A general pause is spent by the note after it

A rest needs a dry room put a hall under the ninth rung’s arithmetic and found that a bar of silence loses 40 per cent of its value in a shoebox concert hall. Its last paragraph named the thing every level track on that page holds fixed:

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.

That is a debt with a prediction attached, and the prediction is the natural one. A rest after a full tutti ought to fall from a reference the whole ensemble has built; a rest after a thinning texture falls from one that was already coming down; and the difference between those two ought to be the quantity that tells a composer whether to write the general pause before the diminuendo or after it.

The collection has both halves of that arithmetic — a ladder that prices a scoring part by part and a ladder that prices a silence — so the question needs no room, no corpus and no listener that is not already here. Joined, they answer it, and they refuse almost every word of the prediction. The ensemble is worth nothing to speak of. The order is worth a great deal. And the reason is neither of the things the debt named.

The reference the ensemble builds, and what it buys #

Take the debt’s prediction on its own terms first, because it is testable in one line and the answer settles the shape of everything after it.

A general pause is priced by how far a listener’s running impression of loudness has fallen when the sound comes back. That impression is the slow half of the pair of smoothers the loudness ladder uses, taken from Glasberg and Moore’s time-varying loudness model, and its release is two seconds. During a silence it decays toward the background, and the decay is exponential in sones.

An exponential decay has no level in it. Falling by a fixed proportion per second means the value it started from cancels, and the whole of what a loud passage buys is a background further beneath it to fall toward before the decay flattens out. So the prediction is true in direction and very nearly nothing in size.

Forty decibels of passage buy 1.04 decibels of rest. And the ensemble does not command forty decibels: a page has two decibels and a player has sixty measured exactly this and found that adding parts to a chord at a fixed level barely moves the loudness, because the added parts land in critical bands the chord already occupies.

Run that measurement over the part counts this rung uses, and the collection’s two loudness models disagree about it in a way worth stating plainly. Summing sones across critical bands — the model every closure and loudness figure on this site is drawn with — a texture of three parts, six, nine and twelve produces 79.36, 80.30, 79.49 and 79.41 decibels: a range of under one decibel, and not even monotone. Integrating Zwicker’s excitation pattern instead, the same four textures produce 85.05, 88.47, 89.96 and 91.08: six decibels, monotone, and compressive as it should be.

The disagreement is not a defect and it is not a tie. The band model groups components by single linkage, so a chain of partials each less than a critical band from the next becomes one band however long the chain — and a dense texture is precisely the input that makes the chain long. The count of occupied bands falls from six to three as the texture grows from three parts to twelve, which is the mechanism visible: the model is merging the ensemble faster than the ensemble is adding to it. That limit is stated where the function is defined, and the excitation version exists to be the same quantity without it.

Either way, the answer to the first half of the debt is the same. The ensemble’s whole contribution to the reference runs between 0.05 decibels — the distance from three parts to twelve under the banded model, which wanders by 0.94 on the way and arrives almost where it started — and 6.03 under the other. Six decibels of passage buy about a sixth of a decibel of rest. A general pause after a tutti and a general pause after a trio are worth the same thing to within a fraction of a decibel, and no reading of the collection’s own machinery makes them differ by enough to notice.

The two orders differ only in what comes last #

So the ordering question survives the refusal of its stated mechanism, and it survives with a much larger answer than the mechanism would have given.

Set the two gestures side by side. Both hold a twelve-part texture, both write a diminuendo of one dynamic mark over four seconds, both write a bar of silence, and they differ in nothing but sequence. Read at the instant the final chord arrives, one is 13.06 decibels below the reference and the other is 6.00.

The mechanism is in the recovery, and it is a property of the listener rather than of the music. The slow smoother is asymmetric: its release is two seconds and its attack is ninety-nine milliseconds, a ratio of twenty. Sound that stops takes seconds to leave a listener’s impression; sound that starts is in it almost at once.

A silence written before a diminuendo therefore does exactly one thing. It drives the reference below the level the diminuendo settles at, and the diminuendo then hauls it back up to that level in a tenth of a second. The reading at the arrival is the diminuendo’s depth and nothing else — 6.00 decibels against a depth computed by an entirely different route, out of two part counts and a written mark. The silence has vanished from the arithmetic. Lengthening it does not help: at a bar of silence the reading is 6.00, at two and a half seconds it is 6.00, and at five seconds it is 6.00.

The surprising consequence is one further step along. A diminuendo written after a general pause is worse than no diminuendo at all. A bar of silence with the final chord straight out of it reads 8.12 decibels; the same bar of silence followed by the diminuendo reads 6.00. The four seconds of quiet music have taken 2.12 decibels away and contributed nothing, because the pause had already got the reference lower than the diminuendo can hold it. At three and a half seconds of silence the loss is 17.71 decibels. That is not a refinement of the ordering rule, it is a sign change: on this arithmetic the second-best thing a composer can do with a general pause is to leave the diminuendo out.

A tenth of a second #

The number that decides all of this is how quickly the resumption spends the silence, and it is much smaller than anything anybody writes.

Ninety-seven milliseconds is a demisemiquaver at a moderate tempo, and there is no note value short enough to protect a general pause from the music that follows it. A demisemiquaver at the previous dynamic, standing between the silence and the arrival, costs half the silence; a quaver costs all but a hundredth of it.

The boundary the curves separate at is the useful statement, because it is a rule a composer can apply without a calculation. The pause has driven the reference 8.1 decibels down. Whatever comes next re-establishes the reference at its own level, so the pause survives only in the part of it that is below whatever comes next. If the resumption is quieter than the silence had already reached, nothing is spent — the reference is still falling when the resumption starts, and it goes on falling. If the resumption is louder, the surplus is gone, and gone in a tenth of a second.

That is why the arithmetic does not distinguish an ensemble from a trio, and it is worth saying why in one line. The size of the ensemble sets the level the reference falls from, and the fall is a ratio, so that level cancels. What does not cancel is the level the reference is dragged back to, and a composer sets that with a marking rather than with a scoring.

What a diminuendo has to be worth to follow a silence #

The rule turns into an exchange rate with one constant in it. A silence drives the reference down at ten times the base-two logarithm of e divided by the release, which is 7.21 decibels a second and contains no level, no ensemble and no room.

A general pause of a bar has spent 8.66 decibels by the closed form, and 8.12 measured on a level track that has a hall’s background under it. To keep them, the music between the silence and the arrival has to sit more than eight and a half decibels below the texture the silence was cut out of.

No scored diminuendo in this collection reaches that. A texture thinning from twelve parts to three is worth 0.05 decibels by critical-band summation and 6.03 by excitation pattern, so it protects between a hundredth of a second and eight tenths of one. The ordinal ladder of six written marks does better and not by much: one mark protects 0.83 seconds, two protect 1.66, and only a drop through the whole range from fortissimo to pianissimo protects a general pause of four seconds.

There is a further reason to reach for a marking, and it is a correction rather than an argument. The catalogue of closing gestures the sixth rung of this ladder draws is the collection’s only attempt at a scored diminuendo, and two of its four textures are generated from a rule whose bar index is reversed: the gesture called a thinning to one part in fact grows from one part to six across the closing bars, and the one called a thinning into a full final chord grows from three parts to six before its last chord. Nothing above uses them, because the scored diminuendo here is a part count set directly. But it means the reading that rung publishes for the second of those, 0.95, is a reading of a texture that thickened; drawn the way its own name describes it the reading is 0.999, and the ending that arrives below the running impression is the full final chord alone. So the four gestures that ladder has drawn contain no thinning at all, which is the strongest available reason to price a scored diminuendo directly, as the figures above do, rather than to read one off them.

So the prescription is sharper than the debt expected and it is about notation rather than about scoring. A composer who wants a silence and a diminuendo both, and wants the silence to count, writes the silence last — or writes a fall of at least a mark and a half into the bars after it, which is a large gesture to spend on undoing a smaller one.

Which computation produced the numbers #

The texture is a tonic triad scored for a stated number of parts, realised every way the ensemble ranges admit, sampled evenly across that enumeration up to ninety-six ways, each part given a string spectrum at 62 decibels, every partial placed in its critical band and the loudnesses summed. That is the arithmetic the loudness ladder built for a page, lifted out for a part count rather than for a scheme, and averaged over realisations because the quantity wanted is a property of the texture rather than of one arrangement of it. The mean is converted back to a level so that the smoothers see the units they were built for.

The level track is that texture for eight seconds, a diminuendo of a stated depth for four, and a stated number of seconds at the hall’s background of 25 decibels, sampled every ten milliseconds. Everything downstream is the ninth rung’s unchanged: the instantaneous sone value, the fast and slow smoothers in series, the slow one’s output read at the instant the final chord arrives, and the conversion into an equivalent diminuendo at ten decibels per doubling of sones, which is Stevens’s power law with ISO 226 underneath it turning a level into a phon. The control is the same texture held for the same total time with neither device in it. Two figures depart from that in ways that change nothing: the one drawing the two plans on a time axis opens with four seconds rather than eight so the gesture fills the frame, and the reference has settled long before either ends; and the one measuring the resumption samples every five milliseconds rather than every ten, because the quantity it is about is a tenth of a second wide.

Two loudness models compute the texture’s level and the essay reports both. The first groups components by single linkage within a critical band and adds sones across groups; the second integrates Zwicker’s excitation pattern along the Bark axis, which is the same quantity without the grouping’s edge. The disagreement is the largest quantity on this page — eleven decibels in the tutti’s level — and it changes the ordering result by 0.16 decibels, because the fall is a ratio.

The two orderings are the same segments in a different sequence, so they last the same time and hold the same silence, and the figure that draws them asserts as much before it draws anything.

Where the model stops #

A diminuendo is not a step. The gesture here drops to its new level and holds it, and what a composer writes is a slope. A slope reaching the same depth over the same seconds ends at the same place, so the arrival reading is close — but the reference during a slope is never as low as during a step of the same depth, so a real diminuendo protects a silence slightly less well than this says.

The parts are all one instrument at one level. A dynamic mark changes what a note is on every instrument except an organ flue pipe, so a texture playing softer is a different spectrum and not the same spectrum turned down. The scored diminuendo here is a count of parts and the written one is a gain, and neither is what an orchestra does.

The tutti is a chord and not a tutti. Twelve parts on a triad is a texture, and an orchestral tutti has doublings, percussion, and parts in registers this enumeration never reaches. The unison case — the same note played by more and more people — is in the collection and spans 19.7 phons for ninety players over one, which is three times the excitation model’s span for a texture and still buys only half a decibel of rest.

The background is asserted. Twenty-five decibels is a quiet hall, and the reference falls toward it rather than toward nothing, which is why 8.12 is measured where 8.66 is the closed form. A noisier hall makes every silence worth less and every ordering effect smaller.

And the room is switched off. The tenth rung showed that a hall fills a short silence and takes 40 per cent of a bar of it. Everything here is dry, so every number in it is the most a composer could hope for.

What the picture cannot show #

It cannot show the silence being heard as a silence. All this prices is a reference falling. Whether a bar of nothing is heard as a pause or as an ending is a threshold at about three and a half seconds, and it is not in any of these curves.

It cannot show what a listener counts. A listener counting bars through a general pause is doing something no smoother does, and a pause of an odd number of bars is not the same object as an even one.

Nor can it show what the players do in the pause. A written silence in every part is a release rather than a switch, and the last chord’s own decay is added to it — which is a short diminuendo written into the instruments and not into the score.

It cannot show a conductor. The length of a written pause in performance is a decision, so every reading above is an option rather than a measurement, and a conductor lengthening a pause is moving along the steepest curve on the page.

And it cannot show the harmony. An ending is five components with no total, and nothing here touches the three that are made of chords. A silence in the wrong place costs seven decibels of one component out of five.

What this makes of the whole ladder #

There is a general statement underneath, and it is better than the particular one.

The reading was a step response swept the tempo and the deceleration and got the same answer to five significant figures from both, and swept the length of the closing gesture and got a move that was a function of bars rather than of seconds. The ninth rung sorted them: a manipulation that moves things about inside a continuous stream returns nothing, because the slow smoother’s output at the end of a stream is set by the level of the stream rather than by its schedule, and only a manipulation that changes the input returns anything. A silence is the largest of those.

This rung sharpens that into something a reader can carry. A listener’s running impression at any instant is set by the last few tenths of a second before it, unless those tenths are silent — and if they are silent, it is set by how long they have been. Everything the ladder has swept follows from that one sentence. The tempo does not matter because it moves events inside a stream. The deceleration does not matter for the same reason. The gesture length does not matter because a gesture is a stream. The silence matters because it is the only thing that removes the input altogether. And the order matters because the sentence has “last” in it.

That also explains a result on the orchestration ladder that looked like a curiosity. A subito piano is a rate, not a level found written dynamics that cannot be produced at all: asking the running impression to fall further than the release allows drives the correction to silence and it still misses. The correction it wanted was a general pause, and it was refused one because a passage has to keep sounding. Closure is the one place in music where the sound is allowed to stop, which is why the device exists there and nowhere else.

Whose music, and when #

The written general pause — everybody stops for a bar, then the final gesture — is a classical and romantic orchestral commonplace, and its canonical placement is exactly the one this arithmetic prefers. In Haydn and in Beethoven the silence sits immediately before the arrival: the fermata, the bar of nothing, the chord. Nothing intervenes, and on these numbers nothing may.

The other placement is a nineteenth-century one and it is easy to name. Bruckner’s symphonies are full of general pauses that separate blocks rather than announce arrivals — the music stops, and then it resumes, often quietly, and continues for many bars before anything closes. The arithmetic says those pauses are not doing what a Beethoven pause does, and the honest reading is that they are not trying to: a pause used as a boundary between paragraphs is a rhetorical device about form rather than a dynamic device about loudness, and this ladder prices only the second.

That distinction is a prediction rather than an observation, and it is falsifiable in a way that does not need a corpus of scores. Two performances of the same passage, one with a quaver of sound between the pause and the arrival and one without, should differ by about eight decibels of apparent dynamic and by nothing else — and conductors who add a breath before a final chord, or take one away, are running that experiment constantly without a number attached to it.

What the arithmetic will not support is a claim about which composers knew this. The device is old, the model of the listener is from 2002, and a correlation between a good instinct and a one-pole filter is not evidence about anybody’s intentions.

Where this ladder goes next #

Eleven 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; the room takes 40 per cent of that back; and now the texture, which turns out to be worth nothing and to make the order worth seven decibels instead.

What is owed after this is the attack side of the same asymmetry, which is arithmetic this collection already has and is the only part of the smoother this ladder has never used. Every number on this page and the two before it is about a reference coming down — a release of two seconds, a silence, a diminuendo. The same smoother has an attack of ninety-nine milliseconds, and the one place a closing gesture uses it is the final chord itself, which arrives as a step into a reference that has fallen. Nothing on this ladder has ever asked how long that chord has to last to be heard at its own loudness rather than at the running impression’s. The arithmetic is available: sweep the final chord’s duration from a demisemiquaver to a fermata, read the fast smoother’s output against the slow one’s at the moment the chord ends, and find the length at which the arrival stops growing. It would say whether a short final chord — the two-note tonic stamp that ends a great many classical movements — is heard as loudly as a held one, and the collection has every constant the calculation needs. It is arithmetic, and it needs no room, no corpus and no listener that is not already here.