Bach's inventions

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101 Bach’s Inventions: Figuration, Register, Structure, and the “Clear Way to Develop Inventions Properly” olli väisälä Bach’s 15 two-part Inventions are viewed from the Schenkerian perspective, focusing on three main topics: (1) the significance of figuration (design) and register as structural determinants, (2) the enlargement of characteristic surface figures, and (3) the significance of the Ursatz for ac- tual musical events. It is argued that these three topics pertain crucially to the musical organization in the Inventions and also offer significant viewpoints for the justification of Schenkerian analysis in Bach research. Keywords: Bach, Inventions, Schenkerian analysis, figuration, register, enlargements B ach’s 15 two-part inventions belong to the very best known repertoire of Western art music. Written for pedagogical purposes, they still hold a standard position in both keyboard and counterpoint pedagogy. They have also frequently served as analytical examples of contrapuntal devices, motivic development, and form. 1 Several have doubtless also considered their harmony and voice leading, but they have not featured prominently in the most significant Schenkerian literature. The present study aims to fill in this gap, suggesting that the Inventions constitute an instructive corpus also for giving a “strong foretaste” of the clarity and imaginativeness of Bach’s voice-leading structures. Three general topics, each of which has significance for Bach’s music and its analysis, will be my focus. The first deals with the role of figuration and register as structural determi- nants and as analytical criteria. The second topic is Bach’s tendency to develop characteristic foreground figures by en- larging them as larger structural frameworks. The third ad- dresses the relationship between the “axiomatic” background of Schenkerian analysis, the Urlinie, and the actual musical events. The first of these topics addresses the means by which the composition expresses the multilevel structures that Schenkerian analysis seeks to elucidate. While each level de- pends on the fundamental principles of harmony and voice leading, I will be emphasizing that the determination of structural weight—which element belongs to which level— is largely based on quite different factors, such as figuration In this paper, I shall allude to Bach’s preamble to the Inventions ac- cording to the translation of the Wiener Urtext Edition (ed. Erwin Ratz and Karl Heinz Füssl, trans. unknown): “A sincere guide, in which lovers of keyboard music, and particularly those desiring to learn, are shown a clear way not only (1) how to play faultlessly in 2 parts, but also, upon further progress, (2) how to treat three obbligato parts correctly and well; and at the same time not only to be inspired with good inventions but to develop them properly; and most of all to achieve a cantabile manner of playing and to gain a strong foretaste of composition.” I am indebted to Lauri Suurpää for his valuable comments on a draft of this article. 1 See, for example, Derr (1981) for discussion on such aspects.

Transcript of Bach's inventions

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101

Bach’s Inventions: Figuration, Register, Structure,and the “Clear Way to Develop Inventions Properly”

olli väisälä

Bach’s 15 two-part Inventions are viewed from the Schenkerian perspective, focusing on threemain topics: (1) the significance of figuration (design) and register as structural determinants,(2) the enlargement of characteristic surface figures, and (3) the significance of the Ursatz for ac-tual musical events. It is argued that these three topics pertain crucially to the musical organizationin the Inventions and also offer significant viewpoints for the justification of Schenkerian analysisin Bach research.

Keywords: Bach, Inventions, Schenkerian analysis, figuration, register, enlargements

Bach’s 15 two-part inventions belong to the verybest known repertoire of Western art music. Writtenfor pedagogical purposes, they still hold a standard

position in both keyboard and counterpoint pedagogy.They have also frequently served as analytical examples ofcontrapuntal devices, motivic development, and form.1

Several have doubtless also considered their harmony andvoice leading, but they have not featured prominently in

the most significant Schenkerian literature. The presentstudy aims to fill in this gap, suggesting that the Inventionsconstitute an instructive corpus also for giving a “strongforetaste” of the clarity and imaginativeness of Bach’svoice-leading structures.

Three general topics, each of which has significance forBach’s music and its analysis, will be my focus. The first dealswith the role of figuration and register as structural determi-nants and as analytical criteria. The second topic is Bach’stendency to develop characteristic foreground figures by en-larging them as larger structural frameworks. The third ad-dresses the relationship between the “axiomatic” backgroundof Schenkerian analysis, the Urlinie, and the actual musicalevents. The first of these topics addresses the means by whichthe composition expresses the multilevel structures thatSchenkerian analysis seeks to elucidate. While each level de-pends on the fundamental principles of harmony and voiceleading, I will be emphasizing that the determination ofstructural weight—which element belongs to which level—is largely based on quite different factors, such as figuration

In this paper, I shall allude to Bach’s preamble to the Inventions ac-cording to the translation of the Wiener Urtext Edition (ed. Erwin Ratzand Karl Heinz Füssl, trans. unknown): “A sincere guide, in which loversof keyboard music, and particularly those desiring to learn, are shown aclear way not only (1) how to play faultlessly in 2 parts, but also, uponfurther progress, (2) how to treat three obbligato parts correctly and well;and at the same time not only to be inspired with good inventions but todevelop them properly; and most of all to achieve a cantabile manner ofplaying and to gain a strong foretaste of composition.”

I am indebted to Lauri Suurpää for his valuable comments on adraft of this article.

1 See, for example, Derr (1981) for discussion on such aspects.

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and register.2 The second and third topics identify twosources for the musical contents of such structures: piece-spe-cific figures, on the one hand, and the general striving for theUrlinie descent, on the other.

While none of these topics is by any means original, aspecial attention to them is, I believe, highly beneficial forour understanding both of Bach’s musical organization andof the value of Schenkerian analysis in its illumination.While several analysts—Rothgeb (1971) and Oster (1961)being classic examples—have pointed out the impacts of fig-uration (or design) and register for structural clarification,the significance of these factors seems not always to be fullyrealized in analytical practice (as I have discussed in Väisälä2008). Motive enlargements are, of course, a familiar topic inthe Schenkerian literature (see, for instance, Burkhart 1978),and my contribution in this respect is confined to illuminat-ing its particular significance for the present repertoire.Finally, while Schenkerian analysts are always concernedwith the relationship between the Urlinie and the actualmusic, the existing literature would seem to lack illuminationof this topic based on systematic studies of well-defined butsufficiently extensive corpuses of basic repertoire. The 15Inventions provide an opportunity for outlining how such astudy may be conducted.

I shall proceed by first discussing selected analytical ex-amples for illuminating the first two topics in sections 1 and2. In section 3, I shall take a more comprehensive look atthe third topic. However, since a single Invention may, ofcourse, pertain to all three topics, it would be impractical tokeep the topics strictly separate in the discussion. Hence,

while the examples of section 1 serve primarily to illustratethe structure-determining significance of figuration andregister, I will also pay attention to the structures they helpto determine, including the background level. Similar inter-twinement of topics is evident in all the sections. Figurationand register will retain their significance as analytical crite-ria throughout the analyses, including those focusing on fig-ure enlargements (which are not based on the desire to findsuch enlargements). Finally, besides illuminating Bach’s mu-sical language, I will suggest that these topics also offer sig-nificant arguments for defending the value of theSchenkerian approach in Bach research. Such argumentswill be outlined in section 4.

abbreviations and analytical symbols

The article indentifies each Invention by key in boldface:major keys are given in majuscule; minor in minuscule. Theabbreviations are used both in the text and in the analyticalexamples. There are two symbols used for special purposesin the analytical examples: an asterisk denotes voice-leadingrelationships clarified by parallelism of figuration; and anaccent sign (>) denotes special emphasis pertinent for struc-tural hierarchy.

contrapuntal design and form

Although I shall concentrate on the voice-leading struc-ture rather than contrapuntal design or form, some notionsconcerning the latter two aspects will be necessary for mydiscussion. Most Inventions fall into two basic types with re-spect to thematic style and the nature of the opening imita-tion. Inventions of Type 1 (C, D, d, e, G, and a) open with astatement of a short theme figure in the right hand alone oraccompanied by a simple left-hand gesture to establish thetonic. This statement is followed by imitation in the lefthand. Inventions of Type 2 (E�, E, f, A, B�, and b) open witha longer theme combined with a countersubject, followed by

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2 I discuss this in greater length in Väisälä 2008, in which I identify foursignificant structural determinants that are logically independent of thenorms of harmony and voice-leading: design (including figuration),register, meter, and rhetorical/gestural devices. All these factors pertainto the present analyses, even though the greatest explicit attention ispaid to the first two.

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another statement with inverted voices.3 A special variant ofType 2 is g, in which two countersubjects alternate in the-matic statements. The remaining Inventions, c and F, openwith canons, but, on the basis of the right-hand materialpreceding the left-hand entrance, F comes closer to Type 1and c to Type 2 (despite its one-voice opening).

A central formal consideration concerns recapitulation.Most Type 1 Inventions (C, d, e, and a) lack an unequivocalrecapitulation, if we define this as requiring the concurrentreturn of the structural tonic and of the theme at the originallevel. However, d and a contain salient “recapitulation ges-tures,” theme statements at the original level but prior to thestructural tonic return.4 Most Type 2 Inventions (E�, f, A, B�,and b), along with G and c, close with a short unequivocalrecapitulation: a tonic return marked by a thematic state-ment or two in the original key.5 Such statements often in-volve adaptations with crucial structural significance; hencesuch sections actually do much more than just “recapitulate,”but I shall call them recapitulations for the sake of simplicity.A longer recapitulation, containing a counterpart for all partsof the opening section, is to be found in D and E. In thesecases, the stretch leading to the dominant in the openingsection is transposed at the lower fifth in the recapitulationso as to redirect the music to the tonic. Finally, a comparablelower-fifth transposition, but without any preceding return atthe tonic, is evident in F and g.6

Each invention subdivides according to thematic and keydesign. Sections are usually (but not always) concluded by acadence and opened by a thematic statement in a key thatdiffers from that in the beginning of the preceding section.In middleground readings, measure lines are employed to in-dicate sectional borders. Another significant kind of formalentity is the retransition, by which I mean any passage lead-ing from the last strongly tonicized non-tonic scale degree tothe tonic return.

1. figuration and register as structuraldeterminants

a: retransitional v�–� prolongation clarified by uniform figuration, with a “failed”recapitulation gesture as a digression

The retransitional events in a offer an excellent introduc-tion to the structure-determining impact of figuration, alsoillustrating the participation of register in structural clarifica-tion. A central analytical problem in this Invention concernsthe structural functions of its two recapitulation gestures(measures 18 and 22); Example 1(a) provides an annotatedscore. For a meaningful solution to this problem, let us ex-amine how each of these gestures relates with surroundingfiguration.

The first recapitulation gesture (measure 18) occurs afterfour sequential measures (measures 14–17), which follow thecadence to V (E minor). While this recapitulation gesturemomentarily interrupts the sequential pattern, the right-handpart of this pattern is resumed immediately after this gesture(measure 19). At this point, the left hand also participates in

transposition of the entire passage, in g the correspondence is variedin several ways, rendering it perceptually less salient. These cases alsorelate differently with structure, since measure 26 in F begins a largesubdominant prolongation but measure 13 in g only prepares for suchprolongation (Examples 6[c] and 7[c]).

3 In B�, the countersubject is slightly varied in these two statements. In b,the countersubject is first presented as an unfigurated skeleton.

4 In a, there are two such gestures, the latter of which (measure 22) is ac-companied by the I6 of a I6–V–I auxiliary cadence and thus anticipatesthe tonic while not yet establishing it (Example 1[a]).

5 In several cases (E�, f, b, G), the effect of the tonic return is softened bya local I6 whose bass represents the goal 3 of a large-scale 5–4–3 top-voice progression. In most cases (E�, b, and G), the 3 returns shortly toits due top-voice location above I.

6 In F, measures 4–12 correspond to measures 26–34. In g, measures 1–3correspond to measures 13–15 and measures 5–9 correspond to mea-sures 163–212. However, whereas F shows a straightforward lower-fifth

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similar figuration, adding to the vigor of this resumption.Similar right-hand figuration continues for the next twomeasures, leading to a semi-cadence V in measure 22.

Figuration suggests thus that the “attempted” recapitula-tion in measure 18 “fails” and is only a temporary digressionwithin an otherwise unified pattern connecting the V� inmeasure 13 with the V� in measure 22.7 Registral featuressupport such a perception. As circled in Example 1(a), astepwise descending line is formed by the high points of thetop voice, except for measure 21, in which the C4 occurs oneoctave “too low.” (The D5–C4 relationship is clarified by dis-sonance treatment.) Since the first of these high points, B�

5,is not supported by the V� but represents a chromatic passingtone, the descent may be interpreted as a filled-in octavefrom an implicit B5 to B4. As for the bass, the effect of therecapitulation gesture (measure 18) is weakened—apart from

the local 63 position of the “I” chord—by the bass’s sudden

withdrawal from the low register. The G�2 at the end of the

sequence of measures 14–17 creates an expectation of an A2,which is only fulfilled in measure 21. The bass motion con-tinues to D�

2, which suggests a registral connection and aneighboring relationship with the original E2 (measure 14).In all, the low register notes form an unfolding figure inwhich E2–G�

2 is answered by A2–D�2.

Example 1(b) clarifies the structural derivation of thispassage. The “I” of the “failed” recapitulation gesture has itsbasis in the neighboring motion B–C–B above the retransi-tional V�–� prolongation.8 Example 1(c) shows glimpses oftwo other Inventions—d and e—indicating that such “I”chords are not unique to a.9 The octave descent from 2 to 2(F�

5–F�4) in e resembles, in certain respects, that in a and might

be viewed as a simplified model for it. On the other hand, d,is another interesting example in which the “I” is exploited

7 The notion of the first recapitulation gesture as a digression relates sug-gestively to the compositional process of a, as this gesture is among thematerial that Bach interpolated to the music after the preliminary ver-sion of a (found in Clavierbüchlein vor Wilhelm Friedemann Bach). Inmy analysis, the two versions of this Invention show a basically similarstructure, as opposed to Neumeyer (1981), who regards the first reca-pitulation gesture as a structural tonic.

8 According to this reading, the C bass of the first recapitulation gestureis a transferred upper-voice tone. This function is concretized by thecorrespondence between the left-hand figure of measure 19 and theright-hand figure of measure 21.

9 Whereas the “I” is subordinate to �IV7 (= VII7 of V) in a, it forms themain support for the neighboring tone in e and d.

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for creating a non-structural recapitulation gesture; I shallreturn to this example in section 2 below in connection withExample 5.

My identification of the first recapitulation gesture in a as“failed” is not meant to belittle its significance for the retran-sitional events. While figuration and register support perceiv-ing this gesture as a digression within a dominant-framedspan, it associates strongly with the upcoming tonic and alsobuttresses the feeling of the tonic key (which, characteristi-cally, arrives much before the structural tonic harmony). Bothof these contrasting aspects are significant for the musical ef-fect, but a much more meaningful picture of prolongationalrelationships emerges if we follow the clues given by figura-tion and register and do not regard this gesture as a structuraltonic.10 Example 1(d) illustrates how the retransitional eventsrelate with the overall structure. The significance of theB5–B4 octave descent is clarified by its registral associationwith an earlier salient coupling C5–C6. Together these cou-plings elaborate degrees 3 and 2 in the 5-Urlinie.

The second recapitulation gesture (measure 223), in con-trast to the first, is followed by no return to the preceding

figuration. As if to repel any such possibility, the figurationpattern of the theme is extended so as to occupy five quarters(measures 223–233). The remaining music also sustains aconstant motion in invariant time values in each hand, creat-ing the impression of a single relentless sweep up to the finaltonic. As indicated in Examples 1(a) and 1(d), such an im-pression may be aptly described in terms of the auxiliary ca-dence I6–IV–V–I.11 Since the bass of this I6 lies an octavelower (C3) than that of the first recapitulation gesture (C4),register also contributes to the impression of the second ges-ture as the more decisive of the two. As indicated inExample 1(d), the top voice of this auxiliary cadence de-scends from 5 to 1, repeating the earlier 5–2 motion in acompressed form and delaying the eventual arrival at 1. The

10 In addition to Neumeyer (1981), Travis (1976) and Adrian (1985) regard this gesture as a structural tonic.

11 The details of this reading may call for some clarification. Readingthe I6 as the governing harmony at the second recapitulation gestureis based on its occurrence at the third beats of measures 22 and 23,the framing points of the uniform right-hand figuration and strongpoints in the perceived meter (notwithstanding the measure lines).The ensuing IV chord (measure 234) is marked, despite its metricalweakness, by the change in right-hand figuration and by the lack ofregistral continuation in the bass at the subsequent strong beat. Thereturn to the IV (measure 244) is underlined by the extreme registralposition of its bass (D2).

example 1(c). Comparable retransitional V �– � progressions

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overall structure may thus be viewed as a variant of the inter-rupted structure, whose second branch is weakened by itsshort duration and by the 63 position of its opening tonic.

The relationship between figuration and structure in thisintroductory example illustrates a basically familiar analyti-cal principle. As formulated by John Rothgeb (1971, 231),“changes in surface design usually coincide with crucialstructural points, and accordingly such changes must begiven the most thoughtful attention in deriving or verifyingan analysis.” The first recapitulation gesture suggests achange in surface design, but the suggestion is quickly can-celled, depriving this gesture of structural decisiveness. Onlythe second gesture effects a permanent change in design,signaling a decisive break with the dominant ’s prolonga-tional orbit. Another way to formulate Rothgeb’s principleis by saying that spans of uniform figuration—i.e., thosewithout changes in surface design—tend to correspond withspans relevant to prolongational hierarchy. This principle isof utmost significance for Bach analysis and, as the presentexample suggests, it is powerful enough to apply even when

the uniformity of figuration is interrupted by a temporarydigression.

This example also illustrates the structural significance ofregistral proximity and extremity. The bass-line unfolding(E2–G�

2, A2–D�2) is supported by both proximity and ex-

tremity, as is the large-scale top-voice coupling (C5–C6,B5–B4). However, whereas figuration or design provides, Iwould suggest, a virtually unfailing “key to structure” inBach, we should bear in mind that registral positions involveconsiderable flexibility. When other factors are sufficientlystrong in determining the structural position of an element,the registral presentation of that element is freer. For exam-ple, since the structural position of the dominant in measure22 is clarified by the expectation created by the preceding °7chord on D�

2 and by the radical change in design, the loca-tion of that dominant (E3) an octave higher than expecteddoes not jeopardize structural clarity. In this case, leaving theE2 implicit in measure 22 is aesthetically motivated by theway it adds to the freshness and effectiveness of the explicitD�

2–E2 motion in the closing cadence (measure 25).

example 1(d). Overall sketch

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e�: parallelism of figuration and register clarifying top-voice motions

Connections of figuration may clarify voice leading evenwhen involving more widely dispersed details, especiallywhen supported by extreme registral position. Blendingstructural with musical notation (the latter only in upperstave), Example 2 illustrates how the cooperation of figura-tion and register highlights top-voice motions in E�.The be-ginning of the Invention theme centers on 3 (G4, measures1–3), but an “open position” arpeggiation G4–E�

5–B�5 leads

the top voice to a higher 5, a persistent tone in the counter-subject (measures 5–8; Example 2 shows only the beginningof the countersubject). Despite the local V harmony, the ex-treme register and the persistence of this B�

5 attract attentionso as to suggest Kopfton status. This suggestion is borne outby subsequent top-voice events, in which C6 (measure 12)and A�

5 (measure 25) are highlighted through similar high-register figuration, creating stepwise connections with the B�

5and confirming its role as the point of departure for top-voice

motions. Of these high-register tones, C6 occurs within acountersubject statement but A�

5 is marked by the freer em-ployment of similar figuration. Hence, while the B�

5–C6 rela-tionship might be viewed as a by-product of the thematic dis-position, the appearance of similar figuration to mark A�

5suggests more expressly an impulse to clarify voice-leading re-lationships as its motivation.

B�5, C6, and A�

5 participate in a 5–6–4–3 framework, com-pleted at the beginning of the short recapitulation (measure27) and thus spanning most of the Invention. Whereas thebeginning of this framework is registrally uniform, the 4–3motion involves a registral drop from A�

5 to G4 (more locallythis motion is transferred to the bass in measures 26–27).Such a registral drop has significant implications for the struc-ture, since it undermines the definitiveness of this initial pro-gression, suggesting that it does not yet form part of theUrlinie proper. And indeed, a more satisfactory 4–3 Urliniedescent, leading to 2–1, follows in the recapitulation once theextended Invention theme has climbed emphatically back tothe obligatory high register (A�

5 in measure 31). The crucial

example 2. E �: register, figuration, and structure

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significance of these registral events for structural perceptionbecomes evident if we consider the effects that a high 3 (G5)would have at the beginning of the recapitulation. A 4 (A�

5)surrounded by two such high 3s would likely be heard as aneighbor of 3 and could not function as an Urlinie tone.Analogous registral events are evident in the bass, whichmoves away from the original low register (E�

2) at the begin-ning of the recapitulation (E�

3) and returns there so as to un-derline the Urlinie 4 (F2).

The structure of E� is particularly noteworthy because itexemplifies a structural pattern whose variants recur in severalother Inventions. I shall return to E� for discussing these re-lated cases in section 3, in connection with Example 14.

d: parallelism establishing registral discrepancyand reconciliation

Figuration and register do not always work neatly in co-operation, but parallelism of figuration may also help to es-tablish a voice-leading connection between registrally non-correspondent elements. As illustrated by brackets and smallasterisks in Example 3(a) (measures 1–5), occurrences of thetheme figure and its variant establish a registral discrepancybetween a low 3 (F�

4) and a high 2 (E5) at the opening of D.This discrepancy is a prominent feature of this Inventionand has crucial ramifications for its unusual structure.

Apart from the registral discrepancy, this 3–2 motionshows other unusual features that highlight the character ofthe 2 as a surprise element. 2 and the accompanying toni-cized dominant arrive exceptionally early and withoutpreceding modulatory procedures. The dominant key is es-tablished only afterwards through a repetitious four-measurepassage (measures 5–8) with a dominant pedal activated byoctave leaps, a passage with uncharacteristically simple tex-ture for Bach.

The large asterisks in Example 3(a) (measures 5 and 47)indicate a parallelism on a larger scale, one between this oc-tave-leap passage and its correspondent in the recapitulation

(measures 47–50). For considering the structural implica-tions of this parallelism, it should first be noted that the for-mer passage begins a large but relatively straightforwardprolongation of V and 2, extending all the way to the begin-ning of the recapitulation (measures 5–42; Example 3[b]provides details of the interpretation). At the end of thisprolongation, the F�

4–E5 seventh is recomposed on a retran-sitional V6

4-5-3 framework, reminding us of the original dis-

crepancy (measures 39–42). Now the ascending seventhF�

4–E5 naturally creates an expectation of D5 to effect achange in direction in the melodic motion and to satisfy thetop voice’s aspiration towards the higher octave. And whilethe recapitulation begins with the low 3 (F�

4, measure 43),which suggests a structural interruption, an ascent to D5follows shortly (measure 47), pointing to a registral E5–D5connection over the intervening low 3. On the large scale,the E5–D5 connection is supported by the parallelismshown by the large asterisks, as the D5 is featured by thereappearance of the octave-leap texture that originally es-tablished the E5.

Design and register thus suggest hearing the D high-lighted by the octave leaps in measures 47–50 as represent-ing 1 in a large-scale 3–2–1 pattern. Such a hearing im-plies that the features pointing to the interrupted structureas the basis of this Invention should not be taken quite atface value. To be sure, the recapitulation section can beheard as embodying the second branch of the interruptedstructure. The octave leaps on 1 are followed by tworegistrally uniform small-scale 3–2–1 progressions inboth registers pertinent to the initial F�

4–E5 discrepancy(F�

5–E5–D5 in measures 51–54 and F�4–E4–D4 in mea-

sures 56–59), as if to correct the discrepancy. The former3–2–1 is accompanied by a deceptive cadence and the latterby the final authentic cadence, which completes the secondbranch. However, the perception of 1 as being already statedbefore these progressions offers an unusual—and aestheti-cally satisfactory—perspective for hearing them. Instead ofintroducing 1, they confirm an earlier 1. Such a hearing

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example 3(a). D: measures 1–12 and 38–59, annotated score

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � �� �� � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � �

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� � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � ���

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � �

��

~

*

3̂ (F�4)

*

2̂ (E5)

5 10

*

*

38

retrans.

(F � 4 E5)

43

recap.

3̂ (F�4)

45

1̂!

*

50

3̂ (F�5) 2̂(E5) 1̂(D5)

55

3̂ (F�4) 2̂ (E4) 1̂(D4)

V 64

V

V 6I

*

(A: I IV 42(V “I” )

I) V 53 I

IV V VI (I) IV I

( )

642(V “I” ) 64

2(V “I” )

40

is supported by features that help to associate the octave-leap 1 (measures 47–50) with the final 1: the reappearanceof the octave leap at the end of the first 3–2–1 progression(measure 54) and the general similarity between the twoprogressions.

Example 3(b) offers more systematic explanation of thestructure. Graph (i) adapts for D major Schenker’s notation(1979, Fig. 21b) for the interrupted structure. A key featureof this idea is the subordinate relationship of the secondbranch to the final 1. Graph (ii) expresses the same idea with

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a slightly different notation, adding an inner voice. A latentproperty of this inner voice is the anticipation of the final 1at the outset of the second branch. While such an anticipa-tion usually has no particular significance, register anddesign bring it to the fore in D. As a result, the anticipated 1may be perceived as overriding the 3–2–1 motion in structuralpriority, as indicated in graphs (iii) and (iv) (with somewhat

different notations).12 Besides being supported by designand register, such a perception also yields aesthetic rewards.The octave-leap texture in measures 5–8 and 47–50, whichin itself may appear strikingly unsophisticated for Bach,

12 While it is controversial whether the conception of the second branch assubordinate to the first is cogent in all instances of the interrupted

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� � �� (iv)

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5 43 47 5 24 35 39 43 47 51 54

(i) (ii) (iii)3̂ 2̂ 3̂( 2̂) 1̂

Schenker's conception ofinterruption (FC: Fig. 21b)

Another notation forsimilar conception

antic.

ant.

2̂ 1̂!

*7th

( ) ( )6th

recap.*7th

* N7th

(reminder)retrans.

*recap.

Inlarge-scale parallelismhighlight the anticipation of

D, register and

1̂.

*

*

*(v)

5 9 12 19 24 33 35 39 43 47 51 54

**

** * * *

*

*

**

***64

53 *

64VI VI 5

3V I V I

I V

(A: I IV“I6”V I)

V 5

(b: V 42

651�

I6VII6(I II6“I6”V) I)

VI V7 V I VI (I) IV “I6”V I

( )( ) ( )

example 3(b). Derivation of structure

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structure it is interesting to note that the two Inventions that are ana-lyzable as embodying variants of the interrupted structure show con-crete features supporting such subordination. In a, the second branchis brief and starts from I6; in D, the anticipated 1 overshadows theKopfton’s return.

turns out to be a vehicle in the subtle and witty poetics oftwo unexpected Urlinie events. Moreover, since both D5 andD4 appear as representatives of Urlinie 1 in this reading, theinitial registral discrepancy is satisfactorily reconciled. TheD5 fulfills the aspiration of the F�

4–E5 motion towards thehigher octave, whereas the D4 remains in touch with theoriginal register. By contrast, if we based our interpretationon an ordinary interrupted structure, only D4 would representthe final 1.

Graph (v) proceeds to the foreground reading. Amongother things, it shows several more instances in which repeti-tions of the thematic figure suggest certain voice-leadingconnections (small asterisks). A noteworthy phenomenon,bearing on several subsequent analyses, is the transfer of suchconnections from the treble to the bass in cadences (measures9–12, 21–24, 35–38, 51–54, 56–59). As illustrated in Example3(c), this formula involves the motion from a local 3 to its in-complete neighbor 4 in the bass, which concurrently functions

as the bass of IV or II6. 4 initiates a passing motion 4–3–2 inhemiola rhythm, in which the passing 3 supports an appar-ent I6 and 2, harmonized by V, returns to its correct positionin the upper voice.13 As the subsequent examples will testify,such “I6” chords, whose bass tones represent passing upper-voice 3s, appear frequently as intermediate elements betweenIV or II and V in Bach’s cadences. The awareness of theirnon-tonic function is essential for Bach analysis.14

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������ � � � �� � � � � � � ������� ���� �� � � � � �

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3̂IN4̂( )

P2̂ 1̂3̂ 3̂ 3̂4̂( ) 2̂ 1̂ 3̂

*

4̂( 3̂) 2̂ 1̂

INP*

56

IN P

IV 642(V “I )”I IV8 7 V I I V I I IV 64

2(V “I )” V I

example 3(c). Cadential formula with “I6” between IV and V

13 For illustrating the general formula, Example 3(c) shows the openingchord as I, although, as evident from Example 3(b), graph (v), thechord in measure 56 may in its actual context be perceived as a paren-thetical event between VI and IV.

14 Characteristics of such progressions include a relatively emphatic IV(or II), a relatively weak “I6,” often approached through a local V4

2, anda V–I cadence. Psychologically speaking, the expectations created bythe emphatic subdominant are not satisfied by the weak V4

2– “I6” pro-gression but only by the strong V–I. As in Rothstein 2006 (268 ff.),such “I6” chords may be described as inverted cadential 6

4s, especiallywhen their metric position is clearly strong in relation to the subse-quent dominant. However, when the 6

3 occupies the second beat in ahemiola rhythm, as in the present case, the aptness of this description issomewhat more questionable. Incidentally, this kind of cadential for-mula is not limited to Baroque music; for a hemiolic example similar toExample 3(c), see Beethoven, Sonata in F Minor, op. 2 no. 1, secondmovement, measures 5–8.

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e: supreme lucidity—with a turn to obscurity

As illustrated in Example 4(a), the opening section of E(measures 1–20) offers especially lucid illustration of thestructure-determining impacts of register and figuration.The outer-voice frameworks are clarified by extreme registralposition. The top voice presents an ascending arpeggiation1–3–5 (E5–G�

5–B5), with a passing �4 (A�4) occurring in a

lower register. The bass descends stepwise from I to V(E2–D�

2–C�2–B1); B1—the only note below C2 in the

Inventions—confirms the registral logic by its appearance atthe very end of the section.15

As for figuration, two rhythmic elements, syncopationand dactyls—hereafter S and D—help to identify units ofstructural significance. Following the pattern set by thetheme, the S element signals the beginnings of such unitsand the D element signals their ends. Hence after the se-quential two-bar SD units in measures 9–12, which lead thebass from E via D� to the C� in measure 13, the pattern islengthened to four bars (SSSD) in measures 13–16, delayingthe arrival at the goal B (Example 4[a], graph [ii]). The im-pact of figuration and register for structural perception be-comes especially evident by considering the status of the Bmajor chord at the downbeat of measure 15. This is preciselythe moment at which the preceding two-bar rhythm wouldlead us to expect the E–D�–C� motion to reach its goal.However, the occurrence of this B within the lengthened SSSDpattern and its high registral position (above its immediate

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1 9 13 17 1 8

(i)

10

(ii)P

15

S D S D S DS S

G�5̂

F� E

F�D�B

20

Cf. theme in B major

25

delayed tensionG� F� E D� F� B

release

53I V (B: II (“I”) 6

5 V 42 I6 V I )

� � 42

66 � �� �� �(etc.)

P

example 4(a). E: first section (measures 1–20)

15 That Bach wrote this B1 even though this tone was not always readilyavailable in the contemporary keyboard instruments may testify to theimportance of registral logic for his musical thinking.

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surroundings) suffice to dispel any effect of goal arrival.Rather, this B functions as harmonic support for the passingmotion E5–D�

5–C�5, which prolongs the C� minor harmony

(II in B major).Register and figuration also strongly pertain to the analy-

sis of the second section (measures 21–42); see Example4(b). In terms of key relationships, the G� minor passage inmeasures 29–32 is a crucial event for preparing the estab-lishment of III at the end of this section. However, key areasand prolongational spans do not necessarily or even usuallycoincide. In fact, instances in which a new key area and a

prolongational span begin concurrently (as in measure 5 ofD, shown in Example 3) are rather exceptional. In modula-tion the new key’s tonic frequently appears in a structurallysubordinate position prior to its actual establishment, as ex-emplified by the above-discussed B major chord (or by the Aminor chord of the first recapitulation gesture in a). In thepresent case, register and figuration, as well as the lack ofpreparation, suggest a relatively weak structural position forthe first G� minor chord (measure 29). Its lowest note, G�

3, isa sixth above the preceding bass note B2, which enables theG� minor triad to be perceived as emerging from 5–6 motion

� � � � � � �� �� � � � � � � � � � � � � � � � � � � � �� � � � � � ! � � � � � � � � �� � � � � � � ! � � � �� � � �� � � � � � � � � � � �� � � � � � � � � � � ! � � � � � �

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30

S S S DTheme Digression

D D D

�N

S“Main discourse” continues

D S D

35

S D D DDecisive breakaway

(to D � in m. 41)

!

25

example 4(b). Measures 25–39, annotated score

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above B. An emphatic return to the registral vicinity of theB2 follows in measure 39, in which the C�

3, supporting aNeapolitan sixth, is marked by the first halt in the left hand’seighth-note motion since measure 29, by hemiola rhythm(measures 39–40), and by the ensuing cadential progression.These features suggest a bass-line connection between theinitial B2 and this C�

3, relegating the intervening G� minorchords to a lower structural status.

Considerations of right-hand figuration support and re-fine such a reading. As evident from Example 4(b), the firstfour measures of the G� minor passage (measures 29–32)comprise dactyls only (DDDD), deviating from the preva-lent norm of S beginnings for structurally significant units.After this passage, the S element is resumed, and three SDunits follow (measures 33–38). Somewhat reminiscent of thefirst recapitulation gesture in a, this DDDD passage suggestsa change in surface design but fails to establish it, whereasthe upcoming Neapolitan sixth on C� (measure 39) is markedby a more decisive abandonment of the S element. The di-gressive character of the DDDD passage is strengthened fur-ther by stepwise connections between the preceding themestatement in B major (measures 25–28) and the ensuing SDunit (measures 33–34). As we can see in Example 4(b),

such relationships concern both the S element (B5–A5; seecircles) and the D element (F�

5–E5; see beams), and also thebass, in which the association between the bracketed figuresin measures 25 and 33 supports the chromatic connectionbetween B2 and B�

3 despite the registral difference.Following the clues given by figuration leads to a reading inwhich the composing-out of the B major harmony continuesafter the DDDD digression up to the emphatic Neapolitansixth; for systematic illustration, see Example 4(c). The lead-ing upper-voice progression descends a fifth from F�

5 to B4and is consistently articulated by the D element. The higherstratum of notes, shown by circles in Example 4(b), has sub-sidiary significance for voice leading, but by echoing thechromatic head-motive B5–A�

5–A�5–G�

5 of the B majortheme statement it strengthens the sense in which this state-ment extends its influence into the subsequent events despitethe destabilization of the B major key.16

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21 39 42 21 36 38 39 42 21 25 30 35 40

(i) (ii) (iii)B A� (A�) G

(cf. m. 25)

P

B A� A�G�

N

N

S S S D D D D D S D S D S D D D D

B5

B5

6

6

example 4(c). Second section (measures 21–42), structural derivation

16 The G�5 in measure 35 resolves to the F�

3 in measure 36, which standsfor an inner-voice tone despite its low registral position. This low posi-tion enables this F�

3 to associate registrally with the subsequent F 3and G�

3, highlighting the 5–6 motion in measures 36–38.

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While the concluding section (measures 43–62) consti-tutes a formal recapitulation, it shows nothing of the lucidcorrelation between register and structure that character-izes the opening section. Through inverted counterpoint(partly at the twelfth), the strong-beat elements of the SDand SSSD units in measures 9–16 are shifted from theregistral extremes to the middle register in the correspond-ing passage of measures 51–58. As illustrated in graph(iii) in Example 4(d), the highest tones do not form anUrlinie descent but neighbor motion around the 8(E5–F�

5–E5–D�5–F5). Figuration, however, proves once

again its reliability as a key to structure. At the end of thelast E major theme statement (measure 50), the D elementis transferred from the right hand to the left hand, carry-ing along 5. In the subsequent passage (measures 51–58),the S and D elements remain in the left hand, which,

owing to these high-profile rhythms, may be heard as at-tracting the primary melodic attention.17 Figuration helpsthus to direct the listener’s attention to the left-hand pro-gression 5–(6–5)–4–3, in which the 4–3 motion is high-lighted through the lengthened SSSD pattern (measures55–58) and through the absence of right-hand activity atthe decisive moment (measure 58–591). The 3, in turn,supports a “I6” whose function is similar to those discussedabove in connection with the cadences of D. It is a passingtone in a 4–3–2 progression, in which 2 is implied by the

17 There is also a less evident way in which figuration supports the 5–6

connection in measures 50–51: the right-hand dactyls in measure 50outline a syncopated B4–G�

4–E4 figure that corresponds to the left-hand C�

4–A3–F�3 figure in measure 51.

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From:

2̂ 1̂

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cadential dominant (measure 59, beat 3).18 The comparisonof graph (ii) with the cadences in D (see especially Example3[c]) illustrates how the harmonic structure of the conclud-ing section is modeled on the cadential pattern.19

These observations suggest that while the Kopfton 5 isestablished in the opening section by a lucid 1–3–5 high-register arpeggiation, the rest of the Urlinie takes the more

obscure form in which the 4–3 motion appears in the bassand the 2 is omitted altogether.20 Graphs (i)–(iii) inExample 4(d) elucidate the derivation of the structure from amore normal registral setting. The transformations leadingfrom the “normal” setting to the actual music are unusuallyradical, which relates with the curious, only “minimally con-clusive” musical effect of the conclusion. In particular, thefinal cadence (measure 59)—missing 7 as well as 2 and fea-turing a registrally high rhythmically weak dominant, B3—sounds deliberately understated.21 Example 4(e) graphs theoverall structure, showing actual registers. It also sheds somelight on the significance of the 8 (E5) that covers the Urlinieprogression, indicating by a dotted tie its relationship withthe prominent 8 of the opening.

The obscurity of the Urlinie descent should not be con-fused with structural ambiguity. While the bass-line articu-lation of 4 and 3 and the omission of 2 makes this descentless prominent, it leads unambiguously from 5 to 1 and is,as far as I can see, the only plausible Urlinie candidate inthis piece.

* * *

By the above four examples, I have attempted to present asample of ways in which figuration (or design) and register

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18 Despite the apparent similarity between the cadences to B major inmeasures 17–18 and to E major in measures 59–60, I have interpretedthem differently. Whereas the bass D� in measure 17 already representsthe third of the upcoming I of B major, the bass G� in measure 59 is apassing 3 between 4 and 2 (cf. Examples 4[a] and 4[d]). This differencein interpretation is supported by differences in design and register.Whereas measure 17 is marked by a clear-cut change in design, measures58–59 are connected through a tie, which makes the latter sound as acontinuation to the preceding events. The perceptually salient unfoldingF�

5–A4–G�4–E5 (measures 58–62) also discourages perceiving the “I6”,

occurring in the midst of the F�5-A4 interval, as a real tonic. Finally, the

prominence of the registrally isolated E2–D�2–C�

2–B2 (+B1) bass line inthe first section overshadows the local dominant in measure 17, but thefinal cadential dominant (m. 59) is not similarly overshadowed.

19 Metrical relationships differ, however, from the cadences in D. The “I6.”in measure 59 of E is clearly strong in relation to the subsequent V,which makes it more justified to describe it as an “inverted cadential 64.”

20 For discussion of bass-line articulation of the Urlinie, see Wen 1999. Asregards my notating the implicit 2 below the bass V in Example 4(d),graphs (ii) and (iii) (instead of showing 2 as returning to its normalupper-voice location), I have been partly motivated by visual simplicity.However, this notation also conveys the idea that the left hand at-tempts, as it were, to express both structural outer voices in one melodicline and one register, but the impossibility to do this for the V and 2necessitates the omission of 2.

21 In considering the aesthetic motivation behind such features, it may benoted that the weak Urlinie articulation and the concomitant lack ofregistral connections with preceding events enhance the character ofthe concluding section as a self-contained entity, reflecting the generaltendency in this Invention towards sharp sectionalization.

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118 music theory spectrum 31 (2009)

relate with structure. The significance of figuration is evidentboth in determining small-scale structural units and in es-tablishing large-scale connections. Registral proximity andextremity often also support such connections. However,voice-leading connections may also transfer from a registerto another if this is clarified through other factors, such asfiguration or dissonance treatment. Such transfers may havevarious compositional purposes. In E�, for example, theregistral drop in the initial 5–6–4–3 progression helps toclarify structural hierarchy by undermining this progressionin relation to the eventual high-register Urlinie descent.However, registral transfers may also be employed for im-parting a special character to the Urlinie progression, as ex-emplified in two different ways by D and E.

Before turning to further examples for illustrating mysecond main topic, figure enlargement, it is worthwhile topoint out some manifestations of this phenomenon in E.One such feature has already been mentioned in reference toExamples 4(b) and 4(c): the B major theme statement is fol-lowed by a hidden repetition of its initial B5–A�

5–A�5–G�

5figure. Moreover, as illustrated in Example 4(c), graph (i),the Neapolitan sixth in measure 39 enables the permutationof this chromatic figure (B–A�–[A�]–G�) to form the top-voice framework of the entire second section, whereas thecounterpointing B–C�–D� bass line reproduces the beginningof the countersubject (measure 25). The vertical alignment inExample 4(a), graph (ii), illustrates how the subsequent fig-ure of the B major theme statement, G�

5–F�5–E5–D�

5 (mea-sure 26) is enlarged in anticipation in the first section. Therelationship between the small and the large is reinforcedby a delay–release pattern manifest in the rhythmic or-ganization on both scales. The associational network is fur-ther strengthened by the surface occurrences of theG�

5–F�5–E5–D�

5 figure in measures 14–15 (summarizing thelarger figure) and in measures 35–36 (marking the return tothe B major harmony after the B–A�–A�–G� enlargement).Finally, as shown in Example 4(d), graph (ii), theC�

4–B3–A3–G�3 line in the concluding section (measures

51–59) relates analogously with the E major theme state-ment; in fact, the section in its entirety might be conceivedas a kind of enlargement of the theme.

2. figure enlargement

d: syncopation as a key to structure

The opening section of d (measures 1–18) offers a modelexample of a concentrated development of a musical idea bysurface repetitions, culminating in the enlargement of thevery same idea. Before entering this example, it should beacknowledged that Wayne Petty (2006), working indepen-dently, has recently presented an excellent discussion of fig-ure enlargement in this section. The following descriptionagrees essentially with Petty’s, but I will attempt to con-tribute to our analytical understanding by relating thisfeature to larger structural perspectives. Moreover, my inter-pretation of the harmonic structure in the opening sectiondiffers substantially from Petty’s.

From the Schenkerian perspective, the opening of d posesthe problem of identifying the Kopfton. As illustrated inExample 5(a), two plausible candidates appear: 5 (A4) of theopening theme figure and the higher 3 (F5, measure 4) that

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bach’s inventions: figuration, register, structure, and the “clear way to develop inventions properly” 119

closes the opening imitations and starts the ensuing se-quence.22 3 seems at first to be supported by its higher regis-ter, but, as shown in Example 4(b), the sequential descent inmeasures 7–14 shifts the focus back to the vicinity of theoriginal 5 by leading the top voice to the syncopated B�

4(measures 14–15). As a new surface rhythm, this syncopa-tion helps to mark B�, but it also creates a special associationwith the opening. As shown in Example 5(b), the theme

implies a syncope figure on B�. Similarly implied syncopessaturate the music, but measures 14–15 become a specialmoment by turning the implicit into explicit at the originallevel. In the ensuing cadence to III (F major), the A–B� con-nection is completed to form an enlargement of the themefigure’s top voice A–B�–A–G–F as the framework of the en-tire section. The passing A is transferred to the bass of the Fmajor 6

3 chord in measure 16, another example of a local “I6”whose bass fulfills such a passing function.23

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22 According to Forte and Gilbert (1982, 264), the Kopfton finds its pri-mary expression in the A5 in measure 5. Despite the high register ofthis A5, measure 5 is, I would suggest, an unlikely location for such adecisive event because measure 7 is marked by a much stronger changein design and also because measure 5 is characterized by voice-leadingtendencies that are only fulfilled in measure 7 (see Example 5[a]).

23 Features such as left-hand figuration ensure that this F major 63 is heardas part of the process leading to the III (m. 18), rather than already rep-resenting the III, as Petty (2006, Example 2) shows it. Another featurein my analysis that deviates from Petty’s is the reading of a G bass assupporting the syncopated B�. This reading relies on the expectations

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120 music theory spectrum 31 (2009)

The concentration of relationships in this example ischaracteristically Bachian. Not only is a figure enlarged, butthe crucial detail in the enlargement—the syncopated B�—also bears a special foreground relationship with the originalfigure. Moreover, additional features enhance the associationbetween the small and the large. The sequential passage inmeasures 7–14 subtly sustains the memory of the B�–A–G–Fline in an inner voice.24 Furthermore, the sequential F5–B�

4descent (measures 7–15) is anticipated at the start of the se-quence (measures 7–8), as is the accompanying D3–G3 bassmotion (see brackets in Example 5[b]).

The A–B�–A–G–F framework in the opening section pointsto 5 as the melodic starting point or Kopfton. As illustrated bythe overall sketch in Example 5(c), this suggestion is borneout by the subsequent events. The second section balancesthe opening A–G–F motion by approaching A from above(C–B–A, measures 18–38). References to the higher 3 (F5)remain, however, an important element in the music. Asshown in Example 5(d), an annotated score of the conclud-ing events, the last of such references is made by the recapit-ulation gesture in measures 44–461. As observed above(Example 1[c]), this gesture does not yet represent the struc-tural tonic return; its bass tone (“I”) functions within a largerbass-line descent that leads from the tonicized A minor tothe C� in measure 47, a goal underlined through the ensuinghemiolic cadential pattern. Despite its non-structural status,the recapitulation gesture functions as a reminder of theoriginal F5 and directs attention to the subsequent E5 andD5, which cover Urlinie tones 4 (G4) and 3 (F4) at the uppersixth (measures 47–49). Most remarkably, the 4 is madestand out through another explicit syncope—the second onein the piece—confirming the key role of this rhythmic devicefor the structure of d.

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created by the chain of parallel tenths at the downbeats of measures 7,9, 11, 13. When the syncopated B� arrives one eighth-note earlier thanexpected (measure 143), coinciding with a G, I find it easy to perceivethe tenth G–B� as continuation to this chain.

24 According to Example 5(b), it is not obvious whether B�, A, G, and Fin measures 8, 10, 12, and 14 are related through voice-leading or func-tion as incomplete neighbors (echappée tones) of preceding inner-voicetones. However, as indicated by Petty (2006), these tones may also beperceived as implicitly suspended in the subsequent measure and thusforming a voice-leading line.

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bach’s inventions: figuration, register, structure, and the “clear way to develop inventions properly” 121

g: the expansive descending tetrachord

g offers another fascinating example in which the deter-mination of Kopfton links with figure enlargement and fea-tures an implied element turning explicit. The pertinent fig-ure is the descending tetrachord G–F–E�–D or 8–7–6–5. Asillustrated in Examples 6(a) and 6(b), the tetrachord occursfirst as a chromaticized form in the bass (measures 1–2), atypical passus duriusculus figure serving as the first counter-subject (even though the concomitant 6–5 upper-voice mo-tion is less typical).25 The second countersubject, given inthe right hand of measures 3–4, with its conspicuous ascentto �7 (F�

5), is more unusual. As indicated in Example 6(b),F5 points to a preceding implicit G5 as its point of origin.Consequently, the 8–7–6–5 progression may be perceived asalso residing implicitly in the second countersubject.

The implicit G5 (8) is made explicit at the downbeat ofmeasure 5. As the melodic peak tone and as the endpoint ofthe opening imitations, it attracts a stronger focus than anypreceding top-voice element. After the attainment of this G5,statements of the 8–7–6–5 progression appear at various lev-els, as indicated in Examples 6(b) and 6(c). The most remark-able of these statements spans most of the Invention (mea-sures 1–20). The connection between this enlargement andthe foreground progression in measures 3–4 is reinforced bysimilar harmonic support I–V�–IV for the 8–7–6 top-voice

25 While the right-hand figure at the end of measure 1 may appear to ex-press a seventh chord (V6

5 ), there is no satisfactory resolution for theapparent seventh (C5), which implies that it has a more unusual antici-patory function.

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26 Schenkerians disagree on how rare 8-Urlinien are in Bach. In Väisälä2008, I argue that several pieces for which Schenker and his followershave suggested 8-line readings are more meaningfully interpreted asembodying 5-Urlinien.

122 music theory spectrum 31 (2009)

stretch in each case. At the end of the piece a 5–4–3–2–1 linefollows, completing an octave descent, a rare specimen of the8-Urlinie.26

f: the ascending arpeggiation and stepwise descent

Example 7(a) illustrates the precision in which figurationand register elucidate figure enlargement in the opening sectionof F. Two basic elements of figuration appear at the outset:the ascending arpeggiation in eighth-notes and the scalardescent involving sixteenths. As shown by beams above andbelow the stave, the initial figure in each element is enlargedthrough occurrences of that very element. The repetitions ofthe ascending arpeggiation figure articulate an enlargement

of the opening F–A–(F)–C motion in a higher octave,whereas the descending scalar passages participate in the en-largement of the descending F–E–D–C motion.27 With oneexception, the highest note in each surface statement takespart in these enlargements. This exception is the G5 in mea-sure 8, whose status as subordinate embellishment (neigh-boring tone) is clarified through dissonance treatment. Acrucial event for triggering the F–E–D–C descent is the re-harmonization of F by the V7 of C major in measure 7,which forces F to resolve to E, and this resolution occursonly after the G5, at the second beat of measure 8. (Apartfrom the change of harmony, this point is marked by the firstchange in the canonic technique.)

A third element of figuration appears in measures 4–6 ina sequence of descending thirds. Adding to the remarkabledensity of interlevel connections, this pattern of descending

27 Forte and Gilbert’s analysis (1982, 206–07) shows the F–E–D–C linebut not the F–A–C arpeggiation. (They indicate the 5 as establishedalready in measure 5 by the metrically weak C6.)

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Example 7(c) sketches the overall structure. After thetonicized dominant (measure 12), I4

3� (measure 25) effects afleeting return to the tonic and directs the focus to the sub-sequent subdominant (measure 26).29 This subdominant ishighlighted in the formal design by initiating a kind of reca-pitulation, based on the lower-fifth transposition of measures4–12. Through this transposition, the F5–E5–D5–C5 pro-gression of the opening section becomes B�

4–A4–G4–F4,which, in the absence of the ascending arpeggiation, assumestop-voice status and takes care of the Urlinie descent.

bach’s inventions: figuration, register, structure, and the “clear way to develop inventions properly” 123

28 Such octave doublings of structural frameworks occur frequently in theInventions. Compare, for instance, the opening section of E (Example4[a]) and the B�–C–A�-G framework in E� (most clearly shown inExample 17), which also involves the expression of a passing tone (B�,measure 22) in the lower octave.

thirds, A5–F5–D5–B4, is also enlarged over the course of thefirst section, as indicated by circles in Example 7(a).

Example 7(b) clarifies the structural position of these ele-ments. Reminiscent of the opening section of E, the top voicepresents a high-register arpeggiation towards the Kopfton,1–3–5 (F5–A5–C6), which is elaborated by a passing �4 in alower register (B4). As the notation in Example 7(b) makesevident, the F–A–C arpeggiation is doubled at a lower octave,and the passing B connects registrally with these doublings.28

The descending fourth (F4–E5–D5–C5) represents an innervoice. The large descending-third pattern (A5–F5–D5–B4),which brings about the transfer from the high 3 to the low �4,is formed by the combination of these voices.

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29 The I43� is emphasized by its bass’s extreme low register (C2). The I4

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3 chords formed by the combination of non-simultaneousbass and upper voices appear in my readings of the Prelude in C Minorfrom WTC II (Example 8, measure 25), E� (Example 17, measure 22),and A (Example 18, measure 17).

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bach’s inventions: figuration, register, structure, and the “clear way to develop inventions properly” 125

To gain additional perspective on such structure, letus briefly look at some related examples, illustrated inExample 7(d). The opening phrase of the Allemande fromFrench Suite VI provides an interesting small-scale corre-spondent.30 Just like F, this phrase presents a 1–3–5 arpeg-giation accompanied by I–V, a fleeting return to the tonic

as an applied dominant to IV, and a 4–3–2 motion abovethe prolonged IV; the prolongation of the IV also takes asimilar form in both cases (IV–V6

5–“I”–II6).31 Such corre-spondences between small- and large-scale structures arenot without interest even when they involve separate andunrelated works. Pedagogically speaking, small-scale corre-spondents, in which structural relationships are readily per-ceptible, may be helpful in making large-scale structuresmore accessible. Moreover, clear foreground occurrencesof a structural framework confirm its position among thecomposer’s resources, buttressing the notion that theframework may also have guided large-scale organization.

A largely similar overall structure is to be found in thePrelude in E Major from The Well-Tempered Clavier I. Just asin F, the initial 1–3–5 arpeggiation is highlighted throughoccurrences of an arpeggiating surface figure, and thus repre-sents figure enlargement.32 Another common feature with Fis a concluding section starting from IV and based on alower-fifth transposition of the opening section. However,telling differences between the two pieces become evident inthe prolongation of the IV. In the Prelude, a IV–�IV7� frame-work supports Urlinie tones 4 and �3(!), the latter being pre-pared by an extraordinary �VI. This level shows the firstglimpse of the extreme character difference between robustdiatonicism and delicate chromaticism that separates theInvention and the Prelude despite their structural conver-gences.33

b: the diminished fourth as a source of bass-line drama

While the preceding examples of figure enlargementmostly focus on upper-voice events, bass-line figures may

31 Another example of such a prolongation of IV occurs in measures234–24 in a (Example 1[a]).

32 Alegant and McLean (2001) also discuss this enlargement.33 For a more detailed analysis of this Prelude, see Väisälä 2008.

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I8 (V) 7 IV5 ( V I�IV5 �) �IV7� �

Prelude in E (overall structure

WTC I ),

overall structure

example 7(d). Points of comparison

30 For Schenker’s analysis of this phrase, see Schenker 1979, Fig. 109, d1.

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126 music theory spectrum 31 (2009)

also be enlarged, and such enlargements may have significantexpressive consequences. A particularly dramatic example ofbass-line enlargement is b, in which a figure with a dimin-ished fourth, D–A�–B, occurs both as a foreground charac-teristic and in the structural framework. For illustrating theexpressive implications of different bass lines, a point ofcomparison is offered by the Prelude in C minor from TheWell-Tempered Clavier II; see Example 8. A common featurein the large-scale structures of b and this Prelude is that aheavily tonicized III does not lead to a strong root-positionV, as is most often the case. However, the Prelude is charac-terized by smooth, conjunct bass motion and by the con-comitant laconic expression on both the small and the largescale, whereas the diminished fourth imparts a sense of vio-lent dramaticism to b.34

Examples 9(a)–(c) survey details featuring the D–A�–Bfigure. As shown in Example 9(a), it is introduced by theopening bass line (whose figured form becomes the counter-subject); A� substitutes for C� as a connective between D andB. While C� does appear at the beginning of measure 2, theparallelism of right-hand figuration brings out the D–A�–Bframework rather than D–C�–B.35 Example 9(a) also showshow another important source of expressivity, the chromaticA�–A� relationship, is subtly introduced at the joint of the firsttwo thematic statements. The implicit A� of the half-cadenceV leads to the explicit A� of the tonicized V.

is the F-minor chord in measure 17, which Wintle reads as beginningthe prolongation of IV but which I see as subordinate to the motionfrom III to the emphatic V of IV in measure 18, whose top-voice G5corroborates the significance of the G4–G5 coupling and also theKopfton status of 5 (Wintle reads 3 as Kopfton).

35 This parallelism also supports reading the I6 chords in measure 1 asconnected, whereas Schenker (1996, 35, Fig. 3) indicates the latter I6 asa passing chord. (Schenker’s Figure is internally inconsistent, however,since some of its graphs show the I6 chords as connected.)

34 Apart from bass-line features, Example 8 shows two enlargements ofthe opening G–A�–G–F–E� upper-voice figure, the larger of which hasbeen previously presented in Wintle 1986, Example 10. However, myanalysis differs in several respects from Wintle’s. A consquential detail

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a) Opening b) Overall sketch

5 9

14 17 18 19 23 26 28

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(smooth, scalar progression: laconic expression)

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IV7�VII6I IV I

51

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example 8. Bach, Prelude in C Minor from The Well-Tempered Clavier II

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bach’s inventions: figuration, register, structure, and the “clear way to develop inventions properly” 127

Examples 9(b) and 9(c) show two returns to the tonic,both of which have the character of abrupt dramatic turns.They also share common features of pitch relationships. Thefirst return (measures 7–8, Example 9[b]), occurring after

the opening thematic statements, implants the D–A�–B fig-ure into a dominant prolongation (V�–�). The diminishedfourth is given a pungent character by the way it serves to“resolve” the seventh between D4 and C�

5 (brought about bya bold use of counterpoint at the twelfth). The A�–A� rela-tionship features as a cross relation between the outer voices(see dotted arrow). The second return (Example 9[c]) is themain return in the large-scale form and coincides with a pairof recapitulatory thematic statements. Just as in the firstreturn, a tonicized V (measure 17) precedes the D–A�–B bassfigure—which now occurs on two scales—producing theouter-voice A�–A� cross relation. The interpretation of pro-longational relationships is, however, much more complicatedfor the second return. One might again consider reading theD bass (measure 18) as subordinate to dominant prolonga-tion, spanning from the V� in measure 17 to the V6

5 at theend of measure 20. However, the D bass is now too stronglymarked by emphasis, design, and register for being relegatedto such a subordinate role. Design connects this D bass withthe other framing point of the pair of recapitulatory state-ments, the B bass of the eventual root-position tonic (mea-sure 21), suggesting that the D represents the tonic triad inan anticipatory fashion.

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* *

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Dim. 4th heightensdramaticism andexpressivity

A� BDD A� BD

I 6 II V65 I IV6 V� �

example 9(a). b: opening

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example 9(b). First tonic return

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128 music theory spectrum 31 (2009)

While the D bass of measure 18 seems thus to representthe tonic, it is not preceded by anything that would nor-mally prepare a structural tonic return: no dominant with araised leading tone appears. Such preparation follows onlyafter the D bass through the D–A�–B figures. The small-scale

D2–A�3–B3 figure (measure 18) dramatically transforms the

chord on D, which momentarily sounds like a D majorchord, into I6. The V6

5 of the larger D2–A�2–B2 figure in turn

prepares for the root-position tonic. The unusual temporalorder—a representative of I preceding the preparation for

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example 9(c). Second tonic return (recapitulation) and conclusion

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bach’s inventions: figuration, register, structure, and the “clear way to develop inventions properly” 129

I—suggests an unusual reading based on temporal displace-ments, as illustrated in graphs (i) and (ii) in Example 9(c). Asshown by slanted lines in graph (i), the A�

3 and B3 of thesmall-scale D2–A�

3–B3 figure may be perceived as belatedinner voices, which “should” have coincided with E2 and D2,respectively, so as to create a normal approach to the I6 fromV4

2. But these E2 and D2 may in turn be understood as antici-patory upper voices in relation to the A�

2 and B2 of the even-tual V6

5–I motion, as indicated in graph (ii). Through thisreading, it is possible to account both for the connection be-tween the D and B basses as representatives of the tonic andfor the intervening A� as part of the dominant that prepares forthis tonic. The pattern of nested temporal displacements issupplemented by the combination of two thematic statementsin inverted counterpoint: the E and D of the former statementare liable to be perceived as bearing an anticipatory relationshipwith the A� and B of the latter (graph [i]).36

The events in b provide another indication of Bach’simaginative and nuanced art of retransition and recapitula-tion. In this case—in contrast to a and d—there is no recapit-ulation gesture subordinate to dominant prolongation, butthe dominant and tonic areas show a temporal overlap. Whilemy reading of temporal relationships is exceptionally com-plex, it seems to me that the complexity is required by the ex-ceptional nature of the events leading to the unexpected tonicreturn. Moreover, the multilevel occurrence of the displace-ment pattern also offers certain kind of support for the reading.The readily perceptible small-scale displacement (measures

17–18) speaks to the significance of such a pattern as a com-positional resource, suggesting that a related pattern may alsounderlie the tonic return as a whole.

Example 9(d) depicts the overall structure. The large-scalebass line in measures 1–21, B–D–(A)–A�–B (I–III–�VII–V6

5–I),combines two central sources of expressivity, the D–A�–B fig-ure and the chromatic A�–A� relationship. The A major har-mony in measure 14 sounds first just as a dominant answer tothe preceding D major statement but then unexpectedly be-gins the process that leads to the tonic return with temporaldisplacements. The unexpected character of the tonic returnis also evident in durational proportions. The first sectionlasts 11 measures and the second section (from the cadence toD to the recapitulation) only 6, whereas in most Inventionsthe second section is about as long or somewhat longer thanthe first. Perceiving a voice-leading relationship between the�VII in measure 14 and the V6

5 that eventually leads to theroot-position I (measure 21) is supported by the return of asimilar registral setting, as evident in Example 9(d). As re-gards the Urlinie, the temporal-displacement figures involvethe consecutive presentation of the 4–3 motion in several reg-isters. The 3 ascends from the low D2 to higher octaves untilreaching the D5 of the obligatory register in the final cadence,as indicated by circles in Example 9(d). The last occurrence ofthe D–A�–B figure forms the top voice of this cadence, A�substituting for C� as Urlinie 2 in a fitting conclusion to thisInvention.

c: figure enlargements overshadowing urlinie descent

As the final example in this section, I shall present someobservations of figure enlargement in the first, and perhapsbest known, of the Inventions. Whereas the preceding ex-amples have concentrated on figures introduced at the verybeginning—in the theme or the countersubject—the overalldesign of C is crucially affected by enlargements whose point

36 The parentheses in Example 9(c) indicate that the anticipatory E–Dmotion in the large-scale displacement figure and in the latter of thesmall-scale figures could be removed without affecting the structure,since this motion also occurs in its due place above the A�–B bass mo-tion (these unfolding figures represent the type shown in Schenker1979, Fig. 43d4). By contrast, the belated A� and B in the first of thesmall-scale figures cannot be removed and are indicated in their “cor-rected” location in graph (ii).

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130 music theory spectrum 31 (2009)

37 One might question whether the original inversion figure (measure 3)is actually based on the A–F–E framework since the A might also beinterpreted as a neighbor to the immediately following G. However, I

of origin is the inversion figure appearing after the openingimitations. As illustrated in Example 10(a), the inversion fig-ure beginning with A5 (measure 3), is taken up at its originallevel after the cadence to A minor (measure 15). This re-statement triggers enlargements of the characteristic A–F–Eframework at three structural levels and in three registers, asindicated by letter names in Example 10(b).37

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example 10(a). C: Introduction and resumption of the inversion figure starting from A5

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bach’s inventions: figuration, register, structure, and the “clear way to develop inventions properly” 131

illustrated in Example 10(b), the 2 occurs in an octave belowthe obligatory register. Moreover, the structural top voice iscovered not only by the A4–F4–E4 inner-voice figure but alsoby a C5–B4–C5 neighboring figure in the highest voice.38

The latter figure also represents figure enlargement, as it re-produces the original counterpoint of the theme (measure 1,latter half ). The Ursatz is further weakened by the brevity ofthe structural dominant, which appears at the last possiblemoment, one quarter before the final tonic.

* * *

The above examples have illustrated that the significanceof the Urlinie for actual musical events is widely variable. In

find it more appropriate to read this G as a passing tone in accordancewith the original statement of the theme figure (in which the corre-sponding D is clearly a passing note). Such a reading also agrees withwhat is typical in similar figures in Bach: when an accented sixteenth-note is followed by a leap and three stepwise moving sixteenths, thesecond of these stepwise sixteenths is usually a passing tone.

For illuminating the artistic motivation behind suchevents, it should be observed that the original statement ofthe inversion figure is strongly marked by both design andregister (A5 being the highest tone in the music until measure14). Such prominence, I would suggest, creates a compellingimpulse for the composer to develop this idea, but an imme-diate development through enlargement is made impossibleby the lack of harmonic support for A5 above the tonic har-mony (Example 10[a]). Only the cadence to A minor (mea-sure 15) provides the long-awaited opportunity for the artisticimpulse to flourish.

The concentration on figure enlargements towards theconclusion of this Invention is so strong that it almost en-tirely overshadows the articulation of the Urlinie descent. As

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example 10(b). Overall structure and figure enlargement

38 Larson (1983) suggests that the V of the first cadence (measure 7) startsa dominant prolongation extending all the way to the final cadential V;in this case the Urlinie 2 would be articulated in the obligatory register(D5). However, the large-scale design, characterized by the pervasiveparallelism between the first two sections (measures 1–7 and 7–15) andby a clearly contrasting concluding section (measures 15–22), clarifiesthat the V in measure 7 is subordinate to a I–V–VI progression.

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39 This model is similar to one of Neumeyer’s (1987) “three-part Ursatz”models.

40 The 1–2–3 ascent is somewhat shaded by the appearance of a high5 (B5) in measure 2, which is also present in the two subsequent

132 music theory spectrum 31 (2009)

some Inventions, such as E� or D (Examples 2 and 3), theimpulse to articulate the Urlinie descent, or to give it somespecial character, appears as a central source of musical con-tent. At the other end of spectrum, C bears little or no traceof such an impulse. The next section is devoted to closer andmore comprehensive discussion of this crucial topic.

3. structural perspectives: the predictive power of the urlinie

My emphasis on figuration and register as analytical cri-teria involves an aspect of empirical orientation in analysis.In section 1, I focused explicitly on the support offered toSchenkerian readings by concrete compositional featuressuch as figuration (design) and use of register. This relateswith my discussion in Väisälä 2008, in which I attempt toshow how the consideration of four types of analyticalcriteria—in addition to design and register, these includemeter and rhetorical/gestural emphasis—helps to clarify theemergence of Schenkerian structures (multilevel patterns ofharmony and voice leading) in several Bach examples. Suchobservations evoke the crucial epistemic question about theexplanatory power of Schenkerian concepts. In principle, onecould approach this question empirically by assessing the ex-tent to which these concepts are supported by compositionalfeatures such as those listed above. While a comprehensivetreatment of this question goes beyond the scope of the pre-sent article, I shall focus on one aspect of it, concerning thebackground top-voice paradigm, the Urlinie. According tothis paradigm, the overall top-voice structure consists of thechoice of one of the tonic triad’s tones as a governing ele-ment (Kopfton) and of a stepwise descent from this tone tothe tonic. I will examine whether this paradigm has predic-tive power for concrete compositional features in the corpusof the 15 Inventions. This examination involves two parts.First, I shall outline criteria for determining the governingtop-voice tone during the opening events in each Invention.

Second, I shall examine whether the initial determination ofgoverning top-voice tone is reflected in the remaining eventsso as to manifest special compositional attention to elementsthat form a stepwise descent from this governing tone to thetonic.

5 or 3? a comparison between d and e

For an introductory example that illustrates what may beunderstood by the predictive power of the Urlinie, let us re-turn to the structure of d and compare it with that of e. Asexplained in the analysis of d, both a lower 5 (A4) and ahigher 3 (F5) appear as significant focal points in thisInvention, but the former takes precedence and assumesKopfton status. Decisive for Kopfton establishment is the factthat the conclusion of the first section establishes a stronglinear connection with the 5 rather than the 3, through theenlarged A–B�–A–G–F figure.

As illustrated in Example 11, a lower 5 (B4) and a higher3 (G5) also occur as focal points in e, an Invention whoseharmonic structure is strikingly similar to that of d.However, the 5 and 3 assume the reverse order of structuralimportance in these two Inventions. A convenient way tocompare their structures is offered by a three-voice modelthat superimposes a 3-line over a 5-line, shown as graph (i)for each case.39 In both d and e, the Invention theme startswith the lower 5, whereas the higher 3 is marked as the end-point of the opening imitations and as the starting point for theensuing sequence. The motions towards this 3 are somewhatdifferent in the two Inventions, since the 5 and 3 are con-nected through arpeggiation (A4–D5–F5) in d, whereas a1–2–3 Anstieg above the original 5 is discernible in e.40

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While the 1–2–3 motion already hints at the reinforcementof the 3 at the expense of the 5 in e, more decisive confir-mation for Kopfton determination is given by the subse-quent linear connections. In d, as already observed, suchconnections link with the opening 5 (A4–B�

4–A4–G4–F4),but the tones that “should” follow the high 3 (E5 and F5) failto occur. Conversely in e, the corresponding higher strandof voice-leading (G5–F�

5–G5) is featured as the explicit

top-voice framework of the opening section, but the lower 5finds only implicit continuation. The comparison also yieldsadditional illustration for Bach’s tendency towards figure en-largement (see brackets in Example 11). The enlargement ofthe A–B�–A–G–F figure in d has already been discussed, butthe G–F�–G framework in e also reflects a detail in theopening thematic figure. While the figure starts from 5, 3 isactivated through the G–F�–G neighboring figure, hinting atits upcoming significance.

The opening sections of d and e point thus to 5 and 3, re-spectively, as governing top-voice tones. In each case, suchan initial bias finds correspondence in the remaining events,which privilege the elements of either the 5-line or the 3-line according to the initial choice. In d, the higher 3-line

bach’s inventions: figuration, register, structure, and the “clear way to develop inventions properly” 133

measures. However, the 1–2–3 motion is emphasized by the gesturesleading to the strong beats of measures 2 and 3, and the primacy of 3is further confirmed by the strong-beat parallel tenths in measures3–5.

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3 7 11 13 15 20 22

5̂!(8(5(5 5) 8)

5)(5 5)

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(not present)

[ ] [ ] [ ] [ ][ ]

cf. theme

5̂(8(5(5 5) 8)

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A shared three-voice model

[ ]

example 11. Comparison between d and e

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134 music theory spectrum 31 (2009)

materializes only sporadically, but e focuses strongly on 2 ofthis line, elaborating it by an octave progression F�

5–F�4

(measures 15–19) and by the extended passing motion2–1–7, with each tone explicitly present (in contrast to d, inwhich the high 7 is omitted).41 Another noteworthy differ-ence concerns the treatment of 4 in the lower 5-line.Reflecting its status as an Urlinie tone, 4 in d is brought outby the prominent syncopated rhythm and by the subsequenthemiolic cadential progression (Example 5[d]). By contrast,the inner-voice 4 in e (measure 20) is not marked by equallyoutstanding means.42

survey of kopfton establishment

The comparison between d and e illustrates how the ini-tial establishment of the governing top-voice tone correlateswith the subsequent emphasis on elements forming a de-scent from that tone to the 1. To the extent that such corre-lation is generally evident, we may speak of the predictivepower of the Urlinie for musical events. For testing such pre-dictive power in the present corpus, it is first necessary totake a look at the ways of Kopfton establishment in the 15Inventions.

In my analyses, 5 serves as the Kopfton in ten Inventions(d, E, E�, F, f, G, A, a, B�, b); 3 appears in four (C, c, D, e);

and 8 in one (g). I argue that in all cases there are clearlyidentifiable factors through which the Kopfton is establishedas the governing top-voice tone during the initial events (ap-proximately within the opening section).

Most of the pertinent factors are evident in the abovecomparison of d and e. Temporal and registral location aretwo basic factors. Of the plausible tonic-triad members, thefirst and the highest are the privileged candidates for thegoverning top-voice tone. If these criteria conflict with eachother—a lower tonic-chord tone is followed by a higherone—two additional considerations pertain to the choicebetween them. The first of them concerns the events lead-ing to the appearance of the higher tone. If it is attainedthrough a clearly goal-oriented progression (such as anAnstieg), or otherwise heard as realizing pre-existing ten-dencies, this will support the impression of its governingposition. The second consideration concerns subsequentevents.43 If these events elaborate the higher tone or formvoice-leading connections with it, this will corroborate itsgoverning status. If, on the other hand, the higher tone isquickly abandoned in favor of prominent voice-leading con-nections with the lower tone—as in d—the higher tone willbe perceived just as a temporary digression from the govern-ing lower voice-leading strand.

While these principles do not amount to rigorous rulesfor the determination of the governing top-voice tone—butrather represent “preference rules,” to cite Lerdahl andJackendoff ’s (1983) notion—they offer sufficient grounds forthe illumination of this aspect in the present corpus. Let usstart our survey from the majority of the Inventions with 5as Kopfton. Example 12 presents an overview of the mostsignificant upper-voice events in the opening sections of

41 The articulation of the concluding 1 in the lower register has signifi-cant expressive consequences in e but is clearly insufficient for cancel-ing the structural priority of the 3-line.

42 While the three-voice model is useful for illustrating the relationshipbetween d and e, the present comparison does not support identifyingit as a “three-part Ursatz” with two Urlinien of equal status, since oneor the other upper voice is clearly predominant. In d, the tones in thehigher 3-line are, strictly speaking, transferred inner-voice tones.Such transference is concretized at the opening of d, in which scaledegree 3 is transferred first from F4 to F3 and thence to F5 (Example5[a]).

43 To avoid circular logic, Urlinie events must naturally be excluded fromconsiderations of Kopfton establishment, since we are concerned withthe correlation between these two aspects.

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bach’s inventions: figuration, register, structure, and the “clear way to develop inventions properly” 135

these Inventions, vertically aligning corresponding phases offormal design.44

In f, B�, and b, there is no conflict between the temporaland registral criteria. 5 appears at the outset and the subse-quent events show either no ascent to higher tonic-chordtones (B�), or an ascent to another 5 (f and b).45

In E�, E, F, A, and a, the conflict between temporal andregistral criteria is resolved in favor of the latter: the topvoice proceeds from an initial 8 (=1) or 3 to a higher 5,which assumes Kopfton status. What may be identified asthe most archetypal goal-oriented motions are those having1 as their point of departure: the Anstieg 1–2–3–�4–5 in Aand the arpeggiations 1–3–5 in E and F.46 The “open-posi-tion” arpeggiation 3–8–5 (G4–E�

5–B�5) in E� is less arche-

typal, but the greater registral supremacy of the 5 at leastpartially compensates for what is perhaps lost in goal-ori-entedness. In a, finally, the effect of 5 is strongly influencedby another kind of preparation. While the upper-voiceframework at the very beginning is 1–2–3 (as implied bymeter and by contrapuntal relationships with the bass), 5 is

����������

� � � � � ���� � � � � ���� ��

���� � �� � �� ��

� � �� � �� ��

���� � � � � � � �� � � � �� �� � �

��� ��� �� �� � ������ � �� � � �� �

�� � � � ���� �� �� � � ���

��

d

E

E

F

f

G

A

a

B

b

� �� �� �� �

� ��

( )

openingimitations

sequenceand othermodulatoryevents

beginning ofsecond section(or the arrivalat the tonic ofthe second key)

187

179

144

� 95

63

6

128

129

21917

14412

9

example 12. Establishment of 5

44 The formal design of B�� deviates from the most common pattern inwhich opening imitations are followed by sequential and other kind ofmodulatory material. In B��, a sequential passage appears immediatelyafter the first thematic statement (measures 4–5) leading to the secondstatement. Example 12 shows all these events (measures 1–8) under therubric “opening imitations.”

45 At the surface, these cases are not completely straightforward. In B�,the priority of F5 is not immediately obvious in the figuration of mea-sure 1, but is clarified by the subsequent small-scale connections(F–E�–D in measures 1–4); measures 6–9 further accentuate the F5. Inb, the F�

4 occurs as the center of voice-leading activity in measures 1–3despite the higher B4, and the primary status of 5 is corroborated by theappearance of F�

5 in measure 5. The octave ascents in b and f differfrom each other in that the higher register is primary (obligatory) in band the lower in f, as clarified by subsequent connections.

46 The priority of motions ascending from 1 to the Kopfton is evident inSchenker 1979, §§120–6.

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featured as the registral frame (E4–E5) of the right-handmaterial, and its more decisive appearance in measure 3 isthus heard as realizing a pre-existing tendency. In all theseexamples, the high 5 is supported by its function as the pointof departure for subsequent voice-leading motions (for Eand F, see Examples 4[c], 4[e], and 7[c]). In particular, inthe two Inventions in which the approach to the 5 is less ar-chetypal, E� and a, 5 is confirmed by elaboration through aneighboring 6.

In d and G, the conflict between temporal and registralcriteria is resolved in favor of the former. The 5 at the outsetis followed by a higher tonic-chord member (3 or 8), but thelatter is approached in a less than maximally goal-orientedway. In both cases, the subsequent descent to a prominentneighboring 6 confirms the governing status of 5.47

Similar considerations apply to Kopfton determination inthe remaining five Inventions. In c and D, the 3 appears atthe outset and is not overridden by any subsequent ascent toother tonic-triad members.48 C and e show a 1–2–3 Anstiegtowards the Kopfton, which is also confirmed by subsequentlinear connections (see Examples 10[b] and 11). In g, theKopfton 8 (measure 5) is not approached through an espe-cially clear goal-oriented progression, but it is supported byregistral supremacy, by the realization of the preceding im-plicit 8, and by the subsequent small-scale 8–7–6–5 motions

(see Example 6[b] above). Moreover, the preceding events inthis Invention do not show a clear focus on any other of thetonic-triad tones to compete with the 8.49

requirements for the 4 and 3 in a 5-urlinie

Having taken a look at Kopfton establishment in the 15Inventions, let us proceed to examining the articulation ofthe remaining Urlinie tones. In considering the requirementsfor the satisfactory articulation of an Urlinie degree, it is im-portant to allow for both harmonic support and explicit em-phasis. Roughly speaking, the weaker harmonic support thereis for an Urlinie degree, the more necessary it is to bring it outthrough special features of design, register, duration, or otherkind of emphasis, in order for it to be experienced as a majorstructural element.50 Consider the dissonant Urlinie 4 in d,the syncopated G in measures 46–47, which represents apassing seventh in a prolonged V8–7–VI progression(Example 5[c]–[d]). Not any kind of passing seventh wouldbe equally satisfactory as an Urlinie 4; the syncope and thehemiolic cadential progression are essential for making this 4stand out. By contrast, Urlinie 2, always supported by the

136 music theory spectrum 31 (2009)

47 Neumeyer and Tepping (1992, 72–74) read the Urlinie from 8 in G,because “[o]nly from 8 (g′′) does subsequent voice leading follow [. . .],”which, of course, neglects the 5–6 connection. 6 and the accompanyingIV are strongly underlined as the starting point of an extensive sequen-tial descent in parallel tenths (measures 4–10); cf. Example 14(e).

48 In D, the initial 3 (F�4) is followed by an ascending motion, but this

leads to 2 (E5) in a way that confirms rather than challenges thegoverning position of the 3 (Example 3[a]). In c, the large-scale as-cent from 3 to 5 (Example 14[g]) is not clearly goal-oriented incharacter, and the high 5 (G5, measure 13) is abandoned as soon asit is attained.

49 An ascending arpeggiation D4–B�4–G5 is discernible as formed by

explicit tones in measures 1–5 of g.50 While the notion of harmonic support and emphasis (or prominence)

as complementary analytical criteria is, I believe, a principle tacitly ob-served by most mainstream Schenkerians, not all theorists agree on it.This is evident, for example, in David Beach’s (1990) and Joel Lester’s(1992) discussion of the opening of Mozart’s Sonata in G Major, K.283. Lester, advocating a 3-line reading, reproaches Beach’s 5-linereading, in which a cadential 6

4 supports 3, for arguing “from promi-nence to background status” (202). In my view, however, a conceptionof structural weight as partly dependent on prominence is stronglypreferable, since—as the present discussion will make evident—thisgreatly enhances the predictive and descriptive power of Schenkeriantheory for actual compositional features.

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bach’s inventions: figuration, register, structure, and the “clear way to develop inventions properly” 137

51 While the conclusion in each of these Inventions is characterized bythe weakness of the structural V (the cadential dominant), this V is stillan indispensable part of the overall syntax.

52 The issue of what constitutes a satisfactory Urlinie 4 may also be illu-minated by a couple of examples that I find as unsatisfactory in this re-spect. Forte and Gilbert (1982, Ex. 213) locate the 4 in d in measure45, and Neumeyer (1982) locates the 4 in a in measure 20. In neithercase are there any special compositional features to distinguish these 4sfrom ordinary foreground passing notes.

53 Another possibility is an unfolding in which 3–5 is answered by 4–3–2.For an example of such a structure, one might consider the Prelude inB� Major from The Well-Tempered Clavier I.

structural V, is much less dependent on explicit emphasis—or even existence. The experiential weight of the all-impor-tant harmonic event will substitute for a weak or lacking ar-ticulation of the Urlinie tone, as exemplified by E and C(Examples 4[d]–[e] and 10[b]).51

Since the 5–4–3 stretch of the 5-Urlinie is unsupported atthe background level, explicit emphasis pertains to the satis-factory articulation of this stretch in general. Moreover, sincethis stretch is the element that distinguishes the 5-Urliniefrom the 3-Urlinie, it offers an optimal testing ground forthe predictive power of the Urlinie. While it is difficult todefine exactly what counts as sufficient emphasis for anUrlinie degree, the predictive power of the Urlinie for thisstretch may be assessed by comparing cases in which either 5or 3 is initially established as the governing top-voice tone.52

Both 5-Urlinie and 3-Urlinie structures may involve 4–3motions leading to 2–1, but only in the former are such mo-tions an integral part of the fundamental structure. In 3-Urlinie structures, the role of 4 is subsidiary; it may functionas a complete or incomplete neighbor of 3 or as an inner-voice passing tone, as in e (Example 11).53 To the extent thatsuch 4–3 motions tend to be more consistently present andmore strongly articulated in pieces in which 5 is initially es-tablished as governing, this testifies to the predictive powerof the Urlinie.

Before surveying the Inventions from this point of view,some observations on middleground harmonies are neces-sary for refining our notions of the ways in which the re-quirement of explicit emphasis for Urlinie degrees 4 and 3 de-pends on harmonic support. Example 13 illustrates theharmonization of these Urlinie degrees in the ten 5-UrlinieInventions, transposing the structures to C major or minor. Inmost cases (Example 13[a]–[b]), the primary harmony sup-porting the 4 is V7 in root form or inversion, although the 4 isoften prepared by a consonance. The requirement of explicitemphasis is indispensable for such dissonant 4s but less so forthe subsequent consonant 3s. If the dissonant 4 attracts suffi-cient attention, the expectation of its resolution will also guar-antee attention to the 3. In d, for example, the 3 is rather un-marked (Example 5[d]). If, on the other hand, the 4 receivesconsonant support through a I–VII–III progression (Example13[c]) or through a pre-dominant IV or II6 (Example 13[d]),the requirement of explicit emphasis is somewhat weaker forthe 4. In the latter case, in particular, the salient 4 in the bassmay substitute for top-voice emphasis.54 On the other hand,the 3, which in such cases is basically a passing tone abovethe IV, must be brought out by some special means. In both5-Urlinie Inventions based on this harmonic pattern (E andF), such means include the local harmonization of the 4–3motion by chords that mimic the V7–I motion.

the expression of the 4: comparisons of b�, e�, a, f, g, b, and c

In the most common structural type of 5-UrlinieInventions (Example 13[a]), 4 occurs as a dominant seventhand is resolved to the 3 on the tonic. The six Inventions that

54 6 frequently substitutes for 4 as a top-voice tone above IV or II6, help-ing to avoid unwanted outer-voice octaves. For small-scale examples,see the auxiliary cadence at the end of a (measure 234, Example 1[a]and Example 1[d]) and the cadences in D (Example 3).

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138 music theory spectrum 31 (2009)

harmonized by I–V

� � � � �� � � � � � � � � �� � � ���� � � � � � � � �� � � ��� � � �

�� � � � �� � � �� � ��� � � � � � �� � � ��� �� �

� � � � ���� � ��� ��

� � � � �� � ���� � � � �� ���

� � �� � � � � � � � �� � ���� � � � �� � ���

���� � �

� � � � �� � ����� ��

( )( )

f A G E B� b d a

7(a) –4̂–3̂ –I with different inversions

43

7 43

6 43

6

harmonized5̂(b) –4̂–3̂

8by V 7–VIharmonized5̂(c) –4̂–3̂

by I–VII–III

I8 7

V II III V I

E F

harmonized by IV–V or II–V4̂(d) –3̂–2̂

I II6 V I I II6(V42“I6”)V I I IV5 6 V I I IV5 (V6

5“I”) 6 V I

� ��65

example 13. Harmonic support for 5-Urlinien

represent this structural type (B�, E�, A, f, G, b) also sharethe formal characteristic of closing with a short recapitula-tion. Example 14 presents an overview of the structures inthese Inventions, adding a 3-line case of a similar formaltype, c, for comparison.

For a starting point for structural comparisons, the readermay review my analysis of E�, as shown in Example 2 above.In this Invention, the span from the opening to the recapitu-lation already expresses a 5–4–3 top-voice progression (withan embellishing 6), whose decisiveness is, however, under-mined by a registral drop towards the 3. As indicated inExample 14(a)–(d), B�, A, and f show a similar preliminary5–4–3 descent (with or without an embellishing 6) charac-terized by a registral drop, which in f takes the form of atransfer to the bass. In these four Inventions, the Urliniedescent only occurs in the recapitulation after a return tothe obligatory register. However, B� (Example 14[a]) devi-ates from E�, A, and f by restating 5 at the beginning of the

recapitulation, which simplifies Urlinie expression by makingthe 5–4 connection readily evident for short-span hearing. InE�, A, and f (Example 14[b]–[d]), there is no such restate-ment, and Urlinie 4 relates directly with the initial 5.55 Theneed to clarify such a relationship, and to prevent the impres-sion of the 4 as a neighbor of 3 gives additional reason to rein-force the 4 at the expense of the preceding 3. In E�, A, and f, 4is marked by an emphatic return to the obligatory register,which also brings about the expressive culmination in theseInventions. In f, the most emphatic articulation of 4 actuallyoccurs one octave above the obligatory register (B�

5), a featurewhose expressive implications will be taken up presently.

In both E� and f, the registral drop of top-voice that un-dermines the initial 5–4–3 progression and the subsequent

55 In f, the 5 (C5) is restated at the beginning of the recapitulation (mea-sure 29), but the listener’s attention is directed to the 4–3 dissonanceresolution that transfers to the bass.

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bach’s inventions: figuration, register, structure, and the “clear way to develop inventions properly” 139

��

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� � � �� � ��� �

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�'

� � �� �� � �

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6

a)

b)

c)

d)

B�

E �

A

f

4̂( ) 3̂( ) 5̂ 4̂ 3̂ 2̂ 1̂

12 17

recap.

Basic pattern:

(A preliminary 5–4–3 line iscompleted at the beginning of the recap.The lower register of 3 clarifies thepreliminary (non-Urlinie) status ofthis 5–4–3 line. A restatement of the 5and Urlinie descent in the obligatoryregister follow in the recap.)

5̂ 4̂( ) 3̂( ) 4̂ 3̂ 2̂ 1̂

12 23

recap.

recap.

(Similar, but without the restatementof 5. The return to the obligatoryregister by an emphasized 4 createsan effect of fulfillment.)

31

9 18

recap.

recap.

(Ditto.)

5̂ 4̂( ) 3̂( ) 4̂ 3̂ 2̂ 1̂

14 17 29

recap.

5̂ 4̂( ) 3̂( ) 4̂ 3̂ 2̂ 1̂

(Otherwise similar, but the returnto the obligatory register, one octavelower than the emphasized 4, createsa tragic effect.)

recap.

recap.

^ ^ ^

^ ^ ^

^

^

^

^

^

example 14. Structural comparison between B�, E�, A, f, G, b, and c

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140 music theory spectrum 31 (2009)

��

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�� �

� � � � � ���'� �� �

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��

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��

������

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( )

( )

( )

( )

e)

f )

g)

G

b

c

Basic pattern: 'recap.

5̂ 4̂! 5̂( )4̂ 3̂ 2̂ 1̂

4 10 14 18 27 29

recap.(The 5–4–3 line leading to therecapitulation assumes Urliniestatus owing to special emphasisand registral circumstances. The 3in this line is one octave higherthan the obligatory register, creatingan effect of extra joyousness.)

recap.

recap.

(The 4–3 Urlinie motion anticipatedin the bass through a temporal-displacement figure.)

11 13 22 23

recap.

N

(A 3-Urlinie structure with aneighboring 4 leading to the recap.)

���

recap.

^ ^

^ ^ ^

^

^

example 14. [continued]

rise to highlight the Urlinie 4 are accompanied by analogousregistral features in the bass: a withdrawal from and a returnto the low register. In f, the tonic at the beginning of the re-capitulation is further weakened by its 6

3 position (with the

transferred 3 in its bass), whereas the subsequent Urlinie 4 isunderlined by a quick descent to the low C2, featuresbrought about by the modification of the countersubject.Such registral events are not evident in the bass line of A,

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but the emphasis on the Urlinie 4 is all the stronger, involv-ing, among other things, a restatement of the 4–3 motion inthe bass through the kind of temporal-displacement figurethat was encountered in the above analysis of b.

In all these Inventions (A, E�, and f), the recapitulationshows significant modifications of the theme or the counter-subject or the general setting, with the effect of emphasizingUrlinie 4—suggesting that Bach’s compositional procedureswere influenced by the striving for a clear Urlinie progres-sion. In B�, structural clarification does not call for equallyextreme emphasis on the 4 owing to the restatement of the5. Nevertheless, even in this case the theme is recomposed soas to considerably reinforce the emphasis on the 4.56

Special compositional attention to the Urlinie 4 is also ev-ident in G (Example 14[e]), but this relates with the formalscheme in a way that contrasts with all the preceding exam-ples. Once again, a 5–4–3 progression (with an embellishing6) leads to the recapitulation. However, now the 3 that closesthis progression is not undermined by a registral drop, butrather reinforced by a temporary rise to a higher register(B5). The preceding 4 is underlined by the unique texturalelement of the trill, which appears in both hands, concretiz-ing the transference of the 4 to the bass (measures 20–24).Owing to these features of design and register, the initial5–4–3 motion assumes a much more emphatic and decisivecharacter than in the preceding examples and becomes partof the Urlinie proper.

The most extraordinary way of presenting and reinforcingthe 4–3 Urlinie motion is evident in b, as analyzed above(Example 14[f ]; cf. Example 9[c]–[d]). Whereas the preced-ing cases involve features that clarify the occurrence of thismotion either within the recapitulation (Example 14[a]–[d])or leading to it (Example 14[e]), the temporal-displacement

figure in b enables an unusual both/and solution to this struc-tural problem.

The relatively prominent neighboring 4 in the 3-Urliniestructure of c (Example 14[g]) offers a relevant point ofcomparison for Urlinie 4s. Since this neighboring 4 leads to3 at the beginning of the recapitulation, the formal positionof 4 is comparable to that of the Urlinie 4 in G. However,despite the relative prominence of this neighboring 4, it isnot marked by anything equally outstanding as the trills thathighlight the Urlinie 4 in G; hence this comparison supportsthe predictive power of the Urlinie.

To supplement these structural considerations, it is worth-while to present some notes on the expressive significance ofthe 4–3 motion. This motion is especially laden with expres-sion in E�, A, and f, in which it coincides with the reestablish-ment of the obligatory register. In E� and A, in which theobligatory register is the highest in use (for scale degrees 5, 4,and 3), the 4–3 motion in this register creates a sense of vic-torious fulfillment. The expressive effect is quite different in f,which is characterized by the unfulfilled striving towards theoctave above the obligatory register. Example 15(a) illustrateshow such a striving is manifest in the preceding events. In thetheme, the diminished fifth E5–B�

5 occurs prominently oneoctave higher than the surrounding voice-leading events(Example 15[a], measure 3). These high-register notes triggera large ascending arpeggiation C5–F5–A�

5–C6 (measure1–14), harmonized by a most expressive chromaticized 5-6

5motion (see Example 15[b] for reduction). However, neitherthese nor any other high-register elements are able to estab-lish the higher register structurally, since a return to the loweroctave follows in each case. The high 4 (B�

5) of the recapitu-lation represents the last desperate manifestation of the high-register striving, as it were, after which the 4–3 motion cre-ates a tragic effect of resignation by effecting the return to thelower obligatory register.57

bach’s inventions: figuration, register, structure, and the “clear way to develop inventions properly” 141

56 The left hand presents the original theme, whereas the right hand fol-lows in a free stretto. 4 (E�

5) is featured in several ways in measures173–192.

57 In G, on the other hand, the temporary visit above the obligatory regis-ter for underlining 3 (B5) creates the effect of unexpected joyousness

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142 music theory spectrum 31 (2009)

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� � ������ �� ���� �� � � �� � � ��� � �� � � ��� �� � �

�� �� � ��� � � ��� � �� �� �� ��� � ���� ��

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~

IN

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5

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6

10 15

N

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P53( 6

453)

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42

example 15(a). f: opening section (measures 1–17)

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The Prelude in E� minor from The Well-Tempered Clavier Iand the Prelude in C minor from The Well-Tempered Clavier IIoffer interesting related examples that illustrate a relationshipbetween register and expression. As evident from Example 16,the E� minor Prelude has many points of contact with f. Thetragically unfulfilled striving towards the higher register is evi-dent both at the opening (Example 16[a]), which closely re-sembles that of f, and in the large-scale structure (Example16[b]). The expression of Urlinie 5 and 4 involves elaboratecoupling (B�

4–B�5, A�

5–A�4), which is followed by the closing

return to the low obligatory register.58 The C minor Prelude,as graphed in Example 8(b) above, offers a contrasting example.

In this case, the striving towards higher register—evident inthe couplings of 5, 4, and 3 (G4–G5, F5–F4, E�

4–E�5)—eventu-

ally leads to the presentation of the final Urlinie degrees in thehigher octave (D5–C5), imparting a sense of defiance to theconclusion of this Prelude, in deviation from the resigned end-ings of f and the E� minor Prelude.59

Examples 17–18 depict the structures of E� and A inmore detail. They also offer additional illustration of en-largements of bass lines and harmonic patterns. While the

bach’s inventions: figuration, register, structure, and the “clear way to develop inventions properly” 143

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5

9 14 17 25 29 31

1̂2̂3̂

*

recap.

*

51

6I5

�� II � 42 V6 4

27� VI 76 I V I

( )

example 15(b). Overall sketch

(Example 14[e]). The different expressive effects of high-register ele-ments in f and G relate with the difference in stability between scaledegrees 4 and 3. In f the unstable 4 (on V7) resolves to a lower stable 3,whereas in G the stable 3 is articulated in the high register.

58 My analysis of this Prelude deviates in several respects from that pre-sented by Wen (1999, Example 11). The most significant difference isthat Wen shows 1 occurring already in measure 29, which I find as pre-mature given the weight of the subsequent harmonic events.

59 While most of the present examples enable a clear and unequivocaldetermination of the obligatory register, some cases are less straight-forward in this respect, since they show Kopfton establishment andUrlinie completion occurring in different registers. Whereas thePrelude in C Minor (Example 8[b]) and D (Example 3) are charac-terized by the top-voice’s fulfilled upward aspiration, F (Example7[c]) and E (Example 4[e]) highlight Kopfton establishment by aregistral position higher than that of the rest of the Urlinie. It is notalways clear whether such cases are better described by identifying ei-ther the beginning or the end of the Urlinie as being in disagreementwith the obligatory register or by adopting a more flexible conceptionof obligatory register.

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structures are largely similar, the accompaniment of thelarge-scale 5–(6)–4–3 motion shows differences that reflectthose in the opening surface progressions (Examples17[b]–[c] and 18[b]–[c]). E� is characterized on both scalesby the calm and steady 5–6 5–6 motion on the neighboringfigure E�–F–E� (this figure is also evident in several otherdetails). A, by contrast, features more dynamic fallingthirds.60

the 5–4–3 stretch in general

The analyses of d, B�, E�, A, f, G, and b show that in eachcase in which Urlinie 4 appears as a dominant seventh, it ismarked by some special features of design and/or register.We have seen that such features are less necessary for the ar-ticulation of the subsequent 3 because it attracts the listener’sattention through dissonance treatment.61 Of the remaining

144 music theory spectrum 31 (2009)

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60 A falling-thirds pattern is also discernible in measures 1–2 of E�(E�

5–C5–A�4–F4), but this is only a detail in the I-II connection.

61 If the 4–3 resolution is complicated by registral transfers, as in G(Example 14[e]), there is more reason to bring out the 3.

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(Examples 4[e] and 7[c]), which in the case of F is stronglymarked by the formal design. In E, Urlinie 4 and 3 occur inthe bass but the rhythmic circumstances—the higherprominence of the left-hand rhythms and the right hand’s

bach’s inventions: figuration, register, structure, and the “clear way to develop inventions properly” 145

the metrically supported bass line C–B–A–G–F in measures 3–6 andby the right hand’s suspension–resolution pattern that follows themodel of measure 4. Moreover, a 6–

56

–4 pattern is not uncommon in theprolongation of II in Bach; for a simple example, see Fugue in C Majorfrom The Well-Tempered Clavier II, measures 40–41.

62 The interpretation of the details in approaching the III in a is by nomeans straightforward. As shown in Example 1(d), I read measure 5 as6–5

6–4 suspensions above A and G, prolonging the II of C major. The D4

that resolves the first suspension represents Urlinie 4. While no 64

chords occur at the surface, this reading is supported by the strength of

5-Urlinie Inventions, a already presents the 4–3 Urlinie mo-tion within the opening section, underlining it by theVII–III cadence (Example 1[d]).62 In E and F, finally, the 4is underlined by a pre-dominant II6 or IV harmony

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example 17. E �: sketches

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immobility at the moment of the 4–3 motion (measures58–591)—are optimal for bringing them out. In F, theUrlinie stretch 4–2–3–1 is effected by the unvaried trans-position of the initial 8–7–6–5 progression and is similarlybrought out by register and design (cf. Example 7[a]).63

These observations suggest that in the ten Inventionsin which the 5 is initially established as the governingtop-voice tone satisfactory Urlinie 4 and 3 also occur. By

comparison, those Inventions in which the 3 is initially es-tablished as governing (C, c, D, e) lack occurrences of 4 and3 that would qualify as equally satisfactory Urlinie tones, ifthey occurred after an initial 5. Two pertinent comparisonshave already been presented: the inner-voice 4 in e and theneighboring 4 in c are less strongly marked than the Urlinie4s in d and G, respectively (Examples 11 and 14, [e] and[g]). Another pertinent comparison may be made betweenthe 4–3–2 progression in the final cadence of D, in whichthe 4 is an incomplete neighbor of 3 (Example 3[a], measures57–58), and the 4–3–2 Urlinie stretch in E (Example 4[d]).While these progressions are based on a similar voice-leading

146 music theory spectrum 31 (2009)

63 To be precise, the emphasis on Urlinie 4 is actually somewhat weakerthan on the opening 1 (=8), since measures 1–3 are left out from thetransposed recapitulation.

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the role of the 8: kopfton or cover tone?

In g, the sole Invention in which 8 is initially establishedas the governing top-voice tone, a clearly articulated 8-Urlinie follows. This one example thus would seem to suggestthat the 8-Urlinie has predictive power similar to that of the5-Urlinie. However, whereas the ten 5-Urlinie and the four3-Urlinie Inventions form reasonably large samples for sug-gesting more general conclusions, one 8-Urlinie Invention is,of course, quite inadequate in this respect. In fact, a look atBach’s works outside the present corpus reveals that there is afairly common structural type contradicting such predictivepower. In this structural type, a prominent 8 occurs at theoutset and is never superseded by other tonic-chord membersunder the criteria outlined above; however, instead of begin-ning a convincing Urlinie descent, the 8 functions in a neigh-boring motion that covers registrally the actual Urlinie.65

As shown in Example 19, the Prelude in F Major fromThe Well-Tempered Clavier I exemplifies such a structure.The 8–7–8 neighboring figure covers each degree of the3–2–1 Urlinie (Example 19[b]). Kopfton 3 is never estab-lished above the initial 8 (F5) through a goal-oriented pro-gression or in an otherwise prepared way. Instead, thePrelude is characterized by a descending pattern,F–D–B�–A, which proceeds from 8 to 3 both at the opening(Example 19[a]) and on the large scale (Example 19[b],graph [ii])—our final example of figure enlargement. 3 israised to a higher register at the end of the large-scale pat-tern (A5, measure 16), but this registral event has the char-acter of unprepared extra emphasis and is thus not compa-rable to the goal-oriented Kopfton establishment inInventions such as C or E (Example 19[c]).66 Nevertheless,

bach’s inventions: figuration, register, structure, and the “clear way to develop inventions properly” 147

65 Schachter (1994) discusses one example showing this kind of covering 8.66 On the other hand, the progression in measures 123–163 of the Prelude

is very similar to that in measures 20–29 in G (Example 14[e]). In bothcases, a voice exchange within II is underlined by trills, and, after a V4

2,another voice exchange within I raises the 3 to a higher register.

64 From a more general viewpoint, the occasional occurrence of stronglyarticulated 4–3 motions in 3-Urlinie pieces does not, of course, negatethe predictive power of the Urlinie, as long as such motions occur moreconsistently when the 5 is initially established as governing.

pattern, rhythmic features bring out the 4 and 3 in E in away for which there is no counterpart in D. In C, finally, thesomewhat similar 4–3–2 progression is even more weakly ar-ticulated (measure 213–4; Example 10[b]). On the basis ofthese comparisons, the present corpus testifies unequivocallyto the predictive power of the Urlinie for the 5–4–3 stretch.64

the 2

Since Urlinie 2 may be weakly articulated or implicit, itholds less predictive power for actual musical events than dothe 4 and 3 of the 5-Urlinie. Consequently, the 3-Urlinieholds less predictive power than does the 5-Urlinie. Thereare, however, two meaningful senses in which we can speakof the predictive power of the 2 and which justify its use asan analytical concept. The first is statistical: of the fifteenInventions, twelve present the 2 explicitly in the highestvoice (and obligatory register). The second is experiential:the cases in which 2 is implicit—exceptions from the rule—will have special experiential effects, as demonstrated by the“understated” cadence in E (Example 4[d]), by the rhetoricaleffect of the substitution of D–A�–B for D–C�–B in b(Example 9[c]–[d]), and—at least arguably—by the less thanmaximal conclusiveness of the ending in C.

When the structural dominant arrives early, the elabora-tion of 2 may become a significant source of musical con-tent, which is more typical for 3-lines than for 5-lines. Inthe Inventions, extensive elaboration of 2 occurs in two ofthe four 3-Urlinie pieces (D and e; Examples 3 and 11) butin only one of the ten 5-Urlinie pieces (a; Example 1). Suchexamples testify to the significance of 2 as a compositionalresource, even though the extent to which this resource is ex-ploited is greatly variable.

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cases such as the F major Prelude offer some additional per-spective for the weak Urlinie articulation in these twoInventions. All three pieces show different manifestations ofa covering 8 at the beginning and at the conclusion, eventhough the intervening events show a determined ascent tothe Kopfton in C and E.

The F major Prelude exemplifies Ursatz articulation at itsweakest not only because of the covering 8–7–8 motion but alsobecause of the extreme rhythmic weakness of the structuraldominant (Example 19[a], measure 18, sixth eighth-note). Theexperiential effect of such weakness is readily evident: it is as ifthis Prelude hastened to make just some kind of minimally ade-quate ending so as to be ready for entering the Fugue.

Examples such as this Prelude suggest one aspect of qual-ification for the principles of Kopfton determination as dis-cussed above. 8 is not quite on an equal footing with othertonic-triad members as a Kopfton candidate. Even when it isfavored by both the temporal and the registral criterion, itmay function as a cover tone rather than the Kopfton; theawareness of such an option would appear significant forBach analysis.

148 music theory spectrum 31 (2009)

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As expressed by the title of his essay, Dreyfus seesSchenkerian analysis as representing “figments of imagina-tion” conditioned by an “organicist” ideology, instead of de-scribing any aspect pertinent to Bach’s composition. Terminghis agenda as “intentionalism,” Dreyfus (1996, 171) observesthat there is no historical evidence that Bach or his contem-poraries intentionally arranged his music according to theconcepts on which Schenker based his analyses. Nor canSchenker’s analyses provide such evidence, since they arebased on the “method of ‘seek and ye shall find’” (181). Thisexpression invokes the charge of circular reasoning that hasoften been made against Schenkerianism, namely, thatSchenker’s demonstration of the prolongational structures’adherence to the norms of harmony and voice leading isbased on searching for elements that fit such norms.

Starting from the last of these issues, I would suggest thatthe most efficient way to counter the charge—and danger—of circularity is to recognize the significance of analytical cri-teria such as figuration (design), register, meter and gestural/rhetoric emphasis, the first two of which comprised my firstmain topic. Since these criteria involve compositionalfeatures that are logically independent of the norms of har-mony and voice leading, they enable us to justify aSchenkerian reading in a clearly non-circular way. Goingback to my first example, the V�–V� prolongation in mea-sures 14–22 of a is supported by the cooperation of unifiedfiguration with the use of register, which brings out theE2–G�

2–A2–D�2 bass unfolding. Moreover, the dominant

bach’s inventions: figuration, register, structure, and the “clear way to develop inventions properly” 149

67 This is made evident, for example, by the following statement: “Itmight well seem curious to pay special attention to an Austrian musictheorist writing some seventy years ago if it were not for the fact thatHeinrich Schenker’s ideas have gained a remarkable and unprecedentedfoothold in the world of Anglo-American musical analysis.” (Dreyfus1996, 170.)

4. on the justification of schenkerian analysis

In addition to illuminating Bach’s voice-leading struc-tures and the means of their expression, the topics of thepresent paper involves viewpoints that help defend the valueof Schenkerian analysis for Bach research, and I shall closethis paper by summing up such viewpoints. As an example ofa writing that expresses misgivings about Schenkerian analy-sis in relation to Bach, we may cite Laurence Dreyfus’s essay“Figments of the Organicist’s Imagination” (Dreyfus 1996,Chapter 6 [169–88]). While Dreyfus’s discussion is based ona single analysis by Schenker, that of Bach’s Fugue in Cminor from The Well-Tempered Clavier I (Schenker 1996,Chapter 3 [31–54]), he makes it evident that he intends hiscriticism as generally pertinent to the Schenkerian move-ment.67 Hence, while it would be out of place to discussSchenker’s analysis of the C minor Fugue here, it is possibleto suggest some ideas about how to encounter criticism suchas Dreyfus’s—which is largely representative of anti-Schenkerian arguments in general—by referring to the pre-sent analyses.

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prolongation accompanies a B5–B4 top-voice motion (withpassing B�

5 substituting for B5), which associates with themost salient high-register feature in the previous events, thecoupling C5–C6 at the beginning of the second section (mm.6–9), thus supporting the larger connection from 3 to 2.Hence my interpretation of both the dominant prolongationand its position in the larger context does not rely on “fig-ments of imagination,” “organicist” ideology, or circular rea-soning, but on prominent compositional features. The hy-pothesis that prolongational patterns pertained to Bach’scomposition helps to explain the emergence of such features;abandoning the study of such patterns—i.e., Schenkeriananalysis—would risk losing an important aspect of the wayin which Bach shaped his music. a is, of course, no isolatedexample in the above analyses. Similar justification is charac-teristic of them, as has been made explicit in a number ofcases (and is implicit in others). More comprehensiveexaminations of the extent to which Schenkerian analysis canbe justified through non-circular analytical criteria—and howstrongly such criteria are manifest in the existing Schenkerianliterature—cannot be undertaken here, but this issue wouldseem a promising area of future studies with crucial epis-temic implications for Schenkerian research.

While I will not speculate on Bach’s intentions, I wouldnote that I find the question of intentionality less crucial tothe cogency of analytical concepts than does Dreyfus, sincethere is no reason to assume that composers’ conscious in-tentions cover all significant aspects of their work. For apoint of comparison, one may consider the role of syntaxin language. Competent speakers of a language do not, as arule, intend to use a certain kind of syntax, yet syntax iscrucial for the way in which they shape their sentences.Analogously, prolongational patterning may be crucial forthe way in which a composer shapes his music regardless ofthe extent of intentionality involved. As regards Bach, theplausibility of this hypothesis is enhanced by the fact that therelationships in prolongational organization rely on basicprinciples of harmony and voice leading that were intimately

familiar to him. Since Bach undoubtedly was capable of ap-plying such principles without conscious effort for surfacepatterns, it is not unrealistic to assume that his creation oflarger patterns may have utilized a similar capability.

My second main topic, figure enlargement, also has cer-tain implications for the question about the compositionalsignificance of multilevel organization for Bach. If Bach’smusic shows a tendency to develop surface figures by en-larging them into large-scale patterns, this points to thecompositional significance of the latter. To be sure, weshould be cautious in assessing the significance of figure en-largements, since apparent enlargements might also arise aschance products, especially when involving common-placefigures such as filled-in intervals. However, several featuresreinforce the connections between the small-scale originalsand their enlargements in the above examples, making itunlikely that they would occur by chance. First, an enlarge-ment is often triggered by a prominent surface repetition, orseveral repetitions, of the small-scale figure (see especiallythe analyses of d, F, and C). Second, the connection be-tween the small and the large is often reinforced by otherfactors, such as harmonic support; in g, for example, the en-largement of the 8–7–6–5 descent (G5–F5–E�

5–D5), whichitself might be regarded as commonplace, also involves theless usual I–(V)–IV harmonic support for the 8–7–6 (occur-ring both in measures 3–4 and on the larger scale in mea-sures 1–15; see Example 6). In addition, comparisons be-tween largely comparable structures, such as d and e(Example 11) or E� and A (Examples 17–18), show devia-tions that accurately reflect the differences in the initial sur-face material.

My third main topic, Urlinie articulation, relates, ofcourse, directly with the justification of Schenkerianism. Bysurveying the ways of Kopfton establishment and the articu-lation of subsequent Urlinie degrees, I have attempted toshow that there is empirical support for the compositionalsignificance of this Schenkerian background concept, con-cerning especially the 5–4–3 stretch of the 5-Urlinie.

150 music theory spectrum 31 (2009)

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g: 6A: 12, 13, 14, 18a: 1(a), (b), (d), 12, 13B�: 12, 13, 14b: 9, 12, 13, 14

works cited

Adrian, Jack. 1985. “J. S. Bach’s Invention in A Minor: ARe-View.” In Theory Only 8.7:15–27.

Alegant, Brian, and Donald McLean. 2001. “On the Natureof Enlargement.” Journal of Music Theory 45.1:31–71.

Beach, David. 1990. “The Cadential Six-Four as Support forScale-Degree Three of the Fundamental Line.” Journal ofMusic Theory 34.1:81–89.

Burkhart, Charles. 1978. “Schenker’s ‘Motivic Parallelisms’.”Journal of Music Theory 22.2:145–75.

Derr, Ellwood. 1981. “The Two-Part Inventions: Bach’sComposers’ Vademecum.” Music Theory Spectrum 3:26–48.

Dreyfus, Laurence. 1996. Bach and the Patterns of Invention.Cambridge MA: Harvard University Press.

Forte, Allen, and Steven E. Gilbert. 1982. Introduction toSchenkerian Analysis. New York: Norton.

Larson, Steve. 1983. “On Analysis and Performance: TheContribution of Durational Reduction to the Perfor-mance of J. S. Bach’s Two-Part Invention in C Major.” InTheory Only 7.1:31–45.

Lerdahl, Fred, and Ray Jackendoff. 1983. A GenerativeTheory of Tonal Music. Cambridge MA: MIT Press.

Lester, Joel. 1992. “Reply to David Beach.” Journal of MusicTheory 36.1:199–206.

Neumeyer, David. 1981. “The Two Versions of J.S. Bach’s A-minor Invention, BWV 784.” Indiana Theory Review4.2:69–99.

———. 1987. “The Three-Part Ursatz.” In Theory Only10.1–2:3–29.

Neumeyer, David, and Susan Tepping 1992. A Guide toSchenkerian Analysis. Englewood Cliffs: Prentice Hall.

bach’s inventions: figuration, register, structure, and the “clear way to develop inventions properly” 151

To what extent these results are generalizable to Bach’senormous output is a question that cannot be addressed inthis paper. Suffice it to say that if the Inventions testify tothe structure-determining roles of figuration and register, tothe multilevel presentation of musical ideas, and to the sig-nificance of the Urlinie (especially the 5-Urlinie) as pertain-ing integrally to Bach’s compositional faculties, such facultiesare certainly exploited outside the present corpus. It is, how-ever, not impossible that the Inventions show exceptionalclarity in some aspects of structural articulation because ofBach’s outspoken pedagogical intent to show a “clear way [. . .]not only to be inspired with good inventions but to developthem properly.” There is, of course, no reason to assume thatBach consciously conceived of structural levels in a way re-sembling Schenker’s theoretical formulations. However, if Iam right in suggesting that all the evidence points to thepertinence of such levels for Bach’s way of composing, it isconceivable that his quest for exemplary clarity may alsohave affected the ways in which foreground ideas, or “inven-tions,” are reflected and developed by middleground pat-terns. In any case, whatever Bach might have consciously in-tended as constituting the “proper development ofinventions,” the organization in his music enables us to un-derstand this phrase in a much less superficial sense thanwhat is often realized.

appendix: a list of examples treating each invention

C: 10, 19c: 14D: 3d: 1(c), 5, 11, 12, 13E�: 2, 12, 13, 14, 17E: 4, 12, 19e: 1(c), 11F: 7, 12, 13f: 12, 13, 14, 15G: 12, 13, 14

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Oster, Ernst. 1961. “Register and the Large-ScaleConnection.” Journal of Music Theory 5.1:54–71. Reprintedin Readings in Schenker Analysis and Other Approaches, ed.Maury Yeston, 54–71. New Haven: Yale University Press,1977.

Petty, Wayne. 2006. “Thoughts on Schenker’s Treatment ofDiminution and Repetition in Part III of Free Compo-sition, and its Implications for Analysis.” In Structure andMeaning in Tonal Music: Festschrift in Honor of CarlSchachter, ed. L. Poundie Burstein and David Gagné,73–78. Hillsdale NY: Pendagron Press.

Rothgeb, John. 1971. “Design as a Key to Structure in TonalMusic.” Journal of Music Theory 15.1–2:230–53. Reprintedin Readings in Schenker Analysis and Other Approaches, ed.Maury Yeston, 72–93. New Haven: Yale University Press,1977.

Rothstein, William. 2006. “Transformation of CadentialFormulae in the Music of Corelli and His Successors.” InEssays from the Third International Schenker Symposium, ed.Allen Cadwallader, 245–78. Hildesheim: Olms.

Schachter, Carl. 1994. “The Prelude from Bach’s Suite No. 4for Violoncello Solo: The Submerged Urlinie.” CurrentMusicology 56:54–71.

Schenker, Heinrich. 1979. Free Composition. Trans. ErnstOster. New York: Schirmer Books.

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152 music theory spectrum 31 (2009)

Music Theory Spectrum, Vol. 31, Issue 1, pp. 101–152, ISSN 0195-6167,electronic ISSN 1533-8339. © 2009 by The Society for Music Theory. All rights reserved. Please direct all requests for permission to photocopy or reproduce article content through the University of California Press’s Rights and Permissions website, at http://www.ucpressjournals.com/reprintinfo.asp. DOI: 10.1525/mts.2009.31.1.101

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Contributors

eytan agmon teaches in the Department of Music of Bar-Ilan University, Israel.

matt baileyshea is Assistant Professor at the University ofRochester.

fernando benadon is Assistant Professor of Music atAmerican University.

guy capuzzo is Associate Professor of Music at theUniversity of North Carolina-Greensboro.

richard domek is Professor of Music at the University ofKentucky.

yayoi uno everett is currently Associate Professor atEmory University in Atlanta, GA. Her research focuses onthe analysis of postwar art music through the perspectives ofcultural studies, semiotics, and East Asian aesthetics. Her re-cent publications include “Gesture and Calligraphy in theLate Works by Chou Wen-chung” in Contemporary MusicReview 26.5–6 (2007) and The Music of Louis Andriessen(Cambridge University Press, 2006).

c. catherine losada is Assistant Professor at the College-Conservatory of Music, University of Cincinnati.

wayne marshall is the Florence Levy Kay Fellow inEthnomusicology at Brandeis University

timothy r. mckinney is Associate Professor of MusicTheory at Baylor University.

david pacun is Associate Professor of Music Theory atIthaca College. His research interests include Brahms,Music Theory Pedagogy, Disabilities Studies, and InterwarJapanese Music.

olli väisälä is Senior Assistant of Music Theory at theSibelius Academy, Helsinki.

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