Root Versus Linear Analysis of Chromaticism: A Comparative Study of Selected Excerpts from the Oeuvres of Chopin

October 1, 1996

Chopin's nineteenth-century chromaticism can be problematic for analysts—although Chopin's music is tonal, some passages defy chord analysis. At times Chopin's non-harmonic chromaticism simply masks the harmonic progressions; at other times extreme chromatic harmonies prevent analysis with traditional Roman numerals. In the past few decades alternate analytical systems have provided fresh interpretations of Chopin's chromaticism. In particular, Schenkerian analyses have explained Chopin's chromaticism in terms of voice leading. From this perspective chromatic harmonies are understood as the composing-out of conjunct linear movement. Schenkerian analyses have shown organic function in chromatic passages where traditional Roman numerals appear meaningless. However, root-based analysis can still offer fresh insight to chromaticism if not limited to Roman numerals. The purpose of this essay is to offer root analysis in passages with extreme chromaticism that might, at first, seem better suited to linear analysis. Specifically, the present essay analyzes chromaticism from three of Chopin's oeuvre—Prelude in A minor, Op. 28, No. 2; Etude in E major, Op. 10, No. 3, and Polonaise-Fantasie in vol36id52 major, Op. 61—in response to published Schenkerian analyses of the same passages.


Prelude in A minor, Op. 28, No. 2

Of the three works to be considered in this study, Chopin's Prelude in A minor, Op. 28, No. 2 has the least chromatic harmony. Cheryl Noden-Skinner, in the Fall 1984 issue of this journal, offers a Schenkerian analysis of the A minor Prelude and states that the A minor prelude "exhibits one of Chopin's most inventive uses of ambiguous tonality" which is achieved "through linear movement which forms non-functional harmonies."1 The Schenkerian analysis "illustrates [Chopin's] expression of the dominant of A Minor throughout the entire prelude."2 The tonal ambiguity is made clear in the Schenkerian analysis because it is E minor that is supported at a higher structural level, not the tonic A, until the end of the prelude.

Although this prelude may have "ambiguous tonality," the chromaticism within the prelude does not prevent traditional harmonic analysis. The vol36id52 of the opening measure is simply a chromatic lower neighbor to the pitch B within the context of E minor harmony (Example 1).


Example 1.



Since the prelude is brief, a harmonic reduction of the entire prelude is included in Example 2.3

Example 2.



From any harmonic approach of the foreground, it is obvious that the tonic A minor is not established in the opening measures because there in no leading tone (vol36id52) nor any tonal-defining progression (i.e. cadence). However, traditional harmonic analysis reveals that there are three tonal regions within the prelude and all have cadential progressions that "tease" the ear by means of interruption. The first phrase (mm. 1-7) cadences in G major, but the satisfaction of resolution (implied on beat 1 of m. 7) is immediately taken away with the introduction of pitch E.

The second phrase (mm. 8-13) begins as a transposition of the first in the tonal region of D major, but the anticipated cadence does not occur (thus, the "tease"). Instead, the second phrase ends deceptively: the root of the A major chord (V) becomes the fifth of a half-diminished seventh chord (mm. 10-11). The change in harmony may be viewed as a linear-derived motion since there are two common tones and the other two tones progress by half-steps. However, there is a harmonic basis as well. If the vol36id52 half-diminished seventh chord of measure 11 is interpreted as a dominant substitute (i.e. an incomplete dominant ninth), the implied root progression between mm. 10-11 would be A-B, the same root movement as the common V-vi deceptive cadence (in the context of the prevailing D major tonality; see Example 3).


Example 3.



Since the following tonal region is not yet established at m. 11, the vol36id52Ø7 chord can be understood as viiØ7/V in the key of A major (since in literature viiØ7/V is used primarily in the major mode)—minor modality is not clear until m. 15 (since modal mixture does allow for viio7/V). Otherwise, the vol36id52 of mm. 11-12 can be considered a suspension into the functional viio7/V in A minor—either interpretation suggests an implied B root which is the usual root of resolution in a deceptive cadence (in the immediate tonal context).

The third and final phrase (mm. 14 ff.) teases the ear because the tonic six-four (m. 15) does not immediately resolve into the dominant (the preceding Fr6 sets up the tonic six-four as a cadential six-four, which creates the expectation for a resolution into the dominant). Although the tonic six-four chord "does not function as a resolution"4 on the immediate level, the harmony does eventually come to the dominant (m. 21 beat 3; marked by the arrow in Example 2) so the tonic six-four chord can also be understood as functioning as an interrupted resolution of the dominant at a higher level. The interrupted resolution of the dominant (between the tonic six-four and dominant) functions like the closing passage of a concerto movement in which resolution of the closing dominant is interrupted by the cadenza inserted after the arrival on the cadential six-four. The unaccompanied melody, structural location (before the final cadence) and slowing tempo (slentando) support an interpretation of measures 17-21 as an inserted "cadenza" interrupting the resolution of the tonic six-four into the dominant. Within the intermediate passage, a diatonic progression of pre-dominant chords is implied (as indicated in Example 2) when considering the harmonic implication of the melody.

The three phrases can be found in the Schenkerian graph by following the sequencing of the descending fourth motive, but harmonic movement is revealed via the unfolding of A minor so the principal pitches of the Urlinie are independent of the phrase structures (the scale degrees vol36id52-vol36id52-vol36id52-vol36id52-vol36id52 are found in measures 3, 4, 17 [last eighth], 21 and 23 respectively). On the other hand, a traditional harmonic analysis emphasizes the similarity between phrase structures and harmonic movement: each phrase uses a tonic six-four to approach the dominant (at different tonal levels). Both analyses make it clear that the home tonality is not confirmed until late in the prelude. All of the harmonic progressions (with the possible exception of the deceptive resolution in the second phrase-unit) are typical of common-practice harmony, thus the harmonies formed can be discussed as functional (and thereby not meaningless).


Etude in E major, Op. 10, No. 3

Chopin's Etude in E major, Op. 10, No. 3, contains considerable chromaticism in the middle section of this tripartite composition. Robert Parks has an illuminating Schenkerian analysis of measures 21-54.5 Of particular interest is the chromaticism that begins in measure 38 with a succession of fully-diminished seventh chords. Parks views the passage from mm. 38-41 as the arrival on the implied vol36id52 in the bass which "serves to prolongate the neighbor note motion around the vol36id52 in the soprano;" the implied vol36id52 is approached by a stepwise ascent in the bass from the vol36id52 that begins the "B" section.6 The chromaticism is part of the prolongation of the harmony II 7 (vol36id52) that begins in measure 22.

Although an analysis with Roman numerals might not be illuminating here, a harmonic analysis with implied roots (the missing "root" of a dominant ninth) offers an interesting harmonic perspective.7 Root analysis, based upon Chopin's spelling of the harmonies, reveals that this "chromatic" passage contains only diatonic roots and, with the exception of the first chord of measure 38, three implied roots are present: E, vol36id52 and B (Example 4a).


Example 4a.



The vol36id52 diminished chord at the beginning of measure 38 can be represented by two implied roots: vol36id52, which shows the fifth progression from the previous chord; and E (allowing for enharmonic spelling, which is common in music literature of this period), which shows the relationship to the following chord (the harmonies of the first two chords in m. 38 are enharmonic equivalents, thereby establishing a consistent pattern of fundamentals that continues into the next two measures).

Significant to the understanding of this etude is the observation that the progression of implied roots of mm. 38-41 directly reflects the progression of roots that begin the "B" section (Example 5).


Example 5.



Also, the implied roots on beat 1 of measures 38-40 reflect the same progression at another structural level (Example 4b).


Example 4b.


The structural level of Example 4b is made explicit in the music because the dynamics, phrasing (i.e. slurs) and directional contours of each measure drive towards beat 1 (the immediate repetition of the harmony at the beginning of each measure also creates a slight agogic stress).

A somewhat contracted and less explicit structural level of the E-vol36id52-B progression continues into measure 42 (Example 4c):


Example 4c.


the implied root E again receives metric stress (on beat 1 of m. 41) as well as some phrase emphasis (E is the end of a 1-measure slur), but there is no agogic emphasis; vol36id52 (the last sixteenth of m. 41) receives harmonic stress because it is the first major-minor seventh chord of this passage, thus vol36id52 is an actual and not implied root (the preceding vol36id52o7 suggests a strong fifth progression via an implied vol36id52 root, but it is enharmonic with vol36id52o7 and is probably heard in that context); the B of measure 42 is also an actual root (not implied) and receives obvious metric, dynamic and agogic stress.

E, vol36id52 and B remain the only roots, whether actual or implied, through measure 54 (beat 1), but do not consistently appear in the same order. Nevertheless, the chromaticism from mm. 38-54 can be analyzed with just three diatonic roots and, from this perspective, can be understood as a development (or prolongation) of the opening progression of the "B" section. In contrast, the Schenkerian analysis shows the same chromaticism as a prolongation of a single harmony.


Polonaise-Fantasie in vol36id52 major, Op. 61

Chopin's Polonaise-Fantasie is a long and complex composition with abundant chromaticism. One of the most thorough analyses of the chromaticism of this piece is a Schenkerian analysis from William YaDeau.8 However, because of the limited scope of this paper, comparative analysis will be restricted to larger harmonic movement and sample chromatic passages.

The role of step vol36id52 in the Polonaise-Fantasie is obvious—the opening progression is i-vol36id52 (in the context of vol36id52 minor) and vol36id52 (written as B major) is the tonal region in the middle macrosection. YaDeau sees the tonal region B major as a prolongation of the lowered mediant degree of vol36id52 major which was established in the introduction through emphasis of the tonic minor. The B major section has a "limited vol36id52-vol36id52 descent" which "serves to de-emphasize the tonal prominence and durability of the mediant step, and to reinforce vol36id52 (=vol36id52 ) as vol36id52 of the Polonaise-Fantasie's initial step of the Urlinie."9 This perspective emphasizes the unfolding of a single tonality, vol36id52 , throughout the composition. The pitch B also has prominence at the Middleground level in the introduction (mm. 10 ff.) and at the Foreground level (mm. 80 ff.): in the introduction the essential harmonic motion is i-V with B as a continued arpeggiation of the minor tonic; at measure 80 B is part of the chromatic bass descent to scale degree vol36id52 (m. 92, V/V in the overall perspective of vol36id52 ).10

The overall harmonic movement to B major in the middle of the Polonaise-Fantasie can also be viewed as a large-scale realization of the harmonic motion of i-vol36id52 of the introduction. This view emphasizes the expansion of scale degree vol36id52 as a harmonic root. In the introduction, vol36id52/B serves as a pedal point from measures 9(beat 3)-16 (albeit with sixteenth-note ornamentation; Example 6).


Example 6.


The principal harmony of mm. 10-11 is a D fully-diminished four-two chord with vol36id52 dominant-seventh chord for the final eighth. The progression can be analyzed in the context of vol36id52 minor (mm. 7 ff. is a transposition of the beginning at the level of the minor dominant). The vol36id52 dominant-seventh chord first serves as the enharmonically spelled augmented-sixth chord progressing to the "incomplete" dominant-ninth (viio7), but is reinterpreted as V 7 in the region of E major (m. 11, last eighth). Throughout measures 12-16, B can be analyzed as the root or implied root—although the tonal region is E major, the E sonority only appears in the cadential six-four position, thus as an extension of the dominant. If one accepts this interpretation of the harmony, B is the principal harmonic diversion of the introduction in terms of time accent. Although i-V may represent the larger, ultimate harmonic motion of the introduction, vol36id52 is the harmony of agogic stress and represents a realization of the opening progression i-vol36id52.

The next section in the Polonaise-Fantasie that has a prominent B bass is within the "B" section (mm. 66 ff.) at measure 80 (Example 7).


Example 7.


The "B" section is built around a repeating sixteenth-note pattern that metrically and agogically emphasizes the harmonies vol36id52 (mm. 66 ff.) then C (mm. 72 ff.). The principal harmony of measures 76-79 is a C dominant-seventh chord, suggesting F minor in the immediate context. However, instead of resolving to F, the C dominant harmony dissolves (via C 9/eo7 and briefly vol36id52o) into an E six-four chord (m. 80); but if one considers the larger harmonic movement (primarily C 7 to E six-four), the resolution suggests a reinterpretation of the function of C 7 as an enharmonic Gr6 (in the new tonal region of E). The bass (B) of measure 80 is metrically and agogically stressed for 2 measures. The B bass has a relatively short emphasis in this section, but its presence is accentuated by the context: it is the only appearance of the "B" theme over a six-four bass (compare with mm. 66, 72, 84, and 86); and the resolution has an immediate element of surprise since V 7-I resolutions have been relatively regular in this section (avoidance of dominant-tonic resolutions is characteristic of this work as a whole). Both the introduction and section "B" introduce E major through a B bass and utilization of the enharmonic V 7/Gr6 sonority as a pivot (although the enharmonic functions are reversed), and both can be viewed as a larger realization of the opening i-vol36id52. No less significant to the understanding of this work is that Chopin introduces vol36id52 at the approximate mid-point in three different structural units (the introduction, the "B" section, and the entire work).

Enharmonic use of V 7/Aug6 sonority is relatively common in Chopin's music and has a double-employment function (to borrow a term from Rameau) suggesting two roots (or implied roots) a tritone apart.11 Thus, the C 7 sonority preceding measure 80 could be represented by the "roots" C and vol36id52, showing the chord's relationship to the preceding F minor and succeeding B (Example 8).


Example 8.


Some of the most chromatic passages can be understood in this context. For instance, the Polonaise-Fantasie has a chromatic passage at the end of the "A" section (mm. 55 ff.) that does not lend itself easily to analysis with Roman numerals. YaDeau's analysis shows this passage as a descending chromatic progression created by the octaves in the outer voices.12 Linear understanding of this passage is perhaps most accessible, but root analysis is possible when understanding the tritone duplicity of the dominant-seventh sonority. The harmonic reduction in Example 9 shows that the roots or implied roots suggest movement by fifth.


Example 9.


The enharmonic interpretation of the harmonies in Example 9 as Aug6 chords means that the chords are not in the common position (there is actually no augmented sixth but a diminished third), but this same position is present in other Chopin works (see m. 5 in his G minor prelude and m. 2 in his op. 53 Polonaise). The apparently non-functional use of chromatically descending dominant-seventh chords can be understood as a functional progression via the double-employment of roots, and as part of the regular harmonic vocabulary of Chopin.


* * *

In summary, Schenkerian analyses of Chopin's music offer fresh perspectives on Chopin's chromaticism, but demonstrating the linear origins to chromatic passages does not necessarily supersede root-based analysis. In the case of Chopin's Prelude in A minor, Op. 28, No. 2, or the key relationships in the Polonaise-Fantasie, Op. 61, linear analysis and traditional Roman numeral-analysis can comfortably coexist, emphasizing different aspects of the same harmonies. In the case of the more chromatic excerpts from Chopin's Etude in E major, Op. 10, No. 3, and Polonaise-Fantasie, Roman numerals appear to offer little insight; in these cases Schenkerian analysis is particularly successful in showing organic meaning to the chromaticism. However, if harmonic analysis is not confined to the Roman numeral system, chromaticism can still be discussed in terms of root (or implied root) movement and, as this essay suggests, still be understood as functional.

1Cheryl Noden-Skinner, "Tonal Ambiguity in the Opening Measures of Selected Works by Chopin," College Music Symposium 24, No. 2 (Fall 1984), 28-34. Quote from p. 31.

2Ibid., 31-32. Refer to Noden-Skinner's Ex. 2. Her graphic analysis reproduces, for the most part, Schenker's own analysis, with the exception of omitting a few interior pitches and slurs, as well as altering the rhythmic values. Compare with Fig. 110, a3 in Heinrich Schenker, Free Composition: Supplemental Musical Examples, ed. and trans. by Ernst Oster (New York: Longman, 1979).

3Robert Collet analyzes measures 10-16 of this prelude and refers to the chromaticism as "characteristic chromatic auxiliary notes." See his "Studies, Preludes and Impromptus," The Chopin Companion: Profiles of the Man and the Musician, ed. Alan Walker (New York: W. W. Norton, 1973), 140. Since this prelude is approachable even to undergraduate music students, the purpose of including a complete analysis is not to show originality, but to refute Noden-Skinner's observation that "chord labels would prove meaningless" (p. 32).

4Noden-Skinner, op. cit., p. 32.

5Robert Parks, "Voice Leading and Chromatic Harmony in the Music of Chopin," Journal of Music Theory 20, No. 2 (1976), 189-214.

6Ibid., 202-212.

7The present analysis of the E major etude, as well as the following analysis of the Polonaise-Fantasie, are adapted from my dissertation: An Application of Grundgestalt Theory in the Late Chromatic Music of Chopin: A Study of His Last Three Polonaises, Univ. of North Texas (Ann Arbor: UMI, 1994).

8William YaDeau, Tonal and Formal Structure in Selected Larger Works of Chopin, Diss., Univ. of Illinois at Urbana-Champaign (Ann Arbor: UMI, 1980).

9Ibid., p. 149.

10Ibid., p. 142; p. 171, Fig 22.

11I am basing my interpretation of roots on the theories of Simon Sechter as found in The Correct Order of Fundamental Harmonies, 1853, 4th ed. Trans. C. C. Muller (New York: Wm. A. Pond & Co., 1880). Sechter, a leading teacher of harmony and counterpoint in Vienna during the life of Chopin, views the root of augmented-sixth chords (whether present or implied) as the supertonic scale degree (thus a major third below the upper note of the augmented sixth; see p. 180).

12YaDeau, op. cit., pp. 151, 171, fig. 22.

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