The Musician's Guide to Theory and Analysis
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Arnold Schoenberg wrote several times, beginning in 1923, about composition with twelve tones. These essays, and many others, were gathered together and edited by Schoenberg's pupil Leonard Stein in a volume called Style and Idea (Berkeley and Los Angeles: University of California Press, 1975; reprinted 1984). His 1941 essay "Composition with Twelve Tones (1)" gives a short history of what he called the "emancipation of the dissonance" (pp. 217-18):

The first compositions in the new style were written by me around 1908 and, soon afterwards, by my pupils, Anton von Webern and Alban Berg. From the very beginning such compositions differed from all preceding music, not only harmonically but also melodically, thematically, and motivally [sic]. . . .

After many unsuccessful attempts during a period of approximately twelve years, I laid the foundations for a new procedure in musical construction which seemed fitted to replace those structural differentiations provided formerly by tonal harmonies.

I called this procedure Method of Composing with Twelve Tones Which are Related Only with One Another.

This method consists primarily of the constant and exclusive use of a set of twelve different tones. This means, of course, that no tone is repeated within the series and that it uses all twelve tones of the chromatic scale, though in a different order.
Schoenberg was aware of other composers who were likewise experimenting with methods of nontonal composition. One of these, composer Josef Matthias Hauer, particularly annoyed Schoenberg. He wrote several short essays on "Hauer’s Theories" (also reprinted in Style and Idea), expressing such opinions as "All the laws so apodictically set forth by Hauer-laws based on the principle I expressed-are wrong. . . . he does not know them and I certainly have more inkling of them than he has, otherwise I could not have expressed the principle."