The shofar awakens the covenant of deveikus

עלה אלקים בתרועה ה' בקול שופר.

"Elokim has ascended with a teruah, Hashem with the sound of the shofar." (Tehillim 47:6)

The Sefas Emes reads the two Names in this passuk as two different aspects of the day: the teruah (the broken blasts) corresponds to the Name Elokim, the attribute of din, while the clear sound of the shofar corresponds to the Name Havayah, the attribute of rachamim.

דהתקיעות הם לעורר ברית אבות כמ"ש במד' בחודש השביעי ירחא דשבועיא שנשבע הקב"ה לאברהם אע"ה לזכור העקידה לכפר על בנ"י.

For the shofar blasts come to awaken the bris of the Avos, as the Midrash says: "In the seventh month" — the month of the oath — that Hashem swore to Avraham Avinu to remember the Akeidah and atone for Bnei Yisrael.

The blowing of the shofar reawakens the covenant Hashem made with our forefathers, recalling the Akeidah so that its merit secures kapparah (atonement) for Klal Yisrael.

ובוודאי אין עיקר השבועה בעבור החוטאים.

And certainly the essence of the oath is not for the sake of the sinners.

The covenant was not given primarily as a mechanism to rescue those who fail.

רק השבועה וכריתת ברית לזרעם להיות להם דביקות בהקב"ה כמ"ש ואתם הדבקים בה' אלקיכם חיים כולכם.

Rather, the oath and the forging of the bris was for their offspring, that they should have deveikus (cleaving) to Hashem, as it is written: "And you who cleave to Hashem your Elokim are all alive today." (Devarim 4:4)

The true purpose of the covenant is positive — to grant Bnei Yisrael an enduring bond of attachment to Hashem, which is itself the source of their life.

ולכן מצד הברית מוחל עונותיהם ומסייע להם לחזור ולהתדבק בו.

Therefore, on account of the bris, He forgives their sins and assists them to return and cleave to Him.

Because the goal is deveikus, when a person sins Hashem clears away the aveiros so that the original bond can be restored — the forgiveness serves the reconnection.

אבל כשהדור צדיקים אז עיקר הברית שיתעלו ויתדבקו בו ית' למעלה למעלה.

But when the generation is righteous, then the essence of the bris is that they should rise higher and higher and cleave to Him, may He be blessed.

For a generation of tzaddikim, who have no aveiros needing kapparah, the covenant operates on its higher level — lifting them to ever-greater levels of deveikus.

והיא עיקר השופר.

And this is the essence of the shofar.

This ascent into deveikus, rather than mere atonement, is the deepest meaning of the mitzvah of shofar.

עלה אלקים בתרועה תיקון העונות.

"Elokim has ascended with a teruah" — this is the rectification of sins.

The teruah, tied to the Name Elokim (din), accomplishes the tikkun (repair) of aveiros — the lower, corrective function of the day.

אבל ה' בקול שופר הוא עיקר הברית כנ"ל.

But "Hashem with the sound of the shofar" is the essence of the bris, as explained above.

The clear tekiah, tied to the Name Havayah, expresses the covenant's true purpose — the ascent into deveikus.

ולכן יש תר"ת.

This is why the order of blasts is tekiah-teruah-tekiah (a clear blast, a broken blast, and a clear blast).

The broken teruah (atonement) is bracketed on both sides by the unbroken tekiah (the covenant of deveikus) — the bond surrounds and contains the repair.

בחי' אברהם אהבה זה כח המצות עשה.

This is the aspect of Avraham, ahavah (love) — this is the power of the mitzvos asei (the positive commandments).

The first tekiah corresponds to Avraham, whose middah was love; love expresses itself as the positive mitzvos, the active drawing-near to Hashem.

ובחי' יצחק המל"ת ולכן היא הכפרה ומחילת עונות.

And the aspect of Yitzchak is the mitzvos lo saaseh (the negative commandments), and therefore this is the kapparah and the forgiveness of sins.

The broken teruah in the middle corresponds to Yitzchak, whose middah was yirah and gevurah; this is the dimension of restraint, of holding back from aveirah, which is where atonement takes place.

והשופר דיעקב הוא התורה שכולל מ"ע ומל"ת:

And the shofar of Yaakov is the Torah, which includes both the positive and the negative commandments.

The final tekiah corresponds to Yaakov, whose middah was emes and Torah; Torah unites both the positive drawing-near of Avraham and the protective restraint of Yitzchak into one complete whole.

Summary: The verse "Elokim has ascended with a teruah, Hashem with the sound of the shofar" reveals the two functions of the shofar on Rosh Hashanah — the broken teruah effects the tikkun and atonement of aveiros, while the clear tekiah expresses the true essence of the bris of the Avos: deveikus to Hashem. The tekiah-teruah-tekiah pattern mirrors Avraham (love / positive mitzvos), Yitzchak (restraint / atonement), and Yaakov (Torah, which unites them all).