you can have pure intervals, just not a whole string of them(no pun intended. yeah, right!) stacked on top of each other. you can tune all the octaves(say "C"), and the 5ths to them, but if you try to tune the 5ths of the 5ths, bingo! trouble. the math just doesn't add up. the 5ths of the 5ths would be the seconds and ninths of the octaves, and the frequencies wouldn't quite add up. you are left with a hole. or too much, I forget.
Well-Tempered tuning alters all of the tunings so that all the intervals work pretty well; all your major thirds are fairly major like, etc.
it's a huge and deep topic, and if you get a chance to hear a group play with Just Intonation, it's really quite interesting. it can be hard to accept to our compromised ears, however.
this of course, really only deals with music from Western Civilization, as other cultures use different tuning systems, with different numbers of notes between the "octaves".
