The green primary decides how sharp the blue is

A gamut margin moves each corner of a display’s colour solid inward, and how far it moves a corner, per unit of margin, says how sharp the corner is in the unit the margin is written in — the measure a margin costs a press its corners introduced on a printing press and a tolerance unit rounds the corners carried into the unit a print buyer writes. A quantiser keeps the blue a spike found ΔEITP — the difference built for high-dynamic-range television on the ICtCp encoding — making a display’s blue its sharpest corner, about three times a margin, where an appearance model flattens the blue by its hue found CAM16-UCS making it nearly a face. The account was that a corner is sharp when the steps that move it inward point nearly the same way, and that the two primaries that can be added to a blue both raise its L and M signals and leave its S alone.

A violet primary is no spike to the quantiser tested that account by moving a monochromatic blue primary from 490 nm to 440, red and green held at BT.2020’s 630 and 530. The spike peaked at 485 nm and faded towards the violet, because of one asymmetry between the two added primaries. Adding red always raises a blue’s L signal by the larger fraction. Adding green does so only while the blue carries enough M signal of its own; a violet primary carries almost none, so adding green raises its M by the larger fraction, and from about 454 nm down the two steps point opposite ways along ICtCp’s L–M axis.

That turn depends on two ratios: the blue’s own L to M, and the green’s. The essay’s closing section predicted that the turn follows the green — a 545 nm green, carrying relatively more L, keeps adding green L-ward down to about 445 nm, while a 515 nm green turns it near 470 and dulls even BT.2020’s own blue. If the turn did not move, the blue’s own ratio would decide it alone, and a display maker choosing a green would have no say in what a ΔEITP margin costs at the blue.

The turn follows the green, and the green sets the spike #

With the red at 630 nm and a monochromatic green moved from 505 to 555 nm, the blue at which adding green’s step crosses ICtCp’s L–M axis falls from 466 nm with a 505 nm green to 461 with a 515 nm green, 454 with the census’s 530, and 446 with a 545 nm green; with a 555 nm green adding green is L-ward for every blue from 440. The sharpest blue is the 485 nm one for every green, and its corner moves 2.39 times a margin against a 505 nm green, 4.37 against 530 and 8.19 against 555. Moving the red from 615 to 690 nm, green held at 530, leaves the turn at 453.70 nm to within a hundredth and the sharpest corner within a twentieth. CAM16-UCS’s 485 nm corner rises too, from 1.16 to 1.92, under half of ΔEITP’s at every green.

• The prediction holds in direction and overshoots in size at the short end. A 515 nm green moves the turn seven nanometres from the census’s, not sixteen; a 545 nm green moves it eight, as predicted.
• The green decides the height, the blue decides the place. Every green leaves the spike at 485 nm and scales it by a factor of more than three across the range.
• The red is a bystander. It points L-ward for every blue, and how far L-ward hardly matters.

The five curves are the same sweep of blues against five greens, and they share a shape. Each rises from the violet, peaks at 485 nm and drops a little at 490; each is the census’s curve stretched upward. At 440 nm they span only 1.18 to 1.57 times a margin; at 485 nm, 2.39 to 8.19. The violet end of every display is blunt whatever the green, and the long blues are as sharp as the green lets them be. A display maker who chooses a 545 nm green over a 530 nm one has bought a blue corner half again as sharp in ΔEITP, ×6.29 against ×4.37, without moving the blue at all.

Where the turn falls #

The turn moves monotonically with the green — 466, 461, 456, 454, 451, 446 nm for greens at 505, 515, 525, 530, 535 and 545 — and the prediction’s two points sit on either side of the curve. At the long end it is right: a 545 nm green turns at 446 nm, predicted 445. At the short end it is too generous: a 515 nm green turns at 461, not 470. The curve is nearly straight: the turn moves about half a nanometre for every nanometre the green moves, between 0.49 and 0.55 in every interval from 505 to 545. The prediction’s long end assumed that rate; its short end assumed twice it.

The dashed line beside it is the turn read the other way: not by where adding green’s step crosses ICtCp’s L–M axis, but by where adding green raises the blue’s L and M signals by the same fraction. The two lie within two and a half nanometres of each other at every green, the axis crossing slightly later for all but the 545 nm green. The turn is a ratio crossing, and the ICtCp encoding’s quantiser and rotation shift it only a little — which is the account’s claim, tested where it could have failed.

Why the green moves it #

Adding a little green to a blue raises each of its cone signals by the green’s own signal in that cone, and the fraction by which it raises each is the green’s signal over the blue’s. Whichever cone the blue has least of, relative to the green, gets the larger fractional rise. A violet primary has a little L and almost no M; a green at any wavelength carries plenty of both; so adding green raises the violet’s M, of which it has almost none, by a larger fraction than its L. A 485 nm blue carries a good deal of M, and adding green raises its L by the larger fraction instead.

Where the two fractions cross depends on the green’s own L to M. A green at 505 nm sits close to the M cone’s peak and carries more M than L by a wide margin; every blue must carry more M of its own before L can win, so the crossing moves towards the longer blues. A green at 555 nm carries more L than M, close to the L cone’s peak, and even a violet primary’s small M cannot outpace the L that green brings; the crossing leaves the range at the violet end. With the handle at 555 the two curves never meet; at 505 they cross at 464 nm.

The prediction’s short end overshot by assuming the turn would move faster on that side, as if a shorter green changed the balance more than a longer one. The census finds an even rate on both sides. Both ratios that meet at the turn change with wavelength — the green’s, and the blue’s that it is compared against — and the crossing moves as their combination does; why that combination comes out so close to linear over fifty nanometres of green is a fact the census reports and does not explain.

Why the green sets the height #

The same ratio sets the angle between the two added-primary steps, and the angle sets the sharpness. At the 485 nm blue, adding red and adding green lie 65 degrees apart with a 505 nm green and 21 degrees apart with a 555 nm one; at 440 nm, 127 and 79. The longer the green, the more its step points L-ward like the red’s, the more nearly the corner’s inward directions coincide, and the further a margin must push the corner before every one of them is satisfied.

The census’s rank correlation between angle and sharpness was −0.98 along the blues; across the greens the same relation holds at the 485 nm blue, every narrower angle a sharper corner. What the angle cannot explain is why the spike stays at 485 nm for every green. At each green the angle is narrowest at 480 to 485 nm and widens again at 490; the blue that brings adding green’s step closest to adding red’s is a property of the blue’s own L, M and S, and moving the green rescales the angles without moving their minimum. The green decides how high the spike stands; the blue decides where it stands.

The red moves nothing #

The red primary is the natural control: a third primary that also feeds the blue’s L and M, moved across a range as wide as the greens’. From 615 to 690 nm it leaves the turn at 453.70 nm to within a hundredth of a nanometre and the sharpest corner between ×4.27 and ×4.48. The turn is untouched because it is defined by adding green alone. The height barely moves because adding red raises the blue’s L by a much larger fraction than its M for every red in the range — a red carries almost no M — so its step points along ICtCp’s L–M axis whichever red it is, and only the green’s step swings.

A display maker’s three choices, then, are not equal choices at the blue corner. The blue primary’s wavelength decides whether the corner is a spike at all and where it peaks, the green decides how sharp it is and where it stops being one, and the red, within the range a display would use, decides almost nothing. A narrow primary buys a disagreement found the primaries’ widths setting how far observers disagree about a display’s colours; at the blue corner in ΔEITP their positions matter unequally as well.

The appearance model follows, gently #

The appearance model was the unit that kept the blue a face, and it does not entirely ignore the green. CAM16-UCS’s 485 nm corner moves 1.16 times a margin against a 505 nm green, 1.30 against 530 and 1.92 against 555 — a rise of two thirds where ΔEITP’s more than triples. At every green it moves under half as far as ΔEITP’s, and the gap widens with the green: at 505 nm the two units disagree by a factor of two, at 555 by more than four.

That gap is the census’s earlier finding at another scale. CAM16-UCS’s hue scale among the blues spreads the two added-primary steps apart, as the appearance-model essay found, and a longer green closes them less in its space than in ICtCp’s. Two tolerance units that disagree about a display’s blue by a factor of two disagree by four on a display with a long green, so a gamut-mapping margin chosen in one and checked in the other will look generous or harsh depending on the green as much as on the blue.

What a display maker should take from this #

A margin written in ΔEITP costs a long-green display far more at the blue than a short-green one. With the blue near 470 to 485 nm, where most displays put it, a 545 nm green makes the blue corner move half again as far per unit of margin as a 530 nm green, and a 515 nm green two thirds as far. A colour engine that keeps colours a fixed number of ΔEITP units inside a display’s blue corner will carve away more of the blue on the first display than on the second, for no reason any viewer would recognise.

The violet escape works with any green. With the blue at 440 nm the corner stays under 1.6 times a margin for every green in the census, and at 445 nm under 1.5 for every green to 545, because the violet’s lack of M splits the two added-primary steps however L-heavy the green. A display that wants a blue corner ΔEITP treats as a face can have one by moving the blue, not the green — at the costs the violet essay counted elsewhere, in CIELAB and ΔE₀₀, which run the other way, and where a display’s blue is sharp where the rotation is not had already found a display’s blue corner sharp.

How the displays were built and read #

Each display has three monochromatic primaries — chromaticities read off the CIE 1931 colour-matching functions at each wavelength — balanced to D65 at 320 cd/m². The red sits at 630 nm, or at 615, 645, 660 or 690 for the control; the green at 505, 515, 525, 530, 535, 545 or 555; the blue at every five nanometres from 440 to 490. With the green at 530 and the red at 630 the construction is the violet census’s to a part in a billion.

At each blue corner the three inward directions — adding red, adding green, dimming the blue — are carried into ICtCp through BT.2020’s matrices, the perceptual quantiser and ICtCp’s rotation, and into CAM16-UCS under an average surround. A corner’s move per unit of margin is the distance from the corner to the nearest point at least one unit inside every face of the cone its three directions span, as in the earlier corner essays. The turn is the blue, interpolated between the census’s five-nanometre steps, at which adding green’s unit step changes sign along ICtCp’s L–M axis; the second reading is where adding green raises L and M by equal fractions.

What this does not reach #

Monochromatic primaries. Every primary here is a single wavelength. Real displays’ primaries have widths — a few nanometres for a laser, twenty to forty for quantum dots and organic emitters — and a broad green’s L to M is an average over its spectrum.

One white and one peak. The white is D65 at 320 cd/m². The violet essay found a brighter display sharpening the long blues and dulling the violet; how the green’s effect scales with brightness is not computed.

A corner, not a gamut. The margin cost at one corner says nothing about how a longer green changes the cost along the blue–cyan edge, where the green and blue primaries mix.

Still open: whether a broad green keeps the turn #

A display’s green is not a line. A quantum-dot green twenty-five nanometres wide and an organic-emitter green forty wide carry, at the same peak, a mixture of the L to M ratios on either side of it — and because the ratio changes faster on the long side of 530 nm than on the short, a broad green’s ratio is not its peak’s.

The calculation is this census with the green and blue primaries given Gaussian spectra two, twenty and forty nanometres wide at the same peaks, red held narrow at 630 nm, the turn and the 485 nm blue’s height read as here. The prediction is that widening the green moves the turn towards the violet and raises the spike, as a longer green does, because the long side of a broad green’s spectrum carries more L than the short side removes; and that widening the blue barely moves the turn, because the blue’s L to M changes slowly near 470 nm. If a broad green lowers the spike instead, the census’s monochromatic primaries overstate how sharp a real display’s blue is in ΔEITP, and the margins quoted for line-primary displays are not the margins for real ones. A broad green sharpens the blue, a broad blue dulls it makes that calculation.

Which knob a result belongs to #

The habit here is about where to look once a mechanism has been located.

The violet essay found the spike’s turn in a crossing of two ratios, the blue’s and the green’s, and studied the one it had moved. The natural next question was the other ratio, and it gave the expected answer about the turn. It also gave an answer to a question nobody had asked: the green, not the blue, sets how sharp the corner is, and the spike’s height had been a property of BT.2020’s green all along.

The failure mode is to attribute a quantity to the variable a study happened to sweep. The violet census reported the spike as ×4.37 at 485 nm, which is true of a 530 nm green and of no other. The test is to move each input the mechanism names, and the ones it does not — here the red, which the mechanism says should not matter, and which does not.