A shading changes two things at once

A damask has two tones: the satin face and its complement, warp where the design is and weft where it is not. A jacquard designer who wants more than two has to find them somewhere, and there is only one warp and one weft to find them in.

The answer is a shading: a series of intermediate weaves between the satin and the sateen, each with one more warp mark per end than the last. On an eight-end repeat there are seven of them, and a designer uses them the way a printer uses a halftone.

The tone steps are exactly even. Everything else about them is not, and the two facts together are why shaded damask looks the way it does.

How the steps are built #

The satin with move m on n ends is a set of intersections (i, mi). Shifting it by j picks gives a coset: the set (i, mi + j), which has exactly one mark in every end and in every pick, just as the satin does.

There are n cosets and they partition the repeat: every intersection belongs to exactly one. So a union of k of them has exactly k warp marks in every end and every pick, whichever k are chosen.

That makes the tone exact. The fraction of warp on the face is k/n, with no averaging, no approximation and no dependence on which cosets were taken. A step of one eighth is one eighth.

It also makes every step one cloth, which is not a foregone conclusion: a draft in which every end and every pick interlaces at least once, and does so equally, cannot fall into layers, and this collection’s criterion confirms it at every step.

And it leaves everything else free #

Which k cosets are chosen does not change the tone at all and changes almost everything else by a factor of several.

At the midtone of an eight-end repeat there are seventy choices of four cosets. They all have exactly half the intersections warp-up. Their longest floats run from one to four: take the alternating cosets 0, 2, 4, 6 and the result is a plain weave; take the consecutive cosets 0, 1, 2, 3 and the result is a 4/4 twill.

A plain weave and a 4/4 twill are the same tone and are otherwise about as different as two cloths of one repeat can be — in firmness, in lustre, in setting limit, in drape, in how they wear.

So a shading’s tone is fixed and its cloth is not. That is the freedom the designer has, and it is invisible in the thing the design is specified in.

A shading has to be a chain #

Not every sequence of tone steps is usable, and the constraint is easy to miss.

A shading is used to grade one region of a design into another, so two adjacent tones sit side by side in the cloth with a boundary between them. If the darker tone’s marks do not contain the lighter tone’s, then crossing the boundary turns some intersections from warp-up to weft-up as well as the other way — and that shows as a line the design did not draw.

So the steps must be nested: each one the previous one plus a coset. That turns the choice from “seven independent selections” into “an order in which to add the cosets”, which is n! orderings, and the shading is determined by it.

Two orderings matter in practice.

The spread chain adds cosets as far apart as the remaining choices allow, which is what a greedy shortest-float rule produces: 0, then 4, then 2, then 6, then the odd ones. The consecutive chain adds them in order — 0, 1, 2, 3 — and is the classical twill-shaded damask.

The lustre profiles, which are the finding #

Both chains pass through the same seven tones. Neither passes through the same lustre.

The spread chain’s longest float runs 7, 3, 3, 1, 3, 3, 7. Its ends are satins and its midtone is a plain weave, so the lustre scale is a V where the tone scale is a straight line. Its firmness — interlacings per intersection — runs 0.25, 0.50, 0.75, 1.00, 0.75, 0.50, 0.25, which is the same fact counted the other way.

The consecutive chain’s runs 7, 4, 3, 4, 5, 6, 7. That is flatter and it is not symmetric, because stacking cosets in order shortens the warp float and lengthens the weft float at the same time, and which of the two is longest changes hands at the midtone.

A spread shading grades evenly in tone and dramatically in lustre. A consecutive shading grades evenly in lustre and holds a long float in every midtone. Which is wanted is a design decision, and the tone scale cannot make it.

The practical reading is that a spread shading is the one to use where the surface should read as a smooth greyscale under diffuse light, and a consecutive shading is the one to use where the design is meant to be seen in a raking light and the lustre should carry the modelling rather than fight it. A damask table linen — flat, seen from above, lit from a window — is nearly always shaded the consecutive way.

The ends of the range are not cloth #

There is one more constraint and it is this collection’s own criterion applied to a design problem.

A tone of zero is every intersection weft-up. A tone of one is every intersection warp-up. Neither is a fabric: nothing interlaces, the two systems lie on top of one another, and the criterion reports as many layers as there are threads.

So a shaded damask cannot reach pure warp or pure weft, however many shafts it has. The palest and darkest tones an n-end shading can print are 1/n and (n − 1)/n, and on eight ends that is 0.125 and 0.875 — a contrast ratio of seven, not infinity.

That is asserted by feeding the machinery the two ends and requiring it to refuse them, which is the form this collection’s negative results take. And the two extremes that are cloth are complements of each other, so a shading is symmetric about its midtone, which is why the spread chain’s float profile is a symmetric V.

The consequence for a designer is a real one: more shafts do not buy more contrast, they buy more steps between the same two ends. Going from eight ends to sixteen takes the extremes from 0.125 and 0.875 to 0.0625 and 0.9375 — a contrast ratio of fifteen rather than seven — which is an improvement, and a much smaller one than doubling the shaft count suggests.

The two chains’ firmness, which runs opposite to their lustre #

The float profile is one reading of the same drafts; the firmness is another, and putting them side by side says something the float alone does not.

The spread chain’s firmness — interlacings per intersection — runs 0.25, 0.50, 0.75, 1.00, 0.75, 0.50, 0.25. It peaks exactly where the float bottoms out, which it must: a plain weave is the firmest cloth of any order and the least lustrous.

The consecutive chain’s runs 0.25, 0.375, 0.50, 0.50, 0.50, 0.375, 0.25 — flat across the whole middle. So a consecutive shading holds its firmness at half through five of its seven steps and a spread shading swings from a quarter to one and back.

Firmness decides setting: a firmer weave interlaces more, each interlacing costs room, so a firmer weave has to be set more openly. A spread shading therefore contains steps that want quite different setts, all of them woven at one sett because they are all in one cloth.

That is a real strain on the fabric and it is the strongest practical argument against the spread chain. A midtone plain weave woven at the sett a satin wants is a plain weave woven far too densely, and it will be tight, hard and prone to reed marks. A consecutive shading, whose firmness barely moves, has no such problem.

So the two chains trade against each other twice, and the second trade points the other way from the first: the spread chain is smoother in tone-to-lustre and worse in setting, and the consecutive chain is the reverse.

What a shading costs the loom #

A shading is a jacquard construction, and on a jacquard every end has its own hook, so the shafts do not constrain it. But the same series drawn on a dobby has a cost, and it is worth knowing what it is.

Each tone step of an n-end shading needs n shafts, exactly as the satin does — every end of the repeat lifts on a different pattern of picks, so no two can share a shaft. So a shading costs nothing extra in shafts over the satin it is built from.

What it costs is in the lifting plan. A satin on n shafts is a straight draw with a stepped treadling; a shading is n different lifting plans that have to be stored and selected between, which is exactly what a dobby stores and what a jacquard exists for.

So a shading is free on a dobby and impossible on a treadle loom, and the boundary between the two is a storage question rather than a geometric one — which is the same conclusion the harness reached from the other direction.

What was counted, and how #

The enumeration is over all 2ⁿ subsets of cosets, filtered to size k, which at eight ends is 254 drafts and is the whole cost.

The tone is exactly k over n, asserted to 1e-15 at every step, because it is exact.

Every step is one cloth, asserted through this collection’s criterion at every step of the chain.

The chain is nested, asserted rather than assumed: the greedy shortest-float rule happens to produce a nested chain and it did not have to, so the check is on the result.

The midtone of a spread shading is less lustrous and firmer than its ends, which is the finding, asserted as two orderings on the solved chain rather than as the sequence 7, 3, 3, 1 — because the sequence is a fact about eight ends and the ordering is a fact about shadings.

The consecutive chain’s profile is flatter than the spread one’s, asserted on the range of the two float sequences.

And the ends of the tone range are refused. A tone of zero or one is fed to the machinery and must complain, and the two extremes that are cloth must be complements — asserted as their tones summing to exactly one.

Sixteen ends, where the choice widens #

Everything above is on eight ends, which is the commonest damask repeat. At sixteen the same structure holds and the numbers get much larger.

There are fifteen tone steps rather than seven, and the number of drafts at the midtone is sixteen choose eight, which is 12,870. The extremes move to 1/16 and 15/16. And the spread chain’s midtone is still a plain weave, because the alternating cosets of any even satin give one.

What changes qualitatively is the granularity of the choice. At eight ends the greedy rule has only a few sensible orderings; at sixteen there is real room to design the lustre profile — to make it flat, or V-shaped, or asymmetric so that the highlights and the shadows of a figure have different surfaces.

That is what a jacquard designer working at high shaft counts is actually doing, and it is not usually described as designing a lustre profile. It is described as choosing weaves, which is the same thing without the axis drawn.

Where the model stops #

Tone is taken as the fraction of warp on the face, which is the standard assumption and is not quite true. Two cloths with the same fraction do not look equally light: a long float catches more light and reads brighter than the same area broken up, which is precisely why the lustre profile matters. So the “exactly even” tone scale is exactly even in area and not in appearance, and quantifying the difference needs a reflectance model this collection does not have.

The surface step is not computed here. A region of long floats stands proud of a region of short ones — a figured cloth has a step in its surface, and it is a ratio of interlacing rates — so a shading has a relief as well as a tone, and it is largest where the float profile is steepest.

Only satin-coset shadings are enumerated. A designer could in principle build a tone series that is not a union of cosets, which would break the exactness of the tone and might buy something else. The space of those is enormous and is not searched.

And nothing here is about more than one repeat. A shading in a real design covers a region many repeats across, and how the eye integrates a boundary between two tones over that region is a question about scale that a rectangular block raises and does not settle.

The longest float is a proxy, and the crown line is the quantity #

The two chains’ lustre profiles are compared above through the longest float, and the longest float is a proxy. What decides specular area is the crown line — how much flat plateau the surface carries — and computing it for the two chains says something the float count cannot.

A run of length L presents L − 1 spacings of plateau between its two turns, so a thread with k marks on the face in R runs contributes k − R, and a draft’s total is the warp’s runs plus the weft’s. That is cloth.js’s crownLine, counted cyclically like every other measure there, and checked against surface.js’s geometric version — which measures the same plateau in millimetres and returns this number times the pitch.

Every end of a consecutive chain carries its marks in one run, whatever the move, because cosets j and j + 1 put their marks on adjacent picks of the same end. So each end contributes k − 1 and the warp half of the crown line is fixed at n(k − 1).

The weft half is not fixed, and it is where the move enters. Pick p carries a mark at the end i with mi + j ≡ p, so consecutive cosets land on ends spaced m⁻¹ apart — adjacent only when the move is 1 or n − 1. On an eight-end repeat m = 3 has m⁻¹ = 3, and the marks in a pick are scattered three ends apart rather than lying together.

So the crown line runs 6, 5, 4, 4, 4, 5, 6 along the consecutive chain of the eight-end satin, and 6, 6, 6, 6, 6, 6, 6 along the consecutive chain of the 1/7 twill. Flat in one case and merely flatter in the other, and the difference is the base rather than the chain.

The spread chain’s runs 6, 4, 2, 0, 2, 4, 6 at every move — a perfect V reaching exactly zero at the plain-weave midtone, which is the same V the float profile shows and is here as an exact quantity rather than as a longest run.

Which is a stronger comparison than the float supports #

A spread shading grades in tone and swings its lustre from full to none and back. Its midtone reflects no specular area at all, so under a directional light the middle of the ramp goes dark independently of its tone, and the tone scale and the visible scale part company.

A consecutive shading holds two thirds of its lustre across the whole range — six down to four and back — where the spread chain loses all of it. The float profile makes those two look much closer than they are.

And the longest float was the wrong proxy in the direction that matters. The consecutive chain’s floats run 7, 4, 3, 4, 5, 6, 7 and look thoroughly uneven; its crown line runs 6, 5, 4, 4, 4, 5, 6 and is nearly level, because shortening the warp float lengthens the weft float by most of the same amount and the crown line counts both. Much of the variation in the longest float is real and carries no lustre with it.

And it says which chain a shading wants #

Add each coset next to one already in the set and the lustre barely moves; add them apart and it collapses towards the midtone. That is a rule a designer can act on rather than a choice between two named chains, and it holds at every move.

And it prices the strain this essay raises against the spread chain. The spread chain’s midtone is a plain weave, which is the firmest cloth of any order and wants the openest sett — so its lustre collapse and its setting strain are the same event, reached at the same step, for the same reason. A shading that keeps its crown line keeps its firmness, and the two objections to the spread chain are one objection.

The generalisation #

A design parameter that is specified exactly usually has a companion that is not specified at all.

The tone here is exact by construction, and specifying it fixes nothing about the cloth beyond the count of marks. Everything a weaver would care about — float, firmness, setting limit, surface — is left free, and the seventy drafts at the midtone span it.

That is a pattern worth watching for. A specification names the quantity somebody was thinking about, and the quantities nobody was thinking about are exactly the ones that then vary without anybody deciding. The remedy is not to specify more; it is to know which of the free quantities matter and to choose them on purpose, which for a shading means choosing the chain.

Who found it, and when #

Shading is old jacquard practice and every design manual has a plate of it. That the steps are built from satin cosets is how it is taught, and the consecutive-coset version — the twill shading — is the standard one.

That the extremes cannot be reached is implicit in any weaving: nobody has ever tried to weave a cloth with no interlacings. Stating it as a bound on a shading’s contrast ratio, and computing where the bound sits, is this collection’s.

So is the comparison of the two chains’ lustre profiles, and the observation that the spread chain’s midtone is a plain weave — which is obvious once it is drawn and is not, as far as this collection can tell, anywhere written down as the reason a shaded damask goes matt in the middle.

Where the ladder goes next #

The tone is exact, the lustre is measured here, and there is a third scale nobody has specified: the cloth’s own thickness. A firmer weave is pressed harder at every crossing and finishes thinner, so a tone ramp is a valley — and which base the shading is built on decides whether the valley is eighty micrometres deep or exactly nothing.

Sideways, back to the water, where a cloth that presses on itself supplies a force this collection’s central criterion has needed since it was written and has never had without a measurement.