in the context of photography (and film, and generally speaking putting pictures onto recording medium), a "defect" in the response of the recording medium.

Let us suppose that we have a light source that projects on film a light spot of "intensity" i (now, I will waffle terribly on the great and beautiful and complex discipline of photometry. Bear with me, bare with me even).
A light meter tells us "125", which means "Dude, if you want your 100 ASA film to turn 18% gray in that spot, you need to keep your shutter open for 1/125 of a second". Now, suppose you make the light source half as strong. The light meter will dutifully want you to expose for twice as much time, to wit, one sixtieth of a second.

Still with me ? Good. Now, what we have here is a reciprocal relationship: in other words, the fraction

Amount of light
--------------- = Darkening of the film
Exposure time
is what the lightmeter believes in. This is usually explained with a bucket analogy: a fireman hose will fill a bucket in less time that a thin pipe.

Enter physics. It turns out that, due to the mechanism that turns film dark, if the intensity of your light source is small enough, the above relationship breaks. What happens is that, at low light intensities, you need more exposure time.
On the other hand, the relationship breaks down also for very short exposure times, but this happens only at times on the order of one ten-thousandth of a second, for most modern films.

So, how do you deal with reciprocity failure if you are into, say, nocturnal photography or astrography, where exposure times can be seconds or even minutes ?
You carry around manufacturer tables, and you bracket a lot.

One last thing to remember is that the different layers in color films may have different amounts of reciprocity failure, which means that for very long exposure you will have increasing color shifts. Which you may want to correct with filters. That have a filter factor: that, in turn will lengthen your exposure, leading to more color shift: that way lays madness.
Just live with the color shifts: after all, if all you wanted was faithful rendition of nature you wouldn't be doing half hour exposures on pushed Velvia, would you ?

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