Two honest bars convict a lying root

A late root is believed because the record is kept by the reading found the one bass no learning listener can survive. It puts the chord’s fifth on the barline and its root at the half-bar, so the witness every earlier listener trusted most — the bass’s root — votes for the wrong half of the bar in every bar. A listener that keeps a record of how often each witness agrees with the reading it holds should, in principle, find the liar out. None did, and the reason was that the record is kept against the listener’s own reading. In the two bars of ground before the tune, the root is the loudest witness and it makes the reading; from then on it agrees with the reading it made, the honest melody disagrees, and the record promotes the liar and silences the truth.

That essay closed by naming the obvious experiment. If the reading formed around the true barline before the root could make it, the record would work the right way round: the melody would agree with the reading and the root would disagree, and the root’s credit would fall instead of rise. So give the passage one honest cue at its very start — a drum stroke on the first downbeat — and see how many bars of truth a listener needs before a lying root, heard in every bar afterwards, can no longer take the reading away.

The answer is two, for one listener out of four. The other three show why the answer is not simply a number of bars.

The passage and the four listeners #

The passages are the four hundred the earlier essays used: one chord a bar, a walking bass on every crotchet of the kind a walking bass marks the bar by the note it is on set out, a tune of crotchets on chord tones with a long note in half its bars, and that long note on the downbeat for a tune that marks the barline by where it lingers. The bass states every root late. The only new element is a drum: a stroke on the downbeat of each of the passage’s first k bars, counted from the first bar of the ground, and silence after it. The passages are seeded exactly as they were without a drum, so at k = 0 every number of the earlier essay is reproduced, and every value of k is the same four hundred passages.

Each witness votes in each bar for the barline or the half-bar, or abstains: the bass’s root, the bass’s motion, the tune’s long notes, and the drum when it plays. A listener weighs the votes by its record of each witness — how often it has agreed with the reading the listener held when the vote was cast — starting from a credit worth four bars of record. The four listeners are the earlier essay’s:

• no habits, crediting every witness at three in four;
• trusts the root, crediting the root at three in four and the motion and melody at a coin toss — the habit the habit worth bringing is to distrust the tune found best whenever the root is honest;
• doubts the root, the opposite habit;
• chooses among all three, keeping all three sets of habits and weighing them by how well each predicts the votes, as a listener who knows the style needs no introduction first set out.

Every listener credits the drum at three in four.

What the drum does to each listener #

The first figure has the whole result in it, and the four curves are four different stories.

The listener that doubts the root needed no help: it heard the tune against the bar in 74 per cent of bars with no drum, because it never gave the root a vote. One stroke takes it to 98 and two to 100.

The listener choosing among all three goes from nothing with no drum to 10 per cent with one stroke, 93 per cent with two and 100 with three or more.

The listener with no habits rises slowly and stops: 5 per cent at two bars, 37 at three, 58 at four, and 74 once the drum plays through the whole ground and most of the tune. It never reaches four bars in five.

And the listener that trusts the root barely moves at all: 6 per cent at three bars, 25 at six, 50 when the drum plays in every bar of the passage. It is the habit that won the earlier comparison of habits, and against a late root it cannot be rescued by a drum, however long the drum plays.

The interesting cases are the two in the middle: why two strokes and not one, and why two strokes rescue one listener completely and another hardly at all.

Why one stroke is not enough #

Read the passages bar by bar and the difference between one stroke and two is a single bar of ground.

A listener’s reading is a running score for each half of the bar, and each bar adds its weighted votes. In the first bar the drum votes for the barline and the late root votes against it. The choosing listener credits the drum at three in four and the root, averaged across its three habits, at rather less — one of the three doubts the root — so the drum wins the first bar and the reading forms on the barline, in every passage.

In the second bar, with one stroke, the drum is silent and the root votes alone. One bar of root is worth more than the margin one bar of drum-over-root left behind, so the reading flips to the half-bar before the tune has entered. From there the earlier essay’s story runs unchanged: the root agrees with the reading it now holds, the melody disagrees, and the record turns against the melody. Ten per cent of the passages recover something; most recover nothing.

With two strokes, the drum outvotes the root in the second bar as well, and the reading is still on the barline when the tune arrives. That is the whole difference. The record scores each vote against the reading held when the vote is cast, so what matters is not how many honest votes are cast but whether the reading is right at the moment the honest tune begins to be scored. One stroke builds a reading that is erased before the melody is heard. Two build one that is still standing.

The same record, turned round #

The earlier essay’s figure of the record showed the root’s weight doubling and the melody’s falling to nothing. The same quantity with two honest bars at the start is its mirror image. By the time the tune has played its first bar the root’s weight has fallen from 0.7 to 0.2, by its second to nothing, and the melody’s has risen from 0.7 to 1.5 by the end of the passage. Nothing about the rule has changed. A witness is credited when it agrees with the reading held, and discredited when it disagrees; the drum decided which reading was held when the record started counting, and everything else followed.

That is the most useful thing the computation shows, because it says exactly what the record is. It is not a detector of liars. It is an amplifier of whatever reading is in place when the evidence starts to arrive. Seeded with the truth it convicts the liar as efficiently as, seeded with the lie, it convicts the honest witness. The earlier essay found the record perverse; it is not perverse, it is faithful, and what it is faithful to is the first reading.

Two bars of drum hold the barline; a dozen bars of bass do not found two bars of drum sufficient by a different route, for a listener with a fixed cost of changing its mind, a bass that was honest and a tune that syncopated. The two numbers come from different models and their agreement may be no more than a coincidence. What the two cases share is the shape of the requirement: the drum has to hold the reading against the strongest competing witness for long enough that the competing witness’s next bar cannot undo it.

Why the choosing listener, and not the others #

The record turns round for every listener once the reading is right. What differs is whether a listener can act on it.

The choosing listener keeps three sets of habits, and its belief in each rises or falls with how well each predicts the votes against the reading held. With the reading made by the root, the habit that doubts the root predicts worst, and its share of the belief falls from a third to 7 per cent. With the reading made by the drum, the same habit predicts best — it expected the root to be unreliable, and the root is disagreeing every bar — and its share rises to 75 per cent by the end of the tune. So the choosing listener does not merely discount the root through its record. It changes which set of habits it is listening with, and the habit it changes to is the one that survived the late root unaided in the earlier essay.

The listener with no habits has only its record — the per-witness record a listener who weights its witnesses needs eight bars of bass introduced. The record does turn against the root once the reading is right, but the root’s credit starts at three in four, the same as the drum’s, so in the ground the two votes cancel exactly. A listener that credits the root and the drum alike is held at a tie by the drum alone, however long it plays: nothing forms a reading until the melody enters and breaks the tie. Its recovery is therefore slow and partial, and it never passes three bars in four.

The listener that trusts the root is worse placed still. It credits the root at three in four and the melody at a coin toss, so in the tune the root outvotes the melody in every bar; while the drum plays it ties the root, and when the drum stops the root has the bar. It holds the barline exactly as long as the drum is there to cancel the root, and not a bar longer.

So the earlier essay’s conclusion — that no single habit serves every bass — survives, and acquires a sharper form. A listener that can switch habits needs only two bars of truth to find the habit a late-root style requires. A listener that cannot switch needs the truth to keep arriving.

How many honest bars each tune needs #

The tunes above all mark the bar, so the melody is an honest second witness from the moment it enters. A tune that syncopates is not: a syncopated melody needs its phrases to land found it leading a listener to the wrong half of the bar three times in four, and syncopation is a number about the metre is the reason it can only be measured against a metre already known.

For the choosing listener the threshold moves by a single bar. With a tune that marks the bar it is two. With a tune whose long notes fall half on the downbeat and half on the half-bar it is three, at 92 per cent, and with a tune that syncopates every long note it is also three, at 86 per cent, and four bars of drum give all of them every bar.

That a syncopated tune can be heard against the bar at all, over a root that also votes for the half-bar, deserves a word, because no witness in the tune is telling the truth. With the reading set on the barline by three bars of drum, the root and the syncopated melody both disagree with it in every bar, and the record discounts both: the root to nothing by the second bar of the tune, the melody from 0.7 to a twentieth. The listener has not been persuaded by the tune. It has stopped listening to the witnesses that are lying and has no others, and the reading stays roughly where the drum left it.

Roughly, and not exactly. The melody’s last scrap of credit and the bass’s motion, which votes against the barline in one bar in ten, go on pushing, and the share of passages still on the barline falls bar by bar through the tune — from all of them in its first bar to two in three by its eighth. That is a reading being worn away rather than defended, and a longer syncopated tune would lose it.

The listener with no habits manages none of this. Under a marking tune it reaches 74 per cent; under a half-and-half tune 35; under a syncopated one 4. Its record discounts the liars as well, but the tie in the ground means its reading never forms on the barline in the first place.

Whether the reading is ever taken back #

The lead asked how many bars of truth a listener needs “before a lying root, heard every bar afterwards, can no longer take the reading away”. Eight bars of tune is not long enough to say no longer. Twenty-four is a fairer test.

The choosing listener, given two bars of drum, holds the barline in about 94 passages in a hundred from the sixth bar onward and is still there at the twenty-sixth. The root has voted against it in every one of those bars, and its record has kept the root at nothing throughout. Once the record has convicted the root, a lying root heard every bar cannot take the reading back, because each of its votes is another disagreement on the record.

The rest are lost early. In the tune’s first bar about one passage in ten flips to the half-bar, while the root still carries a fifth of its starting credit and the record has not yet finished turning against it; a few of those come back as the melody’s long notes accumulate, and the others stay lost. The figure shows them as the small dip after the tune enters.

Against that, the listener that trusts the root shows the opposite pattern exactly: a tie for as long as the drum plays, and the half-bar from the first bar without it. Eight bars of honest drum bought it nothing that lasted.

Where this sits in practice #

A style whose bass keeps the root off the barline almost always has something else that marks the downbeat clearly — a drum, a chord stab, a count-in. The earlier essay named that as the obvious objection to its pessimism. This computation agrees, and it says what that something has to do: be there for the first two bars, so that the reading is formed before the tune starts to be scored. It does not have to be there afterwards.

That also says something about the order in which a band starts a tune. A count-in is an honest cue that comes before any music. A drum fill into the first bar is one. A bass player entering alone, with the root late, in front of a band that has not yet played, is the one start the arithmetic says will mislead a listener who learns — and the reading that start forms will last, because the record will defend it.

The repertoire is the same as the earlier essay’s: walking bass lines that push the root off the downbeat, anticipations and inversions in pop, funk and dance-band playing. The claim is not about how any of those styles is actually heard. It is about what a listener that learns from agreement with its own reading needs before a lying witness is heard.

The model the counts rest on #

Four hundred passages at each drum length, seeded as in the earlier essays, of two bars of ground and eight or twenty-four of tune; one chord a bar; a walking bass on every crotchet with every root moved to the half-bar and the fifth on the barline; a melody of crotchets on chord tones with a long note in half its bars, on the downbeat for a marking tune. The drum votes for the barline in each of the first k bars and abstains after. A listener keeps two running scores with no cost of switching between them, and per-witness records against the reading it held when each vote was cast, with a starting credit worth four bars; the choosing listener weighs its three sets of habits by how well each predicted the bar’s votes, flattened two per cent a bar towards even.

What the arithmetic cannot settle #

A real listener’s habits are not three. The choosing listener has exactly the habit the late root requires among its three. A listener whose habits did not include doubting the root would behave like the listener with no habits, and the two-bar threshold would be gone. What this shows is the value of having the right habit available, not that listeners do.

The credits are conventions. The drum is credited exactly as the root is by two of the listeners, and that equality is what produces their tie in the ground. A listener that trusted a drum a little more than a bass — which is plausible for most listeners in most styles — would form the reading in the ground and recover the way the choosing listener does. The size of that preference is not known, and nothing here measures it.

And none of this is a measurement of a listener. The model is an account of what a particular kind of learning does with a particular kind of evidence. Whether listeners keep anything like a record of their witnesses, and whether it is kept against the reading or against something else, is a question for an experiment, not for this arithmetic.

Still open: whether the drum has to come first #

Every drum here plays the passage’s first bars. A drum that arrives later — two strokes in the middle of the tune, after the root has already made the reading and the record has promoted it — has to undo a reading rather than form one, against a root whose credit is by then twice its starting value. The computation is this one with the drum’s two bars moved along the passage, and it would say whether a listener misled for several bars can be recovered by the same two strokes that would have prevented the error, or whether the price of a late correction rises with every bar the lie has been believed.