The yellow stays sharp because it leans

A margin costs a press its corners shrank a coated press by a safety margin — kept only colours at least that far inside what the press can print — and found its corners paying several times the margin: the solid yellow, the sharpest, moved nearly four times as far inward as a colour on a flat face. A tolerance’s own unit rounds the corners restated the margin in ΔE₀₀, the unit a print buyer uses, and read each end colour’s move as the opening angle of a cone. In CIELAB every corner was sharper than a cube’s 71 degrees. In ΔE₀₀ every one opened past it, except the solid yellow, which opened only from 30 degrees to 59.

The reading offered was about direction. At the solid yellow a ΔE₀₀ unit is about five CIELAB units along chroma, about two across hue and about one and a half along lightness. A spike pointing straight out in chroma would be shortened far more than it is narrowed, and would open wide. If the yellow’s tip also pointed up the lightness axis, ΔE₀₀ would shorten it less. That is a statement about the tongue’s direction, and it can be measured without any margin at all. The prediction was that the axis leans towards lightness by thirty to forty-five degrees, and that ΔE₀₀’s scaling along the tilted axis accounts for the 59-degree opening to within ten degrees. If it did not, the yellow’s persistence would belong to something else — the formula’s hue weighting, which is near its lowest at yellow.

A smaller lean that still does the work #

The press’s light yellow tongue points out along the yellow’s own hue and leans 23 degrees towards lightness — less than predicted. A cone about that axis, rescaled by ΔE₀₀’s lengths at the tip, opens to 49 degrees at a margin of two, within ten of the 59 the margin found, and to 51 at four, against 57. About an axis straight out in chroma it would open to 71 and 73. ΔE₀₀’s own weights at the yellow do not sharpen it: its lightness weight, large at L 89, and its hue function, low at yellow, both round the corner, by thirteen and nine degrees. At every other end colour the same account falls more than ten degrees short of how round the margin found the corner.*

• The prediction half holds. The lean is smaller than predicted and still accounts for the yellow’s opening within the stated ten degrees.
• The alternative is backwards. A low hue weight rounds the yellow; it does not keep it sharp.
• The account is particular to the yellow. The other corners round more than rescaling a cone can explain, and something about their shape does the rest.

The tongue, measured #

The press is the collection’s coated CMYK press at a 320 per cent ink limit, held as a set of one-unit CIELAB cells, and the solid yellow is its most saturated light corner, at L* 89 and a chroma of 91. The tongue is the printable cells near it. Within twelve CIELAB units of the yellow there are 386 of them, and the axis is the direction from the yellow to the centre of those four to twelve units back — the direction the tongue runs into the body of the press, reversed.

It points out along the yellow’s own hue — within five degrees of it, and wholly outward in the chroma plane — and leans up 23 degrees towards lightness. In the yellow’s hue plane the tongue narrows to its tip from below and from inside, and its upper edge, the light yellows of thinning ink on paper, is flatter than its lower edge, where cyan and black would begin to enter. The tip points out and up.

The lean does not depend on how far back the tongue is read. Taking the centroid of the cells three to ten units from the tip gives 23.2 degrees; four to twelve, 23.1; five to fourteen, 23.1; six to sixteen, 23.5; eight to twenty, where the tongue has widened into the body of the press, 24.3. The tongue is straight over its first twenty CIELAB units, which is itself worth knowing: a corner whose axis wandered would not be a cone at any scale, and the account below would have nothing to rescale.

Twenty-three degrees is less than the thirty to forty-five the prediction named. It is not a small lean for what ΔE₀₀ does with it, as the next figure shows. The prediction reasoned from how steeply ΔE₀₀ weighs lightness against chroma at the yellow, and needed a steep lean to explain a large effect; the effect turns out to need less, because the chroma stretch at the yellow is so large — five to one against lightness — that even a moderate lean takes a third of it off the tongue’s length.

What a ΔE₀₀ unit is at the yellow #

ΔE₀₀ stretches CIELAB very unequally at the yellow, which is to say that its unit there is an ellipsoid rather than a sphere — the property three numbers for one ellipse measured three ways for any space’s discrimination contours. One ΔE₀₀ unit is 5.09 CIELAB units along chroma, because the formula divides a chroma difference by one plus 0.045 of the chroma, which at 91 is about five. Across hue it is 1.88: the hue difference is divided by one plus 0.015 times the chroma times a hue function, and that function is at 0.65 at yellow, near its lowest. Along lightness it is 1.59: the lightness difference is divided by a weight that grows away from L* 50, and at 89 it is 1.59.

Along the tongue’s axis a unit is 3.25 CIELAB units. The axis is mostly chroma and partly lightness, and a unit along it is a compromise between the two lengths, weighted towards the shorter by how steeply the axis leans. A tolerance has a grain found ΔE₀₀’s tolerance a long thin shape everywhere in colour space; at the yellow the grain lies along chroma, and a tongue leaning out of the chroma plane cuts across it.

Why a lean keeps a corner sharp #

A corner’s opening in a unit depends on how that unit stretches the corner along its axis against across it. A cone stretched along its axis more than across looks narrower; stretched across more than along, wider. ΔE₀₀ shortens the yellow most along chroma, so a cone whose axis is chroma is shortened most along its length and opens wide — 71 degrees, from 30 in CIELAB. A cone whose axis leans out of chroma is shortened less along its length, and its opening grows less: about the measured axis, 49.

The margin found 59. The account comes within ten degrees; the pure-chroma account, twelve beyond it in the other direction. At a margin of four ΔE₀₀ the account gives 51 against 57; at one it gives 42 against 63, where a single unit’s margin is a cell or two of the press and a cone is a poor description of so small a tip. So the lean is what keeps the yellow the sharpest corner in ΔE₀₀, as proposed, even at 23 degrees.

The alternative was that ΔE₀₀’s hue weighting does it, since its hue function is near its lowest at yellow and a low hue weight means a short ΔE₀₀ unit across hue. Put the hue function at one, as at an average hue, and the account’s opening falls from 49 to 40: the low hue weight makes a hue unit shorter, which shrinks the cross-section more than the length and widens the cone. Put the lightness weight at its mid-lightness value of one, and the opening falls to 36. Both of ΔE₀₀’s peculiarities at the yellow round it. Only the lean keeps it sharp.

At three margins #

The account follows the margin as it grows and stays below it. The found opening falls a little with the margin, from 63 to 57 degrees; the account rises a little, from 42 to 51. The two meet within ten degrees at two and four. The chroma-axis account sits above both at every margin, 62 to 73.

That the account under-predicts at every margin is worth noticing. A cone rescaled by the metric at its tip is the best case for a sharp corner: the tongue’s own flanks curve as they run back into the press, and ΔE₀₀’s lengths change as the colour moves away from the tip — chroma falls, and with it the stretch along chroma. Both round the corner at the scale a margin works at, and neither is in a cone. The yellow is the one corner where the account comes close.

At every other corner #

Everywhere else the margin rounds the corner far more than the account. The eight other end colours, at a margin of two, open to between 79 and 140 degrees; a cone about each one’s measured axis, rescaled by ΔE₀₀ there, opens to between 45 and 80. Every one is more than ten degrees short, most by thirty or more.

So the question the earlier essay asked — why does the yellow stay sharp? — turns out to be half of a question. The yellow stays sharp partly because its tongue leans, and the account that says so is an account of a cone. The other corners are rounder than their cones would be, which means their shapes near the tip are not cones at the scale of two ΔE₀₀ — they round off within a few CIELAB units, as corners of a voxel press made of overlapping inks do — and the yellow’s is. The fifth ink buys a corner found a press’s gamut growing, ink by ink, in the region each ink reaches; the corners are where those regions meet, and the solid yellow is a single ink on white paper, whose tongue runs a long way before another ink enters.

Which way the tongues lean #

The solid yellow’s lean is its own. The chromatic end colours at mid-lightness — the greens of yellow and cyan, the red of magenta and yellow — lie within thirteen degrees of flat. The two half-black yellows lean down by forty and forty-five degrees, their tips pointing into the dark, where black ink pulls the tongue towards the floor of the press. Only the solid yellow, at the top of the press, leans up.

That is why a margin stated in ΔE₀₀ treats it differently from everything else. A lean up out of chroma is available only to a colour whose tongue runs towards white, and on a CMYK press that is the yellow: the one chromatic ink light enough for its full-strength colour to sit near the paper’s lightness. The paper is the ceiling the tongue leans towards, and the paper is the white point is why a press’s colours are so often reported against it.

What a print buyer should take from it #

A ΔE₀₀ margin is not uniform in what it costs a press’s corners, and the yellow is the exception in a particular direction. It is the boundary’s side of what a press is charged for the direction it barely moves found for a press’s variation: a tolerance’s axes are not the press’s, and where they cross decides the bill. A buyer stating “two ΔE₀₀ inside the press” asks for the solid yellow to be given up about twice as far as the margin, while most corners give up a little more than the margin itself. What one number accepts found a single ΔE₀₀ value accepting very different sets of colours in different places; the yellow tongue is where a margin in that number accepts least of the press.

In numbers: at a margin of two ΔE₀₀ the solid yellow moves about four ΔE₀₀ inward, which at the yellow is nearly sixteen CIELAB units — a visibly duller yellow on the sheet. The red of magenta and yellow, at the same margin, moves about two and a half ΔE₀₀. A buyer who wants the press’s yellow kept would do better to state a margin along the tongue’s axis than a margin in a unit that happens to be long along chroma and short along lightness exactly there.

The reason is geometric, not the formula’s hue weighting. A tolerance formula with a different hue weight at yellow would not change it; a press whose yellow sat lower in lightness, or whose tongue ran flatter, would.

How the tongue was measured #

The press is the collection’s coated press at a 320 per cent ink limit, printable cells of one CIELAB unit. The end colours and their margins are the earlier essays’: each colour moved to the nearest cell of the press eroded by a margin in CIELAB or in ΔE₀₀, the move over the margin read as the opening of a cone, twice the angle whose sine is its reciprocal. The tongue at an end colour is the printable cells whose centres lie within twelve CIELAB units of it; its axis runs from the colour to the centroid of those four to twelve units away, reversed; its tilt is that axis’s angle above the chroma plane. ΔE₀₀’s local lengths are from its second derivatives at the colour, by finite differences of 0.02. The account treats the corner as a circular cone about the axis with the CIELAB opening, maps it into ΔE₀₀’s local coordinates, and reports twice the narrower half-angle; the counterfactuals change one length at a time — the axis to pure chroma, the lightness length divided by the lightness weight, the hue length rescaled to a hue function of one.

What this leaves out #

A corner is not a cone, and the account is a cone. The yellow’s close agreement is evidence its tongue is nearly one at the scale of two ΔE₀₀; the other corners’ disagreement is evidence theirs are not, and the account says nothing about what they are instead.

ΔE₀₀’s lengths are taken at the tip. Across a margin of two ΔE₀₀ at the yellow the colours reached lie up to ten CIELAB units inward, where chroma is lower and ΔE₀₀ stretches chroma less; a margin sees an average of lengths the account takes at one point.

One press. A press with a stronger yellow or a brighter paper would move the tip up in lightness and the lean with it.

Still open: whether the other corners round at the scale of their inks’ overlap #

At the eight other corners the margin finds more rounding than a rescaled cone allows. The candidate is that those tongues are not cones at all at the margin’s scale: they are edges where two inks’ full strengths meet, rounded off within a few CIELAB units by the way coverage trades one ink for another near the limit.

The calculation is each tongue’s cross-section measured at successive distances back from its tip, its width against distance, with the opening a cone would give set against the width’s actual growth. The prediction is that the yellow’s cross-section grows linearly from the tip — a cone — while the two-ink corners’ cross-sections grow fast over the first two or three units and then slowly, the shape of a rounded tip, and that the length of that rounded stretch in ΔE₀₀ matches the extra rounding each corner showed. If so, the margin’s rounding at those corners is the press’s own, already present in CIELAB and exposed by ΔE₀₀’s scale.

A reason offered for one case must be tested on the others #

The habit is about how much a confirmed explanation explains.

The lean was offered to explain one corner’s behaviour, and it did: a measured axis, a standard rescaling, and a number within ten degrees of the one to be explained. It would have been easy to stop there. Asked of the other eight corners, the same account missed every one by more than ten degrees, and missed them all in one direction.

The failure mode is to test an explanation only where it was proposed. The yellow’s agreement says the account captures what is special about the yellow. The others’ disagreement says the account is not a general account of how ΔE₀₀ rounds a press — and that is the half of the finding that the next question comes from.