The doup end pays for the crossing

Four essays have explained why a leno holds. Its doup end crosses under its partner between picks and comes up on the other side, which the warp-and-weft matrix cannot even describe. The crossing wraps the pick through half a turn however openly the cloth is set, so no plain weave at any sett grips as hard; the holes cannot drift, because friction that has no sett in it does not fall away as the cloth opens; and the ends are linked, which no other woven structure’s are.

None of those essays asked what the crossing costs. The half turn that holds the pick is made by a thread going somewhere its partner does not: across, down, under and up the other side. That journey is a length of yarn, the doup end has to supply it at every crossing, and its partner does not. Over a piece of cloth the difference adds up to metres, and it decides how the loom must be dressed.

The length is exact geometry once the ends’ sizes and the crossing interval are stated, and it has one feature worth knowing before any number: it falls with the square of the distance between crossings. A gauze crossing at every pick pays heavily; a curtain leno crossing at every other pick pays a quarter as much for half as many crossings.

Both ends share the crimp, and only one changes sides #

A leno pair lifts together. On every pick both ends of the pair go into the same shed, so both pass over or under the same picks, and both take up exactly the same crimp. Whatever the ground weave costs in warp, it costs both ends of a pair alike.

What differs is the crossing. Between one crossing and the next the standard end runs straight along the cloth. The doup end starts beside it on one side, passes beneath it, and comes up beside it on the other. That is all the doup end does that its partner does not, and so it is the whole of the difference in the yarn they use.

Two straight legs per crossing #

Take the shortest path the doup end can follow. At one crossing it lies beside its partner, touching, its centre offset sideways by half the sum of the two diameters. Halfway to the next crossing it is directly beneath its partner, touching again, its centre the same half-sum below. At the next crossing it lies beside its partner on the other side.

Between those three points the path is two straight legs, each the diagonal of a box whose sides are half the crossing interval, the sideways offset and the depth. For ends 0.25 millimetres across, crossing every millimetre, each leg is the diagonal of 0.5, 0.25 and 0.25 millimetres, and the two together run 1.225 millimetres where the standard end runs 1.000.

The doup end uses 22.5 per cent more yarn. A little over half of that is the sideways journey alone: a doup end that could cross without dipping under its partner would still use 11.8 per cent more. The dip beneath adds the rest, because the path is a diagonal in three directions, not two.

The extra falls with the square of the crossing interval #

Stretch the crossings further apart and the legs lie flatter. For a crossing interval several times the yarn diameter the extra fraction is very nearly twice the sum of the squared offset and squared depth, divided by the square of the interval — for touching ends of one diameter, four diameters squared over the interval squared. Double the interval and the extra falls to a quarter.

So a leno crossing at every other pick, with picks still a millimetre apart, uses 6.1 per cent more doup yarn instead of 22.5. Nothing about which pick the crossing falls on enters: a crossing every other pick at one millimetre is exactly the same path as a crossing every pick at two.

The other direction is severe. At half a millimetre between crossings the doup end uses 73 per cent more yarn than its partner, and at three-quarters of a millimetre 37 per cent; spread to three millimetres, 2.7 per cent. The fine gauzes that cross at every pick at close pick spacing are the constructions in which the doup end does most of the work of a whole second warp.

The approximation also says what the yarn does to it. The offset and the depth are both set by the ends’ diameters, so finer ends pay less in proportion: 0.15 millimetre ends crossing every millimetre use 8.6 per cent more doup yarn, not 22.5. And any clearance between the ends — a doup end that does not hug its partner — lengthens both legs: a twentieth of a millimetre of clearance raises the 22.5 per cent to 31.

Because every length scales together, the fraction depends only on how many diameters apart the crossings are. A coarse leno of half-millimetre ends crossing every two millimetres pays exactly what a fine one of quarter-millimetre ends crossing every millimetre pays, 22.5 per cent, since both put their crossings four diameters apart. A curtain net of 0.2 millimetre ends at two millimetres between picks, crossing at every pick, puts them ten diameters apart and pays 3.9 per cent; crossing at every other pick, twenty diameters apart, it pays one per cent. So the rule a designer can carry is a count rather than a length: the doup’s extra yarn is about four over the square of the number of diameters between crossings, and the constructions that pay heavily are the ones whose crossings are only a few diameters apart.

A shared beam runs a pick short in millimetres #

A warp beam delivers one rate. If standard and doup ends come off the same beam, the loom is paying out equal lengths of both while the cloth consumes 22.5 per cent more of one. The same arithmetic priced two layers of a double cloth: a plain face over a five-end satin back differs by twelve percentage points of take-up, twelve metres over a hundred-metre piece, and that already has nowhere to go.

The gauze crossing every millimetre is nearly twice that. Over a hundred-metre piece its doup ends need 22.5 metres of warp more than its standard ends. On a shared beam the difference reaches one pick spacing of slack — a millimetre — after 4.4 millimetres of cloth. If instead the beam is braked hard enough that the doup ends cannot go slack, the shortfall has to come out of their length: spread over the 1,200 millimetres of free warp between the fell and the back rest, it reaches the one per cent strain the shed is budgeted to after 53 millimetres of cloth, and it goes on rising at the same rate for as long as the loom runs. Crossing every other pick, the difference is six metres per hundred and a pick of slack after 16.5 millimetres, which is longer but not a length a loom can wait out.

So a leno needs its doup ends fed separately, and the reason is a length, not a tension. Weaving practice puts doup ends on their own beam or their own weighted let-off; the geometry says how much faster that beam has to turn, and that no shared beam, however well tensioned, can be right for more than a few millimetres.

The grip halves per length while the yarn cost quarters #

The doup end’s extra yarn is the price of the grip, and the two can be set side by side. Each crossing wraps its pick through half a turn, so at a friction coefficient of 0.3 each crossing holds 2.57 times the tension it is given, whatever the sett. Along the cloth, the grip available per unit length goes with the number of crossings in that length — one over the crossing interval.

The yarn cost goes with one over the interval squared. So doubling the crossing interval halves the crossings along a length of cloth and quarters the extra doup yarn, and the yarn paid per crossing halves as well. A designer who spaces crossings twice as far apart buys each remaining crossing’s grip at half the price.

That is the arithmetic under the curtain leno’s usual choice of a crossing at every other pick rather than every one. It gives up half the crossings, and the pick is still held by a half turn at every crossing that remains — the property every earlier account of the leno rests on, and one that does not weaken because the crossings are fewer. What weakens is how many picks lie between crossings unheld, and that is set by how open a cloth can be before a pick slides between two crossings, which is the drift a leno was chosen to prevent.

A shared beam cannot be tensioned into working #

The two readings of a shared beam are the same fact from its two ends. Let the beam turn freely and the doup ends go slack at the rate their partners consume yarn; brake it until they cannot go slack and the doup ends stretch instead. Neither is a setting a weaver can find between the two, because the difference is not a tension that could be balanced but a length that accumulates.

Crossing every other pick, the one per cent budget lasts 198 millimetres of cloth instead of 53; with picks two millimetres apart crossing every other pick, 774 millimetres; at three millimetres, about a metre and three quarters. None of those is a piece. The warp strain a loom can tolerate is set by the shed and has nothing to spare for a second, cumulative strain on a quarter of the ends, so even the gentlest of these constructions exhausts it in the first metre or two of a hundred-metre piece.

That is also why the difference cannot be absorbed by crimp interchange, the mechanism that lets a woven cloth redistribute length between its warp and weft under load. Interchange moves a length that already exists in the cloth from one system to the other. The doup end’s extra length has to arrive from the beam, pick after pick, and no redistribution inside the cloth can supply a length that was never paid out.

A mock leno pays nothing and holds nothing #

There is a construction that looks like a leno and costs none of this. A mock leno groups its ends and floats its picks so that the cloth opens into the same pattern of holes, woven on ordinary shafts with no doup heddle and no end crossing any other. Every end takes the same path as its neighbours in its group, and the warp comes off one beam without complaint.

It also holds nothing the way a leno does. With no crossing there is no half turn, and its picks are gripped only by the crimp of an open weave, which is exactly the grip that falls away as a cloth opens. The mock leno is the control in the comparison this essay has been making: the same appearance, no doup yarn, and no grip beyond what friction at a shallow weave angle provides.

So the doup end’s extra yarn is not an inefficiency of the leno mechanism that a better loom could remove. It is the grip, measured in metres. A crossing that did not cost the doup end a diagonal would not wrap the pick, and the criterion that decides whether a cloth hangs together could not tell the two apart in any case — which is why the price and the property have to be counted separately, as a length and as a friction, and why only the length decides how the loom is dressed.

What was counted, and how #

The doup end’s path over a crossing interval is taken as two straight legs from a point beside its partner to a point beneath it and on to a point beside it on the other side, with the sideways offset and the depth each equal to half the sum of the two diameters plus any stated clearance. The standard end is taken as straight between crossings, and the crimp both ends share over the picks is left out of both.

The extra fraction was confirmed positive and falling at pick spacings from half a millimetre to four; unchanged when every length is doubled; identical for a crossing every other pick at one spacing and every pick at twice it; larger with the dip than with the sideways travel alone; and within ten per cent of the squared-interval approximation once the interval is six times the offset. The beam figures are the extra fraction times the piece length, and the pick spacing divided by the extra fraction.

The grip per crossing is the capstan on half a turn, the relation the earlier leno essays establish; only its ratio to the yarn cost is new here.

Where the model stops #

The legs are straight and the standard end is. A real doup end bends smoothly round its partner rather than meeting it at corners, which shortens its path slightly, and it pushes the standard end aside, which moves some of the difference onto the partner. The straight-leg figure is therefore an upper bound on the difference, and a close one once the crossing interval is several diameters.

The crimp is shared only when both ends lift together. Leno constructions in which the standard end stays down while the doup end works — half-gauzes, or grounds with different sheds for the two ends of a pair — give the ends different crimp as well, and that difference adds to or subtracts from the crossing’s.

Nothing here is about the moment of crossing. Many leno looms give the doup end slack as the crossed shed forms, with a device called an easer, and that is a per-pick event at the heddle rather than an accumulation over the piece. Carrying a doup end five millimetres sideways at a heddle three hundred millimetres from the fell lengthens it by about a hundredth of what the shed itself stretches the front shaft’s ends, which is far too small a path change to need a device of its own. What an easer does relieve is not at the doup at all: it is the kink the crossed shed puts between the doup and the end’s own back heddle, which this path does not contain because it follows the end only between crossings.

And yarn stretches. A few per cent of the doup’s extra length can be taken up elastically for a few picks, which is what lets a leno start at all on a badly set loom, and which fails in the millimetres the beam figures give.

What an easer is relieving, and what a trace would still settle #

The accumulation over the piece is a length and a second beam is its answer. The per-pick slack an easer gives is a different quantity, and the obvious geometric candidate — the doup end’s sideways displacement at the heddle — comes out a hundredth of the shed’s own extension. The length it does give back is at the next heddle along: in the crossed shed the doup lifts the end while its own back heddle holds it down, and that kink is more than ten times the shed.

The measurement that would confirm it is a tension trace on a single doup end through one pick cycle, with and without its easer: a spike at every crossed shed, of the size that kink predicts, and none at the open one.

Who worked it out #

Leno and gauze are among the oldest open constructions, and the doup heddle that lets one end cross another is old enough that its origin is not recorded. That doup ends are fed from their own beam or their own tension is prescribed in weaving manuals for leno work generally. The straight-leg length, its square-law dependence on the crossing interval, and the comparison of yarn cost with grip per crossing were computed directly.