A float limit leaves one row-free satin

Most satins still have a diagonal is the essay this one continues and it ended by naming its own missing constraint. A satin’s interlacings lie on a lattice; every lattice has a shortest step; the marks line up along it; and the only way to avoid a row is for two shortest steps to tie in length. Over the orders from five to forty that happens at twelve of the thirty-five, and the essay closed by saying that the order is capped from the other side — that a float limit decides how long a float a yarn and a finish will tolerate, and that on a yarn stopping at eight or twelve ends a satin with no row was never on the table.

Applying it is one inequality. An n-end satin floats over n − 1, so a float limit of F admits orders up to F + 1 and no further, and the cross is a matter of reading one census against a ceiling.

The answer is sharper than the phrase “never on the table” suggests.

Twelve orders, and where they are #

The row-free orders are 5, 10, 13, 15, 17, 24, 25, 26, 29, 34, 35 and 37. They are not a pattern anybody would guess from the numbers and they are not spread evenly: one of them is below ten, three below fifteen, five below twenty.

A float limit of eight admits four orders in all — five, seven, eight and nine, since a ten-end satin floats over nine — and exactly one of them is row-free.

It is the five-end satin.

Which is the satin everybody already weaves #

That the arithmetic’s one survivor is the commonest satin in the trade is worth pausing on rather than passing over, because it can be read two ways and only one of them is right.

It is not evidence that the trade optimised for rows. The five-end satin is common for a dozen reasons that have nothing to do with lattices: it needs five shafts, it is the smallest regular satin there is, it is what a dobby of ordinary size can carry, and its float of four sits comfortably inside almost any yarn’s limit.

It is evidence that the row question never had room to bite. A designer working at a float limit of eight has four orders to choose between, and whichever they choose for the other reasons, the row-free one is also the smallest, the cheapest in shafts and the least demanding on the yarn. The constraint that would have forced a choice is dominated by three others that point the same way, so the trade arrived at the row-free satin without ever having to ask about rows.

That is a different and more interesting conclusion than either “the weavers knew” or “the weavers were lucky”. The arithmetic says the question was unavailable: to face a real choice between a row-free order and a better one on other grounds, a designer needs a float limit above about twelve, and a limit above twelve is a filament yarn and a modern finish rather than a worsted.

The share never reaches a half #

Raising the limit admits more orders and admits more row-free ones, and it does not admit them faster.

Above a limit of six the row-free share is always a minority, running from 25 per cent at a limit of eight to 42 at sixteen and back to 34 at thirty-nine. It does not converge on anything, because whether an order is row-free is an arithmetic accident of its own lattice rather than a property that becomes commoner with size.

The two very short limits are the exception and they are the uninteresting kind. A limit of four admits one order — the five-end — and it happens to be row-free, so the share is a hundred per cent of a sample of one. A limit of six admits two and the share is a half. Neither is a finding about satins; both are findings about a small sample, and reporting them as shares would be the mistake the cloth-counting arithmetic warns about.

What a designer actually chooses between #

Put the two censuses together and the practical statement is short.

At a float limit of eight — a worsted, an ordinary cotton, anything staple — a designer chooses among five, seven, eight and nine ends. The five is row-free; the seven, eight and nine are not. A designer who cares about rows has one option and it costs them nothing.

At a limit of twelve — a good staple yarn with a soft finish — the choice widens to eight orders, three of them row-free: five, ten and thirteen. Now the question is real, because ten and thirteen carry longer floats and more lustre than five, and choosing one of them for its lustre keeps the row-free property while choosing eleven or twelve loses it.

At a limit of sixteen — a filament — five row-free orders are available among twelve, and the row question is a genuine design constraint operating alongside the others.

So the row-free condition is a constraint that switches on with the yarn. That is an unusual shape and it is the essay’s practical result: the same arithmetic gives no choice, a real choice and a rich choice at three yarn qualities, and a designer working in silk faces a question a designer working in worsted does not have.

The orders that are lost, and what each would have cost #

It is worth naming the row-free orders a float limit throws away, because the list is short and the losses are of very different kinds.

Ten and thirteen are lost to a limit of eight and recovered by twelve. Those are the two a designer would most regret: a ten-end satin floats over nine and a thirteen over twelve, both well inside what a filament will carry, and both are row-free at their best move. They are the orders where the limit is actually doing something, in the sense that the yarn and nothing else decides whether they are available.

Fifteen and seventeen need a limit of sixteen, which is a long float in any staple yarn and is ordinary in silk. Their floats are fourteen and sixteen, and a cloth with a sixteen-thread float is a cloth whose surface is nearly all one system — which is what a satin is for, and is also what makes it snag.

And twenty-four upwards are lost to any limit a cloth is woven at. A twenty-four-end satin floats over twenty-three, which at twenty-eight threads to the centimetre is eight millimetres of unbound thread. Nothing wears that. So five of the twelve row-free orders are unreachable in any cloth, and the census’s tail is a statement about lattices rather than about weaving.

The seven orders a limit of twenty-four admits are the whole practical list, and three of them — five, ten, thirteen — are inside anything a mill would weave. That is the row-free satin vocabulary, and it has three members.

Why a float limit is a ceiling and not a preference #

The inequality is worth stating carefully because it is the only place this account’s arithmetic touches the physical world, and it is doing so through a quantity nothing here derives.

A float limit is a constraint a yarn and a finish impose. A long float snags, abrades and slips, and the float decides most of what a cloth does about all three. The number a designer works to — nothing longer than four, nothing longer than eight — is a rule of practice with a mill’s own experience behind it rather than a derivation.

And it is a ceiling on the order, exactly. An n-end regular satin puts one mark in each end and each pick, so every end is on the face at n − 1 picks in n and floats over n − 1 of them. There is no move, no arrangement and no irregularity that shortens it: a satin’s longest float is its order less one, always. So the ceiling is not a rule of thumb applied to a distribution; it is an equality.

That is why the cross is so clean. One of the two constraints is an exact property of the matrix and the other is a number from a mill, and the census is the first crossed with a stated value of the second.

What the earlier account’s ranking says about the survivors #

Which satins are worth weaving ranked the moves within each order by how far apart the interlacings sit, and found the traditional counter the best move at the orders a weaver mostly uses. That ranking and this one are about different things and it is worth keeping them apart.

The spread ranks moves within an order. It asks, of the several moves an order admits, which scatters the marks furthest — and the answer is usually the counter.

Row-freeness ranks orders. It asks, of the best move an order has, whether its lattice happens to have two shortest steps — and it is a property of the order rather than of a choice.

So a designer has two decisions and only the first is theirs. The order is chosen for shafts, lustre and the float limit; the move is then chosen for spread; and whether the result has a row was settled by the order before the move was picked. That is why this account’s finding is about what is available rather than about what to do: there is no move that rescues a row-bearing order, and the earlier account established that by showing the rows are a lattice property.

The shafts run the other way and nearly cancel it #

There is a second ceiling on the order and it comes from the loom rather than from the yarn, and it happens to fall in almost the same place.

An n-end satin needs n shafts, and a dobby’s harness has a depth that a stated tolerance on warp strain bounds — one per cent buys thirteen. So a shaft loom of ordinary depth admits satins to about thirteen ends, which is the same ceiling a float limit of twelve gives, arrived at by a completely different route.

That is a coincidence and it is worth saying so, because it is the kind that gets mistaken for a design. Nothing connects a warp-strain tolerance to a yarn’s snagging: one is about how far back a shaft stands and the other about how long a thread may lie unbound. They agree because a satin’s order is the same number in both — its shafts and its float plus one — and because the two independent limits happen to be near twelve.

What follows from the agreement is practical. A designer on a shaft loom does not have to check the float limit and one on a jacquard does, because a jacquard abolishes the shaft ceiling entirely and leaves the yarn’s. So the row-free orders past thirteen — fifteen, seventeen and the rest — are jacquard questions by construction, and on a jacquard the only thing stopping them is the float.

What it says about the habit of censusing to forty #

This account is the second time a census here has run to an order well beyond anything weavable, and the two cases give opposite answers about whether that was worth doing.

Which satins are worth weaving ranked moves from five ends to forty and found the traditional counter best at the orders a weaver uses and failing first at thirteen. That census earned its tail: the failure at thirteen is inside a filament’s reach, and knowing where a rule of thumb stops being right is worth the enumeration that finds it.

The row-free census runs to forty and seven of its twelve findings are past any float limit. The orders 24, 25, 26, 29, 34, 35 and 37 are row-free and unweavable, and a reader given the list of twelve without the ceiling would take away a much rosier impression of how available the property is than the arithmetic supports.

So a census’s range is part of its result and ought to be reported with it. Twelve of thirty-five sounds like a third; one of four is the number a weaver has. Both are true of the same enumeration, and the second is the one about cloth — which is this collection’s standing preference, arriving as a caution about how to quote its own tables.

What was crossed, and how #

Each order’s regular satins are found from the moves coprime with it and not one or one less; each move’s lattice is reduced to its shortest pair of vectors by the ordinary lattice reduction; the best move is the one whose shortest vector is longest, and the order is row-free when that move’s lattice has two shortest vectors of equal length. The float limit is applied as n ≤ F + 1, from the exact statement that an n-end satin floats over n − 1.

Six things are checked. There are thirty-five orders from five to forty that admit a regular satin, which is the six-end exception counted rather than assumed. Twelve of them are row-free, which is the earlier account’s figure recomputed here rather than quoted, so that a change in either essay’s arithmetic would show as a disagreement. A float limit of eight admits four orders, and exactly one of them is row-free, and it is the five-end. A longer float admits at least as many row-free orders at every step, which is the monotonicity a slip in the ceiling would break. And above a limit of six the row-free orders are always a minority.

The float limits are inputs and are the trade’s; everything else is the lattice.

Where the cross stops #

The float limit is a single number and a real one is not. A mill’s rule depends on the yarn, the sett, the finish and what the cloth is for, and the same cloth may carry a longer float in the warp than in the weft. The ceilings here are a sweep over plausible values rather than a claim about any of them.

The limit is applied to the satin alone. A satin ground carrying a figure has the figure’s own floats to consider, which can be longer than the ground’s, so a cloth’s float limit may bite before its satin’s order does.

And nothing here is about seeing. Whether a row is objectionable, at what sett, in what yarn, to whom, is a question about a visual system, which this collection said from the beginning it does not have. The row-free condition is a statement about where the closest interlacings lie and not a statement about appearance — and the earlier account said so at greater length.

Still open: whether a row-bearing satin can be dented out of its row #

The row is a property of the lattice in the draft, and a draft’s lattice becomes a pattern in the cloth only through the setts. A row runs along a lattice direction, its visible angle is that direction’s angle scaled by the ratio of the two setts, and a row lying at an awkward angle is more visible than one lying at a comfortable one.

So there is a sett at which a row’s direction coincides with something else, and the question is whether any of them helps. The reed’s own grouping is periodic, a colour order is periodic, and a row is periodic; a row-bearing satin dented so that its row falls on the reed’s own period would have the two periodic marks on top of each other rather than beating against each other, which is either a great deal better or a great deal worse and the arithmetic of the two is the same least common multiple.

That is a computation this collection is equipped for — it has the lattice, the denting arithmetic and the beat — and it has not been run. If it helps, a row-bearing order becomes usable at a stated denting, and the twelve row-free orders stop being the whole of the answer.

Who worked it out #

The float limit is trade practice and its application to satins is in every account of designing for one. That a satin’s longest float is its order less one is elementary. The row-free orders are the earlier account’s, from the lattice reading built there; crossing the two, and finding that an ordinary float limit leaves one row-free order and that it is the one the trade uses, was done here.