A cadence that votes must not vote as the drum

A listener who hears a walking bass state every root on the half-bar comes to believe the half-bar is the barline, and an honest drum entering later has to talk them out of it. The essays on this listener have priced every way of clearing the ground for the drum. A silent break has to outlast the reading’s lead, which before bar 9 means more than twenty bars of silence. A fill cannot end a reading, but it can win the next one — after the reading has been ended. And a cadence has to empty the reading, not weaken it found that a cadence removing ninety-eight per cent of the listener’s conviction left a two-stroke fill winning one passage in a hundred, where an emptied reading let it win every one.

That last essay found the reason in a single stroke. A fill’s first stroke, landing in a reading that still held the half-bar however faintly, was scored as a disagreement, and the drum — whose record the fill shared — arrived with half its credit, to lose the next bar to the late root. In an emptied reading the same stroke was scored against nothing.

The cadences in that essay did one thing: they shrank the reading’s lead. A real cadence does something else at the same moment. It arrives, and it arrives on a downbeat. A perfect cadence’s tonic chord lands where the tune’s long note and the true barline put it, so the cadence that loosens the listener’s grip on the metre is also, in itself, a vote for the barline. That essay ended by asking whether that vote could stand in for the stroke the fill has to waste.

It can, under one condition, and the condition is the essay’s whole result: the listener has to hear the cadence as a new kind of witness, not as one it already has a record for.

Three ways to hear the same vote #

The model is the earlier essays’ listener exactly. It holds a running score for each of two readings, the barline and the half-bar, and every bar four witnesses vote: whether the root sounds, whether the bass moves, whether the tune’s long notes fall, and whether a drum strikes. Each witness’s vote is weighed by its record of agreeing with the reading held when it voted — the bookkeeping a listener who weights its witnesses needs eight bars of bass introduced — and the listener weighs three styles of music — one whose tunes mark the bar, one whose tunes syncopate, one whose bass keeps the root off the barline — by how well each predicts what it hears.

The cadence enters at bar 9. Both running scores are multiplied by the share it leaves, as before. Then its arrival casts one vote for the barline, and that vote has to be weighed by somebody’s record. There are three candidates. The cadence’s tonic arrives with the tune’s long note, so it might be heard as one more of the tune’s votes, weighed by the tune’s record. It might be heard as the drum, if the cadence is hit with the band. Or it might be heard as what it is, a harmonic arrival, a witness of a kind the listener has not yet had occasion to doubt, with the starting credit a style gives the tune’s long notes and no record behind it.

With one fill stroke after it, the silent cadence and the cadence heard as the tune leave the drum nothing until the reading is emptied, and little then. The cadence heard as the drum does exactly what the silent cadence did with two strokes: nothing short of emptiness, every passage at emptiness. The cadence heard as its own witness has a different shape altogether. It gives the drum 90 per cent of passages with a fiftieth of the lead left standing, 93 per cent with a hundredth, and half with three-hundredths. The cliff that sat at zero has moved to two per cent.

The tune’s credit is spent #

Why the tune’s reading of the cadence is worthless is in the record. The tune here is honest: its long notes fall on the barline. But the listener has spent eight bars hearing those long notes disagree with the half-bar reading the late root established, and each disagreement went into the tune’s record. By bar 9 the listener has also mostly decided that it is hearing the style whose tunes syncopate, in which long notes say nothing about the bar — the habit the habit worth bringing is to distrust the tune found a listener does well to arrive with, and here the habit that turns against the barline. A listener who knows the style needs no introduction; this one is learning it, and by bar 9 it has learned the wrong lesson from an honest tune, because the reading it was learning against was dishonest. A cadence heard as one more of the tune’s long notes inherits both judgements: its vote weighs 0.64 of a starting vote before bar 3, 0.24 before bar 5, 0.027 before bar 9 and 0.002 before bar 13.

The cadence heard as its own witness starts with a clean record but not with full credit. It takes the starting credit each style gives a tune’s long notes, so the same style-weighing that has written off the tune halves the cadence too — 0.47 before bar 5, 0.34 before bar 9, 0.29 before bar 13. A third of a vote, in a reading that still leads by a fiftieth of twenty-one votes, is enough to tip it.

The drum’s reading of the cadence carries a full vote. The drum has not yet played, so its record is untouched and its credit is the starting one, and that makes its failure the most instructive of the three. An honest drum has to come first found that a drum entering after the reading is set spends its credit arguing with it; a full vote is exactly what is at risk when the drum’s first word is spoken into the wrong reading.

Heard as the drum, a cadence is one more stroke #

The lines and the rings coincide at every point, and not approximately: a cadence heard as the drum, followed by n strokes, gives exactly the same passages as a silent cadence followed by n + 1. The reason is that the cadence heard as the drum is a fill stroke in every respect the listener tracks. It votes at the drum’s weight, it is scored against the reading held at that moment, and it is booked to the drum’s record. If the reading still holds the half-bar, it is booked as a disagreement, and the drum enters with half its credit, exactly as the fill’s first stroke left it in the earlier essay.

So the lead’s hope — that the cadence’s vote could be the first stroke the fill otherwise has to waste — is met literally and to no purpose. It is the first stroke, and it is wasted in the same way. The stroke was never wasted because it was the first; it was wasted because it was the drum’s, cast into a reading that would convict it, and the drum then had to live with the conviction.

That puts the result of the earlier essay in a sharper light. A late root is believed because the record is kept by the reading: whoever holds the reading when a witness votes decides whether the vote counts for or against that witness. A vote for the barline cast into a half-bar reading damages whoever cast it. The only vote that can be cast there safely is one by a witness with nothing further to lose.

How much of the old reading a cadence may leave #

Across the passage, the cadence heard as its own witness never needs more of the reading removed than a silent one, and at every boundary after the first it needs less. Before bar 3, where the lead is under three votes, a fill can overturn it whatever the cadence does, and every version tolerates half the lead left standing. Before bar 5, with a lead of 6.8 votes, a silent cadence and two strokes tolerates a twentieth; a cadence heard as its own witness tolerates a fifth with two strokes and a tenth with one. Before bar 9 the silent cadence needs the reading emptied and the voting one tolerates a fiftieth. Before bar 13, with a lead of 38.5, the silent cadence never gives the drum nine passages in ten with two strokes, and a cadence heard as its own witness does it with a hundredth of the lead left.

The cadence heard as the drum sits exactly one stroke to the right of the silent cadence at every boundary, which is the previous figure’s identity again.

These are still small shares. A cadence leaving a fiftieth of a listener’s conviction is a strong cadence by any account. What the voting cadence changes is not that it has to be strong but that it no longer has to be total. An emptied reading is a knife-edge — a cadence that removes all but the last hundredth fails — and a fiftieth is a target a real ending could plausibly reach.

How credible a cadence has to be #

The cadence heard as its own witness was given the starting credit of a tune’s long notes, which is a choice, and the listener’s opinion of cadences as metrical evidence is not something the arithmetic can supply. So the credit is swept.

With a fiftieth of the lead left, almost any credit will do: a cadence believed sixty times in a hundred to mark the bar gives the drum 91 per cent of passages, and seventy in a hundred gives 97. Only a cadence no better than a coin toss gives nothing. With a tenth of the lead left the cadence needs to be believed about eighty times in a hundred, and with a fifth about ninety-five. The trade is steep and simple: the more of the old reading a cadence leaves, the more the listener has to trust it. A cadence that is both strong and trusted — a perfect authentic cadence in a style where cadences reliably land on the downbeat — could leave a fifth.

Without the fill #

If the cadence’s own vote is worth a fill stroke, the last question is whether a voting cadence can do without the fill altogether. Early in the passage it can. Before bar 3 a cadence heard as its own witness, with no fill, gives the drum every passage with as much as three-tenths of the lead left; the silent cadence gives it 44 per cent at emptiness and nothing otherwise. Before bar 5 the voting cadence alone gives 66 per cent at emptiness and 45 per cent with a hundredth left, where the silent cadence gives 7 per cent at best.

Before bar 9 it gives nothing. A third of a vote, alone, tips the sliver and then faces the root’s two-vote verdict at the next bar with no stroke of the drum’s own to follow it; the reading goes back to the half-bar. The fill and the cadence divide the work: the cadence’s vote is safe because it carries no record, and the fill’s stroke after it is safe because it lands in a reading the cadence has already turned. Neither alone is enough once the late root has eight bars of credit behind it.

What this says about the endings of sections #

Between them, the four essays on clearing the ground for a drum say something general about how a section boundary works on a listener’s metre. The boundary has to empty, or nearly empty, the conviction the previous section built — silence cannot do that in any reasonable length, a cadence can. Whatever then speaks first into the emptied reading owns it, and whatever speaks first into a reading that is not quite empty is convicted by it.

The voting cadence adds a refinement that makes the arithmetic resemble practice more closely. Popular music marks section boundaries with exactly the combination this computation favours: a cadence, a hit on its downbeat by the whole band, and a fill. The hit on the cadence is not the drum alone — it is everyone at once, a kind of event the listener has heard only at boundaries. The cadence essay before this one modelled a fill heard as the whole band’s hit, a witness of its own whose record dies with the bar, and found it worth a great deal just short of an emptied reading — 78 per cent of passages with a hundredth of the lead left, against two per cent for the same fill heard as the drum. This essay finds the same distinction decisive for the cadence itself, and finds it cheaper: one vote at the cadence does what a whole bar of fill heard as an event did. Whether listeners actually keep separate books for the drum and the band’s downbeat hit is a question about listeners, but the arithmetic says that the boundary works much better if they do.

The listener, restated #

Everything not stated is the earlier essays’. Four hundred seeded passages: two bars of introduction, then twenty-four bars of a walking bass that states every root on the half-bar under a tune whose long notes mark the barline; four witnesses weighed by their records; three styles weighed by prediction; an honest drum on every downbeat from the bar after the boundary.

The cadence multiplies both scores by the share it leaves, then votes once for the barline, weighed by the record named and scored against the reading held at that moment. Heard as the tune, the vote is weighed and booked as one of the tune’s; as the drum, as the drum’s; as its own witness, it is weighed at its starting credit with no record, and nothing is booked because the witness never votes again.

The starting credit of a cadence heard as its own witness defaults to each style’s credit for the tune’s long notes, so a listener who has written off the tune as syncopated discounts the cadence in the same proportion. The figure on credibility sweeps it.

What a model of the metre cannot say #

How much of a listener’s conviction a cadence removes. The share is still a free parameter. The closure essays rank cadences against each other, and nothing maps that ranking onto this listener’s running scores.

Whether a cadence is heard as its own kind of event. The result rests on the listener treating the cadential arrival as a witness without a history. A listener who hears the cadence’s downbeat simply as the tune’s — which is how it is notated — gets nothing from it here.

What the passages are. Every result is about one constructed passage, chosen because it makes the late root hard to dislodge. A tune that syncopated, or a bass that sometimes stated its root on the barline, would give the drum an easier time and the cadence less to do.

Still open: a cadence that also ends the records #

Every cadence here leaves the witnesses’ records as they were: the late root goes into the next section with eight bars of agreement behind it, and that credit is what the drum’s first strokes have to beat. A cadence is heard as the end of something, and the obvious thing that might end with it is the listener’s book on each witness — a new section, a new song, a fresh start. A break rescues a late drum only by ending the reading found that wiping the records alone does little while the reading stands. The computation is this cadence with both its share of the reading and a share of every witness’s record removed together, and it would say whether a cadence that ends the books as well as the reading can let a silent boundary hand the next section to the drum — and whether the two together need less of either than each needs alone.