New music says only that it is new

A repeat sign loses a listener that a refrain does not followed a listener whose attention lapses in the middle of an eight-minute piece and comes back a few seconds or a few tens of seconds later with every count gone. What they hear on returning tells them where they are, provided the material they hear occurs once in the piece; if it occurs more than once they must guess which occurrence it is from how long they think they were away. The result sorted forms in a way no count of repetitions would. A rondo, half of whose sections are the same refrain, placed a returning listener almost perfectly, because an episode always stood between two refrains and a lapse long enough to skip one was rare. A song whose sections were repeated immediately misplaced them often, at the join inside each repeat. And the through-composed piece, with no material heard twice, placed every returning listener exactly.

That listener knew the piece. They knew every section’s material before it arrived, so coming back into the middle of the fifth section they heard the fifth section’s music and knew it for the fifth section. The essay’s last paragraph named the harder case: a listener hearing the piece for the first time, who has only the sections heard so far to go on. Coming back into music they have never heard, they know only that it is music they have not heard.

That changes the ranking of the forms in one place, and the place is instructive. The rondo, the sonata, the song form and the strophic song lose almost nothing to a first hearing. The through-composed piece loses its entire advantage, because every section of it is new, and new music says only that it is new.

Four listeners and one lapse #

The listener is the one this whole group of essays has counted with. They keep a count at every level of the form — bars of two seconds, phrases of sixteen, periods of sixty-four, sections of a hundred and twenty-eight — each count mending the one below when it slips, as only a lapse of attention keeps a form’s top vague set it up. Each count is a timing, and a timing is only as fine as a proportion is only as fine as its two durations found — a few per cent over a second, a quarter or more over minutes. Most of their failures are small and mended. A tenth of their lapses take every count at once. Such a lapse starts at a moment the listener knows, since every count was running until then, and lasts a time they do not know, exponentially distributed about a mean: ten seconds in the earlier essay, twenty here, and swept from two seconds to a minute below.

On returning, the listener hears some music and chooses the section they are most probably in: the section where a lapse of that typical length would most probably have ended, among those whose music could sound like what they now hear. If they choose wrongly, every count resumes a whole number of sections out.

Four listeners differ only in what they know about which sections sound like what.

The listener who knows the piece is the earlier essay’s. What they hear names its section.

The listener told the plan knows the sequence of sections — A B A C, which are alike and which are not — but has not heard the music. A section they have already heard, they recognise; a section they have not, they hear only as new.

The listener who knows the style knows that pieces of this kind take one of five forms — through-composed, rondo, sonata with its exposition repeated, a thirty-two-bar song played twice, strophic — each with eight sections of a minute, and weighs them equally except as the sections heard before the lapse rule some out.

The listener who knows nothing of the style takes each later section to be new music half the time and a return of something already heard otherwise.

The last three are three first-time listeners with more or less knowledge to bring. None of them can identify a section of music they have not yet heard.

What a stranger keeps #

The first figure’s surprise is how much of the familiar listener’s recovery the first-time listeners keep. With lapses of twenty seconds the rondo reads 56.2 proportions for the listener who knows it and 54.6 for the one who knows only the style. The sonata reads 61.0 and 57.8, the song form 35.3 and 34.5, the strophic song 25.9 and 25.5. Even the listener with no knowledge of the style keeps close to nine tenths of each of the four, where in the through-composed piece they keep under four fifths.

The reason is that in these forms most returns are to music the listener has already heard. Coming back into a rondo’s refrain, a first-time listener knows it for the refrain just as surely as a familiar one does, because they heard it at the start. What they cannot always say is which refrain it is, and that is exactly the question the familiar listener could not always answer either: the earlier essay found that the refrain is protected by the episode between returns, not by any knowledge of the refrain’s position. A first-time listener is protected by the same episode.

A strophic song is the extreme case. Every verse sounds like the first, so the familiar listener’s knowledge is worth nothing there — knowing the piece tells them only that every section is the same, which a stranger finds out in the first minute. The two listeners are misplaced at the same rate, 19.3 and 19.8 per cent of returns with twenty-second lapses, and the strophic song’s poor standing belongs to its structure, not to anybody’s ignorance of it.

The form that needs to be known #

The through-composed piece is the exception, and it is a large one. At lapses of ten seconds or less the first-time listener is barely worse placed than the familiar one: most lapses that short end in the section where they began, and the listener recognises the music they were hearing before. From twenty seconds the two lines part. At thirty seconds a familiar listener tells apart 72.0 proportions and a first-time one 41.6; at a minute, 51.5 and 18.6.

The cause is the thing the through-composed piece does that no other form does: everything in it after the first minute is new. A familiar listener coming back into the fourth section hears the fourth section’s music. A first-time listener coming back into the fourth section hears something new, and so does one coming back into the fifth, and a lapse that has carried them across a boundary into new music has left them with no way of saying how many boundaries it crossed. They must fall back on the lapse’s typical length, which is the one thing the earlier essay found every returning listener in an immediate repeat forced to do.

So for a first-time listener a through-composed piece is a strophic song played in unfamiliar music. Its sections sound unlike each other, but they all sound alike in the one respect a lapse leaves the listener able to judge: none of them is anything heard before. Two sections of new music are two identical landmarks, just as two verses are.

The crossing points are what matter for the ranking. Among the familiar listeners the through-composed piece leads every form at every lapse length. Among first-time listeners it leads the rondo by less than three proportions at forty-five seconds, and at a minute the first-time listener in it, at 18.6, is worse placed than a familiar listener in the rondo, at 19.2. The form that is easiest to come back to for someone who knows it is, for someone who does not, one of the pack.

Where a first hearing loses its place #

Drawn across the piece, the first-time listener’s risk is a sawtooth that has the earlier essay’s shape in every section. It is low early in a section, where a lapse usually ends before the next boundary, rises to a fifth or a quarter at each section’s end, where a lapse of typical length crosses a boundary and may cross two, and drops at the boundary. The familiar listener’s line lies along the floor: they are never misplaced, because each section’s music names it.

The last section is safe for the first-time listener too, and for a plain reason. A lapse that begins there can end only in it, or outrun the piece, so there is nothing to confuse it with. The peak in the first section needs a second look, because it is the style entering the arithmetic. Late in the first minute a first-time listener has heard only the opening material. A lapse ends, and they hear the opening material still going. For the familiar listener that settles it: the piece is through-composed, and the opening does not recur. The first-time listener cannot rule out that the piece is a song form or a strophic song whose opening has just been repeated, and a lapse of forty seconds is more likely to have crossed the boundary than not. More than a third of the time they guess that it has, and resume their counts a minute out.

That is the one place in the whole piece where knowing the style actively misleads, and it lasts until the second section is heard and two of the five forms are ruled out. The style narrows as the piece goes: after the first three sections only the through-composed piece is consistent with what has been heard, and from then on the first-time listener who knows the style knows the form. What they still cannot do is recognise its music.

A plan is not a memory #

The natural remedy is a programme note. A listener told before the performance that the movement is through-composed in eight sections of a minute each knows the plan exactly, and ought to be placed as the familiar listener is.

It gives back almost nothing. With lapses of forty seconds the through-composed piece reads 29.5 proportions for the listener who knows the style, 29.8 for the one told the plan and 66.1 for the one who knows the piece. Twenty seconds: 61.4, 62.7 and 75.7. The plan repairs a tenth of what a first hearing costs at twenty seconds and a hundredth at forty, and what it does repair is the first-section confusion above, which knowing the plan removes because it rules out the song form and the strophic song from the start.

What the familiar listener has and the planned one lacks is not a plan but a memory of the music. It is the difference between knowing that the fifth section is new and knowing what the fifth section sounds like, and only the second can tell a returning listener that they are in the fifth rather than the fourth or the sixth. A return has to be remembered made the same point about a single return: a repetition helps only as much as the original is still held, and here the original has not been heard at all.

The sonata shows the plan doing something real. Told that the exposition is repeated before a development, a first-time listener who returns into a subject they have heard knows at which places that subject can sound, which the style alone left open among five forms, and the plan repairs more than half of the sonata’s small first-hearing cost: at forty seconds, 28.0 to 30.6, against 32.7 for the familiar listener. A plan helps where the question is which occurrence of music already heard this is. It cannot help where the question is which piece of music never heard this is.

Whose recovery belongs to whom #

The last figure turns the comparison into a single quantity: for each form, how much of what a returning listener recovers depends on having heard the piece before. With lapses of forty seconds, three fifths of the through-composed piece’s recovery belongs to familiarity; a sixth of the sonata’s; a twelfth of the rondo’s; and next to none of the song form’s or the strophic song’s.

That ordering follows the number of distinct sections each form has — eight, four, four, two and one — until the last two, which are both near nothing. The sonata sits above the rondo because two of its four distinct sections, the development’s, appear only once and late, where a lapse into them lands in music nobody has yet heard. A form’s navigability is partly in its structure and partly in the listener’s memory of its material, and the more different music a form contains, the larger the second part. Repetition is what puts navigability into the structure, where a stranger can use it.

The same distinction came out, at a much smaller scale, in two landmarks in a cycle help only if they can be told apart. There, two recognisable points in a gong cycle were worth a great deal if the listener could tell them apart and very little if they sounded alike, and alike points spaced evenly were worse than one. A first hearing turns every unheard section of a form into an alike point: each is recognisable as something new and as nothing more particular. A through-composed piece heard for the first time is eight sections that a returning listener can tell apart from what came before and not from each other, which is the alike case exactly. The refrain, meanwhile, is a landmark whose identity was supplied in the first minute, and a late dancer needs the landmarks, not the rhythm found the same economy in a dancer joining a dance part-way: what orients a newcomer is what can be recognised, and what can be recognised is what has been heard.

A reversal of the usual advice #

The earlier essay drew a conclusion about writing repeats: an immediate repeat misplaces a returning listener, and a refrain with other music between its returns does not. For a first-time listener that conclusion stands, and a second one joins it. Repetition separated by contrast serves a stranger almost as well as a familiar listener; unbroken novelty serves only the listener who already knows the music.

That puts a small arithmetic weight behind two familiar observations about music written to be heard once. Music for a single hearing — a dance at a gathering, a song in a ritual, much of the popular repertoire — leans heavily on returning material, and a return is shorter than its first hearing found that it costs such music remarkably little in what a listener has to take in. And through-composed forms have a reputation for rewarding repeated hearings. On this arithmetic the reputation has a mechanical component: a through-composed piece is the form in which a listener who drifts is most dependent on already knowing it, so it is the form in which a second hearing buys most.

How the four listeners are scored #

The pieces are 480 seconds of eight one-minute sections, labelled by their material: ABCDEFGH, ABACABAD, ABABCDAB, AABAAABA and AAAAAAAA. The counts, the slips and the ordinary lapses are those of checkpoints sharpen the middle of a form, not its top and the essay that followed it, with a tenth of lapses taking every count. A lapse of attention begins at a known point and lasts an exponentially distributed time. On returning, the listener hears the section’s material as a letter already heard or as new; the familiar listener hears the letter itself. Each candidate section from the one the lapse began in onward is weighted by the chance that a lapse beginning there would end inside it, times the chance, under what the listener knows, that that section would sound as this one does. For the listener who knows the style that second chance is the share of the five forms still agreeing with the sections heard before the lapse whose section at that place would sound so; for the listener who knows nothing it is one half for new music and one half shared among the heard letters. The heaviest candidate is chosen. A wrong choice leaves the counts out by the sections between, and its square enters the whole-piece blur as in the earlier essay. Everything is integrated on a grid of lapse starts and lengths, not sampled, and the familiar listener’s numbers agree with the earlier essay’s to the last digit.

What the arithmetic leaves out #

A stranger’s memory of the lapse’s length. None of the listeners has any sense of how long they were away. A first-time listener who knows they drifted for a long time could discount the nearest section, and that sense would help most in exactly the place the through-composed piece fails.

Anything that makes new music place itself. In the arithmetic, new music is new and nothing more. In a real through-composed piece, new music usually carries its position: a development sounds like a development, a coda like a coda, a climax like a climax, and a first-time listener who knows the style’s rhetoric would be placed by that rhetoric even in music never heard. Every section here is an anonymous letter, which is the pessimistic case for the through-composed piece and the reason its loss is an upper bound.

Learning while listening. A first-time listener who comes back into the fifth section and is not sure whether it is the fifth or the sixth will be told at the next boundary, if the next section is a refrain. The through-composed piece never tells them. A cycle that says where it is measured how fast that kind of correction arrives in a rhythmic cycle, where it can take several strokes. The arithmetic scores the choice at the moment of return, as the earlier essay did, and not what the rest of the piece corrects.

Equal sections, and five forms. The style’s prior is five forms, equally weighted, each of eight equal sections. A real style’s forms differ in length and proportion, which is more information for a first-time listener than a prior of letters gives.

Which pieces, and what the claim is about #

The five forms are the skeletons of the earlier essay, minus the minuet and trio, whose ten sections of forty-eight seconds a listener would tell from the others by the length of the first section alone. The claim is about the arrangement of heard and unheard material in time: that a form whose returning material is protected by contrasting material between serves a first-time listener nearly as well as a familiar one, and that a form of continual novelty serves the first-time listener as poorly as a form of continual repetition once lapses run to half a minute. It is not a claim about how listeners respond to through-composed music, whose other resources the arithmetic does not model. Whether real first-time listeners, lapsing in real pieces, are misplaced where the sawtooth says, and how much a style’s rhetoric rescues them, is a question for listeners and not for this arithmetic. A form is sharp at the bottom and vague at the top began this sequence with a vague top; the vagueness a lapse leaves at the top, it turns out, depends on who is listening.

Still open: new music that says where it is #

The anonymous letter is the arithmetic’s harshest assumption, and it is the one that decides the through-composed case. A development is recognisable as a development by its instability; a recapitulation as a return; a coda by its cadences piling up. If a first-time listener hears each section’s rhetorical function — opening, continuing, intensifying, closing — even when they have not heard its material, then new music is not only new: it carries a coarse label of where in a piece it belongs. The computation would give each section a function drawn from a small set, recognisable at a stated rate even on a first hearing, and ask how much of the through-composed piece’s lost recovery that label restores — which would say whether the forms that serve a first-time listener are the ones that repeat, or the ones whose sections each announce what part of the piece they are.