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The Amida prayer of musaf on Rosh HaShana is unique in that it had three blessings between the introductory and concluding berakhot. These three blessings - referred to by the Gemara as malkhuyot, zikhronot and shofarot (blessings over God's monarchy, His remembrances and the shofar), make up the longest Amida of the year.
Indeed we recite no fewer that ten verses for each of the three blessings,, and when the Gemara asks what each of these ten verses correspond to, Rabbi Levi answers that, ‘they correspond to the ten expressions of praise (הילולים – i.e. forms of Hallel) as expressed by David in Sefer Tehillim’. What this suggests is that the Malchiyot, Zichronot & Shofarot verses are a subtle form of Hallel which are framed around the themes of God as king, God as rememberer, and the symbolism of the shofar.
However, later on the Gemara notes that the collection of Tehillim known to us as Hallel is not, in fact, recited on Rosh Hashanah.
But why? According to Rabbi Abahu, this is because ‘the ministering angels said before the Holy One, Blessed is He: “Master of the Universe! Why do the Jews not recite song (שירה - which is understood to refer to Hallel) before You on Rosh Hashanah and Yom Kippur?” He replied, “Is it possible that the King is seated on the Throne of Judgement, and the Books of the Living and the Books of the Dead are open before Him, and the Jewish people are going to say songs?”.
What this suggests is that due to the weightiness of the days of Rosh Hashanah and Yom Kippur, it would be improper to recite Hallel on these days.
According to the Yerushalmi, ten zikhronot are suggested by the ten expressions of repentance in the first chapter of Isa (1:16-18), and ten shofarot commemorate the sacrifices brought during musaf of Rosh HaShana in the Temple, each of which was accompanied by the sounding of the shofar.