A count is not an estimate

The essay that priced a proportion as a ratio of two estimates found that a proportion is a ratio of two estimates and is blurred by both. The second found a different measure — what a listener has to store rather than how long the clock ran — and that measure is an estimate too. Its closing paragraph named a third route which is not an estimate at all, and asked what it costs.

A listener who has induced a four-bar hypermetre knows a verse was eight bars long, and knows it exactly. Not approximately, not within a Weber fraction: eight. The count is a different kind of knowledge from the other two, and the whole question is what makes it stop being exact.

Two failures, pulling opposite ways #

A count fails twice over and the two failures have nothing to do with each other.

A slip is one unit miscounted — a bar heard as two, or two heard as one, which is what an eight-bar phrase that accelerates does to a listener deliberately. Slips are independent from unit to unit, so they accumulate as a random walk: over $N$ units the count’s standard deviation is $\sqrt{Np}$ units, and its relative error is $\sqrt{p/N}$. That falls as the section gets longer. A section of four bars miscounted by half a bar is a twelve per cent error; a section of a hundred bars miscounted by one and a half bars is one and a half per cent.

A lapse is the count lost altogether — attention elsewhere, a metrical reinterpretation, a passage whose bars cannot be told apart. The chance of surviving $N$ units is $(1-q)^N$, which falls geometrically. That rises as the section gets longer, and a long enough section is certain to lose its count.

A listener who has lost the count has not lost the section: they are back to timing it. So what “counting” delivers is a mixture — the slip error where the count survives, and the timing error where it does not — and a mixture of a falling term and a rising one has a floor.

At a slip rate of two per cent a unit and a lapse rate of one per cent, the floor sits at four units against a timing judgement over minutes, at ten units against several seconds, and at twenty-one against a single second. The shorter the sections being compared, the finer the clock they are compared against, and the further a count has to run before it is worth having.

Where counting beats the clock, exactly #

The lower boundary has a closed form and it is the one number in this essay that needs no enumeration. A count beats the clock when its slip error falls under the Weber fraction:

$$\sqrt{p/N} < w \qquad\Longleftrightarrow\qquad N > p/w^2$$

For a slip rate of two per cent against a timing judgement over minutes, that is $0.02/0.375^2$, or fourteen hundredths of a unit — which is to say, always. Against the finest timing anybody does, a single second at seven and a half per cent, it is 3.6 units.

Below that boundary counting is worse than timing, and that is not a defect of the model. A miscount of a one-unit section is an error of a whole unit — a hundred per cent — where timing the same second is wrong by seven. Counting is a bad way to know the length of one bar and a very good way to know the length of sixteen.

The upper boundary has no closed form and is more interesting. Counting stops being worth much when the lapse rate has eaten the survival probability, and that happens at very nearly the same length whatever the timing band is: the gain falls back under a quarter at 98 units against a one-second judgement, 101 against seconds, and 102 against minutes. The timing fraction sets how much counting is worth; the lapse rate alone sets how long it stays worth anything.

What a proportion costs by each route #

A proportion is two durations, so each route pays twice.

The first essay’s two headline numbers were eleven proportions between 1:1 and 3:1 when both durations are timed as finely as anyone times a second, and two when they are judged from memory over minutes. Counting both sections gives 7.7 at four units each, falling to 3.1 at sixty-four.

So a listener counting a phrase knows a proportion of minutes about as finely as a listener timing a couple of seconds knows one of seconds. That is not a small correction to the first essay. It says the discouraging figure that essay ended on — two distinguishable proportions across the whole range a form uses — is the figure for a listener who is not counting, and that a listener who is counting has three or four times as many.

Which route a form actually offers #

The three routes are not alternatives a listener chooses between. Each is available under conditions the music decides, and the conditions are rarely all met at once.

Counting needs a unit. It needs an induced hypermetre or at least a steady bar, which is to say a passage whose metre is unambiguous and stable. A rubato passage, a transition, a fermata, an accelerando: each of those either removes the unit or makes two consecutive units unequal, and a count of unequal units is not a duration.

Timing needs nothing and gives little. It is always available and its Weber fraction over minutes is the one number here that is genuinely bad.

Storage needs the piece to have returns in it. The second essay’s measure applies to a form that repeats, and it says nothing about a through-composed one.

So a movement that modulates its metre, or that is built of sections in different tempi, has taken the count away exactly where a proportion between those sections would need it. The proportions a form makes hardest to hear are the ones between sections it has separated by a change of pulse, which is a claim about what a tempo change costs beyond its own effect, and it is the counterpart of what a process that enumerates its own form gains by never changing pulse at all.

Where the three routes disagree #

The useful thing about having three measures is not that one of them is best. It is that they disagree, and where they disagree is where an analysis and a listener part company.

Take a rondo whose refrain occupies three fifths of the clock. By the clock the refrain is the larger part. By storage it is much the smaller, because a return costs a listener almost nothing to keep. By count it is neither: it is however many bars it is, and that number is the same on its first appearance and its third.

So the count is the only one of the three that treats a repeat as a length. Timing does too in principle and cannot in practice, because the timing of a stretch of minutes is too blurred to distinguish anything; storage explicitly does not, and that was the second essay’s whole finding. A listener with a count has a measure on which a refrain and a verse of equal bar-length are equal — which is what the score says and what the other two measures deny.

That is the shape of the disagreement and it decides which claims about proportion are about what. A claim that a form is balanced in bars is a claim about the counted measure, and it is available to a listener only where the count survives. A claim that a form is balanced in seconds is available to nobody over minutes. A claim that it is balanced in what a listener carries is available whenever the form has returns in it, and it will disagree with the score wherever it applies.

Why a listener would be counting anyway #

There is one more reason to take the counted route seriously, and it is not about proportion at all.

A listener tracking a metre is doing it for reasons that have nothing to do with the length of sections. They are predicting where the next downbeat falls, which is what makes a syncopation a syncopation; they are hearing what a tonic costs in seconds accumulate on strong beats rather than weak ones; they are parsing a phrase. The metre is being maintained regardless, and a count of bars is very nearly free once it is.

That matters because the alternative reading of this essay is that counting is an effortful, deliberate act nobody performs while listening to music — in which case the numbers above describe a laboratory task. The reply is that the expensive part of counting is the metre and the metre is paid for elsewhere. What a count adds on top is a running total, and a running total is the cheapest thing a mind that already has a pulse can keep.

Which is also why the lapse rate is the parameter that matters most. A total is cheap to keep and easy to drop, and dropping it costs nothing at the moment it happens — a listener who has lost count of the bars notices nothing until they are asked. That asymmetry is why the count can be both nearly free and quite unreliable, and it is why the upper boundary of a hundred units is set by lapse rather than by effort.

What a slip rate of two per cent is #

The two rates are the parameters of this essay and neither is measured here, so what they rest on should be plain.

A slip rate of two per cent a unit means a listener miscounts one bar in fifty. That is chosen to be conservative rather than fitted: it makes a sixteen-bar count wrong by half a bar on average, which is about what a musician who has lost track of a repeat would report, and every conclusion above weakens if it is smaller.

A lapse rate of one per cent a unit gives a count a half-life of about seventy units. A listener who can count a hundred bars without once losing the thread is doing something the model calls impossible, and a listener who loses it every twenty bars has no count worth the name at thirty.

Raising both rates moves the floor and does not change its existence, which is the useful robustness: the finding is that counting has a best length, not that the best length is four.

The two rates are not independent of each other #

One simplification above is doing more work than it looks, and it is worth pulling out before the arithmetic is quoted anywhere.

Slip and lapse are treated as separate events with separate rates, and in a listener they are two readings of one process. A count is maintained by attention, and attention wanders by degrees rather than by switching off: a listener half-attending miscounts more often and is more likely to lose the thread entirely. So the two rates rise together, and a model with one attention parameter behind both would give a sharper floor than this one does, in a narrower range of section lengths.

The direction of that error can be stated even though its size cannot. Correlating the two rates makes counting better than modelled where attention is good and worse where it is bad, so the mixture drawn here is the average over a listener’s attention rather than the experience of any particular moment. A listener attending closely to a sixteen-bar section has a count nearer the slip curve than the mixture; one attending loosely has one nearer the timing line.

That reading also explains something the flat model cannot: why counting feels binary rather than graded. A listener asked how long a section was either has a number or does not, and the mixture curve is a description of how often the first happens rather than of what either answer is like.

Which computation produced the numbers #

A slip is modelled as a symmetric error of one unit with probability $p$ per unit, independent across units, so the count’s variance after $N$ units is $Np$ and its coefficient of variation is $\sqrt{p/N}$. A lapse is modelled as an independent event of probability $q$ per unit that destroys the count, with survival $(1-q)^N$.

The two are combined as a mixture over outcomes rather than as a sum of errors: with probability $(1-q)^N$ a listener has the count and its slip error, and otherwise they have a timed estimate at the Weber fraction of the band the section’s duration falls in. The coefficient of variation of the mixture is the square root of the probability-weighted variances, which assumes the two outcomes have the same mean — a listener who has lost the count is not systematically wrong, only vague.

A proportion of two counted sections is a ratio of two independent estimates, so their log spreads add in quadrature, exactly as the first essay adds two timed estimates. The number of distinguishable proportions between two ratios is the log of their ratio divided by one criterion’s worth of spread, which is that essay’s own measure kept unchanged so the numbers can be set beside each other.

The timing bands are the: about seven and a half per cent for a single interval near a second, fifteen for several seconds judged without counting, and thirty-seven and a half for a stretch of minutes judged afterwards. They are ranges in the source and midpoints here.

Where the model stops #

A slip is not symmetric. Losing a bar and gaining one are not equally likely: a listener who loses the thread is more likely to under-count, because the failure mode is missing an event rather than inventing one. An asymmetric slip adds a bias to the count, which this model has none of, and a bias does not average out over a proportion.

And a lapse is not independent of the music. Counting fails where a passage is hard to parse, and a passage is hard to parse at exactly the places a composer has made interesting — a transition, a developmental passage, a false return. So the lapse rate is not a constant of the listener; it is a function of where in the form the listener is, and it is highest in the sections whose length is least regular.

The count and the clock are treated as independent. A listener who has lost the count has been attending to something, and the something may well have cost them the clock too.

What the picture cannot show #

It cannot show a listener counting in two units at once. A phrase is four bars and a section is four phrases, and a listener tracking both has two counts with different rates. Whether the coarse count is more robust because it has fewer units, or less because each of its units is harder to hold, is a question with an answer and it is the next essay’s.

Nor whether a listener counts at all. Everything here prices a count that is being kept. A great deal of listening is not metrical in that sense, and nothing in this collection says what share of it is.

And it cannot show the score. A count is exact against the written bar and the written bar is a notational object. A phrase is a number of seconds before it is a number of bars, and a passage whose bars are written in five is not counted in fives by most listeners.

Still open: whether a hierarchy of counts beats a single one #

The count priced here is flat: one unit, run from the start of a section to its end. A form is not flat, and a listener who has a phrase level and a section level above it is not counting one thing.

A hierarchy changes both failure rates and it changes them in opposite directions, which is why the answer is not obvious. A count of four sections is four units instead of sixty-four, so its lapse chance is tiny — but each of its units is a section, and knowing where a section ended is the very judgement the count was supposed to supply. A count of bars has reliable units and too many of them.

What would settle it is the same arithmetic run over a tree rather than a line: a slip and a lapse rate at each level, a section’s length as the product of the counts above it, and the question of whether the errors at the levels are independent. If they are, a two-level count of sixty-four bars is a count of four and a count of sixteen, with a combined error well under either — and the best depth would be a number rather than a preference. If they are not, because a lapse at the bar level takes the phrase level with it, the hierarchy buys nothing and a listener is counting one thing after all.