A cloth slips at its least-interlaced thread

A weave’s firmness is its interlacing count: every time a thread changes face it bends round the thread it crosses, the bend takes room and the crossing grips. Plain weave changes face at every crossing and scores one; a 2/2 twill scores a half; an eight-end satin an eighth. The number predicts how closely a cloth can be set, how crisp it feels and how readily its threads slide, and it is quoted as a single figure for the whole weave.

For the weaves it was built on, one figure is exact. Every thread of a plain weave, a twill or a satin interlaces as often as every other, so the average thread is every thread. For almost any other draft it is not. A stripe, a figure, a broken twill or an irregular draft has threads that interlace more and threads that interlace less, and the average describes a thread that is not in the cloth.

A cloth does not fail on average. A thread pulled through the cloth beside a seam or out of a cut edge is held by its own crossings and nobody else’s, and the one with fewest goes first. So the number that decides whether a cloth slips is the least-interlaced thread’s, and for most drafts it sits a long way below the average — in every four-by-four draft but plain weave, at the lowest value a thread can have and still be part of the cloth.

The average thread is a thread nobody wove #

A thread’s interlacings round a repeat are always an even number, because a thread that changes face has to change back before the repeat starts again. A thread that interlaces at all has at least two. In a repeat four picks long that leaves exactly two values, two and four: a thread either floats once and returns, or alternates at every pick.

So a four-by-four draft’s threads each interlace twice or four times, and its average can be anything between. Plain weave has every thread at four. A 2/2 twill has every thread at two. Every other draft mixes them, and its firmness number — its total interlacings over its crossings — lands somewhere in the middle, describing a thread that interlaces three times, or two and a half, or three and a half, which no thread in the repeat does.

Take the satin stripe of the essay on where heddles go: forty ends of plain weave and eight of an eight-end satin, repeated. Its firmness number is 0.87, close to plain weave’s one, and it describes the cloth’s forty plain ends reasonably well. Its eight satin ends interlace at 0.25 per crossing, a quarter of that. The number on the specification sheet belongs to the stripe’s ground; the stripe itself is four times weaker than it says.

Seven drafts in ten have a thread below their own average #

The four-by-four drafts can all be counted, and the count says how ordinary the stripe is. Of the 22,874 four-by-four drafts in which every thread interlaces, 15,920 — seven in ten — have a thread that interlaces less often than their own average thread, in the warp or the weft. Only 6,954 have every thread alike in both directions.

The shortfall comes in three sizes. In 11,328 drafts the weakest thread interlaces at four fifths of the average in its direction; in 4,048, at two thirds; in 544, at four sevenths — two interlacings against an average of three and a half.

And one fact covers all of them. In every one of the 22,874 drafts except plain weave’s two, some thread interlaces exactly twice a repeat. Drafts whose totals run from sixteen interlacings to twenty-eight — firmness numbers from a half to seven eighths — all have a weakest thread at the same floor. Measured by the thread that slips first, the whole family of four-by-four drafts except plain weave is equally firm, and exactly as firm as a 2/2 twill.

One draft, four threads, two firmness numbers #

A single draft shows how the average hides it. In a four-by-four draft whose rows are an alternation, its opposite twice over and a single mark, the warp ends all interlace twice a repeat. Three of the picks interlace four times — every crossing — and the fourth pick twice.

The weft’s average is therefore seven eighths a crossing, and the draft’s firmness number, both directions together, is 0.69. The fourth pick interlaces at a half. It is the least firm thread in the cloth by a wide margin, it runs the full width, and nothing in the one-number description suggests it is there.

This is the shape of a figure that is not a stripe as well: a figured cloth combines two weaves in blocks, and every thread that passes through a block of the looser weave carries that weave’s interlacing rate across it, while the cloth’s average is weighted by how much of each block there is.

The grip is exponential in the thread’s own interlacings #

Counts become forces through friction, and friction at a crossing is not additive. A thread gripped in a seam is held by a capstan at every crossing it turns round: the tension it can resist grows by the same factor at each turn, so the pull-out force is exponential in the number of turns along the gripped length. That is why a satin gives a seam a small fraction of a plain weave’s grip rather than a quarter of it.

The grip model computes that force for a weave’s average thread — the interlacings per thread averaged across the system. Run instead on each thread’s own interlacings, the same model gives each thread its own force, and for the satin stripe the difference is large. At the model’s sheeting geometry, a friction coefficient of 0.3 and a gripped length of two millimetres, the stripe’s satin ends resist 0.136 of the force its average thread would. Measured against its own plain ends, the satin ends hold a tenth.

The average overstates the weakest thread by more the longer the grip. Over one millimetre the satin ends grip at a fifth of the average thread; over six, at under two hundredths; over ten, at two thousandths. The exponent multiplies the difference in interlacings by the gripped length, so a difference in counts that looks moderate becomes, over a seam allowance, a difference of orders of magnitude.

Where a cloth slips and where it frays #

The practical consequences follow the weakest thread wherever it lies. A seam slips before it breaks: the cloth’s threads beside the stitching slide out of the weave while the stitches still hold, and in a striped cloth they slide in the stripe, where the ends are held least. A seam in the plain ground of the satin-stripe cloth is a plain-weave seam; the same seam crossing the stripe opens along the stripe.

A cut edge frays by the same mechanism, a thread escaping from the crossings between it and the cut. An edge cut across a figured cloth frays furthest where the cut passes through blocks of the looser weave, and a frayed edge on a striped cloth is ragged in stripes. Neither is visible in the cloth’s firmness number, and both are visible on the first garment made from it.

The same reading applies to a herringbone’s point. Its warp ends interlace alike, but its picks do not: the picks that cross the reversal in step with the twill interlace at a quarter of their crossings where the others interlace at a half, and the herringbone’s weakest picks grip at 0.57 of its average pick over two millimetres. A herringbone is firm along its warp and has a weaker pick at every turn of the pattern.

A stripe’s weave sets its weakest grip, and its firmness number barely notices #

The stripe is where a designer has the choice, so the per-thread grip is worth running on the alternatives. Keep the forty-end plain ground and change only the weave of the stripe.

With an eight-end satin stripe the cloth’s firmness number is 0.870 and the stripe’s ends interlace at a quarter of their crossings; over two millimetres they grip at 0.10 of what a plain end beside them grips. A five-end satin stripe barely moves the firmness number, to 0.876, but its ends interlace at two fifths of their crossings and grip at 0.19 of a plain end — nearly twice the eight-end stripe’s. A 2/2 twill stripe, firmness 0.911, grips at 0.27; a 2/1 twill stripe, firmness 0.949, at 0.44.

Across the four stripes the firmness number changes by less than a tenth, and the weakest ends’ grip changes more than fourfold. The number that is printed hardly distinguishes them and the number that decides the seam separates them completely. A designer choosing a stripe weave for its lustre, and checking the cloth’s firmness against a specification, will see a satin stripe pass that a seam across it will fail.

Over a longer grip the separation widens again. At six millimetres the eight-end satin stripe’s weakest ends hold 0.018 of the stripe’s average thread and the 2/1 twill stripe’s hold 0.17 — a factor of nine between two cloths whose firmness numbers are within a tenth of each other.

The edge is where a weak thread has no second side #

One thread in a cloth is weaker than any count inside it suggests, because it is gripped from one side only: the end at the selvedge. Inside the cloth a thread has crossings on both sides; at the edge the weft turns round the outermost end and comes back, and whether that turn holds depends on whether the edge end changes face between the two picks. A selvedge holds only where its edge end changes face, and the weaves whose threads interlace least — the long-float twills and every satin — are exactly the ones whose edge ends cannot, at any width.

So the least-interlaced thread argument has two places to bite. In the body of the cloth it picks out the weakest band, and at the edge it picks out whether the weave can hold its own selvedge at all. The four-by-four census that finds a thread at the floor of two interlacings in every draft but plain weave also finds, over the same 22,874 drafts, that four in ten cannot catch every weft turn at either edge whatever the width.

Firmness as a minimum #

None of this makes the firmness number wrong for what it was made to predict. The closeness a cloth can be set at depends on how much room all its bends take together, which is a sum and so an average: a stripe of satin in a plain ground can be set nearly as closely as plain weave, because most of its bends are plain weave’s. The average is the right number for the jamming sett.

It is the wrong number for slippage, fraying and pull-out, which are failures of one thread at a time. For those the number that matters is the fewest interlacings any thread in a given direction makes per crossing, and it should be quoted beside the average whenever a draft’s threads differ. For the four foundation weaves the two coincide. For a stripe they differ by a factor of three or four; for the four-by-four drafts, by up to seven fourths; and because the grip is exponential, the difference in force is always much larger than the difference in the numbers.

The band that slips first tears best #

The minimum does not replace the average for every failure, and one failure runs the other way. Tearing needs threads free enough to bunch at the tip of the tear, so that several share the load where one alone would break, and a thread with fewer interlacings bunches more readily. A satin tears better than a plain weave of the same yarn for exactly the reason it slips worse.

Inside a striped cloth the two orderings sit in the same band. A tear run across the stripes slows where it meets the satin, because the satin ends gather at its tip, and runs freely through the plain ground, where each end meets the tip alone. A seam pulled across the same stripes opens first in the satin. So the stripe is simultaneously the cloth’s weakest band for slippage and its strongest for tearing, and the cloth’s single firmness number, sitting between the two bands, predicts neither.

That is the case for quoting both numbers rather than the minimum alone. The average predicts the cloth’s sett and its handle, the minimum predicts where it slips and frays, and the maximum spread between bands predicts where a tear will stop. For a plain weave, a twill or a satin all three are one number and the single figure on a specification is enough. For anything with bands, blocks or irregular threads it is not, and the draft already contains every thread’s count — the specification is simply reading the wrong statistic from it.

What was counted, and how #

Every end’s and every pick’s interlacings were counted round the repeat, cyclically, for all 22,874 four-by-four drafts in which every thread interlaces, and compared with the average in each direction. The census was confirmed to find a thread at two interlacings in every draft but the two plain weaves, to find as many drafts with a weak warp end as with a weak pick, and to find the same floor across every total from sixteen interlacings to twenty-eight.

The grips use one model throughout: a cloth’s geometry supplies the crimp angle and the contact force, a thread’s interlacings along its gripped length supply its turns, and a capstan at each turn gives the pull-out force. The model’s average thread uses the system’s mean interlacings; the per-thread version uses each thread’s own. In a weave whose threads interlace alike the two were confirmed identical, a satin stripe’s ends were confirmed to grip exactly as a satin’s ends do, and the shortfall was confirmed to grow with the gripped length.

Where the model stops #

A thread is not gripped only by its own crossings. Its neighbours press on it, a float lies against the floats beside it, and a stripe’s satin ends are wedged between plain ends that hold them sideways. The per-thread grip is the capstan on the thread’s own turns and a lower bound on what the cloth around it adds.

The crimp angle is the cloth’s, not the thread’s. A satin end in a plain ground bends round its picks at the angle the plain weave’s geometry sets, which is steeper than a satin’s own. The model takes one angle for the whole cloth, and a stripe’s satin ends are gripped somewhat harder than the per-thread figure says — though still exponentially less than the plain ends beside them.

And a stripe is not the only way threads differ. Uneven yarns, a sett that varies across the width and tension differences between ends all make some threads weaker than others. The interlacing census counts only what the draft decides.

Still open: whether a firmness minimum predicts where cloths fail #

The claim is testable on cloth. Cut strips across a striped or figured cloth, measure the force to pull a single thread out of each band, and compare the bands’ forces with their interlacing rates. The prediction is that the ranking follows the least-interlaced thread in each band exactly and the forces follow it exponentially, and that a seam-slippage test run across the stripes fails first in the loosest band.

The same test on a set of four-by-four drafts of equal firmness number and different minima would say whether the minimum alone predicts the weakest band, or whether the neighbours’ support measurably shifts it. Neither has been done here.

Who worked it out #

The interlacing count as a measure of firmness is Ashenhurst’s, from the setting rules of the 1880s, and it was always a count for regular weaves. The capstan equation is Euler’s. Counting every draft’s weakest thread against its average, and running the grip on each thread’s own interlacings, was done directly.