A servant girl sent for barm comes back with a tale of mishaps from the young men in the street. The tumbling rhythm and comic misfortune are the whole point, and the title's irony is part of the joke.
From Mother Goose / Nursery Rhymes (traditional). See the whole collection.
Trip upon trenchers,
And dance upon dishes,
My mother sent me for some barm, some barm;
She bid me go lightly,
And come again quickly,
For fear the young men should do me some harm.
Yet didn't you see, yet didn't you see,
What naughty tricks they put upon me?
They broke my pitcher
And spilt the water,
And huffed my mother,
And chid her daughter,
And kissed my sister instead of me.
About this rhyme
Barm is a genuine old word for brewer's yeast, used in baking. · Rhyme's content (broken pitcher, teasing, a stolen kiss) is comic rather than sad, matching the 'title is the joke' reading.
"A Melancholy Song" is not really melancholy at all, the title is the joke. A girl is sent out for barm (brewer's yeast, for baking) and comes back to report, half put out, that the young men teased her, broke her pitcher and kissed her sister instead of her.
An old comic rhyme dressed up in a serious-sounding name.
Public domain. Text from The Real Mother Goose (Blanche Fisher Wright, 1916), via Project Gutenberg. View the source edition