A brocade weft floats as far as the next figure

A damask adds nothing to the cloth. Its figure is the ground turned over, the same satin in the other face, so it weighs and wears the same everywhere. A brocade adds a thread. Its pattern weft is a third system, laid on the face over the figure and nowhere else, and that is why a brocade can carry a colour a damask cannot and why a monochrome weave runs out of tones long before a brocade does.

The added weft has to get from one figure to the next. There are two ways. It can be woven only across each figure, on a small shuttle or bobbin that turns back at the figure’s edge — broché, which puts no pattern weft anywhere it is not wanted and costs a hand movement for every figure on every pick. Or it can be thrown across the full width like any other pick — lancé — which the loom does for nothing and which leaves the weft lying somewhere between figures.

It lies on the back, and on the back it floats. Unbound, a lancé weft floats as far as the next figure. Bound, it needs ends dropped under it, and a dropped end brings it to the face. On a satin ground there are exactly two places in every repeat where that can happen without showing, and where they fall is decided by the ground’s move — which is also what decides how smooth the ground is, and the two decisions disagree.

The matrix holds a brocade weft as one more pick #

A pile is a third thread system that the warp-and-weft matrix cannot hold, because it stands away from the cloth. A brocade weft does not stand away. It passes over or under each warp end like any other pick, so it can be written as an extra row: an extra pick after every ground pick, a nought where it lies on the face over the warp, a one where the warp covers it and it lies behind.

Over the figure it is on the face, held down every few ends by an end raised over it — the binding that stops the pattern weft floating loose on the front. Between figures it is on the back, covered by every end. And the matrix then answers the question it always answers, whether the whole is one cloth. It is, as long as the pattern weft meets a figure somewhere on its pick: the figure’s bindings tie it in. A lancé pick that crosses the whole width without meeting a figure is, by the same count, not part of the cloth at all, and the connectivity reports two.

That is the first thing the encoding says, and it is the obvious one. The second is the float.

Unbound, the back float is the gap #

With nothing dropped under it, the pattern weft on the back runs from the edge of one figure to the edge of the next. The float is the gap, exactly, whatever the ground weave: twenty-eight ends in the drawing, which at twenty-four ends to the centimetre is 11.7 millimetres of loose weft on the back of the cloth. A spot figure a centimetre across repeated every five centimetres leaves ninety-six ends and four centimetres.

That is the float the float limit exists to forbid, placed where nobody sees it until a finger or a heel catches it. A back float is not safe for being on the back. It is where a garment rubs its lining and a furnishing fabric rubs its frame, and the damage a float takes goes with its length whichever face it is on.

The trade has a blunt answer for some cloths: cut it. A clip-spot fabric is woven lancé and the back floats are sheared off after weaving, leaving each figure’s pattern weft as a short tuft at its edges — the same knife that makes a corduroy out of a float, used this time to remove one. It works where the figure is bound firmly enough to hold its cut ends, and its cost is plain arithmetic: in the drawing the figure covers twelve ends in forty, so seventy per cent of every pattern pick is woven only to be cut away.

The other answer is to bind the float, and binding has a cost that is less obvious and more exact.

A dropped end shows, unless the ground already drops it #

To bind a back float an end is dropped under the pattern weft for one pick. At that point the pattern weft crosses over the end instead of behind it, and a speck of it comes to the face.

On a warp-faced satin, the face is warp everywhere except at the satin’s own interlacings, one per pick in every n ends, and those are hidden: each is a single sunken end with a long float pressing over it on either side. A binding point placed on the same end, on the pick beside a ground interlacing, sinks into the same place. The end is already dropped for the ground pick; dropping it for the pattern pick too extends the sink by one pick, under the same floats. A binding there is hidden. A binding anywhere else is a new sunken point in a warp surface, and a speck of pattern weft shows — one end interlacing where its neighbours do not, the same kind of mark a bound point-tie turn leaves down the middle of a cloth. That is the criterion this account uses, and it is stated as an assumption below.

It makes the hidden places countable. A pattern pick lying between ground picks i and i + 1 can hide under the end the satin drops on pick i and the end it drops on pick i + 1. The satin’s interlacing steps across by its move from one pick to the next, so those two ends are m apart — and the next pair, one repeat along, is n − m further. Every pattern pick has exactly two hidden places per repeat, alternately m and n − m ends apart, and bound at all of them it floats at most the larger gap less one.

On the eight-end satin on three the gaps are three and five, so the hidden-bound back float is four ends: 1.7 millimetres at twenty-four ends a centimetre, comfortably inside a three-millimetre rule, with nothing on the face.

The ground’s move sets the float, and the smoothest move sets it longest #

The hidden float depends on the move and on nothing else: the larger of m and n − m, less one. It is shortest when the move is near half the order and longest when the move is small.

A satin chosen for its surface wants the opposite. The move that spreads a satin’s interlacings furthest sits near the square root of the order, and past eight ends the square root is well short of half. So at every order with more than one pair of moves, the move that makes the smoothest ground and the move that binds a brocade shortest are liable to be different moves.

At five, eight, ten and twelve ends there is only one pair of moves to choose between, so there is no disagreement. At seven ends the two moves weave equally smooth grounds and bind at four and three. At eleven the smoothest grounds, on three and four, bind at seven and six, and the move that binds shortest, five, weaves a ground whose closest interlacings are √5 apart against √10. At thirteen the smoothest ground, on five, binds at seven, and the shortest binding, on six, again comes with interlacings √5 apart.

Sixteen ends is where it matters most, because sixteen is where the hidden float grows past the limit. The smoothest sixteen-end grounds, on three and five, leave hidden floats of twelve and ten ends — 5.0 and 4.2 millimetres. The move of seven leaves eight, 3.3 millimetres, and weaves a ground whose closest interlacings are √8 apart instead of √10, whose interlacings sit more than twice as close along its diagonal as across it.

So a designer choosing a sixteen-end satin ground for a brocade is choosing between a smoother ground and a shorter back float, and no move gives both. The satin that is best in a plain cloth is the worst of the three here.

Every end of float shorter than that is paid for on the face #

Below the hidden float, binding costs visible points, and the number is exact. A stretch of G ends between hidden places, held to a float of L, needs ⌈G/(L + 1)⌉ − 1 more binding points, and each of those shows.

On the eight-end satin on three, the two stretches are three and five ends. Holding the back float to three or to two costs one visible point in every eight ends of gap, on every pattern pick; holding it to one costs three. At twenty-four ends and twenty-two picks to the centimetre, with a pattern pick after every ground pick, a limit of two puts sixty-six specks of pattern weft in every square centimetre of gap. That is a speckled ground, and for a pattern weft of a contrasting colour it is a different cloth.

The three-millimetre rule is seven ends at this sett. The eight-, twelve- and thirteen-end grounds meet it with hidden binding alone, since their hidden floats are four, six and seven. All three sixteen-end grounds miss it, and each needs the same single visible point in every sixteen ends to reach it — which is the case where the choice of move stops mattering for the float and matters only for the surface.

The count also says where the visible points must go. They cannot be spread evenly across the gap, because the hidden places are not evenly spread: they fall in the longer of the two stretches, the one n − m ends wide, and on a small move every visible point on a pick lands in the same stretch. On the sixteen-end ground on three, the stretches are three and thirteen ends, and every binding the three-millimetre rule demands falls in the thirteen.

A binding warp takes the choice away from the ground #

Everything above follows from one coupling: the pattern weft is bound by the ground’s own ends, so where it can be bound invisibly is wherever the ground happens to drop them. Remove the coupling and the problem goes.

That is what a lampas does. A lampas has two warps. The main warp weaves the ground with the ground weft; a second, finer binding warp does nothing but hold down the pattern wefts, on the face over the figure and on the back between. The binding warp’s ends are not the ground’s ends, so its binding points are placed by its own weave — a plain or a twill of its own, at whatever interval the designer wants — and dropping one of them puts nothing into the ground’s face at all.

The price is the second warp. It has to be warped, drawn through its own heddles or harness cords, and usually carried on its own beam, because an end that interlaces with the pattern wefts on a different rhythm from the ground’s can take up at a different rate. A jacquard that already gives every end its own hook makes the binding warp’s lifting easy. It does nothing about its warping.

So the three constructions sort by where they pay. Broché pays in hand work at every figure. Lancé on a satin ground pays in back float or in face specks, with the ground’s move setting the exchange rate. Lampas pays in a second warp and removes the exchange rate altogether.

What was counted, and how #

Every brocade here was woven as a matrix: a warp-faced satin ground tiled across the width, an extra pick after each ground pick, face over figures bound every four ends away from the figure’s edges, back between them. Back floats were measured as the runs of covered cells on the pattern picks, cyclically across the repeat, and the connectivity was taken on the whole matrix.

The unbound back float was confirmed to equal the gap at four gap widths. Hidden binding — dropping only ends the ground drops on either adjacent pick — was confirmed for every satin move at five, seven, eight, ten, eleven, twelve, thirteen and sixteen ends to leave a back float of exactly the larger of the move and its complement, less one, with no binding point off a hidden place. For every limit below that, at every one of those moves, the fewest bindings that meet the limit were placed by walking each back float and dropping an end whenever the limit was reached, and the visible count was confirmed against the stretch formula. A lancé pick meeting no figure was confirmed to make the matrix two cloths.

The millimetres assume twenty-four ends and twenty-two picks to the centimetre and a three-millimetre float limit, the figures the damask argument uses.

Where the model stops #

The hidden-binding criterion is an assumption. A binding point beside a ground interlacing extends a sunken end from one pick to two, and whether the long floats either side cover two picks as well as they cover one depends on the ground’s sett, the floats’ length and the yarns’ stiffness. A strongly contrasting pattern weft may show even there; a close-set silk ground may hide a binding anywhere. The count is exact given the criterion, and the criterion is the part that would have to be looked at.

The pattern weft is taken at one pick per ground pick. Brocades throw it after every second or third ground pick as often as after every one, which changes how many pattern picks lie between which ground picks and so which ends are adjacent. The argument transfers — two hidden places per repeat per pattern pick — but the adjacent pairs change.

The pattern weft takes no room. An extra pick thickens the cloth and pushes the ground picks apart where it lies on the back, which alters the ground’s float lengths in millimetres and its cover. None of that is in a matrix of noughts and ones.

And only warp-faced satin grounds are counted. A twill ground has interlacings in lines rather than scattered, and its hidden places line up with them; a weft-faced ground hides a pattern weft differently, among its own weft. Both are the same question on a different matrix.

Still open: whether brocade binding sits where the count puts it #

The count makes a prediction that can be checked on existing cloth. A lancé brocade on an eight-end satin ground, turned over and examined under a glass, should show its pattern weft bound on the back at ends lying beside the ground’s own interlacings, three and five ends apart, with back floats of four or less — and a brocade on a sixteen-end ground should show either longer back floats or bindings that break through on the face.

If historical brocades bind elsewhere, the hidden-binding criterion is wrong, and the weavers knew a better one. If they bind where the count says, the ground’s move was being chosen with the pattern weft in mind as well as the surface. Neither examination has been made here.

Who worked it out #

Brocade, lampas, and the distinction between lancé and broché wefts are old: figured silks bound with a separate binding warp were being woven in China and Iran about a thousand years ago, and European silk weaving inherited them. The vocabulary used here is the one textile historians settled on in the twentieth century for describing such cloths structurally, which names the binding warp and the two kinds of pattern weft precisely because the structures differ. The arithmetic of hidden places on a satin ground, and its conflict with the smoothest move, was computed directly.