A listener who learns the plan keeps only the far pair's immunity

A listener hearing a through-composed piece for the first time — eight one-minute sections, every one of them new music — who loses count for a while and comes back has nothing in the music to say where they are, because new music says only that it is new. A section that says what it is for places a first hearing gave each section a rhetorical function — opening, continuing, intensifying, closing — and let the returning listener recognise it. With four functions cycled, the listener got back 85 per cent of what a first hearing costs. Two functions far apart outlast four that blur then let the listener mistake a function for its neighbour in that arc, and found the cycle keeping less than half its gain at one confusion in ten, while opening and closing alternated, sharing no neighbour, lost nothing to confusion at all.

Both essays gave the listener something no first-time listener has: the plan. They knew which section carried which function, so a recognised function told them which sections it could be. A real first-time listener would have to infer the plan from what they had heard — the functions of the sections before the lapse — and an inferred plan is uncertain exactly when it matters, early in the piece. The question the earlier essay left was whether the far pair’s immunity survives that listener, since the immunity seems to depend on knowing that the piece has no continuing sections.

It survives. Most of the rest does not.

What the told listener had #

It is worth seeing the earlier result again before taking the plan away, because the shape of what is lost is set by it.

With the plan in hand, the arrangements were ranked by how far apart they put the same function. The cycle puts each function four sections from its next appearance, beyond almost any lapse, and wins with exact hearing; the blocks put it in the very next section and lose; the far pair puts it two sections on, which is close, but its two functions can never be confused for each other, so it holds its value as hearing worsens. Every one of those properties is a property of the plan. The question here is how much of each survives when the listener has to work the plan out from the piece.

A listener who knows only what plans are possible #

The new listener comes to the piece with a short list of plans a piece of this kind might have: the five arrangements the earlier essays compared — four functions cycled, the same four in blocks of two, three cycled, opening and closing alternated, opening and continuing alternated — each equally likely to begin with. The piece’s true plan is always one of them. Before a lapse the listener has heard every section from its start, and its function with it, so the plans still possible are those agreeing with every function heard so far. Coming back into unheard music and recognising its function — or a neighbour of it, with the stated chance of confusion — the listener weighs each candidate section by how long lapses usually last and by the chance, averaged over the plans still possible, that the candidate’s function would be heard as what was heard.

Everything else is the earlier essays’: lapses of forty seconds on average, a tenth of them taking every count at once, and the measure of a first hearing the proportions a 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.

Told the plan and hearing every function exactly, the cycle gave 60.8 proportions told apart, against 29.8 with no functions and 66.1 for a listener who knows the piece. Learning the plan, it gives 44.2 — more than half of its gain over no functions is gone, with no confusion at all. The blocks fall from 45.0 to 39.1. Opening and closing alternated fall from 39.5 to 37.1, and stay at 37.1 at every rate of confusion. The learner keeps the far pair’s immunity.

Where learning costs, and why it is early #

Drawn across the piece, the cost of learning is almost entirely at the start. A lapse that begins in the first minute misplaces the learner 11.6 percentage points more often than the listener who was told the plan; one that begins in the second minute, 2.6 points more; one that begins anywhere later, no more at all. After two sections the learner knows the plan as well as the listener who was told it.

The reason is the plans themselves. Every one of them begins with an opening, so after the first section all five remain and the listener knows nothing about what comes next. The second section’s function narrows the field to one plan for three of the true plans and to two for the cycle and the opening–continuing alternation, which share a start; the third section settles it everywhere. So the learner is the told listener with a blind first minute and a half-blind second.

That would be a small cost if lapses in the first minute only reached the next section. They do not. A lapse of forty seconds on average has a long tail, and a lapse that begins in the first minute can carry a listener anywhere in the second, third or fourth. Told the cycle’s plan, a listener returning on an intensification knows it is the third or seventh section and takes the third; not told, it knows only that some plan puts an intensification here, and the plans disagree about where — the third or seventh, the fifth or sixth, the second, fifth or eighth, or nowhere. The early lapses are exactly the ones that need the plan most, because the stretch of piece they can reach contains the most sections the listener has not yet heard.

Seen that way, the cycle’s large loss is the price of what made it valuable. Its labels were worth most because the same function was four sections away, beyond a lapse’s reach; but that is information about the plan, and a listener without the plan has none of it at the moment the lapse begins.

A lapse early reaches furthest into the unknown #

The first minute’s weight has a general form that the earlier essays met from other directions. A repeat sign loses a listener; a refrain does not found that what places a returning listener is whether only one section within a lapse’s reach could sound as it does. For the listener who knows only the kind of piece, the sections within reach of a lapse are unheard sections, and the question whether one of them is the right one depends on the plan. Early in the piece every section within reach is unheard and every plan is still open, so the music’s function, however exactly recognised, narrows the candidates only as far as the plans agree.

Later, two things have changed. The plan is known, and fewer of the sections within reach are unheard: by the sixth minute only three sections remain ahead, and a lapse into any of them is placed by the told listener’s arithmetic. Checkpoints sharpen the middle of a form, not its top found the top of a form’s hierarchy the hardest level to mend after a lapse; here the start of a first hearing is the hardest point to mend, for the parallel reason that it is where the listener holds the least of the structure in memory.

So a first hearing is not uniformly fragile. It is fragile at its beginning, where the listener has heard least and the piece could still be any of several kinds, and steadily more robust as the plan reveals itself. A return is shorter than its first hearing measured how much of a returning section a listener already knows; the learner’s arithmetic says the same about the plan, which a listener knows almost entirely by the end of the second section and not at all at the end of the first.

Why the far pair keeps its immunity #

The far pair loses 2.4 proportions to learning — the same blind first minute — and nothing more as confusion rises. The reason is that its immunity never depended on knowing the plan. Opening and closing are the two ends of the arc and have no neighbour in common, so a confused opening is heard as a continuation and a confused closing as an intensification. Under the far plan those confusions are harmless: the far pair’s sections are only openings and closings, so a heard continuation can only have been an opening and a heard intensification only a closing, and the listener told the plan reads them back exactly.

The learner, in its first minute, entertains plans that do contain continuations and intensifications, and a heard continuation might then be a real one. But in the first minute it is placing itself among plans that all begin with an opening and that the second section will separate, and in the computation the confusions it suffers there cost it nothing beyond what exact hearing already costs: whether it hears the true function or its neighbour, its first-minute choice among sections is the same. From the second section it knows the far plan, and the arithmetic of the told listener applies. So the immunity is a property of the arc and the far plan together, and a listener who learns the plan acquires it as soon as it acquires the plan.

That answers the earlier essay’s question. The immunity depends on the piece having no continuing sections, as it seemed to, but the listener does not need to be told that: two sections are enough to find it out.

The learner’s speed has a precedent. The earlier essays’ listener who knows the style of a piece, but not the piece, narrowed a through-composed form’s plan of materials — which sections are new and which return — within three sections, and the function plans here are settled within two or three as well. In both cases the structure a listener needs is revealed at about the rate the piece states it, and the cost of not being told is paid while it is being stated. A plan with fewer shared beginnings would be learned faster: had no two of the five plans shared their first two functions, the second minute’s extra misplacement would vanish, and the cost of learning would be the first minute alone.

Where the far pair overtakes, for a learner #

With exact hearing the cycle is still the best plan for the learner, at 44.2 against the three-cycle’s 43.9. Confusion changes the order sooner than it did for the told listener. The learner’s cycle falls below the far pair’s 37.1 at about one confusion in seven, where the told listener’s cycle held out to nearly one in five. The three functions cycled hold out longer, to about one confusion in four, a little later than the told listener’s 22 per cent.

At one confusion in five the three-function cycle is best for both listeners, 39.8 told and 37.4 learning, with the far pair just behind at 39.5 and 37.1. The earlier essay said that at one confusion in five the far pair was the best of the five arrangements; its own figure gives the three-function cycle 39.8 there, three tenths ahead, and the far pair overtakes it only at about 22 per cent. The sentence has been corrected. The general shape it described stands: as confusion grows, arrangements whose labels need exact hearing give way to the one that needs none.

How much expecting the plan buys back #

A listener need not think all five plans equally likely. A listener who knows the style might expect one shape of piece more than others, and the question is how much a correct expectation is worth.

The returns come in steps, because the listener chooses the most probable section and an expectation changes its choice only when it tips one plan over another. For the cycle, belief up to two-fifths in the true plan changes nothing — 44.2 — because in the first minute the other four together still outweigh it; at three-fifths it gives 47.6, and at 95 per cent 57.5, close to the told listener’s 60.8. For the far pair, a tenth is already enough to recover the learner’s 37.1, and three-fifths recovers the told listener’s 39.5. For the blocks, too little expectation is actively harmful: a listener who puts only five per cent on the true plan, and so expects one of the cycles, falls to 32.1, below what learning with an even prior gives.

So the style a listener brings is worth most to the arrangement that the plan made most valuable, and it has to be a confident style to be worth anything. A listener who half-expects the cycle gets nothing from the expectation; one who expects it with near certainty gets back nine-tenths of what being told it gives.

The model the figures rest on #

The plans are five, a small and particular list; a real listener’s repertoire of plans is larger and less sharply drawn. A larger list would lengthen the blind start — more plans would share their first two functions — and so raise the cost of learning for every arrangement; it would not change the far pair’s immunity once its plan is known.

Functions heard before a lapse are heard exactly. The listener hears each section from its start, and the earlier essays’ confusion applies only to recognising a function on returning into unheard music. A listener who also confused functions while attending would infer the plan more slowly and sometimes wrongly.

The lapses and the counting are the earlier essays’: lapses lasting an exponentially distributed time about a mean of forty seconds, a listener choosing the most probable section, and proportions told apart from the nested counts.

What these figures cannot show #

Whether listeners keep a list of plans at all. The model’s listener reasons about plans; a real one may simply expect certain functions to follow others, which is a different kind of knowledge with a different blind spot. Only a lapse of attention keeps a form’s top vague found the top of a form vulnerable to lapses whatever the listener knows, and this essay finds the start of a form the most vulnerable part for a listener who knows only its kind.

And whether the functions are as sharply heard as the model’s. A function recognised one time in a hundred wrongly and one heard as a vague tendency are both modelled here as categories with a confusion rate. The far pair’s immunity rests on the categories being distinct at their extremes; a listener who heard functions as points on a continuum might find even opening and closing blurring at the edges.

Still open: whether the first minute can be told #

The whole cost of learning is a blind first minute, and composers have a way of filling it: an opening that announces the shape of 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 is a way of giving a listener the plan before the first lapse. The computation is this one with the prior set by such a cue, partially reliable, and the first section’s plans narrowed by it, and it would say how reliable an announcement of the plan has to be before it recovers the cycle’s value — and whether a listener who is told the plan wrongly does worse than one told nothing, which, from the blocks’ fall below the even prior, seems likely.