Three mistakes and the shape each one leaves

A loom does not store a weave as a matrix. It stores it as three separate objects, and the matrix is what those three produce between them: the drawing-in says which shaft each end is threaded through, the lifting plan says which shafts rise for each pick, and the tie-up says which shafts each treadle or each lever actually lifts.

That factorisation is the whole economy of a shaft loom — it is why the harness does not grow with the repeat — and it has a consequence nobody designs for: one mistake is not one mistake. Where it is made decides how much cloth it touches, and the three answers differ by more than three orders of magnitude.

The claim #

On a fifty-metre piece a metre and a half wide, one error puts 110,000, 3,600 or 6,187,500 intersections wrong, depending only on which of the three objects it was in.

The ratio between the extremes is 1,719 to 1. Nothing about the mistakes differs — each is one entry of one table set to the wrong value — and nothing about the yarn, the weave or the loom enters. What differs is how many intersections each object controls.

The argument, which is a counting argument #

A piece of this cloth holds 3,600 ends and 110,000 picks: 396 million intersections, each of them the product of one threading entry, one lifting entry and one tie-up entry.

A threading error touches one end at every pick. The end is threaded once, at the start, and stays on the wrong shaft for the whole warp — so it does whatever that shaft does, for 110,000 picks. One line down the piece, one end wide.

A lifting error touches one pick across every end. The pick is made once and is over. One bar, 3,600 intersections, across the full width.

A tie-up error touches every intersection where that shaft’s ends meet that treadle’s picks. The shaft carries a fraction of the ends — one eighth of them, on an eight-shaft loom — and the treadle is used once every repeat, so a fraction of the picks. The product is 450 ends by 13,750 picks: six million intersections, spread as a lattice over the whole cloth.

So the three objects have completely different reach, and the reach is exactly the fraction of the cloth each entry controls. A drawing-in entry owns a column. A lifting entry owns a row. A tie-up entry owns a sublattice, and a sublattice is a two-dimensional object where the other two are one-dimensional.

The one nobody sees happening #

The three errors also differ in how they are discovered, and the ordering is the reverse of their sizes.

A threading error is visible at the loom. The end is in the wrong heald, and a weaver checking the draw-in can find it by counting. It also announces itself in the cloth almost at once: the line appears with the first repeats.

A lifting error is visible in the cloth immediately. A bar is a bar, and a stopped loom is inspected.

A tie-up error is invisible. Every end is in its correct heald. Every pick is made in whatever shed the harness opened. The loom is running exactly as it should, doing precisely what it was told, and the cloth coming off it is a different cloth from the one that was designed — a coherent, plausible, well-made fabric with a different weave. There is nothing to notice unless somebody compares the cloth against the design.

That is why this error is worth an essay. It is not a fault in the ordinary sense at all: it produces a cloth with no defect in it, and the defect is in the relationship between the cloth and the specification.

What the wrong cloth is #

Since a tie-up error produces a different weave rather than a damaged one, the natural question is: which weave?

The answer depends on the draft, and the interesting part is that it need not be a weave at all in the sense this collection cares about. Changing one tie-up entry changes one shaft’s contribution to one pick, which is a row-and-column pattern of reversals in the matrix — and a reversal is exactly the operation that can split a cloth into layers.

So a tie-up error can produce a fabric that does not hang together, over the whole piece, from a loom running perfectly. That is the worst outcome available anywhere in this ladder, and it arrives from the smallest possible cause.

What was counted, and how #

The three counts are computed from the cloth’s own construction and the loom’s own numbers — the sett gives the ends and picks in a piece of stated size, the shaft count and repeat give what fraction of them each tie-up entry reaches — and the assertions are about the ordering rather than the values, because the values move with every one of those inputs.

A tie-up error must put more intersections wrong than either of the others, asserted over the computed counts rather than assumed from the shape of the argument. That is what would fail if the sublattice were being counted as a line.

A threading error must beat a lifting error at any piece longer than it is wide, which is the aspect-ratio result the fault-geometry essay establishes, arriving here as a special case.

And the ratio between the extremes must be in the hundreds or more, asserted as an inequality so that a change in the loom’s shaft count cannot quietly turn the argument into one about a factor of three.

What each shape looks like to a reader of the cloth #

The counts say how much cloth each error touches. What a buyer sees is decided by the shape as much as by the count, and the three shapes are read very differently.

A line down the piece is the most conspicuous mark a cloth can carry. It is a single continuous edge running the whole length, with nothing to break it up, and the eye finds a straight line in a texture faster than anything else. This is why a drawing-in error, which is only the middle of the three by area, is the one a weaver is most afraid of.

A bar across the piece is conspicuous and short. It is equally continuous, but it lasts one pick and then the cloth is correct again — so it is a defect with an end to it, and a cutter can work around it.

A lattice is the least conspicuous of the three by a long way, even though it touches the most cloth. Its wrong intersections are spread over the whole piece at the period of the shaft and the period of the repeat, so what it produces is a change of texture rather than a mark: the cloth looks like a slightly different cloth. Nothing in it is a line, an edge or a discontinuity.

That inverts the ordering completely. By area: tie-up, threading, lifting. By conspicuousness: threading, lifting, tie-up. The fault that touches most of the cloth is the one least likely to be reported, and it is also — by the argument of the previous section — the one nobody can see happening at the loom.

A lattice does have one property that gives it away, and it is the one this collection has an instrument for. It is periodic in both directions at once, at the shaft’s period across and the repeat’s period down, so its energy sits at exactly two spatial frequencies. A periodic disturbance beats a random one of the same size by the square root of how much cloth is in view — so the way to find a tie-up error is to stand well back from a large piece, where every other kind of variation has averaged itself away and the lattice has not.

How the loom count changes the answer #

The tie-up error’s reach depends on the number of shafts, and it depends the wrong way round from most things about a loom.

More shafts means a smaller fault. Each shaft carries fewer ends, so a wrongly tied one touches fewer of them: at sixteen shafts the lattice is half the size it is at eight, and at a jacquard — where every end is its own shaft — a tie-up error touches a single end and the whole distinction collapses.

That is worth stating as a design property rather than as an accident. A jacquard has no tie-up errors of consequence, because it has no tie-up: the harness is the design, and an error in it is an error in one end. The economy that shaft looms buy by factorising a draft into three small tables is paid for by the fact that an error in the smallest of those tables is the largest fault the machine can make.

The fourth object, which is the design itself #

There is a fourth place a mistake can be made and it belongs in the comparison, because it is the only one whose reach exceeds the tie-up’s.

The design can be wrong. A draft that separates into layers, a float longer than the cloth can carry, a repeat that does not tile: each of those is an error in the object all three tables are derived from, and it puts every intersection of the piece wrong — 396 million of them, sixty-four times the tie-up error’s reach.

That sounds like a category difference rather than a comparison, and it is worth resisting the temptation to treat it as one, because the error looks exactly the same at the loom as a tie-up error does. In both cases every thread is in its correct place, the machine runs perfectly, and the cloth is not the cloth that was wanted. The only difference is which document is at fault.

Reading the table downwards, the visibility falls as the reach rises. That is not a coincidence and it is the whole practical lesson: the errors a loom announces are the small ones, because announcing an error requires something to go wrong mechanically, and the large errors are the ones where nothing does.

It is also the argument for the check this collection exists to make. A design that does not describe one cloth passes every mechanical test a loom can apply, weaves perfectly, and produces fabric that comes apart in the hand — and the criterion that catches it is arithmetic performed on the design rather than on the cloth.

Where the model stops #

The tie-up is treated as a table. On a modern dobby it is, and on a treadle loom the tie-up is cords, which can slip, stretch or be tied to the wrong lam — the same error with a mechanical cause. On a machine with an electronic dobby there is no tie-up at all: the lifting plan addresses the shafts directly, which merges two of the three objects and removes one class of error entirely.

A fault is counted as a wrong intersection and intersections are not equally wrong. An intersection that should have been warp-up and is weft-up is a visible reversal; one that changes a float from four to five is barely anything. The counting above weights them alike, which is right for a comparison of reach and wrong for a comparison of appearance.

The piece dimensions carry the answer. Every ratio here is a statement about a fifty-metre piece a metre and a half wide, and each of the three counts scales differently with those: the threading error with the length, the lifting error with the width, the tie-up error with both.

And no repair is priced. A threading error can be corrected by re-drawing one end, which takes minutes; a tie-up error is corrected by re-tying one cord, which takes seconds — and neither of those has anything to do with how much cloth has been made in the meantime, which is the whole cost.

The three counts are one formula #

The rule of thumb the next section states — that the smallest table controls the largest region — is exact rather than approximate, and the exact form makes the three counts one line.

An entry’s reach is the piece divided by the size of its own table.

The piece holds 396 million intersections. The lifting plan has 110,000 entries, one per pick, and 396,000,000 ÷ 110,000 is 3,600. The drawing-in has 3,600 entries, one per end, and the quotient is 110,000. The tie-up has 8 × 8 = 64 entries, and the quotient is 6,187,500. All three of the essay’s numbers, from one division.

And the fourth row falls out with them. The design is a table of one entry, so its reach is the whole piece — which is what the table says and is now a consequence rather than a separate observation.

That identity is what makes the generalisation exact. In any representation where a result is the product of several lookups, each table’s entries partition the output between them, so an entry controls precisely its table’s share. Compactness and blast radius are the same number seen from two sides.

Which means the tables are equally dangerous and the entries are not #

Follow the identity one step further and it says something the essay’s ordering does not.

Suppose each entry of each table is equally likely to be wrong — the same probability p per entry, which is a fair model of a person filling in tables. Then the expected number of wrong intersections from a table is its entry count times p times the piece over its entry count, which is

p × the whole piece, for every table alike.

The three tables contribute exactly equal expected damage. The tie-up is 1,719 times worse per entry and has 1,719 times fewer entries, and the two cancel to the last intersection.

So the tie-up’s danger is not that it delivers more damage on average. It is that it delivers the same damage in rarer, larger doses — and that matters only because of what the unit of loss is. A fault condemns a piece, not an intersection: a lifting error spoils one bar of one piece and a tie-up error spoils the whole of it, and the piece is what gets thrown away.

The tie-up is the dangerous table because cloth is sold by the piece, and the arithmetic says so without any appeal to how conspicuous the faults are. Against a buyer who priced cloth by the flawless intersection, all three tables would be equally worth checking.

And the shaft count enters at one of two exponents #

The claim that more shafts make a smaller tie-up fault is right, and its rate depends on what the extra shafts are spent on — which is worth separating because the two cases are the two reasons a weaver adds shafts.

Extra shafts on the same weave — a straight draw carried over more shafts than the repeat needs — leave the treadle count where it was and divide the ends among more shafts. Reach falls as 1/S: doubling the shafts halves the lattice, which is the essay’s figure.

Extra shafts spent on a longer repeat divide the ends among more shafts and the picks among more treadles. Reach falls as 1/S²: doubling the shafts quarters the lattice.

At sixteen shafts weaving a sixteen-pick repeat the tie-up error touches 1.5 million intersections rather than the essay’s 3.1 million, and by a jacquard — every end its own shaft, every pick its own lift — the quotient has reached one.

So the harness’s economy and its fragility are the same curve read twice. A loom that stores a design in S² entries makes faults of size 1/S², and the whole progression from four shafts to a jacquard is a trade of storage against blast radius at a fixed product.

The generalisation #

When a specification is stored as a product of smaller tables, an error’s cost is the size of the region that table entry controls — and the smallest table controls the largest region. That is the transferable shape, and it is exactly backwards from the intuition that a big table is a big risk.

The reasoning is worth stating as a rule of thumb: check the small tables first. In any factorised representation — a lookup table times an index, a configuration times a schedule, a lookup of a lookup — the compact object is the one whose entries are reused most, and reuse is what turns one wrong entry into a lattice of wrong outputs.

The second lesson is about faults that produce valid output. A threading error makes a defective cloth; a tie-up error makes a correct cloth of the wrong design. Nothing checking the output for defects will find the second, because there are none. Only a comparison against the specification will, and that comparison is exactly the check that gets skipped when the machine is running well.

Who found it, and when #

The three-object factorisation of a shaft loom is as old as the shaft loom and is how every weaving text presents a draft. That the three carry different consequences when they are wrong is known in practice — a weaver checks the tie-up before starting and the threading as the cloth appears — and appears not to be written down as a quantity.

What this collection adds is the counting: the reach of each error as a fraction of the piece, the ratio between them, and the observation that the ratio moves with the shaft count in the direction that makes compact machines fragile. The last of those has a modern consequence: an electronic dobby, which addresses shafts directly and has no tie-up, has removed the largest fault class its predecessor had, and did so for reasons of convenience rather than of quality.

Where the ladder goes next #

The last rung of this ladder asks the question a weaver actually asks when a fault has happened: will anybody see it? Testing the trade’s answer — that a busy weave hides a fault — against every draft there is gives a result that disagrees with it, and disagrees for a reason about float structure rather than about pattern.

Sideways, the tie-up error’s mechanism is the elementary reversal whose consequences the mispick census counts, and the fault’s cost in cloth is settled by the geometry of the piece.