An announced plan pays the cycle and punishes the blocks

A listener who learns the plan keeps only the far pair’s immunity took away something every earlier essay on this question had given its first-time listener: the plan. A piece of eight minute-long sections, each carrying a rhetorical function — opening, continuing, intensifying, closing — is easier to find one’s way back into after a lapse of attention if the listener knows which sections carry which function. A listener who has to infer the plan from the sections heard so far loses most of that help, and loses it almost entirely in the first minute, because every plan a piece of this kind might have begins with an opening and nothing distinguishes them until the second section.

The cost, then, was a blind first minute, and that essay ended by naming the remedy composers have. An opening can announce what follows: a slow introduction that presents the piece’s materials in the order they will matter, a title, a programme note, the conventions of a genre. Each gives the listener the plan before the first lapse. Each can also be wrong, or be read wrongly. The question is how reliable an announcement has to be to be worth having, and whether a listener told the wrong plan does worse than one told nothing.

The learner’s prior, and what the earlier essay found #

The model is the earlier essay’s. The listener entertains five plans — four functions cycled, the same four in blocks of two, three functions cycled, opening and closing alternated, opening and continuing alternated. Lapses of attention average forty seconds, with a long tail; a tenth of them take every running count at once. Coming back into music it has not heard and recognising its function, the listener weighs each candidate section by how long lapses usually last and by the chance, over the plans still possible, that the candidate carries that function, and picks the likeliest. The measure of a first hearing is the number of proportions the listener can tell apart across the whole piece, from the nested counts a form is sharp at the bottom and vague at the top set up.

The figure is the earlier essay’s last result: belief in the true plan, given for free. For the cycle it pays in steps — nothing until three-fifths, 57.5 at ninety-five per cent, against 60.8 told and 44.2 learning. The steps are the listener’s choices changing: an expectation matters only when it tips one candidate section over another.

An announcement is that belief with a price on it. It names a plan, and it may name the wrong one.

An announcement trusted as far as it deserves #

The simplest announcement names the true plan with some reliability and, when it is wrong, one of the four others at random. A listener who knows how reliable such announcements are puts exactly that share of its prior on the plan named and divides the rest evenly — the calibrated listener. At a reliability of one in five the announcement is no better than a guess and the listener’s prior is even, which is the learner.

For four cycled functions, an announcement is worth nothing up to forty per cent reliability: 44.2, exactly the learner’s. At sixty per cent it gives 46.6, and from there to eighty it barely moves. At ninety per cent it jumps to 56.1, at ninety-five to 56.8, and at ninety-nine to 60.6, within a fifth of a point of being told. Half of what being told the plan is worth to the cycle arrives only when the announcement is right nine times in ten.

The steps have the same cause as before. The listener chooses the likeliest section, and an announcement changes the choice only where the named plan’s weight overturns the combined weight of the plans it disagrees with in the first minute. Below ninety per cent, belief in the named plan is not enough to outvote the four alternatives in the cases that matter most — lapses that begin early and run long — and so the listener chooses as if nothing had been said.

For the far pair, opening and closing alternated, an announcement can never be worth much, because there was little to win: learning cost the far pair only 2.4 points, and an announcement right ninety-nine times in a hundred recovers 2.3 of them. For the blocks it is worse than nothing in the middle range: at fifty and sixty per cent the blocks fall to 37.3 and 37.8, under the 39.1 a listener told nothing achieves. They need seventy per cent before an announcement helps at all.

A wrong announcement, believed #

The second half of the question was whether a listener told wrongly does worse than one told nothing. It depends on which wrong plan it is told.

Some wrong announcements are disasters. The blocks, told with ninety-five per cent confidence that the piece is three functions cycled, fall from 39.1 to 22.9. The far pair told it is four functions cycled falls from 37.1 to 22.9 as well. The cycle told it is in blocks falls from 44.2 to 35.0. In each case the named plan and the true one agree about the first section and disagree about where the second function comes, so a listener lapsing in the first minute is sent confidently to the wrong place.

Others are harmless, and one helps. The cycle told it is opening and closing alternated, or opening and continuing alternated, scores 45.0 — a little above being told nothing. The cycle told it is three functions cycled scores 47.9, more than most correct announcements below ninety per cent give it. That plan puts the intensification where the true plan puts it in its first half, and the listener lapsing into it is steered correctly for the wrong reason. A wrong plan that agrees with the true one about the first two minutes is, for a listener who lapses only then, not wrong.

So the answer to the question is yes, a listener told wrongly can do much worse than one told nothing — sixteen points worse — but the size of the damage is a property of the pair of plans, not of the error. The earlier essay’s suspicion, which rested on the blocks falling below the even prior at a weak expectation, was right about the blocks and too general about the rest.

Where a wrong announcement costs #

Drawn across the piece, every effect of an announcement, good or bad, is in the first two minutes. The listener told the true plan is misplaced 10.9 percentage points less often than the listener told nothing in the first minute, 2.6 less in the second, and identically after that. The listener told the piece is in blocks is misplaced 40.2 points more in the first minute — nearly every lapse that begins late in the first section is resolved into the wrong place — and not at all more afterwards.

The reason is the same as for learning. Once the second section has been heard, the plans that disagree with it have no weight left whatever the announcement said: the prior is multiplied by zero for every plan the music has ruled out. An announcement is an opinion about the first two minutes, overruled by evidence as soon as there is any. That puts a ceiling on what any announcement can do and a floor under what any can cost, and it is why a confident wrong announcement can hurt so much without hurting for long: the whole of its damage is concentrated where the listener has nothing else.

A section that says what it is for places a first hearing found the music’s own functions doing that work from the second section on. An announcement does it for the first section, which is the one stretch of the piece the functions cannot help with, and it does it only by being believed.

Why the first minute carries so much #

It is worth asking why a single minute can carry sixteen points of a whole piece’s worth, because the answer is the reason announcements exist at all. Only a lapse of attention keeps a form’s top vague found that the large proportions of a form — the ones a listener judges across minutes — are lost to lapses rather than to the slow drift of counting, because a lapse throws away a whole running count at once. A lapse that begins in the first minute and runs long carries the listener deep into music it has never heard, and new music says only that it is new: without a plan, the music’s novelty is no guide to which section it is. The function the listener recognises on returning is the only guide, and in the first minute a function means something only if the plan is known.

Later lapses are cheaper for two reasons the earlier essays separated. Fewer of the sections within a lapse’s reach are unheard, so the checkpoints a listener has already passed do some of the placing; and the plan has been learned, so a function points at the right section. A repeat sign loses a listener; a refrain does not found that what places a returning listener is whether only one section within reach could sound as it does. An announcement is a way of making that true from the start rather than from the third section, which is why its value, unlike almost everything else in these essays, is concentrated entirely at the beginning of the piece.

That concentration is also why the figures can be so sharp. The first minute is an eighth of the piece, and its lapses carry a fixed share of the piece’s errors; everything an announcement does, good or bad, is done inside that share. A wrong announcement that sends forty per cent more first-minute lapses astray can cost sixteen points because those lapses were already the most consequential in the piece.

How far to trust it #

A calibrated listener trusts an announcement exactly as far as it deserves. That is not the best a listener can do, because the payoff is not smooth: the listener’s choice changes only in steps.

At sixty per cent reliability, the cycle does best when the listener believes the announcement almost completely: 53.4 at ninety-nine per cent trust, against 46.6 at calibrated trust and 44.2 told nothing. The blocks do best at seventy per cent trust, 40.2, and worse than nothing — 37.4 — at ninety-nine. The far pair does best at half, 38.8, and worse than nothing when the listener believes more than four-fifths. At other reliabilities the pattern holds: at forty per cent the cycle still gains most from near-complete belief, 49.7, while the blocks are best served by ignoring the announcement altogether; at eighty per cent the cycle’s best is 57.1 at near-complete belief and the blocks’ 41.6 at seventy per cent trust; at ninety-five per cent the cycle and the blocks both gain most from near-complete belief, and the far pair is best served by a trust of half.

So credulity is a property the piece rewards or punishes, not a virtue of the listener. A piece built as a cycle rewards a listener who believes what it is told about the plan, even when what it is told is often wrong, because a correct announcement is worth a great deal and a wrong one, for most wrong plans, costs little. A piece built in blocks punishes the same listener, because its wrong announcements are the costly kind and its correct ones are worth little. A listener cannot know which kind of piece it is hearing until the second section — which is when the question stops mattering.

Every arrangement, announced #

Across all five arrangements, an announcement right nineteen times in twenty, trusted that far, is never worse than none and moves each arrangement most of the way to being told. Four cycled functions go from 44.2 to 56.8 against 60.8 told; three cycled functions from 43.9 to 50.9 against 51.4; the blocks from 39.1 to 42.5 against 45.0; the two alternations from 37.1 to 39.2 against 39.5. At sixty per cent the gains are a point or two at most, and the blocks lose 1.3. The cycles, which the plan made most valuable in the first place, have most to gain from being told it, and the arithmetic of the earlier essays holds: the arrangement whose labels are worth most is the one whose labels depend most on knowing the plan.

The model under the announcement #

Five plans and a fixed listener. The announcement names one of the five plans the listener already entertains, and a wrong one is any of the other four alike. A real announcement — a slow introduction, a title — is more likely to be wrong in some ways than others, and a listener whose list of plans was longer or shorter would be differently helped.

Lapses and hearing as before. Forty-second lapses on average, recognised functions heard exactly, a listener choosing the likeliest section, eight sections of a minute each.

The measure is the earlier essays’. Proportions told apart across the whole piece is a count of how many distinguishable ratios of section lengths survive the listener’s timing error and its lapses — a number, as a count is not an estimate put it, of categories rather than a precision. A difference of sixteen in it is large: more than half the distance between a listener with no help from the functions at all and one told the whole plan. A difference of one is at the edge of what the measure resolves, and the gains and losses of a point or two reported above for the alternations are best read as no change.

The announcement is believed once and then overruled. The listener updates its belief only by ruling out plans the music contradicts. It does not lower its trust in the announcer when the announcement turns out wrong, because there is nothing left to use the trust on: by then the plan is known.

What an announcement cannot be shown to do here #

What an introduction actually conveys. The figures treat an announcement as naming a plan. A slow introduction conveys materials and, perhaps, an order; whether a listener extracts the plan of functions from it, and how reliably, is a question about listeners and pieces that no arithmetic here answers. A return is shorter than its first hearing is a reminder that material heard once is recognised later; whether an introduction’s material is recognised as a forecast of order is a different question.

Genre as an announcement. A listener who knows the genre has, in effect, a standing announcement, right as often as pieces in the genre follow its plan. The figures say what that is worth at each reliability; they do not say how often any real genre’s pieces keep to its plan.

Still open: an announcement that the music confirms gradually #

Every announcement here names a whole plan, and the music confirms or overrules it all at once at the second section. A slow introduction is not like that. It plays the functions one after another, and the listener can match them against the piece as it arrives — the opening as the introduction’s first gesture, the continuation as its second — so that confidence in the announced plan grows section by section instead of being given at the start. A listener who lapses during the first minute would then be guided by an announcement half-confirmed. The computation is this one with an announcement whose weight rises with each section that agrees with it, and it would say whether a gradual confirmation buys the cycle’s first minute more cheaply than a trusted title does, or whether, since the first minute is exactly the stretch the music has not yet confirmed, it buys nothing at all.