blend modes by what they do
Twelve modes, four groups, and one line of arithmetic each. Read it as a reference and skip to the group you need; the formula in each entry is the thing that predicts how the mode will fail.
Blend modes are usually taught as a list in the order a menu happens to put them, which is why most people know three of them and guess at the rest. They are far easier to hold in your head grouped by what they do to brightness, because within a group the members differ only in how hard they push. Everything below uses b for the base frame and l for the layer, on values scaled between nought and one, and applies to the red, green and blue channels separately.
Darkening
Every mode here has white as its identity value: wherever the layer is white, the base comes through untouched. None of them can make a pixel brighter than it already was, which is why they are the modes for ink, type, shadow and anything photographed on paper.
Multiply result = b x l
The workhorse of the darkening group, and the mode that behaves most like a physical process: two transparencies on a lightbox. Two mid-grey frames multiply to a quarter of full brightness, which is why a multiply of two ordinary photographs is always darker than either and usually needs the opacity pulled well back.
Darken result = min(b, l)
Not an average of any kind. At every pixel it picks one of the two frames whole and discards the other, so the result carries hard boundaries wherever the two cross. Useful for dropping dark type onto a photograph; poor for anything that needs to look continuous.
Colour burn result = 1 - (1 - b) / l
The aggressive one. A division by the layer's value means that as the layer approaches black the result plunges to black, and it clips long before the slider runs out. Set the opacity first and the mode second.
Lightening
The mirror image, with black as the identity: wherever the layer is black, nothing happens. None of them can darken. Light behaves this way in the physical world, which is why every effect that imitates light — flare, sparks, projection, double exposure — lives in this group.
Screen result = 1 - (1 - b)(1 - l)
The complement of multiply, and the only combination of two frames that adds light with a guarantee of never exceeding white. Two mid-grey frames screen to three quarters. This is the default for a double exposure for exactly that reason.
Lighten result = max(b, l)
Screen's blunt cousin: it keeps whichever frame is brighter and throws the other away. Where screen lifts the shadows of both frames slightly, lighten leaves the darker frame completely untouched, which makes a texture look applied rather than exposed through.
Colour dodge result = b / (1 - l)
A division by the complement of the layer, so the highlights race to white while the shadows barely move. It is the mode for making something glow, and it is almost unusable above about 40 per cent on two photographs.
Contrast
These pivot around middle grey. Below it they behave like a darkening mode, above it like a lightening one, and the interesting question is which of the two frames gets to decide. Overlay lets the base decide; hard light lets the layer decide; soft light lets the layer decide gently.
Overlay result = hard light with the frames swapped
Hard light with the two frames exchanged, which is an odd way to describe a mode until you have pressed X and watched overlay turn into hard light in front of you. The base decides, so the base's own contrast is what the result inherits.
Hard light result = b x 2l below half, 1 - (1 - b)(2 - 2l) above
The layer decides. Below middle grey it multiplies, above it it screens, and the transition at the midpoint is abrupt. Good for a layer that is genuinely graphic — a scan, a texture with real blacks and whites — and harsh on a photograph.
Soft light result = b bent toward or away from sqrt(b)
The same idea with the corners taken off: the curve bends toward the square root of the base rather than doubling, so the result is a nudge rather than a shove. This is the mode to reach for when a texture should be felt and not seen.
Comparative
No tonal agenda at all. These three either replace, subtract or invert, and they are as useful for checking whether two frames are identical as they are for making pictures.
Normal result = l
The only mode with no arithmetic worth printing, and the only one where the opacity slider tells the whole story. Everything else on this page is a reason to move away from it.
Difference result = |b - l|
Subtraction with the sign thrown away. Two identical frames give black everywhere, which makes it the fastest way to check whether two files really are the same picture. On two frames made from different positions it gives noise, and the noise is a true report of the misalignment.
Exclusion result = b + l - 2bl
Difference with the midtones flattened. Where difference is brutal in the middle of the range, exclusion softens it toward a half-inversion, which reads as a colour effect rather than as an error signal.
Why the formula matters more than the name
Two of these modes contain a division, and both of them clip. Three of them contain a minimum or a maximum, and all three produce hard edges where the frames cross. Four are products or complements of products, and every one of those pulls two ordinary photographs away from the middle of the range — down for the products, up for the complements. None of that is a quirk of an implementation; it falls directly out of the line of arithmetic printed beside the name, which is why the arithmetic is printed at all.
Four modes that a layers application offers are absent here: hue, saturation, colour and luminosity. They are non-separable — each one reads all three channels of both frames at once and rebuilds a colour — so none of them can be written as a line like the ones above, and none of them is in this build. A reference that listed them would be describing something you cannot press.