Interlacings and firmness

Plain weave is the firmest structure there is and the one that has to be woven most openly. Those two statements sound as if they contradict each other, and the reason they do not is one sentence: a thread that changes face has to bend, and a bend takes room.

Counting interlacings #

An interlacing is a place where a thread changes face — where a run of warp-up meets a run of warp-down. Counting them is the complementary operation to counting floats: every float is bounded by two of them, so the two numbers are two readings of the same structure.

For a repeat, the count is made round the torus, like everything else here. Walk each column and add one wherever consecutive picks differ; walk each row and do the same. The total is the interlacings per repeat, and dividing by twice the number of intersections gives a firmness between zero and one that can be compared across weaves of different repeat sizes.

Plain weave scores one: every intersection is a change. An eight-end satin scores an eighth. A 2/2 twill scores a half. The number falls monotonically as the floats lengthen, which is what it means for the two to be readings of the same thing.

Why a bend takes room #

Now the mechanism, which is worth doing carefully because everything else follows from it.

Consider a warp end passing over one weft and under the next. Between those two crossings it must travel from one side of the weft plane to the other, so it is displaced by roughly the sum of the two yarn diameters over the distance between the crossings. That displacement has to happen somewhere, and the somewhere is horizontal space.

Now crowd the ends closer. The distance available for each transition shrinks, so the thread has to make the same displacement in less room — which means bending harder. Eventually the bend is a right angle and the thread is running vertically between crossings, at which point the neighbouring threads are touching and nothing further can be pushed in. The cloth is jammed.

A weave with fewer interlacings has fewer transitions to find room for. Its threads run straight for several crossings and only have to move at the ends of a float, so the same number of threads occupies less width — or, equivalently, more threads fit in the same width.

The count that follows #

Turning that into a number requires a model of the yarn, and the simplest useful one is a circular section of fixed diameter — which is what Peirce assumed in 1937 and what every later model relaxes differently.

On that model, the width a repeat must occupy is the threads themselves plus the room their transitions demand, and the maximum sett follows directly. The figure above is that calculation for a yarn a fortieth of an inch across: plain weave jams at about twenty threads per inch, a 2/2 twill at about twenty-seven, an eight-end satin at about thirty-two.

Sixty per cent more threads, in the same yarn, from nothing but the weave. That is not a marginal correction; it is the dominant term in how heavy a cloth is for its fineness.

The classical version of this calculation is Ashenhurst’s, from the 1880s, and it is still the form quoted in mill practice. Various refinements exist — Brierley’s, which handles unbalanced weaves better, and Peirce’s own geometry, which is more careful about what the thread actually does at a crossing. They disagree with each other by ten or twenty per cent and agree completely about the ordering, which is the useful part.

The two directions again #

A subtlety that a single firmness number hides.

An interlacing costs room to both threads involved, but not equally. A warp end changing face has to travel round a weft pick, and the pick has to travel round the end. In a balanced weave those demands are symmetric and the single number is fair. In an unbalanced one they are not.

A 3/1 twill has warp floats of three and weft floats of one, so the weft changes face at every intersection while the warp changes at one in four. The weft is therefore doing nearly all the bending, and the constraint on how densely the cloth can be set falls almost entirely on the warp direction — which is why denim is warp-dense and weft-open, and why its two thread counts are so different.

This is the second reason a single thread count is a poor description. A fabric quoted as “200 count” may be 120 ends and 80 picks or 100 and 100, and in an unbalanced weave those are quite different cloths made from quite different amounts of yarn.

Firmness is not strength #

The word invites a confusion worth heading off.

A firm cloth is one in which the threads are held where they are. Push a plain weave sideways and it resists; the crossings grip, and a thread cannot slide relative to its neighbours without a great many other threads moving too. Push a satin and it gives, because each thread is gripped rarely.

That is a statement about mobility, not about breaking. And the two run opposite ways in one important case: a satin, whose threads can slide, is harder to tear than a plain weave, because at a tear tip the threads bunch and share the load rather than breaking one at a time. The float essay works that through.

So firmness buys stability of shape, resistance to threads being pulled out, resistance to seam slippage, and a crisp hand. It does not buy tear strength, and it costs drape.

What the interlacing count predicts about crimp #

There is a second consequence of the same mechanism, and it is measurable rather than descriptive.

A thread only deviates from straight where it changes face. So the extra length a thread carries — its crimp — is bought entirely by interlacings, and a weave with a quarter of the interlacings has roughly a quarter of the crimp.

That ordering is not asserted here; it comes out of measuring the drawn path against the width it spans, and the figures on this page report what they measured. The ordering is plain weave, then twill, then satin, in the same order as everything else.

It has a practical consequence people meet without knowing why. A plain-weave shirt gives a little at the shoulders and a satin lining does not, because a plain weave has crimp available to be pulled out and a satin has almost none. The give is a rearrangement rather than a stretch, and it is bounded by exactly how much crimp the weave put there in the first place.

The three things firmness decides #

Seam slippage. A seam pulls on the threads next to it, and in a loosely interlaced cloth those threads can migrate away from the stitching, leaving a gap. Satins and loosely set twills are prone to it and plain weaves are not, which is why lining fabrics need generous seam allowances and why upholstery is rarely satin.

Fraying. A cut edge frays when threads slide out of it, which needs mobility and friction working against each other. A firm cloth frays reluctantly; a satin frays enthusiastically.

Handle. A firm cloth feels crisp and a loose one feels fluid, and a great deal of what people mean by fabric quality is this one axis. It is worth knowing that it is available in any fibre, since it is structure rather than material.

Firmness and the cloth that falls apart #

One connection worth drawing, because it links this essay to the check the site is built around.

The census of every four-by-four draft found that the ones describing more than one cloth cluster at long floats — which is to say, at low interlacing counts. Nothing separable appears among the drafts whose longest float is one or two.

That is not a coincidence. Integrity is a question about whether threads are tied down, and interlacings are exactly the places where tying down happens. A structure with very few of them has few opportunities to bind its threads into one system, so it is the structure most at risk of binding them into two.

So firmness is not only about handle and setting. At the extreme of low interlacing, it is about whether the cloth exists at all — and the extreme is exactly where a designer chasing lustre or drape is working. That is why every weave figure on this site prints its layer count beside its float and interlacing numbers rather than treating integrity as a separate concern.

The setting a weave actually gets #

Maximum sett is a ceiling rather than a specification, and cloths are rarely woven at it.

A fabric set at its jamming point is as dense as it can be, which sounds desirable and often is not. It has no room to shear, so its bias is dead and it drapes badly; it is stiff; it takes a great deal of yarn; and it is hard to weave, because the reed has to force the picks into a space that barely accepts them.

Real setts are usually quoted as a percentage of the maximum — sixty to eighty per cent for clothing, higher for a cloth that must be windproof or waterproof, lower for something meant to drape. The maximum is a reference point that makes those percentages comparable across weaves, which is exactly what a raw thread count cannot do.

That is another reason thread count is a poor measure: the same count is a loose cloth in a satin and an impossible one in a plain weave, and the number alone does not say which.

Where the number came from #

Ashenhurst’s rule is Victorian mill mathematics, and its history is a good illustration of what a practical theory is for.

Thomas Ashenhurst was a Bradford weaving instructor, and his setting theory of the 1880s answered a question a mill actually had: given this yarn and this weave, how many ends should be put in the reed? Before it, the answer was experience — a foreman’s judgement, unwritten, unteachable and unavailable to a new cloth. Ashenhurst turned it into arithmetic that a student could apply to a weave nobody had made.

The theory is crude by modern standards and it is still taught, because it has the property good engineering rules have: it is wrong by a consistent margin, in a known direction, for a comprehensible reason. It treats the yarn as a circle of fixed diameter and assumes it does not flatten, so it overestimates the room a crossing needs and gives a maximum sett below what a mill can achieve. Knowing that, a practitioner applies a correction and gets on with it.

Peirce’s 1937 geometry is the more careful treatment, and it is the one that made the subject quantitative: circular sections still, but with the thread path modelled as arcs and straight lines, and with the relations between crimp, spacing, diameter and cloth thickness worked out properly. The later racetrack and elliptical models relax the circular assumption in different ways and disagree with each other by ten or twenty per cent.

All of them agree on the ordering in this essay, which is why the ordering can be asserted here and the absolute numbers cannot be. The model that produced any number is named wherever one appears, because two sources quoting different setts for the same cloth are usually not disagreeing about the cloth.

Firmness at the sett each weave can actually reach #

The three things firmness decides are all listed above at one construction, and that is not how cloths are made. A satin is not woven at a plain weave’s sett; it is woven at its own, which is up to sixty per cent denser — and the extra threads bring extra crossings with them.

So the quantity that decides seam slippage and fraying is not the interlacing rate but how many gripping crossings a thread meets per centimetre of its own length, which is the sett times the rate. Working it out at each weave’s own ceiling changes the ordering’s size considerably:

Compare the last column with the interlacing rates it came from — 1, ½, ⅖, ¼. The eight-end satin’s disadvantage falls from a factor of four to a factor of two and a half, and the recovery is exactly the sett ratio, because that is the only thing that changed.

That is worth stating as a rule rather than a table:

a weave’s firmness at its own densest setting is its interlacing rate times its setting ceiling, and the two partly cancel.

They cancel because both come from the same place. A weave with few interlacings has few places to grip and room for more threads, and the room is bought with precisely the interlacings it gave up. Nothing is free in either direction, and the net is a real but much smaller spread than the interlacing count alone suggests.

Which corrects two of the three practical claims #

The three consequences the essay lists all inherit the correction, and two of them are usually stated in the uncorrected form.

Seam slippage. A fully-set satin resists thread migration two and a half times worse than a fully-set plain weave, not four times. That is still a large factor and it is a different design allowance.

Fraying. Same ratio, same reason — a thread escaping a cut edge has to disengage from every gripping crossing along its length, and there are 40 per cent as many in a satin rather than 25.

And tear strength runs the other way with the same number. Tearing needs threads to be free enough to bunch at the tip, so the resistance goes as the reciprocal: a satin should tear about two and a half times as well as a plain weave of the same yarn, which is comfortably inside the two-to-three-fold advantage the trade reports.

One number, three orderings, two of them corrected and one of them predicted.

The caveat is that a count of crossings is not a grip. This collection’s own arithmetic makes the hold at a crossing exponential in the wrap angle through a capstan, and a satin’s lower crimp means a smaller wrap at each of its fewer crossings — so the true spread is wider than 2.5 and narrower than 4, and where it lands depends on a coefficient of friction. The sett compensation is exact and the crossing-count proxy is not, which is the usual division here between what geometry settles and what friction decides.

Where the model stops #

Three limits, and the first is the one that matters most.

This is geometry, not mechanics. The jamming calculation says when threads touch. It says nothing about the force needed to get them there, nothing about how much they flatten when they do — and real yarns flatten a great deal — and nothing about friction. A mill’s practical maximum is below the geometric one for exactly those reasons.

Circular sections are a fiction. A yarn in cloth is not round; it is squashed where it is gripped and rounder where it floats. The racetrack and elliptical models take that seriously and give different numbers. Where this site quotes a sett it says which model produced it, because the alternative is a permanent disagreement between sources that are each internally consistent.

Finishing moves the answer. Cloth shrinks when it is washed and relaxed, and a fabric can leave the loom below its jamming point and arrive at the shop above it. The sett on the loom and the sett in the hand are different numbers.

What else the count decides #

Two consequences of the interlacing count that belong elsewhere on the site and are worth naming here, because both of them run in the direction a reader would not guess.

Bending stiffness. A cloth bends most easily when its threads can slide past one another, and interlacings are what stops them. So the weave with the most interlacings is the stiffest and the one with fewest is the limpest — plain weave against satin, in the same yarn at the same sett. The ordering is predictable from the matrix and the magnitude is not, because the resistance is friction. Bending stiffness and the drape coefficient takes it up.

Section shape. A thread is squashed flat by the threads crossing it, and it is squashed harder the more often they cross. So the interlacing count feeds back into the yarn’s own geometry, which is one more reason a cloth’s thickness cannot be computed from its yarn count alone — and why the section models disagree about thickness far more than about anything else.

Where the ladder goes next #

The geometric side of this is how close threads can be set, where the models are compared rather than assumed.

The measurement it is usually confused with is thread count, and the quantity that does mean what thread count is taken to mean is cover.

And the mechanical consequence of leaving room in the cloth is the bias, which needs space between the threads and disappears when there is none.

What the pictures here cannot show. Every figure on this page is a structure with an assumed yarn diameter, and none of them draws a real yarn. Flattening, hairiness, twist and friction are all absent, and they are between them the reason a mill’s numbers differ from a geometric model’s.