In fact, since the middle-age, composers have always used the same pitch-based structure that they slowly made evolve, perfecting the rules of harmony, the structure of their compositions and possibilities in orchestration.
However, the early 19th century witnessed the birth of a music that is not only about rhythm and melody, notably with the Futurist movement in Italy or the american composer Charles Ives whom "Symphony number four" (1910-1916) revisits a classic compositional form with an impression of total musical fragmentation, a juxtaposition of themes harmonically and rhythmically different.
This is the context in which Arnold Schoenberg develops the idea of atonality in its music, starting with the concept of "schwebende tonalitat" or fluctuating tonality.(camridge companion) In his book "Harmoniliehere" he states : "If the key is to fluctuate, it will have to be established somewhere. but not too firmly; it should be loose enough to yield. Therefore it is advantageous to select two keys that have some chord in common."(reference)
Another concept elaborated by Schoenberg during the same period is the idea of non-repetition, in "Erwartung"(1909) for instance that the american musicologist Phillip Friedheim describes as "Schoenberg's only lengthy work in an athematic style, where no musical material returns once stated over the course of 426 measures."(reference)
Eager to organize atonal music, Arnold Schoenberg established a method that uses tone row and their inversion, retrograde or retrograde inversion. "The essay "Composition with Twelve Tones," published a year before his death in the first edition of Style and Idea (1950), was his most substantial public statement."(reference)
The most significant and direct follower of Schoenberg is one of his pupil in the second viennese school, Anton von Webern. Even if his work follows on from Schoenberg and uses the same dodecaphonic technique, Webern is regarded by many as an even more radical composer.
In Five Movements for string orchestra (1929) Webern differentiates himself from Schoenberg by composing a very short piece, that leaves behind repetition of theme and motive.(reference cambridrge 2201) The second movement of "Piano Variations" is a good example of Webern's different approach to twelve-tone method , it "uses only three dynamic markings and five varieties of rhythmic cell." (1002 modern music)