A yellowing base leaves the ink where the glow put it

A brightened paper glows: its optical brightener absorbs ultraviolet and emits blue, so that under daylight the sheet returns more blue than falls on it. A pale tint printed on such a sheet covers some of the brightener, and the glow beneath the ink is lost, so the tint’s hue turns away from the ink’s own even for a reader fully adapted to the paper. A light ink can cancel the glow at one strength found that printing each pigment diluted, over more of the sheet, leaves less of the brightener exposed per unit of colour and at one strength cancels the turn; one light ink serves three papers within two degrees found that strength falling along nearly straight lines in the share of each sheet’s luminance that is glow.

A brightener bleaches with use. An ink for a fading sheet is set for its middle age bleached the premium sheet by a quarter, a half and three quarters, holding its base paper as it was, and found every pigment’s cancelling strength rising back along the census line as the glow fell, a shade under it — by as much as 0.013 for the orange at half bleach — and the office sheet’s ink, set for a glow the premium sheet passes through, keeping every residue within 1.4° once any of the brightener had gone.

The likeliest cause of the shade under the line seemed to be the base. The three census sheets differ in their base papers as well as their brighteners — the light sheet’s base yellower, at a dip depth of 0.17 against the premium’s 0.13 — and a sheet that has lost its brightener in daylight has had its base yellowed by the same light. The prediction was that raising the premium sheet’s yellowing in step with its bleaching, to 0.1433, 0.1567 and 0.17, lifts the strengths onto the census line within 0.01 at every stage, and that the office ink still keeps every residue within one and a half degrees. If the yellowed sheets had moved off the line by more than the bleached ones, the base’s colour would enter the cancelling strength on its own account, and a printer would have to read a sheet’s whiteness as well as its glow.

The yellowing moves nothing #

On every yellowed sheet every pigment’s cancelling strength lies within 0.0023 of the merely bleached sheet’s, and its gap to the census line at the sheet’s own glow within 0.0006 of that sheet’s gap. The orange sits 0.0025, 0.0132 and 0.0098 under the line at a quarter, a half and three quarters, against 0.0019, 0.0126 and 0.0103 with its base held. The office ink leaves the yellowed sheets’ worst residue at 1.38°, 1.13° and 1.38°.

• The office ink’s half of the prediction holds, at exactly the residues it left on the merely bleached sheets.
• The strengths’ half fails. The yellowing does not lift any strength towards the line; its largest effect on a gap is six ten-thousandths, a twentieth of the gap it was meant to close.
• Neither alternative named in advance happens. The yellowed sheets move off the line by no more than the bleached ones; they move not at all.

On the scale of the figure, where the five pigments’ strengths run from 0.43 to 0.85, the yellowed sheets’ points sit on top of the merely bleached ones’. The orange, yellow and blue rise back as the glow falls, each a little under its line at the middle stage; the red rises less; the green, whose strength hardly depends on the sheet at all, stays at 0.435 to 0.437 throughout. A printer reading this figure would set every ink for its sheet’s glow and ignore its base, and would be right to.

Why a yellower base does so little #

The base’s yellowing is a small change beside what the brightener does to the same sheet. Bleaching the premium sheet from a quarter to three quarters takes its b from −12.7 to −7.3; raising the base’s yellowing to the light sheet’s adds 0.12, 0.23 and 0.31 to those values at the three stages.* The yellowing is a dip in the base’s reflectance at the short-wave end, falling away over sixty-odd nanometres into the visible, so most of it lies where the eye is least sensitive and where the brightener’s emission is strongest; it moves the sheet’s colour a sixteenth as far as the bleaching does.

It moves the glow even less. The yellowed sheet’s glow is 0.76, 0.64 and 0.43 per cent of its luminance, against the bleached sheet’s 0.77, 0.65 and 0.44. A yellower base reflects a little less of the brightener’s own blue emission and absorbs a little more of the violet the brightener would have converted, and those two effects nearly cancel at this size. The cancelling strength follows the glow, the glow barely moves, and so the strength barely moves either.

So the yellowing that real photo-ageing brings at the rate this census models is too small to see in the ink. A base that yellows much further — a groundwood stock, or a sheet kept in sunlight for years — would be a different paper, and this calculation does not reach it. What it does say is that for a premium stock fading towards a lightly brightened one, the base’s change is a correction of a few ten-thousandths on the strength.

The gap was never the base’s #

Every pigment whose strength depends on the sheet sits under the census line at every stage, the orange and the violet deepest at half bleach, at −0.0132 and −0.0130; the red, magenta, blue, cyan and yellow at −0.0036 to −0.0096 there. The green sits a few ten-thousandths above. The yellowing changes none of these by more than six ten-thousandths, in either direction. The gap’s size and its pattern across pigments are what they were before.

If the base were the cause, the gap should have shrunk as the base approached the light sheet’s. It did not, and so the cause has to be something the bleached sheets share with the census sheets whatever their base. The obvious candidate is the line itself.

A chord across a curve #

Read against their own straight line, the three census sheets do not lie on it. The light sheet sits above it by 0.0086 for the orange and the premium sheet by 0.0075, and the office sheet, between them in glow, sits under it by 0.0161 — deeper than any bleached sheet. The violet’s three are +0.0081, −0.0153 and +0.0072; the yellow’s +0.0059, −0.0110 and +0.0051; the blue’s +0.0035, −0.0066 and +0.0031. Every pigment whose strength moves with the sheet shows the same pattern: the ends above, the middle below.

That is what a straight line through three points of a sagging curve looks like. The cancelling strength falls with the glow, and falls more slowly as the glow grows — the curve is convex — so the line through its two ends and its middle lies above it between the ends. The bleached sheets lie at glows between the light sheet’s and the premium’s — two of them between the office and the premium, the last between the light and the office — where the line is above the curve, and they sit under it by less than the office sheet does because none of them sits at the middle of the sag. The gap is deepest at half bleach for every pigment, the stage whose glow, 0.65 per cent, lies nearest the office sheet’s 0.57.

An ink for a fading sheet is set for its middle age read the census line as a fit to three sheets and the bleached sheets’ shade under it as a departure. It was a departure from the line and not from the curve, and the office sheet’s own departure had been on record since one light ink serves three papers within two degrees called the lines nearly straight. Nearly was 0.016.

The sag is easy to size from the census sheets alone. For the orange, the cancelling strength falls by about 0.37 for each percentage point of glow between the light sheet, at 0.33 per cent, and the office, at 0.57; between the office and the premium, at 0.84, it falls by about 0.18 — half as steeply. Drawn straight from the light sheet to the premium, a chord passes 0.024 above the office sheet; a least-squares line through all three points splits that sag, leaving the middle two thirds of it below and each end a third of it above — −0.016, +0.009 and +0.008, which is what the census line shows. The halving slope accounts for the office sheet’s place and the bleached sheets’ shade without anything else in the model.

Why the strength should flatten as the glow grows is a question about the ink’s job, and the census reads the fact without tracing it. The light ink is set where the tint’s lost glow is balanced by spreading the pigment over more of the sheet, and both sides of that balance depend on how much brightener the ink leaves exposed. Stochastic dots hold the glow at every tint found the residue rising with the screen’s ruling at every coverage, because a finer screen exposes the brightener differently at its dot edges; the cancelling strength is the same quantity read the other way, and nothing requires it to be linear in the glow.

The inks hold as before #

Printed with the office sheet’s light inks the yellowed sheet’s worst residue from five to twenty per cent is 1.38° at a quarter bleached, 1.13° at a half and 1.38° at three quarters; with the fresh premium sheet’s own inks it is 1.75°, 1.80° and 2.37°. Each value matches the merely bleached sheet’s to the hundredth of a degree. The office ink, set for a glow the fading premium sheet passes through on its way down, keeps every residue under one and a half degrees at every stage; the fresh sheet’s own ink passes two degrees once three quarters of the brightener has gone.

The reason the office ink serves the whole life is unchanged. A light ink set for a middle glow overshoots a little on a fresh sheet and falls short a little on a spent one, and the worst residue over a sheet’s life is set by how far the life’s two ends lie from the middle. The yellowed base moves neither end.

Pigment by pigment at a five-per-cent tint, the office ink leaves the orange −1.38°, −0.70° and +0.56° over the three stages, the cyan +0.64°, +0.32° and −0.24°, the red −1.14° to +0.03°; six of eight pigments cross nought between a quarter and three quarters bleached, and the green and magenta, which barely depend on the sheet, stay on one side. That is the middle-age ink at work: a residue of one sign early, of the other late, and near nought where the sheet’s glow passes the office sheet’s. The figure is the merely bleached sheet’s to the hundredth, and the sign changes happen where they happened.

The rim was never most of the glow and a thin dot edge lets the glow back in treated the ink film itself — its edges and how much brightener they expose — and found the edge’s effect small beside the film’s coverage. The base’s yellowing joins it: a second effect on the sheet that the cancelling strength is insensitive to, because the strength is set by the glow under the ink and almost nothing else.

What a printer should take from this #

Set a light ink for the sheet’s glow and ignore the base. A premium stock that ages towards a lightly brightened one yellows its base by about as much as the light sheet’s is yellower than the premium’s, and that moves no cancelling strength by more than a few thousandths.

Read a straight line through three sheets as a chord. The cancelling strength falls with the glow along a convex curve; a line through three stocks overstates the strength between them by up to 0.016. For a sheet whose glow falls between two census stocks, the nearer stock’s strength is a better guide than the line.

The office ink still serves a premium sheet’s whole life, within one and a half degrees, yellowed or not.

How the sheets were built and read #

Each sheet is the census’s model: a base paper whose reflectance is a level less a short-wave dip of stated depth falling away with a sixty-two-nanometre decay, a brightener whose loading sets how much ultraviolet it absorbs and how much blue it emits, and the two combined under a wide D65 into a reflected and an emitted spectrum. The premium stock’s loading is reduced by a quarter, a half and three quarters; on the yellowed sheets its base’s dip depth is raised from 0.13 by four hundredths times the share bleached over three quarters. A pale tint is a uniform film of a pigment diluted to a strength, its coverage spread by Yule–Nielsen with an exponent of three, the brightener under it shielded by the ink vehicle’s ultraviolet absorbance; the residue is the tint’s hue in Oklab against the full-strength ink’s, for a reader adapted to the sheet by a Bradford transform. The cancelling strength is found by bisection on the residue’s sign at a ten-per-cent target. The census line is the least-squares line in glow through the light, office and premium sheets’ strengths.

What this does not reach #

Heavier yellowing. The base’s dip is raised only as far as the light sheet’s, which is what the census holds; real ageing over years in sunlight goes further, and lignin-bearing stocks yellow faster and differently.

A yellowing that changes the brightener. Photo-yellowing products can quench a brightener’s emission as well as tint the base; that is a change in the glow, and the census models the glow only through the loading.

The curve’s shape between the stocks. The chord is read through three census sheets and three bleached stages, which shows its sag but does not trace the curve.

Still open: whether the strength is one curve in the glow #

The census sheets and the bleached and yellowed sheets all sit where a single convex curve in the glow would put them, the office sheet furthest under the chord, though the sheets differ in their base papers and their brighteners. If the cancelling strength depends on the glow and on nothing else about the sheet, a premium sheet bleached until its glow is the office sheet’s should need the office sheet’s ink.

The calculation is the premium stock bleached until its glow equals the office sheet’s 0.57 per cent and the light sheet’s 0.33 per cent, every pigment’s cancelling strength read on each and set against the office and light sheets’ own. The prediction is that each matches within 0.003 for every pigment — the curve is one curve whatever the stock — so that the office sheet’s place under the chord is a property of the glow. If the bleached premium sheets differ from the census sheets at the same glow by more than the yellowing moved them, the strength depends on how the glow is made — on the brightener’s spectrum against the base’s — and the glow alone does not set the ink.

A residual explained by the wrong difference #

The habit here is about where a small residual comes from.

The bleached sheets sat under a fitted line, and the obvious difference between them and the sheets the line was fitted to was their base paper, which had been held fixed when real ageing would change it. Changing the base tested that explanation and removed it: the residual stayed to the ten-thousandth. Reading the fitted sheets’ own residuals then showed the line passing above its own middle point by more than any bleached sheet fell below it.

The failure mode is to explain a residual from a fitted line by the most visible difference between the new points and the fitted ones, before reading the fitted points’ own residuals from the line. The test is to plot every point’s distance from the line, the fitted ones included, and ask whether the new points’ residual is a departure from the line or the line’s departure from the curve.