## About Gamma

When calibrating display devices, the notion of "gamma value" quickly becomes a topic for discussion. Various numbers are often bandied about as if they have a well known and accepted meaning, but it turns out that gamma values are not a very precise way of specifying real world device behavior at all.
A "gamma" curve is typically thought of as an ideal power curve, but no real world device has the necessary zero output at zero input to be able to match such a curve, and in general a display may not exactly reproduce an idealized power curve shape at all. The consequence of this is that there are countless ways of matching a real world curve with the ideal gamma power one, and each different method of matching will result in a different notional gamma value.
Argyll's approximate specification and reading is simply the gamma of the ideal curve that matches the real 50% stimulus relative-to-white output level. I think this is a reasonable (and simple) approximation, because it matches the overall impression of brightness for an image. A more sophisticated approximation that could be adopted would be to locate the idea power curve that minimizes the total delta E of some collection of test values, but there are still many details that the final result will depend on, such as what distribution of test values should be used, what delta E measure should be used, and how can a delta E be computed if the colorimetric behavior of the device is not known ? Some approaches do things such as minimize the sum of the squares of the output value discrepancy for linearly sampled input values, and while this is mathematically elegant, it is hard to justify the choice of device space as the metric.
There are many other ways in which it could be done, and any such approximation may have a quite different numerical value, even though the visual result is very similar. This is because the numerical power value is very sensitive to what's happening near zero, the very point that is non-ideal. Consider the sRGB curve for instance. It's technically composed of a power curve segment with a power of 2.4, but when combined with its linear segment near zero, has an overall curve best approximated by a power curve of gamma 2.2. Matching the 50% stimulus would result in yet another slightly different approximation value of about 2.224. All these different gamma values represent curves that are very visually similar.
The result of this ambiguity about what gamma values mean when applied to real world curves, is that it shouldn't be expected that there are going to be good matches between various gamma numbers, even for curves that are very visually similar, unless the precise method of matching the ideal gamma curve to the real world curve is known.