Checkpoints sharpen the middle of a form, not its top

A form is sharp at the bottom and vague at the top took an eight-minute piece, halved it seven times, and asked at each level how many proportions between 1 : 1 and 3 : 1 a listener could tell apart. Timed, the answer is almost the same everywhere — 2.1 at five levels and 5.2 at the two shortest — because the precision of a duration judgement changes in steps rather than smoothly. Counted in two-second units, the answer climbs from 2.2 at the top to 12.2 at a level of fifteen seconds. The top of a form is vague either way.

That essay named its own strongest simplification. Each level was judged on its own, as a single count in one unit, so a listener who lost the count anywhere in an eight-minute stretch was back to timing all eight minutes. A listener who has counted four phrases of four bars has, in principle, also counted the section — and if the phrase count survives when the bar count lapses, the lapse can be mended. Whether that works depends on something that is a question about cues rather than arithmetic: whether the two counts are kept by the same thing, so that losing one loses both.

The arithmetic can say what hangs on that question, and it turns out to hang a great deal on the middle of a form and very little on its top.

The count the earlier essay used #

The single count fails two ways. A slip miscounts one unit, with a probability of two per cent a unit, and slips add as a random walk, so the relative error of a surviving count shrinks as the section gets longer. A lapse loses the count altogether, with a probability of one per cent a unit, and a listener who has lost it can only time the section. At the top of an eight-minute piece — 240 two-second units — a count survives with probability 0.99 to the 240th power, which is 9 per cent. So 91 per cent of listeners are timing the section, at the Weber fraction of a judgement made over minutes, and that is why the counted column turns vague at the top.

A lapse mended at the last phrase #

Add a second count. The listener is also counting phrases, every sixteen seconds, and a phrase count is kept by something other than the pulse — a cadence, a return, a change of texture, the cues where a phrase ends found a boundary detector reading. When the unit count lapses in the middle of the fifth phrase, the listener knows it is the fifth phrase and knows how many units a phrase holds, and has only to estimate how far into that phrase the lapse came.

That estimate is a timing judgement, but of part of one phrase rather than of the whole section. Taken as uniform over a phrase and blurred by the Weber fraction of a sixteen-second judgement, a mended lapse adds an error smaller than one unit, where an unmended one adds an error of a third of the whole section.

A mend can fail two ways. The phrase count itself lapses, at the same one per cent per event, so it may already be lost when it is needed. And the lapse that lost the unit count may have lost the phrase count with it, with a probability called here the shared share — zero when the two are kept by unrelated cues, one when both are kept by the same pulse. With every lapse shared, a checkpoint mends nothing, and the arithmetic reduces exactly to the earlier essay’s single count; the figure above checks that before it draws anything.

What independent checkpoints buy #

With the two counts failing independently, the whole column changes shape.

The level of a minute goes from 4.1 distinguishable proportions to 19.4. The level of two minutes goes from 3.1 to 13.2, and of four minutes from 2.5 to 7.1. The half-minute level, which the single count put at 5.4, reaches 18.1. The two shortest levels, of fifteen seconds and seven and a half, are not changed at all, because a section shorter than a phrase has no checkpoint inside it.

The reason the middle gains so much is visible in what happens to the counts.

At a minute, a quarter of listeners lapse, and with checkpoints nearly all of them are mended: only 1 per cent end up timing the section, against 26 per cent with a single count. Since the error of a timed minute is about fifteen times the error of a counted one, removing that quarter is what takes the level from four proportions to nineteen. A checkpoint turns the most expensive failure into one of the cheapest.

The top stays vague, and the checkpoints are why #

The whole piece gains much less: from 2.2 to 3.9. The single count had a worse problem there and the checkpoints fix most of it — the share of listeners reduced to timing falls from 91 per cent to 29 — and it still does not make the top of a form sharp.

The 29 per cent is the phrase count’s own lapses. Eight minutes is thirty phrases, and a phrase count with one per cent lapses per phrase has a fair chance of being lost by the middle of the piece, at which point the next unit lapse cannot be mended. So the checkpoints move the vagueness up one level rather than removing it. And because an unmended listener times the whole piece at the Weber fraction of a judgement made over minutes, a quarter of listeners in that state is enough to hold the level near four proportions, however exactly the others have counted.

The decomposition makes the point exactly. At the top, the error of a listener’s estimate is a mixture of three variances weighted by how many listeners end up in each state, and the three are wildly unequal. The untouched listeners carry a slip error of under one per cent. The mended ones carry that plus about two mends, each a fraction of a unit, which together add less than half a per cent. The listeners who lost the count past mending carry thirty-seven and a half per cent. Weighted by 9, 62 and 29 per cent of listeners, 99.8 per cent of the top level’s error variance belongs to the 29 per cent who lost the count, and the relative error of the whole-piece estimate comes out at twenty per cent, set almost entirely by them. At the half-minute level the same split is 49 per cent, which is why that level’s precision is a matter of counting and the top’s is a matter of forgetting.

That is a better answer to the earlier essay’s question than either of the outcomes it predicted. It proposed that if lapses were independent, the column would climb toward the bottom instead of turning, and “the vague top of a form” would turn out to be an artefact of assuming a listener counts in one unit. It is partly an artefact. At the half of the column above a minute, most of the vagueness was the single unit, and a second level of counting removes it. At the very top it is not, because a count of anything, kept for eight minutes, lapses.

A little shared failure costs most of it #

The shared share is exactly the number nobody has, so the figure sweeps it.

The curves are steep at the left and flat at the right, and the steepness is the finding. At a minute, sharing 5 per cent of lapses takes the level from 19.4 to 12.9. Sharing 10 per cent takes it to 10.4, and a fifth of lapses shared leaves 8.0, which keeps 3.9 of the 15.3 proportions the independent checkpoints bought. By half shared the level is at 5.4, only 1.3 above the single count.

The reason is that a mended lapse is cheap and an unmended one is very expensive, so the value of the checkpoints lies almost entirely in the lapses that stay unmended — and every shared lapse is one. Moving the shared share from zero to a fifth multiplies the unmended share many times over, and that share is what sets the level’s precision.

So the question the earlier essay called clean has a sharp answer in one direction and a soft one in the other. If a listener’s phrase count is kept by cues entirely independent of the pulse, the middle of a form is far sharper than a single count says. If the two share even a small fraction of their failures — which is what one would expect of a listener whose attention wanders, since a wandering attention loses both at once — most of that sharpness is gone, and the earlier essay’s figures stand nearly as drawn.

No one checkpoint spacing sharpens every level #

The phrase length has so far been sixteen seconds — in the range a phrase is a number of seconds found a listener’s present allows — and that choice is not innocent.

Every level wants a different spacing. The whole piece is sharpest with a checkpoint every sixty-four seconds, at 6.6 proportions, and so is the four-minute level. The two- and one-minute levels want sixteen, the half-minute wants eight, and the two shortest levels want four.

There is a trade inside each column. Short checkpoints make a mended lapse cheap, since there is little of a phrase to estimate, but there are many of them, so the checkpoint count itself lapses more often over a long section. Long checkpoints are hardly ever lost, but a lapse mended at one leaves a long stretch to time. The best spacing for a level balances those, and it scales with the level: between a quarter and an eighth of the level’s length for the middle of the form, and longer at the top, where the checkpoint count’s own survival is what matters.

A listener with one checkpoint count can sharpen one band of a form’s levels. A listener who wants the whole hierarchy sharp needs a count at every level — bars, phrases, periods, sections — each mending the one below, and the arithmetic above says that is worth a great deal only if every one of those counts fails independently of the others.

Which computation produced the numbers #

The piece is 480 seconds, halved seven times, counted in two-second units with the earlier essay’s rates: slips of 2 per cent and lapses of 1 per cent per unit. The Weber fraction of a timed judgement is the midpoint of the first essay’s band for the duration judged — 7.5 per cent near a second, 15 per cent over several seconds, 37.5 per cent over minutes — and a mended lapse’s position is timed at the band of the checkpoint spacing.

A count at a level ends in one of three states. Untouched, with probability 0.99 to the power of its units, it has the slip error. Lost past mending — a lapse that either shared the phrase count’s failure or found the phrase count already lost — it has the timing error of the whole level. Mended, it has the slip error plus, for each expected mend, the variance of a position timed uniformly over one checkpoint interval. The level’s relative error is the probability-weighted mix of those variances, as in the earlier essay, and a proportion’s spread is $\sqrt{2}$ times its log-error. The phrase count’s chance of being lost when a mend is needed is taken at the section’s midpoint.

Where the model stops #

One checkpoint count, not a hierarchy of them. The sweep over spacings says a listener would want several levels of checkpoint, and only one is modelled. A second checkpoint count would mend the first one’s lapses, and the top of the form would sharpen further by an amount that depends, again, on how independent the counts are.

A mend is instantaneous and exact about which phrase it is. A listener who has lost the bar count re-anchors at the next cadence, which comes some seconds after the lapse, and may miscount the phrase itself. Both would add error to a mend and neither is here.

Lapses are evenly spread. A real listener lapses at predictable places — the middle of a long repetition, a transition with no landmarks — and the phrase checkpoints are strongest exactly where the music is most articulated. Lapses that cluster where checkpoints are weakest would make every number above smaller.

Repetition is not in it. A return is shorter than its first hearing found a listener stores a repeated section as less than its clock length, and a count of something already heard may lapse less, or more, than a count of something new.

And the rates are stated. Two per cent slips and one per cent lapses per unit are the earlier essay’s assumptions, carried forward so the columns can be compared, and they are not measurements.

What no count of proportions can show #

Whether listeners count at all. A hypermetrically aware listener can count phrases, and whether ordinary listeners do so across an eight-minute movement is not settled by anything here. A count is not an estimate introduced counting as a route a listener has, not one a listener takes.

The shared share. The one number this essay is about is the one it cannot supply. It is measurable in principle — a listening experiment that asks for both a bar count and a phrase count across a long passage, and records how often one is right when the other is wrong — and the figures here say which range of it matters: everything happens between none shared and a fifth.

Whether a proportion is what anybody hears. Nineteen distinguishable proportions at the level of a minute is a statement about discrimination, and 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.

Whose forms #

The levels here are the ones Classical form has names for — the bar, the phrase, the period, the section, the movement — and the phrase checkpoint is a cadence — often the half cadence that an ending that exists so a bigger one can described as built to arrive without settling. Forms that articulate their phrases strongly and regularly, with a cadence every four or eight bars, are exactly the ones in which a phrase count is most likely to be kept by a different cue from the pulse. On this arithmetic those are the forms in which a golden section is a coin toss at the top and a real, audible quantity a level or two down. Through-composed music with few articulations offers no independent checkpoint, and its middle levels should be as vague as its top.

Still open: the listener with a count at every level #

The sweep over spacings found that a single checkpoint count sharpens one band of levels and leaves the rest, and that the whole piece wants checkpoints nearly as long as its own sections. The natural extension is a listener counting at every named level at once — units, phrases, periods, sections — each count mending the one below it when it lapses, each with its own lapses, and each sharing some fraction of its failures with its neighbours.

That is a recursion the arithmetic above already has one step of, and the question it would answer is specific. With every level mending the one below and the shared share held at a few per cent, does the top of an eight-minute form reach the precision of its middle — so that the vague top is purely a consequence of listeners counting too few levels — or does the compounding of small shared failures up four levels leave it vague however many counts are kept? The earlier essays predicted the second, this one weakened the prediction, and the recursion would decide it.