How did 12-tone music change the use of sonata form in Arnold Schoenberg’s String Quartet No. 4?

In Arnold Schoenberg’s String Quartet No. 4 (1936), the composer adapted the 12-tone method to the traditional outlines of sonata form.1

Because 12-tone music is atonal and deliberately avoids establishing a sense of key, the work cannot rely on the Classical sonata principle of key contrast—the tension and resolution between different tonal centers that historically defined the exposition, development, and recapitulation. Instead, Schoenberg’s application of the form in this quartet is based on the use of contrasting themes. By fitting his 12-tone row into these structural divisions, he maintained the outward formal architecture of the sonata while replacing its traditional harmonic foundation with a melodic and harmonic basis derived from the 12-tone series.123