3. The Hebrew Calendar
As it exists today, the Hebrew calendar is a lunisolar calendar that is based on calculation rather than observation. This calendar is the official calendar of Israel and is the liturgical calendar of the Jewish faith.
In principle the beginning of each month is determined by a tabular New Moon (molad) that is based on an adopted mean value of the lunation cycle. To ensure that religious festivals occur in appropriate seasons, months are intercalated according to the Metonic cycle, in which 235 lunations occur in nineteen years.
By tradition, days of the week are designated by number, with only the seventh day, Sabbath, having a specific name. Days are reckoned from sunset to sunset, so that day 1 begins at sunset on Saturday and ends at sunset on Sunday. The Sabbath begins at sunset on Friday and ends at sunset on Saturday.
3.1 Rules
Years are counted from the Era of Creation, or Era Mundi, which corresponds to -3760 October 7 on the Julian proleptic calendar. Each year consists of twelve or thirteen months, with months consisting of 29 or 30 days. An intercalary month is introduced in years 3, 6, 8, 11, 14, 17, and 19 in a nineteen-year cycle of 235 lunations. The initial year of the calendar, A.M. (Anno Mundi) 1, is year 1 of the nineteen-year cycle.
The calendar for a given year is established by determining the day of the week of Tishri 1 (first day of Rosh Hashanah or New Year's Day) and the number of days in the year. Years are classified according to the number of days in the year (see Table 3.1.1).
| Deficient | Regular | Complete | ||
| Ordinary year | 353 | 354 | 355 | |
| Leap year | 383 | 384 | 385 |
| 1. Tishri | 30 | 7. Nisan | 30 |
| 2. Heshvan | 29* | 8. Iyar | 29 |
| 3. Kislev | 30** | 9. Sivan | 30 |
| 4. Tevet | 29 | 10. Tammuz | 29 |
| 5. Shevat | 30 | 11. Av | 30 |
| 6. Adar | 29*** | 12. Elul | 29 |
* In a complete year, Heshvan has 30 days.
** In a deficient year, Kislev has 29 days.
*** In a leap year Adar I has 30 days; it is followed by Adar II with 29 days.
| Deficient (haser) month: a month comprising 29 days. |
| Full (male) month: a month comprising 30 days. |
| Ordinary year: a year comprising 12 months, with a total of 353, 354, or 355 days. |
| Leap year: a year comprising 13 months, with a total of 383, 384, or 385 days. |
| Complete year (shelemah): a year in which the months of Heshvan and Kislev both contain 30 days. |
| Deficient year (haser): a year in which the months of Heshvan and Kislev both contain 29 days. |
| Regular year (kesidrah): a year in which Heshvan has 29 days and Kislev has 30 days. |
| Halakim(singular, helek): "parts" of an hour; there are 1080 halakim per hour. |
| Molad(plural, moladot): "birth" of the Moon, taken to mean the time of conjunction for modern calendric purposes. |
| Dehiyyah(plural, dehiyyot): "postponement"; a rule delaying 1 Tishri until after the molad. |
The months of Heshvan and Kislev vary in length to satisfy requirements for the length of the year (see Table 3.1.1). In leap years, the 29-day month Adar is designated Adar II, and is preceded by the 30-day intercalary month Adar I.
For calendrical calculations, the day begins at 6 P.M., which is designated 0 hours. Hours are divided into 1080 halakim; thus one helek is 3 1/3 seconds. (Terminology is explained in Table 3.1.3.) Calendrical calculations are referred to the meridian of Jerusalem -- 2 hours 21 minutes east of Greenwich.
Rules for constructing the Hebrew calendar are given in the sections that follow. Cohen (1981), Resnikoff (1943), and Spier (1952) provide reliable guides to the rules of calculation.
3.1.1 Determining Tishri 1
The calendar year begins with the first day of Rosh Hashanah (Tishri 1). This is determined by the day of the Tishri molad and the four rules of postponements (dehiyyot). The dehiyyot can postpone Tishri 1 until one or two days following the molad. Tabular new moons (maladot) are reckoned from the Tishri molad of the year A.M. 1, which occurred on day 2 at 5 hours, 204 halakim (i.e., 11:11:20 P.M. on Sunday, -3760 October 6, Julian proleptic calendar). The adopted value of the mean lunation is 29 days, 12 hours, 793 halakim (29.530594 days). To avoid rounding and truncation errors, calculation should be done in halakim rather than decimals of a day, since the adopted lunation constant is expressed exactly in halakim.
| Lunations | Weeks-Days-Hours-Halakim | |
| 1 | = | 4-1-12-0793 |
| 12 | = | 50-4-08-0876 |
| 13 | = | 54-5-21-0589 |
| 235 | = | 991-2-16-0595 |
Lunation constants required in calculations are shown in Table 3.1.1.1. By subtracting off the weeks, these constants give the shift in weekdays that occurs after each cycle.
The dehiyyot are as follows:
(a) If the Tishri molad falls on day 1, 4, or 6, then Tishri 1 is postponed one day.
(b) If the Tishri molad occurs at or after 18 hours (i.e., noon), then Tishri 1 is postponed one day. If this causes Tishri 1 to fall on day 1, 4, or 6, then Tishri 1 is postponed an additional day to satisfy dehiyyah (a).
(c) If the Tishri molad of an ordinary year (i.e., of twelve months) falls on day 3 at or after 9 hours, 204 halakim, then Tishri 1 is postponed two days to day 5, thereby satisfying dehiyyah (a).
(d) If the first molad following a leap year falls on day 2 at or after 15 hours, 589 halakim, then Tishri 1 is postponed one day to day 3.
3.1.2 Reasons for the Dehiyyot
Dehiyyah (a) prevents Hoshana Rabba (Tishri 21) from occurring on the Sabbath and prevents Yom Kippur (Tishri 10) from occurring on the day before or after the Sabbath.
Dehiyyah (b) is an artifact of the ancient practice of beginning each month with the sighting of the lunar crescent. It is assumed that if the molad (i.e., the mean conjunction) occurs after noon, the lunar crescent cannot be sighted until after 6 P.M., which will then be on the following day.
Dehiyyah (c) prevents an ordinary year from exceeding 355 days. If the Tishri molad of an ordinary year occurs on Tuesday at or after 3:11:20 A.M., the next Tishri molad will occur at or after noon on Saturday. According to dehiyyah (b), Tishri 1 of the next year must be postponed to Sunday, which by dehiyyah (a) occasions a further postponement to Monday. This results in an ordinary year of 356 days. Postponing Tishri 1 from Tuesday to Thursday produces a year of 354 days.
Dehiyyah (d) prevents a leap year from falling short of 383 days. If the Tishri molad following a leap year is on Monday, at or after 9:32:43 1/3 A.M., the previous Tishri molad (thirteen months earlier) occurred on Tuesday at or after noon. Therefore, by dehiyyot (b) and (a), Tishri 1 beginning the leap year was postponed to Thursday. To prevent a leap year of 382 days, dehiyyah (d) postpones by one day the beginning of the ordinary year.
A thorough discussion of both the functional and religious aspects of the dehiyyot is provided by Cohen (1981).
3.1.3 Determining the Length of the Year
An ordinary year consists of 50 weeks plus 3, 4, or 5 days. The number of excess days identifies the year as being deficient, regular, or complete, respectively. A leap year consists of 54 weeks plus 5, 6, or 7 days, which again are designated deficient, regular, or complete, respectively. The length of a year can therefore be determined by comparing the weekday of Tishri 1 with that of the next Tishri 1.
First consider an ordinary year. The weekday shift after twelve lunations is 04-08-876. For example if a Tishri molad of an ordinary year occurs on day 2 at 0 hours 0 halakim (6 P.M. on Monday), the next Tishri molad will occur on day 6 at 8 hours 876 halakim. The first Tishri molad does not require application of the dehiyyot, so Tishri 1 occurs on day 2. Because of dehiyyah (a), the following Tishri 1 is delayed by one day to day 7, five weekdays after the previous Tishri 1. Since this characterizes a complete year, the months of Heshvan and Kislev both contain 30 days.
The weekday shift after thirteen lunations is 05-21-589. If the Tishri molad of a leap year occurred on day 4 at 20 hours 500 halakim, the next Tishri molad will occur on day 3 at 18 hours 9 halakim. Becuase of dehiyyot (b), Tishri 1 of the leap year is postponed two days to day 6. Because of dehiyyot (c), Tishri 1 of the following year is postponed two days to day 5. This six-day difference characterizes a regular year, so that Heshvan has 29 days and Kislev has 30 days.
3.2 History of the Hebrew Calendar
The codified Hebrew calendar as we know it today is generally considered to date from A.M. 4119 (+359), though the exact date is uncertain. At that time the patriarch Hillel II, breaking with tradition, disseminated rules for calculating the calendar. Prior to that time the calendar was regarded as a secret science of the religious authorities. The exact details of Hillel's calendar have not come down to us, but it is generally considered to include rules for intercalation over nineteen-year cycles. Up to the tenth century A.D., however, there was disagreement about the proper years for intercalation and the initial epoch for reckoning years.
Information on calendrical practices prior to Hillel is fragmentary and often contradictory. The earliest evidence indicates a calendar based on observations of Moon phases. Since the Bible mentions seasonal festivals, there must have been intercalation. There was likely an evolution of conflicting calendrical practices.
The Babylonian exile, in the first half of the sixth century B.C., greatly influenced the Hebrew calendar. This is visible today in the names of the months. The Babylonian influence may also have led to the practice of intercalating leap months.
During the period of the Sanhedrin, a committee of the Sanhedrin met to evaluate reports of sightings of the lunar crescent. If sightings were not possible, the new month was begun 30 days after the beginning of the previous month. Decisions on intercalation were influenced, if not determined entirely, by the state of vegetation and animal life. Although eight-year, nineteen-year, and longer- period intercalation cycles may have been instituted at various times prior to Hillel II, there is little evidence that they were employed consistently over long time spans.