The German-registered private bus tipped over as it drove into a bend on the A2 highway in central Switzerland and came to a rest on its side, police in the canton (state) of Uri said. The bus was carrying 13 people - two drivers and 11 members of the Marcus Miller Band, including Miller.
Over his career, the bassist has worked with jazz greats such as Miles Davis, Dizzy Gillespie and Wayne Shorter, according to his website. He won two Grammys, his first coming in 1991 for Best Rhythm & Blues Song ("Power of Love") along with Luther Vandross and Teddy Vann, and the second came in 2001 for Best Contemporary Jazz Album ("M2").
The band was on its way from Monte Carlo to the Dutch town of Hengelo, the next stop on the American band's tour, where it was due to perform Monday.
The driver who was at the wheel at the time of the accident sustained fatal injuries. Police spokesman Karl Egli said the 12 passengers were injured and taken to hospitals, but none had life-threatening injuries.
Miller was discharged from the hospital later Sunday, as were fellow band members Alex Han and Kris Bowers, but some other band and crew members were being kept in hospitals overnight, according to a post on Miller's official Facebook page.
The cause of the accident was not immediately clear. Police believe no other vehicles were involved.