The Blue Danube

One of the most memorable treatments of ^5 and ^6 is in the first waltz of An der schönen, blauen Donau. Johann Strauss, jr's opus 314 was published in 1867, at the height of his success as a dance composer and conductor and not long before he turned to composing operettas. The design is standard: five waltzes, each of which consists of two strains with both repeated to create a double reprise (waltz-trio-waltz-trio) effect, plus a substantial introduction and a coda that quotes all but the last waltz. The tonal design of the set is also not unusual:

See the piano score for waltz 1, first strain, below. The circles show the expanded treatment of ^5 and ^6, including the "last second" retreat from a tonic with added sixth at the very beginning of the third system. The rectangles sketch my view of the design, which is based on the registral pattern established immediately, then shifted into the upper octave midway through the third system. I would read the strain as derived from a proto-background ^1-^5.

Schenker's desire for a unified, teleological tonal motion leads him to mis-read the first waltz in Free Composition: he maps onto the piece a waltz-trio-waltz design with the expected interruption covering the second strain (which is in A major). In fact, the waltz ends in A major because both strains, not just the first, have their reprise (the alternativo design already common in Mozart's generation [Note 19 June 2017: see my comment on this in the updated blog post for 3 March 2010]).

William Rothstein uses the first strain as a study object for his exploration of the meaning of the term "phrase." Along the way, Rothstein presents a foreground reduction with bar lines and four-bar "phrase" markings, two levels of durational reductions (after Schachter), and a "standard" Schenkerian graph. He describes the figures of this last as follows:

The first sixteen measures are ultimately static, ending where they began, with only minimal motion along the way. The gradual ascent that follows picks up from this same point, without which it would lack a firm beginning. The complete tonal motion therefore comprises the entire excerpt: all of it is to be performed "...figuratively, in a single breath."

The most striking feature of this larger motion -- and what elevates this unpretentious waltz to high artistic rank -- is the broad melodic arpeggiation from the initial ^5...up to the climactic ^3....One can see how this large arpeggiation mirrors and fulfills the smaller arpeggiations of the several upbeat measures. The ascending sixth [A4-F#5] in mm. 27-28 summarizes the entire ascent in a burst of melodic energy....The concluding melodic descent then gradually dissipates the accumulated tension.

Rothstein's reading does a good job of following the broad contours of registral (and dynamic) change through this first strain. It certainly improves on Schenker's reading, which ignores the melody of the inner voice to choose ^3 from the repeated quarter note figures in the upper voices; when these figures cease in m. 24, the voice-leading moves into the arpeggio-based melody. This in itself, although peculiar, is not damning because there is indeed a distinct change of texture (and orchestration) at mm. 24-25, but Schenker flattens out the large registral shape that Rothstein shows so clearly -- Schenker locates the first Urlinie note at the beginning (m. 3-4) and describes the broad figure reaching F#6 (and then dropping back to F#5 for the cadence) as unfolding, although the far more likely label would be coupling, or middleground registral doubling.

Note: establishing the "priority" text for this famous composition is something of a problem, believe it or not. An der schönen, blauen Donau was issued first in May 1867 in an arrangement for solo piano; from August to October of the same year, editions appeared for orchestra, men's chorus and piano, piano four-hands, and violin and piano. In all, between May 1867 and August 1869, fifteen versions were published. The preponderance of early public performances were the version for men's chorus, whose original text is a light inducement to enjoy the Carnival season: it begins "Wiener seid froh," to which the upper-voice ornaments (in the tenors) answer "Oho, wie so?" This banter continues till all the voices join forces for the final phrase and cadence, "Was nutzt des Bedauern, des Trauern? Drum froh und heiter seid!" (Very roughly: What good is sadness? Have fun!")

Reference:

Rothstein, William. Phrase Rhythm in Tonal Music. New York: Schirmer Books, 1989.

LINE, N, and Westergaard's Tonal Theory

In the MTS article, I present a small set of transformations that are applied to the intervals of the proto-backgrounds. I didn't realize it at the time, but Peter Westergaard had already done something very close to what I intended (an earlier, less careful formulation of it was behind the list of forms generated by a "criterion of simplicity" [Neumeyer 1987]). As Stephen Peles describes it:

Westergaard's set of generative operations that determine allowable within-line relations is small. Indeed, Schenkerian theorists will no doubt be struck by the comparatively parsimonious nature of Westergaard's typology of tonal transformations. A whole battery of Schenkerian operations, such as arpeggiation, unfolding, motion from an inner voice, and the like, collapse into a single operation category in Westergaard's system: namely, the arpeggiation operation. (77-78)

Westergaard's order of presentation in the section "Linear Operations and Constructs" (35-37) of his book Introduction to Tonal Theory would have served very nicely as a way to present the intervals and these transformations. He has four classes (he calls them "structures" (37)): (1) rearticulations (that is, repeated notes), which, coincidentally, introduce a whole range of rhythmic-metric issues, but which also lead to (2) neighbors; similarly, (3) arpeggiation [or, two consonant notes originally understood as simultaneous but arranged successively] leads to (4) step motion. The four of these "together with [the rhythmic devices of anticipation and delay can be used] to compose and to understand tonal lines."

What appeals to me is the simplicity of the pairings and the neat symmetry of the two groups, something implied in my Ex. 10 (the 30 backgrounds (301) but not made explicit: rearticulation --> N; arpeggiation --> LINE.

Westergaard does not introduce equivalents of my transformations INV or ADDINV; instead, he moves directly to chapters on species counterpoint and eventually introduces Schenker-derived operations he calls "doubling, borrowing, exchange, and transfer" (289).

References:

Westergaard, Peter. Introduction to Tonal Theory. New York: Norton, 1975.

Neumeyer, David. "Thematic Reading, Proto-backgrounds, and Transformations," Music Theory Spectrum 31/2 (fall 2009): 284-324.

Neumeyer, David. "The Urlinie from ^8 as a Middleground Phenomenon." In Theory Only 9/5-6 (1987): 3-25.

Peles, Stephen. "An Introduction to Westergaard's Tonal Theory." In Theory Only 13/1-4 (1997): 73-94.

Proto-backgrounds (comparison and evaluation)

Over the past month, and one at a time, I have presented readings based on the nine proto-backgrounds. (Refer again to the introductory post; clicking on the label "proto-background" in the sidebar will bring up all the entries in a single window.) In many of the individual entries, comments offer preliminary assessments. This post compares all the readings, which seem to form two (non-exclusive) groups: (1) teleological (or end-oriented), (2) focused on one or both of the principal right-hand melody notes and yielding N or LINE-derived backgrounds.

^1-^1, ^1-^8. Of the first version, the unison ^1, I wrote that "Given the alto's strong focus on ^3 and the soprano's equally dogged emphasis on ^5, a reading generated from ^1 might seem counter-intuitive, but it [effectively conveys] the teleology in the 8-bar antecedent." The unison and octave (^1-^8) readings are very closely related, the main difference being the level to which the octave A4-A5 in bar 9 is assigned (whether proto-background or "background"). My assessment was that I "prefer [the octave] version, if only because it takes the teleological bias (which will be a factor in any analysis of this waltz involving ^1) and pushes it to the max, making for a more consistent interpretation overall." Both readings do a good job of aligning formal design and pitch elements at comparable levels.

^3-^8, ^1-^5. The reading from ^3-^8 takes this further, as "the prominent sixth of the closing cadence [that it represents] is unavoidable; . . . the most sharply teleological of the nine, [this hearing of the waltz requires that] everything must be read 'backwards' from the voicing of the final right-hand chord." By contrast, the end-orientation of the ^1-^5 hearing is limited to phrase level, where ^5 shows up immediately but ^1 only arrives at the end.

^3-^3, ^1-^3, and ^5-^5. Like the unison and octave readings, the unison ^3 and ^1-^3 are closely related: "unison ^3 focuses attention on the alto voice but differs from ^1-^3 in delegating its repeated linear path to later levels." Thus the third level ("first middleground") of the unison reading highlights INV transformations, but the same level in the ^1-^3 analysis features layered lines. At the same time, ^3-^3 and ^5-^5 are related in that each forces intense focus on one of the two right-hand melodic notes throughout. Both align very cleanly with the large units of the formal design, but the ^5-^5 hearing is more dramatic in the chromatic shift of its primary tone (rather than a secondary voice) for the C# major section and therefore offers a very direct expression of the most distinctive feature of this waltz.

^3-^5, ^5-^8. This is the odd couple, in that I characterized the former as the most obvious, one might say "natural," hearing, but the latter as a misreading.

As to which of the nine readings is the "correct" one -- or even which of them I favor -- I will first refer the reader to the MTS article's discussion of Lewin's assessment of four Schenkerian readings of a Schumann song and his ultimate choice among them. Like Lewin, I will say that all of the nine analyses of D779n13 are possible in the sense that they are coherent on their own terms and I can -- sometimes with a little effort -- hear the waltz as each interprets it. Finally, though, I would choose ^3-^5: the strongly teleological readings seem out of sync with a waltz whose sections move unpredictably. The immediacy of the suspension chain does drive forward at phrase-level, but the C# major section even breaks that up (note that there is only one suspension in each unit, not a chain). Similarly, the analyses that isolate either ^3 or ^5 (or ignore them, as in the unison ^1 and octave) are much harder to hear than the one that combines ^3 and ^5 (soprano and alto, male and female dancers).

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"Any background analysis does crucial work in specifying just what some metastable hearing of the piece is" (Lewin, 167). Analysis using the proto-backgrounds as initial structures for generative hierarchies is highly dependent on the choice of the initial or highest-level figure. Such figures are what Lewin calls "metastable": not universals but acting pragmatically as if they were for the sake of the work of analysis or interpretation (see discussion in my MTS article). As I have noted in earlier posts and in web essays, they are, in fact, identical in function to the themes that a reader engenders to gather and guide reading and interpretation of a poem, story, play, or other text. For examples of themes in the analysis of literary works, see the sections Rebecca, Genre Clerk, and Blac Danse in my PDF essay Proto-backgrounds in Traditional Tonal Music: link.

Reference:

Neumeyer, David. 2009. "Thematic Reading, Proto-backgrounds, and Transformations." Music Theory Spectrum 31/2: 284-324.

Lewin, David. Studies in Music with Text. New York: Oxford University Press, 2006.

Proto-background 9: the fourth ^5-^8

Also see the proto-background introduction.

The last of the proto-backgrounds implausibly demotes the alto voice and gives all the attention to the overall shape of the uppermost voice. Although some interesting insights emerge from this focus on the fourth, still the reading overall seems forced, a deliberate misreading.

Schubert uses the ^5-^6-^7 Leerlauf most blatantly to end the first waltz of the Valses nobles. It can also be found in the first strain cadence of D783n16, and, in more elaborate form, in D734n1, in D924n7, and in the last of theValses sentimentales.

In the graphic below, the second level shows other figures in addition to the basic opposed neighbors, E-E#-E, A-G#-A; these could, alternatively, be described as a WEDGE.

In this version, the second staff shows simple registral shifts generating later-level INV transformations, plus the transposition of E5-A5 to generate the C# major section. The third staff elaborates all of these intervals by means of lines.

This conception of the piece traces the interval/line relationship more abstractly, as a series of Transpositions in the first staff, to which the countermanding (and insinuating) line is added. This latter, nevertheless, traces the same fourth down (A-G#-Gnat-F#-E), and in the end notes also reaches a vertical interval of that fourth again.

Proto-background 8: the sixth ^3-^8

Also see the proto-background introduction.

The sixth ^3-^8 often seems more comfortably regarded as a transformation of ^1-^3 through INVersion or ADDINV (in the first case, the sixth replaces the third; in the second case, the sixth is added onto the third).

In D779n13, however, the prominent sixth ^3-^8 of the closing cadence is unavoidable; it generates a foil to the previous reading's static ^3-^5: the new reading is the most sharply teleological of the nine, as everything must be read "backwards" from the voicing of the final right-hand chord.

In the graphic below, the proto-background is at the top. The second level shows the distribution of the pitches over the form (A5 appears at the end of the first strain), along with the first elaboration by means of neighbor notes (as N-1 in both voices). The third level shows the origins of the C# major section in a temporally displaced (and registrally elaborated) INV transformation. The material below the third level shows later iterations of the same kind of registral play with the basic pitch classes.

For other examples of the ^3-^8 proto-background, see my web essays [NOTE 10 June 2016: both essays named here have been integrated into my PDF essay Proto-backgrounds in Traditional Tonal Music: link] on Beethoven WoO10n2, where ^3-^8, curiously, plays the anti-teleological role among the several readings; and Beethoven WoO10n1, in which registral shifts upward elaborate a neighbor-note pair (no commentary).

Proto-background 7: the third ^3-^5

Also see the proto-background introduction.

One might reasonably object that my systematic working through the proto-backgrounds has just delayed the obvious: the alto clearly moves from ^3, while the soprano is just as obviously based on ^5. That sanguine certainty, however, is undermined when one realizes that a proto-background ^3-^5 means the interval also holds sway at the end of the piece: in other words, this reading is radically anti-teleological. Only three of the nine possible proto-backgrounds support such "beginning-loaded" hearings: the unison ^3, the unison ^5, and ^3-^5.

The second level in the graphic below conveys this curiously static sense: the work unfolds in a leisurely way from a firm initial premise, enfolding the C# major section by means of a simple (chromatic) neighbor note and demoting the cadence (whether rising in its direct sense or falling in its hidden sense). The second level also conceals the reprise's instability, which is duly sorted out and explained in the details of the third level.

Proto-background 6: the octave ^1-^8

Also see the proto-background introduction.

Register plays such a crucial expressive role in D779n13 that the octave, at first glance, would seem to be an appealing starting point for an interpretation. As it turns out, however, the proto-background ^1-^8 simply shuffles the priorities (levels) of the reading from ^1-^1, rather than introducing any substantially new information, as will be obvious if you compare the graphic below with the first hearing: blog entry.

Nevertheless, I am inclined to prefer this new version, if only because it takes the teleological bias (which will be a factor in any analysis of this waltz involving ^1) and pushes it to the max, making for a more consistent interpretation overall.

As a postscript to the reading from ^1-^8, here is a reading that, if possible, goes even further to declare a "universal" space of the octave (see graphic below). This hearing of D779n13 follows from an observation that the traditional octave ambitus of the modal scale continued to exert a considerable force throughout the Baroque era, but not later (Neumeyer 1987; Smyth 1999). Since the practices of European tonal music arose and were solidified in this era, it is not unreasonable to suppose that the octave might be a universal principle for tonal space, with the same status as strict counterpoint, figured bass, and the rhetorical schemata exemplified in the partimenti tradition.

If so, the result might be a fixed octave "background" with secondary motions (represented by arrows here) established in relation to it. The idea is not only retrospective but prospective since it is, of course, related to the device of absolute register in some twentieth-century musics.

References:
Neumeyer, David. "The Urlinie from ^8 as a Middleground Phenomenon." In Theory Only 9/5-6 (1987): 3-25.

Smyth, David. “Schenker’s Octave Lines Reconsidered.” Journal of Music Theory 43/1 (1999):101–33.

Proto-background 5: the fifth ^1-^5

Also see the proto-background introduction.

The fifth space ^1-^5 re-introduces a teleological element into the reading, but by no means so radically as when we hear the background as ^1-^1. Here, the upper part of the interval receives attention at the beginning, but the interval as a whole is only defined at the end of the first phrase (see the third staff below). The contrast between upper and lower voices in the right hand, thus, is more striking even than in the reading ^3-^5 (entry to be posted later this week), in that the definition or the concrete presentation of each background tone is situated at opposite ends of the phrase. And the teleological hearing of the alto voice is a good mirror of the listening experience for a string of suspensions, which constantly push forward, ahead, towards a goal, the resolution of the last suspension in the series (here, the 4-3 suspension that brings the secure arrival of ^1).

The fifth space also coordinates nicely with the C# major section (in fact, more simply and directly than any other reading): a WEDGE transformation pulls the notes apart by a half step, to G#-E#, then its inverse pushes them back together again for the reprise/ending. (WEDGE here can be understood as inversion about C/C#; it is a variant of Lewin's transformation W -- see 1987, 124 ff.; see also 2006, 332 ff.)

In a performance of the waltz, of course, the WEDGE is distorted as the G# is G#5, not G#4. We would include the registral shift in a fourth staff/level (not shown here).

References:

Lewin, David. Generalized Music Intervals and Transformations. New Haven: Yale University Press, 1987.

Lewin, David. Studies in Music with Text. New York: Oxford University Press, 2006.

Proto-background 4: the third ^1-^3

Also see the proto-background introduction.

The third-interval rooted in the tonic is a rich source of linear figures, although, like the ^3-^2-^1 of Schenkerian analysis, these figures almost seem too obvious; they have a clichéd or generic feel about them, as if someone were to say that the theme of a story, poem, or film was "love." In the graphic below, I have followed this line in interpreting the second level or "middleground" as a neighbor note figure aligned with the harmony of opening-plus-final-cadence (I------V7-I).

More interesting are the details in the third and fourth level. Interval inversion converts the initial third (in the alto) to a sixth in the cadence, then ultimately inverts the inversion, so to speak, in the truncated reprise. (Strictly speaking, I suppose, the second motion inverts upwards (as shown), then shifts the resulting third down an octave.) The fourth level shows further details of the registral play.

The tonal space of the third ^3-^1 is also used in Schachter's traditional Schenkerian reading of the waltz, of course. I have reproduced the background (strictly speaking, first middleground) form an earlier posting along with my notation of the first strain.

Schachter favors the (inner-voice) descent A-G#-G-natural-F# in the contrasting middle. My own reading of the second strain is close to his but I prefer to emphasize the mixture of E-E#-E.

Differences between the traditional Schenkerian reading and the proto-background can be suggested by aligning both versions with the consequent of the 16-measure theme (mm. 11-18). (b) is the proto-background; (c) is Schenkerian.

Level (c) allows a simplification of the voice leading that is not available without overt doubling in level (b). On the other hand, the bias against register (or, to put it another way, in favor of single-octave solutions) means that (c) must cut across the clearly articulated parallel sixths in the cadence.

Proto-background 3: the unison ^5

Also see the proto-background introduction.

Of course, the unison ^5 forces attention away from the alto (lower right-hand voice) to the soprano, and it also (that is, like the unison ^3 from yesterday's post) aligns itself very cleanly with the formal design. The second line in the graph shows a simple harmonic transformation with one harmony for each section: first strain, contrasting middle, and reprise. In Riemannian terms, this is (L-followed-by-P) followed by the inverse (or, P-followed-by-L). L turns A major into C# minor, and P makes the latter C# major.

The third line fills in a few details for the first strain. Note that the symmetries in the harmonic patterning extend to the outer melodic voices, with the two neighbor-note figures. The inset takes this a little further by understanding the "essential" chromaticism of getting-into and getting-out-of the contrasting middle symmetrically, as well: E5 breaks up to E#5 at the beginning, but G-natural slumps down to F#5 at the end.

Proto-background 2: the unison ^3

Also see the proto-background introduction.

The unison ^3 focuses attention on the alto voice but differs from ^1-^3 (future post) in delegating its repeated linear path to later levels. As the "foreground" figure in a previous posting showed, the potent voice leading clichés invoked by the suspensions in the alto voice and the essentially stationary ^5 in the soprano are subverted at the last moment in the cadence. I have rewritten that figure below in terms of ^3 rather than ^1 (this is a thumbnail; click on it for a larger image):

Details of the first section (bars 1-18). The first line is a "middle-level" version of the second line above, relevant only to the first strain. The second line shows how the intervals unfold. Note especially the 3-6 INV pair nested inside the main pair. The third line gives even more detail, focusing on lines and the cover tone E5.

The reading with a proto-background unison ^3 can be rewritten in more traditional Schenkerian notation (the C# major section is not included here -- it's the empty space in the middle of the graph):

Obviously, one has to allow for the possibility of a background/middleground neighbor note in order to make this work. I write about that issue briefly in the MTS article (291, 297fn30) in connection with Schenkerian readings by Arthur Komar.

Proto-background 1: the unison ^1

Also see the proto-background introduction.

It would have been better, perhaps, to start with a simpler reading, such as ^3-^3, ^1-^3, or the obvious ^3-^5, but the unison ^1 has the advantage of shifting the interpretive ground rather abruptly and thus emphasizing the variety induced by registrally based proto-backgrounds.

NB: The graphics are thumbnails; click on them to see the originals.

Given the alto's strong focus on ^3 and the soprano's equally dogged emphasis on ^5, a reading generated from ^1 might seem counter-intuitive, but it does a very good job of conveying the teleology in the 8-bar antecedent. After the left hand's "oompah" introduction establishes key and meter, the right hand figures unfold over unstable harmonies; the first point of stability is at the end, when the alto reaches ^1 over I (bar 9). The notation in the treble staff reflects that with unstemmed closed notes for ^5 and the line from ^3 but an open note for ^1.

The consequent repeats the harmonic progression with variants in the eighth-note groups, but its ending is a surprise as a line rises from ^5 to the upper octave (A5), overwhelming the placid repetition of the descent in the alto. The idea, then, is to make a REGistral shift, but the register of A4 is doubled -- it doesn't disappear -- and so I call the transformation ADDINV, which adds above a given interval its inverse (here, the octave above the unison).

The register change is not made directly by A4-A5, but at a later level ("foreground") by G#4 -- see the graphic below:

The density of this figure is preferable to a reading that simplifies the passage through reduction to schematic voice leading over the given harmonies.

Returning to the main graphic, I have bracketed the octave with its internal fifth (which receives direct melodic emphasis in bar 17) and then noted how the progress of the second strain takes this framework and transposes it upward twice. (These could have been labeled as diatonic transformations T2 & T3, respectively.) This is an elaboration, however: on the larger scale, the entirety of the second strain repeats (or maintains the result of) ADDINV.

Proto-backgrounds (introduction)

My essay "Thematic Reading, Proto-backgrounds, and Transformations" is about to appear in Music Theory Spectrum 31/2: 284-324. For that reason, several early blog posts will read the Schubert waltz in terms of the proto-background construct developed there. A PDF essay with explanation and examples may be found here: proto-background. [Link updated 10 June 2016.]

In a generative hierarchical theory of traditional European tonality (such as Schenker or Lerdahl and Jackendoff), intervals precede lines. Taking the view that intervals are therefore the proper content of the earliest level(s), I build a set of 9 proto-backgrounds within the octave: three unisons, three intervals above ^1, two above ^3, and one above ^5. For the key of D779n13, these would be:

In the essay, I use an informally applied transformational language to elaborate the proto-background intervals, but any kind of notation for linear analysis (including any of the several "dialects" of Schenkerian notation) would be appropriate, too. Some transformations work directly with intervals (ADDINV and DIVision), others with stepwise formations (LINE, Neighbor, and their directional inverses).

I will post a series of entries, each of which is a reading based on one of the proto-backgrounds. A tenth entry will take up the question of comparison and evaluation.