Roll over, Beethoven, you're streaming now; Primephonic offers classical music fans million-plus tracks at their fingertips.

A new streaming service devoted exclusively to classical music has launched in the U.S., Britain and the Netherlands.

Called Primephonic, the platform claims to have nearly all classical music ever recorded, with more than 1 million tracks available at the push of a button.

Catalogs on offer include those from Warner Classics, Sony Classical, Universal Music Classics, Deutsche Grammophon, Decca and EMI, in addition to more than 400 other labels worldwide. Users can search by composer, title, artist, musical period and genre; background...