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:

- The ORDINARY YEAR consists of twelve months. The first month, and all other odd-numbered months, are 30 days in length; the second month, and all other even-numbered months, are 29 days in length.
- The LEAP YEAR is an ordinary year, except that the twelfth month is 30 days in length.
- The LONG YEAR consists of 13 months. The twelve regular months of the year are of the same length as in an ordinary year, and one of them is followed by a 30-day intercalary month.

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:

- The NORMAL CYCLE is 19 years long. It consists of the following types
of years, in order:
- An ordinary year
- A long year, with an intercalary month following the fourth month.
- A leap year
- An ordinary year
- A long year, with an intercalary month following the first month.
- An ordinary year
- A long year, with an intercalary month following the ninth month.
- An ordinary year (a leap year in LEAP NORMAL CYCLES)
- An ordinary year
- A long year, with an intercalary month following the sixth month.
- A leap year
- An ordinary year
- A long year, with an intercalary month following the third month.
- An ordinary year
- A long year, with an intercalary month following the eleventh month.
- A leap year
- An ordinary year
- A long year, with an intercalary month following the eighth month.
- An ordinary year

- The SHORT CYCLE is 11 years long. It consists of the following types
of years, in order:
- An ordinary year
- A long year, with an intercalary month following the fifth month.
- A leap year
- An ordinary year
- A long year, with an intercalary month following the second month.
- An ordinary year
- A long year, with an intercalary month following the tenth month.
- An ordinary year (a leap year in LEAP SHORT CYCLES)
- An ordinary year
- A long year, with an intercalary month following the seventh month.
- A leap year

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:

- Short cycles;
- Stretches of seventeen normal cycles, in which all are leap normal cycles, except the second, sixth, ninth, twelfth, and sixteenth;
- Stretches of nine normal cycles, in which all are leap normal cycles, except the second, fifth, and eighth (these are always located at the beginning or end of the group, to make stretches of eighteen normal cycles between groups);
- A special stretch of seventeen normal cycles, in which all are leap normal cycles, except the second, fifth, eighth, tenth, thirteenth, and sixteenth;

The types of groups are:

A LONG GROUP consists of:

- A stretch of nine normal cycles
- A short cycle
- A stretch of seventeen normal cycles (which is a special stretch of seventeen normal cycles in a SPECIAL LONG GROUP)
- A short cycle (a leap short cycle in a LEAP LONG GROUP or a SPECIAL LONG GROUP)
- A stretch of seventeen normal cycles
- A short cycle
- A stretch of nine normal cycles

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

- A stretch of nine normal cycles
- A short cycle (a leap short cycle in an EARLY SHORT GROUP)
- A stretch of seventeen normal cycles
- A short cycle (a leap short cycle in a LATE SHORT GROUP)
- A stretch of nine normal cycles

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:

- A long group
- An early short group
- A long group
- A special long group (a leap long group in a LEAP ROUND)
- A long group
- A late short group
- 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.

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):

- Ordinary Year: 354 days. Change: -11.242199
- Leap Year: 355 days. Change: -10.242199
- Long Year: 384 days. Change: 18.757801
- Normal Cycle: 9 ordinary years, 3 leap years, 7 long years: -0.601781 (0.0866)
- Leap Normal Cycle: 8 ordinary years, 4 leap years, 7 long years: 0.398219 (0.0866)
- Short Cycle: 5 ordinary years, 2 leap years, 4 long years: -1.664189 (-1.5041)
- Leap Short Cycle: 4 ordinary years, 3 leap years, 4 long years: -0.664189 (-1.5041)
- Long Group: 36 leap normal cycles, 16 normal cycles, 3 short cycles: -0.285179 (-0.0091)
- Leap Long Group: 36 leap normal cycles, 16 normal cycles, 2 short cycles, 1 leap short cycle: 0.714821 (-0.0091)
- Special Long Group: 35 leap normal cycles, 17 normal cycles, 2 short cycles, 1 leap short cycle: -0.285179 (-0.0091)
- Short Group (Early or Late): 24 leap normal cycles, 11 normal cycles, 1 short cycle, 1 leap short cycle: 0.609287 (0.0228)
- Round: 4 long groups, 1 special long group, 2 short groups: -0.207321
- Leap Round: 4 long groups, 1 leap long group, 2 short groups: 0.792679

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:

- R: 0 LG 372912 ESG 623834 LG 996746 SLG 1369658 LG 1742570 LSG 1993492 LG (2366404)
- LR: 0 LG 372912 ESG 623834 LG 996746 LLG 1369859 LG 1742571 LSG 1993493 LG (2366405)

Displacements of short cycles and stretches within a group:

- LG: 0 S9 62457 SC 66473 S17 184448 SC 188464 S17 306439 SC 310455 S9 (372912)
- LLG: 0 S9 62457 SC 66473 S17 184448 LSC 188465 S17 306440 SC 310456 S9 (372913)
- SLG: 0 S9 62457 SC 66473 SS17 184447 LSC 188464 S17 306439 SC 310455 S9 (372912)
- ESG: 0 S9 62457 LSC 66474 S17 184449 SC 188465 S9 (250922)
- LSG: 0 S9 62457 SC 66473 S17 184448 LSC 188465 S9 (250922)

Displacements of cycles within a stretch:

- S17: 0 LNC 6940 NC 13879 LNC 20819 LNC 27759 LNC 34699 NC 41638 LNC 48578 LNC 55518 NC 62457 LNC 69397 LNC 76337 NC 83276 LNC 90216 LNC 97156 LNC 104096 NC 111035 LNC (117975)
- S9: 0 LNC 6940 NC 13879 LNC 20819 LNC 27759 NC 34698 LNC 41638 LNC 48578 NC 55517 LNC (62457)
- SS17: 0 LNC 6940 NC 13879 LNC 20819 LNC 27759 NC 34698 LNC 41638 LNC 48578 NC 55517 LNC 62457 NC 69396 LNC 76336 LNC 83276 NC 90215 LNC 97155 LNC 104095 NC 111034 LNC (117974)

Displacements of years within a cycle:

- NC: 0 OY 354 LYI4 738 LPY 1093 OY 1447 LYI1 1831 OY 2185 LYI9 2569 OY 2923 OY 3277 LYI6 3661 LPY 4016 OY 4370 LYI3 4754 OY 5108 LYI11 5492 LPY 5847 OY 6201 LYI8 6585 OY (6939)
- LNC: 0 OY 354 LYI4 738 LPY 1093 OY 1447 LYI1 1831 OY 2185 LYI9 2569 LPY 2924 OY 3278 LYI6 3662 LPY 4017 OY 4371 LYI3 4755 OY 5109 LYI11 5493 LPY 5848 OY 6202 LYI8 6586 OY (6940)
- SC: 0 OY 354 LYI5 738 LPY 1093 OY 1447 LYI2 1831 OY 2185 LYI10 2569 OY 2923 OY 3277 LYI7 3661 LPY (4016)
- LSC: 0 OY 354 LYI5 738 LPY 1093 OY 1447 LYI2 1831 OY 2185 LYI10 2569 LPY 2924 OY 3278 LYI7 3662 LPY (4017)

Displacements of months within a year:

- OY: 0 30 59 89 118 148 177 207 236 266 295 325 (354)
- LPY: 0 30 59 89 118 148 177 207 236 266 295 325 (355)
- LYI1: 0 .30. 60 89 119 148 178 207 237 266 296 325 355 (384)
- LYI2: 0 30 .59. 89 119 148 178 207 237 266 296 325 355 (384)
- LYI3: 0 30 59 .89. 119 148 178 207 237 266 296 325 355 (384)
- LYI4: 0 30 59 89 .118. 148 178 207 237 266 296 325 355 (384)
- LYI5: 0 30 59 89 118 .148. 178 207 237 266 296 325 355 (384)
- LYI6: 0 30 59 89 118 148 .177. 207 237 266 296 325 355 (384)
- LYI7: 0 30 59 89 118 148 177 .207. 237 266 296 325 355 (384)
- LYI8: 0 30 59 89 118 148 177 207 .236. 266 296 325 355 (384)
- LYI9: 0 30 59 89 118 148 177 207 236 .266. 296 325 355 (384)
- LYI10: 0 30 59 89 118 148 177 207 236 266 .295. 325 355 (384)
- LYI11: 0 30 59 89 118 148 177 207 236 266 295 .325. 355 (384)

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

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 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.