The parent nobody measured

The seventh rung of this ladder priced what a floor on execution does to the statistics of swing timing, using Pearson’s formulas for a truncated normal. The eighth found that a floor on execution does not truncate — a drummer who cannot play a ninety-millisecond gap produces a hundred, and the attempt goes into the recording — so the sample is censored, the refused mass piles on the wall rather than vanishing, and with the parent’s mean sitting on the wall the bias is exactly half what had been quoted.

It also proposed a test. The censored and truncated samples very nearly agree about the spread and differ by a factor of two in the mean, but their third moments are 1.64 and 0.99 — so the skew, it argued, is the quantity that says which process produced a set of measurements.

Its own first caveat was this:

A normal parent is an assumption and it is doing work. The exactness of the half is a property of the normal’s symmetry, and there is no reason to think a player’s intended timing is symmetrically distributed.

That is right, and the sweep it asks for produces a result in the opposite direction from the one the caveat expects. The normal assumption does no work at all on the number the ladder is built on, and all of the work on the number it proposed as the test.

The number the ladder is built on is a theorem #

With the parent’s mean exactly on the wall, the censored sample’s mean is the expected value of the parent when everything below its own mean is replaced by that mean. Written out, the bias is the average amount by which the parent exceeds its mean — and the average amount by which a distribution exceeds its mean is equal to the average amount by which it falls short, because the two are what makes it the mean.

So the censored bias at the wall is half the parent’s mean absolute deviation, exactly, for every distribution there is.

That is one line of algebra and it decides the whole question, because a mean absolute deviation is famously stable as a fraction of a standard deviation. It is 0.798 for a normal, 0.866 for a uniform, 0.764 for a logistic, 0.707 for a Laplace, 0.736 for an exponential. Half of those is 0.399, 0.433, 0.382, 0.354, 0.368.

So the eighth rung swapped in the right formula for a second reason it did not know about. The censored answer is nearly independent of a distribution nobody has measured; the truncated answer is a strong function of it. A correction that has to be applied without knowing the parent’s shape should be the one that does not need it, and by luck rather than by argument that is the one this ladder now uses.

The relation between the two is exact as well and it explains the factor of two. The truncated mean is the censored one divided by the probability of clearing the wall, so the ratio is one over that probability — one half for any symmetric parent, and something else otherwise. For the right-skewed exponential, whose mass is mostly below its own mean, it is 2.718; for the reflected one, 1.582.

And the number it proposed as the test does not survive at all #

The skew was chosen as the discriminator because it separates the two rules by 0.646 on a normal parent — 1.641 censored against 0.995 truncated — against a difference in the spread of only four per cent.

Across parents it moves much further than that.

The overlap is not a matter of ranges nearly touching. There are specific pairs that are indistinguishable and one of them sits next to the eighth rung’s own numbers.

A censored sample from a parent skewed left by nine tenths has a skew of 1.007. A truncated sample from a normal has 0.995. They differ by 0.012, which no sample size resolves, and the eighth rung’s test would read the first as the second — concluding that the attempts below the floor were absent when in fact they were piled on it.

Nor is that pair contrived. A censored uniform gives 0.930, less than a truncated normal’s 0.995, so a parent with no tails at all and a wall that pushes half its mass onto one point produces less right skew than a smooth parent with the same mass simply removed. A censored normal gives 1.641 and a truncated logistic gives 1.540, a tenth of a unit apart. Every one of those confusions is between two entirely plausible descriptions of a drummer.

The sample size the test would need makes the point from the other side. The standard error of a sample skew is about the square root of six over the number of observations, so resolving 0.646 at three standard errors takes 130 notes — and that is the arithmetic for a parent whose shape is known. With the shape unknown the test is not underpowered; it is unidentified, and no number of notes fixes it.

Where the factor of two goes #

The eighth rung was careful to say that the factor of two is a point value, holding exactly where the parent’s mean sits on the wall and nowhere else. It gave the ratios at other wall positions on a normal parent: 1.41 a spread below, 2.58 half a spread above, 3.45 a spread above, 6.51 two spreads above.

Every one of those is also a function of the parent.

So a study that applied Pearson’s correction across a range of tempi was not merely doubling its correction and distorting its shape, as the eighth rung established. It was doing so by a factor that itself varies between two and five with a distributional shape the study did not measure and could not have.

The censored correction has no such sensitivity, which is the same finding as before arriving at the practical question. Half a mean absolute deviation is between 0.35 and 0.43 spreads whatever the parent, and a spread above the wall the censored bias is between 0.04 and 0.14 — small in absolute terms and small in its variation.

The statistic that needs no parent #

If the third moment cannot say which sample a recording is, something has to, because the two hypotheses make genuinely different predictions and one of them is the difference between a player’s timing being a limit and being a choice.

The answer is in the eighth rung’s own description of censoring and it is not a moment at all. A censored sample has an atom: a share of its observations at exactly one value, the floor. A truncated sample has none. Not a small one — none, for every parent, at every wall position, by construction.

Across the eight parents the atom holds between 0.368 and 0.632 of the sample when the wall is on the mean, and 0.12 to 0.21 when it is a spread below. Its size depends on the parent. Its existence does not, and the test is existence.

Thirteen notes, at a millisecond of timing resolution, decide a question this ladder has been arguing about for three rungs. That is a smaller demand than anything it has previously recorded, and the reason it is smaller is that a nonparametric test on a discrete signature costs far less than an estimate of a third moment.

The window matters and it matters in the direction that helps. A narrower window is a stronger test — thirteen notes at three milliseconds against forty-seven at ten — because the truncated sample’s share of a narrow window shrinks toward nothing while the censored sample’s does not. The only thing that forces a wide window is a soft wall, which is what the curve is about.

Why the atom is the thing a wall actually is #

It is worth saying why the count works when the moment does not, because the reason is not statistical cleverness.

A moment is a summary of a distribution, and a summary of a censored sample mixes two ingredients — the shape of the parent above the wall, and the mass the wall put at one point — into one number. Two different mixtures give the same number, which is what the overlaps above are. The atom is not a summary. It is the wall’s own signature, present when the wall exists and absent when it does not, and no shape can produce it because a continuous density puts zero probability on any single value.

That is also why the test is the one an experiment should have been running from the start. The milliseconds that make a groove are unnotatable but they are not unmeasurable, and a study that reports a mean and a spread has thrown away the only feature that distinguishes a limit from a preference — before any analysis, in the act of summarising. The whole account of swing as a falling ratio rests on summaries of exactly that kind.

What a soft wall costs, and it is the honest limit #

The eighth rung named the soft wall as its most damaging caveat, on the grounds that a real motor limit is unreliable rather than impossible, so the two hypotheses are the ends of a continuum rather than alternatives.

That is right and the arithmetic puts a number on it. A wall soft over a few milliseconds smears the atom into a narrow spike, and a spike is still a spike: up to about a third of the player’s own timing spread the test barely notices, and between a third and two thirds it degrades from fifty-eight notes to three hundred and sixty-six. At a softness equal to the timing spread the censored and truncated samples have literally the same share in any window, and the two hypotheses have stopped being different claims about the world.

That is the right place for a test to fail. A wall as soft as the noise around it is not a wall; it is a mild preference, and there is nothing there to detect.

Which of the eighth rung’s findings survive #

That rung revisited three published claims and this sweep revisits its revisiting.

“Swing timing becomes more consistent at fast tempi.” A claim about the spread, and the eighth rung found the two rules very nearly agree about it — 0.584 censored against 0.603 truncated, an understatement of about forty per cent under either. Across parents the censored spread runs from 0.359 to 0.775 of the parent’s, so the understatement is between 22 and 64 per cent rather than about 40. The direction is safe and the size is not. There is one instructive extreme: truncating an exponential at its own mean leaves an exponential, because the exponential has no memory, so its spread is not understated at all.

“The mean short-note duration is a stable constant.” A claim about the mean, and this is the one the half-mean-deviation theorem makes safe. Whatever the parent, the observed mean is biased up by between 0.35 and 0.43 spreads with the intended value on the wall, which for a fifteen-millisecond spread is between five and six and a half milliseconds. The eighth rung quoted six. It was right to within the whole range of plausible parents.

“The swing ratio falls smoothly with tempo.” A claim about the shape of a relationship, and it is where the exposure is. The bias enters at every tempo differently, the ratio between the two corrections varies with the wall’s position, and that variation is itself a function of the parent — between two and five at a wall one spread below the mean. So a corrected swing curve carries an unmeasured shape in it at every point, and there is no way to compute the correction without the histogram this ladder does not have.

Which computation produced the numbers #

Every parent is standardised to mean zero and unit spread by construction, and the standardisation is checked rather than assumed: each density is integrated with the wall pushed far out of the way, and the resulting mean and spread are asserted to be zero and one. A comparison of shapes that had quietly compared scales as well would look exactly the same.

The moments are quadrature over the parent with one rule about the mass below the wall — moved to the wall, or discarded — so the censored and truncated cases differ by one line and nothing else, which is the claim being made about them. The mean absolute deviation is computed from the same grid, so the half-deviation identity is a comparison of two numbers from one integration rather than a formula checked against itself.

The pile-up test’s shares are closed-form: the mass of the parent between the wall and the window’s edge, plus, for the censored case, the share of the piled mass that a blur of the stated width leaves inside the window. The sample size is a normal approximation to a one-sided binomial comparison at one per cent significance and nine tenths power.

Nothing here is fitted and nothing is drawn from a recording. The only empirical numbers are the ones the ladder already carries: a floor at a hundred milliseconds and a motor spread of fifteen.

What the picture cannot show #

Eight parents is not all parents. They span skews from −2 to +2 and tails from none to exponential, which covers what a timing distribution plausibly is, and a bimodal parent is not among them. A player alternating between two targets would produce one, and every number here would be wrong about it.

The wall is still a conjecture. The sixth rung established that the short note holds a roughly constant absolute duration close to the tempo window’s fast edge; it did not establish that the second causes the first. If the constant has some other cause, this is a correct analysis of the wrong picture, which was true of the two rungs before it as well.

A sample is not a distribution. The deviations in a performance are an offset and a pattern rather than draws from anything, and two players correct toward one another as they go. A pile-up test on a real recording would be counting notes that are not independent, which changes the thirteen and does not change the existence of the atom.

And a millisecond of resolution is an assumption about the analysis, not about the player. An onset detector has a resolution of its own, and a detector whose resolution the performance sets is a real object rather than a hypothetical one. If the extraction quantises at five milliseconds, the atom is smeared by five milliseconds before anybody looks at it, and the fifty-eight-note figure is the relevant one.

Whose playing, and when #

The hundred-millisecond figure comes from jazz drumming and the swing-ratio literature is overwhelmingly about the post-1940 American ride-cymbal pattern. Everything above inherits that.

What this sweep changes about the corpus question is its size. The eighth rung asked for a few hundred short notes with their durations, from two traditions, at three tempi, and called it the smallest thing the ladder had ever asked for. It is smaller than that. The distinguishing feature is a spike, not a moment, and a spike needs tens of observations rather than hundreds — provided the timings are reported at their full resolution rather than as a mean and a spread, which is the actual obstacle and has been all along.

Where this ladder goes next #

Nine rungs. Swing is a ratio and not two to one; the deviations that make a groove are unnotatable; they split into an offset and a pattern; two players correct toward each other; the deviations sit inside categories; the categories have tempo ranges and the swing constant may be the tempo window’s fast edge; three published statistics are biased if it is; the bias is half what was quoted because a floor censors rather than truncates; and the shape behind the wall changes the test far more than it changes the correction.

What is owed is the histogram, and after nine rungs it can finally be specified rather than wished for. Between thirteen and sixty short notes from one player at one fast tempo, with each duration reported to a millisecond, decides whether the hundred-millisecond constant is a floor or a target — and the statistic is the count of observations at the sample’s minimum, not any moment of it. That is a smaller ask than any this ladder has recorded and it is a different one: not a corpus, but a single published figure at full resolution. Every study whose data would answer it has been run; what has never been printed is the distribution rather than its summary.