The release is on the wrong side

The previous rung compares two states — the ensemble before an entrance and the ensemble after it — and a listener never has a state. They have a reading, and a reading is an integral over some window of the recent past.

This anchor carries three of those windows and they differ by two orders of magnitude. The roughness window is a few cycles of whatever fluctuation the texture’s own roughest pair produces, which for a chord in the middle of the keyboard is around a twentieth of a second. The short-term loudness reading has a time constant of twenty-two milliseconds. And the long-term reading, the one a written dynamic is an instruction to, has a release of two seconds.

The eighth rung of this anchor ended by asking whether that two-second release makes an entrance on a loud chord audible at all. It does not, and the reason is not the one the question was expecting.

An asymmetric smoother has two answers and only one of them is two seconds #

The long-term loudness reading is a one-pole smoother with two time constants rather than one. Going up its constant is 99 milliseconds; coming down it is 2.0 seconds. That is not an implementation detail, it is the published model, and it is there because a listener’s impression of loudness does not decay as fast as the sound does — a chord that stops is still in the impression for a second or more, which is exactly the fact the fourth rung of this anchor is built on.

Two seconds is therefore the answer to a different question from the one the eighth rung asked. It is how long a departure takes to be received. An entrance is a rise, and a rise is tracked by the attack constant, which is twenty times shorter.

So an entrance into a five-part texture is half received by the long-term reading in 90 milliseconds, and essentially complete inside a third of a second. There is no pace at which a passage is written that could hide it. The eighth rung’s worry was well-founded about the mechanism and pointed at the wrong side of it.

Three readings, three arrival times, one order #

Being received quickly is not the same as being received first, and the three windows do not agree about when the entrance happened.

The short-term loudness reading has half the entrance at 15 milliseconds. The roughness reading has half at 25. The long-term reading has half at 90. Six times separates the first from the last, and it separates them at every ensemble size on the axis.

What that means for the previous rung’s result is worth stating carefully, because it is the point of putting a clock on it. That rung found that an entrance into a texture of five or more is not a loudness event and is a colour event. This one adds that the colour event also arrives sooner — not because roughness is a faster physical quantity, but because the window a listener integrates it over is shorter. The two findings are independent and they compound: the part of an entrance that survives is also the part that arrives first.

The reading that depends on the ensemble, and it is not the obvious one #

The flat lines in that figure are the result rather than a defect of the drawing.

A one-pole smoother’s time constant is a property of the smoother. It does not know how many parts are in the sound, so an entrance into a nonet is received in exactly the same 90 milliseconds as an entrance into a duo, and the only thing that has changed between the two is how much there is to receive — which is what the previous rung measured and is a fall of a factor of forty-eight.

The roughness window is the exception, and it is the exception for a reason with content in it. That window is not a fixed number of milliseconds; it is a fixed number of cycles of the fluctuation the texture itself produces. A denser texture has more pairs of partials inside a critical band, and its median pair beats faster, so its window is shorter. Across the sizes drawn it falls from 57 milliseconds to 29, and the half-reception time falls with it from 25 milliseconds to 10.

So a large ensemble receives the colour of an entrance sooner than a small one does, and receives its loudness no sooner at all. Nothing in the two published models was arranged to produce that; it falls out of one of them having a window set by its own signal and the other having a window set by a constant.

A fourth clock the chord already had #

There is one more time in the problem and it belongs to the entering note rather than to the reading of it.

A note’s perceptual centre lags its physical onset by an amount set by how fast the instrument speaks, and three players starting together do not sound together. The eighth rung measured that spread across arrangements and found it runs from 14 to 19 milliseconds.

Set beside the three windows above, that number is the same size as the fastest of them. An entering oboe’s perceptual centre falls some 15 to 25 milliseconds after the players start, and the short-term loudness reading has half the change at 15. The entrance’s own onset and the fastest reading of it are the same clock, to within the accuracy either is known to.

That is a coincidence rather than a mechanism, and it is worth naming as one. The perceptual centre is a property of an envelope crossing a criterion; the short-term smoother is a property of a model of loudness integration. They are measured by different experiments and they happen to land in the same tens of milliseconds. What follows from it is only this: there is no interval during which an entrance has begun and no reading has moved. The moment the note is audible at all, the fastest reading is already following it.

Why a score can place an entrance to the beat and not to the reading #

A conductor’s beat resolves to perhaps twenty milliseconds and an ensemble’s own asynchrony is of that order, so the whole of the spread this rung has been measuring sits inside the noise of a real performance — for the fast readings.

The long-term reading is different, and this is where the arithmetic reaches the page. Ninety milliseconds is a demisemiquaver at a moderate tempo. It is longer than any asynchrony an ensemble produces and shorter than any note it plays. So an entrance is a loudness event that begins on the written beat and is complete before the next subdivision, which is precisely the behaviour a score assumes when it writes an entrance on a downbeat and expects the arrival to be there.

An entrance written on an off-beat is received the same way at the same speed. Nothing in the reading is metrical. What is metrical is the listener’s expectation, which this collection prices on the metre anchor and not here.

The same three windows over a whole passage, and the entrance is the easy case #

An entrance is one transition into a texture that is otherwise still. A passage is a sequence of them, and the anchor already has a figure of what the two windows do to a five-chord phrase.

On this passage at 1.2 seconds a chord, the roughness contrast between the chords is a factor of 4,146 and all of it survives the window. The loudness contrast is a factor of 2.80 and 83 per cent of it survives. That difference is the fifth rung’s result and it is the same asymmetry this rung has been measuring, seen from the far end: the window that is short keeps everything and the window that is long does not.

Sweeping the pace does not change the answer, and the reason is the release again. The roughness window is 37 milliseconds and every pace on the axis is longer than that, so nothing is ever lost. The loudness reading loses a sixth of the contrast at 4.5 seconds a chord and a fifth at 0.2, which is a range of five per cent across a factor of twenty in tempo.

A passage’s tempo is nearly irrelevant to what a listener receives of it, and its transitions are not. That is the same sentence as this rung’s, because a transition is where a smoother’s constants are visible and a steady state is where they are not.

A dynamic mark that cannot be produced, and an entrance that always can #

There is one place in this anchor where the release does refuse something outright, and it is worth setting the entrance against it.

A written subito piano after a full chord is a demand that the running impression fall faster than two seconds, and the only lever available is to play the next chord quieter. Past a certain drop the correction runs to silence and the impression still misses — the fourth rung’s word for it is unreachable, and it is a property of the release.

An entrance has no unreachable case at all. Every entrance this rung has drawn is complete in the long-term reading inside a third of a second, at every ensemble size and every dynamic, because the constant that governs it is 99 milliseconds. The smoother refuses one direction and not the other, and a score’s vocabulary reflects it: there is a mark for a sudden quiet and it is famous for being hard to bring off, and there is no corresponding mark for a sudden loud, because a sudden loud is free.

The crescendo the players are not making #

The fourth rung’s problem is to find levels that make the running impression arrive at a written mark, and everything it can adjust is a level. An entrance adjusts the impression without adjusting anybody’s level, and the constant that governs it is short enough that the adjustment is immediate.

That makes an entrance a lever on the same quantity the dynamics are a lever on, and a faster one. The loudness ladder measured the exchange rate: the dynamics are in the score already prices how much loudness a texture produces by adding parts against how much a player produces by changing level, and the two are the same kind of quantity in the same units.

Read forward, that says a composer can write a crescendo by scoring rather than by marking, and the impression will follow it in a tenth of a second whatever the marks say. Read backward, it says something less comfortable: a passage that thins while its players are marked to hold their level is making a diminuendo nobody wrote and nobody can be asked to stop. The impression falls because parts left, and the correction available to the remaining players — playing louder — is bounded by what they can play.

The bound is not symmetric either, and it is the same asymmetry once more. Adding a part raises the impression in 99 milliseconds; removing one lowers it over two seconds. So the scoring lever is fast upward and slow downward, exactly as the level lever is, and for the same reason: neither of them is the sound, both of them are read through one smoother.

An entry that is stopped before it is finished #

There is a consequence of the ordering that is easier to hear than anything else on this page, and it belongs to a rehearsal rather than to a score.

A player who enters by mistake and stops at once has sounded for perhaps fifty milliseconds. The roughness reading has half of that change at 25 milliseconds and the short-term loudness reading at 15, so both have taken it up. The long-term loudness reading has 90 milliseconds as its half-time and will have received well under half of it before the sound stops — after which it decays over the release, which is two seconds.

So a false entry that is caught immediately is received almost entirely as a change of colour, briefly, with almost no change of level and a small smear afterwards. That matches what a conductor reports about such a moment — that something went momentarily wrong with the sound rather than that it got louder — and it is a prediction rather than a description, because nothing in the three models was arranged to produce it.

The same arithmetic says the reverse case is not symmetric. A player who misses an entry in a texture of five or more removes nothing a listener can detect in loudness, by the previous rung’s census, and removes the colour they would have added — which arrives, or fails to arrive, in the twenty-five milliseconds the fast window takes. A missing entrance is a colour that never came, and it is not a hole in the sound.

What the pictures cannot show #

The three readings are drawn as shares of their own eventual change, which is what makes them comparable and is also what removes the thing that matters most. A reading that arrives quickly at a change of 0.32 phons has arrived quickly at nothing. The previous rung supplies the magnitudes and this one supplies the times, and neither figure can show both, because a phon and a squared pascal have no exchange rate and the collection has said so at every rung of this anchor.

Nor does any of this say what a listener notices. A half-reception time is a property of a model of integration. Whether a listener attends to the moment their roughness reading moved rather than to the moment their loudness reading did is a question about attention, and the collection has no model of that anywhere.

The roughness window here is computed from the median fluctuation rate of the whole texture, which is the convention the fifth rung established. A listener attending to the entering part specifically would have a window set by the fluctuations that part is in, which are between its partials and the ensemble’s rather than among the ensemble’s own. That is a different number and it is not computed here.

And the entrance modelled is instantaneous. A real player does not go from silent to steady in no time; a horn takes tens of milliseconds to speak and a bowed entry can take a hundred. Every half-reception time here is therefore a lower bound, and for the fastest reading it is a lower bound that the instrument’s own attack could double.

Whose music this describes #

The practice is orchestral scoring where an entrance is placed to a beat, which is European art music from about 1750 onward and every notated tradition since. What the arithmetic supports about it is the placement and not the dynamic.

A score marks an entering part with its own dynamic, and the treatises discuss how loud an entering horn should be relative to the strings it enters over. That instruction is doing less than it appears to: at five parts and above, the difference between marking the entrance mezzo-forte and forte is a difference in something under the limen, while the colour arrives whatever is marked and arrives first. The mark that does work is the one on the parts already sounding, which is the correction the fourth rung of this anchor computes and which a conductor asks for in rehearsal in exactly those terms.

Where this is likely to be wrong is in music whose entrances are not into a steady texture at all — a fugal exposition, where each entrance is into a texture that is itself changing, or a concerto grosso’s alternation, where the ensemble is switched rather than grown. Both are common and neither is what this rung computed.

The change run the other way #

Every figure here runs one direction: a part arrives. The smoother’s two constants say the other direction is not the same journey backwards, and the difference is the factor of twenty this rung opened with.

The next rung runs it. A part leaving is half received by the long-term reading in 1.43 seconds against the entrance’s 90 milliseconds, and its colour is received in the same 25 milliseconds as the entrance’s was — so a departure is a change whose colour arrives at once and whose level takes most of two seconds, which is the reverse of nothing that has been drawn here and is a shape no entrance has. What follows from it is a claim about the page: a written diminuendo under a departing part is not softening the exit, it is doing the smoother’s release for it on a clock the smoother would otherwise take two seconds over.