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

^{[4]}

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#
Basics

## Rows

### ‘Operations’

### The ‘Prime’ form

### Retrograde form

### Inversion form

### Retrograde inversion

## Examples

### A Fake Example

### A Real Example

Now we’ve got the basic idea, let’s see how this works in a real musical context, taking as our example the row-form in Elisabeth Lutyens’ *Motet (Excerota 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 12-tone technique stipulate that:

X. Twelve-Tone Music

Mark Gotham and Brian Moseley

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, 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 pitch classes*.

This style of composition is commonly associated with a group of composers (sometimes dubbed the ‘Second Viennese School’) whose figurehead was Arnold Schoenberg and whose members included Anton Webern and Alban Berg.^{[1]} But twelve-tone compositional techniques and the ideas associated with it 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 import but subsidiary consideration.

Twelve-tone music is based on *series* (sometimes called a *row*) 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 option to chose 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 favoured and several composers favour 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 semi-tones. This will be familiar enough from other, tonal contexts, but note that we’re obviously 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 semi-tones (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.

12-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 12 pitch classes. As such, a row produces a collection of 48 forms in what is called a **‘row class’**.

The prime form of the row 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 seem equally salient at the beginning, 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 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 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, are 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-sharp (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-flat (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.

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

Serial real by openmusictheory

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,

**P**always appears along the top row_{0}*left to right*- Because
**R**is an 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 an ‘exceptional’ example right from the beginning. Dallapiccola’s *Piccola Musica Notturna*, (literally ‘little night music’, 1954) certainly features in the canon of well-known 12-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 Debussy (as a ‘night time’ complement to Debussy *Prélude à l’après-midi d’un faune* perhaps?) as to the ‘strict’ serialists.

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

So there is range of approaches to making music with the basics of the 12-tone technique. At a minimum, a twelve-tone row will be used somehow in the construction of:

**Themes.**That said, note that serial themes are not always (or even often) exactly 12 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.

- The 'First Viennese School' (by this logic) was the Haydn-Mozart-Beethoven triumvirate. ↵
- 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, modulo 12. ↵
- This is the row form given by Luyens in the BL Add. Ms. 64789. manuscript (f.48b). Credit and thanks to Laurel Parsons for providing this. ↵