A NEW TONAL SYSTEM?

Chapter:

CHAPTER 7 Social Validation

Source:

MUSIC IN THE EARLY TWENTIETH CENTURY

Author(s):

Richard Taruskin

Among Bartók’s most impressive achievements were his six string quartets, all of them very elaborate major works, and all very different from one another. They were composed between 1908 and 1939, a period that encompassed virtually his entire mature career in Europe. Bartók’s intense cultivation of the genre, one of the emblems of the European “classical” tradition, attests to his concerns both for synthesizing the particularly national with the universal, and for conceiving the universal in terms of tradition and advancement in equal measure.

The Fourth String Quartet (1928), completed six years earlier than Music for Strings, Percussion, and Celesta, is often looked upon as the culmination or far out point of Bartók’s maximalistic explorations. It brings his preoccupation with symmetry to a peak that encompasses two musical dimensions: both the “vertical” dimension of harmony, as in the works we have already considered, and the “horizontal” dimension of form as it unfolds in time. In it he deployed for the first time the all-encompassing symmetry of “bridge form,” as he called it, meaning the casting of the constituent sections in a movement (like the nocturnal slow movement of the Music for Strings, Percussion, and Celesta) or even the constituent movements in a full-length “classical” composition like a string quartet, in the form of a palindrome.

Like several of the works that followed it (including the Fifth Quartet, the Second Piano Concerto, and the Concerto for Orchestra composed in America), the Fourth Quartet contains five movements, in which a unique central movement is flanked fore and aft by neighbors of similar character, while the outer movements draw on a common fund of thematic or motivic material. But when representing the form schematically (as in his preface to the Fourth Quartet), Bartók did not designate the sections simply as ABCBA as one might label the sections of a rondo, but rather ABCB′A′, to denote his concern that there be a dynamic forward momentum as well as a sense of return, as in the classical sonata form. To quote László Somfai, the leading Hungarian Bartók scholar, despite all its “quasi-geometrical symmetry,” Bartók’s bridge form “is not static: it does not return to its origins but progresses towards a cathartic outcome,”¹⁴ which of course implies a drama.

In the Fifth Quartet and the Second Concerto, the middle movement is a wild scherzo flanked by slow movements (the first lyrical, the other stunned and immobile). In the Fourth Quartet (as in the Concerto for Orchestra) the arrangement is just the opposite: the central movement is an almost motionless “night music”. In the Quartet this central movement is replete with a Hungarian speech-song intoned by the cello that first alternates, then joins, with nature sounds (prominently including bird calls) against a backdrop of impassively sustained harmonies. The flanking movements in the Fourth Quartet and the Concerto for Orchestra are scherzos. In the Quartet each scherzo is a coloristic tour de force as their performance directions declare: Prestissimo, con sordino in the fore position, Allegretto pizzicato in the aft. (Here Bartók first used the technique of snapping the string against the fingerboard—designated with —that all string players now call the “Bartók pizz,” although it had been anticipated as early as 1673 by the Austrian violinist composer Franz von Biber to imitate cannon-fire in a Battalia for strings.)

The relationship between the outer movements in the Quartet is most vividly grasped by taking a peek at their respective last pages, which are self-evidently similar (Ex. 7-18). The concluding phrases, marked pesante (weighty), are virtually identical. The last movement, however, is no mere reprise of the first. The similar conclusions are reached by differing trajectories; and even on the last pages a closer look reveals telling contrasts of detail. All the motifs on the last page of the quartet (Ex. 7-18b) are diatonic, while those at the end of the first movement (Ex. 7-18a) begin as segments of the chromatic scale and approach the diatonic conclusion only at the very end. The chromatic segments may be found in the finale, too, and dominate its middle, so that the final movement makes a longer, more systematic and decisive approach to a diatonic conclusion that the first movement only briefly foreshadows. Thus even at this preliminary, fairly superficial stage we may discern a sense of ongoing motivic development and resolution linking the outer movements.

But the shared material that unites them actually lies, so to speak, beneath the motivic threshold. A very quick spot survey of the finale will prepare us for a close analysis of the first movement, the longest and most elaborate of the five. (The possibly misleading fact that there are more than twice as many measures in the finale should be balanced against its faster tempo, its two-beat rather than four-beat measure length, and its much lower density of detail.) The finale’s very first harmony (Ex. 7-19a) can be our starting point. It contains a thrice-doubled C and a G that is duplicated even more amply (six times in all; though like the C it is sounded in only three octaves). This strongly reinforced fifth is obviously being projected as a “normative” sonority, analogous to a tonic. And indeed there is no measure that does not contain it somewhere until m. 57. But its absence is fleeting. It reasserts itself dramatically in m. 75, and is thereafter again omnipresent for a while, but in conjunction with its counterpart at the tritone, the fifth F♯/C♯. That already rings a Bartókian bell (and a Stravinskian one as well, if we recall Petrushka from chapter 3).

But let us return to Ex. 7-19a and take note of the two “foreign” pitches, F♯ and D≭ They are the equivalent of the F♯/C♯ just described, but their unusual spelling on their first appearance alerts us that something unusual yet strangely familiar may be afoot. A sharped note tends upward, a flatted one downward; so again we may have a case of neighbors scraping, as it were, the insides of an interval. This assumption is confirmed in a long passage beginning in m. 11 (Ex. 7-19b), in which the cello adds both neighbors simultaneously (the note-spelling now looking especially bizarre) to the viola’s normative fifth in the same register.

In Ex. 7-19c, moreover, which begins at m. 31, the neighbor harmony is transferred to the inner voices, while the outer ones play a theme that consists of the fifth and its neighbors strung out as a sort of arpeggio (identified, when sighted in the Bagatelles, as the “Z-tetrachord,”), in which semitones alternate with fourths (or, as Bartók sometimes insists in his spelling, with augmented thirds). In m. 37 another variant appears, in which the lower G is replaced by a G♯ and the tune approaches a “Hungarian” pentatonic mode. At the pickup to m. 48 yet another variant appears, this time accompanying the normative fifth (now weirdly spelled with a B♯) and the neighbor tones (spelled with an ordinary C♯ in place of the weird D≭). The changed spellings suggest a reversal of the perspective. Now it is the original normative interval, spelled as the augmented third G/B♯, that is playing the role of double neighbor to a newly normative F♯/C♯.

So C(B♯)/G and F♯/C♯(D≭) are in a kind of harmonic stalemate. That much, as the reference to Petrushka reminds us, is nothing new. We’ve seen it before in Ravel, too, and it even arises occasionally in the work of Rimsky-Korsakov, since it arises “naturally” out of octatonic relationships based on the /0 3 6 9/ circle of minor thirds. (When C/G is at “0,” F♯/C♯ will be at “6”.) Indeed, Bartók confirms this common heritage at m. 44, when A and E (at “9”) take over briefly as the normative fifth, providing a halfway house between C/G and F♯/C♯. Indeed, almost the entire pitch content of the first section of the movement can be referred to the same octatonic scale that provided the basic tonality in the second tableau of Petrushka—namely, the one that can be constructed out of triads on C, E≭, F♯, and A, and that can be represented in numbers (with C as “zero”) as /0 1 3 4 6 7 9 10/.

Of the “missing” pitches /2 5 8 11/ (=D, F, G♯A≭, B), only the G♯A≭ figures prominently in the music we are surveying, first as a complementary neighbor to G in Ex. 7-19b, and later, in a single melodic appearance (m. 38), as the result of a transposition. The rest appear before m. 90 only as embellishments to the accompanying ostinato rhythm: the F in the cello as a neighbor to E at m. 44; the D only as part of a folk-primitivistic slashing-the-open-strings effect in the viola (also at m. 44). The B never appears at all, which makes its sudden prominence, beginning in the viola part at m. 90, such an event (Ex. 7-19d). Marking a sudden modulation, just as in traditional “tonal” music, it serves to articulate an important formal division.

But something else is happening as well. The superimposed fifths (or “Z-tetrachord”) C/G + C♯D≭/F♯, whether appearing as an ostinato harmony or as a melodic phrase, are “overdetermined” in this music. It is a subset of the octatonic scale, as it would be in Stravinsky or Ravel, but the alternating use of each constituent fifth as a neighbor scraping the insides of its counterpart recalls the thoroughgoing symmetrical arrays that are uniquely Bartok’s. If we were to continue the pattern implied by the progression from C/G to D≭/F♯ (that is, adding a semitone to the bottom and subtracting one from the top), the next interval to appear would be D/F, followed by D♯/E, an axis pair.

But we’ve been there before: it is all laid out in Ex. 7-15, our “representative ‘odd’ array” expressing “sum 7.” Notice now how our two “Z-related” fifths function within the array: as “contiguities,” or immediate successions on either side of the reciprocal axis—first “forward,” so to speak, with C/G before D≭/F♯, then “backward.” Another way of expressing the relationship would be to note that they exchange places on opposite sides of the reciprocal axis. The C/G and D≭/F♯ are in second place (±2) following each axis pair respectively, and the D≭/F♯ and C/G are in third place (±3).

There is yet a third way of viewing their relationship. Turn back to p.393 and look at Ex. 7-15c, which has not been discussed or even mentioned up to now. It simply shows the two halves of Ex. 7-15a superimposed. The familiar Z-tetrachord now comes as a discrete harmony in positions 3 and 4 (its two forms being related, so to speak, by inversion: two fourths superimposed at the tritone vs. two fifths). But the same harmony also comes at the ends, as the sum of the two axis pairs (again expressed in inversion: two semitones superimposed at the tritone vs. two major sevenths). The remaining harmony, at positions 2 and 5, is a diminished seventh chord that sums up the /0 3 6 9/ symmetry associated with the octatonic scale. For a final demonstration of the “overdetermination” of Bartókian symmetry, note that the Z-tetrachords at positions 1 + 3 (or 4 + 6, their inversions) sum up the contents of the octatonic scale laid out a few paragraphs above, and the chord at positions 2 and 4 is the complementary collection of “missing” pitches.

From this we may draw one final “theoretical” conclusion (or rather, make one more a priori generalization) before going back to the music. Just as two different Z-tetrachords appear in Ex. 7-15c, a summary of the harmonic relations of a single “odd” (or dual-axis) symmetrical array, so any one Z-tetrachord will appear in two such arrays. And just as the two Z-tetrachords within a single array stand at the distance of a minor third (easiest to see in Ex. 7-15c if position 1 is compared with position 4, or 3 with 6), so the two arrays between which a single Z-tetrachord can function as a pivot or bridge will have “sum” numbers that differ by 6 (representing the tritone).

Thus, in Ex. 7-20 (following a demonstration first published by the Bartók scholar Elliott Antokoletz¹⁵), each of the Z-tetrachords in Ex. 7-15c is written out four times, in permutations that show how it may be laid out around four different dual axes of symmetry, each located at a distance of a minor third from its neighbors. The index or “sum” number of each axis semitone is entered so as to confirm the observations made in the previous paragraph, which link Bartók’s symmetrical scheme with the transposition routines already associated with “octatonicism” in the works of Stravinsky, Ravel, and Rimsky-Korsakov, and, behind them all, Bartók’s revered compatriot Franz Liszt. An even simpler demonstration of that association is the bare fact that the sum of the two Z-tetrachords in question, or any two found in the same symmetrical array, is the octatonic scale itself (see Ex. 7-20c).

But now it is time, once and for all, to ask why they are called “Z” tetrachords. The answer to that question, along with a peek at Bartók’s maximalism at its far out point, will emerge from a brief but comprehensive stab at analyzing the first movement of the Fourth Quartet. The very first measure (see Ex. 7-22) contains the telltale clue (as if we needed one by now) that the movement is based on a symmetrical array. A glance at the two violin parts, moving out from a semitone or “compound semitone” (minor ninth) to an augmented second, tell us that, and also specify the semitone E/F as one of the axes. Ex. 7-21a shows the whole array, set out in two tiers like Ex. 7-15c.

The first phrase in Ex. 7-22 (mm. 1–2) is constructed almost entirely out of the intervallic relationships presented in Ex. 7-21a. Besides the semitone E/F and the augmented second E≭/F♯ in the violins (positions 1 and 2 in the top staff of the array), there is the sixth C/A in the cello (position 5) and the concluding simultaneity B/B≭ in the second violin and cello (position 6). Position 3 (the fourth G/D) also comes as a simultaneity in m. 2 (cello and first violin). The second phrase (mm. 3–4) replays it all on the lower staff of Ex. 7-21a, beginning with the two violins on A/C (position 2) and B/B≭ (position 1), and then the viola entry on A≭ against the first fiddle’s C♯. The viola then proceeds to A/C (position 2) expressed as a sixth, to mirror the cello’s sixth in m. 1.

At the pickup to the fifth measure, the cello repeats the last three notes played by the first violin, but in typically reversed order, asserting a “horizontal” axis of symmetry to go with the vertical ones we have been investigating. Its descending third is then imitated by the other three instruments, each entering a semitone above the last and holding the final note so that a maximally dissonant cluster, which sets the tense or angrily agitated tone that will persist throughout the movement, is built up and tied over the bar.

That cluster, however, is no “mere” cluster. It is one of the most important motives in the movement, so important that it was christened “X” or “the X set” by George Perle in a very influential article, “Symmetrical Formations in the String Quartets of Béla Bartók,” published in 1955.¹⁶ That article, one of the first to unlock the secrets of Bartók’s symmetrical arrays, made the X-set famous along with its counterpart the Y-set, which follows immediately on the second eighth of m. 6. As the “X-set” was a four-note cluster encompassing three semitones, so the “Y-set” was an equally elemental particle, a cluster of three whole tones. Since both sets contained four notes, they are now more commonly known as the X and Y tetrachords.

Both, obviously, are intervallically (hence inversionally) symmetrical. The middle semitone of the X-tetrachord is its dual axis of symmetry; the whole-tone cluster has an unplayed single axis between its middle pair of notes. Thus, as Ex. 7-23 shows, the axis of symmetry for the first X-tetrachord in the quartet (m. 5) is C♯/D, while the axis for the first Y-tetrachord (m. 6) is C♯D≭. And as the cello part in m. 7 shows, the characteristic motif whose progress from chromatic to diatonic is in effect the story of the quartet, originates as a rhythmicized “horizontalization” of the X-tetrachord, initially played for maximum dissonance against an accompanying Y-tetrachord. But the Y-tetrachord is just as frequently “horizontalized.” Many of the longer melodic phrases in the movement can be parsed into alternating X’s and Y’s (see Ex. 7-23c).

The passage in mm. 7–11 of Ex. 7–22 pits the two symmetrical hexachords one against the other with increasing speed, while in mm. 11–12 the quartet pairs off into two duos at the octave, demonstrating the symmetry of the X-tetrachord by playing it against its inversion from a common starting point at C♯D≭, the axis of the Y-tetrachord. But of course these inversionally equivalent motives are also palindromic, having a pitch succession that is the same reading front to back or back to front. Just as in Schoenberg’s atonal utopia, in Bartók’s symmetricalized musical space there is “no absolute up or down, right or left.” Bartók, too, has “emancipated the dissonance,” and with similar effect. But his musical utopia is not “pantonal.” There are definite normative harmonies, which can be departed from, returned to, progressed between, and embellished; and so, despite everything, a sense of pitch hierarchy is maintained.

And there is a sense of closure at m. 13, brought about in a way that recalls the ending of Schoenberg’s Erwartung: a cluster covering the tritone B≭–E, which amounts to two conjunct X-tetrachords interlocking on their common point of origin, C♯D≭, and filling in the space of the original Y-tetrachord (see Ex. 7-23d). That space is exactly half of the total available musical space, leaving the other half in reserve for “complementation,” thus allowing for further harmonic movement.

The air having been cleared by the cluster chord and the following two-beat rest, the music at m. 14 has the effect of a new theme. It is none other than what we have already encountered as the main theme of the last movement, the one that, together with its transpositions, exhausts an octatonic scale. So it does here. Note well the outer limits of the theme at its two transpositions: in the second violin, beginning at the pickup to m. 16, the limits are defined as the tritone C♯/G; in the first violin, beginning at m. 17, the limits are the tritone G♯/D.

The same limits are described “inside out” (i.e., D/A≭) by the viola, beginning at the pickup to m. 19. On repetition, the viola connects the A≭ to a high D≭ that comes down via a G, thus combining in one line the two tritones expressed by the two violin parts. The first violin takes a cue from this at mm. 20–21, with a line that turns the viola’s tritones inside out again, combining the tritone G♯/D going up with the tritone G/C♯ coming down. In m. 22, the first violin plays a variant that is entirely confined to the two tritones (compare mm. 31–34 in Ex. 7-19c from the last movement).

But this configuration, of course, is none other than our “Z-tetrachord,” and now we have an inkling into the origin of the term. It was coined by Leo Treitler, an eminent music historian who was then a graduate student in composition at Princeton University, in an article of 1959, “Harmonic Procedure in the Fourth Quartet of Bela Bartók.”¹⁷ Treitler demonstrated the functional equivalence of this harmony, which (as we have already noted) is inversionally symmetrical at multiple axes, with the “X” and “Y” tetrachords already named by Perle. This cumulative process of discovery, in which the work of various scholars contributed toward the elucidation of Bartók’s symmetrical arrays and the tonal system to which they gave rise, was one of the notable detective stories of twentieth-century musicology.

Once discovered as a discrete harmony, the Z-tetrachord revealed fascinating properties. As a look back at Ex. 7-21 will confirm, the Z-tetrachord that Bartók gradually introduces over the span of mm. 18–22 in the first movement of the quartet is deeply embedded in the structure of the “sum 9” array, where it is labeled “Z₂.” It is the sum of the intervals in positions 3 and 4, whether one reads the top staff, the bottom staff, or the two staves as a simultaneity. It is, in effect, the inversional-cum-palindromic fulcrum or balance-point of the array. And the X-tetrachord is just as deeply embedded in the array: it is the sum of the intervals in positions 1 and 2, as well as positions 5 and 6 in any one staff. And the content of the tetrachord as it is first displayed as a harmony (m. 5) corresponds exactly to the intervals in positions 1 and 2 on the upper staff of the “sum 3” array (Ex. 7-21b).

Now that we know the story of the tetrachords’ christening, and are armed accordingly with some insight into their shared properties, we can appreciate what happens in the long passage beginning in m. 48 (Ex. 7-24a), characterized by dogged sequences of mirror writing and by one of the most widely imitated aspects of Bartók’s maximalistic quartet manner: grinding glissandos in contrary motion. The latter take over in m. 75 and drive the movement to a peak of harmonic tension. Every harmony from m. 75 to m. 93 (see Ex. 7-24b for its beginning) consists of one of the three symmetrical tetrachords. As Treitler noted in his article of 1959 (building on an observation that Perle had published four years previously), “the z-group is related to the y as the latter is to the x; i.e., the y expands to the z [as the x expands cadentially to the y].” But when Z proceeds to Z, the relationship can be described as “octatonic complementation”: the two will always be found in complementary positions in a symmetrical array, their sum being an octatonic scale (see Ex. 7-25, an analytical reduction of Ex. 7-24b).

Beginning at m. 83, the “horizontalized” version of the X-tetrachord begins insinuating itself into the chordal texture, leading to a conspicuous reprise ten bars later of the movement’s opening phrases, juxtaposed with horizontalized X-tetrachords and mirror glissandos of even greater compass than before (Ex. 7-26). One recognizes in this section the return of many of the opening gestures, including the modified reintroduction of the Z-tetrachord. (See Ex. 7-27 for an illustration of the process, in which some extra notes are interpolated that can be referred equally to the octatonic or to the “folk” pentatonic scale.) It is surely no accident that the Z-tetrachord in violin II now corresponds to the sum of the axes in Ex. 7-21b.

Citation (MLA):

Richard Taruskin. "Chapter 7 Social Validation." The Oxford History of Western Music. Oxford University Press. New York, USA. n.d. Web. 8 Dec. 2024. <https://www.oxfordwesternmusic.com/view/Volume4/actrade-9780195384840-div1-007006.xml>.

Citation (APA):

Taruskin, R. (n.d.). Chapter 7 Social Validation. In Oxford University Press, Music in the Early Twentieth Century. New York, USA. Retrieved 8 Dec. 2024, from https://www.oxfordwesternmusic.com/view/Volume4/actrade-9780195384840-div1-007006.xml

Citation (Chicago):

Richard Taruskin. "Chapter 7 Social Validation." In Music in the Early Twentieth Century, Oxford University Press. (New York, USA, n.d.). Retrieved 8 Dec. 2024, from https://www.oxfordwesternmusic.com/view/Volume4/actrade-9780195384840-div1-007006.xml