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From this answer on white balancing a LED light I understood there's a thing called Colour Rendering Index (CRI). I understand it has something to do with how well lights reproduce the sun's colour spectrum, and I understand the effect, but how is the CRI determined?
And what are the artificial lights that have a typically high or low CRI?

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This is really well-explained in the Wikipedia article on CRI. In short, though, it is a standard from the International Commission on Illumination (CIE) for rating the degree to which a lighting source shows colors faithfully. A score of 100 represents an ideal "blackbody" source, like an incandescent bulb (or the sun), and scores less than that are worse. A typical awful warm-white fluorescent might score in the 60s; a better fluorescent tube would be in the 70s or 80s. A monochromatic light source like a sodium vapor streetlight might even have a negative score.

The score is computed by comparing the rendition of a small number of pre-selected color patches to their appearance under that ideal light. SRGB approximations of these patches are like this:

sRGB color swatches approximating the patches used in measuring CRI

There are 8 low-saturation patches evenly distributed around the range of hues, and then the four perceptual primaries with somewhat stronger (not extreme) saturation, and then finally two colors meant to represent "complexion" and "foliage".

The key problem with the CRI standard is simply this selection of color patches. The colors are weakly saturated and are particularly lacking deep reds and purples. And, despite the "complexion" sample, there's no real representation of skin tones. (I'm of European descent, and I spend most of my time indoors, and that doesn't even match my skin very closely.) The result is that a light can show impressive numbers but be terrible for portraits, flowers, fresh food, and so on.

There's a newer set of samples in the updated "R96a method" version of the test, but this is not widely used (partly because existing fluorescent bulbs would tend to score lower), and while that revision adds a few more improvements (like additional reference light sources), it doesn't address all of the problems. NIST (National Institute of Standards and Technology) has proposed a replacement called CQS, for Color Quality Scale, but that doesn't seem to have much uptake either.

So, what lights are good and which are poor?

Traditional incandescent and halogen bulbs are great, as is the gigantic ball of plasma which lights the daytime sky. The xenon arc lamps used in flash tubes are also extremely good, because with high current density they approach ideal blackbody radiation.

Anything which depends on fluorescence will be less good — that means fluorescent bulbs and white LEDs. That's because these inherently have peaks only in certain parts of the spectrum, and while their light is perceived as white (of a certain color temperature), whole chunks of color can be missing. More expensive lamps (both fluorescent bulbs and LEDs) use multiple gasses and phosphors with different characteristics to provide better coverage. Theoretically, an LED matrix could include a mix of elements of different types to get even closer, although one would have to be careful to avoid mottled lighting.

All of this is why I'm excited about developments in light-emitting capacitor technology, also known as "FIPEL". This is a broad-spectrum light source which is very efficient and low-temperature. (The brightness isn't up to what would be useful yet, though.)

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