A half-drop wastes less, cut in the right order

A patterned cloth cut into matched drops loses cloth between them. A repeat has to fit the panel, and the panel is cut put a number on it: every drop must start at the same place in the repeat, so the cloth lost after a drop of length $L$ is $(-L) \bmod r$, half a repeat on average and up to a whole one, which is why the trade orders an extra repeat a drop.

That account had one place in the repeat for a drop to start, because it was about a straight match: the pattern runs level across the seam, so every width starts at the same height in the design. A great many furnishing cloths and wallpapers are not sold that way. They are sold as half-drops, in which every other width starts half a repeat lower, and they have a reputation for being the awkward match.

The reputation is half right. Cut in the obvious order, a half-drop loses exactly what a straight match loses. Cut in the right order it loses less — a third less on a pair of curtains, and half as much on a long job — because a second place to start is a second remainder, and a cutter with two remainders can choose the smaller.

A half-drop is a lattice read across a width #

The first thing to settle is what makes a cloth a half-drop at all, because it is not a property of the design alone.

What a repeat repeats found that a pattern’s true unit is a lattice of translations, and that the rectangle on point paper is a choice made by whoever drew it. A half-drop design is one whose motifs step across by some distance and down by half a repeat at the same time — a lattice whose step across carries a drop, of the kind weaves read as plane patterns are classified by. Two widths sewn side by side are related by however many of those steps one width holds.

So the offset at a seam is the drop times the motif steps a width holds, read within the repeat. A half-drop design with one step to the width is a half-drop at the seam; with two it is a straight match, because two half-drops make a whole repeat; with three it is a half-drop again. A quarter-drop design printed two steps to the width is a half-drop at the seam. A third-drop printed two steps to the width starts each width two thirds of a repeat lower, which is still a third-drop.

That has a practical edge, of the kind a motif drawn at the wrong shape on purpose has at the loom: the design as drawn is not the design as it reaches the cloth. Whether a curtain maker is facing a half-drop is decided by the width the cloth was printed at as much as by the designer’s lattice, which is why the match is printed on the ticket rather than inferred from the pattern. Everything below takes the seam’s offset as given, as the ticket gives it.

Two places to start #

A cutter cutting a job of matched drops from one roll cuts them one after another. After each drop, the roll is wound on to the next place where the next drop may start, and the cloth passed over is lost. How much is lost depends only on where the last drop ended and where the next one must begin:

$$\text{gap} = (\varphi_2 - \varphi_1 - L) \bmod r,$$

where $\varphi_1$ and $\varphi_2$ are the two drops’ starting places in the repeat. A straight match has one starting place, so every gap is $(-L) \bmod r$ — call it $u$. A half-drop has two, A and B, half a repeat apart. A gap between two drops of the same class is still $u$. A gap between drops of different classes is

$$v = \left(\tfrac{r}{2} - L\right) \bmod r,$$

and since the offset is half a repeat, it is the same going from A to B as from B to A.

The figure is the whole argument in one case. Four drops of 250 centimetres, a 64 centimetre repeat: two curtains of two widths each, an ordinary furnishing job. Two hundred and fifty centimetres is six centimetres short of four repeats, so a straight match loses six centimetres at each gap. A half-drop’s unlike gap is $(32 - 250) \bmod 64$, which is 38.

Cut in the order they hang, a half-drop crosses between classes at every gap, because the widths alternate as they hang, and loses 38 centimetres three times. Cut with the two A drops together and the two B drops together it crosses once and loses 6 + 38 + 6 = 50. Same cloth, same drops, same match at every seam once hung: 114 centimetres of waste against 50.

In the order they hang, a half-drop costs what a straight match costs #

The natural way to cut a curtain is in the order the widths will hang — first width, then the one that goes beside it, matched to it — and for a half-drop that order alternates the classes at every cut. Every gap is $v$.

Now $v$ is only $u$ moved along by half a repeat. A drop length chosen without regard to the repeat leaves $u$ anywhere between nothing and a repeat with equal likelihood — a population argument about the drops a cloth will be cut to, not about any one of them — and the same is true of $v$. So a half-drop cut in hanging order loses half a repeat a gap on average and up to a whole one, exactly as a straight match does. On any particular drop length the two differ — at 250 centimetres the half-drop is far worse, at 200 it is far better — and averaged over drop lengths they are the same.

That is where the reputation comes from, and it is a reputation earned on particular drops. A curtain maker who meets a half-drop at an unlucky length loses more than the straight match would have lost and remembers it; at a lucky length the half-drop loses less and nobody notices. The arithmetic says neither experience is about the match. Both are about the drop length, and in hanging order the two matches are equally expensive.

The second saw-tooth is low wherever the first is high #

What the hanging order throws away is visible as soon as both remainders are drawn against the drop.

The two saw-tooths are interleaved. Wherever $u$ is large, $v$ is small, and the smaller of the two is never more than half a repeat. A straight match is stuck on one saw-tooth. A half-drop has both under it at every drop length, and the only question is how many of its gaps it can put on the lower one.

That depends on the order of the cuts and on nothing else. A gap between like drops costs $u$ and between unlike drops $v$. A job of $n$ drops has $n - 1$ gaps between them; with $c$ changes of class along the order, it costs

$$(n - 1 - c)\,u + c\,v.$$

The number of changes is the cutter’s to choose, within limits. It cannot be nought, because both classes have to be cut; it is at most $n - 1$ when the classes are equal in number, by alternating them. So the best order is one of two, and which one is decided by a single comparison: if $u \le v$, keep like drops together and change class once; if $v < u$, alternate. Nothing in between is ever better, because the cost is linear in the number of changes.

The order that is best, found by searching every order #

That rule is short enough to be wrong in an interesting way, so it was not trusted. For hangings of two to eight widths, every distinct order of cutting the drops was walked — up to seventy of them for eight widths of a half-drop — and the cheapest taken, at drop lengths every seven millimetres across a whole repeat. The search and the closed form agreed exactly at all 644 combinations. The waste in hanging order agreed with $(n - 1)v$ at every one.

Averaging the closed form over drop lengths is then a short integral. Where $u < r/2$ the cutter keeps like drops together and pays $u$ at every gap but one; where $u \ge r/2$ it alternates and pays $v = u - r/2$, which is below half a repeat. Taken over a whole repeat of drop lengths, the average cloth lost a gap is

$$\frac{r}{4}\cdot\frac{n}{n - 1}.$$

At two widths the half-drop saves nothing, because one gap admits no choice: there is one unlike gap and it has to be paid. At four widths — a pair of curtains of two widths each — it loses a third of a repeat a gap where a straight match loses half, a third less cloth. At six it is three tenths, and in the limit of a long job a quarter of a repeat, half what the straight match costs. Every point on the figure is the exhaustive search’s, and every one sits on the curve within the millimetre the drop lengths are stepped by.

The guarantee falls with it #

A cutting room does not order cloth against an average; it orders against a guarantee, which for a straight match is a whole repeat a drop. The best order lowers that too, by the same factor.

The worst a half-drop’s best order can do, per gap, is

$$\frac{r}{2}\cdot\frac{n}{n - 1},$$

exactly twice its average, which is the same ratio of guarantee to expectation the straight match has. It is reached just short of the drop lengths where $u$ crosses half a repeat, where keeping like drops together pays nearly half a repeat at every like gap and one unlike gap near a whole repeat.

For a pair of curtains that is two thirds of a repeat a gap instead of a whole one. On a 64 centimetre repeat and four drops the allowance is 128 centimetres for the job rather than 192, which is the figure a careful order could be placed against — provided the cutter will cut in the right order. If the cutter will not, the allowance is what it always was.

A pair of curtains at seven drops #

The averages hide the shape of what happens at any one length, and the shape is the useful part: it says how much is at stake when the hanging order is the wrong one.

At 137, 160, 200 and 274 centimetres the unlike gap is the smaller, so the order the widths hang in is already the best and nothing changes. At 160 centimetres — exactly two and a half repeats — the unlike gap is nothing at all, and the whole job is cut without a centimetre lost, which a straight match of the same repeat could not do at that drop. At 183, 228 and 250 the like gap is the smaller, and hanging order is wrong by 64 centimetres every time — exactly one repeat. Over the seven drops the hanging order loses 6.00 metres of cloth, the best order 4.08.

That 64 is not a coincidence of this job. When $u \le v$ the two remainders differ by exactly half a repeat, $v = u + r/2$, so every like gap the cutter fails to use costs half a repeat more than it needed to. Four drops have two like gaps available in the best order, and two halves of 64 centimetres is 64.

The straight match’s marks are a warning against reading too much into one job. Over these seven drops it happens to lose more than the half-drop in either order, 6.96 metres, because several of the seven are unlucky lengths for a straight match. Over every drop length in a repeat it loses exactly what the half-drop loses in hanging order. The claim is about the average and the guarantee, not about any seven drops.

A third-drop, and where the saving ends #

The same arithmetic applies to any drop match, and the general case says what the saving really is.

A third-drop’s widths start at three places in the repeat, a third of a repeat apart, so three saw-tooths lie under it, and the lowest of three is never more than a third of a repeat. With enough drops of each class the cutter can run the whole job along whichever of the three is lowest — keeping like drops together, or stepping through the classes in one direction — and the average falls towards a sixth of a repeat. On nine widths it is already 0.19.

The general statement is short. A drop match with $k$ classes of width cuts like a straight match of a repeat $k$ times shorter — in the limit of a long job, for a cutter free to choose the order — because the places a drop of some class may start are spaced $r/k$ apart along the roll. A third-drop even offers a choice on two widths, which a half-drop does not: cutting the lower width first or second changes the gap from $(r/3 - L) \bmod r$ to $(2r/3 - L) \bmod r$, and the better of the two averages 0.28 of a repeat against a half.

What a cutter has to know, and when #

The rule asks for one comparison before the first cut, and both numbers in it are available at the cutting table.

Lay the cloth out, find where an A drop starts, measure off one drop, and see how far it is to the next place an A drop could start — that is $u$ — and to the next place a B drop could start — that is $v$, and it is $u$ plus or minus half a repeat. If the A start is nearer, cut all the A drops in a row and then all the B drops. If the B start is nearer, alternate. For two widths there is nothing to decide; at four the choice is worth a sixth of a repeat a gap on average, and in a long job a quarter.

A cutter working from two rolls, one for each class of width, is following one of the two orders without choosing it: each roll sees only like gaps, so it pays $u$ at every gap, which is the right choice when $u$ is the smaller and the wrong one by half a repeat a gap when it is not. The comparison decides between that habit and alternating on one roll.

This is the same shape as the divisor argument in the width, one stage later, and the same accounting a figure’s cost to the loom is: a design pays for its scale at every stage that has to accommodate it, and here one stage can pay less by deciding first. There, a designer who knew the warp could choose a repeat that fitted it. Here, a cutter who knows the drop can choose an order that fits it, and the choice is free, which a repeat never is.

Why the ticket cannot say it #

A furnishing ticket carries the repeat and the match, which are the seller’s to know. The saving needs the drop, which is the buyer’s, and it needs the number of widths, which is the job’s. So the waste a half-drop costs is not a property of the cloth on the ticket, exactly as the panel essay found of the straight match’s: a quantity decided by two parties who never meet at the cutting table.

What can be said on the cloth’s behalf is the guarantee. A straight match needs a repeat a drop to be safe. A half-drop needs two thirds of one a gap on a four-width job, and half a repeat a gap plus a little on a long one, if the order is chosen — and a cutting room that orders cloth against the straight match’s allowance for a half-drop is paying for an order it does not use.

What was counted, and how #

Lengths are whole millimetres throughout, because a remainder is only exact in integers. For each hanging the widths’ starting places are $j$ times the seam’s offset, read within the repeat, for $j$ from nought to the number of widths less one. Every distinct order of cutting that multiset of starting places is walked, and each order’s waste is the sum of its gaps between consecutive drops; the cloth before the first cut and after the last belongs to the roll, not the job, and is not counted. That is why a job of $n$ drops has $n - 1$ gaps where the panel essay’s per-panel count gave $n$.

The averages are over every drop length a millimetre apart across one whole repeat of lengths — the uniform remainder that the panel essay’s half a repeat rests on — and the curves of averages and of worst cases are drawn from drop lengths two millimetres apart. The search and the half-drop’s closed form were required to agree exactly at 644 combinations of hanging and drop; the averages to fall within a millimetre of $(r/4)\cdot n/(n - 1)$ and the worst cases of $(r/2)\cdot n/(n - 1)$ at two, four, six and eight widths; two widths to show no gain at all; and a third-drop’s best order to beat a half-drop’s at six and nine widths. The seam offsets were required to give a straight match for a half-drop at two steps a width and a third-drop at three, and a half-drop for a quarter-drop at two.

Where the arithmetic stops #

Every drop is the same length. A bay window with drops of two lengths, or a job mixing curtains and a valance, has a richer set of gaps and the same search applies. A bay window wastes less than a wall counts them, and the mixture turns out to help.

The roll’s own ends are free. The first drop is taken to start wherever its starting place first comes round, and what is left after the last drop is taken back to stock. A roll that starts at an awkward place in the repeat costs up to a repeat before the first cut, in either order, and a short remnant may not be stock at all.

The cutter is free to order the cuts. Flaws in the roll, a fault’s position on the bolt, a need to cut from two lots that drift apart down a drop, or a workroom that cuts and seams as it goes can each fix the order, and then the hanging order’s cost is the one paid.

The match is exact. A seam is taken to match when its widths start at their places in the repeat to the millimetre, as the panel essay took it, and the slop a real seam allows is not modelled. A skew in the weft moves every width’s starting place across its width and is a separate error, added to whichever order is cut.

And nesting is not in it. A cutting room laying several jobs on one roll can take a short piece from another job’s gap; how much of the half-drop’s saving survives that is a packing question and is not computed.

Still open: how often a cutting room is free to choose #

The saving is real arithmetic and it is conditional on one freedom: that the drops of a job can be cut in any order from one roll. How often that freedom exists in a workroom — how often a job is cut whole from one roll, as against from several, or in hanging order because the widths are matched to each other as they are cut — is a question about practice that no count answers. A record of a season’s half-drop jobs, the order each was cut in and its waste against the best order’s, would say how much cloth the habit of hanging order actually costs, and whether the reputation of the half-drop is the reputation of the match or of the order.

Who found it, and when #

The half-drop, the straight match and the extra repeat a drop are old and universal, and the half-drop is commonly treated as the more awkward of the two to cut. The modular arithmetic of the straight match was written out in the panel essay. Reading the seam’s match as the design’s lattice across one width, showing that a half-drop in hanging order costs exactly what a straight match costs, finding the best order’s closed form and checking it against a search of every order, and deriving the $r/(2k)$ limit for a drop of $k$ classes, appear to be this account’s.