Two functions far apart outlast four that blur

A section that says what it is for places a first hearing followed a listener hearing a through-composed piece for the first time: eight one-minute sections, every one of them new music. When such a listener loses count for a while and comes back, new music says only that it is new found that nothing in it tells them which section they have reached. The essay gave each section a rhetorical function — opening, continuing, intensifying or closing — and let the returning listener recognise it. If the listener always recognised the function, the arrangement of functions decided how much of their place came back. Four functions cycled, O C I Z O C I Z, gave back 85 per cent of what not knowing the music costs, at lapses of forty seconds on average. The same four in blocks of two, the shape of a movement that sets out, develops, intensifies and closes, gave back 42.

The reason the cycle won was a matter of reach. A lapse runs only forward, and a label helps a returning listener only if no other section within a lapse’s reach carries the same one. Under the cycle the next section with the same function is four sections on, beyond almost every lapse; under the blocks it is the very next section. The measure the essays use, proportions a listener can tell apart, rests on the nested counts that a form is sharp at the bottom and vague at the top set up, and only a lapse of attention keeps a form’s top vague: the function labels are a way of repairing the top level after exactly such a lapse.

That essay named its most generous assumption plainly. A function is recognised correctly or not at all. Real functions blur into their neighbours. A continuation can sound like a development; a retransition like a close; the third section of a piece that is building up may be heard as continuing what came before or as intensifying it. The essay predicted that confusion would lower every arrangement’s recovery, and the cycle’s furthest, since its value rests on its labels being exact.

The prediction is right, and the size of the fall is the point.

Confusing a function with its neighbour #

The listener, the piece and the lapses are the earlier essay’s. Eight sections of a minute, counted at levels from two seconds to two minutes; a tenth of lapses of attention take every count at once and last an exponentially distributed time about a mean, forty seconds unless stated. On returning, the listener chooses the most probable section among those a lapse could have reached, weighing each by how long lapses usually last and by whether its music — and, where recognised, its function — would sound as it now does. The listener knows the plan of functions: which section carries which.

The one change is how a recognised function is heard. The four functions are taken to lie along an arc, opening, continuing, intensifying, closing, and a function is confused only with its neighbours on it. With a stated chance a recognised function is heard as a neighbour, split evenly between its two neighbours if it has two. Otherwise it is heard as itself. Opening can be heard as continuing, intensifying as continuing or closing, and so on; opening is never heard as closing.

The listener knows these chances too, and uses them properly. Hearing intensifying, it does not insist the section intensifies; it weighs each candidate section by the chance that the section’s own function would be heard as intensifying, and chooses the most probable. With no confusion this is exactly the earlier essay’s listener, and the figures reproduce its numbers: 60.8 proportions for the cycle, 45.0 for the blocks, 29.8 for a listener with no labels, 66.1 for one who knows the piece.

The cycle’s advantage is the first thing to go #

The cycle’s recovery falls steeply. At one confusion in twenty it is 47.7, already down by thirteen proportions; at one in ten, 44.1; at one in five, 39.0; at one in two, 34.9. As a share of what a first hearing costs, from 85 per cent with exact labels to 39 per cent at one confusion in ten and 25 at one in five.

The other arrangements fall less, because they had less to lose, and fall in the same way. The blocks go from 45.0 to 35.3 at one in ten and 33.0 at one in five. Three functions cycled, O I Z O I Z O I, from 51.4 to 41.9 and 39.8. Opening and continuing alternated go from 39.5 to 35.3 and 31.8.

And one arrangement does not fall at all. Opening and closing alternated, O Z O Z O Z O Z, stay at 39.5 at every rate of confusion. At one confusion in five the cycle, which was half again as good with exact labels, is now just below it, and only the three functions cycled are still ahead, by three tenths; the far pair overtakes them at 22 per cent. (Corrected: this sentence first called the far pair the best of the five at one confusion in five, which the hero figure’s own numbers contradict. A listener who learns the plan as it goes found the slip.)

Why the ends of the arc are immune #

Any two functions alternated give the same recovery when they are recognised exactly. An alternation tells a returning listener only whether they are in an odd section or an even one, and two labels do that whichever two they are. Under confusion, which two they are decides everything.

Opening and continuing are neighbours, and each is confused with the other. A heard continuing may be either, and the label no longer says odd or even reliably: the recovery falls to 31.8 at one confusion in five, barely above no labels at all. Opening and intensifying are one apart. Each is confused only with functions the piece does not use — continuing for opening, continuing or closing for intensifying — but they share continuing as a neighbour, so a heard continuing could be either, and only that outcome is ambiguous. The recovery falls more slowly, to 37.4. Opening and closing are the two ends, and share no neighbour. Opening is mistaken only for continuing and closing only for intensifying, and neither occurs in the piece. A listener who knows the plan and hears continuing knows the section is an opening section heard slightly wrong, and reads it straight back. Every confusion is recoverable, and the recovery stays at 39.5.

That is the general rule the arithmetic contains. A label survives confusion when what it is confused with is not itself one of the labels. Four functions cycled use the whole arc, so every confusion lands on a function the piece uses, and every confusion costs. Two functions at the ends use only the arc’s extremes, and the middle of the arc becomes a margin of error that a listener can read through.

It is the same rule two landmarks in a cycle help only if they can be told apart and a gamelan’s beating names its goal only if each goal beats at its own rate found for the marked points of a gong cycle, read one step further. There a landmark helped in proportion to how surely it could be told from the others. Here a function label helps in proportion to how surely it can be told from the others after the listener’s errors, and an alphabet of labels that leaves gaps between them is told apart more surely than a full one.

How long the lapses are decides the trade #

The answer depends on the lapses. A section that says what it is for places a first hearing found that the arrangement worth most depends on how far a lapse usually runs, because a label is worth what a lapse of that length could otherwise mistake. With short lapses, a returning listener rarely crosses more than one boundary, and an alternation of two functions says which of two neighbouring sections they are in, which is all that is needed. The cycle’s four functions add almost nothing, and the smallest confusion takes them below the alternation: at twenty-second lapses, one confusion in a hundred. At thirty seconds, nine in a hundred. At forty, eighteen; at fifty, twenty-eight; at a minute, thirty-nine.

With long lapses the cycle’s extra functions are what tell a listener which of several sections a lapse has carried them across, and they are needed enough to survive substantial confusion. At ninety-second lapses the cycle is better than the far pair at every rate of confusion drawn, up to one in two.

So the earlier essay’s lead had a two-part answer. Whether a piece is better served by fewer, more distinct functions placed far apart or by many fine ones placed well depends on two quantities a composer does not control: how often a listener confuses neighbouring functions, and how long a listener’s attention wanders. For an attentive listener who drifts briefly, the far pair wins almost at once. For a listener who drifts for a minute or more, the cycle wins unless confusion is common.

Confusion costs more than missing #

The earlier essay let recognition fail in a different way: sometimes the listener hears no function at all, and the music is only new. Set the two failures side by side and they are not alike. Recognised three times in four and never confused, the cycle restores 45.4 proportions. Recognised every time and confused one time in ten, it restores 44.1. A tenth of confusion costs more than a quarter of silence. A missed function leaves the listener where they would have been without labels. A confused one sends them the wrong way with confidence, and the listener who knows the confusion rate can only discount it, not undo it.

That is also why the confused cycle runs so close to the far pair across the figure. Confused one time in five, the cycle’s recovery against recognition rate runs within a few tenths of the far pair’s at every rate. Four labels confused a fifth of the time carry about as much information about place as two labels never confused.

Where in the piece the confusion strikes #

Drawn across the piece, the confusion’s cost has the sawtooth shape of every earlier map of this kind. The risk of misplacement is low early in each section and climbs towards its end, where a lapse of typical length crosses into the next section. With exact labels the cycle’s sawtooth is flattened almost to nothing. Confused one time in five it comes back: the risk climbs to between ten and eighteen per cent near the end of each section, and over the piece 4.7 per cent of returns are misplaced. The far pair, confused just as often, misplaces 1.5 per cent, because the confusions it suffers are all read back.

Every arrangement, exact and confused #

Over every arrangement the earlier essay drew, confusion compresses the differences. Recognised exactly, the eight arrangements span from 34.4 (two functions in halves) to 60.8 (the cycle). Confused one time in five they span from 31.7 (an arc in long runs) to 39.5, and the order is different. Opening and closing alternated are at the top. The cycle comes second at 39.0; four functions there and back, O C I Z Z I C O, third at 37.6; the familiar blocks of two near the bottom at 33.0.

The blocks’ fall is worth noticing, because they are the shape most movements actually have. Their four functions use the whole arc, as the cycle’s do, and their repeated pairs gave them less to start with; confusion takes them almost back to the listener with no labels at all.

What this says about real forms #

The arithmetic does not recommend writing pieces that alternate openings and closings, and it would be odd to read it that way. What it says is narrower. The functions that are hardest to confuse are the ones that are furthest apart, beginnings and endings. If a first-time listener’s sense of where they are in a piece rests on rhetorical functions at all, it rests most securely on those, and least on the fine distinctions in the middle of a piece’s arc between continuing and intensifying. A form whose sections carry strongly contrasted functions — an introduction that is unmistakably introductory, a coda unmistakably closing — gives a lost listener landmarks that survive their own errors. A form that develops by fine gradations gives them landmarks that do not.

If listeners unfamiliar with a style do hear its beginnings and endings more clearly than its middles, this arithmetic gives that a second reason beyond attention: the extremes of a piece’s arc are the labels that survive being misheard. Nothing here measures how listeners hear, and the claim is only that the arithmetic would reward such a habit. The form a first hearing cannot have found that a piece’s large-scale period cannot be known until it is nearly over; contrasted beginnings and endings are the part of a form that does not have to wait for that, and on this count they are also the part that tolerates error.

Checkpoints sharpen the middle of a form, not its top found the counts a listener keeps mending one another in a form’s middle levels and leaving its top to the occasional lapse. Function labels were the earlier essay’s mender for that top level. Confusion makes them a weaker mender, and the far pair the one that does not weaken.

The model under the counts #

Everything is the earlier essay’s model, with confusion added. Eight sections of sixty seconds; counts at two, sixteen, sixty-four and one hundred and twenty-eight seconds; slips at two per cent, lapses at one per cent, a tenth of lapses taking every count; lapses exponentially distributed about the stated mean. Each section carries one of four functions along the arc opening, continuing, intensifying, closing. On returning into unheard music the listener recognises the function always unless stated, and hears it as a neighbour on the arc with the stated chance, split evenly between two neighbours; it knows the confusion and the plan, and chooses the section with the greatest product of lapse probability, material likelihood and the chance the section’s function would be heard as heard. Every outcome is weighted rather than drawn, so each figure is one calculation.

What the confusion assumes #

Confusion only with neighbours. A listener might mistake an opening for a close, especially across the middle of a long piece. A confusion matrix that reached every function would remove the far pair’s immunity, which rests entirely on its confusions landing on unused functions; it would still lose less than the cycle, but not nothing.

The arc as a line. Opening, continuing, intensifying and closing are taken to be ordered, with each confusable only with the next and the previous. Real rhetorical functions are not a line, and their confusions are a question about listeners that no arithmetic here answers.

And the listener knows its own confusion rate. A listener who did not know it — who trusted every heard label — would be worse off than any figure here, and the far pair’s immunity would vanish, since it depends on reading an unused function as the error it must be.

Still open: a listener who learns the plan as it goes #

Every listener here is told the plan of functions. A real first-time listener would have to infer it from the style and from the sections heard so far. The earlier essays found the style listener narrowing a through-composed piece’s plan of materials within three sections, and functions would presumably take as long or longer. A listener learning the plan of functions while also confusing them faces both problems at once: an early section heard as intensifying may be a real intensification or a misheard continuation, and the plan inferred from it inherits the doubt. The computation is this one with the plan replaced by a prior over plausible plans, updated by the functions heard before each lapse. It would say whether the far pair’s immunity survives a listener who does not know that the piece has no continuing sections — which is exactly the knowledge the immunity depends on.