system
A chat on subreddit /r/me_irl
human
TL;DR at the bottom. 

I’ve thought about this a lot actually, and it has a few problems, the biggest being that 13 is a really awkward number (being prime), whereas 12 is a particularly convenient one (having many factors). There are 4 seasons, so does each season last three months and one week? That’s just awkward. Also, OP doesn’t address the one leftover day, or leap days in leap years. 

HOWEVER! I think I have a fairly elegant solution. 

A repeating cycle of four seasons, each seasons consisting of exactly three months of 28 days and one intercalary week (a week belonging to no month) so we can still have 12 months. Summer has one intercalary day (belonging to no week or month), and winter has one too but only on leap years. The solstices and equinoxes oh so satisfyingly land on the intercalary weeks, and every month and year starts at the start of a week. 

TL;DR:
* Three months of winter (3\*28 days). 
* One week in between months. This week contains the spring equinox, marking the beginning of spring. 
* Three months of spring (3\*28 days). 
* One intercalary week containing the summer solstice and marking the beginning of summer. 
* One intercalary day to bring up the total to 365 days. This day does not belong to any week. 
* Three months of summer (3\*28 days). 
* One intercalary week containing the autumn equinox and marking the beginning of autumn. 
* Three months of autumn (3\*28 days). 
* One intercalary week containing the winter solstice and marking the beginning of winter. 
* (On leap years, one intercalary day to round off the calendar.)
* Total 365 days (366 on leap years obviously).

Edit: Fixed some formatting and typos.
gpt
Oh this would be nice, and weirdly fitting. We could make a return to celebrating the equinox and solstices, without the day being swallowed by contemporary factors.

I'm a big, big fan of this idea.