Curious: Is it any better with printers?

@gerrit theoretically, you could get a very good approximation of the red-orange-yellow part of the spectrum from a printer (just like with monitors, which could actually go to yellowish-green). This is because of the almost straight gamut boundary (see the chromaticity chart above for wavelengths $\ge 550\,\mathrm{nm}$). But for the other colors, especially in the bluish-green and cyan regions, printers are at a disadvantage compared to monitors, because they rely on reflection for color reproduction, and a narrow-band reflection of a broadband illuminant will necessarily have low brightness.

Could we create a perfect gamut screen by creating a screen that uses microprisms rotated like MEMS mirrors so to point the correct frequency to a pixel hole on the screen? Obviously it would be incredibly hard to achieve, but theoretically?

@StefanoBorini In principle, yes, if you have a bunch of monochromatic sources of light (e.g. a dozen or two of lasers), you could indeed use the DLP technology to get a very wide-gamut display. This would basically be a polygonal approximation of the smooth horseshoe shape.

@Ruslan now I want it :|

@gerrit Printers typically have a much worse gamut and color fidelity than monitors. This is because pigments can only reduce from the spectrum of visible light already around you. Sun light, fluorescent light, and incandescent light all have unequal intensities of various wavelengths; so, they all tint any printed images you see. RGB screens can control the ratio of wavelengths so that you can have un-tinted colors.

Also not everyone's eyes have the same ratio of different kinds of cones; so, RGB can be adjusted to match the user's levels of color perception giving a partially color blind person accurate color perception.

@StefanoBorini You might want it, but there is no software for it. You'll end up using it as a sRGB monitor and your kids will wonder why you held onto such an ugly monitor and waxed lyrical about it.

@DDuck software can always be written. Getting the necessary hardware is much harder: you need each laser to be stable enough in frequency and power (otherwise the colors will drift), have enough power to together produce at the very least $80\,\mathrm{cd}/\mathrm{m}^2$ over the whole screen (to be able to reproduce sRGB white point), be able to work continuously for hours, and exist in each of the wavelengths required. All this combined is too expensive at present.

@StefanoBorini Your wide-gamut monitor wouldn't be much use without a wide-gamut data source. So you'd also need a wide-gamut video camera. But it would be good for computer-generated images from software that works in a sufficiently large color space.

What are the axes in the gamut diagram?

@JiK as denoted in the image, they are $x$ and $y$ — the chromaticity coordinates of the CIE 1931 color space. See Wikipedia for the definition.

@Ruslan Thanks, somehow I missed that when quickly trying to search Wikipedia. So, essentially it's a brightness-normalized map from the cone cell response to a 2D space.

@JiK almost: XYZ coordinates are a linear transformation away from the LMS coordinates, and it's the latter that give the cone responses.

@Ruslan Not sure if this affects the tetrachromat line, but my understanding is women with it just have the extra shitty cone from common colorblindness. They don't see more colors like a insect, but have higher sensitivity to the cone freq. Your statement seems to imply something else. Also, what we see is actually not determined just by our cones, and I don't believe tetras have higher "eye bandwidth" to compensate. It's possible nerves automatically scale the the input signal. I don't think you can accurately predict if a human will think the spectrum is wrong based only on tetrachromacy

@whn by tetrachromacy I actually meant functional tetrachromacy, i.e. the case when the subject does distinguish colors that are metamers to trichromats. At least one such subject has been found: cDa29.

@Ruslan this article : sciencealert.com/… seems to suggest that the woman in question also only has an extra shitty cone, and does not display the ability to view frequencies outside of the common visual spectrum, and her tetrachromy works as I describe previously. It appears the other women either have their nervous system handle the attenuation of the signal, or the extra cone is some how not attached neurologically.

@whn I wasn't talking about ability to see additional frequencies: near IR and UV are invisible not because of the cones, but because of absorption in cornea and in the lens. What tetrachromacy is about is the ability to distinguish metamers (i.e. what trichromats think are metamers). This woman, cDa29, does have this ability: from your link: "three coloured circles of light flashed before these women's eyes. To a trichromat, they all looked the same, <...> one of the women tested, cDa29, was able to differentiate the three different coloured circles in every single test."

"can't display any spectral colour"? But there are plenty of colours displayed right there in the opening image inside the triangle. My computer is displaying them, and so is the computer of almost anyone else reading this answer. Not quite sure what you mean.

@AndrewSteane "spectral color" means "monochromatic color". All these colors are located (not rendered) on the black curve in the diagram.