A fill cannot end a reading, but it can win the next one

A silent break has to outlast the reading’s lead let a listener wait through bars of silence before a late honest drum, charging the reading it held a vote for every downbeat that failed to come. The silence ended the reading only after as many bars as the reading had votes in hand: three bars after the opening two bars of bass, eight before bar 5, twenty-one before bar 9. And where it did end the reading it left the listener’s record of whom to trust untouched, because nothing in a silence votes, so the late root’s long record of agreement came through any length of silence intact and convicted the drum anyway. Before bar 5, twelve bars of silence ended every reading and the drum still won only 7 per cent of passages.

Its last question was about the obvious thing an arranger puts in a break instead of silence. A drum fill in the bar before the drum’s entry, accented on the true downbeat, is a witness that speaks during the break. Its strokes would be scored against the reading held — still the half-bar — and the record would then convict the fill as it convicts the late drum, unless the fill were strong enough to turn the reading within its own bar. Can a fill do in one bar what silence needs twenty to do, or is it simply a late drum that arrived a bar early?

It cannot do it in one bar, and it is convicted faster than a late drum is. But that turns out to be the wrong comparison, because a fill does something silence cannot do at all, and does it in the place silence leaves open.

The listener and the fill #

The listener is the choosing listener of the earlier essays. It weighs four witnesses — the bass’s root and the bass’s motion, which a walking bass marks the bar by the note it is on introduced, the tune’s long notes, which a melody marks the barline by where it lingers did, and the drum — each by a record of how often it has agreed with the reading held when it voted, which a listener who weights its witnesses needs eight bars of bass set up; it keeps three sets of habits, three styles, and weighs them by how well each predicts what it hears, as a listener who knows the style needs no introduction arranged. Its reading is a pair of running scores, one for the true barline and one for the half-bar, and with no cost of switching the margin between them grows without limit. A walking bass whose root lands on the half-bar in every bar builds that margin on the wrong side from the first bar, which is why an honest drum has to come first: two strokes on the opening downbeats hold the barline, and the same two strokes from bar 5 are convicted.

The fill is a bar of strokes in the break before the drum’s entry, each one a vote for the true barline. They are heard one at a time, and each is treated exactly like any witness’s vote: weighed by the credit its record has earned, added to the barline’s score, and recorded as agreeing or disagreeing with the reading held at that stroke. That last part is what decides everything. The stroke is right — the barline is where the fill puts it — but the listener does not know that; it knows only whether the stroke agrees with what it currently hears.

There are two ways to hear a fill. A fill played by the drummer, on the drums, may be heard as the drum: the same witness, sharing one record, so that whatever the fill earns or loses the drum earns or loses too. Or it may be heard as a witness of its own — a hit by the whole band, say — whose record lasts only for its bar. Both are run here.

Each stroke halves the fill’s credit #

The first figure is the arithmetic of a fill speaking against a reading it has not yet turned.

At the drum’s starting credit of three to one, the first stroke is worth one vote. It is recorded as a disagreement, which takes the fill’s estimated reliability from three in four to three in five, and the second stroke is worth 0.37 of a vote. That one is recorded as a disagreement too, and the reliability is now one half — a coin toss, whose vote weighs nothing. Every stroke after that is silent in the only sense that matters: it is heard and it counts for nothing. A fill at three to one can cast 1.37 votes against a held reading, however long it goes on.

Raising the credit helps slowly. At nine to one the fill can cast 3.28 votes, at ninety-nine to one 6.24, at nine hundred and ninety-nine to one 8.43. The reading’s lead, against which those totals have to be set, is 2.9 votes before bar 3, 6.8 before bar 5, 20.7 before bar 9 and 38.5 before bar 13. Run on the passages themselves, a fill on its own turns half the readings before bar 3 at a credit of 2.9 to one — the drum’s own credit, near enough — and before bar 5 at eight to one. Before bar 9 it needs odds of about 7,200 to one, and before bar 13 no credit short of certainty is enough.

So the lead’s first alternative is answered at once. A fill cannot do in one bar what silence needs twenty bars to do, and the reason is not that it is short. It is that a listener who scores every vote against the reading it holds lets a dissenting witness talk itself out of credit. The record is a mechanism for silencing a witness that keeps disagreeing, and a fill that arrives to disagree is exactly what it was built to silence.

Heard as the drum, the fill convicts the drum #

Before bar 3 a fill can turn the reading, because there the lead is under three votes. What happens then depends on how the fill is heard.

One stroke turns nothing. Two strokes turn the reading in 86 per cent of passages — the first stroke’s vote and the second’s 0.37 together exceed the lead in most of them — and from then on any further stroke agrees with the reading it has made. Heard as a witness of its own, the fill hands over a reading on the barline, the drum enters with a clean record, and 86 per cent of passages end on the barline: every passage whose reading the fill turned.

Heard as the drum, the same two strokes leave only 7 per cent. The reading has turned, but the drum’s record now carries the two disagreements the fill earned on the way, so the drum enters at a coin toss with a weight of nothing, and the late root wins the reading back in its first bar. The fill was right and it cost the drum its credit. Three strokes leave 33 per cent, four 56; only at five strokes, when three agreeing strokes after the turn have rebuilt the record, does the drum-heard fill catch up with the other.

This is the lead’s second alternative confirmed with a twist. A fill heard against the old reading is indeed a late drum that arrived a bar early, and it is convicted the same way — faster, in fact, because a fill is several votes inside one bar where the drum is one vote a bar. The twist is that the fill can be its own undoing even when it wins. A fill that turns the reading and stops has spent the drum’s credit on the turn. A fill that turns the reading and keeps going earns the credit back. The figure says how much going on is enough: about as many strokes after the turn as were spent before it.

After the silence, the fill sets the next reading #

All of that was a fill in place of silence. The figure the whole question turns on is a fill after it.

Before bar 5, silence alone leaves the drum 5 per cent of passages after eight bars and 7 per cent after twelve, however many readings it ends, because the record survives. Put a bar of eight fill strokes after the same silence and the drum wins 99 per cent after eight bars and every passage after twelve. Before bar 3 the silence alone never gives the drum more than 44 per cent; with a fill after two bars of silence it gives all of them. Before bar 9, where the silence needs about twenty-one bars to end a reading, twenty-four bars of silence and a fill give the drum 99 per cent, against nothing from the silence alone.

The fill does not need to be large. Turning the dial to two strokes changes almost nothing once the silence is long enough: after eight bars before bar 5 the drum wins 97 per cent.

That is the reverse of what the fill could do alone, and the reason is the rule that made it powerless there. A silence that outlasts the reading’s lead ends the reading completely: both scores are set equal and the listener holds nothing. The first stroke of a fill heard then is not scored against any reading — there is none to disagree with — and its vote makes one, on the barline. Every stroke after it agrees with that reading and is recorded as agreeing. So the fill arrives at the end of the break with a reading it made and a record it improved, and when the drum enters it enters in agreement with both.

Whoever sets the reading turns the record #

The figure that shows why this rescues the drum completely, and not just its first bar, is the one that follows the weights.

After silence alone the reading is empty when the music resumes, and the next bar’s votes decide what fills it. The late root votes with a weight of 1.52, earned over the bars before the break; the drum with 1.00, its starting credit; the tune’s long notes with about a quarter of a vote, having been scored against the half-bar for as long as it stood. The root wins, the new reading is the half-bar again, and the drum is scored as disagreeing with it from its first stroke. By bar 8 its weight is 0.09 and the root’s is still climbing: 2.11 at bar 13. The silence ended the reading and the same reading came straight back, because the record decided what came back.

After the fill the order is the other way round. The reading is on the barline before the root votes again, so the root’s next vote is scored as a disagreement, and so is the one after it. Its weight falls from 1.52 to 0.86 at bar 6, 0.21 at bar 8, 0.08 at bar 9 and nothing by bar 10; the drum’s rises from 2.09, already raised by the fill’s agreeing strokes, to 2.63. By bar 13 every passage is on the barline.

The record was never the obstacle in itself. The obstacle was which side of it the drum came in on. The earlier essay concluded that a silent break could at best end the reading and never touch the record, and that the forgetting of whom to trust had to come from somewhere else. The arithmetic here says that forgetting is not needed. A record of agreement with the reading held is as good a weapon for the barline as against it, and the side that holds the reading when the witnesses next vote is the side the record serves. Silence empties the reading; whatever speaks first into the empty reading owns the record from then on.

That ties this question to a late root is believed because the record is kept by the reading, which found the same mechanism working the other way: a root that is wrong from the first bar is believed because it was there first, and every honest witness after it is recorded as disagreeing. A fill after a silence is a way of being there first a second time.

What the fill does to the silence’s length #

The fill makes the break’s outcome far better. It makes the break only a little shorter.

Before bar 3 the silence alone needs three bars to end half the readings, and with a fill none: a fill alone turns 86 per cent there. Before bar 5 it is seven bars alone and four with the fill; before bar 9, twenty-two and eighteen; before bar 13, forty and thirty-six. The fill takes three or four bars off the silence everywhere, because it can push a reading over that has been worn down to within a vote or two, and that is all it can do against a reading still standing.

So the silence is still doing the hard part. Two bars of drum hold the barline, a dozen bars of bass do not found the same asymmetry from the other side: what comes first decides, and what comes later has to outweigh it. A fill turns a break that ends the reading into one that wins, but it does not turn a two-bar break into one that ends the reading. Before bar 9 the break still has to last eighteen bars. A listener whose conviction accumulates without limit is not reset by any break an arranger writes, fill or no fill; what the fill changes is what happens after a reset, if one comes.

What this says about breaks and fills #

The arithmetic separates two jobs that a break before a new texture is usually expected to do together.

One is to clear what the listener was hearing. Silence can do that, but only slowly, and only for a listener whose conviction is capped — the earlier essay’s listeners with a cost of switching lost their readings to a handful of bars. A fill cannot do it at all against a reading of any strength, because it is silenced by the very record it is trying to correct.

The other is to say what comes next, and that a fill does almost perfectly, provided the listener has nothing in hand when it starts. Two strokes are enough. Whether the fill is heard as the drum stops mattering once the reading is empty, since none of its strokes is then recorded as disagreeing.

Fills in popular music typically come at the end of a section, after the last phrase has ended and often after the harmony has cadenced — at the moment the listener is most likely to be between readings rather than inside one. On this account that placement is not a convention. It is where a fill is worth anything. A fill dropped into the middle of a groove the listener is confident about is heard against the groove and talks itself out of credit; the same fill after a cadence or a stop is heard into a gap and sets the next bar. The claim is about drum fills in song forms built from repeated sections with a stable groove, the repertoire the walking-bass passages here stand for; it is not a claim about fills in music whose metre is meant to stay ambiguous.

The model under the figures #

Everything not stated here is the earlier essays’ model: four hundred seeded passages of two bars of walking bass with the root on the half-bar and twenty-four bars of a tune whose long notes mark the barline; four witnesses voting each bar, weighted by records kept against the reading held, with starting credits worth four bars of evidence; the choosing listener’s three styles weighed by prediction and flattened two per cent a bar; a silent break charging the held reading one vote per failed downbeat, and a reading charged down to its rival’s level abandoned with both scores set equal.

The fill is a bar of strokes inserted after the silence and before the drum’s first bar. Each stroke is one vote for the barline at its witness’s current weight, recorded against the reading held at that stroke and used to update the styles’ shares like any vote. Heard as the drum, it shares the drum’s record; heard as its own, it starts from the drum’s credit and its record is discarded at the end of the bar. A fill sounding through the bar meets every expectation, so no failed downbeat is charged during it.

What the fill cannot show #

How a real fill is heard. A fill is not a row of identical votes for the barline. It has strokes on many subdivisions, accents that may or may not fall on the true downbeat, and a shape — usually a run ending in a crash — that marks its end. Treating its accented strokes as votes for the barline is the simplest model in which it can be right or wrong; a fill that also marked the half-bar would split its votes.

Whether listeners treat a fill as the drum. The two versions differ greatly before a reading is ended and not at all after. Which one listeners are closer to is a question about how they attribute sounds to sources, and the figures say only where the answer matters.

Whether a cadence clears the reading. The rescue depends on the reading being empty when the fill starts. Here only a long silence empties it. If a strong cadence or a section boundary makes listeners hold their metric reading less firmly — as a break rescues a late drum only by ending the reading asked — then a fill after a cadence might win with no silence at all, and the long silences here are an upper bound.

Still open: a cadence as the thing that empties the reading #

Every rescue here needed a silence long enough to outlast the reading’s lead, and those silences are far longer than any break in a song. What empties a listener’s reading at a real section boundary, if anything does, is more likely to be the cadence before it: a harmonic arrival that the listener hears as an ending, after which the metre of what follows is open. The harmony essays have a measure of closure, and the computation is this one with the silence replaced by a cadence whose strength, on that measure, sets how much of the reading’s lead it removes — and it would say how strong a cadence has to be before a two-stroke fill after it can hand the next section to an honest drum, which would turn the question of breaks into a question about endings.