The cast returns where black is steep

A little black ends the cast rounded the start of black generation with a toe and followed a four-ink inverse table’s error down the grey axis. Above the start the grey is three chromatic inks and the table prints a faint warm cast, about 0.0125 in C*ab; below it the cast ends once black supplies a tenth to a sixth of the darkening, because inside a toe the table’s chord runs along the trade of black against the other three inks and errs in lightness rather than hue. It also found a smaller cast coming back just below where each toe joins the straight part of its curve — 0.0013 with a toe of ten units, 0.0025 with twenty, 0.0040 with thirty, up to a third of the three-ink cast.

The explanation it offered was the toe’s curvature. A parabolic toe bends black at a constant rate and then stops bending all at once where it meets the straight part. While black bends, the chromatic inks bend against it and the chord’s error is a trade; when black stops bending, nothing swamps the chromatic inks’ own unequal bends, and the cast returns. If that is right, the returned cast is set by the jump in black’s second derivative at the join: a toe that eases its curvature to nothing should not bring the cast back, and a kink put deliberately into a straight black curve should bring one of its own.

Neither the jump nor the kink #

A toe whose curvature is nothing at both ends returns exactly the parabola’s cast in every segment below the join, at every width from 5 to 30. Over nineteen black curves the returned cast has a rank correlation of 0.18 with the curvature jump, 0.97 with black’s slope on the straight part and 0.99 with how far cyan climbs above its level at the start. A kink put into straight black casts more than both its neighbours in one of twelve cases; in eleven the neighbour on the side where black became steeper casts more than the other. A cubic toe that eases its curvature down returns 1.14 thousandths of C*ab at a width of thirty where the parabola returns 3.96.

• The prediction fails. The curvature at the join decides nothing; removing the jump leaves the cast unchanged to the last digit.
• The returned cast follows steepness. It is set by how fast black has to rise once the toe is done, and how far the chromatic inks climbed while it was being held back.
• A kink casts on its steep side, one segment away from itself.
• A toe’s shape still matters — but through how much black it withholds, not through how smoothly it ends.

Three toes of one width #

The census uses the same four-ink grey ramp as the essays before it, from L* 92 near the paper to L* 14 in a deep shadow — not far above the L* 2.4 that a black that is not black found to be the deepest a four-colour press can print — separated by a strategy that begins black at L* 60 and reaches ninety per cent black at the floor. At every lightness black is fixed by the strategy and cyan, magenta and yellow are solved to hit a neutral grey with that black. A seventeen-node table holds the separations at evenly spaced lightnesses and interpolates between them, and its cast is the mean chroma of the grey it prints in each segment.

The toe comes in three shapes, each joining the straight part with the same slope. The parabola is the earlier essay’s: black’s curvature is constant across the toe and stops dead at both ends. The eased cubic starts with twice the parabola’s curvature and lets it fall in a straight line to nothing at the lower end, so its join is smooth in curvature and its start is not. The smoothstep has a curvature that rises and falls back to nothing at both ends. All three are then scaled so that black still reaches the same coverage at the floor.

The figure shows what that scaling does. Under the parabola and the smoothstep, black enters its straight part half the toe’s width behind where a kinked start would have it; under the eased cubic, only a third. While black is held back, the chromatic inks carry the darkening and cyan keeps rising below the start — by 0.085 of coverage under the parabola, 0.093 under the smoothstep and 0.053 under the eased cubic — before black takes over and it turns down. Below the toe’s end the parabola and the smoothstep are the same curve: their black is identical there, and so is every coverage solved from it.

Where each toe stops bending #

The curvature plot is the variable the prediction named. The parabola’s curvature is a flat step, 1.25 thousandths of coverage per unit of lightness squared across the toe and nothing on either side of it: a jump at L* 60, where the toe begins, and another at L* 40, where it ends. The eased cubic’s curvature falls in a straight line from 2.3 thousandths at the start to nothing at the join, so it has one jump, at the top, and none at the bottom. The smoothstep’s rises from nothing to a peak of about 1.9 and falls back to nothing, and it has no jump anywhere.

If the returned cast were set by the jump at the lower end, the parabola would return a cast and the other two would not. A lattice has no derivative warned that a table’s errors come from what happens between its nodes; a jump in curvature between two nodes is exactly the kind of event that ought to show there. It did not have to be curvature, though — the chord between two nodes misses a curve by an amount set by how the curve bends between them, but which curve bends is the question, and black is not the only ink the chord interpolates.

The cast, segment by segment #

Below L* 40 the parabola and the smoothstep draw one line. The segments from 38.4 down to 18.9 cast 2.53, 2.17, 1.14 and 0.44 thousandths under both, identical to the last digit, because the table’s nodes there hold identical separations. The jump that one of them has and the other does not has no effect on anything below it. Inside the toe the two differ — the smoothstep’s slower start leaves more of the three-ink cast in the first segment below L* 60 — but that is the part of the ramp a little black ends the cast already explained.

The eased cubic returns less. Its segments below L* 40 cast 1.52, 0.94 and 0.36 thousandths, and between the toe’s end and L* 20 the returned cast averages 0.94 against the parabola’s 1.94. The eased toe has no jump at its lower end, but neither has the smoothstep, so the absence of a jump cannot be what made the difference. What the eased toe does differently is visible in the first figure: it holds black back by a third of its width rather than a half, so its straight part begins with more black already printed, and it rises at 22.9 thousandths of coverage per unit of lightness rather than 25.0.

The kinked start, with no toe at all, returns 0.41 thousandths over the same stretch: its black is the least steep of all, rising at 19.6 thousandths a unit.

Across six widths #

The parabola’s returned cast grows steadily with the toe’s width: 0.89 thousandths at five units, 0.94 at ten, 1.29 at fifteen, 1.94 at twenty, 2.82 at twenty-five and 3.96 at thirty. The smoothstep lies on the same line at every width. The eased cubic’s barely grows: 0.50, 0.89, 0.89, 0.94, 1.05 and 1.14. At thirty units the eased toe returns 3.5 times less cast than a parabola of the same width.

The largest single segment tells the same story. At thirty units the parabola’s worst segment below the toe casts 3.96 thousandths, which is the 0.0040 the earlier essay reported, and the eased cubic’s worst casts 1.14. At twenty, 2.53 against 1.52. The earlier essay’s numbers are reproduced exactly; what changes is what they are attributed to.

Width alone does not decide it either. A parabolic toe of fifteen and an eased toe of twenty-five hold black back by about the same amount — seven and a half units and eight and a third — and return 1.29 and 1.05 thousandths. What they share is the slope black must take afterwards, 23.4 and 23.9 thousandths a unit.

What the returned cast follows #

Over the kinked start and eighteen toes, the returned cast ranks with cyan’s climb at 0.99 and with black’s slope on the straight part at 0.97. Its rank correlation with the jump in black’s curvature at the join is 0.18. The points form one rising curve with the three shapes interleaved along it, the smoothsteps sitting just to the right of the parabolas of the same width because they climb a little further inside the toe and then follow the same black below it.

The mechanism the correlation describes is the chromatic inks’ round trip. While a toe holds black back, the grey still has to darken at the rate the ramp asks for, and cyan, magenta and yellow do it: they rise past the coverages they had at the start. Once the toe is done, black has to make up the ground it gave away — it is scaled to reach the same coverage at the floor — so its straight part is steeper than a kinked start’s, and the chromatic inks have to come back down at the rate black takes the darkening from them. They do not come down equally. Their curves bend on the way, the chord between two nodes misses each of them by a different amount, and because black is straight there the misses are not a trade against black: they print as hue. The separation is not unique is why a grey can be made so many ways; this is the cost of changing, along the ramp, which way it is made.

So the returned cast is not the absence of black’s curvature. It is the chromatic inks’ descent, and the steeper black is, the faster and more unequally they descend.

A kink casts one segment away #

The prediction’s second half was a kink: a straight black curve given a sudden change of slope partway down should bring back a cast of its own, at the kink. The segment holding the kink casts more than both its neighbours in one case of twelve, and there by three hundredths of a thousandth — 0.66 against 0.63 and 0.49 at L* 30 with the slope raised by a quarter. Everywhere else a neighbour casts more.

Which neighbour is decided by steepness. A kink that raises black’s slope below it makes the segment below cast: with the slope doubled at L* 40 that segment casts 2.53 thousandths where the straight curve cast 0.55, and doubled at L* 30, 3.96 where it cast 0.17. A kink that lowers the slope below it — which, because the curve is rescaled to reach the same floor, makes black steeper above — makes the segment above cast: 2.77 against 0.41 at L* 40 with a factor of 0.8. In eleven of the twelve cases the neighbour on the steep side casts more than the other. The kink itself is an event between two nodes of the table, and like the jump in curvature it leaves no mark of its own.

The one number that does not fit is instructive in its own way. At L* 30 the doubled slope’s segment below casts 3.96 thousandths, exactly what a thirty-unit parabolic toe casts at the same lightness — and for the same reason: with those settings the two curves are the same straight black below L* 30.

What a separation tool should be told #

Smooth curvature is not the advice. A black-generation curve whose second derivative is continuous everywhere returns the same cast as one whose curvature jumps, if the two withhold the same black. The earlier essay’s suggested remedy, easing the toe’s curvature at its lower end, works in the eased cubic — but because that toe withholds less, not because it ends smoothly, and a smoothstep that eases both ends and withholds as much as a parabola is no better than the parabola.

The advice is about slope. Every stretch of the grey axis where black rises faster than a plain straight start would have it rise is a stretch where the chromatic inks are being retired quickly, and a table interpolating across it casts. A toe that holds black back far is paid for twice: once inside the toe, where the separation errs in lightness and the segment where black begins becomes the table’s worst, and again below it, where black must hurry to catch up. At thirty units that second payment is up to a third of the three-ink cast; with an eased toe, under a tenth.

How much it matters on paper is the same question a little black ends the cast left: casts of a few thousandths of C*ab sit well under a press’s own variation. The point of pricing them is to know what a curve’s shape is doing to the table, so that a curve chosen for other reasons — shadow detail, ink limits, the fourth ink’s actual job — can be chosen knowing where it moves the table’s error.

How the curves were built #

Black is none above L* 60 and rises to ninety per cent at L* 14. A toe of width T replaces black’s first T units with a rise s(d) at depth d below the start: the parabola $d^2/(2T)$, the eased cubic $d^2/T - d^3/(3T^2)$ and the smoothstep $T(x^3 - x^4/2)$ with $x = d/T$, each reaching slope one at d = T and continuing straight; the whole is then scaled by 0.9 over its value at the floor. A kink multiplies a straight curve’s slope by a factor below a stated lightness, again rescaled to the floor. At each lightness the three chromatic inks are solved by damped Newton iteration to print L*, 0, 0 under D50 with the strategy’s black, to 0.002 in CIELAB — a tolerance that moves an individual segment’s cast by up to about five ten-thousandths, less than any difference reported here. The table has seventeen nodes from L* 92 to 14, interpolates the four coverages linearly and is tested at 241 evenly spaced greys; a segment’s cast is the mean C*ab it prints. The returned cast is the mean over segments lying wholly between the toe’s end and L* 20, which leaves out the floor’s own cast; cyan’s climb is its highest coverage below the start less its coverage at the start, found on a half-unit grid; the slope is black’s rise from the toe’s end to L* 20 per unit of lightness.

What this leaves out #

The grey axis is one line through a table of three dimensions. A chromatic colour near the neutral axis is separated by the same strategy and interpolated between the same kind of nodes, and whether its cast behaves this way is not tested here.

The strategy is one family. Black generation in practice is also shaped by an ink limit, which caps the four coverages’ sum in the shadows and bends the chromatic inks down for a reason of its own. An ink limit makes the chromatic inks descend faster in exactly the stretch this essay says matters; it would add to the returned cast, and by how much is a separate calculation.

The printing model is the collection’s, a Neugebauer model whose inks are built from stated absorption bands. It carries the chromatic inks’ unequal bends because it carries their unequal absorption bands, and a real press’s bends would differ in size; the ordering by steepness should not.

Still open: whether an ink limit returns a cast of its own #

Most separations cap the total coverage in the shadows, and where the cap binds the strategy has to take chromatic ink out to make room for black. That is the chromatic inks’ descent again, forced by a constraint rather than by a curve, and it begins exactly where the cap is first reached.

The calculation is this census with a total-ink limit of 300 and 260 per cent applied to each strategy, the chromatic inks re-solved under the cap, and the table’s cast read from where the cap binds to the floor. The prediction is that the returned cast grows with how steeply the cap forces the chromatic inks down — a lower limit, reached higher on the ramp, casting more — and that it ranks with the chromatic inks’ rate of descent as the toes here did. If it does, the useful single number for a separation’s shadow is the fastest rate at which it retires a chromatic ink, whatever retires it.

A cause at the join is a claim about the join #

The habit is about testing a mechanism by removing the thing it names and nothing else.

The returned cast was explained by the jump in black’s curvature where the toe ends, and the explanation fitted every observation made: the cast came back where the jump was, and grew with the toe. The test was a toe with no jump at its end, and it had to be built so that it changed the jump and as little else as possible. The smoothstep did that; below the toe it is the parabola exactly, and it returned the parabola’s cast exactly. The eased cubic, which the prediction also expected to cure the cast, changed the jump and the amount of black withheld together, and it did return less — which, alone, would have seemed to confirm the prediction.

The failure mode is to test a mechanism with an intervention that changes two things, and credit the one the mechanism named. The eased toe was a real improvement for the wrong reason; only the toe that removed the jump alone could say so.