When we think of counterpoint, we’re usually thinking of the "imitative" kind, where melodies (or bits of them) echo one another. This is certainly the kind most of us encounter first, when we learn as kids to sing “Row, Row, Row Your Boat” or “Frere Jacques” as a round. It’s also the kind we encounter most often: sometimes on a cell phone these days, whose ringtone may well be the opening of Bach’s Two-Part Invention in F. (Click on the "Play" symbol in the upper left of the score to hear it.)
But there’s also a non-imitative kind of counterpoint, a kind where different (often very different) melodies are made to fit together. Maybe the most familiar example of this sort of counterpoint is also found in Bach: his setting of the “Wachet Auf” chorale (again, click on the the "Play" symbol to hear it), in which he overlays a stirring Lutheran hymn with an almost ostentatiously dissimilar melody–one of the all-time ear worms, as it happens–of his own invention. (Think this trick is simple? You try writing a tune-for-the-ages that has to work contrapuntally with an existing one.)
My first encounter with non-imitative counterpoint was a revelation (not that I was old enough to have called it that). One day the director of our grade school chorus introduced us to a pretty little song called “Inch Worm” (music and lyrics by the great Frank [Guys and Dolls] Loesser: it's in his score for the movie musical Hans Christian Andersen). Our director divided the chorus in half, and taught one half the song's main melody ("Inch worm, inch worm...", sung by Danny Kaye in the clip) and the other half the song's haunting countermelody (which you hear first in the clip, sung by the class of children: “Two and two are four…” etc.). The sinuous combination of melody and countermelody still gives me the beauty-chills.
It was only many years later that I realized Loesser’s full brilliance in contriving this counterpoint of differing melodies. The overall meter of the song is 3/4 (i.e., waltz tempo), into which the lyric of the main melody fits perfectly: “Inch worm __, inch worm __, measuring the marigolds...”. But the lyric of the countermelody doesn’t fit into 3/4, at least not comfortably; the meter leads to unnatural stresses: “Two and two ARE four __ , four and four ARE eight __ , eight and eight ARE sixteen, sixteen and sixTEEN are thirty two…”). But the lyric of the countermelody (along with the countermelody itself) works perfectly in 2/4 (alternating stressed with unstressed syllables): "Two and two are four __ , four and four are eight __ , eight and eight are sixteen, sixteen and sixteen are thirty-two…". Which is to say that this seemingly innocent ditty is not only polyphonic but polymetric! But here’s the real kicker: what is the lyric of the (2/4) countermelody talking about but measuring marigolds by multiples of two! The mind-boggling musico-verbal wit of this connection--a duplicity (not to say duple-icity) of the benignest kind--reduces one to saying of Loesser, as a certain Schumann said in announcing his first acquaintance with the music of a certain Chopin, “Hats off, gentlemen: a genius.” (As if one didn’t think this of Loesser already.)