A halftone is a luminance object

A printed tint is not a colour. It is a lattice of solid dots and bare paper, and a reader sees a tint because their eye cannot resolve the lattice. Everything the trade knows about screens — the rulings, the angles, why yellow gets the awkward one — is a set of rules about what the eye can and cannot resolve, and the rules are stated as though the eye had one channel.

It has three, they run out at completely different frequencies, and asking the same question of each of them settles something the trade has always done and never explained.

The claim #

A halftone screen is a luminance object, and the two chromatic channels have lost it long before any press runs.

• The luminance channel resolves a screen to 55 cycles per degree; the red–green channel to 15 and the blue–yellow to 10. A factor of about four in ruling, and fifteen in area.
• At 26 cycles per degree — an ordinary screen at reading distance — only the luminance channel can see it at all. The other two are below threshold at every angle and every coverage.
• Rotating the screen matters more to the chromatic channels, not less. The angle changes the luminance response by 1.33×, the red–green by 2.30 and the blue–yellow by 4.55. All three agree that forty-five degrees is quietest.
• And all three agree because they are the same filter with different cutoffs, which is the reason the trade’s angles were never in danger from this question.

What was expected #

This site had already computed a tint at the edge of a page — a halftone away from the centre of gaze, with the threshold elevation and the frequency shift in one calculation rather than multiplied together. That essay closed by naming what it had not done: the three channels have three eccentricity constants and three cutoffs, so a chromatic screen should behave differently off axis, and the guess offered was that it would have no best angle.

The guess was wrong in the direction as well as the size. A channel whose filter is falling steeply where the screen sits is more sensitive to a shift along the frequency axis, not less — and rotating a screen is exactly such a shift, because the response to a two-dimensional lattice depends on where its harmonics land relative to the oblique effect. So the chromatic channels care about the angle by a factor of two to four, against the luminance channel’s factor of one and a third.

What makes the angle a luminance decision in practice is not that the chromatic channels do not care. It is that at any ruling worth printing they cannot see the screen either way, so their opinion is about a picture that is not there.

Recording that correction is the point of writing the guess down in the first place. A prediction that is not checked is decoration.

Where each channel gives out #

The measurement is a screen’s loudness: the contrast left after that channel’s own filter, divided by the threshold at the frequency it survives at. Above one it is a texture a reader can see; below one it is a tint.

The reading is taken through the screen’s harmonics rather than by rasterising it, and that is not a detail. A rasterised screen scaled past its grid’s Nyquist frequency draws its own alias, which is a different pattern with a different answer — the earlier peripheral calculation could not be done the rasterised way at all, because an eccentricity factor of nine puts a press ruling past the grid immediately.

The ceilings are 54.7, 15.4 and 10.3 cycles per degree. They are not the channels’ acuity limits, which are 50, 12 and 8 — a screen is not a sine wave, and its harmonics carry it a little past where a grating of the same fundamental would give out.

A screen ruling in lines per inch turns into cycles per degree through a viewing distance, and at thirty centimetres a 150-line screen is about 30 cycles per degree. So an ordinary printed page sits above every chromatic ceiling and below the luminance one, which is exactly the condition for the screen to be invisible as colour and just about invisible as lightness.

Why yellow is the ink nobody worries about #

The trade’s screen angles are 15, 45 and 75 degrees for cyan, magenta and black, with yellow at 0 or 90. The standard explanation is that three inks at thirty degrees apart minimise the moiré between them and there is no fourth thirty-degree slot, so yellow takes what is left.

That explanation is complete as far as it goes and it leaves out why the arrangement is tolerable. Putting an ink at zero degrees is putting it where a screen is loudest — the axis, not the diagonal — and doing that with cyan or black would be visible.

Yellow’s dots differ from white paper mostly in the blue, which is to say mostly in a chromatic direction and hardly at all in lightness. A yellow tint therefore presents its lattice to the channels that cannot resolve it and barely presents it to the one that can. The ink that gets the bad angle is the ink whose screen the eye is least equipped to see.

That is a prediction rather than a rationalisation, and it has a testable form: a screen printed in two inks of equal lightness and different hue should be invisible at rulings where a black screen of the same ruling is obvious. Duotones made of two mid-tone colours are the case, and printers describe exactly that behaviour — the screen “disappears” in a way a black screen of the same ruling does not.

Off axis, where the chromatic screen goes first #

The shortfall the two curves describe can be read as a number rather than as a walk, and at two rulings it is two quite different numbers.

At a ruling of ten cycles per degree — coarse, roughly a screen-printed poster at reading distance — all three channels are above threshold at the fovea: 34.5, 5.2 and 1.1 multiples of threshold.

Two degrees out the luminance channel is at 8.3, the red–green channel at 0.009 and the blue–yellow at 0.066. The chromatic channels are gone at two degrees; the luminance one survives to ten.

That is the peripheral prediction the earlier essay wanted, in the form it turned out to take: not that the chromatic channels lose their best angle, but that they lose the screen entirely, and much closer to the fixation point. A coarse screen is a texture in the middle of a reader’s gaze and a flat tint everywhere else, and the transition happens within a couple of degrees for its colour and within ten for its lightness.

What was computed, and how #

The screen is a square-dot lattice: a cell is inked where a stated spot function is below the coverage. That is what a screen is — the dots are pixels either on or off, not a picture of a dot — and it means the harmonics are the harmonics of a real halftone rather than of a sinusoid.

Each channel’s response is the two-dimensional contrast sensitivity function with the oblique effect in it, and the three channels differ in exactly one thing: their cutoff. The chromatic ones are low-pass with one constant each; the luminance one is band-pass with a solved shape. Nothing else about them differs, which is what makes the comparison a comparison.

The eccentricity half uses the substitution the earlier essay established: a screen of ruling r seen through a filter whose cutoff has been divided by s is a screen of ruling r·s at the fovea. So no second filter is introduced and no second set of landmarks, and the threshold elevation is applied on top with each channel’s own constant.

assertAScreenIsALuminanceObject requires the luminance ceiling to be several times either chromatic one and requires an ordinary press ruling to fall between them. assertTheAngleMattersMoreToColour requires the correction to hold — the chromatic swings larger than the luminance one, and all three agreeing about which angle is quietest.

The two readings, and which one this uses #

There are two ways to ask how much of a screen survives the eye’s filter, and they do not agree at the frequencies that matter.

The rasterised reading builds the lattice on a grid, filters it in the plane, and measures what is left. It is the honest way to ask the question about a picture, and it has a hard limit: a lattice finer than half the sampling rate is not a screen on that grid, it is an alias of one, and the alias moves with the angle. A 45-sample-per-degree grid cannot carry a 26-cycle-per-degree screen at all.

The harmonic reading takes the screen’s own Fourier components — the fundamental in two directions and their combinations — rotates them, and asks the filter about each. It has no grid to alias against, so it works at any ruling, and it is what the ceilings above are computed with.

Choosing the second is the same lesson the audit of this site’s own claims produced from a different direction: a filtered signal read one way and read another can give different answers, and which reading is appropriate is a property of the question rather than a matter of convenience. Here the question is about frequencies, so the reading is in frequencies.

Who found it, and when #

The screen angles are older than any of the theory. Halftone screening was patented in the 1880s, colour halftone in the 1890s, and the thirty-degree spacing was arrived at empirically by printers who could see the moiré and rotated the screens until it went away. Yellow’s position at zero was arrived at the same way: it was tried there, and it looked acceptable.

The contrast sensitivity function arrived much later — Schade in the 1950s, the sustained programme of measurement through the 1960s and 1970s — and the chromatic channels’ spatial limits later still, because measuring them needs an isoluminant stimulus and producing one needs the observer’s own luminance function. Mullen’s 1985 measurements are the ones the numbers in this file rest on.

So the practice preceded the explanation by a century, which is the usual order. What the explanation adds is not permission to keep doing what worked but the ability to say when it will stop working: at a coarse enough ruling, or a short enough viewing distance, the chromatic channels come back and the angle assignment starts to matter to them.

The three channels, in one table #

Two columns of that table are the essay and the other two are checks. The ceilings sit above the acuity limits in every row, which is what a screen’s harmonics carrying it past its own fundamental should do — if one row had come out below its limit something would be wrong with the harmonic reading rather than with the eye. How far above is not the same in every row, and the pattern is the subject of the next section.

A fine ruling is where the two accounts of eccentricity part company most, since a screen already near threshold has less room to survive being magnified.

The ceiling column is a better check than it is read as #

The table’s first two columns are described as a check that passes because the ceilings sit above the acuity limits “by about the same proportion”. They do not, and the way they fail is a stronger check than the way they were supposed to pass.

The ratios are 1.094, 1.283 and 1.288. The two chromatic channels agree with each other to three parts in a thousand; the luminance channel is fifteen per cent below both.

That is exactly the split the method section describes and does not claim as evidence. The two chromatic filters are the same function with one constant changed, so a criterion applied to both has to give the same ratio — and it does, to a third of a per cent, which is a much tighter agreement than “about the same proportion” would notice. The luminance filter is a different shape, band-pass with a solved dip rather than low-pass with one constant, so its harmonics carry it a different distance past its own fundamental, and it comes out somewhere else.

So the column is not three near-equal numbers with one loose one. It is two identical numbers and a third, and the identity of the first two is the check that the two chromatic channels really are one filter at two cutoffs rather than two filters that happen to have been given similar constants.

The ratio between the two accounts is the number worth quoting, and at a fine ruling it is larger than at a coarse one.

And the angle factors are measured below threshold #

The correction this essay exists to record — that rotating a screen matters more to the chromatic channels than to the luminance one, by factors of 2.30 and 4.55 against 1.33 — is measured at a ruling of 26 cycles per degree. That is 0.48 of the luminance ceiling, 1.69 of the red–green ceiling and 2.52 of the blue–yellow one.

Two of the three channels cannot see the screen at that ruling at all. Their angle factors are therefore ratios between two quantities that are both below threshold, and a ratio between two sub-threshold numbers is a property of how steeply a filter falls rather than a statement about anything visible.

The essay says as much, in the sentence about their opinion being about a picture that is not there — and the arithmetic makes it sharper than that. The ordering 1.33 < 2.30 < 4.55 is very nearly an ordering of how far past its own cutoff each channel is, at 0.48, 1.69 and 2.52 of the ceiling. A filter evaluated deeper into its tail returns a larger ratio between two points on that tail, more or less regardless of what the points are, because the tail is steep and getting steeper.

Which means the correction to the earlier guess is a correction about the filters rather than about screens. The earlier essay guessed the chromatic channels would have no best angle; the answer is that they have the same best angle and a larger apparent sensitivity to it, and the larger sensitivity is an artefact of asking a channel about a frequency it has lost. Both the guess and the correction are statements about a signal neither channel receives.

The measurement that would settle it is the one at a ruling all three can see. The essay has it — at 10 cycles per degree all three are above threshold at the fovea, at 34.5, 5.2 and 1.1 multiples — and the angle sweep is not reported there. Running it at that ruling would give three angle factors that are all about something visible, and it is the number the earlier guess was actually about. At 26 cycles per degree the honest report is that the luminance channel’s factor of 1.33 is the only one of the three that means anything, and that it is the only one the trade’s angles were ever a decision about.

And every row agrees about forty-five degrees. That is not obvious in advance: the three filters have different shapes, and a diagonal being quietest for all of them means the oblique effect is doing the work rather than any channel’s particular passband.

Where it stops #

The three channels are given a common absolute threshold, which is the largest simplification in the spatial model and is stated there rather than hidden: chromatic thresholds are measured in cone-contrast units that are not commensurable with a luminance Michelson contrast. So the ratio between two chromatic ceilings and the ratio between a chromatic ceiling and the luminance one inherit that assumption.

What survives it is the ordering and the rough factor, because the ordering comes from the cutoffs and the cutoffs are three separate measurements. A different anchoring would move all three ceilings together and leave a press ruling on the same side of all of them.

The screen here is one ink on paper, presented to all three channels at once because a real dot differs from paper in lightness and in colour together. Nothing here decomposes a particular ink’s dot into its three channel contrasts, which is what would be needed to turn the yellow argument from a prediction into a number.

And moiré is absent. Two screens at different angles beat against each other at a frequency neither of them has, and whether that beat is visible is a question about the same three channels — with the important difference that a beat can be at a far lower frequency than either screen, which is precisely where the chromatic channels are sensitive.

Where the ladder goes next #

The moiré question is the obvious one and it inverts this essay’s answer. A screen is high-frequency and the chromatic channels cannot see it; a beat between two screens is low-frequency and they can. So a colour moiré should be visible where a colour screen is not, and the three-channel calculation would say at what ruling difference.

The other direction is a real ink. Taking cyan, magenta, yellow and black at a stated coverage, decomposing each dot-against-paper contrast into the three channels, and computing the loudness of each ink’s screen separately would turn the yellow prediction into four numbers — and would say whether the trade’s angle assignment is the one those four numbers recommend.