Transformation vs. Prolongation in Brahms's "In der Fremde"
In a recent paper presented at the annual meeting of the Society for Music Theory, Frank Samarotto discussed a specific point of contrast between the transformational approach to tonal harmony, in the form of Neo-Riemannian triadic transformational theories, and the prolongational approach, as exemplified by Schenkerian theory.2 He mentioned that many of the typical Neo-Riemannian transformations are indeed fairly common chord successions in tonal music. Although he initially questioned the applicability of Neo-Riemannian approaches to music in the common-practice style (since they draw upon analytical principles more typically associated with post-tonal theory), Samarotto was primarily interested in identifying moments in the music where "transformational steps might be leading to a real conflict with prolongational coherence."3
As he pointed out, this conflict has to do with the very different generative principles at play in these two theories. Specifically, transformational theories tend to focus on successive chord connections, without reference to a structural "tonic" chord. In contrast, prolongational approaches to music, such as Schenkerian analysis, focus on the relationship of foreground and middleground elements with a structural harmony in the background. However, if the procedures and techniques that are studied by Neo-Riemannian transformational theory were to be applied to a single chord, would this constitute a tonic prolongation?
Johannes Brahms's setting of Joseph Eichendorff's poem "In der Fremde," Op. 3, no. 5, presents an intriguing situation, in which aspects of both Neo-Riemannian transformational theories and prolongational theories of tonal music can be used in order to reveal a significant compositional idea in the song.4 After a brief discussion of important features of each theoretical approach, I will suggest a way that these theories might be synthesized to help explain a specific musical situation, which I refer to as "common-tone prolongation." Following this discussion, I will present an analysis of "In der Fremde" that draws upon these theories, and I will show that this musical interpretation resonates with significant aspects of the poetic text for this song. The purpose of this paper is thus twofold: on the one hand I will present an argument for how elements of transformational and prolongational theories of tonal music might be synthesized to reveal instances of common-tone prolongation, and on the other hand I will illustrate the viability of this concept through a close reading of Brahms's song.
In his book Generalized Musical Intervals and Transformations, David Lewin defines musical transformation as "something one does to a Klang [sonority] to obtain another Klang."5 In this respect, transformational theories are largely concerned with a series of operations applied to sonorities (such as consonant triads or diatonic seventh chords). Significantly, these transformations occur successively to various triads, one after the other, without necessarily being related to an overall referential chord or tonal center. This type of musical organization can be seen in Richard Cohn's analysis of the Scherzo of Beethoven's Ninth Symphony (Example 1).6 Cohn shows a series of alternating transformations, first "relative," and then "Leittonwechsel," that are applied to successive triads. The result is a kaleidoscopic harmonic odyssey in which nearly all of the major and minor triads are sounded.
Example 1. Beethoven, Symphony no. 9, II, mm. 143-176 (after Cohn 1997, fig.16).
This passage demonstrates the emphasis on common tones between adjacent triads, a central concern for the Neo-Riemannian theories of Cohn, Lewin, and others. Indeed, the three typical Neo-Riemannian transformations, relative, parallel, and Leittonwechsel,7 when applied to a consonant triad, will result in another triad that shares two common tones with the original, with the other voice moving a minimum distance, either by half- or whole-step. Such is the case in the Beethoven excerpt. Let us note that these transformations occur successively, and are not related to an overall tonal center. This has led some theorists to describe this type of organization as stepping outside the realm of tonality. Cohn, for example, traces the development of Neo-Riemannian theory as arising "in response to analytical problems posed by chromatic music that is triadic but not altogether tonally unified . . . " and contends that "some nineteenth century triadic music [is] to some extent tonally disunified, and, to that extent, `post-tonal'" (emphasis added).8
Prolongational approaches to tonal music, especially Schenkerian analysis, take a single basic triad as the generator of a hierarchy of foreground and middleground elaborations. In this respect, every foreground and middleground event is derived from a single background sonority. Implicit in the concept of prolongation is the notion that the triad being prolonged is not present in the foreground at every given moment in a musical setting.9 For example, in the Scherzo of Beethoven's Seventh Symphony the primary-tone A is not literally present above the tonic F major at the beginning of the movement (Example 2). Nevertheless, it is understood as the goal of the initial ascent, and thus still functions as the primary-tone for the movement.10
Example 2. Beethoven, Symphony no. 7, III, opening (after Schenker, Der freie Satz, fig. 37b).
It seems that these two theories stand at opposite ends of a spectrum, one concerned largely with common tones between chord-to-chord successions, the other with relationships between events in the foreground and a fundamental harmony in the background. Aspects of both theories seem persuasive in their own right, if for different reasons. How can one synthesize these two approaches into a useful and analytically persuasive third approach? Moreover, what possible goal would be served by this union of aspects of transformation and prolongation?
As mentioned above, the gulf between these two theories lies in the fact that the concern for overall tonal coherence so central to the Schenkerian approach is not at all required by transformational theory. Furthermore, the emphasis on successive harmonic connections, without reference to an overall tonal center, leads directly to the disunity pointed to by Cohn, who prefers the term "triadic post-tonality" for such music explained by Neo-Riemannian analysis.11
However, if the emphasis is shifted from successive chord-to-chord connections to associations between a single triad and the harmonies most closely related to it by the three typical Neo-Riemannian transformations, an interesting synthesis of transformational theories and prolongational theories will arise. The two common tones between a given tonic chord and its relative chord could be harmonically supported by both of these harmonies in a given musical setting. And this holds true for common tones between the tonic and the parallel or Leittonwechsel chords. Such a musical situation would be strongly analogous to the concept of prolongation, especially the notion that the prolonged entity (in this case the tonic chord) would not be literally present at every moment in a given musical setting, despite the fact that two of its members would be. One might even go so far as to describe this musical effect as "common-tone prolongation."12
A situation quite similar to the one I have just suggested can be seen in the Trio of Beethoven's Seventh Symphony (Example 3). The pitch A appears in three major triads, as the root of A major, the third of F major, and the fifth of D major, and Beethoven uses the pitch A to bridge the harmonic distance between the keys of F major and D major at the onset of the Trio. Returning to the opening of the Scherzo (Example 2), we see the prolongation of the pitch A as a common tone between the distantly-related keys of F major and A major. Indeed, the prolongation of the pitch A as a common tone between these three keys emerges as an important compositional idea of this movement. It should come as no surprise that Beethoven used this pitch in this way, given that the overall tonic of the Seventh Symphony is the key of A major.
Example 3a. A region within a Neo-Riemannian Tonnetz.
Example 3b. Beethoven, Symphony no. 7, III, opening of Trio.
From a purely transformational perspective, the two modulations mentioned above—from F major to A major, and from F major to D major—might be regarded as compound transformations upon the Neo-Riemannian Tonnetz.13 That is to say, the modulation from F major to A major that opens the Scherzo can be schematically represented as a combination of the Leittonwechsel and parallel transformations. Similarly, the move from F major to D major to open the Trio can be thought of as a simultaneous occurrence of the relative and parallel transformations. In both cases, this use of compound transformations results in a chromatic mediant relationship, in which the two keys are separated by a major or minor third, and both keys are major (via the parallel transformation that is coupled with either the Leittonwechsel or relative transformation, respectively).
The prolongation of the pitch A as common tone between these three distantly-related keys charges them with a great purpose and mission, like a compass whose needle points North no matter which direction one is traveling. But if two common tones were to be maintained between the tonic and its transformationally-related chords, would not the sense of prolongation be all the more powerful? Finally, what if a composer chooses to employ all three typical Neo-Riemannian transformations in a given piece, abruptly returning to tonic after each transformation? Would this not constitute a significant compositional idea, a union of transformation and prolongation? Brahms's setting of "In der Fremde" suggests precisely this.
Brahms's "In der Fremde"
"In der Fremde," the first of the op. 3 songs that Brahms composed (1852), shares many interesting similarities with Schumann's celebrated version from his Liederkreis, Op. 39. Both songs are set in the key of F-sharp minor, both end on a F-sharp major chord, and both share interesting motivic details.14 Furthermore, Brahms's setting uses Schumann's altered version of Joseph von Eichendorff's poem, which has been discussed by many scholars, and quite recently by Lawrence Zbikowski.15
Example 4. Text and Translation.
In der Fremde (Eichendorff)
Aus der Heimat hinter den Blitzen rot,
Wie bald, ach wie bald kommt die stille Zeit,
In Foreign Lands (trans. Zbikowski)
From home, behind the red lightning
How soon, oh how soon will that quiet time come,
The poem focuses on the notion of alienation, the narrator's profound sense of abandonment, and the prospects of living an anonymous, nomadic life. This is evidenced by the fact that the narrator's father and mother are long dead, and she is unknown in both her homeland and in her final place of rest.16 Zbikowski identifies the "discomfort of foreign surroundings" as the overriding theme for this poem.17 For Zbikowski, this feeling of discomfort is reflected musically by the "oppressiveness" of the tonic F-sharp minor, which returns abruptly after each harmonic excursion.18 Moreover, each of the four phrases in the song presents an excursion from the tonic in which the process of tonic-chord transformations generates the tonal motion.
Phrase 1, mm. 5-8
Brahms's setting can be regarded as a modified strophic structure, with the two poetic stanzas set by four musical strophes which begin identically but end differently each time. This type of modified strophic form was certainly not uncommon within the nineteenth century, and Brahms's own famous "Wie Melodien zieht es mir," Op. 105, no. 1, can be seen as representing this formal archetype. A four-bar piano introduction plants the motivic seed (Vorimitation) for the first vocal phrase (the first two lines of the poem), which establishes the pitch A over the tonic F-sharp minor harmony in m. 5; this pitch moves to its neighboring tone B over the V7 chord on beat 3. (ex. 5). The neighboring tone B is embellished with a rising third-progression, and unfolds to the pitch G-sharp on beat 4 over the V7 chord. The neighboring tone eventually finds its way back to the A on beat 3 of m. 6, once again over the tonic chord, preceded by a falling third-progression. Thus, mm. 5-6 can be seen as prolonging the pitch A over the tonic triad.
Examples 5. Brahms, "In der Fremde," mm. 5-8.
In m. 7 the melody moves to the pitch C-sharp over the cadential six-four chord of the relative key, A major. A similar neighboring motion occurs in these measures, as the pitch C-sharp is embellished with its neighboring tone D, preceded again by a rising third-progression, which unfolds to the pitch B on beat 4, over the V7 chord of A major. The neighboring tone D falls back to C-sharp in m. 8, over the A major chord. Thus, mm. 7-8 can be seen as prolonging the pitch C-sharp over an A major chord, as shown in Example 5b.
The net result of this first phrase is a harmonic motion from the F-sharp minor tonic to A major, a relative transformation. This is shown in transformational terms in example 5c, which provides a simple harmonic reduction of the first phrase. Note that the two common tones between the tonic and its relative chord, A and C-sharp, are the prolonged melodic pitches in Example 5b. These pitches bridge the gap between prolongation and transformation, in that they are the common tones between the tonic triad and its goal-chord, A major. This function is further emphasized by Brahms's piano accompaniment in m. 9. The right-hand figuration alternates between the common tones A and C-sharp as the harmony hollows out, leaving only these pitches to usher the return back to the tonic F-sharp minor in m. 10.
Phrase 2, mm. 10-16
After the initial move to the relative major, the harmony returns abruptly to the tonic F-sharp minor in m. 10 (Examples 6). Measures 10-11 are structurally equivalent to mm. 5-6, with the pitch A embellished by its neighboring tone B. In mm. 12-16, as the voice rises to the pitch F-sharp, with the emphatic 5-to-1 motion bracketed in example 6a, the accompaniment makes the sudden and unexpected move to D major, undercutting and erasing any sense of arrival gained by the vocal part. The effect is that of a sudden, unexpected plot-twist in the harmonic story, denying what should be a strong arrival on the tonic. The pitch F-sharp is itself embellished with its upper neighbor tone G-natural, which again is reached by a rising third-progression and unfolds to the pitch E over the V7 in D major. Example 6b shows the prolonged melodic pitches A and F-sharp for this second phrase.
Examples 6, Brahms, "In der Fremde," mm. 10-13.
The result of this second phrase is a harmonic motion from the F-sharp minor tonic to the chord of D major (Example 6c), in which the two prolonged pitches for the second phrase, A and F-sharp, are the common tones between the tonic and its goal-chord, D major. The overall effect is that of a transformation of the tonic to its Leittonwechsel chord. Furthermore, the gap from D major back to the F-sharp minor tonic is bridged in the same way as before. In mm. 15-16 of Brahms's piano accompaniment, the right-hand arpeggio figure again alternates between the pitches A and F-sharp, the common tones between the two keys, bridging the gap between D major and the return to F-sharp minor in m. 17.
Phrase 3, mm. 17-21
Like the first phrase, the third phrase moves from the tonic to its relative major (ex. 7). In fact, this phrase is very similar to the first vocal phrase, with one slight change: the C-sharp in m. 21 is embellished with a descending third-progression, which occurs twice, once in the piano, then again in the voice. Furthermore, the repeated third-progression has the effect of a hypermetrical expansion, with the addition of a full extra measure to the four-bar basic phrase. Note that the elongated vocal pitch E, which lingers above the piano's third-progression in m. 21, occurs on the word "über" (above [me]). This slight musical change points up an interesting moment of word-painting in the song: the soaring vocal E, and more importantly the accompanying hypermetrical expansion, corresponds to the moment in the poem that the speaker pauses and observes the forest canopy, and the forest's "beautiful solitude."
Example 7. Brahms, "In der Fremde," mm. 17-21.
Phrase 4, mm. 22-28
Whereas the first three phrases effectively venture away from the tonic to other keys, the final phrase remains in F-sharp (Example 8a). Measures 22-23 are similar to the previous phrases, with the pitch A being embellished by its neighboring tone B. In mm. 23-24 the music takes a turn toward the parallel major, as the prolonged melodic pitch A gives way to its chromatically-inflected counterpart A-sharp on the third beat in m. 24 (Example 8a, dashed arrow).
Example 8. Brahms, "In der Fremde," mm. 22-28.
Several interesting harmonic and motivic features occur in this last phrase, giving it the greater weight appropriate for the ending of this song. First, the leap from C-sharp to F-sharp in mm. 24-25 (Example 8a) is an emphatic cadential gesture that is nevertheless undercut by a harmonic "plot-twist," the arrival on an "eight-six" chord above the tonic root, rather than on true tonic harmony. This ushers in a descending-fourth progression in the melody, bounded by the F-sharp in m. 25 and the C-sharp in m. 28 that closes the song (Example 8b). The pitch F-sharp is embellished in m. 25 by its upper neighboring tone G-natural (Example 8c), which returns to F-sharp on beat 2 in m. 26. This neighboring motion is overlapped with a similar neighboring figure in mm. 26-28, with the pitch C-sharp embellished by its neighboring tone D. The song ends on an F-sharp major chord; the net harmonic result of the entire piece is a motion from the tonic to its parallel chord, as shown in Example 8d. Furthermore, the common tones between the tonic and its parallel chord (F-sharp and C-sharp) are the boundary pitches of the descending fourth-progression that ends the song. As mentioned above, the F-sharp in the vocal part in m. 25 is harmonized not by a tonic triad, but by an eight-six chord above tonic. This chord is very unstable, and functions much in the same way as the cadential six-four chord. In fact, the entire end of the song seems to be focused on this harmony, giving way finally to the F-sharp major triad in m. 28.
A common musical-poetic interpretation of the so-called "Picardy Third" cadence is that of the heroic conclusion, the earned victory, or the triumph over adversity. But to take these associations as the default signifiers of the Picardy Third is to paint it with too broad a brush, which is inevitably bound to miss wondrous and subtle shades of meaning. In this particular case, the final cadence on F-sharp major sounds as though the poem's narrator simply runs out of energy, and momentarily accepts her nomadic existence.
A middleground overview of the song (Example 9) illustrates the prolonged melodic pitches and the harmonic transformations discussed above. Three important aspects of the song emerge clearly: (1) each phrase departs from the tonic and returns abruptly at the beginning of the next phrase; (2) all three typical Neo-Riemannian transformations, parallel, relative, and Leittonwechsel, are applied to a single chord, the F-sharp minor tonic; and (3) the prolonged melodic pitches are the common tones between the tonic and its transformationally-related goal chords for each phrase. It must be understood that the song clearly expresses the key of F-sharp minor, and is tonally unified, despite the fact that it makes numerous attempts to explore other keys. This unity is due in large part to the prolongation of common tones between the tonic and its transformationally-related chords, as described above.
Example 9. Brahms, "In der Fremde," Middleground Overview.
Theorizing Common-Tone Prolongation
A Tonnetz representation aptly illustrates the common tone relationships between the tonic and its three transformationally-related chords (Example 10). It should be noted that in Riemann's theory of harmonic functions, the three transformationally-related chords, along with the F-sharp minor tonic itself, are all understood as representing tonic function in this song. That is to say, each of the parallel, relative, and Leittonwechsel-related triads could serve as a substitute for the tonic triad itself, without changing the underlying harmonic function.19 In this sense, one might even go so far as to describe this situation as "function prolongation."20
Example 10. A Tonnetz Representation.
Whereas the musical idea of Brahms's "In der Fremde" arises from diatonic third relationships between the tonic and its transformationally-related chords, the Scherzo of Beethoven's Seventh Symphony can be characterized as emanating from chromatic third relations. A single tone, the pitch A, is maintained between two chromatic third key relationships (brought about by the use of the compound transformations discussed above). In contrast, Brahms draws upon diatonic third key relationships, in the form of the relative transformation in the first and third phrases, and the Leittonwechsel transformation in the second phrase. The use of singular transformations allows for the prolongation of two common tones for each phrase between the F-sharp minor tonic and its transformationally-related goal chords. It is significant to note that in both cases, a central tonic harmony is clearly emphasized, and the prolongation of the common tones between it and the transformationally-related keys forms the central musical idea.
Relationship to the Text / Poetic Meaning
Let us return now to the text of the song, in order to see how this musical interpretation resonates with the poem's meaning. As mentioned above, Zbikowski maintains that the oppressiveness of the tonic F-sharp minor represents the "feeling of discomfort of foreign surroundings." This rings true, especially considering the abrupt return to F-sharp minor following each harmonic excursion. However, I would like to suggest a different poetic interpretation, one that draws upon the familiar metaphor of tonic key as "home." As mentioned above, this poem is about alienation and abandonment, brought on by the fact that the narrator is unknown in her place of origin. This is especially prevalent in the fourth line, in which the narrator complains: "no one there knows me any more." Significantly, it is not the narrator who has changed, but rather her homeland. In effect, the narrator's home has been transformed into a foreign land, through the passage of time and her absence. This is reflected by the very title of the poem, which reads "In Foreign Lands" rather than "The Foreigner." The emphasis is placed on the difference of the narrator's surroundings, not the narrator personally.
This poetic interpretation resonates with the musical analysis I have sketched in this essay. The prolongation of common tones between the tonic and its transformationally-related chords in each phrase reflects the unchanging nature of the narrator, despite the fact that her homeland has been transformed around her. Consider examples 6a and 8a: the bracketed rising- fourth gesture from C-sharp to F-sharp is undercut each time by a harmonic arrival on a chord other than tonic. What should be an emphatic expression of tonic affirmation is transformed into the narrator's desperate attempt to cling to any shred of home that she can manage.
Moreover, the end of the song sounds no more conclusive than any of the previous harmonic deflections, because the final cadence is on the tonic major, signifying that the narrator's home has irrevocably been transformed. It is as if the narrator comes to the slow understanding that her desire to return home is not to be realized. The end of this song is met with the recognition that the homeland has indeed been lost, or at least transformed into a foreign land.
In Samarotto's view, the use of Neo-Riemannian transformational theory as an explanation for typical harmonic motions in common-practice music is not appropriate, and might be overlooking aspects that a Schenkerian prolongational reading reveals with greater precision. His position unequivocally demonstrates the differences between the incommensurate analytical and philosophical underpinnings of these two approaches. In the present essay, I have argued that there is much to be gained by examining the relationship between the two theoretical approaches, and that the similarities between common tone retention and the notion of prolongation can provide a heuristic method for explanation of music that falls between the cracks of the analytical machinery for each approach when taken individually. My idea of common-tone prolongation describes the musical idea of Brahms's "In der Fremde" especially well, due to the return to tonic after each harmonic excursion. Furthermore, this type of musical organization resonates with aspects of the text and poetic meaning. By shifting the emphasis from connections between successive harmonies to common-tone relationships between a tonic chord and its transformationally-related harmonies, one can experience a significant and interesting compositional idea in this song: the synthesis of aspects of transformation and prolongation of tonic function. This analytical approach seems to be a fruitful one that might prove useful in explaining other musical situations which bridge the gap between transformation and prolongation.
Bernstein, David W. "Nineteenth-Century Harmonic Theory: the Austro-German Legacy." In The Cambridge History of Western Music Theory, edited by
Tho mas Christensen, 778-811. New York: Cambridge University Press, 2002.
Bribitzer-Stull, Matthew. "The A-flat_C_E Complex: The Origin and Function of Chro- matic Major Third Collections in Nineteenth-Century Music." Music Theory Spec- trum 28 (2006): 167-190.
Cadwallader, Allen and David Gagne. Analysis of Tonal Music: A Schenkerian Ap- proach, 2d ed. New York: Oxford University Press, 2007.
Cohn, Richard. "Neo-Riemannian Operations, Parsimonious Trichords, and their
Tonnetz Representations." Journal of Music Theory 41/1 (1997): 1-66.
______. "Introduction to Neo-Riemannian Theory: A Survey and a Historical Perspec- tive." Journal of Music Theory 42 (1998): 167-180.
Dahlhaus, Carl. Studies on the Origin of Harmonic Tonality. Translated by Robert Gjerdingen. Princeton: Princeton University Press, 1990.
Forte, Allen and Steven E. Gilbert. Introduction to Schenkerian Analysis. New York: Norton, 1982.
Hunt, Graham G. "David Lewin and Valhalla Revisited: New Approaches to Melodic Corruption in Wagner's Ring Cycle." Music Theory Spectrum 29 (2007): 177-196.
Kopp, David. Chromatic Transformations in Nineteenth-Century Music. New York: Cambridge University Press, 2002.
Lewin, David. Generalized Musical Intervals and Transformations. New Haven: Yale University Press, 1987.
Platt, Heather Anne. "Text-Music Relationships in the Lieder of Johannes Brahms." Ph.D. diss., City University of New York, 1992.
Roig-Francoli, Miguel. Harmony in Context. New York: McGraw-Hill, 2003.
Samarotto, Frank. "Treading the Limits of Tonal Coherence: Transformation vs. Prolongation in Selected Works by Brahms." Paper presented at the annual meeting of the Society for Music Theory, Madison, Wisconsin, 8 November 2003.
Sams, Eric. The Songs of Johannes Brahms. New Haven: Yale University Press, 2000.
Schenker, Heinrich. Der freie Satz. Vienna: Universal Edition, 1935. Translated and edited by Ernst Oster as Free Composition. New York: Longman, 1979.
Stark, Lucien. A Guide to the Solo Songs of Johannes Brahms. Bloomington: Indiana University Press, 1995.
Temperley, Nicholas. "Schubert and Beethoven's Eight-Six Chord." Nineteenth- Century Music 5 (1981): 142-154.
Wuensch, Gerard. "Hugo Riemann's Musical Theory." Studies in Music from the University of Western Ontario 2 (1977): 108-124.
Zbikowski, Lawrence M. Conceptualizing Music: Cognitive Structure, Theory, and Analysis. New York: Oxford University Press, 2002.
1Earlier versions of this paper were presented at the annual meetings of the Music Theory Society of the Mid-Atlantic, the Florida State University Music Theory Forum, and the Indiana University Symposium on Research in Music Theory. I am grateful to the members of these societies, particularly professors Frank Samarotto and Robert Hatten, as well as the anonymous reviewers of this journal for the numerous comments and suggestions they offered on earlier drafts.
2Samarotto, "Treading the Limits of Tonal Coherence."
4This song is discussed briefly in Stark, A Guide to the Solo Songs of Johannes Brahms, 15-16; and Sams, The Songs of Johannes Brahms, 36-37. Longer discussions appear in Zbikowski, Conceptualizing Music, 243-286; and Platt, "Text-Music Relationships," 298-309.
5 Lewin, Generalized Musical Intervals and Transformations, 177.
6Cohn, "Neo-Riemannian Operations," 33-37. For a concise explication of this specific analysis, see Roig-Francoli, Harmony in Context, 869-871.
7In Neo-Riemannian theory, the "relative transformation" maps a triad onto its relative major or minor (for instance from C major to A minor), maintaining two common tones in the interval of a major third. The "parallel transformation" maps a triad onto its parallel major or minor (for instance from C major to C minor), maintaining two common tones in the interval of a perfect fifth. For the Leittonwechsel transformation (commonly translated as "leading-tone exchange"), two common tones are maintained in the interval of a minor third, and the moving voice "exchanges" the root of a major triad with its leading-tone. The traditional roots of the two triads lie a major third apart (for instance from C major to E minor). For a more detailed explanation of these basic operations, see Cohn, "Introduction to Neo-Riemannian Theory."
8Cohn, "Introduction to Neo-Riemannian Theory," 168.
9Forte and Gilbert, Introduction to Schenkerian Analysis, 142.
10Schenker, Der freie Satz, 45. This analysis is based largely on Schenker's figure 37, which he uses as an example of the initial ascent.
11Cohn, "Introduction," 168.
12This notion is certainly not without precedent. See Cohn, "Neo-Riemannian Operations," 42-46, in particular his discussion of "ternary generators and LPR-groups." Cohn's analysis of the excerpt from Verdi's Il Trovatore, in which the pitch A-flat is maintained between six different tonal areas, is especially similar to what I am suggesting here. For more recent investigations of the relationships between Neo-Riemannian transformational theory and Schenkerian analysis, see Bribitzer-Stull, "The A-flat-C-E Complex" and Hunt, "David Lewin and Valhalla Revisited." The term "common-tone prolongation" is quite similar to Kopp's notion of "common-tone tonality" as expressed in Chromatic Transformations in Nineteenth-Century Music, though his approach is more specifically focused on the analysis of chromatic third relationships.
13In Neo-Riemannian theory, the Tonnetz (or "network of tones") is a two-dimensional conceptual diagram illustrating the musical pitch space created by the network of relationships between musical pitches. The typical Neo-Riemannian Tonnetz, as illustrated in examples 3 and 10, is organized by three vertices, one moving from left to right, one moving from northwest to southeast, and one moving from southwest to northeast. The intervals between pitches on these vertices correspond to the intervals between the notes of a consonant triad: the perfect fifth, the minor third, and the major third, respectively. From this, triads (and more importantly the parsimonious transformational operations that lead from one triad to the next) can be efficiently illustrated. For a more detailed explanation of the Tonnetz and its geometry, see Cohn, "Introduction to Neo-Riemannian Theory."
14For a detailed comparison along these lines, see Platt, "Text-Music Relationships," 299-301.
15Zbikowski, Conceptualizing Music, 272-273.
16Regarding the gender of the poem's subject / speaker, see Zbikowski, Conceptualizing Music, 272-273, which traces the genesis of this poem within Eichendorff's novella.
18Ibid., 282. See the conceptual integration network in Zbikowski's fig. 6.5.
19See Dahlhaus, Studies on the Origin of Harmonic Tonality, pp. xi-xv; and Bernstein, "Nineteenth-Century Harmonic Theory," 778-811.
20This notion is quite similar to Heather Platt's idea of "suppressed dominants," especially the idea that the initial tonic is prolonged throughout an entire composition without arriving at the expected structural dominant to support an Urlinie descent. See "Text-Music Relationships," 298-358.