The page can write a diminuendo and not a climax

A page has twelve phons, and its entries always said so found that the number every essay on endings had leaned on — that adding parts to a final chord moves its loudness by under two decibels — was an artefact of how the loudness had been counted. The grouped-band model sorts a dense chord’s partials into a handful of groups, and an eight-part chord’s sixty-four partials into two, so the more parts it is given the fewer bands it thinks they occupy. Counted along the ear as an excitation pattern, the same ending built up from one part to eight gains twelve phons, rising at every step.

That essay counted chords that had been sounding long enough to reach their full loudness. It ended by pointing at the essay that had computed a whole passage’s dynamic curve from its texture alone and passed it through the two running smoothers — the dynamics are in the score already — on the grouped model. The same computation on the excitation pattern, applied to the four ending textures the closure essays compare, would say how much of a written ending’s dynamic shape is on the page before anybody plays it, and whether the closure component that had been declared unreachable from a score is, after all, mostly there.

The answer splits cleanly in two, and the split is the finding.

Four endings, from the page #

The four endings are the closure essays’ own. A thirty-two-bar song form runs at a texture of six parts, every part a string tone at 62 decibels, until its last six bars. Then the texture either thins to one, losing a part a bar; or holds and puts a full final chord of three more parts on the last bar; or stays unchanged; or thins then full, losing a part a bar and arriving at a final chord two parts thicker than the texture began. Each bar’s parts are realised every way their ranges allow, the partials of each realisation are summed as an excitation pattern, the mean loudness is turned into a level the way the phon is defined, and the bar-by-bar levels are passed through the short-term smoother — which follows the music within a fifth of a second — and the long-term one, which is the listener’s running impression and comes down over seconds, as loud is relative, and it comes down slowly established.

The thinning ending is a staircase down, a step a bar, each step larger than the last: the short-term loudness falls from 20 sones to 10 over the last six bars, and the running impression trails behind it, so that when the final bar arrives what is sounding is 0.69 of what the listener has been hearing — four and a half phons under the impression. The fuller final chord rises at the last bar to just above the unchanged texture: 1.004 of the impression on arrival, a twentieth of a phon. The ending that thins and then arrives full climbs from the bottom of its staircase to a final chord at 1.014 of an impression the thinning has lowered: under two tenths of a phon.

So on the excitation pattern the four endings are in the order anyone would expect: the thinning quieter, the unchanged texture level, the fuller chords louder. That alone is a correction.

The two endings the bands had backwards #

On the grouped bands, the fuller final chord arrives at 0.952 of the running impression and the thinning-then-full ending at 0.929: both endings that add parts arrive quieter than what preceded them. That is the result a final chord is not made loud by adding to it reported, and it was a striking one — that a composer who thickens the final chord makes the ending softer. It came from the grouping. A nine-part chord’s partials are folded into fewer bands than a six-part chord’s, and each band is converted to loudness as one intense sound, so the denser chord counts as quieter. On the excitation pattern the fuller chord is louder, by a little, which is what it physically is.

The thinning ending is the right way round on both models, but its size is not. Bar by bar, the difference is plain.

On the pattern the thinning ending runs 83.3 phons in the full texture, then 82.5, 81.6, 80.1, 78.2 and 73.2 at its single last line: ten phons from the page alone. In grouped bands the same bars wander between 79 and 81 and drop to 76.2 only at the very end: four. The full final chord gains 1.4 phons over the bar before it on the pattern and loses eight tenths in bands; the thinning-then-full ending leaps six phons from its thinnest bar to its final chord on the pattern, and barely moves in bands.

That ten-phon thinning is the first half of the answer the lead asked for. The page essay found that a page spans twelve phons against forty-seven for a player taking a four-part chord from 40 decibels to 100. A thinning written into the last six bars takes away ten of those twelve. A diminuendo of that size is on the page before anybody plays it, and a listener’s running impression, trailing behind, hears the last bar four and a half phons under what it expected — which is the closure component of “arriving quieter” in the arithmetic of a rest is a diminuendo, here produced by the texture rather than by silence.

Adding is cheap and taking away is dear #

The second half of the answer is in the asymmetry between the endings that subtract and the endings that add.

Three more parts on a three-part texture gain 3.3 phons; on four parts 2.2; on six 0.9; on eight 1.2. The thinning loses 7.3 phons from three parts, 8.1 from four, 9.6 from six, and 3.6 from eight — where dropping a part a bar over six bars only reaches three parts and so never takes away the last, most valuable ones.

Every bar shows the same thing: taking loudness away from a texture is several times more effective than adding it. The reason is the one the page essay found when it priced each entry. The first part into an empty ear is worth nearly five phons; the second about one and a half; every part after the fourth between half a phon and one and a half. Loudness grows with the number of parts in the way a sum of spreading excitation patterns must — quickly at first, slowly once the ear is full. So a texture already full gains little from more, and loses a great deal when it is thinned all the way down, because the last parts removed are the expensive first ones.

A composer who wants an ending to arrive loud by texture is working on the flat part of that curve. A composer who wants it to fade is working on the steep part.

What reaches the listener at the moment of arrival #

The question the closure essays asked was not how loud the final chord is but how it stands against what the listener has been hearing at the moment it arrives. That depends on the tempo, because the running impression has a time constant and the bars do not.

The thinning ending arrives six phons under the impression with half-second bars, five and a half with one-second bars, four and a half with two-second bars and three at five seconds a bar. The slower the music, the more time the running impression has to follow the staircase down, and the less of a drop is left at the end — but at every tempo it is a plain diminuendo.

The fuller final chord arrives three tenths of a phon above the impression with half-second bars and nothing above it from three seconds a bar. The thinning-into-full ending does best: a phon above the impression at half-second bars, six tenths at one-second bars, two tenths at two, nothing at slower tempi. On the grouped bands, both endings that add parts arrive nearly a phon under at every tempo.

So the most a page can make a final chord stand out by texture alone is about a phon, and only by doing two things at once: first lowering the listener’s impression with a thinning, then arriving with more parts than the texture had, quickly enough that the impression has not caught up. At a moderate tempo even that is two tenths of a phon. A climax — a final arrival heard as louder than what the music has been doing — is not something the page can write by adding parts. It has to be played.

With a four-part texture the numbers shift without changing their shape. The fuller final chord arrives at 1.007 of the impression, the thinning into full at 1.023; the thinning ending reaches its single line two bars before the end, so the impression has more time to follow it and the last bar arrives at 0.949. A thinner texture gives the added parts more room, and still under a third of a phon at arrival.

Why the thinning-into-full ending is the only one with an arrival #

It is worth being exact about why one ending manages a phon and the others a twentieth, because the reason says what a composer could do about it.

An arrival contrast is a ratio of two smoothed quantities: what is sounding a moment after the final chord begins, and what the listener has been hearing. The fuller final chord changes only the first. The texture before it has been steady for thirty bars, the running impression is sitting exactly on it, and the final chord rises above it by the 0.9 phons three more parts can add to six. The short-term smoother then takes a fifth of a second to rise, so at the moment of reading only part of even that small step has arrived.

The thinning-into-full ending changes both. Its last five bars lower the short-term loudness by five phons, and the running impression follows it down, more slowly, and has not caught up by the final bar. The final chord then jumps six phons, from the bottom of the staircase to eight parts, and lands above an impression that is still part-way down. The contrast is the gap between where the impression has got to and where the chord is, and it is largest when the bars are short, because then the impression has had the least time to fall and the jump has been prepared by the steepest descent.

That is the arithmetic of a final chord stands out for a twentieth of a second, which found that a general pause before a final chord is cashed in only partly because the chord’s own loudness rises faster than the impression can follow it. Here the thinning plays the part of the pause. A composer who wanted the most a page can give would thin the texture quickly in the bars before the end and arrive full, which is something like what a written subito tutti after a thinning does — and even that, by this count, buys a phon.

The surprising part #

The expectation going in was that correcting the count would rescue the loud ending, since the grouping had been found to undercount dense chords. It does rescue its direction. It does not rescue its size, and the reason is not in the grouping at all but in how loudness grows: a full texture is on the flat part of the curve, and the ear does not much care that more is added to it.

That connects the ending question to an older finding about what makes an ending an ending: the closing components were sorted by which could be read off the page and which needed a performance, and the dynamic one was put in the second group for lack of a model. With a model, it splits. The falling dynamic of an ending — the diminuendo, the thinning to a single line, the morendo — is a property of the page, and a large one. The rising dynamic — the climactic final chord — is not, and no amount of doubling makes it so. A performer who plays the last chord louder is supplying the half of the ending’s dynamic the score cannot hold, and one who plays a thinning ending without a hairpin is still playing a large diminuendo.

What the model assumes #

Every part at one level. All four endings hold each part at 62 decibels, so a thinning loses players and a fuller chord gains them without anyone playing louder. That is the point — the question is what the page does — but it is also why the climax is missing: a fuller final chord written with a fortissimo is a different instruction, and the arithmetic here prices only the first half of it.

The string tone and the realisations. Each bar is the mean loudness over up to ninety-six realisations of its parts in their ranges, every part the collection’s string spectrum. The page essay found the excitation-pattern gain from one part to eight lies between twelve and eighteen phons for every instrument spectrum tried there, so the shape of these results should survive a change of instrument; their size would move.

The smoothers’ time constants. The short-term smoother’s fast attack and the long-term one’s slow release are the ones every loudness essay has used. The arrival is read three twentieths of a bar after the final chord begins, as the tempo essay on closure did; read later, every arrival contrast shrinks towards nothing, because the impression catches up.

Steady parts. Every note is held for its whole bar at a steady level, as in the staccato essay’s baseline before it gave notes lengths. Real endings have articulations and attacks, and a final chord struck is louder in its first moments than held — which is a performance property again.

What the curves cannot show #

Whether the thinning is heard as quieter or as emptier. Ten phons is a large change in loudness level, and a thinning is also a loss of voices, colour and roughness. The page essay raised the same doubt: a listener may attend to the voices going rather than to the level falling. The arithmetic says the loudness falls; it cannot say that is what is noticed.

Rooms. Every curve here is the direct sound with no hall. A reverberant room adds its tail under every bar and slows the fall of a thinning, so a thinning in a cathedral would arrive less far under the impression than it does here; the closure essays priced that for silences, and the same arithmetic would apply.

And the repertoire. The four endings are schematic gestures over a song form, chosen because the closure essays compared them. Real endings combine texture with harmony, register and written dynamics, and the claim here is only about the share of an ending’s dynamic that texture alone can carry: most of a fade, almost none of a climax.

Still open: the room under a thinning ending #

The thinning ending arrives four and a half phons under the listener’s impression with no room at all. A hall’s tail sounds under every bar and holds up the short-term loudness as parts drop out, by an amount that depends on the reverberation time and the bar’s length. The closure essays have the room model — a rest needs a dry room used it for silences — and the computation is this one with each bar’s level carried on through the hall’s decay before the smoothers see it. It would say how much of a written diminuendo survives in a concert hall and a cathedral, and whether a thinning ending is, like a general pause, a gesture that needs a dry room to be heard as written.