Corduroy is a cut float

The float has been the central quantity of this site since its foundation. Lustre, drape, snagging, abrasion, tear, the closest a cloth can be set — every one of them turned out to be a function of how far a thread runs on the face before it goes under something.

Corduroy is the fabric where the float becomes the surface itself, geometrically and literally, and the arithmetic is one line.

The one line #

A float spanning f ends at a sett of s ends per inch has length f/s. Cut it at the midpoint and each leg is f/2s long. Stand the leg up and that is the pile height:

height = f / (2s)

Six ends at sixty per inch is a tenth of an inch, which is 1.27 mm — an ordinary fine-wale corduroy. Twelve ends at the same sett is 2.54 mm, a heavy cord. Six ends at a hundred and twenty per inch is 0.63 mm, a velveteen.

Two things about that formula are worth dwelling on, because both are the sort of thing a fabric description gets wrong.

The sett is in it. A pile height quoted as “a six-end float” is not a pile height. The same float count at twice the sett gives half the pile, and a designer moving a corduroy from a coarse yarn to a fine one without changing the float count halves the surface without meaning to.

Nothing else is in it. Not the yarn’s thickness, not the ground weave, not the number of binding picks, not how tightly the cloth is beaten. Those all matter to the fabric and none of them appears in the height.

The wale, which is the other half of the surface #

Corduroy is not a uniform pile; it is a set of ridges with valleys between them, and the ridges are called wales. The wale count is the number a fabric is sold by — an eight-wale cord, a twenty-one-wale needlecord — and it comes out of the same geometry.

A cutting line runs down the middle of each float. Between one float and the next there are some binding ends where the pile weft is caught into the ground and not floating. So the distance from one cutting line to the next is the float plus the binding, and

wales per inch = s / (f + b)

At sixty ends per inch with a six-end float and one binding end, that is 8.6 wales per inch. Widen the float to twelve and it falls to 4.6 while the pile height doubles — which is the whole design space of the fabric in two numbers moving oppositely.

That opposition is the reason corduroy comes in the range it does. A fine-wale cloth has short pile and many ridges; a wide-wale cloth has tall pile and few. There is no corduroy with tall pile and many wales, because both are read off the same float and they read off it in opposite directions.

What holds it in #

The float is cut in the middle, so each leg is anchored where the pile weft was bound into the ground — and the anchorage is exactly the question the previous rung is about.

Here it is a real weakness, and the trade knows it as the fabric’s characteristic failure. A corduroy leg is held by however many picks the pile weft passes under at the binding point, which in the simplest construction is one, and the wrap angle is correspondingly small. Corduroy sheds. It sheds most where it is abraded most, which on a garment is the seat and the knee, and the bald patches on old corduroy are the pile withdrawing rather than wearing off.

The constructions that resist it bind the pile weft into the ground more firmly — under more picks, or with the ground ends grouped so that the binding point is pinched — at the usual price in pile density. The exchange is the same one the previous rung computed for V and W, arriving in a weft-pile construction rather than a warp-pile one.

The float can be eliminated, and then the label gives the pile #

The two expressions share a parameter, so it can be removed. Writing the float in terms of the height and substituting into the wale count gives, after rearranging,

height = half the wale pitch, less half the binding width.

In symbols, one over twice the wales per inch, minus the binding ends over twice the sett. The float count is gone.

That is worth having because a corduroy is sold by its wale count and never by its float. At sixty ends per inch with a single binding end, 8.6 wales gives half a pitch of 1.477 millimetres less 0.212, which is 1.265 — the same figure the float route gives, by a road that uses only what is printed on the label.

Two consequences fall straight out.

There is a ceiling on the pile, and it is half the wale pitch. The binding term is subtracted and cannot be negative, so no corduroy of a given wale count can carry a pile taller than half the distance between its ridges. An 8.6-wale cord is capped at 1.48 millimetres whatever else is done to it, and a 21-wale needlecord at 0.60. A fabric cannot be both finely waled and deeply piled, and this is that statement in its strongest form — not a design tendency but a bound, because the two legs of one float have to fit between two cutting lines.

And the binding term says what a fine cord costs. It is the binding width over twice the sett, so it matters most when the sett is low. Raising the wale count without raising the sett drives the pile towards zero long before the arithmetic runs out — which is why needlecords are woven on finer, more densely set yarn rather than by simply cutting more often, and why the two decisions are not separable.

Why the leg is not quite half a float #

Two idealisations sit in the one line and they pull in opposite directions, which is worth stating because it is why the formula is good rather than exact.

The float is taken as straight. It is not: it crimps slightly over the ends it passes over, and it turns down at each end to reach its binding point. So its true length exceeds f/s, and the height is underestimated.

The leg is taken as standing perpendicular. It does not, entirely: a cut leg leans, and a pair of legs from adjacent floats support one another in a way that a single leg does not. So the height above the ground is less than the leg’s length, and the height is overestimated.

Both are small at ordinary float lengths and neither is negligible at long ones, and they do not cancel in any principled way. The model returns the leg’s length and says so; calling it the pile height is the approximation.

What the finishing does #

A corduroy off the loom does not look like corduroy. The cut legs lie flat, the wales are barely distinguishable, and the surface is closer to a ribbed flannel than to a cord.

What makes the fabric is what happens next: it is brushed to lift the legs, singed to remove the fibres that stand out sideways, and often steamed or heat-set to hold the wales upright and separate. The characteristic round-topped ridge is a finishing shape rather than a woven one, and none of it is in the geometry above.

This is the sharpest example on this site of a fabric whose appearance is decided after the loom, and it is the reason the field that follows has an entire field waiting for it. A model that computes the pile height from the float is computing the height of a leg, and how that leg ends up standing is a different subject with different machinery.

Velveteen, which is the same construction without the cutting lines #

Corduroy’s near relative makes the point about the wale precisely. Velveteen is also a cut weft pile on a woven ground, and the difference is that the floats are staggered rather than aligned, so the cutting lines do not form ridges and the surface is uniform.

Everything above about the height survives unchanged, because the height is a property of one float and the cutting of it. Everything about the wale disappears, because there are no cutting lines to space. Two fabrics, one formula in common and one not, and the difference between them is entirely how the floats are arranged relative to one another rather than what any float does.

That is worth noticing as a general point about this field. The pile geometry and the pile arrangement are independent, the matrix describes neither, and a description that gives only one of them has not specified a fabric.

How much yarn the pile costs #

The height is the number a hand judges and the mass is the number an accountant does, and the second follows from the same geometry with one more step.

Every float becomes two legs whose total length is the float’s length, so cutting rearranges the yarn without consuming any. The pile mass over one wale is therefore the pile weft’s linear density times the float length, and the area that wale occupies is its width times whatever length of cloth one pile pick accounts for.

The consequence is worth stating plainly: the pile mass per unit area does not depend on how the float is divided between height and wale count. A twelve-end float at a given sett puts the same weight of yarn on the surface as two six-end floats, arranged as one tall ridge instead of two short ones. Yarn cost is a function of how much pile weft goes in, and nothing about the wale structure changes it.

That is a genuinely useful separation, because it means the design decisions divide cleanly. How much yarn to spend is one decision, made by choosing the pile weft’s density and how often it is inserted. What shape the surface takes is a second, made by choosing the float length, and it is free.

Why a corduroy is warm #

The one property of the fabric that follows from the geometry without any chemistry is thermal, and it follows because the pile is mostly air.

A pile standing off the ground traps a layer of still air whose depth is the pile height, and still air is a very good insulator — better, by a wide margin, than any textile fibre. So the insulating value of a pile fabric scales with its height, and a wide-wale corduroy is warmer than a needlecord in the same yarn at the same weight for exactly this reason.

The qualification is that it holds only while the pile stands. Crushed pile is not a trapped air layer, and the difference between a new corduroy and a worn one at the knee is largely a difference in how much air the surface is holding. This is the same argument that makes a loop-pile carpet warm and a flattened one not, and it is one of the very few places where a structural quantity computed on this site maps onto a performance a wearer notices directly.

The float limit, arriving from the other side #

There is a constraint on corduroy’s float that has nothing to do with pile and everything to do with the cloth it sits in, and it is one this site already computed for a different reason.

Before the knife, a corduroy is a fabric with a very long weft float in it. Designing to a float limit is about exactly that constraint in ordinary cloth: a float long enough to snag, to abrade unevenly, or in the extreme to leave a thread that never interlaces at all — at which point the integrity criterion refuses the draft outright.

A twelve-end float at four picks to the repeat is a long way past what any ordinary cloth would tolerate, and a corduroy is only allowed it because the float is not going to stay a float. The knife arrives, and what would have been a fault becomes the product.

That is a slightly unusual relationship between a constraint and a design, and it is worth naming. Every other float on this site is bounded above, and here the bound is suspended for a construction that intends to destroy the float before anyone sees it. The bound that does still apply is the ground’s: the binding picks either side of the float have to hold, and if they do not the criterion catches it in the ordinary way.

What was counted, and how #

The height is computed from the float count and the sett by the one line above, with the end spacing taken as the reciprocal of the sett. The wale spacing is computed from the float plus the binding ends. The pile mass per unit area is the pile weft’s linear density times the float length, divided by the area one wale occupies.

The figures draw the leg at the same scale the ground pitch is drawn at, so a longer float visibly produces a taller pile rather than merely printing a larger number beside an unchanged picture. That sounds like a small thing and is the difference between a figure that demonstrates the relation and one that asserts it.

Two assertions run across the sweep of float lengths and let-off ratios. Corduroy’s height rises with the float and with nothing else; terry’s rises with the ratio and with nothing else. Both are monotonicity checks, and their value is that they would fail immediately if either construction’s parameter had leaked into the other’s formula — which is the specific error the two constructions invite, since they are drawn by one generator and computed side by side.

The float, on both sides of the boundary #

There is something worth pausing on in the fact that a float — the quantity this site has counted more often than any other — turns out to decide a fabric that has no matrix.

The float is a property of the matrix. It is defined as a run of consecutive picks over which one end stays on the face, it is measured cyclically on the repeat, and it is exactly the kind of thing the encoding was built to decide. The corduroy’s pile weft, before it is cut, is a perfectly ordinary float in a perfectly ordinary cloth, and every one of this site’s tools applies to it.

Then a knife is run along it, and the fabric that results has three thread systems and no matrix at all — and the float length is still the quantity that decides its surface, because it was measured before the cut.

That is a slightly unusual position for a quantity to be in. It belongs to the encoding, it survives an operation that leaves the encoding, and it does so because the operation is applied to an object the encoding had already described. The general lesson is small but real: a construction outside the matrix may still be specified inside it, if what happens outside is something done to a matrix rather than something woven instead of one.

Terry, on the next rung, is not like that. Its pile is never a float at any stage, so no quantity crosses the boundary and the whole description has to be built afresh.

Where the model stops #

The ground weave does not appear and does matter. A firmer ground grips the binding point better and resists the pile being pulled through it. The height formula is blind to this and the anchorage question is entirely about it.

Nothing here models the cut. The knife is taken as passing exactly through the midpoint. A real cutting machine runs a circular blade along a guide wire laid under the float, and a misplaced wire gives legs of unequal length — which is a visible fault and a common one.

And the pile is taken as standing. Which, as above, it does only after finishing.

Who found it, and when #

Corduroy’s origins are disputed in the way textile origins usually are, and the folk etymology from corde du roi is generally thought to be an invention. Fustians of this general family — cut weft piles on a cotton ground — were made in Lancashire from the eighteenth century, and the cutting was hand work with a knife guided along the float until cutting machinery arrived in the nineteenth.

The geometry needs no discoverer, and that is worth saying. Half the length of a cut float standing up is not an insight; it is what a cutter sees happen. What this essay adds is only that the relation ties the fabric’s surface to the quantity this site has been tracking all along, so that a corduroy’s pile height is a float length in disguise — and the next rung’s fabric, which looks like a close relative, has no float in it whatever.

Where the ladder goes next #

Terry is the contrast case: a pile fabric whose height is set by a let-off ratio and in which the float length does not appear. Beyond it, velvet woven face to face and cut apart is the construction where the integrity criterion stops describing the fabric and starts describing the manufacture.