Mathematics of the Jewish Calendar/The sixty-one types of cycle

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The sixty-one types of cycle

Given the exact molad of the first year of a cycle, you can compute the molad of each year of the cycle hence their year types.

There are 61 possible sequences of years in a cycle.

This can be shown by considering every possible initial molad. Note that since the molad increases by 6939 days, 16 hours, 595 chalakim over a cycle, and 595 is of course a multiple of five, the molad at the start of any cycle must have a number of chalakim ending in either 4 or 9. Also, we may discard multiples of seven days since this makes no difference to the year type. Thus we can start with 7 days, 0 hours, 4 chalakim; by the postponement rules, this gives us Rosh Hashana on Monday and a defective year.

A full discussion and details of all 61 types are given in the book by Burnaby (see Mathematics of the Jewish Calendar/Further reading).

Most cycles have length 6,939, 6,940 or 6,941 days. Only type 61 (with an initial molad of 7 days, 16 hours, 689 chalakim to 7 days, 17 hours, 1079 chalakim) has 6,942 days. Such cycles are quite rare. If the present rules had always existed, the cycles starting in 2908 and 3155, i.e. 854 BCE and 607 BCE, would have been of this type. The next starts in 6765 (3004 CE).

Starts of cycles of length 6942 are separated by either 13 cycles (247 years), 190 cycles (3610 years) or 203 = 190+13 cycles (3857 years). As will be noted later, the calendar repeats approximately, although not exactly, after 13, 190 or 203 cycles.

When tabulating the Jewish calendar over a long period, it is much quicker to work with a whole cycle at a time than with individual years.