There is no D65 lamp

Every display standard written since the 1990s names D65 as its white point. Every viewing booth in a printing works has a lamp in it labelled D65. The first of those is a specification and the second is a claim, and the gap between them is the subject of this essay.

The claim #

D65 is defined by a chromaticity, a chromaticity is three numbers, and a spectrum on this site’s grid has eighty-one degrees of freedom. Matching the definition therefore constrains three of them and leaves seventy-eight free — so a “D65 source” is a family of spectra with almost nothing in common, and how far apart its members are is measurable.

The measurement has a standard shape, and it is not a colour difference. Two sources that agree on a white point by construction cannot be separated by comparing their whites; what separates them is what they do to everything else. So the quantity is a metamerism index: build pairs of reflectances that match exactly under the reference, look at them under the candidate, and report how far they come apart.

What the standard actually pins #

D65 is not a lamp and its temperature is not 6500 K — both facts are already on this site, and the first is usually met as an aside. Stated fully:

• The CIE publishes a spectral power distribution for D65, reconstructed from three measured daylight basis functions at 6504 K.
• Display standards pin D65 by its chromaticity, (0.3127, 0.3290), not by that spectrum.
• No physical source is required to reproduce either. A lamp sold as a D65 simulator is required to be close, and closeness is graded.

The third point is where the interesting arithmetic is, because “close” has to be defined for an object with eighty-one degrees of freedom by somebody who can only measure a few.

Building a source that satisfies the definition and nothing else #

The construction is deliberately crude, because a crude one that passes is the whole argument. Five Gaussian emission bands at 450, 490, 530, 590 and 630 nm, 22 nm wide, with five weights. Three of those weights are used to hit the target chromaticity — two constraints, since chromaticity is two numbers, plus the overall scale that does not matter — and the remaining freedom is spent on nothing in particular.

The solve lands on the target to 8.5 × 10⁻¹⁰ in chromaticity ratios: closer than any instrument in existence could distinguish, and closer than the difference between two units of the same instrument model. Its correlated colour temperature comes out at 6505 K with a Duv of 0.0032, which is D65’s own — the daylight locus is not the Planckian locus, so D65 sits slightly off it too.

Twenty-eight of the eighty-one bands in that source carry less than two per cent of its peak. A third of the visible spectrum is, for practical purposes, missing, and every white-point measurement anybody could make says the source is D65.

What the pairs do #

The test is the one the CIE uses and the reason it uses it: pairs.

The metamer machinery constructs a pair by taking a base reflectance and adding a metameric black — a spectrum that integrates to exactly zero against all three colour-matching functions under the stated illuminant. The pair therefore matches under D65 to machine precision, by construction rather than by fitting, and any difference measured under another source is entirely that source’s.

Five such pairs, under the five-band simulator:

The mean is 4.98. On a grading scale of the shape the CIE uses — A below 0.25, B below 0.5, C below 1, D below 2, E above — this source is a grade E simulator, which is the bottom of the scale, and it is a perfect match for D65 by chromaticity.

How much structure is too much #

The index is not a verdict on the idea of a simulator; it is a measurement of one, and widening the bands makes the source smoother without moving its chromaticity at all:

Every row hits D65’s chromaticity to about a part in a billion. Every row has the same correlated colour temperature to the nearest kelvin. The rows differ by a factor of seven in what they do to a metameric pair, and the only thing that changed is how much structure the spectrum has.

The five rows are a curve rather than five points, and fitting it gives the design rule the sweep is really about.

The index falls exponentially with the band width, halving every 28 nanometres. A fit of the form $8.8\,e^{-w/40.7}$ reproduces all five rows to within eight per cent — and the exponential is the shape one would expect, since a broader Gaussian’s Fourier transform is narrower and the metameric black it fails to integrate away is a wave of stated frequency.

Extrapolating gives the widths each grade needs: about 60 nanometres for grade D, 89 for C, 117 for B and 145 for A.

And that is where the construction runs out. The five bands sit at 450, 490, 530, 590 and 630 nanometres, which is spacings of 40, 40, 60 and 40. A band 117 nanometres wide is three times the spacing between its neighbours; at 145 it is nearly four. Long before either, the five Gaussians have merged into a single broad hump and the source has stopped being a five-band source in any meaningful sense — it is one wide emitter with a slight ripple on it.

So a five-band architecture cannot make a grade A simulator. It can approach grade C by ceasing to be five bands, and grade B by ceasing to be anything but a broad continuum. That is not a limitation of these particular five wavelengths; it is what the exponential means. Every doubling of quality costs 28 nanometres of width, the source has only about 180 nanometres of visible band to spend, and the two run out at each other.

Which is a compact explanation of why real daylight simulators are built the way they are. Filtered incandescent takes a smooth blackbody and removes what it does not want — starting from a continuum and subtracting, rather than starting from bands and widening. Multi-phosphor fluorescent uses a large number of broad emitters rather than a small number of narrow ones. Both are answers to the same arithmetic: the index is bought with smoothness, smoothness is bought with width, and width is bought by having fewer, wider things or by starting with something that has no structure at all.

The extrapolation is worth its caveat. Grade B and grade A sit thirty and sixty per cent beyond the widest width measured, and an exponential fitted over five points does not have to hold there — indeed it cannot, since the index must approach a floor rather than zero, because even a perfectly smooth source that is not D65’s own spectrum leaves some residue. What the extrapolation establishes is not the number but the collision: the widths the grades demand and the spacings the architecture has are the same size, and no refinement of the fit changes that.

The same generator draws the pair of lamps this collection uses elsewhere, and setting a simulator beside them is what says how far a simulator is from an ordinary source.

The two indices measure different things and both are needed. A fidelity index asks how a set of ordinary reflectances renders; a metamerism index asks whether pairs that a standard calls identical stay identical. A source can do respectably on the first and terribly on the second, because ordinary reflectances are smooth and metameric pairs are exactly the samples that are not.

What it costs on real samples #

Metameric pairs are constructed objects, so it is worth asking what happens to ordinary ones. Twelve smooth test reflectances, rendered under D65 and under the 22 nm simulator, with full adaptation to each source’s white:

• mean difference ΔE00 2.38
• worst 5.32

That is not a subtle laboratory effect. It is larger than any paint contract’s tolerance, on smooth samples with nothing tricky about them, under two lights that a colorimeter reports as the same illuminant.

The booth is the instrument #

A viewing booth exists to settle disagreements. Two parties with two samples put them under a named light and decide whether they match, and the whole procedure rests on the light being the same light in both buildings.

What the grading makes visible is that this is a claim about spectra held to a tolerance nobody writes on the enclosure. Two booths from different makers, both honestly labelled D65, both calibrated to a chromaticity, can disagree about a metameric pair by several units of colour difference — and a metameric pair is not an exotic sample. It is what happens whenever a substitute pigment is used to match a specified colour, which is most of what colour quality control is for.

The other half of the booth is the instrument that certified it, and it has its own limit. A spectroradiometer reports the light through a slit of finite width, so a source with narrow structure is measured convolved with the instrument’s own bandpass — and the narrower the simulator’s bands, the more of its structure is smoothed away in the very measurement used to approve it.

What was computed, and how #

The reference. D65 as this site computes it: daylight(6504), the CIE’s three-eigenvector reconstruction, on 81 bands at 5 nm.

The simulator. Five Gaussians of stated centres and width. The weights are solved by a damped least-norm iteration on the two chromaticity ratios $X/Y$ and $Z/Y$, with a non-negativity clamp; the function throws if the solve ends more than 10⁻⁶ from the target, so a figure drawn from it cannot be quietly wrong about the one thing it claims.

The pairs. makeMetamer projects a smooth candidate onto the null space of the 3 × 81 system matrix built from the reference and the 1931 2° colour-matching functions, then scales the result to keep both members inside [0, 1] as reflectances. The residual under the reference is 9 × 10⁻¹⁴ ΔE00, which is the arithmetic’s floor rather than a tolerance.

The index. Each pair’s two members are rendered under the candidate, converted to CIELAB against the candidate’s own white — full adaptation, the most generous assumption — and differenced with CIEDE2000. The mean over five pairs is the index; the grading bands are stated in the code and are of the same shape as the CIE’s rather than identical to them, because the CIE’s are defined on its own five published pairs, which this site does not hold.

The assertion that guards it is written to fail in both directions: it requires the pairs to match under the reference to better than 10⁻⁶, so a broken construction cannot pass, and it requires the index under the simulator to exceed a floor, so a construction that produced no separation would be reported as a failure rather than as good news.

Where the model stops #

The pairs are constructed, not published. The CIE’s metamerism index is defined on five specific metameric pairs, chosen to sample the space of practical failures. The pairs here are built from smooth sinusoids projected onto a null space, so the index computed is of the same shape as the standard’s and is not its value. What survives the difference is the ordering — smoother sources score better — and the order of magnitude.

Only visible-range metamerism is measured. The CIE’s index has a second half for the ultraviolet, and it exists because optical brighteners are in nearly every white paper and fabric: a sample that fluoresces converts ultraviolet into blue, so a simulator with the wrong ultraviolet content mis-renders it whatever its visible spectrum does. This site’s spectra begin at 380 nm, so that half cannot be computed here at all, and every number above is a floor.

And the adaptation is complete. Both members of every pair are judged against their own source’s white. Incomplete adaptation would add a cast on top of the separation rather than removing it.

The observer is the 1931 2° one, and the choice is load-bearing here. A metameric pair is metameric for an observer: the null space it was built from is the null space of one set of colour-matching functions, so the same pair under the same two lights, judged by the 10° observer, does not match under the reference either. The index would then be measuring two failures at once. Every number on this page is under one observer for that reason, and a booth serving a laboratory required to use the 10° functions is being graded against a pair that does not match for it.

The generalisation #

The shape of this argument is the site’s own thesis arriving in the place where standards live.

A specification is a projection, and a projection has a null space. Three numbers pinned, seventy-eight free. Everything that lives in the free directions is invisible to the specification and visible to anything that is not the specification — which, in the case of a light source, is every surface it falls on.

This is the fourth time the same null space has appeared on this site from a different direction, and the four together are the argument for treating it as a governing fact rather than a curiosity:

• The eye’s, where two spectra match because three integrals agree.
• The camera’s, where two colours a person can separate become one file value.
• The press’s, where four inks and three numbers leave a one-parameter family, and every member is a metamer of the others.
• And this one, where a light is specified by what a colorimeter can read of it.

The practical rule that falls out is short. A source is specified by a spectrum or it is not specified. A chromaticity, a correlated colour temperature and a rendering index between them constrain perhaps five of eighty-one numbers, and a supplier meeting all three has agreed to almost nothing about what the light will do to a sample.

Who found it, and when #

The D series was standardised by the CIE in 1964, from Judd, MacAdam and Wyszecki’s principal-component analysis of measured daylight — three basis functions fitted to hundreds of measured skies.

The problem of simulating it in a booth is as old as the standard. The CIE’s method for grading simulators dates from the 1970s and is deliberately built on metameric pairs rather than on a spectral comparison, for a reason worth stating: a spectral comparison would have to say how much disagreement at each wavelength matters, and the answer depends on the samples. Pairs answer that question empirically — they are the samples that expose the disagreement — and the index inherits the same limitation, since a different set of pairs is a different index.

The practical consequence is standing industry practice. Printing and textile colour work specify the booth’s simulator grade alongside the illuminant, exactly because a grade E source and a grade B source are both “D65” and only one of them can settle a dispute about a match.

Where this goes next #

The obvious next question is whether a source could be built that is a good simulator by construction rather than by luck, and the answer is a design problem with a stated objective: minimise the metamerism index subject to hitting the chromaticity. Nothing here does that, and the sweep above shows what the objective would trade against — structure buys efficacy, and the efficacy ceiling is exactly what a broad smooth spectrum gives up.

The sharper question is upstream of lighting entirely. Every specification on this site that names a colour rather than a spectrum has the same null space, and the ones that matter most are the ones nobody thinks of as specifications: a hex code, a profile’s white point, a camera’s target. The next field along asks a related question about the grid those spectra are integrated on, which is a fourth thing a standard leaves unstated and a fourth place a number can move without anybody’s arithmetic being wrong.