Colour Science & Perception
what the eye settles, and what it cannot
Concept

Gamut — where it appears

The set of colours a device can actually produce, whose boundary is where a reproduction has to start lying. It is a property of a device and a state of that device, so it moves with the black point, the substrate, the room and the age of the lamp.

Named by 47 essays across 9 fields — each of them below, with the objects they name alongside it.

The CIE 1931 chromaticity diagram with its unreachable region marked. The spectral locus encloses every chromaticity a human eye can see. Cells inside the sRGB triangle are drawn in their own colour; the 85 per cent outside it are hatched, because no value this display accepts is the colour belonging there.

Most of this diagram cannot be shown

The chromaticity horseshoe is the canonical illustration of colour science, and nearly every printed copy is filled edge to edge with colours the page cannot produce. The honest version marks them, and the marking covers most of the picture.

matching · Gamut Two different spectra that are the same colour. Two reflectance curves differing by 92 per cent RMS, and the two patches they produce under D65: identical to ΔE00 = 6.2e-14, which is arithmetic noise rather than a small number. Both patches are inside the sRGB gamut, so neither has been clipped into agreement.

Two spectra, one colour

Metamerism is usually described and almost never demonstrated. It does not have to be — the metameric black space is enormous, so a matching pair can be constructed to order, verified, and then made to come apart by changing the light.

eye · Cones A gamma probe: which grey matches a half-white dither?. The striped block on the left is half white and half black, so it carries half the luminance of white. Stand back until the stripes blur and find the patch that matches it. On an sRGB display the answer is code 188, not 128 — code 128 has only 22 per cent of white's luminance.

The display is an unknown

This site is displayed on the very apparatus it is about, and it knows almost nothing about that apparatus. Two figures here stop assuming and ask instead — a probe for the transfer function and a probe for the gamut.

limits · Limits sRGB, Display P3 and Rec. 2020 compared on the chromaticity diagram. Three nested triangles inside the horseshoe. sRGB covers 74 per cent of the area P3 covers. Rec. 2020's red and green primaries sit on the spectral locus itself, within 0.000 and 0.002 of it, meaning they are monochromatic.

What a gamut costs

Three primaries reach a triangle and the visible region is not a triangle, so something has to give. Widening the primaries helps, has a price in precision and compatibility, and runs into a limit that is geometric rather than technological.

matching · Gamut The CIE 1931 colour-matching functions. The three functions that turn a spectrum into three numbers. They are all positive, which is why XYZ exists — the RGB functions they were derived from are not. ȳ is by construction the luminous efficiency function, which is why luminance comes out of Y.

Why colour is exactly three-dimensional

Matching every wavelength with three primaries requires, for some wavelengths, a negative amount of one of them. That physical awkwardness is why the colour-matching functions were transformed into XYZ, and why the horseshoe is curved.

eye · Cones The sRGB transfer function, and the gamma 2.2 curve it is not. Code value against relative luminance. The sRGB function is piecewise — a short linear segment near black, then a 2.4 power law with an offset — and it is close to but not the same as a plain 2.2 power law. Half-way along the axis of stored values sits at 21 per cent luminance, and half the luminance of white is at code 188.

A hex code is not a colour

Six hexadecimal digits identify three numbers. Turning three numbers into a colour needs a colour space, a transfer function, a white point and a display, and leaving any of them unstated is the everyday version of every confusion in colour management.

matching · Gamut How far a match comes apart when the observer changes. A broad source and a three-primary source, solved at each primary width so the pair is an exact tristimulus match for the CIE 1931 observer. The pair is then handed to the 1964 observer, and the gap between them is plotted. For the observer they were built for the gap is arithmetic noise at every width. For the other it grows as the primaries narrow, reaching 0.012 at 10 nm — and displays have been getting narrower for twenty years.

Whose eyes

The standard observer is an average over seventeen people, and no reader is it. What that costs was small when displays were broad and grows every time the primaries get narrower.

limits · Limits The 1976 uniform chromaticity diagram, with the unreachable region marked. The u′v′ diagram: the same spectral locus and the same sRGB primaries as the 1931 picture, projectively transformed. Straight lines stay straight, so mixtures and the gamut triangle survive; what changes is the distribution of area, and the green region that dominates the 1931 diagram is much reduced. 8% of the cells sampled inside the locus are reachable at Y = 0.55; the rest are hatched.

The diagram was replaced in 1976

The CIE knew the 1931 diagram was badly distorted and published a better one. Half a century later almost every chromaticity plot in print is still the old one, and both are still printed filled edge to edge with colours no display can show.

matching · Gamut Dominant wavelength as a construction on the diagram, against equal-energy E. A ray is drawn from the white point through each sample and continued until it leaves the diagram. The sample at (0.28, 0.52) leaves through the spectral locus at 537 nm, at excitation purity 0.43. The sample at (0.36, 0.19) leaves through the line of purples, so it has no dominant wavelength at all and is written 544c nm — the crossing on the opposite side, marked with a c. The hatched region is colour this display cannot show and is not drawn as though it could.

Not every colour has a wavelength

The question every reader arrives with is which wavelength a colour is. For close to a third of the directions on the chromaticity diagram the honest answer is that there is none, and the construction colorimetry offers instead is a statement about a diagram rather than about light.

matching · Gamut Chromaticity-triangle area against CIELAB volume, both relative to sRGB. Two ratios for each space, both against sRGB. The upper bar is the area of the primary triangle on the CIE 1931 diagram; the lower is the volume of the gamut solid in CIELAB, computed by tetrahedral decomposition of the RGB cube at 24 cells per axis. Display P3 is 1.36× sRGB by area and 1.50× by volume; Rec. 2020 is 1.89× sRGB by area and 2.26× by volume. The chromaticity diagram divides luminance out, so its triangle is a projection along the axis the eye is most sensitive to — and a coverage percentage quoted on it is a statement about the shadow.

The triangle is a shadow

A display's gamut is drawn as a triangle on the chromaticity diagram, and coverage is quoted as a percentage of that triangle's area. Chromaticity has luminance divided out, so the triangle is a projection along the axis the eye cares most about — and the ratios computed on the solid are not the ratios computed on its shadow.

limits · Limits sRGB, Display P3 and Rec. 2020 compared on the chromaticity diagram. Three nested triangles inside the horseshoe. sRGB covers 74 per cent of the area P3 covers. Rec. 2020's red and green primaries sit on the spectral locus itself, within 0.000 and 0.006 of it, meaning they are monochromatic.

Six numbers make a space

An RGB colour space is eight numbers — three primary chromaticities, a white point, and a transfer function — and everything else about it is derived. Deriving it rather than copying the matrix is the difference between having a colour space and having a table somebody else computed.

matching · Gamut Distinguishable colours in sRGB, counted under two difference formulae. The gamut volume divided by the volume of a ΔE = 1 ellipsoid, integrated over the solid because that ellipsoid changes size and orientation from place to place. Under the 1976 formula the answer is 195,720; under ΔE2000 it is 41,819 — 4.68 times fewer, from the same solid and the same lattice. Both assume perfect packing, which nothing achieves, so each is an upper bound rather than a count of anything. The gap between them is the result: "how many colours are there" is a question about a metric before it is a question about vision.

How many colours are there

Sixteen point seven million counts code values in a file format. Ten million distinguishable colours is a volume divided by the size of a just-noticeable difference — and the two difference formulae this site implements disagree about that size by a factor of nearly five.

difference · Metric What no surface can be more colourful than. The MacAdam limits at 4 lightnesses under D65, each computed by sweeping two-transition reflectances over the whole band and keeping those that land at the target luminance factor. This is a physical bound rather than a gamut: a reflectance above 1 is a surface that emits, so no pigment anybody invents will ever put an object colour outside these curves. The boundary shrinks steeply as the surface lightens, from 0.310 at Y = 0.1 to 0.028 at Y = 0.9 — a very light surface has almost no room to be colourful, and that is physics rather than pigment chemistry. Drawn against it is sRGB at the same luminance factor rather than as a primary triangle, because a triangle is what a display can reach at some luminance and the bound is what a surface can reach at one; matched properly, sRGB covers 36% at Y = 0.1, 40% at Y = 0.3, 40% at Y = 0.6, 21% at Y = 0.9. The faint triangle is the familiar figure, kept only to show how much it misleads.

No surface can be that colourful

There is a hard bound on object colour that no pigment will ever move, and it follows from a reflectance being at most 1. Its boundary is generated by two numbers, it shrinks by a factor of eleven from dark to light — and measured against it properly, sRGB reaches 40% of what a surface could be at mid lightness while Rec. 2020 reaches 106%.

scene · Scene The best a camera profile can do on the surfaces it was fitted to. Per-surface ΔE00 after the best 3 × 3 from raw to XYZ, fitted on 12 surfaces at chroma 0.7 and tested on those same 12 surfaces — mean 1.45, worst 2.58. This is the most favourable measurement it is possible to make of a camera and it is the one usually published.

No matrix is right everywhere

The best three-by-three this sensor admits, fitted and tested on the same twenty-four surfaces, leaves a worst case of ΔE00 2.85. A control sensor built to satisfy Luther's condition reaches ten to the minus seven under the identical computation, which is what makes the first number a measurement.

imaging · Capture What a press and a display can reach at L* = 50. A slice through both solids at lightness 50, with the press dashed and the display solid. The boundaries cross: the press reaches past sRGB in 15 of 48 directions and falls inside it in 31. That is the shape of every conversion between them — colours are lost in one direction and gained in the other, and the picture cannot show the gained ones, because it is being displayed on the gamut that cannot reach them. Those are hatched.

Neither gamut contains the other

Converting a picture to CMYK is described everywhere as a reduction, as though the press were a smaller version of the screen. Measured as solids, the press reaches eleven per cent of its volume outside sRGB while sRGB reaches fifty-four per cent of its volume outside the press — and in the cyans the press wins by seventy per cent of chroma.

applied · Delivery The solids, by volume. CIELAB volume, all measured by the same cell count with the same extrapolation, and quoted against sRGB because that is the comparison the estimator supports. The four-colour press reaches 49 per cent of the sRGB solid's volume — and contains colours sRGB does not, which is why "half the gamut" is the wrong summary of a ratio of one half.

The fourth ink is not for colour

Black is in every process set and it extends the gamut by more than a third — and at every lightness above about forty it extends the chroma boundary by exactly nothing. What it buys is depth at the bottom, a neutral that holds its hue when the press drifts, and a way to put a full-strength dark on a sheet that could not carry three inks' worth of it.

applied · Delivery What happens to a tone the destination cannot print. Source lightness along the bottom, destination lightness up the side, for a print whose black is L* = 8. Clipping is the flat segment on the left — 4 of 10 levels in the bottom twenty points of the scale arrive as the same number, and nothing downstream can recover them. Black point compensation is the straight line: all 10 levels survive as different numbers, and every tone in the picture has moved to pay for it.

A black that is not black

The deepest colour a four-colour press can make is L* 2.4, and against its own paper that is a contrast of about 320 to 1 — less than a cheap display manages in a dark room. Everything a printed picture does in the shadows is done inside that range, and the two ways of fitting a picture into it destroy different things.

applied · Delivery A gradient of one hue, mapped into a press's gamut 2 ways. Chroma asked for along the bottom, chroma delivered up the side, for a ramp at lightness 55 and hue angle 25°. A colorimetric intent follows the diagonal until the press runs out and is flat afterwards — the flat part is a gradient arriving as a single colour. The perceptual intent is under the diagonal from the start, which is the price of never going flat.

No mapping preserves everything

More than half the sRGB solid is outside a press's gamut, so something has to be done with it, and there are two things that can be done. One leaves every reproducible colour exactly where it was and delivers a gradient as a flat area. The other keeps the gradient and moves every colour that needed no help by about three.

applied · Delivery A named ink, and the closest four-colour build of it. The ink is one pigment chosen for its own spectrum; the four-colour build is three chosen for everything. The four-colour build reaches it to ΔE00 = 0.37 under D50 and drifts to 4.46 under a tungsten lamp, because the match is metameric — the two curves cross 4 times rather than coinciding anywhere.

A brand colour is an ink

A named colour is a jar of pigment with a spectrum, and a four-colour build of it is three inks arranged to integrate to the same three numbers. Of 156 constructed single-pigment inks, 15 can be matched at all — and those fifteen drift by a median of 3.8 colour differences under a tungsten lamp, because the match was never a match.

applied · Delivery The solids, by volume. CIELAB volume, all measured by the same cell count with the same extrapolation, and quoted against sRGB because that is the comparison the estimator supports. The four-colour press reaches 49 per cent of the sRGB solid's volume — and contains colours sRGB does not, which is why "half the gamut" is the wrong summary of a ratio of one half.

The fifth ink buys a corner

Orange, green and violet are sold as the answer to a four-colour press's gamut, and each one adds seven to nine per cent of volume. All three together add 23 per cent — almost exactly the sum, with no interaction — and every bit of it lands in the region of the ink that was added, at lightnesses where that ink is dark.

applied · Delivery One pair of colours, blended in four spaces. Every row starts and ends at the same two colours and visits different ones in between. The chroma of the middle falls furthest in linear light, by 83 units below the straight line between the endpoints' chromas — the grey dead zone that every blue-to-yellow gradient has and that no endpoint mentions. Any colour a route visits that this display cannot show is hatched rather than clipped.

A gradient is a path

Two colours fix the ends of a blend and nothing else. The middle is decided by the space the interpolation happens in, and four spaces in daily use put the halfway point of one ordinary gradient as much as thirty-seven units of colour difference apart.

matching · Gamut One adapting colour, two answers. The left patch is what was stared at. The middle is the afterimage the cone-gain arithmetic predicts at 15 per cent adaptation; the right is the inverted code values. They are 22.1 ΔE00 apart. The gains that produced the middle patch are 0.95, 1.06, 1.69 on the long, medium and short cone classes — the reciprocal of what each class had been receiving, taken 15 per cent of the way.

An afterimage is an adaptation

The demonstration everybody gives is an inverted image, which is a statement about a file format. Running the receptoral arithmetic instead puts the afterimage of a saturated red sixty degrees of hue away from the inverse — and outside what any display can show.

brain · Appearance The colourfulness the sRGB cube reaches, against the light in the room. The display is the same display throughout and the signal is the same signal. What moves is the adapting luminance, which enters the appearance model and nothing else. The furthest colourfulness the cube reaches falls from 131 to 57, a factor of 2.29. A gamut in CIELAB cannot show this at all, because CIELAB has no light level in it — which is why every gamut percentage in circulation is quoted without one.

The gamut shrinks in the dark

Every gamut number here is a property of a device and of nothing else, which is why the same display has the same gamut in a studio and in a cinema. It does not. Turn the room down four decades and the solid loses a fifth of its reach and two thirds of its volume, with the signal unchanged.

limits · Limits Several sources on one scale. planck, led, narrowband, each normalised to its own peak and drawn on shared axes with the colour each produces beside it. Every one of these is sold as white light and every one is ordinarily described that way; what they have in common is a chromaticity, and very little else.

A screen is a poor lamp

A display's white is a light. Shone on a surface it renders colour worse than a fluorescent tube — and the wider the gamut, the worse it gets, because the narrow primaries that buy a large triangle are exactly the ones that leave holes in the spectrum.

light · Light How much colour a display delivers, against how bright the room is. The appearance solid of the whole code cube, with the room's reflected light added to every code value and the surround ratio computed from the room's own white against the display's. A 300 cd/m² panel delivers most at 200 lux, and the curve falls on both sides: darker costs the surround, brighter costs the black. At the 32 lux the softproofing standards specify, the solid is 89 per cent of its best — which is not a criticism of the standard, since it is written for matching a screen to a print viewing booth rather than for delivering the most colour.

A display in a room is a smaller display

Two results here turn the room's light in opposite directions and neither knows about the other. A room has one light level and does both at once — so there is a brightness at which a display delivers the most colour, it is not at either end, and it scales with the panel.

limits · Limits The eleven basic colour terms, at their quoted centroids. Each patch is the CIELAB centroid quoted for that term, converted to a stimulus and drawn — except blue, whose focal colour is outside the sRGB gamut and is therefore hatched rather than clipped, which is the rule for an unreachable colour everywhere else and applies here too. The centroids are rounded to 5 units in each coordinate, and moving them by that much either way leaves the same term outside.

A colour has a name

Every quantity here is a number, and the question a reader arrives with is what colour something is called. The eleven basic terms of English divide the space into territories that differ by a factor of five, the metric accounts for fifteen per cent of that, and one of the eleven focal colours cannot be shown on this page at all.

brain · Appearance A colour-order system's chips, sorted by what they would be called. A regular lattice in lightness, chroma and hue — the idealisation of a swatch book, and regular by construction because a person has to be able to find a page. Named, it comes apart: 1320 chips inside the gamut divide into 198 for grey and 66 for blue. And 22 per cent of one-step moves in the lattice change the name, so a page of a swatch book is not a page of a vocabulary.

A catalogue is not a vocabulary

A colour order system is a regular lattice, because a person has to be able to find a page. Sorted by what its chips would be called, that lattice comes apart into piles differing by a factor of three — and a fifth of all one-step moves in it change the name.

matching · Gamut The hue circle cut into names, at L 60 and C 40. Left, the arcs each name claims, drawn at the colour of their midpoints; right, the same arcs measured in ΔE00 by integrating the difference along the ring rather than in degrees. The widest is 5.3 times the narrowest in degrees and 4.5 times in colour difference, so the metric accounts for 15 per cent of the inequality and no more. Only the eight chromatic terms compete on this ring: at this chroma the achromatic three would otherwise take the region where no basic English term sits, which is a defect of the model and is named in the essay.

There is no word for that colour

The naming model built this phase gives a saturated cyan the name green, calls part of a chromatic ring grey, and puts one of the eleven focal colours outside what a display can show. Each failure is a measurement rather than a disclaimer, and together they say what a vocabulary is that eleven points and a distance are not.

limits · Limits What a fourth primary actually buys. All three displays are floating-point-exact matches for the reference observer, so no colorimeter can tell them apart. The bars are the 95th percentile of what two hundred other eyes report. Held to the same gamut floor of 1.4× sRGB, the four-primary design leaves the population 2.6 times closer together than the three-primary one. That is what the extra emitter is worth, and it is not more colour — the gamut is held fixed while it is measured.

A fourth primary is a design

A display's fourth emitter is sold as more colour. Optimised instead against how far apart two hundred eyes are about its white — with the gamut held fixed so it cannot cheat — it buys agreement, and two and a half times closer together than three primaries reaching the same area, and the wavelengths it chooses are not the ones anybody would pick.

matching · Gamut Which of these paints the display can show, and to how many people. Each row is a real surface under D65, and the bar is the share of 120 observers for whom a non-negative mixture of this display's three primaries reproduces it. The question has no observer-free answer: the paint is a reflectance, the primaries are emission spectra, and whether one matches the other is a fact about somebody's cones. A dot marks the rows the 1931 observer calls displayable. 2 of them are rows some real people cannot see, and 4 more go the other way.

A gamut has a population

Whether a display can reproduce a paint is a fact about somebody's cones, so the boundary of a gamut is not a curve but a band. On a laser projector, ten of twenty-eight boundary surfaces are ones the standard observer calls displayable and some real people cannot see — and the wider the gamut, the wider the band.

matching · Gamut What a proof on an unbrightened sheet cannot reach. The paper white of a heavily brightened stock beside the paper white of an unbrightened one, both under an ultraviolet-included instrument. They are ΔE00 9.8 apart and 10.3 of that is in the blue-yellow axis. A proofing system on the unbrightened sheet can print towards the brightened one only by adding ink, which makes the paper darker rather than bluer; it cannot add light at 435 nanometres because it has no brightener and its own lamp is the one in the room. This is why a soft proof and a hard proof of the same job disagree about the white, and why the disagreement is in one direction.

A proof cannot glow

A proof is a different sheet of paper pretending to be the production one, and the pretence works by adding ink until the two agree. It cannot work for a brightened stock, because the direction the proof has to travel is towards more blue at the same lightness and every ink a proofer owns moves it towards less light instead.

applied · Delivery Outside the set of colours a reflecting surface can be. How far each stock sits from the boundary of the object-colour solid, as a fraction of the bound. The line at zero is the boundary: a perfect diffuser sits exactly on it, and every reflectance ever made sits to its left. The pale marker is the sheet measured with the ultraviolet excluded and the dark one with it included. Two of the six cross the line — they are brighter, in a direction that can be written down, than any reflecting surface of their colour could be. This is a proof rather than a hull: for each sample a direction is found in which the largest value any reflectance can reach is computed exactly, and the sample exceeds it.

A white that is not a reflectance

The object-colour solid is the hardest boundary in colorimetry — the set of tristimulus values any reflecting surface can produce, with no assumption about pigments in it at all. A coated press stock under a measurement standard's own lamp sits 1.5 per cent outside it, and a heavily brightened one 4.0, and with the ultraviolet removed both come back inside.

limits · Gamut The locus and the triangle, drawn on one of the diagrams. The spectral locus and the sRGB triangle in CIE xy (1931) — the default of the discipline, and the default used here. The triangle covers 33.6% of the enclosed area here. Nothing about the observer or the display has changed between this picture and any other in the family; the coordinates have, and the coordinates are what an area is measured in.

The diagram has no area

A chromaticity diagram is a projective picture of a three-dimensional space, and the freedom colour matching leaves in the observer acts on it as a projective map. Straight lines and mixture ratios survive that; area, distance and angle do not — so half of what the diagram is used to say is a statement about the paper.

matching · Gamut What share of the diagram the sRGB triangle covers, in twelve published coordinate systems. Each row is a chromaticity diagram somebody has printed, and each bar is the fraction of the enclosed visible area that the sRGB triangle covers in it. Every row describes exactly the same observer and exactly the same gamut. The answer runs from 8.5% to 38.4%, a factor of 4.52, because area is not preserved by the projective maps that carry one of these diagrams to another. The familiar "about a third" is a fact about CIE xy.

Two thirds is not a property of the eye

It has been said from the beginning here that about two thirds of the chromaticity diagram cannot be shown on a screen. The figure is right, on the diagram it was measured on, and across twelve published diagrams the same triangle covers anything from 8.5 to 38.4 per cent of the same locus. Counting stimuli instead gives an answer that does not move.

matching · Gamut Which of this collection's own claims survive a change of basis, and which are about the paper. Nine sentences this site says, sorted by whether they mean the same thing after the observer's three curves are replaced by a nonsingular combination of themselves. 5 of the nine do. The four that do not are not thereby wrong — they are statements about a chosen set of coordinates, and they are true of those coordinates. What they cannot be is statements about the eye, which is how every one of them is usually read.

Which of these is a convention

Nine ordinary sentences from this collection, put through one test — do they mean the same thing after the observer's three curves are replaced by a nonsingular combination of themselves? Five survive and four do not, and none of the four is wrong, because each is a statement about a set of coordinates being read as a statement about an eye.

limits · Limits A display's primaries, scored as the adaptation basis they are. Four primary sets ranked by the mean ΔE00 an adapted observer is left with when the white point moves — which for a display is a gain on R, G and B, and so a von Kries adaptation in the inverse of its own primary matrix. sRGB leaves 2.36, as much as scaling XYZ directly and therefore as much as having no cone basis at all. Rec. 2020 leaves 1.09, better than every published adaptation transform fitted to corresponding-colour data. Nobody chose that: it is what wanting a wider gamut does to a primary's spectral selectivity.

The gamut race chose the basis

Twenty years of arguing about how much of the diagram a display should cover has produced primaries whose inverse is a better adaptation basis than any transform ever fitted to corresponding-colour data. On the invariant count of what those displays can actually show, the same twenty years produced nothing at all.

matching · Gamut A primary's tolerance is a shape, and part of it is unreachable. The CIE chromaticity diagram with the spectral locus drawn, and three closed regions marking where each primary of a display designed for its own inverse can sit while the adaptation cost stays within 1 per cent of its best. None of them is round: the widest runs 15.1 times further one way than another, so a single tolerance figure for a primary is the average of a shape the shape never takes. 47 of the 144 boundary directions leave the region a real primary can occupy, which is a second constraint the objective knows nothing about — the cost does not rise there, and the primary cannot go there.

A tolerance is a region

How far a display's red primary can move before its adaptation behaviour costs anything is not a distance. It is a closed region on the chromaticity diagram, fifteen times longer one way than another, and about half of it lies outside the area a real primary can occupy — where the objective does not rise and the primary cannot go.

applied · Delivery What a display's red primary is allowed to be, under four requirements at once. A close view of the chromaticity plane around one designed primary, 0.101 units across. Four outlines: the set of positions the primary can take before each of four requirements gets one per cent worse — how well a gain in the display's own basis undoes a change of light, how much of the diagram the three primaries enclose, how many real surfaces fall inside them, and whether a light of that colour exists at all. The shaded region is where all four hold. It is 5% of the smallest outline's area, because the outlines are long and thin and cross at an angle rather than nesting. adaptation holds 42% of its boundary, gamut holds 10% of its boundary, realisable holds 48% of its boundary.

A primary is chosen for four things

A display's primaries have to adapt well, cover the diagram, hold the surfaces anybody photographs, and be colours a light can actually have. Drawing all four tolerance regions around one primary shows that no single requirement decides where it can go, and that one of the four never decides anything.

matching · Gamut The same tolerance, in the two numbers somebody actually sets. The plane a maker of a single-peak emitter works in: peak wavelength across, full width at half maximum up. Each marker is a candidate emitter whose chromaticity falls inside the colorimetric tolerance drawn for this display's green primary. They occupy a narrow band — peaks from 528 to 535 nanometres, a span of 7, against widths from 25 to 45 — so a tolerance stated as a region in chromaticity becomes ±3.5 nanometres of peak and a great deal of latitude in width. 2.0% of the 2501 candidates land inside at all: most of a region drawn in chromaticity is a colour no single-peak emitter makes.

A tolerance in the wrong coordinates

A display primary's tolerance is written as a region in chromaticity, because that is where the colorimetry lives. Nobody has a knob for chromaticity. What a maker of an emitter sets is a peak wavelength and a bandwidth, and the map between the two is so anisotropic that on the red primary its condition number is over eleven thousand.

matching · Gamut Where a stated appearance has no stimulus under it. The chroma the inverse can still return a light for, at lightness 50, all the way round the hue circle. Below the curve a stated appearance corresponds to a stimulus; above it the formula still returns three numbers and one of them is negative. Over a lattice of 10488 appearances spanning the whole space a specification is written in, 10.8 per cent are of that kind, and only 44 per cent are colours a display could show.

An appearance is not always a stimulus

CIECAM16's inverse is a closed form that returns three numbers for every lightness, chroma and hue it is handed. Whether those three numbers are a light it does not ask, and over a lattice spanning the space a specification is written in, 10.8 per cent of them are not — one tristimulus value negative, or a relative luminance above the white's. Only 43.8 per cent are colours a display could show at all.

brain · Appearance Which appearances a surface can have, lightness by lightness. The same lattice of lightness, chroma and hue a specification is written in, 10488 points, inverted under daylight with the observer adapted to it. Each row is one lightness, split into three shares: appearances a reflecting surface can have, appearances that are a light but that no surface can return, and appearances with no light under them at all. Over the whole lattice the first is 66 per cent, the second 23 and the third 11. At J 90 a surface can have 38 per cent of the row.

A third of the appearance box is no surface

An appearance specification is written as a lightness, a chroma and a hue, and the model's inverse turns any such triple into three numbers. A tenth of the space turns into something that is not a light at all. A further quarter turns into a light no reflecting surface can return, because a surface cannot give back more than all the light at any wavelength. So a third of the space a paint, a print or a dye is specified in cannot be made from paint, print or dye, and at lightness 90 nearly two thirds cannot.

brain · Appearance The straight line between two colours, and the formula's own shortest path. Five gradients seen from above, in the a and b plane of CIELAB: the straight line between the two colours in grey, and the shortest path under ΔE₀₀'s own local metric in colour. The lightness coordinate bows too and is not drawn. red to green saves 3.6 per cent and leaves the straight line by 13; blue to yellow saves 7.5 per cent and leaves the straight line by 21; cyan to magenta saves 3.9 per cent and leaves the straight line by 10; black to white saves 0.0 per cent and leaves the straight line by 0; red to blue saves 0.3 per cent and leaves the straight line by 4 CIELAB units at the widest. Black to white is the flat case: its shortest path is the straight one.

The straight line is not the shortest gradient

A colour difference formula says what a small step costs at every colour, and that is enough to ask which path between two colours is shortest. It is not the straight line: between red and green the shortest path bows thirteen CIELAB units away, saves four per cent — and stays inside sRGB on every step where the straight line leaves it. The formula's own answer for the two endpoints, meanwhile, is neither length.

matching · Gamut What a slope limit costs the object-colour solid, direction by direction. For each transition width, how much of its ideal reach the solid keeps: the median direction, the tenth percentile, and the worst. At twenty nanometres the median keeps 0.997 and the tenth percentile 0.985; at eighty they keep 0.946 and 0.766, and at 160 0.826 and 0.417. The worst direction falls from 1.00 at five nanometres to 0.20 at 160, with directions reaching under five units beyond black set aside. The cost is in a corner only at widths sharper than an ordinary pigment's.

The limits assume a pigment that switches instantly

The hardest boundary in colorimetry is reached by reflectances that jump between nought and one at a wavelength, and no material does that. Constrain the jump to take twenty nanometres — a sharp dye — and the median direction of the object-colour solid loses under half a per cent of its reach. Constrain it to eighty, an ordinary pigment, and the median loses five per cent, the tenth percentile nearly a quarter, and seven directions in ten lose more than one. The cost is in a corner only for chemistry sharper than paint.

limits · Limits What a gamut charges a gradient, against what the relaxation's noise is. For each gradient, how much longer the best path that stays inside sRGB is than the free shortest path, as a percentage of the free one. The shaded band is the relaxation's own noise, measured by relaxing the same free path from two different starting points: 0.42 per cent at its worst. Every excess is inside it. So holding a gradient inside a display's gamut costs nothing measurable in the metric's own units, on any of these pairs — including the ones whose free path is outside at most of its points.

A gamut charges a gradient nothing

On red to green the shortest path happened to stay inside sRGB where the straight line did not, and that was called a coincidence of mechanism. Made into a measurement it is not a coincidence: constraining a path to stay inside the gamut costs less of the metric's own length than the relaxation's own noise, on every gradient tested — including the ones whose free shortest path is outside at fifteen of its twenty-one steps. The gamut's boundary and the metric's cheap region point the same way, and the price of the constraint is nothing.

matching · Gamut Six starts for every held gradient, and where each one lands. Each row is one gradient held inside a gamut, relaxed from six starts: the straight line, the free shortest path, and the straight line bent towards and away from the neutral axis and up and down in lightness. Each dot is how much longer than the free shortest path that start's result is, on a logarithmic scale from a hundredth of a per cent to a thousand; the ring marks the best. The first 10 rows are gradients between colours on a coated CMYK press's boundary whose straight line leaves the press: their best routes cost a median of 0.12% and at most 1.1%, and on 5 of them some start lands at more than twice the free length. The next 3 are the display gradients held inside sRGB, whose best cost at most 1.3% and whose starts spread by at most 4.2%. The last 6 are press gradients that never leave, where no start is trapped.

A press makes the cheap route a search

Holding a gradient inside a display's gamut cost nothing measurable, and the essay that found it credited the gamut's convexity. A press's gamut is not convex: 216 of 630 straight lines between colours on its own hue ring leave it. Holding gradients inside the press still costs little at best, a median of a tenth of a per cent. But the best route depends on where the relaxation starts, the free path is the best start on one gradient of ten, and on half of them some start is trapped at more than twice the free length. On the display no start is trapped.

matching · Gamut The same gradients, priced by a penalty and by a projection. Each row is one gradient held inside a coated press from six starts. The pale dots are the penalised relaxation — a free step, with a price for leaving the press — and the dark ones are the projected relaxation, which takes the free step and then moves each point to the nearest printable colour. Across is how much longer than the free path each result is, logarithmic. On the 10 gradients whose straight line leaves the press, the penalty leaves 9 starts at more than twice the free length and the projection leaves none. The projection's best route costs a median 0.02% against the penalty's 0.13%, and its spread across starts is 0.95% against 132.5%.

A projection has no reason to detour

Holding a gradient inside a press by penalising the excursion turned the gamut's price from a number into a search: on five of ten crossing gradients some starting point leaves the relaxation trapped at more than twice the free length, and the spread across six starts runs to 271 per cent of the free path. Replacing the penalty with a projection — take the free step, then move each point to the nearest printable colour — leaves no trap on any gradient and a spread of 0.06 to 2.4 per cent.

matching · Gamut How much of a coated press is left at each margin inside its boundary. The share of a coated press's printable volume in CIELAB that lies at least a given distance inside its boundary, for margins from half a unit to 32. A margin of one unit keeps 88%, two keep 80%, four 66% and eight 45%. At every margin up to 30 what is left is a single connected piece; at 31 units, with 0.14% of the volume left, it first splits, into a core of 623 cells and 2 fragments of one or two cells — the deepest point is 32.7 units inside, so what splits is the last crumb of the core, not a waist.

A margin costs a press its corners

A gradient held inside a coated press by projection was predicted to tear if the press were first shrunk by a safety margin, at any pinch narrow enough for the shrinking to cut. The press has no such pinch: shrunk by any margin up to thirty CIELAB units it stays in one piece, and projected routes on it are neither trapped nor torn. What a margin costs is concentrated at the corners — the solid yellow moves nearly four times the margin to get inside, like the tip of a 31-degree spike.

matching · Gamut

Named alongside it

The objects these essays reach for when they reach for this one.

SpecificationDisplay gamutPrimariesCIELABStandard observerChromaticitySpectral locusColour managementGamut mappingObject-colour solidToleranceCIEDE2000

All concepts