Schenker gave a straightforward definition of the divider at the fifth in a passage (1923) which is one of his earliest references to the concept:1
|In Vol. II3 [Free Composition], I call “divider at the upper or lower fifth” the upper or the lower fifth of a chord, that by leap puts itself in the service of a passing motion or a neighbour note. The divider is thus nothing else than a leaping passing motion and the chord brought forth by it is but a passing or a neighbouring harmony.2||Die Ober- oder Unterquint eines Klanges, die springend sich in den Dienst eines Durchganges oder einer Nebennote stellt, nenne ich in II3 ein Ober- oder Unterquintteiler. Der Teiler ist somit nichts anderes als ein springender Durchgang […] und der mit durch ihn hervorgebrachte Klang eben nur eine Durchgangs- oder Nebennotenharmonie.|
The “leaping passing motion” is a particular type of arpeggiation.3 I understand Schenker’s description as referring to something like this:
where the bass merely arpeggiates the tonic chord, passing through its upper fifth, while the upper voices fill the space with two passing notes. There results an apparent harmony, (V). This is the case at the end of bar 3 of Haydn’s F major Sonata, Hob. XVI:29:
In such a case, the fifth of the bass arpeggiation “puts itself in the service” of the passing (or neighbouring) motion in the upper lines while remaining itself only a passing leap in the chord that it elaborates.
It seems to me likely that, in the development of Schenker’s ideas, the Quintteiler has been the model for the concept of Baßbrechung and that discussions of the divider shaded away behind discussions of the Ursatz. Schenker’s writings were meant mainly for readers familiar with his theories: they probably knew what a divider at the fifth was. Schenker apparently did not change his mind about the divider at the fifth in the final version or Der freie Satz and I believe that, if he came to less often refer to the concept, it merely was because he expected his readers to know enough about it4. The dominant of the Ursatz, considered at the deepest background level, is itself a divider at the fifth, because the Baßbrechung is an obvious case where the 5th of the chord “puts itself in the service of a passing motion” in the Urlinie. More generally, any “real” dominant (or other local 5th degree, e.g. II as V/V) at a given level is, at a deeper level, the mere divider of the chord that it elaborates.
This, which Schenker takes for granted in Der freie Satz, nevertheless requires additional consideration in the case of the interruption, because the elaborational value of the Vth degree is less obvious in this case. He writes:
Schenker’s statement that “the bass like the Urlinie aims at only one arpeggiation, namely by the division of the triad at its fifth” seems to be true of any primal structure, and I think that what he means here is that the term ‘divider’, although obsolete (because it is obvious) to denote primal structures in general, nevertheless should be retained in the case of an interruption. This does not mean that it is no more valid for other cases, merely that it is no more necessary.
Oster’s translation in Free Composition is insidiously biased. He fully reorganizes the second sentence of the text above as follows:
|« In order to avoid this danger, and to indicate more clearly the prolongational significance of this dominant at the first level [my italics], I recommend the use of the term dividing dominant, or simply divider. »|
So doing, he unfortunately conveys the idea that “the prolongational significance” in question is proper to “this dominant”, i.e. to the dominant at the interruption, while Schenker merely said that the “first” dominant, i.e. the dominant belonging to the first level of prolongation, should be termed ‘divider’ in order to remind the unicity of the primal structure. Oster also creates a possible confusion between the “dividing dominant” (a term that Schenker does not propose) and an “interrupting dominant” (which Oster does not dare propose, but which he may have had in mind).
This, once again, raises the question of the meaning of Prolongation which, in Schenker’s German, refers to the extension of the laws of strict writing, two-voice counterpoint, to freer writing. When he writes “the first level as prolongation”, he merely indicates that the first level corresponds to a first prolongation of these laws – perhaps meaning, in this particular case, that what appears as a dominant at the first level merely was a divider at the deepest background. When speaking of “the prolongational significance of this dominant”, Oster makes use of the word in another meaning (originating in Salzer’s Structural Hearing) for which Schenker would have used Auskomponierung (“elaboration”): while Schenker may have wanted to convey the idea that the first-level dominant originated as a Teiler, Oster tries to describe the special case where the first-level dominant becomes a “dividing dominant”.
Oster’s translation of § 279 increases the confusion. I reproduce the German text of the 1935 edition between my own translation (at the left) and Oster’s (at the right):5
Schenker merely means that the V of the first level can also be used in the foreground as a transferred divider, and figure 130 (see below) merely illustrates cases where local dividers participate in the elaboration of the chord that they divide.
Oster deforms Schenker’s text in several apparently unimportant ways:
– The translation of der übertragene Teiler (“the transferred divider”) as “the applied dividing dominant, the applied divider,” seems an over-interpretation;
– While the term Ein Sonderfall merely carries the idea of an exception, Oster’s expression “takes a special place” might mean somewhat more;
– The reference to § 89 ff. is moved from the mention of the transferred divider to that of “the characteristic interruption of the voice leading”, and the reference to § 192 disappears.6
– Schenker’s mention of auch jenseits einer Gliederung is unclear, but does not seem to possibly mean “even when no such division is involved”; on the contrary, Schenker probably refers to elaborations that contain a local interruption.
* * *
What Schenker wants to stress, especially in § 89, is that in the particular case of an interruption, the true Vth (the divider) at the first level of the “prolongation” (i.e. the transformation of the Ursatz into the first level) is the Vth of the interruption itself, not that of the second phrase. He confirms in § 90 that the interruption “has the effect of a delay, a retardation on the way to the final goal, ”. This confirms that the goal of the interruption is not the I that immediately follows, but the final one. Once again, the question does not arise in the case of an uninterrupted composition because the Bassbrechung in that case leaves no doubt as to where the divider is; in an interrupted work, on the other hand, there are two instances of V, which both point to the final I.
Schenker usually graphs interruptions with an uninterrupted bass beam, as follows:
stressing where the Vth divider is at the first level and showing that its goal is the final I. In this example, I take the beams in the bass to indicate at what level each of the two V chords is a divider – the second V chord indeed also is a divider, but at a middleground level. In the American usage today (and often in my own, I am afraid), interruptions are graphed like this:
which I now consider mistaken, or at least confusing.7
A particular case of divider, which may further clarify its definition, is that of the “pendular” movement I–V–I, where V immediately returns to the I from which it derives. An early example proposed by Schenker himself is the opening phrase of the 2d mvt. of Beethoven’s 5th, of which Schenker writes: “Thus locked between the two tonic harmonies, however, the V functions more as the divider of I at the upper fifth than as an essential harmonic degree”. (Der Tonwille 5, p. 33; W. Drabkin transl., vol. I, p. 203.)
In this case, the V chord already is a divider at the foreground level, and therefore cannot take on a more functional role at a later level, as this is the last one. But the situation of this divider is not basically different from that of the structural V at the background level, where it merely divides the primal tonic chord (that which is sometimes dubbed the “chord of nature”).
Such a “pendular” situation may also give rise to the divider at the fourth, as in this example from Chopin’s Prelude op. 28 n. 10 (not given by Schenker) which presents dividers both at the fourth and the fifth.
Schenker did not pursue the idea of the divider at the fourth, probably because it could not as easily be related to a mere elaboration of the chord within which it is inserted (the 4th does not belong to that chord). The divider at the fourth may be considered a case where 64 neighbour notes are made consonant by a projected bass.
The American conception, advocated among others by Bill Rothstein, appears to be that there are “real” dominants, with a strong dominant function, and less real ones which are better called “dividers”.8 My reading of Schenker is that any dominant (especially those with a real dominant function) can be seen at a deeper level to have only a “prolongational significance” (or, better said, an “elaborational significance”) and must (or can) be termed “divider” at that level. The possibility to read a dominant as a divider depends on its being imbedded within an elaborated chord rooted a fifth lower. In other words, while the American conception sees a difference between dominants at a given level (some being “real”, others mere “dividers”), I see the difference between levels: a strongly functional dominant at a given level may appear as a mere divider at a deeper level. The level at which a chord reduces to a mere divider is the last one at which it is to be recorded: it disappears at deeper levels together with the compositional elaboration to which it belongs. The only divider that has no occasion to disappear is that of the primal structure, by lack of a deeper level (unless one admits the “chord of nature” itself as some sort of underground level – as Schenker himself may have done).
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In Fig. 130 of Der freie Satz, in § 279 discussed above, Schenker gives the following graph:
Fig. 1301 illustrates that the divider is a kind of projection of the fifth of the chord. Schenker writes: “the transference of the divider is based on its belonging to the Klang as upper fifth, as its fifth-projection”; it is in that sense that it is its “divider”. And he adds: “As can be seen in 2, the characteristic (Merkmal) of the interruption should not be missing”, probably meaning that the transferred divider must retain its effect as an interruption, even although it is not followed by a return to I. Such cases are often referred to in the American Schenkerian literature as “back-relating dominants”, even although the whole point is that, at the level at which they appear, these transferred dividers are mere projected fifths, not true dominants.
In Der freie Satz, the only cases where Schenker does not draw a continuous bass beam are those in Fig. 22 a and b. In both examples, the V chord at the interruption is followed by an unexpected harmonic progression.
The first case is the choral Ich bin’s, ich sollte büssen (see also Fünf Urlinie Tafeln, I). Starting from the V chord of the interruption in bar 6, Schenker reads an ascending 4th progression, E♭–F–G–A♭:
In none of his graphs for this choral (in Five Graphic Analyses and in Der freie Satz) does Schenker beam a Baßbrechung I–V–I. He does however indicate the V of bar 6 as the main dominant, by notating in as a half note. He writes in Der freie Satz, § 88, that “there is an interruption [of the Urlinie], although the bass against it performs the 4th progression E♭–A♭”. This appears as a case where V at the interruption remains a mere divider and never is raised to the status of a true structural dominant in subsequent levels.
The second case is Aus meinen Thränen, n. 2 of Schumann’s Dichterliebe. The example below again starts from the V chord of the interruption, here in bar 9, and continues until the return of I, in bar 13:
Schenker writes: “the bass performs a descending arpeggiation of a fifth through the third, without eliminating the interruption”.
Both cases, especially the Schumann one, somehow correspond to the case in Fig. 130.2. They are instances of what Schenker calls a “transferred divider” – by which he means “transferred to a lower level”. In both cases, V at the interruption can be understood as “a projected fifth” with respect to I at the beginning of the piece. And the reason why Schenker does not beam the interruptions in the bass of the two examples of Fig. 22 is that these interruptions do not belong to the first level and do not form Baßbrechungen properly speaking – they are but transferred dividers.
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In my correspondence with Allen Cadwallader, we discussed the the interruption in Bach’s Sarabande of BWV 1012. My answer may be summarized in the following deep middleground graph, reproduced here without comment, but to say that the interruption (at bar 8) is inscribed within an overall I–IV–V–I movement, much as in example 1302 of Der freie Satz.
1 The present note stems from a correspondence that I had in 2011-2012 with Allen Cadwallader while he was preparing with David Gagné his paper read at the 5th Schenker symposium in Mannes in 2013, published since as A. Cadwallader and D. Gagné, “The Evolution of the Quintteiler Concept in Schenker’s Published Writings”, Music Theory Spectrum 38, 2016, pp. 109-117. They rightly mention an earlier usage of the term by Schenker, in Der Tonwille 2 (1922), p. 31 of the German edition. Unless otherwise stated, all translations in this note are my own. Return to text
2 Der Tonwille 5, p. 4, note *. Dubiel’s translation (Tonwille, vol. I, p. 176) reproduces Schenker’s inverted German construction: “The upper or lower fifth of a chord, presenting itself by leap in the service of a passing motion or a neighbour note, I call an upper- or lower-fifth [divider] in Freier Satz” (the word “divider” is accidentally missing in Dubiel’s translation); but this transforms the “normal” complexity of Schenker’s German into an excessively complex English construction. Dubiel also says that the promised account in II3 is to be found in Erläuterungen (Tonwille, vols. 9 and 10, and Das Meisterwerk in der Musik, vols. I and II), which is only partly true: Schenker gives there no more than two very sketchy examples, without a word of explanation. Return to text
3 The “leaping passing tone” is mentioned in Kontrapunkt II (1922), p. 177, where Schenker explains that “the dissonant nature of the passing tone cancels the consonant effect of the leaping interval” and that “the passing note ensnares the consonant leap into the realm of its own dissonance” (transl. J. Oster and J. Thym, p. 181-182) – probably meaning that in such cases the consonant leap (often in the bass) fails to produce a separate harmonic degree and results only in what I have termed an apparent harmony. The case obviously is even clearer when the bass is only implied. As will further appear below, the apparent harmony at a given level may become more “real” at later levels. Return to text
4For William Rothstein (“Articles on Schenker and Schenkerian theory in The New Grove Dictionary of Music and Musicians, 2nd edition”, JMT 45/1, 2001, pp. 218-19), the word ‘divider’ is one of those that “may have two or more distinct meanings at different points in Schenker’s career”. Rothstein believes that the term changed meaning between vol. III of Das Meisterwerk in der Musik and the later works, Fünf Urlinie Tafeln and Der freie Satz. Cadwallader and Gagné, in their MTS article referred to in note 1 above, also describe an evolving conception of the idea of divider in Schenker’s writing; but they concede relying on the “standard English translations” of Schenker’s works. What I’d like to show here, is that there is no incompatibility, nor even inconsistency, and little evolution between Schenker’s various usages of the term. It is true that Schenker’s examples of 1935 are more complex than those of 1922-1923, which merely indicates that he refined his notion of ‘divider’. Return to text
5 The German text of the second edition, 1956, is somewhat different: Neben dem Klang auf der V. Stufe (V-Klang) im Ursatz (§ 15, sowie Fig. 9-11) und den V-Klängen im Mittel- und Vordergrund als Dominanten übertragener Kadenzen (§§ 277, 278) stellt der übertragene Teiler (§ 89) einen Sonderfall innerhalb der V-Klänge vor. Oster’s translation of this sentence, although he does not say so, appears based on the 1935 text rather than on the 1956 one. Return to text
6 § 89, quoted above, is about the half cadence; it really is § 90 that describes the interruption in the voice leading and the resulting retardation in reaching the goal, the final I; § 192 is about the articulation (Gliederung), i.e. the interruption at later levels. Return to text
7 Oster’s statement, in footnote 7 to § 90 of Free Composition, that “Schenker’s graphic presentation of the interruption scheme is inconsistent”, is unfair. The mention of “a carry-over from earlier times” is unjustifiable. Unless I am mistaken, Schenker did not draw beams in his graphs before vol. III of Das Meisterwerk in der Musik, and did not use them there (in the analysis of Beethoven’s Third Symphony) to graph primal structures. The first beams for primal structures, in Fünf Urlinie Tafeln, are reasonably consistent with the usage of Der freie Satz, where Schenker always beams the bass under the interruption with an uninterrupted beam (Figures 22 a and b are special cases to be discussed below). The case of the beam for the Urlinie is less clear: Schenker usually interrupts it at the interruption, at ^2, but not always; this is not done arbitrarily, but a discussion of his reasons is beyond the scope of this note. It must be mentioned, however, that the Urlinie beams of examples 352, 461, 471, 533 and 762, uninterrupted in the German edition of Der freie Satz, have tacitly been interrupted at ^2 in Free Composition. Return to text
8 See note 4 above. Return to text