A listener who weights its witnesses needs eight bars of bass, not twelve

Two bars of drum hold the barline; a dozen bars of bass do not built a listener who reads where the bar begins one bar at a time. Every witness — the bass’s root, the bass’s motion, the melody, and a drum where there is one — casts a vote between the written barline and the half-bar, and the listener keeps a running total with a cap on how much confidence it can bank. Fed a syncopated tune over one chord a bar and a walking bass, a listener that never forgets needed about twelve bars of harmony and bass before the tune to hear it against the bar in nine bars out of ten. A drum on the downbeat cut that to two.

Every witness in that listener had one vote, always. Its last section named what a real listener plainly does instead. Over a piece a listener learns which witnesses to believe: that in this style the bass walks, so its motion says little about where the bar begins, and that the drum is right about the downbeat. The question it left was what happens when the listener weights each witness by how often that witness has agreed with the reading it currently holds.

The answer has a headline and a warning. The headline is in the first figure: weighting brings the twelve bars down to eight, and it turns a tune that syncopates every bar — for which the earlier listener found no ground a song would use sufficient — into one that eight bars of ground carry to nine in ten. The warning is that the same learning, given nothing to learn from first, learns the syncopation’s lesson instead.

A record for every witness #

The listener is the earlier one — a running total with a cap on the confidence it can bank, the same shape as the key-finder how much evidence a modulation needs gave a cost for changing key — with one change. Each witness keeps a record: in every bar in which it voted and the listener already held a reading, the vote either agreed with that reading or disagreed. The weight given to the witness’s next vote is read from the record, by the rule that gives each of several independent advisers the vote its reliability deserves:

$$r = \frac{\text{agreed} + n\,p}{\text{agreed} + \text{disagreed} + n}, \qquad w = \frac{\log\!\big(r / (1 - r)\big)}{\log\!\big(p / (1 - p)\big)}.$$

Here $r$ is the witness’s estimated reliability and $w$ its weight. Before a witness has been heard it is credited with a reliability $p$ of three in four, worth $n$ = four bars of evidence, so every weight starts at exactly one — the earlier listener’s vote. A witness that agrees with the held reading more often than three times in four earns more than a vote; one that agrees less earns less. At a coin toss the weight is zero.

The log-odds form is not arbitrary. It is the weighting that the theory of voting juries gives for combining independent voters of known accuracy: Nitzan and Paroush showed that a panel decides best when each member’s vote counts in proportion to the log of the odds that the member is right. The listener here does not know those odds; it estimates them from the record, and the record is kept against the listener’s own reading, which is the only standard it has.

Two further choices complete it. An abstention — a witness whose vote is nought in a bar, as the bass’s motion often is — is not recorded either way. And below a coin toss the weight can either stop at zero, which ignores an unreliable witness, or go negative, which reads a reliably wrong witness backwards. The figures use the first. The second is taken up below, because it is the literal version of a phrase musicians use.

With learning switched off, every weight stays one and the listener is the earlier one exactly: the same passages, generated from the same seeds, give the same shares to the last digit, 40, 57, 71 and 93 per cent after none, two, four and twelve bars of ground. Every difference in this essay is the weighting.

What the ground teaches #

The first figure’s gain is made in the ground, before the melody enters, and the second figure shows how.

The bass’s root votes for the barline far more often than against it, because a walking bass must sound the new chord’s root on the downbeat — the reason the chords mark the barline by changing there where a bar-by-bar reading of the chords alone found them a weak witness; when it abstains it is because it sounds the root again at the half-bar. Once the listener holds the barline, the root agrees with it bar after bar and its weight climbs: 1.22 after four bars, 1.51 after eight, 1.67 after twelve.

The bass’s motion goes the other way. A walking bass marks the bar by the note it is on found that such a bass moves less at the barline than half a bar later, because it cannot move onto a root it is already sounding. So the motion votes for the half-bar about as often as for the barline, and its record settles near a coin toss. Its weight falls to 0.85 after four bars and 0.51 after twelve. The earlier essay’s lead asked whether a listener who has heard a few bars of walking bass would stop counting the motion altogether, and the answer is: halfway. The motion is not wrong often enough to be ignored and not right often enough to be trusted.

That is what makes each bar of ground worth more. Over twelve bars of ground the root’s vote averages +0.49 — it votes for the barline in about half the bars and abstains in most of the rest — and the motion’s averages −0.11. Counted alike, those bars bank 0.38 of a vote each. Weighted, the same bars bank 0.56 on average, and by the end of the ground, with the root at 1.67 and the motion at 0.51, about three-quarters of a vote a bar. The ground does not have more in it. The listener has learned to hear what was in it.

Then the melody enters. Its long notes fall on the half-bar, the texture a syncopated melody needs its phrases to land built, so in half its bars it votes against the barline the listener already holds, and in the other half it abstains or votes weakly for it. Its record goes bad fast: its weight halves in two bars and is down to 0.12 by the eighth. The syncopated melody is discounted as a witness about where the bar is. That is one precise sense in which a syncopation is heard as syncopation, and it is a different sense from the one syncopation is a number about the metre priced, where a syncopation’s size is the metrical weight a note skips: the listener has not stopped hearing the long notes on the half-bar, but has stopped letting them say where the bar begins.

The consequence is larger than a smaller loss. Counted alike, the eight syncopated bars after twelve bars of ground spend 0.11 of a vote each, on average, and the listener’s lead drains through them. Weighted, the same bars add 0.70 of a vote each on average, because the root keeps voting for the barline at its enlarged weight while the melody’s vote against it shrinks toward nothing. Once the listener has learned whom to believe, the syncopated tune does not wear the barline down. The bass underneath it keeps building it up.

With nothing to learn from first #

The same mechanism, given no ground, runs in reverse.

With the melody present from the first bar, the listener’s first reading is whatever the first bar’s votes say, and in most passages the syncopation wins it: the melody’s long note on the half-bar plus the motion’s vote outweigh the root. From then on the reading held is the half-bar, and the witness that disagrees with it is the root. So the root is discounted — its weight falls to 0.63 by the eighth bar — and the melody, which agrees with the reading it helped create, keeps its vote at 0.94.

That is the other half of what a learning listener is. It trusts whatever agrees with what it already believes, and what it believes first is set by whatever it heard first — a stronger form of the stickiness a metre has to be able to change its mind found in a tracking oscillator. With no ground, learning buys the barline nothing at all: 39 per cent of the melody’s bars against the barline, against 40 for the listener counting alike. It has learned the wrong lesson thoroughly, which the earlier listener had no way to do.

The same capture can happen inside the ground itself, and the essay returns to it below: in a small share of passages, the bass’s motion wins the first bar or two of ground, and a listener that learns quickly then trusts the motion and distrusts the root for the rest of the passage.

Why the gain is largest for a tune that syncopates every bar #

The first figure’s dashed curves are the harder case the earlier essay found hopeless: a melody with a long note on the half-bar in every bar. Eight such bars spent about five votes in the earlier listener, and sixteen bars of plain ground, at 0.41 a bar, did not bank enough to survive them nine times in ten; the listener counting alike reaches 85 per cent at sixteen. Weighting reaches 74 per cent after four bars of ground, 90 after eight and 98 after sixteen.

The gain is larger here than for the gentler melody, and the reason is on both sides of the ledger. The ground banks more per bar, because the root is trusted and the motion discounted, exactly as before. And the syncopation spends less per bar, because a melody that disagrees with the held reading in every bar loses its weight faster than one that disagrees in half of them. Counting alike, the harder melody costs about five times as much as the gentler one. Weighted, most of its later bars cost almost nothing, because by then it has no vote left to cast.

The earlier essay’s conclusion for this melody was that the bass alone is not a ground at any length a song would use, and that something that votes harder is needed — a drum, or the melody’s own phrase ends. Weighting is a third thing that votes harder, and it is supplied by the listener rather than the band. Eight bars of vamp is a long introduction, but it is a length songs do use.

Weighting and memory are not the same repair #

The earlier listener had one parameter that did most of the work: how much confidence it could bank. A listener that could bank only two votes never passed 54 per cent however long the introduction, because once its bank was full further ground bought nothing. Weighting changes that ceiling.

A listener that can bank two votes rises, weighting, from 65 per cent after two bars of ground to 84 after sixteen, where counting alike it sat at 53 whatever it heard. One that banks four rises from 71 to 94, against a ceiling of 73. That is a larger change than weighting makes to the listener who never forgets, and it is worth saying why.

The bank caps the difference between the listener’s two running scores. It does not cap the size of a single bar’s evidence. A listener that trusts the root at 1.7 and the melody at 0.1 fills its bank faster in the ground and loses less from it per syncopated bar, so the cap is reached with more margin and eroded more slowly. Weighting is a substitute for memory. A listener that cannot hold a large lead can still hold the barline if the evidence it adds and subtracts is sized by who is saying it.

The converse also holds. A listener with a perfect memory gets comparatively little from weighting after a long introduction, because it had already banked more than any syncopation could spend. After sixteen bars the weighting listener is at 94 per cent and the one counting alike at 97. That small reversal is the capture of the ground by its own first bars, which the next figure isolates.

How fast to learn #

The credit a witness receives before it has been heard, four bars’ worth in every figure so far, sets how fast the listener learns. A credit of one bar lets a single disagreement halve a witness’s weight; a credit of sixteen bars needs a long record to move it.

The fast learner is ahead after a short ground, at 74 per cent after two bars against the slow learner’s 64, because it has already given the root its extra weight by the time the melody enters. After a long ground it is behind, at 90 per cent against 97 after sixteen bars.

The reason is visible in the ground itself. The fast learner’s reading during the ground stops rising at about nine passages in ten and stays there, while the slow learner’s climbs to 98. In roughly one passage in ten, the bass’s motion wins the opening bars of the ground — the root abstains, or the first chord happens to have the bass moving on the downbeat — and the fast learner, crediting the motion within a bar or two, starts distrusting the root before the root has had a chance to be right. Nothing later in the ground undoes that, because every later bar is judged against the reading the capture set. The slow learner is not moved by the first bars and is right by the end.

So there is no best speed of learning, only a best speed for a given length of introduction. A listener built for two-bar introductions is fast, and pays for it on the long ones; one built for long introductions is slow, and hears short ones badly. Real listeners presumably differ in exactly this, and the model says the difference would show most clearly in how they hear the first bars of a song with a short introduction.

A drum shortens it again #

A drum on the downbeat was already the strong remedy, taking the earlier listener to 90 per cent after two bars — the marked downbeat that a dancer who comes in late needs. Weighting takes one drummed bar to 87 per cent and two to 99. The drum does most of its work by settling the reading early and firmly, and the weighting then lets the root, which agreed with that reading throughout, carry more of the load once the drum stops.

That is the version of this listener closest to what the styles concerned actually do. A count-in or a bar of hi-hat settles the barline, the bass confirms it for a bar or two, and by the time the tune syncopates the listener has already decided which of its witnesses to believe. The earlier essay found that the drummer’s first two bars make a short introduction long enough. With a listener who learns, the first bar nearly does.

Reading the syncopation backwards buys nothing #

The phrase “hearing the syncopation as syncopation” invites a stronger reading than discounting. A listener that knows the melody always places its long notes against the bar could, in principle, read every long note on the half-bar as evidence for the barline: a reliably wrong witness is as informative as a reliably right one, once its bias is known. The log-odds rule does exactly that if its weights are allowed to go negative.

It makes no difference. With negative weights allowed, the share of the melody’s bars heard against the barline changes by at most two points anywhere in the first figure, and mostly not at all. The reason is in the melody’s record. It disagrees with the held reading in about half its bars and abstains in most of the rest, so its estimated reliability falls toward a coin toss rather than below it, and its weight falls toward zero rather than through it. A melody whose long notes are on the half-bar in half its bars is uninformative about the barline, not anti-informative, and a listener that ignores it has extracted everything it has to give.

A melody that syncopated every bar with complete regularity would be different, and its record would go negative. But that melody is also the one whose syncopation is most easily reheard as a displaced barline, and a listener that reads it backwards is committing to the written bar on no evidence but the melody’s consistency. Whether real listeners do that is the kind of question the model can pose and not answer.

Whose listening #

The repertoire is the one the earlier essays named: ragtime, swing, bossa nova, the popular song whose phrases arrive early. The claim is about a listener who knows nothing of the style in advance and learns within one piece. A listener steeped in swing arrives with the root already trusted and the motion already discounted — a repertoire’s knowledge, which knowing every metre is slower than knowing none found can cost a listener as well as help it. In this model that is a large credit for the root and a small one for the motion before a note is played. That listener would need very little ground at all — which matches the common observation that a practised listener hears a syncopated tune against the bar immediately, while one unfamiliar with the style is easily turned around by it.

What the picture cannot show #

Whether listeners keep records. The rule is the best way to combine witnesses of estimated reliability. It is not established that listeners estimate reliability, and the evidence that they weight metrical cues by experience is about styles learned over years, not about witnesses learned within a piece.

A witness’s record in other pieces. Every witness here starts at one vote. The credit a listener brings from earlier pieces is, on this model, the single largest influence on how much ground a piece needs, and it is set by hand here, not measured.

Correlated witnesses. The log-odds rule assumes the witnesses are independent. The bass’s root and its motion are two readings of one line, and they are not independent: a bass that sounds the root on the downbeat is constrained in whether it moves there. A listener that learned their correlation would weight them differently again.

Two readings. As before, the only contest is between the written barline and the half-bar, which the metre leaves tied in this texture, as a texture in which nothing says where the bar begins arranged.

Still open: a listener that remembers where each witness was right #

Every witness’s record here is a single tally, so a witness that is right in the ground and wrong in the melody — the motion, which agrees more often before the tune enters than after — gets one averaged weight for both. A listener could keep a record per section instead, trusting the bass’s motion in an introduction and discounting it under a tune, or per position, trusting the root at the barline and not at the half-bar. The computation is the same record kept separately for each context, and it would say whether a listener that learns where a witness is reliable can hold the barline through a tune that begins with no introduction at all — which is the one case this essay found learning makes worse.