Printing does not change the sign

A tint in paint turns the other way tinted eight pigments to the same lightness two ways — stirring in white pigment, and adding white light — and found that seven of the eight turn the hue in opposite directions, by up to 27 degrees apart. The reason is in the reflectance: diluting a pigment’s absorption with white raises its curve most where it absorbs least, so its absorption edge moves, where an additive mixture lifts the whole curve and leaves the edge where it was.

It then named the case between them.

A real halftone sits between the two. Light entering the paper between dots scatters sideways and emerges under a dot, so a halftone tint absorbs more than the additive average; that optical dot gain is a partial move from light mixing towards pigment mixing. The question is whether it moves the hue as well as the lightness.

It does. How far is the interesting part, and how far turns out not to depend on the ink.

A third of the way, and the same third for every pigment #

At a Yule–Nielsen factor of two — the far end of what a coated sheet is fitted at — a halftone tint’s hue turn has travelled between 26 and 44 per cent of the way from the additive ideal’s to the paint ideal’s, and seven of the eight pigments fall between 26 and 37.

• A factor of one reproduces the additive tint exactly, to within half a degree at every pigment, which is the check that the halftone model reduces to the ideal it should.
• The journey is nearly a straight line in the factor. The largest departure from proportionality across all eight pigments is under a tenth of the distance.
• Only the violet changes the direction of its turn inside the range a real sheet occupies, at a factor of 1.69.
• The orange would need a factor of 3.2 and the green 24. Neither is a paper.
• So a halftone tint turns the way an additive tint turns, by about two thirds as much. The sign belongs to the light mixture, and dot gain buys a size rather than a direction.

What a halftone is, as a reflectance #

A halftone tint is ink dots on paper, and the naive model of it is an area average: a coverage a of the ink’s reflectance plus 1−a of the paper’s. That is the additive ideal exactly, and it is wrong for a reason that has nothing to do with chemistry.

Light that enters the paper between two dots does not necessarily leave between them. Paper scatters, so some of that light travels sideways within the sheet and emerges under a dot, passing through the ink on the way out but not on the way in. It has been filtered once rather than twice, and it is missing from the light that leaves the uncovered paper. The result is that a halftone absorbs more than its area average — optical dot gain, which is a fact about the sheet and not about how large the dots are.

The Yule–Nielsen relation is how that is usually written. The ink’s and the paper’s reflectances are each raised to the power 1/n, averaged by area, and the result raised back to the power n. At n = 1 it is the area average and at larger n it bends towards the darker of the two, by an amount that stands for how much of the light makes the sideways trip. A coated sheet is usually fitted between about 1.4 and 2; an uncoated sheet, which scatters more, higher.

The three tints have the same lightness and three different curves. The additive tint lifts the pigment’s whole curve towards the white’s and leaves its half-height crossing at 570 nanometres, where the pure pigment’s was. The paint tint raises the reflectance most where the pigment absorbs least, and the crossing moves to 545. The halftone’s crossing sits at 562 — between them, and much nearer the additive one.

That ordering is the whole result in one measurement, and the hue turn follows it.

It is worth being clear why the crossing is the right thing to watch rather than the curve’s height. Paint is not a filter established that a pigment mixture is not an average of reflectances, and the visible consequence is where the absorption edge sits: a colour’s hue is decided by where its reflectance changes, and its chroma by how far it changes. Two tints of the same lightness that differ in where their edge sits will differ in hue, and that is what the three curves above do — the additive one keeps the pigment’s edge, the paint one moves it 25 nanometres towards the short wavelengths, and the halftone moves it 8.

The census of edges is the earlier essay’s and it is the reason the halftone’s position can be predicted rather than searched for. Every pigment with a single edge shows the same pattern — the paint tint’s crossing moves and the light tint’s does not — and the size of the move is what the two ideals’ hue turns are made of. A model that lands a reflectance between the two, as Yule–Nielsen does, will land its edge between the two crossings, and its hue between the two turns.

The turn, for every pigment #

The halftone mark sits between the two bars at every pigment and touches neither end. The orange’s two ideals are −13.9 and +13.5 degrees — 27 degrees apart, and opposite — and its halftone lands at −6.0, still on the additive side. The cyan’s are +14.2 and −8.2 and its halftone is at +6.4. The blue’s are +14.0 and −0.7 and its halftone is at +10.1.

The prediction the earlier essay made was that a halftone tint of an orange should turn less towards red than the ideal additive tint, and could turn towards yellow at high gain. The first half is right and the second does not happen. The orange turns less — 6.0 degrees instead of 13.9 — and it never crosses, because a factor of two is not enough and the factors that would be are not papers.

The journey belongs to the paper #

The striking thing about the eight results is how alike they are.

Eight curves, all rising from nought at a factor of one, all ending between 26 and 44 per cent, and every one of them nearly straight. The spread at the end is under a fifth of the distance travelled, across pigments whose ideals are 0.4 degrees apart in one case and 27 in another.

That is worth stating as a property rather than as a coincidence. How far along the road between the two ideals a halftone sits is set by the sheet’s optical gain and not by the ink on it. Which is, incidentally, the reason a single Yule–Nielsen factor per paper is a usable thing to fit at all: if the factor’s effect on hue depended strongly on the ink, a press profile would need one per ink and the practice of quoting one per substrate would be a fiction.

It also means the result generalises past the eight pigments here. A new ink’s halftone tint will sit about a third of the way from its own additive ideal to its own paint ideal, and both of those are computable from its reflectance.

What it would take to cross #

Since the journey is straight, the factor that would carry a pigment past the crossing can be read off rather than searched for.

Only the violet’s crossing lands inside the range a coated sheet occupies, at 1.69. The yellow needs 2.4, the orange 3.2, the cyan 3.4, the magenta 3.6, the red 4.5, the blue 6.1 and the green 24.

The pattern is not about the pigments’ chemistry. A pigment crosses early when its additive turn is small compared with the gap between its two ideals, because then there is little to undo; the violet’s additive turn is −1.9 degrees against a gap of 6.8, which is 28 per cent, and a halftone at n = 2 travels 34. The orange’s is 13.9 against 27.4, which is 51 per cent, and no realistic sheet travels that far.

So the rule a press operator could carry is a ratio rather than a factor. If an ink’s additive hue turn is less than about a third of the distance between its two ideals, a halftone tint of it will turn the other way; otherwise it will not. Both quantities come out of the ink’s measured reflectance and the paper’s, before anything is printed.

The size of the turns matters as much as their sign, and it is small. Six degrees of hue on a pale tint is not a large shift — it is around one unit of colour difference on these colours at these lightnesses — so nothing here says a profile that treats a halftone as additive is badly wrong. What it says is that the residual error has a direction, the same direction for every ink on a given paper, and a systematic error of one unit that always points the same way is the kind a profile’s own fitting will absorb into its lookup table and then carry into every extrapolation past the samples it was fitted on. A profile interpolates light is where that kind of absorption was measured from the other side.

How the tints were matched #

Each halftone is computed at the ink coverage that gives the same luminance as the paint tint of the same pigment, found by bisection. Matching luminance rather than coverage is what makes the three tints comparable: the question is which way the hue turns for a tint of a given lightness, and a coverage that is right at one factor is a different lightness at another.

The paint tint is the single-constant Kubelka–Munk mixture at 95 per cent white by concentration, as in the earlier essay; the additive tint is the mixture of the same two reflectances at whatever proportion matches that luminance. The hue turns are in Oklab against the pure pigment’s, and the edge positions are half-height crossings, which is the measure that does not move under an additive mixture — a tint in paint turns the other way records why the steepest-slope measure was rejected.

The white is a flat reflectance of 0.9, which stands for a white pigment rather than for paper. That is the one substitution worth naming, because a halftone’s uncovered area is paper and paper is not a white pigment: it fluoresces, it is not flat, and its reflectance is usually above 0.9. All three would move the numbers and none changes the ordering. The fluorescence is the largest of the three and the least like a reflectance at all — the lamp that stopped emitting ultraviolet is where what a brightened sheet does to a measurement was priced, and a brightened stock’s uncovered area returns more blue light than arrives at it.

What this leaves out #

Mechanical dot gain is absent. A printed dot is larger than the dot that was asked for, because ink spreads and the plate and blanket add their own. That changes the coverage and so the lightness, and the tints here are matched on lightness, so it enters as a shift along an axis this essay holds fixed rather than as a change to the result. It would matter to anybody reading a coverage off these figures: the 0.31 coverage the orange’s halftone needs is the effective coverage, and the value a plate would be made at is smaller.

The lightness match is exact and the chroma is not. The three tints agree in luminance by construction and differ in chroma, with the paint tint the most colourful of the three at every pigment — a finding the earlier essay reported and this one inherits. So a reader comparing the three marks is comparing hue angles of colours that are not equally saturated, which is the right comparison for the question and is not a comparison of three colours a specifier could swap.

One Yule–Nielsen factor for all wavelengths. Paper’s scattering is not spectrally flat and neither is its absorption, so a fitted factor is a compromise across the spectrum. A wavelength-dependent factor would bend each tint’s curve differently and would be a different model, not a refinement of this one.

The pigments are constructed reflectances. They are analytic bands and sigmoids rather than measured inks, chosen in the earlier essay so that each has a describable edge. A real process ink has a more complicated curve with a secondary absorption, and the fifth ink buys a corner is where a real ink set’s curves were used for a different purpose; its effect here would be to give some pigments more than one edge, which the half-height measure reports as one and should not.

And the ink is one ink. A real halftone tint of a process colour is often made from two inks overprinted, and the fourth ink is not for colour is where the separation decisions that produce those overprints were priced. An overprint’s halftone is a Yule–Nielsen mixture of four states rather than two, and nothing here says the journey is the same fraction in that case.

Still open: whether an uncoated sheet crosses #

The one thing this result turns on is that a real sheet’s factor stops around two. An uncoated sheet scatters far more and is routinely fitted above it — values from 2.5 to 5 appear in the printing literature, and newsprint higher still.

The computation is this one at those factors, with a paper reflectance rather than a white pigment, and the prediction is sharp: the journey stays straight, so the yellow crosses somewhere near 2.4, the orange near 3.2 and the cyan near 3.4. A newsprint halftone of an orange would then turn its hue the way a paint tint turns it, and a coated one would turn it the other way — the same ink, the same coverage, and opposite hue shifts on two papers.

That is a testable statement about print and it is not an exotic measurement: two sheets, one ink, a step wedge and a spectrophotometer. The quantity to record is the hue angle of each tint against the solid ink’s, and the prediction is that the two papers’ curves diverge with falling coverage and cross the solid’s hue angle at different places — or that the coated one never crosses at all.

A compromise between two models is a model, and it has its own parameters #

The habit is about what sits between two idealisations.

Two models disagreed about the sign of an effect, and the real case was described as lying between them. That description is true and it is not a prediction: between is a line, and where on it a real case sits is a further question with its own answer. Here the answer turned out to be a single number that is nearly the same for every ink, which is the best possible outcome — it means the intermediate model has one parameter and the parameter belongs to the substrate.

The move is to write the intermediate model down rather than to describe the interval. It costs one function, it reduces to each ideal at an end of its range, and the check that it does is the first thing to run — because a model that does not reproduce the case it was built between is not between them.

There is a second thing this shape rewards. Because the intermediate model reduces to each ideal at an end of its range, it comes with two free checks — and the one at n = 1 caught a real error: an earlier version matched the halftone on coverage rather than on luminance, and at n = 1 it did not reproduce the additive tint, because the coverage that gives the right area average is not the coverage that gives the right lightness. It was off by several degrees, and being off named the cause. A model is a claim about what can be known is the standing form of that argument; a reduction check is the cheapest instance of it, and a model built between two others always has one.

The failure mode is leaving a real case described as “somewhere between”. Both ideals are computed, the real one is not, and the question of which sign a practitioner will actually see stays open when it is one bisection away from being answered.