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

Halftone — where it appears

A mosaic of full-strength dots whose areas rather than whose strengths carry the tint, so its colour is a spatial average. Its tone scale is not the area coverage, because the dot prints larger than it was asked to and light scatters sideways under the sheet.

Named by 34 essays across 5 fields — each of them below, with the objects they name alongside it.

A halftone tint, its four regions, and what they average to. At 40% and 30% coverage the sheet is a mosaic of 4 fully-inked regions, not a mixture of anything. Their areas are the product of the coverages — Demichel's rule, which holds because the screens are rotated to make it hold — and the patch's reflectance is the area-weighted average of theirs, raised to the Yule–Nielsen exponent n = 1.8. Averaging the regions' CIELAB coordinates instead, which is what mixing means to almost everybody, lands ΔE00 = 0.76 away.

A halftone is not a mixture

Forty per cent cyan and thirty per cent magenta are not stirred together anywhere. They are laid down as dots, and the sheet is a mosaic of four fully-inked regions in proportions the two coverages fix — which is why the colour is an area average of four spectra rather than a blend of two, and why it lands nowhere near where mixing would put it.

applied · Delivery Two mechanisms, one ramp, and the patch that separates them. A cyan ramp printed by a press with 10 per cent mechanical gain and a Yule–Nielsen exponent of 2.4. Fitting a mechanical gain at n = 1.0 gives 23 per cent and fitting one at n = 2.4 gives 10 per cent; both reproduce the ramp to under 0.66 of a lightness unit, and the curves lie on top of one another. The two-ink overprint below was in neither fit, and there they are ΔE00 = 3.20 apart.

A dot is larger than it was asked to be

Two quite different things make a printed midtone darker than its coverage says — ink spreading under pressure, and light scattering sideways inside the paper. Both bend the tone curve the same way and neither touches its ends, so the measurement every press is characterised by cannot say which happened. A patch that nobody measures can.

applied · Delivery 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 How wrong a profile is between its entries. A device profile is a lattice of measured patches and an interpolation. At every node of every grid drawn here the error is exactly zero, which is why a profile checked at its own patches always looks perfect. Sampled between the nodes, a 3-step grid is worst by ΔE00 = 2.93 and a 17-step grid by 0.30. Both are honest tables of the same press.

A profile is a table

An ICC profile does not contain a model of the device it describes. It contains a lattice of patches that were printed and measured, and everything between them is interpolation — which is exactly zero at every patch, so a profile checked against its own target reports perfection for ever.

applied · Delivery The process inks as reflectance. Each ink is a sum of Gaussian absorbance bands with stated centres, widths and peak densities, and the reflectance shown is the substrate's times the square of the ink's transmittance, because light crosses the film going down and coming back. The first band of each ink is what it is for; the rest are what is wrong with it. Magenta's unwanted absorption in the blue-violet reaches an optical density of 0.46 against 0.95 for the band it exists to have — 48 per cent of its own strength, absorbing exactly where a saturated blue needs light to survive.

The paper is the white point

A printed colour is reported against the sheet it sits on rather than against the light in the room, and the choice is a switch in every colour engine. Flipped one way, the same separation on coated stock and on newsprint is a 3.04 colour difference; flipped the other, it is 0.13 — and the two answers are about different questions.

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 What separates the separations, and under which light. 11 separations of a single colour, differing only in how much of it is carried by black rather than by the three chromatic inks. Under D50 they agree to ΔE00 = 0.00 — the solver was asked for that and delivered it. Under illuminant A they spread to 6.75. They are metamers of one another, and the whole family is invisible to any instrument reading a single illuminant.

The separation is not unique

A four-ink press has one more control than a colour has numbers, so most colours can be printed several ways. The alternatives agree to a hundredth of a colour difference under the light the match was made in — and they are metamers of one another, so under a tungsten lamp the same eleven separations of one colour spread by nearly seven.

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 A chromatic line screen at 10.0 cycles per degree. The upper strip is the pattern as delivered and the lower one is the same pattern after each opponent channel has been low-passed at its own cutoff. Against a flat field of the same mean, the delivered pattern differs by up to ΔE00 14.22 and the filtered one by 7.77 — a ratio of 1.8. Every number quoted elsewhere for a difference of this kind is the first one.

A difference has no size

A colour difference formula answers a question about two large patches seen side by side. Applied to a pattern, it reports fourteen units where the eye is left with less than one — and the ratio depends on nothing but how finely the difference is spread.

difference · Metric A screen at 45°, 8 c/°, and where its energy sits. Left, the pattern. Right, its power in the frequency plane with the zero frequency at the centre and the edges at the sampling limit of 23 cycles per degree, on a logarithmic scale over five decades. The closed curves are the visual system's own sensitivity at 5, 25, 60 per cent of its peak; they are not circles, because sensitivity is lower on the diagonals than on the cardinal axes by a factor of 2.0 at high frequency. Energy inside a curve is seen; energy outside it is not, whatever its size.

A pattern has a direction

Every spatial claim here is a claim about a frequency, and a frequency has no direction in it. Turning a printed screen forty-five degrees makes it exactly twice as quiet with nothing else changed — and the same rotation does nothing at all to a chromatic one.

eye · Cones A halftone tint away from the centre of gaze, two ways. The upper curve raises the threshold by the E2 rule and leaves the filter alone, which is the multiplication the last phase guessed at. The lower one also moves the cutoff, because eccentricity magnifies the whole spatial scale — implemented as the substitution that makes it exact, a screen of ruling r seen through a filter whose cutoff has been divided by s being a screen of ruling r·s seen at the fovea. The horizontal line is threshold. The screen is visible where you are looking and gone by 3°, while the product prediction has it visible across the whole page.

A tint at the edge of a page

The last phase left this join open and guessed at its answer — how visible a halftone tint is away from the centre of gaze should be the product of two effects it had measured separately. It is not the product. At five degrees the guess is fifty-three times too generous, and by twenty it is out by eight orders of magnitude.

difference · Metric Every filtered claim in these essays, read at a point and read as components. Each row is a comparison one of the essays makes. The bar is the ratio between the two readings — how many times larger the component answer is than the point answer, or the reverse — on a logarithmic scale. 5 of 7 disagree by more than half again, and 4 disagree about the direction of the effect rather than merely its size. The three marked as noisy are the ones with a noise field on one side of the comparison, and they are the three largest.

The list nobody made

The last phase found that reading a filtered signal at a point asks a question its thresholds were never fitted to, made it a standing rule, and admitted that nobody had gone back through the site to see which claims it touched. Here is the list. Every claim with noise on one side of it moves — and so do two that have no noise in them at all, which the rule said would not.

limits · Limits How fine a screen each channel can see. A halftone screen at 35 per cent coverage, 45 degrees, seen by each of the eye's three spatial channels. The bar is the ruling at which it drops below that channel's own threshold. The luminance channel is still seeing it at 55 cycles per degree; the two chromatic ones have lost it by 15 and 10. The dashed line is an ordinary press ruling at reading distance, and only one channel is above it. A halftone is a luminance object, which is why the ink whose dots are nearly the lightness of the paper is the one nobody worries about.

A halftone is a luminance object

The eye's three spatial channels resolve a screen at 55, 15 and 10 cycles per degree, so at any ruling a press actually runs only one of them can see it at all. The corollary corrects a guess already published here — rotating a screen matters more to the chromatic channels, not less — but only at rulings so coarse that nobody prints there.

difference · Metric What a halftone measures, against what it looks like. A 50 per cent screen, measured by an aperture that averages the patch and seen by an eye that blurs it first and compresses afterwards. At a coarse ruling the two disagree by ΔE00 11.1, and the patch looks 14.3 lightness units darker than it measures — the average of a concave function is below the function of the average, which is Jensen's inequality and not an illusion. The gap falls under a unit above 30 cycles per degree. That is a viewing distance, and the correction a press applies for dot gain has no distance in it anywhere.

Measured with an aperture, seen with an eye

A spectrophotometer averages a halftone patch and reports one reflectance. An eye blurs it first and compresses afterwards, and the average of a concave function is below the function of the average — so a resolvable screen looks darker than it measures, by fourteen lightness units at a coarse ruling and by nothing at a fine one. The correction a press applies is the same size and has no viewing distance in it.

applied · Delivery The best possible 3×3, and the patches it makes worse. Each row is one patch printed on a brightened sheet, measured under both conditions. The pale bar is how far apart the two measurements are; the dark bar is what is left after the best least-squares 3×3 over the whole set has been applied. It leaves 23 per cent of the mean, and — the part a mean hides — it makes 4 patches worse than doing nothing. The solids are the ones it damages: the ink blocks the ultraviolet, so a solid barely disagrees between the two conditions and the correction has no business touching it. A matrix has no way to apply itself only where the paper is showing.

A tolerance cannot cross a condition

If two measurement conditions disagree by seven units, the obvious repair is a correction matrix fitted between them. The best least-squares 3×3 over seventeen printed patches leaves 23 per cent of the disagreement and makes five patches worse than doing nothing — because the term it is trying to remove is proportional to how much paper is showing, and no linear map on three numbers can express that.

difference · Metric Which way a tint turns a pigment's hue, made in paint and made in light. For eight pigments tinted nineteen parts white to one of colour, the Oklab hue turn from the pure pigment: of the tint made in paint (upper bar) and of the additive mixture with white at the same luminance (lower bar). For seven of the eight the two point in opposite directions. The orange turns +13.5° in paint and −13.9° in light.

A tint in paint turns the other way

A pale colour can be made two ways: by stirring white pigment into a coloured one, or by adding white light to it — which is what a halftone on white paper and a display both do. At the same luminance the two tints are not the same colour. For seven of eight pigments they turn the hue in opposite directions, an orange by +13.5 degrees in paint and −13.9 in light, and the paint tint is the more colourful of the two every time. The cause is in the reflectance: diluting a pigment moves its absorption edge, and adding light does not.

brain · Appearance A halftone tint, between the two ideals and nearer one of them. For each pigment, how far the hue turns when it is tinted to the same lightness three ways: mixed with white pigment, mixed with white light, and printed as a halftone at a Yule–Nielsen factor of two. Right is towards a longer wavelength. 7 of the eight pigments' two ideals turn the hue opposite ways. The halftone mark sits between them at every pigment and between 26 and 44 per cent of the way from the light ideal to the paint one — so it keeps the additive tint's direction and loses about a third of its size.

Printing does not change the sign

A tint mixed in paint and a tint mixed in light turn seven of eight pigments' hues opposite ways. A halftone sits between the two because light entering the paper between dots emerges under a dot, and the question was where. At the optical gain a coated sheet is fitted at, a halftone has travelled between 26 and 44 per cent of the way from the light ideal to the paint one — the same fraction for every ink — and only one pigment in eight changes direction.

brain · Appearance An orange ink's tints on three papers: which way the hue turns. The hue of each tint of an orange ink, from the solid to a tenth coverage, against the solid on the same paper, in degrees of Oklab hue: on a coated sheet, an unbrightened uncoated sheet and a newsprint. Solid lines are read against daylight's white, as an instrument reads them; dashed lines against the paper's own white, as a reader adapted to the page. At a tenth coverage, read against daylight, the coated sheet's tint has turned -8.7 degrees and the newsprint's 21.8; read against each paper's white, -8.7 and -1.1.

Newsprint turns a tint with its colour, not its gain

A halftone tint turns its hue the way a mixture of light does, by less, because optical dot gain carries it part of the way towards a paint tint. The straight line through a coated sheet's gain predicted that an uncoated sheet's larger gain would carry the orange past the crossing at a factor of 3.2, so that the same ink would turn opposite ways on two papers. It does turn opposite ways — on newsprint the orange's pale tints swing 13 degrees one way where a coated sheet's swing 8 the other. But the gain is not what does it. The road towards paint bends and stops at 60 to 70 per cent of the way, the orange needs a factor of 5.4 to cross, and newsprint's reversal comes almost entirely from the paper being yellow. Read against the paper's own white, as a reader looking at the page is adapted, it goes away.

brain · Appearance How much of the turn a reader sees, adapted part of the way to newsprint. The hue turn of a fifth-coverage tint against its solid ink on newsprint, for five pigments, as the reader's adaptation moves from daylight's white to the paper's. Open circles mark where each tint stops turning the opposite way from the same tint on a coated sheet: orange at 0.97, red at 0.89, magenta at 0.98, cyan at 0.90, green at 0.93. Half-way, the orange still turns 7.3 degrees.

A reader sees the reversal until almost adapted

An orange halftone tint on newsprint turns its hue one way and the same tint on coated paper turns it the other, and the reversal is the newsprint's yellow. An instrument sees it in full; a reader fully adapted to the paper does not see it at all. A real reader is somewhere between, adapted to a mixture of the paper and whatever surrounds it. Half-adapted, the reader still sees more than half of the instrument's turn, and the reversal lasts until the reader is adapted nine-tenths or more of the way to the paper. On a paler uncoated sheet it is gone by just past half-way. A coated sheet does not care.

brain · Appearance A pale orange tint's turn on a brightened sheet, against the reader's adaptation. The hue turn of a fifth-coverage orange tint against its solid on brightened office paper in daylight, as the reader's adaptation moves from daylight's white to the sheet's, beside a sheet of the same colour that does not fluoresce, a white sheet with the same dot gain, a heavily brightened sheet, newsprint and a coated sheet. Unadapted the brightened sheet turns it -41.5 degrees, the coated sheet -8.0 and newsprint 12.7. Fully adapted, the non-glowing sheet of the same colour comes down to -3.2, beside the white sheet's -3.4; the brightened sheet stops at -10.0.

Adapting to a brightened sheet leaves the glow

A pale tint on newsprint turns its hue one way and on coated paper the other, and a reader fully adapted to the paper sees newsprint's reversal vanish, because it was only the paper's colour. On a sheet with an optical brightener the tint turns the coated way and five times as far — 41.5 degrees for a fifth-coverage orange. A reader adapted to the sheet removes most of that, as expected, but not all: ten degrees stay. The sheet's colour adapts away like newsprint's. The brightener's glow does not, because the ink puts it out under the solid and leaves it under the tint, and an added light is not something adapting to a white can divide out.

brain · Appearance What adaptation cannot remove from an orange tint, with and without a thin dot edge. For a reader fully adapted to brightened office paper, the orange tint's hue turn against its solid beyond what a non-fluorescent sheet of the same colour leaves, at coverages from five to sixty per cent, with a uniform ink film and with rims of three, six and twelve micrometres. The residue is largest at the palest tint for every rim, and every wider rim raises it: at a fifth coverage from 6.8° to 7.9°.

A thin dot edge lets the glow back in

A pale tint printed on brightened paper keeps part of its hue turn however fully a reader adapts to the sheet — ten degrees for a fifth-coverage orange where a non-fluorescent sheet of the same colour leaves three — because the ink's vehicle puts out the brightener's glow under the solid and the tint's bare paper keeps it. That census gave every dot a uniform film. A real dot thins at its edge, a thin film lets ultraviolet through, and at the pale end of a ramp most of a dot is edge. The prediction was that the edge would restore the glow under the tint's dots, shrink the residue by a third at a fifth coverage and move the worst of it to the mid-tones. The edge does restore the glow — nine times as much under a small dot with a twelve-micrometre rim. And the residue grows: 6.8 degrees to 7.9 at a fifth coverage, 7.2 to 9.0 at a twentieth. The residue is the tint glowing where the solid does not, and more glow under the tint's dots is more of the difference. The palest tint keeps the most, with or without an edge.

brain · Appearance What adaptation cannot remove from an orange ramp, screen by screen. For an orange ramp on brightened office paper, the hue turn against the solid that a reader fully adapted to the sheet still sees beyond what the sheet's colour accounts for, on conventional screens of 100, 150, 200 and 300 lines and on stochastic screens of 10, 20 and 35 µm dots, every dot with a six-micrometre rim. The 150-line screen falls from 8.1° to 5.7° along the ramp; the 20 µm stochastic screen stays near 9.0°; the 10 µm one rises, from 10.4° to 11.4°. The axis starts at 4.5°.

Stochastic dots hold the glow at every tint

On brightened paper a halftone tint keeps a hue turn that a reader fully adapted to the sheet cannot remove, because the paper's glow survives under the ink where the solid puts it out, and a dot's thin edge lets more of it back. A dot's size at a given coverage is set by the screen. Across conventional rulings from 100 to 300 lines the residue rises at every coverage for every pigment, a finer screen making every dot more edge. A stochastic screen of twenty-micrometre dots holds it flat — the orange within a quarter of a degree from five per cent to sixty, starting where a 300-line screen starts — as predicted. What the prediction did not say is where the switch costs: not the pale tints, by under a degree, but the mid-tones, by three or four, because a conventional screen's mid-tone dots are large and mostly full film. On ten-micrometre dots, all edge, the ramp turns over and the mid-tone keeps more than the pale tint. At any one coverage the residue is set by the dot's diameter alone, whichever screen printed it.

brain · Appearance What merging the spots gives back along an orange ramp. For an orange ramp on brightened office paper, the turn a reader fully adapted to the sheet still sees beyond the sheet's colour, on a stochastic screen of twenty-micrometre square spots kept separate and merged where they touch, beside the round-dot stochastic screen and a 150-line screen. Merged and separate agree until the spots begin to touch near thirty per cent; at sixty per cent the merged screen keeps 6.9° against the separate spots' 8.6°. The axis starts at 4.5°.

Merged spots give the mid-tones half the glow back

On brightened paper a halftone tint keeps a hue turn a reader adapted to the sheet cannot remove, and a stochastic screen's small dots, being mostly thin edge, keep it high along the whole ramp — three times the 150-line screen's cost in the mid-tones. That census kept every dot separate. A real stochastic screen's spots touch and merge, and a shared edge between two spots is no edge of the ink. The prediction was that a merged screen would follow the separate-dot curve to about thirty per cent and then fall towards the conventional screens, giving back half the mid-tone cost by sixty. It does, almost exactly. Square spots twenty micrometres on a side, switched on in a dispersed order, stay apart up to a fifth coverage and start touching near thirty per cent; by sixty, nine in ten touch, the rim has fallen from 84 per cent of the ink to 50, and the merged screen keeps the orange's residue at 6.9 degrees against the separate spots' 8.6 and the 150-line screen's 5.7 — giving back 58 per cent of the cost. Every pigment gets back the same share, 55 to 58 per cent, because the share is set by the geometry of the ink, not by its colour.

brain · Appearance Clustered spots along an orange ramp. For an orange ramp on brightened office paper, the turn a reader fully adapted to the sheet still sees beyond its colour, on stochastic screens of twenty-micrometre spots switched on singly and in clusters of two, three and four, beside 150- and 300-line screens. At five per cent clusters of four leave 8.14° against the single spots' 8.94° and the 150-line screen's 8.12°; at sixty per cent, 6.00° against 6.90° and 5.67°. The axis starts at 5°.

Squares of four print a pale tint like a 150-line screen

On brightened office paper a stochastic screen's twenty-micrometre spots are mostly rim while they stay apart, and the rim lets the paper's glow through, so the palest tints carry the screen's whole cost in hue a fully adapted reader still sees. Merging gives back half of it in the mid-tones, where the spots touch, and nothing in the pale tints, where they do not. A second-order screen switches its spots on in clusters from the first tint, and the prediction was that clusters of four would take the orange's pale-tint residue down by about a degree, to within half a degree of a 300-line screen, while the sixty-per-cent tints stayed where the merged first-order screen leaves them. Squares of four do better in the pale tints than predicted and differ where predicted to agree. At five per cent they take the orange from 8.94 degrees to 8.14 — 0.02 above a 150-line screen and 0.81 below a 300-line one — and every pigment lands within a tenth of a degree of the 150-line screen. At sixty per cent they keep the orange 0.9 degrees below the first-order screen, where pairs and Ls come back to within a quarter. From ten per cent on, the 150-line screen pulls ahead.

brain · Appearance Growing clumps follow the 150-line screen along the ramp. For an orange ramp on brightened office paper, the turn a reader fully adapted to the sheet still sees beyond its colour, on single stochastic spots merged where they touch, on squares of four, on thirteen clumps grown from those squares as the tint darkens, and on a 150-line screen. The growing clumps start as the squares, 8.14°, and follow the 150-line screen from there: 7.22° against 7.09° at thirty per cent, 5.65° against 5.67° at sixty. The axis starts at 5°.

A growing clump prints like the screen with as many dots

On brightened office paper a halftone's ink lets the paper's glow through at its thin rim, and the share of the ink that is rim decides how much hue a fully adapted reader still sees turned. A stochastic screen of squares of four prints the palest tint like a 150-line screen and falls behind it from ten per cent, because its clusters multiply where a conventional dot grows. A hybrid grows each cluster as the tint darkens, the number of clumps held, and the prediction was that it keeps the squares' pale tint, reaches within a tenth of a degree of the 150-line screen by thirty per cent, and stays slightly above it through the mid-tones. With thirteen clumps on the census's tile it follows the 150-line screen within two tenths of a degree at every coverage, within a twentieth at twenty per cent, and dips just below it at sixty. The reason is a count. Thirteen clumps on a tile 640 micrometres across are as many dots per area as a 143-line screen; twenty-six follow the 200-line screen and fifty-two the 300-line. A clump's irregular place on the tile does not matter to the glow; how many clumps share the ink does.

brain · Appearance Most of the residue is not at the rim. At each coverage of the orange ramp, the residue a single clump on the tile leaves — the least rim the tile can hold — beside thirteen clumps'. The single clump keeps 92 per cent of the thirteen clumps' residue at five per cent and 94 at sixty: the rest of the turn is the tint's bare paper glowing beside ink that does not.

The rim was never most of the glow

On brightened paper a halftone tint keeps a hue turn a reader fully adapted to the sheet cannot remove, and five essays have chased the part of it that lives at the ink's thin rim — merging spots, clustering them, growing them — down to a rule: the clump count sets the residue, and only coarser patterns lower it. The proposal was a screen coarse in the pale tints and finer as the tint darkens, and the prediction that six clumps on the tile beat the 150-line screen by more than half a degree below twenty per cent. They beat it by 0.11 to 0.17. Scheduled six, thirteen and twenty-six clumps save a third of a degree at ten per cent and give back a fifth at sixty. The reason is the floor. One clump on the tile — the coarsest pattern it can hold, nearly all its rim gone — still leaves 7.48 degrees of an orange's 8.14 at five per cent and 5.30 of 5.65 at sixty. What screening can move is the rim's tenth; the other nine tenths is bare paper glowing beside ink that does not.

brain · Appearance A lighter ink swings the residue through nought. For a five-per-cent tint on brightened office paper, printed instead with the same pigment diluted to a share of its strength over more of the paper — matched in colour on a non-fluorescent sheet — the hue turn a reader fully adapted to the sheet still sees, signed, against the dilution. The orange's -7.2° at full strength passes nought near 0.54 and reaches 10.3° at a quarter.

A light ink can cancel the glow at one strength

A pale tint on brightened paper keeps a hue turn a reader fully adapted to the sheet cannot remove, and nine tenths of it survives every screen, because it is the tint's bare paper glowing beside ink that puts the glow out. A light ink — the same pigment diluted, printed over more of the paper — was the proposed lever, and the prediction that a quarter-strength ink lowers a five-per-cent orange's residue by a third. It does not lower it; it reverses it, from −7.2 degrees to +10.2. Somewhere between, the residue passes through nought, and for every one of eight pigments there is a strength that cancels it: 0.41 for the red, 0.54 for the orange, 0.80 for the blue and the cyan at a five-per-cent tint. The cancelling strength moves by under a tenth from five per cent to forty, so one ink per pigment keeps every pale tint within a degree and a quarter to twenty per cent, where full-strength ink leaves up to nine. The lever is real; it has a setting, and the light inks printers carry are diluted past it.

brain · Appearance One ink overshoots on the light sheet and falls short on the premium. Every pigment printed at the strength that cancels its ten-per-cent residue on the office sheet, its coverage matched to each sheet's own five-per-cent target: the residue a reader adapted to each sheet still sees, signed. On the office sheet it is near nought; on the light sheet seven of eight pigments' residues have turned past nought the other way, and on the premium sheet all eight keep the full-strength ink's sign, by under 1.8 degrees.

One light ink serves three papers within two degrees

On brightened office paper a pale tint keeps a hue turn even for a reader fully adapted to the sheet, and each pigment has one light-ink strength — the pigment diluted and printed over more of the paper — at which that residue vanishes. The prediction for a printer with several stocks was that the cancelling strength falls as the brightener loading rises, and that the office sheet's ink leaves a premium sheet's pale tints within two degrees and a lightly brightened sheet's within one. The strength does fall, at every step and for every pigment — by a fifth for the orange and the yellow, by under one per cent for the green. The office ink keeps the premium sheet within 1.8 degrees, as predicted, but the light sheet only within 1.8 too: it overshoots there, turning seven of eight pigments' residues past nought the other way, and falls short on the premium sheet for all eight. Either way it removes more than four fifths of what the full-strength ink leaves.

brain · Appearance The fresh sheet's ink holds until half the brightener is gone. On the premium sheet as its brightener bleaches, the worst residue any of eight pigments leaves from five to twenty per cent: printed at full strength, with the fresh premium sheet's light inks, and with the office sheet's. The fresh premium inks stay under two degrees to half bleached (25%: 1.75°, 50%: 1.80°, 75%: 2.35°); the office inks, 1.80° on the fresh sheet, stay under one and a half once any of the brightener has bleached.

An ink for a fading sheet is set for its middle age

A light ink cancels a pale tint's residual hue turn on brightened paper at a strength that falls along a straight line in the sheet's glow, and a brightener bleaches with use, so a sheet's glow falls over its life. The prediction was that as the premium sheet's brightener bleaches, every pigment's cancelling strength rises back along that line, and that the fresh sheet's ink stays within two degrees until half the brightener is gone. Both hold. Bleached by a quarter, a half and three quarters, the premium sheet asks for strengths within 0.015 of the census line at its glow, always on or just below it; the fresh premium ink errs by 1.75, 1.80 and 2.35 degrees at worst. But the office sheet's ink, set for a glow the premium sheet passes through on its way down, keeps the bleaching sheet within 1.4 degrees at every stage after the first. An ink set for the sheet's middle age outlasts one set for its first day.

brain · Appearance The census line is a chord across a curve. For four pigments, the cancelling strength less the census line's at each sheet's glow — the light, office and premium sheets and the three bleached and yellowed premium sheets — in order of glow. The light and premium sheets sit above their own line and the office sheet under it, the orange by −0.0161, deeper than any bleached sheet: the line through the three census sheets is a chord across a sagging curve, and the bleached sheets sag beneath it as the office sheet does.

A yellowing base leaves the ink where the glow put it

A light ink cancels a pale tint's residual hue turn on brightened paper at one strength for each pigment, and as a premium sheet's brightener bleaches that strength rises back along the straight line in glow through three census sheets — a shade under it, by up to 0.013. Real photo-ageing yellows the base paper as well, and the prediction was that a base yellowed in step lifts the strengths onto the line within 0.01 and that the office sheet's ink still holds every residue within one and a half degrees. The ink holds, at 1.38°, 1.13° and 1.38°. The strengths do not lift: no pigment's moves by more than 0.0023, and its gap under the line by more than 0.0006. The yellowing raises the sheet's b* by 0.31 at most while the bleaching raises it by five. And the gap was never the base's: the office sheet itself sits 0.016 under the line, deeper than any bleached sheet. The line is a chord across a sagging curve.

brain · Appearance Every stock's ink lies on the premium stock's curve. Five pigments' cancelling strengths at a ten-per-cent target against the sheet's glow: the premium stock at eleven brightener loadings from a tenth of its own to all of it, the light, office and premium census sheets, and the premium stock bleached until its glow equals the office's and the light's. Every census sheet lies on the premium stock's curve, and the bleached premium sheets on the census sheets — the orange at 0.5581 and 0.6461 against 0.5578 and 0.6456.

The glow alone sets the light ink

A light ink cancels a pale tint's residual hue turn on brightened paper at one strength for each pigment, and that strength falls with the share of the sheet's luminance that is glow — along a curve, not the straight line through three census stocks, which runs as much as 0.016 above it. The prediction was that the strength is one curve in the glow whatever the stock: that a premium sheet bleached until its glow equals the office sheet's, or the light sheet's, needs that sheet's ink within 0.003 for every pigment. It needs it within 0.0007, though the two papers differ in base, in whiteness and in how much of the brightener their ink shields. Traced across eleven loadings of the premium stock, the curve runs through every census sheet within 0.003, sags under every chord, and falls five to nine times as steeply on a dim sheet as on a bright one for the orange, red and violet.

brain · Appearance A longer glow moves the cool inks; a shorter one moves them all. For each pigment on the premium stock at the office sheet's loading, the change in its cancelling strength at a ten-per-cent target when the brightener's emission moves from 435 nm to 450 and to 420, the same photons absorbed and emitted. At 450 nm the blue moves by +0.176 and the violet by −0.058, the orange by +0.0011; at 420 nm the orange by +0.038 and the violet by +0.077.

The glow's colour sets the blue ink and spares the orange

A light ink printed on brightened paper turns in hue, because the paper's glow passes through the ink's open area differently from the ink's own film, and a cancelling strength undoes the turn. Every sheet so far glowed with one brightener, and its ink was set by the glow's amount alone. The prediction was that a brightener emitting fifteen nanometres longer, nearer the blue and violet inks' hues, moves their strengths by more than a hundredth and the orange's and red's by under a fifth of that. It does: the blue's strength rises from 0.81 to 0.98, because a glow at the blue's own hue barely turns a blue tint, the violet's falls by 0.06, and the orange moves by a thousandth though the glow's share of the sheet's luminance nearly doubles. A brightener emitting fifteen nanometres shorter moves every ink, the orange by 0.04, and no curve in the glow's luminance reads any of them.

brain · Appearance The glow's pull reads every ink its luminance cannot. For each pigment, the largest miss over the thirty sheets, each left out, of lines in the glow's share of luminance, the share and the chromaticity y, the pull's Oklab a and b, and the pull with its squares and product. The last misses by at most 0.0139; the share alone by up to 0.249, the blue's.

The glow's pull on the white sets every ink

A light ink on brightened paper is diluted to cancel the hue turn the paper's glow gives its tints, and on sheets that all glow with one brightener its strength is a single curve in the glow's share of luminance. Move the brightener's emission and that curve fails by up to 0.21. The prediction was that two numbers — the share and the glow's chromaticity — read every ink within a hundredth. Across thirty sheets, eight of them a grid of emission peaks from 420 to 455 nm, they miss by up to 0.07, and the share alone misses the blue by a quarter. The two numbers that work are the glow's pull: how far, in Oklab, it moves the paper's own reflection to the white an adapted reader sees. Read with their squares, they set every ink within 0.014 — and the warm inks need least strength just short of where that pull is strongest.

brain · Appearance

Named alongside it

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

Chromatic adaptationHueOptical brightenersFluorescenceProcess inksSubstratePaper whiteQuality controlSpecificationContrast sensitivityOptical dot gainSpatial frequency

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