For students who want to find an easy online matrix generator, you can check out: http://composertools.com/Tools/matrix/MatrixCalc.html 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 cre...