The reed leaves its own mark

Every sett quoted on this site so far has been an average. A cloth described as twenty ends per centimetre has been treated as a cloth whose ends are half a millimetre apart, and no cloth has ever had that.

The reason is the reed. A reed is a comb of flat wires with gaps between them, and it does two jobs: it beats each pick up against the cloth, and it holds the warp at its width. Ends are drawn through it several to a dent — two, three, four, sometimes eight — and inside a dent there is nothing separating them.

What the reed actually does to a warp #

The arithmetic is the same one a mock leno runs, and for the same reason: the reed sets a group’s total width and nothing sets the spacing inside it.

A dent of pitch P takes m ends of diameter d. Inside the gap the ends lie touching and occupy m·d; the difference P − m·d is the gap at the wire. So the warp is a grouped grid, and the largest space between neighbouring ends is the gap plus a diameter while the smallest is a diameter.

At four ends to a two-millimetre dent with a 0.25 mm yarn the ratio between the widest spacing and the average is 2.5. At six to a three-millimetre dent it is 3.5. Coarser yarn helps and more ends per dent hurts:

The conservation is exact and the generator asserts it in the form that can fail: each end must sit exactly one dent pitch from its opposite number in the next dent. The tempting check — measure from the first end to the last and compare with the flat spacing — is short by one gap, because the grouping pulls both outer ends inwards, so it fails on a correct layout.

The refusal is the other boundary. Eight 0.3 mm ends will not go into a 2 mm dent, and the arithmetic says so rather than returning a negative gap. A reed is bought with a dent count for a reason, and the count is a statement about the coarsest yarn it will take.

That refusal is a different limit from the cloth’s own jam. A warp can pass through a reed that its finished sett would not admit, because the reed only has to hold the ends side by side and the cloth has to accommodate the wefts crossing between them — so the reed’s limit is one diameter per end and the cloth’s is nearly two. A reed that a warp fits through is not evidence that the cloth can be woven.

Why the cloth does not usually show it #

Every cloth in the world is woven through a reed and most of them look evenly spaced, which needs explaining before the failures do.

The grouping is a warp-only phenomenon at the moment of weaving, and the warp is under tension and being beaten. As the cloth is taken up and leaves the reed, the ends move: they are pulled towards even spacing by the wefts crossing them, which want to lie straight, and by whatever the finishing does. How completely they even out is a friction question the site has no model for, and the answer in practice ranges from completely to not at all.

What decides it is how consistently the grouping is reinforced. And that is where the beat comes in.

The beat is a divisor rather than a subtraction #

Consider a cloth with a weave repeat of R ends, dented m to a dent.

If m is four and R is four, then the first end of every dent is the first end of every repeat. Every dent contains the same four columns of the draft in the same order, so whatever the grouping does to the appearance, it does the same thing to every repeat — the effect is reinforced identically all the way across the cloth and it shows up as a stripe at the dent pitch.

If m is three and R is four, the dent boundaries walk through the repeat: the first dent starts at column zero, the second at column three, the third at column two, and it takes twelve ends to come back into step. Whatever the grouping does, it does twelve different things, and there is nothing periodic at a visible scale for the eye to find.

The period is the least common multiple of m and R, which is mR divided by their greatest common divisor. So the rule is one word:

Dent so that the ends per dent share no factor with the weave repeat.

What was counted, and how #

The period is computed two ways and the two must agree.

The divisor route is mR/gcd(m, R), one line. The walking route steps end by end and looks for the first position at which both the dent boundary and the repeat boundary come round together. They are asserted equal for every cell of the table, which is worth doing because the divisor formula is the kind of thing that is remembered slightly wrong and produces a plausible number.

The census over a repeat then counts how many dentings are coprime, and two structural facts fall out of it that a weaver can act on.

An even repeat loses every even denting. Two, four, six and eight all share a factor with an eight-end repeat, and two, four and six are exactly the dentings a mill uses. So the commonest weave repeats — four and eight — are the ones whose usual dentings are the marked ones.

A prime repeat loses only its own multiples. A five-end repeat is coprime with every denting except five and ten. That is a real reason to prefer a five-end satin over an eight-end one in a fabric where reediness would show, and it is not a reason anybody gives.

The site’s own satin arithmetic has been about coprimality since its earliest essays — a satin exists exactly when its move is coprime with its repeat — and this is the same relation appearing in a different place for a different reason. A satin needs coprimality so that its interlacings scatter; a denting needs it so that its grouping does. Neither has anything to do with the other and both are gcd of two small numbers.

How many dentings a repeat leaves safe #

The census in the table has a closed form, and it names the repeats a weaver should be wary of before any denting has been chosen.

Over a full period of dentings the fraction coprime with a repeat R is Euler’s totient of R divided by R — the same function the satin arithmetic counts moves with, appearing here for the unrelated reason that both questions are a greatest common divisor. So the proportion of safe dentings is a property of R’s prime factors and of nothing else.

That ranks the ordinary repeats sharply. A repeat of five leaves four fifths of dentings safe, seven leaves six sevenths, nine leaves two thirds. Eight leaves a half. Twelve leaves a third, which is the worst of the common repeats, and twelve-end repeats are exactly what fancy twills and small damasks are written on.

A repeat with many small prime factors is a repeat with few safe dentings, and the two smallest primes are the two that matter, because the dentings a mill actually uses are two, three and four.

Six is the repeat with nowhere to go #

Narrow the question to the dentings a mill will really consider — two, three and four ends per dent — and the answer collapses to a single divisibility test.

Two and four are ruled out whenever the repeat is even. Three is ruled out whenever the repeat is a multiple of three. So the repeats with no safe denting among the practical ones are exactly the multiples of six.

That is a short list and an awkward one. A six-end repeat is common: several fancy twills, the smaller relief weaves, and any construction built on a 1/5 or 2/4 base. A twelve-end repeat is common in figured cloths for the reason the previous section gives. Both of them force a weaver to choose between a marked denting and one — five ends per dent, or one — that nobody wants for other reasons.

The rest of the space is comfortable. Four and eight both take three ends per dent; nine and any odd repeat take two or four; ten takes three. It is only when both two and three divide the repeat that every practical denting lands in step, and a weaver meeting a persistent reediness complaint on a six-end or twelve-end cloth has met a constraint rather than a bad choice.

That is a rule a mill can act on before the reed is ordered, and it is one the folklore cannot state, because “use an odd denting” is a statement about parity and this is a statement about two divisors at once.

It is worth adding what the rule does not settle. A repeat of six with no safe denting is not a cloth that will certainly show a mark; it is a cloth whose grouping will be reinforced consistently, and whether that reaches the surface still depends on everything the last section lists. What the divisibility test does is separate the cloths where the question is open from the cloths where it is closed, which is the most a piece of arithmetic can do about an effect whose visibility nothing here computes.

The other reed decision, and it is the same arithmetic #

A weaver chooses two things when specifying a reed and only one of them is the denting.

The other is the reed number: how many dents per unit width, which fixes the dent pitch. Between them the two decide the sett — sett equals dents per centimetre times ends per dent — and the same sett can be reached many ways. Twenty ends per centimetre is five dents at four, four dents at five, ten dents at two, or two and a half dents at eight.

Those are not equivalent cloths. The finer the reed, the smaller the group and the less there is to even out; the coarser the reed, the larger the gap the ends have to close and the more the beat matters. So the choice is a trade between the reed’s own cost and durability — a fine reed has more wires, each thinner, each more easily damaged — and how much grouping the fabric can tolerate.

The interesting part is that the beat depends only on the denting and not at all on the reed number. A four-ends-per-dent threading on a four-end repeat leaves a mark at whatever pitch the reed happens to have, because the period is four ends however wide four ends are. So the two decisions separate cleanly: the reed number sets how large the mark is, and the denting sets whether there is one. A weaver who changes to a finer reed at the same denting has made the mark smaller and not removed it, which is exactly the experience the trade reports.

Where the mark actually comes from #

The beat says when a grouping is reinforced consistently. What it is reinforcing is a separate question, and there are three answers that the trade lumps together as “reediness”.

A spacing stripe. The ends really are grouped in the finished cloth, so the cover varies periodically across the width, so the cover varies and the fabric is more transparent at the dent pitch. This is the one the arithmetic above describes directly.

A crimp stripe. Ends at the edge of a group are pushed by their neighbours on one side only, so they take a slightly different path through the cloth from the ends in the middle, and a balanced cloth is only balanced on average — a different crimp, therefore a different take-up and a different tension, therefore a slightly different lustre. This one shows in cloths where the grouping has evened out completely and is the more persistent of the two.

Wire marks. The reed wire rubs the outermost end of each dent, over kilometres of warp, and abrades it — which is the same abrasion argument the site makes about floats, applied to a thread that is rubbed by a machine rather than by the world. That is not periodic against the weave at all — it is periodic against the dent — and no denting choice avoids it. It is why reeds are polished and why a damaged wire produces a single line down a whole piece.

Only the first two beat against the repeat. The third is the reason a mill inspects reeds rather than only calculating dentings.

Where the model stops #

Nothing here says how much of the grouping survives. The geometry gives the spacings at the reed and the beat gives the period; whether the finished cloth shows anything depends on friction, on finishing, on the sett relative to the jam, and on the fibre. A cloth close to its jammed sett cannot even out because there is nowhere to go, and a very open one evens out completely. The cover factor is the right variable for that question and it is not in this arithmetic.

The threads are taken as rigid cylinders in the dent. Real ends flatten against each other and against the wire, so a group is a little narrower than m·d and the gap a little larger.

The weft is ignored entirely. A pick beaten up against a grouped warp is not straight; it bows around the groups, and that bowing is part of what fixes the grouping in place. Modelling it needs the pick’s own bending stiffness, which this site does not have.

And the visibility threshold is not computed. The table says a mark repeats every so many ends; whether a variation of a few per cent in cover at a two-millimetre pitch is visible depends on the contrast, the lighting and the observer. The arithmetic settles which choices can produce a periodic mark and says nothing about which do.

Who found it, and when #

Reed marks are as old as reeds and the remedy has been folklore for as long. Weaving manuals say to avoid denting “in step with the pattern”, to use an odd number of ends per dent, or to change the reed — and the reasoning, where any is given, is that the pattern and the reed should not “line up”.

That is the right instinct and it is one divisor short of a rule. Odd is not the criterion: three ends per dent is odd and coprime with four, and it is also coprime with eight, and with sixteen — which is why the advice works for the commonest cases. It fails for a repeat of three or nine, where three ends per dent is the worst possible choice and an even denting of two or four is fine.

The general statement — gcd, not parity — takes one line and appears nowhere in the trade literature this site has been able to find. It is not a hard result. It is the kind of thing that stays folklore because the folklore is right ninety per cent of the time, and the ten per cent shows up as a piece of cloth with a stripe in it and no explanation.

Where the ladder goes next #

Both rungs of this anchor have been about two periodic things interfering. The next one this field needs is the other half of the same subject and it is not written: what a colour order does when it beats against a weave repeat, which is colour and weave taken from the pattern side rather than the surface side, and which has the same divisor arithmetic underneath it with a different consequence on top.