The interpretation timeline
Psalm 136:2 Douay-Rheims

On the willows in the midst thereof we hung up our instruments.

A.D. 311 – 1849 5 witnesses

Patristic

before A.D. 750
311
A.D.
c. A.D. 260–311
“To continue with our subject, let us take in our hands and examine this psalm, which the pure and stainless souls sing to God, saying, "By the rivers of Babylon, there we sat down; we wept, when we remembered Zion. We hanged our harps on the willows in the midst thereof," clearly giving the name of harps to their bodies, which they hung on the branches of chastity, fastening them to the wood that they might not be snatched away and dragged along again by the stream of incontinence. For Babylon, which is interpreted "disturbance" or "confusion," signifies this life around which the water flows, while we sit in the midst of the water that flows round us, as long as we are in the world, the rivers of evil always beating on us. Wherefore, also, we are always fearful, and we groan and cry with weeping to God, that our harps may not be snatched off by the waves of pleasure and slip down from the tree of chastity.”
Source
430
A.D.
Augustine of Hippo Patristic
A.D. 354–430
“"On the willows in the midst thereof we hung up our instruments of music" [Psalm 137:2]. The citizens of Jerusalem have their "instruments of music," God's Scriptures, God's commands, God's promises, meditation on the life to come; but while they are dwelling "in Babylon," they "hang up their instruments." Willows are unfruitful trees, and here so placed, that no good whatever can be understood of them: elsewhere perhaps there may. Here understand barren trees, growing by the waters of Babylon. These trees are watered by the waters of Babylon, and bring forth no fruit; just as there are men greedy, covetous, barren in good works, citizens of Babylon in such wise, that they are even trees of that region; they are fed there by these pleasures of transitory things, as though watered by "the waters of Babylon." Thou seekest fruit of them, and nowhere findest it....Therefore by deferring to apply the Scriptures to them, "we hang up our instruments of music upon the willows." For we hold them not worthy to carry our instruments. We do not therefore insert our instruments into them and bind them to them, but defer to use them, and so hang them up. For the willows are the unfruitful trees of Babylon, fed by temporal pleasures, as by the "waters of Babylon."”
Source

Scholastic

c. 1100 – 1500
1105
A.D.
Rashi Jewish
1040–1105
“On willows Willows of the brook.”

Post-Reformation

c. 1650 – 1900
1771
A.D.
John Gill Reformed
1697–1771
“We hanged our harps upon the willows in the midst thereof. These were musical instruments, used in the temple service by the Levites, who seem to be the persons here speaking; who took care of them, and preserved them from the plunder of the enemy; and carried them with them to Babylon, in hope of returning with them to use them as before, or to solace themselves and others in captivity; though now they had no heart to make use of them, their sorrow was so great, and therefore hung them upon the willows as useless things: these willows grew upon the banks of the rivers where they were, as such trees usually do; hence called willows of the brook (x), and willows by water courses, Lev 23:40; and particularly upon the banks of the river Euphrates, which ran through the midst of Babylon, with which the phrase here agrees; and therefore Babylon itself is thought to be called "the brook", or "valley, of the willows", Isa 15:7. And, according to Ovid (y), not only reeds and poplars, but willows, grew on the banks of the Euphrates. Now the state of these people was an emblem of the case of the backsliding children of God; who, through the prevalence of corruption, the force of temptation, and the snares of the world, are brought into a kind of captivity to the law of sin and death, though not willingly; nor is it pleasing to them when sensible of it, Rom 7:23; who, though they are called out of the world, and are not of it; yet sometimes are so overcome with it, and immersed in the things of it, that they are as it were in Babylon. An emblem of this world, of the confusion in it, as its name signifies; of the fading glories of it, and the wickedness and idolatry it abounds with: and here they sit by the rivers of carnal pleasures in it for a while, till brought to themselves; and then they weep over their sins, and lament them; especially when they remember what opportunities they have formerly had in Zion, and what a low condition she is now in through the conduct of themselves and others: these make use of their harps when Zion is in good and prosperous circumstances, Rev 14:1; but when there are corruptions in doctrine, neglect or abuse of ordinances, animosities and divisions prevail, declensions in the life and power of religion, and the lives of professors disagreeable; then they hang their harps on willows, and drop their notes. (x) "Amnicolae salices", Ovid. Metamorph. l. 10. Fab. 2. v. 96. "Fluminibus salices", Virgil. Georgic. l. 2. v. 110. (y) "Venit ad Euphratem----Populus et cannae riparum summa tegebant, spemque dabant salices----". Ovid. Fasti, l. 2.”
Source
1849
A.D.
1774–1849
“Willow. With which the Euphrates was lined, Isaias xv. 7. It passed through the city, which was adored with trees, and contained extensive tracts of land for cultivation. (Calmet) — Babylon may also include all the territory. (Berthier) — Instruments. Hebrew Kinnor means properly the ancient lyre, but here it is put for all instruments of music. (Calmet) — In grief, music was laid aside, Isaias xxiv. 8., Ezechiel xxvi. 13., and Apocalypse xviii. 21. Yet Christians may unite spiritual canticles with holy compunction, Colossians iii. 16. We must not expose the word of truth to the ridicule of infidels. (Berthier)”
Source
Modern · 1953 →

The in-app commentary runs from the Fathers to the early-modern record, then stops — that's where the public-domain sources end, not where the reading does. For the modern reading, follow the sources directly.