IX. Twelve-Tone Music

Mark Gotham

Key Takeaways

When approaching twelve-tone music, it’s easy to get bogged down simply identifying row forms and lose sight of the bigger picture. A list of row forms used in a twelve-tone work is similar to a list of keys in a tonal work—useful, but not enough on its own to be called an analysis. This chapter takes on two iconic works of early twelve-tone music and attempts to connect the rows to wider issues about the work to consider.

Webern: Symphonie Op. 21 (1925)

Look no further than the title of this work for your first can of worms! Why would Webern choose to call this a symphony? What constitutes a symphony; is an atonal symphony an oxymoron? Apart from anything else, doesn’t it usually involve requirements with respect to the key relations? Scholars have a range of reactions to this question:

  • “In choosing the most resonant of classical titles Webern stressed the extent to which it could still be relevant to a work in which only certain structural principles remain valid.” (Whittal 1977, 163)
  • “There is little or nothing in its formal procedures to compare with those of the traditional symphony.” (Taruskin 2010, 728)

Keep these questions in mind as we consider the nuts and bolts of the work.

Row form

Symphony, Op.21 by FourScoreAndMore

Example 1. The row of Webern’s Symphonie (Op. 21) along with the trichordal and hexachordal divisions.

Webern frequently chooses what you might think of as “neat” , and this work is no exception (see Example 1). The row breaks up neatly into two equivalent that are instances not simply of the same but of set 6-1 specifically: half a chromatic scale. In short, each fills the total chromatic collection of half the twelve-tone space.

Further, those hexachords are each set out with one instance of [0,1,3] and one of [0,1,4]. Altogether, the four trichord cells map out as [0,1,3], [0,1,4], [0,1,4], [0,1,3]. These two trichords are further linked by their shared melodic shape: each involves a third (major or minor) and a semitone.

Here is the row matrix, with the symmetry of P0 and R6 highlighted by showing the first six notes of each in bold. [1]

I0 I9 I10 I11 I7 I8 I2 I1 I5 I4 I3 I6
P0 A F♯ G G♯ E F B A♯ D C♯ C D♯ R0
P3 C A A♯ B G G♯ D C♯ F E D♯ F♯ R3
P2 B G♯ A A♯ F♯ G C♯ C E D♯ D F R2
P1 A♯ G G♯ A F F♯ C B D♯ D C♯ E R1
P5 D B C C♯ A A♯ E D♯ G F♯ F G♯ R5
P4 C♯ A♯ B C G♯ A D♯ D F♯ F E G R4
P10 G E F F♯ D D♯ A G♯ C B A♯ C♯ R10
P11 G♯ F F♯ G D♯ E A♯ A C♯ C B D R11
P7 E C♯ D D♯ B C F♯ F A G♯ G A♯ R7
P8 F D D♯ E C C♯ G F♯ A♯ A G♯ B R8
P9 F♯ D♯ E F C♯ D G♯ G B A♯ A C R9
P6 D♯ C C♯ D A♯ B F E G♯ G F♯ A R6
RI0 RI9 RI10 RI11 RI7 RI8 RI2 RI1 RI5 RI4 RI3 RI6

Overall, the row is retrograde equivalent, which is to say, if you play it backwards (R), you have a transposed version of the original (P). When we have equivalences of this kind, there are no longer 48 distinct row forms. Here we have pairs of equivalent rows, and so there are 24 distinct forms (48 / 2).

Webern brings out this symmetrical row by overlapping the ends of row forms with the beginning of the next.

Movement 1

Webern describes this as a “double canon in contrary motion” (where for “contrary motion,” read “inversion”). You could think of the overall form as [:A:][:BA’:] as follows:

  • A: From m. 1 to the double bar (mm. 23-25).
  • B: palindromic: m. 35 as midpoint of m. 25 (Cl, m. 26 VC) — m. 42 (VC, m. 43 Cl).
  • A’: “row recapitulation” from m. 43 (rows only, not motivic rhythm, etc.).

Does that remind you of something symphonic? The repeat markings and the material distribution are loosely suggestive of the and repeats in works, or at least .

So that might be chalked up in favor of the symphonic reading. On the other hand, the extensive symmetry of the row doesn’t end there. Bailey (1991) has described the middle section especially as a symmetrical “tour de force.” Recapitulations and cyclic forms are one thing, but for Western classical music, serious adherence to symmetry is a peculiarly 20th-century concept. Speaking about a chordal version of this issue, the British composer Jonathan Harvey once described the symmetrical strategy of moving the bass into the middle as “our revolution” (1982, 2).

Another key consideration at odds with the notion of sonata form is the extensive canons in each section. For instance, in the opening, there’s a double canon between pairs of parts (1 and 2; 3 and 4) as follows:

1 P0 Horn 2 – Clarinet – VC ; elides with VC – Clarinet – Horn 2
2 I0 Horn 1 – B.Cl – Vla; elides with Vla – B.Cl – Horn 1
3 I8 Harp – VC – 2nd Vln – Harp – Horn 2 – Harp – Horn 2 – etc.
4 P4 Harp – Vla – 1st Vln – Harp – Horn 1 – Harp – etc.

Notice the similar timbral sequences in the two pairs. For instance, in the first pair, we have a horn part, then a clarinet part, and finally a lower string instrument before returning (yes, you guessed it!) symmetrically back the way we came. Breaking up the melodic line in this way is sometimes called Klangfarbenmelodie (sound color melody) and is not unique to the atonalists. Mahler loved to share out melodies this way, for instance (see this book’s section on Orchestration). Perhaps Webern’s most iconic example of this technique is his orchestration of Bach’s Ricercar from the Musical Offering.

Movement 2: Variations. “Double Canon with retrograde”

This time, there’s no disputing the title: “Variations” is certainly apt, and the structure is brought out in typically Webernian fashion using all the parameters. Here’s a brief synopsis in note form of what’s going on:

Theme (related to the coda)

  • Divided in 5.5+5.5 measure (so m. 6 is the midpoint)
  • Row I8 in Cl, I2 in other parts.
  • Note the hexachordal combination.

Variation 1 (related to variation 7)

  • Double canon at the quarter note: Vln1 pairs up with VC; Vln2 with Vla.
  • Across those pairs, Vln1 (I3) is R-related to Vln2 (I9); the Vla (P7) is likewise R-related to VC (P1).
  • 6+6 measures (midpoint m. 17 upper parts, bar line 1-8 lower). Rows swap.
  • Bar 11 and 23 overlap neighbors.

Variation 2 (related to variation 6)

  • 6+6 (bar line 29), except…
  • “Free” third part (Hn1) alternating P-8 with I-7? 6, 12, 6 notes.
  • B. 34 overlap.

Variation 3 (related to variation 5)

  • 1+4+1+4+1.
  • Midpoint in b. 39.
  • Symmetrical melodic figure in each part, and sixteenth-note motion.

Variation 4

  • Webern describes this as the “midpoint.”
  • 5+1+5 (b. 50 midpoint).

Variation 5 (related to variation 3)

  • Four-note cells: Vla, VC: [f,f♯,g,g♯]; Vlns: [b,c,c♯,d]; Harp: [E♭,E,B♭,A]
  • = Pitch repetition. Total chromatic, but not really of a serial kind. Stravinsky-esque.
  • Symmetrical melodic figure in each part, and sixteenth-note motion.

Variation 6 (related to variation 2)

  • Canon at the quarter note: bcl, cl.
  • “Free” third part (Horn 1).
  • Midpoint in m. 73.

Variation 7 (related to variation 1)

  • Triple canon: VC-vln1; vla-vln2; cl-bcl.
  • Distinguished by timbre, rhythm.
  • Midpoint in m. 83.

Coda (related to the theme)

  • Divided (like the theme) into 5.5 + 5.5 measures (with m. 94 as the midpoint).


So you’ll have noticed some recurring themes here, among which perhaps the most all-pervasive is the symmetry Webern adopts from the internal structure of the row, right up to the organization of whole movements.

Does this make it thoroughly modern? Or does the symmetry contribute to a new kind of goal-directed (teleological) music typical of at least 19th-century music since at least Beethoven? How do we feel about the occasional direct historical precedent like the entirely symmetrical Minuet and Trio in Haydn’s 47th symphony?

Most importantly, what is the aural effect of all this symmetry? Did the Harvey quote above make you bristle? There’s a good, natural, acoustical reason why composers have historically tended to build up chords from the actual bass after all. Similarly, we can’t really “hear” linear symmetry (hear in reverse) the same way that we can see symmetry in a painting, for instance. That said, Webern has gone to considerable lengths (at least in places) to illuminate the structure of the work. For Cook, “everything [in the Symphonie] is designed to make the series audible” (1987, 12). So is Webern taking on a brave challenge or a fool’s errand?

Perhaps we should consider this alongside other 20th-century “symphonies with a twist.” Think of Stravinsky’s many “Symphonies” (“in C,” “of Psalms,” “of Winds”), none of them numbered in the traditional way. If nothing else, these works seem to speak of a conviction to the musical traditions these composers inherited, just as Schoenberg was so keen to locate his apparently radical, modernist works in that tradition, and particularly as heir to the work of Brahms.

Webern: Konzert Op. 24 (1934)

Webern’s Konzert (Concerto) raises many of the same technical and wider musical issues as the Symphonie. It has a similarly “neat” row (Example 2), and a similarly suggestive title, apparently alluding to a long-standing musical tradition.

 The row

Concerto, Op. 24 by FourScoreAndMore

Example 2. The row of Webern’s Konzert (Op. 24) along with the trichordal and hexachordal divisions.

Once again, we have two equal (this one is sometimes called the “magic hexachord”) and a meaningful division into four trichords. This time, those are all instances of the same : [014]. This structural division is made abundantly clear in the first few measures, in which the row is set out in its four parts in separate instruments, pulse values, and registers. There follows a fermata. You couldn’t hope to a see a clearer row “exposition.”

But that’s not all. If you permute the order of those trichord cells, you can get other, related row forms:

  • Obviously, reversing the order of the cells can give you a P-R pair (P = 1234; R = 4321).
  • Additionally in this case, the order 2143 gives you I, and so 3412 is an RI.

That being the case, we have a set of four equivalent rows, and thus only twelve distinct row forms this time. For instance, P0 is the same as RI7 starting that rotation from the seventh note, as shown by the bold in the matrix below. This all amounts to a particularly clear and determined level of coherence.

Here’s the row matrix. Again (and perhaps more surprisingly this time) this arrangement does not correspond to the allocation of P0 by Webern in the sketches:

I0 I11 I3 I4 I8 I7 I9 I5 I6 I1 I2 I10
P0 B A♯ D D♯ G F♯ G♯ E F C C♯ A R0
P1 C B D♯ E G♯ G A F F♯ C♯ D A♯ R1
P9 G♯ G B C E D♯ F C♯ D A A♯ F♯ R9
P8 G F♯ A♯ B D♯ D E C C♯ G♯ A F R8
P4 D♯ D F♯ G B A♯ C G♯ A E F C♯ R4
P5 E D♯ G G♯ C B C♯ A A♯ F F♯ D R5
P3 D C♯ F F♯ A♯ A B G G♯ D♯ E C R3
P7 F♯ F A A♯ D C♯ D♯ B C G G♯ E R7
P6 F E G♯ A C♯ C D A♯ B F♯ G D♯ R6
P11 A♯ A C♯ D F♯ F G D♯ E B C G♯ R11
P10 A G♯ C C♯ F E F♯ D D♯ A♯ B G R10
P2 C♯ C E F A G♯ A♯ F♯ G D D♯ B R2
RI0 RI11 RI3 RI4 RI8 RI7 RI9 RI5 RI6 RI1 RI2 RI10


Let’s take a closer look at those cells. Recall how in the Symphonie we had a third (major or minor) and a semitone in each cell? Well, now we’ve settled on a major third specifically, and each iteration of the cell involves that interval (interval class 4) and a semitone (interval class 1) in opposite directions. Across the four iterations in a row, we get each of the four ways of setting this out. In the case of P0, this involves -1+4, +4-1, -4+1, and +1-4.

That being the case (and at the risk of confusing matters), we could think of the P, R, I, and RI versions not just of each row, but of each cell. Continuing to work on the basis of P0, we have:

  1. P (-1+4)
  2. RI (+4-1)
  3. R (-4+1)
  4. I (+1-4)

In this way, each cell is related to every other as follows:

  1 2 3 4
1 = RI R I
2 RI = I R
3 R I = RI
4 I R RI =

It’s hardly surprising given all of this that Webern was a fan of the “Sator Square”: a two-dimensional palindrome that can be read in any direction and still make sense (in Latin at least!). The meaning even seems to pun on the idea of ploughing (“turning”) the field.


That all said, the 4x trichord approach is, of course, not the only way of using the row. It’s focal in the first movement, while the second movement primarily employs 2- and 4-note cells, highlighting interval class 1 (usually set out as a major seventh).


Once again, we have the looming question of a tradition with highly codified expectations. What kind of “concerto” might this be? The idea of a piano concerto finds some support in the distribution of row forms across the work. For instance, that first “exposition” of the row (if you will) falls to instruments other than the piano, and then there’s the fermata and a “second exposition” of the row on the piano alone. Conceptually at least, this is a neat fit for the “double exposition” of concerto first movements—one of the main hallmarks by which concerto-sonata form differs from other contexts.

Are we clutching at straws on the basis of just two statements of the row? Maybe, but then again, Webern is no stranger to the practice of cultivating comparable processes on the very small and large scale, so it’s worth paying attention to these kinds of clues as possible “statements of intention” in the same way that we should take notice of the first highly chromatic event in a Schubert sonata.

In this work, the relationship between piano and “orchestra” continues to reward analytical attention. After those first few measures, the first major section of the movement (which is probably a better fit for the traditional notion of an “exposition”) moves from that initial clear, successive separation of piano from ensemble to a situation where they remain separate in terms of row forms, but sound together (simultaneously). You can analyze the whole work in terms of this relation, and there are many moments where there appears to be both a structural boundary and a change in the relation between piano and ensemble. Examples include the separation in m. 50 and rejoining in m. 63.

Even if you find this line of reasoning compelling, you still face choices at every turn. For instance, you may see the piano’s ubiquity in the slower, quieter second movement (highly reminiscent of the traditional second movement) as a lyrical, dominant, soloistic part, or the opposite – in an accompanimental role. Alternatively, if you reject all of this in favor of an equality among the instruments (matching the equality among the tones, perhaps), then you may find more palatable the idea of reading this as a Concerto … for Orchestra?

As with all music, there’s more than one way to look at this piece, and as with all analysis, we’re much more concerned with a view (“a way” of understanding the work) than trying to find comprehensive solution (“the way”). In short, the analysis of twelve-tone music is just like other forms of analysis: understanding the technical elements is necessary but not sufficient, and there’s always plenty of room for creative vitality.

  1. Note that this is sometimes set out in an alternative format, with P and I the other way around (as, for instance, in Bailey 1991, after Webern’s sketches). These kinds of decisions are often not clearly "better" one way or the other.


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