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We see the mix of red light and green light as yellow light (#FFFF00). The wavelength of yellow light lies between red and green.

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But the wavelength of purple light lies outside of red and blue. Why can we also see the mix of red and blue as purple? Is it real purple?

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The reason that colors combine as they do has everything to do with the response curves of the light-sensitive proteins in your eye. A response curve is just a function that tells you how strongly a particular protein reacts to a fixed amount of light a given frequency (or energy). There are three kinds of these photosensitive proteins (photopsins) in our eyes, one for each of the three different kinds of cones on the retina, and each has a distinct response curve, which you can see e.g. in this image on Wikipedia:

color response curves

When light of a particular frequency comes into the eye, it triggers a certain strength of reaction from each of the three kinds of proteins. For example, light with a wavelength of 580 nm causes the "short" protein (the one that responds most strongly to short-wavelength light) to produce a signal of strength 0.000109, the "medium" protein to produce a signal of strength 0.653274, and the "long" protein to produce a signal of strength 0.969429. It's this set of signal strengths, (0.969429, 0.653274, 0.000109), that triggers the perception of that particular shade of yellow in our brain. (Numeric data come from this site)

But as you might guess, it's possible to "fake" this signal by sending a particular combination of different frequencies of light. For example, you might guess that if you send a combination of 97 parts long-wavelength light and 65 parts medium-wavelength light into the eye, it would produce almost exactly the same set of signal strengths: (0.97, 0.65, 0). In practice you have to be a little more careful than that, because the response curves overlap a bit, but the basic idea that a combination of multiple wavelengths of light can produce the same signal as a single, other wavelength of light, definitely works. This is why red and green combine to produce yellow, for example. It's not because yellow is between red and green in the spectrum, it's because the signal strengths generated when our eye receives yellow light are very nearly the same as the signal strengths generated when it receives a certain combination of red light and green light. Similarly, the signal strengths generated when our eye receives purple (actually violet) light are very nearly the same as the signal strengths generated by a certain combination of red light and blue light.

The above is adapted from a comment I posted on Reddit, and for more information, you might want to look at an earlier comment describing how wavelengths of light (or combinations of wavelengths) get converted to colors.

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I think it's important to add that the spectral colors (i.e. the colors due to only a single wavelength of light) are not all the colors we can perceive. By mixing different wavelengths we can create a multitude of shades that do not exist as spectral colors. For instance, most shades that we might call purple are not spectral colors (except violett and indigo) and can only be achieved by mixing blue/violett and red. This helps when thinking about colors. The spectral colors are all on the bent edge, while those in between needs mixing. –  jkej Jan 15 '13 at 10:01
That's true, but I don't think it's relevant to this question (so I actually took it out - you can see that discussed in my original comment on Reddit). This question is just about differences between spectral colors. –  David Z Jan 15 '13 at 10:03
I think it's most relevant. OP asks about mixing red and blue to get purple. Basically all combinations of red and blue with more blue than red may be called purple. But most of them will be very clearly distinguishable from violett (although violett may aslo be called purple). There is a common misconception that all colors we can perceive exist as spectral colors and this question seems to be influenced by this misconception. Why not dispell that misconception? And why not say clearly that spectral purple (violett) can in fact NOT be achieved by mixing blue and red? –  jkej Jan 15 '13 at 11:28
There seems to be some conflicting data about the exact shape of the response curves; for example, this figure shows a little secondary peak in the L response curve. But also, that has to do with what jkej has been saying, that the purple you see when you look at red+blue is not the same as the purple you see when you look at violet light. –  David Z Jan 15 '13 at 11:50
Also remember that your computer screen can not accurately display spectral violet. It can only blend red, green and blue. So there's no use in trying to find images on the internet that compares spectral violet to various mixes of red and blue. In fact, pure spectral violet is probably a color that we encounter very rarely. If you have access to a violet laser, you could go in to a dark room, shine it on a white surface and experience a color which you might never have perceived before. How we perceive this color might be very individual. –  jkej Jan 15 '13 at 13:31
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