A light ink can cancel the glow at one strength

Office paper is brightened: a fluorescent agent in the sheet absorbs ultraviolet and re-emits it as blue, and a reader adapted to the sheet takes the blue as part of its white. Printed ink puts the glow out where it lies — its vehicle absorbs the ultraviolet before the agent can — and adapting to a brightened sheet leaves the glow found a pale tint’s hue turned by several degrees that no adaptation removes, because a tint glows where its bare paper shows and its solid does not.

Five essays then worked on the edges of the ink, and the sheet itself had been characterised earlier: some paper is brighter than white found a fluorescent sheet to have no reflectance curve at all, its light leaving at wavelengths it never arrived at. A thin dot edge lets the glow back in found a dot’s thinning rim adding to the residue, and the screening essays that followed — stochastic dots hold the glow at every tint, squares of four print a pale tint like a 150-line screen and a growing clump prints like the screen with as many dots — lowered the rim’s part. The rim was never most of the glow put a number on that part: a single clump of ink on the census’s tile, the coarsest pattern a screen can hold, keeps nine tenths of a pale orange’s residue. The floor is the coverage. A tint at five per cent is ninety-five per cent bare paper, glowing.

No screen changes coverage, but an ink can. A light ink — the same pigment at a fraction of its strength, as inkjet printers carry for their pale tints — reaches a pale colour with more of the paper covered, so that less of it glows. The proposal was a light ink at a quarter and a half of the full strength, printed over four and two times the area, and the prediction that the quarter-strength ink lowers the five-per-cent orange’s residue by a third or more, to about five degrees. The alternative was that the residue barely moves, if the floor is set by the colour the tint must reach rather than by how it is laid down.

Not lower: reversed #

A five-per-cent orange tint printed with a quarter-strength orange ink, over the fifteen per cent of the paper that matches its lightness on a non-fluorescent sheet, leaves a residue of +10.2 degrees on brightened office paper where the full-strength ink leaves −7.2 — the turn reversed, and larger than it was. At half strength it leaves +1.0. For every one of eight pigments the residue passes through nought at one ink strength: at a ten-per-cent target, 0.42 for the red, 0.44 for the green, 0.56 for the orange, 0.63 and 0.64 for the violet and magenta, 0.70 for the yellow and 0.81 for the blue and the cyan. That strength rises by under a tenth from a five-per-cent target to forty. Printed at its ten-per-cent cancelling strength, every pigment’s residue stays within a degree and a quarter of nought from five per cent to twenty, against up to nine degrees at full strength, and within five at forty.

• The prediction fails by overshooting. A quarter strength does not reduce the turn; it replaces it with a larger one the other way.
• The alternative fails too. The floor is not set by the colour; a lighter ink moves it through nought.
• The lever has a setting, and it is a different setting for every pigment.

The four curves cross the zero line at four different strengths, and each crosses it once. The orange and the red start below nought at full strength — their tints turned one way — and rise through it as the ink is diluted; the green and the cyan start above and fall. Every curve is steeper at the weak end, which is why the quarter-strength ink overshoots so far: the residue changes more between a quarter and a third of the strength than between a half and full. A printer who chose a light ink by asking only that it be weak enough to print pale tints smoothly would land on the steep part, far past the crossing.

Why diluting reverses the turn #

The residue is a mismatch between two mixtures. A tint is a mixture of two surfaces in proportion to its coverage — bare paper and inked paper — and on a brightened sheet only the bare paper glows. A reader adapted to the sheet discounts the paper’s glow as part of its white, and so sees the inked part of the tint as if it had lost some light it never had: the tint’s hue lies on a line from the sheet’s white towards the ink, and the solid’s hue at the end of that line is shifted by the glow it lacks. That is the full-strength residue: the tint glows where its solid does not, and the reader’s adaptation, tuned to the glowing paper, cannot reconcile the two.

A light ink changes both ends of the line. Its tint covers more paper with vehicle, so less of the tint glows; and its inked part is paler, closer to the paper’s own colour. At the cancelling strength those two shifts meet: the tint’s mixture of glowing and non-glowing surface lands on the line the reader expects, and the turn vanishes. Past it, the vehicle covers so much of the paper that the tint glows less than its colour implies, and the reader sees it turned the other way — the quarter-strength tint has nearly a sixth of its paper under ink where the full-strength tint had a twentieth.

Why the cancelling strength differs so much between pigments the census reports rather than derives. The likeliest account runs through the second shift: a pigment whose ink sits near the paper’s blue glow in hue — the blue, the cyan — would need little dilution before its inked part sits right, and one far from it — the red, the green — much more, which is the order the census finds.

Each pigment’s strength #

The eight pigments fall into three groups. The red and the green cancel near two fifths of their strength, the orange, magenta and violet between a half and two thirds, the yellow near seven tenths, and the blue and the cyan near four fifths. Within each pigment the three targets stack closely: the cancelling strength at forty per cent is higher than at five by 0.05 to 0.14, the red’s the most. A darker target needs a slightly stronger ink, because at forty per cent a weak ink must cover most of the paper to reach the colour at all, and the vehicle has already done most of what it can.

That spread is the practical difficulty. A printer carries one light ink per colour, chosen for smoothness and for how much of a pale area it can cover without visible dots, and typically a quarter to a sixth of the full strength. No pigment here cancels below two fifths. A quarter-strength light cyan, the commonest light ink there is, sits far past the cyan’s crossing at four fifths, and on brightened paper it would turn a pale cyan the other way by more than the full-strength ink turned it.

Along the ramp #

The cancelling strength would be of little use if it held at one tint only. Printed at 0.56, the strength that cancels its ten-per-cent residue, the orange stays within 0.7 degrees of nought from five per cent to twenty and reaches 2.6 at forty, against −7.2 to −6.2 at full strength along the same ramp. The half-strength ink runs parallel, a degree to a degree and a half higher; the quarter-strength ink runs from ten degrees at five per cent to thirteen at twenty, worse at every tint it can print than the full-strength ink, and it cannot print forty at all: covering the whole sheet at a quarter strength falls short of a forty-per-cent tint.

The pale end of the ramp is where the residue mattered most — where the rim essays found nine tenths of it untouchable by screening — and it is where the cancelling ink holds best. At forty per cent a light ink is covering most of the paper and has little left to give; that is also where a printer would switch from the light ink to the full one, as light-ink printers do.

One strength per pigment, across the palette #

At a twenty-per-cent tint the full-strength residues range from −8.9 degrees for the red to +3.3 for the green, and their signs differ: the orange, red, magenta, violet and yellow tints turn one way, the blue, cyan and green the other. Each pigment’s own cancelling ink takes its twenty-per-cent residue under 1.25 degrees, the red’s to 1.2, the violet’s to 0.15, the yellow’s to 0.08. The signs left over are scattered — some tints a little short of cancelled, some a little past — because each strength was set at ten per cent and the twenty-per-cent tint wants a slightly stronger ink.

Nothing about the screen enters here. The census reads each tint as a uniform film over its coverage, with no rim at all — the floor case — and the light ink takes that floor almost to nothing. The rim’s tenth, which the screening essays chased, would ride on top, and a light ink printed through a well-chosen screen would leave about that tenth.

How much more paper a light ink covers #

The coverage a light ink needs rises steeply as it weakens. At its cancelling strength the orange covers 16 per cent of the paper to match a ten-per-cent tint; at a quarter strength it covers 30. Across the eight pigments the cancelling ink covers between a tenth more and twice the target’s area — the blue and cyan barely more, the red and green about twice. That is modest compared with what light inks are usually asked to do, which is to cover enough of the paper that their dots stop being individually visible in the palest tints; a cancelling ink at four fifths of the strength covers too little more to help with that.

So the two purposes a light ink can serve pull apart. Smoothness wants a weak ink and much coverage; the glow wants the ink weakened only to its pigment’s crossing. On unbrightened paper only smoothness matters, and the light inks printers use are tuned for it.

The ink at three strengths #

A light ink here is the same pigment with its absorbance scaled — the colour a thinner layer of the same ink would have — in the same vehicle. At 0.56 of its strength the orange still absorbs most of the blue, a fifth of it reflected where the full ink reflects a twentieth; at a quarter it reflects two fifths of the blue and has become a pale peach. The vehicle is unchanged in all three, and the vehicle is what puts the glow out; that combination — a paler colour carried by a full vehicle — is exactly what makes dilution a lever on the glow and not just on the colour.

What a printer should take from this #

On brightened paper, a light ink’s strength is a colour decision. The pale tints of a full-strength ink turn by up to nine degrees under a fully adapted reader; a light ink can remove that turn entirely, but only at a strength between two fifths and four fifths of the full pigment, different for each colour. The usual quarter-strength light inks reverse the turn instead.

The cancelling strength is stable enough to use. It moves by under a tenth from a five-per-cent tint to forty, so one light ink per colour, at its pigment’s crossing, holds the pale ramp near nought.

It is a property of the paper too. The census’s sheet is one office paper with one loading of brightener, and whiteness is mostly the lamp found how much a sheet’s glow depends on the ultraviolet in the light it is seen by; under a lamp with less ultraviolet the paper glows less and the crossing moves. The brightener is being used up found a sheet’s glow fading as it ages, and a sheet that glows less would cancel at a stronger ink, so the setting drifts with the sheet.

How the tints were computed #

Each tint is the census’s uniform-film tint on the brightened office sheet under D65: its reflected light is the Yule–Nielsen tint, with factor three, of the ink on the sheet’s reflected light, and its glow is the sheet’s glow from bare paper in full and through the ink’s film at the share its vehicle passes, scaled by the ink’s own transmittance. A light ink of strength f has the pigment’s absorbance multiplied by f: its reflectance on the 0.9 white is that white times the full ink’s ratio to it raised to the power f. For each target — a full-strength tint at five to forty per cent — the light ink’s coverage is the one that matches the target’s Oklab lightness on a sheet of the same colour whose brightener does not fluoresce; there the two inks’ tints differ by under 0.012 in Oklab, most by a few thousandths. The residue is the light-ink tint’s hue turn against the full-strength solid on the brightened sheet, read in Oklab by a reader adapted to the sheet’s white through Bradford’s transform, less the target’s turn against the solid on the non-fluorescent sheet. At full strength it is the census’s own residue to a part in a trillion. The cancelling strength is found by bisection between the weakest strength that can still reach the target and full strength.

What this does not reach #

A uniform film. Every tint is read with no rim. The screening essays’ rim adds its tenth on top, and a light ink’s larger coverage changes how much of its ink is rim.

A light ink’s own vehicle. The light ink is the same vehicle with less pigment. A light ink formulated with a different vehicle — less ultraviolet absorption — would put out less glow per unit area, and cancel at a different strength.

A pigment that dilutes simply. Scaling the absorbance assumes the pigment’s colour is a single absorber in a clear vehicle. Real light inks are reformulated, and their hue can drift from the full ink’s as they are diluted.

Still open: whether one ink can cancel two papers #

The cancelling strength depends on how much the paper glows. A printer who prints on several stocks — office paper and a premium sheet with more brightener, or a sheet whose brightener has faded — carries one light ink per colour, not one per paper.

The calculation is each pigment’s cancelling strength on the census’s three brightened sheets — light, office and premium — and the residue each sheet’s tints keep when printed with the office sheet’s cancelling ink. The prediction is that the cancelling strength falls as the brightener loading rises, because a sheet that glows more needs more of its paper covered to bring the tint’s glow to the line the reader expects, and that the office sheet’s ink leaves the premium sheet’s pale tints within two degrees and the light sheet’s within one — so that one ink, set for the middle sheet, serves all three far better than full-strength ink. If the cancelling strength moves by more than a fifth between sheets, a light ink set for one paper is a full-strength mistake on another. One light ink serves three papers within two degrees makes that calculation.

A lever that overshoots #

The habit here is about a lever predicted only by its direction.

The floor came from bare paper glowing, and a light ink covers more paper, so the prediction ran: weaker ink, less glow, less residue. Each step was right. What the prediction did not ask was whether the quantity it was lowering could go below nought — and the residue is a signed difference, a turn one way or the other, that a lever pushing it far enough reverses.

The failure mode is to treat a correction as a dial with one end, turned as far as the hardware allows. Light inks are made weak because weaker is smoother, and on brightened paper weaker is also further past the point that would have corrected the colour. The test is to sweep the lever through its whole range and read the signed result — here, ink strengths from a fifth to full — so that the setting where the error vanishes is found rather than assumed to be the far end.