Whiteness is mostly the lamp

Whiteness is the number a sheet of paper is sold on. It appears on every data sheet, it is what a buyer compares, and it is a single figure computed by a published formula from a measurement anybody can repeat — the paper trade’s answer to where white stops being a point.

Take the ultraviolet out of the instrument and most of it disappears.

The claim #

The CIE whiteness formula measures a sample and a lamp jointly, and on the samples it is used for the lamp is the larger term. Quoting a whiteness figure without a measurement condition beside it publishes half a measurement.

• The formula is a linear fit to visual rankings, published with four inequalities that bound where it is valid.
• An unbrightened sheet of this collection’s base measures W 81.8. The brightened stocks measure 100.5, 113.6, 122.5, 129.0 and 121.3.
• With the ultraviolet removed they measure 84.7, 87.6, 91.6, 93.6 and 85.3. The spread collapses from 47 points to 12.
• So on average 82 per cent of what the scale reports — and 91 per cent on the most heavily loaded sample — is the luminescent term, which is a property of the sample and the lamp and of neither alone.
• And the validity box is closer than it looks. The upper bound moves with the sample’s own luminance, and the laundered shirt sits 9 points below its own limit.

The formula, and why its coefficients look strange #

The CIE whiteness of a sample is

W = Y + 800(xₙ − x) + 1700(yₙ − y)

with (xₙ, yₙ) the chromaticity of the perfect diffuser under the same illuminant, and the tint is

T = 1000(xₙ − x) − 650(yₙ − y)

The coefficients are large because chromaticity coordinates are small. x and y run over a range of about 0.6 across the whole diagram and a white sample moves them by hundredths; Y runs to 100. A coefficient of a thousand or so is what puts a visually comparable step in each term rather than an expression of preference.

What the formula says, once the units are normalised, is that whiteness responds about equally to a sample’s luminance factor and to how far it sits towards blue. That is a claim about human judgement, fitted to rankings of real samples, and it is why a mechanism that moves chromaticity and barely moves Y is rewarded so heavily. A brightener is exactly that mechanism — 0.31 per cent of luminance and 7.4 units of b* — and the formula converts the second into forty points.

The two facts fit together and neither was designed against the other. The formula was fitted to how people rank white samples; brighteners were developed to make sheets look whiter. Both converged on the same axis because that is the axis the judgement runs along.

What survives the ultraviolet being removed #

The decomposition is a subtraction and it needs a control, which is the unbrightened sheet of the same base.

For each brightened stock, take the whiteness it has above that control under an ultraviolet-included instrument — 18.7, 31.8, 40.7, 47.2 and 39.5 points. Then take the same figure under an ultraviolet-excluded one — 2.9, 5.8, 9.8, 11.8 and 3.5.

The second is what the paper is; the first is what the paper and the lamp are together. The ratios come out at 84, 82, 76, 75 and 91 per cent lost, with a mean of 82.

The number is worth reading carefully because it is easy to over-claim from. It does not say that a brightened sheet is not whiter than an unbrightened one — it plainly is, to any observer standing in daylight. It says that the quantity the scale reports is not a property of the sheet, and therefore that two figures produced under different conditions cannot be compared, subtracted, or averaged.

The ranking, which is what the number is for #

A whiteness figure is never used alone. It is used to decide which of two sheets is whiter, and that makes a sharper test available: can the condition change the order?

It can, and among only six samples it does. Office paper and a laundered white shirt swap places between the ultraviolet-included and the ultraviolet-excluded condition, and again between the incandescent condition and the excluded one.

The mechanism is a competition between two properties that the two conditions weight differently. The shirt has a heavier brightener loading and a duller base; the paper has a lighter loading and a brighter base. Under a lamp with ultraviolet the loading dominates. Without it the loading contributes nothing and the base decides.

A buyer choosing the whiter of two samples gets a different answer depending on the supplier’s instrument, with both instruments in calibration. A difference can be budgeted into a tolerance; a reversal cannot, and it is the reversal that makes the condition a specification variable rather than a source of error.

The reversal is in the published numbers, and it has a margin #

The ranking claim is made in words and the table above contains it, which is worth showing because a reversal asserted without a margin is hard to weigh.

Sorting the five brightened stocks by whiteness gives 1, 2, 5, 3, 4 with the ultraviolet included and 1, 5, 2, 3, 4 with it excluded. Exactly one pair changes places, and it is stocks 2 and 5:

Seven and a half points one way and two and a third the other. Neither margin is inside anything that could be called measurement noise on a scale where the whole brightened range is 47 points, so this is two instruments in calibration returning two confident and opposite answers — which is the situation the essay says a tolerance cannot absorb, now with both margins on it.

The asymmetry is itself informative. The reversal is not a near-tie tipping over: the pair is separated by a fifth of the scale’s working range under one condition and a twentieth under the other, and the larger separation is the one that vanishes when the lamp does.

The share lost is orderly, and one stock is not #

The five ratios are quoted as a range and a mean. Sorted by how much whiteness each stock gains over the control, four of the five fall into a clean pattern:

Among the four paper stocks the share lost falls monotonically as the gain rises. The heaviest loading has the smallest proportion of its advantage in the luminescent term, not the largest, because the ultraviolet-excluded gain grows too — from 2.9 to 11.8 points across the four.

That last column is the one worth stopping on, because the essay describes the excluded figure as what the paper is and moves on. A sheet with no excitation reaching it still gains up to twelve points of whiteness over the control, and it gains them in order of loading. Either an unexcited brightener is not optically inert, or the four stocks differ in their bases as well as in their loadings — and the essay’s own limits section says the bases were chosen to span a commercial range, so it is very likely the second.

Which means the 82 per cent is a fraction of the wrong thing to be a statement about brighteners. It is the share of a sheet’s whiteness advantage over one particular unbrightened base that is luminescent, and that advantage contains base differences as well as additive. As a description of the scale it is exactly right — a buyer comparing two data sheets is comparing whole sheets, not additives. As a description of what a brightener contributes it is an underestimate, since some of the non-luminescent remainder belongs to the paper rather than to the chemistry.

The outlier is the same variable as the reversal #

Stock 5 breaks the pattern in both directions at once, and it is the same stock on both occasions.

It has the third largest gain with ultraviolet, 39.5, and the second smallest without, 3.5 — hence the 91 per cent that the claim section leads with. And it is one half of the reversing pair. Both follow from the description the essay gives elsewhere: a heavy loading on a dull base. The dull base puts its excluded whiteness barely above the control, and the heavy loading puts its included whiteness near the top.

So the headline share and the ranking failure are not two findings. They are the same sample’s base luminance seen twice, which is also the variable the validity-box section flags — the shirt has 9 points of headroom against the coated stock’s 48 for the same reason. One stock is carrying three of this essay’s results, and it is carrying them because it is the only one whose base and loading pull in opposite directions.

That is not a weakness of the finding so much as a statement about which samples the effect is largest on. A brightened sheet on a bright base has most of its whiteness in the base; a brightened sheet on a dull base has most of it in the lamp — and it is the second kind that a buyer is most likely to be shown under a favourable light.

The validity box #

The formula is published with bounds, and they are quoted far less often than the formula is.

40 < W < 5Y − 280 and −3 < T < +3, with the 1931 observer under D65. Outside those the formula was not fitted and is not claimed to rank anything.

The lower bound is easy and the upper one moves, which makes the box a shape rather than a range — the same distinction a tolerance is a shape draws for colour differences. 5Y − 280 depends on the sample’s own luminance factor, so a sheet at Y = 90 has a ceiling of 170 and one at Y = 82 has a ceiling of 130. That is not a detail: the heavily loaded samples are the ones with the most brightener and often the ones with the duller base, so the two effects converge on the bound from both sides.

Among the stocks here the coated press stock has 48 points of headroom and the laundered shirt has 9. Nothing leaves the box, and the shirt is the one to watch — the same sheet under unfiltered daylight rather than under the measurement condition would be well past it.

The tint bound is the other half and it is what stops a brightener being pushed indefinitely. Loading more brightener moves the sample along a fixed direction, and past a point that direction takes it out of ±3 — at which stage the sheet is no longer being ranked as a white at all, and, more to the point, is visibly blue-violet rather than white.

What the scale is measuring is not a reflectance at all, which is the formal version of everything above.

The claim that two of these sheets sit outside the object-colour solid is a claim about an observer, so it is worth asking whether the ten-degree one agrees.

Why there are several whiteness formulae #

The CIE formula is not the only one, and the reason is directly upstream of everything above.

Ganz–Griesser whiteness carries an instrument-specific calibration, precisely so that two instruments with different ultraviolet content can be made to agree on a reference set — an admission, built into the formula, that the number depends on the apparatus. Berger and Stensby whiteness are older linear fits with different coefficients. Paper trade practice in some regions uses reflectance at 457 nanometres alone, which is a single-band proxy chosen because the brightener’s emission is there — and which inherits everything a reflectance cannot say about a fluorescent sheet.

Every one of them is a linear function of a measurement made under an unstated lamp, and every one of them therefore inherits this essay’s problem. The differences between the formulae are small compared with the difference between two lamps.

The exception is any specification that pins the measurement condition, which is what the printing standards did in 2009 and what paper standards increasingly do. That does not make whiteness a property of the sheet; it makes it a property of the sheet under a stated lamp, which is the honest form and the only one two parties can argue about.

The place the sheet is looked at moves the number further than any of the measurement conditions do, and that too can be recomputed against the wider-field observer.

What a usable number would look like #

If a single whiteness figure is a joint property of a sheet and a lamp, the obvious question is what would have to be quoted instead — and the answer is short enough to be a data-sheet line.

Two numbers rather than one. The whiteness under an ultraviolet-excluded condition is a property of the sheet and repeats between instruments and across years. The whiteness under an ultraviolet-included one is what a person under daylight would judge. Quoting both makes the luminescent contribution the difference between them, which is the quantity a buyer actually wants and which nobody currently reports.

And a condition, on both. Without one, neither number is comparable with anybody else’s, and the pair is no better than the single figure it replaced.

That is a small change and it is not the change the industry made. What was standardised instead was the measurement condition, which makes two laboratories agree and leaves a data sheet quoting one number under a named lamp. It is a real improvement and it stops short of the useful thing: the decomposition is available from the same two measurements, costs nothing extra to report, and would tell a buyer how much of what they are paying for depends on where the goods will be looked at.

The reason it is not reported is probably that it is unflattering. A premium stock’s headline figure is 122 and its ultraviolet-excluded figure is 92, and the second is the honest description of the paper.

What was computed, and how #

Whiteness is the CIE formula with the perfect diffuser’s chromaticity taken under the same light as the sample, so each row is computed in place rather than referred back to a fixed white. That matters for the incandescent condition, where quoting against a D65 white would fold the lamp’s own colour into the answer.

The split is the sample’s whiteness above the control’s, computed separately under the two conditions and taken as a ratio. Using the control rather than zero is what makes the ratio meaningful: the absolute whiteness of an unbrightened sheet is 82 and none of it is luminescent, so a ratio taken against zero would report a much smaller share and would be answering a different question.

The assertion is that the mean share exceeds 60 per cent, which is a threshold well below the measured 82 and above anything that could be called marginal. It is written that way rather than at the measured value because an assertion set at exactly what was measured fails the next time a stock is added.

Where the model stops #

The stocks are constructed. Their bases and loadings are chosen to span the commercial range rather than measured from real papers, so the exact 82 per cent belongs to this set. What does not depend on the set is the mechanism: the luminescent term is proportional to the excitation supplied, and the excitation is the lamp’s.

The formula is quoted for the 1931 observer under D65 and used here under D50 and under a tungsten lamp, which is outside its stated conditions. That is deliberate and it is what the industry does; the alternative — refusing to compute whiteness under any lamp but D65 — would make the comparison this essay is about impossible to state.

And nothing here is an appearance model. Whiteness is a colorimetric ranking function fitted to judgements, not a prediction of how a sheet looks in a particular room. What a brightened white looks like after the eye has adapted to it is a different question with a different apparatus.

Who found it, and when #

The CIE whiteness formula dates from 1986 and rests on work through the 1960s and 1970s, principally Ganz’s. That fluorescence made whiteness measurement apparatus-dependent was known before the formula was written — it is why Ganz’s own formulation carries an instrument calibration — so the dependence is not a discovery, it is a documented property that gets dropped when the number is quoted.

The measurement conditions that make it statable arrived in 2009 for printing and have propagated unevenly. Paper data sheets in 2026 still routinely give a whiteness figure with no condition beside it.

What is worth putting a number to is the share, because “whiteness depends on the instrument’s ultraviolet” and “82 per cent of the whiteness scale is the instrument’s ultraviolet” are sentences that produce very different behaviour in whoever reads them.

The generalisation #

The pattern is a scale calibrated on a mechanism that the scale’s users then optimise directly.

The formula was fitted to how people rank white samples. Manufacturers then optimised against the formula, which meant optimising the axis the formula weights, which meant brighteners. The scale is still correctly ordered — sheets with more brightener really do look whiter — but the variance it now reports is almost entirely one mechanism’s, and that mechanism is the one with the strongest dependence on the measuring apparatus.

So the scale became a measurement of the thing it was being gamed with. That is not fraud and it is not even a failure of the formula; it is what happens when a fitted ranking function is used as a target. The diagnostic is to decompose the scale’s variance across the population it is now applied to and ask how much of it is one mechanism — which is a computation anybody can do and almost nobody does.

Where the ladder goes next #

If the condition can change the ranking, the natural repair is to convert one condition’s measurements to the other’s, and the best possible 3×3 cannot — because the correction needed is proportional to how much substrate is showing rather than to what the patch reflects.

The other direction is time. A whiteness figure is a measurement of a sheet on a day, and the brightener is being used up by the light that makes it work — so the number has a half-life and a specification that quotes one without a date is quoting a property of a sheet that no longer exists.