Greek calendars are pretty complicated. Each city had its own calendar, and while many cities had similar calendars it was not even certain that an individual city might have only a single calendar. At Athens there were two state calendars and then a sort of agricultural calendar based on the seasons. The festival calendar was pretty bonkers as far as we can tell. It began with the first new moon after the summer solstice. In order to connect the lunar and solar calendar for the festival calendar several complex systems of intercalary days and months probably existed but it's poorly understood. The state (or democratic, due to its foundation after the democracy) calendar was a complex system which divided the solar year into ten months, with no regard for the lunar months. Likewise, the modern notion of years as a fixed chronological sequence did not exist in the Athenian calendar, or any other Greek calendar. The Athenians generally designated their years according to the names of the eponymous archon of that year. 356, the year of Alexander's birth, was therefore the year when Elpines was archon. Fixing dates was therefore rather difficult for much of the Classical Period. Thucydides, for example, dated events by listing a string of local years, including the ephors at Sparta, the archons at Athens, and for heat ever reason the tenure of Chrysis as priestess at Argos. By the Hellenistic Period, under the influence of Timaeus and Eratosthenes, scholars began using Olympiads to date historical events. But this was a scholarly convention, not one that most people recognized and used, in much the same way that AUC was used by scholars but pretty much never by ordinary people, who continued to date according to consular years into late antiquity