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What wider cultural renown was Karlheinz Stockhausen known for?
Karlheinz Stockhausen, along with John Cage, is one of the few avant-garde composers to have succeeded in penetrating the popular consciousness.
How was Karlheinz Stockhausen able to penetrate the popular consciousness
The Beatles included Karlheinz Stockhausen's face on the cover of Sgt. Pepper's Lonely Hearts Club Band. This reflects his influence on the band's own avant-garde experiments.
Why did the Beatles include Karlheinz Stockhausen on the album cover Sgt. Pepper's Lonely Hearts Club Band?
This reflects Karlheinz Stockhausen's influence on the band's own avant-garde experiments as well as the general fame and notoriety he had achieved by that time (1967).

Question: How did he influence the band's avante garde expirements
Self-contained rewrite: How did Karlheinz Stockhausen influence the Beatles's avante garde expirements