Let's do some more examples! This time, we will select a year well beyond the first block of 19 years. Let's pick the year 3001 and use the parts-oriented molad formula. The molad, with respect to the week, for 3001 is then:
remainder of ([actual-based month offset × week-related phase change of a lunar cycle] + first year's molad) divided by 181,440 parts =
([INT (([235 × given year] - 234)/19) × 39,673 parts] + 50,964 parts) MOD 181,440 parts =
([INT (([235 × 3001] - 234)/19) × 39,673 parts] + 50,964 parts) MOD 181,440 parts =
([37,105 × 39,673 parts] + 50,964 parts) MOD 181,440 parts =
1,472,117,629 parts MOD 181,440 parts =
94,909 parts =
3 days + 17,149 parts
Next, in order to determine the year 3001's classification, we evaluate the following:
([235 × given year] - 234) MOD 19 =
([235 × 3001] - 234) MOD 19 =
705,001 MOD 19 =
Since this remainder, 6, is less than 12, 3001 is a regular year. Looking up this year's molad in the non-leap year table gives us the range between Tuesday, 3,444 parts, and Thursday, 3,444 parts (no adjacent year information is needed for this range). So the year 3001 is 354 days long, and Thursday marks the first day of Tishri for that year.
How about the year 3002? Using the molad formula yields 1 day + 745 parts (i.e., Sunday, 745 parts) for that year. Since we already know 3001's remainder, we can conveniently do a "rotational addition of 7" to get the remainder for the year 3002. This remainder is 13, which meets the "12 or greater" qualification. This means that we get to use the leap year table (for a change!), and the molad falls in the Saturday, 12,960 parts-to-Sunday, 15,611 parts range here. So Monday marks the head ("rosh") of this year, and 383 days is its size. But is that remainder really 13 for the year 3002? Here's the proof:
([235 × 3002] - 234) MOD 19 =
705,236 MOD 19 =
How about a more recent year like 5759? The formula produces 2 days + 7,485 parts (i.e., Monday, 7,485 parts) as the molad for that year. The remainder for the classification is 8, which is less than 12, thus we are dealing with a non-leap year. We look up the Sunday, 3,444 parts-to-Monday, 10,309 parts range in the regular year table. And so we get Monday as the Rosh Hashanah day and 355 days as the length for the year 5759. Interested in the next year? The molad for 5760 is 6 days + 17,001 parts (Friday, 17,001 parts), and the classification remainder is 15 (and what does that mean? leap year). The Friday, 15,611 parts-to-Saturday, 12,960 parts range in the leap year table tells us that Saturday is the day of the week for 5760's Rosh Hashanah and that 385 days is the length for that year.
5766 is a rather interesting year to figure out. Its molad is 2 days + 11,676 parts (Monday, 11,676 parts), and its classification remainder is 0 (non-leap). But the real challenge takes place when we go to the regular year table. The applicable range, Monday, 10,309 parts to Monday, 12,960 parts, is dependent upon an adjacent year—that is, we need to know the classification for the previous one! Performing a "rotational subtraction of 7" from 0 (subtracting 7 from, plus adding 19 to, the classification remainder for 5766) gives us 12, thus indicating the previous year, 5765, as a leap year. So we must use the Monday, 10,309 parts-to-Monday, 12,960 parts range that is combined with the additional condition that the previous year is a leap year. This gives us Tuesday as the day of Rosh Hashanah and 354 days as the length of the year for 5766.
More specifically, the molad for 5766 is Monday at 10:48:40 a.m. Checking the detailed molad table for regular years shows a 382DAYS condition, which occurs very rarely. 5766 is one of those years that are subject to such a dechiyyah. (Notice that the 356DAYS condition is considerably less rare.)
Hopefully, these examples should give you a good idea of how to determine the first of Tishri and the size, in days, for any Jewish year.
Would you still like a set of molad tables the express the new moon times of the week strictly in parts? Okay, here are the tables in a specially modified form, with 0 parts marking midnight, the start of Saturday (the highest valid parts value here is 181,439, i.e., Friday, 11:59:56+2/3 p.m.):
|TABLE 7a. Summarized molad table for regular years, using parts only.|
|TABLE 7b. Summarized molad table for leap years, using parts only.|
Keep in mind the Western day format used in these tables.
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