A reed can dent a sateen out of its row and not a satin

A regular satin’s interlacings lie on a lattice, and most satins still have a diagonal because every lattice has a shortest step and the marks line up along it. Only where two shortest steps tie is there no preferred direction and no row. A satin’s row belongs to its sett then moved the lattice from point paper into cloth. There the step along a pick and the step along an end differ in length, and the ties that made twelve orders row-free on paper broke at the first unequal sett. The eight-end satin, rowed on paper, turned out to be row-free at exactly one sett ratio, $\sqrt{5/3}$, and at no other.

A float limit leaves one row-free satin ended on the other thing a weaver controls between the draft and the cloth: the reed. A reed groups the ends several to a dent. The question it left was whether a row-bearing satin can be dented out of its row, and whether a denting that did so would help or would lay a second periodic mark on top of the first.

It can, over a range rather than at a single number, which is the thing the sett could not do. But the range starts at a grouping that a warp-faced satin cannot reach, and what takes the row’s place is a stripe drawn by the reed.

What a reed does to the spacing #

The reed leaves its own mark drew the geometry. Inside a dent nothing separates the ends, so they sit closer together than the average spacing. At the wire between dents a gap opens, and every dent is still exactly its pitch wide, because the reed sets that and nothing else. At the loom, a 0.25-millimetre warp at twenty ends a centimetre, four to a dent, has its ends touching inside each dent and a gap two and a half times the average at every wire.

How much of that survives into a finished cloth is not known. The essay on the reed’s mark said so plainly, and this essay does not know either. So the grouping is carried here as one number, the closing $c$: the ends inside a dent are spaced at $(1-c)$ of the average, and the wire’s gap takes up the rest. With $d$ ends to a dent the gap at the wire is $1 + (d-1)\,c$ average spacings.

At no closing the warp is even and the satin is the lattice every earlier essay drew. At any closing above nought, the marks no longer lie on a lattice. A satin repeats every $n$ ends and the reed every $d$, so together they repeat every $\operatorname{lcm}(n, d)$ ends, and within that period the same step of the satin — so many ends across, so many picks up — has a different length depending on where in its dent it starts.

Which way a mark’s nearest neighbour lies #

The earlier essays measured a row by counting the closest pairs of marks over the whole repeat and asking whether they all point one way. On a lattice that is the same as asking each mark which way its own nearest neighbour lies, since every mark has the same neighbours. Off a lattice the two questions differ. The closest pairs overall are then simply whichever pairs the reed pulled together, and they say nothing about the rest of the cloth.

The question an eye asks is the second one. So each mark is joined to its own nearest neighbour, keeping ties. A row is every link pointing the same way, and it is scattered when the links point several ways. A margin goes with it: for each mark, how much further its nearest neighbour in any other direction lies than its nearest one, averaged over the marks. For an even warp this is the lattice’s own ratio of its two shortest steps, 1.118 for the eight-end satin and exactly one for a row-free satin.

A step inside a dent shortens in proportion to its width #

The eight-end satin on a move of three — the move which satins are worth weaving found best at that order — has two short steps. The row runs along the shorter: two ends across and two picks up, of length $\sqrt 8 = 2.83$ end spacings. The next is three ends across and one pick up, $\sqrt{10} = 3.16$. The row is there because the first is shorter than the second by about a ninth.

Inside one dent, a step across $k$ ends is $k(1-c)$ spacings wide, and its height in picks is untouched. So the three-end step loses width three times as fast as the closing grows, while the two-end step loses it only twice as fast. The three-end step has less height to begin with, so width is more of its length. The two meet where

$$9(1-c)^2 + 1 = 4(1-c)^2 + 4, \qquad (1-c)^2 = \tfrac35, \qquad c = 1 - \sqrt{3/5} = 22.5\%.$$

That is the whole mechanism, and every threshold below is an equation of the same kind. A reed can only reorder steps that fit inside a dent, and it reorders them in favour of whichever step is wider.

Four and five to a dent scatter the row; two and six turn it #

Which marks have a three-end step inside their own dent depends on the dent. With four ends to a dent, the marks in its first and fourth places have one, joining each other, and the two in the middle do not; with five, the marks in four of its five places do; with six, every mark does, in one direction or the other. With three, none does, since a dent of three holds a step of at most two.

So the three dentings behave differently at one and the same closing:

• Four ends to a dent switch half the marks to the new step at 22.5 per cent and leave the other half on the old one. The links point two ways and the row is scattered.
• Five ends to a dent switch four marks in five at the same closing. The links point two ways, and again there is no row.
• Six ends to a dent switch every mark at once. All the links still point one way — the three-end way — and the row has turned rather than gone.

Three ends to a dent reach a scatter by a different route, at a closing of exactly a third, where the row’s step across a wire has lengthened as far as a one-end step inside the dent has shortened: $(2+c)^2 + 4 = (1-c)^2 + 9$. Two ends to a dent never scatter the row at all. The row’s step spans exactly one dent from any start, so it never changes length, and all a closing can do is shorten the three-end step that crosses one wire, until at $3 - \sqrt 7 = 35.4$ per cent it ties. At that one closing the satin is row-free, as it was at one sett ratio. Past it the row has turned.

A range, not a point #

A satin’s row belongs to its sett found that every regular satin is row-free at a few isolated sett ratios and at no interval of them, because a tie between two steps of different shape holds at one ratio only. That made row-freedom something a finished cloth could never be relied on to have. Its sett ratio moves in finishing by more than the width of a tie, which is nothing.

The reed does something the sett cannot. Four or five ends to a dent scatter the eight-end satin’s row at every closing from 22.5 per cent up to the ends touching. This is not a tie. Different marks have different nearest steps, because they sit at different places in their dents, and no small change in the closing brings them back into agreement. A finished cloth whose ends in a dent sit anywhere between a quarter and a half closer than average is scattered, and stays scattered whatever finishing does to it within that band.

This is exactly the property an irregular satin scatters where a regular one lines up found in the irregular satins that survive a range of sett ratios. Their closest steps pointed several ways at once, for reasons built into the arrangement rather than into a coincidence of lengths. A dented regular satin is an irregular satin made by the reed: its marks are no longer related by one translation, and like the irregular ones it can hold a scatter over a range.

Five a dent leaves no reed mark #

The question the float-limit essay left had a second half. A row and a reed’s grouping are both periodic, so a denting chosen to take the row out might land its own period on the row’s and make something worse.

The essay on the reed’s mark gave the rule for the reed’s own beat: dent so that the ends per dent share no factor with the repeat. Four to a dent on an eight-end satin shares a factor of four, and every dent then meets the satin at the same place; the grouping and the weave agree at every dent and a stripe appears at the dent pitch. Five to a dent shares nothing with eight. Its dents meet the satin at each of eight positions in turn, the joint period is forty ends, and nothing repeats at the dent pitch that is not also spread over the forty.

So five ends to a dent scatters the eight-end satin’s row at the same closing as four, and without the beat four brings with it. That is the precise yes the question asked for. It is an ordinary denting as well: five to a dent is in any weaving manual’s table, and at twenty ends a centimetre it means a reed of four dents to the centimetre.

What takes the row’s place #

The drawing at the head of the essay shows what the numbers do not. Once a reed has closed the ends enough to matter, every mark’s nearest neighbour lies inside its own dent. The gap at the wire is wider than any step within the dent, so no link crosses it. The links in a scattered satin form zigzags down each dent, one dent wide, and between them run the lanes at the wires.

That arrangement is a stripe at the dent pitch. With five ends to a dent at a closing of 22.5 per cent the lane at every wire is 1.9 average spacings wide, against 1 in an even warp, and the links an eye follows run along the dents. The reed takes the diagonal away by drawing its own lines in the other direction. Whether that is better than the row depends on which a viewer notices more, a faint diagonal at the satin’s angle or a faint stripe along the warp. The lead guessed that the answer would be a great deal better or a great deal worse. In fact it is a trade of one periodic mark for another, whichever denting is chosen, and the coprime denting only removes the second mark’s beat with the repeat, not the mark.

A warp-faced satin’s ends touch first #

All of this depends on the ends in a dent being able to close up by 22.5 per cent, and whether they can depends on how much room they have. Ends cannot close past touching, and how close can threads be set is the account of what touching costs. The most the closing can be is one minus the warp’s cover, the yarn’s diameter over the average spacing, as where the cover factor comes from defines it.

A warp-faced satin is warp-faced because its warp covers the face. At 62 ends a centimetre — the warp-faced sett the sett-ratio essay used as its example — a 12-tex cotton warp covers 80 per cent of the width, so its ends can close by at most 20 per cent before they touch, short of the 22.5 the scatter needs. A finer or more open warp could get there. But the satin cloths the question is about, where the long warp floats make the row visible at all, are the ones set close enough to cover.

A sateen is the other way round. Its face is weft, and its warp sits open underneath: thirty ends a centimetre of 20 tex covers half the width and can close by up to half. A reed can take a sateen’s row out, and the sateen shows its row on the weft face, where the marks are the warp’s interlacings. So the answer to the question as asked is that a reed can dent a sateen out of its row and cannot dent a satin out of its row, and the difference is the one thing that makes them different cloths: which system covers the face.

Only a weak row can be dented out at all #

The eight-end satin’s row is weak. Its second step is only an eighth longer than its first, which is why a closing of a fifth is enough to reorder them. A satin whose second step is much longer than its first cannot be reordered by any grouping a reed can produce.

The table sorts the orders cleanly. The nine-end and twelve-end satins keep their rows at every denting from two to six, all the way to touching. Their second steps are 1.84 and 1.50 times their first, and no reordering inside a dent closes gaps that large. The seven-, eight- and eleven-end satins, whose rows are weaker — second steps 1.41, 1.12 and 1.14 times the first — can all be scattered at four or five a dent, at closings of 38.8, 22.5 and 20.9 per cent. The eleven-end satin can also be turned and then scattered at six. So a reed can only undo a row that the lattice barely made, and the orders whose rows are strongest are also the ones a reed cannot touch.

Two ends a dent never help #

The commonest denting of all is two ends to a dent. The table’s first column is the one a weaver most needs, and its verdict is uniform. Two ends to a dent never scatter any satin’s row from five to thirteen ends. At best, on the eight-end satin, they turn it through a single tie.

The reason is a symmetry, and it holds at every order. Turn the cloth half round about the point midway between the two marks of one dent. A regular satin’s marks lie on a lattice, and a lattice turned half round about the midpoint of two of its points lands on itself. The reed’s spacing is symmetric about each dent’s centre, so the reed lands on itself too. The half turn therefore carries the first end of every dent onto the second, and every mark sees exactly the same neighbourhood as its partner, turned round. Whatever a closing does to one mark’s nearest step it does to every mark’s, so the links can only swing together. A denting whose places are all alike can turn a row but cannot scatter it, and two is the only number of ends to a dent whose places are all alike. With three, the middle end differs from the outer two; with four, the inner pair from the outer; and those differences are what a scatter is made of.

The reed gives a row as easily as it takes one #

The table’s other half is worse news, and it concerns the one satin the float-limit essay found a designer can actually use without a row.

The five-end satin is the smallest regular satin there is — there is no six-end satin, and four ends admit none — and it is row-free because its two shortest steps, two ends and one pick and one end and two picks, are exactly equal. A tie is fragile, and the reed breaks it. Two, four, five or six ends to a dent give the five-end satin a row at the smallest closing — a hundredth is enough — because each treats the two tied steps differently and one of them always wins. Only three ends to a dent keep it free, and not for any reason of common factors: two, four and six share nothing with five either. With three to a dent the middle end’s nearest link runs one of the two tied ways and the outer ends’ links run the other, so the links still point two ways; the dent has two kinds of place, and here each kind breaks the tie in its own direction. The ten- and thirteen-end satins, row-free on paper, are the same: two ends to a dent give each a row, and a few dentings keep each free.

The row the reed gives is weak at first. At a closing of 20 per cent two ends to a dent give the five-end satin a margin of 1.04, where a tie is 1 and the eight-end satin’s own row is 1.12 — a third of that row’s excess over a tie — and it grows with the closing. But a satin chosen for having no row loses that property at the ordinary denting of two ends a dent, which nothing in a draft, a move number or a sett ratio warns of.

What the drawings do not show #

A dash is not a line, and the drawings show dashes. From the first closing above nought, the gap at every wire makes each mark’s nearest neighbour one inside its own dent, so the links along the row stop at every wire and the row is drawn as a string of dashes one dent long. In the numbers this is invisible — every dash still points the same way — and to an eye a line broken by a gap a few per cent wider than its steps is still a line. The drawings join only nearest neighbours, which is the honest rule and exaggerates the breaks. The scatter at 22.5 per cent is a different thing: the dashes stop pointing the same way.

The cloth relaxes. Everything here takes the reed’s grouping as a fixed pattern of spacings. A finished cloth’s ends move under the weft’s crimp and in wet finishing, and a random error hides and a periodic one shows found that a periodic error in spacing is what the eye sees; a reed’s grouping is periodic by construction.

How the counts were made #

For each order from five to thirteen, the best regular move was taken, as the earlier satin essays took it: the move whose closest marks sit furthest apart at a square sett. The ends were placed at their dented positions over one joint period of $\operatorname{lcm}(n, d)$ ends and $n$ picks, and every mark was compared with every other mark and its images one period away in each direction, to find its nearest neighbours to within a billionth.

The closing was swept from nought to one half, the ends touching for a 0.25-millimetre warp at twenty ends a centimetre, in steps of half a per cent. Each change of regime — row to scatter, row to turned — was refined by bisection to a millionth. The sweep is required to find the closed forms: $3 - \sqrt 7$ at two a dent, a third at three, $1 - \sqrt{3/5}$ at four, five and six, each to a millionth. With no closing, or one end to a dent, every order must come out as its undented lattice, ratio and all. And three things that could each have come out the other way: two ends a dent must never scatter any order’s row, a row whose second step is half as long again must survive every denting, and the five-end satin must gain a row at two a dent and keep its scatter at three.

Who worked out which part #

Denting is ancient practice, and so is the rule that some dentings mark a cloth and others do not. That rule is stated as a greatest common divisor in the essay on the reed’s mark, and a colour order beats the weave it is threaded on found the same divisor governing a second grouping of the same warp. The satin lattice, the row as the lattice’s shortest step, and the isolated sett ratios at which a regular satin is row-free are from the satin essays that came before.

What is added here is the reed as a third lever on a satin’s row, after the move and the sett. A step that fits inside a dent shortens in proportion to its width. Thresholds follow in closed form: $1 - \sqrt{3/5}$, a third and $3 - \sqrt 7$ for the eight-end satin. Dentings split into those that scatter a row over a range of closings, those that turn it through a tie, and two ends a dent, which never scatters any satin to thirteen ends and gives every row-free one a row. And the ceiling a warp’s cover sets on all of it, which a warp-faced satin sits above and a sateen below.

Still open: whether a sateen’s scatter shows #

The whole of the practical conclusion rests on one number: how much of a reed’s grouping a finished sateen keeps. A sateen whose ends in each dent sit a quarter closer than average is scattered at five a dent. One whose ends have relaxed to a tenth still has its row, broken into dashes at every fifth end.

The closing is measurable directly: a pick glass on the back of a finished sateen, the spacings between neighbouring ends read across a few dozen dents, and their pattern set against the reed’s. Crossed with the thresholds here, it would say which sateens a reed has already scattered without anybody intending it. It would also say whether the lane a coprime denting draws at every wire is a better thing to see than the diagonal it replaced, which is a question for an eye and not for a lattice.