Halakhah su Salmi 95:71

And, in truth, not against Him (G–d forbid) is our plaint, but against ourselves; for it [the redemption] is not beyond His powers, viz. (Isaiah 59:1-2): "Behold, the L–rd's hand is not too short to save, and His ear is not too heavy to hear. But your sins have made a separation between you and your G–d, and your transgressions have hid [His] face from you, from hearing." And we find (viz. Sanhedrin 98a) that in the days of R. Yehoshua ben Levi he was told (viz. Tehillim 95:6) that the Messiah would come "today, if you hearkened to His voice" — even though the time of the exile decreed for Israel (one thousand years, corresponding to the one thousand year "day" of the Holy One Blessed be He, had not yet passed [viz. Chida, Petach Eiynayim, Sanhedrin, Ibid.]) In spite of this, the power of repentance would have annulled the decree. How much more so, more than eight hundred years after the end of that [one-thousand-year] "day", [should the Messiah come if we repented]! The fault is ours alone — that with our many sins we do not allow Him to repose His Shechinah in our midst.

(starting at ועכשיו) We gather in shul and we say five chapters of psalms from Lechu Neranena (Ch. 95) until Hashem malach yirgezu amim (Ch. 99) because they are about the future redemptive days. Therefore (like Radak says), a person says to his friend “Let us sing to Hashem,” and then [the next chapter we sing] “Hashem reigns,” that is to say that we’ll fulfill that which is written (Zecharia 14:9) “On that day Hashem will be One and His Name will be One.” For Shabbat is a hint to this time, to “The Day that is Entirely Shabbat,” and then we’ll sing a new song (Ch. 96) to Hashem. For all songs are in the feminine, and the song of the future [redemption] will be in the masculine, like the Midrash explain. Therefore we say these chapters.