A little black ends the cast

A strategy keeps the refit only where black prints took the inverse table a colour engine uses to separate a grey into four inks, filled it exactly from a separation strategy, and asked where it erred and who could correct it. Above the lightness where the strategy begins black, the grey is three chromatic inks, their coverages bend unequally between the table’s nodes, and the table prints a faintly warm grey. Below it, black does most of the darkening and the table errs only in lightness. A refit that must leave the strategy’s black alone can fix a lightness error and not a cast, so it kept the whole of its gain on the black side and almost none on the other — and across five strategies, the share it kept was the share of the ramp printed with black.

That strategy began black with a kink: none above L* 60, then a straight rise. Real black-generation curves begin with a toe, a gentle rise over ten or twenty units of lightness, so that for a stretch the grey is three chromatic inks and a little black. The essay predicted that the toe would behave like the three-ink side — that a little black would not remove the chromatic inks’ unequal bends, so the cast would persist until black carried a substantial share of the grey, and the census’s diagonal would move by the width of the toe.

A tenth of the darkening is enough #

The cast does not survive the toe. At every toe width from nothing to thirty units of lightness, the table’s cast falls below a tenth of its three-ink size in the first segment where black supplies between a tenth and a sixth of the grey’s darkening. In lightness that point moves only from two to twelve units below the start, while the toe widens by thirty. The toe’s cost is elsewhere: it turns the segment where black begins from the table’s most accurate into its least, and a refit on the strategy then keeps a little less of its gain than the share printed with black — short by about one segment’s worth of ramp, not by the toe’s width.

• The prediction fails in direction. Black ends the cast long before it carries most of the grey.
• The cast ends at a share of black, not at a lightness: the toes put it at different lightnesses and at nearly the same share.
• The reason is the direction the table’s chord errs in the toe: along the trade of black for the other three inks, which is a change of lightness, not of hue.
• A smaller cast comes back just below where each toe joins the straight part of the curve, up to a third of the three-ink cast at the widest toe.

Four starts of black #

The strategies are the kinked one of the earlier essay and three with a toe. Each toe is a parabola over the stated width of lightness below the start, joining the straight part with the same slope, and every curve reaches the same black, 0.9, at the ramp’s floor of L* 14 — well above the press’s own deepest colour, L* 2.4, which a black that is not black measured, so the ramp stays inside what the four inks can print. At each lightness cyan, magenta and yellow are solved to make the grey exactly neutral given that black, on the same spectral press model — sixteen overprints on coated paper, Demichel weights, a Yule–Nielsen exponent of 1.8, read under D50.

What the toe changes most is not black but the three chromatic inks. With the kink, cyan turns down the moment black arrives, because black takes over the darkening at once. With a toe of 20 cyan keeps rising for another fifteen units, from 0.34 of coverage at the start to 0.42 at L* 45, because in that stretch black is darkening very little and the three inks are still doing most of it. That is the stretch the prediction was about: three chromatic inks still climbing, with a little black among them.

Where the cast ends #

The table here holds the strategy’s separation at seventeen evenly spaced lightnesses, so each segment is under five units of lightness wide and the cast can be located finely. Its cast is the mean chroma of the grey it prints between two nodes. Above the start every strategy is the same three inks, and the cast is 0.0126 in every segment. It is small in absolute terms, a hundredth of a unit, because a seventeen-node table is fine; what matters is where it goes, and the nine-node table below has the same shape at four times the size.

With the kink it is gone within two units. The segment straddling the start carries a quarter of it and the next carries under a fortieth. With toes of 10, 20 and 30 it ends 7, 12 and 12 units below the start: later, as predicted, but the widening of the toe by thirty units moves the end by ten, and the two widest toes end it at the same segment. Somewhere well inside each toe the cast stops, while black is still rising slowly.

Measured by black’s share #

The prediction was framed in black’s share of the grey rather than in lightness, so the segments are replotted against it. Black’s share at a lightness is how far black alone would darken paper at the coverage the strategy prints there, over how far the grey is darkened in all — a quarter when black by itself would take paper a quarter of the way to that grey. It is not additive with the other inks’ shares, and it is not meant to be; it is a plain measure of how much of the grey is black’s doing.

On that axis the four strategies agree. The first segment below a tenth of the three-ink cast begins where black supplies 0.10 of the darkening with the kink, and 0.11, 0.17 and 0.13 of it with toes of 10, 20 and 30. The prediction put the cast’s end at “a substantial share”, by which the essay meant black carrying much of the grey. It ends at a tenth to a sixth, and whatever the toe, the share at which it ends moves far less than the lightness does.

What the chord crosses in the toe #

The cast is an interpolation error, so its cause is in how each ink’s coverage curve departs from the straight chord a table draws between two nodes. Above the start the three chromatic inks bend a little and unequally — at L* 65 cyan by 0.23 thousandths of coverage, magenta by 0.41 and yellow by 0.34 — and a chord through them prints a grey with a chroma of 0.020. That is the mechanism the earlier essay found, measured here per ink.

Inside the toe the bends are four to nine times larger and the cast is an eighth as large. At L* 50, black bends by 3.7 thousandths — the toe’s parabola has constant curvature, so black’s bend is flat across it — and cyan, magenta and yellow bend the other way by 2.2, 1.9 and 1.9. The chord through those coverages prints a chroma of 0.0026. The inks are bending against each other in a fixed pattern: wherever black’s curve is concave, the three chromatic inks are convex by the amounts that hold the grey neutral as black changes, because that is what the solver does at every lightness. A chord that misses black’s curve misses the three inks’ curves in the compensating direction, and what it prints is one member of the family of neutral separations the separation is not unique describes, displaced along it. The fourth ink is not for colour found a black-built grey moving in lightness, not in hue, under press variation; here the same trade carries the table’s own interpolation error into lightness.

So a little black ends the cast because black’s own curvature takes over the bends. The three inks’ unequal curvature is still there; it is simply swamped by a larger, coordinated bend that errs in lightness. The kinked strategy did the same thing at a single point — the chord across its kink was a trade of black for the three inks, which is why the earlier essay found that segment almost exact.

The cast comes back, a little, where the coordination stops. Just below the point where each toe joins the straight part of its curve, black’s curvature drops to nothing and the three inks’ own unequal bends are no longer swamped. The seventeen-node table’s cast rises there to 0.0013 with a toe of 10, 0.0025 with a toe of 20 and 0.0040 with a toe of 30 — up to a third of the three-ink cast — before falling again. The kinked strategy, whose black is straight all the way down, reaches 0.002 only in its last segment at the floor.

What the toe costs the table #

The finding about the cast is favourable to the toe; what follows is its price. The kinked strategy’s switch segment was the table’s most accurate, at 0.003 on a nine-node table against 0.035 to 0.130 in every other segment, because interpolating across the kink was an exact trade of black for the other inks. With a toe of 10 that segment errs by 0.097 and with a toe of 20 by 0.114, the worst of the nine-node table’s eight. The segment now spans the first ten units of the toe, where black supplies at most about a tenth of the darkening; the three inks are still climbing and bending unequally, black is not yet bending enough to swamp them, and the segment carries a cast of 0.038 against 0.052 on the three-ink side as well as the largest lightness error on the ramp.

The whole table’s mean error barely moves: 0.073 with the kink, 0.081 with a toe of 10 and 0.076 with a toe of 20. A toe does not make the grey axis worse to interpolate; it moves where the difficulty is from the three-ink segments alone to those plus the first segment of the toe.

The refit on the strategy responds as the cast predicts. Above the start it gains 1.02 to 1.17 times, as before, against 2.3 for the free refit. In the segment at the head of the toe it gains 1.52 times against the free refit’s 2.17, keeping roughly two thirds of what the free refit removes — the segment’s lightness error it can fix, its cast it cannot. Below that, it gains what the free refit gains, segment by segment, to within six hundredths.

The census over starts #

The earlier essay’s clearest result was the diagonal: across starts at L* 40, 50, 60, 70 and 80, the share of the free refit’s gain kept on the strategy equalled the share of the ramp carrying black, to within three hundredths. With toes the points fall below the diagonal, by less than the prediction said. With a toe of 10 the four lighter starts keep 0.02 to 0.05 less than their black share; with a toe of 20, 0.06 to 0.14 less. A toe of 20 is 0.26 of the ramp, so if the diagonal had moved by the toe’s width the shortfall would be about a quarter. It is about the width of one nine-node segment, 0.125 of the ramp — the segment at the head of the toe that the refit on the strategy cannot fully correct.

The darkest start is the exception, and it goes the other way. Starting black at L* 40 leaves 26 units of lightness for black to rise through, and a toe of 20 is most of that; there the refit on the strategy keeps 0.40 of the free refit’s gain against a black share of 0.33. The census was built for strategies with a distinct black side, and at this start the toe is the black side. The census offers no clean account of the excess, and it is recorded here as observed.

What a profile maker can do with it #

A toe does not extend the cast. The worry that a gentle start of black leaves a long stretch of three-ink behaviour below the start is unfounded for the shape measured: a tenth of the darkening from black is enough, which for a toe of 20 is reached twelve units below the start. Tables whose separations use a toe can be treated as the earlier essay treated a kinked strategy, with the boundary moved down by that much.

Put a node near the head of the toe. The segment spanning the first stretch of the toe carries the table’s largest error. A lattice that places one node at the start of black and another a few units below it — a profile is a fit between its nodes argued that where the nodes sit matters more than how many there are — splits that segment where its error is concentrated.

Treat the head of the toe as three-ink for refitting. There the correction the table needs is partly a slight imbalance of cyan against magenta and yellow, which a strategy that holds grey balance forbids, just as it forbids it higher up. A profile interpolates light and the inverse table errs dark both found that letting a node be wrong at itself buys accuracy between nodes; the head of the toe is one of the places where that trade needs an imbalance, not a lightness shift.

How the toe was measured #

Everything is as in the earlier essay: the ramp from L* 92 to 14 along the grey axis of the lattice a profile is a table describes, the press model, the table’s linear interpolation of the four coverages between nodes, ΔE₀₀ against the neutral target at 241 greys, and the two refits as coordinate descents on the mean squared error. The toe of width $t$ replaces black’s straight rise $d$ below the start with $d^2/2t$ for $d$ up to $t$ and $d - t/2$ beyond, rescaled so black still reaches 0.9 at the floor; a toe of zero is the kinked strategy exactly. The cast of a segment is the mean of $\sqrt{a^{*2} + b^{*2}}$ over the greys inside it; a bend is an ink’s coverage at a lightness minus the mean of its coverages 2.4 units either side, half a seventeen-node segment. The census is at nine nodes, as before, and the cast’s location at seventeen.

What this leaves out #

The toe is one shape, a parabola joined smoothly to a line. Real curves are drawn by hand in a separation tool or computed from a total-ink limit, and some have a shoulder at the dark end as well as a toe at the light one. The finding should carry to any toe that rises monotonically, since what ends the cast is black’s curvature coordinating the other inks, but a toe with a flat first stretch — black held at a percent or two for several units — would delay it.

The measure of black’s share is one of several. The share of coverage, or of density, would put the threshold at different numbers. What is robust is that the four strategies land together on this axis and apart on lightness.

The grey axis is still one line through a three-dimensional table, and the returning cast at the toe’s lower join is small enough that on a real press it would sit under the press’s own variation. It is reported because it says where the coordination stops, not because it would be seen.

Still open: whether the cast returns where the toe ends #

The cast came back just below the point where each toe joins its straight part, to a third of its three-ink size at the widest toe, and the explanation offered is that black’s curvature stops there, so nothing swamps the chromatic inks’ own unequal bends. If that is right, the returned cast is set by the jump in black’s curvature at the join, not by the toe’s width as such — a toe that ends by easing its curvature to zero over several units, rather than stopping it, should not bring the cast back at all, while a kink at the dark end of a black curve should bring back a cast of its own.

The calculation is this one with toes whose curvature falls smoothly to zero at their lower end — a cubic rather than a parabola — against the same widths, and with a straight black curve given a deliberate change of slope partway down. The quantity worth reporting is the returned cast against the size of the jump in black’s second derivative. If the prediction holds, the useful advice to a separation tool is about the smoothness of its black curve’s curvature, not about the width of its toe.

A threshold is a share, not a place #

The habit is about where to measure a transition.

The prediction was right that the toe delays the cast’s end and wrong about how much, and the reason it could be checked cleanly is that it named the right axis: not “how far below the start” but “how much of the grey black is doing”. Four strategies scattered along lightness gathered on that axis within a few hundredths of each other, which is the sign that it measures the cause rather than a place where the cause happens to sit.

The failure mode is to measure a transition by where it happens. Where it happens depends on the particular curve; what triggers it is a quantity the curve passes through. The cast ends when black supplies a tenth of the grey, and a toe only decides at what lightness that is.