A silent break has to outlast the reading's lead

A break rescues a late drum only by ending the reading let a listener pass through a section boundary keeping a stated share of two things: its reading — two running scores, one for the true barline and one for the half-bar, of which the higher is what it hears — and its record of how often each witness had agreed with the reading it held. A walking bass whose root falls on the half-bar in every bar had built a reading on the half-bar, and an honest drum entering after the boundary could undo it only if the listener came out holding no reading at all. Any trace — a hundredth of the margin — and the drum’s first stroke was scored as a disagreement and convicted.

That essay modelled the break as taking no time. Its last question was what happens when it does. Through a bar or two of silence, a listener who has settled on the half-bar goes on expecting a downbeat once a bar where it hears the downbeat, and the downbeats it expects do not come. An expectation that fails may count against the reading that made it — the kind of evidence a vote-counting listener has no means to represent. So a break might end a reading by itself, through the listener’s own unanswered expectations, without anything in the music having to make it stop.

It can. It takes far longer than any break an arranger writes.

What the earlier break needed #

The figure is where the instantaneous break left things. A late drum from bar 5 needs odds of 260 to 1 with no break, 76 after a break that only wipes the records, 5.3 after one that only ends the reading, and 2.5 — under the three to one it has always had — after one that does both. The reading has to go entirely and the record nearly so. The question here is whether waiting through silence can make either go.

The listener who waits #

The listener is the choosing listener of the earlier essays: it weighs its witnesses — the bass’s root, the bass’s motion, which a walking bass marks the bar by the note it is on established, the tune’s long notes, which a melody marks the barline by where it lingers did, and the drum — by their records, in the arrangement a listener who weights its witnesses needs eight bars of bass introduced, and weighs three styles of music by how well each predicts what it hears, as a listener who knows the style needs no introduction set it up. Its reading is a pair of running totals. Each bar, every witness’s vote, multiplied by the weight its record has earned, is added to the score of the reading it supports and taken from the other; with no cost for changing its mind, nothing caps the totals, and the margin between them is the reading’s lead.

The break is now a stretch of silence some number of bars long, before the drum’s first bar. Through it, the listener keeps expecting a downbeat once a bar at the reading it holds, and each one that fails to come charges the held reading’s score a fixed amount. The amount is stated in the only unit the model has: a witness’s vote at its starting credit, the weight the drum itself carries when it first plays. Nothing votes in the silence, so nothing is added to any witness’s record. And a reading charged down until it is no better than its rival is abandoned — both scores set equal, so that the next bar’s votes make a new reading from nothing, which is the complete ending of the reading that the earlier essay found the drum needs.

That rule makes the arithmetic of the silence very simple. At one vote a bar, a silence ends the reading after as many bars as the reading’s lead in votes.

Before bar 3 — after the two bars of walking bass that open every passage, and before the tune has started — the reading leads by 2.9 votes on average, and three bars of silence end it in 86 per cent of passages, four in all of them. Before bar 5 the lead is 6.8, and six bars of silence end 18 per cent of readings, eight end 88. Before bar 9 it is 20.7, and the silence needs about twenty-one bars. Before bar 13 it is 38.5, and it needs about thirty-eight. No break of one or two bars ends a single reading anywhere after the first two bars of the passage. Doubling the charge halves every length and changes none of the conclusions: at four votes a missing downbeat, a break before bar 9 still has to last six bars.

The lead is a running total #

The reason the silences have to be so long is in what the lead is.

With no cost of switching, the lead is a running total of every bar’s evidence, and the evidence per bar grows. The root’s record of agreement with the reading it made earns it a larger weight each bar, and the honest motion and tune, scored as disagreeing with that reading, earn less, as a late root is believed because the record is kept by the reading found. So the lead rises by about a vote and a half a bar at first and by about six a bar at the end: 4.4 votes after bar 3, 9.8 after bar 5, 24.9 after bar 9, 43.4 after bar 13 and 116 by the end of the tune. A silence charging a fixed amount per bar is racing a quantity that grows faster than the bars do. Every bar of music before a break adds more to the silence it needs than the bar before did.

This makes the most open-minded listener the hardest to reset, which is the surprise in the arithmetic. A listener with no cost for changing its mind is the one the earlier essays used because it is the most responsive: it follows the evidence wherever it goes, and two bars of honest drum on the first two downbeats are enough to hold its reading, as two honest bars convict a lying root found for the witnesses it already had. But having no cost of switching is the same thing as having no cap on conviction. Its lead accumulates without limit, and nothing short of a stretch of silence as long as the section before it can drain it.

A listener with a cost of switching is different, as two bars of drum hold the barline, a dozen bars of bass do not found from the other direction. The cost holds the losing reading’s score within a fixed distance of the winner’s, so the lead stops growing at about the cost plus one bar’s evidence: with a cost of one vote, 5.2 after bar 9 and 7.1 at the end; with a cost of two, 6.2 and 8.1; with a cost of eight, 12.2 and 14.1.

So a listener slow to change its mind is quicker to forget. Before bar 9, a listener with a cost of one vote loses its reading to six bars of silence in 94 per cent of passages; a cost of two needs about the same; a cost of eight needs twelve; and the listener with no cost needs twenty-one or more. None of them loses its reading to two bars, and at four bars only the cheapest-switching listener loses any — one passage in a hundred. The cap on conviction brings the silence needed down from tens of bars to a handful, but not to the length of a break.

What a missing downbeat would have to cost #

The other way to read the same arithmetic is to ask how much an unanswered expectation would have to weigh for a break of ordinary length to work.

For a one-bar break to end half the readings, the single missing downbeat must cost 2.7 votes before bar 3, 6.2 before bar 5, 20.5 before bar 9, 37.9 before bar 13 and 57.9 before bar 17. A two-bar break needs exactly half of each, since the charges simply add: 3.1 before bar 5 and 10.2 before bar 9.

Those numbers have to be set against what a vote is. A vote at starting credit is what a drum stroke on the barline is worth when the drum first plays, or what a long note of the tune is worth before the listener has any record of the tune. For a two-bar break before bar 9 to end the reading, each bar of silence would have to count against the old metre as heavily as ten drum strokes counted for the new one. That is not impossible as psychology — the absence of an expected event can be salient — but it is not a small adjustment to the model either. It says that if a two-bar break really does reset a listener’s sense of where the bar is, the reset is not coming from the listener’s expectations being disappointed at the rate a single piece of evidence would disappoint them.

The pattern has a familiar shape. Every listener in this collection’s metre essays has found that the first bars carry disproportionate weight — the first downbeats a drum plays, the first notes a bass plays on the barline — because everything later is scored against what they established. The silence figures are the same fact seen from the far end. What the first bars established becomes, by bar 9, a lead of twenty votes, and every way of dislodging it has to be measured against that number. The listener that the habit worth bringing is to distrust the tune and its sequels followed, which doubts the root from the start, never builds the lead at all, and for it the question of a break does not arise: it holds the barline through the late root whether or not anything stops.

Ending the reading is still only half #

Where the silence does end the reading, the record is still there.

Before bar 3, three bars of silence end 86 per cent of readings and the drum then wins 38 per cent of passages; four bars end every reading and the drum wins 44 per cent, and no further silence does better. Before bar 5, eight bars of silence end 88 per cent of readings and the drum wins 5 per cent; twelve end every reading and it wins 7. With the records wiped at the same breaks, the drum wins 95 and 96 per cent.

This is the instantaneous essay’s result arriving by a different route, and it was predictable once the silence was modelled honestly. Nothing votes in a silence, so nothing is recorded; the root’s long record of agreement comes through any length of silence untouched, and when the music resumes the drum faces a root credited for every bar it played before. The earlier figure put the price of that at odds of 5.3 to 1 for a drum on bar 5 — above the three to one the drum has — and the silence figure agrees, at about one passage in twenty.

So a silent break, of whatever length, can at best do one of the two things a break has to do. It can end the reading, if it outlasts the reading’s lead. It cannot touch the record, because the record is made of witnesses’ votes and the silence has none. The forgetting of whom to trust, which the earlier essay found the drum also needs, has to come from somewhere else: from time passing, if records fade with time, or from the boundary being heard as the start of something new.

What this says about a break in the music #

An arranger who brings the drums in after a bar of silence is not relying on the listener’s expectations to do the work, at least not on this listener’s. The arithmetic says three things about what must be doing it instead.

The first is that the listener cannot be the one these essays have mostly followed. A listener whose conviction accumulates without limit holds a reading through any silence a song contains. If a two-bar break resets real listeners, real listeners have something like a cost of switching — a cap on how sure they become — and the cap has to be low.

The second is that even with a low cap, a break must last several bars, or each missing downbeat must weigh several times what a drum stroke weighs. Both are claims about listeners that the figures can state precisely and cannot check.

The third is that a break which ends the reading still has to end the record, and silence cannot. If a section boundary makes a listener stop trusting the old bass, it does so because the boundary is heard as a boundary — a change of texture, a cadence, a new instrument — and not because time has passed with nothing in it. That is the earlier essay’s caution, that a fill which keeps the old pulse may be the wrong fill, turned into its positive form: a break that keeps the pulse, a hi-hat or a strummed chord on every beat, meets every expectation and charges nothing, which in this model is exactly the same as no break at all.

The model under the figures #

Everything not stated above is the earlier essays’ model: four hundred passages, seeded as before, of two bars of a walking bass whose root falls on the half-bar and twenty-four bars of a tune whose long notes fall on the barline; witnesses voting each bar for the barline or the half-bar, weighted by records kept against the reading held when each vote was cast, with starting credits worth four bars; the choosing listener’s three styles weighed by their predictions and flattened two per cent a bar. The break is inserted before the drum’s first bar and the passage resumes where it left off. A failed downbeat charges only the reading held at the time, and abandonment sets both scores equal rather than to any particular value, so that the next bar’s votes alone decide what is heard.

What the silence cannot tell #

What a failed expectation is worth. The charge is a free number here, and the figures report what would follow from each value rather than choosing one. Its size relative to a witness’s vote is a question about listeners for an experiment, and the answer may depend on whether the listener is moving to the music.

Whether expectations fade. A listener who expects a downbeat may expect it less firmly as the silence goes on. That would make each successive missing downbeat cost less, and lengthen every silence needed here.

Whether records fade with time. The record in this model changes only when witnesses vote. A listener whose memory of the bass’s reliability decays with time, rather than with evidence, would lose some of its record in a long silence; the figures with the records wiped mark the limit of what that could achieve.

Still open: a fill that votes against the old reading #

Silence cannot end the record because nothing in it votes. A fill can vote. A drum fill in the bar before the drum’s entry, accented on the true downbeat and the true half-bar’s offbeat, is a witness that speaks during the break, and its votes are scored against the reading held — which, at that moment, is still the half-bar. The record would then convict the fill as it convicts the late drum, unless the fill is strong enough to turn the reading within its own bar. The computation is this one with a fill of one bar, at a stated credit, voting in the break’s place, and it would say whether a fill can do in one bar what silence needs twenty to do, or whether a fill heard against the old reading is simply a late drum that arrived a bar early and is convicted the same way.