It doesn't. Quarters just become 13 weeks. They'd mostly correspond to existing quarters as well, give or take a couple of days at each end.

OK, it breaks the assumption that quarters are a whole number of months.

12 is a handy number of months to have in the year. 12 is divisible by 2, 3, 4, and 6. Not by 5, but you can't have everything (unless we had 60 months of 6 or 7 days each... let's not go there).

13 is prime.

I think not bothering with months at all would be better than having a 13 month year.

I like New York in weeks 22 to 25, how about you?

That sounds ridiculous now but it's actually not that bad when you get used to it. Besides, people would likely still keep the month names to refer roughly (if not even exactly) to periods of the year. The real benefit would be in dropping months from the business and commerce world.

I prefer late twenties London.

  I think not bothering with months at all would be 
  better than having a 13 month year.

The phases of the moon are 28 days long. So a 13 month calendar also lines up with the lunar cycle. The more you think about it, you wonder how they ever came to the conclusion to use 12 months instead of 13.

The phases of the moon are about 29½ days long.

(http://en.wikipedia.org/wiki/Moon and see the synodic period (quoted to a tenth of a second even though full moon to full moon can vary by about ¼ day either way).)