Peter J. Wallace writes
“In speaking of a new covenant, he makes the first one obsolete. And what is becoming obsolete and growing old is ready to vanish away” (Heb. 8:13). If the Mosaic covenant is obsolete and vanishing, then why should we sing the Psalms? Aren’t the Psalms the songbook of an obsolete covenant?
It is true that the Psalms are the songbook of an obsolete covenant—in the same sense that the Ten Commandments are the law of an obsolete covenant—and the whole Old Testament itself is an obsolete covenant! And yet, Paul writes that “all Scripture [the whole obsolete testament] is breathed-out by God and profitable for teaching, for reproof, for correction, and for training in righteousness, that the man of God may be competent, equipped for every good work” (2 Tim. 3:16–17). Furthermore, there is not a single sentiment in the Psalms that is not echoed in the New Testament as well.