The harness has a depth

Three essays on this site take a shaft budget as given. One counts how many shafts a draft needs, one shows that the budget does not grow with the repeat, and one works out how to divide a warp among the shafts a budget allows. All three treat the number — sixteen, twenty-four, thirty-two — as a fact about the machine that arrived from outside.

It did not arrive from outside. It is a consequence of the shed, and the arithmetic that produces it is two rungs of the mechanics ladder and one line of algebra.

The strain a shed puts into an end is, to leading order, t²·a / 2(L − a): the square of the half-shed’s tangent, times the shaft’s distance from the fell, over twice what lies behind it. The denominator goes to zero as a shaft approaches the back rest. So the strain is not merely larger at the back of the harness — it grows without bound, and a tolerance on it is a hard distance.

Solve for that distance and count the shafts that fit inside it. On an ordinary broad loom with a shuttle’s shed, a one per cent warp-strain budget buys thirteen shafts.

Solving the bound #

The leading-order strain is small enough to invert in one step. Setting t²·a / 2(L − a) equal to a tolerance ε and solving for a:

a = 2εL / (t² + 2ε)

That is the furthest a shaft may be from the fell. The shafts sit at a known first distance and a known pitch, so the count follows immediately: divide the remaining distance by the pitch and add one.

The closed form is checked against the trigonometry rather than trusted. The exact strain is computed for a harness two shafts deeper than the bound predicts, the shafts inside the tolerance are counted directly, and the two answers are required to agree to within a shaft. They do, which is what makes the algebra above a statement about the machine rather than about an approximation of it.

The numbers on the loom these essays have been using — reed 90 mm in front of the fell, first shaft at 300, pitch 16, back rest at 1,200, half-shed tangent 0.167:

The steepness of that table is the point. Between three quarters of a per cent and one and a half, the harness a loom can carry goes from eight shafts to twenty-one — which is the difference between weaving twills and weaving a small damask.

The bound has a floor and a ceiling, and both are on the loom #

The solved distance is well behaved at both ends of the tolerance, and reading it there gives two numbers the table does not contain.

Below about 0.465 per cent there is no harness at all. The front shaft sits at 300 mm and takes 0.46 per cent whatever anybody does, so a tolerance under that admits no shaft anywhere and the loom cannot weave. The table’s first row — half a per cent buying two shafts — is not the bottom of a gentle slope; it is a fraction of a point above a cliff, which is why that row looks so much worse than its neighbours.

And a tolerance of any size whatever buys fifty-seven. The solved distance approaches the back rest and never passes it, so the shaft count saturates at the whole warp line divided by the pitch: 900 millimetres between the first shaft and the back rest, at 16 mm centres, is fifty-seven shafts and no more. A loom builder offered an infinitely strong warp would still stop there, because a shaft behind the back rest is not a shaft.

Between those two the response is nearly linear at first and then flattens. Going from half a per cent to two buys thirteen times the shafts; going from two to eight would buy less than twice as many again. The whole of the useful sensitivity is in the first two per cent, which is exactly the interval a warp’s own strength puts the tolerance in, and it is why the harness depth of real machines varies so much for such small differences in how hard the warp is driven.

That saturation also settles a question the shed angle raises. Halving the opening at the reed quarters the strain and moves the bound a long way, but it cannot move it past fifty-seven — so a jet loom’s very shallow shed does not buy an unlimited harness; it buys most of the harness the back rest allows, and then stops. The jacquard’s escape is therefore not a matter of degree at any point on this curve.

The square, and what it means for the machine #

The tangent enters the bound squared, so the shed opening is much the strongest thing a loom builder controls.

That has a clean practical reading. A shuttle is a boat: it has to carry a pirn, so it has a height, and the shed has to clear it — thirty millimetres or so on a broad loom. A rapier is a stick with a gripper on the end, and it needs less than half of that. A jet of air needs less again, though it needs the shed to be clean in a way a rapier does not.

So the progression from shuttle to rapier to jet, which is usually told as a story about speed, is also a story about how deep a harness the loom can carry. At a one per cent budget the shuttle loom above gets thirteen shafts; drop the opening to 16 mm for a rapier and the same budget buys thirty-six.

That is a large enough change to move what kind of cloth a machine can make, and it moves in the direction nobody quotes. A rapier loom is sold on picks per minute.

What the jacquard actually escapes #

A jacquard has no shafts. Each end is carried by its own harness cord, running from a hook in the machine above down through a comber board to a mail, and the essay about what that abolishes is elsewhere: it removes the shaft budget and makes the hook budget proportional to the width of the repeat.

What is worth adding here is the mechanical half. The cords all descend from essentially one place. There is no front of the harness and no back of it — every mail sits at the same distance from the fell, so every end takes the same strain, and the quantity this whole essay is about is not merely large or small but absent.

That is a second and independent reason the jacquard can do what a dobby cannot, and it is not the reason usually given. A dobby with two hundred shafts would fail on warp strain long before it failed on mechanism; a jacquard with two hundred thousand hooks has no equivalent failure at all, and its limits are the hook count, the comber board’s density and the machine’s weight.

It also inverts a reading of the two machines that is easy to fall into. The jacquard looks like the extreme end of a continuum — more shafts, more shafts, one shaft per end. It is not on that continuum. Somewhere before the end of it the continuum stops existing, at a distance from the fell that the shed angle decides.

Why the pitch is the lever nobody pulls #

The bound is a distance and the shaft count is that distance divided by a pitch, so the pitch is a full factor in the answer and it appears nowhere in the strain.

That separation is worth stating because it is unusual. Every other term in the calculation — the opening, the reed’s position, the back rest, the first shaft’s distance — changes how hard the warp is worked. The pitch changes only how many shafts fit inside a distance already decided, so halving it doubles the harness at no cost in strain whatever, and the front shaft’s ends are as safe at 8 mm centres as at 16.

A one per cent budget on the loom above buys thirteen shafts at 16 mm and twenty-five at 8. That is the difference between a small dobby and a large one, from a change that no warp end can detect.

What stops it is entirely mechanical and it is worth listing, because the list is short and none of it is about the yarn. A shaft has to be stiff enough not to bow across a broad loom, which sets a depth for its staves. The heddles hanging on neighbouring shafts have to pass each other as one rises and another falls, which needs clearance. And every shaft has to be reached by whatever lifts it, which is the jack-and-lever problem that the usual explanation of a shaft budget starts from.

So the two explanations turn out to bind on two different quantities. The mechanism limits the pitch and the warp limits the distance, and the shaft count is the second divided by the first. A dobby maker who improved the shedding mechanism without changing the shaft pitch would have moved nothing at all, and one who found a way to build thinner shafts would have moved everything — which is the reverse of how the constraint is usually described.

What was counted, and how #

Everything here is arithmetic on the same five loom dimensions used throughout this ladder, stated once so that a figure changing one of them is visibly changing a machine.

The bound is solved in closed form from the leading-order strain and then confirmed against the exact trigonometry, and the two are required to agree to within one shaft. The sweep over shed openings recomputes the whole bound at each opening rather than scaling one answer, because the relation between the opening and the shaft count is not a power law — the tangent squared competes with twice the tolerance in the denominator, and which of the two dominates changes across the range.

Two refusals guard it. A tolerance so small that it does not buy even the front shaft is refused rather than returning zero, because zero shafts is not a loom and the arithmetic that produced it has left the domain. And a harness deep enough to reach past the back rest is refused, because the length behind the shaft has gone negative and every subsequent number would be plausible and wrong.

The sweep over openings also asserts its own direction — a smaller shed must buy a deeper harness — which is the kind of assertion worth having because it fails loudly if the geometry is ever set up back to front.

The number that is not on the specification sheet #

It is worth setting the three quantities side by side, because two of them are quoted everywhere and the third is quoted nowhere.

A loom is sold on its shaft count and its picks per minute. Both are on the plate. The clear opening at the reed is a consequence of the insertion mechanism and of how the shedding motion is timed, and it is not a number anybody advertises — yet it is the one that decides the first of the two, through a square.

That is an unusual arrangement and it is worth naming. The advertised quantity is downstream of an unadvertised one, by a relation steep enough that a small change in the hidden number moves the advertised one by a factor of three. A weaver comparing two machines on shafts and speed is comparing two consequences of a quantity neither datasheet contains.

What a weaver notices instead #

None of this is how the limit presents itself in a weaving shed, and the difference is worth setting down because it is why the arithmetic is not folklore.

What a weaver sees is warp breaks, and they are not distributed evenly. They concentrate at the back of the harness, they get worse as the shed is opened up to clear a bulkier shuttle, and they get worse again on a deep harness. All three of those are the same fact and none of them arrives looking like a fact about geometry: a break is a stopped loom, a mend and a mark in the cloth, and the remedy reached for is a better size on the yarn or a slower speed.

Both of those are good remedies. Sizing raises what the yarn will take and slowing the loom reduces how often it takes it, and a shed’s strain is a cyclic load so the number of cycles is half the problem. What neither does is change the number this essay computes, which is a length divided by a length and is the same at any speed and in any yarn.

That is the practical use of putting a formula to it. The three remedies a weaver has — size the warp, slow the loom, shallow the shed — act on three different terms, and only the third acts on the strain itself. It acts on it as a square, which makes it much the strongest of the three and the one least often reached for, because the shed is set by the insertion mechanism and is not thought of as adjustable at all.

Where the model stops #

This is a bound from one failure mode, not the only bound. A dobby has a mechanism, and the mechanism has limits: how many jacks a box can drive, how much lift a cam can produce, how heavy a deep harness becomes and how fast it can be moved. Any of those can bind first on a particular machine. What this essay says is that the warp-strain limit exists, is computable, and lands in the same range as the shaft counts real dobbies have — not that it is what any particular manufacturer stopped at.

The tolerance is an argument and not a derivation. One per cent is a plausible cyclic strain budget for a sized cotton warp already held at a working tension, and it is stated rather than computed, because computing it needs the yarn’s fatigue behaviour and this site has no model of a yarn’s stiffness at all. Every number here is quoted with the tolerance that produced it, and the shape of the table — the steepness — is what the essay rests on rather than any single row.

The shafts are assumed evenly pitched from a fixed first distance. Real harnesses stagger their lift so that back shafts rise a little less than the clean-shed condition demands, which trades a slightly less clean shed for less strain and is exactly the compromise the arithmetic above predicts somebody would want. Modelling it would move the numbers and not the argument.

And the strain is still a strain. What it costs in force needs the yarn, which is the same gap the previous two rungs record.

The generalisation #

This is a case of a limit that is usually attributed to one part of a system belonging to another, and the diagnostic is that the limit moves when the part nobody was looking at is changed.

The shaft count reads as a property of the shedding mechanism. It moves when the insertion mechanism changes — because insertion sets the opening, the opening sets the shed angle, and the angle enters the bound squared. Nothing about the jacks or the cams has changed at all.

Systems that have this shape are common and the misattribution is stable, because the part that appears to set the limit is the part whose number is quoted. A loom is sold by its shafts and its picks per minute; the opening at the reed is not on the specification sheet, and it is what decides one of the two.

Where the ladder goes next #

This closes the shed ladder’s mechanical half and leaves one thing open that it cannot reach: a force. Everything here is a strain, and turning a strain into a tension needs a yarn model this site does not have anywhere — the same shortfall the tensile ladder records at its own top and the tear essays record from the other side.

Sideways, what a shaft budget is for is the harness does not grow and what a dobby stores, which are the two combinatorial halves of the same budget; how to spend it once it is fixed is where the heddles go; and the machine that removes it is the jacquard. Further out, the same shed geometry decides how much warp a piece of cloth consumes and how wide the reed must be to make it, which is the setting field’s take-up arithmetic.