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, resulting in a sense of free-flowing but non-traditional music. The lack of a clear harmonic resolution was not a rejection of music's structure but more of an exploration of its possibilities when removed from traditional tonal concepts.

Serialism, also pioneered by Schoenberg, takes this idea of abandoning tonal centers a step further by establishing a systematic way of composing with all twelve tones of the chromatic scale, often referred to as "twelve-tone technique". This system attempts to completely organize all aspects of musical composition, not just the notes, but also rhythm, dynamics, and articulation. The core principle of serialism is the "tone row" or series, which is a pre-ordered arrangement of all twelve pitches. This row is then manipulated in various ways (e.g., retrograde, inversion, retrograde-inversion) to create the melodic and harmonic material of a piece. Unlike atonal music, where the avoidance of tonal centers is a more free approach, serialism seeks to impose structure and logic on atonal material, trying to create a complex systematic approach. An early example of this technique is Schoenberg's "Piano Suite, Op. 25." In this piece, the use of the twelve-tone row isn't just a collection of notes but is used to generate the musical form and harmony of the piece. The row isn't just a tool for generating pitches but is also used as a basis for rhythmic and dynamic material, showcasing the total control of serialism.

The influence of atonality and serialism on 20th-century musical experimentation is profound. Atonality, with its freedom from tonal constraints, opened the door to new modes of expression and the exploration of new sounds. It was not a rejection of musical structure but more a shift in the concept of the structure itself. Serialism offered a systematic approach to organizing atonal material and it demonstrated that new forms and structures could be created from systematic logic. While serialism itself went through many changes (such as the total serialism of the post-WWII era), it also showed that musical parameters could be systematically controlled and used to build complex and interesting structures. These ideas influenced subsequent experimental movements, like the development of post-serialism, which allowed for more freedom while still using the basic building blocks of the twelve-tone row and the possibilities for manipulation of musical parameters. It also influenced the development of electronic music and computer-assisted composition which use the same logical principles as serialism.

Both serialism and atonality demonstrated that music did not have to be tied to traditional harmonic or structural ideas, and it was not tied to a single mode or method of composing. They provided composers with new tools and frameworks for musical expression, pushing the boundaries of what was considered music, and influenced generations of composers who sought to explore new sonic possibilities. These developments paved the way for many avant-garde approaches by creating a new playing field for what was considered musical, and what was not. The development of atonality and serialism is a crucial part of understanding the development of modern experimental music.