This paper examines the two intercalation schemes found in the Mesopotamian astronomical compendium MUL.APIN from the beginning of the first millenium BCE, and the lunar theory that they imply. It demonstrates that the two schemes do not agree with each other. Two intercalation rules in the second scheme use the conjunction of the moon and the Pleiades. This paper concludes that the intercalation rules are based on the assumption of a 28-day ideal sidereal month. These rules work with a triennial cycle of intercalating one additional month every third lunar year. Two similar intercalation schemes from other compositions, likewise dating from the beginning of the first millenium BCE, are known: a seventh- century intercalation scheme from Babylonia that also assumes a 28-day ideal sidereal month and an intercalation scheme from an unpublished astronomical commentary that, like the scheme in MUL.APIN, uses a triennial cycle. Previous scholars believed that discrepancies exist between the dates of the conjunctions of the moon and the Pleiades across all three schemes. However, this paper proposes that the astronomical assumptions of the three schemes are identical.