Why the warp shrinks more

Ask anyone who has shrunk a shirt which way it went and the answer is up the body rather than across it. The trade knows the same thing in numbers: for an ordinary cotton fabric, warpwise relaxation shrinkage runs two to four times the weftwise figure.

The tempting explanation is that the warp is somehow different — a harder-twisted yarn, a sized yarn, a stronger yarn, since it has to survive weaving. All of that is true and none of it is the reason.

The two systems have entirely different histories #

Consider what happens to each thread on its way into the cloth.

A warp end is wound onto a beam under tension. It is unwound under tension by the let-off, passes through the drop wires, the healds and the reed under tension, is held under tension while the shed opens and closes for every pick, and is pulled forward by the take-up under tension. From the moment it is beamed to the moment the finished cloth is cut off, it is never slack. Weaving a metre of cloth at fifty picks per centimetre means five thousand shed cycles, each of them a further working of a thread that is already being pulled straight.

A pick is inserted in a few milliseconds. It arrives across the shed, is beaten up by the reed, and is thereafter held only by the friction of the warp ends crossing it and by the temples gripping the two selvedges. It has never been under any tension worth the name in the length direction of its own thread.

So the warp arrives at the finishing works with most of its crimp missing, and the weft with rather little. That asymmetry is a property of the machine, not of the fabric, and this is the reason it survives every change to the yarn or the weave: any loom pulls the warp and does not pull the weft.

What the arithmetic makes of it #

The previous rung reduced shrinkage to one line: (c_relaxed − c_loom)/(1 + c_relaxed). The relaxed crimps are what the cloth wants, and for a balanced plain weave they are equal. So if the two directions differ, the difference is entirely in c_loom, which is entirely in the machine.

At a relaxed crimp of 12.79 per cent in both directions:

The weft column does not move, because the weft’s tension was held at a quarter throughout. The ratio at the right is the observed two-to-four, arrived at by varying one number that has nothing to do with the cloth.

The ratio is exactly the ratio of the tensions, in this model, and it is worth seeing why: with equal relaxed crimps, both shrinkages have the same denominator and the same c_relaxed, so the quotient is the quotient of the two tension fractions. The model says the warp-to-weft shrinkage ratio is the warp-to-weft crimp-removal ratio, with the fabric cancelling out of it entirely.

That is a strong claim and it is a claim the model makes rather than one this site is asserting about real cloth. What makes it interesting is that it is falsifiable in an obvious way: it predicts that two fabrics of very different construction, woven on the same loom at the same settings, should show the same shrinkage ratio and different shrinkage magnitudes.

The case where it reverses #

The prediction has a corollary that is easier to check, because the trade already knows the answer.

There is a construction where the weft shrinks more, and it is exactly the one where the machine’s asymmetry is inverted. A fabric woven with an unusually high warp sett and a low pick density — many ends, few picks — puts most of the crimp in the weft on the loom, because the weft is the system doing the bending. Relaxing it lets the warp take crimp back from the weft, and the width can come in more than the length.

Rather than argue this from the mechanism, the arithmetic will produce it: set the weft tension above the warp’s and the inequality turns over.

This is the shape of assertion this site prefers, and the reason is recorded in an earlier mistake here: the check in relaxationRange does not say the warp shrinks by more than the weft, which is a statement about the defaults. It says a system whose crimp was removed harder gives more back, conditional on which tension is larger, so exchanging the tensions exchanges the inequality and the assertion still holds. An assertion written against the numbers in front of it would have failed here and would have looked like a real defect.

The second asymmetry, which is not a tension #

There is a mechanism the crimp arithmetic cannot see, and honesty requires it be named here rather than at the end.

The temples hold the cloth out to the reed width, and they hold it by gripping the selvedges. That is a width constraint applied to the fabric, not a tension applied to the weft yarn, and it does two things at once: it stops the cloth contracting as the picks are beaten up, and it does so unevenly across the width, because the grip is at the edges and the middle is held only by the cloth’s own stiffness.

The consequence is a fabric whose weftwise shrinkage varies from selvedge to centre. It is a well-known nuisance — it produces the bowed weft that shows up as a hem that will not hang straight — and it is entirely outside a model that treats the fabric as one repeat.

So the weftwise numbers in this essay should be read as an average across the width of a cloth that does not actually have one value. The warpwise numbers do not have this problem, because the warp tension is applied to every end individually and is far more uniform.

The stenter, which puts some of it back #

There is a machine between the loom and the customer that works against everything this essay describes, and leaving it out would misrepresent what a finished fabric has been through.

A stenter is a long heated chamber through which the cloth is carried on two chains of pins or clips gripping its selvedges. The chains diverge, so the cloth is held out to a set width while it dries; and the cloth can be fed in faster than the chains run, which is overfeed, so it is also allowed to contract lengthwise by a set amount.

Both of those are direct interventions in the crimp. Holding the width out takes weft crimp out. Overfeeding puts warp crimp in.

So a stenter is a machine for setting the two crimps to chosen values, and a finisher uses it to place the fabric where the specification wants it. That is why a finished cloth’s dimensions are a decision rather than a consequence — and it is also why a fabric can be stretched to a width it will not hold, which is the commonest cause of a piece that meets its specification at the mill and fails it after one wash.

The stenter cannot exceed the ceiling either. It can put crimp back only up to what the geometry allows, and a cloth pulled wider than its thread length permits is a cloth being stretched at the yarn rather than at the crimp, which it recovers from immediately.

Why the two directions are measured differently #

A last practical asymmetry, and it is about the measurement rather than the fabric.

Warpwise shrinkage is measured on a specimen with marks along its length, and the length is the direction the cloth is long in. A metre of marked length is easy to obtain from any piece.

Weftwise shrinkage is measured across the width, and a full-width specimen is the entire piece. So the standard specimens are small squares — typically half a metre — marked in both directions, and the weftwise figure they give is a local measurement in a fabric that, as the temple discussion above says, does not have one weftwise value.

The warpwise number is therefore both larger and better determined, and the weftwise number carries a variability the single figure does not show. Any comparison of the two should carry that, and a specification that sets a tighter tolerance on the width than on the length is asking for something the measurement cannot reliably confirm.

What was counted, and how #

The rows above come from relaxationRange, which solves the relaxed state once and then computes each row’s loom state from it. Two assertions run at every row.

The first is the conditional inequality already described: where the warp tension exceeds the weft’s, warpwise shrinkage must exceed weftwise. It is conditional in the source, so a call with the tensions exchanged tests the other branch rather than failing.

The second is the ceiling — no row may exceed c/(1 + c) — which is the next rung’s subject and is checked here because a bound is worth checking wherever the quantity it bounds is produced.

And the whole family is asserted monotone: the harder the warp was held, the more it gives back. That one is nearly trivial and is included because the alternative to checking a trivial thing is discovering later that a sign was wrong.

Where the model stops #

The tension fractions are inputs and nothing here derives them. A real value would come from measuring the crimp in a cloth still on the loom and again after relaxation, which is a laboratory operation this site cannot perform. The model turns an unmeasured input into a prediction about a ratio, which is a fair trade only if the ratio is what gets used.

Equal relaxed crimps are an assumption. The clean result — that the shrinkage ratio equals the tension ratio — depends on the two relaxed crimps being equal, which is what a balanced cloth in a symmetric yarn gives. An unbalanced cloth divides its crimp unevenly and the ratio picks up a factor this essay does not carry.

Sizing is ignored entirely. A cotton warp is usually sized before weaving and desized afterwards, and the size stiffens the yarn while it is on the loom — which changes how much crimp the tension can remove. Removing the size is itself a finishing operation and it happens before the fabric is measured.

And the fabric is assumed uniform across its width, which the temple discussion above says plainly it is not.

What the unbalanced case does to the ratio #

The clean result — that the shrinkage ratio is the tension ratio with the fabric cancelling — is stated as conditional on equal relaxed crimps, and the condition is doing more work than it looks.

In general the two shrinkages are Tᵢcᵢ/(1 + cᵢ), so the ratio carries a factor of c₁(1 + c₂)/c₂(1 + c₁). For a balanced cloth the two crimps are equal and it is one. For an unbalanced one it is not, and it is not close.

The force balance fixes the crimp division at a crimp-height ratio of (B₂/B₁)(p₂/p₁)³, and crimp goes very nearly as the square of the crimp height, so

c₁ ÷ c₂ = [(B₂/B₁)(p₂/p₁)³]²

— a sixth power of the sett ratio.

For a denim set two ends to every pick in equal counts that is (1.45)⁶, which is 9.3. So the warp has nine times the weft’s relaxed crimp, and the shrinkage ratio is the tension ratio times about eight and a half.

Which turns the proposed test into two tests #

The essay offers a falsification: weave two very different constructions on one loom and check that the shrinkage ratio is the same for both. On this arithmetic that test would fail, and it would fail for a reason the model itself predicts.

So the test splits cleanly.

Two balanced constructions on one loom must give the same shrinkage ratio. That is the model’s clean claim and it is falsifiable as stated.

A balanced and an unbalanced construction on one loom must give ratios differing by the sixth power of the sett ratio. That is a much sharper prediction, because it names a number rather than an equality, and it is the one worth running: the sixth power is a very strong dependence and a factor of eight is not something a measurement can mistake for scatter.

A model that survives the first and fails the second is a model with the right mechanism and the wrong crimp division, which is a diagnosis rather than a refutation.

And it explains the mills’ own practice #

The sixth power also accounts for something the essay’s conclusion leaves slightly awkward.

The essay argues that the asymmetry lives in the machine, so a mill can characterise its looms once and apply the result across its range — and mills do hold allowances per loom and per setting. They also adjust them by fabric family, which on a purely machine-based account they should not need to.

The correction says why. The machine sets the tension ratio and the construction sets a sixth power on top of it. A balanced shirting and a warp-dense drill on the same loom at the same settings have shrinkage ratios an order of magnitude apart, so an allowance held per loom alone would be badly wrong on half the range.

Both halves of the practice are right and they are doing different jobs: the per-loom figure captures the tensions, and the per-family adjustment captures the crimp division. Neither is a fudge, and the second has an exponent.

The caveat is the crimp-height-squared step, which is Peirce’s small-crimp approximation and softens at large crimps — so the sixth power is an upper bound on the sensitivity rather than a value. What is not in doubt is the direction and the order: the construction’s contribution to the shrinkage ratio is a high power of the sett ratio, and it is larger than the machine’s.

Why it matters where the asymmetry lives #

Locating the asymmetry in the loom rather than in the fabric has a practical consequence, and it is the reason this rung is worth its own essay.

If the asymmetry were a property of the cloth — of the warp yarn’s twist, say, or of the sizing — then it would be addressable by the fabric designer, and different fabrics would need different allowances. Since it is a property of the machine, it is very nearly the same for everything woven on that machine, and a mill can characterise its looms once and apply the result across its whole range.

That is exactly what mills do. Shrinkage allowances are held per loom type and per setting, not per fabric, and a fabric’s own contribution enters through its relaxed crimp — which is a matter of weave and sett and is the part a designer controls.

The cloth that is measured is not the cloth that was woven #

One more consequence follows from putting the asymmetry in the machine, and it is the one that costs money.

A piece of cloth is sold by length. It is measured on a piece-measuring machine, under a small tension, in whatever state it happens to be in when it is measured — and that state is somewhere between the loom’s and the relaxed one, depending on what has been done to the piece in between. The same piece measured before and after finishing gives two different lengths, and neither is wrong.

The trade handles this with a vocabulary that is more careful than it first appears. Loom-state length is what the take-up counted. Grey length is the piece as it arrives at the finishing works, having lost some of its crimp in handling. Finished length is what is sold. The difference between the first and the last is the mill’s yield loss, it runs to several per cent, and it is a real cost that appears nowhere in the fabric.

The asymmetry means most of that loss is warpwise, which is the direction the cloth is sold in. A width loss costs a little at the cutting table; a length loss comes straight off the invoice.

What a shrinkage figure on a label is really about #

Since the two directions differ, a single figure cannot describe a fabric, and the standards say so: a shrinkage specification quotes two numbers and the procedure that produced them.

The procedure matters as much as the numbers, and the reason is in this essay’s mechanism. A test that wets the specimen and dries it flat measures one thing — and wetting is a second mechanism entirely. A test that tumbles it measures another, because tumbling supplies the mechanical action that lets the yarn move past its own friction and reach the state it wants rather than the first state it can get to. Friction is what makes relaxation incomplete, and every procedure in the standards is, at bottom, a specification of how much mechanical energy is put in to overcome it.

That is worth carrying, because it explains why the same fabric can honestly be quoted at two very different shrinkages by two laboratories following two standards. Neither is measuring wrongly. They are measuring how far along a path the fabric has been pushed, and the path has no natural end short of the fully relaxed state.

Who found it, and when #

The observation is as old as woven cloth and the tailor’s rule that predates any measurement of it — allow for the shrink up the length, not across — is the practical form.

The mechanism was put on a footing in the same period as the rest of this field’s arithmetic. Once crimp could be measured on the loom and off it, the asymmetry stopped being folklore and became a table, and the tables in a mill’s shrinkage allowances are the direct descendants.

What is not standard is putting the ratio in this form and noticing that the fabric cancels out of it. That is a consequence of doing the arithmetic symbolically rather than tabulating measurements, and it is the sort of thing that is obvious once the algebra is on the page and invisible in a table of results.

Where the ladder goes next #

Both columns of the table have a bound they cannot pass, and it is the same bound in each: the crimp itself. The next rung shows why, and finds that it is the one result in this field with nothing fitted or measured in it at all.