Concepts, Terms, and Basic Information

Serialism is a compositional process based on an ordered group of pitches. Usually 12 pitches are ordered into a series, but some composers (e.g. Stravinsky) have used fewer or, by including duplications, more pitches. Twelve-tone serialism represents each pitch class only once; pitch classes may be notated in any octave (octave equivalence) and by any spelling (enharmonic equivalence).

While there are a great number of things to think about when analyzing serial music, here are a few suggetions:

1. Do not forget to think first and foremost that this is music--listen, think, and analyze carefully such parameters as form, rhythm, texture, motivic structure, register/range, and so on.
2. Examine the original tone row, Po, for all of its characteristics: intervallic structure, etc.
3. Always be concerned about themusic, go through the entire piece or movement and determine which row forms are used.
4. After figuring out and labelling the row forms used, the next task is to consider how these row forms are projected. In other words, how are they dispersed through the texture of the work? For example:
   • If the prevailing texture is simply two voices, does the composer use a separate row form in each line, or are the row forms intertwined somehow and distributed among both lines?
   • If the prevailing texture is some sort of melody and accompaniment, is there one row form in themelody and a different one in the accompaniment, or do the melody and accompaniment share the same row form as they proceed?
   • If the instrumentation is fairly complex (a lot of instruments playing, as in a symphony), trace how the row forms are distributed with regard to instrumentation. Some pieces (particularly Webern's) will almost appear to be organized into "clumps" of pitches, but there really is a reason.
5. Finally, onemust consider why the particluar row forms are used. Does it have to do with transposition somehow? What interesting things would different transpositions and inversions create?

Definitions:

1. Original Tone Row: This is called Po (that is, the prime form of the row, transposed zero half step). It is the original succession of 12 pitch classes upon which a particular work is based. The original Po is an ordered set of pitch classes (note how this differs from pitch-class sets, which were unordered collections of notes).
2. Forms of a 12-tone Row: There are four (usually distinct) forms which a tone row may take ("row forms" are sometimes called "row transformations").
   • Prime: the forward succession; symbolized P
   • Inversion: the form of P with the intervals inverted; symbolized I
   • Retrograde: the backword version of P; symbolized R
   • Retrograde Inversion: the retrograde of an I form; symbolized RI
3. Transposition: This simply means that any P, I, R, or RI form may be transposed. The levels of transposition are denoted by transposition numbers or index numbers.
4. Index Numbers: Each row form (P, I, R, RI) may be transposed to begin on any of the 12 pitch classes. Index numbers indicate the distance inhalf steps above the zero (0) pitch class.
   • Po is the untransposed original prime form; P1 is the original prime form transposed up a half step; P5 is the original prime form transposed up a perfect fourth, and so on.
   • Io stands for the inverted row whose initial note is the same pitch class as the initial note of the Po form. I1 is the inverted form transposed up a half step; I7 is the inverted form transposed up a perfect fifth, and so on.
   • Therefore, P and I are always indexed by their first note.
   • The untransposed retrograde versions of Po and Io are indicated as Ro and RIo. This is simply because Ro is really Po stated backwards, and RIo is really Io stated backwards. So, R5 is simply P5 stated backwards, RI8 is simply I8 stated backwards, and so on.
   • Therefore, R and RI forms are always indexed by their last note.
5. Matrix: The matrix of row forms is a 12 x 12 arrangement of all 48 forms of the tone row [that is, 4 forms (P, I, R, RI) x 12 transpositions each]. Each row (left-right) indicates a P form at some transposition. Each column (top-bottom) indicates an I form at some transposition. The primary diagonal (upper left-lower right) will always contain the same pitch-class.
6. Order Numbers: Let's say that you can do all of the above, and you now want to examine a piece to see how a composer made use of the row transformations. What you do is go through and find all the row forms used (say, Po, I4, P8, and RI7). Then you need to number the notes within the individual transformations. You might expect to number them from 1 to 12, but actually they are numbered 0-11. Note carefully: while order numbers run from 0 to 11 for P and I forms, they run from 11 to 0 for R and RI forms.