Designing to a float limit

A weaving specification almost always carries a line that looks like a detail and is not: no float longer than four. Sometimes three, sometimes six, occasionally eight for a lining that will never be rubbed. It is the accumulated answer to everything in the previous four rungs of this ladder — the snag, the freed length, the pill — reduced to a single integer a designer can check.

What the line does not say is what it costs. It reads as a rule about a drawing, as though it merely forbade a few obviously bad ideas. Counted, it is a statement about how much of the design space still exists, and the answer depends sharply on the size of the repeat.

Why the fraction falls #

A twill is a sequence of runs, alternately over and under, summing to the repeat. On four ends the sequences are short and so are the runs: there is nowhere for a long float to hide. On twelve ends a sequence can put seven picks into one run and five into the rest, and the enumeration counts every such arrangement as a distinct twill because it is one — a different cloth, with a different face.

So the space of twills grows with the repeat, and the sub-space inside a float limit grows more slowly, because the limit is a fixed number while the repeat is not. The fraction is a ratio of the two and it falls monotonically, which is asserted while the figure draws rather than observed afterwards.

What makes this worth stating is the direction it runs. A designer moves to a larger repeat precisely to get more choices — that is what a larger repeat is for — and the float limit takes a growing share of them away. The two effects work against each other, and at twelve ends a third of the catalogue is already gone.

The satins are decided by a different arithmetic #

For twills the limit removes a fraction. For satins it removes almost everything, because a regular satin’s float is fixed by its order.

An order-$n$ regular satin has one interlacing per pick, so each end is on the face for $n-1$ picks and under for one. The float is $n - 1$, with no freedom at all: choosing the order chooses the float.

A float limit of $k$ therefore permits satins of order at most $k + 1$, and then the coprimality condition removes some of what is left. A limit of four allows orders up to five; four admits no satin at all, so the only regular satin inside a float limit of four is the five-end one. A limit of three allows orders up to four and there is nothing. A limit of seven allows five, seven and eight.

That is a curious inversion. The satin theorem is generous as the order grows: primes admit many moves, and thirteen has five distinct satins. The float limit is a ceiling from the other end. The two constraints meet in the small numbers, and the reason five and eight are the satins of the world is that they are the two orders where both conditions are comfortable at once.

It also explains a piece of terminology that otherwise looks like an evasion. Manuals distinguish a regular satin from a sateen or an irregular one, and treat the irregular kinds as second-best. Under a float limit the distinction has teeth: an irregular arrangement is not obliged to give every end the same float, so it can put a shorter float where a regular satin’s arithmetic would have insisted on the full order less one. What is bought with the irregularity is not an extra order — it is a float shorter than the order would otherwise force, and that is exactly the currency the limit is denominated in.

What a designer actually does: binding points #

A jacquard figure is not chosen from a catalogue of twills. It is drawn — a shape, shaded, on a repeat that may be hundreds of ends across — and the weave underneath it is whatever the shading calls for. A large area of ground in one weave and a figure in another, with the figure’s areas sometimes very large.

Inside a large figured area the float would run the width of the area, which is enormous. What prevents it is binding points: intersections deliberately reversed to tie a long float down, scattered through the area at intervals no longer than the limit.

The cost of a binding point is exactly the cost of an interlacing. Each one interrupts the float, so it interrupts the highlight and dulls the surface slightly; each one adds a bend, so it takes room and lowers the sett the cloth can reach; and each one is visible if the eye can find a pattern in where they fall.

That last is the reason the binding points in a well-made damask are placed on a satin arrangement rather than a regular grid. The problem of scattering interruptions so that no line forms is precisely the satin problem, and it is solved with precisely the satin’s answer, at a different scale and for a different reason. Nobody had to invent a second method.

A float limit is a sett limit wearing different clothes #

The binding points have a cost the section above described qualitatively, and it can be made exact — at which point the float limit turns out to be a constraint on a quantity from an entirely different ladder.

Binding points scattered on a satin arrangement of order n put one interlacing in every n intersections, because that is what a regular satin is. And the float between them is n − 1. So a float limit of k forces a binding arrangement of order at most k + 1, which forces an interlacing density of at least 1/(k + 1).

That is a floor on the interlacings per unit area of a figured cloth, and interlacings per unit area is the quantity firmness and maximum sett are computed from. Every interlacing is a place where a thread turns, a turn takes room, and the room is what caps how densely the cloth can be set.

The third row is the standard case and the second is the interesting one. Order four admits no regular satin, so a limit of three cannot be met by the satin method at all and has to be met by an irregular arrangement — which is the argument for irregular sateens made from the other end, and it says why they are needed at four and six specifically rather than being a general convenience.

Read as a design consequence: tightening a float limit by one raises the minimum interlacing density and therefore lowers the maximum sett the figured areas can be woven at. A cloth specified with floats of three cannot be set as closely as the same cloth at floats of four, whatever the yarn — so the abrasion resistance the tighter limit buys is paid for partly in the cover the cloth can reach, which is the other quantity abrasion depends on.

That is a genuine circularity in a specification and it is not usually noticed, because the two numbers live in different documents. The float limit is on the design sheet and the sett is on the weaving particulars, and nothing carries the constraint between them.

The same arithmetic explains a piece of practice that otherwise looks like superstition. A damask’s ground and figure are usually the same order of satin, one warp-faced and one weft-faced, and the reason offered is that the two then reflect light oppositely. That is true and there is a structural reason underneath it: two satins of one order have the same interlacing density, so the two areas take up at the same rate and lie flat against one another. Two areas of different interlacing density in one cloth take up differently, and the cloth cockles at the boundary — which is the figured weaver’s commonest fault and is a float limit’s arithmetic arriving somewhere nobody was looking for it.

What was counted, and how #

The twill enumeration is exhaustive rather than sampled. Every way of writing a twill on the repeat is generated — every composition of $n$ into an even number of positive runs — and the results are reduced by the two operations that leave the cloth unchanged: starting on a different pick, and turning the cloth over. What is left is the number a designer chooses between, which on eight ends is twenty-one rather than the sixty-four ways of writing one down.

Each surviving class is then built as an actual draft and its longest float is measured cyclically off the matrix, so the float in the table is the float in the cloth and not the largest run in the sequence — those can differ where the sequence wraps.

Two things are asserted while the figure draws. The fraction inside the limit must never rise as the repeat grows, which is the shape of the claim; and the smallest repeat must be entirely inside the limit while the largest is not, so that the figure is showing a constraint that actually binds somewhere in its own range. A limit that excluded nothing would draw a row of full bars and say nothing at all.

There are two limits, and specifications usually give one #

A draft has warp floats and weft floats and they are different numbers. A three-and-one twill has warp floats of three and weft floats of one; turned over it has the reverse. A single line reading no float longer than four is silently a constraint on the larger of the two, and that is very often not what was wanted.

The two systems are not symmetric in any of the ways that make the limit matter. They are usually different yarns — the warp stronger, more highly twisted and sized to survive the loom — and they are set at different densities, so a float of the same length in picks is a different length in inches depending on which system it belongs to. And on a warp-faced cloth the warp takes very nearly all the rubbing, so a weft float of six on the reverse of a denim is of no consequence whatever while a warp float of six on the face is a defect.

A specification that means what it says therefore carries two numbers, and the reason it usually carries one is the same reason the limit is an integer rather than a length: one number is easier to check, and the check is being done by somebody who is not the designer.

Checking it, and the mistake the naive check makes #

Verifying a float limit on a drawing is a run-length scan: walk each end down the picks, count consecutive cells of the same state, and report the longest. It is the first thing anybody writes and it has a bug in it.

A repeat tiles. An end whose run of warp-up begins three picks before the bottom of the repeat and continues four picks after the top has a float of seven in the cloth, and a scan that stops at the edge of the repeat reports two floats — one of three and one of four — and passes it. The float has to be measured cyclically, by rotating the column until it starts at a change of state and only then measuring.

This site measures every float that way, in one function that every figure calls, and the reason it is worth mentioning here rather than treating as an implementation detail is that the failure is invisible in exactly the situation a float limit exists for. The drafts most likely to straddle the join are the ones with long runs, which are the ones the limit is meant to catch, so the naive scan is wrong precisely where correctness matters and right everywhere it does not.

Where the model stops #

The float limit is a proxy and the essay would be dishonest not to say so.

What actually matters is whether a float can be lifted, and that depends on the float’s length in inches, on how well its neighbours support it, and on what will be rubbing against it. A float of six in a densely set fine cloth may be perfectly safe where a float of four in a slack coarse one is not. The same conversion appeared two rungs back: a float in picks becomes a length on the cloth only after the sett is known.

The limit survives as an integer because an integer can be checked mechanically on a drawing and a length in inches cannot be checked until the cloth is specified. It is a rule chosen for auditability rather than for correctness, which is a perfectly respectable reason and worth recognising as the reason.

The second thing the limit cannot see is where the float is. A long float on the back of a backed cloth is out of the way and harmless; the same float on the face is the defect. Applying one number to both faces of a two-faced construction is the commonest way a specification asks for something it did not mean.

The constraint read the other way #

Everything above treats the limit as a loss. It is also the reason the small repeats are as rich as they are, and that is worth a paragraph because it explains an oddity in the historical record.

Machinery capable of large repeats arrived late. A drawloom could produce a figured repeat of some size at enormous cost in labour, and the Jacquard mechanism of 1804 made it cheap. What did not change is the float limit, because it comes from wear rather than from machinery — and so the enormous new design space arrived already reduced by a constraint that had always been satisfied automatically at four and eight ends.

That is why the design vocabulary of figured weaving is so heavily built on satin grounds and binding arrangements rather than on the exotic long-float weaves the new machinery made possible. The machinery removed the mechanical constraint and left the physical one standing, and the physical one is the one that had been doing the shaping.

Who worked it out #

Nobody, in the sense of a discovery with a date. The float limit is a piece of accumulated practice, and the interesting thing about its history is how consistent the number is across places and centuries that had no contact.

Weaving manuals from the nineteenth century onward give it as a rule of the workshop, usually without a reason and usually as a range rather than a value: floats of three or four for cloth that will be worn, more for hangings and linings, and a caution about anything longer. The same range appears in the technical literature of the mid-twentieth century, now attached to abrasion data, and in modern jacquard design software as a checkable constraint with a numeric field.

What was never written down, as far as this site has found, is the arithmetic of what the constraint costs — the fraction of the design space it removes at each repeat. It is not a difficult enumeration and it needs no more than a list of compositions and a float count. The likeliest reason it is absent is that the question only becomes natural once weaves are being enumerated rather than chosen, which is a very recent way to think about the subject and is how this whole site reads it.

Where the ladder goes next #

This is the top of the float ladder as it stands. Below it are the float itself, lustre, abrasion and tear, and the constraint here is what all four of them come to in a specification.

The obvious next argument sideways is how many twills a repeat admits, which is the enumeration this essay borrows its denominator from, and which satins are worth weaving, which is the same question of choosing well inside a constrained set.

What the pictures here cannot show. The bars are fractions of a catalogue, and a catalogue is not a preference. Nothing here says the excluded twills were the good ones or the bad ones; the enumeration counts them and cannot rank them. A designer who loses two-thirds of the twills on twelve ends may have lost nothing worth having, and no figure on this page can say either way.