The answer is B: Arnold Schoenberg.
Explanation for each option:
A. Alban Berg - While Alban Berg was a significant composer and a student of Arnold Schoenberg, he did not establish the twelve-tone system. Berg did, however, use and develop the twelve-tone technique in his compositions.
B. Arnold Schoenberg - Correct. Arnold Schoenberg is credited with developing the twelve-tone technique, also known as dodecaphony, in the early 20th century. This system treats all twelve tones of the chromatic scale as equal, avoiding the traditional hierarchy of pitches found in tonal music.
C. Henry Cowell - Henry Cowell was an innovative American composer known for his experimental techniques, such as tone clusters and extended piano techniques, but he did not establish the twelve-tone system.
D. Nadia Boulanger - Nadia Boulanger was a renowned French composer, conductor, and teacher who influenced many 20th-century composers. However, she did not develop the twelve-tone system.