Only a lapse of attention keeps a form's top vague

Three essays have now asked how finely a listener can judge the proportions of a form, and each has found the top of the form vaguer than its middle. A form is sharp at the bottom and vague at the top timed and then counted each level of an eight-minute piece and found about two distinguishable proportions at the top however it was done. Checkpoints sharpen the middle of a form, not its top gave the listener a second count of phrases that could mend a lapse in the count of bars, and the minute-long level went from four proportions to nineteen while the whole piece went only from two to four. Thirty phrases, it found, are long enough to lose a phrase count as well.

That essay ended on the obvious extension. A listener who counts phrases can also count periods, and sections, and each count can mend the one below it. The question was whether a listener counting at every named level at once reaches the precision of the middle at the top — so that the vague top is only a consequence of counting too few levels — or whether small shared failures, compounding up four levels, leave it vague however many counts are kept.

The answer is neither of the two outcomes the question offered, and it turns on a distinction the question did not draw. It is not how much failure the counts share that decides the top of a form. It is what shape the sharing takes.

The top becomes the sharpest level #

Give the listener four counts: bars of two seconds, phrases of sixteen — inside the span a phrase is a number of seconds found a listener’s present allows — periods of sixty-four and sections of a hundred and twenty-eight. Each lapses at the same one per cent per event that every essay in this sequence has used, and each, when it lapses, is mended by the nearest count above it that is still running. Let every count fail independently of the others, for the moment, and read every level of the piece again.

The column turns over. The whole piece, which a single count put at 2.2 distinguishable proportions and a single checkpoint at 3.9, is now at 76.4. The four-minute level is at 54.0, the two-minute at 38.1. The minute-long level, at 19.4, is exactly where the phrase checkpoint had put it, because no count longer than a phrase fits inside a minute. The two shortest levels have no checkpoint inside them at all and are unchanged.

The reason the top becomes the sharpest level rather than merely a sharper one is the arithmetic a count is not an estimate started from. A count fails in two ways. A slip miscounts one unit and slips add as a random walk, so their relative cost falls with the square root of the section’s length — at two hundred and forty units, two per cent slips leave an error of nine tenths of one per cent. A lapse loses the count and its chance compounds, and it was the lapses that made every long count useless. Mend nearly every lapse and only the slips are left, and the slips favour long sections. The vague top of a form was never a property of long durations. It was a property of long durations counted by a listener who could not recover a lost count.

What the counts above the phrase are for #

The counts above the phrase do something less direct than it looks.

At the top of the piece, nine listeners in a hundred never lose their bar count, ninety-one lose it at least once and have it mended, and effectively none are left timing the piece. With the phrase checkpoint alone, 28.5 per cent were. Almost every one of those mends is made by the phrase count, not by the counts above it. What the period and section counts do is keep the phrase count itself alive: with nothing above it, the phrase count was lost by the middle of the piece fourteen times in a hundred, and when it was lost the next lapse of the bar count had nothing to fall back on. With a count above it, a lost phrase count is re-anchored at the next period and is running again when the bar count next needs it.

So the upper levels of a hierarchy are not a finer tool for measuring the form. They are the maintenance of the tool that does the measuring, and the essay that stopped at one checkpoint had found the one mender that does most of the work and none of the ones that keep it working.

A shared failure must climb to reach the top #

Everything so far has the counts failing independently, and the earlier essay showed that assumption is where most of a checkpoint’s value lives: with a fifth of the bar count’s lapses also taking the phrase count, three quarters of what the checkpoint bought was gone. The obvious fear is that a hierarchy of four or eight counts compounds that — that a small shared failure, met at every level, adds up to a large one at the top.

It does the opposite, if sharing means what it meant there. The earlier essay’s shared lapse was a failure that took two neighbouring counts at once, because both were kept by the same cue. In a hierarchy that is a cascade: a lapse of the bar count takes the phrase count with it some share of the time, a lapse that took the phrase count takes the period count with it the same share of the time, and so on up. To leave a listener with nothing to mend from, a cascade has to reach the top count. With a tenth of lapses cascading and eight levels, the chance of that is a tenth to the seventh power.

The two upper curves climb with every level added, and they climb to nearly the same place. Independent counts take the whole piece from 2.2 to 84.5 proportions over eight levels; counts with a tenth of their lapses cascading take it to 83.8. The cascade costs something at intermediate depths — at four levels it leaves 22.7 where independence leaves 81.3, because four levels is not yet enough climbing to make the top out of reach — but by six levels the difference is under three proportions and by eight under one.

A hierarchy is its own defence against correlated neighbours. Each added level is one more step a shared failure has to climb, and the chance of climbing all of them falls geometrically while the cost of adding a level is one more thing to count.

One cause for every count #

There is a second shape of shared failure and it behaves nothing like the first.

A listener whose attention wanders does not lose the bar count and then, with some probability, the phrase count. They lose everything at once — the bar, the phrase, the period, the section — because all of those were being kept by the same thing, and the thing stopped. Call that a common cause: a share of lapses that take every count together. No level can mend such a lapse, because there is no level left running to mend it from.

The two lower curves above are that case, and they are flat. With a twentieth of lapses common to every count, the piece reaches 6.2 proportions at four levels and stays at 6.2 however many more are added. With a tenth, it reaches 4.5 and stays there. The fourth level — the phrase — is the last that buys anything, because it is the one that mends the ordinary lapses, and nothing above it can touch the common ones. The floor is exact arithmetic: a tenth of one per cent of two hundred and forty units is a common lapse somewhere in the piece for 21 per cent of listeners, and those listeners are timing eight minutes from memory, at a Weber fraction of about thirty-seven per cent. They set the level’s precision as completely as the 91 per cent who lost a single count did.

Put the two shapes on one axis and the finding is plain. At a tenth of lapses shared, cascading counts at every level keep nine tenths of everything the hierarchy could buy, and common-cause counts keep a twentieth of it. The two curves meet only at the ends — when nothing is shared, and when everything is, since a cascade that always climbs and a cause that always takes everything are the same failure. Between those, the same number means entirely different things, and the number the earlier essay swept was a number of neither shape: with only two counts in play, a cascade and a common cause cannot be told apart.

One lapse in a hundred #

The common-cause curve falls hardest at the far left, and that is where the surprise is.

With one lapse in a hundred taking every count at once, the whole piece falls from 84.5 proportions to 13.3, and the sharpest level of the form is no longer the top but the minute, at 23.5. At two in a hundred the top reads 9.5; at five, 6.2. By then the shape of the column is exactly the shape every earlier essay drew — a peak somewhere in the middle and a vague top — at a higher overall precision.

The mechanism is the one the checkpoint essay found and it is sharper here. A lapse that is mended costs a fraction of a unit; a lapse that is not costs a third of the section. Two such unequal costs mix so that a tiny share of the second dominates. At a hundredth of lapses common, only 2.4 per cent of listeners reach the end of the piece with no count running — and those 2.4 per cent contribute more than thirty times the error variance of the other 97.6 per cent combined.

That turns the question the earlier essay called clean into a different and cleaner one. Whether a listener counts at three levels or eight hardly matters; whether their counts are kept by different cues or by the same one matters only a little, once the hierarchy is deep. What decides whether the top of a form is sharp or vague is whether any of the listener’s lapses are lapses of attention, and the answer is almost certainly yes. Attention wanders on a timescale of seconds to tens of seconds; an eight-minute movement at one per cent lapses a unit has two or three lapses in it, and it would be remarkable if none of them were the kind that takes everything.

What an unreliable mender costs #

The model so far treats a mend as exact about which phrase, period or section the listener is in, and that is generous. A phrase count can slip as a bar count can, and a phrase count that has slipped hands its slip to the bar count it mends — a whole phrase wrong, sixteen seconds, where a bar-count slip was two.

It costs about half. With the upper counts slipping as often as the bar count does, two per cent an event, the whole piece reads 46.1 proportions rather than 84.5; at ten per cent, 23.6. What it does not cost is the shape. At every slip rate drawn, the whole piece is still sharper than the minute-long level, and still ten times sharper than a single count. A miscounting mender makes each mend expensive; a common-cause lapse makes a mend impossible, and the difference between an expensive repair and no repair is the whole difference between the two columns.

The surprise is that forgetting looks like duration #

The vague top of a form has always been read as a fact about how long it is. A movement is minutes long, minutes are timed badly, and every theory of large-scale proportion has had to explain why listeners seem to hear something that timing cannot deliver. The arithmetic above says the vagueness is not about the minutes. A listener with a count at every level and no lapses of attention would hear the proportions of a whole movement more finely than those of a single phrase — for the same reason a long exposure is less noisy than a short one.

What produces the vague top is the one failure a hierarchy cannot repair, and it happens to scale with duration. A lapse of attention is a hazard per second; the longer the stretch, the likelier one falls in it; and the longer the stretch, the more a listener who suffered one has to time from memory. So forgetting produces exactly the signature that a Weber fraction produces, and the two cannot be told apart from how precision falls with length. They can be told apart only by what fixes them. A worse clock is fixed by counting. A wandering attention is fixed by nothing the listener does — only by something the music does to bring the place back.

How a remembered level is scored #

The piece is 480 seconds, halved seven times, as in the essays before this one. Counts at every level lapse one per cent per event; the unit count slips two per cent per unit. For a section of a given length only the counted levels shorter than it can mend inside it. A lapse at level k is mended by the first level above it whose count is running; the chance that a level’s count is running when a mend is asked of it is taken at the section’s midpoint, as the checkpoint essay took the phrase count’s, and it is computed from the top level down, because each level’s chance of being lost for good depends on whether the levels above it can re-anchor it.

A mend at level m re-estimates the position inside one level-m interval by timing it, at the Weber fraction of a judgement that long, which adds a variance of the Weber fraction times the interval, squared, over three. A cascade takes the next level up with probability ρ at each step; a common-cause lapse takes all of them. The section’s relative error is the probability-weighted mix of the slip error, the mend errors and — for listeners with nothing left to mend from — the timing error of the whole section, and a proportion’s spread is $\sqrt{2}$ times its log-error, as before. With two levels the recursion is the checkpoint essay’s arithmetic term for term, and its figure checks that before it draws anything.

Where the recursion is too kind #

A mended upper count is taken to be right. A phrase count that was itself mended by the period count carries the period count’s timing error, and that error passes into the next bar-count mend it makes. The figure on miscounting menders prices the nearest version of this, and it halves the top’s advantage; the full recursion of errors on errors is not computed.

A common-cause lapse is permanent. A listener whose attention returns does not stay lost for the rest of the piece; they begin counting again at the next thing they recognise. The model gives them nothing to recognise. That is the largest simplification here, and it is the one that decides whether the vague top is permanent or temporary.

Lapses are evenly spread. A listener’s attention is likelier to go in a long repetition than at a cadence, so common lapses cluster exactly where the counts would most need to be kept.

And the rates are stated, not measured. One per cent lapses and two per cent slips per event are the sequence’s working assumptions, kept so that the columns compare. The share of lapses that are lapses of attention is the one number this essay rests on, and it has no measured value at all.

What discrimination leaves out #

Whether anyone counts at four levels. A trained listener can say which bar of which phrase of which section they are in. Whether an ordinary listener keeps anything like four counts through a movement is not settled by any of this, and the figures say only what such a listener would have if they did.

Whether a proportion is heard. Seventy-six distinguishable proportions across a whole movement is a statement about discrimination between two sections’ lengths. A proportion is only as fine as its two durations was careful to say that telling proportions apart is not the same as hearing a form as proportioned, and a golden section is a coin toss with six coins showed how easily an analysis finds a proportion the listener could never have told from its neighbours.

Which lapses are which. The distinction the essay turns on — a lapse that cascades against one that takes everything — is a distinction between two causes of the same observable event, a count lost. An experiment that recorded only whether each count was right at the end could not separate them; one that recorded when each count went wrong, across several levels at once, could.

Whose forms #

The four levels here are the Classical ones: bar, phrase, period, section, in the ratios eight, four and two that a periodic sonata movement or a dance form nearly always has. Those forms mark every level with a cadence of its own weight — the half cadence at a phrase, the perfect cadence at a period, the change of key at a section — which is what makes each count keepable by its own cue, the cues where a phrase ends found a boundary detector reading, and a cascade of shared failure unlikely. Music whose levels are not marked by distinct cues, such as a through-composed movement or a long passage of a refrain that returns unchanged, gives a listener fewer independent counts and more reason to lose them together. On the arithmetic above, that is not a difference of degree at the top of the form but a difference of kind.

Still open: what brings a lost listener back #

A lapse of attention ends. The listener comes back — a few seconds later, or a phrase — and has lost every count, and what they can do next depends entirely on the music. If the next thing they hear tells them where they are, the lapse costs the few seconds it lasted and nothing more; if it does not, they are timing the rest of the section from nothing.

That makes the next question one about the form rather than about the listener. A late dancer needs the landmarks, not the rhythm found that a dancer joining an additive metre part-way needs only the onsets that mark a bar, and a listener returning from a lapse needs the same thing at a larger scale: a landmark that names its section. A return of the opening theme names one; a refrain that comes back five times names only that it is a refrain, and the listener has to time their way to which. The computation that follows gives the common lapse an end, lets the listener re-anchor at the next landmark the form provides, and asks how much of the vague top that recovers — and whether the forms that repeat their material most, which are the easiest to follow, are the ones that least tell a returning listener where they are.