A bent ramp takes more ink and dulls more

A four-ink press prints its dark colours under a total-ink cap, and where a colour’s inks would exceed it, black is raised and the chromatic inks lowered until the total fits. A profile stores those separations at its nodes and interpolates between them, and a profile is a fit between its nodes is the reminder that wherever the separation bends between two nodes, the colour printed there is not the colour asked for — for dark saturated colours, it is duller.

Easing the cap reaches the deep shadow and no higher lowered the cap on a straight ramp from L* 20 to the floor and found the dulling below L* 20 falling by two thirds, at no cost in the rays’ deepest colour. A higher ramp trades one shadow for another started the same ramp at L* 26 and 32 to reach the high-binding colours’ dulling above L* 20, and found it reached — the upper stretch’s dulling fell by more than half — while the deep shadow’s cure came undone in the same measure, and every group did best with the ramp at L* 20.

Those ramps were straight, and a straight ramp from L* 32 spends most of its ease where the low binders have no dulling. A ramp bent at L* 20 could spend a little above and most below. The prediction was that a ramp easing a twentieth of the cap between L 32 and 20 and the remaining three twentieths between 20 and the floor cuts the high binders’ total below the straight ramp’s by a fifth — their upper stretch losing a third of its dulling while the deep shadow keeps nearly all its cure — and that the low binders lose under a tenth of their gain.* The alternative was that the bent ramp does no better than the straight one, which would mean the deep shadow’s cure needs every part of the ease it gets.

More ink, more dulling #

With the cap eased by a fifth at the floor on the bent ramp, the cap is at or below the straight ramp’s at every lightness, for every cap. Yet the bent ramp leaves every ray, knee and ease with more dulling than the straight ramp from L 20. For the eight colours binding above L 26 it cuts the dulling between their binding point and L* 20 from 0.0062 to 0.0052, a sixth, and raises their dulling below L* 20 from 0.0039 to 0.0072; their total rises from 0.0052 to 0.0060. The brown’s twenty-five low binders, left with 0.00006 by the straight ramp, keep 0.0039 — two fifths of their held dulling.**

• The prediction fails at every step. A sixth off the upper stretch, not a third; half the deep cure lost, not nearly none; the low binders lose most of their gain.
• The alternative holds, and more strongly than it was put. The deep shadow does not merely need its share of the ease; it needs the ease delivered steeply.
• Taking more ink is not what cures the deep shadow.

Split at L* 20, the bent ramp’s effect on the high binders is two moves in opposite directions. Above L 20 their dulling falls from 6.18 thousandths to 5.20; below it, it rises from 3.85 to 7.16*, two thirds of the way back to the held cap’s 11.27. The upper stretch gains about what a twentieth of ease spread over twelve units of lightness could be expected to buy, given that the straight ramp from L* 26, with a tenth there, had bought more than half.

The deep shadow’s loss is the surprise. The bent ramp eases the cap there by more than the straight ramp at every lightness — at L* 20 by a twentieth where the straight ramp has not begun, at the floor by the same fifth — and the deep shadow’s dulling nearly doubles.

The two caps #

Drawn together, the two caps differ in two ways. The bent ramp starts falling at L 32 and reaches 209 per cent at L 20, where the straight ramp is still at 220; below L* 20 both fall to 176 per cent at the floor, the straight ramp by 44 points over six units, the bent ramp by 33.** The bent cap lies at or under the straight one everywhere; what it gives up is steepness, in exchange for an earlier start.

So the census has three facts that fix where the deep shadow’s cure comes from. The straight ramp from L* 26 eased less below L* 20 than the straight ramp from 20 and cured less there; that was compatible with the amount of ink mattering. The bent ramp eases more below L* 20 than the straight ramp from 20, and cures less. What the straight ramp from L 20 has that both others lack is the steepest fall below L 20** — the whole fifth spent over six units — and the deep shadow’s cure tracks that steepness.

Every group #

The blue ray’s dulling is 9.71 thousandths held, 4.89 on the straight ramp and 6.39 on the bent; the brown’s high binders 8.29, 5.19 and 5.98; its low binders 10.15, 0.06 and 3.94. The low binders, all of whose dulling lies below L* 20, are the clearest: the bent ramp eases their whole stretch more than the straight ramp does, and leaves them with forty times its dulling.

The blue ray, whose cases all bind below L* 26, is the same story at a smaller scale. Its dulling, nearly all in the deep shadow, rises by three tenths on the bent ramp.

Arch by arch #

Test lightness by test lightness, the bent ramp’s arches — one between each pair of nodes, its height the separation’s bend there — are lower than the straight ramp’s above L* 20 and taller below it. Below L 20 they stand between the straight ramp’s and the held cap’s*, nearer the held cap’s at the deepest segments. The bend a node sees is set by how the separation curves across it, and a cap falling steeply below L* 20 bends the separation one way at the ramp’s start and straightens it below; a cap falling gently bends it less at the start and leaves more of the held cap’s curvature in the deep shadow.

That is a reading of the arches, not a decomposition of them. The census measures the dulling and the cap; how the chromatic inks curve under each ramp, segment by segment, it has not computed.

Black under the two ramps #

Under the straight ramp, black at this case’s nodes climbs by about four hundredths a node above L 20 and by eight or nine below it, reaching 92 per cent at the floor; under the bent ramp it climbs faster above L 20 and by about seven hundredths a node below it, reaching the same 92 per cent.** The straight ramp’s black turns sharply at L* 20; the bent ramp’s turns less. If the dulling followed black’s turn, the straight ramp would be the worse of the two; it is the better.

Easing the cap reaches the deep shadow and no higher found the eased cap raising black faster than the held one and the dulling falling anyway, and concluded the cure was not black’s steepness. The bent ramp confirms it from the other side: black’s turn is not where the dulling comes from. The chromatic inks, which a steeply falling cap pushes out of the deep shadow quickly and a gently falling one slowly, are the likelier carriers of the bend.

Four ramps on one line #

The census has now read four ways of spending the same fifth, and they can be set side by side on the one property the bent ramp isolates. Below L* 20 the straight ramp from L* 20 lowers the cap by the whole fifth over six units of lightness, about 3.3 per cent of the cap a unit. The bent ramp lowers it by three twentieths over the same six units, 2.5 per cent a unit. The straight ramp from L* 26 has a tenth left to spend there, 1.7 per cent a unit, and the ramp from L* 32 a little under seven per cent, 1.1 per cent a unit.

Ordered by that slope, every group’s dulling falls into line. The high binders’ deep shadow is 3.85 thousandths at the steepest, 7.16 on the bent ramp, 9.33 from L 26 and 10.38 from L 32; the brown’s low binders 0.06, 3.94, 6.27 and 8.44; the blue ray 4.89, 6.39, 7.52 and 8.48.** Not one group breaks the order, and nothing else the four ramps carry sorts them the same way. Ordered instead by how much the cap is lowered below L* 20 in total, the bent ramp would sit first, ahead of the straight ramp from L* 20, and it sits second.

The three straight ramps could not have shown this. On them the amount below L* 20 and the slope below L* 20 move together — a higher start leaves less to spend and spends it more gently — so the higher ramp was free to be read either way. The bent ramp is the one case where they part: more ink taken below L* 20 than the straight ramp from 20, at a gentler slope. The deep shadow sides with the slope.

Four ramps are four points, and four points in a line are a pattern rather than a law. What would make it a law is in the closing section.

Where the upper stretch’s cure must come from #

The bent ramp was the last way of reaching the high binders’ upper stretch with the cap without the deep shadow paying for it, and it pays. Every ramp read so far that lowers the cap above L* 20 lowers the slope below it, because the ease at the floor is fixed and part of it has been spent before L* 20.

That leaves the remedy shared nodes belong in the deep shadow found for the other end of the problem. A node placed between a high binder’s binding point and L* 20 shortens the arch that stretch carries without touching the cap at all; the upper stretch’s 6.18 thousandths on the straight ramp is a bend between nodes, and nodes are what shorten bends. The cap and the nodes then each do the job the census has found them good at — the cap straightens the deep shadow, the nodes the stretch above it.

There is a reason the ink policy cannot do both. An ink limit takes the shadow’s cast away showed the cap’s work is to move a colour’s separation from chromatic inks onto black; once that move has started, the separation’s shape between two nodes is set by where the move started and how fast it goes. A cap that falls gently from L* 32 starts the move early everywhere, and the high binders, which were already moving, gain a little above L* 20 and lose the steep finish below it.

The knee and a tenth #

Through the fifteen-unit knee and at a tenth’s ease, the bent ramp is worse in all four cases: the blue’s knee dulling at a fifth is 7.93 thousandths straight and 8.65 bent; the brown’s 4.75 and 6.74. The margin is least on the blue at a tenth’s ease, 10.05 against 10.40, where neither ramp cures much and where they differ least.

A knee in the ink limit quiets the shadow found the knee spreading the cap’s approach over fifteen units of lightness. Under a knee the cap’s bend is already softened, and the difference between a steep and a gentle ramp shrinks — but does not reverse.

What a profile builder should take from this #

Ease the cap below L 20 by a fifth, steeply, and nowhere else.* Of four ramps the census has read — straight from L* 20, 26 and 32, and bent at 20 — the straight ramp from L* 20 leaves every ray and every group of colours with the least dulling.

Do not spend the ease to reach the high binders’ upper stretch. Every way of spending some of it above L* 20 costs the deep shadow more than it buys above; that stretch belongs to nodes placed between L* 20 and the binding points, as the earlier essay argued.

Judge an ink policy by its slope where the bend is, not by its total. A ramp that takes more ink everywhere can dull more, if it takes it gently where the separation needed a sharp turn.

How the tables were built and read #

The press, rays, strategies and caps are the knee census’s. On the bent ramp the cap at each lightness is the case’s cap times one less the ease times a share that is nought above L* 32, rises linearly to a quarter at L* 20 and to one at the floor; the straight ramp’s share is nought above L* 20 and rises linearly to one at the floor. The ease reached at the floor is a tenth or a fifth. Each separation is the census’s — black raised until the total fits, hard or through the fifteen-unit knee — on 65 even nodes from L* 92 to 14, read at 481 test lightnesses; the dulling is the chroma asked for less the chroma printed, averaged below each case’s binding point and, for the high binders, split at L* 20. Every node printed its colour within the eased cap, and every floor at L* 14.

What this does not reach #

One bent shape. A quarter of the ease above L* 20 is one split; a twentieth would cost the deep shadow less, and the census has not read how small a share above L* 20 stays harmless.

Slope as a cause. The census finds the cure tracking the cap’s steepness below L* 20 across four ramps; it does not vary the steepness alone with everything else held, which is what would make it a cause.

Two rays. Colours whose dulling lies mostly above L* 20 would weigh the bent ramp’s gain there more heavily.

Still open: whether a steeper ramp than the straight one cures more #

If the deep shadow’s cure tracks how steeply the cap falls below L* 20, a ramp that falls more steeply still — the whole fifth spent over the last three units above the floor, from L* 17 — should cure more, and one spent over twelve units, from L* 26 with a sharper end, less.

The calculation is the census with the cap eased by a fifth on straight ramps from L 17 and from L 18.5 to the floor, every case read on 65 even nodes, hard and through the knee, beside the ramp from L* 20.** The prediction is that the ramp from L* 17 leaves the low binders’ and the blue’s dulling lower than the ramp from L* 20, and the high binders’ no higher, because it is steeper still where the bend is. If the ramp from L* 17 dulls more, the cure needs the cap to fall across the whole deep shadow rather than steeply, and L* 20 is not one choice among many but where the deep shadow’s bend begins.

The amount was not the lever #

The habit here is about which property of an intervention is the active one.

Every earlier ramp was described by how much it eased the cap and where, and the natural reading was that easing more in the deep shadow cures more. The bent ramp eases more in the deep shadow than the ramp it was compared with, and cures less, which removes amount as the explanation and leaves the property the two ramps differ in below L* 20 — how fast the cap falls there.

The failure mode is to describe an intervention by one property and attribute its effect to that property, when every intervention compared also differs in another. Two ramps that both ease more and fall more steeply cannot separate the two; the test is to build the case that splits them — here, one that eases more and falls less steeply — and read which property the effect follows.