An honest drum has to come first

Two honest bars convict a lying root gave a listener a passage whose walking bass puts every root on the half-bar instead of the barline, and then gave it a drum on the first two downbeats. For a listener that keeps several sets of habits and chooses among them, those two strokes were enough: the tune that followed was heard against the true barline in 93 per cent of its bars, and a lying root heard in every bar for twenty-four bars afterwards could not take the reading back.

That result had a condition hidden inside it. The drum played the passage’s first bars — before the root had voted at all. In a band that is the count-in or the fill into bar one. But a drum can also enter late: a bass and a singer start a tune, and the kit comes in on the second chorus, or a percussionist joins after the intro has already been played. Then the drum is not forming a reading. It is contradicting one, and the root it contradicts has by then been agreeing with that reading for several bars and has the record to show for it.

The question the earlier essay left was whether the same two strokes can undo what they would have prevented. They cannot, and neither can any number of strokes after them.

The passage, and what has changed #

Everything is as it was in the earlier essay except where the drum plays. The passages are the same four hundred, seeded identically: one chord a bar, a walking bass on every crotchet with the root moved to the half-bar and the fifth on the barline, two bars of that bass alone and then a tune of crotchets on chord tones whose long notes fall on the downbeat. Every witness votes in every bar for the barline or the half-bar: the bass’s root, the bass’s motion, the tune’s long notes and the drum when it plays. A listener weighs each vote by its record of that witness, which counts how often the witness agreed with the reading the listener held when it voted, starting from a credit worth four bars of record.

The choosing listener — the one that keeps three sets of habits and weighs them by how well each predicts what it hears, as a listener who knows the style needs no introduction set it out — is the one the earlier essay found rescued by two strokes, so it is the one followed most closely here. The tune is twenty-four bars long rather than eight, because the question is whether a reading is ever recovered, and a short tune cannot say never.

The drum now plays from a chosen bar onward: either two strokes, as before, or every downbeat to the end of the passage. Its starting credit, three in four in every earlier essay, is now also a variable. A listener that trusts a drum more than a bass is plausible in most styles, and the earlier essay named that preference as the thing it did not know.

One bar late, and nothing #

At the credit the drum has always had, the answer is total. A drum on every downbeat from the first bar to the twenty-sixth holds the reading on the barline in every passage. A drum on every downbeat from the second bar to the twenty-sixth — twenty-five honest strokes, missing only the first — holds it in none. From the third bar: none. From the fifth: none.

The bar-by-bar picture shows that the late drum does not even dent the reading. There is no bar, in any of the four hundred passages, in which a drum that began on the second downbeat brings the listener back to the barline. One bar of root, voting alone, is enough to form the reading on the half-bar; and once it is formed, everything that disagrees with it is a witness that disagrees.

That is a stronger result than the earlier essay’s. Then, one stroke was not enough because the next bar’s root erased the margin it had made. Here twenty-five strokes are not enough, and nothing erases anything: the drum votes for the barline in every bar it plays, at full weight to begin with, and loses every bar.

What happens to the drum’s record #

Follow the weights and the mechanism is visible. By the fifth bar the root has agreed with the held reading four times, and its weight has risen from 0.67 to 1.52. The melody, which entered at bar 3 and voted for the true barline, disagreed with the held reading twice, and its weight has fallen from 0.67 to 0.24. Then the drum enters, at the full weight its starting credit gives it. It votes for the barline. That is a disagreement with the reading held. In its second bar its weight is 0.37; in its third, nothing. The root, meanwhile, is at 1.68, then 1.82, and by the end of the passage 2.95.

The record treats the late drum exactly as it treated the melody. It is a witness that disagrees with the reading, and the record has no other way to judge a witness. A late root is believed because the record is kept by the reading found that this is what promoted the root in the first place; the earlier essay showed the same rule convicting the root when the drum came first. It is faithful to the first reading, and here the first reading is the root’s.

There is a simple piece of arithmetic under the speed. A witness’s credit is its agreements plus four bars’ worth of its starting credit, divided by its votes plus four. A vote carries positive weight only while that credit is above a half. A drum that starts at three in four and disagrees in every bar falls to a half after exactly two disagreements, since three is a half of six. So a drum credited at three in four has two bars in which to change the reading, and no more, however long it then plays. The figure’s 1.00, 0.37, 0.00 is that arithmetic: full weight on its first stroke, a third of it on the second, and silence from the third.

In general a starting credit of p worth n bars of record buys n(2p − 1) bars of patience with a witness that keeps disagreeing. At three in four and four bars, two. At ninety-nine in a hundred, just under four. No credit short of certainty buys more than four. A late drum is therefore not one more witness that eventually wins by persistence. It is a witness on a clock, and its two to four bars of patience have to be enough to turn a reading that the root has been building since the first bar.

Every bar multiplies the price #

The clock explains why the answer depends so sharply on when the drum comes in. In the drum’s few bars of patience, its weight has to outvote the root’s, and the root’s grows with every bar the drum has missed.

For the choosing listener, a drum starting on the first bar needs to be trusted at odds of 2.1 to 1 — less than the three to one it has always had, which is why two strokes there were enough. Starting on the second bar, it needs 4.3 to 1; on the third, 14 to 1; on the fourth, 42 to 1; on the fifth, 260 to 1; on the sixth, nearly 1,900 to 1. From the seventh bar no credit short of certainty is enough. Each bar the root has been believed first multiplies the odds the drum needs, by two at first and by six or seven later, and the multiplier itself grows.

The other two listeners that learn from a record behave the same way, a little worse. The listener with no habits needs three to one on the first bar, nine to one on the second, 36 on the third, 125 on the fourth, a thousand on the fifth, and nothing suffices from the seventh. The listener that trusts the root runs out a bar earlier, at the sixth, because its starting credit for the root was higher, so the root reached a commanding weight sooner.

That is the price the earlier essay’s lead asked about, and it can be stated plainly. A late correction does not cost a fixed number of extra bars. It costs a multiple of the trust a listener must already place in the corrector, and the multiple compounds bar by bar. Somewhere around the sixth bar the trust needed passes anything a sensible listener could hold, and the error becomes permanent.

Playing on does not help #

The obvious response to a drum that came in late is for it to keep playing. The earlier essay found that for the listener with no habits, eight bars of drum did more than two. For a reading that has already formed, persistence is worth very little.

Two strokes starting on the second bar need odds of 6.1 to 1; every downbeat from the second bar to the end needs 4.3. From the third bar, 25 against 14; from the fourth, 195 against 42; from the fifth, nearly 3,000 against 260. The drum that keeps playing always needs less than the drum that stops, and the saving is real at the later bars, where it divides the odds by five or ten. But at every starting bar the saving from twenty more bars of drum is smaller than the saving from starting one bar earlier with two strokes.

The patience arithmetic says why. The bars the drum plays after its first two to four are bars in which it carries no weight; it has already been silenced, and its votes are recorded and ignored. What the extra bars can do is keep voting during the bars of patience, which matters if two strokes left the reading close but not over. Everything after that is a drum talking to a listener that has stopped listening to it.

The one listener the timing does not touch #

Across all four listeners of the earlier essays, two strokes at a later bar rescue nobody who was not already safe. The listener with no habits and the listener that trusts the root were not rescued by two strokes even at the start, and nothing changes for them. The choosing listener is the one that loses everything: 94 per cent of the tune’s bars on the barline with the strokes on bars 1 and 2, none with them on bars 2 and 3.

The exception is the listener that doubts the root, which the habit worth bringing is to distrust the tune and its sequels kept as the mirror-image habit. It gives the late root no vote to begin with, so the root never forms a reading for the drum to contradict. The bass’s motion and the tune’s long notes — 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 honest witnesses here — carry it to the barline on their own, and the drum adds to a reading it already mostly holds: every bar of the tune on the barline with the strokes at the start, 87 per cent with them at bar 5, 77 per cent at bar 9. For that listener the drum is never a rescue, only a reinforcement, and a reinforcement can come at any time.

So the earlier essay’s conclusion now has a sharper form. The choosing listener needed two honest bars to find the habit the late root requires. It needs them before the root has voted, because the habit it finds is chosen by how well each habit predicts the votes against the reading held, and a reading made by the root makes the habit that trusts the root look best. The listener that already holds the right habit does not need the timing. The listener that has to discover it does.

A slow learner forgives more and is rescued less #

Everything above rests on one convention: a witness’s starting credit is worth four bars of record. That number decides how fast a witness is promoted or convicted, and therefore how fast the root takes the reading and how fast the drum is silenced. A listener whose credits were worth more bars would learn more slowly from agreement, which is what a listener who weights its witnesses needs eight bars of bass found to be a cost when the bass was honest.

The slower the record, the longer the drum may wait. With credits worth eight bars, a drum starting on the fifth bar needs 62 to 1 rather than 260; with sixteen, 17 to 1; with thirty-two, about 8 to 1. A listener whose credits are worth thirty-two bars will take a correction on the ninth bar at 55 to 1, where the four-bar listener would take none at any odds. Its patience with a disagreeing witness is n(2p − 1) bars, and a large n means a long clock.

The same slowness costs it the early drum’s whole benefit. Two strokes at the very start bring the four-bar listener 94 per cent of the tune’s bars on the barline. The eight-bar listener gets 87, the sixteen-bar listener 79, and the thirty-two-bar listener 70. A record that moves slowly is slow to be captured by the root, slow to convict the drum, and slow to be convinced by either. There is no setting of the record that both rewards an early correction fully and forgives a late one. The two are the same parameter read in opposite directions.

The connection to another kind of first impression #

The shape of the result has a familiar counterpart outside music, and it is worth naming because it is the same arithmetic. A reader who scores each new source of information by whether it agrees with what that reader already believes will find early sources decisive and late corrections impotent, and will grow more confident in the original error with each source that repeats it. That is a description of anchoring, or of what is sometimes called confirmation, and here it is not a bias added to a sensible learner. It is what the sensible learner of these essays does when its only test of a witness is agreement with its own reading — the rule that worked so well when the reading started out right.

Two bars of drum hold the barline; a dozen bars of bass do not found two bars of drum decisive by a different route, a listener with a fixed cost for changing its mind. That listener would behave differently here: a cost of switching is the same whenever the switch comes, so a late drum would simply have to overcome it, with no clock. The record’s version is the one that makes timing matter, and it is the version these essays have used since the record was introduced.

Where this sits in practice #

A band whose bass keeps the root off the barline, and whose drums come in late, is a real band: an arrangement that opens with bass and voice and brings the kit in for the chorus is common in many styles. The arithmetic does not say listeners to such an arrangement hear the metre wrong for the rest of the song. It says that a listener who learns its witnesses only by agreement with its own reading would, and that the drum’s late entry is not the remedy it looks like.

What would be a remedy is anything that marks the barline before the root has voted alone: a count-in, a pickup in the voice that lands on the downbeat, a guitar chord on bar one. The finding is that the honest cue has to arrive in the first bar, not that it has to be a drum.

The model the numbers rest on #

The model is that of the earlier essays, unchanged: four hundred passages seeded as before, two bars of ground and twenty-four of a tune whose long notes mark the bar, every root late, the listener’s two running scores with no cost of switching, records against the reading held when each vote is cast, the choosing listener’s three sets of habits weighed by their predictions and flattened two per cent a bar. The drum votes for the barline on the downbeats of a chosen set of bars and abstains elsewhere. Its starting credit is set for every set of habits alike. The credits needed are found by bisection on whether more than half of the passages end on the true barline. The share is not all or nothing at the margin — a drum credited at nine to one and starting on the third bar recovers 29 per cent of passages — so the threshold is where half of them are recovered.

What the arithmetic cannot settle #

Whether a listener’s record is kept against its own reading. Everything here follows from that one rule. A listener that judged its witnesses against each other — the drum and the root disagree, so one of them is wrong — rather than against its reading would treat the late drum as evidence about the root, and the timing would matter far less. Which rule, if either, describes a listener is a question for an experiment.

How much more a drum is trusted than a bass. The figures show the answer across credits from three to one to thousands to one. No measurement here says where a listener in a given style sits on that axis, and the bar at which a late drum stops helping moves by two or three bars across that range.

And what a listener who hears the band stop and restart does. A real late entry often comes with a break — a pause, a fill, a change of texture — that might make a listener start a new reading rather than defend the old one. The model has no notion of a fresh start, and any listener that has one is outside it.

Still open: whether a break in the texture resets the record #

A listener’s record here runs unbroken from the first bar. A real arrangement marks its sections, and a late drum usually enters at one of those marks, often after a bar of silence or a fill. If a listener discounts its record at a section boundary — keeping its habits but treating the evidence before the boundary as old — the drum would enter against a root whose credit had partly fallen back to its starting value, and the patience arithmetic says it could then win in two bars, as the first-bar drum does. The computation is this one with the record’s counts shrunk by a stated factor at the bar the drum enters, and it would say how much forgetting a section boundary has to cause before a late entry works as well as an early one — which turns the arranger’s habit of bringing the drums in after a break into a number.