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I came up with this independently years ago. It'll never catch on for the idiotic reason that you can't subdivide 13 like you can 12. 13 is a prime number, while 12 can be divided easily by 2, 3, 4, and 6. 12 is like the whore of simple math.
Yeah, but that only matters for months. We could instead just use weeks since there are 52 weeks per year, so a quarter would be 13 weeks instead of 3 months. It would be easier to determine how many weeks there are in a span of a couple months because it's not variable, or any number of months because they're just multiples of 4. I know a lot of people would be turned off by the system because the number 13 comes up so often and people are superstitious but it really would make things easier imo.