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Technically, the Harvest Moon is the Full Moon closest to the September equinox around September 22. Most years it is in September, but around every three years, it is in October. The Harvest Moon is the only Full Moon name which is determined by the equinox rather than a month.
September: Corn Moon
The Full Moon in the month of September in the Old Farmer's Almanac is called Harvest Moon, which corresponds with the Old English/Anglo-Saxon name, while other names are Corn Moon or Full Corn Moon or Barley Moon.