Atonal Theory and Matrix
The atonal system was a large break from the traditions of before. Western music first began with simple intervals in the time of Pythagorus. His system marked intervals like a unison and an octave as the most tonal (and pleasing to the ear). In the middle ages, musicians used church modes (Gregorian Chant ex.: http://www.youtube.com/watch?v=_MbDqc3x97k). In the Baroque Era, composers began to use what is now considered major and minor scales. True western harmony was born.
When Arnold Schoenberg experimented with atonality, he developed a complex musical (and mathematical) system. Essentially, a composer would arrange the twelve notes of a scale in a specific row, and then compose using those rows (in that order alone) to eliminate any sense of tonality. The retrograde, retrograde inversions, etc. of the tone row were created using a matrix like the one shown above.
Eventually, composers began incorporating nonpitch elements like rhythm, dynamics, and expressions into the matrix. Schoenberg's works were revolutionary. However, while contemporary composers learn this style of composition, generally, most composers do not use strict atonal music theory in their works in the United States, while other nations, such as Taiwan, Japan, and Korea still actively use the atonal system.
Brief video on Schoenberg: