A display's red takes the reds' move and leaves the pair

At the gamut’s edge the reds move as far as the violets took two colour-difference units to the boundary of a BT.2020 display. At every fifteen degrees of hue and three lightnesses it found the most saturated base colour whose two one-unit pairs — one stepping in lightness, one in chroma — the display could still show, and asked how far each unit moved the balance between the two pairs as the display brightened from a 1.5 to a 10,000 candela white. CAM16-UCS moved every colour by about the same factor. ΔEITP moved a yellow-green towards chroma by ×0.54 and a dark red at lightness 25 and hue 30 towards lightness by ×1.64, with the red-purple at hue 315 beside it at ×1.61.

The reds’ move was the surprise, and the explanation offered for it was a single number. ΔEITP passes three cone-like signals through the perceptual quantiser separately, and the hue disagreement is the quantiser’s had found that how far a colour moves follows how unequal its three signals are. The dark red at BT.2020’s edge has its L signal well above its M — further apart than almost anything a display can make — and it can because BT.2020’s red primary is a spectral line at 630 nanometres, far out on the long-wave side of the locus. A real display’s red is less extreme. The prediction was that moving the red primary in would lower the reds’ move faster than any other hue’s, while the yellow-greens and red-purples held, so that on a P3 display the top of the range would go back to the red-purples — and that the forced-choice experiment’s pairing would then depend on the display it was run on.

The reds’ move is the red primary’s; the pair is not #

Three parts of the prediction hold and its consequence does not. As the red primary moves in along the locus from 630 to 605 nanometres, the dark red’s move falls from ×1.64 to ×1.40 at 615 and ×1.25 at 605 — the largest fall of any colour in the census — while the yellow-green’s holds to within a hundredth. From 625 nanometres in, the top of the range is the red-purple at hue 315, and it rises rather than holding, from ×1.61 to ×1.68, because BT.2020 had put it past the chroma at which its own move peaks. Display P3’s red primary is a 615 nm line to within three ten-thousandths in chromaticity, and its reds move exactly as the 615 nm display’s do. But the pair the forced-choice experiment should use is the same yellow-green and red-purple on every display, and it gets slightly cheaper as the red moves in.

• The reds’ move belongs to the display. It falls with the red primary, and so does every hue from 330 round to 45 degrees; nothing else in the census moves by more than a hundredth and a half.
• P3 is the 615 nm display wherever red does the work. Its differences lie in the green and the blue.
• The red-purple at 315 degrees was being held back by a display too wide for it.
• The experiment does not need the display in its specification for its pairing, only for its numbers.

The figure carries the result. Each line follows one base colour as the red primary moves in, the base always at the most saturated colour that display can show at its hue and lightness. The two reds fall steadily. The red-purple at hue 345 falls with them. The red-purple at hue 315 rises and crosses above the dark red between 630 and 625 nanometres. The yellow-green does not move. The hollow circles at 615 nanometres, Display P3’s values, sit on or beside each line at that wavelength.

Six displays that differ only in their red #

The census is the earlier essay’s without change except in one place. At lightness 25, 50 and 75 and every fifteen degrees of CIELAB hue, a base colour is walked out in chroma until one of the five colours it involves — the base and the two members of each pair — asks one of the display’s primaries for less than nothing or more than the white takes. At that chroma both units’ moves are computed exactly as before: the ratio of the lightness pair’s difference to the chroma pair’s on a 10,000 candela display, capped where any signal meets the quantiser’s peak, over the same on a 1.5 candela display.

What changes is the display. BT.2020’s green and blue primaries and its D65 white are kept, and its red primary is moved along the spectral locus to 625, 620, 615, 610 and 605 nanometres, so each display is still a laser or quantum-dot display of the kind BT.2020 describes, with a less extreme red. The seventh display is Display P3, the primaries of most wide-gamut phones and monitors — the space in which a hex code is not a colour found the same six digits naming a different colour than in sRGB — whose green and blue are also narrower than BT.2020’s. On BT.2020’s own primaries the census reproduces the earlier one, every base’s edge chroma to the last digit.

Display P3’s red primary lies on the spectral locus at 615 nanometres to within three ten-thousandths in chromaticity. It is specified as the point (0.680, 0.320), and the 1931 observer puts a 615 nm line at (0.6801, 0.3197). That was not designed into the census, and it makes P3 an unexpectedly clean test — the same diagram most of this diagram cannot be shown drew to mark what no display reaches, with its reddest corner now a question of which line: for every colour whose edge the red primary sets, P3 and the 615 nm display are the same display, and any difference between them is the green’s or the blue’s.

Where the edge moves #

Moving the red primary in from 630 to 615 nanometres pulls the display’s edge in over one stretch of hue and nowhere else. At lightness 25 the edge comes in from hue 315 round to hue 30 — from chroma 96 to 87 at 315, 65 to 56 at 345, 70 to 59 at 30 — and every hue from 45 to 300 degrees keeps its edge where it was, because there the green or blue primary sets it. At lightness 50 the stretch is the same; at lightness 75 the red primary sets no edge at all, since a light colour runs out against the green and blue first.

Display P3 draws the same line through the reds and a lower one through everything from the greens to the violets: its green primary is less saturated than BT.2020’s, and its blue sits a little further in. At lightness 25 its edge at hue 180 is chroma 27 against BT.2020’s 45.

This is the whole of what the displays change. ΔEITP’s move at a base colour is a function of that colour and of the two display whites; the display only decides which colour is the base, by deciding how far out along each hue the census can go.

The reds fall and nothing else does #

Moving the red primary from 630 to 605 nanometres lowers ΔEITP’s move most at the dark red at hue 30, by 0.38, and every base whose move falls by more than a tenth lies between hue 330 and 45 at lightness 25 or 50. That stretch is exactly the stretch whose edge the red primary sets. Outside it, every base’s move changes by under a hundredth and a half, except one, which rises.

The fall follows the mechanism the earlier essay named. At the dark red’s edge on BT.2020 the quantised L signal stands 0.057 above M; with a 615 nm red, where the edge is at chroma 59 rather than 70, it stands 0.047 above; at 605 nanometres, 0.037. A red further in on the locus is a red whose light the L and M cones share more evenly, and the most saturated red a display can make from it has a smaller L-minus-M difference. The reds moved as far as they did on BT.2020 because BT.2020’s red primary let them be that unequal, and a display that cannot make them so unequal cannot move them so far.

The yellow-green, whose move is the S signal alone, holds. On the six red primaries its lowest move runs from ×0.544 to ×0.548. On Display P3 it is ×0.563, a little higher, because P3’s green primary brings the yellow-green’s edge in from chroma 74 to 70. CAM16-UCS’s moves stay in the narrow band they occupied before on every display, from ×0.74 to ×0.89 — the same movement for every colour, which a lit room brings the units’ medians together traced entirely to what the viewer is adapted to.

The red-purple was past its own peak #

The one base that rises is the red-purple at lightness 25 and hue 315. Its move goes from ×1.61 on BT.2020 to ×1.63 at 625 nanometres, ×1.66 at 615 and ×1.68 at 610 and 605, and it is the largest move in the census on every display whose red lies inside 630 — on Display P3 too, at ×1.68. The prediction had it holding.

The reason is that its move is not largest at its most saturated. Held at lightness 25 and hue 315 and walked out in chroma with no display in the question, ΔEITP’s move rises to a peak of ×1.682 near chroma 80 and falls after it, to ×1.61 at chroma 96. BT.2020’s edge at that hue lies at chroma 96: the display lets the red-purple be more saturated than suits its move. Moving the red primary in pulls the edge back — to 87 at 615 nanometres, 82 on P3, 76 at 605 — through the peak, and the move rises to meet it.

The dark red’s curve has no peak in the same range. Its move rises with chroma all the way to BT.2020’s edge, so every display that stops the red sooner lowers it. The two curves together are the whole of the figure at the top: one colour whose move a narrower display takes away, and one whose move a narrower display gives back, crossing between 630 and 625 nanometres.

It says something about the edge census’s own claim. The earlier essay read the top of the range as a pair of colours, a red and a red-purple, sharing ×1.6. On BT.2020 they did; but the red-purple was there despite the display, and on any real display it is there alone.

The purples next to red follow it #

Between the red-purple at 315 and the reds, the purples at hues 330 and 345 behave like the reds. At lightness 50 and hue 345 the move falls from ×1.49 on BT.2020 to ×1.37 at 615 nanometres and ×1.25 at 605, and its edge chroma from 105 to 75. These are purples made mostly with the red primary, and their L-minus-M difference goes down with it, 0.053 to 0.035.

So “the red-purples hold” was half right in a precise way. The purple nearest blue, whose three signals are unequal because its S signal stands far above L and M, holds and rises; the purples nearest red, whose inequality is L against M, fall with the reds. What decides whether a hue’s move belongs to the red primary is which of the two signal terms makes it large, which is the two-term account the hue disagreement is the quantiser’s gave, read across displays instead of across hues.

The experiment’s pair does not depend on the display #

The prediction’s consequence was about the experiment the units disagree in the dim proposed: a forced choice across a 15 and a 5,000 candela display, in which observers say which of a colour’s two pairs looks more different, and the two units predict different answers. The edge essay had found that pairing two base colours — the one whose move ΔEITP puts furthest below CAM16-UCS’s, and the one it puts furthest above — separates the units’ predictions 1.8 times as far as the single violet base, and so needs 0.31 of its observers. If the reds had been that pair, a display without BT.2020’s red could not have run it.

They were not the pair, on any display. The base colour that stands furthest above CAM16-UCS is chosen by the ratio of the two units’ moves, not by ΔEITP’s move alone, and the dark red, which CAM16-UCS also moves further than most colours, never wins it. On BT.2020, on every display along the sweep and on P3, the pair is a yellow-green at hue 90 and the red-purple at lightness 25 and hue 315. Its gap grows from 0.61 on BT.2020 to 0.65 at 605 nanometres, and it needs between 0.28 and 0.31 of the violet stand’s observers — two observers at the stated spread of individual judgements, on every display.

So the experiment’s design survives the display. What the display does decide is the number each unit predicts, and so the answer an experimenter should expect: a result reported from a P3 monitor is a result about P3’s edge colours, which on the red side are 615 nm colours. A specification needs the primaries for that reason, and not to choose the pair.

What a display’s red is worth #

The earlier essay’s finding about the reds was true of BT.2020 and is a finding about BT.2020. On a display with Display P3’s primaries — which is to say on almost every wide-gamut screen in use — the dark red at hue 30 moves ×1.40, not ×1.64, and it is nowhere near the top of the range; the red-purple at hue 315 is, at ×1.68, alone.

What a gamut costs described widening primaries as a trade in precision and compatibility for reach. This census adds a quantity the trade changes that was not in it: how differently two colour-difference formulas behave at the display’s own edge. A laser red at 630 nanometres lets a display make colours at which ΔEITP and CAM16-UCS disagree by more than on any smaller display; moving that red in removes the disagreement at the reds and leaves it everywhere else. Primaries chosen for their inverse found that a display’s primaries can be chosen for a property other than reach at almost no cost in gamut; this is one more property a choice of red decides without anyone having asked it to.

How the census was taken #

The base colours, pairs and moves are those of at the gamut’s edge the reds move as far as the violets: at lightness 25, 50 and 75 and hues every fifteen degrees, a base colour and its two pairs one CIEDE2000 unit apart, in lightness and in chroma; each unit’s ratio of the lightness pair’s difference to the chroma pair’s; and the move, that ratio with a 10,000 candela white — capped where the pair’s largest signal meets the quantiser’s peak — over the same with a 1.5 candela white. A display is a 3 × 3 matrix built from its primaries’ chromaticities and a D65 white, and a colour is inside it when its three drives lie between 0.0001 and 0.9999 of the white’s; the edge is found by bisection in chroma to forty halvings. The swept red primaries are the 1931 observer’s chromaticities of spectral lines at 625, 620, 615, 610 and 605 nm; BT.2020’s own is its specified (0.708, 0.292). The pair and its observers use the earlier essay’s arithmetic — a 15 and a 5,000 candela white, the gap between the two units’ predicted log ratios, and observers in proportion to the inverse square of the gap.

What this leaves out #

A display’s primaries are not lines. A quantum-dot red is a band some tens of nanometres wide and a phosphor red is broader still; its chromaticity sits inside the locus, and a band and a line at the same chromaticity are the same primary for this census, which reads colours through the 1931 observer alone.

The observer is one observer. A gamut has a population is the general form of the warning: narrow primaries are where observers disagree most about what a display shows, and the reds at a 630 nm edge are exactly the colours a real observer’s L and M cones would place differently from the standard observer’s.

Both whites are D65. The moves depend on the displays’ luminances and not on their white’s chromaticity in this model, but a display calibrated to another white would move every edge.

Still open: whether the red-purple’s peak is a property of the quantiser #

The red-purple’s move rises to a peak near chroma 80 and falls beyond it, while the red’s rises to the edge. Nothing in the display explains the peak; it is a property of the colour’s three signals on their way through the quantiser. The candidate is that beyond chroma 80 the red-purple’s S signal is so far above L and M that on the bright display it reaches the quantiser’s flat upper stretch, where the curve compresses the step between the two pair members and the move stops growing.

The calculation is the move against chroma for every hue at lightness 25, with each base’s highest quantised signal on the bright display beside it, and the question whether every peak lies where some signal crosses a stated height on the quantiser’s curve. The prediction is that the red-purples and violets from 285 to 315 degrees all peak when their S signal on the bright display passes about 0.75 of the quantiser’s range, and that the reds, whose largest signal is L and never gets that high at lightness 25, do not peak at all. If so, the top of the range on any sufficiently wide display is set by the quantiser, not by the primaries — and a display wide enough to push a colour past it gives back less than a narrower one.

Change the thing a finding was attributed to #

The habit is about testing an explanation that names a cause outside the calculation.

The reds’ large move was explained by BT.2020’s red primary, and the explanation was already inside a single census on a single display: nothing in it varied the primary. The only test was to vary it — keep everything else and move the red along the locus — and the explanation passed. But varying it also showed two things the single census could not: a colour whose move the same primary had been suppressing, and that the experiment the finding seemed to threaten never depended on it.

The failure mode is to accept an attribution that the data could not have contradicted. A finding measured on one display is attributed to that display’s properties by argument; the argument is only as good as the census that varies the property it names.