Extra bit:
The Atlantic crossing can be rough, especially in winter, and I've just read a suggestion that the last day of September is about the turn of the weather, so any ship's captain delaying sailing across the Atlantic Ocean until then would be at risk of bad weather, and thus his crew would be more at risk on the crossing.

Gut instinct says it was a sea shanty used to mark time by sailors on rigged trading ships crossing the Atlantic, and taught by old salts to their grandchildren and turned it into a game that may or may not have origins in the rope moving through ship fittings, as a line of children move under the arns of another, echoing the movement of the rope.

And generations later, a child in a dentist unwittingly recalls her genetic origins in song. Nice.