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I am currently fascinated by the Hebrew calendar, and how it relates to the historically fraught question of the date of Easter. Originally, of course, Christian Easter and the Jewish Passover were supposed to occur within the same week. Until recently, I had always believed that the technical reasons for the two festivals diverging were:

  • Christians insisted on celebrating Easter on a Sunday, whereas the Jewish Passover is not tied to any particular weekday.
  • Christians fixed the date of the vernal equinox on 21 March, at least for the purposes of the computus, and this caused further discombobulation with the introduction of the Gregorian calendar.

But are these the only reasons why the timing of the two festivals diverged? I had assumed so, until I read the other day that the Passover does not actually always fall on the first full moon after the vernal equinox; in fact, in around 20% of years, such as 2016, it falls on the second full moon after the vernal equinox instead.

After the destruction of the Second Temple in 70 AD, the vast majority of Jews adopted a reform in which a system of mathemical rules - essentially following the 19-year Metonic cycle - were used to fix the calendar. Before that time, the day of the month was determined by visually observing the phase of the moon, and the date of the new year - and thus the date of the Passover - was tied to physical confirmation of the ripeness of the barley crop. If the barley is not ripe enough yet, wait another month. Once the mathematical rules were adopted, intercalary or "leap" months were instead added according to a fixed cycle, and the addition of leap months in such a fashion is what sometimes causes the Passover to fall slightly out of sync with the vernal equinox in some years.

But are the mathematical rules adopted by the Jewish community around 70 AD simply the best anyone could do given the constraints? Could we do any better today?

Putting this all together, is it possible to construct a calendar with the following parameters?

  • The calendar is predetermined according to a set of mathematical rules.
  • The day of the month is tied to the phases of the moon.
  • Each year consists of an integer (but not necessarily constant) number of months.
  • The first full moon after the vernal equinox falls on the same date each year.
  • It is acceptable, where necessary, to make certain assumptions and approximations - e.g. regarding the exactness of the Metonic cycle, the constancy of the motions of heavenly bodies, etc - such that any discrepancies would take at least a few millennia to be felt.
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  • $\begingroup$ The Jewish calendar is designed around a lunar cycle, the Christian’s a solar cycle. The Jewish changes from a basic lunar calendar are because Passover is supposed to be in spring. If not for said changes, Passover would end up like Ramadan, falling 11 days back each time and being equally likely to be anywhere in the year. The Christian changes from a perfect solar calendar simply because haveing 23 hrs, 59 minutes, 58 or so second days would be an annoying, hence the leap years Considering they are different religions on different calendars why bother trying to make Easter+Passover overlap. $\endgroup$
    – Starship
    Commented Oct 22 at 0:18
  • $\begingroup$ Since we can predict the phase of the moon quite accurately well into the future, the answer is obviously "yes". The only question would be how accurate vs how computationally intensive you want it to be. $\endgroup$
    – Greg Miller
    Commented Oct 22 at 3:21
  • $\begingroup$ In your proposed lunisolar calendar, how much drift is acceptable in the date of the vernal equinox? BTW, the Callippic cycle is slightly more accurate than the Metonic. $\endgroup$
    – PM 2Ring
    Commented Oct 22 at 5:12
  • $\begingroup$ @Starship The date of Easter is conceptually the first Sunday after the first full Moon on or after the vernal equinox, making Easter a lunisolar concept. I wrote "conceptually" because of the computus. This became particularly problematic using the pagan Julian calendar, which doesn't quite have leap years correct. Getting the date of Easter correct was very important to the Church; Easter and Passover were intended to overlap. This eventually resulted in the change from the pagan Julian calendar to the Christian Gregorian calendar, which does a better job with leap years. $\endgroup$
    – David Hammen
    Commented Oct 22 at 11:18

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Sure, set the start of the year to "the day of first full moon observed in Jerusalem after the moment of vernal equinox as computed in the DE440 ephemeris published by the JPL (NASA)". Each month then counts days until the next full moon (as computed in DE440).

The ephemeris is correct to a few km for thousands of years, and good enough to be visibly correct for longer than that.

The DE440 ephemeris is good enough that this is essentially the same as just observing the full moon. It accounts for lots of perturbations from big ones (The sun's gravity and the fact that Earth isn't round) to little ones (General relativity, interactions between the lunar core and mantle) It is many thousands of times more accurate than metonic cycles and so on. Of course, a computer is needed to run the model, it is far to calculation intensive to do by hand. The mathematical rules are expressed in the form of a rather complex computer code.

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  • $\begingroup$ The math for DExxx actually could be done by hand, it's just a few dozen multiplications and additions. It's a large dataset because each block is only valid for 32 days, so you need a lot of them to cover long time scales. Not saying I'd actually want to do it, though. $\endgroup$
    – Greg Miller
    Commented Oct 22 at 18:16

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