Dear Calendar People

In an earlier note I mentioned the idea of the Pre-Islamic calendar using the Octaeteris (8 years with 3 leap months) during the 200 years it was lunisolar. This would fit in well with the idea of each leap month occurring after each regular
month in turn as suggested in

https://en.wikipedia.org/wiki/Nasi%27#As_lunisolar_intercalation .

The error would be about one and a third months over the 200 years of use.

I then thought of a way of correcting the Octaeteris that could be used in such a calendar. It is to add one 12-month year without a leap month.

Four Octaeterides have 32 years with 12 leap months equivalent to 33 lunar years. If one correction were added to these, we’d have 33 years with 12 leap months equivalent to 34 lunar years.

If one in every seven Octaeterides has a correction, seven Octaeterides would equal three Metonic cycles (7*8+1=57 years of 7*3=21 leap months).

Here I show years ‘s’ for 12-month year and ‘L’ for a year with a leap month for seven Octaeterides with the middle (fourth) one corrected by inserting ‘s’ year before last year

above three Metonic cycles

ssLssLsLssLssLsLssLssLsLssLssLssLssLssLsLssLssLsLssLssLsL

ssLssLsLssLssLssLsLssLssLsLssLssLssLsLssLssLsLssLssLssLsL

These two differ by the placing of only 4 of the 21 leap years, of which 2 are one year early and two are one year late.

I reckon the jitter of the corrected Octaeterides by taking the ‘ssLssLssLssLss’, which begins with the corrected 4^{th} octaeteris. These 14 years have 4 leap years compared to 14*7/19 = 5.15789…. on average for 14 years. So
the jitter is 1.15789… months or exactly 22/19 months, which is 4/19 months more than the 18/19 months jitter of the Metonic cycle.

The 4/19 suggests that the above alignment with 4 leap years not synchronised is the best alignment one can get.

Karl

16(01(29