A part that leaves is not a part that arrives

The previous rung found that the long-term loudness smoother has two constants rather than one, and that the two-second figure this anchor quotes everywhere is the wrong one for an entrance. Ninety-nine milliseconds is the right one, and it is what makes an arrival prompt.

Two seconds is the other direction, and the other direction is a score’s other half. Players leave as often as they arrive, and every treatise that discusses entries at length says almost nothing about exits — which turns out not to be an oversight in the treatises.

The entrance is half received in 90 milliseconds and the exit in 1.43 seconds. Nothing about the sound is asymmetric: the same two states, the same two spectra, the same levels, run in the two possible orders. The whole of the factor of sixteen is in the smoother.

A window has no direction and a filter does #

The two readings behave differently because they are different kinds of object, and the difference is not a detail of implementation.

The roughness reading is a causal rectangular window — the last thirty-seven milliseconds, averaged, with nothing after them. A window like that receives a step in exactly the length of the window whichever way the step goes, so an exit and an entrance are received identically, and the two faint curves in the drawing lie on top of each other because there is nothing to separate them.

The loudness reading is a one-pole smoother with two coefficients, chosen by whether the input is above or below its own current output. That is not an arbitrary choice; it is what makes the model reproduce the thing it was fitted to, which is that a listener’s impression of loudness persists after the sound has gone. A room falling silent does not become quiet in the listener at the speed it becomes quiet in the air.

So the asymmetry is a claim about hearing rather than an artefact, and the claim has been in this collection since the loudness ladder’s third rung. What is new here is only that a scoring decision runs into it, and that it runs into it in the direction a score cares about.

What the asymmetry does at the pace of real music #

A factor of sixteen between two time constants only matters if a passage moves fast enough for it to.

At three tenths of a second a chord, an entrance is 94 per cent received by the end of the chord after it and a departure is 12 per cent. At 1.2 seconds — an ordinary crotchet at an ordinary tempo — the two are 100 per cent and 44. At four and a half seconds a chord, which is slower than almost anything, the departure has still only reached 89 per cent.

An entrance is complete inside one chord at every pace on the axis and a departure is complete inside one chord at none of them. That is the sentence the rung exists for, and its consequence is that an exit is smeared across the music that follows it in a way an entrance never is.

The roughness readings, drawn faintly, are complete at every pace in both directions. Thirty-seven milliseconds is short against three tenths of a second and enormously short against four and a half.

The mark a score puts under a departing part #

A score does not usually let a part stop. It writes a diminuendo under it, and then stops it.

The arithmetic above says what that mark is doing, and it is not what it looks like. A diminuendo lowers the part’s level over some interval, so the input to the smoother falls gradually rather than stepping; and the smoother’s release only binds when the input falls faster than the release can follow. A diminuendo whose length exceeds two seconds is followed exactly, and the impression arrives at the departure with the part already gone from it.

So a written diminuendo under an exiting player is not softening the exit for the ensemble’s sake. It is doing the smoother’s release for it, on a clock the smoother would otherwise take two seconds over — which is why the mark is so nearly universal and why nobody writes the corresponding mark under an entering part. There is nothing for it to do there: the attack constant is already faster than any crescendo a player can produce.

The closure ladder found the same object from the other side and did not have this name for it.

A rest is a diminuendo prices exactly this — a silence read through the same release — and finds that a silence is worth twenty-four decibels of dynamic component in an ending, more than anything else the ladder swept. The two rungs are the same arithmetic applied to different music. Closure removes everything and reads the fall as an ending; orchestration removes one part and reads the fall as a mistake that has to be covered.

Whether the departure is a colour change too #

The previous rung’s census says an entrance into a texture of five or more is a colour event and not a loudness one. A departure ought to be the same statement backwards, and it very nearly is.

The census is a difference between two states and knows nothing about direction, so the magnitudes are the same: a player leaving a sextet takes away 0.90 phons and 11.9 units of roughness, exactly what a player joining a quintet brings. What differs is only what a listener does with those two changes over time.

Combined with the clocks, that gives the shape of a departure precisely. Its colour goes at once, in the same twenty-five milliseconds an entrance’s arrives. Its loudness goes over most of two seconds, and there is under a phon of it to go. So a departure from a large ensemble is a change of colour followed by nothing — the level component is both small and slow, which are the two ways a change can fail to be an event.

That is a sharper statement than the entrance’s. An entrance in a large ensemble is a colour event that arrives first and a loudness non-event that arrives second. A departure is a colour event that arrives first and a loudness non-event that never quite arrives at all.

What an orchestrator is choosing between #

Set the two together and the practice has an arithmetic under it that it does not state.

Bringing a player in is an event with a place. It is received in a tenth of a second by every reading a listener has, so it can be placed to a beat and will be heard where it was placed. Taking a player out is not an event with a place at all: its loudness component takes longer than most bars and its colour component is the only part of it that has a moment.

Which says that the moment of an exit is carried entirely by its colour, and that a score which wants the exit to be heard as happening has to make the colour change large. Removing the instrument whose spectrum is least like the rest — the one whose partials the others do not have — is what makes a departure legible, and removing a doubling of something already present is what makes it invisible.

That is a rule an orchestrator would recognise and it is usually given the other way round, as advice about how to hide a part leaving: take out the doubled line, never the distinctive colour. The arithmetic agrees and says why, which is that the loudness component of the departure is under the limen either way and the colour component is all there is.

The part of a departure that is an addition #

There is a third thing an exit does that neither of the two readings above can see, and it points the other way.

Every component of every part sits above or below the threshold the other parts put over it, and the entering player contributes to that threshold like everybody else. So when it leaves it takes its own share of the mask away, and some of what the remaining parts were playing becomes audible that was not.

Counted on this texture, the entrant covers between one and five of the ensemble’s own components — between 2.5 and 13.3 per cent of what the rest of it has audible. The share falls as the texture thickens for the reason the masking anchor keeps finding: a dense texture masks itself so thoroughly that there is little left for one more part to take out of service.

So a departure is a small subtraction and a smaller addition at once, and both are largest in a small ensemble. What that says about the practice is a caution rather than a result: an exit from a quintet does not merely remove a colour, it uncovers one, and which components come back depends entirely on where in the spectrum the departing player was. Nothing on this anchor has ever had to think about that, because until three rungs ago nobody could leave.

Two thousand milliseconds is a bar #

It helps to put the release next to the units a score is written in, because the number is easy to read as small and it is not.

Two seconds is a whole bar of common time at a moderate tempo, and rather more than one in most fast music. So a part leaving on the first beat of a bar is still substantially present in the listener’s impression when the bar ends, and a passage that thins over four bars is being read through a smoother that is always about a bar behind what the page says.

The consequence for a score is the one this collection keeps arriving at from different directions. A written dynamic is an instruction to the impression rather than to the air, and the impression is a bar behind on the way down and a beat behind on the way up. A thinning texture is therefore a diminuendo the page did not write, delivered late; a thickening one is a crescendo the page did not write, delivered on time.

Which is not a small correction to how a passage reads. The loudness ladder’s fourth rung already found that a score’s texture carries most of its dynamic shape without any marks at all, and this rung adds the clock to it: the texture’s contribution arrives at a different time from the marks’, and by a bar when the texture is thinning.

Whose music, and where the practice put the mark #

The scoring practice this describes is European orchestral writing from about 1780 onward, and its evidence is typographic rather than acoustic: the near-universal diminuendo hairpin or morendo under a part about to stop, and the near-total absence of any corresponding mark under a part about to start.

That asymmetry in the notation is very old and it is not usually explained. The explanations that are offered are about the players — a part that stops abruptly is harder to make sound intentional, an entering player needs no help because the attack does the work — and both are true and neither predicts the size of the asymmetry or its direction from anything.

The release does. It says the mark under a departing part is doing two seconds’ worth of work that would otherwise be done by a listener’s smoother in a way the composer cannot control, and that the mark under an entering part would be doing none, because the attack constant is already ten times faster than the shortest crescendo an orchestra can play. A convention with an arithmetic under it that nobody stated is the most this rung can claim, and it is a claim that can be checked in scores rather than argued about.

Where it should not be pushed is music in which parts do not stop at all. A continuo texture, an organ registration held through a movement, or a drone tradition has no exits in this sense, and nothing on this page describes them.

The exit that takes everybody, which another ladder already priced #

There is a limiting case of a departure and it has a name in the repertoire: everybody leaves at once. The closure ladder’s eleventh rung is about exactly that, and its result reads differently once this rung’s asymmetry is in view.

That rung’s general statement is that 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 that a silence is the only manipulation that changes the input rather than rearranging it. Everything the closure ladder swept — the tempo, the deceleration, the length of the closing gesture — moves events around inside a stream and leaves the impression where it was.

An exit is the partial version of that manipulation and it is partial in a specific way: it removes some of the input rather than all of it. So it is subject to the same release and produces the same slow fall, scaled by how much of the sound left. A general pause removes everything and the impression falls to nothing over two seconds; one player leaving a sextet removes under a phon and the impression falls by under a phon over the same two seconds.

The two are the same device at two ends of one axis, and the axis is what fraction of the sound stops. That also explains why a general pause is a rhetorical event and a single exit is not: the closure ladder measured the silence as worth twenty-four decibels of an ending’s dynamic component, more than anything else it swept, and an exit from a large ensemble is worth a fraction of a phon. The release is the same; what differs is what it has to work on.

Which suggests the reading a score would actually use. If a departure is meant to be heard as a departure and the ensemble is large, the arithmetic offers two routes and no third: take out enough parts that the loudness component clears the limen, or take out the one part whose colour the others cannot supply. A score that does neither has written a change that a listener has no reading capable of receiving.

What the pictures cannot show #

The exit modelled here is instantaneous, and the section above is about the mark that makes it not be. Nothing here computes a departure under a written diminuendo — the claim about the mark is an argument from the release’s time constant and not a figure — because doing it properly needs a level track through the departure, which is the fourth rung’s machinery pointed at a scoring whose player list changes, and that is not built.

The two readings are drawn as shares of their own change, so the drawing says nothing about size. A departure that is 12 per cent received after three tenths of a second is 12 per cent of under a phon, which is a quantity below the limen being received slowly.

And the release constant is one number from one published model. Two seconds is what Glasberg and Moore’s time-varying loudness model uses, it was fitted to particular experiments, and every result on this page scales with it. A release of one second would halve the factor of sixteen; a release of four would double it. The direction of the asymmetry is not in doubt and its size is one citation deep, which is a different kind of confidence and should be read as one.

Where this leaves the ensemble as a variable #

Three rungs have now made the ensemble something a score can move. It changes both objectives, in opposite directions and by very different amounts; a listener receives an arrival in a tenth of a second and a departure over two; and the colour component carries the moment in both directions.

What none of them has asked is where. Every entrance drawn on these three rungs happens at a fixed place — a texture is sounding, a player joins, and which chord it happens over has not been a variable at all. A score chooses that, many times a page, and the choice is not free: the ensemble already sounding puts a masked threshold over whatever register the entrant takes, and that threshold is set by the voicing rather than by the dynamic. The next rung sweeps it and finds that the moment is worth as much as the register — up to twenty-nine decibels of margin between the best chord of a passage and the worst, with the quietest chord of the five turning out to be the worst place in it to bring somebody in.