Apostel studied with Schoenberg and Berg. He was honored many times for his compositional work, including the Major Austrian State Prize in 1952 and the Major Monaco Composition Prize in 1968. He started out as a lyrically Romantic composer and progressed through atonal expressionism to absolute serialism. His Paralipomena dodekaphonika is an example of his later work. Apostel believed in a connection between music and the visual arts and often tried to translate images into music in his compositions. The initial notation for his pieces were frequently graphic in nature and these sketches contained all the elements necessary for a complete composition.