Two layers need two beams

Every rung of this anchor so far has treated a double cloth as a matrix problem. A tube, a double-width cloth and two separate cloths are the same draft; a stitch is one reversed intersection; an interchange joins along the boundary it draws. All of it is combinatorics and none of it has any length in it.

The loom has length in it. A double cloth’s two layers are two warps, and the question of whether they can come off one beam is not a question about connectivity at all.

The rate a layer eats its warp at #

A warp end is longer than the cloth it crosses, because it goes over and under. The excess is the crimp, and a layer weaving one metre of cloth therefore takes 1 + c metres of warp off its beam.

That number is the whole of the problem, and it is a property of the weave and the sett rather than of anything the weaver can adjust. Peirce’s geometry, asked about a thread that bends at its own average rate rather than at every crossing, gives for a quarter-millimetre yarn set at half a millimetre:

The spread is enormous and it is the same spread the float decides everything else with: a plain weave turns at every crossing and a satin turns once in eight, so a plain weave’s thread has to travel eighteen times as much extra distance.

And the arithmetic that follows has no tolerance in it #

Put a plain face over a satin back. The face wants 1.1435 metres of warp per metre of cloth and the back wants 1.0203. A beam delivers one length to everything wound on it.

Over a hundred metres of cloth that is 114.35 metres of warp against 102.03 — a gap of more than twelve metres. There is nowhere for twelve metres of slack to go.

The tempting response is to look for a tolerance: how much difference can the loom absorb? The answer is that the question is the wrong shape. A back rest and a let-off compensate tension, over a distance of a few centimetres between the beam and the fell of the cloth. They do not compensate accumulated length, and the mismatch here accumulates with every pick woven.

So the right measure is not a percentage but a distance: how much cloth can be woven before the mismatch reaches one pick spacing, which is already a visible fault. At a twelve per cent difference and a half-millimetre pick spacing, it is four millimetres.

Which leaves exactly one construction that can share a beam #

Run the table and the pattern is stark. A plain face over a 2/2 twill is 11.1 points apart; over a five-end satin, 12.3; over an eight-end satin, 13.6. Even two satins — five-end over eight-end, which look like nearly the same cloth — are 1.25 points apart and lose a pick spacing in forty millimetres.

The only pair at zero is two layers of the same weave at the same sett.

That is not a curiosity, it is exactly the construction the trade weaves on one beam: a doubled shirting, a two-ply blanket, a tube of plain weave, a double-width plain cloth folded at the selvedge. Every one of those is the same cloth twice, and every one of them is woven from a single warp with alternate ends going to alternate layers.

And it is the same set of cloths that has to be stitched, because it is the set with no pattern in it and therefore no interchange to join it. The construction that escapes the beam problem is the construction that cannot escape the stitching problem, and the two are the same fact seen from opposite ends: a double cloth is either one cloth doubled, or two different cloths.

The sett does it too, and that is the part that surprises #

The rule is usually stated as being about weaves, and it is not.

Take two layers of plain weave — the same weave, the same yarn — and set one twenty per cent more openly than the other. The crimps come out different, because crimp is a function of the spacing as well as of the bending rate, and the two layers are back to needing two beams.

That is a much more common situation than two different weaves. A backed cloth’s back layer is nearly always set more openly than its face, because the back is there for weight and warmth rather than for appearance, and the yarn is coarser and the ends fewer. So a backed cloth needs two beams for a reason that has nothing to do with its two weaves, and the design manuals’ framing — “a double cloth of two different weaves needs two beams” — understates the rule by leaving the setts out.

The general statement is the honest one: any two layers that differ in anything the crimp depends on need two beams, and the crimp depends on the weave, the sett, the yarn diameter and the tension. Two layers agreeing in all four are one cloth woven twice.

Why the two layers cannot help each other out #

A reader who knows crimp interchange will look for a way out here, and it is worth saying why there is not one.

Within one cloth, crimp is a shared budget. Pull it warpwise and the warp crimp falls while the weft crimp rises, because the two systems are wrapped round each other and neither thread stretches — so a cloth extends by moving its crimp rather than by stretching anything, and the total thread length is conserved through the exchange.

Two layers of a double cloth are not wrapped round each other. That is the whole content of the construction: the face’s ends cross the face’s picks, the back’s cross the back’s, and the only places the two meet are the stitches and the interchanges. A face layer that is short of warp cannot take the length out of the back layer’s crimp, because the back layer’s crimp is holding the back layer’s own weft.

So the two budgets are separate, and there is no mechanism by which a shortfall in one is met from the other. The layers can be pulled tighter or slacker against each other — that is what two let-offs are for — but the rate at which each consumes warp is set by its own geometry and is not negotiable from outside.

The exception is the stitch, and it is a small one. At a stitching point the two layers really are interlaced, so a stitch does couple the two budgets — at one intersection per repeat, against the hundreds of intersections that set the crimp. The coupling is real and it is three orders of magnitude too weak to matter, which is a satisfying place for a rule to come from: the same sparseness that makes a stitch invisible makes it mechanically negligible.

What happens if a mill tries one beam anyway #

The failure has a shape and a name, and knowing it makes the arithmetic feel less abstract.

The layer with the lower crimp needs less warp per metre of cloth, so on a common beam it receives more than it wants and goes slack. Slack warp does not shed cleanly: the ends sag into the shed, the shuttle catches them, and the cloth acquires floats and broken ends where the weaving is otherwise perfect.

Before that, and much earlier, it goes soft. A slack layer beats up loosely, its picks crowd, and its own sett drifts away from the design’s — which changes its crimp, in the direction that makes the mismatch slightly smaller. The cloth does compensate, a little, and it compensates by becoming a different cloth from the one specified.

That is the honest reason a single-beam attempt is not simply impossible. It fails gradually, by degrading the loose layer into whatever construction happens to balance, and a weaver watching the face of the cloth may not notice for some distance. The crimp ratio a cloth settles at is not a measurement but a state, and a mismatched double cloth is a cloth being pushed towards a state nobody chose.

What a second beam actually costs #

The remedy is old and it is worth pricing, because the price is what decides which cloths get made.

A second beam is a second warp: separately warped, separately sized, separately mounted, separately let off, and separately tensioned. On a loom built for it that is a fitting; on one that is not it is a rebuild. A mill that owns single-beam looms cannot weave a face-and-back double cloth on them at all, and that is a capital constraint rather than a technical one.

It also doubles the setting-up. The two warps have to be run out together at the right ratio, and the ratio is the crimp ratio computed above — which means the ratio is a property of the design and has to be reset whenever the design changes. A mill running one construction can set it and forget it; a mill running short pieces of many constructions is resetting two let-offs against each other for every piece.

Which is why the commercially common double cloths are the ones with the smallest crimp difference. A face and back of the same weave family at similar setts is a cloth a single-beam mill can very nearly weave and a two-beam mill can weave without fuss; a plain face over a satin back is a construction that exists in design books and rarely in warehouses.

That is a real explanation of an observed distribution, and it is not the one usually given. The manuals explain the rarity of extreme face-and-back combinations by appearance and by the difficulty of hiding the stitching. The arithmetic says the binding constraint is the ratio of two let-offs.

A number for the backed cloth, since that is the common case #

The two-setts figure above is the construction a mill actually runs, so it is worth having its number rather than the extreme pair’s.

A plain face at half a millimetre over a plain back at 0.62 mm — the same weave, the same yarn, the back set a quarter more openly — gives crimps of 14.35% and 8.80%, a difference of 5.55 points. That is 5.5 metres of warp over a hundred-metre piece, and one pick spacing of slack after eleven millimetres of weaving.

Eleven millimetres rather than four. It is nearly three times as forgiving as the extreme pair and it is still nothing: a loom weaves eleven millimetres in a couple of seconds.

So the common case is not a marginal case. A backed cloth is the double cloth a mill is most likely to have on the loom, its two layers differ in nothing but their sett, and it needs two beams as absolutely as a plain-over-satin does. The rule has no comfortable middle in it, which is why it reads in the manuals as a flat statement rather than as a calculation — and why nobody has ever needed the calculation to obey it.

What the calculation adds is the ability to answer the question the other way round. Given two layers, how different may their setts be before the beams have to be separated? The answer is: by nothing that survives a metre of weaving, so the question does not arise. And that is worth knowing, because it is the sort of question a designer trying to save a beam would otherwise spend a day on. The sett owns the pitch and the yarn owns the height — and here the sett owns the beam as well.

What was counted, and how #

The crimps come from texture.js’s regionCrimp, which is Peirce’s geometry handed a bending pitch rather than a thread spacing — the same model every crimp on this site is computed from, unchanged. What is supplied here is the bending pitch of each weave, and it is the average: an end that changes face k times in a repeat of n picks bends once every n/k spacings.

The average and not the longest float, and that is a modelling choice worth naming. A thread with one long float and several short ones bends at its average rate over the repeat, and using the longest float would have given a satin a much lower crimp than it has. Which is right depends on how the tension distributes along the thread, and this collection does not have that.

The take-up is the crimp and nothing else. A real loom’s take-up also has to feed the cloth’s own contraction and the weft’s insertion, and those are common to both layers. Only the difference between the layers matters here, and the common parts cancel.

The proportionality is asserted rather than assumed. The gap at ten metres and at a hundred metres is computed separately and required to differ by exactly ten, which is what makes it an accumulation with no settling value. An offset would have shown up as the two being equal.

And the negative case is asserted as hard as the positive. Two layers of one weave at one sett must come out at a difference of nothing — not “small”, nothing — because the whole rule is that this one pair escapes and every other does not. A version of this that only checked the failing pairs would have been consistent with every pair failing.

The same arithmetic at a different yarn #

Everything above is a quarter-millimetre yarn at half a millimetre. Both numbers move the crimps and neither moves the conclusion, which is worth checking rather than asserting.

At a fifth-millimetre yarn set at 0.55 mm the plain weave’s crimp falls from 14.35% to 7.04% and the eight-end satin’s from 0.79% to 0.41%. Every number is smaller and the gap is smaller with them, so the failure distance roughly doubles — from four millimetres to eight.

Eight millimetres is not a reprieve. The rule survives the change of scale intact, because it is a rule about a ratio of two rates and both rates move together. What would break it is a construction in which the two layers’ crimps moved in opposite directions with the yarn, and there is no such construction: a finer yarn crimps less in every weave.

That is the check worth having, and it is the one that turns “these particular numbers are large” into “the conclusion does not depend on these particular numbers”.

Where the model stops #

Tension is not in it. A warp under more tension has less crimp, and the two beams of a real double cloth are deliberately run at different tensions for exactly that reason — a tighter back pulls the face into relief and is how a matelassé gets its quilted look. So the crimps computed here are the cloth’s own relaxed geometry and a weaver has a lever on them that this calculation does not include. What the lever cannot do is close a twelve-point gap.

The pick spacing used for the threshold is the cloth’s, and it is a stand-in. “One pick spacing of slack” is a legible amount of slack rather than a measured failure threshold; the real threshold depends on the shed, the beat-up and how forgiving the yarn is. The distances quoted are therefore the right order of magnitude and not measurements.

Both layers are given the same yarn. A real backed cloth’s back is coarser, which changes its crimp again — usually upward, since a thicker thread has further to travel round its crossings, so the gap between a fine face and a coarse back is generally larger than the same weaves at one diameter would suggest. Not computed.

And nothing here is about the weft. The two layers’ wefts are inserted separately and take up separately too, and a double cloth’s weft crimps differ by as much as its warp crimps do. That does not need two beams — the weft comes off a shuttle rather than a beam — but it does decide the two layers’ widths, and two layers wanting different widths on one loom is a problem this collection has not looked at.

Who found it, and when #

That a double cloth needs two beams is trade knowledge of the plainest kind: it is in every manual, it is in every mill’s setting-up sheet, and no weaver has ever been surprised by it.

What the manuals give is the rule and not the quantity. “Two beams are needed when the two cloths take up differently” is what they say, and the reader is left with no way to tell whether a particular pair takes up differently enough to matter, or how much cloth could be woven before it did.

Putting numbers on it is this collection’s, and two of them are worth carrying. The first is the distance: four millimetres, for the extreme pair, which is why the rule is absolute rather than a matter of judgement. The second is the sett: the rule is about anything the crimp depends on and not about the two weaves, which means a backed cloth of one weave at two setts needs two beams and the manuals’ phrasing does not cover it.

Where the ladder goes next #

The repeat says a two-layer draft can hold as many cloths as half its ends, and this rung says the loom needs a beam for each of them that differs. Those two ceilings are nowhere near each other: an eight-end repeat admits four layers and no loom in the world carries four warp beams and four shuttle boxes with a shed deep enough to separate them — so what a repeat allows and what a loom can weave are two different numbers, and only one of them has ever been the binding one.