by seconds. The ninth rung sorted them: a manipulation that changes when things happen inside a continuous stream returns nothing, because the smoother’s output at the end of a stream is set by the level of the stream and not by its schedule, while a manipulation that changes what the final chord is measured against returns something. A silence is the largest of the second kind.

The room belongs to that account rather than sitting beside it. It is a fourth manipulation of the same kind as the rest — it changes the input rather than the schedule — and it changes it in the opposite direction. A rest lowers the bottom of the step; reverberation puts the bottom back up.

That makes the whole ladder’s arithmetic a single statement with two terms. The arrival reading is the size of a step, the composer sets its bottom with the length of a silence, and the room sets a floor under how far down that silence can reach. Neither of the other two parameters anybody swept was ever in the expression.

Which computation produced the numbers #

The level track is a steady six seconds at 78 decibels, a stated number of seconds of reverberant decay, and three seconds at 78 again, sampled every twenty milliseconds. The decay starts at 78 and falls at sixty decibels per reverberation time until it reaches the hall’s background of 25, where it stays.

The running impression is the loudness ladder’s own pair of one-pole smoothers in series, with the long release at two seconds, applied to the instantaneous sone value of the track. The reading is the long smoother’s output at the instant the sound returns, and the equivalent diminuendo is how far that has fallen from its value with no gap, at ten decibels per doubling of sones.

The articulation figure is the same smoother on a different input: a passage at 80 decibels sounding a stated fraction of each inter-onset interval, with each gap filled by the same reverberant decay, and the quantity is the difference in reported loudness between that and the same passage sounding continuously.

Every room is one number, its reverberation time, taken from the room ladder’s own table of rooms.

Where the model stops #

A tail is not a straight line in decibels. A real decay is straight only after the early reflections have passed, and the first hundred milliseconds of a hall’s response is a set of discrete arrivals rather than a slope — which is the part of the response that matters most for the shortest gaps, and therefore for the articulation figure.

The tail starts at the wrong level. Here it starts at the level the passage was sounding, which is true for a listener beyond the critical distance and increasingly false as the seat gets closer to the players. Near the front the direct sound dominates, the reverberant field is well below it, and a rest is worth more than this says.

Reverberation time is one number for a room and it is not. A hall is longer at low frequencies than at high ones, and the loudness smoother is fed a single broadband level here.

The background is asserted. Twenty-five decibels is a quiet hall, and the tail’s own length depends on how far it has to fall. A noisier hall shortens every tail and makes a rest worth less still.

And the orchestra does not stop instantly. A general pause is a written silence in every part, and what actually happens is a release: strings lift bows, winds stop the air, and the last chord’s own decay is added to the room’s.

What the picture cannot show #

It cannot show the audience. The loudest thing in many silences is a hall full of people, and coughing is not a decaying exponential.

Nor can it show early reflections. A shoebox hall’s first lateral reflection arrives in twenty milliseconds and is nearly as loud as the direct sound, which is a bump in the tail this model draws as a slope.

It cannot show the players’ response to the room. Musicians shorten their notes in wet halls, which is the same correction by another route and would change the input to every figure here.

Nor can it show the score. A general pause of an odd number of bars feels different from an even one, and a listener counting through a silence is doing something no smoother models.

It cannot show a fermata. The length of a written pause in performance is a conductor’s decision, so every value above is an option rather than a measurement.

And it cannot show the harmony. The room already decides how fast the chords may change, and a hall that erases a rest is the same hall that leaves the previous chord sounding under the next one.

Whose music, and when #

The reverberation times are the room ladder’s, and they are drawn from published measurements of real buildings of each kind. The passage is generic and the levels are generic.

The historical range is worth stating plainly because it is the interesting part. Music written before about 1700 was largely written for rooms above three seconds, and a written silence in it is worth under half of what the same silence is worth in a hall — which is one reason the earlier repertoire ends its phrases with harmony and cadence rather than with air. The concert hall of the nineteenth century is a drier room and a louder one, and both changes make a rest worth more: a bigger orchestra raises the level the reference falls from, and a shorter tail lets it fall.

That is a correlation with two mechanisms behind it, and it is still a correlation. What would make it evidence is a measurement in two rooms rather than an argument about two centuries.

Where this ladder goes next #

Ten rungs. An ending is five components with no total; half the cadences withhold some of them; the performance slows on a curve; nothing in the statistics announces a stop; a silence becomes an ending after three and a half seconds; the dynamic component is two quantities of different size; the tempo did not decide it; neither did the deceleration or the gesture length; the silence did, by twenty-four decibels; and now the room, which takes 40 per cent of that back at the length a composer writes it.

What is owed after this is the texture the silence is cut out of. Every level track on this page is a single steady loudness, and the thing a general pause interrupts is an orchestra with a number of parts in it — which this collection can price, because the orchestration ladder computes the sounding loudness of a scoring part by part. A rest after a full tutti falls from a reference the whole ensemble built; a rest after a thinning texture falls from one that was already coming down, and the difference is exactly the quantity that says whether a composer should write the pause before the diminuendo or after it. That is arithmetic the collection has both halves of, and it needs no room and no listener that is not already here.