Twelve-tone composition involves using all twelve pitch classes roughly equally. That means it (usually) isn't appropriate to look for a key, mode, tonic pitch, or other tonal elements.

Composers often use a fixed ordering of the twelve pitch classes called a row, but also adapt it in various ways, notably through:

Transposition (T)

Inversion (I)

Retrograde (R)

Retrograde inversion (RI)

In practice, there is a great variety of how composers approach the task of composing with twelve tones.

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Twelve-tone music is most often associated with a compositional technique, or style, called serialism, though these terms are not equivalent:

“Serialism” is a broad designator referring to the ordering of things. In television, for instance, a serialized show is one in which the episodes are aired in a specific order to tell a continuous story. Serialism in music involves putting musical elements in some kind of order, whether they are pitches, durations, dynamics, or something else. So note that not all serial pieces use a twelve-tone row.

"Twelve-tone composition" refers more specifically to music based on orderings of the twelve pitch classes.

This style of composition is commonly associated with a group of composers (sometimes called the "Second Viennese School") whose members included Arnold Schoenberg, Anton Webern, and Alban Berg.The "First Viennese School" (by this logic) centers on Haydn, Mozart, and Beethoven. But twelve-tone compositional techniques and the ideas associated with them have been influential for many composers, and serial and twelve-tone music is still being written today. Much of this music shares similar axioms, which we outline in the following chapters, but it's important to stress that composers have used these basic ideas to cultivate a wide range of different approaches, and that the emphasis for most composers is on the music, with the technique as an important but subsidiary consideration.

Rows

Twelve-tone music is based on a series (sometimes called a row) that contains all twelve pitch classes in a particular order. This order is not the same in each piece—in fact, there are 479,001,600 rows to choose from!This number comes from the mathematical expression 12! (read: "12 factorial"), which means 12 × 11 × 10 … × 2 × 1. Some of these row forms have been used in multiple works, as they contain properties that several composers may favor, while many others have never been used at all.

Operations

There are four main ways in which composers move a row around without fundamentally changing it. We call these "operations" (in the mathematical sense rather than the medical one).

Transposition (T). Take all the pitches and move them up or down by a specified number of semitones. Transposition is a familiar process from tonal music, but note that (as in set theory) we're always working in transposition by semi-tones here and never diatonic steps.

Inversion (I). Reverse the direction of the intervals: rising intervals becoming falling intervals, and vice versa. Again, this is just like melodic inversion in other contexts, and once again, we're only dealing with exact inversion, preserving the interval size in terms of semitones (not using diatonic inversion or generic intervals here).

Retrograde (R). Reverse the order of pitches so the last comes first, and vice versa. This, too, has a precedent in tonal music with the "retrograde" (also known as the "crab" or "cancrizans") canon, for instance, though it's a lot rarer in tonal music than transposition and inversion.

Retrograde inversion (RI). As the name suggests, this really involves combining two of the operations described above: the retograde and the inversion. The order in which you do those operations does matter, but we'll return to that later on.

Twelve-tone rows that can be related to each other by transposition, inversion, and/or retrograde operations are considered to be forms of the same row. Unless a row has certain properties that allow it to map onto itself when transposed, inverted, or retrograded, there will be 48 forms of the row: the four types—prime (P), inversion (I), retrograde (R), and retrograde inversion (RI)—each transposed to begin on all of the twelve pitch classes. As such, a row produces a collection of 48 forms in what is called a row class.

A fake example

To get a sense of the basic operations the composers perform on tone rows, let's start with a fake example: an ascending chromatic scale starting on C (Example 1). Composers tend to prefer more interesting tone rows, but we'll start with this simple case for illustration. Row forms also don't usually commit to placing pitches in a specific octave, but we'll set it out in musical notation and with treble and bass clefs to show the inversions nice and clearly.

Example 1. A chromatic scale in four forms: prime, retrograde, inversion, and retrograde inversion.

Prime form

The prime form of the row (top left in Example 1 above) is the main form to which all other forms are related. In some pieces, one form of the row will clearly dominate the texture. If that is not the case, we generally choose the most salient row at the beginning of the work and label it P (for "prime"). If more than one row seems equally prominent at the beginning, then simply choose one (flip a coin!). The decision of which to call “prime” is not always important, but it's useful to allocate a single row form to serve as a point of reference.

Any row form that is the same as, or a strict transposition of, that opening prime form is also a prime form. Once you have labeled the main prime form at the beginning of the piece, any subsequent row that is an exact transposition of that row is prime. Likewise, any row that exhibits the same succession of ordered pitch-class intervals is also a prim