A satin's row belongs to its sett

Most satins still have a diagonal, and the reason is a lattice. A regular satin’s marks sit on the points of a lattice, every lattice has a shortest step, the closest pairs of marks all run along it, and a run of closest pairs pointing one way is a row. Only a tie between two equally short steps avoids it, and twelve satin orders from five to forty ends have one. An irregular satin scatters where a regular one lines up: with no lattice, its closest pairs can point several ways, and at seven and nine ends one does exactly that at the best spread.

Every one of those results was taken on point paper, where an end and a pick are the same length. The account that found the irregular scatter ended by naming the correction that could change the answer rather than the numbers: a cloth is not square. A cloth set at twenty-eight ends and twenty-six picks a centimetre has its marks on a rectangle about eight per cent from square, and a tie in the draft’s metric need not be a tie in the cloth’s.

It is not, almost anywhere. And the correction runs both ways: satins that are row-free on paper acquire a row, and satins with a row on paper lose it at particular setts.

A lattice step measured in cloth #

Measure lengths in units of the end spacing. A step of x ends and y picks then has length $\sqrt{x^2 + a^2y^2}$, where a is the sett ratio: ends a centimetre over picks a centimetre. On point paper a is one. A warp-faced satin set at sixty ends and forty-six picks has a sett ratio of 1.30; a balanced sheeting set at twenty-eight by twenty-six, 1.08.

Two steps tie at a sett ratio a exactly when $x_1^2 + a^2y_1^2 = x_2^2 + a^2y_2^2$, which has a solution in a only if the steps differ in shape in the right way — and a tie that holds at every sett ratio needs $x_1^2 = x_2^2$ and $y_1^2 = y_2^2$: two steps that are mirror images of each other across a thread.

That one condition decides everything below. A tie between mirror images survives any sett. A tie between any other pair of steps is a coincidence of one particular ratio.

A quarter turn is not a symmetry of a cloth #

The reason the paper ties are fragile is a symmetry that point paper has and cloth does not. A square grid is carried to itself by a quarter turn, and the seventeen groups a draft can have include the ones with fourfold rotation for exactly that reason. The five-end satin’s two shortest steps are one such quarter turn of each other, and so are the steps of every other square-type row-free order: their tie is the quarter turn.

A cloth set at unequal setts is a rectangular grid, and a rectangle has no quarter turn. Its symmetries are the half turn and the two mirrors across the threads. So the only ties a cloth guarantees are ties the half turn and the mirrors guarantee — and the half turn only pairs a step with its own negative, which is the same direction. What survives an unequal sett is exactly what survives the loss of the quarter turn: mirror images.

That is also why the diagonal is not a thread in the sense a weaver sometimes speaks of it. A satin’s diagonal is a direction in the lattice of its marks, and a lattice direction’s length depends on the metric the lattice is measured in. A thread’s direction does not.

The row-free orders gain a row off a square sett #

The twelve row-free regular orders tie steps that are not mirror images. The five-end satin on a move of two ties the step of one end and two picks against the step of two ends and one pick back — the same shape turned a quarter, which on point paper is the same length and in cloth is not.

At a sett ratio of 1.077 — an ordinary balanced cloth — the step along the picks is stretched and the step along the ends is not, so the five-end satin’s closest pairs all point one way and it has a row. The same is true of all twelve row-free orders at a ratio of 1.02, a sett two per cent from square, which is inside the tolerance a loom holds a cloth to.

Row-freedom in a regular satin is a property of point paper, and of a woven cloth only if its setts are equal to within a thread.

How strongly the new row shows #

A row that appears because a tie breaks is a weak row at first, and its strength is measurable: the next-shortest step over the shortest. At a tie the two are equal and the ratio is one; the further above one, the more the marks prefer a single direction.

The five-end satin’s ratio is 1.044 at a sett ratio of 1.075 and 1.309 at 1.6. The ten-end goes from one to 1.442 over the same range. A warp-dense five-end satin has a clear row, which is what a square-drawn account of it would never predict: at a sett of sixty ends to forty picks its next-closest marks are 26 per cent further away than its closest, all of which lie along one direction.

The orders with a row on paper move the other way first. The eight-end satin’s ratio falls from 1.118 at a square sett, and the eleven-end’s from 1.140, as the sett grows warp-dense — because stretching the picks lengthens their shortest step and brings their second step level with it.

A rowed satin loses its row at one exact sett #

Falling ratios reach one. The eight-end satin on a move of three is row-free at a sett ratio of exactly $\sqrt{5/3}$ = 1.2910, where its step of one end and three picks and its step of three ends and one pick back come to the same length in cloth. The eleven-end satin does the same twice in range, at 1.2649 and 1.5275, and the thirteen-end at 1.3693.

That answers the question the irregular account ended on — whether a satin’s row is something a weaver can dent away — with a precise yes and a practical no. A sett of 1.291 ends per pick is squarely in the range warp-faced satins are woven at: 62 ends against 48 picks is 1.292. But the tie is an equality. A cloth set at 1.28 or 1.30 has a row again, weak but present, and a sett ratio is not a number a finished cloth holds to three decimal places.

Across the orders to seventeen, every row-free sett ratio for a regular satin is a single isolated number. None is an interval, because a regular satin’s lattice never contains two closest steps that are mirror images of each other — a mirror pair would need the move number to satisfy 2x·m ≡ 0 modulo the order for a short step, which no satin move does.

The sett a cloth ends at is not the sett it is woven at #

A weaver who set a satin at exactly 1.291 on the loom would not find it there in the finished cloth. A motif is drawn at the wrong shape on purpose because the two setts move in finishing, and in opposite directions: the warp relaxes more and the cloth closes up along its length, so the sett ratio a cloth ends at is not the ratio the reed and the take-up gave it.

That account found a balanced cloth’s aspect moving by about three per cent and a warp-dense one’s by sixteen. Either is far more than the width of a tie, which has no width at all. So a regular satin’s row, where the loom made it vanish, returns in finishing; and a row-free square cloth acquires a row. The practical statement is not that rows can be dented away but that how strongly a satin’s row shows is a property of the finished sett, readable off the ratio curve above, and it is the finished cloth’s ratio that has to be put into it.

A scatter that survives needs mirror images #

The irregular satins are where row-freedom can survive a range of setts, because an irregular satin is free to have closest steps that are mirror images, and mirror images stay equal whatever the ratio.

At seven ends the scatter is lost at the first per cent. The irregular seven-end satin’s closest steps on paper point four ways — one-two, two-one, two-minus-one and one-minus-two — and those are two mirror pairs of different shapes. Off a square sett one pair is shorter than the other; the best spread in cloth is then reached only by regular satins, and they have rows.

At nine ends the scatter survives. From a sett ratio of 1.01 to 1.5, eight of the fourteen satins at the best spread in cloth scatter their closest pairs, with no more than three in five pointing one way, and every one of them does it with mirror-image steps. Their scatter is weaker than on paper — three fifths in one direction rather than a quarter — and it does not depend on the sett.

At ten ends a scatter appears where none was. The ten-end regular satin’s paper tie breaks at once, and by a sett ratio of 1.3 two irregular ten-end satins at the new best spread scatter with mirror-image steps, half their closest pairs each way.

What the correction does to the earlier count #

A float limit leaves one row-free satin: of the orders under a float of eight, only the five-end is row-free, which is the satin the trade already uses. That result is a statement about point paper, and the correction changes it in both directions.

Off a square sett the five-end satin has a row — weak at an ordinary balanced sett, clear at a warp-dense one. The eight-end satin, under the same float limit, has no row at one sett ratio in the warp-dense range and a weakening one around it. So under a float limit of eight the row-free candidates in cloth are not “the five-end satin” but “the five-end at a square sett, the eight-end at 1.29, and nothing over a range” — and the one satin that scatters over a range of setts, the irregular nine-end, floats eight and just clears the same limit.

That is a more useful answer for a designer than the paper one, because it names the cloth. Which satins are worth weaving was argued from spread and from the diagonal on point paper, and both arguments carry over — but the diagonal part carries over only with a sett attached, and at the sett ratios satins are actually woven at, the ranking by row strength is not the paper ranking. A warp-faced satin of eight ends set close to 1.29 has the weakest row any regular satin can have under that float limit, and an irregular nine-end satin has a scatter that no finishing can take away.

What the row does to the shine #

A satin is woven for its floats, and a satin shines because its long floats lie flat and parallel and return light along a narrow lobe. The row is a second, coarser order laid over that: a line of interlacings running across the floats at the lattice’s shortest step. Where the row is strong, the cloth’s lustre is broken into bands along it; where the marks scatter, the breaks are spread evenly and the lustre reads as a single sheet.

So the sett ratio moves a satin’s look in two ways at once. It changes the float lengths in cloth, which is the lustre’s own geometry, and it changes how strongly the interlacings line up, which is the row. A warp-dense satin set for a longer, flatter float and a brighter face also sets its row’s strength by the curve above — strengthening it for a five-end satin and, near 1.29, weakening it for an eight-end one. There is no six-end satin with a move number to choose from, and at every order that has one the move is chosen on paper; the choice that matters for the finished face has a sett in it that the move number cannot see.

What was counted, and how #

For regular satins, every lattice step within a few repeats is enumerated — each x from minus to plus the order with its picks congruent to x times the move — and lengths are taken as $x^2 + a^2y^2$ in units of the end spacing. The shortest step and the shortest step independent of it give the row’s strength as their ratio. The best move at a sett ratio is the one whose shortest step is longest. Tie ratios are found from every pair of candidate steps, $a^2 = (x_2^2 - x_1^2)/(y_1^2 - y_2^2)$, and kept only if the two steps are the shortest at that ratio and the move is still the best there.

For all satins of five to eleven ends — every arrangement of one mark per end and per pick with no two touching, up to where the repeat starts — the closest pairs are found on the torus with the pick coordinate scaled by the ratio, and a satin scatters when its closest pairs point more than one way. The census at a sett ratio of one was confirmed to reproduce the square-sett lattice move for move; every row-free order to forty ends was confirmed to have a row at a ratio of 1.02; the eight-end satin was confirmed row-free at one ratio between 1.2 and 1.35; nine-end irregular scatter was confirmed at four ratios from 1.02 to 1.3 with every scattering satin’s closest steps mirror images; ten-end mirrored scatter was confirmed at 1.3; and seven-end scatter was confirmed lost at 1.02.

Where the lattice stops describing the cloth #

Marks are points. A satin’s interlacing is a short break in a float, and a thread’s crimp and the yarn’s width make a mark an extended shape whose own aspect may differ from the sett ratio.

The yarn’s twist lies across the marks. A satin’s apparent diagonal is read partly off the twist of its floats, which runs at an angle of its own, and nothing here computes how an eye weighs the two.

A ratio of 1.04 may not be seen. The strength of a row is a ratio, and where an eye stops seeing a line in closest marks four per cent nearer one way than the other is a question about perception, not lattices.

Still open: how close to one an eye stops seeing a row #

The whole correction turns on the ratio curve, and the curve’s practical meaning depends on one threshold no calculation gives: the ratio above which a row is visible. A set of cloths woven from one yarn in one satin at sett ratios stepping from 1.00 to 1.30, shown to observers under a single raking light with the question of which way the satin’s diagonal runs, would put a number on it. With that number, the curves above become a statement about which finished satins read as rowed — and the eight-end satin’s $\sqrt{5/3}$ becomes either a sett worth aiming for or an equality with no visible consequence.

Who found it, and when #

Satin moves, their diagonals, and the practice of setting satins warp-dense are old weaving knowledge, and lattice reduction under a quadratic form is standard number theory. Measuring satin lattices at a cloth’s sett ratio, finding that every paper-row-free regular order acquires a row off a square sett while rowed orders become row-free at isolated ratios, and that only mirror-image irregular satins keep a scatter over a range, was done here.