During the Babylonian exile, which started in 586 BCE, Babylonian month names were adopted, which are still in use. The Syrian calendar used in the Levant (Mediterranean) region shares many of the names for months as the Hebrew calendar, such as Nisan, Iyyar, Tammuz, Ab, Elul, Tishri, and Adar, indicating a common Babylonian origin.
The Hebrew months do not correspond directly to the western calendar months. The Jewish calendar is a "lunisolar", based on twelve lunar months of 29 or 30 days, with an intercalary lunar month (a leap month) added 7 times every 19 years, so that the Biblical holidays remain in the proper season (for example, Passover must be in the Spring). This is why some years we have Adar I and Adar II (Adar II is this leap month that is added).
Each month begins on a new moon and is called "Rosh Chodesh" (the head of the month). The term new moon doesn't refer to the scientific definition (which refers to the moon's invisible phase), but to the first siting of the crescent moon as it emerges from its invisible phase. In Biblical and Talmudic times, the day of Rosh Chodsh was determined by the Sages of the Sanhedrin on a month-to-month basis, based on the testimony of witnesses who had seen the new moon. Several holidays such as Pesach and the first day of Shavuot always fall on the full moon (which would be the 15th of the month). In approximately the eight century, the beginning of the months were fixed by calculation rather than siting of the new moon.
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