In 1924, Herbert Eimert’s Atonale Musiklehre was the first published text to describe a systematic approach to composing atonal music. It contains significant contributions to the discourse on the early development of twelve-tone composition. While Eimert uses the term “atonal” to describe his compositional approach, his definition of atonality demands that all twelve tones be present with none repeated, and that they present as complexes not ordered rows. Eimert’s discussion of atonality differs from others of the same period because he focuses on vertical sonorities and introduces “interlocking complexes”, wherein two separate statements of the aggregate can overlap by one pitch or by a set of pitches. Interlocking complexes are an important feature of Eimert’s string quartet Fünf Stücke für Streichquartett, which was published in 1925 and composed at the same time as Atonale Musiklehre was written. In the foreword to Atonale Musiklehre, Eimert clarifies that he is not the originator of the concept of atonality, rather that he absorbed the ideas of Josef Matthias Hauer and Jefim Golyscheff. Twelve-tone complexes appear first in Golyscheff’s 1914 String Trio. He refers to them as “twelve-tone duration complexes” and labels them in the score. As the name “duration complexes” implies, there are …