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Space exploration was big at the time, and was probably an influence on this. Pink Floyd was making music with a similar sound.
The psychedelic sound reflected the times. It was the summer of love (1967).
Mick Jagger got the idea for this while in jail on drug charges.
The Stones played this on their Steel Wheels tour in 1989. A show in Atlantic City was broadcast with this song shot in 3D, which viewers could see using those goofy glasses.
Various echo effects and drum sounds were added in overdubbing.
Brian Jones played the Mellotron, an early synthesizer. He died in 1969 when he drowned in his swimming pool.
'90s Psychedelic group The Brian Jonestown Massacre recorded a tribute to the Stones' Psychedelic period (and this song) called Their Satanic Majesties' Second Request. (thanks, Bertrand - Paris, France)
Mark Arm of Mudhoney
When he was asked to write a song for the Singles
soundtrack, Mark thought the Seattle grunge scene was already overblown, so that's what he wrote about.
Dean Friedman - "Ariel"
Dean's saga began with "Ariel," a song about falling in love with a Jewish girl from New Jersey.