A missing end is a fault the length of the piece

Two accidents, the same size. An end breaks in the warp and is not pieced up, so the cloth is woven with a thread missing. A pick is laid in the wrong shed, or laid twice, or missed, so one row of the cloth is wrong.

One thread in each case, and the same amount of wrongness per intersection. What they cost is not remotely the same.

The claim #

A warp fault condemns thirty times as much cloth as a weft fault of the same size, and the thirty is the aspect ratio of the piece.

Not a property of the loom, the yarn, the weave or the fault. A fifty-metre piece a metre and a half wide has an aspect ratio of thirty-three, and a warp fault costs thirty-three times a weft fault because it runs the long way. Weave the same cloth in five-metre lengths and the ratio is three; weave it in hundred-metre lengths and it is sixty-seven.

This is the reason a broken end stops the loom and a bad pick often does not, and it is the reason every grading scheme prices the two differently — usually without saying that the price is arithmetic rather than judgement.

The argument, which is two multiplications #

A fault occupies a band. The band’s width is set by how many threads are involved, and its length by which direction those threads run.

A three-thread warp fault at twenty-four ends per centimetre is 1.25 mm wide and fifty metres long: 0.063 m². A three-thread weft fault at twenty-two picks per centimetre is 1.36 mm wide and 1.5 metres long: 0.0020 m². The whole piece is 75 m², so the first is a twelve-hundredth of it and the second a thirty-seven-thousandth.

Neither number is large. That is the first surprise, and it points at the thing that actually decides what a fault costs.

A fault does not condemn its own area #

The area a fault occupies is nearly irrelevant, because cloth is not sold by the flawless square centimetre. It is cut into pieces, and a fault condemns whichever piece it lands in.

So the cost of a fault is the area of a garment piece, not the area of the fault — and it follows that the same fabric, with the same faults, wastes different amounts depending on what is being cut from it.

With faults falling at random at a rate λ per square metre and pieces of area A, the fraction containing at least one is $1 - exp(-\lambda A)$, and the area lost is that fraction of the whole. Two limits are worth having:

• Small pieces waste almost nothing. As A goes to zero the loss goes to λA per piece, which vanishes: a fabric cut into postage stamps loses only the stamps that actually contain a fault.
• Large pieces waste everything. As A grows every piece contains a fault, and the loss tends to the whole cloth.

So a fault rate is not a property of a cloth on its own. It becomes a cost only when multiplied by what the cloth is for, and the same roll can be excellent for shirts and unusable for coats.

What the fault does to the cloth around it #

The area arithmetic says what a fault costs. It says nothing about what the fault is, and here the missing end has a property the mispick does not.

A broken end is a missing constraint, and the picks it was holding are now held further apart. The neighbours close up, the pick floats across the gap, and the longest float in the cloth grows — from one to three end widths in a plain weave, which is a threefold change in the quantity this collection has spent a whole ladder on.

That is the second reason a broken end stops a loom and a mispick does not: a mispick is a mark, and a missing end is a structural change that continues for the whole length of the piece. What that change does to the weave is a question with an exact answer over every draft there is, and the answer is the next rung of this ladder.

What was counted, and how #

Every number above is computed from the cloth’s own construction rather than quoted. The band a fault occupies comes from the sett — three threads at twenty-four ends per centimetre is 1.25 mm and no judgement enters — and the areas are that band times the piece’s own dimensions.

Three things are asserted while it runs.

• A warp fault condemns more than ten times a weft fault, asserted as an inequality with a factor in it rather than as a value, because the value is the aspect ratio and changes with the piece. What would fail is a band computed in the wrong direction, which is the one mistake this arithmetic invites.
• A larger cut piece is likelier to contain a fault, at every step of the sweep. Trivially true, and it is the check on the exponential being the right way up.
• Each fault condemns a larger multiple of its own area as the pieces grow, also at every step — which is the same statement read as a cost rather than as a probability, and would fail if the loss were being divided by the wrong area.

The fault rate is an input, not a result. Nothing in this collection predicts how often an end breaks; that is a question about yarn strength, warp preparation, humidity and machine condition, and none of those is structure. What is computed is what a given rate costs, which is the half that belongs here.

From a fault rate to a yield #

A cloth is bought against a fault rate — so many points per hundred square metres, in whatever scheme the trade uses — and a cutter needs a yield. The conversion between them is the exponential above and it is worth tabulating, because the two ends of it are so far apart.

At a rate of 0.4 faults per square metre:

The cloth is the same in every row. A supplier and a buyer can agree completely about the fabric’s quality, in a number both of them measured, and disagree entirely about whether it is usable — and neither is wrong, because the number they agreed on does not contain the piece size.

That is an argument for a specification carrying two numbers rather than one: the fault rate, and the largest piece the cloth is intended to yield. The second is the buyer’s to state and is the one that decides everything.

And it puts a value on where the faults are. Two rolls with identical rates, one with its faults evenly spread and one with them clustered into a few metres, are not equally good — the clustered roll condemns fewer pieces, because two faults in one piece cost what one does. Every grading scheme in use counts faults and none of them counts clusters, so the roll that is genuinely more usable grades identically.

Why the selvedge is the exception #

One place in the cloth breaks the arithmetic above, and it is the place that matters most to a cutter.

A fault at or near the selvedge condemns nothing extra, because it is not used in a garment anyway — a few centimetres at each edge are trimmed off before any piece is cut. So a fault’s cost depends on where across the width it falls as well as on its direction, and a broken end within the trimmed band is free.

That gives a genuinely odd consequence: the cost of a broken end depends on which heald it was in. Two identical accidents, one at end 40 and one at end 400 of a two-thousand-end warp, cost nothing and a full length of cloth respectively.

It also gives a design decision. A weaver who knows a warp is prone to breaking in a particular region — a badly wound beam, a worn heald shaft — can arrange the marker so that the trimmed or seamed parts of the garment fall there. That is not a repair. It is a rearrangement of the cost, and it is available immediately and exactly.

The three ways a warp fault ends #

A broken end does not necessarily run the whole piece, and which of three things happens decides whether the arithmetic above applies at all.

It is caught by the stop-motion and pieced up. Every loom carries a drop-wire on each end, and a broken end releases its wire and stops the machine. The fault then runs from where it broke to where the loom stopped, which is a few picks — a matter of millimetres, and a fault two orders of magnitude smaller than the full-length case. This is what the whole stop-motion mechanism exists to buy, and the factor it buys is the length of the piece divided by the stopping distance.

It is not caught, and runs to the end of the piece. That is the case the area arithmetic describes, and it happens when the wire hangs up, when the end breaks behind the drop-wires, or when the end does not break at all but merely runs slack — a tight or slack end, which is a fault of exactly this shape and which no stop-motion detects, because nothing has broken.

It is caught by the weaver and mended afterwards. The fault exists in the cloth but is repaired: the missing end is darned in by hand, which costs time and leaves a mark that may or may not be acceptable.

The middle case is the one worth dwelling on, because it is the one this collection’s own machinery has something to say about. A slack end is structurally identical to a missing one as far as the picks are concerned: it is not holding them, so the floats grow exactly as though it were absent — and what remains holding them is friction at the crossings rather than any change of side. It differs only in that the thread is still there to be seen — which means the fault is invisible on the point paper, invisible to a stop-motion, and fully present in the cloth.

That is the shape of defect this collection keeps finding: an omission rather than an error, where everything that exists is right and something that should be doing work is not.

Counted in panels, the multiplier is not thirty #

The headline compares two areas, and area is the wrong currency once the essay’s own second argument has been made: a fault condemns the piece it lands in, so what a cutter loses is panels. Recount in panels and the multiplier changes, because a fault’s length is now divided by a panel dimension rather than by nothing.

A panel 900 millimetres long and 400 wide, cut with its length along the warp as the grain requires, tiles the fifty-metre piece into 55 rows of 3¾ — call it 208 panels.

A broken end runs the length of the piece, so it crosses all 55 rows and condemns one panel in each: 55 panels, a quarter of the bolt. A mispick runs the width, so it crosses 3¾ columns and condemns 3 or 4.

The ratio is fifteen, not thirty-three. And the missing factor has a name: the panel’s own aspect ratio, 900 over 400, which is 2.2 — and 33 ÷ 2.2 is 15.

So the refined statement is that the multiplier is the piece’s aspect ratio divided by the panel’s, and the essay’s thirty-three is that quantity for a square panel. Every real garment panel is longer than it is wide, and every one of them is cut with its length along the warp, so the multiplier in a cutting room is always smaller than the one in the area arithmetic — by exactly the amount the panels are long.

That is not a comfort. Twenty-six per cent of a bolt condemned by one unnoticed broken end is a worse number than any fraction of a square metre suggests, and it is the number that decides whether a roll is usable.

What clustering is worth, in the same units #

The other correction the essay flags is that faults are not Poisson, and the direction is known — clustering helps — but not the size. It can be estimated with one parameter.

If faults arrive in clusters of mean size c rather than singly, then the clusters arrive at a rate λ/c, and a panel is spoilt by a cluster falling in it whatever the cluster contains. So the faulted fraction is 1 − e^(−λA/c) instead of 1 − e^(−λA), which is the same expression with the rate divided by the cluster size.

At the essay’s rate of 0.4 per square metre and a two-square-metre coat back, singly-arriving faults spoil 55 per cent of panels. In clusters of three, 23 per cent. In clusters of five, 15.

Clustering by three more than halves the condemned area at an unchanged fault rate, and no grading scheme in use can see the difference — which turns the essay’s aside into a sizeable commercial statement. Two rolls that grade identically can differ by a factor of two in yield, in the direction that the worse-looking roll, with its faults obviously bunched into a few metres, is the better buy.

It also says what a buyer should ask for that nobody asks for. Not a lower fault rate, which is expensive and is the mill’s constraint, but the positions: a fault map costs nothing to record on a modern inspection frame, and it converts the whole of this arithmetic from a probability into a certainty, since a marker can then be laid to put the faults where the trimmings fall.

That last is the same move the selvedge section makes, generalised. A fault whose position is known is not a fault about which anything need be probabilistic, and every expression in this essay with an exponential in it is an expression about ignorance rather than about cloth.

Where the model stops #

Faults are treated as falling at random and they do not. A broken end is likelier where the warp is already weak, and a loom that makes one mispick is likelier to make another; both are clustered rather than Poisson. Clustering makes the picture better for the cutter, not worse, because two faults in one piece condemn only one piece — so the Poisson figures above are a pessimistic bound.

The fault is treated as full-length. A broken end is often noticed and pieced up, so the fault runs from where it broke to where it was found, which may be a metre rather than fifty. That converts the warp fault’s arithmetic into the weft fault’s and is the whole reason loom stop-motions exist.

Nothing here prices repair. A mend, a re-weave, a piece cut out and the roll re-joined — the last of which needs a seam that will hold: all are ordinary practice, all cost time rather than cloth, and none of them is in the area arithmetic.

And the condemned area assumes the fault is found. A fault that survives inspection costs a returned garment rather than a square metre of cloth, which is a different order of magnitude entirely and is not a fact about structure.

The generalisation #

The cost of a defect is set by the granularity of what is cut from the material, not by the size of the defect. That is the transferable statement, and its practical form is a question worth asking of any material sold in quantity: what is the unit that gets thrown away?

If that unit is much larger than the defect, the defect’s own size does not matter at all and the arithmetic is entirely about how often units contain one. If the unit is comparable to the defect, the defect’s size matters and its shape does too. The two regimes want opposite responses: in the first, reduce the number of defects; in the second, reduce their extent.

And the second lesson is that an aspect ratio can be a cost multiplier. A defect that runs along a material’s long dimension costs its length divided by its width more than one that runs across, and nothing about the defect itself is responsible. Wherever a material is made in long pieces — cloth, paper, film, strip, wire — the same factor of thirty is sitting there, deciding which of two identical accidents is a disaster.

Who found it, and when #

The asymmetry between warp and weft faults is universal knowledge and is built into every fabric-grading scheme: the four-point system and its relatives all penalise a defect by its length, which prices a warp fault at many times a weft fault automatically. Nothing here is a new observation about faults.

What is worth setting down is the reason the schemes are shaped that way — that the multiplier is the piece’s aspect ratio, and therefore travels with the way the cloth is made rather than with the way it is graded — and the second-order point that the cost is decided by the marker rather than by the fault at all, which the grading schemes cannot express because they grade a roll before anybody knows what will be cut from it.

Where the ladder goes next #

The fault’s cost is settled; its structure is not, and the next rung asks what a missing end actually does to the cloth by putting it to every draft there is: what a missing end does to the weave, where the answer turns out to be an outcome the criterion has never had to report before.

Sideways, the shape of a fault depends on where in the loom the mistake was made, and the three places differ by four orders of magnitude in how much of the cloth they touch.