A Commentary, Critical, Practical, and Explanatory on the Old and New Testaments, by Robert Jamieson, A.R. Fausset and David Brown  at sacred-texts.com
psa 81:1Gittith--(See on Psa 8:1, title). A festal Psalm, probably for the passover (compare Mat 26:30), in which, after an exhortation to praise God, He is introduced, reminding Israel of their obligations, chiding their neglect, and depicting the happy results of obedience. (Psa. 81:1-16)
our strength-- (Psa 38:7).
psa 81:2unites the most joyful kinds of music, vocal and instrumental.
psa 81:3the new moon--or the month.
the time appointed--(Compare Pro 7:20).
psa 81:5a testimony--The feasts, especially the passover, attested God's relation to His people.
Joseph--for Israel (Psa 80:1).
went out through--or, "over," that is, Israel in the exodus.
I heard--change of person. The writer speaks for the nation.
language--literally, "lip" (Psa 14:1). An aggravation or element of their distress that their oppressors were foreigners (Deu 28:49).
psa 81:6God's language alludes to the burdensome slavery of the Israelites.
psa 81:7secret place--the cloud from which He troubled the Egyptians (Exo 14:24).
proved thee-- (Psa 7:10; Psa 17:3) --tested their faith by the miracle.
psa 81:8(Compare Psa 50:7). The reproof follows to Psa 81:12.
if thou wilt hearken--He then propounds the terms of His covenant: they should worship Him alone, who (Psa 81:10) had delivered them, and would still confer all needed blessings.
psa 81:11They failed, and He gave them up to their own desires and hardness of heart (Deu 29:18; Pro 1:30; Rom 11:25).
psa 81:13Obedience would have secured all promised blessings and the subjection of foes. In this passage, "should have," "would have," &c., are better, "should" and "would" expressing God's intention at the time, that is, when they left Egypt.