Post-Tonal Theory (Atonality)

The very point of Post-Tonal Music Theory is to make all the notes and chords equal: there is no hierarchy. Instead, notes/chords can be described in relative terms which are unique to that particular occurrence of the note/chord: whether one note is higher than another, whether one chord has more notes than another, whether one chord is more widely spaced than another, etc.

There is no "tonic" chord or note; Instead music is written freely without any central "theme". One way of composing using atonality is the twelve-tone composition method.

In a twelve-tone composition, every note can be accounted for as being a member of the original series or one of its permutations, providing unity to the piece as a whole. Additionally, a twelve-tone series is a repository of intervals and can be seen as an outgrowth of atonal music with its emphasis on interval over chord or scale. The basic premises of twelve-tone music are as follows:

All twelve notes of the chromatic scale must occur
No note can be repeated in the series until the other 11 notes of the chromatic scale have occurred (exceptions include direct repetition of a note, trills, and tremolos)
The series can be inverted, retrograded, and the inversion can be retrograded
The order of notes in a series remains fixed, without reordering.

Inversion: This method is a bit trickier. It involves taking the melody or a part of it, and then inverting it ("upside down" order). For example:

If the original melody goes up a minor 2nd, then we'll go down a minor 2nd in the inversion.

Retrograde: Play the melody in reverse order and see if it sounds good. This can be paired with Inversion to create "retrograde inversion".