A leno's slackener eases the partner, not the doup

One weight under a leno pair ends on one end found that the partner — the end a leno’s crossing end winds about, threaded through one ordinary heddle and lifted on no pick — is an ordinary end in every respect that matters. Its eye grips, and the grip adds five per cent to its front span on whichever shaft it hangs, exactly as it does for every other end on that shaft. It needs nothing an ordinary end does not need.

That finding rested on one simplification, and the essay said so. The doup’s loop was taken not to touch the partner. In a bottom-doup leno it is not clear of it. The doup is a half heddle whose loop passes beneath the partner on every pick, and the crossing end rides in that loop. When the doup lifts the crossing end on the far side of its partner for the crossed shed, the loop’s leg has to rise past the partner it lies under, and if the doup’s standard holds the loop taut, the leg presses up on the partner’s underside and lifts it.

A lifted partner has a second turn on its path, a few millimetres in front of its own heddle. The question left open was whether that mattered, and the answer is that it matters a great deal more than the earlier essay guessed, in a place no easing bar can reach — and that the device a leno loom already carries for its doups is the one that cures it.

A loop is a third eye, six millimetres from the second #

The partner untouched is two straight spans. It runs 300 millimetres from the fell down to its heddle eye, held 50 millimetres below the centre line in the bottom of the crossed shed, and 900 millimetres from there back up to the back rest. It turns once, at its eye, through 13 degrees.

A loop that lifts it changes the picture in a small region with large consequences. The loop meets the partner where the doup hangs, which on the doup’s own shaft is a few millimetres in front of the partner’s heddle; six is taken throughout. Lifted four millimetres there, the partner now runs from the fell up over the loop, drops four millimetres in six to its heddle eye, and climbs from the eye to the back rest. It is three spans and two turns, and both turns are sharp.

The partner turns 31 degrees over the loop and 43 at its heddle eye, where untouched it turned 13. A four-millimetre lift is not much of a displacement on a warp a metre and a fifth long, and it is not much of a stretch either: the path is 6.62 millimetres longer than the straight line from fell to back rest, against 5.53 with the loop clear, about a fifth more. The bend is what is large. Seventy-four degrees of turn in six millimetres of warp is a sharper corner than anything else on the partner’s path, and sharper than either eye on the crossing end, whose kink a heddle eye lets the kink through priced at 69 and 64 degrees spread across 64 millimetres.

The solution method is the one used for the crossing end. Yarn passes from one span to the next only through a turn, and a thread is gripped where it turns: it slides round a turn of angle θ only when the tension on one side has reached $e^{\mu\theta}$ times the tension on the other. The shed is opened in small steps from a straight warp, because which way yarn last slid through each turn is part of the state, and at each step each turn is either holding or sliding at its limit. The loop grips at the same coefficient as the heddle eye, 0.3, unless stated otherwise.

The lift lands on the front span #

The hero figure is the answer at every lift from nought to eight millimetres. With the loop clear the front span carries 1.07 newtons and the back span 1.00, the partner the earlier essay described. Lifted four millimetres, the front span carries 1.48 newtons, the short span between loop and eye 1.26, and the back span 1.00. At eight millimetres the front span reaches 1.90.

Two things in that are not what a reader of the crossing-end calculations would expect.

The first is where the tension goes. On the crossing end the steep span between the two eyes was the one at risk; that is where the length is taken, and it is where a capstan’s grip ought to trap it. On the partner the short span between the loop and the eye is the steep one, and it is not the tightest. The front span is. The reason is which span the shed stretches. The partner is pulled down at its heddle by fifty millimetres, and that drop is paid for mostly in the long front span from the fell; the yarn to relieve it has to come from the back span, through the eye and then over the loop, two turns in series. The two capstans multiply. At a four-millimetre lift the front span carries 1.474 times the back span, which is $e^{0.3 \times 1.29}$ — both turns sliding at their limit, one after the other, and their angles added.

The second is the back span. It carries 1.00 newtons with the loop clear and 1.00 with the loop lifting four millimetres, and moves by at most a few hundredths of a newton anywhere under six. The whole premium sits in front of the heddle. An easing bar acts on the back span, and a bar that answers to tension — the dead weight a leno easer should be a light weight recommended — feels only the back span. It cannot see the loop at all.

Most of the premium is friction #

How much of the 0.41 newtons is the bend, and how much is the grip? The model separates them by switching the friction off, one contact at a time.

With nothing gripping anywhere the touched partner is one tension through all three spans, its whole extension read through the yarn’s modulus: 1.12 newtons, a tenth over the frictionless untouched partner’s 1.02. That is what the bend costs as length, and it is small, because the extra length is a millimetre on a metre and a fifth.

Let the heddle eye grip and keep the loop smooth, and the front span rises to 1.32. The eye’s own turn has grown from 13 degrees to 43, because the short span arrives at it steeply from above, and a turn three times as large is a capstan three times as strong in the exponent. Let the loop grip as well and the front span reaches 1.48. Of the 0.41 newtons the loop adds to the untouched partner, 0.36 is friction, and more than half of that is the heddle eye doing more than it did before.

So the gripping partner pays four times what the bend alone would charge. That ratio is the reason the earlier essay’s reassurance does not survive. It expected a second turn to “raise the partner’s tension a little in the crossed shed”, which is right about the length and wrong about the grip; a second turn close to the first does not add a little, it multiplies.

At a lift of one millimetre the front span already carries 1.16 newtons, eight per cent over the untouched partner; at two, 1.25. The untouched partner was five per cent over the 1.02 newtons that every easing calculation on this loom has held the crossing end under, and that was the price of being an ordinary end. At a millimetre and a half of lift it is eighteen per cent over, 1.20 newtons, more than an ordinary end carries on any of the four shafts nearest the doup.

Moving the heddle back helps, to a point #

The untouched partner did not care where its heddle hung. On every shaft from the doup’s back to the twelfth its eye added the same five per cent, and its tension rose with the shaft only because the back shaft works hardest: a heddle further back has to drop further to make the same opening at the reed.

The touched partner cares a great deal, because the loop does not move with the heddle. The loop hangs where the doup is. Each shaft the partner’s heddle moves back puts sixteen more millimetres between the loop and the eye, and a four-millimetre lift over twenty-two millimetres is a far gentler bend than over six.

One shaft back, the touched partner’s front span falls from 1.48 newtons to 1.24, and the loop’s premium over an untouched partner on the same shaft from 38 per cent to 11. Two shafts back it is 1.23, with a premium of seven per cent. From there the premium keeps falling — four per cent at four shafts, one and a half at twelve — but the shed’s own cost rises faster, and the tension climbs again. The touched partner is least two shafts behind the doup, at 1.23 newtons, still fifteen per cent above the untouched partner on the doup’s shaft.

That is a real choice a loom has. A leno harness puts its doup at the front because the crossing is easiest there, and the partners can be drawn through any free shaft behind it. The earlier essay found the choice immaterial; with the loop touching, the first free shaft behind the doup is worth most of what moving can buy. It is not worth all of it. Nothing about the heddle’s position removes the loop.

A bar can give the length, but not on its own terms #

The other way to relieve a tight span is to give it yarn. An easer behind the harness hands length to the back span, and the eye and the loop pass some of it forward.

Both curves fall straight, because the eyes are sliding and the front span loses yarn in proportion. 3.3 millimetres behind the harness brings the touched partner back to the 1.07 newtons it would carry with the loop clear. That is a small length — a twentieth of what the crossing end needs at the same moment — and it says the loop’s premium is not a stubborn one. Give the partner a little yarn and it is gone.

The difficulty is getting the partner that little without the rest. A bar under the crossing ends alone gives them their seventy millimetres and gives the partners nothing; the shared-bar essay showed that a bar under both gives each the same length, and the partner can take only eleven millimetres of the crossing end’s seventy before it hangs. The loop changes that onset only slightly — the touched partner hangs after 12.0 millimetres rather than 10.9 — so the shared bar’s verdict, which turned on fifty millimetres of difference, stands. And a dead weight on the partners alone could not find the 3.3 millimetres, because the back span it would answer to reads 1.00 newtons whether the loop is pressing or not.

What is wanted is something that acts on the partner only when the loop is lifting it, and only by the amount of the lift. A loom has one.

The slackener takes the lift off the loop #

A slackener lets the doup’s standard drop slightly as the crossed shed forms. It is ordinary leno equipment, and its usual explanation is that it eases the doup end through the crossing: the loop has to rise past the partner and carry the crossing end to the far side, and a standard held rigidly makes that crossing tight.

The loop hangs from the standard, so whatever the standard drops, the loop drops; and the loop’s lift on the partner is exactly what a taut loop does that a slack one does not. The model therefore takes the drop off the lift, millimetre for millimetre, and lowers the doup’s eye — and with it the crossing end’s kink — by the same amount.

On the partner the effect is complete and immediate. Each millimetre of drop takes about a tenth of a newton off the front span, and at four millimetres the loop is clear and the partner is exactly the untouched end again, 1.07 newtons, as the model is required to reproduce. On the crossing end it is slight. The lower doup shortens the kink, and each millimetre of drop saves about a millimetre of the 67.9 an easer has to give; four millimetres saves 4.1, six per cent, and six millimetres saves nine.

That is the reverse of the device’s reputation. A slackener does help the doup end, but by a few per cent of what the doup end needs, and the easer does the rest. What a slackener does completely, for a few millimetres of travel, is free the partner from the loop. It is the partner’s easer. And unlike a bar behind the harness it acts at the right place — the loop itself — and only in the crossed shed, when the loop is lifting.

Lowering the whole shed narrows none of it #

A device that lowers the doup lowers the crossing ends in the crossed shed, and the crossing ends form the top of that shed. Every millimetre the doup drops is a millimetre off the top line at the doup, and about three tenths of one at the reed, ninety millimetres from the fell, where the shuttle has to pass. A loom with a thirty-millimetre shed and a six-millimetre slackener would expect to lose about 1.8 millimetres of it.

It does not lose it while the loop is touching. The partners form the bottom of the crossed shed, and the loop’s lift was raising them: four millimetres at the loop is 1.22 at the reed, a bottom line pushed up into the shuttle’s path. The slackener lowers the loop as it lowers the doup, so the bottom line falls with the top. Measured at the reed, the top falls 0.300 millimetres for each millimetre of drop and the bottom 0.306, the slight excess because the loop sits six millimetres nearer the fell than the doup’s eye. The clear opening is unchanged — fractionally wider — until the loop is clear. Only beyond that does the top line go on falling alone, and the shuttle starts to pay, 0.58 millimetres at a six-millimetre drop.

So a slackener set to the loop’s lift costs the shed nothing, and the arithmetic says how to set one: drop the standard by exactly as much as the loop would otherwise lift the partner, and no further. Less leaves the partner carrying part of its premium; more takes clear opening from the shuttle and gives the crossing end only another per cent and a half of easing for each further millimetre.

What the model assumes #

The partner is the eyes model with one more turn. It is three straight spans — fell to loop, loop to heddle, heddle to back rest — of 25 tex cotton on the ordinary broad loom used throughout the leno essays, its heddle on the doup’s shaft unless stated, 300 millimetres from the fell, held down by the centre-closed shed to the bottom line. The loop is a point six millimetres in front of the heddle, gripping at 0.3 like the eye. The shed opens in 120 steps, and at each the two turns are relaxed alternately until neither moves; a short span between two close turns is stiff, and this takes many more rounds than the crossing end’s longer kink.

The loop’s lift grows in proportion to the opening and reaches the stated rise at the full shed. With the lift at nought, or a slackener’s drop equal to the lift, the model is required to reproduce the two-span partner exactly; with no friction anywhere, one tension equal to its whole extension through the modulus; with friction, each turn holding within its own capstan and the front span carrying the most, by at most the two capstans together. Every millimetre of lift from nought to eight is required to raise the front span; every shaft the heddle moves back is required to take some of the loop’s premium off; and the touched partner, closed again, is required to return to its rest.

The slackener drops the loop and the doup’s eye together. The crossing end’s easing is found as before: the length behind the harness that leaves no span of it above an ordinary end’s tension on the doup’s shaft at the full shed, with the doup’s eye lowered by the drop. Each line’s fall at the reed is its fall at its own contact, scaled along a straight sheet to the reed’s ninety millimetres.

What the calculation does not know #

It does not know how far a real loop lifts its partner. That depends on the doup’s construction — a cord, a wire, a steel half heddle — on how tightly its standard holds it, and on how far the partner itself gives, which here is not at all; a real loop may stretch, and a real partner’s heddle eye has some play. The lift is swept from nought to eight millimetres rather than computed, and every number in this essay that depends on it is stated against a four-millimetre lift. The shape of the result does not depend on it. A bend near a gripping eye multiplies through the capstan whatever its size, and a slackener that matches the lift removes it whatever the lift is.

The coefficient at the loop is assumed equal to the heddle eye’s. A polished steel doup would grip less; a cord doup might grip more. The bar figure brackets the loop’s share between nothing and the full 0.3.

And the slackener’s drop is assumed to come off the loop and the doup’s eye in equal measure. A slackener that let the loop down more than the eye, or less, would divide its effect between the two ends differently — though since its effect on the crossing end is a few per cent either way, the conclusion that it is the partner’s device is the robust one.

Nor does the model follow the rubbing. The loop presses on the partner’s underside with 0.76 newtons at a four-millimetre lift, every crossed shed, while 0.86 millimetres of partner slides across it and back. That is a wear load on an end the earlier essays treated as idle; abrasion takes the hairs first describes what such rubbing does to a yarn’s surface, but how many picks it takes to matter is a question about the yarn and the doup’s material that this geometry cannot answer.

Who worked out which part #

The bottom doup, the doup’s standard and the slackener are all ordinary leno practice, described in the standard weaving manuals, and so is the habit of drawing the partners through a shaft close behind the doups. The capstan is Euler’s and Eytelwein’s. The shed is an extension gives the partner’s path and what the shed costs in newtons its tension; the eyes model, the easer and the shared bar are this account’s earlier leno essays.

What is added here is the loop as a third eye on the partner; the finding that its premium lands in the front span, multiplied through two capstans, while the back span an easer feels stays where it was; the heddle position that minimises it; and the reading of the slackener as the partner’s easer, which clears the loop exactly at a drop equal to the lift and, until then, costs the shed nothing.

Answered since: whether the slackener can follow the shed #

Everything above is read at the full crossed shed, and a slackener acts all the way through it. The loop’s lift is assumed to grow in step with the opening, and the slackener’s drop with it, so the two cancel at every fraction. A real slackener is a cam or a weighted lever with its own timing, and the doup rises on its own.

If the drop lags the lift, the partner carries part of its premium through the middle of the shed even though it is clear at the top, and that is exactly the error a cam easer gives its slack too early found in the crossing end’s easer, run the other way. If it leads, the crossing ends are lowered before the loop has risen, and the shed pays early for nothing. The calculation is the same three spans with the drop and the lift given separate schedules, and it would say how closely a slackener’s motion has to track the doup’s to keep the partner clear at every fraction of the shed rather than at the last.

A leno’s slackener only has to arrive runs the two schedules apart. For the partner the drop need not track the lift at all, only arrive by about three quarters of the shed, and lateness is paid for in rubbing at the loop.