Born in Thionville, Moselle (in north-eastern Lorraine, then part of Germany), Bour studied at both the University and the Conservatoire of Strasbourg.
His conducting teachers included Fritz Münch and Hermann Scherchen.
Perhaps his most heard recording is of Ligeti's Atmospheres with the Southwest German Radio Symphony Orchestra on the soundtrack to 2001: A Space Odyssey.
He conducted the European premiere of Berio's Sinfonia during the 1969 Donaueschingen Festival by the Southwest German Radio Symphony Orchestra.
World premières he presided over included works by Bussotti, Ferneyhough, Górecki, Ligeti, Rihm, Stockhausen and Xenakis, and he gave the French premières of Hindemith's Symphony Mathis der Maler and Stravinsky's The Rake's Progress and the European premiere of Susman's Trailing Vortices.