Application
Compositional applications of these theories are numerous, but in the present context of "Post-tonal" music the most important is Serialism. In this system, certain notes are chosen then written in an order e.g. E-F♯-C-B♭-G-F. (Usually there is no repetition, but this is not always observed.) These notes are then used as the basis for a composition by playing them in the original order, in reverse order (Retrograde), in "upside down" order (Inversion i.e. upward intervals now go down, and vice versa), or both (Retrograde Inversion or "Reversion" ), and then transposed up or down. Chords can also be formed out of the "Series" and these can be treated to similar techniques. Schoenberg used these methods in what has become known as Twelve-tone technique. In this, all unique twelve notes of the musical scale are played once and once only in a specified order. The serial techniques described above are then applied. 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. See also Formula composition which describes techniques used by Karlheinz Stockhausen.
Aside from Serialism, other forms of compositional technique arose such as those based on chords utilizing fourths rather than the more traditional thirds (see Quartal and quintal harmony and Synthetic chord), those based on other mathematical processes (see Schillinger System and those based on specific scales (or "modes": see Hexatonic scale, Heptatonic scale, Octatonic scale and Synthetic scale). 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.
Read more about this topic: Post-tonal Music Theory
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