IX. Twelve-Tone Music

Basics of Twelve-Tone Theory

Mark Gotham and Brian Moseley

Key Takeaways

Twelve-tone composition typically involves using all twelve roughly equally. That means it isn’t (usually) appropriate to look for a key, mode, tonic pitch or other tonal elements.

Composers often use a fixed ordering of the 12 pitch classes called a , but also adapt it in various ways, notably through:

  • Transposition (‘T’),
  • Inversion (‘I’),
  • Retrograde (‘R’), and
  • Retrograde-Inversion (‘RI’).

In practice, there is a great variety of how composers approach the task of “composing with twelve tones.”

A Real Example

Now 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’ Motet (Excerota Tractati Logico-Philosophici), Op.27.[4]

Serial real by openmusictheory

Enter the Matrix

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,

  • P0 always appears along the top row left to right
  • Because R0 is exactly the same as P0 in reverse, you already have R0 also on that top row, by reading from right to left.
  • I0 begins on the same pitch as P0, so we set that out in the other direction: down along the first column, top to bottom.
  • RI0 is to I0 as R0 is to P0, 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:

I0 I11 I3 I7 I8 I4 I2 I6 I5 I1 I9 I10
P0 0 11 3 7 8 4 2 6 5 1 9 10 R10
P1 1 0 4 8 9 5 3 7 6 2 10 11 R11
P9 9 8 0 4 5 1 11 3 2 10 6 7 R7
P5 5 4 8 0 1 9 7 11 10 6 2 3 R3
P4 4 3 7 11 0 8 6 10 9 5 1 2 R2
P8 8 7 11 3 4 0 10 2 1 9 5 6 R6
P10 10 9 1 5 6 2 0 4 3 11 7 8 R8
P6 6 5 9 1 2 10 8 0 11 7 3 4 R4
P7 7 6 10 2 3 11 9 1 0 8 4 5 R5
P11 11 10 2 6 7 3 1 5 4 0 8 9 R9
P3 3 2 6 10 11 7 5 9 8 4 0 1 R1
P2 2 1 5 9 10 6 4 8 7 3 11 0 RI0
RI2 RI1 RI5 RI9 RI10 RI6 RI4 RI8 RI7 RI3 RI11 RI0

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

From Theory to Practice

In general then, the basics of 12-tone technique stipulate that:
  1. Pitch classes are played in the order specified by the row
  2. 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.

12-tone serial, but not so strict

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.


Serial but not 12-tone

Likewise, we also get music that’s clearly not 12-tone serial, but which uses strict serial techniques. Listen to The Lamb by John Tavener:


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 & bid thee feed / By the stream & o’er the mead”) with the second half as a a strict retrograde of the first:

  • “Gave thee life & bid thee feed” (prime): G, B, A, F♯, E♭, F, A♭,
  • “By the stream & 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).

What do we know?

So there is a wide range of approaches to making music with the basics of the 12-tone technique and its 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 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.
  1. Coming soon!

  1. The "First Viennese School" (by this logic) centres on Haydn, Mozart, and Beethoven.
  2. This number comes from the mathematical expression '12!' (read: '12 factorial') which means 12 x 11 x 10 ... x 2 x 1.
  3. Or, equivalently, or 9 semitones up, modulo 12.
  4. 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.


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