How to Keep Track of Heliocal Risings in Ancient Mesopotamia

The need for precision arises from the fact that a city state started to grow to encompass tens of thousands of subjects. It's simply not feasible to plan in advance with so many people without some means of synchronization. Most of the month names in ancient Mesopotamia that don't refer to agriculture refer to a festival. If you want everyone to stop working to enjoy a festival on the same day, you need some way to keep track of the days.