I learned the powers of 2 from "Inchworm", and one of xkcd's What If? articles suggested that "Seasons of Love" is a good way to remember that a year has five hundred twenty-five thousand six hundred minutes. Perhaps, as a public service, someone should popularize a song enumerating the days in each month or describing minor scales.
--- One man's converging evidence is another man's pilpul.
Not a song (at least that I know of), but could be set to music:
Thirty days has September, April, June, and November. All the rest have thirty-one, Excepting February alone, And that has twenty-eight days clear, And twenty-nine on each leap year.
Also, here's one about mitosis our teacher had us learn in middle school: https://www.youtube.com/watch?v=4W8kupKFCPk
Then here's this one from Happy Days about blood flow through the heart: https://www.youtube.com/watch?v=gIXcWE0bTwY
And of course, you've probably heard of School Rocks, right? --- Fame is but a slow decay. -Theodore Tilton
When thinking about "Seasons of Love," I actually start thinking about the rest of the verse. "525,600 minutes. How do you measure, measure a year? In daylights, in sunsets, in midnight, In cups of coffee In inches, in miles, in laughter, in strife."
So I thought it'd be a cool project to pick something and actually measure a year in it, keeping track of, say, cups of coffee, or miles traveled. Of course, I'm not the first person to think of such a thing:
http://feltron.com/ar05_01.html
As for songs, I don't really have an good mnemonic ones. Maybe Barenaked Ladies' "Crazy ABCs"? Well, it does have the word "mnemonic" in it.
http://www.youtube.com/watch?v=nQLv7zrJk9U --- "To truly live, one must first be born." ~ Evan [aX] Paper Mario Social: The Safe Haven of GameFAQs. (Board 2000083)