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A Luni-Solar Calendar

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.

Types of years

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.

Types of cycles

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).

Types of groups

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:

  1. A stretch of nine normal cycles
  2. A short cycle
  3. A stretch of seventeen normal cycles (which is a special stretch of seventeen normal cycles in a SPECIAL LONG GROUP)
  4. A short cycle (a leap short cycle in a LEAP LONG GROUP or a SPECIAL LONG GROUP)
  5. A stretch of seventeen normal cycles
  6. A short cycle
  7. A stretch of nine normal cycles

A SHORT GROUP, which is either an EARLY SHORT GROUP or a LATE SHORT GROUP, consists of:

  1. A stretch of nine normal cycles
  2. A short cycle (a leap short cycle in an EARLY SHORT GROUP)
  3. A stretch of seventeen normal cycles
  4. A short cycle (a leap short cycle in a LATE SHORT GROUP)
  5. A stretch of nine normal cycles

The Round

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:

  1. A long group
  2. An early short group
  3. A long group
  4. A special long group (a leap long group in a LEAP ROUND)
  5. A long group
  6. A late short group
  7. A long group

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.

An Epoch for the Calendar

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.).

Some Real Luni-Solar Calendars

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
------------------------------------------------------------------------------------------------
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

The Lunar Haab

In general, the lunisolar calendars are based on the Metonic cycle of 19 lunar years - 12 normal years of 12 months and 7 leap years of 13 months:

 19 * 365.242199    = 6939.60178
235 *  29.530588853 = 6939.68838

The ancient Egyptians, who used a vague year of 365 days without any leap years, used a similar cycle to keep in step with that year, of 25 lunar years - 16 normal years of 12 months and 9 leap years of 13 months:

 25 * 365           = 9125
309 *  29.530588853 = 9124.95196

Note that here the discrepancy is .05 of a day instead of .08 of a day, so this is a more accurate cycle for its purpose, even if fundamentally less useful.

This calendar was described, using the Carlsberg IX papyrus as a source, in The Calendars of Ancient Egypt by Richard A. Parker.

His reconstruction of the 25-year cycle proceeds as follows:

                      L     L        L        L        L     L        L        L        L
                      1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
                     --------------------------------------------------------------------------
Thoth       30  30    1 20  9 28 18  7 26 15  4 24 13  2 21 10 30 19  8 27 16  6 25 14  3 22 12
Paophi      30  29    1 20  9 28 18  7 26 15  4 24 13  2 21 10 30 19  8 27 16  6 25 14  3 22 12
Hathor      30  30    * 19  8 27 17  6 25 14  3 23 12  1 20  9 29 18  7 26 15  5 24 13  2 21 11
Choiak      30  29   30 19  8 27 17  6 25 14  3 23 12  1 20  9 29 18  7 26 15  5 24 13  2 21 11
Tybi        30  30   29 18  7 26 16  5 24 13  2 22 11  * 19  8 28 17  6 25 14  4 23 12  1 20 10
Mechir      30  30   29 18  7 26 16  5 24 13  2 22 11 30 19  8 28 17  6 25 14  4 23 12  1 20 10
Phamenoth   30  29   29 18  7 26 16  5 24 13  2 22 11 30 19  8 28 17  6 25 14  4 23 12  * 20 10
Pharmouthi  30  29   28 17  6 25 15  4 23 12  1 21 10 29 18  7 27 16  5 24 13  3 22 11 30 19  9
Pachon      30  30   27 16  5 24 14  3 22 11  * 20  9 28 17  6 26 15  4 23 12  2 21 10 29 18  8
Payni       30  30   27 16  5 24 14  3 22 11 30 20  9 28 17  6 26 15  4 23 12  2 21 10 29 18  8
Epiphi      30  29   27 16  5 24 14  3 22 11 30 20  9 28 17  6 26 15  4 23 12  2 21 10 29 18  8
Mesore      30 (29)  26 15  4 23 13  2 21 10 29 19  8 27 16  5 25 14  3 22 11  1 20  9 28 17  7
Epagomenes   5  29          4        2                       5        3        1

In the first three columns, I give the name of each month of the Egyptian vague year, and then the number of days in that month, and then the normal number of days of a lunar month beginning in that month.

In the case of a lunar month beginning in Mesore, it will be 30 days long if it ends in the Epagomenes of the same year, but only 29 days long if it ends in Thoth of the next year.

A leap year can originate from two causes: a 29-day lunar month can begin on the first day of the month in the vague year, causing another month to begin on the thirtieth day of the same month, or a month can begin in the Epagomenes.

The chart of the 25 years in the cycle shows the dates in each month of the vague year where a lunar month begins. An asterisk indicates where lunar months begin on both the 1st and the 30th days of a given month of the vague year.

As can be seen from the chart, when lunar months begin on the first day of a month of the vague year, their length is governed by the normal length applicable to the preceding month; this delays the occurrence of the first days of two lunar months within a single month of the vague year by one month. This is similar to the case of Mesore; when the lunar month beginning in Mesore begins early enough so that it will cause a leap year by having the next month begin in the Epagomenes, it is lengthened to 30 days. This rule can avoid ambiguous cases in the calendar.

In the 23rd year of the cycle, however, that rule is not followed, because following it would delay the dual month from Phamenoth to Pharmouthi, and both of these months have 29 days as the normal length of a lunar month beginning within them, and so, presumably, the usual benefit from the delay is not gained.


Our Gregorian calendar usually has one leap year every four years, and sometimes omits the leap year entirely rather than having some leap years at five year intervals, thus introducing odd numbers in the calendar. When one is adding a whole month, rather than just a day, to the calendar, simplicity may have to take a back seat to minimizing the maximum deviation of the calendar from the seasons.

However, the idea is tempting to use four of these Egyptian cycles in a century, and then add one extra leap year occasionally to keep the calendar in step with the tropical year instead of the vague year.

A conventionalized luni-solar calendar of this type might run as follows:

01 04 06 09 12 15 17 20 23
26 29 31 34 37 40 42 45 48
51 54 56 59 62 65 67 70 73
76 79 81 84 87 90 92 95 98

would be the digits ending each leap year, with the year ending in 99 occasionally being a leap year as well.

Four 25-year cycles adding up to 36499.80782 days, this is short of 100 tropical years by 24.4120777 days. This means that one extra leap month in a century would be the usual case, so the century should perhaps instead have leap years like this:

01 04 06 09 12 15 17 20 23
26 29 31 34 37 40 42 45 48
51 54 56 59 62 65 67 70 73
76 79 81 84 87 90 92 95 97 99

with the year ending in 97 not being a leap year in one in every six centuries, or, as 5.7693708 would be the more exact figure, perhaps not being a leap year four times in every 23 centuries.

A Solar Calendar

A structure with cycles of this type, to reduce the discrepancy between the calendar and its astronomical ideal, by delaying leap days instead of omitting them, has been used in one real calendar.

A proposed algorithmic version of the Persian calendar alternates between cycles 29 and 33 years in length, which begin with four non-leap years followed by a leap year, and then continue with groups of three non-leap years followed by a leap year.

However, at the end of a period of 2820 years, consisting of groups of one 29 year cycle followed by three 33 year cycles, the final cycle is made 37 years long, instead of replacing a 29 year cycle by a 33 year cycle, and distributing the cycles evenly by following some 29 year cycles by four (instead of three) 33 year cycles. Thus, the principle of gradualism is not followed consistently into this higher level, but instead the suddenness found in our Gregorian calendar appears at that level.


I believe this would work out as follows:

Instead of 21 cycles of 29,33,33,33 followed by one cycle of 29,33,33,37, there would now be a total of 21 cycles in 2,820 years; four cycles of 29,33,33,33,33 and seventeen cycles of 29,33,33,33.

And, so, to follow the pattern of the rest of the calendar, first the pattern

29,33,33,33,33, 29,33,33,33, 29,33,33,33, 29,33,33,33, 29,33,33,33, 29,33,33,33

would occur once, and then the pattern

29,33,33,33,33, 29,33,33,33, 29,33,33,33, 29,33,33,33, 29,33,33,33

would occur three times.


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