A press is charged for the direction it barely moves

A colour tolerance written as one ΔE₀₀ is, pulled back into tristimulus values, a long thin shape, and a tolerance has a grain found that its thin direction points nearly the same way across the whole sRGB cube: X up and Y down, the direction of a*. It priced drifts of stated shape one at a time against that grain, and closed on the question the grain leaves open. A press does not drift one shape at a time. It varies its four ink densities, its dot gain and its trapping together, sheet after sheet, and what matters is how much of that variation lies along the one direction the tolerance does not forgive.

Most of a press run lies the loose way, and the tight sliver pays #

A press’s sheet-to-sheet variation runs almost entirely along the direction a colour tolerance forgives, and the small part that does not is priced out of all proportion to its size.

• Along the tolerance’s tightest axis a press puts between 1.5 and 2.0 per cent of its variation, and that axis pays between 29 and 34 per cent of the price, averaged over the patches, under four stated mixes of press variation.
• Along the loosest axis it puts 79 to 83 per cent of its variation, and that axis pays 28 to 39 per cent — about the same as the sliver.
• On a blue overprint the tightest axis pays between 76 and 90 per cent of the price for between 0.7 and 3.6 per cent of the variation.
• The largest single component of a press’s variation lies within 13 to 17 degrees of the white on the median patch and carries 89 to 99 per cent of the variation; the rest carries about four to six and a half times its share of the price.

A covariance, taken apart along the tolerance #

The press here is the four-colour model the collection’s printing essays are built on: four inks as absorbance bands, Neugebauer primaries with a double pass and trapping, Demichel areas — because a halftone is not a mixture but an area average of the overprints — a mechanical dot gain and a Yule–Nielsen exponent. Thirty patches are printed on it, the same thirty a soft proof was matched against: every combination of none, forty per cent and solid in cyan, magenta and yellow, a fifty per cent black tint, a stone grey and a skin tone. Bare paper carries no ink and no press setting moves it, so twenty-nine patches vary.

Each patch is printed three hundred times with the press settings drawn afresh for every sheet, and read under D65. The press varies in seven ways: each of the four solid densities on its own, all four densities together, the dot gain, and the trapping of an ink laid on wet ink. How much each varies is a statement rather than a measurement, and there are four such statements, called mixes here: an even mix; inking alone, with every density varying independently and nothing else; a gain-led press, whose plates or blankets vary its dot gain most; and a trap-led press, whose ink-on-ink transfer varies most. None is a named press. They bracket the shapes a press’s variation might take, so that anything true under all four is a statement about the tolerance rather than about a mix.

The three hundred readings of one patch form a cloud around the nominal print, and a cloud has a covariance: three variances and three correlations in tristimulus values. What one number accepts described the tolerance at the same patch as a quadratic form — the local metric, whose three principal axes are the tolerance’s tightest, middle and loosest directions and whose three stiffnesses say what a unit of movement along each costs. The expected squared colour difference of the cloud is the metric applied to the covariance, and in the tolerance’s own axes it is a plain sum: each axis’s stiffness times the variance along it. So both the variation and the price split exactly into three shares, one per axis, and the shares can be compared.

Pricing through the local metric is linear in the covariance, which is what allows the split. It is also an approximation to the full formula, and it is checked against it: every sheet is also priced directly in ΔE₀₀, and the root-mean-square of those three hundred differences agrees with the metric’s price to within one per cent on the median patch under every mix.

The white carries the variation #

Under the even mix a patch puts, on average, 1.5 per cent of its variation along the tightest axis, 15.1 per cent along the middle one and 83.4 per cent along the loosest. The price splits 29.3, 34.9 and 35.8. The axis holding five parts in six of the variation pays a little over a third of the price, and the axis holding three parts in two hundred pays nearly three tenths.

The other three mixes move the numbers and not the shape. A gain-led press puts 1.8 per cent along the tightest axis and pays 29.3 per cent there; a trap-led press 1.6 and 28.8. Inking alone is the mix that reaches furthest along the tight axis, at 2.0 per cent of the variation, and there the tightest axis pays 34.4 per cent and the loosest only 27.6, less than the sliver.

The reason the variation lies the loose way is physical, and it follows from what each source does to a patch’s spectrum. A thicker film of every ink, a larger dot and a better trap all do the same kind of thing: they cover more of the sheet with more absorption, across the broad bands where the inks absorb, so the patch reflects less light of roughly the colour it already had. That scales its tristimulus values together, which is the direction of the white, and the grain found the white to be the direction a tolerance is loosest along. The press’s largest component lies 13 degrees from the white on the median patch under the even mix, and 87 degrees from the tolerance’s tightest axis, which is as near perpendicular as three-dimensional directions usually get.

The sliver is priced by the stiffness #

The tight axis pays so much because of how much stiffer it is, and the right-hand panel of the opening figure is that statement drawn.

On the blue overprint the tolerance is 28 times narrower along its tightest axis than along its loosest, so a unit of movement along the tight axis costs 28 squared — nearly eight hundred — times as much in squared colour difference. Drawn in tristimulus units, the three hundred sheets make a long streak along the loose axis and hardly move up. Rescaled so the tolerance is a circle, the same sheets stand up: the small movement along the tight axis has become larger on the page than the long one across. In the tolerance’s own units the cloud is taller than it is wide, and the height is the price. That is 1.3 per cent of the variation and 84 per cent of the price on that patch. It is also most of the reason the patch fails: under the even mix nine sheets in ten land more than one colour difference from the nominal print and four in ten more than three, with the median sheet at 2.47.

A size is not a direction found that two departures cannot be composed without the angle between them in the local metric. The same object appears here in its statistical form. A covariance is a population of departures, the metric weights each direction by its stiffness, and a direction that carries almost none of the departures can still carry most of the price if it is stiff enough. The ratio between the stiffest and the softest axes is the whole of the effect: had the tolerance been round, the shares of price would have equalled the shares of variation exactly.

Which patches the tight axis owns #

On five of the twenty-nine patches the tightest axis pays more than half the price. The blue overprint with forty per cent yellow leads at 84 per cent, the three-ink solid follows at 69, yellow with forty per cent magenta at 68, the plain blue overprint at 65 and a forty per cent magenta tint at 51. At the other end the fifty per cent black tint pays nothing along its tightest axis, because a black tint’s variation is all lightness — the property the fourth ink is not for colour credits black with, a neutral that holds its hue when the press drifts.

The blue overprints lead for a reason that can be read off the inks. Cyan absorbs in the red and magenta in the green, and on the press each is laid by its own inking unit. On the blue overprint with forty per cent yellow a one per cent rise in cyan’s density takes X down more than Y, and the same rise in magenta’s takes Y down more than X. When the two units agree, those opposite tilts cancel: both densities up by one per cent moves the reading 89 degrees from the tight axis and costs 0.22 colour differences. When they disagree they add: cyan up and magenta down moves it 65 degrees from the tight axis and costs 0.81 — nearly four times as much for the same two one-per-cent changes. On the three-ink solid the ratio is 1.00 to 0.31. The tolerance is charging for the two units’ balance rather than for the level of either.

Yellow with forty per cent magenta is the counter-example, and it shows the tight axis can be paid by one source alone. On that patch there is no cyan to balance, and the source that pays is the dot gain: 56 per cent of the variation and 71 per cent of the price, where magenta’s own density is 21 per cent of the variation and 12 of the price. A forty per cent magenta tint on a solid yellow ground is a tint, and a tint’s colour moves with its dot. So two different mechanisms put a patch at the top of the list — two units disagreeing on a solid overprint, and a single tint’s gain — and the list cannot be predicted from the inks alone.

What the mix changes, and what it cannot #

When the only variation is independent inking, the tight axis’s average share of the price rises from 29 to 34 per cent, and on the blue overprint with yellow to 89.9 per cent, for 3.6 per cent of the variation. That is the balance argument at its strongest. With no common density, no gain and no trapping, nothing moves every ink together, so a larger share of what happens is one unit against another.

A trap-led press shows the opposite case on the three-ink solid. There the largest component of the variation — how well each ink transfers onto the one beneath — pays 74 per cent of the price. Under inking alone the largest component on the same patch pays 7 per cent. Which part of a press run pays for the three-ink solid depends on how the press varies; that the tight axis is a sliver of the variation does not. Under all four mixes, on every patch, the loosest axis holds at least seven and a half times the variation of the tightest; the closest the two come is a red overprint on the trap-led press.

Which part of the press pays #

Splitting by source instead of by axis says the same thing in the press’s own terms. Under the even mix, dot gain is 33 per cent of the variation and 32 per cent of the price: a gain change mostly lightens or darkens, and the tolerance is indifferent to it in proportion. All four densities moving together are 21 per cent of the variation and 14 per cent of the price, priced at two thirds of their share, because a common change of inking is a change along the white. Cyan’s density moving alone is 13 per cent of the variation and 20 per cent of the price, half again its share, because cyan varying against magenta is the tight direction. Yellow’s density alone is 7.5 per cent and 10.0; magenta’s 17 and 17; trapping 7.6 and 7.1; black 0.7 and 0.7.

Part of why cyan is charged more than magenta shows on the blue overprint. Neither ink’s density alone moves the reading far towards the tight axis — cyan’s change lies 74 degrees from it and magenta’s 82 — but cyan’s is the nearer, and it costs 0.46 colour differences per per cent against magenta’s 0.37. Averaged over every patch the premium comes from many such geometries, and no single angle explains the ranking; what holds everywhere is the rule the next section tests, that a change the units share is cheap and a change one unit makes alone is not.

Together is cheap under every mix #

A press whose trapping varies most changes the ranking and not the rule. Trapping becomes the largest source, at 32 per cent of the variation and 35 per cent of the price, with dot gain at 23 and 21. All four densities moving together are 18 per cent of the variation and 12 per cent of the price — priced at two thirds of their share again. Cyan’s density alone is 8.9 per cent and 12, yellow’s 6.2 and 7.8, both above their share; magenta’s 12 and 11 is just below, and black’s is under one per cent of either.

So who pays follows who varies, which a press can change, and the rate at which each source is charged follows which way it moves the reading, which it cannot. A change that every unit shares lies along the white and is charged at a discount under both mixes. A change that one unit makes alone has a component across the white and, for cyan and yellow, is charged a premium under both.

What a press control is watching #

A press is usually controlled by density: a densitometer reads each solid through a filter matched to its ink and the operator holds each within limits. That is a control of each unit’s level. What the tolerance charges for on the patches it is hardest on is the balance between two units, and a level control reaches the balance only indirectly — by holding each unit near its aim tightly enough that their difference is small too.

The argument here does not show that density control is wrong. It shows where its effort is spent. Most of the variation a density control removes is along the white, where the tolerance was paying about a third of the price for five parts in six of the variation. The part a tolerance prices hardest is a difference between units, and on a blue overprint one per cent of the variation carrying four fifths of the price is a strong argument for controlling that difference directly — reading the overprint itself rather than the two solids that make it — wherever the job’s colours live near cyan and magenta together.

A dot is larger than it was asked to be showed that dot gain moves a tint’s colour a long way; here it is a third of a press’s variation and a third of its price, because nearly all of what it does is along the white. A larger dot is a darker tint of nearly the same colour. The tolerance forgives that direction, and the press run’s largest source of variation is one the tolerance charges exactly in proportion.

A tolerance with its grain, read against a press #

A delivery tolerance is three tolerances already, a tolerance is a shape whichever way its components are written, and the grain essay suggested a fourth parameter: the ratio between the allowance along the white and along a*. The press run says what that ratio would be for. A supplier who knew the covariance of their own process could compute, before printing, what share of the price each axis will pay and on which patches, and could put control effort where the price is rather than where the variation is.

What they would need is the covariance, and that is the part nobody computes. A pressroom already measures every sheet it samples; the readings are tristimulus values or can be made into them; the covariance of those readings per patch is three lines of arithmetic. Split along the local metric’s axes it says directly how much of that process’s variation lies along the one direction the tolerance does not forgive, and what it costs — which is the whole of what the four stated mixes here can only bracket.

What was computed, and how #

Each patch is printed on the four-colour model with its settings drawn per sheet from seeded normal distributions: each solid density with a standard deviation stated as a fraction of the nominal density, a common density factor applied to all four, the dot gain’s tone value increase at half coverage about its nominal 0.16, and the trapping factor about its nominal 0.86, clipped to sensible limits. The even mix is five per cent on each density, three on the common factor, 0.02 on the gain and 0.03 on the trap; inking alone keeps the five per cent densities and removes the rest; the gain-led and trap-led mixes lower the densities to three per cent and raise the gain to 0.04 or the trap to 0.06. Each draws every source on every sheet, so a run restricted to one source is the same sheets with the others held still.

The readings are tristimulus values under D65 on the scale where a perfect white has Y of one. The local metric at each patch’s nominal reading is the quadratic form of ΔE₀₀ for small tristimulus steps, found by finite differences, and its principal axes and stiffnesses by the closed-form eigen-decomposition. The press’s own principal components are those of the sample covariance.

Stated mixes, and one formula #

The mixes are the weakest link and are named as such. A press whose variation is dominated by something not modelled here — register, which moves colours at edges; ink-water balance, which moves densities in a correlated way the common factor only approximates; paper variation, which moves the white under every patch — would lie differently. Register and paper are both, like the common factor, mostly changes along the white for a patch in its interior, so the qualitative claim should survive them. That is an expectation rather than a computation.

Everything is priced in ΔE₀₀ at the default parametric factors, whose weights bend the grain away from a* on saturated reds. The textile setting of the three constants nobody quotes halves the lightness weight, which would loosen the white further and move more of the price onto the other two axes. Another formula would give different stiffness ratios and different shares; a formula with no weights would put the tight axis on a* exactly and make its stiffness larger still.

Variation and price are two different spreads #

The tempting summary of a process is its variation — a standard deviation in density, a spread of readings, a capability index — and the tempting summary of a tolerance is its size. Neither says what the process costs against the tolerance, because that depends on how the two are oriented against each other.

The move is to express the process as a covariance in the same coordinates the tolerance is a quadratic form in, and split both along the tolerance’s axes. The shares of variation and the shares of price are then two columns of three numbers, and where they disagree is where control effort is misplaced. A press can hold nearly all of its variation where a tolerance forgives it and still pay most of its price for the sliver that is not, and nothing in a spread of readings or a size of tolerance says which.