Twelve-Tone (Serial) Music
This page was last edited on 15/08/05
| What is a Tone Row? | Transposing the Prime Row | The Retrograde | The Inversion |
| The Retrograde Inversion | The Matrix | How to fill in the Matrix (not on this page) | GCSE Composition |
What is 12-tone music?
Look at this piano keyboard.
It has a repeating pattern of black and white keys. In each repeating unit or octave , there are 5 black keys and 7 white keys. So there are 12 pitches in an octave. One unit gives us what is called a chromatic scale :- c c# d d# e f f# g g# a a# b. Two adjacent pitches (e.g. f# and g) are a "semitone" apart. A tone is the distance between two pitches separated by one pitch (e.g. d and e, which is separated by d#). [There are other ways of writing the same chromatic scale, but natural signs are not readily available in most fonts.]
We are now ready to look at the twelve-tone system. In this system all 12 pitches are equally important. Music written using the twelve-note system does not use an ordinary chromatic scale in the way that music in, say, a major key depends upon an ascending and descending series of notes. Instead, the 12 pitches are put into a particular order in the way which will be described below.
The most fundamental 'rule' of 12-tone music is this:-
Once a note has been used, it cannot be used again
until the other 11 pitch
'names' have occurred.
This rule applies no matter in which octave a pitch happens to be used.
What is a tone row?
In order to write a piece of twelve-tone music, the composer decides upon the order in which the pitches are to occur. This order, called the row , is extremely important. It will be repeated, not only in exactly its original form, but it will be subject to a variety of transformations. The original ordering of the twelve tones is referred to as the Prime (P) form of the row. This can be employed in a composition simply by repeating it over and over again. [See MUSICAL EXAMPLE ONE] Click Bookmark one to return here.
There are various ways of arriving at a tone row and some rows are thought to have less potential than others. These issues will be discussed later on. For now we simply want to place the 12 pitches in some sort of order. We can do this on music paper, but using numbers can be clearer and probably less confusing.
First, we need to represent each pitch by a number : "c" is zero "c#" is 1 and so on until "b" is 11. Like this:-
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FIG. 1 |
c | c# | d | d# | e | f | f# | g | g# | a | a# | b |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
Let's use the row g d# b a# a c e c# f f# d. This would be written as 7 3 11 10 8 9 0 4 1 5 6 2. Our prime row in table form is:-
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FIG. 2 |
g | d# | b | a# | g# | a | c | e | c# | f | f# | d |
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7 | 3 | 11 | 10 | 8 | 9 | 0 | 4 | 1 | 5 | 6 | 2 |
We can use this to compose a piece of music. Look again at MUSICAL EXAMPLE ONE. Click Bookmark two to return here (There is a 'plan' although it takes a while to find it!) Notice that it is possible to use chords . However, the prime row is not usually used by itself to create a piece of music. The row is likely to appear in various guises.
You may be interested to know that there are a great many possible tone rows. There are 12 pitches and 12 'positions' to fill up. If I had all the 12 notes on slips of paper in a bag and drew one out at a time there would be 12 choices for the 1st note. With only 11 pitches left, there would be only 11 possibilities for the 2nd note, 10 for the 3rd and so on. The number of tone rows in the world is 12! (or factorial 12) which is 12x11x10x9x8x7x6x5x4x3x2x1. That is 479001600.
What else can you do to a tone row?
Tone Rows can be changed by transposition. All pitches of the row can be increased or decreased by any number of semitones, as long as every pitch is changed by the same amount.
Now, the row above, might logically be called P7, meaning that it begins with 'g', the pitch numbered 7. However, the relationships between various transpositions of the row are the same whatever version is deemed to be the original one. So, it is actually a lot easier to call the row in FIG. 2, P0 (0 = zero). [It is important to understand this is not PO where O = the letter 'o'!]
FIG. 3 (below) is a table of all the transpositions of the row in FIG. 2. In row P1, the notes of P0 have been raised by one semitone and those in P2 have been raised another semitone (that means, P2 is a tone above P0). We therefore have our row starting on each of the 12 possible pitches.
| P0 | 7 | 3 | 11 | 10 | 8 | 9 | 0 | 4 | 1 | 5 | 6 | 2 |
| P1 | 8 | 4 | 0 | 11 | 9 | 10 | 1 | 5 | 2 | 6 | 7 | 3 |
| P2 | 9 | 5 | 1 | 0 | 10 | 11 | 2 | 6 | 3 | 7 | 8 | 4 |
| P3 | 10 | 6 | 2 | 1 | 11 | 0 | 3 | 7 | 4 | 8 | 9 | 5 |
| P4 | 11 | 7 | 3 | 2 | 0 | 1 | 4 | 8 | 5 | 9 | 10 | 6 |
| P5 | 0 | 8 | 4 | 3 | 1 | 2 | 5 | 9 | 6 | 10 | 11 | 7 |
| P6 | 1 | 9 | 5 | 4 | 2 | 3 | 6 | 10 | 7 | 11 | 0 | 8 |
| P7 | 2 | 10 | 6 | 5 | 3 | 4 | 7 | 11 | 8 | 0 | 1 | 9 |
| P8 | 3 | 11 | 7 | 6 | 4 | 5 | 8 | 0 | 9 | 1 | 2 | 10 |
| P9 | 4 | 0 | 8 | 7 | 5 | 6 | 9 | 1 | 10 | 2 | 3 | 11 |
| P10 | 5 | 1 | 9 | 8 | 6 | 7 | 10 | 2 | 11 | 3 | 4 | 0 |
| P11 | 6 | 2 | 10 | 9 | 7 | 8 | 11 | 3 | 0 | 4 | 5 | 1 |
FIG. 4 (below) shows the numbers from FIG. 3 converted into notes. Of course, you can just use notes if you like, but you will see why numbers can be helpful shortly.
| P0 | G | D# | B | A# | G# | A | C | E | C# | F | F# | D |
| P1 | G# | E | C | B | A | A# | C# | F | D | F# | G | D# |
| P2 | A | F | C# | C | A# | B | D | F# | D# | G | G# | E |
| P3 | A# | F# | D | C# | B | C | D# | G | E | G# | A | F |
| P4 | B | G | D# | D | C | C# | E | G# | F | A | A# | F# |
| P5 | C | G# | E | D# | C# | D | F | A | F# | A# | B | G |
| P6 | C# | A | F | E | D | D# | F# | A# | G | B | C | G# |
| P7 | D | A# | F# | F | D# | E | G | B | G# | C | C# | A |
| P8 | D# | B | G | F# | E | F | G# | C | A | C# | D | A# |
| P9 | E | C | G# | G | F | F# | A | C# | A# | D | D# | B |
| P10 | F | C# | A | G# | F# | G | A# | D | B | D# | E | C |
| P11 | F# | D | A# | A | G | G# | B | D# | C | E | F | C# |
The Retrograde
The prime row can also be altered by reversing its order. This is done simply by reading the notes of the row from last to first. This transformed version is called the Retrograde (R). The retrograde of P0 is called R0 and it is 2 6 5 1 4 0 9 8 10 11 3 7.
Transposition and Retrograde transformations can be combined. See if you can work out the Retrograde of P8 which will be R8. Write it down as numbers and then notes by reading the tables above from right to left.
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R8 notes |
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The Inversion
The most difficult transformation to understand is the inversion ( I ). Described simply, it is the upside-down version of the prime row. If all notes in a prime row were taken from the same octave, a move from a lower note to a higher note would become, in the inverted form, a move from a higher note to a lower one. This is why numbers are easier to use. The process of converting our P0 into I0 using notes works like this.
The Retrograde Inversion
All the inversions can be transposed or, if you prefer, all the transposed 'P' rows can be inverted. The inversions can also be used backwards, in Retrograde to give Retrograde Inversions. FIGS. 5 and 6 show these rows.
| I0 | G | B | D# | E | F# | F | D | A# | C# | A | G# | C |
| I1 | G# | C | E | F | G | F# | D# | B | D | A# | A | C# |
| I2 | A | C# | F | F# | G# | G | E | C | D# | B | A# | D |
| I3 | A# | D | F# | G | A | G# | F | C# | E | C | B | D# |
| I4 | B | D# | G | G# | A# | A | F# | D | F | C# | C | E |
| I5 | C | E | G# | A | B | A# | G | D# | F# | D | C# | F |
| I6 | C# | F | A | A# | C | B | G# | E | G | D# | D | F# |
| I7 | D | F# | A# | B | C# | C | A | F | G# | E | D# | G |
| I8 | D# | G | B | C | D | C# | A# | F# | A | F | E | G# |
| I9 | E | G# | C | C# | D# | D | B | G | A# | F# | F | A |
| I10 | F | A | C# | D | E | D# | C | G# | B | G | F# | A# |
| I11 | F# | A# | D | D# | F | E | C# | A | C | G# | G | B |
| RI0 | C | G# | A | C# | A# | D | F | F# | E | D# | B | G |
| RI1 | C# | A | A# | D | B | D# | F# | G | F | E | C | G# |
| RI2 | D | A# | B | D# | C | E | G | G# | F# | F | C# | A |
| RI3 | D# | B | C | E | C# | F | G# | A | G | F# | D | A# |
| RI4 | E | C | C# | F | D | F# | A | A# | G# | G | D# | B |
| RI5 | F | C# | D | F# | D# | G | A# | B | A | G# | E | C |
| RI6 | F# | D | D# | G | E | G# | B | C | A# | A | F | C# |
| RI7 | G | D# | E | G# | F | A | C | C# | B | A# | F# | D |
| RI8 | G# | E | F | A | F# | A# | C# | D | C | B | G | D# |
| RI9 | A | F | F# | A# | G | B | D | D# | C# | C | G# | E |
| RI10 | A# | F# | G | B | G# | C | D# | E | D | C# | A | F |
| RI11 | B | G | G# | C | A | C# | E | F | D# | D | A# | F# |
The Matrix
(Find out how to fill in a matrix here)
All of the above transformations can be shown in a single chart called the matrix . Notice that the rows do not occur in the order 0 to 11 as they have above. (FIG. 7)
| I 0 | I 8 | I 4 | I 3 | I 1 | I 2 | I 5 | I 9 | I 6 | I 10 | I 11 | I 7 |
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| P 0> | G | D# | B | A# | G# | A | C | E | C# | F | F# | D |
< R 0 |
| P 4> | B | G | D# | D | C | C# | E | G# | F | A | A# | F# |
< R 4 |
| P 8> | D# | B | G | F# | E | F | G# | C | A | C# | D | A# |
< R 8 |
| P 9> | E | C | G# | G | F | F# | A | C# | A# | D | D# | B |
< R 9 |
| P 11> | F# | D | A# | A | G | G# | B | D# | C | E | F | C# |
< R 11 |
| P 10> | F | C# | A | G# | F# | G | A# | D | B | D# | E | C |
< R 10 |
| P 7> | D | A# | F# | F | D# | E | G | B | G# | C | C# | A |
< R 7 |
| P 3> | A# | F# | D | C# | B | C | D# | G | E | G# | A | F |
< R 3 |
| P 6> | C# | A | F | E | D | D# | F# | A# | G | B | C | G# |
< R 6 |
| P 2> | A | F | C# | C | A# | B | D | F# | D# | G | G# | E |
< R 2 |
| P 1> | G# | E | C | B | A | A# | C# | F | D | F# | G | D# |
< R 1 |
| P 5> | C | G# | E | D# | C# | D | F | A | F# | A# | B | G |
< R 5 |
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| RI 0 | RI 8 | RI 4 | RI 3 | RI 1 | RI 2 | RI 5 | RI 9 | RI 6 | RI 10 | RI 11 | RI 7 |
In all, the prime row can be altered in 47 ways, so that a composition based on a 12-tone row may actually use up to 48 different 'rows'. However, certain rows (e.g. palindromic rows) produce matrices in which several of the transformed rows are identical.
The row (7 9 11 1 3 5 6 4 2 0 10 8), for example, can be transformed into only 23 unique rows. In this row, "5 6" is the axis of symmetry in the row. Notice that each jump on the left half of the row is identical to the corresponding jump on the right. This feature of palindromic rows was employed by Alban Berg in extremely clever and complex ways.
Every note in a piece of twelve-tone music must be derived somehow from the prime row. However, in his 12-tone compositions, Berg often mixed twelve-tone music with non-twelve-tone music. He would also use more than one prime row within a single composition. He also used other procedures which need not concern us at GCSE level.
In composing the opera "Lulu", Berg assigned a separate row to each of the principal characters of the opera. Thus purely abstract note progressions have dramatic meaning. This is of course very similar to Wagner's leitmotiv, the musical phrase that represented a character, object, or state of being. Berg used true leitmotiv in both of his operas. The row transformations found in Berg's twelve-tone music themselves have dramatic significance.
12 tone composition for GCSE
Candidates should compose a piece based on a 12 note row. The chosen row should be used to generate melodic and harmonic material according to a set of predetermined rules.
Examples of briefs
Compose the theme music for a TV or Video Science programme about Mirrors.
Compose the music for a Ghost Story or film, perhaps using storyboards to help you plan the sections.
Compose a Study for an unusual ensemble of instruments to help the players test and improve their rhythmic skills in performance.
Compose a piece for Electronic sounds (from keyboards perhaps for a small group of players) to suggest the future or to go with a Science Fiction play or film.
Compose a set of Variations.
Compose a piece which uses various kinds of canon.
Compose a piece for String Quartet which exploits all the 'special' ways of playing string instruments for one or all of the players.