The courante, corrente, coranto and corant are just some of the names given to a family of triple metre dances from the late Renaissance and the Baroque era. Modern usage will sometimes use the different spellings to distinguish types of courante (Italian spelling for the Italian dance, etc.), but in the original sources spellings were inconsistent. Courante literally means running. From Wikipedia
My fingers fly down the scale passages in the Courante from Bach's French Suite VI. It’s like running down a hill --- fast a little out of control… like flying. A friend of my mine recently celebrated a milestone birthday with a Contra dance party. We ended the night of wonderful dancing with a simple version of the Flying Scotsman. It was exhilarating to sash-say down the line of dancers and back again. I was out of breath from running and laughing...just like playing the scales in the Courante.