A hall leaves a thinning ending most of its diminuendo

The page can write a diminuendo and not a climax computed four ending textures from the score alone — no player getting louder or softer, every part held at the same level, and only the number of parts changing. The loudness of each bar came from summing its partials on the excitation pattern over every way its parts could be voiced, and two smoothers turned that into what a listener has: a short-term loudness that follows the sound closely, and a running impression that lags it, released over a couple of seconds, as loud is relative and it comes down slowly set out.

The ending that thins — a part dropping out each bar over the last six — was the one with a real dynamic. It arrived about four and a half phons under the listener’s impression, a diminuendo written into the score. The endings that added parts arrived barely louder than what came before; a page could take loudness away but could not make a climax.

All of that was computed in no room at all. An ending is heard in a hall, and a hall’s tail sounds under every bar. A rest needs a dry room put a hall under a general pause and found the tail taking back a large share of the silence: a bar of rest keeps 60 per cent of its value in a concert hall and 22 per cent in a cathedral. The lead was that a thinning ending might be the same kind of gesture — a reduction in the sound that the room quietly makes up — and that it too might need a dry room to be heard as written.

The arithmetic says it is a different kind of gesture, for a reason that is simple once it is drawn.

The endings in a hall #

The room enters as a reverberant field. Every bar’s sound feeds it, it follows the direct sound with a time constant set by the reverberation time, and at a seat well back it stands six decibels above the direct sound once a note has been held. So a two-second concert hall raises the whole curve — every bar is louder than in a dry room — and smooths each step down so that the level glides into the next bar rather than dropping.

What matters is not the curve’s height but the contrast at the arrival. In the concert hall the thinning ending arrives 3.83 phons under the listener’s running impression, against 4.46 dry: it keeps 86 per cent of its written diminuendo. The fuller final chord arrives 0.13 phons over the impression, against 0.05, and the ending that thins and then lands on a full chord arrives 0.43 over, against 0.17. Turn the dial to a cathedral’s eight seconds and the thinning arrives 2.34 under; to no room, 4.46.

A thinning keeps far more than a rest #

The three ways of putting the room in bracket the answer. A reverberant field six decibels over the direct sound is a seat well back in a hall. One equal to the direct sound is a seat at the critical distance, where the two balance. And the rule the rest essay used — the level the listener hears may fall no faster than the hall decays, sixty decibels per reverberation time — treats the tail as a floor rather than as added energy.

They agree on the shape. Up to a reverberation time of about a second, a club or a small opera house, the thinning keeps all of its contrast under every rule. In a two-second concert hall it keeps between 86 and 100 per cent; in a four-second stone church between 67 and 90; in an eight-second cathedral between 52 and 66. The floor rule is the most forgiving in ordinary halls and the energy rules in the slowest, but at no reverberation time do they differ by more than a quarter of the contrast.

The dashed line is a two-second general pause in the same rooms, computed the rest essay’s way — a bar at this scheme’s tempo, and so a little longer than the bar of rest quoted above, which is why it keeps more. It keeps 75 per cent in the concert hall and 32 per cent in the cathedral. At every reverberation time a hall takes less from a thinning ending than from a bar of rest — in the concert hall, a tenth or less where it takes a quarter.

Why: a thinning steps down slowly #

The rest and the thinning differ in how far the sound falls and how fast. A general pause drops the sound from playing level to the hall’s background, fifty decibels or so, in an instant; the tail then has to fall that whole distance before the silence is silent, which takes most of a reverberation time, and in that time the listener’s impression has heard the tail rather than the rest.

A thinning drops a part at a time. On the excitation pattern, six parts to five is half a decibel, five to four nine-tenths, four to three a decibel and a half, three to two nearly two. Only the last step, two parts to one, is large: five decibels. A hall’s tail falls sixty decibels in a reverberation time, so in the three tenths of a second before the arrival is read a concert hall’s tail has fallen nine decibels — further than any step of the thinning — and a stone church’s four and a half. The tail has caught up with every step before the listener registers it, except the last step in the slowest rooms. A cathedral falls two and a quarter decibels in those three tenths of a second, so it softens the five-decibel last step and only that.

So the arithmetic answers the lead’s analogy with a ratio. What decides whether a room erases a reduction is the size of the step against how far the tail falls before the step is judged. A rest’s step is fifty decibels and outruns every hall. A thinning’s steps are a decibel or two and are outrun by every hall but a cathedral.

What the smoothers see #

Drawn as the input to the smoothers, the room’s effect is almost entirely a constant. The concert hall lifts every bar by seven decibels — the six of the reverberant field and a little more as it adds to the direct sound — and the thinning’s steps ride on top of that lift nearly unchanged: 83.3 to 73.2 dry, 90.3 to 80.2 in the hall, a ten-decibel fall in both. The cathedral lifts the bars by the same seven decibels and blurs each step into a glide that is still gliding when the next bar arrives; its last bar settles at 81.4 rather than 80.2, which is the decibel and a bit of softening the steps figure predicted.

A constant lift does not change a contrast, or not much. What the listener judges at the arrival is a ratio between two smoothed readings of the same level — the note’s own loudness, which rises with a time constant of about 22 milliseconds, and the running impression, which follows with one of about 99, as a final chord stands out for a twentieth of a second measured them — and a lift added to both nearly cancels. Not exactly: the floor rule, which smooths the steps without lifting the level, leaves a concert hall’s thinning all of its contrast, while the two energy rules, which lift and smooth together, take a tenth or so — so the smoothing that comes with the lift is the likelier source of what a concert hall takes, though the two have not been separated. The running impression and the short-term loudness are both lifted, so their ratio at the arrival — the quantity the ending essays have used throughout — moves only as far as the steps themselves are smoothed. That is why the result is robust to how the room is put in: every way of adding a hall lifts the level, and only the slowest smooth the steps.

The room helps the endings that arrive #

The same mechanism runs the other way for an ending that adds. When a full chord arrives after a thinning, the direct sound jumps at once, while the reverberant field built under the thin bars is low and takes a while to build up to the new level. The short-term loudness follows the direct sound’s jump; the running impression is held back by its own lag. In a dry room the thin-then-full ending arrives 0.17 phons over the impression. In a concert hall it arrives 0.43 over, two and a half times as much, and the gain holds to the cathedral, where it is 0.40. The fuller final chord without a thinning before it rises from 0.05 to 0.13.

That is a real effect and a tiny one. The ending essay found that a page could not write a climax, and a hall more than doubles the climax a page can write without bringing it to half a phon. A fuller chord heard in a hall stands out a little more than it would in a studio. It does not stand out.

Whatever the texture before #

The thinning’s dry contrast depends on the texture it starts from, for reasons the earlier essay explained: from six parts it reaches a single part on the last bar, from eight it ends on three and its last step is small, and from three or four it reaches one part early and the impression catches up before the end. The hall changes none of that ordering. A concert hall leaves 0.19 of 0.18 phons from three parts, 0.67 of 0.63 from four, 3.83 of 4.46 from six, 1.20 of 1.30 from eight. In a cathedral, the two textures that thin to one part early come out with slightly more contrast than dry — the tail of the fuller bars is still fading when the last bar arrives, and the impression has been lifted by it — while the two that end on their largest step lose about a third and a half.

The rule, applied to the other written dynamics #

The comparison of step against decay is general enough to say something about gestures this essay did not compute, and it is worth stating as a prediction rather than a result. A part entering or leaving changes the level by about as much as one step of the thinning — a part entering is not a change of level priced a single entry at under a phon on average — so a hall should leave entries and exits as written. A subito piano is a single large drop, ten or fifteen decibels, smaller than a rest’s but larger than any step of a thinning; a subito piano is four seconds longer in the bass found the listener’s impression taking seconds to follow it even dry, longest in the bass. A concert hall’s tail falls nine decibels in the time a listener takes to register a change, so the rule puts a subito piano in a two-second hall between the two cases here: blunted, but not erased, and more blunted in a church.

The page’s own dynamics — the curve the dynamics are in the score already drew from part counts alone, with no performance in it — are made almost entirely of small steps, because a texture changes a part or two at a time. On that evidence they are the part of a score’s dynamic shape that a hall disturbs least, and a performer’s sudden changes the part it disturbs most.

What the hall changes, and what it does not #

Put together, the figures say that a thinning ending is a gesture a concert hall leaves nearly whole, and a general pause is one it erodes. The difference is not in the rooms but in the gestures: a rest is a single large step, a thinning a series of small ones, and a hall’s decay is fast compared with a small step and slow compared with a large one.

That has a consequence for performance practice that the arithmetic can only point at. An ending that relies on silence — a general pause before the last chord, a long rest after it — is the kind a resonant room spoils, and a general pause is spent by the note after it priced what that silence buys when it works; a performer in a resonant room who lengthens a rest is, on this arithmetic, buying back what the tail took. An ending that relies on a written thinning of the texture needs no such adjustment except in the most resonant spaces, and there only its last step suffers. The diminuendo that the page writes by subtraction is the one dynamic gesture of the four that travels from room to room.

The model, restated #

The bars are the earlier essay’s. Each bar’s loudness is the mean over up to ninety-six voicings of its parts, every part a string spectrum at sixty-two decibels, summed on the excitation pattern and turned into a loudness level the way a phon is defined.

The room is added three ways. As a reverberant field in power that follows the direct sound with a time constant of the reverberation time over 13.8, and sits six decibels over it in the steady state, or level with it; or as a floor under the level, falling at sixty decibels per reverberation time, the rule the rest essay used. The room is added to the level after each bar’s loudness is computed, so it is treated as a 1 kHz tone of equal loudness rather than as the bar’s own spectrum carried on; a tail with the bar’s spectrum would add loudness on the excitation pattern rather than as an equivalent tone, which would change the size of the lift and not, on the evidence of the constant lift, the contrast.

The arrival is read three tenths of a second into the final bar, as before. Read later, every contrast shrinks, because the impression catches up.

What the curves cannot show #

A hall’s colour. Real halls decay faster at high frequencies than at low, so a tail is darker than the sound that fed it, and a thinning that takes away high parts first would be blurred differently from one that takes away low parts. The model’s tail has no spectrum.

Where the listener sits. The two energy rules stand for two seats. A seat close to the stage hears more direct sound than either and would see even less of the room; a seat in the gallery of a cathedral could hear a reverberant field twelve decibels over the direct sound, which the figures do not reach.

What a performer does about the room. The figures hold every part at one level. Players in a resonant room adjust — release earlier, phrase longer — and those adjustments are performance, which every essay on loudness here has left out on purpose.

Still open: a thinning that takes the high parts first #

The tail here has no spectrum, and a real hall’s does: the high frequencies die faster than the low, so a hall’s reverberant field is a darker sound than the music that made it. A thinning ending chooses which parts to drop, and the score usually decides. One that drops the upper parts first leaves the tail’s own darkness to fill the gap; one that drops the bass first takes away exactly what the tail holds longest. The computation is this one with the reverberant field carried band by band at each band’s own reverberation time, fed by each bar’s spectrum, and it would say whether the order in which a score thins its parts decides how much of the diminuendo survives a hall — and whether, in a resonant room, a thinning from the top down is the one that sounds as written.