Longmuir formed the Bay City Rollers with his brother Derek, and they went on to find huge success in the 1970s.
Many have taken to Twitter to mourn the loss of the music great.
The Bay City Rollers became teen sensations in the UK and the US with hits like Bye Bye Baby and Shang-a-Lang.
At the height of their success, the band was deemed responsible for 'Rollermania.'
The group's best-known lineup, which featured Les McKeown and guitarists Eric Faulkner and Stuart Wood as well as the Longmuirs, came together early in 1974. According to legend, the group chose its name by throwing a dart at a map, which landed on the US city of Bay City, Michigan. At the peak of their success the group had their own television show and a fanbase known as the 'Tartan Horde.'
Our thoughts are with Alan's family and friends.