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A calendar usually comes with a leap rule that determines when, how often, how many and where leap items have to be inserted into the common year in order to keep it in sync with the seasons.

The leap item can be any calendaric unit:

Intercalary items can be considered a special kind of leap item that occurs every year, i.e. the leap cycle is one year.

A common leap cycle – sometimes also called solar cycle – for a single leap day in ancient calendars, e.g. the Julian calendar, is 4 years, because 365.25 days approximates the length of the solar year much better than 360, 365 or 366 would.

The Gregorian calendar has a leap cycle of 400 years containing 97 leap days (and leap years), which results in a mean year length of 365.2425 days. Its rule of determining whether a given year is leap is easy enough to remember and calculate in ones head, but in doing so it does not distribute leap days as evenly as possible across the cycle, therefore nominal dates of equinoxes and solstices deviate more than necessary from their actual occurrence. Some alternative leap rules work better than that.

The table below shows some leap cycles that have a whole number of weeks and so can be used by a leap week calendar.

Possible leap cycles
years days weeks leap days leap weeks mean days/year mean weeks/year lunations
400 146097 20871 97 71 365.2425 52.1775 4947.311…
231 84371 12053 56 41 365.(24) 52.177489… 2857.071…
293 107016 15288 71 52 365.242321… 52.177474… 3623.903…
896 327257 46751 217 159 365.2421875 52.177455…

11081.966…


A leap cycle with c years which contains both, a whole number d leap day years (365/366 days) and w leap week years (364/371 days), must conform to the formula c = 7·wd and should fulfill 0.24 ≤ d/c = 7·w/c - 1 ≤ 0.25 to approximate the solar year.

Further reading[]

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