Where the heddles go

How many shafts a draft needs is a question about the repeat: count the distinct columns, and that is the answer. The site has computed it since its earliest essays and it is exact.

It is also not the question a weaver asks next. A repeat is four ends wide or eight or forty-eight, and a warp is two thousand ends wide, and what has to be decided before a single end is drawn in is which shaft each of those two thousand hangs on. The shaft count does not settle it, and nothing in the draft does either.

The compound-cloths field recorded that nothing on the site computed the distribution and left it open. This closes it.

What a shaft carries, and why it matters #

A shaft is a frame with heddles hanging in it, one per end, and it is lifted and lowered once per pick — the first thing on this site that is not the cloth, and the first that has a mass. Everything about how heavily it is loaded follows.

Weight. A shaft with three times its neighbour’s heddles is three times as heavy to lift, so it accelerates more slowly and reaches the top of its shed later — which on a fast loom is a timing error rather than an inconvenience.

Friction. Every end in the shed rubs against its neighbours and against the heddle eye, and a heavily loaded shaft opens its shed against more of that.

Wear. The heddles on a heavily loaded shaft do more work per pick and the ends through them see more abrasion, so a stripe fabric’s ground ends break more often than its figure ends and nobody attributes it to the threading.

None of that is visible in a draft. Two drafts needing four shafts each can hang perfectly evenly and three to one, and the drawings are equally plausible.

Where the imbalance comes from #

It is a property of the column multiplicities of the repeat, and of nothing else.

Group the ends of a repeat by their columns: two ends whose columns are identical must lift together, so they belong to the same class, and the shaft count is the number of classes. The load on a class is how many ends of the warp fall into it, which is the class’s multiplicity in the repeat times however many repeats the warp holds.

A straight draw puts each class in exactly once, so every shaft carries the same, and a 2/2 twill or an eight-end satin threads perfectly evenly. A pointed draw duplicates its turning ends, so a 1/3 herringbone pointed at five has multiplicities of four, two, two and two, and the heaviest shaft carries 1.6 times the average.

A stripe is where it becomes severe. A satin band of eight ends beside a plain ground of forty needs eight shafts for the band, one end each, and two for the ground, twenty ends each. That is a multiplicity ratio of twenty to one before a weaver has made a single decision.

The freedom, and it is the only one #

The threading is not quite determined by the draft, and the gap is the whole of what a weaver can do about the problem.

Two ends whose columns are identical must be lifted together. They need not hang on the same shaft: a second shaft carrying the same lifting plan will lift them just as well, and then the class’s heddles are shared between the two.

So a budget larger than the shaft count buys nothing in what can be woven and something in how evenly it hangs. The question becomes an allocation: give each of the k classes at least one shaft out of a budget S, and the heaviest shaft carries the largest of ⌈n_c / s_c⌉ over the classes. Minimising that is the whole problem.

The algorithm is a greedy — hand the next shaft to whichever class is currently heaviest — and it is checked against exhaustive search rather than trusted, because a greedy that is merely plausible is how a wrong number gets a figure drawn round it. Every allocation of the budget over the classes is enumerated for the cases the essays draw, and the greedy’s answer must equal the best of them exactly.

Two things in that table are worth naming. The first twelve shafts halve the worst load and the last four buy nothing at all — the returns fall away exactly as the reciprocal in the ceiling function says they must. And the optimum allocation is not symmetric at every budget: at twenty shafts the greedy gives seven and five, which is exhaustively verified to be as good as any split, because ⌈20/7⌉ and ⌈20/5⌉ are three and four and no other split beats four.

What was counted, and how #

Two enumerations.

The allocation is checked exhaustively for every case drawn, by walking every composition of the budget over the classes with a bound that prunes any branch already worse than the best found. That is not a sampling check; it is the whole space.

The census runs the four-by-four sweep — every draft in which each end and each pick interlaces — threaded across a warp of twelve hundred ends.

The answer is unexpectedly stark. Of the 22,874 drafts, 17,634 thread perfectly evenly and 5,240 do not — and every single one of the 5,240 is uneven by exactly the same ratio, 1.5.

That is not a coincidence and it is worth a sentence. A four-end repeat has at most four classes; a draft threads unevenly exactly when its four ends fall into fewer than four classes with unequal multiplicities; and on four ends the only way to do that is three ends in one class and one in another, or two and two, which is even. Three to one gives a heaviest load of three quarters of the warp against a mean of half, which is 1.5. There is no other possibility at this repeat, so the census has exactly two rows.

The stripe is where every quantity on this site collides #

A stripe fabric is one draft in which different bands of ends interlace by different rules, and building one out of the site’s existing pieces makes several things visible at once.

The repeat becomes wide. A satin band of eight beside a plain ground of forty is a forty-eight-end repeat with eight picks, because the pick count is the least common multiple of the bands’ and the end count is their sum. Almost everything the site measures is a property of the whole repeat, so a stripe’s float census, symmetry and plane group are those of a forty-eight-end cloth.

The weight varies across the width. The satin band has less crimp than the plain ground, so a square centimetre of stripe weighs measurably less than a square centimetre of ground — which is why a heavy stripe fabric is often a different sett in the two bands rather than a different weave at one sett.

The shrinkage varies too. Different crimp means different relaxation, which is exactly the relief mechanism: a bold satin stripe on a plain ground puckers in finishing unless the setts are adjusted, and the trade calls the result a defect in one fabric and a cloqué in another.

And the threading is the one this essay is about, which is the only one of the four that has no cure in the cloth’s design at all. A designer can change the setts to fix the weight and the shrinkage; nothing about the design fixes the heddle load except buying shafts.

Why the returns fall away, and where the knee is #

The table’s shape — twelve shafts halving the worst load and the last four buying nothing — is a property of the ceiling function rather than of the cloth, and reading it says where to stop buying shafts before any allocation is computed.

A class of n ends spread over s shafts has a worst load of ⌈n/s⌉, which falls as one over s and does so in steps. So the marginal shaft is worth n/s(s+1), which is the whole return: the first extra shaft on a class of a thousand halves it, the tenth takes off nine, and the hundredth takes off a tenth of one.

That gives a rule that needs no search. The budget stops being worth spending when the heaviest class’s load per shaft approaches the second-heaviest class’s, because after that the greedy alternates between them and each shaft buys half as much. On the stripe above, the two ground classes have a thousand ends each, so they alternate from the start and every pair of shafts buys one step — which is why the table’s improvements come in pairs and why an odd budget gains nothing over the even one below it.

The steps also explain the flat stretch at the end. Between twenty-eight and thirty-two shafts the worst load sits at a hundred, because ⌈1000/10⌉ and ⌈1000/14⌉ are a hundred and seventy-two and the other class is still at a hundred — so the improvement waits until both classes can step together. A budget spent in the flat part of a staircase buys nothing at all, and the staircase’s treads get longer as the budget grows.

So the practical instruction is to compute the load, choose a target, and buy the smallest budget that reaches it — never a round number, and never one shaft more than a step. The allocation is exact and the staircase is what makes an approximate budget wasteful in a way an approximate anything else would not be.

There is a second reading of the staircase and it is the one that decides whether to bother at all. The steps are steps in the worst load and not in anything a weaver measures, so a loom bought two shafts short of a step performs exactly as one bought at the previous step, and nothing about running it says so. A drawing office comparing two harnesses on their shaft counts is comparing a quantity whose effect is piecewise constant, and the pieces are wide near the top of the range — which is exactly where the harnesses are expensive.

The same is not true of shafts bought for pattern, where every one buys a distinct column and the return is linear. So the two uses of a shaft budget differ in their return as well as in their purpose, and a loom’s single quoted number is describing two quantities with two different shapes.

Which shaft goes where, which is a different question #

There is a second decision the draft does not make and it is not an allocation. Having decided how many heddles each shaft carries, a weaver has to decide which position each shaft occupies — front, nearest the weaver, or back.

The trade’s rule is to put the heaviest shaft at the front. The reason is geometric: the shed is a triangle between the fell of the cloth and the harness, and a shaft further back has to lift its ends further to clear the same shed, because it is further from the apex. Putting the many-heddle shaft where the lift is shortest minimises the total work.

That is a real optimisation with an exact geometry behind it and this essay does not compute it, because it needs the loom’s own dimensions — the distance from fell to harness, the shed height, the heddle eye position — none of which is a property of the cloth. It is recorded as not done and it is the natural next thing.

The connection to what a dobby stores #

A dobby chain holds lifting plans, one per pick, and the site’s harness census established that the pattern width is bounded by the threading rather than by the shaft count — which is why a sixteen-shaft dobby is not limited to a sixteen-end repeat.

The allocation above is the same economics from the other side. Extra shafts bought for balance rather than for pattern do not extend what can be woven at all: they carry duplicate lifting plans, so the dobby’s chain has the same distinct rows it always had, and the loom’s pattern capability is untouched.

There is a second reading of the same fact and it is the encouraging one. Because balance shafts carry duplicate lifting plans, adding them costs nothing in the dobby’s chain: the chain stores distinct rows and the duplicates are not distinct. So the extra shafts are cheap in the machine’s memory and expensive only in its frames, which is the opposite of how shafts bought for pattern behave.

So a loom’s shaft count is spent on two different things and the trade quotes one number for both. A twenty-four shaft dobby weaving a stripe may have ten shafts of pattern and fourteen of ballast, and its capability is a ten-shaft loom’s. Nothing in a specification distinguishes them.

Where the model stops #

Every end is taken as identical. A real stripe fabric often uses a different yarn in the figure from the ground, so the loads are not comparable in weight even when they are comparable in count. The arithmetic counts heddles and a shaft lifts mass.

The selvedge is ignored. Selvedge ends are usually threaded on their own shafts or on doups and are excluded from the pattern’s allocation entirely, so a real threading has a few classes the draft knows nothing about.

Nothing here is a force. The essay says a shaft carries more heddles and therefore more of everything; how much lifting force that is, and whether it matters at a given loom speed, needs the shed geometry and the warp tension, and this site has neither.

And the allocation assumes shafts are interchangeable. On a real loom they are not — the front and back positions differ, as above, and some looms have a fixed number of shafts that can carry a given lifting plan. The allocation computed is a bound on what is achievable rather than a threading plan.

Who found it, and when #

This is folklore rather than literature. Weaving manuals say to distribute the heddles evenly where possible, to double up a heavily loaded shaft, and to put the heaviest shaft at the front; none of them says how to choose the doubling, because with four or eight shafts and a hand loom there is not much choice to make.

The problem becomes real on a dobby weaving a figured cloth on a broad warp, and there the decision is made by the drawing office from experience. The allocation is small enough that experience does well: with two heavy classes and a budget, the obvious split is usually optimal, and the greedy above confirms it.

What experience does not supply is the shape of the return. The table above says the first few extra shafts are worth a great deal and the rest are worth almost nothing, and where the knee sits depends on the multiplicities in a way that is not intuitive — at twenty-eight shafts the worst load reaches a hundred and the next four shafts buy nothing at all. A drawing office deciding whether to buy a wider harness has a number for that now and did not before.

Where the ladder goes next #

This anchor has now taken the harness from what it can weave, through what a dobby stores, to how the ends hang on it. What remains is the shed itself — the triangle between the fell and the harness, which decides the shaft order, the lift each shaft needs and how much a warp is abraded on the way through. That is the loom’s own geometry rather than the cloth’s, and it is the first thing on this site that would need a machine’s dimensions.