A higher ramp trades one shadow for another

A four-ink table stores a separation at each of its nodes and interpolates between them, and where the separation bends — as it does when a dark colour’s inks meet the press’s total-ink cap and black is raised to keep the total under it — the colour printed between nodes comes out duller than the colour asked for. Shared nodes belong in the deep shadow found that dulling, for colours whose cap binds high, carried mostly by the deep shadow below L* 20, and easing the cap reaches the deep shadow and no higher lowered the cap there instead of adding nodes.

The ease worked where it reached. Eased by a fifth on a straight ramp from L 20 to the floor, the deep-blue ray’s dulling on even nodes halved, the brown ray’s fell to an eighth, and the floor still printed at L 14.** The eight colours whose cap binds above L* 26 kept three fifths of theirs: their deep-shadow dulling fell by two thirds, but between their binding points and L* 20 — six to eleven units of lightness — it stayed exactly where it was, since the cap there was unchanged.

The natural next ramp starts higher. A ramp from L* 32 begins easing the cap above every colour’s binding point, so it lowers the cap through the high binders’ whole stretch of dulling, not just the bottom of it. The prediction was that a ramp from L 32 halves the high binders’ dulling, while costing the low binders a little of what the L 20 ramp gave them**, because the same fifth of ease, spread over eighteen units of lightness instead of six, is weaker in the deep shadow. The alternative was that the high binders’ dulling does not move with a higher ramp, in which case the bend in their upper stretch is the strategy’s, and no total-ink policy reaches it.

The bend is the cap’s in both stretches #

With the cap eased by a fifth at the floor on ramps from L 20, 26 and 32, every ray, knee and group of colours is left with the least dulling, or within a fiftieth of the least, by the ramp from L 20. For the eight high binders a ramp from L* 26 cuts the dulling between their binding point and L* 20 from 0.0062 to 0.0023, and lets their dulling below L* 20 return from 0.0039 to 0.0093: their total stays at 0.0052, and from L* 32 rises to 0.0061. The brown’s low binders, left with almost no dulling by the ramp from L* 20, keep 0.0063 from L* 26 and 0.0084 from L* 32, against 0.0102 with the cap held. No ramp costs the floor its lightness.**

• The prediction fails. A higher ramp does not halve the high binders’ dulling; it leaves it where it was or makes it worse.
• The alternative fails too. The upper stretch’s dulling falls by more than half once the cap above L* 20 is eased: that bend is the cap’s.
• The low binders pay most, giving back nearly all their gain.

Drawn against the ramp’s start, the high binders’ two stretches cross. The upper stretch falls from 6.18 thousandths with the ramp at L 20 to 2.31 at 26 and 3.13 at 32; the deep shadow rises from 3.85 to 9.33 and 10.38.* The total, the mean over both stretches weighted by their lengths, is 5.19 from 20, 5.20 from 26 and 6.07 from 32 — a line nearly flat and then rising, under the held cap’s 8.29.

Each stretch answers the cap where the cap is eased. The earlier essay showed the deep shadow answering; this one shows the upper stretch answering too, which settles the question the alternative raised — the high binders’ upper dulling is the cap’s bend, not the black strategy’s. What the census shows instead is that a single ramp has one fifth of ease to spend, and spending it higher spends it away from the deep shadow, where it was already doing as much as it does above.

Every ray prefers the deep shadow #

Over each ray as a whole the ordering is plain. The blue’s hard-cap dulling is 4.89 thousandths with the ramp at L 20, 7.52 at 26 and 8.48 at 32, against 9.71 held; the brown’s is 1.30, 6.01 and 7.87, against 9.70.* Through the knee the same order holds on both rays. A ramp from L* 20 gives every ray more than twice the benefit a ramp from L* 26 gives.

The reason lies in where the rays’ colours bind. All thirty-three blue cases and twenty-five of the thirty-three brown ones bind at or below L* 26, and for them a higher ramp moves ease from the deep shadow, where all their dulling is, to lightnesses where they have none at all. The eight high binders are a minority of the census, and even for them the higher ramp only trades.

The low binders give it back #

The brown’s low binders keep 0.06 thousandths of dulling with the ramp at L 20, 6.27 at 26 and 8.44 at 32, against 10.15 with the cap held; the blue’s keep 4.89, 7.52 and 8.48.* The prediction called this a small cost; it is most of the gain. A colour that binds at L* 20 has all its dulling between 20 and 14; a ramp from L* 32 has eased that stretch by only the last third of its fifth, and the colour is left with four fifths of its held dulling.

For the brown’s low binders the L* 20 ramp was nearly a cure — some of them printed a hair more chroma than asked — and a ramp from L* 32 undoes almost all of it. A profile builder choosing one ramp for every colour chooses between curing the many and trading for the few.

The ramps themselves #

The three ramps share their end and differ in how they get there. By L 20 the ramp from 26 has already eased the cap by a tenth and the ramp from 32 by thirteen per cent, where the ramp from 20 has not begun*; at the floor all three reach a fifth. Between L* 20 and the floor the higher ramps ease the cap by only the remaining tenth or seven per cent, against the L* 20 ramp’s full fifth.

That is the whole of the trade, drawn. Ease spent above L* 20 is ease not spent below it, and below L* 20 every colour in the census has dulling, while above it only the high binders do. A ramp of any shape spends one budget; the census says the deep shadow is where it buys most for every group, the high binders included.

The knee agrees #

Through the knee, which already softens the cap’s approach over fifteen units of lightness, the order is the same and the differences smaller. The blue’s knee dulling is 7.93 thousandths from L 20, 9.30 from 26 and 9.80 from 32; the brown’s 4.75, 7.86 and 8.98.* A knee in the ink limit quiets the shadow found the knee spreading the cap’s bend; spread already, it leaves less for any ramp to straighten, and less difference between ramps.

The knee’s smaller differences say the same thing from the other side. Under a knee the cap’s bend is already spread over fifteen units of lightness above the binding point, so for a high binder part of it reaches past L* 26 and a higher ramp has something more to work on; for a low binder the knee’s spread puts more of its bend above L* 20, which the deep-shadow ramp cannot reach. Both effects narrow the gap between ramps, and neither reverses it: the ramp from L* 20 is still best on both rays.

A tenth agrees too #

With a tenth’s ease the picture is the same in miniature. The brown’s high binders are left with 7.17 thousandths from L 20, 7.04 from 26 and 7.32 from 32* — the one place a higher ramp does better, by under two per cent — while the low binders and the blue lose ground at every higher start. A tenth’s ease is too small to cure any group, and too small for where it is spent to matter much to the high binders.

Why the deep shadow buys most #

The earlier essays give the reason the deep shadow is the better place for the ease, even for the colours whose dulling it holds least of. Three nodes at the cap’s corner cut the dulling by a third found the bend sharpest just below where the cap binds, and shared nodes belong in the deep shadow found black climbing fastest in the deep shadow, more than twice its strategy’s ceiling under the held total. A cap eased in the deep shadow lowers the target where black’s climb was steepest and the chromatic inks were being pushed hardest along the cap, and the dulling there falls by two thirds for every colour.

Above L* 20 the high binders’ bend is gentler, a long stretch of slow change, and the same ease spread along it cuts it by about the same share while costing the deep shadow its fuller ease. A two-part policy — the cap eased from each colour’s binding point and reaching a fifth by L* 20 — would give both stretches their ease; but a total-ink cap in a profile is a function of the colour’s lightness, shared by every colour at that lightness, and a high binder’s binding point is a low binder’s undamaged highlight.

Nodes for one stretch, ink for the other #

The run of essays has now tried two kinds of remedy on the same dulling, and the census can say which belongs where. Nodes sample the bend: more nodes quarter the dulling and leave the knee a larger share found each doubling of a table’s nodes cutting the dulling about fourfold everywhere, and three nodes at the cap’s corner cut the dulling by a third found nodes placed at a colour’s own binding point cutting it most where the bend is sharpest. The ink policy removes the bend: the eased cap straightens the separation itself, and no number of nodes is needed where the separation no longer turns.

For the low binders the ink policy wins outright. Their whole bend lies in the deep shadow, the L* 20 ramp straightens it, and on the brown ray it leaves them with almost nothing for nodes to sample. For the high binders the two remedies divide the work along L* 20: the ramp below, where it removes two thirds of their deep-shadow dulling, and nodes above, where the ramp cannot go without taking from the deep shadow. Three nodes shared by every colour, placed where the earlier essay placed them, sit in the deep shadow and add nothing once the cap is eased there — the census found the shared nodes and a fifth’s ease together no better than the ease alone. Nodes for the upper stretch would have to sit between L* 20 and the binding points, from 26 to 31, where only the high binders need them.

A profile is a fit between its nodes is the reason a profile builder can afford that. Nodes between L* 20 and 31 cost a table a few more rows and no ink; they are shared by every colour at those lightnesses, and the low binders, whose highlights those lightnesses are, are not harmed by being sampled more finely. The ink policy, by contrast, is shared too, and there the sharing costs: an ease at L* 26 is spent on colours that have no dulling there.

The refit is the third remedy the census has met. A strategy keeps the refit only where black prints found refitting an inverse table’s nodes against their own error halving its miss where black prints — which, under an eased cap, is now most of the deep shadow and, for the high binders, all of their upper stretch. Whether a refit on the eased table reaches the upper stretch the ramp cannot is a calculation the census has not made; it is cheaper than either of the others, since it changes neither the table’s size nor its ink.

What a profile builder should take from this #

Start the ease at L 20.* Of the ramps the census reads, it leaves every ray and every group of colours with the least dulling or within a fiftieth of it.

Do not chase the high binders with the cap. Their remaining dulling is the cap’s, but easing above L* 20 takes ease from the deep shadow where every colour needs it; three shared nodes, or more nodes, remain the remedy for that stretch.

Expect no cost at the floor. No ramp the census reads costs the rays’ deepest colour its lightness.

How the tables were built and read #

The press, rays, strategies and caps are the knee census’s. The cap at each lightness is the case’s cap times one less the ease times the share of the way from the ramp’s start to the floor, nought above the start; the ease reached at the floor is a tenth or a fifth and the start L* 20, 26 or 32. 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.

What this does not reach #

Straight ramps. A ramp that eases slowly from a high start and steeply below L* 20 might keep most of the deep-shadow cure and add some of the upper stretch’s; the census reads three straight ramps only.

Two rays. Colours at other hues bind at other lightnesses, and a hue whose colours mostly bind high would weigh the trade differently.

A floor at L 14.* The rays end well above the press’s deepest black.

Still open: whether a bent ramp keeps both gains #

Three straight ramps spend a fifth of ease along three lengths, and the shortest did best. A ramp that eases gently from L* 32 to 20 and then steeply to the floor would spend a little on the high binders’ upper stretch and most on the deep shadow.

The calculation is the census with the cap eased by a twentieth between L 32 and 20 and by the remaining fifth between 20 and the floor — a quarter of the ease above, three quarters below — every case read on 65 even nodes, hard and through the knee.* The prediction is that the high binders’ total falls below the L* 20 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. If the bent ramp does no better than the straight one from L* 20, the deep shadow’s cure needs every part of the ease it gets, and the high binders’ upper stretch belongs to nodes, not to the cap. A bent ramp takes more ink and dulls more makes that calculation.

One budget, two places to spend it #

The habit here is about a remedy with a fixed amount to give.

The earlier essay found the ease working where it was applied and not where it was not, and the obvious move was to apply it where it was not. That treated the ease as something that could be extended without being taken from anywhere. A ramp to the same floor eases the same total amount however it is drawn, and moving its start up moves ease out of the deep shadow.

The failure mode is to extend a remedy to an untreated region without asking what the extension takes from the treated one. The test is to hold the remedy’s total fixed and read every region under each way of spending it — here, the deep shadow and the upper stretch for every group of colours — before deciding that the untreated region was the one that needed it.