Who is the Greek goddess of music?

There is no Greek goddess of music, but technically there is a Greek god of music, Apollo. There are also the nine Muses, which brought joy to anyone who heard their music. Their names are:

Thalia - Muse of comedy and idyllic poetry

Melpomene [mel-po-men-ee] - Muse of tragedy

Cleo - Muse of history.

Euterpe - Muse of lyric poetry.

Terpsichore - Muse of music and dancing.

Erato - Muse of erotic poetry.

Calliope - Muse of epic poetry and rhetoric.

Urania - Muse of astronomy.

Polyhymnia or Polymnia - Muse of sacred hymns and harmony.

http://www.csun.edu/~hchum001/muses.htm