A tolerance has no light level

A colour difference formula takes two colours and returns a number. One unit in another room asked what happens to that number when the colour of the light changes, built twenty-three pairs at exactly ΔE₀₀ 1.000 under D65, and found them anywhere between 0.64 and 1.57 under the other lights modelled here. The room’s brightness was not among those changes, and it could not be: the formula has no argument for it.

The formula cannot move, and the unit that can moves by two thirds #

A tolerance written in ΔE₀₀ is the same requirement in a cinema and in sunlight, and the difference it stands for is not.

• Every pair is at ΔE₀₀ 1.0000 in all five rooms, to the last digit the arithmetic carries, because the room is not one of the formula’s inputs.
• In CAM16-UCS the same pairs are a median of 0.72 in a cinema, 1.04 in an office and 1.30 in direct sun — a factor of 1.62 between the cinema and an overcast sky.
• Inside one room the twenty-three spread by between 1.47 and 1.78 times, so there is no single number to convert a tolerance in one unit into a tolerance in the other, in any room.
• Seven tenths of the room’s effect is on the chroma part of the difference and a fifth on the lightness part, so a tolerance blind to the room is most blind about colour.

Two units, one of which was told where it is #

The pairs are the ones one unit in another room built: a reflectance walked along a fixed direction until ΔE₀₀ against its partner lands on 1.000, under D65 normalised to a luminance of a hundred. Twenty-three of the candidates reach the target exactly, and they are the same twenty-three throughout.

ΔE₀₀ reads them through CIELAB, whose only reference to a situation is the white point: divide by the white, take cube roots, weight the three differences. Nothing in it names the light level. CAM16-UCS reads them through CIECAM16, which takes an adapting luminance and a surround as arguments, computes a degree of adaptation, a set of induction factors and a nonlinearity from them, and only then produces coordinates — so the same two colours have one ΔE₀₀ and as many CAM16-UCS distances as there are rooms.

The comparison is not a competition between the two units. It is what the second one says about the first, which is that the first has left something out. A distance raised to a power has no length established what the model’s unit may and may not be used for — it is a metric and it does not add along a path — and neither of those properties is at issue here. What is at issue is a term, and the term is the room.

What one unit comes to, room by room #

In a cinema at one candela a square metre with a dark surround, the pairs run from 0.58 to 1.04 with a median of 0.72. In a dim room they run 0.68 to 1.16, median 0.82. In an office at a hundred candelas the median is 1.04 — the two units nearly agree, which is unsurprising, because the power correction that makes CAM16-UCS’s distance was fitted on data gathered in rooms of about that brightness. Under an overcast sky the median is 1.18, and in direct sun 1.30.

So a specification saying “within one colour difference” is asking for about three quarters of a model unit from a cinema and about one and a third from an object in sunlight. The same document, the same sample, the same instrument reading, and a requirement that differs by a factor of 1.8 between the ends of the range and 1.62 between a cinema and an overcast sky.

Which way that cuts depends on what the tolerance is for. If it is a proxy for visibility, then the fixed number is too loose in bright light and too tight in dim — the tolerance permits a difference that grows as the room brightens. If it is a contract about the sample rather than about a viewer, then the room is irrelevant by construction and the drift is somebody else’s problem. Most specifications are written as the first and defended as the second.

The room acts on the chroma, not the lightness #

The effect has a location, and knowing it makes the size easier to believe.

Between a cinema and an overcast sky the chroma-and-hue part of the median pair grows by a factor of 1.71 and the lightness part by 1.21. That is the Hunt effect arriving in a tolerance: colourfulness rises with luminance, which brighter looks more colourful measured directly on the model, and a difference between two chromatic colours is largely a difference in colourfulness.

It also says which tolerances are most exposed. A grey-balance tolerance, whose pairs differ mostly in lightness, moves least between rooms. A tolerance on a saturated colour — a brand colour, an ink, a display primary — moves most. The specification that cares least about the room is the one written on a neutral, and the one that cares most is the one written where the money usually is.

Whether the two rooms agree about which pair is worse #

A tolerance does two jobs: it accepts or rejects, and it ranks. The formula, having no room, has no opinion at all about how these twenty-three compare — they are all exactly 1.000. The model has one, and it is not the same opinion in every room.

The rank correlation between the two orderings is 0.88, and eleven of the twenty-three pairs move three places or more, the largest by seven. So the room does not merely scale the numbers: it reorders a few of them, which is what it means for the room to be a term in the difference rather than a constant multiplying it. If it were only a scale, the ordering would be identical and a single factor per room would convert one unit into the other — which is the arrangement the next paragraph rules out.

No single conversion, in any room #

The obvious repair is a conversion factor: one colour difference is 0.72 model units in a cinema, so multiply. It does not work, and it fails twice over.

It fails first between rooms. Every one of the twenty-three grows from the cinema to the sky, so the direction is common to all of them, but the factor runs from 1.29 to 1.75 around its median of 1.62 — a spread of 35 per cent. A single number per room would have to stand for all of that.

It fails again inside a room. The twenty-three pairs run from 0.58 to 1.04 in a cinema and from 0.98 to 1.44 under the sky — a spread of 1.78 and 1.47 times respectively. A factor that fixes the median leaves individual pairs out by as much as forty per cent. That is not the room’s doing: it is the two units disagreeing about which colours are far apart, which the weighting is the disagreement traced to whether a chroma difference is divided by the chroma it was measured at.

So there are two distinct gaps between a tolerance and what it stands for. One is that the formula has no room, and this essay measures it: a factor of 1.62 across ordinary rooms. The other is that the two units rank pairs differently even in one room, which is older and larger. A specification that wanted to name the appearance difference it is really asking for would have to name the room and abandon the idea that one number in one unit converts into one number in the other.

The same term, already measured twice #

The room’s presence in the model’s unit is not news to this collection; it has been met twice before, from other directions, and both are worth putting beside this one because they bound how far the argument generalises.

The appearance model has no straight piece found that the price of a deviation near black rises without limit in the model and flattens in CIELAB, at a rate the surround sets — the same two units disagreeing about the same thing, at the dark end of the lightness scale rather than across rooms. The gamut shrinks in the dark found a display losing a fifth of its reach and two thirds of its volume when the room is turned down four decades, with the signal unchanged.

Set beside those, the tolerance result is the mild one: a factor of 1.62 rather than of three. That is worth saying plainly, because it bounds the practical claim. The room is a first-order term for a gamut, a first-order term for a black, and a third-of-the-tolerance term for an ordinary pair of colours. A specification that ignores it is not making a catastrophic error; it is making one about the size of the terms a careful delivery budget spends its time arguing over.

What a specification could say instead #

Three repairs, in increasing order of what they ask of the people writing the document.

Name the room. A tolerance quoted beside its viewing condition is not a better tolerance, but it is a checkable one: two laboratories reading the same sample under booths of different brightness are no longer entitled to the same answer, and the document says so. A tolerance cannot cross a condition found the same thing one level down, for the measurement geometry — a correction fitted between two conditions left a quarter of the disagreement behind. Naming the condition is cheaper than correcting for it.

Quote the pair’s own conversion, not a room’s. If what a specification means is an appearance difference, the honest way to write it is in the appearance unit and let the instrument do the conversion sample by sample, since the conversion is 0.58 for one of these pairs and 1.04 for another in the same room. That is a heavier instrument requirement — a spectral measurement and a model rather than three numbers and a formula — and it is what the numbers here say the requirement actually is.

Or accept the formula’s answer as the contract and stop calling it a visibility. A tolerance in ΔE₀₀ is a perfectly good agreement about samples. It is a mean is not a difference’s point in another form: a number that reduces a set of judgements to one figure can be honest and still not be the judgement anybody makes. What it cannot be is both a fixed contract and a statement about what a viewer will see, because the second moves with the room and the first does not.

What was computed, and how #

The pairs come from the tolerance construction in this collection’s adaptation work: a family of reflectances, each walked along one of two fixed directions in reflectance space by bisection until ΔE₀₀ against its partner equals 1.000 to within a millionth, under D65 normalised to a luminance of a hundred. Twenty-three of the candidates converge; the rest are discarded rather than approximated.

Each pair is then read twice. In ΔE₀₀, through CIELAB against that same white — which is what makes every room’s answer exactly 1.0000, and is checked rather than assumed. And in CAM16-UCS, through CIECAM16 at a stated adapting luminance and surround, with the background factor held at 20 and the white at D65, taking the Euclidean distance in J′a′b′ without the power correction, so that the lightness part and the chroma part can be separated and compared.

The five rooms are one candela a square metre with a dark surround, four with a dim one, and a hundred, a thousand and ten thousand with an average one. The sweep behind the first and third figures holds the surround at average and moves only the luminance, so that the surround’s own contribution is not mixed into the trend.

What this does not settle #

Whether the model’s room term is the right size is a question about CIECAM16, not about the tolerance, and this essay inherits whatever the model gets wrong. The Hunt effect it encodes is well attested in direction; its magnitude at the extremes of luminance rests on fewer data than the middle.

The comparison also holds the sample fixed and moves the room, which is the experiment a specification cares about. The other experiment — hold the room and ask what size of difference is just visible in it — is a threshold measurement rather than a suprathreshold one, and a threshold is not a unit is the reason the two cannot be substituted for each other. A tolerance drawn from threshold data would move with the room too, and by a different factor.

And the pairs are constructed rather than sampled from a real process. They span the reflectance family the adaptation census uses, which is broad but not a delivery chain’s own colours.

One more term is held still here and moves in practice. The background against which a sample is judged enters CIECAM16 as its own argument, and every number above is computed against a mid-grey at a luminance factor of 20. A patch is not a scene measured what the other viewing-condition arguments are each worth; the one moved here is the adapting luminance, and it moves it alone so that the trend belongs to it.

Still open: the room a specification is actually read in #

The number that would settle the practical size of this is the distribution of adapting luminance in the rooms where colour decisions are actually made — a press hall’s viewing booth, a design studio’s desk, a warehouse aisle where a sample is checked against a swatch.

The three can be estimated rather than guessed, because an illuminance and a background reflectance give a luminance: a perfectly diffusing grey of twenty per cent returns E · ρ / π. A graphic-arts viewing booth at 2,000 lux is then 127 candelas a square metre, an office desk at 300 lux is 19, and a dim aisle at 100 lux is 6. Put through the same pairs, those come to 1.05, 0.95 and 0.88 model units: a booth against a desk is 11 per cent and a booth against an aisle 19.

So the honest practical claim is narrower than the range of rooms suggests. Between the places a specification is genuinely read, the room is worth about a tenth to a fifth of the tolerance — not the 62 per cent that the cinema-to-sky span implies, and not nothing either. What is missing is the distribution rather than the endpoints: how often a sample is actually judged away from a booth, which is a survey of practice rather than a calculation.

A formula with an argument missing #

The habit is about what a function’s signature declares it cannot know.

ΔE₀₀ takes two colours. Whatever else it might be wrong about, it cannot be wrong about the room in the ordinary sense, because the room is not in the domain — the answer is not a bad estimate of a room-dependent quantity, it is an estimate of something else. The same is true of every function whose arguments are fewer than the situation’s: a gamut with no room in it, an appearance model with no clock in it, a tolerance with no observer in it.

The move is to read the signature before the residual. If a quantity the answer plainly depends on is not among the arguments, no amount of fitting will put it there, and the size of what is missing can usually be got by finding a second model that does take it and asking that one.

The failure mode is treating a missing argument as a small error. It is not an error at all: it is a different question, answered exactly.