Rosh Hashana is "the day of the sounding of the shofar [ram's horn]," according to its biblical definition (Numbers 29:1). However, Yom Kippur, the Day of Atonement and Forgiveness, likewise involves shofar blasts following the poignant ne'ila (closing) prayer and generally serving as the conclusion of the fast.
What is the biblical basis - if any - for the shofar sound on Yom Kippur, and how does it differ from the shofar on Rosh Hashana?
Let us first explore the significance of the Rosh Hashana shofar. Fascinatingly, the sages of the Talmud teach that the biblical "day of the sound of the shofar" refers to the straight (tekiya), broken (shevarim, terua) and straight ram's horn blasts linked to the musaf amida (additional standing prayer). Indeed, the initial custom was to sound the shofar even during the silent amida - something still done in most Sephardi and Hassidic synagogues, but considered too confusing for most Ashkenazim (B.T. Rosh Hashana 33).
Logic would dictate that if the shofar blasts are not considered an "interruption" (hafsaka) of the amida prayer, they must be seen as a part of the prayer. And so Rabbi Joseph B. Soloveitchik explains the true significance of the shofar sounds (based on the rulings of Maimonides and the explication of Rabbi Haim Brisker) as prayer by means of sounds. We pray with words - the verbal formulations of God's Kingship (malchuyot), God's Remembrances (zichronot) and God's rams's horn blasts (shofarot) - and we pray with sounds: the exultant, victorious tekiya shout, the sighing and sobbing shevarim-terua cries, and the concluding, victorious tekiya once again for final emphasis.
The crafted verbal formulation interconnected with the primal shofar sounds provide a powerful message: God is King not only of Israel but of the entire world; there is an architect to creation, and life is not "a tale told by an idiot, filled with sound and fury, signifying nothing." But if God is truly King, then He can rightfully hold us mortals accountable - especially for the fact that His ethical monotheism has not yet been accepted by the world, and not even by the majority of Israel. Hence we express sighs and sobs at our failings. We nevertheless conclude with an exultant shout, since repentance holds out the possibility of forgiveness, reconstruction and repair.
The same is true regarding remembrances. The axiom that there is also a divine plan for history, with a specific function set aside for nations as well as individuals, deserves an exultant shout, engendered by the knowledge that there is a more perfect society toward which we are heading; however, it also occasions sighs and sobs, lest we are not successfully fulfilling our mission, lest we are on the wrong track, in the wrong ball-park.
And finally shofarot, the ram's horn blast which emanated from atop Mount Sinai at the time of the Revelation. Here too, we express the consummate joy of the tekiya with the realization that God has given us His Torah, His formula for a proper and satisfying moral, ethical and spiritual life, which we must learn ourselves and then communicate to the world. Herein lies the means through which we can become a "holy nation and a kingdom of priest-teachers." However, the sighs and sobs emanate from the fact that we ourselves are often found wanting; how can we teach others what we ourselves have failed to learn?
In all these instances, the sound of the shofar is the sound of the Jew, a primal sound emanating from the most essential inner "divine portion," his exultant prayer of gratitude and his beseeching request for strength and discipline to fulfill his mission. Indeed, we pray with words and sounds.
However, there is one crucial difference between the first two instances of malchuyot and zichronot - wherein the sounds emanate from the individual at prayer - and shofarot, wherein the sound initially emanated from God: "God rose up through the sound of the terua, the Lord was in the sound of the shofar." Similarly, at the time of the redemption "all the inhabitants of the world and the dwellers on earth will see, when the banner on the mountains is held aloft; and they will all hear, when [God] will blast the shofar" (Isaiah 18:3), and then again, "And it will happen on that day that the great shofar shall be sounded [by God], and those who are lost in the land of Assyria and scattered in the land of Egypt shall come up and bow down before the Lord on the holy mountain in Jerusalem" (Isaiah 18:30).
Now we can begin to see the difference: the shofar blast on Yom Kippur is not derived from the biblical "day of the broken, staccato sound which is unto you," because the biblical text there relates to the people ("unto you") who are sounding the shofar at prayer, and since on Rosh Hashana the major emotion on this first of the Ten Days of Repentance is that of inadequacy, sighing and sobbing, terua, the very day is biblically defined as a "day of terua."
The shofar blast on Yom Kippur, on the other hand, is derived from the straight and exultant tekiya of Yom Kippur on the Jubilee year, the majestic declaration of "freedom throughout the land," the glorious announcement of redemption. (Leviticus 25:9-11, utilizing the Hebrew word shofar, signifying a beautiful, joyous straight sound). This is proper for Yom Kippur, the day when God promises - and guarantees - forgiveness and purification after five prayers in which we affirm (and request) that our Temple be a House of Prayer for all nations. And even though the Yom Kippur blast nowadays is only a rabbinical reminder of the Jubilee, every traditional Jew awaits the final blast by God, with its inherent vision of universal Revelation - when "He will enable us to hear again before the eyes of all living beings" - the redemptive shofar call to the entire world in the days of the Messiah.