A cadence has to empty the reading, not weaken it

A fill cannot end a reading, but it can win the next one found a drum fill powerless in one place and decisive in another. Played into a reading the listener still holds — the walking bass’s half-bar, which a late root had made convincing — every stroke of the fill is scored as a disagreement with that reading, and the fill talks itself out of credit within two strokes. Played after a silence long enough to end the reading, the same fill speaks into nothing: its first stroke is scored against no reading at all, its vote makes a new one on the barline, and every stroke after agrees with what it made. Two strokes after such a silence hand the next section to the honest drum in every passage.

The trouble was the silence, and a silent break has to outlast the reading’s lead had already priced it. Before bar 9 of the passage the reading’s lead is about twenty-one votes, and a silent break charging one vote a failed downbeat needs more than twenty bars to outlast it. No song stops for twenty bars. That essay ended by naming what does empty a listener’s conviction at a real section boundary, if anything does: the cadence before it, a harmonic arrival heard as an ending, after which the metre of what follows is open.

The question it posed was how strong a cadence has to be. Put in the model’s terms, a cadence removes some share of the reading’s lead — both running scores multiplied down by the same factor, so that the reading still points the same way but with less behind it — and the fill then speaks into what is left. A cadence that removes everything is the long silence in a bar. A cadence that removes most of it might be nearly as good.

It is not. The arithmetic draws a line with almost nothing on one side of it.

A sliver is as good as the whole #

Before bar 9, where the reading leads by 20.7 votes on average, a cadence that leaves half the lead standing gives the drum nothing, as expected. A cadence that leaves a tenth gives it nothing. One that leaves a fiftieth — ninety-eight per cent of the listener’s conviction removed — gives the drum one passage in a hundred. One that leaves a hundredth, two in a hundred. Only the cadence that empties the reading completely gives the two-stroke fill what the long silence gave it: every passage.

Before bar 13, where the lead is 38.5 votes, the line is the same and sharper. Every cadence that leaves anything gives the drum nothing at all; the emptied reading gives it 84 per cent. Before bar 5, with a lead of 6.8, the cliff has a slope: leaving a fifth of the lead gives the drum 26 per cent, an eighth 87, a hundredth 96, none 100. Only before bar 3, where the lead is under three votes and a fill can overturn it outright, does a cadence that removes half the lead already give the fill 92 per cent.

So the answer to the question as posed is that a cadence has to be complete. A cadence strong enough to remove almost all of a listener’s conviction is, for a short fill, not much better than no cadence at all. The interesting part is why, because the fill is doing something on the far side of the line that it plainly cannot do on the near side, and the difference is a single stroke.

The fill turns the reading either way #

The obvious explanation would be that a sliver of lead is still more than two strokes can overturn. It is not that. Leave an eighth of the lead standing before bar 9 and the two strokes turn 97 per cent of the readings to the barline; leave a tenth or less and they turn every one. On the near side of the line the fill wins the reading exactly as it does on the far side.

And then the drum loses it. Of the passages whose reading the fill turned with a fiftieth of the lead left, one in a hundred is still on the barline at the end of the tune. The same turn, made into an emptied reading, holds in every passage. The fill does the same thing in both cases, by the measure of which reading the listener holds when the drum arrives. What differs is something the listener carries that is not the reading.

What the first stroke is recorded as #

The weights show it. The listener keeps a record for each witness of how often it has agreed with the reading held when it voted, and weighs its next vote by that record, as a listener who weights its witnesses needs eight bars of bass set out. By bar 8 the late root has eight bars of agreement with the half-bar behind it — the habit a walking bass marks the bar by the note it is on described, turned against the barline by a bass that states its roots late — and its vote weighs 2.04.

Into an emptied reading, the fill’s first stroke is scored against nothing: with no reading held, there is nothing for it to agree or disagree with, and its record is left alone. Its vote makes the reading, and its second stroke agrees with that reading and is recorded as agreeing. Heard as the drum, the fill hands the drum a record improved by one agreement: the drum enters at bar 9 with a vote of 1.26. Then the late root votes for the half-bar, against the reading the fill made, and is recorded as disagreeing. Its weight falls to 1.01 by bar 11, 0.54 by bar 13 and nearly nothing by bar 17, while the drum’s climbs to 2.26.

Into a reading with a fiftieth of its lead left, the fill’s first stroke is scored against the half-bar, which is still the reading held, however faintly. It is recorded as a disagreement. Its vote is enough to tip the sliver over, and the second stroke agrees with the new reading — but the drum’s record now holds one agreement and one disagreement, and the drum enters bar 9 with a vote of 0.63, half of what it would have had. The root votes for the half-bar at 2.04. That is more than the drum and the tune’s long notes together, so the reading goes straight back to the half-bar, the drum is scored as disagreeing from its first bar, and its weight is gone by bar 13. The root’s climbs back to 1.90.

The whole difference is whether the first stroke of the fill lands in a reading. Emptied, the reading cannot convict the fill. With any lead left, however small, it convicts the first stroke, and the first stroke is the one that costs the drum its standing at the moment the root decides the next reading. A late root is believed because the record is kept by the reading found that whoever holds the reading when the witnesses vote owns their record; the same rule, applied to a reading that is nearly gone, says that nearly gone is still held.

Every sliver costs strokes #

If the trouble is one stroke recorded as a disagreement, a longer fill should be able to pay it back with agreeing strokes after the turn, as the earlier essay found a fill heard as the drum had to.

It can, at a price that rises steeply with what the cadence leaves. Before bar 9, an emptied reading needs two strokes. A reading with a hundredth to three hundredths of its lead left needs three; with a twentieth, four; with a tenth or an eighth, eight; with a seventh or more, more than sixteen. Before bar 13 every step costs more: four strokes from a hundredth to three hundredths, six at a twentieth, eight at a fourteenth, and more than sixteen from a tenth. Before bar 5 the cost never rises above three strokes, and before bar 3 a single stroke is enough after any cadence that leaves less than a third.

The steps come from the same exchange as before. Each stroke recorded as disagreeing has to be answered by enough agreeing strokes that the drum’s record, when the root next votes, is good enough for the drum and the tune together to outweigh it. The more lead the cadence leaves, the more strokes the fill spends turning the reading before any of them can agree with it, and every one of those is a stroke the drum then has to pay back. Later in the tune the root’s weight is larger, so the drum has to arrive with a better record, and the payback is longer — the asymmetry two bars of drum hold the barline, a dozen bars of bass do not found from the other side, where what comes first decides and what comes later has to outweigh it.

A fill of eight strokes is a full bar of quavers — a common enough fill. Before bar 9 it hands the next section to the drum after any cadence that removes nine-tenths of the reading’s lead. That is a real cadence’s work, if a cadence can do it, and a bar of fill is ordinary. So the cliff in the first figure is a property of the shortest fills. What a long fill buys is the ability to tolerate an incomplete cadence.

Heard as its own witness #

The earlier essay ran two versions of the fill: heard as the drum, sharing its record, and heard as a witness of its own — a hit by the whole band, say — whose record dies with the bar. The second cannot cost the drum anything, so it ought to escape the cliff.

It escapes part of it. With a hundredth of the lead left, a two-stroke fill heard as its own witness gives the drum 78 per cent of the passages, against two per cent heard as the drum; with a fiftieth, 67 against one; with three hundredths, 56 against one. Past a twentieth it gives almost nothing either way, and emptied it gives 97, a shade under the drum-heard fill’s 100.

The own-witness fill does better near the line because its disagreeing stroke is charged to a record that is discarded. The drum then enters with its starting credit intact and meets the root on equal terms with the reading on the barline. It does worse than the drum-heard fill when the reading is emptied, because then the drum-heard fill’s agreeing strokes are credited to the drum, and the own-witness fill’s are thrown away. Each version wins where the other’s record does it harm.

The figure leaves the lead’s question harder rather than easier to answer. How listeners attribute a fill — to the drummer, or to the band as an event — was a question the earlier essay said mattered only before a reading had ended. It turns out to matter at exactly the point a cadence is supposed to deliver: a cadence that falls a little short of emptying the reading is worth a great deal if the fill is heard as an event and almost nothing if it is heard as the drum.

A cadence and a silence together #

The silence that the earlier essays needed was long because it had to wear down the whole lead at a vote a bar. A cadence that removes most of the lead leaves the silence much less to do.

Before bar 9, two strokes after silence alone need twenty-four bars. After a cadence that removes half the lead they need twelve; after one that leaves three-tenths, eight; a fifth, six; a tenth, three; a twentieth, two. Before bar 13 the same cadences leave twenty, twelve, eight, four and two bars. Before bar 5, a cadence that removes even half the lead brings the silence needed down from six bars to three.

So the division of labour is clean. A cadence does most of the emptying, a short break finishes it, and the fill speaks into what is left. The break’s job is no longer to outlast a reading built over a whole tune; it is to remove the last few votes, at one a bar. A song section that ends on a strong cadence, stops for two bars and comes back with a fill is doing, on this account, exactly what a listener with a firmly held metre needs done before an honest drum can take over.

What this says about section boundaries #

The earlier essay concluded that fills belong after the end of a section, where a listener is between readings rather than inside one. This one says what being between readings requires. It is not enough for a listener’s conviction to be weakened, even drastically: a short fill that arrives while any of the old reading is still held is recorded against it, and the record is what the old reading’s best witness then uses to take the metre back. The section boundary has to be an ending in the strict sense — an emptying — or the fill has to be long, or heard as an event rather than as the drum.

That gives a precise meaning to an old intuition about cadences and metre. The cadence as evidence treated a cadence as a vote for where a phrase ends; what makes an ending an ending asked what a listener needs before an arrival is heard as closing anything. The model here adds that, for metre, an ending is valuable in proportion to how completely it releases the listener, and that the value is not proportional: the last few per cent of release are worth more than the first ninety. A silence long enough to be an ending found silence doing that releasing on its own; a cadence can do most of it faster.

The claim is about song forms built from repeated sections over a steady groove, the repertoire the walking-bass passages stand for, and about the transition between sections where a new texture enters. It is not a claim about music whose metre is meant to stay ambiguous, nor about cadences within a section, where nothing new is arriving to be heard.

The model under the cadence #

Everything not stated here is the earlier essays’ listener: four hundred seeded passages of two bars of walking bass with the root on the half-bar and twenty-four bars of a tune whose long notes mark the barline; four witnesses, root, motion, the tune’s long notes and the drum, each weighted by its record of agreement with the reading held; three styles weighed by prediction; a fill of accented strokes, each a vote for the barline at its witness’s weight and recorded against the reading held at that stroke.

The cadence multiplies both running scores by one factor. The reading still points the same way after the cadence, with a lead reduced in proportion. That is the simplest way a cadence could loosen a listener’s grip on a metre, and it leaves the witnesses’ records untouched: the cadence is heard as an ending of the music, not as evidence about which witness was right. A cadence that also loosened the records would help the drum more, and a break rescues a late drum only by ending the reading found what wiping the records alone does.

The share is a free parameter. No measurement here says how much of a listener’s metric conviction a perfect authentic cadence removes, or a half cadence, or a deceptive one. The harmony essays’ measure of closure ranks cadences against each other, but mapping that ranking onto a share of a metric reading would be a guess, and the figures report every share instead.

What the cadence cannot be shown to do #

Whether a cadence empties a metre at all. The model assumes a cadence removes conviction about where the barline is. It is at least as plausible that a strong cadence confirms the metre it ends in — it usually arrives on a downbeat — and so adds to the lead rather than removing it. If so, cadences are the wrong tool for a change of metric reading, and the long silences of the earlier essays stand.

Whether a listener’s conviction is a single number. The listener here holds a running score per reading. A real listener might hold a reading firmly for a while and then let it go all at once, which is closer to the emptied case than to any share, or might never fully let go, which is closer to a sliver. The cliff in the first figure says that which of those listeners exists matters more than anything else in the account.

Still open: a cadence that is also a vote #

Every cadence here reduces the lead and votes for nothing. A real cadence arrives on a beat, and in the passages modelled the arrival falls where the tune’s long notes and the true barline put it, so a cadence heard metrically is a vote for the barline in the same act that releases the listener. The computation is this one with the cadence’s arrival scored as a witness — with its own record, or as the tune’s — at the moment the lead is reduced, and it would say whether a cadence that votes as it releases needs the reading empty at all, or whether its own vote can serve as the first stroke the fill otherwise has to waste.