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Contents

Music in the Early Twentieth Century

A LITTLE “SET THEORY”

Chapter:
CHAPTER 6 Inner Occurrences (Transcendentalism, III)
Source:
MUSIC IN THE EARLY TWENTIETH CENTURY
Author(s):
Richard Taruskin

For a final demonstration we can turn to the first of Schoenberg's Sechs kleine Klavierstücke, op. 19 (“Six little pieces for piano,” 1911), a set of tiny, aphoristic piano pieces in an idiom that finally seems purged of all “tonal reference” (Ex. 6-13). No single pitch emerges from the texture with sufficient frequency to suggest itself as a candidate tonic; fifth relations are not salient; major or minor triads are not in evidence, nor are dominant-seventh chords. It would appear that the whole conventional vocabulary of music has been suppressed in favor of a private language.

And yet even if a familiar vocabulary is missing, there still remain vestiges of a familiar syntax. There are significantly recurring harmonies: both in the middle of m. 3 and on the downbeat of m. 5 are chords that could be cumbersomely described (in “tonal” terms) as a minor seventh chord without a fifth and with a major seventh along with the minor seventh. In m. 5 the four notes are F–A♭–E♭–E while the corresponding notes in m. 3 are C–E♭–B♭–B (the last note being the first in the right-hand melodic figure that enters after a sixteenth rest). The same harmony, expressed as an arpeggio or melodic succession, is also heard at the very beginning of the piece in the left hand.

A Little “Set Theory”A Little “Set Theory”

ex. 6-13 Arnold Schoenberg, Sechs kleine Klavierstücke, Op. 19, no. 1

A similar example comes in the middle of m. 2, where we find a distinctive chord—a tritone atop a perfect fourth—that we have already encountered in Stravinsky’s The Rite of Spring, where it was so prevalent that we ended up calling it the Rite-chord. Its appearance in m. 2 is prefigured as the harmonic sum, so to speak, of the last three notes in m. 1 (the F♯, interestingly, being the upward resolution of a downward-leaping appoggiatura). As an arpeggio, it reappears at the beginning of m. 8 in inverted form, the tritone now below the perfect fourth rather than above. Note, too, that the right-hand triplet at the beginning of m. 5 consists of the same intervals compressed into the “best normal order”: flip the E♭ and D into a seventh, put the A in the middle, and the result is another Rite-chord.

For a third example, compare the three-note chord in the left hand in m. 1 with the long-sustained chord in the right hand that lasts from the downbeat of m. 15 virtually to the end of the piece. It may not be immediately apparent that the one is the intervallic inversion of the other, but putting both chords into “best normal order” will make their relationship clear. The closest possible spacing of the notes in the first chord is B–D♯–E, a major third beneath a semitone; a similar operation at measure 15 produces D♯–E–G♯, a semitone beneath a major third. And once we have noticed this much, we can notice the many transposed occurrences of the same harmony (or “intervallic set”): the first arpeggiated right-hand chord in m. 7, the three right-hand thirty-second notes in m. 8 (B♭–D–A) that follow the arpeggiated Rite-chord we have already noted, and so on.

One could go on noting correspondences like these almost indefinitely. Sometimes they resemble traditional contrapuntal devices: in m. 2, for example, the right hand melody (or rather its descending component) is mirrored in diminution by the thirty-seconds in the left hand. At other times they are purely harmonic: at the end of m. 5 (left hand) and in m. 7 (right hand) an augmented triad is the second chord in a pair, suggesting a cadential approach. Elsewhere (m. 3, left hand; last measure, both hands) multiple neighbors suggest cadences, just as they had done in early songs like Erwartung and Der Wanderer, now even in the absence of a tonic. One might even go so far as to suggest that the use of multiple neighbors lends the final chord in the piece the de facto status of a tonic even though it is not (as Schoenberg often put it) a “codified” harmony.

Contextual relationships like these arise, to use Schoenberg's own expression, from “working with the tones of a motive”17 —that is, a group of notes with a distinctive intervallic profile. But the relationships we have been tracing, while numerous and cumulatively impressive, are not the whole story, or even the most significant part of it. The story as it stands shows Schoenberg working, even in a piece of only seventeen measures’ duration, with a great many scattered motives. What justifies such a procedure if we assume (as anyone coming out of the intellectual and artistic traditions in which Schoenberg’s style developed would have had to assume) that a piece of music should be “all of a piece”? Is there any principle by which all the scattered observations we have been making can be related?

For an answer, look once again at the chord in the middle of m. 3 (C–E♭–B♭–B), one of the first musical events we noted in taking the piece apart. If we refer it to Ex. 6-5, from Schoenberg’s Mädchenlied, op. 6, no. 3 (a “tonal” song in E minor), we may note, to our possible surprise, that all of its tones are found there. And that means (if we recall the significance of Ex. 6-5 in the argument of this chapter) that all its tones belong to the “Eschbeg set,” Schoenberg’s musical signature. They are, in fact, the first four notes of the set with their order rearranged (as 2–1–4–3). And if we continue to survey the right-hand melody in m. 3, we encounter all the remaining tones, even including the A that provides the composer’s first initial, so that the set assumes its complete “A. Schbeg” form.

The implications of this observation are far-reaching. To begin with, all of the relationships we have traced to the chord in m. 3—the chord in m. 5, the left-hand melodic phrase at the very outset—are likewise traceable (through transposition) to the “A. Schbeg” (or “Aschbeg”) set. And that also means that any other group of tones that can be derived as a transposed or untransposed subset from the “Aschbeg” set are related (or at least “relatable”) to all the other ones. That includes virtually all the musical configurations we have noted. They are all related to one another, if the “Aschbeg” set is regarded as the nexus.

To demonstrate this we have to put the “A. Schbeg” set in its “best normal order.” In the order of Schoenberg’s name the constituent tones are A–E♭–C–B–B♭–E–G. The closest possible spacing of these tones is G–A–B♭–B–C–E♭–E. If we call G “zero” and count by semitones, the Aschbeg set looks like this: /0 2 3 4 5 8 9/. Now we are ready to compare the other pitch configurations we have noted to the Aschbeg set. The fourth-tritone configuration, noted in m. 1, m. 2, and m. 8 (among other occurrences) reduces to /0 1 6/. This configuration can be “mapped” onto the numerical form of the Aschbeg set by adding 2 to each of its members: /2 3 8/. Each of the resulting numbers can be found in the Aschbeg set.

This can easily be put in more familiar musical terms. All that the numbers really mean is that if you start on the A (= 2), you can construct a fourth-tritone chord out of the notes of the Aschbeg set: A–B♭–E♭, which if spaced B♭–E♭–A is an exact transposition of the chord in m. 3. Similarly, the third + semitone configuration that we traced in m. 1 (left hand) and m. 15 (right hand), when placed in best normal order (/0 1 5/), can be mapped onto the Aschbeg set by adding 3: /3 4 8/(the numbers corresponding to B♭–B–Eb, a transposition of the D♯–E–G♯ in m. 15).

Did Schoenberg do all this “math”? Of course not. He needed only to have the Aschbeg set in mind as his starting point, select subsets from it, and transpose them as he wished. It is we who need the math to demonstrate (or at least plausibly propose) the first premise, namely that Schoenberg had the Aschbeg set in mind as his starting point. There are other ways of showing (or proposing) this as well. The piece actually begins with a very slightly modified Aschbeg set, consisting of the first two notes in the right hand and the left-hand part up to the three-note chord we have already analyzed from another perspective. The sum of all these pitches is “Aschbeg” minus the B♭, and with a G♯ in its place. The G♯ is not really as much of a deviation as it may seem. Its enharmonic equivalent, A♭, is “As” in German: the first two letters of Aschbeg.

There are at least two more complete Aschbegs embedded in the music of op. 19, no. 1. The left-hand part of m. 7, plus the pickup at the end of m. 6 and the first right hand note in m. 8, amounts to Aschbeg transposed down a semitone. The most hidden or “occult” occurrence of the set is found in m. 5. It needs to be illustrated with an example, since it crosscuts three of the polyphonic voices delineated in Schoenberg’s notation. It should be emphasized, though, lest anyone suspect it to be fortuitous, that the pitches involved are all contiguous (see Ex. 6-14).

A Little “Set Theory”

ex. 6-14 “Aschbeg” set in Arnold Schoenberg, Sechs kleine Klavierstücke, Op. 19, no. 1, m. 5

Notes:

(17) Schoenberg, “Composition with Twelve Tones (II)” (1948); Style and Idea, p. 248.

Citation (MLA):
Richard Taruskin. "Chapter 6 Inner Occurrences (Transcendentalism, III)." The Oxford History of Western Music. Oxford University Press. New York, USA. n.d. Web. 23 Apr. 2019. <http://www.oxfordwesternmusic.com/view/Volume4/actrade-9780195384840-div1-006010.xml>.
Citation (APA):
Taruskin, R. (n.d.). Chapter 6 Inner Occurrences (Transcendentalism, III). In Oxford University Press, Music in the Early Twentieth Century. New York, USA. Retrieved 23 Apr. 2019, from http://www.oxfordwesternmusic.com/view/Volume4/actrade-9780195384840-div1-006010.xml
Citation (Chicago):
Richard Taruskin. "Chapter 6 Inner Occurrences (Transcendentalism, III)." In Music in the Early Twentieth Century, Oxford University Press. (New York, USA, n.d.). Retrieved 23 Apr. 2019, from http://www.oxfordwesternmusic.com/view/Volume4/actrade-9780195384840-div1-006010.xml