Two lamps do not average

A room with two lamps in it is the ordinary case. A warm lamp on the desk and a cooler one in the ceiling, a window and a bulb, a shop’s own fittings and whatever comes through the door — almost no interior is lit by one spectrum, and almost every lighting specification is written as though one were.

The claim #

Mixing light is addition. Every quantity a lamp is specified by is a nonlinear function of the spectrum, so none of them mixes. The chromaticity of the sum is not the average of the chromaticities, the correlated colour temperature is not the average of the temperatures, and the colour rendering of the mixture is not between the renderings of its parts — it can be, and here is, better than either.

Each of those three is a separate arithmetic fact, and all three follow from the same line: the operation on spectra is linear and every reported number is a projection.

Addition is the whole model #

There is nothing subtle about the mixing itself. Two sources illuminating one surface deliver the sum of their spectral power distributions, band by band, and light does not interact with light. So

$$E_{\text{mix}}(\lambda) = w_1 E_1(\lambda) + w_2 E_2(\lambda)$$

with the weights set by how much each lamp contributes. Here the weights are luminance shares rather than power shares, because that is what a lighting layout is drawn in and because it makes the mixture of a lamp with itself exactly itself — asserted, to machine precision, since a mixing function that fails that test is broken in a way no figure would show.

Where the chord goes #

Chromaticity is $x = X/(X+Y+Z)$, a ratio of linear functionals, so a mixture’s chromaticity lies on the straight line joining its parents’ — that much is preserved, and it is the reason a chromaticity diagram is useful at all.

The Planckian locus is not a straight line. It is the curve traced by blackbodies from red heat upward, and it is concave towards the top of the diagram, so a chord joining two points on it lies below it. Every mixture of two Planckian radiators is therefore off the locus, on the pink side, by an amount that vanishes at both ends and peaks in the middle:

A Duv of −0.0064 is not a rounding error. Lighting standards for white sources routinely draw the acceptable band at ±0.006, so an equal mixture of two lamps that each sit perfectly on the locus produces light that would fail the specification neither of them fails.

Two more views of the same pair say what the mixture costs on surfaces rather than on the diagram, which is where a lighting designer would meet it.

It averages in mireds, which is the scale the trade already uses #

The claim that a mixture’s colour temperature is not the average of its parents’ is true and slightly too discouraging, because it is an average of something — just not of kelvin.

An equal-luminance mixture of 2700 K and 6500 K comes out at 3953 K. The arithmetic mean of the two temperatures is 4600, which is sixteen per cent high and is the number an intuition supplies. Converting both parents to mireds — a million divided by the temperature, the reciprocal scale on which equal steps are roughly equal visual shifts — gives 370.4 and 153.8, whose mean is 262.1 mireds, or 3815 K. That is three and a half per cent low.

The whole table behaves the same way. Interpolating linearly in mireds predicts every intermediate row to within 1.3 to 4.0 per cent, always slightly low, against errors of up to sixteen per cent for the kelvin average.

So a lighting designer has a usable rule and it costs one reciprocal. Convert each fitting’s colour temperature to mireds, take the luminance-weighted average, convert back, and the room’s colour temperature comes out within a few per cent. Mireds are already the working unit for lighting filters, precisely because a filter’s effect is roughly a fixed mired shift whatever it is applied to, so nothing new has to be learned — the scale that makes filters additive is the scale that makes lamps additive.

And the residual is the same curvature as the Duv. The mired prediction is low on every interior row and lowest in the middle, which is where the Duv is largest; both are the chord cutting inside the locus, read once as a distance off it and once as a shift along it. The two are not independent errors to be corrected separately — they are one geometric fact projected onto two axes, which is why a mixture that is far off the locus is also the one whose temperature the rule mispredicts most.

That gives the rule its own error bar without any further computation. Where the Duv is small the mired average is good; where the Duv is large it is a few per cent low. A pair of lamps a few hundred kelvin apart produces almost no departure and almost no error; a candle mixed with a north sky produces both, and neither the number nor the whiteness of the room can be predicted from the boxes.

It also sharpens the specification complaint at the end. The essay asks for three numbers no product carries — the mixture’s chromaticity, its distance off the locus, and a rendering measure on the sum. Two of the three are now cheap. The colour temperature of the room follows from a weighted mired average of what is already printed on the fittings, and the departure follows from the same geometry to the accuracy the rule is wrong by. Only the third, the rendering index of the sum, genuinely needs the spectra — and it is the one where the mixture can beat both parents, so it is the one worth having.

What does not become cheap is the direction of the departure. For two Planckian sources the chord always falls below the locus, so the room is always pinker than either lamp, and that much is geometry. For two real lamps that are themselves off the locus in different directions the chord can cross it — the narrowband-and-tube sweep above runs from +0.0175 to −0.0035 and passes through zero — so the sign is a property of where the parents sit rather than of the mixing, and no rule of thumb replaces knowing it.

What the number on the box would have to say instead #

A correlated colour temperature is the temperature of the Planckian radiator nearest the source in the 1960 diagram, and “nearest” is a projection: it throws away the distance. That is the whole reason Duv exists as a second number, and this site has drawn the perpendicular before for single lamps sold on their temperature alone.

What the mixture adds is that the departure can be manufactured by the installation rather than by any lamp in it. A specification naming a colour temperature and a rendering index for each fitting has no field in which to write down what the room ends up with, and the room is what people are in.

The rendering index is not between #

The third consequence is the surprising one, and it needs the machinery rather than the geometry.

A fidelity index is a set of colour differences: render a set of test reflectances under the source and under a reference of the same correlated colour temperature, adapt both, and measure how far apart the pairs land. Colour differences do not add, the reference moves with the mixture’s own temperature, and the samples that suffer under one source are not the samples that suffer under another. Nothing in that construction obliges the mixture to score between its parents.

Measured on a mixture of a narrowband source and a triphosphor tube:

The best mixture beats both parents by more than a point. The mechanism is simple once stated: two sources with holes in different places fill each other’s in. A narrowband source has nothing to return in the deep red and a phosphor blend has a trough where the tube’s mercury lines are not; adding them leaves a spectrum with fewer places for a saturated sample to fall into.

The same sweep also crosses the locus: its Duv runs from +0.0175 at the narrowband end to −0.0035 at the tube end, passing through zero at about an 80% share. So among the mixtures of two off-locus lamps there is exactly one that is on the locus, and it is not either parent.

What it does to a surface #

The abstraction has a consequence anybody can see in a shop, and it is worth putting a number on.

Take a mid-red reflectance and compute its colour under each light, adapting fully to that light’s own white — which is the most generous assumption available, since it grants the visual system perfect constancy. Under the 2700 K lamp and under the 6500 K lamp it differs by ΔE00 3.61: not a match by any tolerance in industry. Under the equal mixture it sits 1.88 from the first and 1.79 from the second, which is the honest description of what a two-lamp room does — it makes everything a compromise between two colours the surface has, rather than either of them.

And the visual system makes it worse by being good at its job. Constancy works by estimating a single adapting white for the scene, so a room with two lights produces one estimate that suits neither, and everything is slightly wrong everywhere instead of clearly wrong in one place.

What was computed, and how #

Every number above comes from four steps and no tables.

The spectra. The two parents are Planck’s law at 2700 K and 6500 K, evaluated on this site’s 81-band grid from 380 to 780 nm. The real-lamp pair is this site’s own constructed narrowband source and triphosphor tube — emission lines and phosphor bands at stated positions, not measurements of any product.

The mixture. Each source is scaled so its luminance under the CIE 1931 2° observer is its share of the total, and the two are added band by band.

The temperature and the distance. cct() converts the mixture’s white point to the 1960 UCS diagram and minimises distance to the Planckian locus by golden-section search over temperature. The distance at the minimum, signed by which side of the locus the point falls, is Duv.

The index. fidelity() renders twelve constructed test reflectances under the mixture and under a reference illuminant of the mixture’s own correlated colour temperature, converts both through CIECAM16 to CAM16-UCS, and averages the distances. It is not CIE Ra: the CIE’s eight test samples are tabulated measurements this site does not hold, so the samples here are smooth reflectances of stated form and the index is one of the same shape rather than the standard’s value.

The assertions that guard all this are the interesting part. Three are positive — mixing a lamp with itself changes nothing, a mixture of two Planckian sources leaves the locus, a mixture can render better than both parents — and each is required to fail when handed a case it should reject: two identical lamps, which cannot leave a locus they are already on, and an impossible floor on the departure.

Where the model stops #

There is no geometry here. Two lamps in a room deliver different amounts of light to different surfaces, at different angles, and a mixture with fixed weights is a point in a room rather than a room. The essay’s numbers are what a surface receives where the two contribute equally, and that surface is somewhere specific.

There is no interreflection. A bounce is a multiplication, so light that has already touched a wall arrives carrying that wall’s reflectance, and in a coloured room the actual mixture has more terms than two. That argument belongs to the scene field and it makes every number here a floor.

The constructed lamps are constructions. A ranking between two of them would be a fact about the constructions, which is why the fidelity result above is stated as a mixture can beat both parents rather than as a claim about any product.

And the adaptation is complete. Computing a surface’s colour under each light with full adaptation to that light is the most favourable possible assumption; incomplete adaptation, which is what CIECAM16 actually models, leaves a residual cast that makes the disagreement larger rather than smaller.

What the pictures cannot show #

Three things, and the first is the one a reader is most likely to want.

None of these figures shows the room. A chord on a chromaticity diagram is a set of white points, and a white point is not an appearance — the pink of a Duv of −0.0064 is a cast on everything, visible by comparison and nearly invisible in isolation, because constancy discounts a single illuminant well and a mixture badly. A figure that showed it convincingly would have to show two halves of one scene at once, and the page has one illuminant of its own.

The fidelity sweep is a mean of twelve numbers. An average is exactly the wrong summary for a failure concentrated in one part of the hue circle, which is what a narrowband source’s failure is, and the mixture that scores best is not necessarily the one that renders any particular sample best.

And the lamps are line drawings of lamps. The constructed spectra carry mercury lines because mercury emits at those wavelengths and phosphor bands because phosphors are broad, and they are faithful in structure rather than in detail. Every claim here is about what the structure implies, which is why the results are stated as a mixture can rather than as this product does.

The chord and the locus are both drawn in a diagram defined by an observer, so it is worth confirming that the gap between them is not an artefact of which one.

The generalisation #

The pattern here is not about lamps. It is the standing hazard of any specification written in derived quantities:

A linear operation on the underlying object becomes an unpredictable operation on the numbers reported about it. Spectra add; chromaticities project; temperatures are a nearest point on a curve; indices are averages of distances after an adaptation. Each layer is a defensible summary of the one below it, and each loses exactly the property that would have let the summaries be combined.

The same shape has already appeared twice on this site from other directions. A separation is not unique because a four-ink press has one more control than a colour has numbers, so the specification cannot see which member of the family was used. A camera’s matrix is a fit whose error depends on a sample set nobody quotes. In every case the specification is not wrong; it is incomplete in a way that closes over the operations people actually perform.

The practical form for lighting is one sentence: a specification that names a colour temperature per fitting has specified nothing about the room. What would be needed is the mixture’s own chromaticity, its own distance off the locus, and a rendering measure computed on the sum — three numbers that no product carries because no product is a room.

Who found it, and when #

The additivity of light is Grassmann’s, stated in 1853 as the laws that make colour a vector space, and the whole of colorimetry is built on it.

Correlated colour temperature is Judd’s, in the 1930s, with the projection made well defined by the 1960 UCS diagram — which survives for that purpose alone. Duv followed as the admission that a projection had thrown something away, and it is quoted in standards and almost never on packaging.

The colour rendering index dates from 1965 and its reference-illuminant construction from the same period, including the discontinuity at 5000 K where the reference changes from a Planckian radiator to a reconstructed daylight. The observation that mixtures can outperform their components is old in the lighting industry — multi-channel luminaires are sold on it — and it is usually presented as a design achievement rather than as a fact about the arithmetic, which is what it is.

Where this goes next #

Two directions, and the second is the harder one.

The first is the D65 simulator, which is the same argument run backwards: instead of asking what a mixture does to a specification, it asks what a specification permits — and the answer is any spectrum at all, provided it lands on three numbers.

The second is the room. Everything here is a mixture with stated weights at one point in space, and the actual object is a field of mixtures whose weights vary from wall to wall and whose components have already bounced. That is the scene field’s machinery, and running a two-lamp installation through it would produce a map of colour temperatures rather than a number — which is what a lighting designer is really specifying, and what no current standard has a place to write down.