The parameter that did not decide the answer

A final chord is not made loud by adding to it is the closure ladder’s sixth rung, and it ends with an unusually specific warning about itself:

Every number here rests on a bar being two seconds against a smoother whose release is two seconds, and those two being comparable is the whole reason the contrast gets spent. A sweep over tempo would say at what speed a diminuendo into a final chord starts working, it needs nothing this ladder does not have, and it is the kind of parameter that has embarrassed this collection before by turning out to decide the answer.

The sweep costs a loop. Across a tenfold change of tempo — from a bar of half a second to a bar of five, which is four hundred and eighty beats a minute down to forty-eight — the reading of a two-bar closing gesture moves by four per cent, against a twenty-three per cent gap between the two gestures it is supposed to be distinguishing.

The parameter did not decide the answer. Finding out why is half the rung; the other half is that the figure drawn to answer the question turns a second dial as well, nobody named it, and it moves the reading five times as far.

What the premise actually was #

The warning has an argument inside it and the argument is where the error is.

The loudness ladder’s third rung established that a listener’s running impression of loudness has a fast attack and a very slow release — about a tenth of a second up and two seconds down, in the published model this collection uses. That release is the reason a contrast can be spent: a diminuendo followed by a loud chord only works if the running impression has come down far enough by the time the chord arrives, and it comes down over about two seconds.

The sixth rung then reasoned: a bar is two seconds, the release is two seconds, so the two are comparable and the tempo must matter.

The mistake is in what is being compared. The release is not racing a bar. It is racing the whole diminuendo, which is six bars in that rung’s own figures — twelve seconds at its own tempo, three at the fastest tempo anybody plays a closing phrase, and thirty at the slowest. Every one of those is comfortably longer than two seconds.

So the running impression has finished following the diminuendo down long before the final chord arrives, at every tempo. What is left at the arrival is the last bar’s own jump, which is a fixed number of decibels between two textures and does not know what a second is.

Where the tempo does appear, and how little #

The curves are not perfectly flat and two different things are in the residual, of quite different sizes.

The small one is where the measurement is taken. The reading is sampled a fixed fraction of the way into the last bar, which is seventy-five milliseconds at the fast end and three quarters of a second at the slow one, so the fast end catches the running impression before the short-term smoother has finished its own hundred-millisecond attack and the slow end catches it after the long-term smoother has begun its two-second release. Both raise the ratio, one at each end, and the reading is therefore lowest in the middle: on the sixth rung’s own six-bar gesture it runs 0.780, 0.764, 0.758, 0.764, 0.780, 0.803 as the bar goes from half a second to five. That is a six per cent bowl and it is a property of the sampling point rather than of the ending.

The large one is not a residual at all. The three curves of one gesture are the same instruction over two, four and eight closing bars, and they are nowhere near each other. Two bars of thinning read 0.84 to 0.88; four read 0.94 rising to exactly 1.00; eight read 0.83 rising to exactly 1.00. The closing length moves the reading by twenty per cent where the tempo moves it by four, and the sixth rung’s warning did not mention it.

So the honest reading of the sweep is: the tempo enters this quantity only through where the sample is taken, and the number of closing bars enters it through something else entirely. That is a stronger statement than “the effect is small”, and the second half of it is what the rest of this rung is about.

What the closing length is doing, which is not what it looks like #

The obvious reading of a twenty per cent spread across closing lengths is that a longer gesture builds more contrast. It is the wrong way round, and the arithmetic says so plainly: the longest gesture is the one that reads 1.000, which is no contrast at all.

A texture instruction that drops a part a bar, starting from six, reaches a single line after five bars and can go no further. Over six closing bars it arrives at one part exactly at the final bar, which is where the sixth rung read it. Over eight it arrives at one part two bars early and then holds — so the last two bars of the piece are identical, there is no change at the arrival to measure, and a listener whose impression has had time to settle on the single line hears the final bar as a continuation.

That is why the eight-bar curve climbs to exactly 1.000 and why it climbs with the tempo. At half a second to the bar the gesture is four seconds long, the release is two, and the impression is still carrying the fuller bars when the piece stops — 0.833. At five seconds to the bar the same gesture is forty seconds, the impression has followed every step of it down, and the arrival is nothing. The tempo is not acting as a tempo there. It is acting as the number of seconds an already-finished gesture has been given to be absorbed in, which is exactly the mechanism the sixth rung proposed and could not find.

The four-bar curve is the same story with the sign of the gesture reversed: four bars of thinning stop at three parts, and three parts is as loud as six on this model, so the closing bars are almost a flat texture and the reading rises to 1.000 as soon as the impression has time to notice. The two-bar curve is the only one that does not reach 1.000, and it is the only one whose last bar differs from the bar before it.

Why this is a result and not a null #

This collection has recorded the same weakness against itself again and again — results resting on an asserted parameter that nobody has turned — and it has lately begun to do something about it, sweeping the internal-noise sigma in the categorical-hearing ladder and the root-motion weights in the tonal-expectation ladder. Both of those found the parameter mattered: the sigma sweep inverted one of the collection’s own headline results, and the weight sweep priced a table that turned out to have a dead entry in it.

This one finds both at once, which is the outcome nobody plans for. The parameter named in the warning is a null and the sixth rung’s finding — that a final chord’s contrast comes from what the texture does rather than from how thickly it is scored — can now be quoted without the qualification it was published with. And the parameter that was never named, sitting in the same figure as a second index, moves the reading five times as far.

That is the argument for sweeping the parameter nobody is worried about. A sensitivity analysis run on the knob somebody is nervous about answers a question about that person; run across the whole figure, it answers a question about the model. The closing length was in every figure the sixth rung drew, fixed at six bars, and it was fixed at the one value where a drop-a-part-a-bar gesture happens to finish exactly on time.

What the sweep also does is correct the ladder’s account of its own mechanism. The rung said the contrast gets spent because the release is comparable to a bar. It gets spent because the release is short compared with whatever part of the ending is already over, which is a different comparison and gives a different intuition: the way to preserve a contrast is not to play faster, it is to arrange for the gesture to still be happening when the last chord arrives.

The subito, which the sweep points at #

If a gesture longer than the release is always absorbed, then the gesture that is not absorbed is one shorter than the release — and that is a device with a name.

A subito piano drops the level within a bar and gives the running impression no time to follow. The loudness ladder measured it directly: it is the passage that buys the most contrast per decibel of any of the devices in that rung’s list, and the reason is exactly the asymmetry between the fast attack and the slow release.

So the closure ladder and the loudness ladder have been describing two ends of one continuum without saying so. A closing diminuendo over six bars spends its contrast; a subito over one bar keeps it; and the boundary between them is the release, at about two seconds, which is between one and four bars depending on tempo. That is where the tempo enters — not in the size of the effect but in how many bars a composer has to work inside before the smoother catches up.

There is a second parameter in the same position — chosen once, never swept — and it turns out to matter more than the one this essay is about.

Counting the partials one at a time overstates the loudness, and worst at the bottom, where a low note’s partials are packed inside a band that is wide in hertz. So the choice of whether to group is doing more work than the parameter swept above — which is the ordinary situation with a sensitivity analysis, and the reason to run one is to find out which knob it was.

What the last bar is doing instead #

If the tempo is not carrying the effect, something is, and the sweep says what: the jump between the last bar’s texture and the one before it, read against a running impression that has already settled.

That jump is a fixed number of decibels because it is a fixed change of part count, and a part count is not a duration. Six parts to nine is about a decibel and three quarters of level and rather less than that of loudness, because a chord is not as loud as its notes — the critical band groups what is added. Two parts to one is a much larger fall and is the ending that scores highest here.

So the arithmetic of a closing gesture is: the running impression sits where the body of the closing phrase put it, and the arrival is measured against that. Everything before the last bar has been absorbed and everything in the last bar has not. It is a two-state model rather than a continuum, and the sweep is what makes that visible — a continuum would have bent.

It also says why the closing length is the parameter that matters and the tempo is not. The reading can only see the last bar, so what the closing length decides is which part of the gesture the last bar is. Two closing bars put the sixth-to-fifth part step there, six put the second-to-first step there, and eight put nothing there at all. A tempo cannot change which step is last; a bar count is the only thing in the model that can.

The one place a tempo term would show #

There is a version of the question where the tempo would matter, and stating it is the useful thing this rung can do for the next one.

The absorbed-or-not boundary is at the release, about two seconds. A gesture longer than that is absorbed and a gesture shorter is not. In bars, that boundary moves with the tempo: at half a second to the bar it is four bars, and at five seconds to the bar it is under half of one.

So a composer writing a closing diminuendo at a fast tempo has four bars to spend before the listener’s running impression starts following them down, and one writing at a slow tempo has none. That is a real compositional constraint and it is tempo-dependent — and it is not what the sixth rung’s reading measures, because that reading is taken at the arrival and by then both cases have settled.

Measuring it needs a different quantity: not the ratio at the arrival but the area between the sounding curve and the heard one over the closing bars, which is how much contrast is available to be spent rather than how much is left at the end. That is one integral over the same two curves, and running it finds the tempo term the arrival reading does not have.

A factor of between three and a half and four across the same sweep the arrival reading crossed at four per cent, and — the part worth having — one pair of the four changes places. The thinning leads at every tempo and the plain loud final chord is second at every tempo, so the two gestures the reading separates most are separated on this measure too. What moves is the bottom pair: at half a second to the bar the diminuendo-into-a-loud-chord has more available contrast than doing nothing, and by two seconds it has less, and by five it is last of the four outright.

Two seconds a bar is not an arbitrary place for that crossing. It is where six closing bars occupy twelve seconds, which is six releases — the point past which the running impression has followed the whole diminuendo down and there is nothing left in it that a listener has not already absorbed. Below it the gesture is partly unspent; above it, entirely spent, and the six bars of thinning have bought less than not thinning at all.

So the sixth rung’s worry was justified about the mechanism and mistaken about which reading would show it. Tempo moves what is left at the arrival by four per cent and moves what was available to spend by a factor of four; a rung that measured only the first was measuring the quantity the tempo can barely reach.

The reordering has a musical statement in it, and it is not a comfortable one for the gesture this ladder keeps reaching for. The diminuendo into a loud final chord is last of the four on this measure at every tempo slower than two seconds to the bar, which is most of the repertoire, and it is last because it stops thinning one bar short of the bar that was worth thinning into. The instruction is not too long; it is aimed one bar wide.

Which computation produced the numbers #

The reading is endingDynamics, unchanged from the sixth rung: each ending is a texture instruction over the closing bars, the dynamic curve is scoreDynamics — count the parts, realise them in their ranges, put every partial in its critical band, sum to sones, convert back to a level — and the running impression is loudnessTrack with the published two-part smoother.

Two arguments are new. secondsPerBar was a fixed two and is now swept; the reading point was a fixed 0.3 seconds into the last bar and is now fifteen per cent of a bar. The second change is deliberate and it makes the comparison harder rather than easier: a fixed absolute reading point would have produced a spurious tempo dependence of its own, since a fast bar would be read proportionally later in it. It has a cost, which the residual above is, and the cost is smaller than the artefact it removes.

The third argument is not new and is the one this essay is really about: the number of closing bars the texture instruction covers, which the figure runs at two, four and eight. The available-contrast table below holds it at the sixth rung’s six.

The area in that table is the integral of the difference between the two smoothers over the closing bars, in sone-seconds — how far apart what is sounding and what is remembered were, summed over the whole gesture rather than sampled at its end. It is the quantity the arrival reading is a single sample of, and it is reported unsigned because a gesture that opens a gap and closes it again has spent the contrast either way.

The realisations are sampled at twenty-four per bar rather than the sixth rung’s ninety-six, because this figure asks for eighteen whole curves. The two counts agree to under a thousandth, which is checked rather than assumed — the quantity is a mean over voicings and converges quickly.

Where the model stops #

A tempo is not only a bar length. Everything here changes the seconds per bar and nothing else, so a fast ending in this model is a slow ending played faster, with the same number of parts, the same voicings and the same texture instruction. A real fast ending is scored differently, and the closure ladder’s third rung is entirely about the fact that a real ending slows down — a ritardando means the last bars are not at the tempo the rest was.

A gesture that runs out is still a gesture here. Drop a part a bar from six and there are five drops available; asked for eight, the instruction spends them and then repeats a single line three times. The eight-bar curve is therefore a five-bar gesture with three bars of silence-but-for-one-voice after it, which is a real way to end a piece and is not the instruction the figure’s legend claims. Nothing in the arithmetic refuses to be asked for more bars than the texture has steps, and the twenty per cent that makes this essay is partly a report on that.

The reading point is a convention. Fifteen per cent into the last bar is a stated choice and the six per cent bowl is a property of it. There is no principled place to read a running impression against an arrival, and the honest version of that is that the arrival is not an instant.

And the release is one number. Two seconds is the published figure for a broadband sound at a moderate level, and it is not a constant of nature: it depends on level, on bandwidth and on what came before. The whole argument here is a comparison of durations against that one number, so a release of one second or four would move the boundary between an absorbed gesture and a preserved one.

What the picture cannot show #

It cannot show a listener who is following the form. Every quantity here is a sensory running average with no knowledge of where it is in a piece. A listener at the end of a movement knows an ending is coming, which is what the fourth rung of this ladder is about, and expectation is not a smoother.

The ritardando is now in, and it turns out not to be a second variable. Lengthening the closing six bars by the third rung’s own kinematic model, down to half tempo at the final bar, moves every arrival reading by at most 0.013 — because to a smoother a ritardando over the closing stretch is that stretch at a slower tempo, which is precisely what the sweep already covered and covered ten times as far. The two things this rung separated are one variable seen twice, and the honest consequence is that a real decelerating ending is at some interior point of a curve that is flat anyway.

And it cannot show a tempo that is not a bar length. Doubling the notes per bar at a fixed bar length changes the dynamic curve through the texture rather than through the clock, and the model would report it as a change of scoring.

Whose music, and when #

The tempi swept run from four hundred and eighty beats a minute to forty-eight, which is wider than any single repertoire uses and is the point: the result is that nothing in the range matters, so the range is drawn wide enough to make that a claim.

The gestures are the ones the sixth rung chose and they are common-practice European: a texture that thins to one part, a full final chord, a diminuendo into a loud arrival. All four are instructions a nineteenth-century score states explicitly and an eighteenth-century one leaves to the performer, which is the notation ladder’s own subject — so a claim about what a written ending does is a claim about a repertoire in which endings are written.

Where this ladder goes next #

Seven 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 strongly asymmetric between adding parts and taking them away; and now the parameter the sixth rung was worried about, turned, and found not to matter — beside a parameter nobody named, in the same figure, that does.

What is owed after this is the deceleration, and the sweep is what makes it worth doing. Holding the tempo fixed across the closing bars is now known to be nearly harmless; letting it move is a different question, because a ritardando lengthens exactly the bars in which the gesture is happening, so a diminuendo that would have been absorbed at a steady tempo is given more of the smoother’s own time to be absorbed in. The third rung has the deceleration curve with its one free parameter and this rung has the dynamic reading, and joining them would say whether a ritardando helps a closing diminuendo or spends it.

And there is a sharper question underneath both, which this rung can state and not answer. Every curve above is explained by the same sentence — the reading sees the last bar and the closing length decides which bar that is — and a sentence that explains a tempo null, a four per cent bowl and a twenty per cent climb at once is either the mechanism or a coincidence. What would separate them is a sweep of the delay at which the reading is taken, because a quantity that is a step read on an exponential has a time constant in it and a quantity that is a shape does not.