A figured warp pays in tension before it needs a beam

A figured warp needs a beam for every share of its figure counted the beams a figured cloth needs when its figure and ground take up warp at different rates. Two ends can share a beam only if their running difference in consumption never exceeds what the loom’s slack absorbs; over a hundred-metre piece any drift at all exceeds it; so the beam count jumped from one to all at the first nonzero crimp difference. A disc twelve blocks across needed four beams for an eight-end satin figure on a five-end satin ground whose crimps differ by a third of a per cent, exactly as many as a satin on plain forty-five times further apart.

It ended by naming what the count left out. The count treats slack as a length and a warp’s tension as something that does not move. A real let-off controls tension, and an end that runs ahead pulls harder and straightens, a feedback that shrinks the drift until the tension difference it costs balances it. How large that tension difference is decides whether the step at nought is real.

That count was right about its model, and its model was right about one thing: a take-up difference that nothing corrects drifts by the same amount every repeat, and two thousand repeats multiply any drift past any slack. What the model did not have was anything to correct it.

It is not. For small crimp differences the warp absorbs them itself, at a tension difference of a few per cent of the working tension, and one beam serves. The step moves to where the ground’s crimp runs out — a few per cent of take-up — and past it no tension will do.

A let-off holds tension, and tension moves crimp #

A warp beam turns to keep the warp at a set tension, not to deliver a set length. Every end wound on it is fed at the same rate, and the ends pull the same length off it per pick. When figure and ground take up at different rates, the ends that spend more of their length in the higher-crimp region want more warp than the beam gives them, and the shortfall appears as tension: those ends stretch a little, and pull harder on everything they pass through.

An end pulled harder changes the cloth it is woven into. Crimp interchange and what it costs is the mechanism: a warp under more tension straightens, handing crimp to the weft, and a straighter warp takes up less length per unit of cloth. So the ends that consumed more consume less once they are tight, and the tension rises only until the ground’s crimp has fallen by exactly the take-up difference. From then on every end consumes alike, the beam’s single feed rate suits all of them, and nothing drifts.

The fixed-slack count was the limit of this feedback with an infinitely stiff cloth. A real cloth is not stiff at all in its first few per cent — the locus gets a force and a cloth has one budget for two directions are this account’s measurements of how little it costs to move crimp from one system to the other — and that softness is what lets the warp regulate itself.

The price is the slope of the ground’s own locus #

The tension difference needed is the force that takes a crimp of Δ out of the ground’s warp, and that force is already a known curve: the slope of the region’s bending energy along its constant-thread-length locus, which is what a cloth’s load–extension curve is in its first few per cent.

Each ground is treated as the earlier count treated it: a region whose threads float over $f$ crossings is Peirce’s plain geometry at $f$ thread spacings, so its locus is a plain cloth’s at a sett of $1/f$. The curves start almost flat and turn up sharply at their ends, and where each ends is how much crimp the ground can give up at all: 4.8 per cent for plain, 3.2 for a 2/2 twill, 1.4 for a 3/1 twill, and half a per cent for a five-end satin. A ground with long floats has little crimp to give, because it had little to begin with.

The yarn’s rigidity enters the force in proportion, and it is known only as a bracket. Every force here is quoted at both ends of the band where a bending test and a washing test agree the yarn sits, which is a factor of three.

Six pairings, three verdicts #

The damask needs nothing. Its figure and ground are one weave and its complement, their crimps are equal, and a damask is the only figure that costs its beam nothing holds with or without regulation.

An eight-end satin figure on a five-end satin ground, crimps 0.31 per cent apart, needs a tension difference of 0.006 to 0.020 newtons an end — one to four per cent of the half-newton the front shaft of an ordinary loom holds. That is inside the unevenness any warp carries anyway, and it is the pairing the fixed-slack count gave four beams. One beam serves, and the ground’s ends run a few grams tighter than the figure’s.

A five-end satin figure on a 3/1 twill ground, 0.90 per cent apart, needs 0.021 to 0.069 newtons: four to thirteen per cent of the working tension. Whether that is acceptable depends on where in its bracket the yarn really sits — at the lower placement it is inside a tenth of the working tension, at the upper just outside — and on how much tension difference the cloth will show. It is the borderline case, and it is the one a mill would test.

An eight-end satin on a 2/2 twill, 3.02 per cent apart, needs 0.30 to 1.0 newtons: from sixty per cent of the working tension to twice it. The warp could in principle absorb it; the ground’s ends would run at up to three times the figure’s tension, which is a streaky cloth and a warp breaking at the back shaft. In practice this pairing needs its beams.

A satin on plain and a twill on plain, 14 and 11 per cent apart, are past the end of the plain ground’s locus. A plain ground has 4.8 per cent of crimp to give, and no tension takes out more than it has. No regulation reaches them, and the fixed-slack count’s beams are exactly right.

The disc shows why regulation is a property of each column rather than of the figure as a whole. A column that crosses the figure for a quarter of the repeat is a quarter figure and three quarters ground, and it settles at a tension between the all-ground column’s and the all-figure column’s — near the ground’s for most of the disc’s edge, near the figure’s through its middle. On one beam the disc is woven as a smooth gradient of tension across the warp, largest between the columns that never enter the figure and those at its widest, and for the eight-on-five satin the whole gradient spans one or two hundredths of a newton.

The step moves, and becomes a band #

So the earlier essay’s step at nought is replaced by something with two edges. Below about half a per cent of take-up difference, the warp absorbs the figure on one beam at a tension difference no weaver would notice. Above the ground’s own crimp budget — a few per cent — no tension does. Between them is a band where one beam works at a price in tension that rises steeply, and whether to pay it is a judgement about the cloth.

That is the shape the trade’s practice has without the arithmetic. Damasks and satin-on-satin figures are woven from one beam as a matter of course; a satin figure on a plain ground is known to need a separate figure beam or to be woven as a brocade, where the figure is extra weft and the warp never enters it. The count’s four beams for an eight-on-five satin were a real consequence of a model with no tension in it, and the trade never paid them.

What a seersucker does on purpose #

There is a cloth that runs this mechanism in reverse and wants the drift. A seersucker is made at the loom: two beams at different tensions, the slack beam’s stripes taking up more warp than the tight one’s, so that when the cloth leaves the loom the slack stripes are longer and pucker.

A seersucker is a figured warp whose tension difference is chosen to be large rather than absorbed. The regulation argument says how large: to hold a take-up difference of Δ between the stripes, their tension difference must be the force along the tight stripe’s locus at Δ — and past that stripe’s crimp budget the difference cannot be held by tension at all, which is why a seersucker needs two beams rather than one beam pulled unevenly. The puckers are the crimp that the regulation would have removed.

The numbers make the point. A plain seersucker stripe can give up at most 4.8 per cent of its crimp before its locus ends, at a tension that climbs past a newton an end on the way; a pucker of ten per cent in length is twice what any tension on one beam could hold, so it has to be fed from a second beam at a second tension, and the two beams’ difference in feed is the pucker. A seersucker is the one figured cloth for which the fixed-slack count’s answer — a beam per region — is the design rather than the cost.

The fixed-slack count had one more feature regulation removes. Under it, a sample could share a beam that a piece could not, because drift accumulates with length and a short enough warp never uses up its slack. That made the beam count a property of the piece as well as the cloth, and it meant a trial weaving could not predict the production run. With regulation the count is a property of the pairing alone: a pairing the ground absorbs shares a beam at any length, and one it does not needs its beams at any length long enough to matter.

How long the warp takes to settle #

A regulated warp does not start regulated. At the first pick every end is at the beam’s tension, and the tension difference builds as the ground’s ends consume their extra length and stretch. The stretch needed is tiny: an end of this yarn stretches by its tension over a stiffness of one or two hundred newtons per unit strain, so a hundredth of a newton is a strain of a few hundred-thousandths to a ten-thousandth — a tenth of a millimetre over a metre of warp between the beam and the fell. The take-up difference supplies that in a few centimetres of cloth.

So the regulation is established within the first repeat or two of the design, long before the fixed-slack count’s drift could have accumulated to anything, and a figured warp on one beam is regulated for the whole piece. The first centimetres of the cloth carry the transient, which is one more reason the start of a piece is cut off.

What the yarn’s placement decides #

Every force above comes as a range, and the range is the yarn’s stiffness bracket narrowed to the band two measurements agree on — a factor of three. For the clear cases the range does not matter: the eight-on-five satin is inside a tenth of the working tension at both ends of it, and the satin on plain is past its locus at both. For the borderline case it decides the verdict. A five-end satin on a 3/1 twill needs 0.021 newtons at the lower placement, inside the tenth, and 0.069 at the upper, outside it.

So the one pairing whose verdict is uncertain is uncertain for a reason this account can name, and it is the same reason the loom hands the crimp to the weft found its switch uncertain: the force that moves a crimp is proportional to the yarn’s rigidity, and the rigidity is known only to where two tests place it. A softer-spun yarn favours one beam; a harder one needs two.

How the forces were computed #

The take-up difference is the earlier count’s: each region’s warp crimp from Peirce’s geometry at its own bending pitch, for a 0.25-millimetre yarn at a 0.5-millimetre spacing, the difference between figure and ground taken as Δ.

The force is the ground’s locus slope at Δ. The ground is built as a plain cloth of the same yarn at a sett of 20 over its float length — ends and picks alike — and the site’s load–extension calculation gives the force per end at each extension along its constant-thread-length locus, from the least-energy state. The force at Δ is interpolated; a Δ beyond the locus’s end is recorded as unreachable. The rigidity is placed at both ends of the band where the cantilever and the washing measurements agree.

What is required of it. A damask must need no tension difference; some pairing with a real difference must fall within a tenth of the working tension; and some pairing’s difference must lie past its ground’s locus. All three hold, and the last two are claims the model could have failed.

What the regulation argument leaves out #

The figure’s side. The ground’s ends tighten and the figure’s slacken; the argument prices the tightening and assumes the slackening costs nothing, which is the soft end of the figure’s own locus and small. A fuller account shares the difference between them in proportion to their stiffnesses.

Friction. A crimp does not move the moment its energy says it should; the crossings hold it, and the most a cloth can give back has a band of states the cloth rests in. Regulation needs the tension difference to overcome that band as well, which raises every force here by an amount the model does not contain.

And what an uneven tension shows. Every threshold above is set against a tenth of the working tension, which is a round number and not a measurement of when a warp’s unevenness becomes visible in the cloth. The borderline pairing’s verdict turns on it.

Who described which part #

Figured weaving on one beam is ordinary practice, and so is the rule that a figure on a very different ground needs its own beam or a supplementary weft. The fixed-slack count was this account’s, and so is its correction.

What is new here is the price. The tension difference that lets a figured warp share a beam is the slope of its ground’s own locus at the take-up difference — a force this account already computes for other reasons — and it turns a step at nought into a band with two computable edges: where the warp absorbs the figure for nothing, and where the ground runs out of crimp to give.

Still open: whether the streaks show #

The borderline pairing needs up to thirteen per cent of the working tension as a difference between columns, and a tension difference between neighbouring ends does something the argument above does not follow: it changes how each end sits in the cloth. A tighter end is straighter and lies lower in the ground; a slacker end stands proud. The difference is a relief, one column high and a figure wide, and a seersucker is that relief made large.

Whether a five-end satin on a 3/1 twill, woven on one beam, shows a visible ridge where its columns change is the question the borderline turns on, and the arithmetic for it is the crimp height at two tensions — the loom-state crimp the tension essay computes — differenced across a boundary. That calculation would replace “a tenth of the working tension” with the tension difference a finished cloth can hide.