Been familiarizing myself with plenty of atonal music over the decades. My approach is just very melodical, but what I lack is the know-how to analyze it like analyzing tonal music. Although writings on 12-tone music often focus on the rows, that's not how most people listen to it. (The scottish composer Jonathan Harvey, studying with both Babbitt and Stockhausen, expressed doubts whether it's even possible to hear the series, or partitions of the series.)
I've tried if I can see, whether different motifs and themes can be outlined by the intervals they span. Example: Berg's String Quartet Op. 3. The first motif spans 8 semitones. The continuation spans 4 semitones, which is conversely the same as 8, and when the motive is repeated, it's 4 semitones lower. The next motif spans 10 (conversely 2) semitones. The accompaniment is starchly differentiated by avoidance of these aforementioned interval spans. Now, I can't make deeper sense of it than that. (I've also tried to hear atonal music, to no avail, in terms of bass movement).
I've tried if I can see, whether different motifs and themes can be outlined by the intervals they span. Example: Berg's String Quartet Op. 3. The first motif spans 8 semitones. The continuation spans 4 semitones, which is conversely the same as 8, and when the motive is repeated, it's 4 semitones lower. The next motif spans 10 (conversely 2) semitones. The accompaniment is starchly differentiated by avoidance of these aforementioned interval spans. Now, I can't make deeper sense of it than that. (I've also tried to hear atonal music, to no avail, in terms of bass movement).