How close can threads be set

Ask how many threads go into an inch of cloth and there are two answers. One is a choice, and the other is a ceiling that no amount of choosing can exceed.

The ceiling is geometric. At some spacing the threads are touching, and there is nowhere for another one to go.

What jamming means #

Take a woven cloth and push the picks closer together. At first they move. Then the warp, which has to bend round each pick, runs out of room to make its bends in, and the cloth resists absolutely rather than stiffly. That state is jamming, and it is where every geometric setting theory tries to locate the maximum.

Two things are jammed at once and they are worth separating. Warp jamming is the ends touching each other; weft jamming is the picks touching. A cloth can be jammed in one direction and open in the other, which is exactly what a warp-faced cloth like denim is.

The condition is that the space taken by the threads themselves plus the space their crossings demand fills the available width. Both terms depend on yarn diameter, and the second depends on how often the thread changes face — which is why the weave enters at all.

Why an interlacing costs room #

The mechanism was set out in the essay on firmness and it bears restating in one sentence: a thread that changes face has to travel from one side of the crossing plane to the other, and that travel needs horizontal distance.

Crowd the threads and the distance available for each transition shrinks. Eventually the thread is running vertically between crossings, the bend is as sharp as the geometry permits, and the neighbouring threads are in contact. Nothing more will go in.

A weave with fewer interlacings has fewer transitions to accommodate, so more threads fit. That is the whole of the weave’s contribution, and it is a large contribution.

Ashenhurst, and what a Victorian rule is for #

The first usable version of this calculation is Thomas Ashenhurst’s, from Bradford in the 1880s, and it is still quoted in mill practice.

Ashenhurst’s insight was to count the crossings. The width a repeat must occupy is the threads in it, each taking one diameter, plus one further diameter for each place a crossing thread has to pass through. Divide the number of threads in the repeat by that width and the maximum sett falls out.

The result is crude. It treats yarn as an incompressible circle, ignores that a gripped thread flattens, and takes no account of how the crossings are arranged rather than merely how many there are. It gives maxima below what a mill can actually weave.

What makes it good engineering rather than bad physics is that it is wrong by a consistent margin in a known direction for a comprehensible reason. A practitioner applies a correction factor and gets a usable number for a cloth nobody has made — which is exactly what a mill needs and what experience alone cannot supply.

Peirce, and what a careful model buys #

Frederick Peirce’s 1937 geometry is the serious treatment, and it changed the subject from rules to relations.

Peirce modelled the thread path properly: circular yarn cross-sections still, but the path taken as circular arcs where a thread wraps a crossing, joined by straight lines where it does not. From that he derived a set of relations connecting thread spacing, yarn diameter, crimp, weave angle and cloth thickness — so that given any few of them the rest follow.

The jammed state falls out as a limiting case rather than as a separate rule, which is the mark of a model that is doing real work. And because the relations are exact within the model, they can be inverted: a mill can ask what sett gives a required cover, or what crimp a required sett implies.

The circular assumption is where it is vulnerable, and everybody has known that since it was published. A yarn in cloth is squashed at the crossings and rounder along the floats, so its effective width varies along its own length.

The two directions are set separately #

A cloth has two setts and they are chosen independently, which the single word “sett” obscures.

Ends per inch is fixed at warping and threading: the warp is wound, threaded through the heddles and the reed, and its spacing is settled before a single pick is thrown. Changing it means rebuilding the warp.

Picks per inch is fixed during weaving, by how hard each pick is beaten up against the cloth already made. It can be changed on the fly, and it is the variable a weaver actually controls.

That asymmetry has consequences. A mill running one warp can produce several cloths from it by varying the picks — heavier and lighter versions of the same fabric — and it is a cheap way to extend a range. It is also why warp-dense cloths are common and weft-dense ones less so: crowding the warp is a decision made once, and crowding the weft costs loom time on every pick.

The two constraints also interact. Crowding the warp leaves less room for the weft to be beaten in, so a very warp-dense cloth has a lower maximum in the weft than the same weave loosely warped — which is why the jamming condition has to be evaluated for the cloth rather than for each direction alone.

Racetrack, ellipse, and the ten per cent #

Two later families relax the circle in different ways.

The racetrack section is a rectangle with semicircular ends: a yarn free to flatten to a chosen ratio and no further. The elliptical section is what it says. Both allow a gripped thread to spread, which means it takes less vertical room and more horizontal room, which means the cloth jams at a higher thread count than Peirce predicts.

The three models disagree by ten or twenty per cent on the same cloth, and the disagreement is not resolvable by being cleverer — it is a disagreement about which idealisation to make. Real yarn flattens by an amount that depends on its twist, its fibre, the tension it was woven under and how the cloth was finished, and none of that is in any of the models.

So the honest position is the one this site takes everywhere: the model that produced a number is named. The ordering across weaves is robust and every model agrees on it; the absolute values are not, and quoting one without its provenance is how two internally consistent sources come to disagree permanently.

Nobody weaves at the maximum #

The ceiling is a reference point rather than a target, and cloths woven at it are usually bad.

A jammed cloth has no room for the threads to move. It cannot shear, so it has no bias and drapes badly. It is stiff. It uses a great deal of yarn. And it is hard to weave, because the reed has to force each pick into a space that barely accepts it, which strains the warp and slows the loom.

So setts are quoted as a percentage of the maximum: perhaps sixty to eighty per cent for clothing, higher for a cloth that must be windproof, lower for something meant to fall in folds. That percentage is comparable across weaves and yarns in a way a raw thread count is not, which is one more reason to prefer it to the number on the packet.

Where a cloth actually sits #

Real cloths occupy a fairly narrow band below the ceiling, and knowing the band is more useful than knowing the ceiling.

A shirting runs at perhaps sixty to seventy per cent of maximum: firm enough to hold its shape, open enough to drape and to breathe. A windproof or waterproof cloth runs at ninety per cent or above, which is where the air permeability falls off sharply. A voile or a georgette runs at thirty or forty, and is deliberately open.

The band is narrow because both ends are bad. Above about ninety per cent the cloth is stiff, expensive in yarn and hard to weave; below about thirty it is not really a fabric. Almost the entire useful world of woven cloth lives in a factor of three, and where a fabric sits in it predicts more about how it will behave than its fibre does.

That is the fourth or fifth time on this site that a single geometric ratio has turned out to organise a whole family of properties, and it is the same trade every time: compliance against firmness, bought with space.

What the ceiling is used for #

Three practical jobs, and they are the reason a Victorian rule survives.

Designing a cloth that does not exist yet. Given a yarn and an intended weave, the maximum sett says whether the specification is possible before anything is warped. A specification above the ceiling is not ambitious; it is impossible, and finding that out on the loom is expensive.

Comparing across weaves. A percentage of maximum is comparable where a raw count is not. Two cloths at seventy per cent of their respective maxima are similarly set, whatever their thread counts say.

Predicting behaviour. Where a cloth sits relative to its ceiling predicts its handle, its drape, its air permeability and how much bias it has, and it does so across yarns and weaves. A cloth at ninety-five per cent is stiff and windproof whatever it is made of; one at fifty is soft and open.

The general shape is familiar from any engineering constraint. The ceiling is rarely the design point, and knowing where it is turns a set of intuitions into a design space that can be reasoned about. That was Ashenhurst’s contribution, and it is why a crude model that is honest about being crude has outlived several more accurate ones.

The two limits that are not this one #

Two other ceilings get confused with jamming and are different.

Cover reaching one hundred per cent means the threads leave no visible gap. That happens before jamming in most weaves, because a cloth can be optically solid while the threads still have room to move — which is why a cloth can look dense and still drape.

Locking is the shear limit rather than the setting limit. It is the angle at which threads jam against their parallel neighbours when the cloth is sheared, and it is a different geometry with a different formula. Its own essay takes it up, and the two are related — a densely set cloth locks at a smaller shear — without being the same quantity.

Where the model stops #

Three limits, and the first is the largest.

It is geometry, not mechanics. The calculation says when threads touch. It says nothing about the force required to get them there, and a cloth can be beaten up harder at the cost of straining the warp. Practical maxima are a matter of what the loom and the yarn will tolerate as much as of what the geometry permits.

Yarn is compressible. Real threads flatten and their fibres migrate, so a jammed cloth is not a rigid packing of circles. Every model that takes this seriously gets a different answer.

Finishing moves the sett. Cloth relaxes, shrinks and is often deliberately compacted after weaving, so the sett in the loom and the sett in the hand are different numbers — sometimes by ten per cent, occasionally by more. A quoted sett should say which.

Why the ceiling is a weave property at all #

It is worth closing on the thing that makes this calculation interesting rather than merely useful.

A naive account would say the maximum thread count is a property of the yarn: threads have a diameter, an inch is an inch, and the answer is one divided by the other. That would give the same maximum for every weave, and it is wrong by a factor approaching two.

The extra factor is entirely structural. It comes from the crossings, and its size is set by how many of them there are — which is the interlacing count, the same number behind firmness, crimp, drape and lustre.

So the sett ceiling is not a yarn fact with a weave correction. It is a weave fact scaled by the yarn, and that ordering matters: change the yarn and every weave’s ceiling moves together, but change the weave and the ceilings move relative to one another. The structure decides the shape of the answer and the yarn decides its size, which is the site’s organising claim turning up once more.

What the trade rule is being compared against #

Ashenhurst’s rule is a bookkeeping argument rather than a geometry, and it is worth knowing how it compares with the geometries that exist.

Peirce’s circular thread model, solved at the jam, gives a spacing of 1.73 diameters for a square plain weave. Ashenhurst’s rule gives 2. So the trade rule is about thirteen per cent more conservative, and in the right direction: a real cloth cannot be woven at the geometric jam, because at the jam the threads have no straight portion left and the loom cannot beat another pick in.

That is the honest way to compare them. One is a weaving limit with an allowance in it and the other a geometric limit with none, and the ratio between them measures how much of a cloth’s construction is machine rather than shape. Peirce against the racetrack runs the geometries side by side and finds they agree closely on the sett and not at all on the thickness.

The ceiling has a closed form and an asymptote at two #

Ashenhurst’s bookkeeping is described above as counting crossings, and the count can be written down. A repeat of n threads needs n diameters for the threads themselves and one more for each place a crossing thread passes through, which is i — the interlacings per thread per repeat. So the repeat occupies n + i diameters and

maximum sett ∝ n ÷ (n + i).

For a plain weave n = 2 and i = 2, giving a half — a spacing of two diameters, which is the number quoted above. For every satin i is two whatever the repeat, so the expression becomes n/(n + 2) and the whole family lies on one curve.

The eight-end satin’s 1.60 is the essay’s own “sixty per cent more threads”, recovered from the formula rather than quoted.

And the last row is the interesting one. No weave can ever double a plain weave’s sett, and the bound is approached and never reached. The reason is immediate once written: a weave with no interlacings at all is a sheet of parallel threads lying at one diameter apart, and a plain weave sits at two, so two is the whole of what structure can buy. Every weave anybody has ever made lies between one and two on this axis, and the commonest ones lie in the first half of it.

Which settles a disagreement between two trade rules #

There is a second rule for the same quantity, and this collection uses it elsewhere: Brierley’s, fitted on cotton, which makes the ceiling proportional to (n/2) raised to the power 0.39. It is an empirical curve where Ashenhurst’s is a count, and the two have no business agreeing.

They agree remarkably well, and then they stop.

Within a tenth of a per cent at five ends, and seventeen per cent apart at twelve. Two independent rules, one counted and one fitted, tracking each other across the whole range of weaves anybody actually uses and diverging exactly where nobody weaves.

That is not a coincidence to be admired and left. It says which of the two to trust outside that range, and the answer is unambiguous. A power law has no ceiling and the physical quantity has one. Brierley’s curve passes two at a repeat of eleven and keeps climbing — at a repeat of a hundred it claims four and a half times a plain weave’s sett, which is a spacing of 0.43 diameters and a cloth whose threads overlap. Ashenhurst’s approaches two and stops, because the count it is made of cannot do anything else.

So the fitted rule is excellent where it was fitted and must not be extrapolated, and the crude Victorian count is the one that is right at the limit. The rule that was derived beats the rule that was measured, precisely once and precisely where no measurements were taken — which is the ordinary division of labour between the two and is worth stating, because a designer reaching for a long-satin construction is reaching into exactly that region.

Where the ladder goes next #

The other geometric ceiling is the locking angle, which limits shear rather than setting and matters wherever cloth has to go round a curve.

The measurement this one is confused with is thread count, and the structural variable behind both is the interlacing count.

What the pictures here cannot show. Every setting figure on this page assumes a circular yarn of one fixed diameter, which is Peirce’s assumption and not a fact about thread. Flattening, twist, hairiness and the tension history of the cloth are all absent, and they are between them the reason a mill’s numbers differ from a geometric model’s.