An ink limit takes the shadows' cast away

The cast returns where black is steep followed a seventeen-node four-ink table down the grey axis and asked what brings a faint cast back below a black-generation toe. It was not the jump in black’s curvature where the toe ends. It was how fast the chromatic inks have to come down once black takes the darkening over: while a toe holds black back, cyan, magenta and yellow carry the darkening and climb, and afterwards they have to be retired again, unequally, at the rate black takes over from them. Over nineteen black curves the returned cast ranked with cyan’s climb at 0.99.

That essay ended on the other thing that retires chromatic ink. Almost every separation carries a total-ink limit, a cap on the sum of the four coverages, because a sheet carrying too much ink will not dry; the separation is not unique put the usual figure between 240 and 320 per cent. Where the cap binds, black has to rise faster than the strategy asked so that the chromatic inks can fall. The prediction was that this forced descent returns a cast of its own — growing with how steeply the cap pushes the chromatic inks down, a lower cap, reached higher on the ramp, casting more — and that the useful number for a separation’s shadow would be the fastest rate at which it retires a chromatic ink, whatever retires it.

The cap replaces the strategy’s shadow, and its shadow is quieter #

The prediction’s half about rates holds and its conclusion does not. A cap does make the chromatic inks descend, and the shadow cast it leaves does rise with the rate of that descent. But measured against what the same strategy prints with no cap, the cap takes the shadows’ cast down in 32 of 33 strategy-and-cap pairs — from 110 thousandths of C*ab to 15 in the strongest whole-shadow case — because what it replaces is worse: a strategy light enough on black to reach the cap is one whose chromatic inks climb towards solid in the shadows, and that climb casts far more than the cap’s descent. And a lower cap casts less, not more.

• Below where it binds, the cap owns the separation. Every strategy under one cap holds the same four coverages there, to within the solver’s tolerance, and prints the same cast.
• A lower cap retires the chromatic inks more slowly, from lower coverages, and its shadow casts less: 12.9 thousandths in the table’s last segment at 200 per cent, 15.9 at 220, 21.1 at 240.
• Where the cap is first reached the table is quiet — as long as the cap is reached inside a segment rather than at a node.
• Black the strategy generates still beats black the cap forces. A heavy-black strategy, which never reaches any cap, casts a sixth as much in its shadows as the best capped one.

The figure is the result for one strategy. With no cap its cast climbs steeply through the last three segments, to 62.6 thousandths between L* 18.9 and the floor. Under each cap the line drops to almost nothing in the segment where the cap is first reached and then settles well below the uncapped line. Everything below is why a cap leaves a smaller cast than the curve it overrides, and what decides how small.

A cap only binds on a strategy light on black #

The census needed rebuilding before the question could be asked, because the strategy every earlier essay on this grey ramp used cannot reach a cap. It begins black at L* 60 and brings it to ninety per cent at the floor, and the most ink it ever prints in total is 167 per cent, at the floor. No press specifies a limit that low. A black that is not black priced a limit in the press’s gamut and found it nearly free down to about 260 per cent and expensive below; this essay prices it in the table’s grey, and needs a separation the limit actually touches.

A cap binds on a strategy that holds black back — one that starts it late, or stops it short, and leaves the shadows to cyan, magenta and yellow. The census uses nine such strategies: black beginning at L* 40, 50 or 60, rising to 0.3, 0.4 or 0.5 at the floor. Their totals peak at 307, 281 and 257 per cent, by where black stops. Against them, five caps: none, 260, 240, 220 and 200 per cent. A cap of 300 would bind on only the three lightest-black strategies, and only in the last six tenths of a unit of lightness above the floor, so the census starts at 260.

Under a cap, every lightness at which the strategy’s own separation would print more than the cap is re-solved: black is raised until cyan, magenta and yellow, solved again to print the grey with that black, bring the four to the cap exactly. Above that lightness the strategy is untouched. Below it the separation changes shape.

For the strategy with black from L* 50 to 0.4, a cap of 220 is reached at L* 21.9. Uncapped, cyan is still rising there, at 0.695, and goes on rising to 0.845 at the floor; black creeps from 0.31 to 0.40. Under the cap, black turns sharply up at the binding point and reaches 0.67 at the floor, and cyan turns down, to 0.576. The fastest rate at which the cap retires a chromatic ink is 16.4 thousandths of coverage per unit of lightness — the descent the prediction was about, and not a gentle one: after a toe twenty units wide the earlier census had black rising at 25 thousandths a unit, and below this binding point it rises at 45 on average.

Below where it binds the strategy has no say #

Drawn for all nine strategies at once, the capped black curves are nine different curves above their binding points and one curve below the lowest of them. Under 220 per cent the binding points fall between L* 24.9 and 18.9, and below 18.9 every strategy prints the same black, 0.67 at the floor, and the same chromatic inks with it.

That is exactly what a cap is, once written down. On the grey axis there is one degree of freedom — the separation is not unique is the essay about it — and a strategy is a rule for spending it. A binding cap is another rule for spending the same freedom, and where it binds it wins: the black is whatever brings the four inks to the cap, and which strategy was in force above no longer enters. Every table node below the lowest binding point holds the same four coverages to within eight hundred-thousandths across the nine strategies, which is the tolerance of the solver that finds them, and every segment there prints the same cast to within 0.4 thousandths.

The practical consequence is simple and is worth saying plainly. In a separation’s deep shadows, the ink limit is the separation setting. The black-generation choices a profile-building tool offers — where black starts, how much of the grey it replaces — describe the shadows only above the point where the limit takes over, and under a cap of 200 per cent the nine strategies, below L* 22.4, have no choice left in them at all.

The cast the cap prevents is the chromatic inks’ climb #

The prediction compared a capped shadow with nothing. The right comparison is the same strategy with no cap, over the same segments, because that is what the cap replaces. Of the 33 strategy-and-cap pairs in which a cap binds, 32 cast less capped than uncapped over the segments the cap reaches, and they sit well below the diagonal in the figure: the uncapped shadows run up to 110 thousandths, the capped ones never above 34. The one exception is the strategy with black from L* 40 to 0.5 under 200 per cent, which casts 9.2 thousandths capped against 8.9 uncapped — the strategy whose own shadow was already the quietest of the nine.

The reason is visible in the uncapped curves. A strategy light enough on black to reach a cap is one in which the chromatic inks carry the shadow, and they carry it by climbing: cyan from 0.60 at L* 30 to 0.85 at the floor in the reference strategy, magenta and yellow with it. That climb is where the uncapped cast comes from. The table interpolates each ink in straight lines between nodes — a lattice has no derivative is the essay about what that costs — the inks’ curves bend more and more unequally as they approach solid, and in the last segment the grey the table prints misses neutral by 62.6 thousandths of C*ab. The earlier essays on this ramp found the table’s cast in a chromatic inks’ round trip, rising and then being retired; the cap cuts the round trip short at its worst point.

So the prediction had the sign of the mechanism right and the sign of the effect wrong. A cap does force a descent, and a descent does cast. It forces it in place of a climb that casts several times more.

A lower cap descends more slowly and casts less #

The table’s last segment, from L* 18.9 to the floor, is the one place every cap below 260 decides outright for every strategy it binds on above it. Its cast there, averaged over those strategies, rises with the cap: 12.9 thousandths at 200 per cent, 15.9 at 220, 21.1 at 240. The prediction expected the opposite order, on the reasoning that a lower cap is reached higher on the ramp and so forces a longer, harder descent.

A lower cap is reached higher, but it does not descend harder. The fastest rate at which it retires a chromatic ink falls with the cap: 15.9 thousandths of coverage per unit at 200 per cent, 16.5 at 220, 17.3 at 240. A cap reached at L* 25 has eleven units of lightness to hand the darkening from the chromatic inks to black; a cap reached at L* 19 has five, and must do it faster. And the lower cap keeps the chromatic inks lower throughout — cyan finishes at 0.47 under 200 per cent and 0.67 under 240 — which puts them further from solid, where their curves bend least.

So the prediction’s rule about rates survives in this narrow form: across caps, the cast in the cap’s own shadow rises with the cap’s rate of descent. It is one of two things that move together, since a lower cap is both slower and lower, and three caps cannot say which of them carries the cast. What the rule does not survive is its intended use. The fastest rate of retirement, read alone, would have ranked a capped shadow above the uncapped one it replaced, because the uncapped shadow retires nothing: its chromatic inks only climb.

The segment where the cap is reached is quiet when the cap is reached inside it #

Each capped line in the first figure dips to almost nothing in the segment holding its binding point, and the dip is too regular to be chance. The segment where a cap is first reached casts less than both its neighbours in 24 of the 33 cases, and the placement of the binding point inside the segment decides which. Wherever the cap is reached more than a quarter of the way down a segment and more than a fifth above its lower node, that segment casts at most 0.43 of its quieter neighbour, and as little as 0.02. The nine that are as loud as a neighbour all have the binding point within a quarter of a segment of a node.

The mechanism is the one a strategy keeps the refit only where black prints found at the other end of the ramp, where black begins. There, the table’s most accurate segment was the one holding the switch, because its chord runs between a node on one side of the switch and a node on the other, and the straight line between two separations of nearly the same grey trades black against the chromatic inks — a move along the separation family, which changes the grey’s colour very little. The cap makes the same kind of corner in the same kind of curve, and a segment whose two ends lie on either side of the corner draws its chord across it. When the corner is near a node the segment’s chord is the strategy’s alone or the cap’s alone, and it casts like its neighbours.

This is a statement about where a table’s nodes fall, not about the cap, and a table’s nodes are not usually placed with its ink limit in mind. It says that the dips in the first figure are not the cap’s doing so much as the corner’s, and that they would move with the grid — a table’s error lives between its nodes, which is what a profile is a fit between its nodes measured for a whole printer profile.

Black generated early casts less than black forced late #

The cap improves on the light-black strategy it overrides, and it is still worse than a strategy that never needed it. Below L* 30, the heavy-black strategy’s worst segment casts 2.1 thousandths, where the cap at 200 per cent casts 13.0 in the last segment and 14.4 in the one above. Both end the ramp with a lot of black, 0.9 and 0.77. What differs is how they get there. The heavy strategy brings black in from L* 60 at a steady rate and cyan never climbs past 0.34; the capped one holds black back until L* 25.6 and then has to push it up at 43 thousandths a unit, and its chromatic inks start their descent from above 0.6.

That is the earlier essay’s finding reached by another route. The cast follows how much chromatic ink the separation carries into the shadows and how fast it then has to hand the darkening to black, and a cap is a way of doing that handover late and fast. It is better than not doing it at all, which is what the uncapped light-black strategy does; it is not as good as doing it early.

What a separation tool should be told #

The ink limit is not a cost to be charged against the shadow’s neutrality. Measured against the separation it replaces, a cap improves the table’s grey in the shadows of every light-black strategy but one, often several times over. A shop that holds its limit for drying reasons is not paying for it in the table’s cast.

Below the binding point, the black-generation settings are not settings. Two profiles built with different strategies and the same limit print the same shadow, and a proofing comparison between them there is a comparison of nothing. The fourth ink is not for colour argued that black’s job is to replace the chromatic inks’ grey; the cap enforces the replacement where the strategy declined to.

If the shadows need to be neutral, generate the black rather than letting the limit force it. A heavy-black strategy that stays under every cap gives a shadow six times quieter than the best capped one, because its chromatic inks never climb.

How much any of this matters on paper is still the question a little black ends the cast left. Casts of a few hundredths of C*ab are far below a press’s own variation, and a table’s errors in lightness — the inverse table errs dark is the essay about those — are usually the larger part of what it gets wrong. The point of pricing the casts is to know what a separation’s shape does to the table’s error before choosing it for other reasons.

How the census was built #

The press, the grey ramp and the table are those of the earlier census: the same four-ink Neugebauer model under D50, a ramp from L* 92 to 14, and a seventeen-node table holding the separations at evenly spaced lightnesses and interpolating the four coverages linearly, tested at 241 evenly spaced greys. A strategy fixes black at each lightness — none above its start, rising linearly to its maximum at the floor — and cyan, magenta and yellow are solved by damped Newton iteration to print L*, 0, 0 with that black, to 0.002 in CIELAB. Under a cap, wherever the strategy’s four coverages would total more than the cap, black is stepped up in hundredths until the re-solved total fits and the last step is then bisected thirty times, so that the total equals the cap to within a few hundred-thousandths; the total falls monotonically as black rises along the grey axis, and every capped separation prints its grey to within 0.002. The binding point is the lightest grey at which the cap is reached, found on a tenth-unit grid. A segment’s cast is the mean C*ab of the greys the table prints inside it; a shadow’s cast is the mean over segments ending below the binding point; the rate of descent is the largest drop in any chromatic ink over half a unit of lightness below the binding point.

What this leaves out #

The ramp is the neutral axis. A dark saturated blue near the cap is separated by the same strategy and meets the same limit, but its separation has no free coverage to trade in the same way, and whether a cap there also quiets the table is not measured.

The cap is a hard corner. Profile-building tools often approach their limit gradually, easing black up over some range of lightness before the total is reached, so that the separation has no kink. A gradual approach moves the corner this essay found quiet into a stretch rather than a point, and what that does to the segment where it begins is the question below.

The strategies are straight lines. Black rises linearly from its start; the toed and eased curves of the earlier census would reach the cap at slightly different points and, below it, would print exactly what these do.

Still open: whether a cap reached gradually stays quiet wherever the nodes fall #

The binding segment here is quiet because a corner in the separation sits inside it, and loud when the corner sits at a node. A real limit is usually approached through a knee a few units of lightness wide, and a knee is a corner spread out. The calculation is this census with the cap approached over a knee of five, ten and fifteen units, black eased up so that the total reaches the cap smoothly, and the binding segment’s cast read against where the knee falls among the nodes. The prediction is that a knee as wide as a segment — about five units on this table — makes the binding segment quiet wherever it falls, because every chord then crosses part of the knee, while a knee much wider than a segment turns the cap back into a steep straight black, whose shadow casts like the heavy strategy’s at the same rate of rise. If both hold, the knee’s width is a choice a profile-building tool can make against its own grid.

Measure an intervention against what it displaces #

The habit is about the baseline a prediction is tested against.

The cap was predicted to return a cast, and the prediction was tested by asking how much cast the capped table prints below the binding point. It prints some — twelve to twenty-one thousandths in the last segment — and read alone that would have confirmed it. The number that decided the question was the cast the same strategy prints there without the cap, because a cap does not add to a separation; it replaces part of one. Against that, the cap’s cast is a large reduction, and the mechanism the prediction named, a forced descent, turns out to be the smaller of two effects moving in opposite directions.

The failure mode is to measure a constraint’s effect against zero when its effect is a substitution. Whenever a rule overrides another rule inside some range, the right comparison is the overridden rule inside that range, and the interesting finding is often which of the two was worse.