Introduction

Schoenberg's Opus 23, no. 3 has a clear subject, with a clear prime transpositional level: Bb4–D4–E4–B3–C#4. The subject, with its thematic registral contour, opens the piece in the right hand. It recurs, transposed up 7 semitones, to open the next section of the piece (m. 6, right hand), and, an octave below its prime transpositional level, in the left hand of measure 6. It recurs, at its prime transpositional level, as a cantus firmus that opens the next section of the piece (mm. 9–10, ruhig). It recurs an octave below its opening level, to open the next section of the piece (mm. 12–13, mf ). It recurs at its prime transpositional level, with Bb and B∂ exchanged in register, to open the next section of the piece (end of m. 16, ruhig). It recurs transposed down 5 semitones, to open the next section of the piece (mm. 18–19, left hand). And finally, after some reprised material, it recurs to open the final sections of the piece (mm. 26–27, right hand), where it is mirrored against itself in an inverted form. The mirror texture continues to the end of the piece.

The subject projects the pitch-class series Bb4–D4–E4–B3–C#4. We shall denote this series of pitch classes (not pitches-in-register) as S0 (S for subject or series; 0 for its prime (pc-)transpositional level, as the pc series Bb4–D4–E4–B3–C#4). S0 projects the unordered pc set ﹛Bs,B,C#,D,E﹜; we shall denote that unordered pentachord by P0 (P for pentachord; 0 for its prime (pc-)transpositional level). The piece is saturated with transposed and inverted forms of P0; some appear with the thematic S-ordering, while others do not.

Traditionally, an analyst seeks a notation that will indicate each of the “transposed and inverted forms” of S0, or P0. Traditionally, one indicates by the twelve symbols Sn (respectively Pn) “S0 (resp. P0), transposed by the pc-interval n.” Traditionally, one labels as “s0” and “p0” some inverted form of S and the correspondingly inverted form of P. The various other inverted forms are then labeled “sn” (resp. “pn”), meaning “s0 (resp. p0), transposed by pc-interval n.” But which inverted form of S (resp. P) is to be labeled as s0 (resp. p0)? Traditionally, one might reply, “that inverted form most characteristically paired (and compared) in the music with S0 (resp. P0).”