The setts a loom can reach

Take any draft, turn it through a quarter turn, and what comes out is a draft. Its floats are the old floats with warp and weft exchanged, its interlacing count is the same number, its layer count is the same number, and the whole four-by-four catalogue maps onto itself. This site’s own census of what a loom can weave is symmetric about its diagonal for exactly that reason: three shafts and four treadles reaches the same 110 cloths as four shafts and three treadles, because the catalogue does not know which direction is which.

The machine knows completely. And the asymmetry is not a small one hiding in a corner of the mechanism — it is in the two most basic numbers a weaver sets.

The warp sett is a comb #

A reed is manufactured at a pitch. Reeds come in a catalogue — in the metric system, dents per ten centimetres at a five-unit step — and a warp is drawn through one at a whole number of ends per dent, which is one, two, three or four for ordinary cloth and above four is crowded.

So the achievable warp setts are the products of a catalogue with a small set of whole numbers, corrected by the contraction the previous rung computed. That is a lattice, not a continuum, and it has two properties a continuum does not.

It has gaps, and they grow. In the range ordinary cloth is woven in the gaps are half a thread per centimetre or so, which is fine enough not to matter. Above the range a single reed can cover on its own, the only way up is more ends per dent, which multiplies the step as well as the value — so the widest gap in the whole reachable range, 4.2 to 84 threads per centimetre, is 2.10 threads per centimetre, sitting between 63 and 65.1 where only four ends per dent can reach.

And it is fixed for the piece. Changing the reed sett means drawing every end of the warp through a different comb. There is no adjustment; there is a re-reeding, which means stopping the loom and doing several hours of work.

The pick density is a gear #

The take-up motion turns the cloth roller through a train of wheels, so the picks per centimetre is a constant of the loom multiplied by a ratio of two whole numbers taken from a wheel set. Two ratios that reduce to the same fraction give the same density, so what a wheel set reaches is a Farey set scaled by the loom’s constant.

A wheel set running from twenty to a hundred and twenty teeth gives 6,340 distinct densities between four and sixty picks per centimetre, and 4,825 of them fall in the eight-to-forty range this essay compares over. The mean spacing there is 0.0066 picks per centimetre, and the largest gap in the whole reachable range is half a pick per centimetre — at the very bottom of it, where the ratios are extreme.

And it can be changed by swapping one wheel, which is a job of minutes.

Ninety-nine to one #

Over eight to forty threads per centimetre, which is where nearly all apparel and household cloth lives, the mean spacing of the reachable warp setts is 0.656 threads per centimetre and of the reachable pick densities is 0.0066. The ratio is 99.

That is a measured number rather than a slogan, and it is worth saying what it depends on. The wheel set is an ordinary one and the reed catalogue is an ordinary one; either could be enlarged. What could not be changed is the kind of thing each is. A reed’s sett is a manufactured pitch times a whole number, and whole numbers are coarse when they are small; a gear ratio is one whole number over another, and ratios of whole numbers are dense. Two ends per dent and three ends per dent are a fifty per cent step apart. Wheels of sixty and sixty-one teeth are one and a half per cent apart.

So the factor of a hundred is a consequence of the mechanisms being multiplicative and divisive respectively, and it would survive any reasonable change to either catalogue.

Crowding the reed makes the lattice coarser #

The reachable warp setts are a catalogue multiplied by a small whole number, and that multiplication has a consequence for resolution that runs against ordinary mill practice.

A metric reed catalogue steps by half a dent per centimetre, so the sett step is half a thread per centimetre at one end per dent, one at two, one and a half at three and two at four. The step is proportional to the ends per dent, and the sett is too — so at any given target sett, the relative resolution is best with the fewest ends per dent the reed stock can reach.

Twenty-four ends per centimetre is available as twenty-four dents at one, twelve at two, eight at three or six at four. All four give the same cloth sett and the same mean spacing of the ends. They do not give the same neighbourhood: the first can be moved by half a thread per centimetre and the last only by two.

That matters because a mill does not usually choose a reed for resolution. It chooses one it owns, and owning a reed for every count in the catalogue is expensive, so the standing practice is to keep a modest stock of coarse reeds and reach the setts wanted by crowding them — three and four ends to a dent — which is exactly the choice that quadruples the step.

So the lattice a mill actually works in is coarser than the catalogue’s, by whatever multiplier its reed stock forces on it, and the cloths most likely to be crowded are the close ones where the gaps are widest to begin with. The beat the denting leaves against the repeat is the other cost of the same decision, and the two are usually weighed against the price of a reed rather than against each other.

The lattice matters only where somebody writes a number down #

Whether the coarse dial is a problem depends entirely on which quantity the specification names, and the two common forms behave completely differently.

A specification naming ends and picks per centimetre meets one of its two numbers exactly and the other approximately, and the approximate one is the fixed one. That is the case the previous section is about.

A specification naming a cover factor, or a weight in grams per square metre, is a different situation altogether. Both of those are functions of both setts, so a warp sett that landed a little away from where it was wanted can be compensated by moving the picks — and the picks can be moved to a part in ten thousand. The fine control absorbs the coarse control’s rounding, exactly, and the specification is met.

What is not recoverable is the balance. Compensating a warp sett by moving the picks changes the ratio of the two setts, which is what balance is, and a cloth that meets its weight by an adjusted pick density is a slightly more weft-dense cloth than the one specified. That shows in the crimp division, in the two directions’ extension, and in everything the interchange decides.

So the lattice is invisible in one specification, visible in another, and displaced into a third quantity in the case where it is compensated. Which of those happens is decided by how the specification was written, and by nobody who knew the lattice existed.

What the matrix does not know #

The comparison matters because the cloth side of this site treats the two directions as interchangeable and is right to.

Balance is the mean of the weave matrix, and exchanging warp and weft complements it. The interlacing count is a sum over both directions. The layer count is a property of a digraph that treats ends and picks alike. The catalogue of four-by-four cloths is closed under transposition, and a loom of s shafts and t treadles reaches exactly as many cloths as one of t shafts and s treadles — 110 either way at three and four, and that symmetry is a fact about the catalogue and not a coincidence of the counting.

None of that survives contact with a machine. The warp is a set of threads held under tension for the whole of weaving, at a spacing fixed before the first pick; the weft is a single thread laid in one at a time, at a spacing that can be changed mid-piece. That asymmetry is the reason the warp shrinks more than the weft, it is the reason the two thread systems’ crimps are unequal off the loom, and it is the reason a weaver’s two setts are decisions of completely different kinds.

What was counted, and how #

Both sets are enumerated rather than described.

The reed set is every product of a catalogue count with one, two, three or four ends per dent, converted to a cloth sett by multiplying by one plus the weft crimp, deduplicated at six decimal places, and sorted. The gaps are read off the sorted list and the widest is reported with the two values it lies between.

The wheel set is every ratio of two wheels reduced to lowest terms, so that two gearings giving the same density are counted once, scaled by the loom’s take-up constant and filtered to the plausible range. Reducing to lowest terms is load-bearing: without it a set of a hundred wheels appears to give ten thousand densities, and it gives 6,340.

The comparison then takes the mean spacing of each set over a common range — the span divided by one less than the count — rather than comparing counts directly, because the two sets do not cover the same span and a count on its own would be answering a different question.

The assertion that guards it is that the weft is the finely adjustable one. It could fail, and it would fail if the two sets had been built the wrong way round, which is exactly the kind of error a plausible-looking chart would not reveal.

What a lattice does to a specification #

The gaps only matter where somebody writes a number down, and that is worth being specific about, because most of the time nobody does.

A mill weaving to its own recipe picks a reed from what it has and records what it got. The lattice is invisible: the sett was never a target, it was an outcome, and the cloth is described afterwards by measuring it.

A mill weaving to a specification is in a different position. A customer’s sett is a number somebody chose, usually round, usually from a table, and there is no reason at all for it to be one the reed catalogue reaches. What happens then is that the nearest achievable value is used and the specification is met to within a tolerance nobody stated, which is fine in the middle of the range and stops being fine near the top where the gaps reach two threads per centimetre.

The asymmetry then does something slightly perverse. A specification’s pick density is met almost exactly, because the take-up will reach it, so the two numbers in the specification are met to completely different accuracies — and the one met precisely is the one that can be changed later, while the one met approximately is fixed for the piece. A reader comparing a measured cloth against its specification will find one number right and one number close, and the reason is not a measurement error.

Where the model stops #

The catalogues are ordinary and are stated. A mill with a wider reed stock reaches more setts; a loom with a finer wheel train reaches more densities. The factor of a hundred is computed from one plausible pair of catalogues, and what does not depend on them is the kind of difference — multiplicative against divisive — which is the part the essay rests on.

Ends per dent is capped at four here and the cap is a convention. Six and eight ends per dent are used for some cloths, particularly heavy ones, and they extend the lattice upward while making the gaps larger still.

The take-up constant is a stand-in. Real looms differ in their gearing and a positive take-up on a modern machine may be electronically controlled, in which case the pick density is effectively continuous — which makes the asymmetry larger rather than smaller and is the direction the trade has moved.

And a lattice is not the same as an obstacle. A weaver who wants 26.3 ends per centimetre and can have 26.0 has lost almost nothing. The gaps only bite at the top of the range and when a specification is written by somebody who does not know the lattice exists.

The generalisation #

Two quantities that a description treats as interchangeable can be produced by mechanisms of entirely different kinds, and the description will not say so.

That is the whole of it, and the diagnostic is worth carrying: the symmetry of a model is a property of the model. A matrix is symmetric under transposition because a matrix is a matrix. It says nothing whatever about whether the two indices are equally easy to control, equally cheap to change, or changeable at all once the process has started — and in this case one of them is fixed for the piece and the other is adjustable pick by pick.

Where this bites is in reasoning that moves between the two descriptions without noticing. A designer who reads off the matrix that warp and weft are interchangeable, and then specifies a cloth by exchanging the two setts, has produced a construction whose warp sett may not exist.

Who found it, and when #

None of this is a discovery. Reed catalogues are published, wheel sets are published, and every weaver knows that changing the picks is easy and changing the ends is not. The reason it is worth writing down is that the comparison is not usually made, because the two numbers are set at different times by different people and never appear side by side.

The transposition symmetry of the weave catalogue is this site’s own and is some years old. Putting it next to the machine’s asymmetry is what turns two familiar facts into a statement: the object is symmetric, the process is not, and the fabric that results carries the process’s asymmetry in every quantity that was ever measured on it.

That last point is the one with consequences. A tabulated fabric property measured warpwise and weftwise is almost never symmetric, and the usual explanations are about tension history and finishing. This adds a smaller and more mundane term: the two setts a fabric was specified with were chosen from sets a hundred times different in resolution, so one of them is very likely the number that was wanted and the other very likely the nearest thing available.

Where the ladder goes next #

The take-up ladder has two rungs and both are about what the machine does to a number the cloth is described by. What neither reaches is a force: the beat-up that fixes the pick spacing is a blow, its energy decides how close the fell can be driven, and this site has no model of it. That is the same shortfall the shed ladder records from the other side.

Sideways, the reed’s other consequence is a mark rather than a width; the quantity both rungs are computed from is the crimp; and what the two setts together decide about a cloth’s surface is the cover factor. Further out, the asymmetry between the two thread systems runs through the whole finishing field, starting with why the warp shrinks more.