A jacquard harness needs three half-spans of height

A shaft loom cannot usefully carry two hundred shafts, and the reason is not its mechanism. Each shaft stands further from the fell than the one in front of it, each has to lift further to make the same opening at the reed, and a stated tolerance on warp strain is a stated depth of harness — thirteen shafts at one per cent. A jacquard escapes that bound completely. It has no shafts: every end has its own hook and its own cord, and every cord comes down from the same machine to a mail at the same distance from the fell. There is no back of the harness to work hardest.

That is true along the warp. Across it, a jacquard harness has a dimension a shaft loom never had. The hooks stand in a machine a few tens of centimetres wide, mounted above the loom. The cords run down from them to a comber board — a perforated board as wide as the cloth, one hole per end — and hang straight from the board to their mails and weights. Between the machine and the board they fan out.

A cord that fans out leans, and a leaning cord does not lift its mail as far as its hook rises. How much less is exact trigonometry, it is worst at the selvedges, and holding it to a stated tolerance fixes how far above the comber board the hooks must be. For a five per cent tolerance the answer is about three times the edge cord’s sideways reach, and for a cloth of furnishing width that is two metres of empty cord before the machine itself begins.

Above the comber board every cord but the middle one leans #

Take a comber board 130 centimetres wide under hooks spread over 40, and suppose each hook feeds the hole in the same position across the board — the first hook the first hole, the middle hook the middle hole, the last hook the last. The middle cord drops vertically. Every other cord leaves its hook and arrives at its hole some distance to the side, and the distance grows steadily outwards, to 45 centimetres at each selvedge: half the difference of the two widths.

With the hooks 150 centimetres above the board, that edge cord leans 16.7 degrees from the vertical. With the hooks 60 centimetres above, it leans 36.9.

The hole in the comber board is what makes the lean matter. It is a fixed point: the cord passes through it and hangs vertically below, so whatever length the cord gains above the board is drawn up through the hole and appears as a rise of the mail. The question is how much length a leaning cord gains when its hook rises.

The mail rises by the difference of two hypotenuses #

Before the lift, the cord above the board is the hypotenuse of a triangle whose sides are the fall H and the sideways reach d. After the hook rises by s, it is the hypotenuse of H + s and d. The mail rises by the difference, √((H + s)² + d²) − √(H² + d²), and nothing else enters.

For a small lift the difference is s times the cosine of the cord’s angle, because only the part of the hook’s motion along the cord lengthens it. For a real lift it is slightly more than that, since the cord straightens a little as it rises. At a fall of 150 centimetres and a reach of 45 the cosine is 0.958 and the exact ratio for a ten-centimetre lift is 0.960: the edge mail rises 9.60 centimetres where the centre mail rises ten.

Two properties make the number usable. It has no scale: double the fall, the reach and the lift together and the ratio does not change, so the answer is a statement about the shape of the mount rather than its size. And it barely depends on the lift. A lift of five, ten or fifteen centimetres changes the height a given tolerance requires by about two per cent, which is why the rest of this essay can quote heights without quoting a hook stroke.

A low mount starves the edges #

Four per cent is a small loss. Lower the machine and it stops being small. At a fall of 60 centimetres the same edge cord leans nearly 37 degrees and its mail rises 8.22 centimetres for a ten-centimetre lift, 82 per cent of the centre’s. The shed at the selvedge opens nearly a fifth less than the shed in the middle of the cloth, on every pick.

A shed that is short at one side is a shed that closes on the shuttle there. The shuttle’s path is set by the reed and race, and the warp sheet has to clear it everywhere across the width; an edge that opens less has either to be given more lift, which over-strains the centre, or accepted as the place where ends are caught and broken.

The loss is flat in the middle and steep at the edges #

Across the board the loss is not spread evenly. Near the centre the reach is small and the cosine of a small angle is almost exactly one, so the loss grows with the square of the distance and is negligible over the middle of the cloth. Towards the edges the angle is no longer small and the loss runs away.

At a fall of 60 centimetres the five-per-cent line is crossed about thirty centimetres from the centre on each side, so more than half the width of the cloth weaves a shed more than five per cent short. At 100 centimetres the edge is 8 per cent short; at 150, 4 per cent; at 250, one and a half. Each extra metre of height buys less than the one before it, because the cosine flattens out as the angle closes.

A five per cent shed needs about three half-spans #

Turn the question round: for a stated tolerance, how high must the hooks be? The answer comes out of the same two hypotenuses, found by bisection on the exact rise, and it scales with the edge cord’s reach — the half-span, half the difference between the comber board’s width and the hook spread.

For a tolerance of five per cent the fall comes out at 2.6 to 3.0 half-spans, rising slowly towards three as the half-span grows. For two per cent it is 4.5 to 4.9; for ten per cent, 1.6 to 2.0. The small-lift rule, d cot θ with cos θ = 1 − tolerance, gives 3.04 half-spans at five per cent and is slightly conservative everywhere, which makes it a safe figure to carry in the head.

In centimetres, with hooks spread over forty: a narrow comber board of 60 centimetres, a reach of ten, needs a fall of 26. A board of 90 needs 71. The 130-centimetre board of the drawings needs 132. A furnishing board of 180 centimetres, a reach of 70, needs 208 — and at a two per cent tolerance, 340.

That is the fall alone. Above the hooks stands the machine that lifts them, and below the comber board hang the lower cords to the mails, the mails to the warp line, and the loom beneath all of it. A furnishing jacquard needs two metres of free cord added to all of that, which is to say a room considerably taller than a room is otherwise built. The silk weavers’ workshops of Lyon’s Croix-Rousse are commonly described as having been built with ceilings around four metres high for exactly this machine, and the arithmetic says why a lower room would have woven narrower cloth or worse sheds.

Height is also load, and the edge holes wear first #

The cord does not pass through the comber board straight. It arrives at the leaning angle and leaves vertically, so it bends through that angle over the edge of the hole, and a cord bent over a fixed edge carries friction by the capstan relation: the tension on the hook’s side is the weight’s tension times e raised to the friction coefficient times the angle.

At a fall of 150 centimetres and a friction coefficient of 0.3, the edge cord’s hook carries 9 per cent more load than the centre cord’s for the same weight below. At 60 centimetres it carries 21 per cent more; at 250, five and a half. With a friction coefficient of 0.4, the low mount’s edge hooks carry 29 per cent more.

So the low mount costs twice at the same place. Its edge ends rise least and their hooks work hardest to raise them, and the cord’s sawing across the hole edge is worst at the angle that causes both. The comber board’s outermost holes, and the cords through them, are where that mount wears out.

The jacquard’s evenness is a property of a tall mount #

The depth argument concluded that a jacquard’s ends all take the same strain, because every mail sits at the same distance from the fell. That is right for the distance and needs a qualification for the lift. The strain a shed puts on an end goes with the square of the end’s lift, so an edge mail that rises 82 per cent as far as the centre’s is strained about two thirds as much — the edge ends of a low mount are spared, and their shed is short.

A jacquard escapes the unevenness of a deep harness by trading it for an unevenness across the width, and how much of that it accepts is set by the height of its mount. On a tall mount both are negligible, which is what the depth argument assumed. On a low one the jacquard reproduces, from side to side, a version of what the back shaft does from front to back — a gradient in how far ends are lifted, decided by the machine’s geometry and invisible in the draft.

The levers are the two widths. A wider hook spread shortens every reach, which is one reason larger machines are not merely more hooks; a narrower cloth does the same. Neither is free, and the room’s height is the lever that remains.

A dobby pays in depth and a jacquard pays in height #

Set the two machines side by side and each pays for its harness in a different dimension. A dobby stores its design as lifts of shafts, and a shaft is a rigid bar as wide as the warp: every heddle on it rises exactly as far as every other, so a dobby has no gradient across the width at all. Its gradient runs front to back, one shaft behind another, and it is bounded by depth. A jacquard abolishes the shafts and with them the depth, and replaces a rigid bar with a fan of cords, which introduces the gradient across the width that the bar never had.

Neither gradient is visible in a draft. The dobby’s is decided by the threading — which shaft an end hangs on — and the jacquard’s by the tie, which hole a hook feeds. Both are properties of the machine that the matrix describing the cloth does not contain, which is why a threading that economises on shafts can look perfect on paper and still work its back shafts’ ends hardest, and why a jacquard design can look perfect and still weave a short shed at its selvedges.

The two also scale differently with the cloth. A dobby weaving a wider cloth needs longer shafts and heavier lifting, and no more depth. A jacquard weaving a wider cloth needs a wider comber board and, by the three-half-span rule, a proportionally taller mount — so a cloth half as wide again asks for half as much height again above the board. That is one arithmetic reason the widest jacquard cloths have tended to be woven with the design turned at the centre: a point tie doubles the width of the figure for a given number of hooks, but it does not shorten a single cord’s reach, because the comber board is as wide as the cloth whatever the tie. Turning the design saves hooks and costs nothing in height; it also saves nothing.

What does shorten the reach is placing the hooks over more of the board. Mounting the machine so that its hook rows run across the loom rather than along it, or using two machines side by side over a very wide cloth, each over its own half of the comber board, both cut the half-span directly — the second halves it, and by the rule halves the height with it. Those are arrangements the arithmetic recommends and a harness builder would weigh against the cost of a second machine and the difficulty of keeping two in step.

A brocade adds one more consideration. A pattern weft bound by a separate binding warp needs that warp lifted too, and on a jacquard its cords come through the same comber board and fan from the same machine, with the same reach and the same loss at the edges. A binding end at the selvedge rises less than one in the middle, which is harmless while its lift is generous and becomes a binding that fails to catch its pattern weft when the mount is low. The height of the mount is therefore a limit on the finer figured cloths as well as on the wider ones — and the widths those cloths are woven to were chosen in rooms whose ceilings had already been built.

What was counted, and how #

Each cord’s rise was computed as the exact difference of two hypotenuses, not as a cosine, and checked to lie above the cosine by less than one per cent and to approach it as the lift vanishes. The centre cord was checked to lift its mail by the full stroke, the rise was checked to fall steadily with reach, and the ratio was checked unchanged when fall, reach and lift were all doubled.

The least fall for each tolerance was found by bisection on the exact rise at half-spans from 10 to 100 centimetres and checked to lie within stated bands of half-spans — 4.6 to 4.95 at two per cent, 2.75 to 3.05 at five, 1.8 to 2.05 at ten — and never above the small-lift rule. The friction factor is the capstan relation on the edge cord’s angle at the stated coefficient.

The hook spread of forty centimetres, the hook lift of ten, and the proportional assignment of hooks to holes are the model’s inputs, not measurements of any particular machine.

Where the model stops #

The tie is assumed to spread one row of hooks evenly over the board. A real harness is tied in one of several arrangements that decide which hook feeds which hole, and a real machine’s hooks stand in a grid several rows deep, which leans the cords fore and aft as well as sideways. Among the ways of assigning a row of hooks to a row of holes, keeping their order makes the longest reach as short as it can be, so for a single row the heights here are lower bounds.

The cords are taken as inextensible and weightless. A long cord stretches under its weight’s pull, and stretches more the longer it is, so a tall mount gives back a little of what its height gains. The mails and weights below the board are taken to follow the cord exactly.

The tolerance is an assumption. Five per cent is a round number for how much shorter an edge shed may be than a central one, not a figure from any weaver’s specification; the heights for two and ten per cent are given so that a reader with a better number can read off its consequence.

And the friction coefficient is a range. Cord on a board’s hole edge depends on the cord, the board’s material, its polish and its wear, and the loads are ratios at stated coefficients rather than measurements.

Still open: how short an edge shed a shuttle forgives #

Everything here turns on one number the arithmetic cannot supply: how much less an edge shed may open before the shuttle strikes the warp there. It depends on the shuttle’s height, the reed’s angle at the moment of passage, the race board, and the speed — and it is measurable on any jacquard by lowering the lift until the selvedge ends start to break.

With that number in hand, the three half-spans become a specification: a mount height for a given cloth width and machine, and a reason to reject a room. Without it they are a scale — the right order of magnitude for why jacquard rooms are tall, and no more than that.

Who worked it out #

The comber board and the fanned harness came to the jacquard from the drawloom, whose harness had the same geometry with a person rather than a machine pulling the cords, and Jacquard’s mechanism of 1804 was mounted over them without changing it. The cosine loss of a leaning cord is elementary and weavers who mounted harnesses dealt with its consequences by adjusting cords and heights. The exact rise, the tolerance heights and the friction at the board were computed directly.