Why don't we say Hallel on Rosh Hashanah?

Question

For most months, on the first day (Rosh Hodesh) we say the Hallel service between Shacharit and Mussaf. For the first day of Tishrei (Rosh Hashanah), however, we do not say Hallel. Why is it not done? Are there some Jews who do say Hallel on Rosh Hashanah?

Answer 1

I once wondered about this and found an answer at Ohr Somayach. They say that Hallel is said with joy and that our focus on judgement precludes this. They cite Rosh Hashanah 32b:

אמרו מלאכי השרת לפני הקב"ה רבש"ע מפני מה אין ישראל אומרים שירה לפניך בר"ה וביום הכפורים אמר להם אפשר מלך יושב על כסא דין וספרי חיים וספרי מתים פתוחין לפניו וישראל אומרים שירה:

Said the angels before G-d, Lord of the Universe, why does Israel fail to utter song before you during Rosh Hashana and Yom Kippur? He replied, When the King sits on the throne of judgment and the books of life and death are before him, can Israel utter song?"

These are the words of R. Abahu. I checked there for more on this but that's all the g'mara says (there) about Hallel on Rosh Hashana. This g'mara (nearly word for word) also appears on Erchin 10b (h/t DonielF), again without further elaboration.

(This explanation does not mean we should only be focused on judgement; it's still a festive day. See the Ohr Somayach article for more on that.)

Answer 2

In the words of Rambam (Hilkhot Megillah 3:6), we do not recite Hallel on Rosh HaShana (or on Yom Kippur) since they are times of repentance, fear, and terror; not times for excessive joy.

אבל ראש השנה ויום הכיפורים, אין בהן הלל, לפי שהן ימי תשובה ויראה ופחד, לא ימי שמחה יתרה

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It should be noted that Rambam is dealing with Hallel on Rosh HaShana in terms of Hallel for holidays, rather than Hallel for Rosh Hodesh, but the reasoning is the same.