Easing the cap reaches the deep shadow and no higher

A four-ink press cannot lay down all four inks at full strength — the paper will not dry — so a profile’s separations are held under a total-ink cap, and where a dark colour’s inks would exceed it, black is raised and cyan, magenta and yellow lowered until the total fits. A profile is a fit between its nodes is the reminder that the profile stores those separations only at its nodes and interpolates between them, and wherever the separation bends between two nodes, the colour printed there is not the colour asked for. For dark saturated colours it comes out duller.

A run of essays has chased that dulling. More nodes quarter the dulling and leave the knee a larger share found it falling about fourfold with each doubling of nodes; three nodes at the cap’s corner cut the dulling by a third placed nodes just below where each colour’s cap binds; and shared nodes belong in the deep shadow found that for colours binding high, most of the dulling was never at the cap’s corner at all. It lay in the deep shadow, L 14 to 20, where black keeps climbing under the held total* — for every colour binding above L* 26, more than one and three quarter times the dulling just below its binding point.

Nodes can only sample a bend. The closing proposal was to remove it: let the total fall below the cap in the deepest shadows, so that the separation there is not forced along the cap. The prediction was that easing the cap by a tenth and a fifth between L 20 and the floor halves the high-binding colours’ dulling on even nodes, more than the shared nodes did, at a cost of a unit or two of L in the darkest colour the press can reach.** The alternative named was that the dulling barely moves, which would mean the deep shadow’s bend comes from the strategy’s black curve rather than from the cap, and nodes are the only remedy.

Halved where it was low, a third off where it was high #

With the cap eased on a straight ramp from its full value at L 20 to a tenth or a fifth less at the floor, the deep-blue ray’s mean hard-cap dulling on 65 even nodes falls from 0.0097 to 0.0064 and 0.0049, the brown ray’s from 0.0097 to 0.0062 and 0.0013. Every eased table prints the floor at L 14.00 and prints every node’s colour within the eased cap. The eight colours whose cap binds above L* 26 — all on the brown ray — keep 86 per cent of their dulling eased by a tenth and 63 per cent eased by a fifth: their dulling below L* 20 falls from 0.0113 to 0.0039, and their dulling between the binding point and L* 20 stays at 0.0062. A fifth’s ease leaves them less dulling than the shared nodes did, 0.0052 against 0.0066; a tenth’s leaves more.**

• The prediction’s cost does not come. No colour the table asks for is lost.
• Its halving fails for the colours it was about and holds for the others: the blue halves, the low-binding browns nearly vanish.
• The alternative fails too. The dulling moves wherever the ease reaches; it is the cap’s, below L* 20.

Read ray by ray, the ease buys most where the cap binds lowest. The blue’s hard-cap dulling falls by a third at a tenth’s ease and by half at a fifth’s; through the knee by a quarter and by two fifths. The brown’s hard-cap dulling falls by a third and then by seven eighths, and through the knee by a fifth and then by more than half. Every one of the blue ray’s thirty-three cases binds below L* 26, between 15 and 25, so its whole stretch of dulling lies in or near the eased deep shadow; the brown’s binding points run from 16 to 31.

The brown at a fifth’s ease is near its floor. Its low-binding cases, whose whole stretch of dulling lies below L* 20, are left with a mean dulling of six hundred-thousandths — some of them now printing a little more chroma than asked — and the ray’s remaining dulling is almost all in its eight high binders.

The ramp #

The ease is the simplest shape that leaves everything above the deep shadow alone. A 220 per cent cap stays at 220 down to L 20 and falls on a straight line to 198 or 176 per cent at L 14**; every cap in the census is eased the same way from its own value, 200 to 260 per cent. Above L* 20 nothing in any separation changes, which is why every colour’s dulling above that lightness is untouched and why the census’s test of the ease is clean: whatever moves, moves because of the deep shadow.

The prediction feared the darkest colours. A cap a fifth lower at the floor asks the press to print the rays’ deepest colour, a neutral at L* 14, with a sixth less ink. It does not need the extra: with black raised and the chromatic inks reduced, every case’s separation at the floor still prints L* 14.00 within a hundredth. The census’s floor is where its rays end, not the press’s deepest black — a black that is not black found a four-ink press reaching L* 2.4 — and whether an ease carried down there would cost darkness is a question this census does not reach.

Where the ease reaches #

Split at L* 20, the high binders’ dulling shows exactly where the ease acts. Below L 20 it falls from 0.0113 to 0.0086 and 0.0039 — by two thirds at a fifth’s ease; between the binding point and L 20 it stays at 0.0063, 0.0062 and 0.0062.** The earlier essay found these colours’ deep-shadow dulling large, and it is; it also found it larger than the dulling just below the binding point, and that stretch, six to eleven units of lightness for these colours, carries most of what is left.

That stretch is out of the ease’s reach by construction, since the cap there is unchanged. Whether a ramp starting higher could reach it, and at what cost, is a separate question; what this census settles is that the ease removes the dulling the cap causes where it is applied, and that for the high binders the cap is not the cause of all of it.

Arch by arch #

Drawn test lightness by test lightness, each case’s dulling is a row of small arches, one between each pair of nodes, its height set by how sharply the separation bends in that segment. Eased by a fifth, the arches below L 20 flatten for both cases; above L 20, the brown case’s arches are unchanged to the last digit.** Some of the low-binding blue’s arches dip just below nought near the floor: the eased separation’s bend there runs the other way, and the table prints a hair more chroma than asked.

The arches’ sameness above L* 20 is the census’s own check on the ramp. A change confined to the deep shadow should change nothing above it, and nothing does; the high binder’s remaining dulling is the dulling it had above L* 20 from the start.

Black rises faster, not slower #

The proposal’s mechanism was that an eased total lets black rise less steeply. The opposite happens: for the brown case binding at L 31.4, black at the floor rises from 77 per cent to 92 with a fifth’s ease*, and at every case on both rays black at the floor is more than a tenth higher eased than held. A lower total leaves less room for cyan, magenta and yellow, and black, which darkens without adding to the colour’s hue, takes over more of the deep shadow.

The dulling falls anyway, so the bend that caused it was not black’s steepness. A plausible reading is that the chromatic inks’ own curvature is what the nodes miss: under a held cap they are pushed down along the cap as black rises, and the separation turns where they are pushed; under an eased cap they leave earlier and more smoothly. The census measures that the dulling falls and black rises; it does not decompose the bend ink by ink.

Against the shared nodes #

The high binders were the shared nodes’ best case, and they set the bar. Held, on even nodes, they lose 0.0083; eased by a tenth, 0.0072; on the three shared nodes, 0.0066; eased by a fifth, 0.0052; eased by a fifth on the shared nodes, 0.0052. A tenth’s ease buys less than three shared nodes; a fifth’s buys more; the two together buy no more than the fifth alone, because the shared nodes sit in the deep shadow, where the ease has already straightened what they were placed to sample.

So the two remedies are not complements for these colours. A profile builder who eases the cap by a fifth has no use for the shared nodes on the high binders; one who cannot ease the cap — a press whose deepest black needs all the ink — still has them.

The knee under the ease #

The cap can be met hard, black raised exactly as far as the total requires, or through a knee that begins easing the chromatic inks out before the cap is reached. A knee in the ink limit quiets the shadow found the knee quieting the cast a four-ink table prints along the grey axis, and off the grey axis a knee trades hue for chroma found it, on saturated colours, buying a steadier hue at the price of more dulling. The knee’s dulling has been the larger number in every essay since, and the shared nodes reduced it less than the hard cap’s.

The ease reduces it less, too: the blue ray’s knee dulling falls from 0.0131 to 0.0101 and 0.0079, by two fifths at a fifth’s ease, where the hard cap’s falls by half; the brown’s from 0.0108 to 0.0087 and 0.0048. A knee already spreads the cap’s approach over fifteen units of lightness, so part of the bend the ease straightens was never sharp under the knee, and the ease has less to straighten. The knee and the ease are two ways of softening the same constraint, and the second buys less once the first has been paid.

What the knee keeps is its hue. The census reads chroma lost; the earlier essay’s case for the knee was that a saturated colour’s hue swings less through it, and the ease, which lowers the cap rather than softening its approach, does not change that argument. A builder who chose the knee for its hue keeps that choice and gets two fifths off its dulling besides.

The ease also leaves room for the refit the census has used before. A strategy keeps the refit only where black prints found refitting a table’s nodes against their own error halving an inverse table’s miss where black prints, and the eased deep shadow is where black now prints most: the refit would be working on a separation with less chromatic ink and a straighter black, which is the kind of separation it did best on.

What a profile builder should take from this #

Ease the total-ink cap below L 20 by a fifth.* It halves the dulling of colours whose cap binds low, nearly removes it on the brown ray, and costs nothing in the darkest colour the census asks the press for.

Do not expect it to fix colours that bind high. Their dulling between the binding point and L* 20 is untouched; the ease removes only what lies below.

Expect black to take over the deep shadow. An eased cap prints the darkest colours with more black and less chromatic ink, which is easier on drying and on grey balance and changes how the shadows respond to a black ink’s drift.

How the tables were built and read #

The press, rays, strategies and caps are the knee census’s: a four-ink press at D50, a deep-blue ray at the hue of solid cyan and magenta and a brown at a dark warm mix, each at chroma (L* − 14)(80 − L*)/60, black laid by each light-black strategy and cyan, magenta and yellow solved to print the colour asked; under a cap, black is raised until the total fits, hard or through a fifteen-unit knee. Here the cap at each lightness is the case’s cap times one less the ease times the share of the way from L* 20 to the floor, nought above L* 20. Where black at full would still not fit, the chromatic inks would be scaled down to the cap and the print’s lightness recorded; no node needed it. Each table holds 65 even nodes from L* 92 to 14, or those and the three shared nodes of the earlier essay, and is read at 481 test lightnesses; the dulling is the chroma asked for less the chroma printed, averaged below the binding point.

What this does not reach #

One ramp. The ease starts at L* 20 and is straight; a ramp starting where a colour binds would reach the high binders’ upper stretch, at a cost the census has not read.

A floor at L 14.* The rays end at L* 14, far above the press’s deepest black at L* 2.4; an ease carried down to the press’s own floor could cost darkness that this census cannot see.

Two rays. A deep blue and a brown; colours at other hues bind at other lightnesses and spread differently across the two stretches.

The cap’s purpose. The cap stands for drying and trapping, which the census does not model; an eased cap is less ink, which a press can only welcome, but the census does not say how much less a press needs.

Still open: whether a higher ramp reaches the colours that bind high #

The ease removed the dulling below L* 20 and left the high binders’ dulling between their binding point and L* 20 alone. A ramp starting at L* 26 or 32 would ease the cap where they bind.

The calculation is the census’s strategies with the same tenth and fifth of ease reached at the floor, on ramps starting at L 26 and 32, every case read on 65 even nodes, hard and through the knee.* The prediction is that a ramp from L* 32 halves the high binders’ dulling, because it now lowers the cap through their whole stretch, while costing the low binders a little of what the L* 20 ramp gave them, since the same ease spread over more lightness is weaker in the deep shadow. If the high binders’ dulling does not move with a higher ramp, their upper stretch’s bend is the strategy’s and not the cap’s, and no total-ink policy will reach it. A higher ramp trades one shadow for another makes that calculation.

A remedy that answers one cause #

The habit here is about a cause and a remedy that share a place.

The deep shadow held most of the high binders’ dulling, and the cap is held there, so the cap looked like the cause and easing it like the cure. Easing it removed the dulling in the deep shadow and showed, by what it left, that the high binders had a second stretch of dulling with a different cause. The prediction had averaged the two stretches and expected the remedy to halve the average.

The failure mode is to judge a remedy by its effect on a total that mixes regions it reaches and regions it does not. The test is to split the effect along the remedy’s own boundary — here, L* 20 — and to read each side: two thirds removed below, nothing above, which is a remedy working exactly as designed on half of the problem.