A Simple Lunisolar Calendar


The following is an outline of a simple lunisolar calendar which I first presented to the CALNDR-L mailing list on October 28, 2002.

Not wishing to reinvent the wheel, I've maintained the Gregorian calendar to keep track of the sun. With the following adjustments the months will keep in sync with the moon.

Except for the sixth month, all months have fixed lengths, and generally alternate between 30 and 29 days. The months and their lengths are as follows:

Alpha:	30 days
Beta:	29 days
Gamma:	30 days
Delta:	29 days
Epsilon:	30 days
Zeta:	29 or 30 days	(Zeta has 30 days in years evenly divisible by 5, except years evenly divisible by 200 or 500)
Eta:	30 days
Theta:	29 days
Iota:	30 days
Kappa:	29 days
Lambda:	30 days
Mu:	29 days
[Nu:	30 days]	(NOTE: Not all years contain the month Nu.)

An important rule in running this calendar is that New Year's Day must always fall in the month of Alpha. (Alpha is, by definition, the month in which New Year's Day falls.) Therefore, if there are 30 or more days left in the year after Mu, a thirteenth month (Nu) has to be inserted before Alpha. This will happen every two or three years, but there will never be any ambiguity: Nu is inserted only if not doing so would result in Alpha ending before New Year's Day.

The calendar can be said to have begun with New Years Day, 2001, which was the eighth day of the lunation begun on December 25th, 2000. So New Year's Day was 8 Alpha, 2001.

That's it. There are only two considerations in running this calendar: add a day to Zeta every five years (unless the year is evenly divisible by 200 or 500), and insert the month Nu after Mu if there are 30 or more days left in the year after the 29th of Mu.

By following these simple rules each month will actually coincide with the appearance of the moon in the sky while the years keep track of the sun.

Accuracy

I've projected this calendar over the next 500 years and have compared the results with those listed on the website www.lunaroutreach.org/phases/phases.dat, which lists the times for all new moons until the year 2730. Over the 500 years tested (from 2001-2500) the calendar goes through 6,184 months, 57% of which actually coincide with the date of the beginning of a real lunar month; 29% begin on the date prior to that listed for the lunar month, and 13% on the date after it. That means that 99% of the 6,184 calendar months over the next 500 years will begin no later or earlier than one date following or preceding the true date.

The difference between the starting dates of calendar months vs the dates of new moons from 2001-2500 (total 6,184 months):

Days' difference	-2	-1	0	+1	+2
Number of months	64	1806	3525	783	6
Percentage	1%	29%	57%	13%	0%

In these 500 years there will be 3281 long months of 30 days, and 2903 short months of 29 days, giving an average of 29.530563 days per month. For comparison, the Islamic calendar, which has 191 long months for every 169 short months, gives a monthly average of 29.530556 days. (An average lunation is actually about 29.530588 days.)

Some members of the CALNDR-L mailing list have helped determine the accuracy of this calendar mathematically. Amos Shapir calculated the average month to be 29.5305735 days, and added that this "represents a drift from the current mean lunation by about one day in 5262 years. Not bad, considering that the Gregorian calendar itself is not going to last unchanged for that long."

Karl Palmen has provided this calculation:

"2000 years have 5*97=485 leap years and 2*194=388 abundant years, so creating an epact drift of
2000*11 + 485 - 388 = 22097.
60,000 years has 30 times the epact drift so has 22097 intercalary months and hence 742097 months altogether.
The mean month is hence 60,000*365.2425/742097 = 29.530573496... which is in agreement with Amos's figure."

On the following page I've included a chart of the Gregorian dates for the first day of each month from 2001 to 2500. (Note that Alpha normally begins in December of the preceding year: 1 Alpha 2001 is in fact 25 December 2000.)

Note also that the calendar is at its most inaccurate in the first century, especially in the first 20 years, when 21 of 248 months start two days before the real lunar month, 150 start one day early, and 76 start on the same day.

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