The email detailed how a bomb attack would be made on CIAPC’s office this week.
Antti Kotilainen, who heads up the association, said he takes the threat seriously.
CIAPC and the Finnish branch of the International Federation of the Phonographic Industry, IFPI Finland, have also asked police to investigate denial-of-service attacks on their websites.
On Monday, Helsinki District Court ordered internet service providers Elisa and Saunalahti to block access to The Pirate Bay, where users exchange computer files, including copyrighted films and music.
The music industry is now seeking similar Pirate Bay blocks for Sonera and DNA internet customers.
"If IFPI gets to block access for TeliaSonera & DNA, We'll tear it down. We'll make it crumble," tweeted AnonFinland, Finnish activists of the group Anonymous, early Wednesday. Later tweets posted in the group's name said that it does not condone the use of physical violence.