A weft stripe is counted in pairs of picks

A check is two stripes: an order of coloured ends down the warp and an order of coloured picks across it, chosen separately and quoted separately. On point paper the two are the same kind of object, a sequence of colours, and a designer can write either one as freely as the other.

A loom does not treat them alike. A warp’s colour order is laid out once, when the warp is made, by putting the right packages in the right places on the creel. From then on every end is simply there, in its colour, for the length of the piece, and the loom never has to do anything about it. A weft’s colour order is thrown, one pick at a time, by a shuttle that carries one colour, crosses the cloth and stays on the side it lands on. Which orders a loom can throw is a question about where its shuttles are, and the answer is not every order.

A shuttle is always on the side it last arrived at #

A shuttle loom throws its shuttle from alternate sides: left to right on one pick, right to left on the next, the picking motion at each side taking its turn. The shuttle that has just crossed is sitting in the box on the far side, and unless something swaps it for another, it is the one thrown back.

Changing colour is swapping shuttles, and a shuttle can only be swapped where there are boxes to swap it in. A box motion is a stack of shuttle boxes at the side of the loom that drops or rises to bring a different shuttle level with the picking stick. Where the stacks are — at one side of the loom, or at both — decides which colour changes are possible and when.

Nothing about the weave enters this. A plain weave, a twill and a satin woven with the same weft order meet the same constraint, because the constraint is on the order in which coloured threads can be delivered and not on what the shed does with them. That is the sense in which a weft stripe is a different object from a stripe partitioning the warp: the warp’s partition costs shafts and the weft’s costs shuttle trips.

Nor does a jacquard help. It gives every end its own hook and leaves the weft exactly as it found it — one thread per pick, from whichever shuttle the boxes have brought to the picking stick — so the most flexible shedding machine there is weaves under the same weft rule as the plainest.

And the pattern a weft order makes on the cloth is not the order. A colour order beats the weave it is threaded on, so what a reader sees repeats at the least common multiple of the two; but the loom meets the order first and the pattern second, and a pattern that would need an odd band to draw is refused before any weave is chosen.

Boxes at one side, so pairs #

Put the boxes at the left, four of them, and a single box at the right.

A shuttle thrown from the left lands in the single box at the right. The next pick must be thrown from the right, and the only shuttle there is the one that has just arrived. So it goes straight back, in the same colour, and only when it is home at the left can the stack drop and a different shuttle take its turn.

Every change of colour therefore happens between a pair of picks, and every coloured band is an even number of picks wide. A band of four is two out-and-back trips; a band of three would need the shuttle to stay at the right after its third crossing, with a shuttle of the next colour waiting there to be thrown, and there is no box at the right to wait in.

The rule is exact rather than a tendency. Across every order of two colours up to ten picks and every order of three up to eight, a loom with boxes at one side can throw an order exactly when every run round its repeat is even — the two conditions pick out the same orders, one by one.

An odd repeat cannot be thrown at all #

The same arithmetic has a consequence for the whole repeat rather than for a run within it.

A repeat of odd length makes the loom’s picking and the colour order slip against each other: the order returns to its start after an odd number of picks, so the second time round every band begins on the other side of the loom. Pairs that lined up with the bands the first time are out of step with them the second, and a band that was thrown in pairs is now split across a boundary.

So no order of odd length can be thrown on a one-sided box loom unless it is a single colour. The census puts that at nought orders out of six at three picks, nought of thirty at five and nought of five hundred and ten at nine, and a designer who counts a weft repeat in odd numbers has written a pattern this loom cannot weave in any arrangement.

What boxes at both sides buy #

Put a stack of boxes at the right as well, and the picture changes in a precise way.

A shuttle still crosses on every throw, and it still has to be on the side it is thrown from. But now either side can swap, so a colour can be picked up at whichever side it is waiting. What the loom needs is simply that each colour is thrown as often from the left as from the right over its own cycle, so that its shuttles come home. A colour thrown twice running from one side needs a second shuttle of that colour waiting there.

That admits odd bands. The order in the figure has runs of one, one, two, one, one and two picks, and it is thrown throughout: each colour is used four times in the repeat, twice from each side, with two shuttles of each colour so that one is always waiting where the next throw of that colour has to leave from.

It still refuses pick and pick. One pick of each colour in turn puts every pick of the first colour on a left throw and every pick of the second on a right one, so the first colour’s shuttles all end up at the right and never return. No number of shuttles fixes that; it takes a picking motion that can throw twice from the same side, a pick-at-will motion, or a loom with no shuttle at all.

How few orders of any length survive #

The two rules can be put to every order a designer could write, and the proportions are the practical result.

At eight picks and two colours, a one-sided loom can throw 28 of the 254 orders there are — eleven per cent — and a two-sided loom 68. Forty of those sixty-eight have odd bands in them, which is the whole of what the second stack of boxes bought.

The shares fall as the repeat grows, and they fall much faster for the one-sided loom. At twelve picks a one-sided loom throws three per cent of the two-colour orders and a two-sided one twenty-two and a half; with three colours at twelve picks the figures are a quarter of one per cent against six and a half. A loom that picks at will throws every one.

Nobody designs by drawing orders at random, and a stripe designer uses a handful of them. What the shares measure is how often a handful chosen for how it looks will include one the loom refuses — and for a weft of any length on a one-sided box loom, the answer is nearly always, unless the handful was chosen in pairs.

Odd repeats go the other way #

A two-sided loom has an unexpected advantage at odd lengths, and it is worth stating because it reverses the one-sided result exactly.

An odd repeat slips against the picking every time round, so over two repeats every pick of the order is thrown once from the left and once from the right. Every colour is then automatically balanced, whatever the order is, and the only question left is whether there are enough boxes for the shuttles. At three, five and seven picks the census finds every two-colour order throwable on a two-sided loom with four boxes a side; at nine picks, 492 of 510, the rest wanting more shuttles than seven boxes hold.

So the loom that can throw no odd repeat and the loom that can throw nearly all of them differ only in where the boxes are.

The boxes also limit the colours #

A loom with four boxes at one side can hold four shuttles, one per box, and at the moment one of them is thrown its box is empty to receive it back. So a one-sided loom throws at most as many colours as it has boxes, and each colour has exactly one shuttle.

With boxes at both sides, shuttles can wait at either, but one box must always be empty on the side a shuttle is about to land. Four boxes a side hold seven shuttles, and an order whose colours need more than seven between them is refused on that count alone — which is where the eighteen nine-pick orders above fail.

The same budget prices the other use of a second weft. A shaded weave gets more tones by throwing two colours rather than by enlarging its repeat, and every extra tone bought that way is a colour change the boxes have to make — so the tones a box loom can add are limited by the same pairs and the same shuttles as a stripe.

A dobby stores lifts and a box chain stores colour changes, and the two budgets are independent. A cloth with a complicated weave and a simple weft order is cheap in boxes; a plain weave with a long weft stripe sequence is expensive in them, and nothing in the draft reveals which.

A tartan is designed for its weft #

Everything so far applies to any check, and it binds hardest on the one check that is defined by using the same order twice.

A tartan’s sett is a single sequence used in the warp and the weft alike. The warp could take any sequence at all. The weft can take only what the loom can throw. So the weft’s rule is imposed on the warp as well: a sett a one-sided box loom can weave is a sett all of whose bands are even, in both directions, whether or not the warping would have cared.

It binds a stripe that has no colour in it too. A shadow stripe is two twists, and in the weft a band of Z-twisted picks beside a band of S-twisted ones is two shuttles exactly as two colours are: the loom cannot tell a change of twist from a change of colour, and a shadow check woven with a band of three picks of one twist is refused on the same count as a coloured one.

That is a constraint on the design, not on the weaving of it. A tartan designed on paper with a band of seven threads is a tartan that cannot be woven on such a loom — not woven slowly, or awkwardly, but not at all — and the fix is to change the design by a thread, which in a tartan changes what the sett is.

The pivot decides it #

A tartan’s sett is written as a half-sett and mirrored, and the mirroring has to decide what happens at the two pivots, where the order turns round. Counted once, a pivot thread is the middle of its block and appears a single time, so a pivot run written as c threads becomes 2c − 1 in the cloth — always odd. Counted twice, it becomes 2c — always even.

The district sett these essays draw has half-sett runs of ten, four, fourteen, four and four. Counted once, its weft order is seventy picks with blocks of 4, 14, 4, 7, 4, 14, 4 and 19; both pivot blocks are odd and a one-sided box loom refuses it. A two-sided loom throws it, needing five shuttles for three colours. Counted twice, the order is seventy-two picks, the pivot blocks are eight and twenty, and a one-sided loom throws it with a shuttle per colour.

The shepherd’s check is starker. Written as a half-sett of six and six, counted once it is two blocks of eleven, and neither shuttle rule can throw it: with only two colours an odd block unbalances both of them at once. Counted twice it is two blocks of twelve and any box loom weaves it. The fine sett in the same table behaves the same way — blocks of three and seven refused by both, four and eight thrown by both.

One thread that nobody sees #

The difference between the two countings is a single thread at the middle of each pivot block — one pick in nineteen, or in eleven — and in the finished cloth it is invisible. Nobody looking at a tartan can tell whether its pivot block is nineteen threads or twenty.

On the loom it is the difference between a sett that can be thrown and one that cannot. A written convention that looks like a question of tidiness is a question of which machines can weave the result, and a register that expanded its pivots one way would describe setts a box loom could always weave, while one that expanded them the other way would describe setts with odd blocks at every pivot.

Whether any register’s convention was chosen with the loom in mind is a question about history that the arithmetic cannot answer. What it can say is that the two conventions are not equivalent, that the choice is not cosmetic, and that the expansion computed here — pivots counted once, which is what the tartan essay uses — produces setts a one-sided box loom refuses every time.

A loom with no shuttle has no rule #

Most weft is now inserted by rapiers, jets of air or water, or projectiles, none of which carries a package across and leaves it there. Each pick is drawn from a stationary package and cut at the selvedge, and a colour selector simply offers the next rapier a different thread. Every order is throwable, with no parity, no balance and no shuttles to count.

The rule has gone because the thing that made it — a package that stays where it lands — has gone. What arrives in its place is a cut end at both selvedges on every pick, which is why such looms weave a leno selvedge to hold them; the colour freedom and the frayed edge are one change seen from two sides.

That also dates the constraint. A stripe sequence woven freely on a modern loom and copied onto a box loom may not be weavable at all, and a sett that has always been woven in even bands may owe its evenness to a machine nobody uses any more.

How the orders were counted #

A weft colour order is written as a sequence of colour indices and read round its own repeat. For the one-sided rule, the order is checked for pairs: whether, starting at one of the two possible offsets, every pick is the same colour as the pick after it, all the way round. That is compared, order by order, with the runs of the order read round the repeat, and the two agree on every two-colour order up to ten picks and every three-colour order up to eight.

For the two-sided rule, each colour’s throws are counted by side over the loom’s own cycle — the order’s length if even, twice it if odd — and its running balance of left throws over right throws is followed through the cycle; a colour is throwable if the balance returns to where it started, and it needs as many shuttles as the balance’s range. The traces in the figures start each colour with exactly that many shuttles, on the sides the balance says, and follow them pick by pick.

Where the model stops #

Alternate picking is assumed throughout. It is how a shuttle loom normally runs, and it is the assumption that makes the rule; a loom with a picking motion that can throw twice from one side is not bound by it, and the census counts such a loom separately as picking at will.

Shuttles of one colour are treated as interchangeable. In practice two shuttles of the same colour carry two packages of the same yarn, which may differ in shade or tension enough to show — which is the same problem a feeder band is in a knit — so the extra shuttles a two-sided loom needs for odd bands are not free in appearance even when they are free in arithmetic.

Nothing here is about speed. A box motion takes time to change and some changes are faster than others; which orders a mill finds worth weaving is a question about that, and the rule only says which it can weave at all. Nor is a shuttle running out of weft modelled, which swaps a shuttle mid-band and is handled by the loom rather than by the design.

Still open: which colour-and-weave effects the weft forbids #

Colour and weave builds its patterns from a colour order in both systems, and several of the classic ones are written with bands of one — a one-and-one order that makes a hairline, a two-and-one that makes a step. Every one of those is a weft order before it is a pattern, and the rule above sorts them into those a box loom throws, those it throws only with boxes at both sides and extra shuttles, and those that need a loom picking at will. Which of the named colour-and-weave effects fall into which class — and whether the ones that need the rarest machine are the ones with the fewest names — is a census that has not been run.