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

^{[4]}

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## 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]}

Serial real by openmusictheory
Here’s a recording of the full piece:https://open.spotify.com/embed/track/1KHlQk1IYYZZocedHyughI
# 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 typically involves using all twelve 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 , 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 , though these terms are not equivalent:

- “Serialism” is a broad designator referring to the ordering of things, whether they are pitches, durations, dynamics, and so on. This extends beyond music as, for instance, in a television series (many episodes, linked together).
- “Twelve-tone composition” refers more specifically to music based on orderings of the twelve .

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 (sometimes called a ) that contains all twelve pitch classes in a particular order. There is no one series used for all twelve-tone music. In fact, there are 479,001,600 distinct options to choose from!^{[2]} Some of these row forms are popular and we end up with several pieces based on the same row. More precisely, it’s fairer to say that some properties of rows are favored and several composers favor rows with those properties, without necessarily going for exactly the same one. In any case, many other row forms 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” (taking that term from the mathematical, rather than medical, sense!).

**Transposition (T).**Take all the pitches and move them up or down by a specified number of semitones. This will be familiar enough from other, tonal contexts, but note that 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, 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” (a.k.a. “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 .

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.

Serial fake by openmusictheory

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 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 course, 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 (p.c. 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.

Serial real by openmusictheory

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

**P**always appears along the top row_{0}.*left to right*- 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, i.e..*bottom to top*

All the transpositions of these row forms appear in the same directions, so the broad structure of a matrix is like this:

↓ Inversion forms read top to bottom |
||

→ Prime forms read left to right |
The rows go in here | ← Retrograde forms read right to left |

↑ Retrograde-inversion forms read bottom to top |

And here’s a real matrix for the Lutyens example discussed above:

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 | RI_{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} |

We’ll take another look at matrices in the Naming Conventions chapter.

- 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 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.

https://open.spotify.com/embed/track/6TuIPzAl86OSluM7LH1B0v

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:

https://open.spotify.com/embed/track/6mJlwt6XPR6p3CD1JjaJq8

The pitches in the melody at the beginning (“Little lamb, who made thee?”) are:

- Soprano: 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 and coextensive with their rows.**Motives.**This is pertinent, for instance, in cases where the row form includes several iterations of a smaller cell (about which more follows in the Row Properties chapter).**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 x 11 x 10 ... x 2 x 1. ↵
- Or, equivalently, or 9 semitones up, [pb_glossary id="1075"]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. ↵

All pitches that are equivalent enharmonically and which exhibit octave equivalence

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

A strategy of putting elements in a particular order. The elements can be any dimension of music: pitch, duration, dynamics, etc....

Non-musical elements can also be serialized, such as the episodes in a television series.

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

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 from one pitch-class to another. Since pitch-classes are collections of all pitches related enharmonically and by octave equivalence, we need to define how to calculate this distance:

if the ordering of pitch classes (pcs) matters, calculate the distance from pc 1 to pc2 in semitones as if you're going up to pc2, regardless of whether the actual pitch of pc 2 is higher or lower then pc1. Calculate the size within an octave.

If the order of the pcs doesn't matter, then just calculate the closest distance between the two pcs in semitones.

In twelve-tone music, the matrix is a 12-by-12 grid that sets out all 48 forms of a row class.