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The core rules say 1CC is 1 Kua year, which is 336 days. With 9 equal segments, and 3 single days between them, that makes each segment 37 days, which doesn't divide into anything smaller.

I can see that this isn't needed though. In fact, with no labour unions or any worker protection, there's no-one to argue for a 'working week'. Perhaps I'm so used to weeks it's weird to give them up. ... And the religious influence on real world days of the week is used in Coriolis for segments instead.

Did I just need to go through that to get comfortable with 9 37-day months?

Did old Persian calendars just have months and nothing smaller?

Well there is no way to break it down evenly into weeks. Maybe 3 week of 12 days with a leap day somewhere?

I'd divide each segment into four quarters, nine days each with 1 day right in the middle that is a special holliday devoted to the icon for that particular segment.