Courante means running or flowing and has got the same route as our english word current (as in water). There’s a French form (courante) and Italian form (courrente). They are the same dance, although the Italian is more spritely and lively and the French is more elegant and restrained. This reminds us that French and Italian baroque are different, but we must also remember that the father of French baroque was Giovanni Battista Lulli. It was also the favoured dance of Charles II and Louis XIV.
