That’s called “Pythagorean tuning” and it’s possible but it’s not especially common. The system we use is a compromise derived from that system. If you use pure 3:2 ratios you get the so-called “wolf tones”, but we don’t get those any more.

The current most common system is equal temperament, which is a type of meantone tuning, where you take the 3:2 ratio and alter it. Equal temperament alters it to 2^(7/12) ≈ 1.4983, and it has the nice property that you get twelve notes and it’s all perfectly symmetrical, it doesn't matter which note you start on.

Before equal temperament, there were various other systems. One popular one was quarter-comma meantone, which took four stacked 3:2 ratios (81:16) and squashed them to equal two octaves and a 5:4 ratio (80:16), which results in a value of 5^(1/4) ≈ 1.4953. You see it in some old organs—which may have, say, separate keys for G# and Ab.

Then there are various non-meantone tunings, various “just” intonations, etc. In just intonation, a chord like C E G will have a 4:5:6 ratio. You can’t do this with all major triads, though, so you end up with some major triads that sound correct and others that sound out of tune. Our modern system is a compromise where all major triads are equally out of tune, rather than having some better and some worse.