A Step-by-step example and some shortcuts.
We'll start off simple, with a year from the first block of 19 years. Let's pick the year 7, which is easily known to be a regular year. Of course, we could also use the method that involves the remainder comparison with 12. In this case y = 7, so 235y - 234 = 1645 - 234 = 1411. Upon dividing this by 19, the remainder is 5, which is less than 12. So 7 is indeed a non-leap year. The integer quotient is 74, and hence the month offset. (Don't forget to hold on to that remainder of 5, in case it is needed in step 4.) This completes Step 1.
And now the hard part—Step 2. We multiply 74 by 29 days + 12 hours + 44 minutes + 3 1/3 seconds to get the days-oriented offset. This gives us:
2,146 days + 888 hours + 3,256 minutes + 246 2/3 seconds =
2,146 days + 888 hours + 3,260 minutes + 6 2/3 seconds =
2,146 days + 942 hours + 20 minutes + 6 2/3 seconds =
2,185 days + 6 hours + 20 minutes + 6 2/3 seconds
Next, we divide this by 7 days (all we're really dividing here is just the 2,185 days, not the smaller units). While the integer quotient for the days is 312, what we're really interested in is the remainder, which—with the smaller units included—gives us 1 day + 6 hours + 20 minutes + 6 2/3 seconds as the phase shift. This concludes Step 2. In Step 3, this phase shift is added to 1 day + 23 hours + 11 minutes + 20 seconds, the first year's molad. The result is:
2 days + 29 hours + 31 minutes + 26 2/3 seconds =
3 days + 5 hours + 31 minutes + 26 2/3 seconds
In other words, this sum amounts to Tuesday, 5:31:26+2/3 a.m. as the new moon time for the year 7. Now that this step is done, let's carry out Step 4.
Remember the remainder of 5 from the first step? Do we need it? No, not at all for a Tuesday. We do not need to check the leap/regular classification for an adjacent year. So we can head straight to the molad table for a non-leap year. Look up Tuesday, 5:31:26+2/3 a.m. It occurs no earlier than Tuesday, 3:11:20 a.m., yet it occurs before Thursday, 3:11:20 a.m. Therefore Thursday marks the day of the first of Tishri in the year 7, a 354-day (mid-sized regular) year.
With Step 4 done, we now know how the months of the year 7 are to be composed!
I will have to admit, all this computing can be a lot of work when dealing with all those units of time (especially carrying the smaller units into the days)! Isn't there a better, quicker way?
Fortunately, yes. Remember the part, that versatile unit of 3 1/3 seconds? Its usage is very popular in Jewish calendar computing. Let's take a deeper look at this unit:
1 minute = 18 parts.
1 hour = 1,080 parts.
1 day = 25,920 parts.
We can express the day's smaller units strictly in parts, as opposed to hours, minutes and seconds. We can also perform "phase math". Now what is that?! It's a different way of carrying out Steps 2 and 3. All we are really looking for in these two steps is phase information. Actually, Step 3 already involves a phased approach to addition if the initial sum is 7 days or greater, because 7 days are subtracted in such a case. It's like going from a particular time of a given week to the exact same time of a different week. But Step 2 is where phasing can be very useful.
We multiplied the size of the lunar cycle in this step. But since we only need the phase shift with respect to the week, we can simply multiply the phase shift along the week that the lunar cycle causes. When we divide the lunar cycle's length by 7 days, what we get for the remainder is:
1 day + 12 hours + 44 minutes + 3 1/3 seconds
This is the change along the week that each lunar cycle causes. For example, if a new moon takes place at 11:00 a.m. on a Monday, the next one takes place on a Tuesday at 11:44:3+1/3 p.m., even though that Tuesday is in a different week. While the actual difference between these two times is a little over 29 1/2 days, the week-sized phase difference is a little over 1 1/2 days. Let us take 74 again, just like before, but this time multiply it by the phase shift. This gives us:
74 days + 888 hours + 3,256 minutes + 246 2/3 seconds =
74 days + 888 hours + 3,260 minutes + 6 2/3 seconds =
74 days + 942 hours + 20 minutes + 6 2/3 seconds =
113 days + 6 hours + 20 minutes + 6 2/3 seconds
When we divide this amount of time by 7 days, we get the remainder of 1 day + 6 hours + 20 minutes + 6 2/3 seconds, just like before (even though the integer quotient for the days unit is 16, that's not important here). Now we could add this remainder to the first year's molad, but since we divided 113 days + 6 hours + 20 minutes + 6 2/3 seconds by 7 days, and this 7-day divisor is the size of a single weekly phase, the remainder that we got is equal to the amount of phase change of what we just divided! In other words, when adding to year 1's molad, we can use that which we just divided, instead of the remainder. This is because we are only concerned with the sum's phase position with respect to the week. This approach can save us a division operation. The division by 7 days is deferred until we reach this sum. Then, after we perform the division here (dividing this sum), we get the correct phase position in the week (i.e., the time of the week) for the given year's molad. So let's add 113 days + 6 hours + 20 minutes + 6 2/3 seconds to the first year's molad, i.e., 1 day + 23 hours + 11 minutes + 20 seconds. This gives us:
114 days + 29 hours + 31 minutes + 26 2/3 seconds =
115 days + 5 hours + 31 minutes + 26 2/3 seconds
Next, dividing this by 7 days gives us a remainder of:
3 days + 5 hours + 31 minutes + 26 2/3 seconds
That's Tuesday, 5:31:26+2/3 a.m., as our result when using "phase math", the same result as what we got earlier when using our original approach.
For performing computations using parts, let us gather some relevant information. The first year's molad, 1 day + 23 hours + 11 minutes + 20 seconds, is equivalent to:
1 day + (23 × 1,080 + 11 × 18 + 6) parts =
1 day + (24,840 + 198 + 6) parts =
1 day + 25,044 parts
The size of a single lunar cycle, 29 days + 12 hours + 44 minutes + 3 1/3 seconds, is equivalent to:
29 days + (12 × 1,080 + 44 × 18 + 1) parts =
29 days + (12,960 + 792 + 1) parts =
29 days + 13,753 parts
The phase change amount for this, with respect to the week is easily derived:
1 day + 13,753 parts
Expressing the molad tables in parts is also useful.
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