A brief introduction to the Twelve-tone system.
Tonality, the musical system used in the West since the 16th century, is only one of the many ways that can be used to write music. If we could get our hands on all the music written by Mankind, most of this music won't be tonal. Gregorian chant, Javanese gamelan, Navajo lullabies, Hindustani ragas... or operas like Lulu, are outside of the tonal system.
Some people, naively, think that there is a kind of mathematical or physical reality that impose some specific chords, or relations among them. But there isn't. It's just a convention.
During the 19th century and the first decades of the 20th, composers like Wagner, Debussy, Mahler,... were going more and more outside the limits of tonality. Schönberg was the first to flatly renounce tonality, and compose "atonal music", breaking with the attractions among notes, and harmonic relations. However, Schönberg was not happy because atonalism was more a denial of tonality, that something new.
Twelve-tone was not created to refine atonalism, but rather to use as the basic stone of the new musical building "atonicism", to take away the tonic, and give all the notes the same importance. Because, how do you give more importance in a musical phrase to some notes, over others?
- By repeating the note more
- By playing it longer
- By placing it in a determined place
- By stressing the rhythm in this note...
Twelve-tone techniques try to avoid these situations, and place all the notes in the same plane.
Looking at the piano keyboard, we can see there is an unit that repeats itself, formed by 5 black keys, and 7 white keys. This unit is an octave and is divided in 12 notes, that are the basis of the Western harmonic system. Together, they are the chromatic scale:
C - C sharp - D - D sharp - E - F - F sharp - G - G sharp - A - A sharp - B
0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11
and the distance between two adjacent notes, is a semitone.
Within the Twelve-tone system, all the 12 notes must be played (in any octave) before you can start again the cycle. When the order in which the notes are played is decided, we get a Twelve-tone row, that gets indentified by its numerical sequence. In this way, the following Twelve-tone row:
D - E - F - F sharp - C - C sharp - B - A - D sharp - A sharp - G - G sharp
is also: 2 - 4 - 5 - 6 - 0 - 1 - 11 - 9 - 3 - 10 - 7 - 8
and it will be for us a prime series, P(2), the 2 is because starts in D, and could be the basis to write one piece of music.
We can use three basic techniques to work with series:
a) Transposition, each note of the prime series moves up or down a fixed number of semitones. If we move down P(2) two semitones, we get 0 - 2 - 3 - 4 - 10 - 11 - 9 - 7 - 1 - 8 - 5 - 6, i.e. a P(0) Twelve-tone series.
b) Retrograde, each note in the prime series invert the original ordering. In our example it will be :
R(2) ==> 8 - 7 - 10 - 3 - 9 - 11 - 1 - 0 - 6 - 5 - 4 - 2
c) Inversion, each notes is replaced by her mirror note in the chromatic scale, taking as center the first note in the prime series. In our example;
I(2) ==> 2 - 0 - 11 - 10 - 4 - 3 - 5 - 7 - 1 - 6 - 9 - 8
Of course all three transformations can be combined.
Also, the composer can choose to present the Twelve-tone row either in the melody, or in the harmony. That means that our P2) can be just played note after note or, maybe, we can start with D, the use a chord with E, F, F sharp and C, continue in the melody with B, another chord using A, D sharp, A sharp and G,...
All this in the octave of your choosing, with the instruments you want, any rhythm .....
It's a system less predictable than tonality, and you can use it to write marvels like Alban Berg's violin concert, Dem Andenken eines Engels.
In the Opera world, there are several very good pieces. Just to name a few:
Lulu by Alban Berg
Moses und Aron by Schönberg
Karl V by Ernst Krenek