Approximately when was European music first written down

By the middle of the 9th century, however, a form of neumatic notation began to develop in monasteries in Europe as a mnemonic device for Gregorian chant, using symbols known as neumes; the earliest surviving musical notation of this type is in the Musica disciplina of Aurelian of Réôme, from about 850.

Music was actually written down as far back as the 6th Century B. C. in ancient Greece and Rome. The writing in this time was a bit more of a reminder on how to sing the songs rather than precise notation but it was certainly the beginning of writing down music in a digestible form.