A colour-and-weave look costs its cheaper order

The finest colour-and-weave effects need the rarest loom, and the reason was an asymmetry between the two thread systems. A warp’s colour order is laid out once, at warping, and costs nothing whatever its pattern — a stripe is a partition of the warp and its shaft cost does not care about colour at all. A weft’s is thrown one pick at a time by shuttles that stay where they land, and a weft stripe is counted in pairs of picks: a loom with boxes at one side can throw only orders whose every run is even; boxes at both sides add the orders that balance across the loom’s own cycle; everything else needs a loom that can pick from either side at will. So an effect’s price was its weft order, and the one-and-one orders — end-and-end, hairline, log cabin — were priced at the top.

That account ended on a route it could not follow. An effect can keep a fine order in the warp and a coarse one in the weft, and then a box loom throws it. The account’s table could not say whether that route mattered, because every effect in it used one order in both systems.

It matters, and more simply than that account put it. The route is not a compromise on the effect. It is the same cloth, and the claim that an effect costs its weft order turns out to be a claim about how a cloth is drawn rather than about the cloth.

A cloth can be woven lying across the loom #

Take any colour-and-weave construction: a weave, a warp order and a weft order. The face shows the warp’s colour wherever the warp is up and the weft’s wherever it is not.

Now weave the cloth the other way round. The threads that ran down the loom run across it and the threads that ran across run down; the old warp order becomes the new weft order, and the weave is turned a quarter with the systems exchanged. The fabric that comes off the loom is the same fabric, turned through a right angle with its face still up.

So a construction has two prices, one for each way it can lie on the loom, and the cloth’s price is the cheaper. For the asymmetric twill above that is the difference between the cheapest loom there is and the dearest. A mill with box looms and an order for that cloth weaves it as drawn; a mill that drew it turned would have decided it needed a machine it did not.

The trade already turns patterns through a right angle for its own reasons. An upholstery cloth railroaded carries its design along the bolt rather than across it, so that a sofa’s full width can be cut from its length without a seam. What railroading does for width, weaving across does for the loom.

Every look the small weaves make #

The question is how often it matters, and it has a definite answer over a stated family.

The family is twelve weaves — plain, a 2/2 hopsack, and 2/1, 1/2, 2/2, 3/1 and 1/3 twills in both hands, since a cloth turned a quarter reverses its twill’s diagonal — and every two-colour colour order of two to six threads that uses both colours and does not repeat within itself: 104 of them. Every weave with every warp order and every weft order is 129,792 constructions.

Many make the same cloth. A look is a coloured face up to where its repeat is started and which colour is which, so that two constructions share a look exactly when a viewer could not tell their cloths apart. The constructions make 4,036 distinct looks, a median of thirty constructions each.

Each look is priced twice. As drawn, by the cheapest weft order among the constructions that make it. Either way round, by the cheapest order in either system — equivalently, the cheapest weft order among the constructions that make it or make it turned.

As drawn, 2,214 of the 4,036 looks — 55 per cent — need a loom picking at will. Either way round, 1,244 — 31 per cent — do. Weaving across moves nearly half the dearest looks down, and it more than doubles the looks the cheapest loom can make, from 690 to 1,262.

Where the looks move to #

The census’s totals hide which way each look moves, and the moves are not symmetric.

Four hundred looks go from the dearest loom straight to the cheapest. They are looks with a single-thread band or an unbalanced run in one system and every run even in the other — fine one way, coarse the other. Five hundred and seventy go from the dearest to the middle: fine one way and odd but balanced the other, the two-and-one step’s arithmetic arriving in a warp. A hundred and seventy-two go from the middle to the cheapest.

And 1,244 go nowhere. Every construction that makes them, and every construction that makes them turned, has an order the one-sided and two-sided rules refuse in the system thrown. These are the looks that are fine, or unbalanced, in both directions at once, and no choice of how the cloth lies on the loom helps them.

The named effects are exactly the ones it cannot help #

The census’s most useful result is about the vocabulary rather than the counts.

Every named effect in the earlier table — end-and-end, hairline, log cabin, tattersall, birdseye, crow’s foot, houndstooth, shepherd’s check, gun club, glen check, and the two steps — uses one colour order in both systems. A cloth with the same order in both systems costs the same order whichever way it lies, so turning cannot change its price. The named effects are therefore, without exception, on the side of the census that weaving across leaves alone.

That is why the earlier table could be right about every entry and still wrong as a rule. Its claim that an effect’s price is its weft order is exactly true of a symmetric effect and of no other kind, and the trade’s vocabulary contains no other kind. The asymmetric looks weaving across rescues are the ones the vocabulary never named.

End-and-end is the clearest case. Its face, turned, is end-and-end again with its colours shifted, and its weft order either way is one-and-one. The finest effect in the vocabulary is priced at the top whichever way the loom is threaded, and that is not a fact about looms but about symmetry: an effect whose two systems carry the same order has only one order to be priced by.

A tartan is the one check that cannot be turned #

The earlier account set a tartan’s sett beside the named effects, and a tartan shows the limit of weaving across from the other side. A check is two stripes and a tartan is one: a tartan’s warp and weft carry the same sett, thread for thread, so its two orders are the same order by definition. Turned a quarter, a tartan is itself, and it costs the same loom whichever way it lies.

What decides a tartan’s loom is instead the one-thread question the earlier account found at its pivots — counted once, the pivot blocks are odd; counted twice, even — and that is a question about how the sett is written, not about which way the cloth lies. So the two cheapenings are complementary. A tartan is made cheap by its writing and never by turning; an asymmetric check is made cheap by turning and never by its writing, because its fine order is fine however it is written.

A plain check — two stripes that are not the same stripe — sits between them. If its warp and weft orders differ, it has two prices and weaving across chooses the cheaper, exactly as for any asymmetric colour-and-weave look; the colours in a check are just a coarser order.

A step can be woven cheaply if its step is in the warp #

The middle class of the earlier account — the two-and-one step, thrown by boxes at both sides because its odd runs balance — moves too, and the move is instructive because it goes the full distance.

A step in the warp and an even order in the weft is thrown by the cheapest loom. The same cloth drawn with its step in the weft needed boxes at both sides and three shuttles. The step is a feature of the look, and its price is a choice of which direction to draw it in.

The same holds for a single-thread line of any kind. An unbroken line is not a clean one, and a line one thread wide is a weft order the one-sided loom refuses outright; laid in the warp it is free. The same holds for a tattersall’s single overcheck thread. The earlier account found that one run of one prices a coarse pattern as a fine one, because the shuttle it needs is on the wrong side after a single pick. A tattersall whose overcheck runs one way only — a stripe of single threads in the warp over an even-banded order in the weft — costs nothing extra at all, laid in at warping, and it is a cloth a box loom throws.

How the share moves with longer orders #

The census is bounded at six threads, and the share of looks needing the rarest loom depends strongly on where the bound is drawn.

At two threads the only orders are one-and-one and every look needs the rarest loom whichever way. From three threads on the two lines separate and stay separated, and they zigzag together: an odd length adds orders whose repeat is odd, many of which balance across a two-sided loom’s doubled cycle, while an even length adds orders with more single-thread runs. At every bound from three to six, weaving across removes between a third and two thirds of the looks that needed the dearest loom — 61 per cent at three, 47 at four, 69 at five, 44 at six.

The zigzag says the fraction is not converging to a tidy number inside this range, and nothing here claims what it does at eight or twelve threads. What is stable is the direction: weaving across always helps a large share of the dear looks and never helps a symmetric one.

Turning does not change the repeat #

One thing weaving across is guaranteed not to move is the size of the pattern. A colour order beats the weave it is threaded on, and the surface repeats at the least common multiple of the order’s length and the weave’s in each direction. Turning exchanges the two directions and takes each multiple with it, so the turned cloth’s repeat is the drawn cloth’s repeat turned, thread for thread.

That matters for one practical reason. A mill that decides to weave a look across rather than as drawn changes its loom and its shuttle count and nothing about the design’s scale, its match across a seam, or the length of cloth one repeat takes. The saving is entirely in the weft insertion, which is where the whole price was.

What a mill actually gives up by turning a cloth #

The argument treats warp and weft as interchangeable, and in a finished colour-and-weave cloth they very nearly are — the effect lives in the order and the weave, and the draft is a matrix that does not care which system is which. On the loom they are not.

A warp is under tension for the whole of weaving and a weft is not, so the two systems crimp and relax differently, and the warp shrinks more in finishing. A cloth woven across puts the fine order in the system that shrinks more, which moves the pattern’s proportions by the difference between two shrinkages. A warp is also sized and a weft usually is not, so a yarn fine enough for a single-thread line in the weft may need a different finish to survive as a warp.

And the cloth’s width is the loom’s, so a design drawn at a certain width across becomes a design at that length along. For a small repeat that is nothing; for a panel design it decides whether the piece can be woven at all, which is the reverse of railroading’s reason.

None of those makes the turned cloth a different look. Each is a reason a mill might choose to throw the dearer order anyway, and the census prices the choice rather than making it.

What was counted, and how #

Each weave is built from its rule: plain, a 2/2 hopsack, and 2/1, 1/2, 2/2, 3/1 and 1/3 twills as Z and as S. Each colour order is a string of two colours of length two to six that uses both and is not a repetition of a shorter string. For every weave, warp order and weft order the face is drawn over the least common repeat — the warp’s colour where the warp is up, the weft’s where it is not — reduced to its own smallest repeat in each direction, and named by its least reading over every starting corner and both colourings. The same face turned a quarter with its face still up is named the same way.

A weft order’s cost is the cheapest of the three insertion rules the earlier account defined that throws it with four boxes a side. A look’s price as drawn is the least weft cost among constructions naming it; its price either way round adds the weft costs of constructions whose turned face names it. The census was confirmed never to price a look higher either way round than as drawn; to move a large share of the at-will looks, some of them to the cheapest loom; to leave one-and-one in both systems at the dearest loom on plain and on a twill whichever way it lies; and to put one-and-one over two-and-two on the cheapest loom as drawn and the dearest turned.

Where the census stops #

Two colours and six threads. Every named effect with three colours, and every order of eight — houndstooth among them — is outside the family. The named effects’ behaviour under turning follows from their symmetry and needs no census; the shares do not extend.

Twelve weaves. Satins, broken twills and larger repeats make looks these weaves cannot, and whether their looks divide the same way is not computed.

A look is taken up to colour exchange. A dark cloth with a light line and a light cloth with a dark line count as one look here. A designer would call them two, and they cost the same loom.

Nothing is judged. The census counts which looks a loom can make, not which anyone would want. The asymmetric family’s existence is a fact about looms; whether its members read as designed cloths or as mistakes is a question about how people look at checks, which is the question the earlier account left and this one does not answer.

Still open: whether asymmetric effects appear in pattern books #

The arithmetic says a mill with only box looms had a cheap way to weave a hairline one way round, and gave its looks no names. That is either because the asymmetric looks read badly, or because nobody drew a colour-and-weave effect with two different orders. The two can be told apart without judging taste. Pattern books and mill sample ledgers record warp and weft orders separately for every cloth; a count of entries whose two orders differ, set against the share of possible looks that are asymmetric, would say whether the asymmetric family was avoided or merely unnamed — and, if it was woven, whether its single-thread order sat in the warp as often as the loom’s economics say it should.

Who described it, and what is new here #

Colour and weave, box looms, and the practice of turning a design along the bolt are all old trade knowledge. The census of two-colour looks over twelve small weaves, priced as drawn and as woven across, the finding that nearly half the looks needing a loom picking at will need a cheaper one turned, and the observation that every named effect lies among the looks turning cannot help because every named effect uses one order in both systems, were worked out here.