Theory
Music written within the tonal system is generally analysed by defining a certain note as the primary or "Tonic" note and the derived triad is the "Tonic" chord. Other notes and chords are subservient to the Tonic and in a strict hierarchy: the "Dominant" note/chord is second in importance, others are lower down still. One example of this style of analysis is called Schenkerian analysis. 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. Instead, notes/chords can be described in terms of their properties and relationships at any particular moment: whether one note is higher than another, whether one chord has more notes than another, whether one chord is more widely spaced than another, and so on. One can also compare and contrast different strings of notes as transpositions (change in pitch) or inversions (change in note order) of each other. These terms are also used compare chords. These methods of analysis have been used for centuries but became more important as music began to lose its tonal basis. One also needs to consider other aspects, such as how two or more simultaneous melodies relate to each other (counterpoint) and the same tools are used for this.
In the later 20th century, analysts started to adapt these tools to the yet more complex music being written. Musical set theory was first elaborated for tonal music but was quickly applied to atonal music since it simply provides concepts for categorizing musical objects (notes, chords, melodies and so on) and describing their relationship, without defining any particular note or chord as "primary". The later Transformational theory uses a similar approach but concentrates on the relationships themselves. There are also theories which attempt to relate pitch and rhythm.
Read more about this topic: Post-tonal Music Theory
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