Final answer:
The method developed by Arnold Schoenberg to organize atonal music is known as the twelve-tone system, a radical innovation in music composition.
Step-by-step explanation:
The systematic method developed by Arnold Schoenberg to organize atonal music is known as the twelve-tone system. Schoenberg's explorations in the realm of atonality were significant because they challenged the traditional scales that had been the basis for Western music since the Renaissance. By employing the twelve-tone system, he created music that did not adhere to a central key, using a scale of twelve tones arranged in a specific order, or series, which becomes the foundation for a composition's melody, harmony, and structure. This method radically departed from conventional tonal music, leading to innovative forms and textures in music composition.