*Motet (Excerpta Tractati Logico-Philosophici)*, Op. 27.

^{[4]}

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#
Basics of Twelve-Tone Theory

# Rows

# Operations

## A fake example

## Prime form

## Retrograde form

## Inversion form

## Retrograde inversion form

## A real example

Now that we’ve got the basic idea, let’s see how this works in a real musical context, using the same layout and taking as our example the row form in Elisabeth Lutyens’s *Motet (Excerpta Tractati Logico-Philosophici)*, Op. 27.^{[4]}
# Enter the Matrix

# From Theory to Practice

In general, then, the basics of twelve-tone technique stipulate that:
### Twelve-tone serial, but not so strict

### Serial but not twelve-tone

## What do we know?

IX. Twelve-Tone Music

Mark Gotham and Brian Moseley

Key Takeaways

- 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.

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.^{[1]} 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.

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!^{[2]} 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.

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.

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 (

). 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.

The prime form of the row (top left in **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 prime form.

Since **P** can be transposed to any pitch-class level, we distinguish them with subscripts. There are multiple common systems for deciding the numbering. The simplest, which we will follow in this book, is to number the row by its starting pitch class. If the prime form begins on G (pitch class 7), it is **P _{7}**; on B (pitch class 11) it is

A retrograde form of the row takes a prime form and exactly reverses the pitch classes. Its interval content, then, is the reverse of the prime forms. Retrograde forms are labeled R followed by a subscript denoting the last pitch class in the row. This will ensure that if two row forms are exact retrogrades of each other, they will have the same subscript.

For example, if a row has the exact reverse interval structure of the prime forms and ends on F♯ (6), it is **R _{6}**, regardless of its first pitch.

A row form that exactly inverts the interval structure of the prime form (for example, 3 semitones up becomes 3 semitones down)^{[3]} is in inversion form. Inversion forms are labeled according to the first pitch class of the row form. An inversion-form row that begins on E♭ (3) is **I _{3}**.

Note that this label is not always the same as the inversion operation that produces it. If you begin with **P _{0}**, the inversion operation and the resulting row form will have the same subscript. Otherwise, they will be different. Take care not to confuse them.

The relationship of the retrograde inversion (**RI**) to the inversion (**I**) is the same as that between retrograde (**R**) and prime (**P**). Retrograde inversion forms reverse the pitch classes of inversion forms and are named for the last pitch class in the row form.

As one final piece of technical, terminological preamble, we introduce the matrix (plural: matrices). This is a neat, compact way of setting out all of the 48 rows in a row class on one 12-by-12 grid. By convention:

- A prime form of the row (
**P**) always appears along the top row_{n}*left to right*. For this explanation, let’s assume we are beginning with**P**._{0} - Because
**R**is exactly the same as P_{0}_{0}in reverse, you already have R_{0}also on that top row, by reading from*right to left*. **I**begins on the same pitch as P_{0}_{0}, so we set that out in the other direction: down along the first column,*top to bottom*.**RI**is to I_{0}_{0}as R_{0}is to P_{0}, so again, we read RI forms along the same axis as I, in the opposite direction:*bottom to top*.

*Motet* from . We’ll take another look at matrices in the Naming Conventions chapter. is the row matrix for the Lutyens

I_{0} |
I_{11} |
I_{3} |
I_{7} |
I_{8} |
I_{4} |
I_{2} |
I_{6} |
I_{5} |
I_{1} |
I_{9} |
I_{10} |
||

P_{0} |
0 | 11 | 3 | 7 | 8 | 4 | 2 | 6 | 5 | 1 | 9 | 10 | R_{10} |

P_{1} |
1 | 0 | 4 | 8 | 9 | 5 | 3 | 7 | 6 | 2 | 10 | 11 | R_{11} |

P_{9} |
9 | 8 | 0 | 4 | 5 | 1 | 11 | 3 | 2 | 10 | 6 | 7 | R_{7} |

P_{5} |
5 | 4 | 8 | 0 | 1 | 9 | 7 | 11 | 10 | 6 | 2 | 3 | R_{3} |

P_{4} |
4 | 3 | 7 | 11 | 0 | 8 | 6 | 10 | 9 | 5 | 1 | 2 | R_{2} |

P_{8} |
8 | 7 | 11 | 3 | 4 | 0 | 10 | 2 | 1 | 9 | 5 | 6 | R_{6} |

P_{10} |
10 | 9 | 1 | 5 | 6 | 2 | 0 | 4 | 3 | 11 | 7 | 8 | R_{8} |

P_{6} |
6 | 5 | 9 | 1 | 2 | 10 | 8 | 0 | 11 | 7 | 3 | 4 | R_{4} |

P_{7} |
7 | 6 | 10 | 2 | 3 | 11 | 9 | 1 | 0 | 8 | 4 | 5 | R_{5} |

P_{11} |
11 | 10 | 2 | 6 | 7 | 3 | 1 | 5 | 4 | 0 | 8 | 9 | R_{9} |

P_{3} |
3 | 2 | 6 | 10 | 11 | 7 | 5 | 9 | 8 | 4 | 0 | 1 | R_{1} |

P_{2} |
2 | 1 | 5 | 9 | 10 | 6 | 4 | 8 | 7 | 3 | 11 | 0 | R_{0} |

RI_{2} |
RI_{1} |
RI_{5} |
RI_{9} |
RI_{10} |
RI_{6} |
RI_{4} |
RI_{8} |
RI_{7} |
RI_{3} |
RI_{11} |
RI_{0} |

- Pitch classes are played
**in the order specified**by the row. - Once a pitch class has been played,
**it isn’t repeated**until the next row.

Those are the basic “rules” of which all twelve-tone composers are at least aware, but as we said at the outset, composers vary widely in what they actually do with this technique in practice. To that effect, let’s take some “exceptional” examples right from the beginning.

Luigi Dallapiccola’s *Piccola Musica Notturna* (literally “little night music,” 1954) certainly features in the canon of well-known twelve-tone works, but note how right from the beginning, and throughout, there is a free and easy attitude to repeating pitches and even motivic figures. There is a row, but it unfolds gradually, undogmatically. This is key to Dallapiccola’s style, to the luxuriant atmosphere of this piece, and to much “serial” music in which some form of deviation from strict practice is extremely common. All told, to my ears at least, this piece has as much to do with the world of Claude Debussy (as a “night time” complement to Debussy’s *Prélude à l’après-midi d’un faune*, perhaps?) as to the “strict” serialists.

Likewise, we also get music that’s clearly not twelve-tone serial, but that uses strict serial techniques. Listen to *The Lamb* by John Tavener. The pitch classes in the soprano melody at the beginning (“Little lamb, who made thee?”) are G, B, A, F♯, G. The soprano then repeats that melody for the second line (“Dost thou know who made thee?”), while the altos sing the inversion:

- Soprano (prime): G–B–A–F♯–G
- Alto (inversion): G–E♭–F–A♭–G

Then the soprano sings a longer tune (“Gave thee life and bid thee feed / By the stream and o’er the mead”) with the second half as a a strict retrograde of the first:

- “Gave thee life and bid thee feed” (prime): G–B–A–F♯–E♭–F–A♭
- “By the stream and o’er the mead” (retrograde): A♭–F–E♭–F♯–A–B–G

Again, we get this melody a second time (“Gave thee clothing of delight / Softest clothing wooly bright”) with the altos now singing the inversion:

- Soprano (prime then retrograde): G–B–A–F♯–E♭–F–A♭ | A♭–F–E♭–F♯–A–B–G
- Alto (inversion then retrograde inversion): G–E♭–F–A♭–B–A–F♯ | F♯–A–B–A♭–F–E♭–G

Clearly this is a highly serial way of writing. Then again, this passage has a very clear modal final on G, and the two parts (soprano and alto separately) can be considered in terms of standard chord modes. (See Diatonic Modes to review.)

So, there is a wide range of approaches to making music with the basics of the twelve-tone technique, and it’s not always clear what counts. Those differences notwithstanding, at a minimum, twelve-tone rows are used somehow in the construction of:

**Themes.**That said, note that serial themes are not always (or even often) exactly twelve notes in length, or complete row statements.**Motives.**This is pertinent, for instance, in cases where the row form includes several iterations of a smaller cell (see the Row Properties chapter for more information).**Chords.**As we’re generally not working within tonal constraints (and even when we occasionally are), there are many different chordal configurations possible. The row’s properties give rise to the particular construction of chords used.

- Chose any row from the Twelve-Tone Anthology that interests you and write out:
- The row matrix with all 48 row forms (i.e., with numbers on the grid as shown above)
**P**_{0, }**R**_{0, }**I**_{0, }**RI**in musical notation_{0 }

- The "First Viennese School" (by this logic) centers on Haydn, Mozart, and Beethoven. ↵
- This number comes from the mathematical expression 12! (read: "12 factorial"), which means 12 × 11 × 10 … × 2 × 1. ↵
- Or, equivalently, or 9 semitones up, [pb_glossary id="4000"]modulo 12[/pb_glossary]. ↵
- This is the row form given by Lutyens in the BL Add. Ms. 64789. manuscript (f.48b). Credit and thanks to Laurel Parsons for providing this. ↵

definition

A group of pitches that are octave equivalent and enharmonically equivalent.

The ordered elements in a serial composition, also referred to as a series. These elements are often pitches, but could be other things such as durations or dynamics.

A strategy of putting elements of music (pitch, duration, dynamics, etc.) in a particular order.

Refers to the ordered elements in a serial composition. These elements are often pitches, but could be other things such as durations or dynamics.

The number of scale steps between notes of a collection or scale.

A collection of all forms of a given row. Most row classes contain 48 versions of a row, but some contain fewer due to duplications of row forms. For example, a prime version of a row may be equivalent to a retrograde version of the row.

The distance between pitch classes from lowest to highest. In other words, pitch class intervals are measured on the clock face, always going clockwise.

A 12-by-12 grid that sets out all 48 forms of a row class.