"[Rosh Hashanah] shall be a day of shofar-sounding for you." (Num. 29:1)

The Baal Shem Tov gave the following analogy to illustrate the effect of blowing the shofar:

Once there was a king who had an only son.

Once there was a king who had an only son. The son was well learned and his father loved him very much. One day, the king and the prince decided that it would be educational for the prince to travel to faraway lands to learn the wisdom and ways of the people who lived there. The king gave the prince an entourage of ministers and servants as well as a large amount of money for this expedition, all so that he advance in his knowledge and wisdom beyond his ability to do so at home, in the king's court.

But as the journey wore on, the prince spent all the money on the luxuries he was accustomed to at home plus other excesses that he indulged in on the way. Eventually, he was left with nothing, and had arrived at a place so far away from home that no one there had ever heard of his father.

Distraught, the prince decided it was time to go home. But he had been away so long that he had forgotten his native tongue, so when he finally made his way back to the capital city of his kingdom, he could not explain to anyone who he was and where he needed to go. He tried to gesture to them that he was the prince, but of course no one paid any attention to him. Finally, when he was near enough to the palace so the king could hear him, he let loose a wordless scream so his father would recognize his voice. The king indeed recognized his son's voice and sent for him, and so they were reunited.

...the Jewish soul is G‑d's child...

So, too, the Jewish soul is G‑d's child; this child was sent into the foreign environment of this material world for its own edification, accomplished by learning the Torah and fulfilling its commandments. But by indulging in the delights of this world, the soul becomes increasingly estranged from its native milieu; it is gradually drawn into an environment that does not recognize Divinity and is not concerned with it, and it eventually forgets the language of holiness and purity.

But at some point, it remembers who it is and cries out to G‑d. This is the wordless blast of the shofar, which utters the innermost voice of the soul in its regret for its past deeds, its longing for its Divine home, and its desire to rededicate itself to its Father. When G‑d hears this cry, it arouses His mercy, and He forgives the soul restoring it to its former intimacy with Him.

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Rabbi Levi Yitzchak of Berditchev also gave an analogy:

A king once set out on a journey that led him deep into a thick forest. At one point, he lost his way and could not determine how to get out. A group of villagers passed by, so he asked them for directions back to the palace. But they did not recognize him, so they did not know if they should help him or not, and moreover, they did not know the way to the palace. Eventually, someone passed by who did recognize that this was the king and who did know the way to the palace, so he escorted the king back home. The king was so impressed with the person's knowledge that he made him his personal advisor.

...the advisor wronged the king in some way...

A long time after this, the advisor wronged the king in some way, and in his anger, the king told his ministers to judge the advisor and declare him guilty of rebellion. The advisor was very upset because he knew what this meant, so he asked the king for one last request: that they both dress themselves in the clothes they wore when they had their first encounter in the forest. The king agreed, and when he put on the clothes he wore then, he remembered at once the tremendous favor the advisor had done him by leading him out of such a hopeless situation. In his gratitude, the king forgave the advisor of his misdeed and returned him to his post.

Similarly, when G‑d wished to give the Torah, he first inquired of all the other nations and none accepted it. It began to look as if no one was interested in fulfilling G‑d's purpose in creation and G‑d had created the world for naught. But then, the Jews accepted the Torah immediately and enthusiastically.

Eventually, our initial enthusiasm waned and we transgressed the Torah's instructions. We therefore sound the shofar to remind G‑d of the day when we first "met" at Mount Sinai and the shofar was sounding as we accepted His Torah. The shofar blast reminds G‑d of how we accepted His Torah unconditionally, and He forgives our misdeeds.

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Renewal

...Rosh Hashanah is a time of renewal, of returning to the origin...

BOTH OF THESE PARABLES revolve around the idea that Rosh Hashanah is a time of renewal, of returning to the origin and drawing new levels of connection from the inexhaustible wellsprings of our relationship with G‑d.

This annual renewal is necessary if life is to retain its freshness and novelty. Every level of Divine consciousness carries its inherent modes of thinking, expression, and action—its own language. If we merely continue developing the same level of Divine consciousness we have been nurturing the past year, we will remain locked in its intrinsic limitations and religious life will begin to seem repetitive and dull. On Rosh Hashanah, G‑d withdraws the Divine energy that sustained creation the previous year and replenishes with new and fresh vitality. It is therefore an opportunity for us to do the same: to make a quantum leap to a new plateau of Divine consciousness that will inspire our lives for the coming year.

Wordless Energy

To do this, however, we cannot rely on words, because words carry specific meanings for us that are limited by the knowledge and experiences we have accrued in our lives. In order to break out of the contextual meaning of our limited modes of expression, we use the blast and wails of the shofar which transcend the confines of language.

In this way, we recapture the innocence and inspiration of a soul newly born and of the Jewish people at Mount Sinai, and this renewed inspiration powers our relationship with G‑d for the coming year.

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[From Torat Shmuel 5637 (Vekachah) 80; Kedushat Levi 96a; Sefer HaMa'amarim Melukat, vol. 1, p. 426.]