The weave decides the sett, and two models disagree about it

There is a maximum number of threads that will go into an inch of cloth, and it is not a property of the yarn alone. A weave that interlaces often forces its threads to bend often, a bend takes room, and the room comes out of the space available for other threads.

That much has been settled on this site twice already — once as a limit and once as a trade-off against firmness. What neither of those said is that there are two quite different ways of computing the number, that they disagree by fifteen per cent on the simplest cloth there is, and that the one which disagrees upward is the only one that can be asked about a satin at all.

The first model: count the intersections #

The rule usually attached to Thomas Ashenhurst, who set it out in the 1880s, is a piece of accounting so simple it can be written in a line.

Consider one repeat of the weave, and one warp end’s worth of width. The repeat contains some number of ends, each of which occupies one diameter. It also contains some number of places where a pick has to pass from one side of the warp to the other, and at each of those the crossing thread needs a diameter of room to get through. So the width the repeat must occupy is

$$w = d\,(\text{ends} + \text{crossings}),$$

and the sett is the number of ends divided by that width. The same accounting with the roles exchanged gives the picks.

It is not a mechanical model and it does not pretend to be. It is a statement that thread takes room and that a crossing takes room, added up, and the whole of the weave enters through one integer: how many times the two systems change places in a repeat.

Its great virtue is that the integer is available for any draft whatever. Hand it a satin, a herringbone, a double cloth or one of the twenty-two thousand four-by-fours nobody has a name for, and it will count the crossings and return a number. Nothing else in this subject generalises so cheaply, and that is why the rule outlived everything that was supposed to replace it.

The second model: solve the geometry #

Peirce’s 1937 geometry is a different kind of object. It writes down where the threads actually are.

Each thread runs straight between crossings and turns through a circular arc around the thread it crosses, so a repeat is described by the thread spacings, the lengths of thread in it, the crimp heights and the weave angles. Adding the closure condition — that the two crimp heights sum to the thickness of the cloth, because the two threads between them fill it — gives a system that can be solved rather than estimated.

Jamming is the state in which the straight portions have vanished entirely and the threads are in contact along their arcs. Imposing that on the equations fixes the weave angle at exactly sixty degrees, which is not an assumption but a consequence, and the thread spacing follows: $p = 2d\sin 60°$, so the sett is $1/(2d\sin 60°)$.

What they say about a plain weave #

Both models can be asked about a plain weave, and the answers are not the same.

In a yarn a fortieth of an inch across, the intersection count gives twenty threads to the inch: two ends per repeat, two crossings, four diameters of width for two threads. Peirce’s jam gives $0.577/d$, which is 23.1 threads to the inch.

Fifteen and a half per cent apart, on the cloth both models were designed for, with the same yarn and the same definition of what jamming means.

The disagreement is not a mistake in either. It is a disagreement about what happens at a crossing. The intersection count charges a full diameter for the crossing thread to pass, as though the two threads sat side by side. Peirce’s geometry lets the crossing thread nest into the space between its neighbours, so it costs less than its own width — and the sixty-degree angle is exactly the statement of how much less.

Peirce is the better model of the plain weave, and it is right to be higher. Threads do nest. A rule that charges full width for every crossing must under-estimate, and it does.

Why only the crude model reaches the satins #

The obvious response is to use the better model everywhere. It cannot be done, and the reason is structural rather than a matter of effort.

Peirce’s closure condition is a statement about a plain weave: every crossing is an interlacing, every thread turns at every crossing, and the two crimp heights fill the thickness. In a twill or a satin a thread runs straight past most of its crossings and turns at a few, so there is no single weave angle, no single crimp height, and the condition that closes the system does not apply.

The standard workaround is to treat a weave of $n$ ends with $k$ interlacings as a plain weave with the spacing inflated by some factor — Brierley’s exponent and its several successors — and every such factor is fitted rather than derived. The moment that step is taken, the “better model” has become an intersection count with more decimal places.

So the honest position is the uncomfortable one. For the plain weave there is a solved geometry and it should be used. For everything else there is an accounting rule, and the numbers in the right-hand column of the first figure are that rule’s numbers, and they are low by something like the fifteen per cent the plain weave shows and possibly by more.

Three different cloths, one number #

The intersection count’s blindness has a sharper demonstration than a percentage, and it is visible in the site’s own table.

A 2/2 basket, a 2/2 twill and a 3/1 twill all return 26.7 threads to the inch in the same yarn. They have the same repeat and the same interlacing count, so the rule cannot distinguish them — it has been given only one number about the weave and there is only one number to give back.

Those are not the same cloth. In a basket the interlacings are adjacent, so the pairs of threads lie side by side and the crossings are grouped; in a 2/2 twill they run on a diagonal; in a 3/1 they are one in four and the cloth is heavily warp-faced. Their handles differ, their drape differs, and there is no reason at all to expect their jamming setts to agree to three significant figures.

They agree because the model was asked a question with one input. That is a limitation worth naming as a limitation and not as a finding: the rule does not claim to see where the interlacings are, only how many there are.

The accounting rule in one line, and the nesting as a discount #

Both models are quoted above as procedures. Written as formulae they become comparable, and the comparison says exactly what the fifteen per cent is.

The intersection count charges one diameter per end and one per crossing, so for a repeat of e ends carrying c crossings the width is d(e + c) and the sett is

1 ÷ (d(1 + f)), where f = c/e is the weave’s interlacing fraction.

One line, one integer ratio, and every weave in the table falls out of it. A plain weave has f = 1 and reaches 1/(2d); an eight-end satin has f = 0.25 and reaches 0.8/d.

That form makes a connection this collection uses elsewhere and never derives. The fraction of full cover a weave can reach is 1/(1 + f) — a half for a plain weave, four fifths for an eight-end satin — which is exactly the pair of fractions the cover-factor ceilings are computed from. The fourteen and the 22.4 on that scale are 28/(1 + f) at f = 1 and f = 0.25, and the count cancels because both the covering sett and the jamming sett go as one over the diameter.

Now put Peirce’s answer into the same form. His jammed plain weave sets at 0.577/d, which is 1/(1 + f) with f = 0.732 rather than 1. So the whole disagreement can be stated as a discount on the crossing charge: a crossing costs 0.732 of a diameter rather than a full one, and 0.732 is 2 sin 60° − 1, which is the nesting written as a price.

That is worth having because it is a single number rather than a percentage attached to one cloth, and it invites the obvious extrapolation:

The right-hand column is an extrapolation and is labelled as one. The 0.732 is derived for a plain weave, where every crossing is a turn and the nesting is the same at every one of them; in a satin most crossings are straight travel and the thread nests differently, or not at all. There is no reason the discount should carry across unchanged and every reason to expect it to shrink as the interlacings thin out, since a thread with nothing to nest between has nothing to save.

So the column is offered as the shape of the correction rather than its size — it says the accounting rule is low everywhere, that the shortfall is largest where the interlacings are densest, and that the ordering across weaves survives either way. Which is the same conclusion the rest of this essay reaches, arrived at with one number instead of two procedures, and with the place a better model would have to start clearly marked.

Which direction jams first #

Both models above have been written as though a cloth had one sett. It has two, and they jam separately.

The accounting rule gives the warp sett from the ends in the repeat and the crossings a pick has to make, and the weft sett from the picks and the crossings an end has to make. On a balanced weave those are the same and the distinction is invisible. On an unbalanced one they are not, and the consequences are the ones a weaver meets in practice.

A cloth is normally set with the warp near its limit and the weft well below, because the warp is fixed by the reed before weaving starts and the weft is put in one pick at a time and can be stopped. So the binding constraint in a real cloth is usually the warp, and the weft sett is a choice. That asymmetry is not in either model; both compute a symmetric pair of ceilings and neither knows which one a loom will run into first.

It also means that “the cloth is jammed” is ambiguous in the same way the twill angle is. A fabric can be warp-jammed and weft-open, which is what a poplin is, and calling it jammed without saying in which direction states a quarter of the situation.

What was counted, and how #

The interlacing counts come from walking the matrix: a crossing is an interlacing when the state at that intersection differs from the state at the next, in the direction being counted, taken cyclically so the wrap is included.

The setts come from the accounting rule with those counts substituted in, and the figures assert the ordering rather than reporting it: the weave that interlaces most must be the weave that sets most openly, and the sequence must be monotone across the whole set. A model that ever reversed that order would be broken in a way no single number would reveal.

Peirce’s jam is solved separately, and its own consistency is checked by putting the solution back through the equations it came from before anything is drawn from it. The sixty-degree weave angle is asserted rather than typed, so a change to the closure condition would show up as a failed assertion rather than as a slightly different number.

Neither model is allowed to borrow from the other anywhere on this site. Where a figure needs a maximum sett across weaves it uses the accounting rule and says so; where it needs a jammed plain weave it uses Peirce and says so. Blending them would produce a third model nobody had validated, which is the failure the section models essay exists to warn about.

What the disagreement means for a specification #

Fifteen per cent is not a rounding error in this subject. It is the difference between a cloth that runs on the loom and one that does not.

The practical consequence is that neither number should be used as a target. A maximum sett is a ceiling, and cloths are woven at some fraction of it — commonly sixty to eighty per cent, depending on what the fabric has to do — so the fraction absorbs the disagreement between the models as long as the same model is used consistently. A mill that changes its rule and keeps its fraction has quietly changed its cloth.

What the number is genuinely good for is comparison. Both models agree that fewer interlacings allow a denser setting, and both agree on the ordering across weaves. The ordering is robust and the absolute value is not, which is a common shape for a geometric model and worth recognising as the shape rather than as a defect.

There is a second thing it is good for, which is spotting an impossible specification. A sett quoted above any of these ceilings is not an ambitious cloth, it is arithmetic that went wrong somewhere upstream — a count written in the wrong system, a diameter taken as a radius, a metric figure read as an imperial one. Every figure on this site that takes a sett refuses one past the jam for exactly that reason, and the refusal has caught more than it has obstructed. A ceiling that is fifteen per cent uncertain is still a perfectly good detector of a number that is out by a factor of two.

Who worked it out #

Thomas Ashenhurst was a Bradford weaving teacher and his Treatise on Textile Calculations and the Structure of Fabrics went through several editions from the 1880s. The rule as it is quoted today — threads plus intersections, divided into the width the diameter allows — is his, and it is stated there as arithmetic for the weaving shed rather than as a model of anything. It was, and is, extremely useful, and it survives in mill practice essentially unchanged.

Peirce’s 1937 paper in the Journal of the Textile Institute, The Geometry of Cloth Structure, is a different exercise entirely: an attempt to describe the thread path exactly enough to derive crimp, thickness and cover rather than to estimate them. It is the foundation of everything that came afterward, and it is confined to the plain weave for the reason given above.

The gap between them has been filled repeatedly and never closed. Brierley’s work in the 1930s introduced a weave factor with a fitted exponent; later authors proposed others; and the modern textbooks give a table of factors by weave with the honest note that they are empirical. Fifty years of effort has produced better numbers and no second solved geometry, which suggests the difficulty is real: a weave with unequal float lengths has no single thread path to solve for, and the moment an average is taken the exactness is gone.

That is a good outcome to record on a site whose method is to compute rather than to quote. The matrix settles what is decidable from the matrix, exactly and without a fitted constant anywhere. The sett is not one of those things, and pretending otherwise would be the over-claim this site’s own rules forbid.

Where the ladder goes next #

Below this rung are the maximum sett itself and the two section models, which is the same disagreement about a different quantity.

Sideways, firmness is the interlacing count read as a handle rather than as a limit, and the cover factor is what a sett becomes once it is divided by the yarn it is set in.

What the pictures here cannot show. A jamming sett is a state no photograph can settle, because a cloth at ninety-eight per cent of its jam and one at a hundred look identical and behave differently on the loom. Every number on this page is a model’s answer, both models are named where they are used, and the fifteen per cent between them is the honest width of the answer rather than an error in either.