A stripe's edge is cleanest inside one dent

An unbroken line is not a clean one proved something about every weave there is. The thread at a line’s boundary has to go under somewhere, so at that crossing the edge retreats to the thread beside it, and no line in any weave has a straight edge. It ended on a loophole. The theorem is about a matrix, and a cloth has one more freedom than its matrix, namely where its ends actually lie across the width. A denser dent at the stripe’s boundary would shorten the retreat in millimetres while the matrix stayed as it was, and a boundary end set at half the spacing would halve the visible swing.

That suggested a cheap repair with a possible sting in it. The calculation finds the repair is less available than it looked, and it finds a better one that nobody had to pay for. Two ends cannot be closer than a diameter, so at ordinary covers a swing cannot be halved. But the reed already groups ends, and whether a line’s edge falls inside a group or across one changes its swing by a factor of two at no cost at all.

What a dent does to the spacing #

A reed is a comb, and it sets the warp by dents. The reed is not the sett worked out how the two differ along the cloth’s length and width. What matters here is the smaller thing: ends passing through one dent are drawn together by it, and ends in neighbouring dents are held apart by a wire. The reed leaves its own mark deals with what that grouping looks like at the scale of a whole cloth, and a reed can dent a sateen out of its row used it to break up a diagonal. This essay asks what it does to one line’s edge.

The description needs one number. With $D$ ends to a dent and a closing $g$, ends inside a dent stand $1 - g$ thread spacings apart, and the gap across a wire is

$$1 + (D-1)\,g,$$

so the mean spacing stays one. The closing moves ends and adds no thread, which is checked across twenty-seven spacings and is the reason the weave and the sett stay as they were.

A boundary’s retreat is then a sum of gaps rather than a count of threads, and which gaps it crosses depends on where the line begins in the dent. Nothing else about the calculation is new.

How close two ends can stand #

The original proposal has a limit that is not in its arithmetic. Two round ends of diameter $d$ on a spacing $p$ cannot stand closer than touching, so $1 - g$ cannot go below $d/p$, which is the warp’s cover $c$:

$$g \le 1 - c.$$

So the shortest swing that any denting can reach, in any weave, is $c$ of the even one.

A halved swing is therefore available only below half cover, and how close threads can be set places most clothing warps well above that. A warp at a cover of 0.7 can lose at most three tenths of its swing. A voile at 0.4 could lose six tenths, and it hardly matters there, since a voile with light showing between its ends has no sharp stripes to begin with.

Then there is a further loss. All of that assumes a finished cloth keeps the whole of the grouping its reed gave it, and it does not. The beat-up and the finishing both spread ends back out. If half of the closing survives, a warp at 0.7 keeps 0.85 of its swing. The share kept is not measured here, nor anywhere this account has found, and it decides how much of everything here reaches a finished cloth. It stays a swept number.

A retreat has to fall inside the group #

The surprise comes from looking at which gaps a swing actually crosses.

A 2/2 twill’s line swings two threads. The commonest reed of all is two ends to a dent, which looks like it should help, and it does nothing whatever. A retreat across two gaps at two to a dent always crosses one close gap and one wide one, whose sum is $(1-g) + (1+g)$, which is two spacings at every closing and every alignment.

At three to a dent there are two ways it can go. A retreat lying inside one dent crosses two close gaps and costs $2(1-g)$. A retreat crossing a wire costs $(1-g) + (1+2g)$, which is $2+g$, and that is longer than if the reed had grouped nothing. So the reed is not a mild improvement that might be missed. It moves the swing in both directions from the even value, by whichever way the line happens to sit, and most lines sit wherever the warping plan left them.

Which raises the question of how many ends to a dent a given line wants. The retreat must fit inside one dent, so the dent must hold the swing plus one end. And the earlier essay supplied a theorem that makes this exact.

The narrowest line fits one dent exactly #

A line’s swing is its narrowest unbroken width less one, in every weave, which was the census’s sharpest result. So the dent a boundary’s retreat needs is exactly as large as the line’s narrowest width. A line drawn at its narrowest, with every one of its ends in a single dent, has both of its boundaries’ retreats inside that dent at once:

$$\text{swing} = (W-1)(1-g).$$

That was tested on every one of the 22,874 interlacing four-by-four drafts. 18,530 draw their line two threads wide and want two to a dent. 4,320 draw it three wide and want three. 24 draw it four wide and want four. On all of them, with no exception, the swing is shortened by the whole of the closing.

The practical reading of that table is pleasant. Most of the named weaves have a line two ends wide, and two to a dent is the commonest reed there is. So a two-end pinstripe in a plain, a 3/1 twill or a satin, on the most ordinary reed in the mill, has an edge of 0.70 spacings if its two ends share a dent and 1.30 if they straddle a wire, nearly twofold at a closing of 0.3, from one decision in the draw-in plan. Nothing has to be bought and no shaft is added. It takes only the drawer-in starting the stripe on the first end of a dent instead of the second.

On a shirting at forty ends a centimetre the spacing is a quarter of a millimetre, so that is 0.18 millimetres against 0.33 at that closing. The earlier account’s rule of thumb was that an eye separates about a tenth of a millimetre at reading distance. So a difference of that size can be visible close to, and it is the sort of difference that gets attributed to a good weaver.

A wider line has to be a multiple of the dent #

A stripe is usually wider than its narrowest line, and that adds a condition. Each boundary has its own retreat, one at each end of the line, and both have to fall inside dents.

A line whose width is a multiple of $D$ can start on the first end of a dent and finish on the last end of another. Its retreats are both inside. A width not divisible by $D$ cannot do that at any alignment, so one edge is across a wire and swings $2 + g$, and it is worse than an even setting at any closing whatever.

So in a three-to-the-dent reed a 2/2 twill stripe of three, six or nine ends has clean edges, and one of four, five, seven or eight has one bad edge. The stripe’s order and the reed have to be drawn together. A designer choosing ends in a stripe does not usually consider the reed at all, and a stripe is a partition of the warp has so far said nothing about the reed either. The width of a stripe in ends has a divisibility condition on it that nothing in its draft can show.

There is a corollary at the level of the repeat. If every stripe in a warp is to have clean edges, the whole colour order has to run in multiples of $D$. A warp at three to a dent with a stripe order of 6, 3, 9, 6 can do it, and one with 4, 2 cannot.

The reed has to agree with the weave as well #

A denting chosen for the line is imposed on the whole width, and the reed’s mark is then a design decision. The reed leaves its own mark worked out when a grouping beats with the weave’s repeat. Three to a dent on a 2/2 twill shares no factor with its repeat of four, so the mark falls on a different thread of the weave at each dent and repeats only every twelve ends. Two to a dent on the same twill shares a factor and shows a stripe at the repeat’s scale.

That is good news for this one case, since the reed that cleans the twill’s line is also the one that hides best in its ground. It does not hold generally. Two to a dent on a plain weave, which is the ordinary arrangement and the one that gives a plain’s pinstripe its clean edge, puts the dent at exactly the weave’s own period, and whether that shows depends on how much of the grouping the cloth keeps. A clean edge and an invisible reed are two separate demands on one number, and at higher closings the ground starts to show the reed before the line reaches its limit.

A line can be crowded alone, and it costs a gap #

The original idea was a little different. It would crowd only the stripe’s own dent, and leave the ground evenly set. That is what a selvedge’s denting does already, so the mechanism exists. It avoids any reed mark in the ground, and it runs directly into the cost the earlier essay worried about.

The room taken from inside the line has to go somewhere, and it goes beside it. Crowding a $W$-end line by $g$ releases $(W-1)g$ spacings, shared between the two gaps at its edges, so each gap beside the line grows by half of that:

$$\text{beside} = 1 + \frac{(W-1)\,g}{2}.$$

For a three-end line crowded until its ends touch, the clear gap beside it exactly doubles, at every cover, since the release equals the plain clear gap and is split in two. At a cover of 60 per cent that is 0.8 of a spacing, or a fifth of a millimetre on a forty-end shirting, running the length of the cloth on each side of the line.

Weft shows in that gap. So whether the repair draws a second line depends entirely on the weft, and the two cases are opposite. With a dark ground and a dark weft, a light line’s halo is the same colour as the ground, and the line simply looks sharper and slightly narrower. With a dark ground and a light weft, it shows as a pale rule along each side of the line, and it could be more visible than the swing it replaced. This is the second line the earlier essay suspected, and it is real for one of those two cases.

The line itself also changes. Its width falls from 2.6 spacings to 1.8 at the touching limit, so it has become almost a third narrower, and its same ends fill that narrower width, which should make it read as brighter. Someone wanting a line of a given weight would use another end, at which point the divisibility condition returns.

The weft cannot do any of this #

Everything so far is about warp lines, and it depends on a reed. A weft line has no reed, since picks are placed by the beat-up at a spacing that is close to even, and no ordinary loom groups picks. So the earlier result that a weft bar’s edge swings further than a warp line’s has no such relief, and a line across the cloth has none of the freedom of a line along it. A pick can be doubled, and a group is one thread for cover and two for bending says what that does, and it changes the swing’s unit without shortening it.

That produces an asymmetry a check would show. The warp lines of a check can be drawn to fit their reed, and the weft lines in the same check cannot be, so a check’s two sets of lines need not have the same edge quality even in a square cloth with a square order. The difference would show where the two cross, and its size depends again on what the finishing keeps.

What the count and the closing were #

The swings are read from the same table of which end is lit at which crossing that the earlier essay built, with ends placed by the dent’s closing instead of on an even grid. Every draft was run twice: even, with the requirement that the swing equal the thread count, and at $W$ to a dent closed by 0.3 with the line on the dent’s edge, with the requirement that the swing equal $(W-1)(1-g)$ to nine figures. The named weaves’ tables take the best and worst over every alignment. Two to a dent on a 2/2 twill must return exactly two spacings at both alignments, three to a dent must beat an even setting on the edge and lose to it off the edge, and thirty ends must keep their mean pitch. At the touching limit the swing must equal the cover, and a three-end line crowded there must have nothing inside and twice the plain gap beside. A dent holding no ends, a closing of all of a spacing, an alignment outside the dent, a cover of one, a cloth keeping more than it was given and a closing past touching are all refused.

Where the model stops #

The closing is one number across a width, and it is the wrong shape. Ends in a dent do not lie evenly close, and the wire’s effect relaxes in the fell, in the cloth and in finishing, by amounts that are not measured here. How much of a reed’s grouping a finished cloth keeps is the largest unknown in this essay and it scales everything.

Ends are round. A flattened end in a soft-twist yarn closes further than a round one, so a touching limit of $c$ is a little too strict, and a real warp might get past it. Whether two ends in a dent roll over each other, and what that does to their lit width, is left out.

The edge is the outer side of a lit end, and a float spreads. A thread’s rectangle is the earlier essay’s stated simplification, and a real float presses sideways over its neighbours, which rounds every step and might hide a difference smaller than itself.

Nothing here is a claim about seeing. A difference in swing is a difference in how far a boundary moves, and whether a viewer sees it depends on the swing’s period, the contrast, the light and the distance. The figure of a tenth of a millimetre is a rule of thumb, as it was before.

Only the warp’s spacing is considered. Picks are taken as even and the weft’s take-up as the same on either side of a crowded line. Crowding the warp locally changes how the weft crimps there, and the earlier essay on a cloth’s ridge at a figure boundary gives reason to think that a local change in setting can show as relief.

Who found it, and when #

Denting a stripe differently from its ground is old practice, and so is crowding a selvedge. Drawing-in to a plan that keeps a colour in one dent is an instruction found in weaving manuals, and it is usually given for its effect on a cloth’s appearance without a reason beyond that. The reed’s mark has an old literature of its own.

The limit at the cover, the fit between a line’s narrowest width and a dent on the whole four-by-four catalogue, the condition on a stripe’s width and the doubling of the gap beside a crowded line are computed here. None of them is known to have been measured.

Still open: how much of the reed a cloth keeps #

Everything above depends on one quantity that nobody appears to have read off a finished cloth, and the argument here supplies a cheap way to read it.

Weave a sample warp with one reed and two identical lines, one on a dent’s edge and one off it. The two have the same weave, the same yarn, the same sett and the same finish, so any difference in their swings is the reed. With a swing of two threads at three to a dent, the ratio of the two swings is $2(1-g)/(2+g)$, which gives the closing the cloth has kept directly, from measurements with a counting glass on one piece of cloth before and after finishing.

That one measurement would say whether a clean edge is worth planning for, how much of a dented sateen’s scatter survives the same finishing, and how much of a reed’s mark is left to see. It would also say whether the difference between the two lines is visible at all, which is the only part that a person looking at the cloth can answer.