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12 Tone Music - Page 1
The Twelve-Tone Method of Musical Construction
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]
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.
12 Tone Music - Page 2
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:FIG. 1
c
0
c#
1
d
2
d#
3
e
4
f
5
f#
6
g
7
g#
8
a
9
a#
10
b
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:FIG. 2
g
7
d#
3
b
11
a#
10
g#
8
a
9
c
0
e
4
c#
1
f
5
f#
6
d
2
We can use this to compose a piece of music. Look at MUSICAL EXAMPLE ONE. (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 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
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
7
8
9
10
11
0
1
2
3
4
5
6
3
4
5
6
7
8
9
10
11
0
1
2
11
0
1
2
3
4
5
6
7
8
9
10
10
11
0
1
2
3
4
5
6
7
8
9
8
9
10
11
0
1
2
3
4
5
6
7
9
10
11
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
9
10
11
4
5
6
7
8
9
10
11
0
1
2
3
1
2
3
4
5
6
7
8
9
10
11
0
5
6
7
8
9
10
11
0
1
2
3
4
6
7
8
9
10
11
0
1
2
3
4
5
2
3
4
5
6
7
8
9
10
11
0
1
12 Tone Music - Page 3
FIG. 4 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
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
G
G#
A
A#
B
C
C#
D
D#
E
F
F#
D#
E
F
F#
G
G#
A
A#
B
C
C#
D
B
C
C#
D
D#
E
F
F#
G
G#
A
A#
A#
B
C
C#
D
D#
E
F
F#
G
G#
A
G#
A
A#
B
C
C#
D
D#
E
F
F#
G
A
A#
B
C
C#
D
D#
E
F
F#
G
G#
C
C#
D
D#
E
F
F#
G
G#
A
A#
B
E
F
F#
G
G#
A
A#
B
C
C#
D
D#
C#
D
D#
E
F
F#
G
G#
A
A#
B
C
F
F#
G
G#
A
A#
B
C
C#
D
D#
E
F#
G
G#
A
A#
B
C
C#
D
D#
E
F
D
D#
E
F
F#
G
G#
A
A#
B
C
C#
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.
R8 numbers
R8 notes
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.
12 Tone Music - Page 4
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
I1
I2
I3
I4
I5
I6
I7
I8
I9
I10
I11
G
G#
A
A#
B
C
C#
D
D#
E
F
F#
B
C
C#
D
D#
E
F
F#
G
G#
A
A#
D#
E
F
F#
G
G#
A
A#
B
C
C#
D
E
F
F#
G
G#
A
A#
B
C
C#
D
D#
RI0
RI1
RI2
RI3
RI4
RI5
RI6
RI7
RI8
RI9
RI10
RI11
C
C#
D
D#
E
F
F#
G
G#
A
A#
B
G#
A
A#
B
C
C#
D
D#
E
F
F#
G
A
A#
B
C
C#
D
D#
E
F
F#
G
G#
C#
D
D#
E
F
F#
G
G#
A
A#
B
C
F#
G
G#
A
A#
B
C
C#
D
D#
E
F
A#
B
C
C#
D
D#
E
F
F#
G
G#
A
F
F#
G
G#
A
A#
B
C
C#
D
D#
E
D
D#
E
F
F#
G
G#
A
A#
B
C
C#
D
D#
E
F
F#
G
G#
A
A#
B
C
C#
F
F#
G
G#
A
A#
B
C
C#
D
D#
E
A#
B
C
C#
D
D#
E
F
F#
G
G#
A
F#
G
G#
A
A#
B
C
C#
D
D#
E
F
C#
D
D#
E
F
F#
G
G#
A
A#
B
C
E
F
F#
G
G#
A
A#
B
C
C#
D
D#
A
A#
B
C
C#
D
D#
E
F
F#
G
G#
D#
E
F
F#
G
G#
A
A#
B
C
C#
D
G#
A
A#
B
C
C#
D
D#
E
F
F#
G
B
C
C#
D
D#
E
F
F#
G
G#
A
A#
C
C#
D
D#
E
F
F#
G
G#
A
A#
B
G
G#
A
A#
B
C
C#
D
D#
E
F
F#
The Matrix
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)
P 0>
P 4>
P 8>
P 9>
P 11>
P 10>
P 7>
P 3>
P 6>
P 2>
P 1>
P 5>
I0
I8
I4
I3
I1
I2
I5
I9
I6
I 10
I 11
I7
G
B
D#
E
F#
F
D
A#
C#
A
G#
C
D#
G
B
C
D
C#
A#
F#
A
F
E
G#
B
D#
G
G#
A#
A
F#
D
F
C#
C
E
A#
D
F#
G
A
G#
F
C#
E
C
B
D#
G#
C
E
F
G
F#
D#
B
D
A#
A
C#
A
C#
F
F#
G#
G
E
C
D#
B
A#
D
C
E
G#
A
B
A#
G
D#
F#
D
C#
F
E
G#
C
C#
D#
D
B
G
A#
F#
F
A
C#
F
A
A#
C
B
G#
E
G
D#
D
F#
F
A
C#
D
E
D#
C
G#
B
G
F#
A#
F#
A#
D
D#
F
E
C#
A
C
G#
G
B
D
F#
A#
B
C#
C
A
F
G#
E
D#
G
RI 0
RI 8
RI 4
RI 3
RI 1
RI 2
RI 5
RI 9
RI 6
RI 10
RI 11
RI 7
<R0
<R4
<R8
<R9
< R 11
< R 10
<R7
<R3
<R6
<R2
<R1
<R5
12 Tone Music - Page 5
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.
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.
12 Tone Music - Page 6
How to fill in a matrix
The quickest way to write down all guises of a row is to use a matrix. In such a matrix the rows are
read as follows:Prime Rows
(P): 0-11
Inversions (I): 0-11
Retrogrades (R):
Retrograde
Inversions (RI):
The series as it is constructed is read from left to right as are all the
transpositions of it.
These are read downwards.
The 12 prime series in reverse order so you just read from right to left.
Each inverted row is sounded in reverse order so you read the inverted
rows (columns) series upwards.
Stage 1
Write P0 in the leftmost box of the second row and then write in your tone row across the same row
of the matrix, using the numbers 0-11 to represent the chromatic scale c c# d d# e f f# g g# a a# b.
Leave the rightmost box empty for now.
Stage 2
Fill in the second column from the left working downwards. This row is, in fact the Inversion of the
original tone row. All moves up become moves down and vice versa. Calculate the difference
between the 1st two numbers of the top row. We have 3 and 2 so the difference is -1. Thus, we add
+1 to 3 to give our next number in the column.
After this we go from 2 to 4, a difference of +2. We need a difference of -2 to give 2. Now, we
meet 4 and 9, a difference of +5. If we subtract 5 from 2 we get -3. By the rules of “Clock
Arithmetic” (Modulo 12) the number we actually have is 9. Briefly (using a line of numbers rather
than a ‘clock’ you start on 2 and move backwards (minus) 5 places. Note that we have 0-11 not 112.
Move left
<etc. 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 etc.
>
We can now finish this column. Here are stages 1 and 2. Leave the bottom box blank for now.
P0
3
4
2
9
5
1
10
7
8
6
0
11
2
4
9
1
5
8
11
10
0
6
7
12 Tone Music - Page 7
Stage 3
We need to label the left most column. P0 starts with the number 3. P1 is one semitone higher,
which is the row which starts with 4. Label this P1. Then find 5 and label this P3 and so on. Use
clock arithmetic to find where P9 is!
Stage 4
The Inversions are labelled across the top. Simply match up the numbers in the ‘P’ column but
replace the ‘P’ with an ‘I’. Clearly, the first column of notes is I0. Likewise, because the next
column has 2 at the top, this must be I11. You may as well also label the R and RI boxes as the
numbers of the rows and columns stays the same.
P0
P1
P11
P6
P2
P10
P7
P4
P5
P3
P9
P8
I0
3
4
2
9
5
1
10
7
8
6
0
11
RI0
I11
2
I1
4
I6
9
I10
1
I2
5
I5
8
I8
11
I7
10
I9
0
I3
6
I4
7
RI11
RI1
RI6
RI10
RI2
RI5
RI8
RI7
RI9
RI3
RI4
R0
R1
R11
R6
R2
R10
R7
R4
R5
R3
R9
R8
Stage 5
All that remains is to fill in the rows. Start with P1. We have already added 1 to the 3 at the top of
the column. Work along the row adding 1 to all the numbers. Then do the same for row P2. You
can either work from row P1 which you have just done and continue to add 1 or, you can always
work from P0 and add the number implied by the row label. Don’t forget to use clock arithmetic
when you add 1 to 11 to get 12. (Subtract 12 to leave zero.)
P0
P1
P11
P6
P2
P10
P7
P4
P5
P3
P9
P8
I0
3
4
2
9
5
1
10
7
8
6
0
11
RI0
I11
2
3
I1
4
5
I6
9
10
I10
1
2
I2
5
6
I5
8
9
I8
11
0
I7
10
11
I9
0
1
I3
6
7
I4
7
8
4
6
11
3
7
10
1
0
2
8
9
RI11
RI1
RI6
RI10
RI2
RI5
RI8
RI7
RI9
RI3
RI4
I5
I8
I7
I9
I3
The completed matrix is set out on the next page.
I0
I 11
I1
I6
I 10
I2
I4
R0
R1
R11
R6
R2
R10
R7
R4
R5
R3
R9
R8
12 Tone Music - Page 8
P0>
P1>
P 11 >
P6>
P2>
P 10 >
P7>
P4>
P5>
P3>
P9>
P8>
3
4
2
9
5
1
10
7
8
6
0
11
2
3
1
8
4
0
9
6
7
5
11
10
4
5
3
10
6
2
11
8
9
7
1
0
9
10
8
3
11
7
4
1
2
0
6
5
1
2
0
7
3
11
8
5
6
4
10
9
5
6
4
11
7
3
0
9
10
8
2
1
8
9
7
2
10
6
3
0
1
11
5
4
11
0
10
5
1
9
6
3
4
2
8
7
10
11
9
4
0
8
5
2
3
1
7
6
0
1
11
6
2
10
7
4
5
3
9
8
6
7
5
0
8
4
1
10
11
9
3
2
7
8
6
1
9
5
2
11
0
10
4
3
RI 0
RI 11
RI 1
RI 6
RI 10
RI 2
RI 5
RI 8
RI 7
RI 9
RI 3
RI 4
<R0
<R1
< R 11
<R6
<R2
< R 10
<R7
<R4
<R5
<R3
<R9
<R8
You then need to convert the numbers back into notes to give the following.
P 0>
P 1>
P 11>
P 6>
P 2>
P 10>
P 7>
P 4>
P 5>
P 3>
P 9>
P 8>
I0
I 11
I1
I6
I 10
I2
I5
I8
I7
I9
I3
I4
D#
E
D
A
F
C#
A#
G
G#
F#
C
B
D
D#
C#
G#
E
C
A
F#
G
F
B
A#
E
F
D#
A#
F#
D
B
G#
A
G
C#
C
A
A#
G#
D#
B
G
E
C#
D
C
F#
F
C#
D
C
G
D#
B
G#
F
F#
E
A#
A
F
F#
E
B
G
D#
C
A
A#
G#
D
C#
G#
A
G
D
A#
F#
D#
C
C#
B
F
E
B
C
A#
F
C#
A
F#
D#
E
D
G#
G
A#
B
A
E
C
G#
F
D
D#
C#
G
F#
C
C#
B
F#
D
A#
G
E
F
D#
A
G#
F#
G
F
C
G#
E
C#
A#
B
A
D#
D
G
G#
F#
C#
A
F
D
B
C
A#
E
D#
RI 0
RI 11
RI 1
RI 6
RI 10
RI 2
RI 5
RI 8
RI 7
RI 9
RI 3
RI 4
<R0
<R1
< R 11
<R6
<R2
< R 10
<R7
<R4
<R5
<R3
<R9
<R8
The whole matrix was derived from the original tone-row series. But using this matrix for musical
composition is not as easy as you might think.
MUSICAL EXAMPLE TWO - “Lex-x-ede”.
Examine the piece “Lec-x-ede” which uses some of the rows from the matrix above. “Lec-x-ede” is
not a very representative 12-tone piece; in fact it is rather contrived because it was composed in an
afternoon. You need to study a few real examples and work more steadily. Read the notes and see
which ‘hints’ “Lec-x-ede” has ignored.
Also consider the following:1. Does “Lec-x-ede” have any interesting ideas?
12 Tone Music - Page 9
2.
3.
4.
5.
Was P0 a ‘good’ row?
Are the instruments used idiomatically?
Comment on the plan.
How many ideas are there?
Points to think about when composing a 12-tone piece.
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Avoid rows which include recognisable tonal scales or arpeggios.
Avoid using too many melodic intervals of the same or similar size because these may
lead to melodic monotony.
You may wish to avoid pairs of notes a semitone apart if you feel that they will sound like
moving from the leading note to the tonic in a major or minor scale.
Unless you actually want your row to contain several examples of a particular interval
(say, 3rds of various sizes) then aim for balanced number of each kind of interval size.
One way of creating a row is to compose a fragment of melody first and then adapt it.
Details of this methods are elsewhere.
Any pitch of the series may be written in any octave and can be spelled enharmonically
(F sharp = G flat).
The order of the notes must remain the same. If you want a different order, use a
different transformation; that’s why they are there!
Try to avoid using a rhythmic pattern in consecutive or nearby measures. However, this
may happen if imitation is being used.
Quite a few serial pieces call for unusual combinations of instruments in Webern’s
music. The instruments, used at the extremes of their registers, frequently play one at a
time and very little at a time. This is known as pointillism.
Webern went further than Schoenberg by avoiding repetition of pitch colour. Sometimes,
each note of a melody is played by a different instrument.
Serial pieces can have very precise instructions about the way the notes are to be played pizzicato, muted, unmuted, what dynamic [There are no such details in “Lec-x-ede”
because it was merely composed as an exercise.]
Serial music often uses canonic principles.