The back shaft works hardest

The previous rung established what a shed does to an end: it lengthens the path, by a fraction that depends on the shaft’s distance from the fell and grows towards the back of the harness. What it did not say is how often it happens to any particular end, and that turns out to be a completely separate quantity living in a completely different object.

An end only takes the strain when it changes sides. If the design leaves an end up for four picks running, the shed opens four times and the end goes up once and comes down once — one excursion, not four. So the number of times an end is dragged through its heddle eye in a repeat is the number of times its column changes value, cyclically.

That number has appeared on this site before under another name. It is exactly twice the number of floats in that column, and summed over the whole matrix it is the interlacing count — the number that decides how firm the cloth feels, how densely it can be set and how much crimp it takes. Nobody chooses it for the loom’s sake. It is chosen for the cloth’s.

The count that is already on the site #

Take any weave and read up an end. A 2/2 twill’s column goes over, over, under, under, and round the repeat that is two changes. A plain weave’s column alternates, which is also two changes in a repeat of two. An eight-end satin’s column is up seven times and down once: two changes again, in a repeat of eight.

So all three of those weaves put the same number of excursions into every end, over quite different repeat lengths, which is not the same as saying they cost the same. The plain weave does its two changes in two picks and the satin does its two in eight, so the plain weave’s end is flexed four times as often per metre of cloth. That is the interlacing rate exactly, and it is the reason a plain weave is a slow cloth to weave as well as a firm one.

What is worth having is the spread, and a rate says nothing about it.

A theorem about which weaves are even #

There is a large class of weaves in which every end is flexed exactly as often as every other, and the class has a clean description.

A shift-rule weave is one whose every column is a rotation of one column. Plain weave, every twill, every satin and every sateen is one, by construction: the rule is that the column slides by a fixed step from end to end. Rotating a column does not change how many times it changes value, so every end in such a weave has the same flex count, whatever the float length and whatever the repeat.

That is a theorem with a one-line proof, and it says something a weaver would not necessarily expect: the whole traditional catalogue of basic weaves spreads the loom’s work perfectly evenly across the warp, and does so for a reason that has nothing to do with anybody having wanted it.

The converse is false and the census says so. A 2/2 hopsack is even without being a shift-rule weave — its columns are not rotations of one another but they happen to have the same number of changes — so evenness is the weaker property. What actually produces a spread is a design that treats different groups of ends differently, and that means a stripe, a cord, a relief weave or a figure.

The product, and the spread it produces #

Now put the two together. An end’s share of the loom’s work is how often it flexes multiplied by how far it is stretched when it does, and the two factors come from different worlds.

The flex count is a property of a binary matrix and has no millimetre in it. The strain is a property of a machine and has no weave in it. The only thing that joins them is the threading — which end goes on which shaft — and a threading is chosen for convenience, for how the pattern reads on the drawdown, or because a straight draw is easy to remember. Nobody threads a warp in order to spread the loom’s punishment.

On a satin stripe over a plain ground, threaded as its own shaft count requires, the product runs 3.74 to one across a single repeat. The worst-off ends are plain-weave ends that landed on a back shaft; the best-off are satin ends at the front. Both extremes are accidents of an allocation nobody made on structural grounds.

What was counted, and how #

The flex count is computed per column, cyclically, from the matrix that draws the figure — not from the interlacing rate, which is an average and would hide the whole finding.

One step is checked rather than assumed, and it is the step that lets a per-end quantity be reported per shaft at all: every end on one shaft has the same column by construction, so it must have the same flex count, and the machinery asserts that end by end rather than taking it on trust. A shaft is a set of ends that do the same thing, and if the threading and the flex counts ever disagreed the factorisation would be wrong.

The census over eight weaves runs both halves of the theorem. Every weave marked as a shift-rule weave must come out even, and at least one weave that is not one must not — because an assertion that only ever confirms is an assertion that has stopped being able to fail. The two halves are separate assertions for that reason.

The product is computed by threading the weave onto the harness its own shaft count requires, front shaft first, and multiplying each end’s flex count by its shaft’s exact strain. The machinery refuses a threading that names a shaft the harness does not have, which is the failure mode of feeding it a weave and a harness that do not match.

Why the product is worth having at all #

It is fair to ask what a load index is for, given that neither factor is a force and their product is in units of nothing in particular.

The answer is that it is the only quantity here that is a function of a decision somebody makes freely. The interlacing rate is chosen for the cloth: a weaver picking a plain weave over a satin is choosing a handle, a cover and a firmness, and the flex count comes with it. The strain profile is the machine’s: it follows from the shed the insertion needs and the harness the design needs, and a weaver does not adjust it. The threading is free. A design needing eight shafts can be threaded onto those eight shafts in many orders, all of them producing exactly the same cloth, and the only thing that changes between them is which ends land at the back.

So the product is a quantity that can be improved without changing anything a customer sees, which is a rare position. A design whose most-flexed ends are on the front shafts and whose idlest are at the back is the same fabric as one threaded the other way round, and the difference between them is the whole 3.74 in the worked case.

Whether it is worth improving is a different question and this site cannot answer it, because the answer needs a warp-break rate and that needs a force.

And there is an exact answer to what the best threading is, which is worth having even without a force to price it, because it turns out to be one line and to be what the trade already does.

The problem is a matching: a list of end-classes, each with its own flex count, against a list of shafts, each with its own strain, one class to a shaft. Minimising the total work over all such matchings is the rearrangement inequality, and the inequality’s answer is that two sequences multiplied term by term are smallest when they are sorted in opposite orders. So the optimal threading puts the most-flexed ends on the least-strained shafts: the highest flex count at the front, in decreasing order to the back. The same pairing also minimises the largest single product, so it is optimal by either reading of what “hardest worked” means.

That is exactly the rule a weaver already follows. “Ground weave at the front” is the special case in which the ground is the class with the most changes of side, which for a plain ground it always is. What the inequality adds is that the rule is not a reasonable heuristic but the exact optimum, that it extends to any number of classes rather than to two, and that it is settled entirely by the order of the flex counts and never by their sizes — so it can be applied without knowing any of the strains, provided only that they increase towards the back, which the previous rung proved they do.

The worked stripe is threaded the other way in places, which is how it reaches 3.74 to one at all.

What an even weave does not buy #

The theorem above says every shift-rule weave flexes all its ends equally, and it is easy to read that as saying such a cloth is easy on its warp. It says nothing of the kind, because the flex count is only one of the two factors.

An eight-end satin flexes every end twice per repeat and needs eight shafts, and eight shafts on the loom these essays have been using run from 0.460 to 0.722 per cent. So a perfectly even weave still spreads the loom’s work 1.57 to one across its own warp, from the strain alone, and no threading can improve it: every end has the same flex count, so every matching gives the same answer and the rearrangement inequality has nothing to work with.

That is the reverse of the stripe’s situation and it is worth putting the two side by side. A striped cloth has an uneven flex count and an uneven strain, so its spread is large and most of it is recoverable by rearrangement. An even cloth has a flat flex count, so its spread is small and none of it is recoverable. The cloth that offers the improvement is the one that looked worst, and the cloth that looked well behaved has no improvement to offer.

The spread grows with the shaft count rather than with the design, so a damask needing twenty-four shafts carries a strain spread of 3.75 to one whatever its columns do — which is very nearly the figure the eight-shaft stripe reaches by being uneven in both factors at once. The two numbers are close and the closeness is a coincidence: one is a harness depth and the other is a combination of a pattern and a threading, and nothing connects them. What can be said is that the opportunity exists, that it costs nothing, and that nobody takes it — because a threading is chosen for how the drawdown reads and for how easy it is to remember, which are both good reasons and neither of them is this one.

Where the model stops #

Flexing and stretching are two mechanisms and this multiplies them as though they were one. An end that is stretched often is fatigued; an end that is dragged through a heddle eye often is abraded. They are different failures with different remedies — sizing helps one and a smoother heddle helps the other — and the product above is a load index rather than a prediction of anything. What it is good for is the ratio between ends, where the units cancel.

The threading is taken as given. A warp can be threaded in many ways for the same shaft count, and the allocation problem that produces — which is a real one, with an exact answer — is where the heddles go rather than this essay. Nothing here optimises anything.

The heddle eye’s own geometry is absent. An end passes through an eye with a shape, at an angle that changes as the shaft rises, and how much of the abrasion happens there rather than at the reed is a question about surfaces that this site has no model for.

And the count is per repeat. Two weaves with the same flex count and different repeats do not flex their ends equally often in time, which is the point made above and is worth repeating because a chart of flex counts invites exactly that mistake.

The generalisation #

The shape of this is a product of two quantities that live in different descriptions of the same object, joined by a mapping that nobody chose for the purpose.

The matrix knows the pattern and nothing about size. The machine knows size and nothing about pattern. The threading is a map between their index sets, and every quantity that is a function of both is therefore a function of an arbitrary choice — which means it can be very uneven without anybody having done anything wrong, and it can be improved by a rearrangement that costs nothing.

That is a common situation and it is usually invisible, because a quantity nobody computes is a quantity nobody notices is uneven. The textile case makes it unusually sharp: the two descriptions are both exact, the mapping is written down explicitly in every weaving draft there is, and the product has never been anybody’s number.

Who found it, and when #

Every part of this is known in the trade in some form. Ground weaves go on the front shafts. Back shafts break more ends. A plain weave is hard on a warp and a satin is easy. Stripes and figured cloths break ends unevenly, and a weaver watching a piece knows which shafts to look at.

What is not usually written down is that the last of those is the product of the first two, and that the two factors are separately exact. The flex count has been on this site since its first essays, under the name interlacings, as a property of the cloth’s firmness. The strain has been here since the previous rung. Multiplying them is the obvious move once both exist, and it is only obvious once both exist.

The one thing here that is genuinely a theorem rather than an observation is the shift-rule result, and it is small enough to state in a sentence: rotating a column does not change how many times it changes value. What makes it worth stating is what it covers — plain, twill, satin and sateen, which is very nearly the whole of what anybody calls a basic weave.

Where the ladder goes next #

The last rung of this ladder turns the strain into a bound. A stated tolerance on warp strain is a stated distance from the fell, and a stated distance is a whole number of shafts — so a loom has an answer to how deep its harness may usefully be, and the answer is thirteen shafts at one per cent on the machine these essays have been using. The shed angle enters squared, so the insertion mechanism decides it.

Sideways, the allocation of ends among shafts once the count is fixed is its own problem with its own optimum; the flex count as a property of the cloth rather than of the loom is firmness; and what happens to all this accumulated history the moment the cloth is released is relaxation.