It revolves around the idea of 'emancipating dissonance', that is, freeing the structure of music from the familiar harmonic patterns that are derived from natural overtones.
Arnold Schoenberg and his pupil Anton Webern proposed a theory on the emancipation of the dissonance to help analyse the general trend and, in particular, their own atonal music.
By the later nineteenth century the higher numbered dominant-quality dissonances had also achieved harmonic status, with resolution delayed or omitted completely.
The greater autonomy of the dominant-quality dissonance contributed significantly to the weakening of traditional tonal function within a purely diatonic context."
However, this form of analysis cannot be applied to atonal music since the very point is to make all the notes and chords equal: there is no hierarchy.
Compositional applications of these theories are numerous, but in the present context of post-tonal music the most important is serialism.
[9] Later composers, such as Jean Barraqué and Pierre Boulez, sought to unify pitch and rhythm by organising the elements into sets of twelve, which resulted in what became known as total serialism.
Olivier Messiaen in his work The Technique of my Musical Language developed what he called modes of limited transposition which displayed a special type of symmetry and which he used in numerous compositions.