Wait! Cancel that! It does play if I do it in 2 steps. This is in Windows 7.

1. Right click on the link above that says "Click to Play"

2. Somewhere else on the screen, probably at the bottom, a little box appears that says CharmBracelet.mid

Click on that, and another box appears that says 'Windows Media Player,' and the music starts itself.

Another challenge surmounted!