X. Twelve-Tone Music


Mark Gotham and Brian Moseley


A Fake Example

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

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]

Enter the Matrix

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,

  • P0 always appears along the top row left to right
  • Because R0 is an 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 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.


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.

  1. The 'First Viennese School' (by this logic) was the Haydn-Mozart-Beethoven triumvirate.
  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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Open Music Theory by Mark Gotham and Brian Moseley is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

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