A colour order beats the weave it is threaded on

The reed leaves its own mark because it groups the warp — several ends to a dent — and the grouping beats against the weave repeat. The arithmetic is a least common multiple rather than the subtraction a watered finish needs, because the two grids are counted in whole threads rather than measured in millimetres, and the rule that came out of it was one a weaver can act on: dent so that the ends per dent share no factor with the repeat.

A colour order is a second grouping of the same warp. It is laid on the same threads, it has a period in whole ends, and it beats against the same repeat. Everything about the arithmetic carries over unchanged.

Nothing about the consequence carries over at all.

Two gratings, one warp #

Colour and weave is usually introduced as a surprise: houndstooth is a perfectly ordinary two-and-two twill and the pattern is nowhere in the draft. That is the surface reading, and it is correct.

The pattern reading is the one this ladder is about, and it starts from the observation that there are two periodic things on the same set of threads. The weave has a period in ends — four for a 2/2 twill, five for a five-end satin — and the colour order has one, and they are laid over one another exactly as the reed’s grouping is laid over the weave.

So the surface is a product of two gratings and its repeat is what a product of two periodic functions always has: the least common multiple. In each direction separately, because the warp’s colour order and the weft’s need not agree.

A colour period of p on a weave of q ends gives a visible repeat of pq/gcd(p, q). Houndstooth is p = 8 against q = 4, so its surface repeats on eight ends; the same twill with two dark and two light is p = 4 against q = 4 and repeats on four.

Where the consequence parts company with the reed’s #

The reed’s version of this arithmetic came with a preference, and the preference was for the coprime case. A denting that shares a factor with the repeat treats every repeat of the weave identically, so the cloth carries a stripe at the dent pitch — the reed mark — and a denting that shares no factor spreads the grouping across every phase of the weave and is invisible.

Turn that over for colour and the sign flips.

A colour order that shares a factor with the weave repeat treats every repeat the same way, so the surface repeats on something small: a check. A colour order coprime with the repeat visits every phase of the weave before it comes back, so the surface repeats on the product: a large scattered figure.

Both are cloths people buy. A gingham is the first and a heather mixture is the second, and neither is a fault. The reed’s beat is a fault because the reed is not supposed to be visible at all; the colour order’s beat is the design, so its period is a specification rather than a defect.

And the coprime case is not available on every weave #

There is a second asymmetry with the reed and it is a practical one.

A weaver choosing a denting can always find a coprime one: the census shows that any repeat admits several, and one end per dent is coprime with everything. A designer choosing a colour order has the same freedom in principle and very little of it in practice, because colour orders are almost always even.

A stripe of dark and light in equal blocks has period 2k, which is even; so does a check. And every weave repeat anybody uses is even too — plain is two, twill and basket are four, the common satins are five and eight. So the gcd is at least two in nearly every combination a designer actually writes, and the visible repeat is at most half the product.

The five-end satin is the exception and it is instructive. Five is odd and coprime with every even colour period, so a two-and-two colour order on a five-end satin repeats on twenty ends where the same order on a 2/2 twill repeats on four. That is a factor of five in the size of the figure, from changing the weave and not the colours.

The bound is a bound, and it is almost always the answer #

Everything above is an upper bound. The least common multiple says the surface cannot need more than pq/gcd; it does not say the surface needs that much, because the face is a function of both gratings and a coincidence in it could repeat sooner.

That is a question to be measured rather than argued, so it was. Every colour order with a minimal period up to eight — 472 of them, each reduced to its own period so that an order written twice over is counted once — was laid on four weaves, the surface built at the bound, and the surface handed to the same minimalRepeat every draft on this site is measured with.

The bound is exact in 468 to 470 of the 472, on every weave tried. The exceptions are two to four per weave and they are all the same kind of thing:

• a constant colour order, where the cloth is one colour and repeats on one intersection;
• plain weave with alternate colours, which repeats on one end and two picks — the log-cabin degenerate, where every end shows the same colour down its whole length;
• a five-end satin with a one-in-five colour order, which repeats on one pick and five ends, because the coloured end lands on the satin’s own float at every pick and the pattern is a plain warp stripe.

So the bound is loose only where the pattern has stopped being a pattern. Every case in which the surface repeats sooner than the arithmetic allows is a case in which the surface has collapsed into a stripe or into a plain colour, and there is nothing left with a period to be shorter than the bound.

That is the honest form of the result. The arithmetic is not a theorem — it can be beaten — and the ways of beating it are exactly the ways of losing the design.

What a designer is actually choosing #

The arithmetic turns into three statements a designer can use, and none of them requires drawing the cloth.

The size of the figure is set by the gcd, not by the colours. Two dark and two light on a 2/2 twill gives a four-end figure; four and four on the same twill gives an eight-end one. The colours are the same colours and the scale has doubled, because the colour period has doubled and the gcd has not.

Changing the weave changes the figure without touching the warping. A warp already threaded four dark and four light produces an eight-end figure on a 2/2 twill and a forty-end figure on a five-end satin. The warp is not rethreaded, the shuttle boxes do not change, and the pattern is five times the size — which is a real production fact, because rethreading a warp is a day and changing the lifting plan is a minute.

And the two directions are independent. The warp’s colour order beats the repeat in ends and the weft’s beats it in picks, and a design with a period of four in the warp and six in the weft on a 2/2 twill repeats on four ends and twelve picks. That is an oblong figure from square colouring, and it is what a great many woollen suitings are.

Which is why the reed and the colour order have opposite rules #

Both gratings are on the same warp and both beat against the same weave repeat, and the advice about them is exactly reversed.

For the reed: pick a coprime denting, so the grouping never lines up with the repeat and the effect vanishes into the cloth. The reed exists to space the warp and any pattern it produces is a fault.

For the colour order: pick the gcd the design wants. A large gcd gives a small check and a gcd of one gives a scattered figure, and both are designs somebody is paying for.

The two rules are the same rule with different signs on the objective, and the reason they can be stated as one rule is that neither depends on anything but the periods. Neither the reed’s beat nor the colour order’s cares what yarn is used, how the cloth is finished or what the colours actually are; both fall out of two integers and a common divisor.

That is the shape of every result on this ladder. A folded taffeta’s watered figure is the difference of two wavevectors, a slub’s diagonals are a fault period against a cloth width, and both are the same subtraction. The reed and the colour order are the same subtraction counted in whole threads, which turns it into a divisor calculation, and the divisor calculation is where a rule a weaver can act on comes from.

Why houndstooth is eight ends and not four #

The commonest colour-and-weave cloth there is makes the arithmetic concrete, and it also shows where the arithmetic stops.

Houndstooth is a 2/2 twill with four dark and four light in both directions. The colour period is eight, the weave repeat is four, they share a factor of four, and the visible repeat is 8 × 4 ÷ 4 = eight ends and eight picks. Not four, which is what a reader who expected the weave to dominate would guess, and not thirty-two, which is what a reader who expected them to be independent would guess.

The doubling is the whole of the pattern. Four dark and four light on a 2/2 twill covers exactly two repeats of the weave per colour block, and the twill’s diagonal moves two ends per two picks — so the second repeat of the weave inside a colour block meets the diagonal at a different phase from the first, and the two halves of the block do not look alike. That difference is the hook on the tooth.

Halve the colour order to two and two and the block is one weave repeat, every block meets the diagonal identically, the surface repeats on four, and the hook disappears: what is left is a small even check with no shape in it. The same twill, the same two colours, half the period, and a completely different cloth — which is a stronger statement than “the pattern is not in the draft”, because here the draft and the colours are both unchanged and only the block length has moved.

The reed is still there underneath both of them #

One thing the parallel with the reed makes easy to forget: the reed has not gone away. A cloth with a colour order in it is dented as well, so there are three periodic things on the warp — the weave, the colour order and the denting — and all three beat.

The cloth’s whole visible repeat is the least common multiple of all three, and the practical consequence is that a denting chosen to be invisible against the weave can be highly visible against the colour order. Four ends per dent on an eight-end weave repeat shares a factor and is a known reed mark; four ends per dent under a four-and-four colour order puts a dent boundary at every colour boundary, which is a much more visible thing — the dent’s slight opening lands exactly where the colour changes and sharpens every stripe edge in the cloth.

Whether that is wanted is a design question with a real answer either way. A crisp gingham wants it and a soft heather does not, and it is decided by a reed plan that nobody thinks of as a design decision. The reed’s own rule is stated against the weave alone, which is correct as far as it goes and is not the whole of what a reed is beating against.

Three gratings, one warp, and only one of the three has a published rule.

What was counted, and how #

The bound is computed and the surface is measured, and the two are separate calls. colourBeat does the arithmetic from the two periods and touches no cloth; colourSurface tiles the weave out to the bound, colours it, and hands the result to cloth.js’s minimalRepeat — which finds a period by trying divisors and then tiles the cell back out to check it, because a plausible wrong answer is the failure mode of any period search.

Two things are asserted on every single call, not on the census: that the surface’s own repeat is no larger than the bound, which would be an arithmetic error, and that it divides the bound, which it must because a period of a period is a period. A surface repeat that did not divide the bound would mean the tiling was wrong rather than the bound.

The census enumerates orders by binary mask and reduces each to its minimal period, so 1100 and 11001100 are one order rather than two, and a constant order is kept rather than skipped — because the constant order is exactly the degenerate case the census is looking for.

And the finding is asserted in both directions. That the bound is exact in over ninety-five per cent of cases is one assertion; that at least one order beats it is another, because a claim that a bound is always tight is a claim this measurement can refute and did. The third is that every order beating the bound has collapsed to a stripe or to one colour, which is what makes the exceptions uninteresting rather than merely rare.

Where the model stops #

Two colours, and they are drawn as filled and empty. Every count here treats a thread as dark or light, and the panels draw the dark ones filled. Two colours of similar value make a far weaker pattern than the figures suggest, and three or more colours are a different arithmetic — the period is the lcm of all the orders involved and the visible contrast is a question about which pairs of colours the eye separates, which this collection has no model for.

The period is not the appearance. A surface repeating on twenty ends and one repeating on four are different cloths, and how different depends on what is inside the repeat. Two orders with the same period on the same weave can produce a crisp check and a muddy mixture, and nothing here distinguishes them; the collision census is the collection’s measure of that and it is about information rather than about scale.

The census stops at a period of eight. Real colour orders run much longer — a tartan sett is a hundred and forty threads — and the arithmetic carries on unchanged while the enumeration does not. What a long order buys is not a longer period but structure inside the period, which is the tartan’s own subject.

And nothing here is about the weft’s phase. The two orders are taken as starting together at the corner of the repeat, and shifting one against the other gives a different surface with the same period. Which phase a design wants is a real choice and the relative origin decides more than it looks like it should.

Who found it, and when #

Colour and weave is as old as coloured warps and the standard treatments — Watson, Grosicki, every design manual — set it out as a construction: here is a draft, here is a colour order, here is the surface. The surfaces are drawn and the periods are not computed, because a designer working on point paper draws the repeat and can see how big it is.

The divisor statement appears not to be written down, and the reason is probably that it is only interesting when it is compared with something. On its own “the surface repeats on the least common multiple” is an observation a designer makes without needing to be told. Set beside the reed, where the identical arithmetic produces a rule with the opposite sign, it says something about both.

The measurement — that the bound is exact except where the pattern has gone — is this collection’s, and it is the kind of check that is worth making precisely because the answer was expected. A bound that is always attained is a bound worth quoting as a period; one that is sometimes loose is a bound that has to be qualified every time it is used, and which of those is true was not knowable without building the surfaces.

Where the ladder goes next #

Every rung of this anchor so far has computed a beat and left it there. A beat is also an instrument: two gratings at a small angle magnify their own difference by one over the angle, so a moiré measures a pitch far more finely than the eye can read one — and the same gain that makes the measurement possible amplifies the cloth’s own irregularity by exactly as much, which is why a watered finish never comes out regular.