A repeat sign loses a listener that a refrain does not

Only a lapse of attention keeps a form’s top vague gave a listener a count at every level of an eight-minute piece — bars, phrases, periods, sections — each mending the one below when it lapsed, and found the whole piece became the sharpest level of the form, at seventy-six distinguishable proportions against two for a single count. What put the vagueness back was a lapse that took every count at once, a lapse of attention rather than of counting. One in a hundred lapses of that kind brought the whole piece down to thirteen proportions; one in ten, to four and a half.

That essay made those lapses permanent, and said so. It had begun from a count is not an estimate, whose count failed in two ways, and from a form is sharp at the bottom and vague at the top, whose vague top it was trying to explain. A listener whose attention wandered was lost for the rest of the piece. But attention comes back. A lapse lasts a few seconds or a few tens of seconds, and then the listener is listening again, with every count gone and one question to answer: where am I? Whether the lapse costs the few seconds it lasted or the rest of the section depends entirely on whether the music can answer that question for them.

It turns out that most forms answer it perfectly, and the one that does not is the one written with one of the most familiar signs in notation. A refrain separated by other music always tells a returning listener which refrain it is. A section repeated immediately does not, and it is the repeat, not the repetition, that costs.

A listener who knows the form #

The listener in this essay is the one the hierarchy of counts already assumed: someone who knows the piece well enough to count its phrases, periods and sections, and who therefore knows its plan — which sections there are, in what order, how long each lasts, and which of them are alike. Such a listener, coming back from a lapse, hears the material of the section they are in. If that material occurs once in the piece, they know exactly where they are, and at the section’s next boundary every count can start again from a known place. The lapse costs nothing but itself.

If the material occurs more than once, the listener has to choose which occurrence this is, and the only evidence is how long they were gone — which is exactly what a lapse of attention does not record. What they do know is where the lapse began, because every count was running until then, and that a lapse is more likely to have been short than long. So the listener chooses the occurrence a lapse of typical length would most probably have reached. If they choose wrongly, every count resumes a whole section out.

Six forms, all eight sections of a minute each except the minuet and trio, which has ten of forty-eight seconds. With lapses that never end, all six read the same 4.5 proportions across the whole piece, because the listener never recovers and the form cannot matter. With lapses that end, they split into two groups that barely overlap.

The through-composed piece, the rondo and the sonata with its repeated exposition recover almost everything: 77.1, 76.4 and 76.6 proportions, against 77.2 for a listener who is always placed exactly. The other three recover much less. The minuet and trio reaches 50.2, the song form 47.7, the strophic song 35.4. The rondo — four of its eight sections the same refrain — is in the first group. The minuet and trio, whose sections each recur only twice or three times, is in the second. How much a form repeats is not what sorts them.

Why a refrain names itself #

The difference is where the repeats sit. A rondo’s refrain sounds four times, but never twice in a row: between any two refrains there is an episode. A listener who lapses in the second refrain and comes back hearing the refrain again is almost certainly still in the second, because to have reached the third they would have had to be absent through an entire minute of episode. Ten seconds is the typical lapse; a minute is six typical lapses end to end, and the chance of one lapse lasting that long is a quarter of one per cent.

A section that is repeated immediately offers no such protection. A listener who lapses near the end of the first A of A A B A and comes back hearing A has two candidates a few seconds apart in time — the rest of the first A, or the start of the second — and nothing in the music to choose between them. The most probable reading is the one a short lapse would give, and a lapse that crossed the boundary is misread every time.

Drawn across the piece, the risk has a shape that could not be read off a count of repetitions. It is zero through most of every section. It rises in the last few seconds of each section that is about to be repeated, to 44 per cent just before the boundary, and falls back to zero at the boundary itself — since a lapse that begins in the second A can only be misplaced into a third A, and in this form there is none adjacent. Where a section is followed by different music, there is no rise at all. A returning listener is misplaced at exactly one kind of place: the join inside an immediate repeat.

The dial lengthens the typical lapse, and the peaks grow — to half the returns just before a repeat at forty-five seconds — while the sections followed by other music stay near zero until lapses are long enough to skip a whole section of it.

The same problem in a smaller cycle #

That shape is familiar from a different scale of music altogether. A cycle that says where it is asked how quickly a listener hearing a rhythmic timeline can tell where in its cycle they are, and found that a symmetric pattern never locates at all: if a pattern looks the same from two positions, no amount of listening tells them apart. A strophic song is that symmetric pattern made of verses. Every verse is the same as every other, so the only evidence of position is elapsed time, and a listener who has lost track of time has lost track of where they are.

The difference between a rondo and a repeated minuet is the difference between the two timelines that essay compared, and the cycle that outruns the memory added the clock that turns a number of steps into a number of seconds — the same conversion a lapse of attention forces here. Rhythm is a circle, and the bar line is a choice is the reason a strophic song and a cycle are one object: both repeat with no marked beginning. The rondo’s refrains sit at positions that the music between them distinguishes; the minuet’s repeated halves sit at two positions nothing distinguishes but the clock. That the same arithmetic governs a two-second bell pattern and an eight-minute movement is the surprise, and it is not a coincidence: both are a listener placing themselves in a sequence by what they hear, with the sequence’s symmetries as the only obstacle.

How long a lapse lasts #

Everything above assumes a typical lapse of ten seconds, and that is a number chosen rather than measured.

The forms with an immediate repeat lose precision from the shortest lapses on. The strophic song reads 58.1 proportions with lapses of two seconds, when every other form without a repeat reads 77.2; even a lapse that short, if it happens a second before a boundary, has a fair chance of crossing it. The forms without one hold their precision until lapses begin to approach the length of the music between two identical sections: the rondo and the sonata stay above 76 up to lapses of ten seconds and fall steeply after twenty. The through-composed piece, with nothing repeated at all, holds longest, above 70 proportions to a mean of half a minute, and loses at a minute only because lapses that long are often still running when the piece ends.

Two crossings on the figure are worth reading. At a mean lapse of a minute, the rondo, at 19.2, has fallen below the minuet, at 26.8. Its misplacements are rarer but worse: a listener who comes back to a refrain after skipping an episode is two sections out, not one. And the song form played twice, at 15.3, is now close to the strophic song, since at lapses that long its B sections are too short a separation to protect the A’s either side of them.

A short repeat is cheaper than a long one #

The length of the repeated section enters twice and pulls two ways.

A short repeated section is misread often. With fifteen-second sections, over a third of returning listeners choose the wrong repetition, because a ten-second lapse crosses a boundary a large share of the time. A four-minute section is misread hardly ever, one return in seventy. But each misreading costs a whole section, and a whole four-minute section is sixteen times as costly as a fifteen-second one, in duration, and two hundred and fifty-six times in squared error. The rate falls roughly as the lapse over the section length and the cost rises as the square of the section length, so the damage grows in proportion to the section length, and the piece is sharper when what repeats is short: 49.3 proportions at fifteen seconds, 25.0 at four minutes.

That is the same trade checkpoints sharpen the middle of a form, not its top found in its spacings — frequent small failures against rare large ones — with the failures now in the form rather than in the counting.

By how much, and how often #

The last figure separates the two quantities that the whole-piece precision mixes.

The two groups fail differently. The forms with an immediate repeat fail often and by one section: the listener is placed in the wrong half of a repeated pair. The rondo and the sonata fail rarely and by two sections: the listener has skipped an entire episode or second subject and landed on the next occurrence of what they left. In the rondo and the sonata a misplacement needs an unusually long lapse; in the others it needs only an unluckily timed one.

What this says about writing repeats #

Repeat signs are the oldest economy in notation, and in the dances and binary movements of the eighteenth century every half of every piece was played twice. A minuet, a gavotte, the movements of a keyboard suite, the exposition of a symphony — each of these marks its first half to be repeated and very often its second. What the arithmetic here says about those forms is narrow, but it is not nothing: of all the ways a form can reuse its material, the immediate repeat is the only one that leaves a listener who knows the piece unable to say where they are after a moment’s inattention. A refrain does it with an episode between; a recapitulation does it with a development between. A repeat sign does it with nothing between at all.

The sonata exposition is an instructive case because it is the repeat that looks most like the minuet’s. In the plan here — first subject, second subject, first subject, second subject, then development and recapitulation — the exposition is repeated, but the repeat is of a pair, and at the level of sections the first subject is never followed by itself. The returning listener is protected by the second subject exactly as a rondo’s listener is protected by the episode. At the level of phrases the same exposition may contain immediate repeats of its own, and those would misplace a returning listener by a phrase rather than by a section.

How a returning listener is scored #

Each piece is 480 seconds, divided into equal sections labelled by their material; two sections with the same letter sound the same. The counts are those of the essay before: bars, phrases, periods and sections of two, sixteen, sixty-four and a hundred and twenty-eight seconds, with slips at two per cent per unit and lapses at one per cent per event, a tenth of which take every count at once. The other nine tenths are mended by the hierarchy exactly as there.

A lapse of attention begins at a known point and lasts an exponentially distributed time of stated mean. On returning, the listener hears the material of the section they are in and considers every section with that material from the one the lapse began in onward. The weight of each candidate is the chance that a lapse beginning where this one did would end inside it; the listener takes the heaviest. A wrong choice leaves the counts a whole number of sections out, and its square is the error that lapse contributes. A lapse still running at the end of the piece is never recovered, and the listener times the stretch from the lapse to the end. The misplacement rates are integrated over the lapse’s start and length on a fine grid, not sampled, and the map across the piece averages back to the whole-piece rate, which its figure checks.

What the arithmetic assumes about the listener #

That they know the form. Everything here is a listener who knows which sections are alike and how long each is. A listener hearing a piece for the first time has no plan to place themselves in; what a return tells them is only that it is a return. That listener’s problem is different and harder, and it is not computed.

That a lapse leaves no trace of its length. A listener whose attention wandered usually has some sense of having been away a long time or a short one, and that sense, however coarse, would help most exactly where the forms here fail — in choosing between two halves of a repeat a few seconds apart.

That sections are recognisable at once. A returning listener here knows the section’s material from the first moment back. Material that takes several seconds to identify moves every misplacement later and every recovery with it.

That repeats are exact. A performer who ornaments a repeat, as eighteenth-century practice expected, makes the second half of a pair distinguishable from the first, and on this arithmetic that is not decoration but a landmark. Written-out repeats with varied endings, first- and second-time bars, do the same at the boundary itself, which is exactly where the misplacements concentrate.

And that sections are equal. Forms whose sections differ in length give the listener another cue — how long ago the last boundary was — and some of the ambiguity of a repeated pair goes with it.

Which forms, and where the claim stops #

The six forms here are skeletons of Classical and Baroque practice, drawn at the level of sections: the rondo as A B A C A B A with a coda; the sonata as an exposition played twice, a development and a recapitulation; the minuet and trio with both halves of each repeated and the minuet played again without repeats; the thirty-two-bar song form of popular song, A A B A, played twice; and the strophic song, a verse over and over. The claim is about the arrangement of identical material in time and about nothing else in those forms — nothing like the proportion an analysis reports, which a golden section is a coin toss with six coins found an analyst can nearly always find somewhere — not their harmony, not their themes, and not whether a listener attends to them in the way the arithmetic assumes. Popular song’s verse-and-chorus forms, which a return is shorter than its first hearing found to be balanced to the bar by the clock and badly unbalanced by memory, alternate their two materials and so fall with the rondo here; their doubled choruses at the end do not.

Still open: the listener who has never heard it #

Every number above belongs to a listener who knows the plan, and that listener is the easy case: knowing the plan turns a return into a place. A first-time listener who loses every count and comes back hearing the opening theme knows only that the opening theme has returned. Whether that is the recapitulation or the repeat of the exposition, whether the development has been missed or not yet begun, is something they have to infer from what they remember of the piece so far and from the conventions of the style.

That makes the next question one about how much of a form’s plan a listener can reconstruct as it goes. A return has to be remembered discounted a repetition by how long ago its original was heard, and a late dancer needs the landmarks, not the rhythm found that a dancer joining part-way needs only the onsets that mark a bar. A first-time listener returning from a lapse is a late dancer at the scale of a movement. The computation that would settle it gives the listener only the sections heard so far and a prior over how forms of the style continue, and asks how much of the precision the familiar listener has recovered a stranger recovers — which would say whether the forms that are easiest to follow on a first hearing are the same ones that are easiest to come back to.