Serialism

Introduction: Composing with Twelve Tones

In the early 1920s, Arnold Schoenberg devised a "method of composing with twelve tones which are related only with one another." Instead of a key, a piece is built from a twelve-tone row (also called a series or tone row): a specific ordering of all twelve pitch classes, each used once. The row supplies melodies, chords, and counterpoint throughout the piece, guaranteeing that all twelve pitch classes circulate evenly and that no single pitch becomes a tonic. Schoenberg's students Alban Berg and Anton Webern adopted the method, and after 1945 composers such as Milton Babbitt and Pierre Boulez extended it to rhythm, dynamics, and timbre.

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The Row and Its Forms

A row can appear in four basic forms, each transposable to any of twelve levels, for 48 row forms in all:

• Prime (P): the row in its original order.
• Retrograde (R): the row backward.
• Inversion (I): every interval turned upside down (an ascending minor third becomes a descending minor third).
• Retrograde inversion (RI): the inversion backward.

We label each form by its first pitch class for P and I: P₇ begins on G, I₂ begins on D. A retrograde takes the label of the prime it reverses, so R₇ is P₇ backward (it ends on G), and RI₂ is I₂ backward. (Some books instead call the first form of the piece P₀, whatever its first note. Always check which convention an analysis uses.)

The Twelve-Tone Matrix

A matrix (or magic square) displays all 48 forms at once. Write P₀ across the top row and I₀ down the left column; each remaining row is a transposition of P, beginning on the pitch class in its first column. Then:

• Read left to right for P forms (labeled at left).
• Read right to left for R forms (labeled at right).
• Read top to bottom for I forms (labeled at top).
• Read bottom to top for RI forms (labeled at bottom).

To fill in row k quickly: each entry is (the row's first pc) + (the P₀ entry above it) − (P₀'s first pc), mod 12. The diagonal from top left to bottom right always repeats P₀'s first pitch class.

Using the Row

• Melodically: the row unfolds as a melody, often split among instruments or registers (Webern's Klangfarbenmelodie).
• Harmonically: consecutive row segments (dyads, trichords, tetrachords) are stacked as chords.
• Contrapuntally: different row forms sound at once in different voices.
• Repetition: a note may repeat immediately, or be sustained or trilled, without "using it up."
• Rhythm, register, dynamics, and texture are free (in Schoenberg's practice), which is where much of the expressive variety comes from.

Invariance and Derived Rows

Some row forms share segments with others: a trichord in one form might appear, reordered or not, in another. These invariants let a composer link row forms smoothly, much like common tones between chords. Webern liked derived rows, built from one trichord or tetrachord and its transformations. His Concerto, Op. 24, uses a row made of four (014) trichords. Some rows are internally symmetrical: the row of his Symphony, Op. 21, is its own retrograde at T₆, so R and P forms coincide in pairs.

Hexachordal Combinatoriality

Divide a row into two six-note halves, or hexachords. A row is combinatorial if some other form of the row has, as its first hexachord, exactly the pitch classes of the original's second hexachord. Played together, the first hexachords of the two forms complete an aggregate (all twelve pitch classes), and so do the second hexachords. Schoenberg built much of his later music on such pairs, typically P with an I form (inversional combinatoriality).

In the model row, H1 = {0, 3, 4, 7, 8, E} and H2 = {1, 2, 5, 6, 9, T}. Forms whose first hexachord equals H2 (and so combine with P₀ into aggregates): I1, P2, I5, P6, I9, PT. This row's hexachords are both members of set class 6-20 (014589), the "hexatonic" hexachord, which is unusually rich in combinatorial partners.

Integral (Total) Serialism

After 1945, composers including Messiaen (Mode de valeurs et d'intensités, 1949), Boulez (Structures Ia, 1952), Stockhausen, and Babbitt applied serial ordering to other parameters: durations, dynamics, articulations, and registers each received their own series. The result, called integral or total serialism, produced music of extreme surface complexity, and provoked reactions in the opposite direction (chance music and minimalism, Chapter 34).

Try It: Matrix Builder

Summary

• A twelve-tone row orders all twelve pitch classes; it appears as P, R, I, and RI, each at twelve transpositions (48 forms).
• Label P and I forms by their first pitch class; R_n and RI_n reverse P_n and I_n.
• The matrix shows all forms: P across, R backward, I down, RI up.
• Rows supply melody, harmony (stacked segments), and counterpoint; rhythm, register, and texture shape the music.
• Invariance, derived rows, and hexachordal combinatoriality (forms whose hexachords complete aggregates) give rows structural power.
• Integral serialism extends ordering to durations, dynamics, and articulation.

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