A broad green sharpens the blue; a broad blue dulls it

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 reading a margin costs a press its corners made of a printing press and a tolerance unit rounds the corners carried into ΔE₀₀. In ΔEITP — the difference built for high-dynamic-range television on the ICtCp encoding — a display’s blue corner is the sharpest it has, and a quantiser keeps the blue a spike traced that to the two primaries that can be added to the blue moving it in nearly the same direction. A violet primary is no spike to the quantiser found the spike peaking at a 485 nm blue and fading towards the violet, where adding green stops raising the blue’s L signal by the larger fraction.

The green primary decides how sharp the blue is then moved the green. A longer green makes every blue sharper and moves the blue at which adding green turns L-ward towards the violet: with the blue at 485 nm the corner moved 2.39 times a margin against a 505 nm green, 4.37 against the census’s 530 and 8.19 against 555. The red, moved from 615 to 690 nm, changed nothing that mattered. Every one of those primaries was a single wavelength.

A display’s primaries are not. A laser’s are a few nanometres wide, a quantum-dot film’s twenty to thirty, an organic emitter’s thirty to forty or more. The prediction was 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. The alternative named was that a broad green lowers the spike, in which case the census’s line primaries overstate how sharp a real display’s blue is in ΔEITP.

The green’s width sharpens, the blue’s dulls #

With each primary a Gaussian spectrum and its chromaticity integrated over the CIE 1931 functions at one nanometre, a green 530 nm at its peak and forty nanometres wide sharpens the 485 nm blue’s corner from 4.37 times a margin to 5.05 and moves the turn from 453.7 nm to 451.2 — the corner of a single-wavelength green at 535.8 nm, though the broad green’s dominant wavelength is 533. Twenty nanometres wide, it gives 4.51 and 453.1. A blue twenty nanometres wide, with a single-wavelength green, moves the turn by a nanometre, to 452.6, and dulls the corner by a tenth, to 3.94. Together, a forty-nanometre green and a twenty-nanometre blue give 4.58 — within a twentieth of the line primaries. The spike stays at 485 nm for every width, and a red forty nanometres wide moves the corner by a fiftieth and the turn by a hundredth of a nanometre.

• The prediction about the green holds. A broad green is a longer green to the blue corner, and longer by more than its colour suggests.
• The prediction about the blue fails where it did not look. The blue’s width barely moves the turn, as predicted, and it costs the spike a tenth of its height.
• The alternative, inverted, is partly true. Line primaries understate a broad-green display’s spike and overstate a broad-blue one’s; a display with both lands close to the lines.

The curves keep the census’s shape — rising from the violet, peaking at 485 nm, dipping at 490 — and stack by width. At the 485 nm blue: 4.37 with three single wavelengths, 4.51 with a twenty-nanometre green, 5.05 with a forty-nanometre green; 3.94 with a twenty-nanometre blue. At the violet end the curves nearly meet, 1.23 to 1.26 at 440 nm, as the green census found every curve meeting there: the violet’s lack of M splits the two added-primary steps however the green is drawn, and a broader primary does not change that.

A broad green is a longer green #

The green census mapped single-wavelength greens from 505 to 555 nm, so a broad green’s effect can be read as an equivalent line. A green two nanometres wide is a 530 nm line to the blue corner; twenty nanometres wide, a 531.3 nm line; forty nanometres wide, a 535.8 nm line. The equivalent wavelength grows roughly as the square of the width, as a symmetric spectrum’s effect on a curving response should.

The broad green’s dominant wavelength — where the line from the display’s white through its colour meets the spectral locus — moves less: 531 nm at twenty nanometres wide, 533 at forty. So the broad green’s hue accounts for about half of its effect on the corner, and the rest comes with the other thing a broad green has lost, its purity: a forty-nanometre green sits well inside the spectral locus, at (0.192, 0.740) against the line’s (0.155, 0.806). Which of the two does what is not separated here; what is measured is that the corner treats the broad green as a line nearly six nanometres longer, and that the angle in ΔEITP between adding red and adding green at the 485 nm blue closes from 37.5 degrees to 33.1, as a longer green’s does.

Why the green’s width moves the turn #

The turn is where adding green to a blue stops raising the blue’s M signal by the larger fraction and starts raising its L. It depends on the green’s L to M: the more L a green carries, the earlier, towards the violet, its step turns L-ward. A broad green carries more L than its peak does, and the turn moves from 453.7 nm to 453.1 at twenty nanometres wide and 451.2 at forty — two and a half nanometres, the same direction and about the same size as the 535.8 nm line’s. The prediction’s reason was that the long side of a broad green’s spectrum carries more L than the short side removes, and the direction and size of the move bear it out; the census does not take the green’s L and M apart side by side to confirm the mechanism itself.

The size of the move follows the width’s square, as the equivalent line did: 0.6 nm at twenty nanometres wide and 2.5 at forty, four times as much for twice the width. A green’s spread pulls on the turn through how its spectrum curves against the cones’ curves, and a symmetric spread’s leading effect on any smooth curve grows as the spread squared, so on that reading a green half again as broad as forty would move the turn about twice as far again — an extrapolation the census does not test. The blue’s width moves the turn less, from 453.7 to 452.6 at twenty nanometres wide, a little under half of what a forty-nanometre green does. The prediction said it barely matters, and for the turn that is about right.

A broad blue dulls the corner #

What the prediction did not look at is the corner’s height. A twenty-nanometre blue lowers the 485 nm corner from 4.37 to 3.94 times a margin, by a tenth, and every long blue by about as much; the violet blues hardly move. The census’s account says a corner is sharp when its inward steps point nearly the same way, and a broad blue opens the angle between adding red and adding green at every long blue: at 485 nm from 37.5 degrees to 40.7, at 470 from 44.6 to 47.5. A broad blue at 485 nm is a mixture of bluer and greener light, and the two added primaries’ fractional steps, set by the blue’s own signals, part a little further.

So a display’s primaries’ widths pull the corner both ways. With a forty-nanometre green and a twenty-nanometre blue the 485 nm corner moves 4.58 times a margin, within a twentieth of the three single wavelengths’ 4.37, and the turn moves to 449.8, further than either width alone. A display maker cannot read a broad-primary display’s blue corner off the line census, but for displays with both green and blue broad the line census is a fair first guess at the height and a poor one at the turn.

Broadened together, the effects cancel on the height and add on the turn #

A real display does not broaden one primary at a time. Three sets of widths stand in for the range: three lines two nanometres wide, as a laser projector’s nearly are; red, green and blue 25, 30 and 20 nm wide; and 35, 40 and 25 nm wide. The sharpest blue corner is ×4.36, ×4.31 and ×4.43 on the three — within three per cent of one another — while the turn moves from 453.7 nm to 451.0 and then 448.8. The broad green’s sharpening and the broad blue’s dulling nearly cancel at the spike’s height; on the turn they push the same way, towards the violet, and add.

That splits the line census’s usefulness cleanly. For the height of a real display’s blue spike, the single-wavelength census is a good guide whenever the green and blue are broadened together, because their two effects offset. For where the spike ends — which blues a ΔEITP margin treats as a face rather than a spike — it is not: a display with broad primaries keeps a sharp corner several nanometres further into the violet than its peak wavelengths alone would suggest. A display maker choosing a 450 nm blue to get a blunt corner would find, with broad primaries, that 450 nm is still on the sharp side of the turn.

The cancellation is a coincidence of these widths rather than a rule. The forty-nanometre green alone sharpens the corner by sixteen per cent and the twenty-nanometre blue alone dulls it by ten; a display whose blue is broader than its green would come out blunter than the lines, and one whose green is much broader, sharper. What carries over is the turn, which every broadening of the green or the blue moves towards the violet, and which the red’s width leaves alone.

The red is still a bystander #

The green census found the red’s position mattering to nothing at the blue corner, because adding red raises a blue’s L by a much larger fraction than its M whichever red it is. Its width matters as little: a red forty nanometres wide moves the sharpest corner from 4.37 to 4.45 and the turn by under a hundredth of a nanometre. A broad red spreads into the orange, where it carries a little more M; its step along ICtCp’s L–M axis shortens by a few per cent and points the same way. Of a display’s three primaries, the red’s spectrum is the one ΔEITP’s blue corner does not care about.

What a display maker should take from this #

A broad green costs a ΔEITP margin more at the blue; a broad blue costs it less; and broadening both moves the end of the spike towards the violet. A quantum-dot or organic display whose green is thirty or forty nanometres wide has a blue corner up to a seventh sharper than a laser display with the same peak wavelengths, and a margin written in ΔEITP will carve away correspondingly more of its long blues. A display whose blue is broad gets about a tenth of that back. The line census’s numbers are a good first guess for a display with all three primaries broad, and an underestimate for one whose only broad primary is the green.

A narrow primary buys a disagreement found narrow primaries making observers disagree about a display’s colours; here the primaries’ widths change how a tolerance unit reads its corners. Both are reasons a display’s primaries should be specified as spectra, not as chromaticities, when a colour engine is going to put margins on them.

How the displays were built and read #

Each primary is a Gaussian spectrum with a stated full width at half height — none for a single wavelength, 2, 20 or 40 nm — at a peak of 630 nm for the red, 530 for the green and every five nanometres from 440 to 490 for the blue. Its chromaticity is its spectrum integrated over the CIE 1931 functions interpolated to one nanometre by Sprague’s formula; with no width it is the functions’ own value at the peak, and with three single wavelengths every corner is the green census’s to a part in a million. The display is balanced to D65 at 320 cd/m².

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 the corner’s move per unit of margin is the distance from the corner to the nearest point a unit inside every face of the cone they span. The turn is where adding green’s unit step changes sign along ICtCp’s L–M axis, interpolated between the blues. A broad green’s equivalent line is the single-wavelength green, interpolated between the green census’s, whose 485 nm corner moves as far; its dominant wavelength is read from D65’s white at one-nanometre steps of the locus.

What this does not reach #

Gaussian spectra. Real primaries are not Gaussian: quantum-dot emission has a long red tail and organic emitters several peaks. The census’s widths are a first approximation to a shape, and the three sets of widths are illustrations of a range, not measurements of particular displays; a display’s own spectra, measured, can be run through the same calculation directly, and its blue corner read off rather than inferred from these.

One green peak. The broad greens are centred at 530 nm; a broad green centred at 520 or 545 sits on a different part of the cones’ curves and may move the corner by more or less than its width suggests.

ΔEITP and its own response. The corner’s sharpness is ΔEITP’s; the green census found CAM16-UCS — which an appearance model flattens the blue by its hue found treating the blue nearly as a face — following the green gently, and its response to width is not computed. Nor is ΔE₀₀’s, in which a display’s blue is sharp where the rotation is not found a display’s blue corner sharp in its own way.

Still open: whether a quantum-dot green’s tail moves the corner more than its width #

A quantum-dot green is not symmetric: its emission falls steeply on the short side and trails on the long side, towards the yellow. If the corner treats a broad green as a longer green because of the L it carries on its long side, a tail should matter more than a symmetric width of the same area.

The calculation is this census with the green primary given an asymmetric spectrum — a Gaussian core twenty nanometres wide at 530 nm with an exponential tail on its long side holding a fifth, a third and a half of its power — the 485 nm corner, the turn and the equivalent single-wavelength green read as here. The prediction is that a tail holding a third of the power moves the equivalent line further than a symmetric forty-nanometre green does, past 538 nm, because all of its spread is on the L side. If the tail moves it less, the corner’s response to a broad green comes from the green’s lost purity more than from its L, and the symmetric width was the better description after all. A green’s long tail sharpens the blue; its short tail does not makes that calculation.

A specification in chromaticities is not enough #

The habit here is about what a display’s primaries are taken to be.

The green census specified each primary by a wavelength, and a display maker specifies each by a chromaticity. Both are a point in a plane, and both leave out the spectrum’s shape. Here the shape changes a corner’s sharpness by a seventh in one direction or a tenth in the other, and a broad green’s effect is nearly twice what its dominant wavelength would suggest.

The failure mode is to carry a result computed for idealised inputs to real ones by matching the one number the two share. The test is to give the idealised inputs the property the real ones have — here, width — and see whether the number they share still predicts the result. A broad green’s dominant wavelength did not; its equivalent line, found by matching the result, did.