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:

• 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.

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:

• The NORMAL CYCLE is 19 years long. It consists of the following types of years, in order:
  1. An ordinary year
  2. A long year, with an intercalary month following the fourth month.
  3. A leap year
  4. An ordinary year
  5. A long year, with an intercalary month following the first month.
  6. An ordinary year
  7. A long year, with an intercalary month following the ninth month.
  8. An ordinary year (a leap year in LEAP NORMAL CYCLES)
  9. An ordinary year
  10. A long year, with an intercalary month following the sixth month.
  11. A leap year
  12. An ordinary year
  13. A long year, with an intercalary month following the third month.
  14. An ordinary year
  15. A long year, with an intercalary month following the eleventh month.
  16. A leap year
  17. An ordinary year
  18. A long year, with an intercalary month following the eighth month.
  19. An ordinary year
• The SHORT CYCLE is 11 years long. It consists of the following types of years, in order:
  1. An ordinary year
  2. A long year, with an intercalary month following the fifth month.
  3. A leap year
  4. An ordinary year
  5. A long year, with an intercalary month following the second month.
  6. An ordinary year
  7. A long year, with an intercalary month following the tenth month.
  8. An ordinary year (a leap year in LEAP SHORT CYCLES)
  9. An ordinary year
  10. A long year, with an intercalary month following the seventh month.
  11. 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).

Types of groups

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:

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):

• 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.).