A weave is a halftone screen with n greys

The analogy is in every account of shading and this ladder’s own first rung uses it: a shaded damask grades its tone the way a printer’s halftone does, by covering more or less of a cell.

The analogy is exact enough to be worth pushing, and pushing it produces a number that ought to be embarrassing. A printer working with an eight-by-eight cell of dots has sixty-five levels of grey. A weaver working with an eight-by-eight repeat has seven.

Where the fifty-eight go #

The two counts are not the same count because a cell of dots and a cell of cloth are asked different questions, and there are exactly two extra questions.

A thread must reach both faces. An end that is warp-up at every pick of the repeat never goes under anything; it lies on the surface for ever, held by nothing, and the criterion this collection is built on reports it as a separable layer of its own. So the marks in the cell cannot be fewer than n — one per pick at least — nor more than n² − n.

That removes the levels 0 to n − 1 and n² − n + 1 to n², which is 2n of them. At eight ends it takes sixty-five to forty-nine.

And every thread must carry the same number of marks. This is the one that costs, and it is not a rule anybody states, because it is invisible until it is broken.

A tone step is used across a region several centimetres wide, and a region is a repeat tiled. If end three of the repeat carries five marks and end four carries three, then every third end of that region is 62% warp and every fourth is 37% — which is a warp stripe running the height of the region at 25% contrast, at the pitch of the repeat. Nobody drew it and it is the most visible thing on the cloth.

So the marks have to be k in every end and k in every pick, and the levels are k = 1 to n − 1. Seven.

The count is exact, and it is König’s theorem that makes it so #

It would be reasonable to expect that requirement to be approximate — a tolerance, a “roughly equal”, a rule that softens at large repeats. It is not.

Read the repeat as a bipartite graph with the ends on one side, the picks on the other, and an edge wherever the warp is on the face. “Every end and every pick carries k marks” is exactly “every vertex has degree k”, and König’s edge-colouring theorem says a k-regular bipartite graph splits into k perfect matchings.

Two things follow and both are stronger than they look.

There is nothing between k and k + 1. A degree is an integer, so a uniform tone is an integer number of marks per thread, so the tone scale is exactly k/n with no intermediate values available at all. That is why the tone steps are exactly even rather than approximately so — the evenness is not a property of the construction, it is a property of there being nothing else there.

And every uniform tone is already a union of parts, whether or not anybody built it that way. The decomposition into permutation matrices that the rung below identifies as a Latin square is not a construction imposed on the problem; it is what uniformity is.

Which constraint costs more, and the answer changes at five ends #

Interlacing removes 2n levels and evenness removes (n − 1)(n − 2). Those cross.

At four ends interlacing costs 8 levels and evenness costs 6, so the criterion that this collection exists to apply is the more expensive of the two. At six ends it is 12 against 20, at eight 16 against 42, and at sixteen 32 against 210.

So the cheap constraint and the expensive one swap places at five ends, and every repeat anybody weaves a shading on is on the far side of the crossing. At the eight-end damask repeat evenness costs two and a half times what integrity costs; at sixteen it costs six and a half times.

That is worth stating because the two constraints feel completely different in kind. Integrity is a hard physical fact — a thread held by nothing falls out of the cloth. Evenness is an appearance requirement, a thing about how a region reads at a distance, and it sounds like the softer of the two. It is the one that takes fifty-eight sixty-fifths of the tone scale away.

What the two failures actually look like #

The three cells in the figure at the top carry the same thirty-two marks and are disqualified in different ways, and both disqualifications are worth seeing rather than accepting.

The printer’s dot is not cloth by a long way. Four of its sixteen threads carry every mark or none, and the criterion reports sixteen separable layers rather than one — the two systems have come apart entirely in the middle of the dot, where the warp lies over the weft with nothing crossing. That is not a marginal failure to be finessed with a finish; the cloth is in sixteen pieces.

The uneven cell is perfectly good cloth. It is built from the tone step by moving two marks sideways within their own picks, so every pick still carries four and every end still interlaces. One cloth, longest float five, nothing wrong with it — and its ends run 3, 4, 4, 4, 4, 5, 4, 4, which is a stripe of 25% contrast at the repeat’s own pitch.

So the expensive constraint is the one nothing enforces. A weaver who breaks the interlacing rule finds out immediately, because the cloth falls apart on the loom. A weaver who breaks the evenness rule gets a sound cloth with a stripe in it, and the stripe is a design fault rather than a construction fault — which is exactly the kind of thing that reaches the finished piece.

The exponent, which is what the trade actually pays #

The way out of a small tone scale is a bigger repeat, and the arithmetic of that is the essay’s practical half.

Tones grow as n − 1 and the cell’s area grows as n². A printer’s tones grow as n² + 1, which is the same rate as the area. So:

• to double a printer’s greys, double the cell’s area;
• to double a weaver’s tones, quadruple it.

Going from eight ends to sixteen takes a damask from seven tones to fifteen and takes the design’s smallest placeable feature from eight ends across to sixteen. At a sett of 24 ends per centimetre that is a step from 3.3 mm to 6.7 mm, and a figure whose outline is drawn in 6.7 mm steps is a figure with visible corners on it.

That trade is the reason a shaded damask looks the way it does. A damask is a coarse picture in fine cloth, and it is coarse because the tone scale is bought in the same currency as the resolution and at a worse exchange rate than a printer pays. What a figure costs the loom is the other half of the same budget, counted in shafts rather than in tones.

The escape a jacquard does not have #

There is an obvious way out and it is worth explaining why it is not taken.

A printer improves a halftone by making the dots smaller rather than the cell larger — a finer screen at the same cell count. The equivalent in cloth is a finer yarn at a higher sett, and it does work: the resolution of a figured cloth is its repeat divided by its sett, so doubling the sett halves the block in millimetres without touching the tone count.

What it does not do is help with tone, because the tone count is a property of the repeat in threads and is untouched by how thick those threads are. Doubling the sett doubles the number of hooks a design of a given width needs, which is a real and expensive change to the machine, and buys no tones at all.

So the two axes are bought from two different budgets and only one of them is the loom’s. Resolution is bought with sett, which costs hooks and yarn. Tone is bought with repeat, which costs resolution. Neither purchase helps the other, and the second is the one that is quadratic.

And the seven are not seven free choices #

The count above is a ceiling on what one repeat holds, and a design gets fewer, because a shading has to be a chain.

Two adjacent tones sit side by side in the cloth with a boundary between them, and if the darker tone’s marks do not contain the lighter’s then crossing the boundary turns some intersections from warp-up to weft-up as well as the other way — which shows as a line. So the tones a design uses have to nest, each one the last plus a part.

That is a much stronger constraint than it looks. There are 70 drafts at the eight-end midtone alone and a chain uses exactly one of them, chosen by the ordering rather than by anything about that tone. Once the decomposition and the order are fixed, all seven tones are fixed — a designer picks a chain, not a set of tones.

The practical consequence is about a design with more than one shaded region. Two regions modelled independently — a highlight on one form and a shadow on another — must use the same chain, or their tones do not nest with each other and any boundary where the two regions meet carries a line. So a jacquard design with a dozen shaded passages has one tone scale for all of them, and the choice of chain that suits the drapery has to suit the face as well.

That is why a shaded damask is a monotone object in a way a printed halftone is not. A printer’s cell can hold any of its sixty-five levels next to any other; a weaver’s seven come as an ordered ladder and the ladder is the design’s, not the region’s.

A second colour multiplies what a second repeat cannot #

The way out of a seven-tone scale is not a bigger repeat, and the trade knew that long before anybody counted anything.

Put two colours in the weft and the visible tone of an intersection depends on which thread is on top and which colour it is. Colour and weave is the general statement of that, and its consequence here is arithmetic: a cell whose warp is one colour and whose weft alternates between two gives three visible states per intersection rather than two, and the levels available multiply rather than add.

And it costs nothing in resolution. A colour order is a property of the warping and the shuttle box, not of the repeat, so the design’s smallest placeable block is unchanged. The second weft costs a shuttle and doubles the picks per centimetre of cloth, which is a real cost in loom time and no cost at all in tone-per-block.

That is why brocade and lampas exist, and why a damask that wants more than a monochrome ramp reaches for a second weft rather than for a bigger repeat. The seven tones are what one colour in one repeat holds; the way past them is out of the matrix entirely, which is where a figure that is not a stripe ends up too.

What was counted, and how #

The three level counts are arithmetic and are checked against an enumeration, because a closed form is exactly the kind of expression that is wrong by one at the ends. The interlaced count is computed as n² − 2n + 1 and separately by counting the mark totals from n to n² − n inclusive, and the two are required to agree.

The crossing at five ends is asserted as a crossing, not as an inequality. Sweeping the orders and requiring evenness to cost more than interlacing at every one of them would have been an assertion about large repeats that fails at four; requiring it to cost more exactly at five and above is the claim, and it is the shape this collection’s own ledger records fourteen times as the defect to watch for — an assertion calibrated on the default rather than on the family.

The three arrangements are constructed and then measured, and none of the three is told what it is. The dot is the k·n cells nearest the centre with ties broken on position so it is the same dot every build. The uneven cell moves the rightmost mark of a pick into that pick’s leftmost gap, which leaves every pick’s count untouched by construction and is the only way to get an uneven cell that is still cloth. The uniform cell is k parts of the cyclic decomposition.

Then all three are put through the same measurements — marks per end, marks per pick, the layer count, the longest float — and the three disqualifications are asserted separately: the dot is not cloth, the uneven cell is cloth and striped, the uniform one is neither. Asserting an ordering instead would have hidden that the two failures are independent, which is the whole finding.

And the exponent is asserted over a range. Every step up in repeat has to grow the cell’s area faster than the tone count, checked between consecutive orders rather than at one pair.

Where the model stops #

“The same tone” means the same fraction of warp on the face, which is not the same appearance. The rung below this ladder starts from says so: a long float catches more light than the same area broken up, so the printer’s sixty-five levels and the weaver’s seven are not even measured in the same units. The comparison is of counts, not of appearances, and a reflectance model would change the ratio in a direction not computed.

The stripe’s visibility is asserted and not modelled. A 25% contrast stripe at the repeat’s pitch is described here as the most visible thing on the cloth, and that is a judgement rather than a measurement: at eight ends and 24 ends per centimetre the pitch is 3.3 mm, which the eye resolves easily at reading distance and not at all across a room. Where the threshold sits needs a contrast-sensitivity model this collection does not have.

Nothing here is about more than one repeat. A design could in principle vary the tone from one repeat to the next — dithering the shading — and buy intermediate tones at the cost of a coarser cell, exactly as a printer does. That is a real construction and it is not counted here; the levels above are what one repeat holds.

And the halftone comparison flatters the printer. A real screen at sixty-five levels is not usable at sixty-five levels either: dot gain, ink spread and the eye’s own response take a good many of them, and a printer quotes an effective count well below the geometric one. What survives the comparison is the exponent, which is a fact about the two constraints and not about either craft’s practice.

Who found it, and when #

The halftone analogy is old and universal; every account of shading reaches for it in the first paragraph, and this collection did too.

The arithmetic under it does not appear to have been done. The reason is probably that the two crafts do not talk: the printing literature counts levels in a cell as a matter of course, because that is what a screen ruling means, and the weaving literature counts shafts and never counts levels at all, because the shading is presented as a construction rather than as a resolution.

König’s theorem is from 1916 and the application here is immediate once the question is asked in graph terms. What this collection adds is the question — how many tones, exactly, under which constraint — and the finding that the answer splits two ways with the appearance requirement costing several times what the structural one does.

Where the ladder goes next #

Four rungs of this anchor have taken the shading apart along one axis at a time: tone, lustre, surface, and now the count of tones itself. What none of them has asked is what happens where two tones meet — the boundary the nesting rule exists to keep clean, which is one intersection wide in the drawing and several threads wide in the cloth, and where the whole apparatus of exact tones runs into a cloth that does not have edges in it.