The shape a wall leaves behind

The sixth rung of this ladder found that the short note of a swung pair holds a roughly constant absolute length of about a hundred milliseconds, and that this length is indistinguishable from the fast edge of the tempo window — which would make it a floor rather than a fitted parameter. The seventh took that seriously and priced what a floor does to the statistics: if the parent distribution’s mean walks down onto a wall, the observed mean is biased upward, the observed spread is understated, and three published findings about swing become artefacts.

The arithmetic it used is Pearson’s, and it is the arithmetic for a truncated normal — writing α for how far the wall is below the parent’s mean in spreads and λ for the ratio of the density to the tail there:

Truncation means the observations below the wall are absent. A survey that only reaches households above a certain income is truncated; the poor households exist and are not in the sample.

That is not what a floor on execution does.

Censored is what a player does #

A drummer aiming for a gap of ninety milliseconds when ninety is unexecutable does not produce nothing. They produce a hundred — the shortest thing they can do — and it goes into the recording like any other note.

So the mass that would have fallen below the wall does not vanish. It piles up on the wall, at exactly the floor value, and the observed distribution is a spike at the floor plus whatever was above it. That is a censored sample, and it is the standard object in survival analysis and in any measurement with an instrument that saturates.

The two differ, and they differ in a way that is easy to state exactly at one point. Put the parent’s mean on the wall. Half the mass is now at a single value, the floor, and the other half is the upper half of a normal, whose mean is 0.798 spreads up. So the whole sample’s mean is

Exactly half the truncated answer, and derivable in one line. Every bias the seventh rung quoted at that point is twice what a floor on execution actually causes.

The third moment, which is what this rung is for #

The seventh rung’s own last paragraph said what it was owed:

Every argument here is about the first two moments of a distribution, and the thing that actually distinguishes a censored sample from a free one is its third — the skew.

That is right, and the correction above makes it sharper: the third moment does not merely distinguish a walled sample from a free one, it distinguishes the two kinds of walled sample from each other, and it does so far more strongly than the first two do.

With the parent’s mean on the wall, the censored sample’s skew is 1.64 and the truncated sample’s is 0.99 — a difference of two thirds, against a difference in the standard deviation of only four per cent (0.584 spreads against 0.603). So the mean is off by a factor of two, the spread is nearly identical, and the skew is the quantity that says which process produced the data.

What is predicted, and what would falsify it #

The useful form is a prediction about a shape, and it is worth stating as something that could come out wrong.

If the hundred-millisecond floor is a floor, then swing-timing data at fast tempi should show: a right-skewed distribution of short-note durations; a pile of observations at or very near the floor value rather than a smooth density running down to it; a skew around 1.6 rather than around 1.0; and a mean that is biased by about 0.4 spreads rather than 0.8.

If it is not a floor — if a hundred milliseconds is simply the value players aim at, for whatever reason — then the distribution should be roughly symmetric, its skew near zero, and there should be observations comfortably below it.

The bias table below adds a third prediction that is easier to test than either, because it needs only two tempi rather than a distribution. The disagreement between the two models is a function of how far the intended value sits above the floor, so a study that measures at a tempo where the wall barely bites and again at one where it bites hard has, in the difference between those two, a quantity the two models put a factor of two apart at one end and a factor of six at the other. A single tempo cannot separate them; a pair can.

Those are different pictures and a histogram of a few hundred short notes would tell them apart at a glance. What this collection does not have is the few hundred short notes. That is the same debt the clave rung recorded, arriving at the same ladder from the other end, and it is the same shape of debt the melody ladder recorded, which is that a distribution cannot be checked against a summary. It is now the fourth thing in this anchor that needs a corpus and cannot be got at without one.

The two samples differ in how far each moves the mean, and the ratio between them is exact: with the parent’s mean exactly on the wall the truncated bias is 0.797 spreads and the censored bias is 0.399. Half of it — because censoring puts the refused mass at one point on the wall while truncation removes it, so the censored sample keeps half the displacement. The factor of two is arithmetic rather than a coincidence of these numbers, which is what makes it usable on a measurement whose wall is somewhere else.

Which of the seventh rung’s three findings survive #

The previous rung named three published statistics that a floor would bias, and the correction here does not affect all three equally.

“Swing timing becomes more consistent at fast tempi.” This one is a claim about the spread, and the spread is the quantity the two models very nearly agree on: 0.584 spreads censored against 0.603 truncated, at the wall. So the seventh rung’s finding stands almost unchanged. A wall understates the spread by about forty per cent under either reading, and the reported increase in consistency at fast tempi is the wall arriving rather than players becoming steadier.

“The mean short-note duration is a stable constant.” This one is a claim about the mean, and it is where the halving lands. A censored process biases the observed mean up by 0.4 spreads rather than 0.8, so the apparent stability is real to a greater degree than the previous rung allowed: the constant is less of an artefact than it said.

“The swing ratio falls smoothly with tempo.” This one is a claim about the shape of a relationship, and it is the one the correction leaves most exposed, because the bias enters at every tempo differently. As the tempo rises the parent’s mean walks down onto the wall, so the bias grows — and it grows on a different curve under the two models, which is worth drawing rather than asserting.

The factor of two is a point value and not a correction factor. It holds exactly where the parent’s mean sits on the wall and nowhere else: a spread above it the truncated formula overstates the bias by three and a half times, two spreads above by six and a half, and a spread below by only 1.4. So a study that applied Pearson’s correction across a range of tempi did not merely double its correction; it distorted the shape of it, most at the tempi where the wall barely bites.

The absolute disagreement between the two runs the other way and peaks below the wall — 0.45 spreads at a gap of −0.73, where a player is attempting something they mostly cannot execute. That is the region where the two models are most distinguishable and it is also the region where the biases are largest under both, so it is the region a study should be sampling if it wants to tell them apart.

Carrying that into the swing ratio itself, on an illustrative model where the intended short note is thirty per cent of the pair: below about 140 to the minute the two agree to two decimal places, because the wall is several spreads away and nothing is being censored. Above it they separate, with the largest gap around 200 — where the truncated reading gives an observed ratio of 1.75 and the censored one 1.93 against an intended 2.33. So under censoring rather more of the observed fall is real, which is the same direction the mean finding moved: the wall is doing about half what the seventh rung charged it with, and the players about half again as much.

So the correction is not uniform. It strengthens one of the previous rung’s findings, weakens another, and changes the shape of the third — which is a fair summary of what happens when a model is replaced by a nearby one rather than by a different one.

It is worth naming what the illustrative swing model is and is not. The thirty per cent is a stand-in for a relationship this collection does not have measured — what a player intends the short note to be at each tempo — and the whole point of the exercise is that the intended value is exactly what a walled sample cannot report. So the two curves above are what the two corrections would do to a stated intention, not a reconstruction of anybody’s. What does not depend on the stand-in is the tempo at which the models separate: it is wherever the intended short note comes within about two spreads of the floor, which for a thirty-millisecond spread is about a tempo and a half below where the floor is reached.

The third finding is about the swing ratio against tempo, which falls from something near three to one at slow tempi toward one to one at fast ones — swing is a ratio rather than a duration, and the short note holding a roughly constant hundred milliseconds is what produces the fall. That constant is the wall this essay is about, seen from the side it was originally measured from.

How the moments were computed, and why not by formula #

Both distributions are integrated numerically rather than evaluated from closed forms, and the choice is deliberate.

Closed forms exist for both. The truncated case is Pearson’s and the censored case is a standard result in the survival literature; the third moments of both are also published. Using them would have made this essay a matter of quoting two sets of expressions and asserting that they differ.

Quadrature instead means one procedure produces every number: a grid over the parent normal, a rule about what happens to the mass below the wall — discarded, or moved to the wall — and then the first three central moments of whatever comes out. The two cases then differ by one line of code and nothing else, which is exactly the claim being made about them, and the arithmetic that produces the mean is the arithmetic that produces the skew rather than a second derivation to be trusted separately.

It also checks itself. At a wall three spreads below the mean both procedures return the parent’s own moments — mean unbiased, spread 0.98 of the parent’s, skew 0.12 — and at the wall the censored mean comes out at 0.399 spreads, which is the one-line hand calculation above. A quadrature that fails either of those is wrong in a visible way.

Where the model stops #

A normal parent is an assumption and it is doing work. The exactness of the half is a property of the normal’s symmetry: half the mass below the mean, and an upper half-normal with a known mean. A parent with any skew of its own gives a different factor, and there is no reason to think a player’s intended timing is symmetrically distributed.

The parent’s spread is asserted. Every figure here uses fifteen milliseconds, which is the middle of the motor-noise band the third rung of this ladder established. The shape of every curve is independent of it — the horizontal axis is in units of that spread — but the milliseconds quoted in the prediction section are not, and a spread of ten or of twenty would move them by a third either way.

A hard wall is an idealisation. Real motor limits are soft — a gap ten per cent below the floor is not impossible, it is unreliable — so the true observed distribution is a blur of the spike rather than the spike, and its skew is somewhere between the censored and the truncated answers. That is the most damaging caveat here, because it means the two hypotheses this essay proposes to distinguish are the ends of a continuum rather than two alternatives, and a measured skew of 1.3 would be uninformative.

And the floor is still a conjecture. The sixth rung established that the short note’s absolute duration is roughly constant and close to the tempo window’s fast edge. It did not establish that the second causes the first, and this rung has assumed it throughout. If the constancy has some other cause — a preference, a stylistic target, a limit of the instrument rather than of the player — then everything here is a correct analysis of the wrong picture.

Whose playing, and when #

The hundred-millisecond figure comes from jazz drumming, and the swing ratio literature it belongs to is overwhelmingly about the ride-cymbal pattern in the post-1940 American tradition. Every number this ladder quotes is from that repertoire.

That matters for this rung in particular, because a floor set by motor limits is a property of human beings and a floor set by an instrument or a style is not. If the wall is motor, the same shape should appear in any tradition that pushes its subdivisions fast — West African bell patterns, Balkan dance metres, drum and bass — and the skew should be the same 1.6 wherever it is measured.

If the wall is stylistic, it should not travel, and the distributions in other traditions should be symmetric around whatever value that tradition uses. That is a much better test than anything available inside the jazz corpus, because it varies the thing the two hypotheses disagree about, and it is a corpus problem rather than a modelling one.

What the picture cannot show #

Which sample a real recording is. A recording of a performance is not a sample from a distribution of intentions; it is one realisation of a player who is also adjusting to what they hear, to the tempo, and to the other players. Two players correct toward each other and the correction is not in this model at all — and the correction is itself a source of skew, because a player who is late has more room to move than one who is early against a wall.

Nor whether a player is aiming at a duration at all. The whole framing takes the short note’s length to be the thing intended and the wall to be a limit on it. A player may instead be aiming at a ratio, or at a place in the bar, or at nothing describable — and the deviations may be an offset and a pattern rather than a distribution, which is what the third rung of this ladder found and which no summary statistic of durations can see.

And it cannot show a wall that moves. The tempo window’s fast edge is a function of the tempo, and a performance’s tempo drifts. A wall that moves during a take smears the spike over a range, which is the same visual result as a soft wall and a completely different mechanism — and separating them needs the timing data with its tempo curve attached, which is more than a histogram.

Where this ladder goes next #

Eight rungs. Swing is a ratio and not two to one; the deviations that make a groove are unnotatable; they split into offset and pattern; two players correct toward each other; the deviations sit inside categories; the categories have tempo ranges and the swing constant may be the window’s edge; three published statistics are biased if it is; and now the bias is half what was quoted, and the third moment is the quantity that says so.

What is owed is not another moment. It is that every rung of this ladder since the fourth has been an argument about what a distribution of timings implies, and this collection has never had one — every figure is a model evaluated at published summary statistics. The corpus needed is small by any modern standard: a few hundred short notes with their durations, from two traditions, at three tempi. It is the smallest thing this ladder has ever asked for and it is the fourth consecutive rung to ask for it.