A brand colour for a population

A brand colour is the largest population any colour specification faces, and the one specification that is chosen freely at the start rather than inherited. That combination makes it the place where an observer allowance would be worth most and where it is least likely to be considered.

The claim #

Observer robustness is a property of a colorant’s spectrum, it varies by an order of magnitude between candidates, and it is computable at the moment a brand colour is chosen.

• The observer term runs from about 0.9 to 8.2 ΔE₀₀ across a family of ordinary surfaces, at the median and the ninety-fifth percentile respectively.
• Two candidates with the same specified colour can differ enormously, because the term depends on the spectrum rather than on the tristimulus values.
• It is computable from the ink’s reflectance and the intended viewing illuminant, with no new measurement.
• And it would be a fourth criterion beside gamut, cost and lightfastness — currently absent from every brand specification this collection has seen.

Why a brand colour is the extreme case #

A brand colour is an ink established the practical position: a brand colour is specified as a spot ink with a reference, it is reproduced in process inks, on screens, on fabric, on plastic and in paint, and each reproduction is a different colorant with a different spectrum matching the same numbers.

Three things make it the extreme case for the observer term.

It is seen by everybody. A specification for a component inside a machine is judged by two engineers; a brand colour is judged by the entire population of a market, which contains the whole spread of lens densities, macular pigments and cone peaks.

It is reproduced metamerically by design. Every reproduction that is not the original ink is a metameric match to it, and a metameric match is exactly what an observer difference breaks — the same fragility a proof has against a press sheet. A brand’s whole visual identity is a family of metamers.

And it is chosen freely. Almost every other colour specification is inherited — a customer’s requirement, a standard’s value, an existing product. A brand colour is picked from a catalogue at the beginning, which is the one moment when a criterion can be applied at no cost.

What decides a colorant’s observer robustness #

The audit’s mechanism gives the answer directly, and it is about spectral shape rather than about colour.

An observer departure is a pairing of the observer’s deviation with the stimulus’s deviation from the adapting white. So a colorant carries a large term when its reflectance departs strongly from flat in the regions where observers differ: the blue below 500 nanometres, the macular band around 460, and the flanks of the long- and middle-wavelength cone sensitivities near 540 to 620.

A colorant whose structure is elsewhere carries little. The interference filter in this round’s surface family, with a notch at 545 nanometres, has an observer term about half the red pigment’s, because its structure sits between the departures rather than on any of them.

Two colorants of identical colour can therefore differ substantially, and the difference is invisible in the specification because the specification is three numbers.

That is the same argument that decides which pairs are metameric, and it is not the same quantity. A pair’s metamerism is about the difference between two spectra; a single colorant’s observer term is about its own deviation from the light. A colour can be robust and still form a fragile metameric pair with its own reproduction.

The worst light matters, not the average. A brand colour’s viewing condition is not a choice, and that changes which statistic matters.

A specification for an industrial component names its viewing condition, so the relevant observer term is the one under that light. A brand colour is seen in a shop under a three-emitter LED, at home under a warm one, in daylight through a window, and on a screen with narrow primaries — so the relevant figure is the worst over the set, not the mean.

Across this round’s five well-sampled lights the mean departure on a red pigment runs from 1.64 to 2.37 and the worst single term from 1.15 to 4.47. So the worst-light figure is about twice the daylight one for a saturated colour, and a specification budgeting for the booth is budgeting for the best case of a distribution it does not control.

That argues for evaluating a candidate under several lights and taking the maximum, which is one extra loop over an existing calculation. It also argues for something a brand can actually control: choosing a colorant whose structure avoids the departures’ regions makes it robust under every light at once, rather than under one.

The narrowband case is worth separating because it is now the commonest condition a brand colour is seen under and it was rare twenty years ago. A three-emitter backlight is in most phones and most televisions; a three-emitter lamp is in a great many shops; and both excite the pigment-peak departure more than any thermal source does.

So a brand identity designed in the era of print and thermal lighting is being viewed, most of the time, under a condition that raises one of its observer terms by half. The viewing conditions moved and the specifications did not, which is a change nobody made and nobody could have anticipated at the time the identities were designed.

That also says something about which reproduction of a brand colour carries the most risk. The printed one is seen under whatever is in the room; the screen one is seen through three narrow primaries and carries the higher term, on top of whatever the display’s own gamut mapping does.

What a specification could carry #

The proposal is the same as the tolerance essay’s and it is cheaper here, because the choice is being made anyway.

A brand colour is specified with a reference ink, a set of process-ink builds, screen values, and tolerances. Adding one number per reproduction — the observer allowance for that colorant under the intended illuminant — costs one integral over data that already exists in the specification’s own spectral measurements.

What it would change is which candidate gets chosen. Two colours a designer considers equally suitable, with equal gamut availability and equal cost, may differ by a factor of two or three in how much the population will disagree about them. That is exactly the sort of tie-breaker a specification process can use and currently cannot see.

It would also make the reproductions comparable. A brand’s process-ink build and its spot ink match numerically and carry different observer terms, and a specification could say which reproduction is the robust one and prefer it where the choice exists.

Where this differs from the usual metamerism advice #

The standing advice about brand colours and metamerism is to specify spectrally, and it is right. This is a different and complementary point that is worth separating.

Spectral specification removes the metamerism between reproductions: if every reproduction matches band for band, they match for everybody under everything. It is the strongest available requirement and it is often impossible, because a screen cannot make an ink’s spectrum and a fabric dye cannot make a plastic’s.

Observer-robust colorant choice reduces the term that remains when spectral matching is impossible. It does not require the reproductions to match spectrally; it requires each of them to sit where observers agree, which is a weaker and more achievable condition.

The two work together. A brand colour chosen in a region where observers agree, reproduced as closely spectrally as each medium allows, carries less observer risk than either measure alone would give.

Two paints that stop matching is the illuminant version of the same problem, and the same two responses apply to it: match spectrally, or choose colorants whose difference is small where illuminants differ.

The one number a specification needs #

That figure reduces the whole recommendation to a quantity, and the quantity is easy.

The horizontal axis is the excitation deviation of the sample from the adapting white — the Euclidean distance between its relative cone excitations and (1, 1, 1). It is computed from the colorant’s reflectance and the illuminant’s spectrum, both of which a brand specification already contains.

The vertical axis is the observer allowance, and the line between them is a coefficient of about 5 ΔE₀₀ per unit of deviation under a smooth light, rising to about 8 under a warm one and about 10 under a narrowband source.

So the specification’s addition is one number per colorant per illuminant, computed by one integral and one multiplication. It is not a research programme; it is a column in a table, and the reason it is absent is that nobody has asked for it rather than that anybody found it hard.

The colours a brand would avoid #

Pushing the argument to a recommendation gives a short and slightly surprising list.

Saturated blues and violets carry the most. They have their structure below 500 nanometres, where the lens and the macular pigment both live, and the two are the largest departures in the round.

Saturated greens carry a good deal. Their structure is on the cone flanks near 540, where a peak shift does most of its work.

Mid-lightness warm colours carry least among saturated options, because their structure is above 580 where only the long-wavelength peak has much to say.

And neutrals carry nothing at all, exactly, which is why a monochrome identity is the observer-robust choice and is not usually chosen for that reason.

That list is a first approximation from the mechanism rather than a computation over a catalogue, and computing it properly over a real ink library is the obvious next step and is not in this round.

The identity is the reason a brand’s typographic and structural elements are never the problem. Black text, a grey rule, a white background: every observer computes the same colour for all of them, exactly, whatever the lamp and whatever the reproduction.

So the observer risk in a visual identity is concentrated entirely in its saturated elements, which are usually one or two colours out of a palette of a dozen. That concentration is convenient — it means the analysis is a two-colour analysis rather than a palette-wide one — and it explains why the problem, when it appears, appears as a complaint about one specific colour rather than about a brand’s reproduction generally.

It also means a brand with a saturated primary colour and a neutral secondary palette carries all of its exposure in one place, and moving that one colour a little towards a region where observers agree would move the whole identity’s risk. One colour usually carries the whole term, which is the best possible news for anybody who has to act on it.

The comparison worth putting in front of a designer is between two saturated colours rather than between a saturated and a pale one. A notch filter and a red pigment are both strongly coloured; under the same warm lamp the first carries departures from 0.36 to 1.50 ΔE₀₀ and the second from 0.79 to 4.47.

That is a factor of three between two candidates neither of which is neutral, and it is exactly the choice a brand makes at the start.

The slope is the number a brand specification would carry, and it is computable from a measured reflectance and a stated illuminant before any reproduction exists.

What was computed, and how #

The departures are this round’s, over forty-two analytic surfaces under six constructed lights, through this collection’s usual CAT16 route. The coefficient of about 5 ΔE₀₀ per unit of excitation deviation is read off the linearity measurement rather than fitted.

Nothing here is computed over a real ink library, and the recommendations by hue family are inferences from where each departure lives rather than measurements of colorants. A real ink’s reflectance has structure the analytic family does not.

The claim that the term is computable from data a brand specification already has rests on brand specifications containing spectral data, which the better ones do and many do not — a specification carrying only L*a*b* values cannot compute it, which is one more argument for the spectral form.

There is a practical objection to all of this that deserves an answer, and it is that a brand colour is chosen for reasons that have nothing to do with colour science. It is chosen to be distinctive, ownable, appropriate to a category and available as a trademark, and a designer told that one candidate carries two colour differences of observer spread and another carries four will reasonably say that the first is the wrong colour for the brand.

That is a complete answer and it does not make the number useless. A criterion that loses most of the time is still worth having when it is free, because it decides the cases where the other criteria are tied — and it converts a later surprise into an earlier decision. A constraint costs what it points at, and an observer allowance points at a small subset of the colour space and leaves the rest alone.

The stronger version of the objection is that the difference between two and four colour differences of population spread may not be perceptible in any situation the brand actually encounters, since brands are rarely seen beside their references. That is very likely true for most of the palette and is not true for packaging, where a shelf holds several production batches side by side.

Where the model stops #

Everything here is about a single colorant’s departure from its adapting white. A brand’s real risk is often about a pair — the spot ink against the process build — and that is an observer-metamerism question with a different and larger answer.

The population is represented by two constructed observers differing in one parameter at a time, not by a distribution. The population machinery would give the acceptance fraction, which is what a brand actually needs, and the two are not numerically comparable.

And nothing here says whether a viewer would notice. Two units of disagreement about a brand colour that nobody sees side by side with its reference is a different situation from two units on a pack sitting beside another pack.

The generalisation #

The habit is about criteria that are free at the moment of choice and expensive afterwards.

A specification is chosen once and lived with for years, and the criteria applied at the moment of choice are the ones that shape everything downstream. A criterion that is cheap to apply then and impossible to apply later is worth identifying, even when its effect is modest, because the alternative is not a smaller effect but a permanent one.

The move is to ask which of a decision’s consequences are still adjustable afterwards. Cost is adjustable, suppliers are adjustable, tolerances are adjustable. The colorant’s spectrum is not, once an identity is built on it.

The failure mode is to apply only the criteria that have established procedures. A criterion with no standard method is a criterion that does not get applied, and this one has no standard method for the reason the tolerance essay gives: it is not a measurement uncertainty and there is no line to write it on.

Who found it, and when #

Observer metamerism’s importance for colorant matching has been understood since the 1970s and is the reason the CIE published a standard deviate observer in 1989. That work is about pairs — will this match survive a different observer — which is the formulation the colorant industry needed.

The single-colorant form used here, as an allowance attached to one colour rather than to a match, does not appear in that literature. It is a straightforward consequence of the same pairing structure and it answers a different question: not whether a match will survive, but how much a population will disagree about a colour that nothing is being matched to.

Where the ladder goes next #

The audit’s terms have been carried into every part of the delivery chain. What is left is the model this collection uses to say how a colour looks rather than what it matches, and its input is the observer’s output: an appearance model takes tristimulus values, and everything upstream of them is in this round.