Of all tunes available in the world, why do we choose a "worker's" song? A song composed by slaves as they served their masters?
Why not something more uplifting?
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In addition to the answer al derech sod based on the teachings of the Hasidic masters, there is another answer al derech ha-peshat. The Gemara (Megillah 14a) teaches us that the sages of the time did not want to institute the reading of the Megillah, because no other prophets had ever added or subtracted from the Torah, and now, for the first time Mordechai and Esther wished to add something new.
It was only when they realized that there is a kal va-chomer argument in favor of instituting the Megillah reading that it was allowed. The kal va-chomer is: מה מעבדות לחירות אמרינן שירה ממות לחיים לא כל שכן--if going from slavery to freedom we sing a song, certainly when going from death to life we should sing. Thus on Purim we sing the song of the freed slaves--"Pick a bale of cotton"--to explain why it is that we can sing the song of the Megillah.
The answer lies in a teaching of the great hasidic teacher, R. Zusia of Anipoli, cited by R. Tzadok of Lublin (Pri Tzadik, Naso 13, translated here):
“As is well known in the name of Zusia of blessed memory that he was once walking along the road and it chanced upon him that a gentile wagon driver, who was transporting hay, fell over. The man requested help, to which he responded, ‘I cannot.’ The gentile said to him, “You can, but you just do not want to.” Regarding this Rebbe Zusia said to himself that it must be an allusion to the fallen hey (referring to the fifth letter in the Hebrew alphabet) represented in God’s name. It has fallen and it is in my power to pick it up, I just don’t want to. And so it is with all occurrences that chance upon a person, for all worldly events are really spiritual allusions.”
It seems that we sing the song to the tune of "Pick a Bale of Hay" to allude to our obligation to lift up the hey of God's name. This is particularly relevant on Purim, because as Chazal on the verse כי יד על כס י-ה teach us, אין השם שלם ואין הכסא שלם עד שימחה שמו של עמלק.
The tune is meant to address a question raised by the very song being sung. Here we are singing about how happy we are now that Adar is here. So why don't we say Hallel on Purim?!?!???11?!?!?!?1one?!?!
The tune reminds us of the Talmud's answer (Megillah 14a), that we are still the slaves of Achashverosh. To remind us of our continued slavery, we sing this phrase to a slave tune.