We are told many meanings of the shofar-blowing. In fact, the leading Jewish sage in the tenth century C.E., Saddia Gaon, listed ten major ones, each with a scriptural basis. Rabbi Sa'adia explained that the sound of the shofar should call to mind:
1) the creation of the world
2) the beginning of the new year
3) the Mt. Sinai experience
4) the inspiring words of the prophets
5) the destruction of the Holy Temple, and
6) the Binding of Isaac.
It should also arouse and increase in us:
7) fear and awe of G-d Al-mighty
8) fear and awe for the Day of Judgment
9) belief in the future ingathering of the exiles and ultimate redemption of Mashiach, and inspire our yearning for it, and
10) belief in the future Resurrection of the Dead.
(His list and attendant verses may be found in English in "Book of our Heritage," among other sources.)
Keep in mind that while all of these ten are true and excellent interpretations, and are good to have in mind before or during the actual moments of the shofar-blowing, we cannot single out one of them or even all of them collectively as the real reason why the shofar is blown on Rosh Hashanah h. The official reason is quite simple; G-d instructed in the Torah that the shofar should be blown "on the first day of the seventh month". But He did not confide in us what this commandment signifies to Him.
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