Stockhausen regarded Kreuzspiel as his first original composition, as opposed to the style-imitation exercises he did as part of his music studies.
[1] According to the composer, it was influenced by Olivier Messiaen's "Mode de valeurs et d'intensités" (1949) and Karel Goeyvaerts's Sonata for Two Pianos (1950), and is one of the earliest examples of "point" music.
Though routinely described (by the composer as well as others) as a "serial" composition, Kreuzspiel does not employ a referential, recurring twelve-tone ordered set.
Rather, it uses constant reordering of twelve-element (linked pitch, duration, dynamic, and—in the original version—attack) sets—a device sometimes called "permutational serialism" (e.g. Howel)[4] It also uses a permutational seven-element system to control register.
These gradually move into the four middle octaves until an equal distribution of pitches throughout the entire range is achieved at the centre of the movement.