Textile Structure
how cloth is put together
Cloth doing a job

A grade charges by the length and a cutter pays by the panel

The four-point system scores a fault by how far it runs — one point to three inches, four beyond nine — and caps a linear metre at four points however many faults it holds. A cutting room pays by how many panels the fault lands in. Below a panel's length every fault costs exactly one panel while its score runs from one to four; above it the panels grow without bound and the score does not move at all. Two fifty-metre pieces built to the same 267 points a hundred square metres lose 33 per cent of their panels and 92.
15 min read 6 figures A mechanism, not a materialExactly this manyMeasured, not claimed

Worth reading first: A missing end is a fault the length of the piece · A repeat has to fit the panel, and the panel is cut · The bias cut and the selvedge.

A missing end is a fault the length of the piece settled what a fault costs, and the answer arrived in two stages. Its area is a band a few threads wide times the piece’s own length or width, which is almost nothing. Its cost is whatever panel it lands in, because cloth is not sold by the flawless square centimetre — it is cut, and a fault condemns the piece it is cut into. Counted in panels, a broken end running a fifty-metre bolt takes a quarter of it.

Between the mill and the cutting room sits a third quantity nobody has looked at yet, and it is the one both parties actually agree the contract on. A roll is bought against a grade, and the grade is a number a scheme produces from an inspection. If the scheme’s number tracked the panels lost there would be nothing more to say.

It does not, and the mismatch is not a matter of calibration. The scheme is a function of a fault’s length and the cost is a function of where it falls, and the two are different functions with different shapes.

The scheme, which is a convention

The four-point system scores each fault by how far it runs: one point up to three inches, two to six, three to nine, and four beyond. It then caps a linear unit of cloth at four points, however many faults that unit contains, and grades the piece by its total points scaled to a hundred square units.

None of those numbers comes from anything about cloth. The bands are a convention of the trade, of long standing and near-universal, and there is nothing in a woven fabric that produces the number four. That is not a criticism — a scheme has to choose something, and a scheme everybody uses is worth more than a better one nobody does. What can be examined is what the convention does, and that is arithmetic.

Below a panel’s length, the scheme charges four to one for nothing

Take a single warp fault, vary its length, and ask both questions.

What the four-point system charges for a fault, and what the cutting room pays. For a single warp fault of each length: the points the four-point system scores it — one up to three inches, two to six, three to nine and four beyond — and the number of 900-millimetre panels it condemns. Below a panel's length every fault condemns exactly one panel while its points run from one to four, so the scheme charges four times as much for a fault that costs the same. Above a panel's length the panels grow without bound and the points stay at four, so the scheme stops charging exactly where the cost starts rising. A ten-metre fault scores 4 and condemns 12 panels. What the chart cannot show is the marker, which decides the panel size and therefore the whole of the second curve.
Fig. 1 For one warp fault of each length: the points the four-point system scores it, and the number of 900-millimetre panels it condemns. Below a panel’s length every fault condemns exactly one panel while its points run from one to four. Above a panel’s length the panels grow without bound and the points stay at four. The two curves cross once and follow each other nowhere.

A fault ten millimetres long and a fault nine hundred millimetres long both condemn exactly one panel, because a panel is nine hundred millimetres long and either fault fits inside one. The scheme scores them one point and four.

So across the whole range in which most faults actually fall, the scheme’s ranking is not merely imprecise — it is a ranking of something that does not vary. A cloth does not mind a hole structurally; what minds it is the marker. A mill that halved the length of every fault it made — by catching its broken ends sooner, which is what a stop-motion buys — would improve its grade by a factor approaching four and would deliver exactly the same number of usable panels.

Above a panel’s length, the scheme stops charging

Past nine inches the scheme has run out of bands, and the cost has not run out of anything.

A fault 1.8 metres long condemns two panels and scores four. One five metres long condemns six panels and scores four. One fifty metres long — which is the broken end this account began with and which changes the weave around it as well as condemning cloth — condemns fifty-five panels and scores four points per metre, which is the cap.

The per-metre cap is what stops the score from going to zero in relative terms, and it is a blunt instrument: a warp fault is scored in every linear metre it passes through, so its total is four times its length in metres whatever it actually is. The scheme’s charge for a long fault is proportional to its length and its cost is proportional to its length, which is the one region where the two agree — and they agree for a reason that has nothing to do with the bands, which have all been exhausted.

The cap discards what a bad metre is made of

The cap’s other half is what it does to a metre with several faults in it, and here it hides rather than distorts.

What the four-point cap discards, against how many faults a metre carries. A single linear metre of cloth carrying from one to twenty-four faults of 60 millimetres each, scored with and without the cap of four points to the metre. 1 a metre: 1 points scored, 1 counted; 2 a metre: 2 points scored, 2 counted; 3 a metre: 3 points scored, 3 counted; 4 a metre: 4 points scored, 4 counted; 6 a metre: 6 points scored, 4 counted; 8 a metre: 8 points scored, 4 counted; 12 a metre: 12 points scored, 4 counted; 16 a metre: 16 points scored, 4 counted; 24 a metre: 24 points scored, 4 counted. Above four faults a metre the scheme stops distinguishing, so a metre with four faults and a metre with twenty-four are the same number — while the panels they spoil are not. What the bars cannot show is the faults' positions, on which the panel count entirely depends.
Fig. 2 One linear metre of cloth carrying from one to twenty-four short faults, scored with and without the cap of four points to the metre. Up to four the two agree. Beyond four the scheme stops distinguishing: a metre with four faults and a metre with twenty-four score the same, while the panels they spoil do not.

Above four faults a metre the scheme is blind. That is by design — the cap exists so that one catastrophically bad metre cannot condemn a whole roll on paper — and the design decision has a consequence nobody states: the scheme’s resolution in the bad direction stops exactly where the cloth gets bad.

The previous account noted that clustering helps a cutter, because two faults in one panel cost what one does. The cap is the scheme’s own version of the same observation, applied to metres rather than panels, and applied without asking what size the panels are. A cap that matched the cutting room would cap per panel, and a scheme graded before anybody knows what will be cut from the roll cannot.

Two pieces of one grade

Put the two effects together and pieces of the same grade can be built with very different yields, which is the essay’s claim and is a construction rather than a supposition.

Two pieces of one grade and two yields. Two pieces of the same cloth, each 50 metres long and 1500 millimetres wide, drawn as the panels they are cut into with the condemned ones shaded — the top 22 rows of panels shown, the pattern continuing down. The first carries one broken end running the whole piece; the second carries short faults at eight to the metre. Both score 200 points, which is 267 per hundred square metres, because the cap of four points to the linear metre is reached by each of them and the second's 400 raw points are cut back to 200. The first loses 33.3% of its panels and the second 92.1%. What the drawing cannot show is where the short faults actually fell, which is the whole of the difference and is what no grade records.
Fig. 3 Two fifty-metre pieces a metre and a half wide, drawn as the panels they cut into with the condemned ones shaded. The first carries one broken end running the whole piece; the second carries short faults at eight to the metre. Both score 200 points, which is 267 to the hundred square metres. The first loses a third of its panels and the second loses ninety-two per cent.

Both pieces grade at 267 points to the hundred square metres. The first reaches the cap in every metre because one fault crosses every metre; the second reaches it because eight faults a metre score eight points and four are thrown away. A buyer holding a specification of, say, forty points a hundred square yards rejects both, and a buyer with a looser one accepts both.

The first loses 33 per cent of its panels and the second 92. They are not the same cloth by any measure a cutter would recognise, and no number either of them carries says so.

That is a factor of nearly three in the quantity the cloth is actually bought for, hidden behind an equality in the quantity it is actually bought against. It is also, notably, the wrong way round from the earlier account’s caution about clustering: there, the surprise was that a roll which grades identically may be better than it looks because its faults are bunched. Here the second roll grades identically and is much worse, because its faults are spread and each one lands in a different panel. The grade is blind to position in both directions, and the two effects do not cancel; they simply mean the grade is uninformative about yield.

What would have to change

The arithmetic points at three repairs and only one of them is cheap.

Score by count rather than by length. The cutting room’s cost is one panel per fault for nearly every fault, so a scheme that scored one point per fault regardless of length would track yield far better than the bands do across the range where most faults fall. It would lose the long-fault case, where length and cost genuinely do move together, so the honest scheme is a count plus a length term that starts at a panel’s length rather than at three inches — a scheme whose first band is the panel, not the inch.

Cap per panel rather than per metre. A cap that discards faults after four in a metre discards exactly what a cutter is paying for when the panels are small, and discards nothing he cares about when they are large. Capping per panel needs the panel size, which the mill does not have.

Or record the positions and let the cutter do the arithmetic. This is the cheap one, and the earlier account already named it: a fault map costs nothing to record on a modern inspection frame, and it converts every expression here from a probability into a certainty. With positions in hand a buyer computes the yield for their own panel size exactly, and the grade becomes a summary rather than a contract.

The third repair is the one that makes the other two unnecessary, and it is worth noticing that it is not a better measurement — it is the same measurement, kept rather than summarised. The four-point system’s whole information loss happens in the step where a list of faults becomes a number, and the number was only ever needed because a list could not be carried from a mill to a cutting room on a piece of paper.

The buyer’s number and the mill’s number are about different things

There is a reading of all this that is less damning than it sounds, and it is worth stating because it is probably the right one.

A grade is a measure of how well a mill is weaving, and what makes a fault in the first place is one row of the matrix in the wrong place or one thread missing from it. Points per hundred square metres tracks fault incidence, it is comparable between mills and between months, and it does not need to know anything about what the cloth is for. As an instrument pointed at a loom shed it is doing its job.

A yield is a measure of what a cutting room can get. It needs the panel size, the marker, the fault positions and the cloth’s width, none of which is the mill’s business.

The mistake is not in either number. It is in a contract that uses the first to promise the second, which is what a points specification in a purchase order does, in the way a thread count is taken for a quality — and which every party to it knows is approximate and none of them knows is uninformative. The factor of three above is the size of that gap for two pieces built deliberately; for two real pieces of the same grade it would be smaller and it is not small.

The mismatch moves with the panel and the scheme cannot

The scheme’s bands are fixed in inches and the cutting room’s unit is not, so the size of the gap between them is a property of what is being cut — and the gap is worst for the smallest panels.

What the four-point system charges for a fault, and what the cutting room pays. For a single warp fault of each length: the points the four-point system scores it — one up to three inches, two to six, three to nine and four beyond — and the number of 400-millimetre panels it condemns. Below a panel's length every fault condemns exactly one panel while its points run from one to four, so the scheme charges four times as much for a fault that costs the same. Above a panel's length the panels grow without bound and the points stay at four, so the scheme stops charging exactly where the cost starts rising. A ten-metre fault scores 4 and condemns 25 panels. What the chart cannot show is the marker, which decides the panel size and therefore the whole of the second curve.
Fig. 4 The same two curves for panels 400 millimetres long — a sleeve, a facing, a pocket. Every fault up to 400 millimetres condemns one panel and scores anything from one to four points, and the panel count only starts rising at four tenths of a metre. The shorter the panel, the sooner the cost curve leaves the scheme behind.
What the four-point system charges for a fault, and what the cutting room pays. For a single warp fault of each length: the points the four-point system scores it — one up to three inches, two to six, three to nine and four beyond — and the number of 2000-millimetre panels it condemns. Below a panel's length every fault condemns exactly one panel while its points run from one to four, so the scheme charges four times as much for a fault that costs the same. Above a panel's length the panels grow without bound and the points stay at four, so the scheme stops charging exactly where the cost starts rising. A ten-metre fault scores 4 and condemns 5 panels. What the chart cannot show is the marker, which decides the panel size and therefore the whole of the second curve.
Fig. 5 And for panels two metres long — a curtain drop, a coat back cut in one. Now every fault the bands distinguish between, from ten millimetres to nine inches and well beyond, condemns exactly one panel of the same value. The four bands rank faults across a range in which the cost is flat, and the cost only moves at lengths the bands have already stopped counting.

Both ends of the panel range make the mismatch worse and they make it worse differently. With small panels the scheme’s bands are too coarse at the bottom and the cost curve rises steeply just above them; with large panels the bands are entirely inside the flat region and the cost curve never rises where the scheme is looking. The one panel size at which the bands would nearly track the cost is the one where a panel is about nine inches long, which is a pocket flap.

That is why the repair suggested above is a scheme whose first band is the panel rather than the inch: the inch is a unit of cloth and the panel is a unit of loss, and a scheme is an instrument pointed at the second while measured in the first.

What a fault costs is decided by how the cloth is cut up. A fault does not condemn the cloth it occupies; it condemns whatever piece is being cut when it lands in one. With faults falling at random at 0.4 per square metre, the fraction of pieces containing at least one is 1 − exp(−λA), which runs from 0.4% for a 0.01 m² piece to 80% for a 4.0 m² one. The consequence is not obvious and is worth stating plainly: the same cloth wastes less of itself when it is cut into small pieces, and the multiple of its own area each fault destroys — 1.00 at the small end and 0.50 at the large — is what a marker planner is actually trading against. Nothing here is about the fault. It is about the pieces.
Fig. 6 And the earlier account’s curve, for the other half of the same arithmetic. The fraction of cut pieces containing at least one fault rises with the piece’s area at a fixed fault rate — 4 per cent for a sleeve, 55 for a coat back — so a fault rate becomes a cost only when multiplied by what the cloth is for. The grade carries the rate and never the multiplier.

Why a scheme measures a length at all

A convention with no derivation behind it is still worth asking about, and the reason for the bands is legible once the question is put the right way round: what could an inspector actually measure in 1960, running cloth past a light at fifteen metres a minute?

A length. A fault’s extent along the piece is visible at speed, it needs a tape and nothing else, and it is the one property of a fault that does not require a decision. Its position across the width needs a second measurement; its severity needs a judgement; the panel it would land in needs the marker, which does not exist yet. So the scheme measures the only quantity a moving inspection can capture without slowing down or arguing.

That is a good reason and it is a reason about the instrument rather than about the cloth — and it explains the shape of the mismatch exactly. The scheme is a function of the one thing that was cheap to measure, and the cost is a function of the one thing that was expensive, which is where the fault sits. Every repair above is a way of spending what is now cheap: an inspection frame records position at no cost at all, and the whole information loss is a habit inherited from a machine that could not.

The same shape recurs whenever a standard outlives its instrument. The quantity in the standard is the one the old measurement produced; the quantity in the decision is the one it could not; and the two are compared for decades by people who assume the first was chosen to predict the second. It was chosen because it could be had.

What was computed, and how

Each fault is scored from its length by the four bands, and a warp fault is scored in every linear metre it crosses. The points are summed metre by metre with the cap applied to each metre, so the discarded points are visible as the difference between the raw and the capped totals; the grade is the capped total scaled to a hundred square metres. The panels are the piece tiled at a stated panel size, cut with the panel’s length along the warp as the grain requires, and a fault condemns every panel it touches — which is the same counting the earlier account uses.

Seven things are checked. The four bands score one, two, three and four, checked at a length inside each. Every fault shorter than a panel condemns exactly one panel, over eight lengths, while its points run from one to four over the same range — the pair that makes the essay’s first claim. Above a panel’s length the points stay at four and the panels grow, which is the second. The cap discards points on a piece with many faults in a metre, which would fail if the cap were not being applied. A full-length broken end and a scatter of short faults grade within two per cent of each other, and their yields differ by more than a factor of two — a pair of results in which an error in either half of the arithmetic would break one of them.

The bands, the cap, the panel size and the piece’s dimensions are inputs. The bands and the cap are the trade’s convention and are quoted; nothing here derives them or claims they should be different numbers.

Where the model stops

The panel size decides everything on one side. Nine hundred by four hundred millimetres is a garment panel of an ordinary size, and a scheme’s mismatch with a cutting room is a mismatch with that cutting room. A roll cut into handkerchiefs and a roll cut into coat backs have entirely different yields from identical faults, which is the earlier account’s result and is the reason a grade cannot contain a yield. A repeat has to fit the panel too, and for the same reason: the panel is the unit and the cloth is not.

The faults are placed rather than observed. The two pieces are constructed to make a point, not measured off a frame, so the factor of three is a demonstration that the gap can be that large and not a measurement of how large it usually is.

Detection is not modelled. An inspection frame runs at a speed and an inspector sees what they see; a fault missed is not scored and is still cut around. Every number here assumes the faults are known exactly, which flatters the scheme rather than the cutter.

And a panel is taken as spoilt or sound. A fault in a seam allowance, under a pocket or in a facing costs nothing — and a selvedge is trimmed off before any panel is cut, which is the same exemption at the cloth’s edge, and a cutter who knows where the faults are places the marker to put them there — which is the third repair above, and is the difference between a panel being touched and a panel being lost.

Still open: how much a fault map is actually worth

Every repair above ends at the same place: record the positions. What none of this computes is what that recording buys, and it is a computable question with a definite answer.

Given a fault map and a panel size, the best possible marker is an assignment of panels to cloth that avoids as many faults as possible — which is a covering problem, and a small one. The interesting quantity is the gap between a marker laid in ignorance and the best marker laid with the map, as a share of the bolt, and it will depend on the fault density, the panel size and how much freedom the marker has to shift.

The prediction the arithmetic already supports is that the gap is largest exactly where the grade is least informative: a roll with faults spread one to a panel gains nothing from a map, because every panel is spoilt whatever the marker does, while a roll with faults clustered gains a great deal. So a fault map is worth most on the roll a grade says is best, which if true is the sharpest thing to be said about the whole arrangement — and it has not been worked out here.

Who worked it out

The four-point system is an industry standard of long standing and its bands and cap are quoted here from the trade. That a grade and a yield are different quantities is understood by everybody who buys cloth and is the reason purchase orders carry both a points figure and a cutting allowance. Computing the two side by side over the same pieces, and finding that the scheme’s ranking is of a quantity that does not vary across the range most faults occupy, appears to be new here.

Where this goes next

The same fault in the warp and in the weft. A 50 m piece 1500 mm wide, with a 3-thread fault in each direction. The width is drawn 9.3 times over scale so that the piece is a rectangle rather than a line, and the two faults are drawn as marks rather than at their own widths, which at this scale are a fifth of a pixel. They have the same cause size — 3 threads — and they condemn 0.063 m² and 0.0020 m² respectively, a ratio of 31 to one, because a warp fault runs the length of the piece and a weft fault runs its width. That ratio is the aspect ratio of the piece and nothing else, so it is a property of how cloth is made rather than of what went wrong. It is why a broken end stops the loom and a mispick often does not, and why the two faults are priced by every grading scheme as though they were different kinds of thing. A missing end is a fault the length of the piece A broken end and a mispick are the same size of accident — one thread — and they condemn areas that differ by a factor of thirty. The ratio is the aspect ratio of the piece and nothing else, which makes it a fact about how cloth is made rather than about what went wrong. What matching a pattern costs a cutting room. The cloth a matched panel needs beyond its own length, against the pattern repeat, for 6 panels of 70 cm. Every panel of a patterned cloth must start at the same phase of the repeat or the pattern breaks at the seams, so a panel's cut length is rounded up to a whole number of repeats. The bar is the allowance a cutting room budgets — a whole repeat a panel, because a panel's length is not a multiple of anything — and the mark is the waste actually expected, which is half a repeat. The two differ by (L + r)/(2L + r), which is between a half and two thirds and is nearer two thirds the larger the repeat. The rows marked in the second colour are the repeats that happen to divide the panel exactly and waste nothing at all, which is what makes the real cost jagged rather than smooth. What the bars cannot show is nesting: a cutting room lays many panels on one length and a short panel can sometimes be taken from another's waste. A repeat has to fit the panel, and the panel is cut The warp's width is fixed at warping and a piece's length is not, so the fitting problem in the two directions is not the same problem. Along the length a repeat has to fit a *panel*, because every panel of a patterned cloth must start at the same phase or the pattern breaks at the seams — and the allowance is a whole repeat per panel. A ten-centimetre repeat costs a seventy-centimetre panel twelve and a half per cent and a sixty-four-centimetre repeat costs it forty-eight. A cloth does not mind a hole. Strength left after a hole, against the width of the hole, for a woven cloth and for a film of the same width. The cloth's threads carry their own load, so it loses exactly the threads the hole removes and the loss is linear. The film is a continuum, so a hole of any size at all costs it the stress concentration factor — quoted from Kirsch's elasticity solution, not derived here — and the size of the hole does not enter. A cloth does not mind a hole A hole in a film costs it two thirds of its strength whatever the hole's size, because a continuum concentrates stress at an edge. A cloth's threads carry their own load and hand almost nothing to their neighbours, so a hole costs exactly the threads it removes — a loss that is linear in the hole, independent of the sett, and zero for a slit along the load.

What links here

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Broken endCondemned areaFaultMarker efficiencyMeasurementSpecification