Referential Collections in Debussy's Danseues

Mike Krzyzaniak

From a birds-eye prospective, Claude Debussy's Danseues des Delphes, L 117 № 1 (1909), is functionally tonal: It begins and ends on B♭ major chords in accordance with its key signature; Each section moves prominently to its dominant; There is a large-scale divergence from and return to B♭, and so forth. The functionality of this piece, however, does not exhibit the fractal-like symmetry of many classical-era compositions. Upon magnification, the functionality breaks down. It is often difficult or pointless to even label individual chords or notes, let alone ascribe functionality to them. Nonetheless, bits of music that are structurally functional in the traditional sense are often smeared together with referential collections, and these collections take on function of their own, independent of whichever individual chords may be used within them.

Measure 1 contains a miniature example of how a referential collection can bridge otherwise severed pieces of functionality. The measure contains a B♭ major chord followed by two augmented triads. The B♭ triad is tonic, and the second of the augmented triads is V+, which resolves again to I on the first beat of m. 2. These functional chords, however, are sundered by the first augmented triad, which is not functional. From a referential prospective, the two augmented triads together comprise WT1, and thus form a single aural unit that excludes the B♭ triads. In other words, the B♭ chords only make sense in the functional context, and the first augmented chord only makes sense in the whole-tone context. The second augmented chord, by virtue of both its functional and referential connotations, bridges them.

The passage from mm. 11 - 20 exhibits a variety more complex uses of referential collections to bridge functional chunks. The passage as a whole prolongs F. Although F triads are not prominent in this section, the passage is preceded by a functional modulation to F, sustains F in the bass for 4 measures, and ends on an F major triad. On the next level of magnification, this passage may be broken into 2 subsections, the first of which (mm. 11 - 14) prolong F explicitly, and the second of which (mm. 15 - 20) begin by prolonging the dominant of F before returning explicitly to F. Upon even deeper magnification, it will be seen that these prolongations are not accomplished through functional progressions from one triad to the next, but instead through progressions of one referential collection to the next.

The first subsection (mm. 11 - 14) prolongs F by gradually moving from F major (Ionian) to F minor (Æolian). This is accomplished through a sequence of F-centered referential collections of which each has one more flat than the previous. The measure before this section (10) is clearly in F major, and adheres strictly to DIA-1. Measures 11-12 use DIA-2 while sustaining a F pedal in the bass, indicating F Mixolydian. The only outlier in these measures is an E♮ in m. 12, which has apparently been used to avoid a diminished triad in a sequence of otherwise diatonic parallel triads. Measure 13 strictly comprises DIA-3. Again, the F pedal is sustained, giving this passage the quality of F Dorian. Measure 4 introduces a D♭, creating DIA-4, or F Æolian. Again, the triad that would be diminished in this collection, g, has been altered to G Major, which produces two outliers, b♮ and d♮ in the penultimate chord of the measure, but the ultimate chord promptly overrides both outliers (i.e. it contains B♭ and D♭). It as though these measures have 'sunken' from F major to f minor through the gradual introduction of successive flats into the implicit key-signature.

The second subsection (mm. 15 - 20; the double bar at the end of m. 16 in my edition appears to be completely arbitrary) extends C as the dominant of F for 3 measures before transitioning back to F. The pitches in measure 15 do not sum up to any referential collection, but they do use a non-functional progression to 'tonicize' C. The chords through this measure (including the first beat of m. 16) are C - A - B - C. This sounds basically like the melodic progression do - la - ti -do in C, which has been harmonized with root-position major triads built on each note. After the previous four measures of F pedal, the C chords are readily heard as the dominant of F. Measure 16 beat 2 through m. 17 are more difficult to analyze on a chord by chord basis, nonetheless referential collections can aid an understanding that these chords prolong the dominant of F. On the one hand, these measures do not explicitly contain any triads (there are a number of arguments one could make about which triads are implied, but it will turn out to be unimportant), and there is not really an unambiguous pitch center. On the other hand, m. 16 starts on a C major triad, m. 17 ends on octave Cs, and the pitch C sounds somewhere in every intervening beat, which makes C a good candidate for a pitch center. Furthermore, these measures contain 6 pitch-classes that could comprise either DIA0 or DIA-1, contingent upon the presence of B♭ or ♮. This collection indicates either C major, or C Mixolydian which is the dominant mode of F. Either way, it is as though the composer is using a diatonic wash of an (almost) entire dominant mode as a substitution for a simple, tertian dominant chord. The ambiguity of which specific mode resolves to essentially the question of whether these measures represent V or V7, which is inconsequential.

The rest of this subsection (mm. 18 - 20) transitions from the dominant side of F, back to F proper, which arrives explicitly in m. 20. Again, this transition is not functional on the level of individual chords, but rather employs a carefully chosen referential collection. The pitches in these measures comprise DIA-3 (the outlier is discussed below). This collection is optimally suited for this particular transition because the third flat (A♭) provides a positive link back to the f minor sonority from which the music departed in m. 14 (the root progression through these measures is simply me - re - do in f minor), yet the d♮ (which might have otherwise been the fourth flat) provides a smoother transition from the collection in the previous measure. In fact, the notes of the melody (played in octaves) in these measures do not, by themselves, stray from the previous collection, and this owes to the d♮. In other words, rather than resolving C to f using a functional V-I progression, the music transitions there using a diatonic collection that kind of splits the difference.

Not only do these measures (mm. 18 - 20) transition between pitch center, but they also transition from a referential language to a functional one. The only outlier to the DIA-3 is the A♮ in m. 20, which produces an F major rather than minor triad. Here, F major is the structural dominant that sets up the return of the home key of B♭ in m. 21. Retrospectively, the g minor in m. 19 can be heard as the predominant (vi) of this cadence. The A♭ in m. 18 does not relate functionally to either the target key of B♭ or the local 'key' of f. So, m. 18 only makes sense referentially, m. 19 is comfortable in either the referential or functional contexts, and m. 20, because of its outlier, makes more sense in the functional context. In this way the functional moment arises gradually from the referential material.

In conclusion, although the functional harmonic structure of Debussy's Danseues collapses under sufficient magnification, referential collections at the smallest meaningful level of analysis support the overall structure. At times, these collections mimic the functionality that may have otherwise been provided with triads. At other times they prolong or transition between harmonies that were set up with more traditional means, and sometimes they pivot between functional and non-functional triads. In none of these cases, however, are the collections in conflict with the overall harmonic goal, rather they always aide it.