A repeat has to fit the panel, and the panel is cut

The rung below fitted a repeat to a warp and found the arithmetic sparse: the repeats that divide a body exactly are its divisors, which is two to eight per cent of the candidates, so a designer choosing a repeat for any other reason gets a broken pattern at the selvedges.

The same question in the other direction has a completely different answer, and the reason is that a warp’s width is fixed at warping and a piece’s length is cut.

The length is free and the phase is not #

A piece can be cut anywhere. There is no arithmetic forcing a length to be a multiple of anything, and a cutter who wants seventy centimetres of cloth takes seventy centimetres.

Unless the cloth is patterned. Then two panels sewn together have to match: the repeat must line up across the seam, or the pattern jumps at the join and the garment is a fault. Matching means every panel starts at the same phase of the repeat, and once that is required a panel’s cut length is no longer free.

So a panel of length L cut from a repeat of r has to be cut at the next whole number of repeats above L, and what is rounded away is waste:

waste = (−L) mod r.

That is a modular arithmetic and it has no geometry in it at all. It does not depend on the panel’s shape, on the cloth’s width, on the yarn or on the weave — only on two lengths and a remainder.

The allowance is a repeat and the waste is half of one #

A cutting room does not budget the waste; it budgets the allowance, and the two differ by a factor worth knowing.

A panel’s length is not a multiple of anything, so the remainder is anything between zero and r − 1 and the room cannot know which until it has the cloth and the pattern. So the allowance is a whole repeat per panel — the guarantee rather than the expectation — and that is exactly what the trade quotes: a furnishing fabric’s requirement carries a “pattern repeat allowance” of one repeat for every drop.

What is actually wasted, averaged over panels whose lengths bear no relation to the repeat, is r/2.

The two differ by (L + r)/(2L + r), which is a half when the repeat is small against the panel and rises towards two thirds as it grows. So the allowance is between one and a half and two times the expected waste, always, and never more.

The waste column is jagged and the allowance column is smooth. A ten-centimetre repeat divides seventy exactly and wastes nothing; a sixteen-centimetre one wastes ten; a twenty-four-centimetre one wastes two. That is the modular arithmetic showing through, and it is why a cutting room quotes the smooth column: the jagged one depends on a coincidence between two numbers nobody controls together.

Which makes a large repeat expensive in a way a small one is not #

The allowance is a whole repeat against a fixed panel, so it is a share of the panel and the share rises with the repeat.

At seventy centimetres — a skirt panel, a chair back, a cushion front — a ten-centimetre repeat costs twelve and a half per cent of the cloth and a sixty-four-centimetre repeat costs forty-eight.

Nearly half the cloth, for a large-repeat design, and none of it is anything a reader ever sees.

That is the reason furnishing fabrics quote their repeat in the same breath as their price, and it is the reason a large-repeat design has to be worth the repeat: the design is competing not only against other designs but against its own consumption. It is the same accounting a figured cloth’s cost to the loom is, one process further along — a design pays for its scale at every stage that has to accommodate it.

And it is why the two directions are different problems #

Set the two rungs side by side and the asymmetry is complete.

Across the width the remainder is a one-off. A warp is warped once at a stated number of ends and the pattern is centred, so the whole piece — a hundred metres of it — carries one broken repeat at each selvedge and pays nothing further.

Along the length the remainder is per panel. Every panel pays its own allowance, so the cost multiplies by the number of panels and a garment of six panels pays six times over.

So the same arithmetic costs a fraction of a per cent in one direction and half the cloth in the other, and the difference is entirely about whether the thing being fitted happens once or many times. That is the same accumulation a crimp mismatch between two layers suffers from and the same one a repeat’s aspect error suffers from: a per-unit cost multiplied by a count nobody was watching.

That also says which direction a designer should worry about. The width’s fit is a cosmetic question — how much of a motif is cut at the selvedge — and the length’s fit is a costing question. They feel like the same problem and they are not even the same kind of problem.

The width has a matching problem too, and it is the third case #

There is one place where the width’s remainder does multiply, and it is worth separating because it is the case that gets confused with the others.

Two panels seamed side by side — a curtain made of two widths, a wide cover made of three — have to match across their vertical seam as well. That is a matching problem in the width, and it behaves like the length’s rather than like the rung below’s: each additional width pays its own allowance in the warp direction, because the two widths have to be cut at the same phase of the cross-repeat.

So the cost of a wide made-up article is the length’s arithmetic applied twice, once down and once across, and it multiplies rather than adding.

That is why a large-repeat furnishing fabric on a wide window is the most expensive thing in the catalogue, and why the trade’s own rule — order an extra repeat per drop and per width — is exactly right and exactly what the arithmetic gives.

What a designer can do about it, which is choose a divisor #

The waste column being jagged is the useful half, because it means the cost is not fixed by the repeat alone.

A repeat that divides the panel wastes nothing. Ten centimetres into seventy is seven repeats and no remainder; the cutting room takes exactly the panel and the allowance is a formality.

So a designer who knows what the cloth is for can choose a repeat that fits: 10, 14, 35 or 70 centimetres for a seventy-centimetre panel, of which the first two are ordinary furnishing repeats.

And a designer who does not know cannot. A dress fabric is cut into panels of every length by every maker who buys it, so no repeat fits and the expected half-repeat is the right number to plan on — which is a population argument rather than an arithmetic one. A furnishing fabric is cut into drops of a small number of standard lengths, so a repeat chosen against those is a real saving.

Which is why the practice differs between the two trades, and the arithmetic says the difference is not a habit. A furnishing designer can fit a repeat to a use and an apparel designer cannot, so the first is worth doing and the second is not.

That is the same shape as the divisor argument in the width with the roles exchanged: there, the warp’s width is a fact and the repeat is chosen against it; here, the repeat is a fact and the panel is chosen against it — and only one of the two parties gets to choose.

What the phase costs when it is not free #

The whole arithmetic above assumes the panel can be cut anywhere along the piece, which is what makes the remainder uniform. There is a case where it cannot, and it is the expensive one.

A one-way design has a top. A directional print, a nap, a pile, a design with a horizon in it — all of them must be cut with the same end upward on every panel, so a panel cannot be turned end for end to nest with its neighbour.

That doubles the waste in the ordinary layout, because nesting two panels head to tail is exactly what a cutting room does to fill a length — and it interacts with the repeat rather than adding to it: a one-way design’s panels all start at the same phase and face the same way, so the cloth between two panels is unusable rather than merely offcut.

A cloth that is both patterned and one-way is therefore the worst case, and it is a common one: a directional floral furnishing is exactly that. Its allowance is a repeat per drop and its nesting is nil.

That is the honest reason the trade quotes furnishing requirements so generously, and it is two constraints rather than one. The cutting room’s own waste is a packing problem and this is a registration problem, and they compound: the packing cannot use the offcuts the matching creates, because the offcuts start in the wrong place.

And the repeat that is a compromise #

Reading the two rungs together gives a designer a repeat’s whole cost, and the two halves point in opposite directions.

A large repeat is better in the width. The remainder at the selvedges is a fraction of a repeat and its fraction is what a reader sees, so a large repeat’s broken edge is a small share of a large motif and reads as a border; a small repeat’s is a large share of a small motif and reads as a mistake.

A large repeat is much worse in the length. The allowance is a whole repeat per panel and the share rises without limit.

So the two constraints are opposed and the optimum is somewhere between, and where depends on what the cloth is for: an apparel fabric cut into many small panels wants a small repeat and a furnishing cut into few long drops can afford a large one.

That is exactly the distribution the trade shows — dress fabrics repeat in centimetres and furnishings in tens of centimetres — and the arithmetic says the reason is not fashion. It is that the two trades sit at different points on the same trade-off, because their panels are different lengths.

What was counted, and how #

The arithmetic is integer and is exact. The cut length is the ceiling of the panel over the repeat, times the repeat; the waste is the difference; the allowance is a repeat a panel. Nothing here is estimated.

The expected waste is stated as a uniform average and the shape of that assumption is worth naming. A panel length drawn without regard to the repeat leaves a remainder uniform on 0 … r − 1 and averages r/2 — which is right for a fabric bought by many makers and wrong for one whose panel lengths were chosen against the repeat. The table above gives both, so a reader can see which case they are in.

The relation between the two is asserted exactly, not approximately: the expected waste is (L + r)/(2L + r) of the allowance, which is a half in the limit of a small repeat and two thirds in the limit of a large one, and the assertion is that the ratio is between those two at every repeat tried. A version asserting “about a half” would have been an assertion about small repeats.

And the rows that waste nothing are asserted to exist, because a sweep in which no repeat divided the panel would produce a smooth column and would hide the whole finding.

The one number a specification should carry and does not #

The arithmetic reduces to a single ratio and it is worth saying what a specification would have to hold for a buyer to compute it.

A furnishing fabric’s ticket carries the repeat. That is one of the two numbers, and it is the one the seller knows.

The other is the panel length, and the buyer knows it. So the calculation is available to whoever puts the two together, which is the buyer, and it is a division and a ceiling.

What is missing is that the ticket does not say the calculation exists. A repeat quoted as “64 cm” reads as a fact about the design’s scale and is also a statement that every drop of this cloth will cost up to sixty-four centimetres more than its own length. Those two readings are the same number and only one of them is what the number looks like.

The apparel case is worse, because there the repeat is often not on the ticket at all — a dress fabric is sold by its fibre, its weight and its width, and its pattern repeat is something a maker measures off the cloth. A quantity that decides up to a third of the material is not in the specification, which is the same complaint a fancy yarn’s overfeed invites and for the same reason: it is a parameter of the process rather than a property of the product, so nobody’s records carry it.

Where the model stops #

Nesting is not in it. A cutting room lays many panels on one length of cloth and a short panel can often be taken from a long one’s waste, so the total for a real garment is below the sum of the panels’ allowances. How far below is a packing problem — the same one a marker’s layout is — and is not computed.

The panels are taken as all the same length. A garment’s panels are not, and a set of unequal panels against one repeat has a much better chance of some of them fitting, which lowers the average.

And the match is taken as exact. A real seam allowance, a real stitching line and a real cloth’s own irregularity mean that a match within a few millimetres reads as a match, and how much slop the eye allows is a question about vision that this collection has no model for. What can be said is that the slop is small against a repeat and does not change the arithmetic.

Nothing here is about the cross-repeat’s phase within one width. The rung below’s centring puts the pattern in the middle of the warp; whether two widths cut from the same piece then match each other across a vertical seam depends on that centring being the same in both, which it is by construction.

Who found it, and when #

The pattern repeat allowance is old, universal and correctly stated everywhere: order an extra repeat per drop. Every furnishing catalogue quotes the repeat, every making-up guide explains the allowance, and no seamstress has ever been surprised by it.

The arithmetic underneath appears not to be written out, and the two things it produces are worth having. The allowance is between one and a half and two times what is actually wasted, always — which is the price of a guarantee and is a number nobody quotes. And the real waste is jagged rather than proportional, so a repeat chosen against a known panel length can cost nothing at all.

The comparison with the width is this collection’s, and it is the part that changes how the problem is seen. The same fitting arithmetic is a cosmetic question in one direction and half the cloth in the other, and the reason is not about cloth at all: it is that one of the two remainders is paid once and the other is paid every time.

Where the ladder goes next #

Three rungs of this anchor have taken a repeat from an infinite pattern to a warp, a piece and a panel, and every one has treated the repeat as a fixed number of threads. It is not fixed: a repeat is a count of ends and picks and the cloth it makes has a size in millimetres, which moves between the loom and the finished piece — and by different amounts in the two directions.