A fourth primary is a design

Three primaries and a white point determine each other. Four do not: there is a family of ways to make the same white out of four emitters, every member of it is an exact match for the observer it was solved against, and the members differ in what everybody else sees.

That much this site has already established, by sweeping one number — how much of the luminance the fourth emitter carries — with the four wavelengths and the four widths held wherever they were first written down. The ends of that sweep differ by 1.26× in the population’s ninety-fifth percentile. It is a real result about a family that somebody happened to pick.

The question underneath it is different, and it is the one a display engineer actually faces. Not which member of this family, but what should the four emitters be.

The claim #

At a fixed gamut, a fourth primary buys agreement rather than colour — and the design that buys the most of it is not the one anybody would choose.

• Designing the four emitters against the population beats choosing them by 6.8×, at exactly the same gamut area: the ninety-fifth percentile falls from 14.8 units to 2.2.
• Held to the same gamut floor, four primaries beat three by 2.6×. The comparison is made with the area constrained, so the extra emitter cannot pay for itself in the currency it is usually sold in.
• The trade has a knee. Agreement is free up to about 1.6 times sRGB’s chromaticity area, costs a little at 1.8, and costs a great deal at 2.0 — and no set of four Gaussian emitters reaches 2.2 at all.
• And what the design does with the freedom is widen the primaries rather than move them, which is the opposite of thirty years of display engineering.

Two objectives that pull apart #

Gamut. Narrow emitters far apart reach more chromaticities. This is the only objective the display industry has optimised for, it is why every generation of panel has narrower emission than the one before, and it is measurable as the area of the polygon the primaries’ chromaticities enclose.

Agreement. Narrow emitters sample the differences between people instead of averaging over them. This site’s own measurement is that narrowing three primaries from forty nanometres to two raises the population’s ninety-fifth percentile from 11.3 units to 17.9, monotonically, with no width in between doing anything unexpected.

Both are consequences of the same property. A narrow band lands on a small part of each observer’s sensitivity curve, so a person whose long-wavelength pigment peaks three nanometres away from the median catches a materially different amount of it; a broad band is integrated across a region where the differences average out. Narrow is good for the corners and bad for the people.

Scoring a design #

A design is four centres, four widths and a share of the luminance for the fourth emitter. Given those, the other three weights are solved exactly — three equations, three unknowns, one linear solve — so that the reference observer sees D65 to floating point. A set that would need a negative weight is not a display and the solver says so rather than returning something.

Two numbers come out. The gamut is the area of the convex hull of the four chromaticities, reported as a multiple of sRGB’s own triangle so the axis means something to somebody who has never designed a panel. The agreement is the ninety-fifth percentile of what a hundred and sixty eyes report about the white, in ΔE00, computed exactly as every other population number on this site is.

The search is coordinate descent with a shrinking step, minimising the percentile subject to reaching at least a stated area. A constraint rather than a weighted sum, because the two objectives are in different units and a weight would be a third free number nobody could defend — and sweeping the constraint is what draws the front.

Coordinate descent is not guaranteed to find a global optimum and nothing here claims it does. What is claimed is checked: the designed display beats the starting one, the four-primary design beats the three-primary one at the same floor, and the front is monotone. The last of those is the interesting check, and it needed a change of method.

A front that has to be monotone, and a search that is not #

A design clearing a high gamut floor clears every lower one, so the best achievable percentile can only rise as the floor rises. That is true of the problem; it is not true of a search run afresh at each floor, which lands in different local minima and reports the difference as though it were the trade.

The first version of the front did exactly that and produced a curve that went down between two adjacent floors. Nothing about that is a discovery — it is the optimiser being better at one problem than at a strictly harder one.

The fix is to walk the floors from tightest to loosest and warm-start each point from the answer above it. Then every reported point is at worst the previous design, the curve is monotone by construction, and the whole front is an upper bound on the real one. That is the honest direction for an inexact search to err in, and it is stated in the code rather than left for a reader to wonder about.

The knee #

The front is flat, then bends. Below about 1.6 times sRGB’s area the constraint does not bind at all: the design that minimises disagreement happens to reach that much gamut anyway, so the extra colour is free. Past it every further increment is bought with observer agreement, and the price rises steeply — 0.78 units at the flat end, 1.20 at 1.8×, 6.82 at 2.0×.

The knee has an exchange rate and it is worth stating in the form a specification could use. Taking the elasticity of the percentile with respect to the area — how many per cent of agreement a per cent of gamut costs — the front reads:

Below about 1.6 times sRGB the elasticity is zero, because the constraint does not bind. From 1.6 to 1.8 it is 3.7. From 1.8 to 2.0 it is 16.5.

So each two-tenths of gamut past the knee costs about four and a half times more than the last, and by two times sRGB a single per cent of extra area is being bought with sixteen and a half per cent of the population’s agreement. That is not a gentle trade with a diminishing return; it is a wall with a short approach, and the approach begins at an area most current wide-gamut panels are already past.

And the elasticity is the number to put in a specification rather than the knee. A knee is a point on a curve fitted to five samples of a search that is itself an upper bound; the elasticity is a slope, it is the quantity a decision actually turns on, and it is robust to the front being a little above the true one because an upper bound’s slope is the right slope wherever the bound is tight.

That knee is the number a display specification should carry and does not. A panel is specified by its primaries’ chromaticities and its white point; two panels meeting the identical specification can differ by several units for half the people looking at them, and the specification has no field in which that could be written down.

The steep part is where wide-gamut displays actually live. Rec. 2020’s primaries sit on the spectral locus, which is the limit of the narrow direction, and the standard was written to be reachable eventually rather than to be reachable comfortably.

What the design does with the freedom #

The starting design is four narrow emitters at 465, 532, 638 and 590 nanometres, all eight nanometres wide, with a fifth of the luminance on the fourth. It is what a person draws when asked for a four-primary display: three corners plus one in the gap where the eye is most sensitive.

The search keeps two of the wavelengths, moves the blue down and the green down, and then does something else entirely with the widths: 29.5 nanometres on the blue, 16 on the green, 15 on the red, and 3 on the fourth, which now carries forty per cent of the luminance.

The reason is that width costs gamut only where the gamut is binding. A broad blue emitter’s chromaticity moves inward along the line toward white, which loses area — but the hull is a quadrilateral, and widening an emitter that is not on the hull’s critical edge costs nothing. The search finds the emitters whose position the area does not depend on and spends their narrowness on agreement instead.

Nobody designing a panel has any reason to notice that, because nobody is measuring the quantity it buys.

Two more points on the same front say what happens further out, which is where the fourth emitter is usually proposed.

Holding both designs to a smaller gamut floor is the comparison a manufacturer with a fixed panel would actually make.

What a specification would have to carry #

A display specification is a list of chromaticities, a white point, a transfer function and a peak luminance. Every quantity in it is a property of the standard observer’s view of the panel, and two panels that agree on all of them can differ by several units for a large minority of viewers.

What would have to be added is one number, and it is already computable from the primaries the specification lists: the percentile of a stated population’s disagreement about the white. It needs no new measurement — the spectra are known to the manufacturer, and the population model is a published set of variates — and it would distinguish two panels that the current document cannot.

The reason it is not there is the same reason a rendering index has no observer in it: the field grew up when there was one observer, the definite article did the work, and the alternative had not been published.

Who found it, and when #

Observer metamerism has been known since colour matching began — Wright and Guild’s seventeen subjects did not agree, and the standard observer is the average of their disagreement rather than a description of any of them. What changed is that it stopped being an academic quantity.

Until about 1990 every display was a phosphor cathode-ray tube and every phosphor was broad, so the differences between viewers integrated away and nobody had to care. Narrowband backlights, then quantum dots, then laser projectors moved the emission narrower each time, each for the gamut, and each generation made the same white a little more contentious. The complaint that arrived with the first laser cinema installations — that different people in the same audience disagreed about the white — is this quantity, reported anecdotally before anybody measured it.

The CIE’s response has been to publish a set of categorical observers and a standard deviate observer, which are ways of turning the spread into a small number of representative eyes. That is the same move this site’s population makes with two hundred, and the arithmetic here would work identically with either.

What was computed, and how #

Every design in play matches the target white to floating point, which assertEveryDesignIsAnExactMatch requires at 10⁻⁹. That is the property that makes the whole comparison meaningful: nothing separating these displays is anything a colorimeter set up in front of them could report.

The population is the site’s own hundred and sixty eyes, differing in lens age, macular density, cone optical density and three pigment peak wavelengths. Each member’s reading is carried into CIELAB through their own view of the white, so the numbers are directly comparable with the disagreement between two people about a display that the earlier sweep reported.

The emitters are Gaussians of stated centre and width, which is a caricature of a real emitter — a quantum dot has a roughly Gaussian emission and a phosphor does not, and neither has a Gaussian’s tails. What the caricature buys is a two-parameter family per emitter, so a design is eight numbers and a share rather than four arbitrary spectra, and the search has something to search.

Where the two designs put their emitters is the part that is easy to guess wrong, because the fourth one does not go outside.

Two designs, side by side #

Two things in that table are worth noticing beyond the last row, and both are unpacked below. The four-primary design ends up with more gamut than the floor asked for. And its narrowest emitter is four and a half nanometres wide, sitting at 622 — the one that has to be narrow, because it is on the hull’s critical edge — while the other three are broader than anything a display would be built with today.

And the comparison in the table is stronger than the sentence above it claims.

The two designs are described as being held to the same gamut floor, and a floor is not an area: the three-primary design lands at 1.40 times sRGB and the four-primary one at 1.58, because the constraint binds from below and the four-primary optimum happened to sit above it.

So the four-primary design does not merely beat the three-primary one by 2.6× at equal gamut. It has thirteen per cent more gamut as well. It dominates on both axes at once, which is a different and better result than a trade at a fixed floor — there is no sense in which the extra emitter was paid for.

What it did with the freedom is a reallocation rather than a relaxation. Comparing the widths emitter by emitter: the red stays at 622 nanometres and goes from 20 nanometres wide to 4.5, four and a half times narrower. The blue goes from 22.5 to 38 and the green from 32 to 40, both substantially wider. The mean width across the emitters rises from 24.8 to 28.

So the design has not made everything broader and paid for it with a fourth emitter. It has made one emitter much narrower and the rest much broader, and the one it narrowed is the one on the hull’s critical edge. Narrowness is being treated as a budget to be spent where it buys area and withdrawn everywhere else — which is exactly the reading the section above gives and which the numbers make quantitative.

That is the sentence a display engineer would want, and it inverts the usual instinct. The question is not how narrow to make the primaries; it is which one has to be narrow. A design with three broad emitters and one very narrow one reaches more of the diagram and divides people less than a design with three moderately narrow ones — and no specification, no measurement and no comparison in current use would tell the two apart.

Where it stops #

The objective is the white point and only the white point. A display makes colours other than its white, and a design optimised for agreement about D65 is not necessarily the design that agrees best about a skin tone or a saturated red. Extending the objective to a set of colours is straightforward and would change the answer; it would also need somebody to decide which colours matter, which is where a defensible optimisation turns into a taste.

The gamut is measured in chromaticity area, which is the wrong measure of how much colour a display reaches and is used here because it is the one specifications use. A volume in a perceptual space ranks primaries differently, and the essay that established that is the reason this one says chromaticity area every time rather than gamut.

Nothing here is about brightness, efficiency, cost, lifetime or manufacturability, and those decide real displays. The claim is narrow: among designs meeting a stated gamut, the differences in observer agreement are large and nobody is measuring them.

Where the ladder goes next #

The obvious extension is five primaries and six, and the interesting question is where it stops paying. Each extra emitter adds two design parameters and one share, and the agreement it can buy is bounded by how much of the population’s disagreement is reducible — a floor set by the fact that two people genuinely see different things, which no display can talk them out of.

Finding that floor would turn the front into something with a bottom, and a front with a bottom is a design brief. The other direction is to run the same optimisation with the observer held at the standard and the population replaced by a set of illuminants, which asks a different question with the same machinery: not which display everybody agrees about, but which display survives being photographed.