First off, most printing today is done with dots. Halftone dots. Oh, you knew that already? Just in case you didn't, have a look at magazine with a magnifying glass. You'll see that highlight areas are made of tiny dots, and shadows are made of dots that are so big they run all together.
Scan from a catalog showing halftoning
A formula to predict the reflectance of a halftone
Way back in 1936, a fellow by the name of A. Murray of Eastman Kodak  was pondering the measurement of halftones. Murray sought out his friend, E.R. Davies  to ask if there were a simple formula. Davies gave him such a good enough answer that Murray decided to publish the results. Naturally, by Stigler's law, the resulting equation isn't known as the Davies equation, but rather, as the Murray-Davies equation.
Contrary to popular belief, the Murray-Davies formula
was not developed by Bill Murray and Geena Davis
was not developed by Bill Murray and Geena Davis
Suppose that you have an area that is halftone dots that cover 20% of the area, something like the area below. Of the light shining on this area, 20% will hit ink, and the remaining 80% will hit paper. If there are 100 photons playing this game, 20 will hit the ink. Suppose further that the ink will reflect 5% of the light that hits it. That means that, of the initial 100 photons in the game, only 1 will reflect back from the ink, since 5% of 20 is 1.
Now, let's look at the fate of the other 80 photons. Suppose that the paper has a reflectance of 80%. That means that 80% of those 80 photons (64) will reflect. All told, 1 + 64 = 65 of the original 100 photons will reflect.
The pattern on my jammy bottoms
Did that all make sense? If so, then congratulations! You understand the Murray-Davies equation. If the various R's in this equation stand for the reflectance of the various things, and A stands for the dot Area, then this simple formula will give an approximation for the reflectance of the halftone:
The celebrated Murray-Davies equation
You may have seen an uglier version of this equation, one involving logarithms or 10 raised to some power. Trust me. IT's all the same equation. People put that kind of complication in just to make themselves sound smart. Sometimes they are actually smart. Especially when I do it. But the point is, the simple equation above is the basic Murray-Davies equation.
Trouble in Paradise
It didn't take long to see that the formula does not do a fabulous job of estimating the reflectance of a halftone. And it didn't take long to find an explanation. If one compares the size of the dots on the plate with the size of the dots on the paper, it is obvious that the dots get bigger when you print. There is dot gain. And if the dots are bigger, then one would expect the reflectance to go up as well.
Clearly if one would like midtones to be the right color, dot gain must be measured and ultimately controlled. The Murray-Davies equation provides a way to measure the dot gain. First, you solve the Murray-Davies equation for the area term, A. This revision of the Murray-Davies equation (shown below) answers the question "How big must the dots be in order to get a certain reflectance?"
You measure the reflectance of the paper, the solid, and the halftone, plug them into the equation below, and you have an indirect measurement of the size of the dots. The difference between this indirectly measured dot area and the area on the printing plate (or the area requested in the digital file) is called the dot gain.
Aside from changing the name from "dot gain" to "tone value increase", this basic formula has survived to this day as a control metric. The image below is a screenshot from the most recent (unpublished, as yet) version of the main ISO standard for printing, ISO 12647-2. The horizontal axis is the tone value (dot area) going from 0% (paper) to 100% (solid ink). The vertical axis is the expected amount of increase in tone value for each of five different types of printing. For printing type A, a 50% tone value in the image file is supposed to print like a 66% halftone.
One of the jobs of the printer is to maintain the press so that these are the tone value increases that are seen on a daily basis. If the press is printing differently (maybe the 50% tone value measures as a 64% rather than a 66%) a look up table (known as a plate curve) is introduced between the digital file and the plate manufacture so that the combined system has the proper tone value increase. As a result, a contact proof can be printed or displayed on a calibrated monitor that accurately predicts what the press will print.
So, it would appear that all is well with the world. The printer has a tool to monitor the way that halftones are printed, and can digitally adjust this to a value that ISO has prescribed. By keeping this constant, the printed product will always be the correct color.
Or so it would appear. Stay tuned for the next installment, where I show that this simplistic view of tone value increase is lacking, particularly when we go beyond the borders of conventional web offset printing with CMYK inks.
 Almost no one knows this guy's first name. It is ironic that his last name has gone down in the history of printing without a first name. Note to the historians: When you put my name down in the history books, please make sure my first name is there. And if you have enough space, I wouldn't mind having my middle name, "the Math Guy" included. I said, "almost no one" because I have found exactly one person, Dr. J. A. S. Viggiano, who has preserved the first name: Alexander. Alexander has that first name on at least 22 patents.
 Guess what? I don't know this guy's first name.