IX. Twelve-Tone Music
107
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 considers two iconic works of early twelve-tone music – Webern’s Op. 21 and Op. 24 – with analysis of both the
- technical details of constructing symmetrical row and composing serial canons, with
- wider issues about the work to consider, such as the meaning of a “Symphony” and “Concerto” in this context.
- Scores may be found on IMSLP.org (Op. 21; Op. 24)
Webern: Symphonie Op. 21 (1928)
Even the title of Anton Webern’s Symphonie Op. 21 raises questions. Why would Webern choose to call this a symphony? If we think of a symphony as having certain types of key relations, then is an atonal symphony an oxymoron? Commentators 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.” (Whittall 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
https://musescore.com/user/30425053/scores/8171934/embed
Webern frequently chooses what you might think of as “neat” row forms, and this work is no exception (see pitch-class set but of set 6-1 specifically: half a chromatic scale. In short, each fills the total chromatic collection of half the twelve-tone space.
). The row breaks up neatly into two equivalent hexachords that are instances not simply of the sameFurther, those hexachords are each set out with one instance of trichord (013) and one of (014). Altogether, the four trichord cells map out as (013), (014), (014), (013). 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 | 9 | 6 | 7 | 8 | 4 | 5 | 11 | 10 | 2 | 1 | 0 | 3 | R0 |
| P3 | 0 | 9 | 10 | 11 | 7 | 8 | 2 | 1 | 5 | 4 | 3 | 6 | R3 |
| P2 | 11 | 8 | 9 | 10 | 6 | 7 | 1 | 0 | 4 | 3 | 2 | 5 | R2 |
| P1 | 10 | 7 | 8 | 9 | 5 | 6 | 0 | 11 | 3 | 2 | 1 | 4 | R1 |
| P5 | 2 | 11 | 0 | 1 | 9 | 10 | 4 | 3 | 7 | 6 | 5 | 8 | R5 |
| P4 | 1 | 10 | 11 | 0 | 8 | 9 | 3 | 2 | 6 | 5 | 4 | 7 | R4 |
| P10 | 7 | 4 | 5 | 6 | 2 | 3 | 9 | 8 | 0 | 11 | 10 | 1 | R10 |
| P11 | 8 | 5 | 6 | 7 | 3 | 4 | 10 | 9 | 1 | 0 | 11 | 2 | R11 |
| P7 | 4 | 1 | 2 | 3 | 11 | 0 | 6 | 5 | 9 | 8 | 7 | 10 | R7 |
| P8 | 5 | 2 | 3 | 4 | 0 | 1 | 7 | 6 | 10 | 9 | 8 | 11 | R8 |
| P9 | 6 | 3 | 4 | 5 | 1 | 2 | 8 | 7 | 11 | 10 | 9 | 0 | R9 |
| P6 | 3 | 0 | 1 | 2 | 10 | 11 | 5 | 4 | 8 | 7 | 6 | 9 | 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 divided by 2).
Webern brings out this symmetrical row by overlapping the ends of row forms with the beginning of the next.
Movement 1
Webern describes the first movement as a “double canon in contrary motion,” where “contrary motion” means the same thing as “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 the midpoint of mm. 25–43 in the clarinet (or mm. 26–42 in the cello).
- 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 Exposition and Development–Recapitulation repeats in sonata form works, or at least rounded binary.
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, 96) 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 (P0 and I0; I8 and P4) as follows:
[table “82” not found /]
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 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 Orchestration). Perhaps Webern’s most iconic example of this technique is his orchestration of J.S. Bach’s Ricercar from the Musical Offering.
Movement 2: Variations; a double canon with retrograde
With the second movement, 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)
- Measures 1–11.
- Phrases divide into two groupings of 5.5 + 5.5 measures; m. 6 is the midpoint.
- Row I8 is in the clarinet; I2 in the other parts.
- Uses hexachordal combinatoriality.
Variation 1 (related to variation 7)
- Measures 11–23; beginning elides with the theme.
- A double canon at the quarter note: violin 1 pairs up with cello; violin 2 with viola.
- Across those pairs, violin 1 (I3) is R-related to violin 2 (I9); the viola (P7) is likewise R-related to cello (P1).
- Phrases divide into 6 + 6 measures; the midpoint is m. 17 for the violins, m. 18 for the viola and cello. The violins swap rows, as do the viola and cello.
- The ending elides with variation 2.
Variation 2 (related to variation 6)
- Measures 23–34.
- Phrases divide into 6 + 6 measures (midpoint at m. 29).
- However, there is a free third part (horn 1). The on-beats play P8 while the off-beats play I7, so the eighth notes are alternating between the two rows.
- The ending elides with variation 3.
Variation 3 (related to variation 5)
- Measures 34–44.
- Phrases divide into 1 + 4 + 1 + 4 + 1 measures; midpoint in m. 39.
- Symmetrical melodic figure in each part, and sixteenth-note motion.
Variation 4
- Measures 45–55.
- Webern himself describes this variation as the midpoint of the movement.
- Phrases divide into 5 + 1 + 5; m. 50 is the midpoint.
Variation 5 (related to variation 3)
- Measures 56–67.
- Pitches are presented in four-note cells:
- Viola, cello: [5, 6, 7, 8]
- Violins: [11, 0, 1, 2]
- Harp: [3, 4, 10, 9]
- These four-note cells create a twelve-tone aggregate, but the composition is not row-based.
- Each part has a symmetrical melodic figure.
Variation 6 (related to variation 2)
- Measures 67–77.
- Canon at the quarter note between the bass clarinet and clarinet parts.
- Free third part (horn 1), the retrograde of the free part in Variation 2.
- Midpoint in m. 73.
Variation 7 (related to variation 1)
- Measures 78–88.
- Triple canon: cello/violin 1, viola/violin 2, and clarinet/bass clarinet. Each line of the three canons is distinguished by timbre and rhythm.
- Midpoint in m. 83.
Coda (related to the theme)
- Measures 89–99.
- Phrases are divided into 5.5 + 5.5 measures, with m. 94 as the midpoint.
Issues
You’ll have noticed some recurring themes in the analysis above; perhaps the most all-pervasive among them 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 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 are reasons based in acoustics for 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, 295).
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 (
), and a similarly suggestive title, apparently alluding to a long-standing musical tradition.The row
https://musescore.com/user/30425053/scores/8171973/embed
Once again, we have two equal hexachords of set class (014589) and a meaningful division into four trichords. This time, those trichords are all instances of the same set class: (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. This is followed by a rest in all instruments. 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 of rows (P = 1234; R = 4321).
- Additionally in this case, the order 2143 gives you an I row, and so the reverse of that (3412) is an RI row.
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.
gives 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 | 11 | 8 | 2 | 3 | 7 | 6 | 8 | 4 | 5 | 0 | 1 | 9 | R0 |
| P1 | 0 | 11 | 3 | 4 | 8 | 7 | 9 | 5 | 6 | 1 | 2 | 10 | R1 |
| P9 | 8 | 7 | 11 | 0 | 4 | 3 | 5 | 1 | 2 | 9 | 10 | 6 | R9 |
| P8 | 7 | 6 | 10 | 11 | 3 | 2 | 4 | 0 | 1 | 8 | 9 | 5 | R8 |
| P4 | 3 | 2 | 6 | 7 | 11 | 10 | 0 | 8 | 9 | 4 | 5 | 1 | R4 |
| P5 | 4 | 3 | 7 | 8 | 0 | 11 | 1 | 9 | 10 | 5 | 6 | 2 | R5 |
| P3 | 2 | 1 | 5 | 6 | 10 | 9 | 11 | 7 | 8 | 3 | 4 | 0 | R3 |
| P7 | 6 | 5 | 9 | 10 | 2 | 1 | 3 | 11 | 0 | 7 | 8 | 4 | R7 |
| P6 | 5 | 4 | 8 | 9 | 1 | 0 | 2 | 10 | 11 | 6 | 7 | 3 | R6 |
| P11 | 10 | 9 | 1 | 2 | 6 | 5 | 7 | 3 | 4 | 11 | 0 | 8 | R11 |
| P10 | 9 | 8 | 0 | 1 | 5 | 4 | 6 | 2 | 3 | 10 | 11 | 7 | R10 |
| P2 | 1 | 0 | 4 | 5 | 9 | 8 | 10 | 6 | 7 | 2 | 3 | 11 | R2 |
| RI0 | RI11 | RI3 | RI4 | RI8 | RI7 | RI9 | RI5 | RI6 | RI1 | RI2 | RI10 |
Cell
Each cell can also be understood as an ordered trichord, like a miniature three-note row. Each iteration of the cell involves interval class 4 and 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:
- P: <−1, +4>
- RI: <+4, −1>
- R: <−4, +1>
- I: <+1, −4>
In this way, each cell is related to every other as shown in
:[table “83” not found /]
It’s hardly surprising given all of this that Webern was a fan of the “Sator Square”: a two-dimensional Latin palindrome that can be read in any direction and still make sense ( ).
That all said, the four-trichord approach is not the only way of using the row. It’s focal in the first movement, while the second movement primarily employs cells of two and four notes, highlighting interval class 1 (usually set out as a major seventh).
Discussion
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 falls to instruments other than the piano, and then there’s the full-ensemble rest 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 sonata form in concertos 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 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 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.
- Bailey, Kathryn. 1991. Twelve-Note Music of Anton Webern: Old Forms in a New Language. Cambridge: Cambridge University Press.
- Cook, Nicholas. 1987. A Guide to Musical Analysis. London: Dent.
- Harvey, Jonathan. 1982. “Reflection after Composition.” Tempo, no. 140, 2–4.
- Taruskin, Richard. 2010. “In Search of Utopia.” In Music in the Early Twentieth Century, chapter 12. New York: Oxford University Press. https://www.oxfordwesternmusic.com/view/Volume4/actrade-9780195384840-div1-012010.xml
- Whittall, Arnold. 1977. Music since the First World War. New York: Oxford University Press.
- Try your own analysis of another work from this time, such as Webern’s Variations for piano, Op. 27, using a similar combination of technical row analysis with contextual discussion.
- Note that this is sometimes set out in an alternative format, with P and I the other way around. Bailey (1991, appendix II) after Webern’s sketches uses this alternative format. These kinds of decisions are often not clearly "better" one way or the other. ↵
A group of pitch classes.
A complex large-scale musical form that can be understood as an elaborate version of rounded binary form with a balanced component. The larger level names are as follows: Exposition (≈A), Development (≈B), and Recapitulation (≈A′). In general terms, the exposition contains two main sections (primary and secondary) separated by a transition. The secondary theme is presented in a non-tonic key in the exposition, and crucially, it is restated in the recapitulation in the tonic key. The exposition and recapitulation often end with a large suffix (closing section).
A type of form that has two core sections. These sections are often called "reprises" because each is typically repeated. There are two main types of binary form: rounded and simple.
A group of pitch-class sets related by transposition or inversion. Set classes are named by their prime forms; for example, (012) is a set class.
Unordered pitch-class intervals; that is, the smallest possible distance in semitones between two pitch classes. Thus, mi2 and ma7 are both IC 1; ma2, mi7, +6 are IC 2; mi3, ma6, +2 are IC 3, etc. The largest interval class is six semitones, because if order is disregarded, the tritone is the largest possible interval.
Key Takeaways
- This is an anthology of a different kind and at a larger scale than the usual provision. Instead of showcasing a few, select examples (as the textbook chapters proper do), the goal here is to provide long lists of cases that at least one analyst sees in terms of the chord under discussion.
- Harmonic analysis is a reductive and subjective task and you should fully expect to disagree with some of the entries included here.
- The idea is to provide minimal curation, allowing you to roam freely among potentially relevant cases from across a broad repertoire, making up your own mind about what counts as an 'real' example.
Throughout this textbook, we have provided short examples of the musical subjects under discussion such as a specific chord or progression. These examples have included both simple, ‘prototypical’ versions to clarify how the idea works in principle, and also moments from real pieces in the repertoire.
Inevitably, these chapters only have space for a few such examples, so this final, ‘anthology’ section seeks to provide many more instances, enabling users to see a wider range of cases and with full context. No textbook or anthology can hope to capture the full range of ways in which these chords are used. Indeed, it’s not always clear whether a moment constitutes a ‘real’ example of the chord at hand. Hopefully, this at least provides space to roam and explore those 'edge cases'.
This first instalment of that vision focuses primarily on harmonic matters and on a corpus of nineteenth century songs encoded as part of the OpenScore Lieder Corpus which releases its transcriptions under the CC0 licence meaning that they can be used for any purpose whatsoever without restriction.[1] The tables below list moments identified as relevant by human analysts. So while much of the grunt work of collating lists and retrieving examples etc. has been automated, the analysis itself has not.[2]
Each of the tables below gives the:
- song's metadata: composer, collection title, song name,
- measure number and Roman numeral (figure and key) for the moment in question.
- URL link through to check out (play, download, etc) the score online.
This page currently includes lists for:
- Augmented Sixth chords
- Augmented Triads
- Common Tone Diminished Sevenths
- Modal Mixture
- Neapolitan Sixth Chords
Please get in touch if you would like to see other chords or progressions represented here. For those interested in the computational side, all source material (scores, analyses, code, etc.) is available here.
Augmented Sixth chords
Click here for this textbook's chapter on this topic.
[table “32” not found /]
Augmented Triads
Click here for this textbook's chapter on this topic.
[table “34” not found /]
Common Tone Diminished Sevenths
Click here for this textbook's chapter on this topic.
[table “85” not found /]
Modal Mixture
Click here for this textbook's chapter on this topic.
This table provides examples of modal mixture which is much more difficult define robustly than is the case for the Neapolitan sixth chords, for example. The list here provides examples where:
- Clearly mixed tone counts towards mixture and clearly non-mixed tone counts against it.
- E.g., F-Ab-C-E in C major has both a clearly mixed tone (Ab, the minor sixth) and a clearly non-mixed tone (E, the major third).
- Clearly clearly non-mixed tones do not necessarily preclude mixture if there are also strongly mixed tones.
- Mixture does not necessarily need to work in relation to all forms of the minor mode.
- "Chromatic" notes (that are not diatonic to the primary mode, shared between major and minor, nor mixed) are neutral
- Enharmonic spelling of the chord matters (so Ab might indicate mixture from c minor into C major, but G# probably does not);
- Secondary Roman numerals
- can indicate mixture (are not necessarily excluded);
- are considered against the home (primary rather than secondary) key. E.g., viio/V is considered against the I, primary tonic rather the V secondary.
Complex eh?! I'll provide a clear explanation and demonstration of all these options somewhere accessible soon. For now, every entry on this this makes at least a reasonably strong case for mixture.
[table “31” not found /]
Neapolitan Sixth Chords
This table includes root position (bII) in addition to first inversion (bII6) chords, as well as seventh chords based on both.
Click here for this textbook's chapter on this topic.
[table “33” not found /]
... and many more
Here, finally, are external links to corresponding anthologies on the same topics for other repertoire collections. These don't get the same focus here as they are slightly more experimental and the all important links to scores online are not currently possible. All the same, "more is more" for this large, exploratory style of anthology.