A section that says what it is for places a first hearing

A listener whose attention lapses in the middle of a long piece comes back with every count gone and must work out where they are from what they now hear. A repeat sign loses a listener a refrain does not found that for a listener who knows the piece, the through-composed form is the easiest of all to come back to: nothing in it is heard twice, so whatever is sounding names its own section. New music says only that it is new then gave the same piece to a listener hearing it for the first time, and the advantage vanished. Coming back into music never heard, the listener knows only that it is new — and every section after the first is new, so a lapse long enough to cross a boundary leaves them unable to say how many it crossed. At lapses of forty seconds on average, a familiar listener tells apart 66.1 proportions across the piece and a first-time listener 29.5. Being told the plan in advance gives back almost none of that: 29.8.

That essay’s harshest assumption was the one it named at the end. Its first-time listener hears new music as anonymous: new, and nothing more. Real sections are not anonymous. A development sounds unstable; a coda’s cadences pile up; an opening sounds like a beginning. If a first-time listener can hear what a section is doing, even without having heard its material, then new music carries a coarse label of where in a piece it belongs, and the question becomes how much of the through-composed piece’s lost recovery that label buys back.

It can buy back nearly all of it, or almost none, with the same four labels. What decides it is the order they come in.

Giving new music a function #

The piece, the lapses and the listener are the earlier essays’. Eight sections of a minute, a listener keeping counts at every level from bars of two seconds to sections of two minutes — the nested counting that a form is sharp at the bottom and vague at the top set up, each count only as fine as its two durations allow — and a tenth of all lapses of attention taking every count at once and lasting an exponentially distributed time about a mean, here usually forty seconds. On returning, the listener chooses the most probable section, among those whose music could sound like what they now hear, given how long a lapse of that kind usually lasts. A wrong choice puts every count a whole number of sections out.

The new listener is the earlier essay’s listener told the plan — the one who knows that the piece has eight sections, all different — with one addition. Each section carries a function, one of a small set. When the listener returns into a section whose music they have not yet heard, they recognise its function with a stated probability, and when they do, they recognise it correctly. When they do not, the music is only new, as before. The listener knows which function each section carries; what they do not know is the music.

That is a generous listener in one respect and an honest one in another. It is generous because it knows the plan of functions, which a real first-time listener would have to infer from the style. The earlier essay found that a listener who knows the style narrows a through-composed piece to its form within three sections, so the plan is what such a listener has for most of the piece. It is honest because recognising a function is not the same as recognising a section: a listener who hears “intensifying” knows only that they are in one of the sections that intensify.

The model passes its two checks. With no function recognised, it is the listener told the plan exactly: 29.8 proportions at forty-second lapses. With eight different functions, one per section, recognised every time, it is the listener who knows the piece exactly, 66.1: a function unique to its section names the section as the music did.

The order, not the number #

The first figure has four arrangements of four functions — or two, in one case — every one recognised more and more reliably from left to right. They separate as soon as recognition is possible at all, and by full recognition they are far apart.

Two functions alternating, opening and intensifying, O I O I O I O I, reach 39.5 at full recognition. An arc in long runs, one opening, three continuing, three intensifying and a close, reaches 38.6. Four functions in blocks of two, O O C C I I Z Z, the familiar shape of a piece that sets out, develops, intensifies and closes, reaches 45.0. The same four functions cycled, O C I Z twice through, reach 60.8 — nine tenths of the way to the familiar listener.

As a share of what not knowing the music costs, the gap between 29.8 and 66.1, those are 27, 24, 42 and 85 per cent. The cycle has the same four labels as the blocks and restores twice as much.

The reason is the one two landmarks in a cycle help only if they can be told apart found in a gong cycle, and that a gamelan’s goal-tone beating found again: a label is worth what it distinguishes. A listener who lapses near the end of the fourth section, under the blocks, and comes back into new music labelled as intensifying, knows they are in the fifth or the sixth section — and a forty-second lapse has carried them past one boundary or two with comparable odds. The label says nothing about the one question the lapse raised. Under the cycle, the next section with the same function is four sections on, and a lapse that long is rare: whatever function they hear, there is only one section within a lapse’s reach that carries it.

Lapses decide which order is best #

The ranking is not fixed. It moves with how far a lapse is likely to carry the listener, which is the length of lapse that matters.

At lapses of ten seconds, two functions alternating restore everything a first hearing costs. A ten-second lapse almost never crosses more than one boundary, and two labels that alternate are enough to tell a section from its neighbour. At twenty seconds they still restore 85 per cent. At thirty, 48; at forty, 27; at a minute, 14 — by then a typical lapse can cross two boundaries, and the section two ahead carries the same label as the one it started in.

The cycled four keep everything to thirty seconds, 85 per cent at forty, 56 at a minute: their labels recur four sections apart, and only the longest lapses reach that far. The blocks of two sit almost level, between 42 and 50 per cent at every length, because their weakness is at the shortest distance — two adjacent sections share a function — and their strength is that the same function does not come back once its block is over.

So the arrangement a piece needs depends on how far its listeners drift. For lapses of a few seconds, almost any alternation does. For lapses of a minute, only an arrangement in which each function’s next appearance is far ahead does much at all.

Seven arrangements and one quantity #

Seven arrangements make the rule visible. At forty seconds, two functions in halves (O O O O I I I I) restore 13 per cent; the arc in long runs 24; two alternating 27; three that never repeat in a row (O C I C I C I Z) 35; four in blocks of two 42; four there and back (O C I Z Z I C O) 64; four cycled 85.

Beneath each is one number: for every section after the first, how many sections on until the next section with the same function, or the end of the piece, averaged. A lapse runs only forward, so that distance is how much of the piece a label covers without ambiguity. It is 1.57 for the halves, 1.71 for the long arc, 1.86 for the alternation, 2.00 for the three, 2.71 for both the blocks and the there-and-back, and 3.14 for the cycle — and the shares rise with it.

The two arrangements tied at 2.71 are the instructive exception. The blocks restore 42 per cent at forty seconds and the there-and-back 64. They have the same average distance but not the same distribution of it: the blocks’ short distances are all at the starts of blocks, where a section is followed immediately by its twin, while the there-and-back’s one short distance is the pair Z Z in the middle and its other distances are long. A label that is ambiguous in four places of eight costs more than one ambiguous in one place, even when the averages agree. The average is a good guide, not a law.

At twenty seconds the alternations do far better than the average distance suggests, because at that length only the next section matters, and alternation never puts two alike in a row. At a minute, the long distances are what count, and the alternations fall to the bottom. The quantity that matters is the distance to the next alike section within the reach of a typical lapse, and that reach is the mean lapse in sections.

Where in the piece the labels act #

Drawn across the piece, the unlabelled listener’s risk is the earlier essay’s sawtooth: low early in each section, high near its end, where a lapse of typical length crosses the next boundary and may cross two. The blocks cut every other tooth, and which ones is worth seeing. In the first section of each block the risk falls from about 22 per cent to 4: a lapse from there runs into the block’s second section, whose function is shared only with a section already heard, and so is not new, or into the next block’s first. In the second section of each block it stays near 18: a lapse from there runs into a block of two unheard sections that carry the same function, and the label cannot say which of the two it reached. Over the piece that halves the risk, from 8.26 per cent of returns to 4.26. The cycle cuts every tooth and leaves 0.03 per cent.

That last figure explains the one puzzle in the hero, which is why the cycle stops short of the familiar listener’s 66.1 at 60.8 when it almost never misplaces anybody. The few misplacements it leaves are lapses long enough to reach the next section with the same function, four sections on, and a listener put four sections out has lost half the piece. The count of proportions a listener can tell apart weighs errors by the square of their size, and three returns in ten thousand that are each four minutes wrong account for most of the five proportions between the two.

What the shapes of real forms say #

Composers have never been asked to arrange a piece’s functions so that a distracted stranger can find their place, and the arithmetic does not suggest they should be. What it does do is sort familiar shapes. A movement that sets out, develops, intensifies and closes, each for a stretch, is the blocks: it gives a first-time listener back about two fifths of what not knowing it costs, whatever their attention is like. A piece in which a refrain-like function returns regularly, as in a rondo, behaves like the alternations: excellent for brief lapses and poor for long ones — and the rondo’s actual refrain, which a first-time listener does recognise after hearing it once, was already the earlier essays’ finding. A piece that never repeats a function, that has a different kind of music in every section, is the cycle at its limit: one function per section, and a first-time listener placed as well as one who knows it.

Checkpoints sharpen the middle of a form, not its top found that a second count mending the first sharpens a form’s middle levels and barely touches its top, and only a lapse of attention keeps a form’s top vague that what blurs the top is the occasional lapse that takes every count at once. A function label is a mender of a different kind: it arrives only when the listener needs it, on returning from such a lapse, and it works at the level of sections, which is the level the counts below cannot repair. The form a first hearing cannot have found that a piece’s large-scale period cannot be known until the piece is nearly over. A function label does not change that; it gives a listener who loses their place a way back that does not depend on knowing it.

This last is the point the earlier essay could not reach. Its through-composed piece had eight sections of new music, and new music says only that it is new. A through-composed piece whose eight sections each do something different — whose fifth section is unmistakably a development and whose sixth is unmistakably a recapitulation-like return — says where it is, even to a stranger. The form that is easiest to come back to for a listener who knows it can be nearly as easy for one who does not, provided its sections differ in kind and not only in material.

The model under the counts #

Eight sections of sixty seconds; a listener counting at levels of two, sixteen, sixty-four and one hundred and twenty-eight seconds, with timing precision falling with the interval timed, local slips at two per cent and lapses at one per cent of counting units, a tenth of lapses taking every count; lapses exponentially distributed about the stated mean. On returning, the listener weighs each later section by the probability that a lapse of that kind ends there and that its music, and where recognised its function, would sound as heard, and chooses the most probable. Recognition of a function is a coin of the stated probability for each return, and the two outcomes are weighted rather than drawn, so that every figure is the same calculation every time. Proportions told apart are counted from the variability the slips and the misplacements together give to the piece’s sections, as in the earlier essays.

What the labels assume #

That a function is recognised correctly or not at all. A listener who heard an intensifying section as a closing one would be worse off than one who heard nothing, and real functions shade into each other. With confusions between functions, every arrangement’s share would fall, and the cycle — whose value rests on labels being exact — would fall furthest.

That the listener knows the plan of functions. A real first-time listener knows the style, not this piece’s arrangement, and would have to learn the arrangement as the piece goes, which new music says only that it is new found the style listener doing within three sections for the plan of materials. Learning functions would take at least as long.

That rhetorical function is recognisable on a first hearing at all. That is the empirical question the whole argument stands on, and nothing here measures it. The figures say how much it would be worth at each rate of recognition, so that a measurement of the rate — how often listeners hearing an unfamiliar movement can say whether a passage is opening, developing or closing — would turn directly into a statement about how lost they get.

Still open: functions that are confused #

Every label here is recognised exactly or not at all. Real functions blur: a continuation can sound like a development, a retransition like a close. The computation is this one with a confusion matrix between functions — each function heard as each other with a stated probability, adjacent functions in the arc most often — and it would say whether the cycle’s advantage survives realistic confusions, or whether a piece is better served by fewer, more distinct functions placed far apart than by many fine ones placed well.