What a terrible calendar do we currently have. One guy offered this one: 28x13=364. In that case we have 13 months each of 28 days. Strict 4 weeks in one month. 365 and 366 are non-calendar holidays.
There are two definitions of seasons. In the simpler one, they start March 1, June 1, September 1, and December 1. (In the other, they are truer to astronomy and start about 20 days later.)
With 13 months in a year, that would be a lot messier.
Since there is symmetry to the year, it makes sense to divide it into a number of pieces that works well with that symmetry. 12 pieces makes sense because it's a multiple of 4.
If you look at the length of days as a function of year, it resembles a sine wave. You wouldn't create a unit of angular measurement where one full circle is 13 units. (You'd have trig identities like sin(x+13/4)=cos(x)!) There would be similar disadvantages if you divided a year into 13 pieces.
Another issue, is that while 28 is neatly divisible, 13 is not. What is half a year? When do you make up the numbers for Q3? etc. It introduces so much issues, that the net-benefit is hardly there.
Except that people can still be born and die and check in and out of hospitals on those days, so you have to be able to record those days, so they immediately get back onto the calendar but in a far uglier and more hacky way than the entire system that you just tried to replace.
Why not 8 days (5 work, 3 day weekend)? Or 6 days (4 + 2)
By the seventh day God had finished the work he had been doing; so on the seventh day he rested from all his work. Then God blessed the seventh day and made it holy, because on it he rested from all the work of creating that he had done.
[0] - https://www.discovermagazine.com/planet-earth/why-are-there-...