White is a region

A lamp does not arrive with a chromaticity on it. It arrives with a number — 2700, 3000, 4000 — and what that number means is not this lamp’s colour temperature but this lamp’s chromaticity fell inside a quadrangle drawn around a nominal one.

The quadrangle is a range of correlated colour temperature crossed with a range of distance off the Planckian locus. Two lamps out of the same box can sit at opposite corners of it, and the distance between those corners is larger than any tolerance anybody would accept between two panels of a car.

The claim #

A nominal white is a region, its corners are far apart, and almost the whole of the difference disappears once anybody has been in the room for a minute.

Three measurements on the 4000 K bin, at an ordinary eight-step tolerance — a five and a half per cent spread in reciprocal temperature and a Duv range of ±0.006:

• The bin spans 3,791 to 4,233 kelvin. Four hundred and forty-two kelvin, on a lamp labelled with one number.
• Its worst corner pair differs by ΔE00 11.26, judged against the bin’s own centre as the adapting white. That is not a subtle difference; it is the distance between two paints nobody would accept as a match.
• And in a room, the same twelve surfaces differ by 0.73 CAM16-UCS units on average and 1.04 at worst between the two temperature ends of the bin, with the observer adapted to each. A unit is about a just-noticeable difference for a surface.

Eleven units on a shelf, one unit in a room. Both are real, both are computed the same way, and almost everything anybody believes about lamp colour comes from the first while almost every lamp is used in the second condition.

Why the corners are far apart #

The two coordinates of a bin are not commensurable, and the second one is the one nobody reads.

Along the locus the bin is stated in reciprocal temperature — equal steps in mired rather than in kelvin — because a hundred kelvin at 2700 K is a visible colour difference and a hundred kelvin at 6500 K is almost nothing. That convention is right and it is why the warm bins are narrow in kelvin and the cool ones wide.

Off the locus the bin is stated as Duv, the perpendicular distance in the 1960 diagram. A lamp at +0.006 is visibly green and one at −0.006 is visibly pink, and both carry the same number on the box — which is why a lamp sold as a D65 simulator is not D65 and why nothing about the label says so. This is the coordinate the whole trade knows about and the specification treats as secondary, and it is worth more of the difference than the temperature range is:

The two axes are not perpendicular where it matters #

The corners are explained above as the diagonal of a rectangle being longer than its sides, and that is right as far as it goes. The table says something more specific, and it is the part a standards committee could act on.

The corners are further out than a rectangle allows. A displacement of 5.18 along one axis and 8.92 along the other, if the two were at right angles, would put the corner at 10.31. It is at 11.26, nine per cent beyond. The other corner is at 10.60 against an orthogonal 10.18, four per cent beyond. Working the angle back out of the two: the temperature displacement and the Duv displacement sit 77 and 85 degrees apart rather than at ninety.

The two coordinates are perpendicular by construction — that is what the 1960 diagram is retained for, and the perpendicular is exactly what Duv means. They are not perpendicular in ΔE00, because the space the tolerance is judged in is not the space the tolerance is drawn in, and a right angle is not preserved between them.

That has a consequence the table only half shows. If the two axes are partly aligned, then two of the four corners are long diagonals and the other two are short ones — and only the long pair is quoted. Working the same angle through the other combinations predicts warm-and-pink at about 9.1 and cool-and-green at about 9.3, against 10.2 and 10.3 if the axes were square. A bin is not a rectangle with four equal corners; it is a parallelogram with a long diagonal and a short one, and the difference between the two is about two units — twice the size of the entire room result this essay is otherwise about.

The Duv range is worth 1.7 times the temperature range #

The second finding in the same table is a number a committee can use directly. Dividing the Duv corners by the temperature corners: 8.92 against 5.18, and 8.75 against 5.21 — a factor of 1.72 and 1.68.

So the two halves of a bin are not balanced. The ±0.006 permitted off the locus costs about seventy per cent more than the ±5.5 per cent permitted along it, and the label carries only the second.

That is a stronger statement than Duv matters and nobody reads it, which is where the essay leaves it, because it says by how much and therefore what would fix it. To make the two axes cost the same, the Duv half-range would have to be ±0.0035 rather than ±0.006 — a tightening of forty-two per cent on one coordinate, with the other left exactly as it is.

Whether that is the right trade is not a question this arithmetic can answer; a wider Duv range may be what makes the bins manufacturable at all, since a phosphor’s thickness varies across a wafer in a direction that moves a diode off the locus rather than along it. What the arithmetic does establish is that the imbalance is a choice rather than an accident of the units, and that a bin drawn to be perceptually square would look nothing like the ones in use.

Both findings point the same way, and it is the way the tolerance-shape essay points: a tolerance drawn as a box in coordinates chosen for measurement, and judged in a space chosen for perception, is loose in the directions where the two disagree and tight where they happen to line up. Here the disagreement is a thirteen-degree lean and a factor of 1.7 in scale, and the two together put the worst corner twenty-two per cent further from the centre than the nearest one.

The corners are worse than either edge, which is what a rectangle in a roughly uniform space does: its diagonal is longer than its sides. That is the same geometric point a tolerance box makes against an ellipsoid, arriving in a specification written eighty years after MacAdam measured the ellipses it should have been shaped like.

And why the room does not care #

The second number needs its own paragraph, because the first one is so large that it invites the wrong conclusion.

Two lamps at the two temperature ends of the 4000 K bin are 442 kelvin apart. Put twelve surfaces under each, adapt an observer to each lamp’s own white, and compute what those surfaces look like: the mean difference is 0.73 CAM16-UCS units and the largest is 1.04. That is at the edge of noticeable and nowhere near the eleven units the two whites differ by when seen side by side.

The reason is chromatic adaptation, and it is the single most powerful mechanism in the subject. An observer in a room adopts the room’s white as their reference and judges everything against it, and a whole-room change of illuminant is very nearly the change adaptation is best at cancelling.

So the bin is a specification about the wrong condition. It constrains how two lamps compare when both are visible at once — a shelf, a fitting with two bulbs in it, a corridor where one has been replaced — and the reader’s actual condition, one lamp in one room, is the one where the bin is loosest and matters least.

What a bin does not specify at all #

The measurement above uses only the two temperature ends of the bin, and the reason is a finding rather than a convenience.

A bin does not specify a spectrum. Its temperature corners can be given real spectra, because a correlated colour temperature on the locus is a Planckian radiator and Planck’s law is a function of one variable. Its Duv corners cannot. A stated distance off the locus is satisfied by infinitely many spectra, and those spectra render surfaces differently by any amount at all — a narrowband source and a broad phosphor can share a chromaticity exactly and disagree about a surface by tens of units.

So the 0.73 above is a lower bound on what a bin permits, and it is quoted as one. The bin constrains chromaticity and nothing else; the rendering is a separate specification, on a separate axis, with a separate number, and the two are printed on the same box in a way that invites reading one as a summary of both.

What a bin would have to say instead #

Written out, the missing fields on a lamp’s label are not exotic. Each is measurable, each is already measured by the manufacturer, and none of them appears:

• Where in the bin. A lamp reported as 4000 K, Duv +0.004 costs nothing extra to state and is the difference between two lamps matching on a shelf and not.
• Which observer. A chromaticity is an integral against a kernel somebody chose. The 2° and 10° functions put the same lamp at different points, and a bin boundary drawn under one is a different boundary under the other — which matters most for exactly the narrowband sources bins were introduced for.
• The spectrum, or a proxy with more than one number in it. The rendering index is one number about a twelve-dimensional failure, and the whole content of what a lamp cannot give back is that one number hides where the failure is.
• And the drive. Nothing on any lamp’s label says whether it is switching on and off a hundred times a second, which is invisible standing still and unmissable on anything moving.

Three of those four are already inside the manufacturer’s measurement file. The fourth is a property of the driver rather than the lamp, and is the only one that would cost anything to add.

What was computed, and how #

The bin’s corners are constructed and then measured back. A chromaticity at a stated temperature and Duv is built by finding the Planckian point at that temperature in the 1960 diagram, constructing the perpendicular there, stepping along it, and converting back. assertTheBinIsWhereItWasPut then runs the site’s own correlated-colour-temperature search on each corner and requires it to recover the temperature to one per cent and the Duv to 5 × 10⁻⁴. That round trip is the check that a superseded diagram has been used correctly in both directions.

The perpendicular’s orientation was wrong first. There are two perpendiculars to the locus and the sign convention is that positive Duv is green. The first version took whichever one the tangent’s rotation produced, which is green at some temperatures and pink at others, and the round-trip assertion caught it immediately by reporting −0.006 where +0.006 had been asked for.

The side-by-side comparison uses the bin’s own centre as the adapting white, which is the fairest reading of two lamps on a shelf under the shop’s own lighting and the only choice that does not make one of the two the standard by construction.

And the adapted comparison uses this site’s twelve test reflectances, which are not the CIE’s samples and do not claim to be — smooth raised-cosine bumps on a grey pedestal, spanning the hue circle at moderate chroma, stated as a construction wherever they are used.

Where the model stops #

The bin’s numbers are the ordinary ones and not a particular standard’s. A five and a half per cent mired spread and ±0.006 Duv is the shape the binning standards take; particular standards differ in both, and every result above is stated as a function of the two rather than as a property of any document.

A real bin is a quadrangle, not a rectangle. Binning standards draw four-sided figures whose sides are isotemperature lines and Duv contours, which are curved in the chromaticity diagram; the corners here are the four combinations of the extreme temperatures and Duvs, which is the same object to the accuracy anything here needs.

The adaptation is complete in the model’s own sense and not in fact. CIECAM16’s degree of adaptation is below one at every light level, which is in the computation; how completely a person adapts also depends on how long they have been there, and that clock is a separate machinery that has not been joined to this.

And no lamp in this essay flickers, which is the axis the essay beside this one adds and which no bin mentions either.

The generalisation #

The sentence worth carrying: a specification that constrains a chromaticity has said nothing about a light.

That is the same sentence the rendering essays arrive at from the other direction, and putting the two together gives the shape of what a lamp’s label actually is: two projections of an eighty-one-dimensional object onto two axes, with a tolerance on each, and no constraint at all on the seventy-nine directions left over.

The surprising connection is with the adaptation result. The eleven-unit corner-to-corner difference is a real number about a real pair of lamps, and it is almost entirely absorbed by a mechanism nobody designed the specification around. So the bin is simultaneously too loose — eleven units is enormous — and irrelevant, because the condition it constrains is not the condition lamps are used in. A tolerance can be both, and it is not obvious from either number alone.

The condition where it does bite is worth naming: replacing one lamp in a fitting. Two lamps of the same nominal temperature, side by side, one new and one three years old, is exactly the shelf comparison — and it is the commonest complaint about lamp colour there is.

Two more families say that a bin is drawn around a point in a space where the lamps are not points.

Who found it, and when #

MacAdam’s ellipses are 1942, and they are the measurement a chromaticity tolerance should be shaped like. That lamp binning is specified with quadrangles instead is a practical decision — a quadrangle is easy to test a measurement against and an ellipse is not — with the cost this essay measures.

Correlated colour temperature and Duv were formalised in the 1930s and 1960s respectively, on a diagram that was superseded in 1976 and is retained for this purpose alone. The isotemperature lines are only perpendicular in the 1960 diagram, so nearest point on the locus is only well defined there.

The binning practice is much newer than either, and belongs to solid-state lighting: incandescent lamps did not need bins because a filament’s colour is set by its temperature, and a diode’s is set by a phosphor thickness that varies across a wafer.

What has not changed is what a customer is told. A lamp carries one number for its white, one for its rendering, and nothing at all for its position inside the tolerance either of them names.

What the pictures cannot show #

They cannot put two lamps in a room. Every swatch on this page is a patch on a screen seen under whatever light the reader is sitting in, which is a third illuminant neither of the two being compared. The eleven-unit figure is a computation about two lights, drawn as two colours.

And they cannot show the adaptation. The one-unit result is a statement about an observer who has been under each lamp long enough to settle. A figure showing two versions of a scene side by side shows neither condition: the reader adapts to the page, not to either lamp, and sees a difference that is neither the shelf comparison nor the room one.

Where the ladder goes next #

The nearest unfinished piece is the bin’s shape. Nothing here compares the quadrangle with an ellipse of the same area, and the comparison would give the number a standards committee would want: how much of what the bin permits is outside a seven-step ellipse, and how much of the ellipse the bin excludes.

The second is the replacement case, which is the one that generates complaints. Two lamps in one fitting, one at each end of a bin, seen simultaneously and adapted to their mixture — which is a third white neither of them is, and which does not sit on the locus. All three pieces exist and nothing has put them together.