Determining the first day of the calendar year, Rosh HaShanah, on 1 Tishri:
The new year starts on the day of the new moon that occurs about 354 days (or 384 days if the previous year was a leap year) after 1 Tishri of the previous year
If the new moon occurs after noon on that day, delay the new year by one day. (Because in that case the new crescent moon will not be visible until the next day.)
If this would cause the new year to start on a Sunday, Wednesday, or Friday, delay it by one day. (Because we want to avoid that Yom Kippur (10 Tishri) falls on a Friday or Sunday, and that Hoshanah Rabba (21 Tishri) falls on a Sabbath (Saturday)).
If two consecutive years start 356 days apart (an illegal year length), delay the start of the first year by two days.
If two consecutive years start 382 days apart (an illegal year length), delay the start of the second year by one day.

Note: Rule 4 can only come into play if the first year was supposed to start on a Tuesday. Therefore a two day delay is used rather that a one day delay, as the year must not start on a Wednesday as stated in rule 3.

When is the new moon?

A calculated new moon is used. In order to understand the calculations, one must know that an hour is subdivided into 1080 ``parts''.

The calculations are as follows:

The new moon that started the year AM 1, occurred 5 hours and 204 parts after sunset (i.e. just before midnight on Julian date 6 October 3761 BC).

The new moon of any particular year is calculated by extrapolating from this time, using a synodic month of 29 days 12 hours and 793 parts.

Note that 18:00 Jerusalem time (15:39 UTC) is used instead of sunset in all these calculations.

 Rosh Hashanah

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