A break rescues a late drum only by ending the reading

An honest drum has to come first gave a listener a walking bass whose root falls on the half-bar in every bar, a tune that marks the true barline, and an honest drum that enters late. A drum on every downbeat from the first bar held the reading on the barline. The same drum from the second bar, playing twenty-five strokes running, held it in no passage at all. The listener keeps a record of each witness — how often it agreed with the reading the listener held when it voted — and a late drum disagrees with a reading the root has already made, so it is convicted within two bars. The odds the drum needed rose from 2.1 to 1 on the first bar to 260 to 1 on the fifth, and from the seventh nothing short of certainty was enough.

That listener’s record ran unbroken from the first bar, and the essay said so as the thing it had left out. Real arrangements are made of sections, and a late drum usually enters at the start of one, after a fill, a held chord or a bar of silence. A listener might treat the boundary as a fresh start. The earlier essay guessed how: by discounting its record of the witnesses, so that the root’s long run of agreement counts for less, and the drum meets a root whose credit has fallen back towards where it began.

That guess turns out to be the wrong half of the listener. A break that discounts the record helps a little. A break that ends the reading helps nearly completely, and nothing else will do.

Two things a listener carries across a boundary #

The listener of these essays holds two kinds of state, and a break could act on either.

One is the reading: two running scores, one for the barline and one for the half-bar, each bar’s weighted votes added to one and subtracted from the other, and the higher of the two is what the listener hears. With no cost for changing its mind, the scores are simply running totals, and after eight bars of the late root they are far apart. (Two bars of drum hold the barline; a dozen bars of bass do not gave a listener such a cost instead, which caps the margin; that listener is a different case and is not followed here.)

The other is the record: for each witness — the bass’s root, the bass’s motion, the tune’s long notes, the drum — a count of how often it has agreed and disagreed with the reading held when it voted. A witness’s credit is its agreements plus four bars’ worth of its starting credit, over its votes plus four, and the weight its vote carries grows with that credit — the arrangement a listener who weights its witnesses needs eight bars of bass introduced. A late root is believed because the record is kept by the reading found that the record is what lets the root take over: every bar it agrees with the reading it made, its credit rises, and the honest witnesses who disagree are convicted.

A section boundary is modelled here as a moment at which the listener keeps a stated share of each. Keep all of both and there is no boundary. Keep none of the record and the witnesses start over at their starting credits — the habits the listener brought, which the choosing listener of a listener who knows the style needs no introduction keeps as weights over three styles, are not touched. Keep none of the reading and the listener hears nothing in particular as the next section begins. The passages are the earlier essay’s four hundred, seeded identically, with the drum entering on the bar after the break and playing every downbeat to the end at the three-to-one credit it has always had.

Forgetting whom to trust is not enough #

The dashed lines sit on the floor of the figure. A break that wipes the witnesses’ records entirely, but leaves the reading standing at half its margin, lets the late drum win no passage — from bar 5, from bar 9, from bar 13. The drum enters, votes for the barline against a reading still on the half-bar, and is convicted exactly as it was with no break at all.

The solid lines are breaks that end the reading too. With none of the record kept, the late drum wins 96 per cent of passages from bar 5, 95 from bar 9 and 93 from bar 13: as well as a drum on the first bar. Keep some of the record and the drum begins to lose. From bar 5 it survives a break that keeps a tenth of the record, at 91 per cent, and falls to 46 at a fifth. From bar 9 it falls from 92 per cent at two hundredths kept to 47 at five hundredths. From bar 13 it is 57 per cent at two hundredths.

Two things have to go, then, and they are not equally hard. The reading has to go entirely. The record has to go almost entirely, and more completely the longer the section before the break was.

Even a trace of the reading is fatal #

That the reading must go entirely is the surprising half. A listener that forgot nine-tenths of its confidence in the half-bar reading would still hear the half-bar; one might expect the drum, the motion and the tune together to turn so faint a margin in a bar. They do turn it. It does not help.

With a break that wipes every record and leaves a hundredth of the reading’s margin, the drum’s first bar after it is heard on the barline in 48 per cent of passages — the honest witnesses outvote the root in that bar, and in about half the passages they outvote what is left of the margin too. In the next bar the share is nothing, and it stays at nothing to the end. The drum won its first bar and was convicted for it.

The weights show why. The record scores each vote against the reading the listener held when the vote was cast, and the first bar after the break is cast against whatever reading came through it. A hundredth of a margin is still a reading. So the drum’s first vote is recorded as a disagreement, and so are the motion’s and the tune’s, while the root’s is recorded as an agreement — in the very bar the honest witnesses were winning. The drum’s weight falls from 1.00 to 0.37 at bar 6, the root’s rises from 0.86 to 1.18, and the reading swings back. By bar 9 the drum carries 0.20 and the root 0.75, the same shape as a late drum with no break.

With the reading ended outright, the first bar after the break is not scored at all: there is no reading for the votes to agree or disagree with. The bar’s votes make the reading — the drum, the motion and the tune outvote the root at their fresh starting credits — and from the second bar on, the honest witnesses are agreeing with a reading they made. The drum’s weight climbs to 1.23, 1.41, 1.57 and 1.92 by bar 12, and the root’s falls to 0.04. The late drum after a complete break is not a weaker version of the first-bar drum. It is the first-bar drum, because the first bar of a passage is the only other bar in which the record has nothing to score against.

That is the real content of the earlier essay’s finding, seen from the other side. A late drum is not convicted because the root has been trusted for longer. It is convicted because its first stroke is judged by a reading the root made, and a break rescues it only if it removes the judge.

What each kind of break is worth #

The credit the drum needs puts all four possibilities on one scale.

With no break the needs are the earlier essay’s: 4.3 to 1 on bar 2, 14 on bar 3, 42 on bar 4, 260 on bar 5, nearly 1,900 on bar 6, and nothing after.

Wiping the records lowers them, but by a factor of three or four rather than of hundreds: 9.8 to 1 on bar 3, 17 on bar 4, 76 on bar 5, 390 on bar 6, about 2,600 on bar 7, and nothing from bar 9. A listener that forgets whom it trusted at every section boundary can still not be corrected by a drum entering on the ninth bar. The fresh starting credits do not help the drum, because its first stroke is still judged against the old reading and convicted.

Ending the reading and keeping the records changes the curve’s shape entirely. The needs rise only gently, 3.1 to 1 on bar 3, 5.3 on bar 5, 9.5 on bar 9: the drum’s first bar is unscored, but it then has to outvote a root whose record of agreement is intact, and that record grows with every bar of the section before.

Doing both makes the drum cheap everywhere: 2.2 to 1 on bar 3, 2.5 on bar 5, 2.7 on bar 9 — below the three to one it has always had, and within a few tenths of the 2.1 a first-bar drum needs. The small rise is what the break leaves alone: the listener’s weights over its three styles, which the late root has been moving all along. By bar 9 the habit that trusts the root holds about two-thirds of the weight, against a third at the start, and the habit that doubts it has fallen from a third to seven per cent.

So the earlier essay’s lead had the mechanism the wrong way round. It expected the record to matter and the arithmetic to turn on how far a root’s credit falls back. The record matters only after the reading is gone; before that, it is not even the obstacle.

How much of the record a break may keep #

Where the reading does end, a break can still leave some of the record behind, and the longer the section before it, the less it may leave.

After two bars of root the break may keep 85 per cent of the record; after three, 34 per cent; after four, 19; after eight, 4.3; after sixteen, 1.6. In bars of the root’s own record that is 0.85 bar, then 0.69, 0.58, 0.30 and 0.23. Less than a bar of the root’s agreement may come through any break, and less still the longer the section, because a longer section has also convicted the tune more thoroughly, and a convicted tune that keeps any of its record carries less weight on the drum’s side.

The patience arithmetic of the earlier essay explains the scale. A witness credited at p over n bars of record has n(2p − 1) bars of patience with disagreement: two, at three in four over four bars. A break that leaves the root a fraction of one bar of record leaves it little more than its starting credit, so the honest witnesses outnumber it from the first bar; a break that leaves it a whole bar has handed it most of a bar of the patience the drum has to overcome.

After a complete break, the tune before it is lost #

A complete break turns a late drum into an early one, but it can only rescue the section after it.

For the choosing listener, a complete break before bar 3 and a drum from there to the end puts 95 per cent of the tune’s bars on the barline; before bar 5, 88; before bar 9, 71; before bar 13, 55. Every bar before the break was heard on the half-bar, and the break does not reach back. Two strokes after the break do nearly as well — 91, 80, 58 and 46 per cent — because once the reading is made, the honest motion and tune hold it without the drum, as two honest bars convict a lying root found for the first two bars of a passage.

The other listeners follow their own habits. The listener with no habits recovers less, 82 per cent after a break before bar 5 and 57 with two strokes. The listener that trusts the root and doubts the tune recovers half the tune with the drum playing on and almost nothing with two strokes, since the drum is the only witness it believes against the root. The listener that doubts the root — the habit the habit worth bringing is to distrust the tune and its sequels mirrored — never needed the drum and holds 89 to 100 per cent of the tune whenever the break comes. For it, the honest bass motion and tune, which a walking bass marks the bar by the note it is on and a melody marks the barline by where it lingers established as witnesses here, were always enough.

What the arranger’s habit is, in these terms #

Bringing the drums in after a break is a common arranging device, and these figures give it an exact condition. It works for a listener of this kind if, and only if, the listener comes out of the break with no metrical reading at all — if the break is long or disruptive enough that the next bar is heard as a beginning rather than a continuation. A fill that leads into the downbeat, a held chord with no pulse, a bar of silence: each is plausibly a way to make a listener stop hearing the old metre, and the figures say that is the whole of what a break must do. Lowering the listener’s trust in the bass, by contrast, which might seem like the point of stopping the bass, buys very little.

It also says that a break that only half works is worse than it looks. A listener who hears a break but keeps a faint sense of where the beat was — the kind of thing a steady tempo across the gap would encourage — gives the late drum nothing, however clean the drum is. In these terms, a fill that keeps the old pulse going may be the wrong fill.

The model the numbers rest on #

Everything is the earlier essay’s model: four hundred passages, seeded as before; 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 downbeat; witnesses voting each bar for the barline or the half-bar; a listener with two running scores and no cost of switching; records kept against the reading held when each vote is cast, 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 a single moment, before the drum’s first bar, at which every witness’s agreements and disagreements are multiplied by one share and both running scores by another. Nothing is played during the break: the model has no bar for it, so a break here takes no time.

What the arithmetic cannot settle #

Whether a real break ends a reading. The figures make the condition sharp — any surviving reading at all fails — and that sharpness is a property of the model’s record, which scores each vote against the reading held. Whether listeners carry a metrical reading through a bar of silence, a fill or a held chord, and which of these ends it, is a question for an experiment, and the answer may differ between a listener who is tapping along and one who is not.

Whether the boundary is heard as one. The model is told where the break is. A listener has to notice that a section has ended, and the cues that mark a section boundary — a change of texture, of instrumentation, of harmony — are themselves things the listener reads with a record of its own.

And the style shares. The break here leaves the listener’s habits alone. A listener who also took a section boundary as a reason to reconsider which style the piece is in would sit somewhere between these results and a listener starting the piece afresh.

Still open: a break that keeps the pulse #

The figures treat the break as taking no time and the reading as a pair of scores that a boundary can shrink. A real break has a length, and a listener waiting through it may go on expecting beats at the old period and phase — which, for a listener that has settled on the half-bar, means expecting the next downbeat half a bar early. If the drum’s first stroke after a break lands where the old reading predicted silence, that may itself count against the old reading, as an expected event that failed to come, in a way the vote-counting listener has no means to represent. The computation is this one with a listener whose reading predicts onsets through the break and is charged for each prediction that fails, and it would say whether a break of one or two bars ends a reading by itself, through the listener’s own unanswered expectations, or only when something in the music makes it stop.