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>Instead of trying to do hacks to get all of these cycles to line up, why not move in the other direction and take the approach of having regular cycles at different frequencies that do not line up?

With that system, how do you refer to a particular date? YYYY-MM-DD doesn't make much sense anymore, since a month can span multiple years, and a day can span multiple months and/or years.

Mesoamerican (mostly known through Mayans, but others used the same calendar) had two ways of doing that.

First was the "Calendar Round", they had two non-synchronous "yearly" calendar of 365 days (Haab') and 260 days (Tzolk'in), giving a date in both calendars provided an exact identification in a repeating cycle (era) of 18980 days (~52 solar years).

Second was the "long count", a monotonically increasing calendar from a root date (think CE/BCE, except including days). A "long count" date is composed of a number of counters mostly in base 20: K'in (day), Winal (20 K'in), Tun (18 Winal), K'atun (20 Tun), B'ak'atun (20 K'atun) (dates have been found with even higher orders, but they're rare, those are the most common). A B'ak'atun unit represents ~394 solar years. This provides an unambiguous and very long term calendar. It's essentially what the UNIX system does, except starting from days (interestingly, a standard 5-units long count fits in just 22 bits, 31 bits [to account for signing] allows for 5.8 million years before wrap-around)

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