No rest balances a louder chord held past a ninth of a second

A louder final chord is a deeper silence and a brighter sound found that, to the smoothers that model a listener’s running sense of loudness, a rest before a final chord and a louder mark on it are the same kind of preparation. Each opens a gap between where the listener’s impression of the passage is and where the next sound is, the chord spends its first few tens of milliseconds closing that gap, and it takes back between sixty and sixty-eight per cent of it whatever made it. The gaps add: a rest and a mark together are simply a deeper preparation.

They differ in one place, and it is the place that essay ended on. A rest lets the impression fall, so a short final stamp after a rest leaves the passage heard as quieter than it was played. A louder mark raises the chord, so a louder stamp leaves the passage heard as louder. Put a rest before a louder chord and there should be a balance — a rest long enough that the passage finishes exactly at the weight of its own tutti, while the chord stands out by the sum of both preparations. That essay found one: a stamp a tenth of a second long and a step louder is balanced by a rest of about two and a half seconds.

The question it left was whether that is a curiosity of one stamp or a surface, and what the surface says about the most common written ending of all: a rest, then a loud final chord. The answer is narrower than the question expected. The balance exists only for a chord so short that it is a stamp and not a chord, and past a ceiling of about a ninth of a second no rest of any length reaches it.

A balance for some stamps and not for others #

Take the tutti of the closing essays, a full orchestral chord at its steady loudness, and end it with a rest and a final tonic struck a step louder — f to ff, which on a struck-string spectrum through the ear’s excitation pattern is 7.12 phons. Read, at the instant the chord is released, where the listener’s running impression stands against the tutti’s.

Every curve falls as the rest lengthens, because a longer rest lets the impression fall further before the chord has to lift it back. The fifty-millisecond stamp crosses the tutti after a rest of 0.29 seconds — under a crotchet at most tempos. The seventy-millisecond stamp crosses after 0.73 seconds. The tenth-of-a-second stamp crosses after 2.47 seconds, which is the balance the earlier essay found.

Then the curves stop crossing. A stamp of 120 milliseconds is still 0.59 phons above the tutti after six seconds of rest and is flattening out; a stamp of 200 milliseconds sits at 4.56 phons after six seconds and is barely moving. Neither of them will ever reach the tutti, and the reason is not that the rest is too short.

A silence without end is the whole of the rest #

A rest cannot do more than let the impression fall all the way to the silence. After six or seven seconds it has done that, since the impression releases with a constant of two seconds, and a longer rest changes nothing. So the question of whether some rest balances a stamp is the question of where a stamp lifted out of complete silence leaves the impression.

Out of silence, the impression climbs toward the chord with an attack constant of ninety-nine milliseconds, and the chord’s loudness is above the tutti’s by the mark. So there is a length at which the climb, starting from nothing, reaches the tutti exactly: 115 milliseconds for a chord a step louder, 67 for two steps and 45 for three. A stamp shorter than that is released before the impression has climbed back to the tutti, and some rest balances it. A stamp longer than that is still being held when the impression passes the tutti on its way up, and no rest of any length can lower the starting point far enough to stop it.

The ceiling can be had in one line, which is a check on the figure and a reason for its numbers. A chord a mark louder than the tutti has a loudness that is, in sones, 2 to the power of the mark over ten times the tutti’s, so the tutti is a share of the chord: 61 per cent for one step, 37 for two, 23 for three. An impression climbing out of silence toward the chord with a ninety-nine-millisecond attack reaches that share after the attack constant times minus the logarithm of one minus the share — 93 milliseconds for one step — and it starts climbing only once the note’s own loudness has arrived, about twenty-two milliseconds late. The sum is 115 milliseconds for one step, 68 for two and 48 for three, within three milliseconds of the stepped computation in every case. The ceiling is the ear’s long attack constant, rescaled by how much louder the chord is, and nothing about the rest or the passage enters it.

That is the ceiling on the balanced ending, and it is a small number. A ninth of a second is a semiquaver at 135 beats a minute. A louder final chord held as long as a quaver at almost any tempo is past it.

Near the ceiling, milliseconds buy seconds #

Below the ceiling, the rest that balances a stamp is not a gentle function of the stamp’s length.

A step louder, a stamp of twenty-nine milliseconds is balanced by seventy milliseconds of rest, and one of sixty-three milliseconds by just over half a second. From there the curve steepens: ninety-two milliseconds needs 1.75 seconds of rest, a hundred and one needs 2.72, and a hundred and eight needs 3.78. In the last sixteen milliseconds before the ceiling, the balancing rest more than doubles. Two steps louder and three, the curves have the same shape, compressed toward shorter stamps.

That shape has a consequence for how the balance could ever be written. A composer writes the rest and the dynamic, and both are fixed on the page; the length of the final chord in performance, if it is short, is fixed by how the players release it. Near the ceiling, a variation of ten milliseconds in that release moves the rest the ending needs by a second or more. The balance is in the release, not in the rest. A composer who writes a rest of two and a half seconds before a ff stamp has written a balanced ending only if the stamp is played at a hundred milliseconds and not at a hundred and ten.

What the balance buys #

The earlier essay’s motive for looking was that the balanced ending would stand out by both preparations at once while costing the passage nothing. It does.

A tenth-of-a-second stamp balanced by its 2.47 seconds of rest stands 14.57 decibels above the running impression at its peak. The same rest with the chord at the tutti’s own level would stand 10.43, and the louder mark with no rest 4.51. The balanced ending stands out by more than either alone and leaves the passage exactly where the tutti left it. The longer the stamp, the longer the rest it needs and the larger the stand: at 110 milliseconds, balanced by 4.36 seconds, the chord stands 21.42 decibels above an impression that has fallen most of the way to the silence.

At the short end the balance is cheap and almost worthless. A fifty-millisecond stamp is balanced by less than a third of a second of rest, and the rest adds a single decibel to the mark’s stand. So the balanced endings that stand out most are the ones closest to the ceiling — exactly where the balancing rest is most sensitive to how the chord is released.

Through the ending, one sample at a time #

The level reading is taken at the instant the chord is released, and it is worth seeing where that instant sits in the whole gesture.

During the tutti the note and the impression are one line. At the rest the note drops away within a twentieth of a second, and the impression begins its fall at the fixed rate a rest is a diminuendo measured, about seven decibels a second at first and slowing as it nears the silence. When the stamp arrives the note jumps to seven phons above the tutti almost at once and the impression chases it, and for the tenth of a second the chord lasts the gap between them is what a final chord stands out for a twentieth of a second called the stand. At the release, the impression has climbed back exactly to the tutti and turns to fall again.

Dragging the rest shows the balance as a crossing. With half a second of rest the impression is released 2.55 phons above the tutti; with four seconds, 0.87 below. Nothing about the stamp changes; only how far the impression had fallen when it began.

A bar’s rest, at any tempo #

The written form of this ending is a bar’s rest and a final chord, and a bar is a length that depends on the tempo. So the useful table is the one that inverts the earlier question: given a bar’s rest at a tempo, how long must the louder chord be to balance it?

The answer hardly moves. At fifty beats a minute a bar’s rest lasts 4.8 seconds and a step-louder chord of 111 milliseconds balances it; at 168 beats a minute the rest is 1.43 seconds and the chord that balances it is 87 milliseconds. More than tripling the rest changes the balancing chord by twenty-four milliseconds. Two steps louder, the range is 65 to 49 milliseconds; three steps, 43 to 33.

In every case the balancing chord is shorter than a semiquaver at its own tempo — at a hundred and twenty beats a minute a semiquaver is 125 milliseconds and the chord is 95. That is a secco stamp, a chord struck and immediately released. The ending that composers write most often — a rest, then a final chord held for a crotchet or with a fermata over it — is far past the ceiling at every tempo and with every mark, and it cannot be balanced. It ends the passage louder than the tutti was, whatever rest is written before it.

The ending belongs to the articulation #

The surprise is where the question lands. The balance was supposed to be a question about two things a composer writes — a rest’s length and a dynamic mark — and it turns out to be decided by a third that the notation barely records: how long a short final chord actually sounds. That is the same finding, made about passages rather than final chords, that a staccato is a dynamic mark made about passages: the loudness a listener hears depends on how much of each note is sounded, and an articulation mark is therefore a dynamic instruction in disguise. Here the articulation of one chord decides whether the rest before it can balance it at all.

It also makes a claim the earlier essay could not about what that common ending is doing. A final ff chord held after a rest does not end the passage at its own weight and cannot; it ends it louder by very nearly the whole of the mark, since a held chord brings the impression all the way up to its own level. The rest does not undo the mark. What the rest does is deepen the chord’s stand for its first fraction of a second. A composer who wants the passage to end at the weight of the tutti and stand out besides has one way to do it, and it is to make the final chord a stamp shorter than a ninth of a second and put a rest of the right length before it — the right length being something no performer could reliably produce.

The arithmetic under the curves #

The tutti is the closing essays’ full orchestral chord, at the loudness of twelve parts at their written level. The final chord is the six-note tonic from C3 to C5, each note a struck string, whose louder marks are 7.12, 14.21 and 21.23 phons above the tutti for one, two and three steps of hammer velocity, as the earlier essay computed them through the ear’s excitation pattern. The two smoothers are the published ones — a short-term loudness with a twenty-two-millisecond attack and fifty-millisecond release, and a long-term impression with a ninety-nine-millisecond attack and two-second release — stepped every two milliseconds.

A steady tutti leaves both smoothers at its own loudness, exactly, and a silence long enough leaves both at the silence’s; the balance reading starts from those states and steps only the rest and the chord. On the same segments it agrees with the earlier essay’s computation over the whole prelude to the last digit, and the first figure checks that it does. The balancing rest is found by bisection on the level left, the ceiling by bisection on the level left after a silence without end, and the balancing stamp for a stated rest by bisection on the stamp’s length — to the two milliseconds the smoothers step in, which is why two neighbouring tempos can share a balancing stamp.

What the smoothers leave out #

The room. Every number here is a dry passage. In a hall, the tutti’s own reverberation holds the impression up during the rest, and a rest needs a dry room found a bar of silence keeps between 22 and 60 per cent of its dry value in real rooms. A stamp in a hall also does not end at its release: its tail goes on, and the smoothers would treat the stamp as longer than it was played.

One frequency. The smoothers run on a loudness computed at one kilohertz, as every closing essay has done. The chord’s spectrum enters only through the mark’s gain.

The first sixty-eight per cent. The chord’s stand is the gap between two smoothers, and a real listener may not register a gap that lasts a twentieth of a second as standing out at all. A silence long enough to be an ending and what makes an ending an ending both treat a loudness stand as one component of closure among several, and nothing here says how much it weighs.

And a mark is a stated velocity. The factor of two in hammer velocity per step is a convention, as the mark that is not a level was careful to say; a different scale would move every ceiling, though not the finding that a ceiling exists.

Which endings this is about #

The ending in question is the orchestral one of the classical and romantic repertoire: a tutti close, a general pause or a bar’s rest, and a final tonic chord marked louder than the passage — or, very often, the same chord with no rest, which a general pause is spent by the note after it priced separately. The arithmetic says something about the written instruction, and about the difference between a final chord played secco and one held; it does not say which endings composers write, and nothing here has counted them. What it does say is that if an ending of this kind is heard as finishing at the weight of the tutti rather than above it, the reason is not the rest.

Still open: whether a hall lets any ending balance #

The dry balance lives below a ceiling of about a ninth of a second, and a hall moves both sides of it. During the rest the tutti’s reverberant tail keeps the impression from falling, so the rest buys less depth; after the stamp the chord’s own tail keeps sounding, so the stamp behaves as a longer one. Both push the same way — toward the region past the ceiling where no rest can balance a louder chord at all.

The computation is the one here with a room in it: the tutti’s decay through the rest, and the stamp’s decay after its release, fed to the same two smoothers, with the reading taken not at the release but at the moment the chord’s tail falls below the silence. The prediction is sharp. In a dry studio the balanced ending is a stamp of about a tenth of a second and a rest of a couple of seconds; in a concert hall with two seconds of reverberation it may not exist for any stamp at all, which would mean that the one ending that finishes at its own weight is an ending only a recording can have.