In a hall the balanced ending is a louder chord

No rest balances a louder chord held past a ninth of a second asked when a passage ending with a rest and then a final chord a step louder finishes exactly at the weight of its own tutti — its listener’s running impression of loudness, once the chord is released, neither above the tutti nor below it. In a dry room there was such an ending, but only a narrow one. A stamp a tenth of a second long is balanced by a rest of 2.47 seconds, and a stamp longer than 115 milliseconds is not balanced by any rest, because even an impression that has fallen all the way to silence climbs past the tutti inside it.

That essay’s lead put a room round the ending, and predicted the worst. 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 like a longer one. Both push toward the region where no rest balances anything, and the prediction was that in a concert hall the ending that finishes at its own weight might not exist at all — an ending only a recording could have.

The reading the lead specified settles it the other way. Read when the chord’s tail has died into the silence, a hall’s tail is a diminuendo the listener follows, and it pulls every ending down. In a small room the prediction holds; in a hall of more than about a second and a quarter, every stamp finishes below its tutti after no rest at all, and the balanced ending is a held chord marked louder than the passage — by an amount the hall sets and the chord’s length hardly touches.

The same two smoothers, with a tail in front and a tail behind #

The listener is the closing essays’ listener: loudness in sones through two smoothers, a fast one that tracks the sound and a slow one that tracks the fast one, and the slow one read as the running impression of the passage. It rises with a constant of ninety-nine milliseconds and falls with one of two seconds. The tutti is the closing essays’ full orchestral chord at its steady loudness, and the final chord is struck a step louder, f to ff, which on the struck-string spectrum through the ear’s excitation pattern is 7.12 phons.

The room is one number, its reverberation time: when a sound stops, the level at a seat beyond the critical distance falls sixty decibels in that many seconds, until it reaches the room’s own silence. During the rest, the smoothers are fed the tutti’s tail instead of silence. After the stamp’s release they are fed the stamp’s tail. The reading is taken when that tail has fallen to the silence, which is when the ending is over for the listener. With a reverberation time of nought this is the earlier essay’s computation exactly, which the figures check to the last digit.

A short stamp in four rooms #

The stamp of the dry balance, a tenth of a second long, behaves three different ways in three rooms.

In a small room, 0.6 seconds of reverberation, the prediction holds. The rest is filled by the tutti’s tail, so after eight seconds of rest the stamp still leaves the impression 0.08 phons above the tutti, and no rest balances it. The tail after the stamp is too short to pull it back down: it lasts about 0.6 seconds, much less than the impression’s two-second release. The room has taken away the depth the rest used to buy and given nothing back.

In a room of one second, the stamp leaves the impression 0.32 phons above the tutti after no rest, and a rest of a third of a second balances it. The stamp’s own tail now lasts about a second, long enough for the impression to follow it partway down, and that fall does the balancing the rest used to do.

In a hall of two seconds, the stamp leaves the impression 4.65 phons below the tutti after no rest, and every rest makes it lower. The tail after the stamp lasts two seconds, and the impression follows it down the whole way.

The tail is a diminuendo #

Watching one ending sample by sample shows both halves of the prediction and why they do not add up the way it expected. Through a one-second rest the dry room lets the impression fall to 6.7 phons under the tutti; the hall’s tail holds it at 3.7 under. That is the first half of the prediction, and a rest needs a dry room had already priced it: a hall takes much of a silence’s value away. At the release the hall’s ending is also higher, 2.5 phons over the tutti against 1.7 dry, because the stamp started from a higher impression. That is the second half.

Then the dry ending stops and the hall’s goes on. For two more seconds the stamp’s tail falls, and the impression, whose release constant is two seconds, follows it. By the time the tail has reached the silence the impression is 5.1 phons under the tutti. The prediction treated the stamp’s tail as lengthening the stamp. To a listener whose impression is a running average, a tail is not more of the chord. It is a diminuendo from the chord to silence, and a diminuendo pulls a running average down.

Why the small room is the one the prediction described #

The three regimes come from comparing two times: how long the room’s tail lasts, which is roughly the reverberation time, and how fast the listener’s impression lets go, which is its two-second release constant.

In a small room the tail is much shorter than the release. During the rest it is long enough to hold the impression up for its first half-second or so, which is most of what a short rest was worth, and after the stamp it dies before the impression has moved much. So the room takes depth away from the rest and gives nothing back after the chord, which is exactly the prediction’s mechanism. In a room of 0.6 seconds, a tenth-of-a-second stamp after eight seconds of rest still finishes 0.08 phons heavy.

In a hall the tail is as long as the release or longer. The listener’s impression then follows the tail most of the way down, and the fall after the chord outweighs whatever the rest lost before it. The crossover lies where the two times are comparable: somewhere between 0.6 and 1.25 seconds of reverberation, which is the band in which a short rest still balances a short stamp, and it is narrow.

That is also why the earlier essay’s ceiling of a ninth of a second barely moves when the ending is read at the release. At the release, the room has had time to fill the rest but not yet to pull the impression down, so only the first half of its effect is visible. Every effect this essay finds on the second reading is in the part of the ending the first reading does not see.

A rest, then a diminuendo that nobody wrote #

The closing essays found early on that a rest is a diminuendo: to a running impression, a silence and a written decrescendo are the same kind of thing, a stretch in which the level falls. The hall adds a second diminuendo, one the composer did not write, after the final chord. A silence long enough to be an ending asked how long the silence after the last note must be before a listener takes the piece to be over. In a hall, the first seconds of that silence are not silent; they are the room’s decay, and the listener’s impression is falling through them.

So an ending in a hall has a shape the score does not show: the tutti, a rest that the room partly fills, the final chord, and a decay the length of the reverberation time. The last of those is the longest, and it moves the impression the furthest. A composer marking a final chord louder than the passage in a live building is, on this arithmetic, compensating for the part of the ending the building writes.

Which reading is the ending #

That turns on when the ending is read, and the two choices give opposite answers.

Read at the release, which is how the dry essays read every ending, the hall does what the prediction said, only gently. The balancing rest grows from 2.51 seconds in a nearly dry room to 3.23 in a room of three seconds, and the ceiling on the stamp stays near a ninth of a second. On this reading a hall makes the balanced ending a little harder and never impossible.

Read as the tail dies, the lead’s reading, the rooms divide into three. Below 0.8 seconds of reverberation no rest balances the stamp at all. At 0.8 and 1 second a rest of 1.82 and 0.32 seconds does. From 1.25 seconds up the stamp finishes light with no rest, so the balanced ending needs something that adds weight, not a rest that takes it away.

The second reading is the one that corresponds to what a listener has when the music is over. In a dry room the two readings are the same instant. In a hall they are two seconds apart, and everything the listener hears in those two seconds is the tail. A final chord stands out for a twentieth of a second found how briefly a chord stands above the impression; this finds that in a hall what follows the chord matters more than the chord.

The length of the chord stops mattering #

The earlier essay’s whole argument was about stamp length: a chord shorter than about a ninth of a second could be balanced, and one longer could not. In a hall, length stops mattering once the chord has lasted half a second. A chord held half a second, one second or three all end where the impression has climbed to the chord and the tail has carried it back down, and that depends only on the room.

Dry, a held chord a step louder leaves the impression at the full 7.12 phons over the tutti, since nothing brings it down. In a room of 0.6 seconds it leaves it at 4.09, at one second at 1.88, at 1.5 seconds at 0.85 under and at two seconds at 3.50 under. The room subtracts a fixed amount from the ending, and the amount grows with the reverberation time: about 3 phons at 0.6 seconds, 5 at one, 8 at 1.5 and 10.6 at two.

How much louder #

That fixed subtraction is what the first figure turns into a marking. To finish at its tutti’s weight in a room, a held final chord must be louder than the tutti by about as much as the room will take away: 1.1 phons at 0.3 seconds of reverberation, 2.7 at 0.6, 5.0 at one second, 8.1 at 1.5, 11.7 at two and 15.7 at 2.5. It barely matters whether the chord is held half a second, one or two.

A dynamic step, f to ff, is 7.12 phons on this spectrum. So a final chord a step louder than the passage balances in a room of about 1.3 seconds, which is a small concert hall, and one two steps louder balances at about 2.3, which is a large one. In a room drier than a second the louder mark overshoots and the passage finishes heavy; in a cathedral even two steps undershoot.

The closing essays began with what makes an ending an ending and have kept returning to the final chord as the thing that has to stand out. In a hall, on this arithmetic, the louder mark written on so many final chords is not only emphasis. It is roughly the correction the room requires for the passage to finish where it was.

What the room model leaves out #

One reverberation time. A hall’s reverberation is longer in the bass than the treble, and the tail of a chord is not one decay but a spectrum of them. The loudness model reads everything at one kilohertz, so the arithmetic takes a single time as the hall’s.

A seat past the critical distance. The tail is the full level of the chord falling at the room’s rate, which is what a listener hears beyond the critical distance. Close to the stage the direct sound stops at the release and only a fraction of the level remains to decay. For such a listener every effect in this essay is smaller, and only the player hears a staccato end says where that transition lies.

The tutti’s tail under the stamp, and the release of the tutti. When the stamp enters, the tutti’s tail is still sounding beneath it. A step louder, the stamp dominates, and the model drops the tail’s contribution; with a quieter final chord that would not be safe.

The same listener. The smoothers’ constants are the closing essays’ and are measured for loudness, not for the sense of an ending. Whether a listener’s feeling of finishing tracks the running loudness through a tail, or stops at the release, is a question about listeners, and the two readings above are its two answers.

Whose endings #

The endings at stake are the loud ones: the orchestral final chord marked ff after a f passage, the organ’s last full chord in a stone building, the choir’s final chord held into a resonant church. A general pause is spent by the note after it found how little a silence buys once something sounds. In a hall the arithmetic here says that the final silence, after the chord, is not a silence either but the room’s diminuendo, and that the marking on the chord before it is what decides where the listener ends up.

The organ is the clearest case, because an organist cannot mark a chord louder by playing harder; the only way to a louder final chord is to add stops. The habit of drawing out the full organ for the final chord of a piece played in a stone church is usually described as grandeur. On this arithmetic, at three seconds of reverberation the room already takes about twenty phons from the ending, more than any change of dynamic a score marks, and a listener’s impression, followed through the long decay, ends below the passage’s unless the last chord is very much louder than anything before it. What remains to the player in such a building is the length of the chord, and the figure on chord length says that past half a second the length no longer helps.

Still open: whether the ending is heard at the release or at the silence #

Everything turns on the moment of reading, and the two readings give opposite answers. A listener might judge the ending when the players stop — the release — and hear the tail as the room, not the music. Or the ending might be judged when the sound is over, with the tail heard as part of it. The computation that would decide it is not arithmetic but a comparison: the same ending recorded in a dry room and in a hall, with the final chord’s marking adjusted until listeners judge the passage to finish at the weight it had. If the adjustment tracks the tail reading, a hall’s louder final chord is a correction; if it tracks the release reading, it is emphasis, and the room is being heard as a room.