I'm not a Cylon but ...

For the last month or so, there's been a tune that creeps up on me when my mind is idle, usually when I'm walking by myself. I know only two lines of the song, but I know them very well, from piano lessons as a child or teenager. It's a jaunty tune, with a simple melody line that might be classical, or might be of an American folk song. I've definitely played it before, I could play it now if I had a piano before me. But I cannot for the life of me call to mind who the composer was or what the song might be. I'm not even sure I could pin it down as 19th or 20th century.

Yes, I've turned to my iPhone, but Shazam isn't so good at matching hummed tunes. A friend recommended SoundHound, but now I feel like I'd rather keep the mystery going, until one day serendipitously I stumble across the piece of music proper (or dig up an old piano score back in Singapore).