An easer gives back the kink the crossed shed puts in

The count of what a doup end pays for its crossing left one device on a leno loom unexplained. Many leno looms carry an easer, which gives the crossing ends slack as the crossed shed opens and takes it back as the shed closes, pick after pick. That count found the obvious reason for it wanting. Carrying a doup end five millimetres sideways at its heddle lengthens it by 0.056 millimetres, where the shed itself lengthens an ordinary end on the same shaft by 5.5 — a hundredth. Nobody builds a mechanism to relieve a hundredth of a shed.

The length an easer gives back is real, and it is one heddle further back. A crossing end is threaded through two heddles, and in the crossed shed one of them lifts it while the other holds it down. Between those two eyes the end is pulled up one side of a steep triangle and down the other, and the triangle is a few centimetres long and more than ten centimetres high.

That is exact trigonometry on the loom the shed was first priced on, with nothing fitted in it, and it is not small. On that loom it stretches the crossing end nearly thirteen times as far as the shed stretches an ordinary end, and it reaches the region where cotton breaks.

A crossing end is threaded through two heddles #

A leno pair is two warp ends, one of which changes sides. The warp-and-weft matrix cannot even describe the change, because the end at one position on one pick is a different end on the next. A loom has to do it with hardware, and the usual hardware is two heddles for the one end.

At the front of the harness is the doup, a half heddle whose loop passes beneath the partner end and holds the crossing end on the far side of it. Some shafts behind the doup is the crossing end’s own ordinary heddle, called the back standard, which holds it on its own side.

In the open shed the back standard lifts, and the end rises on its own side of its partner; the doup is left slack and follows. In the crossed shed the doup lifts instead, and the end rises on the far side. The back standard does not lift in the crossed shed, and it cannot: the end is being carried beneath a partner that is itself down, and an end held up behind the doup would have to dive under that partner to reach the far side.

So in the crossed shed the same thread is raised at one eye and held down at the next, a few shaft pitches apart. That is a different path from any shed a loom opens with shafts alone, where each end is displaced at exactly one place.

The loom the shed was priced on #

The shed was shown to be an extension on an ordinary broad loom: the reed 90 millimetres in front of the fell, the first shaft at 300, the shafts at 16-millimetre centres, the back rest at 1,200, and 30 millimetres of clear opening at the reed for a shuttle. One condition fixes every lift. The same opening has to reach the reed from every shaft, so each heddle eye is displaced from the straight warp line in proportion to its own distance from the fell — a sixth of that distance on this loom.

An end on the front shaft is lifted 50 millimetres, and its path from fell to back rest is 5.53 millimetres longer than the straight line: 0.46 per cent of the free warp. That is the number every shed on this loom is measured against.

Put the doup on the front shaft’s position and the back standard four shafts behind it, 64 millimetres further back. In the crossed shed the doup’s eye is up 50 millimetres. The back standard’s eye is down by a sixth of its own distance from the fell, which is 60.7 millimetres.

Eleven centimetres of climb in six and a half #

Between the two eyes the end climbs 110.7 millimetres over a run of 64. That leg is steeper than sixty degrees, and its length is 127.8 millimetres where the straight line is 64. The kink alone lengthens the end by 63.8 millimetres.

The two outer legs add little. From the fell to the doup the end runs the ordinary shed’s front leg and gains 4.14 millimetres; from the back standard to the back rest it runs down to the line again over 836 millimetres and gains 2.20. The whole path is 70.2 millimetres longer than the straight line, which is 5.85 per cent of the free warp.

The same end in the open shed, lifted at its back standard, is 7.2 millimetres longer — 0.60 per cent, a little more than the front shaft because the back standard is further back. The crossed shed is 12.7 times what an ordinary end on the doup’s shaft takes, and nine tenths of that is the one steep leg between two heddles.

The sideways carry was the wrong eye #

The first candidate followed the doup end along the cloth, between one crossing and the next, where the crossing is a journey of a millimetre or less: across, under the partner and up. On that scale the sideways carry at the doup’s eye is the natural thing to suspect, and it is genuinely negligible.

The crossed shed has to be followed along the loom instead, from eye to eye, where the distances are centimetres. The comparison is not close. The kink is more than a thousand times the sideways carry, and ten times the shed an ordinary end takes on the same shaft.

A back rest six centimetres behind the front shaft #

The shed essay found its strain growing as the shaft’s distance from the fell over the distance left behind it, and it drew the warning out of that denominator: the strain grows without bound as a shaft approaches the back rest, so a harness deep enough would tear its own warp with nothing else happening. On a broad loom that pole is far away, and a one per cent budget buys thirteen shafts before the strain gets there.

A leno reaches the pole with two heddles. In the crossed shed the end’s own back standard is the nearest point that holds it, and it stands 64 millimetres behind the doup instead of 900. A lift of fifty millimetres against a back rest six centimetres away is the shed essay’s hyperbola evaluated nearly at its asymptote.

The back standard does worse than stand still. If its eye merely stayed on the straight line the kink would already cost 21.4 millimetres, nearly four times an ordinary shed. It is pulled the other way, 60.7 millimetres down, so the climb between the eyes is not the doup’s lift but the doup’s lift and the back standard’s fall together — a full shed height across a few shaft pitches.

Seen that way the easer is not an odd attachment peculiar to leno. It is what any harness would need if two of its heddles could pull one end in opposite directions, and leno is the one construction that has to.

Nearly six per cent is a load near breaking #

What the shed costs in newtons turned the loom’s strains into tensions with a yarn’s modulus. A 25 tex cotton yarn at 700 turns a metre takes 112.6 newtons per unit strain and breaks at 3.74 newtons, so the front shaft’s 0.46 per cent is 0.52 newtons, fourteen per cent of breaking load, and a one per cent budget is a working stress close to where a fatigue allowance would be set.

The crossed shed’s 5.85 per cent, through the same modulus, is 6.58 newtons: 1.76 times the yarn’s breaking load. That reading is linear and a yarn this far along its load–extension curve is softer than its initial slope, so it overstates. It does not need to be right by a factor of two. The shed essay quotes cotton breaking somewhere near seven per cent extension, and a crossing end pulled 5.85 per cent at every crossing, on top of the warp’s working tension, is being taken most of the way to it several hundred times a metre.

That is what an easer is for, and the arithmetic says it is not optional. A leno woven on this loom’s shed without one would break its crossing ends at the doup.

Moving the back heddle back helps, and then stops #

The obvious remedy is to move the back standard further from the doup, so that the kink’s run is longer and its leg shallower. It works, and not for long.

With the back standard one shaft behind the doup the crossed shed lengthens the end by 93.6 millimetres, 7.80 per cent. Four shafts back it is the 70.2 already found; eight shafts back, 55.8 millimetres and 4.65 per cent. It keeps falling to a least of 47.7 millimetres, 3.98 per cent, seventeen shafts back — 272 millimetres behind the doup — and then rises again, reaching 6.69 per cent at forty shafts.

The minimum is the clean-shed condition turning on itself. The further back an eye is, the further it has to move, which is why the back of a harness works harder than the front, and it is equally true of an eye that is being held down. A back standard seventeen shafts back gives the kink a run four times as long as one four shafts back, and its eye is 95.3 millimetres down instead of 60.7, so the climb grows by a third while the run quadruples. Beyond seventeen shafts the growing fall wins.

Moving the back standard back has a second price, on the other shed. In the open shed it is the back standard that lifts the end, and a back shaft’s lift is a back shaft’s strain: the open shed passes one per cent at thirteen shafts, and at seventeen it is 1.26 per cent on its own. Pushing the back heddle back trades the crossed shed down against the open shed up, and there is no position at which both sit inside the budget.

At its least the kink is still most of it #

At the best position the doup is still 50 millimetres up and the back standard 95.3 millimetres down, and the leg between them runs 272 millimetres. That leg lengthens the end by 36.4 millimetres of the 47.7; the back leg, which now runs from much further down to the back rest, adds 7.2.

So the least the crossed shed can be made on this loom by moving heddles is 8.6 times an ordinary shed, three quarters of it in the kink, and every position from one shaft to forty is more than eight times.

A shallower shed shrinks it more slowly than it shrinks a shed #

The other lever the shed essay found was the shed itself. Strain goes as the square of the shed angle, so halving the opening at the reed quarters the strain, and that is why looms that need a smaller shed can carry deeper harnesses. For an ordinary end it holds exactly: at a 16-millimetre opening the front shaft takes 0.13 per cent, 0.285 of what it takes at thirty, which is the square of sixteen over thirty.

The crossed shed does not follow the square. At a 16-millimetre opening, with the back standard four shafts back, it is 2.07 per cent — 0.354 of what it was, not 0.285. A shallow leg’s extra length grows with the square of its climb and a steep one’s grows nearly in proportion, and the kink is steep, so narrowing the shed takes it down more slowly than it takes down a shed.

So the ratio of the two climbs as the shed closes: 10.5 at 44 millimetres, 12.7 at thirty, 15.8 at sixteen, 17.1 at ten. The crossed shed comes under one per cent only at about ten millimetres of opening, 0.88 per cent — a third of what this loom’s shuttle needs. A shallower shed is a real help and not a cure.

What remains has to be given back at every crossing #

Moving the back standard and narrowing the shed can be combined, and the combinations are worth setting side by side because every one of them leaves a length.

Seventeen shafts is not a position a loom usually has to spare, so take one, four and sixteen shafts, at the 30-millimetre shed and at sixteen. The best of the six — sixteen shafts back and a 16-millimetre shed — is 1.18 per cent, still over the budget and nine times an ordinary end. The worst, one shaft back at a shuttle’s shed, is 7.80 per cent.

Whatever an easer is made of, the length it has to hand over is fixed by this geometry. To leave the crossing end no worse off than an ordinary end on the doup’s shaft it must give back the crossed shed less the ordinary one: 64.7 millimetres at four shafts and a 30-millimetre shed, and still 12.6 at the best of the six. It has to do it as the crossed shed opens and take it back as the shed closes, because in the open shed the same end wants its length again.

For a gauze crossing at every pick, picks a millimetre apart, that is five hundred crossed sheds in every metre of cloth, and thirty-two metres of easer travel for every metre woven. A device that moves that much, that often, is a large part of why leno is woven slowly.

A large length that comes back and a small one that does not #

The doup end now has two demands on its supply that the loom answers with two different devices, and setting them on one axis shows why neither device can do the other’s work.

The crossing’s own extra yarn is small and permanent. A doup end of 0.25-millimetre yarn crossing every millimetre runs 22.5 per cent further than its partner, so every millimetre of cloth takes 0.225 millimetres more of it, and none of it is ever given back. It is why a leno needs its doup ends fed separately, from a second beam that turns faster.

The kink is large and temporary. It asks for 70.2 millimetres at every crossed shed and hands all of it back at the next open one, so it never grows however much cloth is woven.

The accumulated extra yarn passes the kink after 312 millimetres of cloth, and after a hundred metres it is twenty-two metres against seventy millimetres. A second beam turning faster pays yarn out and can never take it back, so it cannot answer the kink; an easer swings back and forth and adds nothing over a piece, so it cannot answer the accumulation. A leno loom carries both because the crossing costs the doup end two different kinds of length.

A mock leno needs neither device #

There is a construction that makes the same holes and asks for neither length. A mock leno groups its ends on ordinary shafts, so every end is lifted at one eye by one shaft and takes an ordinary shed. There is no doup, no back standard, no kink and no crossing excess.

It pays in grip instead. With no end crossing another, a mock leno’s picks are held only by the crimp of an open weave, which is the grip that falls away as a cloth opens, and its holes are free to wander in exactly the way a leno’s holes cannot. The two devices on a leno loom are what the half turn costs at the loom, in the same way the doup end’s extra yarn is what it costs in warp.

Three straight legs on one loom #

Every length here is taken on the ordinary broad loom the shed essays use, with a centre-closed shed in which each heddle eye is displaced from the straight warp line by a sixth of its distance from the fell. In the crossed shed the crossing end runs in three straight legs: from the fell up to the doup’s eye, from there down to the back standard’s eye, and from there up to the back rest. The extension is the three legs less the straight line; the strain is that extension over the 1,200 millimetres of free warp; the open shed is two legs through the back standard’s lifted eye, and the ordinary shed two legs through the doup’s.

The arithmetic was confirmed to give an end on the doup’s own shaft exactly the front shaft’s shed; to sum its three legs to the whole; to leave every strain unchanged when every distance on the loom is doubled; to put the crossed shed above the open shed and above eight times the ordinary shed at every back standard position from one shaft to forty; and to find the least strictly inside that range, with the curve falling all the way to it. Tensions use the modulus and breaking load of 25 tex cotton at 700 turns a metre from the calculation that put the shed into newtons. The sideways carry is the earlier count’s own figure.

Where the eyes are taken as points #

The end slides freely through both eyes. Every strain above spreads the extension over the whole free warp. A heddle eye has friction in it, and the end turns 69 degrees at the doup and 64 at the back standard, which at a friction coefficient of 0.3 is a capstan of 1.44 and 1.40. If the eyes held completely, the kink’s 63.8 millimetres would be taken up in the 64-millimetre span between them, a strain near a hundred per cent, which cannot happen. The truth lies between free sliding and that impossibility, and it is worse than the figures above.

The eyes are points and the doup’s crossing happens at its eye. A real doup carries the end round its partner over a few millimetres of heddle, and a real eye has a length. Both shift the corners slightly and neither changes a climb of a full shed height across a few shaft pitches.

The shed closes at its centre. A loom whose unlifted ends rest at the bottom of the shed moves a lifted eye twice as far and a held eye not at all, so the climb between the eyes is 100 millimetres rather than 110.7 at four shafts back — a full shed height either way, and the same order of kink.

The tension is linear in the strain, which overstates it at several per cent. The comparison with breaking load is made in the direction that reading cannot reverse.

Nothing here says how an easer is built, where along the warp it acts, or how its motion is timed against the shed. It says what length any easer must give back.

Still open: how much of the kink the eyes let through #

The length is fixed by geometry and the strain is not, because the strain depends on where the length is taken up. If the crossing end slides freely through both eyes the kink is shared by 1,200 millimetres of warp; if the eyes grip, it is concentrated between them. The two capstans of 1.4 decide which, and so does where the easer gives its slack back: an easer behind the harness has to pass that slack through the back standard’s eye to reach the kink at all.

A tension trace on one crossing end, taken in front of the doup and behind the back standard through a single pick cycle with the easer working and with it locked, would say how much of the kink each eye lets through. That is the measurement that would turn the length here into a strain on a real loom.

Who described the harness, and what is new here #

The doup, the back standard and the easing motion that gives crossing ends slack in the crossed shed are described in weaving manuals for leno and gauze work, and the arrangement is old enough that its origin is not recorded. The three-leg path of the crossing end, its size against the shed on a stated loom, the position of the back standard at which it is least, the way it shrinks more slowly than a shed as the opening narrows, and its separation from the doup end’s accumulated extra yarn were computed directly.