Some worthwhile things to remember from the discussion on Webern’s Op. 21 Symphonie, and some clues to helping you figure out Babbitt’s Semi-Simple Variations.
Webern serves as a model for many post-WWII composers, because his use of 12-tone serial composition points the way to using serialism as an organizing feature for more than just pitch, and to basing large-scale (movement) structure on the structure of your row.
In the Op. 21 Symphonie Webern uses a row that is “mirrored” around the interval of 6 (tritone) between row members 6 and 7. The first instance of the row, P-8 is
9 6 7 8 4 5 11 10 2 1 0 3
with ordered pitch-class intervals of
9 1 1 8 1 |+/-6| 11 4 11 11 3
The mirror point is denoted by the | | marks. Moving outward from that point the ordered pitch-class intervals are a reversed and inverted (1 becomes 11, 8 becomes 4, etc). When you complete the matrix for this row, you find that every retrograde is the same as the prime form form of the row that starts on the same pitch class. For example, R-9, which starts on p.c. 3, is the same as P-3. Retrograde Inversions have the same relationship with Inversions. This is an extreme example of invariance, where parts of a row form at specific transposition levels are repeated exactly in other row forms and/or other transposition levels. Invariance is not necessarily a post-1945 technique. It allows for the linking of row forms, providing the composer with a path to move from one row form to another.
The row can also said to be derived. The second half is the result of a permutation, or specific transformation of the first half. Any time you generate a part of the row from some other part (through inversion, transposition, retrograde, etc.), you can say that the row is derived.
Webern uses these derived and invariant relationships in a number of ways. Note that the row mirrors around a 6 (tritone), and that the unordered pitch-class interval between the beginning of the row and the end is a 6. In P-9 (the first prime form of the row), the two pitch classes are A and Eb. To open the first movement, Webern uses four rows, combined into two canons of inversion. P-9 is paired with I-9, which both start on A and end on Eb. I-5 is paired with P-1, which both have the 9 – 3 (A – Eb) tritone as their middle interval class.
The idea of the mirror imparts itself onto the movement form through the use of a mirror-point in the movement, where everything up to that point reverses for the rest of the movement. However, Webern sets up a conflict between the inherent symmetry of the row and movement through his decision to stress four-note groupings of the row at the beginning of the movement. Webern projects these four-note groupings into a three-section movement that fits into a two-reprise form. (||: A :||: B – 1/2A :||)
A final important idea of the Webern Symphonie is the notion of fixed register. Throughout the work you find that register is mostly fixed within a section for a given pitch class. For example, the pitch class F (5) happens for the first time in m. 2 in the harp, at the pitch a perfect fifth below middle C. The next time it happens is in m. 6 in the clarinet, at the same exact pitch level (register). This is followed by the first horn in m. 12, the clarinet again in m. 14, and finally the harp in m. 15, all at the same register. The only pitch classes that “float,” or change their register within a section, are Eb in the first section, A in the second section, and both A and Eb in the third section. This is another example of projecting the row structure onto the large-scale formal design.
Post-1945
Post-1945 composers started using serialism to order other parameters of music, particularly rhythm and dynamics. Babbitt’s Semi-Simple Variations shows one way that this can happen with regards to rhythm.
The work makes use of a 16-item rhythmic series, with each item being a unique division of the quarter note into 16th-note attack points and non-attack points. Although you typically write the series using 16th-notes and 16th-rests, you need to look at attack points (when a note starts) as the organizing feature. In this way, a quarter note, an eighth-note and eighth-rest, and a sixteenth-note and dotted-eighth-rest are all the same pattern of
n-r-r-r
(1/16-note, 1/16-rest, 1/16-rest, 1/16-rest)
If you trace the attack points, the Anthology gives you the first three quarter-note beats of the series (each beat is surrounded by bar lines, which signify beats in this case):
| n-n-n-n | n-n-r-r | r-r-r-n |
Note that I have transcribed the rhythms from the Anthology (and piece) so that the series is always comprised of 1/16-notes or 1/16-rests, as opposed to the notated grouping of rests in the book. This helps in recognizing each rhythmic series member no matter what the localized rhythm may be, and helps in generating the inversion of the prime. Babbitt creates the inversion by replacing attack points with rests, and rests with attack points. (If you are so inclined, you can think of this as a binary operation – 1’s become 0’s, and vice versa.) Therefore, the inversion of the prime starts:
| r-r-r-r | r-r-n-n | n-n-n-r |
The retrograde is just the prime form in reverse, and likewise the retrograde inversion is a reverse of the inversion. This gives four forms of the rhythmic series. Note that transpositions have no match in rhythm.
In the variations, the rhythmic series is spread among all the voices and pitch rows.
Two final aspects of the Babbitt work in regards to pitch and row organization. One aspect is that Babbitt eventually transforms the original row by reordering elements. The other aspect is that Babbitt does not always start his use of a row form at the beginning.
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