Tolerances for spot colors

There is a little bit of a hole in the most recent version of the ISO spec for flexo printing[1]. I will explain the hole, discuss my analysis, and provide a patch for this hole, but first a bit of background.

There is a family of standards called ISO 12647, which all cover printing. In particular, they provide some target color values, along with acceptance tolerances around those colors. In almost all case, those tolerances are in terms of ΔE, with the newer revisions of the standards shifting from ΔE_ab to ΔE₀₀.

Part 6 of this standard is devoted to standards for flexographic printing. Flexo printing is used on such a wide variety of substrates that it is often not possible to nail a particular color. As a result, tolerances based on a ΔE-type color difference are not possible. One has to be content with getting the correct hue.

ISO 12747-6:2012 states target hue angles for C, M, and Y, and provides a ±6^o tolerance. Spot colors are also given a tolerance in terms of hue angle, but it’s ±8^o. Black, however, does not have a hue angle tolerance. This makes a great deal of sense, since black is neutral. The hue angle is undefined.

Tolerances from ISO 12647-6

But, “black” is not the only black. My Pantone book lists six shades of black. Presumably, the target a* and b* values for these inks are also 0, so the hue angle is undefined. This is the question that led to the realization of the hole in the standard: “how can analysis software decide if the ink name ‘midnight’ or ‘inkwell’ is equivalent to black?”[2]

Worse yet, what if the target for a spot color is not quite exactly neutral, but near neutral? If the chroma[3] of a spot color is small, then a tolerance of 8^o might be impossibly tight. For example, in my Pantone book, the ink “Warm gray 5” has an L*a*b* value of {69.61, 2.58, 1.00}. Let’s say the hue of this ink is rotated by 8° to {69.61, 2.70, 1.63} so as to be at the very edge of acceptance in terms of hue. The color difference is a meager 0.38 ΔE₀₀. Holding a printer to a 8° tolerance is pretty tight.

For highly saturated colors, a tolerance of a specified number of degrees around a target hue is perfectly reasonable. But as a color gets closer to a near neutral, it would be preferable to shift over to a ΔE₀₀ tolerance. At what chroma value do you need to shift between a ΔE₀₀ tolerance to a hue angle tolerance?

First naïve approach

I call this the naïve approach, because I am going to make a mistake. It turns out that the mistake is not huge, and it gives some intuitive understanding, so I will leave it in on this first pass. Watch for the mistake.

The figure below shows two colors r₁ and r₂, in the a*b* plane. Both have a chroma of c, but are separated by 8^o. If r₁ is the target hue, then r₂ is at the edge of the 12647-6 tolerance window.

The arc length between the two colors is fairly trivial to compute. I could use the normal trig stuff, but since the arc is so small, I will just estimate the ΔE (which I a straight-line distance) with the arc length.

Thus, if we assume a tolerance of 1.5 ΔE, the crossover is close to a chroma of 10. Below that point, a hue tolerance is overly restrictive.

Second, not so naïve approach

Did anyone catch the mistake I made? Yes, you in the back? Ahhhh… the old accidentally used ΔE_ab instead of ΔE₀₀ mistake! The formula for ΔE₀₀ is just a tiny bit more complicated than the formula for ΔE_ab.[4]  So, I need a slightly less naïve way to look at this problem.

Suppose I picked a color at random, and then rotated it by 8° - that is to say, shifted the hue by 8° without changing L* or the chroma, C*_ab. What is the color difference (in ΔE₀₀) between the original color and the hue-shifted color? Then let’s say you computed that for a whole big bunch of colors just to see how it all played out. You would expect that, the larger the chroma, the larger the color difference, right?

Well, I just happened to have a large collection of colors (11,488 of them to be exact) just laying around from a blog of mine on counting colors. Below is a plot of the color difference caused by an 8° hue shift as a function of chroma.

Color difference caused by an 8° hue shift

Is this cool?

To answer the rhetorical question, yes, it is cool. But the data is a bit sparse down there in the lower left-hand corner around 1.5 ΔE₀₀ where we are looking for our answer. This is not a surprise, since my color database only includes colors on a grid with spacing of 5.

So, I took a little different approach. For my next plot, I looked at points in color space with a chroma of 10. (This is the estimate we came up with by analyzing ΔE_ab.) These are all shades of gray, essentially, with a moderate amount of a color cast to them. I looked at how much color difference there was for each of them when they shifted in hue by 8°. The plot below shows this color change as a function of original hue angle.

The color difference caused by an 8° change in hue, as a function of hue angle

We see that the amount of color change oscillates between 1 and 2 ΔE₀₀, so the average must be somewhere around 1.5 ΔE₀₀. The true average is 1.45 ΔE₀₀. Going back to Table 4 from ISO 12647-6, this is really close to the variation tolerance for spot colors.

For an ink with a target chroma of 10, a tolerance of 8° is roughly equivalent to a tolerance of 1.5 ΔE₀₀.[5] Based on that, I have a simple recommendation for spot colors: If the target chroma is less than 10, then the appropriate tolerance is 1.5 ΔE₀₀. For target chroma of 10 or greater, then the hue angle tolerance of 8° should be used.

Just in case someone wants to repeat this analysis for angles other than 8° or color differences other than 1.5 ΔE₀₀, I provide the useful nomograph below. The t-shirt version should be out just in time for Christmas.

Nomograph for determining the crossover chroma
for various combinations of color difference tolerance and hue tolerance

The red line on the nomograph shows the determination of the crossover chroma between tolerances of 1.5 ΔE₀₀ and 8° of hue. The graph shows a crossover at C* = 10.5. Just between friends, let’s call it 10.

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[1] I am not blaming any of the diligent folks on TC 130 who reviewed this document. I was one of them!

[2] Thanks to Bruce Bachmann and Mike Sisco for this realization.

[3] I am using the word “chroma” to mean C*_ab, which is defined as 

[4] Well, to be fair, the formula for ΔE₀₀ is just a tiny bit more complicated than the formula to get a balanced budget.

[5] One caveat: I have made the assumption that the only change in color is in hue angle.