A silence long enough to be an ending

The five components of a closure are all binary. A cadence’s bass either moves by a fifth or it does not; the melody either lands on the tonic or it does not; the arrival is either metrically strong or it is not. Four of the five are switches.

Silence is not. A gap has a length, the length can be any number of seconds, and somewhere along that continuum it stops being a rest inside a piece and becomes the space after one. That is the only closure component with a threshold to find, so it is worth finding.

The first attempt, which found nothing #

The obvious approach is to take the beat tracker this collection already has and run it into a gap.

A beat tracker carried through three silent beats does not lose lock; it drifts, at a rate set by how wrong its tempo estimate is. That is the mechanism this whole essay measures, and it has no threshold in it — the drift is continuous and what is discrete is the point at which it exceeds half a beat.

Run the same tracker into a longer gap and the answer is the same. It coasts, at the period it had, indefinitely. Two beats of silence and twenty beats of silence produce identical behaviour, because a first-order corrector with no onsets to correct against has nothing to do but continue.

So there is no threshold in the tracker. That is a real finding and it is the reason the second attempt looks where it does: the length of a silence makes no difference to a beat that is perfectly kept, so whatever difference length makes has to come from the beat not being perfectly kept.

Drift, and a limen that was already measured #

A listener carrying a beat through a silence is carrying an estimate. Tempo discrimination is a few per cent — the published just-noticeable difference for a change of tempo is between about two and a half and six per cent over the useful range — so the rate being coasted at is a few per cent wrong.

A period error accumulates one error per beat. Past half a beat of accumulated phase error, the beat that emerges from the silence is nearer the wrong onset than the right one, and the metre that comes out is not the one that went in. Not because anything lapsed: because it drifted onto the offbeat.

Ten beats is the whole of the arithmetic and it is not a fitted number: half a beat divided by five per cent. At the preferred beat rate of about 550 milliseconds that is five and a half seconds.

The five per cent is the one soft number in it, and the threshold is inversely proportional to it — at the two and a half per cent that the same literature’s better listeners reach, the drift takes twenty beats and eleven seconds, and at six per cent it takes eight beats and four and a half. So the tracker’s half of the answer runs from about four and a half seconds to eleven depending on whose tempo discrimination is used, and it is the lower end of that range that agrees with the psychological present. The two literatures meet at the pessimistic reading of the first, which is worth saying plainly rather than letting the agreement look tighter than it is.

And the same seconds, from somewhere else #

A phrase is a number of seconds established the other quantity: the psychological present, the window inside which a series of events is held as one group rather than remembered as a sequence. It runs from about two seconds to about eight, with three and a half as the value most often quoted.

And a phrase is two to eight seconds, so a silence long enough to outlast a phrase is a silence long enough to end one — the two windows are the same window read from either side.

The two bounds have nothing to do with each other. One comes from tempo discrimination and a first-order tracker; the other from studies of temporal integration in memory, with no beat anywhere in them. They land between three and a half and five and a half seconds.

That is the threshold, stated as honestly as it can be: under about two seconds a gap is a rest inside something; over about five it is after something; between them it is ambiguous, and the ambiguity is real rather than a limitation of the measurement.

What a hall does to the first two seconds #

There is a complication that changes the numbers substantially, and it belongs to a different ladder entirely.

A silence in a room is not silent. The last chord goes on sounding, decaying, for as long as the room’s reverberation time — and the reverberation time of the rooms this repertoire is played in is a large fraction of the threshold just computed.

Two seconds is the time to fall sixty decibels, and what matters is the time to fall below the room’s own background — for a fortissimo chord at eighty-five decibels against a hall background near thirty, that is fifty-five decibels, or 0.92 of the reverberation time. Subtracting it from the threshold gives what is left:

In a cathedral the room finishes the piece before any silence begins. The last chord is still audible for more than seven seconds, which is longer than the whole threshold computed above — so by the time a listener has any silence to measure, the ending has already been declared by the drift of their own beat and by the expiry of the present. The gap after a cathedral cadence is not a silence being judged; it is the tail of the chord outlasting the judgement.

Solving for where that happens gives a reverberation time of about six seconds against the upper threshold and 3.8 against the lower. Above about four seconds of decay a room supplies the ending on its own, and above six it does so under any reading of the threshold — which is a property of the building rather than of the music, and it sorts the repertoire’s rooms into two kinds rather than placing them on a scale.

A shoebox hall at 1.8 seconds of audible decay is the interesting middle: the room takes half of the lower threshold and a third of the upper, so a performer there is spending real time on the room and still has silence to spend afterwards. That is the room in which the length of a closing stop is a decision, and it is the room most of this repertoire is played in.

A hall’s own decay is the other clock in the room: at a 2.2-second reverberation time a chord is still audible for two seconds after it stops, so the first two seconds of a silence are not silent and the count does not start where the score says it does.

Two consequences follow and both are audible practice.

A resonant room lengthens every ending, up to a point and then stops. A cadence in a cathedral needs a longer stop after it than the same cadence in a studio, because the first seconds of the stop are being spent on the room — and past about four seconds of reverberation the additional stop buys nothing, because the ending has already been declared while the chord was still sounding. So the lengthening is a real practice in a hall and a formality in a cathedral, and the arithmetic says an organist waiting out a cathedral’s decay is observing the sound rather than making an ending. Performers do this without being told and organists do it most.

And a dry room makes silence available as a device. A general pause is a strong effect in a dry hall and a weak one in a wet one, which is a reason the device is commoner in chamber music than in the repertoire written for large stone buildings. The table puts a size on that too: a studio has 3.2 of the 3.5 seconds available to spend and a church has none, so the device is not merely weaker in a resonant room — it is unavailable, in the sense that the composer has nothing to write into.

The silence that is not at the end #

The same arithmetic covers a case that is not an ending at all, and running it there is what makes the threshold a measurement rather than a definition.

A rest inside a phrase is a gap the metre survives. A breath between phrases is a gap the metre survives with a small deliberate lengthening. A general pause is a gap the metre is meant to survive and the listener is meant to notice surviving. All three are below the threshold, and the composers using them keep them there.

That is the operational content of the whole rung. An ending is a silence long enough that whatever comes next is heard as a beginning. It is not a property of the gap; it is a property of what the gap does to the listener’s metrical hypothesis, and the threshold is where the hypothesis stops being recoverable.

Two predictions come out of that formulation and both match practice.

A silence should be more ending-like at a fast tempo, because ten beats of a fast tempo is fewer seconds. A two-second gap in a quick movement is past the threshold and the same gap in a slow one is not, which is why a general pause in an allegro is a shock and in an adagio is a breath.

And a silence should be less ending-like if the metre is strongly established. More evidence before the gap means a better period estimate, a smaller error, and a longer survivable silence — which is the arithmetic reason a general pause works best after a long stretch of regular metre, and is exactly where the repertoire puts it.

A hall’s decay is also different in every band — long in the bass and short in the treble — so what survives longest into a silence is the low end, and what a listener is counting against for the first second or two is a bass note rather than the chord.

What the notation has for this, which is very little #

A rest has a duration in beats. A fermata has none — it means hold, and how long is the performer’s decision. A caesura means stop, with no indication of how long. A general pause is a bar of rest in every part at once and its length is a bar.

So the one closure component with a continuous scale is the one the notation quantifies worst, and the reason is that its correct value depends on the room, which the score does not know. Everything the notation does quantify — pitches, durations, metre — is room-independent, and this is not.

That is the same shape of gap this collection has recorded twice before: notation has nothing for an envelope, and it hides the fact that a bar is a variable number of seconds. In every case the missing quantity is the one that depends on something outside the score.

The fade-out, which supplies none of it #

There is one ending in common use that has no closure components at all, and it is worth putting beside the others precisely because it is so bare.

A fade-out has no cadence — the material usually continues unchanged underneath it. It has no ritardando; the tempo is constant. It has no silence, because the sound does not stop, it becomes inaudible. And it has no final chord, so four of the five components have nothing to attach to.

What it has is a monotonic decrease in level, which is a signal in exactly one respect: it is a thing that cannot be undone. A piece that has faded to nothing has no way to continue that would not be heard as a new beginning.

The device is a fact about a medium and it has a date. It requires a recording — a live ensemble cannot do it convincingly, and the attempts are conspicuous — and it becomes common in the 1960s, for a reason that is documentary rather than musical: it lets a repeating groove be released as a track without anybody having to compose an ending for it.

Which is a rather exact statement of what an ending is for. A cyclic form cannot cadence, and a fade is the technological answer to that problem: not an ending, but a way of leaving.

Three thresholds, and only one of them is about silence #

It is worth separating the quantities this essay has used, because they are three different things that all come out in seconds and it would be easy to run them together.

The drift bound is about a metre. It says how long a beat can be carried before it is on the wrong side of the grid, and it is a number of beats that becomes a number of seconds only when a tempo is supplied.

The psychological present is about grouping. It says how far apart two events can be and still be one thing, and it is a number of seconds outright, with no tempo in it.

The reverberation time is about a room. It says how long the previous sound remains, and it is a property of the building.

And the levels a silence has to outlast are the metrical ones: the beat goes first, then the bar, then the four-bar group, and a silence that outlasts the largest of them has removed the frame the next event would have been heard in.

The three are independent and they do not have to agree. That they land within a factor of about two of each other is the finding, and it is the kind of coincidence this collection has recorded before: two clarity design targets falling out of arithmetic that has neither in it, and a voice’s carrying frequency landing on the ear’s own resonance. Convergence from unrelated directions is the strongest evidence a number of this kind can have.

Whose silences, and when #

The general pause is a Classical-era device, most famously Haydn’s, and it works there because the metre is strongly established and a listener continues it through the gap — which is the drift arithmetic above, used deliberately and kept well short of the threshold.

Outside that repertoire the picture varies enormously. A gamelan’s ending is a coordinated stop cued by the drum, with the gong ageng’s decay filling the seconds after it; the gong’s own decay is long, and the piece is over while it is still sounding. In much North Indian classical performance the end is a cadential figure landing on the first beat of the cycle, after which the silence is unmeasured and unremarked.

And 4′33″ is worth one sentence, correctly stated: it is not a piece of silence. It is a piece in which the performers do not play, and its content is what the room and the audience do, which makes it a work about the fact that the silence measured in this essay does not exist.

What the picture cannot show #

The tempo limen is a range and the middle of it was used. Five per cent gives ten beats; two and a half per cent gives twenty and six per cent gives just over eight. The threshold in seconds moves with it by a factor of two and a half, which is a large uncertainty and is stated rather than hidden. What does not move is that the answer is a number of beats and not a number of seconds.

The drift model has one error term. A real listener’s coasting is worse than a fixed percentage error: attention wanders, the estimate is noisy from beat to beat, and both make the drift faster. The bound computed here is therefore optimistic — a real metre survives a shorter silence than this says.

The psychological present is not one number and not one experiment. The two-to-eight-second range spans several different tasks, and the collection has used the same range for a phrase, which is a reuse worth being suspicious of. Both uses are about the same underlying claim — that events inside a window are one group — and neither is a measurement made here.

And the reverberation figures are Sabine’s, which has a stated domain and is an average over a room. What a particular listener hears decaying depends on where they are sitting.

Where this ladder has got to #

Five rungs. An ending is five harmonic components with no total; half of the cadences in the repertoire are built to withhold some of them; the performance slows, on a curve with one free parameter; nothing in a piece’s own statistics can announce that it is about to stop; and a silence is an ending somewhere between three and a half and five and a half seconds, with a hall’s reverberation eating the first two.

What is left unmeasured is the loudest cue of all. Every ending in every repertoire is also a dynamic event — a diminuendo, a final chord louder than the rest, a texture thinning — and this collection has no model of dynamics in a form at all. That is the next rung, and it is the one that would need a corpus rather than an arithmetic.