The separation is not unique

A colour arrives at a printer as three numbers and leaves as four. That is one more number than the problem has, and the consequence is the thing this field is really about: there is no such thing as the separation of a colour. There is a family of them.

The claim #

Four inks against three numbers leaves one degree of freedom, so a printable colour is reproduced by a one-parameter family of ink combinations. The members of that family match under the illuminant they were computed for and are metameric pairs everywhere else.

The second half is not an approximation or a side effect. It is forced: two different sets of ink films produce two different spectra, and two different spectra that integrate to the same three numbers under one light are the definition of a metameric pair.

The whole apparatus of printing therefore manufactures metamers, deliberately, thousands of times a second, and calls it black generation.

The family, in ink #

Take a mid blue-grey and ask for every separation of it at black levels from nothing to fifty per cent:

Every row is the same colour under D50, to within the solver’s tolerance of a few thousandths of a ΔE00. The total ink on the sheet falls from 161 per cent to 98 — a saving of nearly two fifths — which is why the family is walked in the first place.

Why they are metamers #

The argument is three lines and it is worth having in full, because it explains why no improvement in ink or press could remove the effect.

A patch’s reflectance is a weighted average of its Neugebauer primaries’ reflectances, and the primaries are products of ink transmittances. Two separations use different primaries in different proportions — one is mostly three-ink overprints, the other is mostly black over lighter tints — so their reflectance curves are different functions of wavelength.

Those two curves are then integrated against three colour-matching functions. Agreeing on three integrals is three equations; the curves have eighty-one degrees of freedom between them. Agreement on three numbers says nothing at all about the other seventy-eight, and the difference between the two curves is a metameric black: a spectrum that integrates to zero against all three matching functions under the stated light, and to something else under any other.

How far apart they come #

The measurement is the family’s spread under a second illuminant, and it depends strongly on which one:

Under D65, two separations of the same colour differ by more than three times a paint contract’s tolerance. Under a tungsten lamp they differ by nearly seven, which is a difference nobody would describe as a match.

And the ordering is not arbitrary. Equal energy is a flat spectrum, so it weights the metameric black uniformly and mostly cancels it; D65 differs from D50 in slope across the band; illuminant A differs enormously, being a Planckian radiator at 2856 K. The further the second light is from the first in shape, the more of the metameric black it reveals — which is the same rule that governs every other metameric failure on this site, arriving here from inside the printing pipeline rather than from a pair of paints.

There is more in that table than an ordering, and it is the part with a consequence for anybody choosing a viewing standard.

Putting each illuminant at its distance from D50 in mireds — the reciprocal-temperature scale, which is the one on which equal steps are roughly equal shifts:

Illuminant A is three and a quarter times further away than D65 and costs only 1.84 times as much. The spread grows roughly as the square root of the distance, so the curve is steepest at the start: by the time the second light has moved as far as one daylight is from another, the family has already given up 54 per cent of everything a tungsten lamp would take from it.

That inverts the way this exposure is usually discussed. The tungsten figure is the one quoted, because it is the largest and the most alarming, and it invites the reply that nobody views print under tungsten any more. The reply is true and does not help: the case that matters is D50 against ordinary daylight, two lights a standards committee would call interchangeable, and it is worth more than half the headline.

It also means the repair everybody reaches for does not work. Moving a shop’s lighting closer to D50 buys back the last part of the exposure rather than the first — going from illuminant A to D65 recovers 3.1 units of the 6.75, and closing the remaining 46 mireds to D50 would recover the other 3.7. The second half of that journey is much harder and worth about as much as the first.

What was computed, and how #

The family is constructed rather than sampled, and the construction is the part worth trusting or not.

For each black level, the three chromatic coverages are solved by Gauss–Newton with a numerical Jacobian against the full spectral halftone model, clamped to the unit cube and damped when a step makes things worse. The target is the CIELAB coordinate of a stated separation, so a solution is known to exist for at least one member; the solver either converges to within a few thousandths of a ΔE00 or reports failure, and a black level whose solve fails or whose total ink exceeds the limit is dropped rather than approximated.

That the members agree under D50 is therefore not a finding — it is what the solver was asked for, and the 0.003 figure is a check that it did its job. The finding is the number under the second illuminant, which nothing in the construction controlled.

The family ends in two places, and both are physical. At the bottom it ends where black reaches zero. At the top it ends where the three chromatic inks reach zero, or where the total ink exceeds what the sheet will carry — so a dark colour has a shorter family than a light one, and a colour near the gamut boundary may have a family of one.

And the freedom is not the same length everywhere in the gamut, which decides where in a job this exposure actually lands.

The family is bounded at both ends by ink rather than by arithmetic: it stops below where black reaches zero, and above where the chromatic inks do or the sheet fills up. Those bounds are far apart for a colour with a large grey component — a mid-tone near-neutral, where all three chromatic inks are substantially engaged and most of what they are doing is making grey — and close together for a saturated one, where the chromatic inks are already carrying the hue and there is little common part to take away.

So the longest families, and therefore the widest metameric spreads, belong to the near-neutral mid-tones. That is the least convenient answer available. It is the region a press is controlled in, the region a grey balance is watched in, and the region where the eye and the difference formula are both at their most sensitive to a cast. The colours with the most freedom to be separated several ways are the colours in which two separations coming apart is most visible.

The saturated colours, where a mismatch would be tolerated more readily, mostly have short families and in places families of one — a colour on the gamut boundary has exactly one separation and no exposure at all. The mechanism concentrates its own error in the region the trade is most careful about, which is a good deal of why it is met as an unexplained disagreement between two shops rather than as a known cost.

Which member gets written down #

A profile has one entry per colour, so a policy has to choose, and the choice is made once and applies to everything printed through that profile for its life.

The choices are made on grounds that are all outside colorimetry. Heavy replacement — as much black as possible — uses less ink, dries faster, costs less and holds its neutrals when the press drifts, because a black grey drifts in lightness where a three-ink grey drifts in hue. Light replacement is smoother in gradations, more forgiving of registration error, and preferred for flesh tones, where a visible black dot structure in a light tint is objectionable.

None of that is visible to a colour measurement, so a profile’s black generation policy is invisible in any verification that measures patches under the profiling illuminant — which is every verification.

The consequence that costs money #

Two jobs printed from two profiles with different policies will match on the press-room measuring desk under D50 and disagree in a shop window.

That is the scenario the next rung of this field is entirely about, and the mechanism is exactly the one measured here. It is not a fault in either job. Both met the specification, both were verified, and the specification named one illuminant.

The related case is a job split across two presses or two runs — a cover printed here and an insert printed there, a reprint six months later with a different prepress version. Matching them under D50 is achievable and is what everybody does; whether they still match under the light the reader will use is not part of the contract.

Where this model stops #

The family here is parameterised by black alone. A real separation policy varies black and the chromatic inks in ways that are not a single parameter — start point, maximum black, replacement percentage — so the real space of separations reproducing a colour is larger than the curve drawn here, and its spread under a second illuminant is at least as large.

The solver’s tolerance is not a press’s tolerance. Agreement to 0.003 ΔE00 is arithmetic; a press holds a solid to a few units and a tint to a fraction of one, so in practice the members of a family differ under D50 as well, by press variation rather than by construction. The point stands: the systematic difference under a second illuminant is larger than the random one under the first.

Fluorescent substrates are excluded, and they would make it worse. A brightened sheet’s apparent reflectance depends on the light’s ultraviolet content, which adds a second, independent illuminant dependence on top of the one measured here.

And the effect is not a defect to be engineered away. It follows from having four controls and three constraints. The only ways out are to use three inks — which costs the depth and the stability the fourth was bought for — or to match spectrally rather than colorimetrically, which requires a target spectrum that a three-number colour specification does not contain.

What would fix it, and why nobody does it #

There is a repair available and it is worth understanding why it stays in the literature.

The freedom exists because the target is three numbers. Give the press a spectrum to match instead, and the problem changes shape entirely: matching eighty-one values with four controls is massively over-determined, the family collapses, and what comes back is a least-squares fit whose residual is a real spectral difference rather than an invisible one. With seven inks it becomes possible to get that residual genuinely small over a useful part of the space, and a match that holds under every illuminant at once is what spectral reproduction means.

Three things stop it. Nothing upstream carries a spectrum: a photograph is three numbers per pixel, a design file names CMYK or Lab, and no widely used format has anywhere to put a reflectance curve. The instrument chain is colorimetric by convention even though every spectrophotometer measures spectrally — the software reduces to three numbers and discards the rest at the moment of measurement. And the specification is colorimetric, so a spectral match would be verified by a colorimetric measurement that cannot see the difference it bought.

The obstacle is not the press. It is that the whole pipeline around it was designed to carry three numbers, and a degree of freedom is only visible to a system that has somewhere to record what was done with it.

The generalisation #

Any system with more controls than constraints reproduces its target along a manifold, and the members of that manifold differ in every quantity the constraints did not mention.

This site has now met the same structure three times, in three fields, from three directions.

A camera has its own metamers: the null space of the sensor’s three curves is not the null space of the eye’s, so pairs the camera merges the eye separates and vice versa. That is the input side of the same algebra.

A metameric match survives a flat wall and breaks in a corner: a bounce squares the reflectance, and a metameric black’s square does not integrate to zero. That is the same null space met by a nonlinearity.

And here it is on the output side: four inks give the pipeline a null direction of its own, and moving along it is free under one light and not under another.

The general lesson is short. A degree of freedom that costs nothing under the measurement being made costs something under a measurement that is not. The engineering question is never whether to use it, since it is usually the only way to satisfy a constraint like an ink limit, but which second measurement to check it against — and the answer for printing is a second illuminant, which is the subject of the next rung.

Who found it, and when #

The freedom itself is as old as four-colour printing and was originally exercised by hand: a skilled etcher deciding how much of a shadow to carry on the black plate is choosing a member of this family without a name for it.

Undercolour removal was named in the 1930s as a photomechanical technique; grey component replacement as a computed policy dates from electronic prepress around 1980, when the separation stopped being a photographic operation and became an arithmetic one. That is the moment the family became explicit, because a program needs a parameter where a craftsman needed a judgement.

That the members are metamers, and that the choice therefore has consequences under other illuminants, is a much later observation and belongs to the colour management era. It is documented in the graphic-arts literature and is not, on the whole, acted upon: the standards still specify a single illuminant, verification is performed under it, and the parameter is chosen for ink cost.

The measurement in this essay is not new. What is unusual is being able to run the whole family and quote its spread, which requires a spectral model of the press rather than a colorimetric one — and the reason most of the industry cannot make this measurement is that its profiles contain three numbers per patch and not a spectrum.

Where the ladder goes next #

This rung sits on the fourth ink, which is where the extra degree of freedom comes from, and on two spectra, one colour, which is the algebra it exploits.

Directly above is what happens when the second illuminant is not a thought experiment: the lamp in the shop decides, where a family verified under D50 is looked at under a retail LED.

Beside it, a brand colour is an ink takes the same metameric failure to the case where the target is a named pigment rather than another separation — and finds that the best possible four-colour build of a spot ink is a metamer of it by construction, with a drift that no press can reduce.