How did the Babylonian 19-year lunisolar cycle work?

The Babylonian 19-year lunisolar cycle was a sophisticated chronological system developed to reconcile the lunar year—consisting of 12 synodic months and approximately 354 days—with the solar year of the seasons.12

Before this system was regularized around 380 BCE, intercalation (the addition of an extra month) was performed haphazardly by royal decree to keep the calendar aligned with agricultural cycles. Under the refined 19-year cycle, Babylonian astronomers established a fixed rule: 7 out of every 19 years included an intercalary month, resulting in a total of 235 months per cycle.231

These extra months were inserted in the 3rd, 6th, 8th, 11th, 14th, 17th, and 19th years of the cycle. Typically, the month Adaru II was added, except in the 17th year, when Ululu II was inserted instead. This mathematical precision ensured that the start of the year, Nisanu 1, remained consistently near the spring equinox, oscillating within a period of 27 days. This system is closely related to the Metonic cycle used in Greek astronomy.213