The fourth ink is not for colour

Three inks are enough. Cyan, magenta and yellow between them absorb across the whole visible band, and the theory of subtractive colour needs nothing else — a fact every account of printing states before adding, without much explanation, that presses use four.

The usual explanations are that three inks cannot make a good black, that black ink is cheaper, and that text needs to be sharp. All three are true. None of them is the measurement, and the measurement is more interesting than any of them.

The claim #

Black adds 37 per cent to the volume of a three-ink gamut and adds nothing at all to its chroma above mid lightness. The entire contribution is depth.

Measured as the maximum chroma reachable at a given lightness and hue, with and without the fourth ink:

Four inks first, three second. At L* 70 and L* 50 the two are identical in three of four directions — black cannot help a colour that is not dark, because adding black ink to a light colour makes it a different colour rather than a more saturated one. At L* 20 the three-ink set has all but vanished, and what remains of it is not a gamut but a corner.

The parting lightness is different for every hue #

The claim above says black adds nothing above mid lightness, and the table it rests on says something more specific that is worth reading out, because it identifies the mechanism rather than the effect.

The two boundaries do not part company at one lightness. Reading the table by column rather than by row:

The order is the order of how light the ink that carries that hue is. Yellow solid is by far the lightest of the three chromatic inks; blue is not an ink at all but cyan and magenta together, which are the two darkest. So the parting lightness is not a property of black. It is a property of how far down each direction the three-ink set can go before it has to start using ink it does not want.

A dark yellow is the case that makes it obvious. To reach L* 70 at hue 90° a three-ink press has yellow, which is already lighter than that, so it must darken with cyan or magenta — and both of them move the hue away from yellow while they do it. Black darkens without moving anything, which is why the fourth ink is worth four units of chroma there and worth nothing at all at 270°, where the inks that make blue were dark to begin with.

That reframes the headline. Black is not a shadow ink; it is a substitute for whichever chromatic ink is currently being spent on lightness rather than on hue — and how soon that spending starts depends entirely on which direction is being asked for. A gamut volume adds those four columns together and reports one number, which is why the 37 per cent looks like a general improvement and is in fact four quite different improvements at four quite different lightnesses.

Why three inks cannot get dark and stay coloured #

The reason is worth deriving rather than asserting, because it explains the whole shape of the table above.

A subtractive set makes a colour dark by absorbing, and it makes a colour saturated by absorbing selectively. With three inks those are the same knob. To reach a low lightness, all three have to be near solid; but the three together absorb across the whole band, so what comes back is nearly flat, and flat is neutral.

The darkest a three-ink set gets here is L* 19.6, and at that point its chroma is 1.3 — it is a dark grey with a slight cast. The four-ink set reaches L* 2.4.

Black ink breaks the coupling. It absorbs flatly and strongly, so it moves lightness without moving hue, which is exactly the degree of freedom the three-ink set is missing. Everything the fourth ink does follows from having one absorber whose job is not selective.

The second reason, which no gamut measurement shows #

The volume argument is the one a colour scientist reaches for. A printer would give a different one first, and it does not appear in any gamut at all.

A neutral grey built from three inks is a balance. Take the grey that 50 per cent black produces — L* 50.1, a* −0.3, b* −0.9 — and ask the three-ink press for it: 45 per cent cyan, 35 magenta, 35 yellow. That combination is neutral because the three are in the right ratio, and a press does not hold ratios.

The magnitudes are similar. The direction is not. A press drifting on the cyan unit turns a three-ink grey blue-green; the same drift on the black unit makes a black-only grey lighter. A cast is visible in a way a density change is not — a grey that has gone slightly green is instantly recognisable, and a grey that is two units light is a grey.

That is the practical argument for putting as much of the neutral axis as possible on the black ink, and it is a robustness argument rather than a colour one.

The third reason, which is arithmetic about liquid #

A sheet can only carry so much ink before it stops drying, and the limit is quoted as total area coverage — the sum of the four percentages. Three hundred per cent is a common figure for coated stock, 240 for newsprint.

A three-ink set spends its entire budget getting dark: the darkest colour it can make is 300 per cent. A four-ink set reaches a far deeper black at 400 per cent and can reach almost the same one under a limit, because the limit costs the shadows rather than the gamut and black is the most efficient way to spend the last of the budget.

Replacing three inks with one where they overlap is called grey component replacement, and it is the same operation viewed as a saving: wherever all three chromatic inks are present, the amount they have in common is a grey, and a grey is what black is for. It reduces total ink, reduces cost, speeds drying, and — the reason a pressroom cares most — makes the neutral axis stable.

It is also the choice that generates the field’s central surprise, because how much to replace is a free parameter, and every value of it produces the same colour under the profiling illuminant.

The neutral axis is where the eye is watching #

The grey-balance argument above deserves more than a table, because it rests on a fact about the observer that the rest of this field can take for granted and this rung cannot.

A cast on a neutral is the most visible colour error there is. The visual system spends most of its effort deciding what the illuminant is and discounting it, and the whole apparatus is calibrated on the assumption that the average of a scene is roughly neutral. A grey that is not grey therefore reads as the light being wrong rather than as the patch being wrong, and the error propagates across everything nearby — which is why a slightly green grey in a photograph of a face is catastrophic and a slightly green leaf is not noticed at all.

There is a measurable version of it in this site’s own difference machinery: ΔE2000’s chroma weighting makes a fixed displacement in a* and b* count for more near the neutral axis than out at high chroma, by design, because that is what the discrimination data say. A three-ink grey drifting by 4.1 units of chroma is being measured in the region where the formula’s weighting is harshest, and it is being seen in the region where the visual system’s is harshest too.

So the printer’s instinct — put the neutrals on one ink and keep them there — is a robustness argument aimed at the one part of the space where robustness is worth the most.

How many ways a grey can go wrong #

The drift table above compares magnitudes and directions. The property it does not show is the one a pressroom actually experiences, and it is a counting argument rather than a colorimetric one.

A three-ink grey is held by three numbers, so the space of things the press can do to it is three-dimensional. Decompose that space and the three directions are not alike:

• all three units drifting together moves the patch along lightness and leaves it neutral — the benign direction, and there is one of it;
• the other two directions are differential — cyan up against magenta, or yellow against the other two — and every one of them turns the grey a colour.

So one of three directions is harmless. A black-only grey is held by one number, and its single direction is the benign one. The comparison is not that black drifts less; the drift table says the magnitudes are within about a quarter of each other. It is that one direction in three is safe on a three-ink build and one in one is safe on a black build.

The consequence compounds in the way variances do. If the four units drift independently — which is roughly what a press does, since they are separate inking systems with separate rollers and separate temperatures — the chromatic error of a three-ink grey accumulates from two independent sources while its lightness error accumulates from the third. The black grey accumulates all of its error in lightness and none anywhere else, no matter how many units are drifting or how far, because there is no combination of one ink’s coverage that has a hue.

That is a stronger statement than the table makes, and it is the reason the practice is stated as an absolute in pressrooms rather than as a trade. Moving a neutral onto the black plate does not reduce the error; it moves all of the error into the one direction nobody can see. The three-ink build’s error is not larger — it is merely pointed somewhere visible, in the one region of colour space where the eye and the difference formula are both at their most sensitive.

How much to replace is a policy, not an answer #

One last thing this essay opens and does not close. Given that black can carry the grey component, how much of it should it carry?

The trade names the extremes. Undercolour removal takes ink out of the neutral shadows only, leaving the light and mid tones as three-ink builds; it is conservative, it was what could be done photomechanically, and it leaves most of the stability problem in place. Grey component replacement applies the substitution across the whole tonal range, up to and including light tints, and takes the total ink down substantially — the family drawn above runs from 161 per cent to 98.

Neither is right, and the reasons for choosing are all outside colorimetry: heavy replacement dries faster and costs less, light replacement is more forgiving of registration error and gives smoother gradations in flesh tones, and shops have house preferences that follow their equipment.

What makes the choice interesting rather than merely practical is that it is invisible in the measurement. Every member of the family is the same colour under the profiling illuminant, so no instrument reading a patch under D50 can say which policy produced it — and that is the fact the next rung is entirely about.

Where this model stops #

The fourth ink here is a smooth flat absorber, and real black inks are not quite. A carbon black has a slight blue reflectance rise at the long end and most process blacks are marginally warm; the model’s black has a stated small structure and is otherwise level. Nothing about the argument changes, and the exact volumes would move by a per cent or two.

The drift model moves one unit at a time. The counting argument above says what happens when several move together, and it is an argument about directions rather than a measurement of one: no combination of drifts was actually searched for the worst case, and a press whose units are correlated — which they are, through temperature and through the substrate — does not sample those three directions evenly.

Registration is not modelled at all, and it is the reason black is used for text. Four separately-registered plates cannot align to better than a fraction of a dot, so a letterform built from three inks has coloured fringes; one built from black alone does not. That is a spatial argument about a printing press, not a colorimetric one, and it belongs to a subject this site does not cover.

Cost is real and is not modelled. Black pigment is the cheapest of the four by a wide margin, and grey component replacement is sold to printers on the ink bill as often as on stability.

And the gamut comparison assumes the same ink limit for both sets. A three-ink press has no reason to observe a 320 per cent limit, since it cannot exceed 300 per cent; the comparison above is therefore slightly generous to the three-ink case, which is the right direction for an argument that concludes the fourth ink is worth having.

Two more slices say that the fourth ink buys nothing anywhere except at the dark end, which is the claim the volumes were summarising.

The generalisation #

A device gains a fourth control not to reach further out but to decouple two things its first three controlled together.

The pattern recurs wherever a small number of channels has to do several jobs. Three cone types collapse a spectrum onto three numbers and cannot separate a change of illuminant from a change of surface, which is why the visual system needs a mechanism — constancy — rather than a fourth cone. A camera’s three channels cannot separate exposure from colour, which is why raw carries them separately. A four-ink press separates lightness from hue.

The measurement to make on any such addition is the one made here: hold the outer boundary and ask what changed inside it. A fourth ink that added chroma would show as a wider gamut at every lightness; this one shows as a deeper one at one end, and as a different answer to the question of how to reach a colour it could already reach. The second is invisible in any gamut plot and is most of the value.

Who found it, and when #

Four-colour process printing is older than any of the theory here: the three-colour patents of the 1890s were followed almost immediately by the addition of a black printer, for the practical reasons above, and it was settled practice long before anybody could measure a gamut volume.

Grey component replacement as a deliberate, computed policy arrived with electronic prepress in the late 1970s and early 1980s, when a scanner’s output could be processed rather than merely exposed. Before that, the amount of black was set by the skill of the person making the separation, and “undercolour removal” — taking ink out of the shadows only — was as far as it went, because it could be done with a mask.

That the choice is a one-parameter family with a colorimetric identity along it, and that the members of the family are metamers of one another, is a modern observation and is the subject of the essay two rungs above this one. It could not have been made while the choice was a craft decision, because a craft decision does not come with a parameter.

Where the ladder goes next #

This rung sits on the halftone and on the gamut comparison, which is where the four-ink solid is measured against a display’s.

Beside it, a black that is not black takes the depth this essay claims for the fourth ink and asks what it actually amounts to — a printed black is L* 2.4 at best and the contrast against the paper is about 320 to 1, which is less than any display in a dark room.

Above, the free parameter this essay introduces becomes the field’s central measurement: the separation is not unique, and its members come apart under a different lamp.