A pick density is a force budget

The take-up ladder found something genuinely asymmetric about a loom. The matrix is perfectly symmetric under exchanging warp and weft — the catalogue of cloths is closed under transposition, and three shafts with four treadles reach exactly the same 110 cloths as four with three. The machine is not.

The setts a loom can reach counted both sides. A reed catalogue at one to four ends per dent reaches 49 warp setts between 8 and 40 threads per centimetre; a change-wheel take-up reaches 4,825 pick densities over the same range. The ratio is 99, and the coarser of the two cannot be changed at all once the warp is drawn in.

That count is a count of what the gear train can select. This rung asks what it can reach.

The claim #

A force ceiling cuts the top off the take-up gear’s range and leaves the fineness of the choice untouched. So the weft direction’s ninety-nine-fold advantage over the warp is an advantage in resolution and not in reach, and the two halves of it behave completely differently when a real machine is put behind them.

There is a second claim underneath it, and it is the more surprising one. Past a certain force the binding limit is not the machine. A loom that can push harder eventually stops gaining anything, because Peirce’s geometry runs out of room before the force does — and the pick density at which that happens is a property of the yarn’s diameter and nothing else.

Two ceilings, and which one is in force #

There are three ways for a sweep of increasing pick density to end, and the distinction matters because they respond to completely different remedies. Two of them are about the cloth and the machine; the third is about the model.

The force ceiling is reached when the beat-up force needed exceeds what the loom has. It moves if the loom is heavier, if the sley swings harder, or — as the previous rung’s expression makes explicit — if the warp is held at a higher tension, since the force is proportional to it.

The geometric ceiling is reached when Peirce’s circular sections cannot be fitted any closer at all. No force helps. It moves only if the yarn is finer or is flattened, which is what a calender does and is one of the reasons calendering exists.

The third is not a ceiling at all. Past a certain pick density Peirce’s geometry cannot divide the crimp in the ratio the model asks for, and a state at some other ratio is not an answer to the question. The machinery refuses it rather than returning it, and reports which of the three ended a sweep — for most cloths in the table it is this one, which is a statement about the model’s reach rather than about the loom’s.

For a sheeting in 25 tex cotton the first two sit at 27.5 and about 28 picks per centimetre, so they are close — but which is binding at any given force is a fact with consequences:

The last two rows are identical, and that identity is the finding. Doubling the loom’s force buys nothing at all, because at a thousand newtons per metre the sheeting has already reached the wall its own yarn diameters put there.

Why the fineness survives and the reach does not #

The catalogue’s 4,825 values are dense: between 8 and 40 picks per centimetre they leave gaps averaging under seven thousandths of a thread per centimetre. That density is a property of how many ratios a hundred-and-one-wheel change train can make, and cutting the range at any point leaves the surviving part exactly as dense as it was.

So a force ceiling at 23 picks per centimetre does not coarsen the choice below 23. It removes everything above.

That is worth holding against the site’s own account of what a sett is for. Thread count is not quality argues that a high count buys less than it is sold as buying; this rung says something adjacent and different, which is that a high pick count is also harder to deliver than a specification implies, and the difficulty is not linear in the number. Two cloths quoted at 22 and 26 picks per centimetre differ by eighteen per cent on paper and by nearly a factor of three in what the loom has to do.

That is a different kind of limitation from the one on the warp side, and the difference is worth being precise about. The reed’s limitation is a resolution limitation: 49 values over the whole range, with gaps of a whole thread per centimetre in places, and no way to get between them. The beat-up’s limitation is a range limitation: everything below the ceiling is available at full resolution and nothing above it is available at all.

A designer meets those two as completely different problems. A warp sett that falls between two reeds has to be rounded, and the rounding is visible in the cloth. A pick density above the beat-up ceiling cannot be rounded to; it simply is not a cloth this loom makes.

Cloth by cloth, and the inversion in it #

The ceiling is not a property of the loom alone; it is a property of the loom and the cloth together, because the force needed to reach a given pick density depends on how near that cloth is to its own jamming point.

Run the same sweep across the table at 500 newtons per metre and something inverts. The openly set, coarse-yarn cloths reach their whole range easily, because they are nowhere near jamming and their crimp elasticity is small. The finely set cloths do not, because they are being woven close to the wall.

So a force budget is hardest on exactly the cloths a fine-yarn mill is trying to make. The cover the cloth ends up with is the quantity that was really being bought, and the last few per cent of it are the expensive ones. A cheesecloth at 9 picks per centimetre needs 27 newtons per metre and could be woven on almost anything; a sheeting at 26 needs 573, which is a real machine.

That is the practical shape of the whole result. A loom’s force capacity is not a general capability, it is a statement about which cloths it can make, and the ones it cannot make are at the dense end of the range — which is where the value is.

The thing that count was really saying #

It is worth going back to the ninety-nine and asking what survives of it.

The claim was that the loom can choose a pick density ninety-nine times more finely than a warp sett, over a catalogue that does not distinguish the two directions at all — a genuine asymmetry between the machine and the cloth. That is unaffected by anything here. The gaps in the take-up catalogue really are a hundredth of the gaps in the reed catalogue.

What does not survive is the reading of that as the weft direction is the free one. It is free in choice and constrained in extent, and the constraint bites exactly where a maker would want to go. The honest summary is that the machine gives the weft a fine dial with a short travel and the warp a coarse dial with a long one, and a cloth that needs a high pick density is asking for the one thing the fine dial cannot give.

The exchange rate between tension and picks, and why it collapses #

The force is proportional to the warp tension, so a mill with a loom it cannot change has one lever left: hold the warp harder. It is worth pricing, because the arithmetic above already contains the answer and it is discouraging.

Read the sheeting’s table as a derivative. Going from 200 to 500 newtons per metre — two and a half times the force, which is two and a half times the tension — moves the reachable pick density from 10.5 to 23.3 per centimetre. Going from 500 to 1,000 moves it from 23.3 to 27.8. Going from 1,000 to 2,000 moves it not at all.

The marginal picks per newton falls to zero at the geometric wall, and it starts falling long before it gets there. The first doubling of tension buys nearly thirteen picks per centimetre; the second buys four and a half; the third buys nothing at all.

That is the shape of every approach to a hard limit, and here the near side of it is expensive for a second reason the same tension controls. A warp end at higher tension breaks more often — it is nearer its own breaking load, the shed is already spending part of its strength, and a warp is a very long specimen with tens of thousands of independent chances to be thin. So the breakage rate rises steeply with the tension while the picks bought by it fall steeply, and the two curves cross well before the wall.

Where they cross is a mill’s decision and it is not a decision about cloth. It depends on how much a stop costs, how many ends the weaver watches and what the yarn is worth — none of which is in any model here. What the arithmetic supplies is the shape both sides of the crossing have: the benefit is concave and the cost is convex, so the optimum is interior, unique, and well below the geometric ceiling.

The practical reading is the one a specification writer needs. The last two picks per centimetre in a dense cloth are not two per cent more difficult; they are a different proposition entirely, and a specification that moves a sheeting from 26 to 28 picks has asked for the part of the curve where tension buys almost nothing and costs a great deal. That is the same warning the force ceiling gives about the catalogue, arriving as a rate rather than as a count.

What was counted, and how #

The catalogue is enumerated rather than quoted: every ratio a change train of a hundred and one wheels can make, reduced to lowest terms so that no gear pair is counted twice, filtered to the range, and sorted. It comes to 4,825 between 8 and 40 picks per centimetre, which is the take-up ladder’s own number recomputed in the same run as everything it is compared with.

The force at each pick density is the previous rung’s: a Peirce solution at that spacing, a central difference for the crimp’s derivative, and the same quantity computed twice by different routes and required to agree to one part in a million.

The sweep steps the pick density by a quarter of a thread per centimetre and stops on whichever ceiling it meets first, recording which one it was — that distinction is the substance of this rung, and a sweep that reported only a number would have lost it.

Two assertions guard the result. The force must rise with the pick density across the whole sweep, which would fail immediately if the sign of the crimp derivative were wrong. And the sweep must reach more than a handful of densities before stopping, so that a cloth whose geometry refuses at the very first step is reported as a refusal rather than silently returning an empty range.

The warp tension throughout is one per cent strain on a 25 tex cotton yarn, and every force scales with it exactly.

Where the model stops #

The frictional resistance is missing and every number here is a lower bound because of it. The reed pushes the new pick past picks that are already gripped by the warp, and that resistance is real and is not modelled. Including it would raise every force and lower every ceiling, so the reachable fractions above are optimistic in a known direction.

A loom’s available force is not on this page. Every row of the table is a horizontal line drawn at a stated height, and which height belongs to a given machine is a question about sleys, cranks and inertia rather than about cloth. The numbers chosen — 200 to 2,000 newtons per metre — bracket what is reported for cotton weaving, and no loom in particular is being described.

The third stopping reason is a model limit and not a physical one, and on six of the eight cloths in the table it arrives before either of the others — so the ceilings reported for those cloths are where the model stops rather than where the loom does.

The take-up catalogue assumes a particular change train. A hundred and one wheels and a constant of two thousand is one machine’s; another train gives another catalogue, and the 4,825 would move. What would not move is the shape of the finding, since the density of any such catalogue is far finer than the range a ceiling leaves.

And nothing here is about what happens past the ceiling. A loom asked for a pick density it cannot reach does not stop; the fell sits further forward and the cloth comes out more openly than the gear says, which is a real and common fault. Predicting how much more openly needs the fell’s equilibrium position, which is a dynamic problem.

The generalisation #

The shape here is a common one and is usually met the wrong way round.

A catalogue of settings is not a catalogue of outcomes. Two independent limits — one on resolution and one on range — apply to the same dial, and quoting either alone gives a misleading account of the machine. A stepper motor with sixteen thousand microsteps and a torque ceiling has exactly this structure; so does a variable-speed drive, a pipette, a furnace controller.

The useful discipline that follows is to ask of any quoted capability which of the two it is. “Ninety-nine times finer” is a resolution claim and is true. “The weft direction is the free one” is a reach claim and is false. They are made of the same number and they are not the same statement, and it took a force to tell them apart.

The second half is about ceilings that stop moving. When two ceilings sit close together, spending on the lower one has a hard stop, and the stop is invisible until the second ceiling is computed. A loom-buyer comparing beat-up force between machines is buying something real up to a thousand newtons per metre on this cloth and nothing at all above it — and no amount of information about the loom would reveal that, because the limit that takes over belongs to the yarn.

Who found it, and when #

The take-up gear’s catalogue is an artefact of nineteenth-century loom practice and its enumeration is this site’s; change wheels are tabulated in every weaving handbook and nobody counts what the table reaches.

The observation that the cloth fell sits where a balance puts it, rather than where the take-up asks, belongs to Greenwood and Cowhig in the 1950s. The practical consequence — that a loom asked for too many picks per centimetre delivers fewer — is loom-room knowledge of long standing and is usually described in terms of the fell moving rather than in terms of a ceiling.

Putting the two together, so that a force becomes a count of reachable settings out of an enumerated catalogue, is this site’s, and it is the sort of thing that becomes possible only when both halves happen to have been computed for other reasons.

Where the ladder goes next #

The beat-up ladder’s own next rung is the missing frictional resistance, which is the crossing force applied at the fell — and which would turn every ceiling on this page from a lower bound into a value.

Sideways, the same warp tension that multiplies the beat-up force is what the shed is already spending of the warp’s strength, so raising the tension to reach a closer sett is a trade against warp breaks that both rungs can now price.

Further out, the geometric ceiling is the jamming argument arriving from a new direction, and the fact that a finish moves it — a calender flattens the yarn and lets the same threads sit closer — is the connection to the finishing field that this ladder has not yet made.