The Tur Orach Chayim (Siman 581/ סימן תקפ״א) writes in the name of Pirkei D’Rebbe Eliezer: Our minhag to blow shofar during Elul flows from a tradition that we have. When Moshe Rabbeinu was commanded to ascend Har Sinai to receive the second set of Luchot, Hashem requested that a shofar be blown in the camp to mark the moment, in order to avoid the earlier mistake of miscalculating Moshe’s expected descent that led to the Chet Ha’Egel. The same shofar that was used to announce Moshe’s going up, the Tur says, was used to elevate Hashem, as it is written in Psalm 47:6: God ascended with a blast, Hashem with the sound of a shofar.
The Tur adds: Therefore, the Sages established blowing the shofar throughout all of Elul in order to motivate the people to do teshuva, as it says in Amos (3:6) Will the shofar be blown in the city and the nation not tremble? …and also to confuse the Satan.
In our source verse, it says they blew shofar only on the day that Moshe Rabbeinu re-ascended Har Sinai. Why then do we blow shofar on every day of Elul?
What is the connection between the tekia (and the announcement that Moses went up to the heavens), and the ascent – so to speak – of the Master of the Universe, also with a shofar?
If we say that blowing shofar serves to awaken the people to do teshuva, as the Tur wrote: Is it possible that a shofar sounds in the city and the nation not tremble? – then why do we not blow during the 10 days of Penitence? The Aseret Y’mei Teshuva are the essence of repentance, days reserved for doing teshuva. How can we not blow during those 10 days?