A tint in paint turns the other way

White turns a hue, and the models part at blue asked three colour models what happens to a display colour’s hue when white light is added to it, and found that all three turn the hue, that they agree about reds and disagree about blues. That is the tint a display makes when it desaturates a colour, and very nearly the tint a halftone makes when it leaves more white paper between its dots.

A painter makes a tint differently, by stirring white pigment into a coloured one, and paint is not a filter is the reason that is a different physical operation. In a paint film, each pigment’s absorption and scattering add in proportion to its concentration, and the reflectance that results is a nonlinear function of their ratio. Why blue and yellow make green followed that nonlinearity through a mixture of two coloured pigments and found the mixture bulging towards green.

The same nonlinearity applies when one of the two pigments is white. So there are two tints of any coloured pigment that have the same luminance and are made of the same materials in a sense — the pigment and something white — and the question is whether they are the same colour.

Opposite turns, and more colour in paint #

At nineteen parts white to one of colour, the tint made in paint and the tint made in light at the same luminance turn the hue in opposite directions for seven of eight pigments. The orange turns +13.5 degrees in paint and −13.9 in light, 27 degrees apart. And at every pigment the paint tint is the more colourful.

• The two colour models agree about the direction: Oklab and CIECAM16 turn the paint tint the same way for seven of the eight pigments, and the light tint the same way for all eight. The disagreement is between two physical mixtures, not between two models.
• The cause is visible in the reflectance. Tinting in paint moves a pigment’s half-height edge — by about 25 nanometres at nineteen parts white — towards the side where the pigment absorbs. Adding white light leaves the edge exactly where it was.
• A long-pass pigment’s edge moving to shorter wavelengths makes it yellower, and the orange, red and yellow paint tints all turn towards yellow; their light tints turn the other way.
• The paint tints keep between 1.3 and 3.6 times the Oklab chroma of light tints at the same luminance.
• The green is the exception: its reflectance is a band with an edge on each side, and in paint both edges move outwards together, widening the band without shifting its centre much, so its paint and light tints turn by a similar small amount the same way.

Eight pigments, two ways to lighten them #

The pigments are eight reflectances of the shapes real colorants have: a blue band at 450 nanometres, a green band at 530, a magenta that absorbs a band around 540, a violet with a blue band and a red tail, and four single-edged pigments — a cyan that reflects below 540, a yellow above 500, an orange above 570 and a red above 610. The white pigment reflects nine tenths of the light at every wavelength.

The paint tint is the single-constant Kubelka–Munk mixture of the pigment and the white at a stated concentration of white — a quarter, a half, three quarters, nine tenths and nineteen twentieths. The light tint is the additive mixture of the pigment’s reflectance with the white’s, in the proportion that gives the same luminance under daylight as the paint tint. The light tint is what a halftone makes when coloured dots and white paper average on the retina, or what a display makes when it adds white to a colour, and it is also the mixture the other essay measured.

The two are compared at equal luminance so that only the colour can differ. For the orange at nineteen parts white, the paint tint has a luminance factor of 68; the light tint that matches it is three fifths white light and two fifths pigment. That is the first sign of the difference in kind: a mixture that is nineteen parts in twenty white pigment is only as light as one that is three parts in five white light, because the coloured pigment goes on absorbing strongly in its own band long after it has been diluted.

The figure above is the result, pigment by pigment. Seven of the eight pairs of bars point in opposite directions; only the green’s point the same way. The widest pair is the orange’s, and the smallest the magenta’s, whose two tints turn by a degree or two either side.

One pigment, from the reflectance up #

The orange shows the mechanism most clearly.

The additive mixture lifts the whole reflectance curve and scales its range. At every wavelength it is the same fixed blend of the pigment’s reflectance and the white’s, so a curve that crossed half way between its lowest and highest values at 570 nanometres still crosses half way at 570 after mixing. The shape is unchanged, the floor is raised, and the edge stays put.

The paint tint does not lift the curve uniformly. In Kubelka–Munk terms, a pigment’s reflectance at a wavelength is set by the ratio of its absorption to its scattering there, and the reflectance changes steeply with that ratio where the ratio is small and hardly at all where it is large. Mixing in white dilutes the ratio at every wavelength by the same factor. Where the pigment absorbs strongly the ratio stays large and the reflectance stays low; where it absorbs only moderately — beside its edge — the diluted ratio becomes small and the reflectance jumps. The reflecting region grows into the absorbing side, and at nine parts white the orange’s half-height edge has moved from 570 to 549 nanometres.

A reflecting edge at a shorter wavelength lets through more of the yellow-green light, and so the paint tint turns towards yellow — +11.4 degrees in Oklab at nine parts white, where the light tint of the same luminance has turned −12.5, towards red and pink.

The edge moves in every single-edged pigment #

The mechanism predicts a direction for each pigment from where it absorbs, and the four single-edged pigments test it.

Every paint tint’s edge moves towards the absorbing side, and every additive mixture’s stays exactly on the line. The yellow’s, orange’s and red’s edges, which rise to longer wavelengths, move towards shorter wavelengths: 500 to 475, 570 to 545 and 610 to 585 at nineteen parts white. The cyan’s edge, which falls towards longer wavelengths, moves towards longer ones, from 540 to 564. All four move by about 25 nanometres, and all four move steadily as white is added, a little at a quarter and most over the last few parts.

The directions of the hue turns follow. The yellow, orange and red paint tints turn towards yellow, by +5.9, +13.5 and +3.2 degrees at nineteen parts white, and their light tints turn the other way, by −4.9, −13.9 and −8.3. The cyan paint tint turns −8.2 degrees and its light tint +14.2. The mechanism states the sign of every one of these turns before the colours are computed, and the colours agree.

Both models see it #

A difference of this size could in principle be a quirk of one colour model’s hue scale. It is not.

Oklab and CIECAM16 trace nearly the same pair of curves for the orange. At nineteen parts white the paint tint has turned +13.5 degrees in Oklab and +13.0 in CIECAM16; the light tint −13.9 and −14.3. The two tints separate from the first step — +1.7 against −2.8 at a quarter white — and keep separating. Across the eight pigments the two models agree about the direction of the paint tint’s turn for seven, disagreeing only for the magenta, whose turn is under two degrees in either.

This is also the sense in which the light tint’s turn is familiar. It is the Abney effect of the additive mixture, which the other essay measured on display colours; for the reds and oranges CIECAM16 and Oklab agree about it closely. What is new is that the paint tint does not share it. A painter adding white and a display adding white start from the same colour and move its hue in opposite directions, and both models say so.

Paint keeps more of the colour #

The second difference is in how colourful the two tints are.

At every pigment the paint tint is the more colourful, by factors from 1.3 for the blue to 3.6 for the yellow. The orange’s paint tint has an Oklab chroma of 0.091 and its light tint 0.050; the yellow’s 0.082 against 0.023, so pale that the light tint is nearly neutral.

The same edge movement explains it. An additive mixture raises the reflectance where the pigment absorbs as much as where it reflects, filling in the absorbed band with white light and diluting the colour uniformly. A paint tint raises the reflectance least where the pigment absorbs most, so the absorbed band stays comparatively dark while the reflected band brightens, and the reflectance keeps more of its shape. The paint tint is lighter where the pigment was already light, which is exactly a more colourful way to be lighter.

That has a direct consequence for anybody matching one kind of tint with the other. A pale tint on a paint chart and the same pigment as a halftone on white paper, adjusted to the same lightness, differ by a large chroma difference and a hue turn in opposite directions; a halftone is not a mixture described the halftone half of that pair, and the two are not interchangeable at any lightness.

A cyan, for the other direction #

The cyan’s edge falls rather than rises, and it moves the other way: from 540 towards 560 nanometres as white is added in paint, extending the reflecting region into the green. Its paint tint turns −8.2 degrees in Oklab — away from blue, towards green — while the light tint turns +14.2. The reflectance makes the pattern plain once more: the additive curve keeps its fall at 540, the paint curve falls later.

What this changes #

Three things for anybody who works across paint, print and screen.

A colour’s pale version depends on how it is made pale. A brand colour is an ink made the point that a specified colour is made of something; its pale versions are made of something too, and of a mixing law as well as a material. Specifying a tint by the colour it came from and the proportion of white is incomplete until the mechanism is named: a paint tint and a light tint of the same pigment at the same lightness can be 27 degrees apart in hue. A brand’s pale version made in paint and the same pale version made as a halftone or on a display need separate specifications, and the choice of which is the reference is a real decision.

Tint ramps generated additively will not predict paint tints, and a ramp tool that interpolates towards white in any colour space is making light tints. A gradient is a path showed that the path from one colour to another depends on the space it is interpolated in; for a path to white it depends first on whether the interpolation is standing in for light or for paint, and no choice of space turns one into the other. A dot is larger than it was asked to be measured how a halftone’s tints depart from their nominal coverages; this adds that even an accurate halftone tint is the wrong model for a painter’s.

And the direction of the difference is predictable from the pigment’s reflectance. A single-edged pigment’s paint tint turns towards the colour its edge moves towards — long-pass pigments towards yellow, a short-pass cyan towards green — while its light tint follows the Abney effect of the additive mixture. A band pigment’s edges move together and its two tints stay close.

How the tints were computed #

Each pigment’s reflectance is an analytic curve on the five-nanometre grid: Gaussian bands for the blue, green, magenta and violet, logistic edges of fifteen-nanometre scale for the cyan, yellow, orange and red, each on a low base. The white reflects 0.9 everywhere.

The paint tint is the single-constant Kubelka–Munk mixture: each reflectance is converted to its ratio of absorption to scattering, the ratios are averaged with the concentrations as weights, and the average is converted back. The light tint is the linear blend of the two reflectances with the white’s share chosen so the blend’s luminance under D65 equals the paint tint’s. Colours are computed under D65 through the 1931 observer; hue and chroma are Oklab’s and CIECAM16’s, the latter adapted to D65 at an adapting luminance of 100 cd/m². A pigment’s edge is the wavelength at which its reflectance crosses half way between its minimum and maximum, interpolated between bands.

What this leaves out #

The single-constant model assumes every pigment scatters alike, and a white pigment is chosen precisely because it scatters far more than most coloured ones. The two-constant model, with separate absorption and scattering for each pigment, would weight the white’s contribution more strongly and should move the edges further for the same concentration; it would not reverse their direction, because the direction follows from diluting absorption, which both models share.

The white is flat. A real titanium white absorbs in the violet and a real zinc white differs again, and either would add a small hue of its own to every tint. The film is also opaque and of fixed thickness; the colour is in the thickness is the reminder that a translucent layer’s colour changes with its depth, and a thin glaze of the same mixture over a white ground would follow neither curve here exactly. The light tint here is an ideal additive blend; a real halftone adds optical dot gain, which darkens tints and moves them part of the way from the additive towards the subtractive behaviour.

And the tints are compared as colours under one light. A paint tint and a light tint that match under daylight are not metamers of each other — their reflectances differ in shape — and under another lamp they would move apart by an amount that is not computed here.

Still open: a halftone between the two #

A real halftone sits between the two ideals. Light entering the paper between dots scatters sideways and emerges under a dot, so a halftone tint absorbs more than the additive average; that optical dot gain is a partial move from light mixing towards pigment mixing.

The question is whether it moves the hue as well as the lightness. The prediction from the mechanism is that it does, because optical gain filters some of the white paper’s light through the pigment and so behaves partly like diluting the pigment rather than averaging it; a halftone tint of an orange should turn less towards red than the ideal additive tint and could turn towards yellow at high gain. The computation is the halftone model with a Yule–Nielsen factor swept from one to two, the hue turn read at each, and the value of the factor at which the turn changes sign recorded for each pigment.

Two mixtures with one name #

The habit is about a word that names two operations.

“Add white” names two physical processes with different mathematics: an average of spectra and a mixture of absorbers. Both lighten, both desaturate, and at the same lightness they are different colours with hues that can turn opposite ways. Describing a tint as “the colour plus white” leaves unstated the only thing that decides which of those colours it is.

The move is to name the mixing law before reasoning about the result, and then to check the prediction at the reflectance — here, whether an edge moves — rather than at the colour, where two different causes can produce similar-looking shifts. The edge test separated the two tints before any colour model was consulted, and both colour models then agreed with it.

The failure mode is to settle a physical question by comparing colour models. Two models agreeing about a hue turn says the turn is in the stimuli; it says nothing about which stimuli are the right ones to compare, and a paint tint and a light tint are both legitimately “the colour with white added”.