Terry needs two beams

A corduroy’s pile is a float cut in half, so its height is a length that was already in the cloth. A towel’s pile is not like that at all. Nothing is cut, no float is involved, and the loops are made of warp that was never in the cloth’s length in the first place — it was pushed in from outside.

The mechanism is two warp beams and a beat-up that comes in two stages, and it is worth describing before any arithmetic because the arithmetic is entirely a consequence of it.

The contrast with the previous rung is sharp enough to be worth stating at the outset. Corduroy’s pile height is decided by a number written on point paper — a float count — and can be read off a draft by anybody who knows how to read one. Terry’s is decided by nothing that appears on point paper at all. Two towels with identical drafts and different beam tensions are different fabrics, and the difference is the one a hand notices first.

Two beams and a loose beat #

The ground warp is held at ordinary tension on its own beam and behaves like any warp: it takes up the length the cloth consumes and no more.

The pile warp sits on a second beam under very light tension and is let off several times faster. Call the ratio r — five to one is a typical towel, seven to one a luxurious one, three to one a light one.

The weaving happens in groups. Two or three picks are inserted and beaten only partway, stopping short of the cloth fell; then the next pick is beaten fully and drives the whole group home together. The ground warp, being tight, holds its position. The pile warp, being slack, has more length in that region than the cloth can accommodate, and the surplus is pushed up out of the plane on both faces.

So the loops are made by surplus, and the surplus is what the let-off ratio provides.

The one line, and it is a different line #

Over one group of g picks at a pick spacing p, the ground warp takes up g·p of length. The pile warp, let off r times as fast, delivers r·g·p. The excess is

excess = (r − 1) · g · p

and it goes into one loop, which has two legs. So

height = (r − 1) · g · p / 2

A five-to-one towel with three-pick groups at forty picks to the inch: the ground takes 0.075 inch, the pile delivers 0.375, the excess is 0.3, and the loop stands 0.15 inch — 3.8 mm. That is a recognisable towel.

Set it against the corduroy formula from the previous rung:

corduroy — height = f / (2s), from a float count and a warp sett.

terry — height = (r−1)·g·p / 2, from a let-off ratio, a group size and a pick spacing.

They share no quantity. Not one. One is a warp spacing and a float count; the other is a pick spacing, a group size and a ratio between two beams. Two fabrics with pile standing off a woven ground, and the parameters that decide their surfaces have nothing in common.

What a towel is actually sold on #

The formula explains a fact about towels that is otherwise mildly puzzling: they are sold by weight — grams per square metre — rather than by pile height, even though pile height is what a hand judges.

The reason is that the two are the same number. The pile fraction of the fabric is

excess / total pile warp = (r − 1) / r

which at a ratio of five is 80% and at seven is 86%. Almost all the pile warp is loop. So the mass of pile per unit area is very nearly the whole mass of pile warp consumed, and the pile warp consumed is r times the ground — meaning a heavier towel is a towel with a higher let-off ratio, which is a towel with taller loops.

Weight, ratio and height are one parameter wearing three hats, and weight is the one that can be measured on a scale by somebody who did not weave it.

The identification is not quite perfect and the imperfection is instructive. A towel’s total weight includes its ground, and two towels at the same let-off ratio in different ground yarns weigh differently while having identical pile. So grams per square metre is the pile height plus a constant, and comparing two towels by weight alone credits a heavy ground as though it were pile — which is exactly the substitution a manufacturer making a cheap towel would want to make, and does.

Why the two constructions had to differ #

It is worth asking why terry could not simply have been made the corduroy way, because the answer says something general about warp and weft.

Corduroy’s pile is a weft float, and a weft float is cheap to make long. The shuttle crosses the whole width anyway; letting it pass over six ends instead of one costs nothing but a lifting plan, and the extra length is already in the pick.

A warp float is not like that. Every warp end comes off a beam under tension, and the beam supplies exactly the length the cloth consumes. To give one warp end more length than its neighbours, that length has to come from somewhere — and on a single beam there is nowhere for it to come from, because all the ends are being let off together.

So a long warp float is not a surplus of yarn; it is the same yarn lying flatter, and it produces a smooth warp-faced cloth rather than anything standing up. This is why every warp pile construction in existence needs a second beam and why no draft on a single beam produces one. It is a constraint about supply rather than about interlacement, and it is completely invisible in the matrix, which knows how threads cross and nothing about where they come from.

The corollary is that the two pile families are not two solutions to one problem. They are solutions to different problems: corduroy asks what to do with an over-long weft float, and terry asks how to dispose of warp that has nowhere to go.

Why a towel dries #

The formula also gives the property the fabric exists for, at least in outline. A loop pile presents a very large surface area of yarn to the water and holds it by capillary action in the spaces between the legs, and both of those scale with the amount of pile — which is to say with r.

That is as far as this site’s machinery goes, and it is not very far. Absorbency is a wetting problem involving fibre chemistry, yarn twist, finish and the surfactant residue from processing, and none of those is a structure. A low-twist pile yarn absorbs far better than a tightly twisted one at identical geometry, and a towel finished with softener absorbs conspicuously worse — an effect large enough that towel manufacturers warn about it, and completely invisible to everything computed here.

So the honest statement is that the geometry sets how much yarn is available and the chemistry decides what that yarn does with water, and only the first is on this site.

The loop that is not cut, and the one that is #

Terry is a loop pile and stays one. Cut the loops and the fabric becomes a velour, which is the same construction sheared — and the shearing changes the properties in a direction worth noting because it goes the way the corduroy essay would predict.

An uncut loop is anchored at both ends. A cut loop is two legs anchored at one end each, so the anchorage per leg is what it was for the whole loop, spread over half as much yarn — and the wrap angle at the binding point is unchanged. Velour therefore sheds where terry does not, and velour towels are famously less absorbent and more attractive, which is a trade nobody makes for performance reasons.

The relevant point for this ladder is that cutting is the only operation that changes the anchorage question, and it changes it in the same way in every pile construction: it converts one strand held at two points into two strands held at one point apiece.

The ratio is a tension, not a setting #

There is a practical point buried in the phrase “let off several times faster” that changes what kind of quantity r is.

A beam is not driven at a chosen speed. It is held back by a brake or a controlled motor, and the warp comes off it because the cloth is pulling. So the ratio between the two beams is not set directly; it emerges from the difference in tension between them, and it is a consequence of how lightly the pile beam is restrained.

That has two effects a formula does not show. It makes the ratio drift as the beams empty — a full beam and a nearly empty one have different effective radii, so a brake set once gives a changing ratio through the piece, and loop height varies from the beginning of a run to its end unless the let-off is actively regulated. And it makes the ratio depend on what the cloth is doing: a heavier beat pulls harder, so a change in pick density changes the pile height even though pick density does not appear in the formula at all.

So r is an operating condition rather than a design parameter, which is a different status from corduroy’s float count. A float count is drawn on point paper and is what it is. A let-off ratio is a number a loom is running at, and keeping it constant is a control problem — which is why terry looms acquired regulated let-off early and why the fabric is one of the more demanding things to weave consistently — a difficulty of the same kind as the one an unbalanced cloth creates for a single beam.

The loop has a base as well as a height #

The height formula divides the surplus between two legs and stands them straight up. They are not straight up: they leave the cloth at the two picks that bind them, which are (g − 1) pick spacings apart, so the loop is a triangle rather than a hairpin.

Each leg is (r − 1)g p/2 long and spans half the base, so

height = ½ √{ [(r − 1)g p]² − [(g − 1)p]² }.

For the standard towel — five to one, three-pick groups, forty picks to the inch — that is 3.76 millimetres against the naive 3.81: a correction of one and a half per cent, which is why the simpler formula survives.

It is not always small. At three to one with the same group the legs are shorter and the base is not, and the correction is five and a half per cent. The splay matters for a light towel and not for a heavy one, which is the opposite of the direction a reader would guess and follows from the base being fixed while the legs grow with the ratio.

Which gives terry a floor #

The interesting part of the correction is where it drives the height to zero, because that is a construction limit the height formula cannot show.

The loop vanishes when the legs are no longer than the base’s halves — (r − 1)g = g − 1 — which gives

r_min = 2 − 1/g.

One and a half at a two-pick group, 1.67 at three, 1.75 at four, approaching two for a long group. Below that ratio the pile warp’s surplus is exactly absorbed by the distance between its own binding picks, and it lies in the plane of the cloth instead of standing out of it.

So there is a minimum let-off ratio below which no terry exists, and it is not zero and not one. A pile beam running at 1.4 times the ground’s produces no loops at any group size; one at 1.8 produces loops only with groups of four or more.

That explains a fact the essay’s linear formula makes look arbitrary: commercial ratios start at three. Three is not the bottom of a continuum reaching down to one; it is comfortably clear of a floor at 1.67, and a mill running near the floor would be making a fabric whose loop height was extremely sensitive to a ratio that drifts as the beams empty.

And it prices the group size properly #

The same expression settles what a larger group buys, which the height formula makes look like a free linear gain.

For a long group the base term stops being negligible and the height goes as

½ g p √{(r − 1)² − 1},

so the effective ratio is not r − 1 but the square root of (r − 1)² − 1. At five to one that is 3.87 against 4 — three per cent — and at three to one it is 1.73 against 2, which is thirteen per cent.

So a larger group buys height linearly and costs a fixed fraction of it, and the fraction is small for a heavy towel and substantial for a light one.

The design consequence is a preference the trade already has. A light towel is better made with a small group and a higher ratio than with a large group and a low one, even though the height formula says the two are equivalent — because the low-ratio route pays the splay penalty and sits nearer the floor where the ratio’s own drift matters most. Two ways to reach the same nominal loop, and only one of them is stable.

Where the loop stands is a separate question #

The formula says how much surplus length there is. It does not say what shape that surplus takes, and the difference matters.

Surplus warp pushed out of the plane of a cloth could form one tall loop, two short ones, or a lumpy irregularity that is neither. What decides it is the drafting: the pile ends are threaded so that they are bound at the outer picks of a group and free at the inner ones, and the free portion is where the surplus can go.

Change that drafting and the same surplus produces a different surface. A three-pick group with the pile bound on the first and third picks gives loops on one face; alternating the binding gives loops on both, which is what a towel wants and what makes it symmetric. A five-pick group gives taller, more widely spaced loops from the same ratio.

So there is a second design axis the height formula does not contain, and it is the axis that decides whether a terry is a towelling, a single-sided velour ground, or a fault. The formula bounds the surface; the drafting shapes it. Stating both is the honest description and quoting the first alone is the ordinary way this fabric gets described.

What was counted, and how #

The height is computed from the let-off ratio, the group size and the pick spacing by the formula above. The pile fraction is the excess over the total pile warp delivered. The loops per inch is the reciprocal of the ground length per group.

Two idealisations are named in the model and both are in the direction of overstating the height. Thread lengths are taken along the cloth with crimp ignored, so the ground warp is credited with less take-up than it really has and the excess is overstated. And the excess is taken to divide equally between the two legs of one loop, which is exact for a symmetric group and not exact for the three-pick group that is standard.

The assertions are monotonicity in each construction’s own parameter, run across the sweep of ratios and float lengths together. Their purpose is not to check the arithmetic, which is a division; it is to check that the two constructions’ parameters have not leaked into one another, which is the specific error a shared generator invites.

Where the model stops #

Crimp is ignored and it is not small. A ground warp in a firmly beaten cloth crimps several per cent, and the pile warp crimps hardly at all because it is slack. Including it would reduce the excess and therefore the height, so the figure computed here is an upper bound rather than an estimate.

The group is taken as symmetric. A three-pick group is not: the loop forms on one side and the geometry of the two legs differs. Whether the loops appear on one face or both is a design choice made in the drafting of the pile ends, and this model does not distinguish them.

The beat-up is taken as perfect. In reality the partial beat and the full beat are set by a mechanism with its own tolerance, and loop height variation along a towel is a standard fault caused by exactly that.

And nothing here says how the loop stands. A loop pushed up out of the plane could go up, down, or lean; which it does is decided by the drafting of the pile ends relative to the ground picks and by the finishing afterwards, and the model computes only how much length is available.

Who found it, and when #

Terry weaving is nineteenth-century in its mechanised form. The construction is traditionally credited to Samuel Holt in Lancashire around 1848, with the loom mechanisms for the double beat-up developed through the following decades; the fabric is named for the loop — terry from the French tiré, drawn or pulled — which is a rare case of a textile name describing the mechanism rather than a place.

What is worth recording is that the invention is a loom invention rather than a draft invention. Corduroy needed a knife; terry needed a second beam and a two-stage beat-up, and no arrangement of a single warp on an ordinary loom produces it. That is the sharpest illustration in this field of something the shaft essays worked out from the other direction: the machine decides what fabrics can exist, and the decision does not always look like a shaft count.

Where the ladder goes next #

Both pile geometries have now been worked out and shown to share nothing. The last rung of this ladder is the construction that unifies pile with double cloth and turns this site’s central check into a description of a manufacturing step: velvet is woven as one cloth and cut into two.