A designed thin place is kinder than an accidental one

Everything this collection has said about yarn irregularity has been about accidental variation. A floor set by counting fibres, an index measuring how far a process exceeds it, an averaging that beats the result down by the square root of the threads in view. All of it assumes the variation is something happening to the spinner rather than something the spinner is doing.

A slub yarn is spun with deliberate thick places — a controlled variation in the drafting, at an amplitude and a spacing the spinner chooses. It is a designed defect, and the design problem is the defect problem with the sign changed.

The number a mill quotes cannot tell them apart #

A spinner specifies evenness with a coefficient of variation, and the CV is a second moment: it is the standard deviation over the mean and it says nothing whatever about the shape of the distribution behind it.

So a slub yarn has a CV, and it is a large one. A base count with thick places two and a half times as heavy over eight per cent of its length has a standard deviation of 0.407 of its base and a mean of 1.12 of it, which is 36.3 per cent — three times the CV of an ordinary carded cotton and about what a very badly spun yarn would show.

The same number describes both yarns and they are not alike in any way that matters. That is the sort of collapse treating a cloth as a population is supposed to prevent, and here the population’s first two moments are the whole of what is recorded.

What separates them is a floor against a tail #

The difference is in the third and every subsequent moment, and it has a shape that can be stated in one sentence each way.

A slub yarn has a floor. It is spun by adding to a base, so its thinnest place is the base, and the base is a stated quantity with the spinner’s own tolerance on it. Test a metre or a kilometre and the thinnest place is the same: 89 per cent of the mean, for the slub above, and never anything below it.

A randomly uneven yarn has a tail. Its thin places are draws from a distribution and the worst of n draws falls without limit as n grows — slowly, as the order statistic requires, but without any floor to stop at. At the same 36 per cent CV a lognormal reaches 56 per cent of its mean over ten gauge lengths, 39 per cent over a hundred, 30 over a thousand and 24 over ten thousand.

So the advantage is not a number. It runs from 1.6 times at ten lengths to 3.7 at ten thousand, and it is a function of how much yarn is being asked about rather than a property of either yarn.

Which reverses the expectation #

The expectation a designer starts with is that a slub yarn must be weak, because it is deliberately uneven and unevenness is what breaks yarns. That expectation is right about the mechanism and wrong about the magnitude.

A yarn breaks at its thinnest place, and a bundle is weaker than its threads for the same reason: the weakest member decides. So the relevant quantity is the minimum, not the mean and not the CV — and the minimum is exactly where the slub’s designed structure is kindest.

A slub yarn of 36 per cent CV is stronger, at its worst place, than a badly spun yarn of 15 per cent CV tested over any reasonable length. The slub bottoms at 89 per cent of its mean; the 15 per cent yarn reaches 78 per cent over ten lengths and 55 over ten thousand.

That is not a small correction to an intuition, it is the opposite conclusion, and it explains something the trade already does: slub yarns are woven as warp, which nobody would do with a yarn whose CV was accidental.

And it says which slub specifications are dangerous #

The floor is base over mean, which is 1/(1 + f(r − 1)) for a slub r times the base over a fraction f of the length. That is a formula a designer can act on, and it points the wrong way from the eye.

The floor falls with the product f·(r − 1) and nothing else. A slub four times the base over eight per cent of the length gives a floor of 81 per cent; the same visual effect made with a slub twice the base over twenty-four per cent gives 81 per cent too. The two look completely different in the cloth and are identical in strength.

So the specification’s two knobs trade exactly against each other for strength and not at all for appearance, and a designer wanting a coarse effect at a safe strength should buy it with amplitude rather than with frequency — a few large slubs cost the same floor as many small ones and read as a much stronger texture.

The CV and the floor are separate knobs, and only one is specified #

The two-parameter form makes something visible that the single number hides completely.

Take two slubs with the same floor. Four times the base over eight per cent and twice the base over twenty-four per cent both give f(r − 1) = 0.24 and a floor of 80.6 per cent. Their coefficients of variation are 65.6 per cent and 34.4 per cent — nearly a factor of two apart, on two yarns that are exactly as strong at their worst place.

Now take two slubs with the same CV. Two and a half times the base over eight per cent gives 36.3 per cent; twice the base over twenty-four gives 34.4. Their floors are 89.3 and 80.6 per cent — a difference of nearly a tenth of the yarn’s count, on two yarns a mill’s records cannot distinguish.

So the number a spinner specifies is nearly orthogonal to the number that decides the outcome. That is a stronger statement than “the CV is a poor summary”: it is not a lossy summary of the floor, it is a summary of something else.

The practical form is a specification a designer could actually write. Quote the base count and the slub ratio, which fix the floor and the appearance between them, and let the CV fall where it falls. What is quoted instead is the resultant count and the CV, from which neither the floor nor the appearance can be recovered.

Why the trade randomises the spacing anyway #

There is a practice that appears to contradict all of this, and it does not: slub yarns are almost always specified with a random or variable slub distribution, and a spinner who delivered them at a regular spacing would be sent them back.

That is not about strength. It is about the cloth, and the other half of this ladder’s neighbourhood has the argument. A periodic variation is not averaged out by the cloth the way a random one is: it beats a random error of the same size by a factor of several at its own frequency, and a slub period that happens to divide the cloth’s width produces stripes down the piece from a fault that is entirely in the weft.

So the spinner randomises the spacing to give up the visibility the periodicity would have bought. The amplitude — which is what sets the floor — stays exactly where it was, and the distribution of positions is what is scrambled.

Which is a nice separation of concerns and worth naming. The amplitude decides the strength and the spacing decides the pattern, they are independent knobs, and the trade turns them in opposite directions on purpose: amplitude regular and large, spacing irregular.

What the cloth does with it, which is a second extreme #

A yarn’s own minimum is what breaks a yarn on a tester. A cloth breaks somewhere else and the arithmetic goes round again.

A woven cloth pulled warpwise fails when the ends in the breaking zone fail, and they do not fail together: a bundle is weaker than its threads because the weakest goes first and its load transfers to its neighbours. So the quantity deciding a cloth’s strength is an extreme over the ends in the zone, and each of those ends is itself a length of yarn with its own thin places.

That is two extremes stacked, and the slub’s floor survives both. Every end of a slub warp bottoms at the same base count, so the extreme over ends is the same number as the extreme over one end’s length: there is nothing for the second minimisation to find. A randomly uneven warp has a fresh tail at every end and the two extremes compound — the weakest place in a hundred ends of a thousand lengths each is the minimum of a hundred thousand draws, not a thousand.

Reading the curve at 10⁵ rather than 10⁴ costs the random yarn another few percentage points and costs the slub nothing at all, which is the same finding one order of magnitude further along. A floor is a floor at every scale and a tail is worse at every scale.

That is the reason a slub warp weaves, and it is worth setting against the intuition once more: the reason is not that a slub yarn is even. It is that its unevenness is bounded, and boundedness is the property the whole extreme-value apparatus is sensitive to.

What was counted, and how #

The random half is spread.js’s own order statistic, unchanged. expectedExtreme integrates the density of the minimum of n draws against the lognormal this site uses for every yarn population, and it has its own check against a simulation elsewhere in that file. Nothing about the random yarn is computed here.

The slub’s own numbers are two-point arithmetic and are exact. A yarn that is base or base times r, with duty f, has mean base(1 + f(r − 1)) and variance f(1 − f)(base(r − 1))², which is the two-point variance and needs no distribution behind it.

The comparison is at equal CV by construction. The slub’s CV is computed from its own specification and then handed to the lognormal, so the two yarns being compared are the two yarns a mill could not tell apart from its own records. Comparing at equal mean rather than at equal CV would have been comparing two different questions.

The finding is asserted as a direction and as a trend, over three slub amplitudes rather than at the one the figures draw. The direction is that the designed floor is above the random minimum at every length; the trend is that the advantage grows with the length, which is the half that says the comparison has no single number in it.

And the guards are fed what they must refuse: a slub with no base, a slub no thicker than its yarn, a slub thinner than it, and a slub occupying none or all of the length.

Why the intuition is wrong, which is worth having on its own #

The expectation that a slub yarn must be weak is not a careless one. It is an inference from a real rule, applied to a distribution the rule was not stated for, and the failure has a shape worth recognising because this collection makes the same move often.

The rule is that a yarn’s strength falls as its irregularity rises, and it is true. Every evenness test in the trade rests on it and so does every quality specification.

The reason it fails here is that the rule is a statement about a family of distributions — the ones a spinning process produces, which differ from each other in scale and not much in shape. Within that family the CV really does determine the minimum, because the shape is fixed and the minimum is a fixed multiple of the standard deviation. A slub yarn is not in the family. Its distribution is two spikes rather than a spread, and the mapping from CV to minimum that holds across the family does not hold for it.

That is a general hazard rather than a fact about yarn. A summary statistic that is sufficient within a family is used as though it were sufficient generally, and the case that breaks it is always the one deliberately constructed to sit outside — which is what “designed” means. The coefficient of variation is the right instrument for measuring a process and the wrong one for describing a product.

The remedy is not a better statistic. It is knowing whether the object in hand is a sample from a process or a thing somebody made on purpose, and those are recorded in different places or in no place at all.

Where the model stops #

Linear density is not strength. Everything above compares thin places by their count, and a yarn’s breaking load at a thin place is not proportional to the linear density there — the fibre count falls, the twist redistributes, and the migration pattern changes. The conversion needs a fibre model this comparison does not use, so the ratios here are ratios of thinness and not of strength.

The slub is treated as two states. A real slub has shoulders: the drafting ramps into it and out of it, so the yarn passes through every count between the base and the thick place. That makes the two-point variance an overestimate of the CV and moves the comparison slightly, in the direction of making the slub look more even than modelled.

The random yarn’s draws are independent and a real yarn’s are not. Thickness is correlated over a fibre length and over a drafting-roller circumference, so a real yarn has fewer independent thin places than gauge lengths — which makes its minimum less bad than the order statistic says, again in the direction of narrowing the gap. Both corrections point the same way and neither closes a factor of three.

And nothing here is about the cloth. A yarn’s minimum is what breaks a yarn; what breaks a cloth is the weakest end in the breaking zone, which is a second extreme over the ends and is the subject the bundle ladder owns.

Who found it, and when #

Slub yarns are old and the spinning of them is a solved industrial problem: the mechanisms are catalogued, the specifications are standard, and every spinner knows that the slub distribution must be randomised.

The comparison with an accidental variation at equal CV does not appear to be made anywhere. The reason is probably departmental: evenness testing is a quality function and fancy yarn is a design function, and the CV of a slub yarn is a number nobody has any use for — it is measured, if at all, to be ignored.

What this collection adds is that the number is the same for both and that it is the wrong number for both. A specification that records a second moment about a distribution with a floor has recorded almost nothing, and the quantity that decides the outcome — the minimum, and whether it moves with the length tested — is not in the record at all.

Where the ladder goes next #

A slub changes the thread’s thickness and leaves it a thread. The other family of fancy yarns changes its length: a bouclé, a gimp and a loop yarn are all made by feeding an effect thread faster than the core it is wrapped on, so the yarn contains more thread than it is long — which is crimp, under a different name, and it gives a fancy yarn the same two-regime extension a woven cloth has.