It was developed, validated, and placed in the public domain in Hebrew year 5766 (Gregorian year 2006) by Dr. Irv Bromberg of the University of Toronto, Canada.
Hebrew calendar solar drift
The Traditional Hebrew calendar is a lunisolar calendar that employs a fixed arithmetic leap cycle for its solar component, and a fixed arithmetic constant-interval cycle for its lunar component (molad). Its calendar mean year is exactly 365 days 5 hours 55 minutes and 25 + 25/57 seconds, but that is 6 minutes and 25 + 25/57 seconds too long compared to the present era mean northward equinoctial year of 365 days 5 hours 49 minutes 0 seconds (mean solar time). Thus the Traditional Hebrew calendar drifts with respect to the northward equinoctial year, presently at the relatively rapid rate of 1 / (6/1440 + (25+25/57) / 86400) = about 224 years per day of drift. For comparison, the Julian calendar presently drifts at the rate of 1 / (11/1440) = about 130.9 years per day of drift. The fixed arithmetic Hebrew calendar started in Hebrew year 4119, so since then it has accumulated a drift of about (5767-4119)/224 = slightly more than 7 days late (on average).
Note, however, that it is not possible for any individual Hebrew date to be "7 days late", because each month starts within a day or two of its molad moment. Instead, presently about 80% of Traditional Hebrew calendar months are "on time" and about 20% of months are "one month late". Typically, the Traditional Hebrew calendar enters the "one month late" state after it "prematurely" inserts a leap month, then it remains "one month late" until a year later when the leap month "should have been" inserted. The current pattern of this alternating state is such that it amounts to the Traditional Hebrew calendar having drifted 7 days later than it was in the era of Hillel II (Hebrew year 4119, Gregorian year 359 AD), as judged by the average timing of either the start of the month of Nisan or the end of the first day of Passover relative to the timing of the northward equinox on a Jerusalem mean solar time clock.
Hebrew calendar lunar drift
The lunar component of the Traditional Hebrew calendar employs a constant interval (molad interval) of 29 days 12 hours and 44 + 1/18 minutes to account for each mean lunar month. Due to tidal forces slowing the Earth rotation rate, however, the length of the mean lunar cycle gets progressively shorter. Consequently, in the present era the traditional molad interval is about 3/5 second too long, but this discrepancy is growing at a progressively faster rate (quadratically). On average the Hebrew calendar's estimate of the mean lunar conjunction is currently about 2 hours late, as judged by the reading on a Jerusalem mean solar time clock.
Rectified leap rule
The cycle of leap years is the primary difference between the Traditional and Rectified Hebrew calendars.
The Traditional Hebrew calendar has 7 leap years per 19-year cycle, and its leap rule is:
This expression inherently causes leap year intervals to fall into uniformly spread sub-cycle patterns of (3+3+2) = 8 years or (3+3+3+2) = 11 years, which alternate 8+11=19 years per cycle.
In 1931 Dr. William Moses Feldman (1880-1939, originator of the term "biomathematics") briefly proposed an improved Hebrew calendar leap cycle of 334 years having 123 leap years and a total of 4131 months per cycle. He derived that improved leap cycle by using a continued fraction approximation of the ratio of the "tropical year" (365 days 5 hours 48 minutes 46 seconds, which is about 14 seconds too short relative to the present-era mean northward equinoctial year) to the "lunar year" (354 days 8 hours 48 minutes 36 seconds, which corresponds to 29 days 12 hours 44 minutes 3 seconds per lunar cycle = about 1/3 second shorter than the traditional molad interval, yet almost 1/3 second too long relative to the present era mean synodic month) . Although he only described his leap rule qualitatively, the arithmetic that exactly reproduces Feldman's leap cycle is:
This expression inherently causes leap year intervals to fall into uniformly spread sub-cycle patterns of (3+3+2) = 8 years or (3+3+3+2) = 11 years, which further group to: 17×(8+11)+11 = 17×19+11 = 334 years. In other words, each Feldman cycle has 17 repeats of the traditional 19-year cycle and one truncated 11-year subcycle. The much shorter mean year of the 11-year subcycle offsets the excessively long mean year of the 19-year subcycles, yielding a net calendar mean year of 365 days 5 hours 48 minutes and 39 + 1/3 seconds (calculated using the mean synodic month that Feldman used). Although the Feldman leap cycle is a substantial improvement over the excessively long Metonic 19-year cycle, it "over-corrects" the drift, being about 20 + 2/3 seconds per year too short for the present era.
Rather than the mean tropical year (which applies only to atomic time), it is the mean northward equinoctial year (measured in terms of mean solar time, as given above) that is the appropriate year length to keep the Hebrew month of Nisan aligned relative to the northward equinox. The use of the correct present-era ratio of the mean northward equinoctial year to the mean synodic month applied to the continued fraction method yields the even more accurate leap cycle of the Rectified Hebrew calendar, which has 130 leap years per 353-year cycle of 4366 lunar months:
This expression inherently causes leap year intervals to fall into uniformly spread sub-cycle patterns of (3+3+2) = 8 years or (3+3+3+2) = 11 years, which further group to: 18×(8+11)+11 = 18×19+11 = 353 years. In other words, the Rectified cycle is the same as the Feldman cycle except that it has one more 19-year subcycle per cycle, making the net calendar mean year 365 days 5 hours 48 minutes and almost 58 seconds, or only about 2 seconds too short per year (calculated using a mean synodic month that is about 3/5 second shorter than the traditional molad interval).
The Rectified Hebrew Calendar also has a progressively shorter molad interval, which closely matches the actual length of the mean lunation interval (mean synodic month), and which explicitly refers its progressive molad moments to Jerusalem's longitude (mean solar time).
On the first day of Tishrei 5766 (autumn 2005) the mean synodic month was about 3/5 second shorter than the traditional molad interval, and was getting progressively shorter by about 27 microseconds per lunar month.
Like the Traditional Hebrew calendar, the mean year of the Rectified Hebrew calendar depends on the sum of the molad intervals, and in the present era amounts to about 365 days 5 hours 48 minutes and 57.6 seconds. This is intentionally slightly shorter than the present era northward equinoctial mean year of 365 days 5 hours 49 minutes 0 seconds, to allow for future tidal slowing of the Earth rotation rate. The Rectified Hebrew calendar mean year will continue to progressively shorten by about 3/2 second per 353-year cycle.
Astronomical calculations suggest that this seemingly minor "tweak" of the molad interval will actually extend the future useful range of the Rectified Hebrew calendar by about 3 millennia!
Rosh HaShanah Postponements
The use of progressively shorter molad intervals necessitates a modification to the way that the Rosh HaShanah postponement rules are handled. The Rectified Hebrew calendar employs novel postponement rules that are logically equivalent to those of the Traditional Hebrew calendar. (When applied to the arithmetic of the Traditional Hebrew calendar with its fixed molad intervals, the modified postponement rules yield identical dates for the full 689472-year repeat cycle of the Traditional Hebrew calendar.) For further information please see the discussion of the Rosh HaShanah postponement rules on the Rectified Hebrew calendar web site.
Reference Meridian of Longitude
The moments of astronomical events such as equinoxes, solstices, and lunar conjunctions must be referred to a specified meridian of longitude. In other words, for the quoted moments to be meaningful and unambiguous, the time zone of the clock of the observer must be specified. Conventionally, the moments of such celestial events are calculated for Universal Time at the Prime Meridian, which is the meridian of longitude that passes through Greenwich, England. Obviously, the Prime Meridian was never the reference meridian for the Traditional Hebrew calendar, but the actual original reference meridian was never specified in the classical rabbinic sources.
It is quite widely assumed, or at least implicitly assumed, that the reference meridian for the Hebrew calendar is the longitude of Jerusalem, Israel, which is about 2 hours and 37 minutes ahead of Universal Time. Consequently, Jerusalem mean solar time was explicitly used for the development and evaluation of the Rectified Hebrew calendar.
More recent astronomical historical evaluations, however, have suggested that the original meridian, in the era of the Second Temple, was midway between the Nile River and the end of the Euphrates River, a longitude that is about 4° east of Jerusalem, or 16 minutes of time ahead of Jerusalem mean solar time. There could have been at least two reasons for choosing that meridian. In the era of the Second Temple that meridian was generally considered to be the center of the civilized world, and it served as the reference meridian for many astronomical calculations, even those that Ptolemy published several centuries later. More importantly for Jews, however, may have been the promise of HaShem to Abram in the Torah, Genesis chapter 15 verse 18: "On that day HaShem made a covenant with Abram, saying To your descendants have I given this land, from the river of Egypt to the great river, the Euphrates River." This territory also corresponds to the full range of the patriarch's travels during his lifetime, as described in the Torah, from Ur to Egypt.
- Feldman, WM. Rabbinical Mathematics and Astronomy, page 208. Hermon Press, New York, 1931.
- The Lengths of the Seasons
- The Seasonal Drift of the Traditional (Fixed Arithmetic) Hebrew Calendar
- The Length of the Lunar Cycle
- The Molad of the Hebrew Calendar (shows why original reference meridian was 4° east of Jerusalem)
- The Rectified Hebrew Calendar (home page)
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