Which composer invented the revolutionary "method of composing with twelve tones related only to each other"?

The correct answer is Arnold Schoenberg.

He was an Austrian composer who was born in the 19th century, but died in the 20th, in 1951. His method of composing with twelve tones related only to each other has become an extremely important method in Western music, and can be found in his Wind Quintet, for example.