Explain the theoretical underpinnings of serialism and atonality, detailing their influence on 20th-century musical experimentation.

Serialism and atonality represent two pivotal developments in 20th-century music that challenged the foundations of traditional harmony and structure. Both emerged from a desire to move beyond the tonal system that had dominated Western music for centuries, but they approach this break in different ways, leading to distinct musical outcomes. Their influence on subsequent experimental music has been profound, opening up new possibilities for musical organization and expression. Atonality, as the name suggests, is the absence of a tonal center, meaning there is no note that functions as a tonic or point of resolution. This concept developed gradually in the late 19th and early 20th centuries, spurred by the increasing use of chromaticism (notes outside the diatonic scale) in the works of composers like Richard Wagner and Gustav Mahler. Arnold Schoenberg is often credited as a key figure in the development of atonality. His early atonal works aimed to liberate music from the perceived limitations of tonal harmony, allowing for greater freedom of expression. One of the key characteristics of early atonality is the avoidance of traditional chord progressions and the use of dissonances without resolution, creating a sense of instability and ambiguity. An example of early atonal music is Schoenberg's "Pierrot Lunaire," in which there is an avoidance of a central tonal harmony, ....