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Atonal Theory and Matrix

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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 created using a matrix like the on…