Ink limit — where it appears
Named by 16 essays across 2 fields — each of them below, with the objects they name alongside it.
The fourth ink is not for colour
Black is in every process set and it extends the gamut by more than a third — and at every lightness above about forty it extends the chroma boundary by exactly nothing. What it buys is depth at the bottom, a neutral that holds its hue when the press drifts, and a way to put a full-strength dark on a sheet that could not carry three inks' worth of it.
A black that is not black
The deepest colour a four-colour press can make is L* 2.4, and against its own paper that is a contrast of about 320 to 1 — less than a cheap display manages in a dark room. Everything a printed picture does in the shadows is done inside that range, and the two ways of fitting a picture into it destroy different things.
The separation is not unique
A four-ink press has one more control than a colour has numbers, so most colours can be printed several ways. The alternatives agree to a hundredth of a colour difference under the light the match was made in — and they are metamers of one another, so under a tungsten lamp the same eleven separations of one colour spread by nearly seven.
The fifth ink buys a corner
Orange, green and violet are sold as the answer to a four-colour press's gamut, and each one adds seven to nine per cent of volume. All three together add 23 per cent — almost exactly the sum, with no interaction — and every bit of it lands in the region of the ink that was added, at lightnesses where that ink is dark.
A press makes the cheap route a search
Holding a gradient inside a display's gamut cost nothing measurable, and the essay that found it credited the gamut's convexity. A press's gamut is not convex: 216 of 630 straight lines between colours on its own hue ring leave it. Holding gradients inside the press still costs little at best, a median of a tenth of a per cent. But the best route depends on where the relaxation starts, the free path is the best start on one gradient of ten, and on half of them some start is trapped at more than twice the free length. On the display no start is trapped.
An ink limit takes the shadows' cast away
A cap on total ink forces black up and the chromatic inks down where it is reached, which is the descent that brought a small cast back below a black-generation toe, and the prediction was that a cap would bring one back too. Measured against what the same strategy prints with no cap, it does the opposite. Below where it binds the cap decides the separation outright — nine strategies print one table there — and in 32 of 33 cases its shadows cast less than the strategy's own would have.
A knee in the ink limit quiets the shadow
A cap on total ink coverage quiets a four-ink table's shadow, and the one segment where the cap is first reached is quiet or loud depending on where the binding point falls among the table's nodes: inside a segment it is quiet, within a quarter of a segment of a node it is not. A real cap is approached through a knee. The prediction was that a knee as wide as a segment would make the binding segment quiet wherever it fell, and that a knee much wider would turn the cap into a steep straight black that casts like the heavy-black strategy. The first half holds only in part: a five-unit knee halves the binding segment's cast and quiets three of the nine loud ones, but those whose binding point sits within a quarter of a segment of a node stay louder than a neighbour. The second half is wrong in sign. A wider knee leaves the whole capped shadow quieter at every step — its mean cast falls from 11.5 thousandths to 3.0 at fifteen units — because a knee spreads the chromatic inks' forced descent, and a slower descent is one the table's straight lines between nodes can follow.
Off the grey axis a knee trades hue for chroma
Along the grey axis a total-ink cap approached through a knee quiets the cast a four-ink table prints between its nodes, because the knee slows the chromatic inks' retirement where black takes over. In a saturated shadow the chromatic inks carry the colour, and the prediction was that a knee would quiet the hue error there in the same way, a fifteen-unit knee halving the worst segment's. On two rays of dark colours, a deep blue and a brown, it does quiet the hue — every wider knee lowers it on both — and halves the worst segment's on the blue, to 0.28 of the hard cap's, but not on the brown, at 0.68, whose inks the knee slows far less. And the hue was never the main error. Between nodes the table prints these colours duller than asked, its chroma error five to eight times its hue error, and every wider knee makes that loss slightly worse, so the table's total error rises as its hue error falls. The cap itself, met hard, halves the blue's error, as it removed the grey's cast; on the brown it changes almost nothing.
More nodes quarter the dulling and leave the knee a larger share
A four-ink table printing a dark saturated colour between its nodes prints it a little duller than asked, and a knee in the ink limit, which quiets the hue error, deepens that dulling by about a hundredth of a unit. The prediction was that doubling the table's nodes quarters the dulling — the square of the spacing — and the knee's added cost with it, so that on a dense table the knee's hue gain is all that remains. On tables of 9, 17, 33 and 65 nodes along a deep-blue and a brown ray, the dulling does fall about fourfold with each doubling, capped or not, to under a hundredth at 65 nodes. The knee's added dulling does not keep pace: on the blue it is 13 per cent of the hard cap's at seventeen nodes, 23 at 33 and 35 at 65, and the knee's whole error, an eighth more than the hard cap's at seventeen nodes, is a quarter more at 65. The knee keeps its hue gain at every density. A denser table shrinks every error and makes the knee's trade worse.
Three nodes at the cap's corner cut the dulling by a third
A four-ink table printing a dark saturated colour between its nodes prints it duller than asked, and at 65 nodes a fifteen-unit knee in the ink limit carries 35 per cent of that dulling on a deep-blue ray. The prediction was that the knee's added dulling sits in the two or three segments just below where the cap binds, and that three nodes placed there take its share back under a tenth. The first half holds: the knee's addition peaks one unit of lightness below the binding point and four fifths of it lies within seven. The second does not. Three nodes placed one, two and three units below the bind cut the hard cap's dulling by a third, 0.0097 to 0.0064, and the knee's by nearly two fifths, where three more evenly spaced nodes cut each by under a tenth — but the knee keeps a quarter of the dulling. The hard cap has a corner, which one node can catch; the knee has a bend spread over a dozen units, which three nodes split but do not straighten. A node at the binding point alone helps only the hard cap, and leaves the knee 41 per cent.
Shared nodes belong in the deep shadow
Three nodes placed just below where each dark colour's ink reaches the press's cap cut a four-ink table's dulling of that colour by a third — but a profile's nodes along lightness are shared by every colour, and the colours' caps bind anywhere from L* 15 to 31. The prediction was that one shared placement keeps two thirds of the own-placement gain on a deep-blue ray and all of it on a brown. Placed once, one to three units below the median binding point, the nodes keep 63 per cent of the blue's hard-cap gain and 92 of the brown's — and the averages hide a split. Every colour whose cap binds more than two and a half units below the shared nodes keeps under a twentieth of its own gain; every colour whose cap binds more than two units above them keeps more than all of it, up to four times. For those colours most of the dulling was never at the cap's corner: it lies in the deep shadow, where black keeps climbing under the cap, and the shared nodes sit there. Spreading the three nodes across the binding points buys nothing three even nodes would not.
Easing the cap reaches the deep shadow and no higher
A four-ink table prints dark saturated colours duller than asked between its nodes, and for colours whose ink reaches the press's total-ink cap high up the dulling lies mostly in the deep shadow below L* 20, where black climbs under a held total. The proposal was to let the total fall below the cap there, and the prediction was that easing the cap by a tenth or a fifth between L* 20 and the floor halves those colours' dulling on even nodes — more than three shared nodes do — at a cost of a unit or two of L* in the darkest colour the press can reach. There is no cost: every eased table still prints the floor at L* 14. Eased by a fifth, the deep-blue ray's dulling halves and the brown's falls to an eighth. But the eight colours that bind above L* 26 keep three fifths of theirs: the ease removes two thirds of their deep-shadow dulling and none of what lies between their binding point and L* 20. And the eased cap does not let black rise less steeply — it makes black rise faster.
A higher ramp trades one shadow for another
Easing a four-ink table's total-ink cap by a fifth on a ramp from L* 20 to the floor halves the dulling of dark colours whose cap binds low, and leaves the colours that bind high with three fifths of theirs, because most of what is left lies between their binding point and L* 20, above the ramp. The prediction was that a ramp starting at L* 32 halves the high binders' dulling, at a small cost to the low binders. It does neither. Started at L* 26, the same fifth's ease does cut the high binders' upper dulling by more than half — the cap was its cause after all — but their deep-shadow dulling comes back to more than twice what the L* 20 ramp left, and their total is unchanged at 0.0052. The low binders give back nearly everything: on the brown ray from almost nothing to 0.0084 with the ramp at L* 32. Every ray, knee and group does best, or within a fiftieth of best, with the ramp at L* 20.
A bent ramp takes more ink and dulls more
Easing a four-ink table's total-ink cap by a fifth on a straight ramp from L* 20 to the floor straightens the deep shadow's bend and halves the dulling of dark colours, and a ramp started higher trades the deep shadow's cure for the upper stretch's. The prediction for a ramp bent at L* 20 — a quarter of the ease spent between L* 32 and 20, the rest below — was that it keeps nearly all of the deep shadow's cure while taking a third off the high-binding colours' upper stretch, their total falling by a fifth. The bent ramp takes at least as much ink as the straight one at every lightness. It dulls every group more. The high binders' upper stretch improves by a sixth; their deep shadow loses half its cure, and their total rises from 0.0052 to 0.0060; the brown's low binders, left with almost nothing by the straight ramp, keep 0.0039. The deep shadow's cure follows how steeply the cap falls below L* 20 — a fifth of the cap over six units, not three twentieths — and not how much ink is taken there.
A steeper ramp cures the shadow by overshooting it
A four-ink press's total-ink cap dulls the deepest colours it binds, and easing the cap by a fifth on a straight ramp from L* 20 to the floor halves the deep blue's dulling. A ramp bent to take more ink above L* 20 dulled every group more, so the cure seemed to follow how steeply the cap falls below L* 20. The prediction was that a ramp steeper still — the whole fifth spent from L* 17 — cures more. Read as the census has read every ramp, by the mean chroma lost, it does: every group dulls less at each step from L* 20 to 18.5 to 17. But the brown's low binders cross nought: their prints come out more chromatic than the colour asked, by as much as 0.07, and read by the error's size the brown ray's mean error more than doubles. The measure that ranked every earlier ramp was signed, and a signed measure cannot see an overshoot. Eased by a tenth, the steep ramp is no worse anywhere and better on three cases of four.
Each dark ray wants its own ease
Easing a four-ink press's total-ink cap below L* 20 cures the dulling of the deepest colours it binds, and a steeper ramp — the whole ease spent from L* 17 — cures more on the signed mean, but eased by a fifth it prints the brown ray's low binders up to 0.07 more chromatic than asked and more than doubles the brown's error read by size. The prediction was that three twentieths from L* 17 is the ease the steep ramp wants: below every earlier setting on both rays. The brown still overshoots, by 0.039, and errs nearly twice as much as on the gentle ramp. The overshoot follows one number — how fast the eased cap gives ink back per unit of lightness — and the two rays want that number set differently. Hard, the blue errs least on three twentieths from L* 17 and the brown on three twentieths from L* 20; the brown's best costs the blue a fifteenth more error, the blue's best costs the brown four fifths more.
Named alongside it
The objects these essays reach for when they reach for this one.
SeparationProcess inksInterpolationThe ICC profileGrey component replacementGamutHalftoneSpecificationQuality controlChromaSubtractive mixtureBlack point compensation