Radak: This refers to the beginning of the redemption. It is as if it says Cho Amar Hashem l'Yakov Ranu Simchah. I.e. sing b'Simchah. Also in other verses, the words are transposed, e.g. "Mekom Sham Kever" (Yechezkel 39:11) means Mekom Kever Sham. "Kol Rabim Amim" (Tehilim 89:51) means Kol Amim Rabim, and similar verses.

Malbim: Initially they will sing due to the Simchah that will be to Yakov (being on top of the Goyim); refer to 31:6:2:3. It is Simchah only for Yakov, the lower, commoners of our nation. Yisrael, the great people, will desire the Shechinah's return to Tziyon.

Rashi: Exult on high towers, so people will hear from afar.

Radak citing Targum Yonasan: Exult overtly in the eyes of all nations.

Malbim: The Simchah will be that servitude will be removed from them, and they will exult that they are on top of the Goyim. The Goyim will honor them, and Yisrael will be at their heads - whereas before, they were disgraced and lowly among the nations.

Radak: The tidings, and praise Hashem!

Radak: Just like You made us hear the tidings, so save Your nation! [Hashem says,] say so, and I will do like your words.

Malbim: Save ¹ the Tzadikim, who want to the true salvation - the gathering of the exiles and their return to Tziyon.