The repeat allows four layers and the loom allows two

A repeat holds at most half its ends as separable cloths, and the bound is tight: a stack of two-end layers on an n-end repeat really does come apart into n/2 pieces, and the site’s own witness reaches the ceiling at every size tried.

That is a fact about matrices. It is not a fact about weaving, and setting it beside two facts that are says something about which kind of ceiling is worth computing.

The harness, which is the first real one #

A layer of a double cloth weaves a weave, and a weave needs as many shafts as it has distinct columns. Stack k layers and the columns do not overlap — an end of layer one is up on a different set of picks from an end of layer two, always — so a k-layer stack needs k times one layer’s shaft count.

And a harness has a size, set not by the mechanism but by the warp. The strain budget is the argument: a shaft’s ends have to survive being lifted into the shed, the strain rises with the shaft’s distance from the fell, and a stated tolerance is therefore a stated number of shafts. On an ordinary broad loom at a one per cent limit it is thirteen.

Divide:

The two columns go in opposite directions. A finer layer weave costs more shafts, so the harness carries fewer layers of it — while the repeat gets larger as the layer weave does, so its own bound rises. The two are never far apart at the coarse end and are four times apart at the fine one.

Where the last row goes #

The eight-end satin row is the interesting one and it is worth stating as a sentence rather than as a table entry.

A double cloth of eight-end satin layers needs sixteen shafts, and the strain budget buys thirteen. It cannot be woven on a dobby at all — not by a mill with a bigger dobby, because the limit is not the dobby, it is what the back shaft does to the warp.

So the fine double damask, which is exactly the cloth a designer would most want two satin layers for, is a jacquard construction by arithmetic. It is usually described as a jacquard construction because it carries a figure, and the figure is a sufficient reason; the shaft count is a prior one, and it holds even for a plain two-layer satin cloth with no figure in it whatever.

The beams, which bind lower still #

The rung below priced the second ceiling. Two layers that differ in anything the crimp depends on eat their warp at different rates, and the difference accumulates with the length woven rather than settling — so they need a beam each, absolutely rather than as a matter of degree.

A loom carries two beams. Some carry three; almost none carry four, and those that do are built for a named construction rather than for general work.

So the practical ceiling on a double cloth with differing layers is two, and the harness’s ceiling of three or six is only reachable by stacking layers that are the same cloth — which is to say by weaving one construction several times over.

That is the shape of the whole thing. The three ceilings do not bind in the same regime:

• at two-end plain layers the beams bind, at two, and both the harness and the repeat allow six;
• at twill layers the beams bind again, at two, against a harness allowing three;
• at five-end satin layers the harness and the beams bind together, both at two;
• at eight-end satin layers the harness binds alone, at one.

The repeat’s bound is never the binding one. It coincides with the harness’s at the degenerate two-end layer and is above everything else at every other row.

And a fourth cost that is not a ceiling #

There is one more quantity and it is worth separating from the three above, because it is a cost rather than a limit.

A k-layer cloth needs k times the picks per centimetre of finished cloth: each layer wants its own pick density, and they are all inserted through the same reed by the same shuttle. So the loom runs k times as long for a metre of cloth, and the mill’s output falls by the same factor.

That is not a ceiling — nothing stops a loom weaving slowly — and it is why the constructions people actually make sit at two layers even where three would be possible. A three-layer cloth is a third slower than a two-layer one at making the same length, on a loom that already needs an extra beam.

It also explains an asymmetry the ceilings do not. Weaving three layers of a coarse plain weave is allowed by the harness, forbidden by an ordinary loom’s two beams, and would take three times as long — so the constraint that stops it is whichever the mill notices first, and mills notice loom time. That is why triple cloths are made where the third layer is wadding: a wadding layer has no weave of its own, needs no shafts and no beam, and adds picks and weight without adding a construction — which is what a backed and stitched cloth is doing one layer down.

Why only the witness reaches the bound #

The repeat’s ceiling is half its ends, and the reason the four-layer witness reaches it is worth spelling out, because it is also the reason nothing else does.

The bound comes from a two-line argument: every component that is a cloth rather than a loose thread contains a cycle, a cycle alternates ends and picks, and a cycle cannot close in two steps — so every cloth in a draft owns at least two ends and at least two picks, and n ends therefore hold at most ⌊n/2⌋ cloths.

A layer that owns exactly two ends is the smallest cloth there is, and the only weave on two ends is plain. So the witness is not one construction among several that reach the bound; it is the only one, and every other layer weave is spending ends the bound would rather it did not.

Put a number on the spending. A stack of k layers of an m-end weave sits on k·m ends, which admits k·m/2 cloths, and delivers k. The fraction of the repeat’s capacity used is therefore 2/m, whatever k is:

• two-end plain layers: 100%;
• 2/2 twill layers: 50%;
• five-end satin layers: 40%;
• eight-end satin layers: 25%.

So the notation’s bound is not merely non-binding, it is unreachable by anything a weaver would call a cloth. A two-end plain layer at a workable sett is a scrim; four of them stacked is a construction with no use anybody has proposed. Every real double cloth is spending half its ends or more on making its layers into fabrics rather than into the minimum the criterion accepts.

That reframes the bound rather than dismissing it. It is a statement about how much structure a repeat can hold, and a layer weave is structure: the difference between 25% and 100% is the difference between a cloth with long floats and a cloth with none. The repeat’s capacity is spent on the layers’ own weaves before any of it is spent on the layers, and the ceiling counts only the second.

Which is a statement about what this collection computes #

The four-layer witness is a real object. It is a draft, it is eight ends by eight picks, the criterion reports four separable cloths, and the count is checked against the bound at every size the site draws it at. Nothing about it is wrong.

It is also unweavable, and the essay that established the bound did not say so — not because it was careless but because nothing in a matrix knows about a beam.

That is the general shape of the risk in a site built on counting matrices. Every bound this collection computes is a bound on the notation, and the notation is more permissive than the machine in every direction that has been checked: the harness does not grow, a beam delivers one rate, a shed opens one hole. A result that says “a repeat can hold four cloths” is true and is not an invitation.

The remedy is not to stop counting matrices; it is to say which ceiling is being computed. This rung’s contribution is the comparison, and the comparison’s finding is that the matrix’s ceiling has never once been the binding one.

What a jacquard changes, and what it does not #

The eight-end satin row says the construction needs a jacquard, so it is worth asking what a jacquard actually removes from the arithmetic.

A jacquard gives every end its own hook, so the shaft count stops being a count of distinct columns and becomes a count of ends in the repeat. A sixteen-end double cloth of satin layers needs sixteen hooks and a machine has six hundred, so the harness ceiling vanishes entirely.

And the strain argument does not vanish with it. A jacquard’s harness cords run from the comber board to the ends and the geometry is different from a shaft’s, but the ends are still being lifted into a shed and the shed still has to open far enough for a shuttle. What a jacquard removes is the shaft-count limit rather than the strain limit — the two were tied together on a dobby because shafts sit at increasing distances from the fell, and on a jacquard they are not.

So the fine double cloth is possible on a jacquard, and the other two ceilings are exactly where they were. The beams still bind at two, and the pick cost is untouched. A jacquard double damask is a two-layer cloth for the same reason a dobby double cloth is, and the reason has nothing to do with the harness.

That is worth having because it inverts the usual explanation. The trade’s account of why fine double cloths are jacquard work is about the figure; the arithmetic says a plain one would need the machine as well, and that the machine buys nothing at all towards the layer count.

The cloth the ceilings actually allow #

Reading the table for what it permits rather than for what it forbids gives a short list, and the list is recognisable.

Two layers of anything, provided the two get a beam each and their combined shaft count is inside thirteen. That covers every ordinary double cloth: plain over plain at four shafts, twill over twill at eight, plain over 2/2 twill at six, and a five-end satin over a plain at seven.

Three layers of a coarse weave, if the loom has three beams, and the third layer will usually be wadding rather than a construction — which costs no shafts and no beam because a wadding end weaves nothing and is simply bound in.

And four layers only as the witness: two-end plain layers, eight shafts, four beams, four times the picks. Every one of those is possible and no two of them are convenient together.

Which is a fair description of the historical corpus. Double cloths are everywhere, triple cloths exist and are nearly always wadded, and four-layer cloths are curiosities. What a figure costs the loom is the same kind of accounting applied to a design rather than to a stack, and it reaches the same sort of conclusion: the constructions that get made are the ones whose several budgets happen to be satisfiable at once.

What was counted, and how #

The shaft count per layer is cloth.js’s shafts, unchanged: the number of distinct columns in the weave. That the stack’s count is k times it rather than something smaller follows from the columns not overlapping, and is checked by running shafts on the stacked draft itself rather than multiplying.

The strain budget is loom.js’s shaftBudget, also unchanged, at its own default of one per cent. That function solves the budget from the leading-order strain and then confirms it against the exact trigonometry, which is worth knowing because the number quoted here inherits both.

The repeat’s bound is evaluated at the stack the harness allows, not at some fixed repeat, so the two numbers in each row are about the same cloth. Comparing a repeat’s bound at eight ends with a harness’s bound at sixteen would have been comparing two different constructions.

And the finding is asserted in both directions. The plain-layer row must have the two ceilings equal — that is where the notation’s bound is reached — and every other row must have the harness’s strictly below the repeat’s. A version of this asserting only the second would have passed on a table in which the repeat was never reached at all, which is a weaker and less interesting claim.

The last row is asserted separately, because “needs more shafts than the budget buys” is the sentence the essay is about and a table row is not an assertion.

Where the model stops #

The strain limit is a stated tolerance and not a measurement. One per cent is the figure loom.js carries and it is a reasonable warp strain; a stronger yarn or a more forgiving mill would take more, and the table’s harness column moves with it. What does not move is the ordering, because every entry in that column is the same budget divided by a different shaft count.

The beam count is a fact about looms and not about weaving. Two is what a general-purpose loom carries; a loom built for a named construction carries what that construction needs. So the beam ceiling is a statement about what a mill has rather than about what is possible, and it is the softest of the three.

Nothing here is about the shed’s depth. A stack of layers has to be separated vertically at the reed for a shuttle to pass, and how far the warp sheet must open for k layers is a real question this collection has not answered — the shed model is written for one cloth and the extra opening a second layer needs is the cloth’s own thickness rather than a second shuttle’s height, which is a much smaller quantity but is not zero.

And the pick cost is stated rather than computed. “k times the picks” assumes each layer wants the same pick density as a single cloth would, which is right for a face-and-back construction and wrong for a wadded one, where the wadding is inserted at a fraction of the rate.

Who found it, and when #

That a double cloth needs two beams and a triple cloth three is trade knowledge, and so is the observation that fine double cloths are jacquard work. Neither is presented as arithmetic anywhere this collection has looked.

The layer bound — half a repeat’s ends — is the site’s own, established a field ago with a witness that reaches it. What is added here is the comparison, and the comparison is uncomfortable in a useful way: the bound the site proved is the one bound of the three that has never decided anything.

What that is worth is a habit rather than a number. A ceiling computed from a notation is a ceiling on the notation, and the interesting question is always which of the several ceilings a real object sits under. This collection is unusually well equipped to compute the wrong one, because counting matrices is what it does.

Where the ladder goes next #

Five rungs of this anchor have taken a two-layer cloth apart: the same draft finished four ways, where a stitch may go, what an interchange joins, what the beams cost, and what the loom allows. What none of them has asked is what the reader gets — a double cloth is twice the material and it is not twice the cloth, and how its thickness, its weight and its warmth compare with a single cloth of the same total yarn is a question about the stack rather than about the draft.