A lamp is two audits at once

Two audits in one round, on the same six lights, and the lights come out ordered differently by each. That is not an inconvenience; it is the clearest available statement of what a lamp’s spectrum decides.

The claim #

A lamp’s exposure to the two audits is decided by two different properties of its spectrum, and the two run in opposite directions with colour temperature.

• The tabulation term follows ultraviolet content, which rises with colour temperature: 0.54 ΔE₀₀ under a 6500 K radiator, 0.07 under tungsten.
• The observer term follows blue poverty, which falls with colour temperature: 1.77 under daylight, 2.37 under tungsten.
• A third property, narrow structure, decides both in a third way — aliasing for the tabulation and the pigment-peak departure for the observer.
• So no single figure of merit describes a lamp for either audit, and correlated colour temperature describes neither.

Three properties, three consequences #

A lamp’s spectrum is an infinite-dimensional object and the two audits between them care about three summaries of it.

How much lies outside the visible band, weighted by the observer’s tails. That is what a truncation costs and it is almost entirely the ultraviolet end, since the observer is identically zero in the infrared and merely small below 380.

How poor the lamp is in the blue relative to the rest. That decides the short-wavelength cone’s relative excitation, which is a ratio of two quantities, and a ratio of two small numbers is what a blue-absorbing filter disturbs most.

How narrow its narrowest feature is. That decides whether the tabulation can represent it at all and, independently, how sharply it samples the cone sensitivity curves.

None of the three is a function of the other two, and correlated colour temperature is a function of none of them. There is no D65 lamp and two lamps at the same temperature can differ arbitrarily in all three.

The anti-correlation, and why it is real #

The first two properties run in opposite directions with colour temperature, and the reason is thermodynamic rather than coincidental.

A blackbody’s spectrum shifts towards shorter wavelengths as it gets hotter. So a hot radiator has more ultraviolet — raising the tabulation term — and more blue relative to red, which improves the conditioning of the short-wavelength ratio and lowers the observer term.

A cool radiator has almost no ultraviolet, so its truncation is nearly free, and it is blue-poor, so its observer term is highest.

That is a genuine trade rather than an accident of the two lights measured, and it holds along the whole Planckian locus. The property that makes a lamp cheap for one audit makes it expensive for the other, and the two effects are of comparable size in the range ordinary lamps occupy.

The consequence is that a collection or a laboratory cannot reduce its total inherited term by choosing a colour temperature. The total is flatter than either component and is roughly two to two and a half colour differences across the range.

The third property, which is not a trade #

Narrow structure is different from the other two because it hurts both audits at once and has no compensating benefit.

For the tabulation it is catastrophic: a feature narrower than the step puts the calculation in the aliasing regime, where the answer depends on where the grid begins and refining does not converge. The fluorescent tube’s step term is 0.83 ΔE₀₀ at the median against 0.06 for daylight, and its origin spread is 3.18.

For the observer it raises the pigment-peak departure, because a narrow source samples the cone sensitivities at points rather than integrating over them. The three-emitter LED’s peaks departure is 2.84 against daylight’s 2.37, and a three-laser projector’s is 3.60.

So a lamp with narrow emitters is worse for both audits, by different mechanisms, with no offsetting property. The two audits agree about narrowness and disagree about temperature, which is a clean summary of everything the round measured about lights.

That has an unwelcome consequence for the direction lighting has moved. Solid-state sources are narrower than what they replaced and the trend is towards narrower still, so the one property both audits punish is the one the industry is optimising for other reasons.

Two independent tests pick out the same property. The origin sweep is the sharpest test for the third property and it is worth using as a diagnostic rather than as a result. A lamp whose computed colour depends on where a five-nanometre grid begins is a lamp with structure the grid cannot hold, and the test takes three lines of code and no knowledge of the spectrum.

The same lamps fail a second test for a different reason. Their pigment-peak departure is high because narrow emitters sample the cone curves at points, and that is measured by a different calculation entirely — two observers rather than two grids.

Two independent tests picking out the same two lamps is what a shared property looks like from two directions, and it is the reason narrowness can be called a property of a lamp rather than an artefact of either audit.

It also gives a cheap screening procedure for anybody assembling a calculation. Slide the grid; if the answer moves, the lamp is narrow, and both audits’ worst terms apply. That is a great deal to learn from one loop.

Which lamp for which purpose #

The three properties give a short and slightly counter-intuitive set of recommendations.

For a collection computing on a truncated grid, a warm source is cheapest — its ultraviolet is negligible and the truncation costs a hundredth of a colour difference. That is the reverse of the usual preference for daylight in colorimetric work.

For a viewing booth, a smooth daylight source is best, because it minimises the observer spread and because the truncation is a property of the calculation rather than of the room.

For a shop or a home, nothing helps: the light is what it is, and the audits’ terms are largest where people actually look at things.

And for anything, a smooth source beats a narrow one of the same temperature on both audits at once. That is the one unambiguous recommendation the round produces about lamps, and it is directly opposed to the efficiency trend.

What a lamp specification carries and what it would need to #

A luminaire is specified by its luminous flux, its efficacy, its correlated colour temperature, its rendering index and increasingly its fidelity and gamut indices.

None of those five is any of the three properties above.

Colour temperature is a chromaticity summarised as a temperature and says nothing about ultraviolet content, blue poverty relative to a blackbody, or bandwidth.

A rendering index is a comparison against a reference of the same temperature over a set of test samples, computed through one observer. It is one observer’s opinion and it says nothing about the spread between observers, which is a different quantity entirely.

Efficacy is a photometric ratio and is if anything anti-correlated with what the audits want, since efficiency is bought by narrowing.

What a specification would need is three numbers: the fraction of the visual product below 380 nanometres, the ratio of blue to total content against a blackbody of the same temperature, and the narrowest emitter bandwidth. All three are computable from a spectrum, none is measured beyond what a manufacturer already measures, and none appears anywhere.

Why the two audits share a set of lights #

The methodological point deserves a paragraph because it is the reason any of this could be said.

The observer audit and the tabulation audit were run on the same six analytic lights and the same forty-two analytic surfaces, deliberately, before either had produced a result. That is a small discipline and it made three things possible that would otherwise not have been.

It let the interaction be computed — the finding that a coarse grid conceals an observer departure entirely, which is the round’s sharpest single result and required both audits on one stimulus.

It let the two terms be compared per light, which is this essay.

And it let the anti-correlation be noticed at all. Two audits on two different sets of lights would have produced two tables that could not be laid side by side, and the trade would have been invisible.

Sharing a stimulus set between audits costs nothing and is the difference between two results and three, which is a cheap lesson and one this collection had not previously written down.

Real hardware sits in four different places. Placing real hardware on the three axes is worth attempting, even roughly, because the constructions are idealisations and a reader has real lamps.

A halogen lamp is a Planck radiator behind a quartz envelope, so it is the tungsten row with slightly more ultraviolet — cheapest of all for the tabulation, worst for the observer, and perfectly smooth.

A daylight-simulating booth lamp is either a filtered tungsten source or a phosphor fluorescent one. The first is smooth and behaves like the daylight row; the second has mercury lines and inherits the narrow-structure penalty, which is a reason to prefer the filtered kind that nobody states.

A modern shop’s LED downlight is a blue pump and one or two phosphors: almost no ultraviolet, so the tabulation is nearly free; one narrow feature at 450 nanometres, which lands on the macular band and raises that departure; and a smooth remainder.

A quantum-dot backlight narrows the green and red as well, which raises the pigment-peak departure and leaves the tabulation alone.

So the four ordinary lamps sit in four different places on the three axes, and no ordering of them survives all three. That is what it means to say a lamp is two audits at once.

What was computed, and how #

The lights are six analytic constructions: two Planck radiators at 2856 and 6500 K, a blue-pumped phosphor LED, a three-emitter LED, a fluorescent tube with mercury lines on a phosphor bed, and a three-laser projector.

The tabulation terms are medians over forty-two analytic surfaces against a tenth-nanometre reference; the observer terms are means over six departures at their literature strengths on a red pigment. Those are different statistics on different quantities and the comparison between them is of magnitudes rather than of numbers.

The claim that the anti-correlation is thermodynamic rather than coincidental follows from Wien’s displacement law and is not separately computed; the two radiators measured are consistent with it.

There is a fourth appearance of the narrowness property and it is the one that explains why its penalty is so large. A tristimulus value is a ratio and the grid appears in both halves of it, so most of a quadrature error divides out — between one and four times on the smooth lamps.

On a lamp with lines narrower than the step there is nothing shared to divide out, because the sample’s sum and the white’s sum miss different lines. So the mechanism that protects every other lamp is absent exactly where it is needed, and the narrow lamps pay their full quadrature error rather than a quarter of it.

One property, four consequences: aliasing, a failed cancellation, a raised pigment-peak departure, and a tabulation that does not converge. That concentration is the reason the round’s recommendation about lamps is as unambiguous as it is.

Where the model stops #

Six constructed lights are not a survey of lamps. A real fluorescent tube has pressure-broadened lines with Lorentzian wings; a real white LED has a phosphor tail; a real daylight is not a Planck radiator and has absorption features the reconstructions carry and these constructions do not.

The three properties are proposed as the ones that matter and are not shown to be sufficient. A lamp with structure in an unusual place — a notch filter over a broadband source, say — could be exposed differently in a way none of the three captures.

And the observer terms are means over six departures, which is a summary that hides which departure a lamp excites. A blue-pumped LED’s mean is unremarkable and its macular departure is the second-highest in the round.

One more consequence follows for how this collection should choose the lights it draws with. Its figure families default to D65 and to blackbody radiators, which is the right default for a collection about colour science and is the expensive default for its own tabulation.

Nothing follows about changing it. The truncation term is half a colour difference, it is comparative in almost every figure and therefore cancels, and switching the collection’s default illuminant to something warmer would be optimising an inherited term at the cost of every argument that depends on daylight being the reference.

What does follow is that any absolute number quoted here under a daylight source carries the term and any quoted under a warm one does not, which is a fact about this collection’s own figures that nothing in their captions says. It joins the other two inherited terms on the list of things a caption could carry and does not.

The generalisation #

The habit is about what a summary of a rich object can and cannot carry.

A spectrum is a function and a lamp specification is half a dozen numbers, so the specification is a projection and the question is which projection. Every summary in use — temperature, rendering index, efficacy — was chosen for a question somebody had, and each is silent about questions nobody had when it was chosen.

The move is to identify which functional of the spectrum each downstream consequence depends on, and to ask whether the existing summaries determine it. Usually they do not, and usually the missing functional is easy to compute and absent for historical reasons rather than technical ones.

The failure mode is to use an available summary as a proxy. A number that correlates with what is wanted is not the number that is wanted, and correlated colour temperature is the standing example: it correlates with everything about a lamp and determines nothing.

A last observation about why the anti-correlation is worth stating rather than merely noticing. A reader who knew only the tabulation audit would conclude that warm light is cheap and cool light is expensive, and would be right about that audit and wrong about the total. A reader who knew only the observer audit would conclude the opposite with the same confidence.

Each of those is a correct conclusion from a complete analysis of one term, and each is misleading. That is the ordinary hazard of an audit that examines one thing carefully: the thing examined becomes the thing that matters, and the trade against everything else is invisible because everything else was held fixed.

The remedy is to run two audits on one set of stimuli, which is what this round did, and the anti-correlation is the payment for having done so.

Who found it, and when #

Wien published the displacement law in 1893 and the temperature dependence of a blackbody’s ultraviolet content follows immediately from it.

The inadequacy of correlated colour temperature as a description of a lamp is thoroughly established and is the reason the CIE has published rendering, fidelity and gamut indices. None of those addresses either of the audits here, because both were unmeasured when the indices were designed.

One last practical note for anybody assembling a lighting specification from this. The three properties are not equally hard to obtain. The narrowest emitter bandwidth is on every LED datasheet; the blue content against a blackbody of the same temperature is one integral from a measured spectrum; and the ultraviolet share requires a spectrum measured below 380 nanometres, which many spectroradiometers do not report because the standards do not ask for it.

So the cheapest of the three to specify is the one that matters for both audits, and the most expensive is the one that matters for only one. That is a convenient ordering and it is worth taking advantage of: a lamp specification carrying nothing but its narrowest emitter bandwidth would be a substantial improvement on one carrying none of the three.

Where the ladder goes next #

Three audits, three subjects, one round. What they have in common is not the subject matter and is worth a final essay: three audits and one shape, which is the round’s real result.