The construction a loom must be set to

A cloth specification is two numbers and two counts: so many ends and so many picks per centimetre, in this yarn and that. Everybody in the chain reads them as a description of the finished fabric, because that is the fabric anybody cares about, and everybody who has to make the fabric knows they are not the numbers the loom gets set to.

This collection has two rungs on that already. The reed’s dents are not the cloth’s ends per centimetre, because the cloth contracts in width as it is woven. The picks per centimetre are not what the take-up motion is driven at, because the cloth contracts in length too. Both are contraction while weaving, under tension, and both are old.

There is a third change and it happens after the cloth has left the loom entirely.

The claim #

A cloth off the loom is not at the least-energy state of its own locus, and the state it relaxes to is at a different sett in both directions.

Two consequences follow and the second is the sharper one.

The first is arithmetic: the finished setts differ from the loom’s by a computable amount, and the two differ in opposite senses. A cloth that shortens has more picks per centimetre and a cloth that widens has fewer ends per centimetre, always, because those are the same statement about one curve.

The second is about what a specification can ask for. The locus is one-dimensional, so the set of states a given pair of thread lengths can reach is a curve, and only one of the two finished setts is free. A specification that quotes both has over-specified the cloth, unless the pair it quotes happens to lie on the curve — and there is no reason for a pair chosen by a customer to lie on any curve.

What relaxes, and why it is not take-up #

Take-up is a contraction under tension while the cloth is being made. The warp is held straight on the beam, the picks are driven in around it, and the cloth that winds onto the roll is shorter than the warp that went in. That is a geometric fact about crimp being put in, and it happens at the fell.

What this rung is about happens later and is a different thing: the cloth, once it is off the loom and once the finishing has let it move, slides along its own locus to the state that minimises its bending energy. That state is not the one the loom left it at, because the loom left it at whatever the beam tension and the take-up between them chose, and neither of those was aiming at a minimum.

This collection established where that minimum is when it gave the locus a force, and recorded a warning at the same time which applies directly here: the minimum is a state at a different sett from the quoted construction. That warning was written to stop a crimp ratio being drawn onto a figure of a quoted cloth. Here it is not a warning, it is the answer — because the change of sett is the relaxation, and it is exactly what is being computed.

The numbers, and the outlier #

For seven of the eight cloths in this collection’s table the shifts are a little over one per cent: the warp sett falls between 0.84 and 1.38 per cent, the weft sett rises between 0.85 and 2.08, and the cloth ends up between one and two per cent shorter and about as much wider.

The poplin is different by an order of magnitude. Its warp sett falls 6.79 per cent and its weft sett rises 10.58; the cloth shortens by 9.57 per cent and widens by 7.28.

That is not an error and it is not a coincidence of which row it is. The poplin is the one construction in the table that is unbalanced in both respects: thirty-two ends against twenty-two picks, and 15 tex warp against 20 tex weft. A cloth whose two systems are that different has its quoted state a long way from its own least-energy state, because equal sharing of the crimp — which is what the quoted state assumes — is a long way from what the two systems would agree on if left alone.

The distance from the quoted state to the relaxed one is a measure of how much the quoted state is an assumption. For a balanced cloth the assumption is nearly right and the shift is small. For an unbalanced one it is not, and the shift is what that costs.

Why both setts cannot be chosen #

Here is the part a designer meets and a specification does not admit.

Two thread lengths fix a locus. Along it, one dimension rises exactly as the other falls, so the states reachable at constant thread length are a curve in the plane of ends and picks per centimetre, not a region. Once the yarn is chosen and the loom has put a certain amount of thread into a certain amount of cloth, the finished construction is a point on that curve.

A customer who asks for 30 ends and 25 picks per centimetre in 20 tex cotton has named a point in a plane. Whether it lies on any curve a loom can produce is a question with an answer, and the answer is usually no.

What a mill actually does in that situation is change something the locus does not contain: the finishing route, the tension it is held at while it dries, whether it is stentered wide or allowed to relax. Every one of those is a way of not letting the cloth reach its own minimum, which is to say of leaving residual excess in it — and this collection knows what residual excess does. It comes out in the wash, a few per cent at a time, over five cycles and then more slowly.

So a specification with two numbers in it is a specification with a shrinkage in it, whether or not it says so.

What the two contractions have in common, and what they do not #

It is worth putting the three changes side by side, because they are habitually run together under one word and they are three different mechanisms.

Width contraction at the loom happens because the warp, held straight in the reed, has to start bending around the picks the moment it leaves the fell. The cloth narrows and the ends per centimetre rise. It is under tension, it is immediate, and it is entirely reversible in the sense that nothing has been dissipated — the cloth has simply taken up its geometry.

Take-up is the same thing in the other direction: the warp thread that goes in is longer than the cloth that comes out, by its crimp, and the ratio is what the take-up motion is set against. Again immediate, again under tension.

Relaxation is neither. It happens after the tension is off, it is a slide along the locus toward an energy minimum, and it is opposed by friction the whole way. That is why it is progressive, why it is not complete, and why it is the one of the three that continues to happen in a customer’s washing machine.

The three are usually collapsed into “the cloth shrinks”, and collapsing them loses the property that matters most: the first two are done when the cloth leaves the loom and the third is not done at all. A mill that has allowed correctly for take-up and width contraction has a cloth of the right dimensions on the roll and a cloth of the wrong dimensions after five washes, and no amount of care about the first two addresses it.

The sett shifts computed here belong wholly to the third. They are what is left after the loom has finished, and they are the part a specification has no vocabulary for.

What was counted, and how #

The relaxed state is the least-energy state of the locus at the site’s own bending rigidities, which this collection computes from the yarn counts and a stated packing factor. The sett shift is asserted to be the same statement as the sett ratio, computed two ways, because a shift and a ratio differ by a sign convention and a sign convention is the sort of thing that quietly inverts a table.

The direction is asserted for every cloth: the warp sett must fall and the weft sett must rise, without exception, because it is a statement about the shape of the locus rather than about any particular fabric. A row that broke it would mean the energy minimisation had walked the wrong way.

The rigidities are a bracket rather than a value, and the relaxed state is computed at the free bound. Moving up the bracket moves the minimum, and the direction of every shift survives, because the shape of the locus does not depend on the stiffness at all — only on where along it the energy is least.

Where the model stops #

The finishing route is not modelled. Every real cloth is washed, dried, stentered, calendered or pressed on its way from the loom to the roll, and most of those hold it at a length while something else happens to it. What is computed here is where the cloth would go if it were left entirely alone, which is the endpoint of every finishing route and the destination of none of them.

The relaxation is treated as complete. A cloth reaches its minimum only if friction lets it, and friction does not: it comes to rest somewhere in a band, several per cent wide, and which point in the band depends on how much it was agitated. So the finished setts here are the centre of an interval rather than a value, and the interval is wider than the shifts themselves for the seven balanced cloths.

And the map is computed forwards only. Given a loom construction this rung says what comes back. The inverse — given a wanted finished construction, what should the loom be set to — is a two-dimensional solve this collection has not built, and it is the calculation a mill actually needs. What can be said without it is that the inverse does not always exist, and that is not a limitation of the solver.

The inverse exists, and it is what over-specifies the cloth #

The limits section says the inverse — given a wanted finished construction, what should the loom be set to — has not been built, and that it does not always exist. The second half needs care, because the way it fails is more useful than the failure.

For a fixed pair of thread lengths the reachable states are a curve, and a point off that curve is unreachable. But a mill choosing a construction chooses the thread lengths too: how much warp goes on the beam and how many picks go in are exactly what the take-up and the let-off set. Two thread lengths, two finished setts — a map from a plane to a plane, and generically invertible.

So the inverse exists. What it costs is the crimps.

At the least-energy state the crimp division is not free: balancing the two systems’ forces gives a crimp ratio of (B₂/B₁)(p₂/p₁)³, with B the bending rigidities and p the two spacings — the balance this collection derives from its own force machinery. Combined with the closure condition, that fixes both crimp heights; the crimps then fix the thread lengths; and the thread lengths fix the loom.

So four numbers a customer already quotes — two counts and two setts — determine the finished state completely, with nothing left for the mill to choose. A specification of two setts is not under-determined and is not over-determined. It is exact, and it has silently specified the crimps as well.

Which gives an existence test in one line #

That makes the failure checkable before anything is warped, because the implied crimp ratio has to be one the geometry admits.

Compute (B₂/B₁)(p₂/p₁)³ and see whether it falls inside the interval of crimp ratios the construction can hold.

Run it on the poplin, which is the table’s outlier. Bending rigidity goes as the fourth power of a diameter and so as the square of the count, giving B₂/B₁ = (20/15)² = 1.78. The spacing ratio is 32/22, cubed is 3.08. The product is

5.5 — the warp asked to take five and a half times the weft’s crimp height, which is eighty-five per cent of the cloth’s whole thickness.

The interval this collection computes for a construction of that closeness stops well short of that; a crimp ratio much above one and three quarters is simply not a state Peirce’s geometry admits. So the poplin’s quoted construction is not an attainable relaxed state at all, and the cloth’s ten per cent slide is the geometry going as far towards it as it can and stopping.

That is a much sharper account of the outlier than the only construction unbalanced in both respects. It is unbalanced in both respects in the same direction, so the two contributions to the crimp ratio multiply rather than cancelling — a square in the counts times a cube in the setts, both above one — and the product leaves the admissible range.

And it says which specifications to refuse #

The test inverts into a rule a mill could apply at the quotation stage.

A construction whose count ratio and sett ratio pull the same way is the dangerous one. A fine warp set densely against a coarse weft set openly multiplies a square by a cube and reaches five or six; the same two yarns with the setts the other way round gives 1.78 ÷ 3.08 = 0.58, comfortably inside.

So the same two yarns can be woven to a perfectly attainable construction or an unattainable one depending only on which system is set closer, and nothing in the specification says which. That is the practical form of everything above: a customer naming four numbers has named a crimp ratio, and whether the cloth exists is a question about that ratio rather than about any of the four.

What it means for a sett to be quoted at all #

If the finished sett is a point on a curve and the loom’s is another point on the same curve, then a sett quoted with no state attached is ambiguous by the width of that curve — which for a balanced cloth is a per cent or two and for the poplin is ten.

A per cent or two sounds small until it is compared with what a sett is used for. This collection computes a cover factor from a sett, a jamming limit from a sett, an interchange budget from a cover, and a weight per square metre from both. A two per cent error in a sett is a two per cent error in a cover, and near the jamming limit a two per cent error in a cover is the difference between a cloth that can be woven and one that cannot.

The poplin makes the point without needing an argument. Quoted at thirty-two ends and twenty-two picks, it relaxes to 29.8 and 24.3. Those are not the same cloth by any measure this collection uses: the warp cover falls from 0.463 to 0.431, the weft cover rises from 0.368 to 0.406, and the interchange budget — which is a function of cover — moves with them.

So the correct discipline is the one this collection has been applying to crimp since its second field, extended to one more quantity. Say which state. A sett in the reed, a sett on the roll and a sett after five washes are three numbers, and the difference between the first and the last is larger than most of the tolerances anybody specifies.

The generalisation #

A specification that names more coordinates than the system has degrees of freedom is not a tight specification, it is an inconsistent one — and the way it fails is by being met approximately, with the residual showing up somewhere else. Here the residual shows up as shrinkage in use, which is a place nobody thinks to look for the consequences of a sett having been over-specified.

The pattern recurs wherever a process is asked for two outputs that a conservation law ties together: a rolling mill’s thickness and width, a drawing operation’s diameter and wall, an extrusion’s two cross-sectional dimensions. In each case the naive specification names both, the process meets one and approximates the other, and the discrepancy is absorbed by a variable nobody wrote down.

The narrower lesson is about which state a number refers to. Every dimension of a fabric is a dimension in a state, and the state is part of the number. This collection has said that before about crimp, cover and thickness. The sett is the last of the four to acquire it, and it is the one every specification quotes.

Who found it, and when #

Take-up and width contraction at the loom are as old as weaving and are calculated routinely; every weaver sets a reed wider than the cloth wanted. Relaxation shrinkage in finishing is equally standard and is the subject of every pre-shrinking process.

Peirce’s geometry is from 1937 and the least-energy state of a constant-thread-length locus is a modern reading of it that this collection built for its own purposes.

What appears to be this collection’s own is the composition: treating the loom-to-finished change as a move along the locus, computing it for a table of cloths, and drawing the consequence that the finished construction is a point on a curve rather than a pair of free numbers. The last of those is the useful half and it is not, so far as this collection can tell, how a specification is usually understood.

Where the ladder goes next #

The two setts moving in opposite directions is a change of aspect as well as a change of density, and a pattern drawn on point paper is drawn in that aspect — so a motif has to be drawn at a shape the loom will not give it.

Sideways, the residual excess this rung leaves behind is what a wash takes out, and it comes out in a series whose shape measures something about the cloth. And the budget the finished cloth has depends on which construction it ended up at, which is a quantity with a maximum in it.