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Every essay — page 19

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What cloth is Weaves Setting and geometry Knits and other structures Mechanics and drape Pattern and colour Compound and figured cloths After the loom Cloth doing a job

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Compound and figured cloths

Double cloth, pile, leno, and the loom that decides what can exist. Where a fabric stops being one binary matrix — a third thread system, a warp order that changes between picks — and where the integrity criterion runs out and friction takes over.

plain over 5-end satin: what each layer's warp does. A warp end of each layer of a double cloth, in section over the same span of cloth, with the length of warp each eats drawn beneath. The crimps come from Peirce's geometry asked about a thread that bends at its own average rate rather than at every crossing, and they are 14.35% for the plain and 2.03% for the 5-end satin. They are not the same, so the two layers consume warp at different rates: a hundred metres of cloth takes 114.4 m of warp from one and 102.0 m from the other. A beam delivers one rate, so the difference has nowhere to go and accumulates with the length woven — one pick spacing after 4 mm. What the sections cannot show is the yarn's thickness, which is exaggerated so the path is legible; the crimps beside them are computed at the real diameter and are not read off these drawings.

Two layers need two beams

A layer weaving a metre of cloth eats one plus its crimp metres of warp, and a beam delivers one rate. A plain face over a five-end satin back differs by twelve percentage points of crimp, which is twelve metres of warp over a hundred-metre piece and one pick spacing of slack after four millimetres of weaving. The difference has nowhere to go and does not settle — so the only double cloth that can share a beam is two layers of the same weave at the same sett.

7 figures · Double cloth
How many layers the repeat, the harness and the beams each allow. Three ceilings on the number of layers a double cloth can have, for five layer weaves. The repeat's bound is half its ends and is a property of the notation. The harness's is the strain budget — 13 shafts on an ordinary broad loom at a 1.0% warp strain limit — divided by the shafts one layer of that weave costs. The beams' is how many warps the loom carries, which is 2. The shortest of each three is marked, and it is the beams at the coarse end of the table and the harness at the fine end; the repeat is never the binding one except at two-end layers, where it happens to coincide with the harness. A double cloth of eight-end satin layers needs 16 shafts and the budget is 13, so it is a jacquard construction by arithmetic rather than by choice. What the bars cannot show is the pick rate: a k-layer cloth needs k times the picks per centimetre of finished cloth and takes k times as long to weave, which is a cost rather than a ceiling and is the reason four-layer cloths are rare even where they are possible.

The repeat allows four layers and the loom allows two

A repeat of eight ends can hold four separable cloths, and the site has a witness that reaches the bound exactly. No loom weaves four. The harness's strain budget buys thirteen shafts and a layer costs its own weave's shaft count, so five-end satin layers stop at two and eight-end satin layers cannot be doubled on a dobby at all — and two differing layers already want a beam each. The notation's ceiling is the only one of the three that is never binding.

6 figures · Double cloth
A designed thin place does not get worse and an accidental one does. The thinnest place a yarn reaches, against how many gauge lengths of it are tested, for a slub yarn and for a randomly uneven yarn of the same 36.3% coefficient of variation. The slub's floor is its base count — 89% of its mean — and it is a horizontal line, because a designed variation has a stated minimum and never goes below it. The random yarn's minimum is an order statistic and falls without limit: 56% over 10 lengths and 22% over 30000. So the advantage is not a number but a function of how much yarn is being asked about, running from 1.6× to 4.0×. What the curve cannot show is the break itself: a yarn's strength at a thin place is not proportional to its linear density there, and the conversion needs a fibre model.

A designed thin place is kinder than an accidental one

A slub yarn and a badly spun one can carry exactly the same coefficient of variation, and the number tells a mill nothing about which it has. The designed variation has a floor — its base count, and it never goes below it — while the accidental one has a tail that falls further the more yarn is tested. At 36% CV the slub bottoms at 89% of its mean and the random yarn reaches 24%, and the gap widens from 1.6 to 3.7 times as the test grows from ten gauge lengths to ten thousand.

5 figures · Fancy yarn
A bouclé at an overfeed of 0.80. A core of 20 tex running straight, an effect thread of 30 tex delivered 80 per cent faster than it, and a binder of 15 tex over the top. The surplus effect thread has nowhere to go but into loops, and their size is not a free choice: a semicircular loop of radius r consumes r(π − 2) of surplus, so at one loop every 3 mm the radius is 2.10 mm — 0.70 of the spacing. The yarn's resultant count is 89 tex and the sum of its three components is 65, because the effect enters multiplied by its own overfeed. What the drawing cannot show is the load path: the core and the binder are at the yarn's own length and the effect thread is longer than the yarn, so a pull on the yarn is carried by 39 per cent of its mass and the loops stay slack.

A fancy yarn has its crimp in the wrong thread

A bouclé is made by feeding an effect thread eighty per cent faster than the core it wraps, so the yarn contains more thread than it is long — which is crimp by this collection's own definition. A cloth's crimp is in the thread that carries the load, so removing it is the cloth's first extension. A bouclé's surplus is in a thread that carries nothing, so pulling the yarn stretches the core at once and the loops never straighten. Thirty-nine per cent of the yarn's mass is on the load path, and the other sixty-one is decoration.

6 figures · Fancy yarn
An 8-end weft-faced satin through a point tie on an end. The 8-end satin on a move of 3, weft-faced, woven through a point tie of 12 hooks that turns on a single end, so 22 ends carry one repeat. The strip above is the tie, hook number against end. Inside the satin the weft floats 7 ends; across the turn it floats 13, outlined, because on some pick the nearest interlacing is 7 ends away on both sides. What the drawing cannot show is how a long float at one line down the cloth behaves in use, which depends on the yarn and the finish rather than on the count.

A point tie nearly doubles the float at the turn

A point tie halves the hooks a symmetrical figure needs by driving every hook's end twice, out and back. It mirrors the ground as well, and a satin is symmetrical about no end: across the turn its weft floats 2n − 3 ends, or 2n − 2 between ends, at every hook count and wherever the satin starts. At eight ends that is thirteen or fourteen against seven — and the turn's other promise, more repeats that fit the width, is not kept either.

7 figures · Jacquard
A lancé weft bound where the 8-end satin interlaces. A warp-faced 8-end satin on a move of 3, with an extra weft thrown after every ground pick and laid on the face over figures 12 ends wide and 28 ends apart. Between figures it is bound only on ends the ground already drops on an adjacent pick, 56 places in the drawn repeat, which leaves back floats of at most 4 and no binding on the face. The drawing is 40 ends by 16 picks and counts as 1 cloth. What it cannot show is whether a binding point counted as hidden is hidden in a real cloth, which depends on the yarns' contrast and on how closely the ground's floats cover it.

A brocade weft floats as far as the next figure

A brocade's pattern weft is a third thread, laid on the face only where the figure wants it. Thrown from selvedge to selvedge it floats on the back across the whole gap to the next figure, unless an end is dropped under it — and on a satin ground a dropped end hides only where the ground already interlaces. That fixes the shortest float that shows nothing, at the larger of the move and its complement less one, and puts the smoothest satin ground at odds with the best-bound brocade.

6 figures · Jacquard
A jacquard harness 130 cm wide under a 150 cm fall. A jacquard harness in front elevation, to scale: hooks spread over 40 cm, cords fanning down 150 cm to a comber board 130 cm wide, and hanging straight from the board to their mails. The centre cord is vertical and lifts its mail by the whole 10 cm hook lift; the edge cord leans 16.7° and lifts its mail 9.60 cm, 96.0% as far, and its bend at the board adds 9% to the hook's load at a friction coefficient of 0.3. What the drawing cannot show is the tie that decides which hook feeds which hole, which in a real mount is not the simple spread assumed here.

A jacquard harness needs three half-spans of height

A jacquard has no front shaft and no back one, so every end takes the same shed — as long as every cord hangs straight. Across the width they cannot: the hooks sit in a machine a few tens of centimetres wide and the comber board is as wide as the cloth, so an edge cord leans and its mail rises by the difference of two hypotenuses rather than by the hook's lift. Holding the edge shed within five per cent of the centre's takes a fall of about three times the edge cord's sideways reach, and that is a height a room has to have.

6 figures · Jacquard
20 tex yarn in one layer and in 2. Sections across the width, to scale, of a cloth of 20 tex cotton at a cover of 0.8, and of the same yarn per area divided into 2 layers two ways: by count, 10.0 tex at the same sett, and by sett, 20 tex at 1/2 of the ends. Divided by count the cloth is 1.41 times as thick with a cover of 0.57 in each layer; divided by sett it is 2.00 times as thick with a cover of 0.40. At the free end of the yarn's stiffness bracket both are exactly as stiff as the single cloth; at the coherent end the first is 0.50 times as stiff and the second 1.00. What the sections cannot show is crimp, which thickens every layer by an amount the weave decides.

A double cloth is only softer if its yarn is set

A double cloth is sold as weight without stiffness: two light cloths in place of one heavy one. Divide the same yarn into two layers and the cloth is √2 or twice as thick, but at the free end of a yarn's stiffness bracket — where an unset yarn sits — its bending rigidity does not move at all, because it is the number of fibres across the width times the stiffness of one. Only a set yarn makes the double cloth the softer, and stitching the layers together pushes it the other way.

6 figures · Double cloth
A doup end crossing every pick at 1 mm. One leno pair of 0.25 mm ends over 2 crossing intervals, with picks 1 mm apart and the doup end changing sides every pick, drawn to scale in plan and in section. The standard end runs 1.00 mm per interval; the doup end, passing from 0.250 mm to one side, down 0.250 mm under its partner and up to the other side, runs 1.225 mm — 22.5% more, 11.8% of it from the sideways travel alone. Over a 100 m piece that is 22.5 m of warp, and on a shared beam a pick spacing of slack in 4.4 mm of cloth. What the drawing cannot show is the crimp both ends share over and under the picks, which the comparison cancels.

The doup end pays for the crossing

A leno holds its picks because its doup end crosses under its partner and comes up on the other side — half a turn at every crossing, whatever the sett. That half turn is also a length. When 0.25 mm ends cross every millimetre, the doup end travels 22.5 per cent further than its partner: twenty-two metres of extra warp over a hundred-metre piece, and a shared beam a pick short within four and a half millimetres of cloth. The extra falls with the square of the crossing interval, which is why a leno is woven from two beams and its crossings are spaced as far apart as the cloth allows.

6 figures · Leno
A bouclé cloth jammed at its loops and at its count, overfeed 0.8. Sections across 36 mm of plain-woven cloth in a bouclé of 89 tex — a 20 tex core, a 30 tex effect overfed 80% with loops every 3 mm, and a 15 tex binder — drawn to scale. The loops stand 2.10 mm off the core, so the yarn's outline is 4.43 mm across while the diameter its count implies is 0.352 mm, 12.6 times smaller. Jammed at the outline the cloth takes 1.13 ends a centimetre and weighs 20 g/m²; jammed at the count, 14.2 and 252 g/m². What the drawing cannot show is where between the two a real cloth jams, which is how far its loops interleave with their neighbours'.

A bouclé is set by its loops and weighed by its count

A bouclé yarn has two diameters: the one its count implies, a third of a millimetre, and the one its loops occupy, over four millimetres. A cloth jams where its yarns touch, and a bouclé's yarns can touch at either — so its jamming sett is a bracket twelve and a half times wide, from 1.1 to 14 ends a centimetre. The weight follows the count wherever in that bracket the cloth is set, which puts the open end at a twelfth of the close end's weight with four per cent of its area covered by yarn.

6 figures · Fancy yarn
The crossed shed, with the back standard 4 shafts behind the doup. A leno's crossing end in section from the fell to the back rest of an ordinary broad loom, in the crossed shed. The doup 300 mm from the fell lifts it 50.0 mm; its back standard 64 mm further back leaves it 60.7 mm down, beneath its partner. Between the two eyes the end climbs 110.7 mm in 64 mm, and the whole path is 70.2 mm longer than the straight line, 5.85% of the free warp, of which 63.8 mm is that one climb. The same end in the open shed, lifted at its back standard, is 7.2 mm longer, and an ordinary end on the doup's shaft 5.5 mm: the crossed shed is 12.7 times the ordinary shed. Vertical scale exaggerated 3 times. What the drawing cannot show is friction at the eyes, which would keep the extra length in the short span between them.

An easer gives back the kink the crossed shed puts in

A leno loom gives its crossing ends slack at every crossed shed, and the obvious reason — the doup carrying its end sideways — is worth a hundredth of an ordinary shed. The length is one heddle further back. In the crossed shed the doup lifts the end on the far side of its partner while the end's own back heddle, a few shafts behind, holds it down, and between those two eyes the end climbs eleven centimetres in six and a half. On a loom that stretches an ordinary end 0.46 per cent, that is 5.85 per cent — nearly thirteen times as much — and no position of the back heddle takes it below four.

7 figures · Leno
Bouclé loops on 3 mm binder spacing at overfeeds of 30%, 80%, 150%, 300%, 500%. Loops of an overfed effect thread drawn to one scale, each between two binder points on a straight core, as the curve of least bending energy with its ends along the core. Overfeed 30%: 1.30 times its base in thread, 1.10 mm tall, a bell, against a semicircle of 0.79 mm; Overfeed 80%: 1.80 times its base in thread, 1.95 mm tall, a bell, against a semicircle of 2.10 mm; Overfeed 150%: 2.50 times its base in thread, 2.92 mm tall, overhanging, against a semicircle of 3.94 mm; Overfeed 300%: 4.00 times its base in thread, 4.83 mm tall, overhanging, against a semicircle of 7.88 mm; Overfeed 500%: 6.00 times its base in thread, 7.25 mm tall, overhanging, against a semicircle of 13.14 mm. The dashed arcs are the semicircles, drawn where they fit. What the drawing cannot show is the binder's own path and thickness, which the loop's feet are taken to sit exactly on.

A bouclé loop is an elastica, not a semicircle

A bouclé's loop was taken to be a semicircle, and the semicircle cannot be right: it meets the core at a right angle, and at eighty per cent overfeed it is wider than the gap it stands in. Solved as what it is — a length of thread leaving the core along the core at two binder points — the loop is taller than the semicircle below an overfeed of two thirds and shorter above it, overhangs at 119 per cent and closes its neck at 559. Nothing of the fibre is in its shape, and the bouclé's jamming bracket at eighty per cent is 11.7 wide rather than 12.6.

5 figures · Fancy yarn
A bouclé loop at 80 per cent overfeed, pressed to four heights. One loop of a bouclé at 80 per cent overfeed on 3 millimetre binder spacing, drawn to scale at 100, 80, 50, 20 per cent of its free height, with the force it pushes back with beneath each. The pressed loop is the same elastica the free one is, with one condition added: at its apex the tangent is along the core again, so a pressed loop is two half-loops each with its rise prescribed. At 100 per cent it stands 1.95 millimetres and pushes with -0.00 millinewtons; At 80 per cent it stands 1.56 millimetres and pushes with 12.19 millinewtons; At 50 per cent it stands 0.97 millimetres and pushes with 10.02 millinewtons; At 20 per cent it stands 0.39 millimetres and pushes with 4.90 millinewtons. What the drawing cannot show is the stiffness bracket: the force is the lower bound, with the fibres free to slide.

A loop has a maximum force in it

A bouclé loop pressed by its neighbour is the same elastica the free one is, with one condition added — at its apex the tangent lies along the core again, so a pressed loop is two half-loops with their rise prescribed and needs no contact solve at all. Solved, the eighty-per-cent loop pushes back with nothing at its free height, twelve millinewtons at four fifths of it, and two and a half at a tenth. A curve with a maximum in it is a softening spring, and the promise this account made — a cloth's position read from its beat-up — cannot be kept, because the whole bracket sits at one pressure.

5 figures · Fancy yarn
A 100-gram bouclé in section, with the depth to its own load path. A bouclé cloth of 100 grams a square metre in section: the core and binder along the bottom, carrying everything, with the effect thread's loops standing 0.31 millimetres above them. The cloth's surface is the loops and nothing else — its loop envelope is 12.6 times its count diameter — so a rubbing surface meets effect thread first and reaches the core only after that depth. 60.7 per cent of the yarn's mass is effect thread on no load path and 39.3 per cent is core and binder. What the section cannot show is the third dimension: the loops of neighbouring yarns lie between these and are pressed by them.

A bouclé wears from the loops down

A cloth loses its strength before its mass, because a woven cloth's crowns are the very threads that carry the tension. A bouclé inverts it exactly. Its surface is its loops, its loops hang off a core that carries everything, and 61 per cent of its yarn is on no load path at all — so a rub takes three tenths of a millimetre of thread that does nothing before it reaches anything that does, and the cloth is ruined to look at while it is still as strong as it was woven.

6 figures · Fancy yarn
A comber board for 60 ends a centimetre, in side elevation. A jacquard's comber board seen from the side, with the fell of the cloth at the left and the back rest at the right. The board carries one hole per end; at 60 ends a centimetre the ends are 0.167 millimetres apart and a cord with a mail on it needs 0.9, so the holes are ruled in 6 rows staggered fore and aft, 6 millimetres apart — a harness 30 millimetres deep. Each row's ends are strained by its own distance from the fell: 0.460 per cent at the front and 0.524 at the back, a spread of 0.0636. What the elevation cannot show is the sideways fan of the cords above the board, which is a separate and much larger geometry.

A jacquard's harness has a depth after all

A jacquard was said to escape the shaft loom's depth entirely, because every mail hangs at the same distance from the fell. Every mail does, if the comber board has one row of holes — and it cannot. At sixty ends a centimetre the ends are a sixth of a millimetre apart and a cord with a mail on it wants most of one, so the holes are ruled in six rows thirty millimetres deep. The escape is real and it is a factor of eight rather than a release, and it closes as the cloth is set finer.

7 figures · Jacquard
What each figure-and-ground pairing costs in differential take-up. For each pairing of a figure weave with a ground weave, the difference between the two regions' warp crimps and the length of figure that difference allows before an end goes slack, taking the loom to absorb 1.2 millimetres. a damask: satin on its own complement: 0.00 per cent apart, no bound; an eight-end satin figure on a five-end satin ground: 0.31 per cent apart, a figure up to 392 millimetres; a five-end satin figure on a 3/1 twill ground: 0.90 per cent apart, a figure up to 133 millimetres; an eight-end satin figure on a 2/2 twill ground: 3.02 per cent apart, a figure up to 40 millimetres; an eight-end satin figure on a plain ground: 14.15 per cent apart, a figure up to 8 millimetres; a 2/2 twill figure on a plain ground: 11.14 per cent apart, a figure up to 11 millimetres. A damask's two regions are the same satin used two ways, so complementation leaves both float lengths where they were and the crimps are identical rather than close. What the bars cannot show is the slack, which is an input and which every length is proportional to.

A damask is the only figure that costs its beam nothing

Figure and ground consume warp at different rates, and the difference accumulates down the length of the figure. An eight-end satin figure on a plain ground puts its two regions fourteen per cent apart, which on a loom absorbing a millimetre of slack bounds the figure at eight and a half millimetres. A damask's two regions are the same satin used two ways, so complementation leaves both float lengths where they were — and their crimps are identical rather than close, to the last bit of a double.

6 figures · Jacquard
The crossed shed's three spans at an eye friction of 0.3. A leno's crossing end in the crossed shed on an ordinary broad loom, back standard 4 shafts behind the doup, with both heddle eyes gripping at a coefficient of 0.3 and a background tension of 0.5 N. The end turns 69 degrees at the doup and 64 at the back standard, capstans of 1.44 and 1.40. From fell to doup it settles at 6.95 N, a strain of 5.73%; from doup to back standard it settles at 9.48 N, a strain of 7.97%; from back standard to back rest it settles at 6.78 N, a strain of 5.57%. With frictionless eyes every span would take 7.08 N; an ordinary end at the doup takes 1.02 N and the yarn breaks at 3.74 N on its initial modulus. Vertical scale exaggerated 3 times. What the drawing cannot show is the tension's fall round each eye, which happens over the eye's own few millimetres.

A heddle eye lets the kink through

A leno's crossing end is pulled up at its doup and held down at its back standard, and the length that costs was priced as if both heddle eyes were frictionless. They grip, and gripping ought to trap the kink between them at nearly a hundred per cent strain. It does not come close. A capstan bounds a ratio of tensions, not a difference, and the spans either side are already stretched, so at a coefficient of 0.3 the span between the eyes takes 9.48 newtons against 7.08 with no friction at all — a third more, not fifteen times more — and an easer has to give back 67.9 millimetres rather than 64.4. What friction changes more is when the length is wanted.

8 figures · Leno
The beams a disc 12 blocks across needs, column by column. A disc 12 blocks across, 12 blocks by 12, figure shaded, with the beam feeding each column of ends written above it, for a figure floating 8 on a ground floating 1, whose warp crimps are 14.15 per cent apart, at 4 mm blocks over a 100 m piece with 1.2 mm of slack. It has 4 distinct columns and 4 distinct shares of the repeat in figure, and needs 4 beams. What the grid cannot show is the cloth's own crimp interchange, which would move tension between the columns instead.

A figured warp needs a beam for every share of its figure

Figure and ground take up warp at different rates, so a figured cloth on one beam is bounded in how long its figure may run, and a second beam is the obvious escape. It escapes only for ends that live wholly in one region. An end that crosses the figure for part of the repeat consumes warp at its own rate, and two ends can share a beam only if they spend the same share of the repeat in the figure and never drift a slack apart inside it. So a round figure twelve blocks across needs four beams, ninety-six blocks across needs twenty-nine, and any crimp difference at all — a third of a per cent will do — costs every one of them over a piece.

7 figures · Jacquard