The Pentateuch (the first five books of the Bible) says nothing about the shofar being blown on the Day of Trumpets. We get that from other parts of the Book. Intriguingly, the blowing of the shofar is mentioned in one place that is very interesting; and that is in Exodus 19 and 20. This is just after the people got to Sinai, and God told Moses to get the people ready.
Now, notice: If you understand the drift, the Israelites did not blow the shofar here. It was evidently blown by angels. The people were trembling. They were not blowing shofars. They were trembling, because this loud shofar was blowing and making them very fearful.
This is something that a human set of lungs could not do. A continuous pealing of this shofar - increasing in strength as it kept on going, and going, and going.
I get a mental picture very similar to what happened at the World Trade Center, when that tower came crashing down; and you saw people rushing away from it - because they did not want to be caught in it!
He wanted to scare them witless - to let them know what He was like. Why? The very next phrase:
The man of God drew near. People full of sin ran away. It is very interesting that here, when He gives the law as a test for Israel, they heard the shofar. What does that say about the Day of Trumpets? There are three things that are happening when the shofar sounds.
So the three things are (1) God is drawing near. (2) There is law that guides the affairs of men - between men and God, and man and man. (3) There is a test, a judgment, a crisis where things could go one way or the other. On which side are you going to fall? I just thought that was interesting that the only place in the whole book of the law (the five books of Moses) where shofar is mentioned - other than in Leviticus 25, about the Jubilee - it has to do with God coming near, giving the law, and passing judgments. Of course, we blow it on the Day of Trumpets; and this is the sort of thing that we are supposed to remember when we hear it - when we keep this day.
First we see the trumpet being used to call people to assembly, and then the trumpet was used to manifest just a tiny particle of the power of God, giving them unbearable evidence that they were in the presence of somebody of awesome magnitude. We can conclude from this, then, that trumpets are intended to be blown announcing the arrival of a king. In this case, it was the Great King.