In short a rule of law emerged; until all twelve notes have occurred none of them may occur again. The most important thing is that each run of twelve notes marked a division within the piece, idea or theme.1
Thus Webern described the compositional process he used in his early atonal works. The "run" through the twelve notes of the chromatic scale is Webern's substitution for the harmonic/melodic progression of tonal music. That each "run" served as a means of marking off boundaries attests to the stabilizing quality attached to the filling out of the chromatic space.
Stability and resolution come about in tonal music principally by means of the cadence. There are other means of achieving stability, such as rhythmic periodicity and phrase structure, but these are tightly bound to tonal direction. Tonal direction finds resolution at the cadence.
In the early atonal works of the Second Viennese School, the unfolding of pitches aims at filling out the chromatic space.2 The completion of this drive toward chromatic saturation serves a cadential function. The unfolding of pitches within the twelve-note span is guided by a premise that opposes both the repetition and omission of pitches. Yet, what is one to make of the occurrence of frequent deviations from such lines of order? Such deviations cannot be viewed solely in terms of local relationships, but have long-range implications that apply both to concrete musical relationships and to musical conventions. These implications lie beyond a particular "rule of law" and lead to the work itself.
In classical/romantic music, many compositions operate beyond the systematized framework of common-practice harmony and tonality and, in some works, the tendency toward the filling out of the chromatic space is present. Where and how does chromatic saturation occur and what is its function in music of the classic/romantic era? Can the understanding of such instances give greater insight into the nature of the atonal works for which this procedure is the norm? These are the issues and questions this article will address.
The concept of perpetual variation, whereby musical elements such as pitch, harmony, or rhythm continually transform and grow, was important to the composers of the Second Viennese School. In his Theory of Harmony, Schoenberg gives the model for a chromatically saturated chord. The chord is established by the rules of classical tonal harmony. However, Schoenberg warns that this chord is not for "fastidious ears".3 By adding four different bass notes to the same diminished-seventh chord, Schoenberg shows how the diminished-seventh chord can be interpreted as four different dominant-ninth chords. The different roots of the dominant-ninth chords form another diminished-seventh chord, and the roots of the chords to which these dominant-ninth chords resolve form a third diminished-seventh chord. The chords of resolution can be either major or minor. Through the concentration of the three diminished-seventh chords, or by what Schoenberg refers to as "thinking faster,"4 a chord containing twelve different notes is constructed. We see from this demonstration an important example of the way in which Schoenberg's thinking grew out of traditional tonal concepts.
Example 1. Arnold Schoenberg, twelve-note chord.5
The roots of Schoenberg's formulation can be found in numerous examples from the literature. The transition passages from Mozart's Piano Sonata in F major, K. 332, show how chromatic completion can function in a tonal work. The passage from mm. 22-40 is clearly set off from both the first and second themes by rests. There is no preparation for this passage, which begins abruptly, forte, with octaves in similar motion in the new key of D minor. The transition passes briefly through the tonal areas of D minor and C minor. The use of the major dominant sets the transition off from the second theme, which follows abruptly in C major. Both D-minor and C-minor chords are preceded by diminished-seventh chords, vii°7/vi = -E-G- and vii°7/v = B-D-F-. But Mozart stops his descent at the minor dominant and the second theme begins, in the key of C major.
The recapitulation of the transition at m. 157 begins in the same way it began in the exposition, but after reaching C minor, it passes through the additional key of minor, the minor subdominant. Tonally this move to the subdominant is common in the closing half of the sonata-allegro movement and provides a loosening of the tension created by the focus on the dominant in the exposition. But this also provides a completion of the diminished-seventh chord succession. Just as the D-minor and C-minor chords were preceded by vii°7, so is the -minor chord (vii°7/iv = A-C--). The diminished-seventh chords descend by whole step in this progression. The complete succession of diminished-seventh chords in the recapitulation contains all twelve pitches and fills in what was omitted in the exposition.
Thus the recapitulation provides tonal/chromatic completion. The reduction of this passage (Example 2) demonstrates how chromatic completion is achieved not only among diminished-seventh chords but also among principal pitches in the outer voices.
Example 2. Mozart, Piano Sonata in F major, K. 332, mm. 157-77.
The direct succession of diminished-seventh chords containing all twelve pitches is a common element in romantic music. In Chopin's Mazurka in C Minor, Op. 56, No.3, such a progression occurs just prior to the first important cadence in the coda. The three diminished-seventh chords substitute for the dominant-seventh chords in a chromatic circle-of-fifths sequence. The diminished-seventh chords descend by half step in this progression, which could also be interpreted as a circle-of-fifths progression of rootless dominant-ninth chords. Because of the rootless nature of the diminished-seventh chord, the tonal location of any point in the progression is ambiguous. The N6-V7-I progression which follows anchors the preceding circle-of-fifths sequence of diminished-seventh chords in the key of C Major.
Example 3. Chopin, Mazurka in C minor, Op. 56, No. 3, mm. 184-89.
Chromatic completion is a prominent shaping element of the opening idea of the Prelude to Act I of Tristan und Isolde. The generating shape in the upper voice is the four-note chromatic -A--B. In the first two phrases of the Prelude, transpositions and inversions of the motive interconnect and, when combined, result in an ascending and descending chromatic line extending from the axis pitch , to D, the tritone above and below.
Example 4. Wagner, Prelude to Tristan und Isolde, mm. 1-7.
Note how the diminished-seventh chord (-B-D-F) is embedded in the chromaticism of ascending and descending half steps. The second phrase provides chromatic completion. As a transposition of the first phrase, phrase two generates the four pitches of the twelve-note chromatic (, G, C, ) that were missing from the first phrase. Phrase three of the Prelude is not an exact transposition of the first two. Of significance in phrase three is the extra pitch in the upper voice line. The directly ascending chromatic line of the upper voice, which began its ascent with the opening -B motive, stops abruptly and dramatically at in m. 11. What follows is a series of octave transpositions of the rising minor second -. Only one pitch is needed to complete the twelve-note span, begun in m. 2 with . When the music resumes its motion in m. 16, the twelfth pitch, G, is provided. However, it is placed on a weak part of the measure and is passed over quickly. Rather than serving as a goal, its function is that of a passing tone to A. Interestingly, it is the appearance of G at the end of the Prelude that constitutes the final close. The G at the end of the Prelude can be viewed as stemming directly from the chromatic line of the upper voice found at the Prelude's opening. As such, it is not surprising that the closing G should be preceded by an .
There is a significant correlation between chromatic completion and tonal resolution in the Adagio from Mahler's Tenth Symphony. This correlation is highlighted twice in the Adagio, each time involving the closure of large sections. The central tonal focus of the Adagio oscillates between minor and major and reaches its striking climax in the dominant-nineteenth chord at m. 206. It is at this point that musical intensity, chromatic saturation, and tonal resolution coincide. The dominant-nineteenth chord is made up of nine different pitches: --B-D-F-A-C--G, and bears immediate resemblance to Schoenberg's twelve-note chord in Example 1. The A that is sustained in the trumpets through m. 213 will act as a leading tone to , the third of major. Underneath the suspended A, in m. 211, the leading tone to , , occurs in the cellos. The descends by half step to the tenth pitch, E, which in turn descends by half step through and D. It is at the double bar (m. 213) that chromatic completion, tonal resolution, and what have been the two main motivic ideas of the Adagio, A- and -D, interlock. A in the trumpet, acting as leading tone to the third of major, ascends by half step to , the eleventh pitch, in the first violins and flutes. At the same time, D descends to , and the dominant-nineteenth chord resolves to I in major. The twelfth pitch, , is the root of the I to which the dominant-nineteenth chord resolves. The resolution to major signals this key as the principal key area of the remainder of the Adagio.
Example 5. Mahler, Adagio, Tenth Symphony, mm. 203-13.
Chromatic completion and tonal resolution coincide for the second time in the closing measures of the Adagio. The closing passage begins with a variation of the opening viola solo in the first violins (m. 256). Shortly thereafter (m. 259) the violins begin a chromatic ascent from scale degrees 1 to 5. Set apart from the ascending chromatic line in the violins are the pitches D and E, orchestrated in the flutes, clarinets, and violas (mm. 259-61). These two pitches are representative of the key of minor and immediately following their occurrence (m. 262) the -minor chord makes a brief appearance, as support to the A in the chromatic line of the violins. After this brief appearance, A rises to , which is held for three measures (263-65). Once the chromatic line reaches its goal, , only two pitches remain to complete the total chromatic— and . The is held for four measures while a dominant-thirteenth chord, containing both and , unfolds beneath it. In m. 277, the , now doubled in the flutes, proceeds melodically to , which functions as a passing tone to (m. 272), the twelfth pitch. The leading tone, , subsequently rises to . Thus the chromatic forces embedded in this passage contribute to cadential structure. That chromatic completion occurs at the two major cadence points in this work gives clear conformation of the relationship.
In the tonal works examined thus far, chromatic completion has played an important contributing role in the cadential structure. Often in the closure of large sections, the music moves on at least two structural levels, one tonal, the other chromatic. In the early atonal works of the Second Viennese School, all parts move in accordance with chromaticism. In these works chromatic completion is no longer just one part of the structure: it is the fundamental goal. Through the observation of chromatic completion in tonal music one gains insight into the reason for its stabilizing effect in atonal music. In the early atonal works of the Second Viennese School, chromatic completion maintains and, by necessity, enlarges its cadential role. Moreover, in the absence of tonality, the registral and timbral properties of pitch take on heightened importance. The fifth movement from Webern's Six Bagatelles for String Quartet, Op. 9, provides a crystalline example of the process of chromatic completion. Through the analysis of voice-leading procedures in movement five, structural implications inherent in the process are revealed.
The generating interval for the fifth movement is the minor second. The piece is divided into three sections, delineated by the principle of chromatic completion. Measures 1-7 constitute section 1. In m. 1, two musical elements are presented: the sustained C-E major third in the outer voices (cello and viola) and the linear major second - in the inner voice (second violin). The two motives combined result in two distinct pairs of inversionally-related pitch-class sets: C--/--E and C--E/C--E. The latter pair of pitch-class sets are a minor second expansion of the former. Motivic organization in this piece is the result of the consistent use of these two pairs of pitch-class sets. The principle of intervallic expansion by the minor second provides the directional force for the unfolding of the chromatic span. In section 1, the twelve pitches unfold in a progression of intervals that contract and expand by the interval of a minor second.
Example 6. Webern, Six Bagatelles for String Quartet, Op. 9, No. 5, registral expansion, mm. 1-7.
Thus registral contraction and expansion in one voice are balanced by the same type of motion in the other voice. Only two pitches prove to be an exception to the rule: G and in m. 6. The adjustment Webern makes is a necessary one. If from the minor sixth - Webern had continued to expand out by the minor second in both directions, he would have eventually run into the octave.
Example 7. Webern, Six Bagatelles for String Quartet, Op. 9, No. 5, minor-second expansion, mm. 4-7.
In order to avoid the octave, Webern simultaneously introduces the G and in the same register. The is sustained past the G and it intersects with the A in m. 7, in the bottom voice. As a result, the final vertical interval in the twelve-note run is the major seventh, an inversion of the generating minor second.
Throughout section 1, various pitch repetitions occur. Webern allows himself the repetition of pitches as long as they recur in their initial fixed register. This allows the unfolding progression of new pitches and the subsequent unfolding of registral space to go undisturbed. In addition, the restriction of each different note to a fixed register increases the sense of a need for the completion of the chromatic total. The process of perpetual variation pervades the work, particularly in regard to pitch repetition. Upon each of its restatements, a pitch undergoes subtle rhythmic and timbral variation.
The repeated C in m. 5, one octave higher than its initial appearance in m. 1, is unusual in this regard. It breaks the orderly expansion of interval and register and its repetition does not conform to Webern's previous handling of repeated notes. Therefore its function cannot be explained on the local level. In specifying an up-bow (the only bowing indication in the piece), Webern clearly intends for this pitch to project beyond its local surroundings. An examination of the second twelve-note run that comprises section 2 of the piece clarifies the appearance of the C in m. 5.
The closing of section 1 and the opening of section 2, which covers mm. 7-11, overlap in m. 7. The beginning of section 2 differs from section 1 in a number of ways. The texture of section 2 is more refracted, in the sense that the elements are broken up and refocused. Pitches are introduced one at a time instead of in pairs. Though the generating interval remains the minor second, the second section makes use of its compound expansions (the minor ninth) as well.
Section 2 begins with the minor tenth G- in m. 7 acting as an upbeat to section 2. This expands the registral space from the previous A- by a major second. Expansion by major second is a deviation from minor-second expansion. In this instance it is a means of avoiding the cross-octaves that would result from continued minor-second expansion.
Example 8. Webern, Six Bagatelles for String Quartet, Op. 9, No. 5, m. 7.
However, in expanding the space by the major second, minor-second expansion is implied in the cross relation of A and , and and G. Registrally the three pitches that follow, , E, and F, mark a return to the work's middle register. These pitches have appeared in close proximity before, in m. 3, where their appearance was preceded by a solo pizzicato D. Their subsequent appearance in m. 9 is this time followed by another solo pizzicato D. The D in m. 9 initiates a major fourteenth (D-) glissando that considerably expands the registral space of this movement. The glissando itself is a telescoping of the total two-octave chromatic, and with the D- appearing at opposite ends of the glissando, the minor-second relationship is literally torn apart registrally.
It is interesting to see how Webern has prepared the entrance of the glissando. The unaccompanied pizzicato D initiating the glissando has appeared before, an octave higher, in m. 2. The D is the only pitch so far to appear unaccompanied. The at the top of the glissando has been registrally prepared by the C in m. 5, a minor second below. In m. 5, the C had seemed out of place, both registrally and in regard to pitch repetition. However, with the appearance of in m. 9, a minor second above the C, the connection between the two pitches becomes apparent. To emphasize the connection Webern overlaps the in m. 9 with the following C a minor ninth below. This C is then followed by a B a minor second below, thus completing a four-note chromatic succession of pitches: D--C-B. The following in the first violin completes another four-note chromatic succession that was initiated in m. 8: E--F-. The final two notes of section 2 and of the second twelve-note run appear simultaneously in m. 11: -A. These are the same two pitches that closed the first twelve-note run in m. 7. The voice crossing that takes place between the A and in mm. 7 and 11 contributes to the closure of both sections.
Example 9. Webern, Six Bagatelles for String Quartet, Op. 9, No. 5, voice crossing, mm. 7 and 11.
As a result of this voice crossing, the gap that was left by the major-second expansion, A- to G-, is filled in. Further emphasis of the connection between these two points of closure is provided by the pitches that appear in m. 12, G and . The G and in m. 12 once again share the same register. This relates to their appearance in m. 6: it was the appearance of G and together in m. 6 that initiated the final A- interval of the section.
The final measure of the piece contains the first four-note simultaneity. Each pitch in the four-note chord receives a different articulation. The use of different timbres and articulations has played an important role in differentiating repeated pitches. Few pitches have appeared in exactly the same way twice. In this sense, the chord in m. 13 can be viewed as a timbral summing up of the piece.
Marking off the final cadence is the interval of a minor sixteenth, -D, in the viola. Initiated by the D- glissando in m. 9, these pitches represent a final minor-second expansion of the registral space. They are as a result the highest and lowest pitches of the work. Here again Webern uses voice crossing to contribute to the sense of closure.
Example 10. Webern, Six Bagatelles for String Quartet, Op. 9, No. 5, registral expansion and voice crossing, mm. 1-13.
The pitch content of m. 13 breaks down into a transposition of each of the two opening pitch-class sets from m. 1: B-D- and -D-E. The intersection between these final pitch-class sets occurs on D. It is this pitch that filled in the gap left between both pitch-class sets in m. 1. D is the final note of the piece and, as has been customary, it appears unaccompanied. Registrally, D has occurred as a polar pitch in three important locations. It is the final pitch, the highest note of the piece, and it initiates the glissando in m. 9. It is also the axis pitch of the whole movement, around which the others revolve.
The careful control of register, along with the use of such contrapuntal techniques as voice crossing, itself a registral operation, contributes to the organization of pitches within the twelve-note chromatic. That these operations most clearly demonstrate themselves at the closing of a twelve-note run attests to the importance of such a completion and contributes further to its stabilizing effect.
In reflecting on the process of composing the Bagatelles, Webern remarked, "Here I had the feeling, when all twelve notes have gone by, the piece is over. Much later I discovered that all this was part of the necessary development."6 It is profoundly instructive to consider Webern's remark both in light of what came before—and what was to come.
1Anton Webern, The Path To The New Music, trans. Leo Black, ed. Willi Reich (Bryn Mawr: Theodore Presser, 1963), 51.
2See Charles Rosen, Arnold Schoenberg (New York: Viking, 1975), 57-62.
3Arnold Schoenberg, Theory of Harmony, trans. Roy E. Carter (Berkeley: University of California Press, 1978), 367.