A motif is drawn at the wrong shape on purpose

A jacquard design is drawn on point paper: one cell per warp end, one row per pick, and a shape built out of cells. This collection has already established what that costs at the boundary — an outline is a staircase, its available directions are quantised, and a diagonal at forty-five degrees on the paper is not a diagonal at forty-five degrees on the cloth unless the two setts happen to be equal.

That last qualification is where this rung starts. The proportions of a motif are set by the ratio of ends per centimetre to picks per centimetre, so a design drawn at one ratio and woven into a cloth at another is a design that has been stretched.

Which raises a question the outline ladder did not ask. Which cloth’s setts? The cloth on the loom and the cloth a customer holds have different ones, in opposite directions, and the design was drawn before either existed.

The claim #

A design’s aspect ratio is a property of the finished cloth, and the finished cloth’s setts are not the ones the loom was set to.

The size of the correction is a property of the construction rather than of the design, which has two consequences worth separating.

The first is that a card cut for one cloth is the wrong card for another. The same design woven on a different construction is a different shape, and the difference is not a scaling — it is a shear of the proportions, so a circle becomes one ellipse on one cloth and a different ellipse on another.

The second is that the correction cannot be built into a house style. It has to be computed per construction, and for a balanced cloth it is small enough to ignore while for an unbalanced one it is not.

Where the distortion comes from #

The finishing relaxation moves both setts and moves them oppositely. A cloth that shortens has more picks per centimetre; a cloth that widens has fewer ends per centimetre. Those are the same statement about one curve, and this collection computes both.

Ends per centimetre and picks per centimetre are the two denominators in a motif’s dimensions. A shape k cells across is k divided by the ends per centimetre; the same shape k cells down is k divided by the picks. So the aspect ratio of anything on the cloth is the ratio of the two setts, and it moves by the product of the two shifts — not by either one alone.

The two shifts multiply rather than cancelling. That is why the distortion is larger than either shift: for the poplin the warp sett falls 6.79 per cent and the weft sett rises 10.58, and the aspect moves by 15.7.

The numbers, cloth by cloth #

For the balanced cottons the distortion runs from 1.7 per cent for the filter cloth to 3.4 for the cheesecloth. A designer drawing a circle at the loom’s numbers on any of them gets an ellipse whose axes differ by a few per cent, which is at the edge of what an eye notices in a small motif and clearly visible in a large one.

For the poplin it is 15.7 per cent. A circle a hundred millimetres across comes back sixteen millimetres out of round, and nobody would call it a circle.

The poplin is the outlier for the same reason it is the outlier in every table this collection builds: it is the only construction here that is unbalanced in both its counts and its setts, and an unbalanced cloth is a long way from its own least-energy state.

That is worth restating as advice, because it inverts what a designer might expect. The cloths that distort a motif most are the ones whose two directions are least alike — which are also the cloths that are chosen precisely because their two directions are unlike, for a warp-faced shirting or a weft-faced upholstery. The constructions on which figuring is most often done are the constructions on which the correction is largest.

Which state a designer should draw in #

The answer is the finished one, and stating it that way makes the practical problem visible: the finished setts are not known until the cloth has been made.

What a mill does in practice is work from experience of the construction — a known contraction for a known cloth on a known finishing route — and the arithmetic here is a way of saying what that experience is a measurement of. It is a measurement of where along the locus the finishing route leaves the cloth, which is not fully determined by the construction and depends on the tension the cloth is dried at.

Two things follow that are worth having.

The correction has a computable floor. However the finishing is run, the cloth cannot go past its own least-energy state without something holding it, so the relaxation shifts computed here are the largest the route can produce. A design corrected for them is corrected for the extreme, and a cloth finished under tension distorts less rather than more.

And the correction is not a scaling. A design can be scaled to any size by cutting a card with more or fewer cells, and no amount of scaling fixes an aspect. The two are independent, and confusing them — correcting a motif that came out too small by adding cells — leaves the distortion exactly where it was.

Two corrections that are usually one #

A designer working to a ruled chart is applying one correction that is really two, and separating them says which part is reliable.

The loom’s part is take-up and width contraction, and it is settled before the cloth leaves the machine. Both are computed routinely, both are under tension, and both are stable: run the same warp on the same loom at the same tension and the same numbers come back. A chart ruled for that part is a chart a mill can trust.

The finishing’s part is the relaxation this rung computes, and it is not stable in the same way. It depends on how far along its own locus the finishing route lets the cloth go, which depends on the tension at which it is dried, how much it is agitated, and whether it is stentered to width. Two lengths of the same cloth through two routes relax to two different points, and their motifs are two different shapes.

The trade’s answer to that has always been to fix the route and treat the correction as a property of the cloth-and-route pair rather than of the cloth. That is exactly right and it is worth knowing why it is right: the quantity being fixed is a position on a curve, and fixing the route is the only way to fix it, because nothing about the construction does.

It also says where a surprise will come from. A design that has been woven satisfactorily for years and suddenly comes out wrong has usually not had its card changed and has not had its construction changed. What has changed is the route — a new stenter, a different drying tension, a wash added — and the motif’s aspect has moved with it.

The distortion a reader can see, and the one a reader cannot #

There is a reason this correction is discussed as a matter of eye and experience rather than of arithmetic, and it is that the eye is very good at some parts of it and useless at others.

A circle is the case where the eye is unbeatable: a two per cent ellipse in a large motif is visible, and everybody who has drawn one knows it. A diagonal is nearly as good, because a line that should meet a corner and does not is obvious.

What the eye cannot do is compare two motifs on two cloths. Each looks acceptable on its own, and the fact that the same card has produced two different shapes is invisible unless the two are laid side by side, which they rarely are. That is where an arithmetic earns its keep: it says that two cloths differing by one per cent in each sett produce motifs differing by two, and that the difference is a property of the constructions rather than of anybody’s drawing.

What was counted, and how #

The relaxed construction is the least-energy state of the cloth’s own locus at the site’s computed bending rigidities, and the sett shifts are read off it. The distortion is the ratio of the two aspect ratios, computed as a ratio rather than as a difference of percentages, because the two shifts multiply.

The direction is asserted for every cloth: the warp sett must fall and the weft sett must rise, without exception, since that is a statement about the shape of the locus. A row that broke it would mean the minimisation had gone the wrong way.

The motif sizes are computed at both constructions from the same cell count, so the figure compares one design in two states rather than two designs.

Where the model stops #

No finishing route is modelled. The relaxed state is where a cloth goes if left alone, and every real route holds it somewhere short of that. So the distortions here are upper bounds on what a route can produce and are not predictions of any particular mill’s cloth.

The relaxation is not complete. Friction leaves the cloth resting in a band several per cent wide, which is wider than the shifts themselves for the seven balanced cloths — so for those, the aspect correction is smaller than the uncertainty in where the cloth came to rest. That is an honest statement of when the correction is worth making: on the balanced cloths it is inside the noise, and on the poplin it is not.

And the design is treated as a shape on point paper. A jacquard design is a shape, a weave assignment and a set of block boundaries, and this collection has separate rungs on what the second and third of those cost. Nothing here computes what the relaxation does to the weaves inside the motif, which change their own crimp as the cloth relaxes and therefore change the surface step at the boundary.

The two ends of the range are worth having beside it, because the correction is worth making at one of them and not at the other.

Where the correction meets the staircase #

The outline ladder and this rung are about the same edge from two sides, and putting them together sharpens both.

An outline on a figured cloth is a staircase, and the directions it can take are quantised: a step of one end and one pick, one and two, two and one, and so on, each making a definite angle that depends on the two setts. That quantisation is a property of the cells, and it does not move when the cloth relaxes — a step of one end and two picks is a step of one end and two picks whatever the spacings are.

What moves is the angle each step makes. At the poplin’s loom construction a one-and-one step makes one angle; at its finished construction it makes another, and the difference is the same 15.7 per cent that distorts a circle. So a designer choosing between two available staircase directions to approximate a wanted line is choosing between two angles that will both have moved by the time anybody sees them.

The consequence is that the coarser the figuring, the more the correction matters. A motif built out of large blocks has few boundary steps and each is long, so an error in the angle accumulates into a visible departure from the wanted line; a finely figured motif has many short steps and the eye integrates them. That runs the opposite way to intuition, which expects a coarse motif to be more forgiving.

The wash is a third state, and on an ordinary cloth it is the larger one #

The essay names two states — the loom’s and the finished cloth’s — and mentions a third in passing. The third can be computed from this collection’s own laundering figures, and on most of the eight cloths it turns out to be the bigger of the two corrections.

A standard cotton loses 7.94 per cent of its length and 2.84 of its width in laundering. Both raise a sett: the ends per centimetre by a factor 1/0.9716 and the picks by 1/0.9206. So the aspect ratio moves by

0.9716 ÷ 0.9206 inverted — a factor of 0.947, or 5.3 per cent.

Set that beside the finishing distortion. For the poplin, at 15.7 per cent, the wash is the smaller of the two. For the seven balanced cloths, whose finishing distortions run from 1.7 to 3.4 per cent, the wash is roughly twice as large.

And the two compound rather than cancelling, because both move the aspect the same way: the finishing lowers the warp sett and raises the weft, and the wash raises both with the weft rising further. On a muslin the product is 0.972 × 0.947 = 0.921 — eight per cent from the drawing to the cloth in the customer’s hands, of which the mill’s own correction addresses under a third.

Which makes the correction a choice rather than a calculation #

That changes what a designer is doing, and the change is not a refinement of the arithmetic but a question about who the motif is for.

A design corrected to the finished cloth is right on the shelf and five per cent out after a wash. A design corrected to the washed cloth is five per cent out on the shelf and right in use. There is no correction that satisfies both, because the cloth occupies two different geometries at two different times and the card occupies one.

The trade’s practice is the first, and it is defensible for the ordinary reason: the cloth is inspected, sold and judged in its finished state, and the customer who washes it has stopped comparing. But it should be recognised as a decision rather than as the answer, and the arithmetic says the decision is worth five per cent — which is well above the level at which a circle stops looking round.

There is one route that removes the choice, and the mill already owns it. A pre-shrunk cloth has had most of its relaxation shrinkage taken out mechanically, so its wash distortion is what remains of a residual under one per cent rather than of a full eight. Sanforising is bought for dimensional stability; it also fixes a motif’s proportions, and that is a reason for it nobody quotes.

And it reverses which correction is worth making #

The essay’s honest conclusion about the balanced cloths is that their finishing correction sits inside the noise of where the finishing route left them. Adding the third state makes that conclusion sharper rather than weaker.

On a balanced cloth, the finishing distortion is two or three per cent, the frictional band it is uncertain within is of the same order, and the wash distortion is five per cent and is not uncertain at all — it is a property of the construction, it happens on every garment, and it happens whatever the finishing route did.

So the correction worth computing on an ordinary cloth is the one the mill has been ignoring, and the correction the mill computes carefully is the one buried in its own process variation. The large, reliable, uncorrected term is the wash, and it is uncorrected because it happens after the cloth has been sold.

The generalisation #

When a shape is specified in units of a lattice, and the lattice’s two spacings change independently, the shape changes and the specification does not. The card is right, the loom is right, and the cloth is wrong, because the shape was never in the card — it was in the card and a pair of spacings, and only one of the two was written down.

The pattern is everywhere a design is expressed in counts rather than in lengths. Pixels on a display with non-square pixels, stitches on a knitted chart, holes on a punched tape, courses of brick against a stretcher bond: in each case the artefact is specified in units of a repeating element and comes out at whatever proportions the element has that day.

The narrower lesson is about which of two things to correct. A distortion of proportions and an error of size look alike in a finished object and are fixed in different places — one in the aspect and one in the count — and a maker who has only one lever will use it on both. This collection can say which is which because it computes the two setts separately rather than a single scale.

Who found it, and when #

Pre-distorting a jacquard design for the contraction of the cloth is standard practice in figured weaving and is as old as the technique; a designer works to a chart ruled for the construction rather than to a square grid. The trade term for it is simply working to the correct ratio, and the ratio is obtained from experience of the cloth.

What this collection adds is where the ratio comes from. Take-up and width contraction at the loom are calculated routinely and account for part of it. The rest is the relaxation to the least-energy state of the locus, which is a modern reading of Peirce’s geometry that this collection built for a different purpose entirely — and it says that the two contributions to the ratio have different characters: one is done when the cloth leaves the loom and one continues in the customer’s hands.

Where the ladder goes next #

The distortion computed here is the aspect ratio’s share of a change that also moves every cover factor, every jamming limit and every interchange budget on the cloth — which is why a sett must be quoted with a state attached.

Sideways, a motif’s boundary has its own arithmetic that the relaxation does not touch, in the staircase a woven outline is and in what a rectangular block costs. And the residual the finishing leaves behind is what comes out in the wash, which distorts the motif a second time and by a smaller amount.