A warp jams where its threads are thickest

The maximum sett of a yarn is one of the oldest computations in the trade. Threads cannot overlap, so the closest they can lie is when they touch; a diameter gives a jammed spacing, and the jammed spacing gives a sett. This collection has done it too, with a proper section geometry rather than the flat-disc version, and got numbers that behave sensibly against the setts weavers actually use.

Except that they behave sensibly by being much too high, and the gap is always explained away the same way — a cloth at its geometric jam cannot be woven, the shed will not open, the reed will not clear, so weavers set below the maximum by a margin that experience supplies.

Some of that margin is not experience. It is arithmetic, and it can be computed exactly.

The claim #

A jammed sett is an extreme value, so it depends on how many threads there are. A warp of a hundred ends of ordinary cotton jams thirty per cent above its mean diameter; one of ten thousand ends of the same yarn jams fifty per cent above it.

That is an uncomfortable statement, because a jammed sett has always been quoted as a property of the yarn. It is a property of the yarn and of how wide the cloth is, and the second is not a small correction: doubling a loom’s width moves the jam by three or four per cent, and the range from a ribbon to a broadloom moves it by twenty.

The argument #

Two ends touch when the sum of their radii reaches the spacing between them. The spacing is fixed — set by the reed and by the take-up, and not the same thing as the sett — so what has to fit is the sum of two neighbouring diameters, and what decides whether a cloth can be woven at a given sett is the largest such sum anywhere across the width.

That is an extreme of a moving sum, and three things follow immediately.

It grows with the width, because more pairs is more chances. The growth is logarithmic and therefore slow, which is the only reason the trade’s practice of quoting one figure per yarn works at all.

It falls with the length of the run, because a run averages its own members. The thickest single end in a warp of a thousand is 61 per cent above the mean; the thickest adjacent pair is 40 per cent above; the thickest run of four is 26 per cent above. A quantity decided by a longer run is a quantity decided by a partly averaged population, and the averaging takes the tail out.

And it is what the reed and the cloth each ask that differ. The cloth jams pairwise: two neighbours touch, or they do not. A reed dent holds two, three or four ends together and asks whether the whole group will pass through a fixed slot — so the reed’s constraint is a run of four, which is a full fourteen points gentler than the cloth’s. The reed is not the binding constraint on a variable yarn, which is the opposite of what a weaver watching threads chafe in the dents would guess.

Why the pair and not the thread #

The choice of run length is the one modelling decision in this essay, and it is worth defending because the answer moves by twenty points depending on it.

A single thread cannot jam. Jamming is a statement about two things not fitting into the room between them, so the smallest object that can jam is a pair, and the quantity that has to clear the spacing is the sum of two radii — one from each of two different threads. The thickest single end in a warp is a fact about that end; it becomes a fact about the cloth only when the end beside it is asked as well.

That is why the pair curve is the one to read for the cloth, and it is also why the pair curve sits well below the single-thread curve: a pair is already a small average, and the thickest end in a warp is very unlikely to have the second-thickest beside it. The extreme of a sum of two is not the sum of two extremes — an identity that does hold for a thickness under a presser foot, because there the two maxima are over different populations, the ends and the picks, and here they are over the same one.

Two objects with the same name and different arithmetic, which is exactly the trap this collection keeps recording: read the bodies, not the names.

What was counted, and how #

Warps of the stated width are drawn from a lognormal at the yarn’s coefficient of variation, every run of the stated length is summed, and the largest is kept — forty warps at each width, so that the number reported is an average of forty worst cases rather than one.

The simulation is checked against the order statistic that would apply if every window were an independent draw. It is not one: consecutive windows share all but one of their terms, so the effective number of tries is smaller than the number of windows and the independent estimate should sit above the simulation. It does, by 0.3 per cent — small enough to say plainly that the overlap between neighbouring windows is not what is going on here, and that the whole effect is the extreme rather than the correlation.

That check is worth its cost precisely because it came out uninteresting. The alternative was a paragraph of hedging about dependence in a moving sum, which would have been true, unquantified, and much larger in the reader’s mind than the third of a per cent it is worth.

What it does to the setts this collection has quoted #

Every jammed sett in this collection was computed from a mean diameter, so every one of them is a sett a warp of one thread could reach.

The correction is a multiplication and it is uncomfortably large. A cloth of two thousand ends at a fifteen per cent yarn cannot be set within forty-three per cent of the sett its mean thread allows, because somewhere across that width there is a pair that will not go. In practice a weaver stops before that, because a cloth right at its jam cannot be beaten up and the shed will not clear — but the arithmetic says that a substantial part of the traditional margin is not craft caution at all.

And it explains a fact about fine yarns that is otherwise odd. A combed, well-made yarn is more even as well as finer, and its lower coefficient of variation moves its jam closer to its geometric value. So two yarns of the same count but different evenness have different maximum setts — the even one can be set closer — even though every geometric quantity in a specification for them is identical. That is a real difference in what can be made, priced by a number that appears on the yarn’s delivery note and never reaches the cloth’s.

The weft has a far larger population than the warp #

There is an asymmetry hiding in this that no geometric account of jamming can contain, because it is about counting rather than about shape.

A warp has as many ends as the cloth is wide: two thousand for an ordinary shirting, ten thousand for a wide sheeting. A weft has as many picks as the piece is long — a fifty-metre piece at twenty-two picks per centimetre has a hundred and ten thousand of them, and every consecutive pair of them is another chance at a jam.

So the two directions are drawing their extremes from populations that differ by a factor of fifty, and the excess follows:

A square-set cloth is not symmetric in this respect at all. The two systems have the same yarn, the same mean diameter and the same nominal sett, and the weft’s worst pair is a sixth further above the mean than the warp’s, purely because there are more picks in a piece than ends across it.

What saves the weft is that its jam is not a hard stop. A pair of thick picks cannot refuse to be woven; they are simply beaten up against more resistance, so the beat-up takes them a little less far and the pick spacing opens locally. The warp has no such escape: an end that will not pass its neighbour has to break, or chafe, or stop the loom. The same arithmetic produces a fault in one direction and a variation in the other, and which of the two it is depends entirely on which system is free to move.

The evenness that buys a sett #

Because the jam sits at a fixed number of standard deviations above the mean, it moves in proportion to the spread — so evenness converts directly into how close a cloth can be set.

A combed yarn at eight per cent can be set within a fifth of its geometric jam; a coarse carded one at twenty cannot come within three fifths of it. Both yarns have the same mean diameter and the same count, so nothing that appears in a construction specification distinguishes them, and the difference between them is a third of the available sett range.

That is a large enough number to change what is worth buying. The premium on an even yarn is usually justified by appearance — fewer thin places, a cleaner surface, less barré — and appearance is a matter of taste and of light. This is not: it is the difference between a cloth that can be made and one that cannot, at a construction the buyer has already specified.

Why a sample loom lies, and by how much #

The width dependence is stated above as a curiosity about broadlooms. It has a much more common consequence, and it is one that every product-development department meets and attributes to something else.

The extreme of a large sample sits about √(2 ln n) standard deviations above the mean, and the standard deviation of a pair’s mean diameter is the yarn’s own coefficient of variation divided by √2. So the excess is

CV × √(ln n) to a good approximation,

which for a fifteen per cent yarn over two thousand pairs gives 41 per cent against the 43 the simulation returns — close enough to trust the scaling and not the third figure, the gap being the lognormal’s own skew, which a Gaussian estimate cannot carry.

Now put two widths through it. A thirty-centimetre sample blanket at the same sett has about 450 ends and jams 37 per cent above its mean diameter. The 1.5-metre production cloth has 2,250 and jams at 42.

Five points of sett, between a sample and the production run of the identical construction in the identical yarn.

That is a familiar and thoroughly annoying industrial fact: a construction proves out on the sample loom, goes to production, and chafes. The usual suspects are the loom, the beam, the sizing and the humidity, all of which are plausible and all of which get investigated. The arithmetic says a share of it is neither — it is that a wider warp is a larger sample of the same yarn, and a larger sample has a worse worst pair.

And the effect cannot be sampled away. Weaving a longer trial on the narrow loom does not help, because the warp’s population is set by its width and not by its length. The only honest trial of a warp’s jam is a trial at the production width.

What each doubling costs #

Because the excess goes as the square root of a logarithm, its increments are regular and shrinking. Differentiating, a doubling of the width adds

CV × ln 2 ÷ (2 √(ln n))

which at two thousand ends and a fifteen per cent yarn is about two points of sett margin, falling to one and a half at ten thousand.

So the penalty for width is real, bounded, and front-loaded: the step from a ribbon to a shirting costs more than the step from a shirting to a sheeting, and a mill that already weaves wide is not being punished much further for weaving wider. That is a comfortable shape, and it is the reason the trade’s habit of quoting one jammed sett per yarn survives at all — the quantity it ignores changes by a couple of points across the whole range of looms in use.

The same expression run the other way says what evenness is worth against width. Since both terms multiply, a yarn one point better in CV buys back about the same margin as halving the loom’s width, at ordinary values. A mill choosing between a finer-spun warp and a narrower cloth is choosing between two ways of buying the same number, and only one of them costs anything per metre.

And the weft’s is a running total #

The pick population grows with the piece rather than being fixed by the machine, which gives the weft’s version of the same statement a different and slightly unnerving shape.

A five-metre trial length carries 11,000 picks and its worst pair sits 46 per cent above the mean. The fifty-metre production piece carries 110,000 and reaches 51.

So the weft’s jam gets worse as the piece is woven, in the sense that the worst pair encountered so far keeps climbing — logarithmically, so slowly, and without limit. A trial length cannot demonstrate the absence of a fault that only the tenth of the run is long enough to contain, and the growth is not a matter of anything wearing out.

That is a genuine limit on what a trial can prove, and it is arithmetic rather than pessimism.

Where the model stops #

A thick place is not a whole thread. The variation used here is between ends, and a real yarn also varies along its own length, so the pair that jams does so at one place down the piece rather than everywhere. A cloth can be woven past its pairwise jam if the jam is momentary — the beat-up simply pushes a little harder for a few picks — and what that costs is a local disturbance in the pick spacing rather than a stoppage. Nothing here prices that, and it is the reason the number above is a bound rather than a prediction of where a loom stops.

The threads are treated as independent. Neighbouring ends come from different bobbins and are genuinely independent; but a yarn’s own thick places recur at the spinning frame’s period, so one end’s thick places are not independent of each other. That does not affect the pairwise jam, and it does affect how often it occurs down the piece.

The section is circular and the diameter is Peirce’s. A flattened thread jams at a different place, and a thick end under a hard beat-up flattens more than its neighbours, which relieves exactly the pair that was binding. So the extreme computed here is the extreme of an unyielding population, and a real warp buys some of the margin back the moment it is pressed.

And nothing here is about the loom’s other limits. A shed that will not clear, a reed that chafes, a warp that sheds fibre: those are the reasons a weaver gives for setting below the maximum, all of them real, and none of them touched by this argument. What is claimed is only that the arithmetic contributes a share of the same margin, and that its share is computable while theirs are not.

The generalisation #

A limit that has to hold everywhere is an extreme, and an extreme is a question about the size of the population. That is the general shape, and it turns up wherever a system fails at its worst point rather than at its average one: the narrowest place in a pipe, the weakest link in a chain, the shallowest patch of a channel, the thinnest section of a casting.

The diagnostic is the same one this ladder keeps returning to. Ask whether the number would change if twice as much of the same thing were included. A mean would not. A jam would, and does, and the arithmetic of how fast is the difference between a specification that transfers between looms and one that does not.

The second lesson is that averaging is a design variable. The reed jams over a run of four and the cloth over a run of two, and that single difference is worth fourteen points of margin. Wherever a constraint can be arranged to apply to a group rather than to an individual, the tail of the population stops deciding the answer — which is why a dented reed is a kinder machine than its geometry suggests, and why the fix for a marginal jam is sometimes to change the denting rather than the sett.

Who found it, and when #

The geometric jam is Peirce’s, from 1937, and everything since has refined the section rather than the argument. That real setts fall well short of it is universally known and universally attributed to weavability.

The extreme-value part appears not to have been written down for a warp, though the ingredients are old: yarn evenness has been measured since the 1940s, and the growth of a maximum with sample size is elementary. What makes it worth the arithmetic here is the direction of the surprise — not that the jam is lower than the geometry, which everybody knows, but that it depends on the width of the loom, which nothing in the geometry can express.

Where the ladder goes next #

The other two extremes in this ladder are the same argument with the maximum somewhere else: a thickness gauge reads the highest crossing under its foot, and at the other end of the distribution a bundle of threads breaks well below its mean thread.

Sideways, a warp set near its jam is a warp whose holes are nearly shut, and the holes at that point vary far more than the threads do — which is where most of a close cloth’s air comes from.