A final chord stands out for a twentieth of a second

A general pause is spent by the note after it found that a listener’s running impression of loudness rises twenty times faster than it falls, so a silence written before a diminuendo keeps none of its value. Every figure in that essay, and in the four before it, read the impression at one instant: the moment the final chord arrives. The chord was treated as a point.

It is not a point. It has a length, and the length is one of the few things about an ending a composer writes down precisely — a crotchet with a fermata, a minim, a quaver followed by rests. What the earlier readings leave out is what happens during the chord, and the smoother that produced every number above has an answer to that as well, because it has two attack constants rather than one.

The note’s own loudness — what the published model calls short-term loudness — rises with a constant of twenty-two milliseconds. The listener’s running impression of the passage, the long-term loudness, rises with a constant of ninety-nine. A final chord that arrives into a fallen impression is therefore heard, for a while, as louder than the passage it ends: its own loudness has arrived and the impression has not caught up. How long that while lasts, and how much of the silence it gives back, is the question here.

The stand, and why it has a peak #

Define the chord’s stand as how far its own loudness sits above the running impression, in the units the closing readings have used throughout: ten decibels per doubling of sones. Before the chord the stand is the depth of the prelude’s fall. The instant the chord begins, both smoothers start to rise toward its level.

The note’s loudness gets there first. Nine tenths of the way in fifty milliseconds is what a staccato is a dynamic mark measured for a note out of silence, and it is the same constant here. The impression starts later in effect, because its input is the note’s loudness rather than the sound, and it rises four and a half times more slowly. So the stand first grows, while the note climbs faster than the impression, and then shrinks, while the impression closes a gap the note has stopped widening.

That makes the stand a curve with a maximum, and the maximum is about a demisemiquaver long. After a bar of silence it comes 54 milliseconds into the chord. A chord released at that instant is standing as far above the impression as any final chord ever will out of that prelude; a chord released later is standing less far, and one held for half a second is standing by six hundredths of a decibel.

The chord that stands above nothing is also worth a sentence. A final chord at the tutti’s own level, arriving straight out of the tutti, has an impression already at its level and an own loudness at the same level. Its stand is zero at every length, which is the arithmetic form of what the reading was a step response found three times over: nothing that happens inside a continuous stream at one level moves the running impression.

The share of the silence a chord can take back #

The stand’s peak is always smaller than the fall that made room for it, and by a nearly fixed proportion.

Across six silences from a third of a second to five seconds, and three diminuendi from three decibels to twelve, the best stand is between 59 and 68 per cent of the fall. A bar of silence buys 8.09 decibels of room and the chord uses 5.14 of them, 63.6 per cent. Three and a half seconds buys 23.69 and the chord uses 14.42, 60.9 per cent.

The share is not exactly constant, and the way it moves is the mechanism. The deeper the fall, the further the impression has to climb, the faster it climbs in decibels per millisecond, and the sooner it overtakes the note’s own rise — so the peak arrives earlier, 86 milliseconds after the shortest silence and 20 after the longest, and it captures a slightly smaller share. The two constants set the ceiling at roughly two thirds; the depth moves it by a few points.

No final chord ever stands above the impression by the whole of what the silence bought. The third of it the chord cannot reach is the part the impression recovers while the note is still arriving — which is the attack-side counterpart of the general pause essay’s result that a resumption spends half a pause in ninety-seven milliseconds. There the resumption was music the composer wrote between the silence and the chord. Here it is the chord’s own first twentieth of a second.

Two quantities that a chord’s length trades #

A stand that peaks early would make the short final chord the better one, and on the stand alone it is. But a closing chord is heard twice by this model, and the second reading goes the other way.

The first reading is the stand: how far the chord is heard above the passage while it sounds. The stamp stands out for 80 per cent of its life. The held chord stands out for 206 milliseconds and then sounds, for more than a second, at exactly the level of the impression it has built — which is to say it no longer stands out at all.

The second reading is the impression the chord leaves behind, which is the passage’s loudness as a listener would report it. The held chord drags the impression all the way up to its own level, 79.4 phons. The stamp releases before the impression has caught up and leaves it at 75.7 — 3.7 phons short of the level the chord was played at.

So the length of a final chord is a choice between two readings of loud. A short chord is heard as louder than the music around it and leaves the passage quieter than it was. A long chord is heard as no louder than the music it has just become and leaves the passage exactly as loud as the chord. The two curves cross at no length: every added millisecond past the peak lowers the stand and raises the impression.

The other stand, after a long silence #

The trade has the same shape after any prelude, and a longer silence stretches both halves of it.

After a silence long enough to be heard as an ending, the stamp spends 96 per cent of its life standing out, and the held chord stands out for about a quarter of a second rather than a fifth. The price of the stamp is larger too: the impression it leaves is 6.3 phons short of the chord, against 3.7 after a bar of silence, because it had further to climb.

The shape of the trade is therefore a fact about the two constants and not about the particular prelude. What the prelude sets is the scale: how much stand there is to have, and how much impression a short chord fails to rebuild.

What a second stamp stands on #

A great many classical movements end not with one chord but with two or three short ones separated by rests — the tonic stamped, a rest, the tonic stamped again. On the arithmetic above that is a very specific device: each stamp takes its stand, and each stamp raises the impression that the next one has to stand above.

The first stamp raises the impression by several phons, and the rest lets it fall at the smoother’s release — 7.2 decibels a second in the limit, the constant a rest is a diminuendo found in every closing reading. So the second stamp stands on whatever the rest left.

Put in note values, the result is concrete. After a quaver rest at 108 beats a minute, 278 milliseconds, the second stamp stands 3.35 decibels above the impression against the first’s 5.14. After a crotchet rest at 72, 833 milliseconds, it stands 5.76 — slightly more than the first, because the rest has taken the impression below where the first stamp found it. The rest that makes two stamps equal is 690 milliseconds, a crotchet at about 87 beats a minute; any shorter and the second stamp is the weaker, any longer and it is the stronger.

Shortening the stamps changes the answer more than lengthening the rest does. A stamp of fifty milliseconds is still at its peak stand, because the peak comes at fifty-four; but it raises the impression much less, so the chord after it has less to overcome. Two fifty-millisecond stamps a third of a second apart stand equally. Two hundred-millisecond stamps need twice that rest.

The length that serves both readings #

The two readings pull in opposite directions, but not at the same rate, and that is what makes a length worth naming.

The stand’s duration stops depending on the chord almost immediately. After a bar of silence a chord is more than a phon above the impression for 30 of its first 50 milliseconds, 80 of 100, 180 of 200 — and then for 206 milliseconds whatever its length, because by then the impression has caught the note and nothing the chord does afterwards separates them again. Every chord longer than about a fifth of a second has bought all the stand it will ever have, and every millisecond after that is spent on the second reading alone.

The impression the chord leaves behaves the other way. It is 4.7 phons short of a fully arrived chord after a stamp of 50 milliseconds, 3.7 after 100, 1.3 after 200 and 0.07 after 500. Most of the climb happens in the first two hundred milliseconds, because the impression rises with a constant of ninety-nine and has covered about seven eighths of any gap in two of them.

Put the two columns side by side and the trade has a knee:

A final chord of about a fifth of a second has nearly all of both. It stands out for nine tenths of its length, and it leaves the passage’s loudness within a phon and a half of its own. Shorter, and the impression it leaves falls away quickly; longer, and the stand is diluted by a held chord that no longer stands out. A fifth of a second is a quaver at 150 beats a minute, or a semiquaver at 75 — which is, in round terms, the note value classical final stamps are written in at the tempos those movements are played.

That is a coincidence of the right order of magnitude rather than an explanation, and the arithmetic cannot say more than that. It says a composer who wanted an ending that is both heard above the music and heard at its own full weight would have to write something close to that length, and it says nothing about whether anyone wanted that.

What the arithmetic does not know about #

The chord is a level, not a sound. The note here switches on at the tutti’s level and switches off, and both smoothers read that step. A real final chord has an attack of its own — a bowed chord takes tens of milliseconds to reach its level and a note is heard after it starts by about that much — and a struck chord decays from the moment it sounds. Both would move the peak later and lower, and a struck chord’s decay would make the held chord’s stand shorter still.

A real ending does not stop dead. Every chord here is followed by silence at a quiet hall’s background, so the impression it leaves is read in a dry room. A rest needs a dry room put a hall’s reverberation under a silence and found it takes 40 per cent of a bar of it back; the same tail would extend a short stamp and raise the impression it leaves, so a stamp’s price is smallest in exactly the rooms where its stand is smallest too.

The chord is at the tutti’s level. A final chord written louder than the passage — a fortissimo stamp after a forte tutti — adds a stand that does not decay, since the impression can never rise past the chord. Nothing above prices that, because the closing essays have all asked what an ending can do at one dynamic, and the answer to a louder final chord is the dynamic itself.

And the smoother is the published one, unfitted to music. The two attack constants are Glasberg and Moore’s, fitted to judgements of amplitude-modulated and time-varying sounds rather than of chords. The ratio between them is what decides the peak and the two-thirds share; a different pair would move both.

What a curve of stands cannot tell a listener #

Whether standing out is what an ending wants. The arithmetic prices a quantity and is silent on its value. A final chord that is heard as louder than the passage and leaves it quieter is a different rhetorical object from one heard as no louder and leaving the passage at full strength, and a composer can want either.

How the stand is heard. Five decibels above the running impression for fifty milliseconds is a contrast the model registers. Whether a listener hears it as an accent, as a sharper attack, or as nothing separate from the chord at all is not in the model. Playing louder is playing earlier found a louder note heard as an earlier one, which is one plausible way the same few decibels could reach a listener without being heard as loudness at all.

The harmony of the stamps. Two tonic stamps and a dominant-tonic pair are the same levels to this arithmetic and very different endings. An ending is five components with no total, and the stand belongs to one of the five.

Whose endings, and how they are written #

The short final stamp is a classical and early romantic habit. The ends of a great many Haydn, Mozart and Beethoven movements are a chord or two of a quaver or a crotchet, with rests between them and a rest after — and a fermata on the last chord is common but by no means universal. The held final chord with a fermata, lasting as long as a conductor cares to hold it, becomes the default later, and in slow movements and choral music it always was.

On this arithmetic the two habits are not two lengths of the same gesture. The stamp is the ending that stands out, spends most of its life doing so, and leaves the passage a few phons quieter than it was played. The held chord is the ending that stands out for a fifth of a second and then becomes the loudness of the whole passage. A composer choosing between a quaver and a fermata is choosing which of the two readings of the smoother the ending is addressed to.

The stamps-with-rests device has a sharper claim on the numbers. At the tempos such endings are played, rests between stamps are a quaver or a crotchet long, and the arithmetic puts the rest that makes two hundred-millisecond stamps equal at 690 milliseconds. A quaver rest at 108 gives the second stamp about two thirds of the first’s stand; a crotchet rest at a slow one gives it more than the first. Nothing here says which the repertoire prefers, and a count of endings by tempo and rest length would settle it.

Still open: a final chord written louder than the passage #

Every final chord in the closing essays sounds at the tutti’s level, and in much of the repertoire it does not: the last chord is marked a step louder than what precedes it, or a sforzando, or a fortissimo after a forte. A louder chord adds a stand that the impression cannot close, since the impression can rise only to the chord’s own level, and it raises both of the readings above at once — the stand while the chord sounds and the level the passage is left at. The mark that is not a level found that a louder dynamic changes a struck note’s spectrum rather than only its level. Joining the two is what would say whether a louder final stamp is heard as a louder ending or only as a brighter one, and both halves of that arithmetic are already worked out.