A tone ramp is a valley, and the satin digs it

A shaded damask is specified in one number per step, and the rung below this one found that the number fixes almost nothing. The tone is exactly k/n; the float, the firmness, the setting limit and the crown line are all left free, and the seventy drafts at an eight-end midtone span every one of them.

That essay measured two of the free quantities and named a third without computing it. The third is the cloth’s own thickness, and it turns out to be the one a reader cannot avoid: a step in tone can be looked at or not, and a step in thickness catches a raking light whether anybody is looking for it or not.

The mechanism is borrowed and only the pairs are new #

Nothing about the physics here is a shading’s. It is the mechanism a figured cloth’s step is made of, applied to a different pair of weaves.

A thread presses on the thread it crosses only where it turns. Between turns it lies on the face and touches nothing. So the pressing a region of cloth receives, per unit area, is the force at one crossing times the number of crossings per unit area — and the second factor is the interlacing rate, which is a property of the matrix exactly, with no yarn in it.

A region that turns less often is pressed less often. Less pressed means less flattened, and less flattened means thicker. That is the whole of it, and its consequence for a shading is immediate, because firmness is exactly the interlacing rate and a shading’s firmness is not constant along the chain.

The spread chain’s firmness is a peak, so its surface is a pit #

The spread chain runs from the satin to its complement by adding cosets as far apart as the remaining choices allow, which is what a greedy shortest-float rule produces. Its firmness — interlacings per intersection — runs 0.25, 0.50, 0.75, 1.00, 0.75, 0.50, 0.25.

That is not a gentle rise. The midtone of a spread shading is a plain weave, the firmest cloth of any order, interlacing four times as often as the satin at either end of the ramp. Four times the crossings means four times the pressing, at the same force per crossing, on the same threads at the same sett.

On a sheeting at half a newton in the end, the cloth’s own thickness at the ends of the ramp is 281 micrometres and at the midtone 197. The ramp sinks 84 micrometres, which is thirty per cent of the cloth’s whole thickness, and it sinks symmetrically because the two ends of a shading are complements and complements interlace identically.

The word for that shape is not a slope. Both ends of the tone range stand at the same height and the middle is below them, which is a groove running along the tone ramp — and a tone ramp is used to model a curved form, so the groove runs along exactly the part of a design that was supposed to read as smooth.

Every boundary is a line, and the sizes are not the tone’s #

A tone ramp is not drawn as a gradient. It is drawn as regions, one tone each, with a boundary between adjacent ones, and each boundary is a step in the surface of the size the difference computes.

The tone changes by exactly one eighth at each of the six boundaries. The thickness changes by 43, 25, 16, 16, 25 and 43 micrometres — nearly a factor of three between the largest and the smallest, on steps the design treats as identical.

The largest steps are at the ends of the ramp, between the palest tone and the next, and between the darkest and the one before it. That is the opposite of where a designer would expect trouble. The midtones are where the modelling happens and where the eye is most engaged; the ends are the highlight and the shadow, and a designer thinks of them as the safe extremes. They carry the biggest step in the cloth.

The reason is arithmetic rather than anything about design. Firmness rises linearly along the spread chain, in equal increments of 0.25, but thickness is not linear in firmness: pressing a cloth is a compression, and the first crossings added to a very open weave do far more flattening than the last crossings added to an already firm one. The equal steps in firmness at the ends of the ramp therefore buy the largest steps in thickness, and the equal steps near the midtone buy the smallest.

The classical chain sags half as far, and two of its boundaries are exactly level #

The consecutive chain — cosets added in order, which is the shading every design manual draws — behaves differently, and the difference is larger than its float profile suggests.

Its firmness runs 0.25, 0.375, 0.50, 0.50, 0.50, 0.375, 0.25. Flat across the middle three tones — and flat means the middle two boundaries of the ramp are exactly level, not approximately. The two weaves either side of each interlace at precisely the same rate, so they are pressed precisely as hard and finish at precisely the same thickness.

The sag is 43 micrometres against the spread chain’s 84, the worst boundary is 24 micrometres against 43, and two of the six boundaries leave no line at all.

So the classical chain wins a third time. The rung below found it flatter in lustre and flatter in setting; it is flatter in surface as well. The greedy shortest-float rule loses on every quantity the float was standing in for, which is the sharpest thing that can be said against optimising a proxy.

And the sag is not the cloth’s, it is the matrix’s #

The numbers above are a sheeting at half a newton in the end. Every one of them is a product of two things — a ratio taken off the matrix and a geometry taken off the cloth — and only the second changes when the cloth does.

Run the same chain through six cloths from a voile to a duck, and the absolute sag runs from 61 to 107 micrometres, which is nearly a factor of two. Divide each by that cloth’s own thickness and it runs 30, 35, 29, 31, 30 per cent, with only the duck falling away to 21.

A shaded damask sags by about a third of its own thickness whatever it is woven from, and the exception is instructive: a duck is thick and heavily jammed already, so there is less compression left in it to give away, and the ratio falls. The rule is not a constant of nature; it is what a ratio of interlacing rates does when the compression curve is nearly the same shape on nearly every cloth.

That is a more useful statement than the micrometres, because a designer does not know what the cloth’s compression curve is and does know roughly how thick the cloth is.

Where the valley actually comes from, which is the move #

Everything above treats the base of the shading as fixed — cosets of the eight-end satin, move three, which is what a shaded damask is. Change the move and one of the two chains changes completely.

On a move of one the consecutive chain’s surface is exactly flat. Not flatter: flat, at every one of the seven tones, with all six boundaries leaving no step.

The reason is the same reason its crown line is constant at that move and not at any other. Cosets j and j + 1 land on adjacent picks of the same end whatever the move, so every end of a consecutive chain carries its marks in one run and the warp-direction interlacings are pinned at two per end. Whether the picks also carry their marks in one run depends on the move: pick p takes its marks from the ends i with mi + j ≡ p, so consecutive cosets land on ends spaced m⁻¹ apart, and those are adjacent only when m is 1 or n − 1.

At move one both directions are pinned, the interlacing rate is 0.25 at every tone, and a constant interlacing rate is a constant pressing and a constant thickness. At move three the picks are scattered three ends apart and the weft interlacings run 16, 32, 48, 48, 48, 32, 16 while the warp’s stay at 16 — which is where the sag comes from and where the two level boundaries in the middle come from too.

Which makes the move a lever nobody knew they were pulling #

The consecutive chain on a move of one is the twill series: 1/7, 2/6, 3/5, 4/4, 5/3, 6/2, 7/1. Every step of it is an ordinary twill, and a twill is what the trade has always called this construction — a twill shading.

So the classical name is exactly right and the classical practice is not, because a shaded damask is built on a satin. The satin exists to destroy the twill’s diagonal, since a visible line across a lustrous face is the thing a damask cannot have; and the diagonal is precisely what was holding the surface level.

The choice is a genuine one and it has never been stated as a choice. A twill base gives a tone ramp with no relief and a visible diagonal. A satin base gives a ramp with a diagonal nobody can see and a groove down the middle of it.

At sixteen ends the lever has a range rather than two settings. The consecutive chain’s sag is 35 micrometres at a move of three, 59 at five and 78 at seven, against a spread chain that sits at 119 whatever the move. A factor of two and a bit, decided by a number that appears in no specification of a shaded cloth, and chosen — where it is chosen at all — for the appearance of the satin at the ends of the range rather than for anything happening in the middle.

The two readings of one rate do not track each other #

The crown line and the relief both follow from the interlacing rate, so it is tempting to treat them as one finding counted twice. They are not, and the way they part company is worth having.

Along the spread chain the crown line runs 6, 4, 2, 0, 2, 4, 6 — linear in the distance from the midtone, because each coset added takes exactly one spacing of plateau off every thread. The surface height runs 0, −43, −68, −84, −68, −43, 0, which is not linear at all: the first coset costs 43 micrometres and the third costs 16.

So the boundary that costs the most lustre and the boundary that costs the most surface are different boundaries, and they are at opposite ends of the ramp. Every step loses the same two spacings of crown line; the steps near the extremes lose two and a half times the thickness that the steps near the middle do.

The reason is that plateau is a count and pressing is a compression. Adding a crossing to a very open weave squeezes a cloth that still has room in it; adding one to a cloth already flattened by three times as many crossings does much less. The count is linear because counting is linear, and the cloth is not.

The practical form of that is a warning about which end of a ramp to worry about. A designer told that a shading loses its lustre in the middle will attend to the middle; the lines left in the cloth are at the ends, where the tone is doing the least work and where nobody is looking. A figure showing by its shine rather than by its step is the same pair of quantities taken between two regions rather than along a chain, and it separates them the same way.

What a finisher can do about it, and what it costs #

The relief is a difference in how much compression each region has left, so anything that spends the compression evenly across the whole cloth removes it.

That is exactly what a calender does. Pressing a cloth between rollers flattens the standing threads hardest, which is to say it flattens the thickest regions hardest, so a pass through a calender takes more off the tones at the ends of the ramp than off the tones in the middle. A heavily calendered damask has a flatter tone ramp than a loom-state one, and the flattening is a real removal rather than a disguise.

What it costs is the thing the damask was for. The compression a calender spends is spent for good — the cloth does not recover it — and the same flattening that closes the relief also closes the difference in specular area between one tone and the next, because a flattened float and a flattened plateau reflect much more alike than two round ones do. A finish that removes the groove removes some of the modelling with it.

Which is a genuine trade rather than a free repair, and it is decided in the finishing room by somebody who has not been shown either curve.

What was counted, and how #

The tone steps and the chains are the rung below’s machinery, unchanged. What is added here is the pairing.

The thickness of each step comes from grip.js’s surfaceStep, which is the function the figured cloth’s own essay is built on. It takes two weaves and a named cloth, computes each weave’s pressing from its interlacing rate, and puts each through the same compression model. Nothing about it is new here and nothing about it was changed; the shading supplies the pairs and it supplies the geometry.

The heights are relative and the steps are absolute. Only differences in thickness are computed, so the profile is drawn from the palest tone at zero. Reading a height off it as an absolute thickness would be reading something that was never measured.

The sag is taken over the profile rather than assumed to be an end against the midtone. A symmetric chain puts its lowest point in the middle, and asserting that in advance would have made the assertion true of the symmetric case only.

The claim about the move is swept rather than checked. Every move coprime to the repeat is built and the flatness recorded, and the assertion is that the consecutive chain is level exactly when the move is 1 or n − 1 — which is a statement about the family. A version of it checked at the default move would have been a statement about one number, and that is the shape of the error the crown-line identity in the rung below was corrected from.

And the guards are fed what they must refuse. A shading on three ends, a tone of nought, a tone step that is the whole decomposition: each has to raise the collection’s own error type rather than merely raise something, which is a distinction that was not being drawn until a guard called an undefined helper and passed the test by throwing the wrong thing.

Where the model stops #

The compression model is one model. surfaceStep presses each weave with the same contact force and puts both through the same curve, so what is being compared is how often the pressing happens and not how hard each individual crossing bears. A satin’s crimp is genuinely smaller than a plain weave’s and its turns genuinely gentler, so the real step is larger than the one computed, by an amount this collection does not have.

Nothing here says whether the step is visible. A groove 84 micrometres deep across a region several centimetres wide is a very shallow groove, and whether it reads depends on the light, on the finish and on the cloth’s lustre — all three of which a calender changes and none of which is modelled. What can be said is that the modelling a shading exists to do is done in raking light, because that is what makes a damask show at all.

And the pressing is taken as uniform within a region. A tone step is one weave over a region many repeats across, so its interior is uniform; its boundary is not, and the two or three repeats either side of a tone boundary are being pressed by a neighbour with a different rate. The transition is therefore softer than a step and its width is not computed.

The relief and the crown line are two readings of one quantity and are not the same reading. Both follow from the interlacing rate, so they move together — but the crown line counts plateau and the relief counts pressing, and a finish that flattens the cloth changes the second far more than the first.

Who found it, and when #

That a figure stands proud of its ground is old observation and old practice: damask is finished by pressing precisely because the two regions take a press differently, and every account of finishing a table linen says so.

That the same mechanism runs inside a shading, between tones of one design rather than between figure and ground, does not appear to be written down. The reason is likely that a shading is thought of as a tonal device — a halftone in threads — and a halftone is a thing that happens on a surface rather than to it.

The dependence on the move number is this collection’s, and it arrived as a correction rather than as a result: the rung below asserted a constant crown line for the consecutive chain, derived that identity from a step that holds only at a move of one, and drew every one of its figures at a move of three. Sweeping the moves to check the correction is what turned up the flat surface, which is the more interesting of the two.

Where the ladder goes next #

The tone is exact because every step is a union of parts that each carry one mark in every end and every pick, and the rung below built those parts out of a satin. A satin is not the only thing that makes them, and at four ends and at six there is no satin at all — so either a four-end shading is impossible or the construction was never about satins. It was never about satins, and what it is about is a Latin square.