A rest is a diminuendo

The reading was a step response is the strangest rung on this ladder. It swept the tempo over a factor of ten and the deceleration over a factor of three and got the same arrival reading to five significant figures from both; it swept the length of the closing gesture over a factor of forty-eight and got an eight per cent move that turned out not to be a function of seconds at all. Its account of that pair is a single sentence — the reading sees how far the last bar stands from the bar before it, and nothing else — and the sentence divides every manipulation into two kinds. A manipulation that rearranges the schedule returns nothing. A manipulation that changes the input returns something.

Its last paragraph names the one that should move it:

What is owed after this is the silence. The one manipulation that should move a running impression is a gap, because two seconds of nothing is one full release and the chord after it arrives against a reference that has fallen.

The two ingredients are already here. The loudness ladder’s smoother has a two-second long release; the fifth rung has the measured threshold at which a gap stops being a pause and becomes an ending.

What a rest costs, in decibels #

Run a passage at a steady level, insert a gap, and read the running impression at the moment the next chord arrives.

A quaver rest at a moderate tempo is worth a decibel. A general pause of a bar is worth eight. And at three and a half seconds — the threshold for a gap being heard as an ending — the reference has fallen by twenty-four decibels, which is a wider range than fortissimo to pianissimo.

That is the manipulation the eighth rung’s account predicts should be the largest, and it is the one nobody swept because it was not a number in the model. A tempo is a number and a deceleration is a number; a rest is a hole.

Which is what a general pause is for #

The device this prices has a name and a repertoire. A general pause — everybody stops, for a bar or two, before the final gesture — is a nineteenth-century orchestral commonplace, and what everybody says about it is that it makes what follows enormous.

This says exactly how enormous, and the mechanism is not about the chord that follows at all. The chord is the same chord. What has changed is the reference it lands against, and the reference has been decaying at the release of a smoother that takes two seconds to fall by a factor of e.

So a composer who writes a bar of silence before a final fortissimo has written a crescendo of eight decibels without changing a dynamic marking, and one who writes three and a half seconds has written one of twenty-four.

A rest is the loudest thing a composer can write, and it is written as nothing.

The same argument runs at the other end of a piece and nobody calls it a device. An opening chord arrives against a reference that has been at the hall’s noise floor for however long the audience has been quiet, which on this curve is the far right of the axis — so a first entry is heard against a reference several tens of decibels below where any later entry lands. The first chord of a piece is louder than the same chord anywhere else in it, and it costs the players nothing.

It also completes the eighth rung’s sorting #

The eighth rung sorted its sweeps into two kinds and this rung supplies the extreme case of the second.

Tempo and deceleration change when things happen inside a continuous stream of sound. The smoother’s output at the end of a continuous stream is set by the level of the stream and not by its schedule, which is why moving the schedule around by factors of ten and of three changed nothing to five figures.

The number of closing bars is not a schedule change and it did not come back empty: it decides which texture the final chord is measured against, and moving it from one closing bar to eight moved the reading by eight per cent. A silence is the same kind of manipulation carried much further. It sets the signal to nothing for a stated duration, so the thing the final chord is measured against is not a thinner texture but no texture at all, and the smoother has an exponential answer to that.

So the eighth rung’s finding and this one are the same statement. The arrival reading is a step response, its only variable is the size of the step, and there are exactly two ways to make a step bigger — raise the top or lower the bottom. Every parameter made of seconds was trying to raise the top by rearranging the schedule and none of them could. The bar count lowers the bottom by a texture; a rest lowers it by everything.

The sizes line up with that reading rather than merely being consistent with it. Eight closing bars instead of one buys eight per cent; a bar of silence buys eight decibels, which is a factor of two and a half on the same ratio. The deepest hole a texture instruction can dig is a single line still sounding, and the deepest a rest can dig is the noise floor of the room.

Where the two thresholds meet #

There is a coincidence in the numbers worth naming, and it is a coincidence rather than a mechanism.

The fifth rung found that a gap is heard as an ending rather than as a pause at about three and a half seconds. That is a measured perceptual threshold from a completely different literature, about segmentation rather than about loudness.

At that same three and a half seconds the running impression has fallen by twenty-four decibels, which is essentially the whole usable dynamic range of an orchestra.

Two unrelated accounts of what a long silence does — one saying the listener has decided the piece is over, the other saying the listener’s loudness reference has run out — arrive at the same duration. Nothing here explains why, and both numbers come from the same two-to-four-second scale that keeps appearing in this collection whenever a listener has to hold something.

That scale is worth watching. It is the loudness release, it is the segmentation threshold, and it is near the upper end of what this collection calls the present.

The size of the effect against a player’s own range #

It is worth putting the twenty-four decibels beside what a performance can do, because a number in decibels is easy to nod at.

An orchestra’s usable dynamic range from a genuine pianississimo to a genuine fortississimo is somewhere around forty decibels in a good hall, and much of it is unusable in practice — the bottom is eaten by the room’s noise floor and the top by what the players can sustain. A composer’s eight markings span that range.

Three and a half seconds of silence is worth twenty-four of those decibels, delivered at no cost to the players, and it can be combined with a dynamic marking rather than traded against one. A fortissimo after a general pause is a fortissimo arriving against a reference two-thirds of the way down the orchestra’s whole range.

That is why the device sounds like more than the sum of its parts, and this is the arithmetic of the excess.

What it says about the five components #

This anchor’s first rung set out five components of an ending with no total: the cadence, the deceleration, the dynamic, the silence and the metrical position. Eight rungs later, three of them have numbers and two did not.

The silence now has one, and it is the largest of them. It is worth more than the dynamic component, because it is the dynamic component, applied through the same smoother by a different route — and it is worth more than the deceleration, which the eighth rung found worth nothing.

That reorders the list. The components a listener is most sensitive to at an ending, in the units this collection can compute, are the cadence and the silence; the tempo and the gesture are free parameters the model cannot see; and the dynamic marking is real but is bounded by what a player can produce, which a rest is not.

A composer with a rest has a larger dynamic than a composer with an orchestra.

The device is available at every scale, and that is unusual #

One more thing the curve says, which the two-second release makes almost inevitable and which is worth stating for musicians rather than for the model.

The relation between silence and equivalent diminuendo is smooth and monotone from a semiquaver rest to a fermata. There is no threshold in it, no minimum useful gap and no saturation until several seconds in. So the device scales: a composer who wants one decibel writes a quaver rest and one who wants twenty writes three seconds, and the same mechanism delivers both.

Very little else in this collection behaves that way. A cadence is a discrete choice between kinds; a dynamic marking is an ordinal scale with about eight steps; a deceleration turned out to do nothing at all. A rest is a continuous control with a wide range and a computable transfer function, and it is written in a notation that records it exactly.

It may be the only continuous dynamic control a score actually has.

Which computation produced the numbers #

The level track is a steady six seconds at 78 decibels, a stated number of seconds at a floor of 25, and three seconds at 78 again, sampled every twenty milliseconds.

The running impression is the loudness ladder’s own pair of one-pole smoothers in series — a short one and a long one, 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, converted at ten decibels per doubling of sones.

The floor of 25 decibels is a quiet hall rather than an anechoic chamber, and it matters: a true silence would drive the reference lower still, and a noisy hall would arrest the fall.

Why a rest does what a diminuendo cannot #

The comparison in the title is exact and it is worth being clear about what it is not.

A diminuendo lowers the level and the smoother follows it down at the same two-second release. So a diminuendo to pianissimo over three and a half seconds moves the reference by very nearly what three and a half seconds of silence does — the smoother does not care how the input got small.

What it cannot do is get there and stay playing. The orchestration ladder found a subito piano unproducible for exactly this reason: solving for the levels that make the running impression arrive at a written marking drives the players toward silence and still leaves the impression above the target. The correction the optimiser wanted was to stop, and it was not allowed to.

A rest is that correction, permitted. It is the one move that reaches the bottom of the smoother’s range without asking a player for a level they cannot produce, and it is why the two ladders’ findings are the same finding read from opposite ends.

Where the model stops #

The hall is not silent. A real general pause in a real room is two seconds of reverberation, not two seconds of nothing, and reverberation is exactly what stops the reference falling. In a dry hall a rest is worth what this figure says and in a wet one it is worth considerably less — which is a prediction about repertoire and rooms that somebody with both could check.

The release is one number. Two seconds is the loudness ladder’s own, taken from published measurements, and every quantity here scales with it.

A steady six seconds is not a piece. The reference the gap falls from is whatever the passage before it built, and a quiet passage leaves less to fall from — so the twenty-four decibels is the value after a sustained forte and is smaller after anything else. The shape of the curve does not move, because the release is a proportion rather than a level.

And there is no attention in the model. A listener in a silence is not a smoother running down; they are a person noticing that the music has stopped, and the fifth rung’s threshold is about that and this rung’s arithmetic is not.

The floor is asserted. Twenty-five decibels is a stated background and the curve’s tail depends on it; the first second, where most of the musical uses are, does not.

What the picture cannot show #

It cannot show the audience. A general pause in a concert is filled with coughing, and the loudest part of many silences is the room.

Nor can it show the ending it is not. A gap in the middle of a piece produces the same fall and means something entirely different, and what makes an ending an ending is five components of which this is one.

Nor can it show anticipation. A listener who knows the piece knows the chord is coming, and the effect this prices is largely one of surprise, which a second hearing does not have.

It cannot show what the players do. A conductor holds a pause for as long as the room will bear, and the duration is a performance decision — which makes this figure a set of options rather than a measurement.

It cannot show the metre. A bar of silence is still a bar, and a listener counting through it is doing something the smoother does not model — which is the other half of what an ending is, and is why a general pause of an odd length feels different from one of an even length.

And it cannot show a rest in the middle. Everything here is a gap before a final gesture; a rest inside a phrase does the same thing to the reference and has an entirely different meaning.

Whose music, and when #

The passage is generic and the levels are generic.

The device has a period. A written general pause before a final cadence is a classical and romantic orchestral gesture — it is all over Haydn, and it becomes a standard rhetorical move in the nineteenth century — and it is rare in earlier music, where the texture is thinner and the halls are more reverberant. Both of those reduce what a rest is worth: a thin texture leaves less reference to fall from, and a reverberant hall stops it falling.

So the model predicts that the general pause should have become effective as orchestras got louder and halls got drier, which is roughly the century it appeared in. That is a correlation with a mechanism attached and it is not evidence, because a great many things changed in that century.

Where this ladder goes next #

Nine 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 is 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 length; and now the one thing that does, by twenty-four decibels.

What is owed after this is the room. 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, and this collection has a room ladder that computes exactly how a level decays after a source stops. Putting a hall’s reverberation time under the loudness smoother would say how much of a rest survives being played in a cathedral — and it would say whether the general pause is a device that needs a dry room, which is a claim about where the repertoire was written as well as about what it does.