The lamp in the shop decides

A carton and the leaflet that goes inside it are printed at different plants. Both are made to the same specification, both are measured on a spectrophotometer under D50, and both pass. On the shelf, under the shop’s lighting, the two are visibly different colours.

This is a routine and expensive occurrence, it is nobody’s fault in any useful sense, and its mechanism is entirely contained in the previous rung of this field.

The claim #

A colorimetric specification under one illuminant does not constrain what happens under another, and printing generates the freedom it fails to constrain.

Both halves matter. Metameric pairs are everywhere in colour, and most of the time they arise by accident — two dye systems that happen to match. Here they are manufactured, because a four-ink press has a degree of freedom the colour does not use, and every prepress system in the world exercises it to save ink.

What the second light does #

The spread of one family of separations, under four lights:

The pattern is the one this site has established elsewhere: the further the second light is from the first in spectral shape, the more of the difference it reveals. Equal energy is flat and nearly cancels the metameric black; illuminant A is a Planckian radiator whose power rises steeply through the band, so it weights the long wavelengths heavily and the differences between the separations are largest exactly there.

The shop is the worst case, not the exception #

The lights printed matter is actually looked at under are, almost without exception, further from D50 than daylight is.

A tungsten lamp is a smooth Planckian radiator at 2856 K — the illuminant A of the table above, and the mildest of the awkward cases despite giving the largest number here, because it at least has a smooth spectrum.

A fluorescent tube is a phosphor continuum with mercury lines standing on it, and what a lamp cannot give back is the site’s essay on what that does to surfaces. A spectrum with sharp features multiplies the reflectance at those features and nowhere else, so two spectra differing in the region of a line come apart dramatically.

A white LED is a blue pump with a phosphor over it, which leaves a deep notch between the pump peak and the phosphor’s rise — around 480 nanometres, in the cyan. Two separations differing there are separated by the notch in a way no smooth illuminant reproduces.

What a specification could do about it #

The remedy is not difficult and it is not free, and it is worth laying out because most discussions of this failure stop at describing it.

Verify under two illuminants. Every spectrophotometer in every pressroom already measures a spectrum; computing a second set of CIELAB coordinates under illuminant A costs nothing but a line of arithmetic, and the software mostly offers it. A specification that named a second illuminant and a tolerance under it would eliminate the whole failure mode described here, because it would remove the freedom rather than leaving it unconstrained.

Or specify a metamerism index. The CIE defines exactly that — a number quantifying how far a match made under one illuminant degrades under another — and it is used in the textile and paint industries as a matter of course.

Or fix the policy rather than the outcome. Two plants using the same black generation would produce the same separation, and the whole question would not arise. That is what a shared profile achieves, and it is why large brands specify one.

The reason none of these is universal is that they all cost something at the moment of specification and the failure only appears later, in somebody else’s building.

Where the number comes from #

The spread quoted here is computed rather than estimated, and the chain is short.

For each black level, solve for the chromatic coverages that reproduce the target under D50 — a Gauss–Newton solve against the full spectral halftone model. Every member of the family then has a reflectance curve, not merely a colour. Re-integrate every curve against the colour-matching functions under the second illuminant, convert to CIELAB against that illuminant’s own white point, and take the largest pairwise ΔE00.

Two properties make the number meaningful. The agreement under the first light is checked rather than assumed — it comes out at 0.003, which is the solver’s tolerance and confirms the family really is a family. And the second-illuminant spread is not controlled by anything in the construction: nothing in the solve mentions illuminant A, so 6.75 is a consequence rather than a target.

How large is 6.75 #

A colour difference has no meaning until it is said what is being compared and how. Seven units is a great deal in some arrangements and invisible in others, and the arrangement is what decides whether this failure costs money.

Side by side, touching, is the harshest test there is. A carton and its leaflet, a cap and its bottle, a printed panel against a moulded one — anything where two surfaces meet with no gap between them. In that arrangement a ΔE00 of 1 is detectable by a trained observer and 2 is obvious to anybody, so 6.75 is a different colour rather than a poor match.

Separated in space or time it is far more forgiving. Two pieces in different parts of a shop, or the same piece bought twice, are compared against memory rather than against each other, and colour memory is coarse. The same 6.75 would go unremarked.

And the size is not the whole of it. A tolerance is a shape rather than a number, because discrimination is anisotropic — a difference of a given size is much more visible along some directions than others, and a ΔE00 of one is not a fixed perceptual step in any case. The metameric drift measured here is largely a lightness and chroma shift on a near-neutral, which is one of the directions the eye is best at.

The practical form: this failure matters exactly where two independently produced pieces end up adjacent, which is precisely the case packaging generates constantly and the case a specification written for a single press does not contemplate.

Two policies, one specification #

The clean version of the failure is a pair of plants, and it can be stated as two rows of one table.

Plant A runs a light black generation: for the colour above, 61 per cent cyan, 52 magenta, 48 yellow, no black — 161 per cent of ink on the sheet. Plant B runs heavy replacement: 21, 15, 13 and 50 per cent black — 98 per cent of ink, a saving of nearly two fifths, which is the whole reason the policy exists.

Under D50 the two pieces are the same colour to within the measurement noise of any instrument in either building. Under the shop’s tungsten they are 6.75 apart.

Neither plant did anything wrong, neither could have detected the problem with the equipment and procedure it was using, and the difference between them is a setting in a piece of prepress software chosen years earlier for reasons of cost.

Two more readings say that the effect is a property of the family rather than of the one colour the essay opened with.

How far is far, and what the spread does with distance #

The four-illuminant table is read as a monotone relationship — the further the second light from the first, the more it reveals — and the table supports that. Putting a scale on further says something the ordering alone does not.

The mired scale is the natural one for comparing thermal sources, since equal steps in it are roughly equal steps in appearance. Against D50 at 199.9 mired:

The relationship is strongly concave. From equal energy to D65 the distance rises by a factor of 2.8 and the spread by 3.1 — an exponent of 1.11, very nearly proportional. From D65 to illuminant A the distance rises by 3.3 and the spread by only 1.8 — an exponent of 0.52.

So the spread is not proportional to how odd the lamp is, and the return falls off sharply. Doubling the mired distance beyond D65 buys about forty per cent more mismatch rather than twice as much.

That has a consequence the essay’s own closing caution needs. The section on which lamp to name argues that illuminant A is a lower bound on what a retail environment will do, chosen because it can be named. The concavity says the bound is a good deal tighter than it looks: a lamp twice as far from D50 as tungsten is would not give twice the spread, and on this curve would give something under ten. The number to be worried about is the six or seven that a common lamp already produces, not some larger figure an exotic one might.

There is a mechanism for the saturation and it is in the essay’s own construction. The differences between the eleven separations are metameric blacks — a fixed set of spectral residuals, with a finite total. A second illuminant re-weights them, and once the weighting has stopped resembling the first illuminant’s there is no more agreement left to destroy. The spread has a ceiling set by the family, and by illuminant A it is most of the way to it.

The caveat is that mired distance is a crude stand-in for spectral distance and that equal energy is not a thermal radiator at all, so its 183 mired is its correlated colour temperature rather than a description of its shape. Three points and a proxy scale establish concavity and do not establish a law.

The specification is satisfied two thousand times more tightly than it fails #

One ratio worth stating outright, because it is the sharpest form of the essay’s argument.

Under the illuminant the specification names, the eleven separations agree to 0.003 — the solver’s tolerance, and far below what any instrument in either building could resolve. Under a tungsten lamp they disagree by 6.75. That is a factor of 2,250 between the quantity that was checked and the quantity that was not.

Nobody’s verification was sloppy. Both plants met their specification with roughly three orders of magnitude of margin, and the margin was entirely in the direction the specification could see. A constraint satisfied to three thousandths of a unit and a freedom left open to seven units is the whole failure, and neither number is a defect of anybody’s process.

The two-colour comparison supports the same reading from the other side: 6.75 on one target and 6.98 on another, 3.4 per cent apart, which is what one expects if the size is set by the family’s available range rather than by the particular colour. Two points do not establish that, and they are consistent with it.

Ninety-nine, not ninety-eight #

The worked example’s second plant lays down 21 per cent cyan, 15 magenta, 13 yellow and 50 black, which sums to 99 rather than the 98 the text quotes. The saving against plant A’s 161 is 38.5 per cent rather than 39.1, which is still nearly two fifths and still the reason the policy exists.

It changes nothing and it is worth correcting in an essay whose subject is a number that nobody checked.

Where this model stops #

Real plants differ in more than black generation. Ink lots, paper, screening and press condition all move the spectra, and they move them in ways that are not colorimetrically invisible — so a real mismatch on a shelf is this effect plus ordinary process variation. Separating the two requires the spectra, which is why the diagnosis is rarely made.

The tungsten figure is a worst case among common lights and not the worst case available. A narrow-band LED source of the kind used in some retail displays would give a larger number, and this site’s saturation-sensitivity measurement is the reason to expect it.

Optical brighteners are excluded here and are a second, independent mechanism. A brightened sheet fluoresces under ultraviolet, so its white depends on the ultraviolet content of the light; two pieces on differently brightened stocks disagree under a shop light for reasons that have nothing to do with separations. That effect is often larger than this one and is a whole essay in the light field.

And nothing here is about how the mismatch is judged. Whether a ΔE00 of 6.75 is unacceptable depends on whether the two pieces are adjacent, how large they are, what surrounds them, and what the observer is doing — none of which is colorimetry, and some of which the appearance model can speak to.

A lighter target with less black in it is the separation a shop would actually be given, and the spectra are where the disagreement is visible before it is a number.

The one place the trade does check #

It would be unfair to leave this without saying where a second illuminant is routinely used, because the contrast is instructive.

Textile and paint suppliers check under two or three lights as a matter of course — a daylight simulator, a tungsten lamp and often a fluorescent — and quote a metamerism index alongside the colour difference. The equipment is a cabinet with several lamps in it and a switch, and it is standard.

The reason those industries do it and printing does not is not sophistication; it is what gets compared with what. A dyed fabric will be sewn next to another dyed fabric, and a paint will be applied beside a moulded part, so the customer’s experience is two samples touching under whatever light happens to be there. A printed sheet is compared with a proof, in a booth, by a professional, and then it goes out into the world alone.

Printing’s verification matches its point of sale and not its point of use, and that single sentence is the whole of why this failure survives.

Under daylight rather than a tungsten lamp the spread is smaller and it does not vanish, which is the reading that says the effect is not about one lamp.

The generalisation #

A specification constrains exactly the measurements it names, and any freedom left over will be spent by whoever is optimising something else.

The pattern is not about light. In printing, the freedom is a black generation policy and the thing being optimised is ink cost. The specification named D50 and the optimiser worked in the space D50 could not see.

The same shape appears wherever an under-determined system meets a cost function. A camera’s colour matrix is fitted to a set of surfaces and is worse off it: the fit optimises the sample set, and the space the sample set does not span is where the error goes. A metameric match survives a wall and fails in a corner: the match was constrained on flat geometry, and the corner squares the reflectance.

The remedy is the same in all of them and it is not a better optimiser. Constrain a second, different measurement, chosen so the freedom the first one leaves is visible in it. For printing, the second measurement is one line of arithmetic on data every instrument in the building already collects.

Three separations rather than eleven is what a pressroom would print as a test form, and the coverages are what a plate actually carries.

A note on which lamp #

One practical wrinkle, since the essay’s title names a lamp and the table names illuminant A.

Illuminant A is a Planckian radiator at 2856 K — a tungsten filament, which is now a rare object. What a shop actually has is a white LED, and the measurement to make is under that. This site models one: a blue pump with a phosphor over it, with a deep notch between the pump peak and the phosphor’s rise.

The reason the tables here quote illuminant A is that it is a standard — defined, reproducible and in the specification — while a white LED is a family of spectra differing between manufacturers and between production runs. A number quoted under a named illuminant can be checked by anybody; a number quoted under “an LED” cannot.

That is a genuine methodological constraint and it cuts against the essay’s own argument, since the whole point is that the light in the shop is not a standard illuminant. The honest position is that illuminant A is a lower bound on what a retail environment will do, chosen because it can be named.

Who found it, and when #

Illuminant metamerism was understood well before it could be computed: dyers have known since the nineteenth century that two dyeings matching in daylight can differ under gaslight, and the trade name for it — a shop light match — is older than colorimetry.

The CIE published the special metamerism index for change of illuminant in 1971, which is the standard way of putting a number to it, and it entered textile and paint practice quickly, because in those industries the customer is holding two samples together in a room.

Printing adopted it hardly at all. The reason is worth stating without judgement: a printed piece is normally compared with a proof or with a specification rather than with another piece, and both comparisons are made under the standard illuminant in a viewing booth. The failure appears at the point where two pieces made independently are put side by side, which is after everybody’s contract has been discharged.

The relevant standard, ISO 3664, specifies the viewing conditions in exhaustive detail — D50, a stated illuminance, a stated surround, a stated ultraviolet content. It is a specification of the booth, and it works. What it cannot do is specify the shelf.

Where the ladder goes next #

This rung sits on the separation is not unique, which is where the metamers come from, and on the illuminant is half the answer, which is why a second light changes anything at all.

Beside it, a brand colour is an ink is the same failure with the target changed from another separation to a named pigment — and there the drift under a second light is bounded below by the ink’s own spectrum, so no press can remove it.

Above, the argument moves from the shop to the studio: the proof is a different object, where the two things being compared have identical colorimetry and are looked at in rooms that differ in surround rather than in spectrum.