The finest colour-and-weave effects need the rarest loom

Colour and weave is the effect a collection of illustrated cloths cannot leave alone: thread an ordinary weave with an order of coloured threads and a pattern appears that is in neither. A houndstooth is a 2/2 twill with four dark and four light in both systems, threaded on a draft that is a matrix like any other, and nothing about its interlacement is unusual at all.

A weft stripe is counted in pairs of picks because of where the shuttles are. A warp’s colour order is laid out once at warping and the loom never thinks about it again; a weft’s is thrown one pick at a time by shuttles that cross the cloth and stay where they land. On a loom with boxes at one side only, the shuttle that goes out must come straight back, so every coloured run is an even number of picks. On a loom with boxes at both sides, odd runs become possible and are paid for in shuttles. A loom that can pick from either side at will has no constraint but the number of colours.

Every named colour-and-weave effect is a weft order before it is a pattern. So each of them belongs to one of those three looms, and which loom is arithmetic. That is the census this account runs, and its answer has an awkward shape: the effects a designer would call simplest are the ones the cheapest machine cannot make.

The warp is free and the weft is not #

The asymmetry is worth stating plainly, because a colour-and-weave effect is usually drawn as a square of point paper with a colour order written along the top and down the side, and the two look alike there.

They are not alike. The warp’s order is a decision taken at the warping mill: threads are wound on to the beam in the stated sequence and every pick that follows sees the same sequence, at no cost in apparatus whatever. A four-colour warp order and a forty-colour one cost the same to weave, and a stripe’s shaft cost is a set union rather than a sum whatever its colours.

The weft’s order is a decision taken on every pick, by a mechanism that has to have the right shuttle on the right side at the right moment. So the whole price of an effect is in one of its two systems, and since a colour-and-weave effect needs its order in both systems by definition, the weft order is the effect’s price.

The classification is the parity, which is why the disputed widths do not matter #

The counts these effects are written at vary between sources and always have. A shepherd’s check is given at four and at six; a houndstooth at four and at eight; a glen check’s bands differ between districts and between mills; a gun club is written at four in three colours and at other sizes besides.

That variation would sink a census that depended on the numbers. It does not touch this one, because a one-sided box loom’s constraint is a constraint on parity. An effect written at four and an effect written at six are in the same class, and an effect written at one is in a different class at every size anybody writes it at.

And no source disagrees about the one thing that matters. Nobody writes a hairline at two threads, because a hairline is a hairline precisely by being one thread; nobody writes a houndstooth at three. The disputed numbers are all even and the undisputed ones are all one, so the classification is stable against exactly the variation the table contains.

That is an unusually comfortable position for a census over trade vocabulary to be in, and it is worth naming as a general habit: when a classification depends only on a residue, the disputes about the value are irrelevant to it. The reed’s own beat against a repeat is decided by a common factor in the same way.

One-and-one cannot be thrown by any shuttle loom #

The sharpest result in the table is the one about the simplest order there is.

End-and-end, hairline and log cabin are all one dark thread and one light, over and over. In the warp that is nothing: two cones at the warping creel. In the weft it is impossible on either shuttle rule, and the reason is a drift rather than a shortage.

A shuttle crosses the whole cloth on every pick, so the side it is thrown from alternates: left, right, left, right, for as long as the loom runs. A colour thrown at every other pick is therefore thrown from the same side every time. Its shuttles arrive on the far side and are never picked from there, and the other colour’s shuttles arrive on the near side and are never picked from there either. Adding shuttles postpones the problem by exactly the number added; it does not solve it, because the imbalance is one shuttle per repeat and repeats do not stop.

So the finest colour order there is needs a loom that can pick from either side at will — a revolving box, or a machine that selects its weft and does not need it back. The effect that costs nothing in the warp costs the most in the weft, and the two facts are about the same one-and-one order.

A tattersall is a hairline that only looks coarse #

The one-and-one result reaches further than the effects that are visibly fine, and the case that shows it is the one that looks least like a hairline.

A tattersall is a wide light ground with a single coloured thread at intervals — an overcheck rather than a check, with nine or a dozen threads of ground between the lines. It reads as a coarse pattern. Its colour order contains a run of one.

One odd run is enough. A one-sided box loom cannot throw a run of one whether it sits between other runs of one or between runs of eleven, because the shuttle it needs is on the wrong side after a single pick. So a tattersall belongs with the hairlines and not with the checks, and the shortest band in its order is the whole of what decided that.

That is the reading a designer would get wrong from the drawing. An effect’s cost is set by its finest feature, not by its general scale, and a large quiet pattern with one fine line in it is priced as a fine pattern.

Two-and-one is cheap and three-and-one is not #

The class that a parity test alone would get wrong is the one the second loom exists for, and it separates two orders a designer would think of as the same kind of thing.

A two-and-one step — two picks of dark, one of light — has an odd run, so a one-sided loom refuses it. Its repeat is three picks, which is odd, so the loom’s own cycle is six; over those six picks each colour is thrown twice from the left and twice from the right. It balances, and boxes at both sides throw it, at a cost of three shuttles.

A three-and-one — three dark, one light — also has an odd run, and it does not balance. Its repeat is four picks, so the loom’s cycle is four, and the dark colour occupies picks one, two and three: two throws from one side and one from the other. Every repeat leaves a dark shuttle stranded, and no number of shuttles fixes a drift.

So the three rules are not a scale of coarseness. They are three different questions — is every run even, does every colour balance across the loom’s cycle, and are there enough boxes — and the middle one is not implied by the first. A designer who knows that odd runs are dear knows half of it; which odd runs are dear is decided by an arithmetic on the repeat’s length that has nothing to do with how the pattern looks.

Only one effect in the table lives in the middle class #

Running the three rules over the whole table gives a distribution that is not what the three-way division suggests.

Six effects are thrown by the cheapest loom: birdseye, crow’s foot, houndstooth, shepherd’s check, gun club and glen check. Every one of them has all its runs even, and the three-colour gun club needs only a third box.

Five need a loom picking at will: end-and-end, hairline, log cabin, tattersall and three-and-one. Four of those are one-and-one orders and the fifth is the unbalanced step.

One sits between them: the two-and-one step.

The middle class is nearly empty, and it is nearly empty in the named effects while being a substantial share of the possible orders. Over all 254 two-colour orders of eight picks, 28 are throwable one-sided, 68 two-sided and all 254 at will — so the middle class holds forty orders that the cheap loom refuses and the dear one is not needed for, which is sixteen per cent of the space and one twelfth of the named vocabulary.

The named vocabulary is concentrated at the two ends of the space, and that is the census’s real finding. It is not that the trade avoided the middle class; it is that the effects which got names are the ones a box loom throws and the ones fine enough to be worth a better loom, and the orders in between — coarse enough to be unremarkable and odd enough to be awkward — were never worth naming.

What the cheap loom would have to change #

An effect a box loom cannot throw can always be approximated by a box loom, and the arithmetic says exactly what that costs.

The nearest even order to one-and-one is two-and-two, which is birdseye. So a mill with box looms and an order for hairline suiting can weave birdseye, and the result is a different named effect with twice the band and a visibly larger spot. The substitution is not a degradation of the hairline; it is a different cloth with its own name, which is why both names exist, and why a line one thread wide is a different object from a line two threads wide in the warp as well.

The nearest even order to a tattersall’s one-and-nine is two-and-eight or two-and-ten, and here the substitution is much less visible: the overcheck line becomes two threads instead of one against a ground of eight or ten, which doubles the line’s weight and leaves the pattern’s scale alone. That is a real design decision with a real answer — a tattersall coarsens gracefully and a hairline does not — and the difference is that the hairline’s fineness is the effect while the tattersall’s fineness is a detail of it.

And there is a third route that the census’s own asymmetry supplies. An effect can keep its fine order in the warp and lose it in the weft. A one-and-one warp on a two-and-two weft is a cloth a box loom throws, and it is neither the hairline nor the birdseye: it is an asymmetric colour-and-weave effect, which the tartan’s whole argument turns on not being. Whether any named effect is of that kind is a question this table cannot answer, because every entry in it uses one order in both systems.

What a pivoted sett does to the same question #

A tartan is the case where this arithmetic has already been run, and it produced a result worth setting beside the effects.

A tartan’s sett is written as a half-sett that reverses at two pivots, and whether the pivot threads are counted once or twice decides the parity of the pivot blocks. Counted once, a pivot run of c threads becomes 2c − 1 in the cloth and is always odd; counted twice it becomes 2c and is always even. The difference is one thread per pivot, which nobody looking at a finished tartan can see.

So a convention about how to write a sett down — and a tartan is one sett read in both systems — decides which machine can weave it, and the convention is not about weaving. The same one-thread question that separates a hairline from a birdseye separates two ways of writing the same tartan — and in the tartan’s case the cheaper answer is available for nothing, because the two cloths are indistinguishable.

The named effects have no such escape. A hairline written at two threads is a birdseye, and everyone can see the difference.

How the effects were sorted #

Each named effect’s colour order is written as a cyclic list of runs, expanded into the pick-by-pick sequence a loom sees, and put to each of the three insertion rules in turn. The one-sided rule asks whether the repeat is even and whether its runs line up with the loom’s pairs. The two-sided rule counts, over the loom’s own cycle — the repeat if it is even, twice it if it is odd — how often each colour is thrown from each side, requires the two to be equal, and prices the shuttles as the spread of each colour’s running balance. The at-will rule asks only whether there are boxes enough for the colours. The cheapest rule that succeeds is the effect’s class.

Five things are checked. Every effect is throwable one-sided exactly when every run is even and there are boxes enough, checked effect by effect rather than stated — which is the claim that makes the disputed widths harmless. No effect with a single-thread band is thrown by a loom with boxes at one side. Every one-and-one order needs a loom picking at will, and there are at least three of them in the table. The two-and-one step is the one entry whose cheapest loom is the middle one, which is the case that would fail if the balance test collapsed into the parity test. And an effect written at four and one written at six are in the same class, checked on the houndstooth against the shepherd’s check, which is the disputed-width claim tested rather than argued.

The colour orders themselves are quoted reference data and not results. The general census over all orders of a stated length is the same three rules run over every word in the space.

Where the model stops #

The orders are the ones the books give, and the books disagree. Every count in the table can be found written differently somewhere, and the essay’s defence is that the classification depends on parity alone rather than that the counts are right. An effect mis-transcribed here at an odd count would be mis-classified, and the only entries at risk are the ones already in the at-will class.

A loom is more than its boxes. A drop-box motion has a limit on how fast it can change boxes, and the shed it throws into is an extension of every end, so an order that changes colour on every possible pick may be feasible and slow, and a mill choosing between two orders is choosing between two speeds as well as two machines. Nothing here counts picks per minute.

Shuttleless looms are outside the model. A rapier or projectile loom with weft selection has no shuttle to bring back and throws anything, which is the at-will rule; what it costs in yarn at the selvedge, and what a tucked or leno selvedge does to a fine colour order, is a question about edges rather than about orders.

And the pattern is not computed. This census prices the order and says nothing about what the order and the weave make of each other — whether two colours collide at an intersection, whether the pattern reads at the sett intended, whether a substituted order produces a recognisable version of the effect. Colour and weave as a two-colour problem is where that arithmetic lives, and the two have not been run on the same construction here.

Still open: whether the asymmetric effects were ever woven #

The census assumes what the vocabulary assumes: that a colour-and-weave effect uses one order in both systems. The arithmetic says there is a whole family it is ignoring — effects with a fine warp order and a coarse weft one, which a box loom throws and which are neither of the two named effects they sit between.

Such a cloth is cheap to weave, easy to draw and has no name in any of the sources consulted here. That is either evidence that it looks bad, or evidence that nobody looked. The drawing is one line of arithmetic and the judgement is not: a one-and-one warp on a two-and-two weft makes a pattern that is a hairline in one direction and a birdseye in the other, and whether that reads as a designed cloth or as a mistake is a question about a visual system rather than about a loom.

If it reads as a designed cloth, the finding is that the box loom’s constraint has a cheap escape the trade did not take. If it reads as a mistake, the finding is that a colour-and-weave effect genuinely needs its symmetry — which would be the first result here about why an effect is symmetric rather than about what its symmetry costs.

Who worked it out #

The colour orders are trade reference data of long standing and appear in every book of woven design. The three insertion rules are a statement of how a shuttle loom works and are equally old as practice. Sorting the named effects by them, the result that a single odd run in a coarse order prices the cloth as a fine one, and the separation of two-and-one from three-and-one by a balance across the loom’s cycle rather than by parity, were computed directly.