Based on the value of 365.242199 days for the mean tropical year, and of 29.530588853 days for the mean synodic month, I worked out a calendar that was based on these values (rather than on astronomical observations of actual new moons) that would, using a standard scheme of alternating between months and years of different lengths, the same way that the Gregorian calendar uses a fixed rule for leap years, remain in step with both the sun and the moon.
There are three basic types of year, although the year with an intercalary month, called the long year, can have that extra month at different positions during the year (thus keeping the other months closer to their appropriate seasonal times, by making the interval between intercalary months either 32 or 33 months long).
The types of years used in the calendar are as follows:
No year is both long and leap; the twelfth month only has 30 days in years with only twelve months.
There are two basic types of "cycle" from which this calendar is built. The most common one is the well-known Metonic cycle, consisting of 19 years, seven of which have an extra intercalary month. After seventeen or eighteen of these cycles in a row, however, there is a sufficient build-up of error due to a difference between 19 tropical years and 235 synodic months, that a second type of cycle, consisting of 11 years, four of which have an intercalary month, is used to restore the balance.
The cycles are constructed to be symmetrical, and for each type of cycle, the months followed by an intercalary month are fixed. This means that the months do not quite coincide as well with their proper seasonal placement as would be the case if some variation were allowed, for example, by changing the positions of the intercalary months in the first few and the last few cycles in a stretch of 17 or 18 normal cycles. However, this calendar is complicated enough!
Here are the types of cycle used in the calendar:
These cycles are chosen so as to be symmetric and smooth. The normal cycle, corresponding to the Metonic cycle, is the most common, but occasional short cycles, one short cycle for every 17 or 18 normal cycles, are required to cope with a slight discrepancy between 19 tropical years (6939.60178 days) and 235 synodic months (6939.6883804 days).
A GROUP is either 35 or 52 years long, and is built from the following items:
The types of groups are:
A LONG GROUP consists of:
A SHORT GROUP, which is either an EARLY SHORT GROUP or a LATE SHORT GROUP, consists of:
Finally, we proceed to the highest-level structure in the calendar. Well, almost. A ROUND is 6,479 years long, consisting of five long groups and two short groups, and one extra day is added to one round in every five to make this calendar just about as accurate as the calculations on which it was based, carried out on my trusty old Texas Instruments SR-56 programmable calculator.
A round is built from the following groups, in order:
Every group of five rounds has one round, the third, as a leap round.
This corresponds very closely to the proper average length of a round, consisting of 6,479 tropical years or 80,134 synodic months, of 2,366,404.207 days.
Incidentally, a group of five rounds with one leap round, and a normal round, both take five days longer than an even number of weeks, thus, it will take a full thirty-five rounds, comprising 226,765 years, for the sun, the moon, and the week all to coincide once again.
Well, it's all very well to describe a complicated calendar. But it is not very useful if it isn't possible to say what year it is by that calendar.
Of course, one could start the calendar at any time one liked. But let us suppose a conventional starting point: let each month attempt to begin approximately on the new moon, and let the year begin at the first new moon on or after the vernal equinox; the conventional "first day of spring" that takes place around March 21st on the conventional calendar.
Thus, the first day of the year on this calendar could, at its extreme earliest, fall on a day when the mean new moon takes place at 00:00:01 AM and the actual vernal equinox takes place at 12:59:59 PM.
Given a tropical year of 365.242199 days, the gain and loss through various types of years and cycles is as follows (in parentheses the gain and loss if we assume the year is composed of ideal lunar months, rather than 29 or 30 day months, is shown):
From examining ephemerides, it appears that a fit may be obtained with a special long group that had started in the year 1495. This places us in a round that started in 1224 B.C.
The following set of tables will allow one to determine the Julian Day Number of the beginning of any month within a round by this system.
Displacements of groups within a round:
Displacements of short cycles and stretches within a group:
Displacements of cycles within a stretch:
Displacements of years within a cycle:
Displacements of months within a year:
Thus: March 20, 2004 (J.D. 2453085) is the first day of the fifth year in a leap short cycle. Thus, that leap short cycle began 1447 days earlier, on J.D. 2451638 (April 3, 2000). This leap short cycle followed a special stretch of 17 long cycles, so it occurred 184447 days after the beginning of a special long group, which started on J.D. 2267191 (March 26, 1495, by the Julian calendar). A special long group occurs 996746 days after the beginning of a round, on J.D. 1270445 (April 16, 1235 B.C.).
The two calendars which use a lunar month, and which have an extra month in some years to keep in line with the seasons, that may be familiar to most people are the Hebrew calendar and the Chinese calendar.
The Hebrew calendar, used for keeping track of Jewish religious festivals, is calculated according to rules which are made more complicated by the need to adjust some months to prevent certain religious holidays from falling on certain days of the week.
This calendar is based on the 19-year Metonic cycle. Leap years occur in the 3rd, 6th, 8th, 11th, 14th, 17th, and 19th years of the cycle, and the added month is always known as ve-Adar; Adar is the 12th month of the year as the months were numbered in the Bible, but today the Jewish New Year is celebrated on Rosh Hashanah, the first of Tishri, the 7th month in the original numbering, and thus Adar is the 6th month in the current numbering. The month corresponding to Nisan in the Babylonian calendar, Nisannu, was the first month of the year in that calendar as well.
The current Metonic cycle in use for the Jewish calendar is as follows:
1 October 2, 1997 5758 8* September 16, 2004 5765 15 September 29, 2011 5772 2 September 21, 1998 5759 9 October 4, 2005 5766 16 September 17, 2012 5773 3* September 11, 1999 5760 10 September 23, 2006 5767 17* September 5, 2013 5774 4 September 30, 2000 5761 11* September 13, 2007 5768 18 September 25, 2014 5775 5 September 18, 2001 5762 12 September 30, 2008 5769 19* September 14, 2015 5776 6* September 7, 2002 5763 13 September 19, 2009 5770 7 September 27, 2003 5764 14* September 9, 2010 5771
The names of the months in the ancient Babylonian calendar, and the corresponding month names in the religious calendar of Judaism, as well as those in the Chinese calendar and the ancient calendar of Athens, are listed below for reference:
Babylonian Jewish Athenian Chinese
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Spring
Aries 1 Nisannu 7 Nisan 10 Munichion 4 Er Yue Pink/Milk/Growing
Taurus 2 Aru 8 Iyyar 11 Thargelion 5 San Yue Flower Moon (Hare)
Gemini 3 Simanu 9 Sivan 12 Skirrophorion 6 Si Yue Strawberry/Mead
Summer
Cancer 4 Du'uzu 10 Tammuz 1 Hecatombaeon 7 Wu Yue Buck/Grain
Leo 5 Abu 11 Ab 2 Metageitonion 8 Liu Yue Sturgeon/Fruit/Corn
Virgo 6 Ululu 12 Elul 3 Boedromion 9 Ch'i Yue Harvest Moon (Wine)
Fall
Libra 7 Tasritu 1 Tishri 4 Maimakterion 10 Pa Yue Hunter's Moon
Scorpio 8 Arahsamna 2 Marheshvan 5 Pyanepsion 11 Chiu Yue Beaver/Oak
Saggitarius 9 Kislimu 3 Kislev 6 Poseideon 12 Shi Yue Cold/Old/Winter
Winter
Capricorn 10 Tebetu 4 Tevet 7 Gamelion 1 Shi Yi Yue Wolf Moon
Aquarius 11 Sabatu 5 Shevat 8 Anthesterion 2 Shi Er Yue Snow/Lenten/Ice
Pisces 12 Addaru 6 Adar 9 Elaphebolion 3 Cheng Yue Worm/Egg/Storm
The Harvest Moon is the Full Moon that comes closest to the Autumnal Equinox, and thus the New Moon that follows it would be the first one on or after the Autumnal Equinox, which is why it is placed at the end of Summer rather than at the beginning of Fall, so that it might be consistent with the months listed here.
The Chinese calendar is based on direct observation, and the intercalary month can occur in any part of the year, as the months are based on the signs of the Zodiac.
In the ancient Babylonian calendar, an extra month was added in the 3rd, 6th, 8th, 11th, 14th, 17th, and 19th years of a 19-year cycle, but while a second Addaru (Adar) was added most of the time, in the 17th year of the cycle, a second Ululu (Elul) was added instead. This somewhat evened out the spacing of intercalary months, keeping the year more closely in harmony with the seasons.
Thus, the time between intercalary months was 3, 3, 2, 3, 3, 2 1/2 and 2 1/2 years. This also made the cycle symmetric; so changing to a second Ululu in the 6th year of the cycle, or replacing the second Addaru in the 8th year of the cycle by a second Ululu in the 9th year would have broken the symmetry again.
Had it been desired to retain symmetry, and stay even closer to the seasons, of course, they could have gone with a second Arahsamna (Marheshvan) in the 6th year of the cycle, and a second Du'uzu (Tammuz) in the 9th year of the cycle, for a spacing between intercalary months of 3, 2 2/3, 2 2/3, 2 2/3, 3, 2 1/2, and 2 1/2 years, but Darius no doubt thought that would be too complicated. (Actually, of course, one could doubt that the idea even crossed his mind.)
In the normal cycle shown above, the extra months are added in the 2nd, 5th, 7th, 10th, 13th, 15th, and 18th years of the cycle, and so the first year of the Babylonian cycle corresponds to the eighth year of the normal cycle, and the first year of the normal cycle corresponds to the thirteenth year of the Babylonian cycle.
Thus, this table shows the possible leap months in a Metonic cycle in these schemes, using the month names of the modern Jewish religious calendar as being somewhat more familiar than those of the ancient Babylonian calendar:
Hebrew Babylonian Modified Normal
3rd year ve-Adar ve-Adar ve-Adar ve-Adar
6th year ve-Adar ve-Adar ve-Marheshvan ve-Kislev
8th year ve-Adar ve-Adar ve-Tammuz ve-Ab
11th year ve-Adar ve-Adar ve-Adar ve-Iyar
14th year ve-Adar ve-Adar ve-Adar ve-Tevet
17th year ve-Adar ve-Elul ve-Elul ve-Tishri
19th year ve-Adar ve-Adar ve-Adar ve-Sivan
The normal cycle and the short cycle, considered in terms of their own starting and ending times as given above, would work like this:
Normal Cycle (19 years) Short Cycle (11 years) 2nd year ve-Tevet 2nd year ve-Shevat 5th year ve-Tishri 5th year ve-Marheshvan 7th year ve-Sivan 7th year ve-Tammuz 10th year ve-Adar 10th year ve-Nisan 13th year ve-Kislev 15th year ve-Ab 18th year vi-Iyyar