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Bach Inventions

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  • 101

    Bachs Inventions: Figuration, Register, Structure,and the Clear Way to Develop Inventions Properly

    olli visl

    Bachs 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

    Bachs 15 two-part inventions belong to the verybest known repertoire of Western art music. Writtenfor pedagogical purposes, they still hold a standardposition 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 Bachsvoice-leading structures.

    Three general topics, each of which has significance forBachs 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 Bachstendency 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 weightwhich element belongs to which levelis largely based on quite different factors, such as figuration

    In this paper, I shall allude to Bachs preamble to the Inventions ac-cording to the translation of the Wiener Urtext Edition (ed. Erwin Ratzand Karl Heinz Fssl, 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 Suurp 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 Bachs musical organization andof the value of Schenkerian analysis in its illumination.While several analystsRothgeb (1971) and Oster (1961)being classic exampleshave 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 Visl2008). 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 Bachs 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

    102 music theory spectrum 31 (2009)

    2 I discuss this in greater length in Visl 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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  • bachs inventions: figuration, register, structure, and the clear way to develop inventions properly 103

    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 failedrecapitulation 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 1417), 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 I6VI 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 543 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 412 correspond to measures 2634. In g, measures 13correspond to measures 1315 and measures 59 correspond to mea-sures 163212. However, whereas F shows a straightforward lower-fifth

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  • 104 music theory spectrum 31 (2009)

    retransition begins

    15

    20

    25

    * * * *

    * *

    V 65

    1st recap. gesture retransition figure resumed

    2nd recap. gesturetheme figure extended

    I 67VII V53 IV

    I6

    7IV V I

    [ ]

    aux. cad.

    example 1(a). a: measures 1425, annotated score

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  • bachs inventions: figuration, register, structure, and the clear way to develop inventions properly 105

    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 D5C4 relationship is clarified by dis-sonance treatment.) Since the first of these high points, B5,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 weakenedapart from

    the local 63 position of the I chordby the basss suddenwithdrawal from the low register. The G2 at the end of thesequence of measures 1417 creates an expectation of an A2,which is only fulfilled in measure 21. The bass motion con-tinues to D2, 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 E2G2 is answered by A2D2.

    Example 1(b) clarifies the structural derivation of thispassage. The I of the failed recapitulation gesture has itsbasis in the neighboring motion BCB above the retransi-tional V prolongation.8 Example 1(c) shows glimpses oftwo other Inventionsd and eindicating that such Ichords are not unique to a.9 The octave descent from 2 to 2(F5F4) in e resembles, in certain respects, that in a and mightbe 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 Clavierbchlein 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.

    13

    N

    2^

    21 22 13 18

    2^

    22 13 18 22

    2^* * * * * *

    *

    ( ) ( )

    ( )*

    V 65 IV7 V V I IV7 V V 65 6I 53 IV7 V

    1st recapgesture

    example 1(b). Measures 1425, structural derivation

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  • 106 music theory spectrum 31 (2009)

    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 asfailed 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 theB5B4 octave descent is clarified by its registral associationwith an earlier salient coupling C5C6. 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 223233). 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 I6IVVI.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 52 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

    e

    V V

    13

    2^

    ( )

    ( )

    (recap

    ( )gesture)

    1^

    5^ 4^ 5^ 4^

    5^ 4^

    15 19 14 15 16 1918 38 46 47 38 4746 38 44 46 47 48

    d

    I I V7 V I V I6 53 V7 V8 (I) 7 V8 (I) 7 V8 (I6 53 7)

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  • bachs inventions: figuration, register, structure, and the clear way to develop inventions properly 107

    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 Rothgebs principleis by saying that spans of uniform figurationi.e., thosewithout changes in surface designtend 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(E2G2, A2D2) is supported by both proximity and ex-tremity, as is the large-scale top-voice coupling (C5C6,B5B4). 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 D2 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 explicitD2E2 motion in the closing cadence (measure 25).

    example 1(d). Overall sketch

    V IV7 V I

    3

    5^ 4^ 3^ 2^ 1^5^ 4^ 3^ 2^( )

    ( )

    ( )( )

    6 9 13 14 18 22 23 25

    retrans.

    * * * ** *

    * *

    1strecap.gesture

    retrans.figureresumed

    2nd recap.gesture

    IV7 V II)VIII

    43(e: III)V

    65

    64(C: II

    I (I6 53 ) aux. cad. to

    I6 IV

    65

    64

    65

    64

    65

    42

    65

    42

    6

    ( )

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  • 108 music theory spectrum 31 (2009)

    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, measures13), but an open position arpeggiation G4E5B5 leadsthe top voice to a higher 5, a persistent tone in the counter-subject (measures 58; Example 2 shows only the beginningof the countersubject). Despite the local V harmony, the ex-treme register and the persistence of this B5 attract attentionso as to suggest Kopfton status. This suggestion is borne outby subsequent top-voice events, in which C6 (measure 12)and A5 (measure 25) are highlighted through similar high-register figuration, creating stepwise connections with the B5and confirming its role as the point of departure for top-voice

    motions. Of these high-register tones, C6 occurs within acountersubject statement but A5 is marked by the freer em-ployment of similar figuration. Hence, while the B5C6 rela-tionship might be viewed as a by-product of the thematic dis-position, the appearance of similar figuration to mark A5suggests more expressly an impulse to clarify voice-leading re-lationships as its motivation.

    B5, C6, and A5 participate in a 5643 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 43motion involves a registral drop from A5 to G4 (more locallythis motion is transferred to the bass in measures 2627).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 43 Urliniedescent, leading to 21, follows in the recapitulation once theextended Invention theme has climbed emphatically back tothe obligatory high register (A5 in measure 31). The crucial

    example 2. E : register, figuration, and structure

    4 ^ 3 ^ 2 ^ 1 ^

    5 9 12 23 25 27 31

    5^ 6^( 4^ 3^ )

    PIN

    (countersubject) recap. (extension of theme)

    PN

    ~~~ ~

    V

    I

    I5

    V I6

    6

    VI II

    II

    (IV)

    V

    V

    43

    42 I

    6

    I VII6 (or V 43) I6

    53

    II6 I

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  • bachs inventions: figuration, register, structure, and the clear way to develop inventions properly 109

    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 (A5)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 (E2) at the begin-ning of the recapitulation (E3) 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 15), occurrences of thetheme figure and its variant establish a registral discrepancybetween a low 3 (F4) 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 32 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 58) 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 4750). 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 542; Example 3[b]provides details of the interpretation). At the end of thisprolongation, the F4E5 seventh is recomposed on a retran-sitional V64-5-3 framework, reminding us of the original dis-crepancy (measures 3942). Now the ascending seventhF4E5 naturally creates an expectation of D5 to effect achange in direction in the melodic motion and to satisfy thetop voices aspiration towards the higher octave. And whilethe recapitulation begins with the low 3 (F4, measure 43),which suggests a structural interruption, an ascent to D5follows shortly (measure 47), pointing to a registral E5D5connection over the intervening low 3. On the large scale,the E5D5 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 4750 as represent-ing 1 in a large-scale 321 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 321 progressions inboth registers pertinent to the initial F4E5 discrepancy(F5E5D5 in measures 5154 and F4E4D4 in mea-sures 5659), as if to correct the discrepancy. The former321 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 unusualand aestheti-cally satisfactoryperspective for hearing them. Instead ofintroducing 1, they confirm an earlier 1. Such a hearing

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  • 110 music theory spectrum 31 (2009)

    example 3(a). D: measures 112 and 3859, annotated score

    ~

    *

    3^ (F4)*

    2^ (E5)

    5 10

    *

    *

    38

    retrans.

    (F 4 E5) 43recap.

    3^ (F4)

    45

    1^!

    *

    50

    3^ (F5) 2^(E5) 1^(D5)

    55

    3^ (F4) 2^ (E4) 1^(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 )

    642(V I )

    40

    is supported by features that help to associate the octave-leap 1 (measures 4750) with the final 1: the reappearanceof the octave leap at the end of the first 321 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 Schenkers 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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  • bachs inventions: figuration, register, structure, and the clear way to develop inventions properly 111

    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 321 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 58 and 4750, 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

    *

    *

    (iv)

    5 43 47 5 24 35 39 43 47 51 54

    (i) (ii) (iii)3^ 2^ 3^( 2^) 1^

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

    Another notation forsimilar conception

    antic.

    3^

    ant.

    2^ 1^!

    *7th

    ( ) ( )6th

    recap.*7th

    * N7th

    (reminder)retrans.

    *recap.

    Inlarge-scale parallelismhighlight the anticipation of

    D, register and

    1^.

    *

    *

    *(v)

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

    **

    ** * * *

    *

    *

    **

    ***64

    53 *

    64VI VI

    53V I V I

    I V

    (A: I IVI6V I)

    V 5

    (b: V 42

    651

    I6VII6(I II6I6V) I)

    VI V7 V I VI (I) IV I6V I

    ( )( ) ( )

    example 3(b). Derivation of structure

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  • 112 music theory spectrum 31 (2009)

    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 theKopftons 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 F4E5 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 (measures912, 2124, 3538, 5154, 5659). 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 432 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 Bachs cadences. The awareness of theirnon-tonic function is essential for Bach analysis.14

    3838

    3^IN4^( )

    P2^ 1^3^ 3^ 3^4^( ) 2^ 1^ 3^

    *

    4^( 3^) 2^ 1^

    INP*

    56

    IN P

    IV 642(V I )I IV8 7 V I I V I I IV 642(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 V42, anda VI cadence. Psychologically speaking, the expectations created bythe emphatic subdominant are not satisfied by the weak V42 I6 pro-gression but only by the strong VI. As in Rothstein 2006 (268 ff.),such I6 chords may be described as inverted cadential 64s, especiallywhen their metric position is clearly strong in relation to the subse-quent dominant. However, when the 63 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 58.

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  • bachs inventions: figuration, register, structure, and the clear way to develop inventions properly 113

    e: supreme luciditywith a turn to obscurity

    As illustrated in Example 4(a), the opening section of E(measures 120) 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 arpeggiation135 (E5G5B5), with a passing 4 (A4) occurring in alower register. The bass descends stepwise from I to V(E2D2C2B1); B1the only note below C2 in theInventionsconfirms the registral logic by its appearance atthe very end of the section.15

    As for figuration, two rhythmic elements, syncopationand dactylshereafter S and Dhelp 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 912, which lead thebass from E via D to the C in measure 13, the pattern islengthened to four bars (SSSD) in measures 1316, 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 EDC motion to reach its goal.However, the occurrence of this B within the lengthened SSSDpattern and its high registral position (above its immediate

    ( ) ( ) ( )

    1 9 13 17 1 8

    (i)

    10

    (ii)P

    15

    S D S D S DS S

    G5^

    F E

    FD B

    20

    Cf. theme in B major

    25

    delayed tensionG F E D F Brelease

    53I V (B: II (I)

    65 V

    42 I

    6 V I )

    42 66 (etc.)

    P

    example 4(a). E: first section (measures 120)

    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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  • 114 music theory spectrum 31 (2009)

    surroundings) suffice to dispel any effect of goal arrival.Rather, this B functions as harmonic support for the passingmotion E5D5C5, 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 2142); see Example4(b). In terms of key relationships, the G minor passage inmeasures 2932 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 keys 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, G3, isa sixth above the preceding bass note B2, which enables theG minor triad to be perceived as emerging from 56 motion

    ! ! !

    ! ! !

    !

    30

    S S S DTheme Digression

    D D D

    N

    SMain discourse continues

    D S D

    35

    S D D DDecisive breakaway

    (to D in m. 41)

    !

    25

    example 4(b). Measures 2539, annotated score

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  • bachs inventions: figuration, register, structure, and the clear way to develop inventions properly 115

    above B. An emphatic return to the registral vicinity of theB2 follows in measure 39, in which the C3, supporting aNeapolitan sixth, is marked by the first halt in the left handseighth-note motion since measure 29, by hemiola rhythm(measures 3940), and by the ensuing cadential progression.These features suggest a bass-line connection between theinitial B2 and this C3, 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 2932)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 3338). 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 2528) and the ensuing SDunit (measures 3334). As we can see in Example 4(b),

    such relationships concern both the S element (B5A5; seecircles) and the D element (F5E5; see beams), and also thebass, in which the association between the bracketed figuresin measures 25 and 33 supports the chromatic connectionbetween B2 and B3 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 F5 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 B5A5A5G5 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

    ( ) ( ) ( )

    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 GN

    N

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

    B5

    B56

    6

    example 4(c). Second section (measures 2142), structural derivation

    16 The G5 in measure 35 resolves to the F3 in measure 36, which standsfor an inner-voice tone despite its low registral position. This low posi-tion enables this F3 to associate registrally with the subsequent F 3and G3, highlighting the 56 motion in measures 3638.

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  • 116 music theory spectrum 31 (2009)

    While the concluding section (measures 4362) 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 916 are shifted from theregistral extremes to the middle register in the correspond-ing passage of measures 5158. As illustrated in graph(iii) in Example 4(d), the highest tones do not form anUrlinie descent but neighbor motion around the 8(E5F5E5D5F5). Figuration, however, proves onceagain 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 5158),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 listeners attention to the left-hand pro-gression 5(65)43, in which the 43 motion is high-lighted through the lengthened SSSD pattern (measures5558) and through the absence of right-hand activity atthe decisive moment (measure 58591). 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 432 progression, in which 2 is implied by the

    17 There is also a less evident way in which figuration supports the 56connection in measures 5051: the right-hand dactyls in measure 50outline a syncopated B4G4E4 figure that corresponds to the left-hand C4A3F3 figure in measure 51.

    ( )

    ( )( ) ( ) ( )( )

    ( )

    ( )

    43 51 55 58 59 47 50 55 60

    (i) (ii) (iii)5^ 4^ 3^ 2^ 1^5^

    4^ 3^ 2^ 1^

    5^

    D

    D 4^

    Cf. theme delay release

    From:

    2^ 1^

    2^ 1^3^

    S D S D S DS S

    42 III

    6 V(V 6)

    53VI II

    6 64 I 5

    3V64 I

    (V42 )6I

    VI II6 quasiI

    example 4(d). Third section (measures 4362)

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  • bachs inventions: figuration, register, structure, and the clear way to develop inventions properly 117

    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 135 high-register arpeggiation, the rest of the Urlinie takes the more

    obscure form in which the 43 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, B3sounds 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

    ( ) ( )( )

    9

    5^

    16 21 39 42 51 55 59

    recap. 4^ 3^ 2^ 1^

    ( )

    example 4(e). Overview

    18 Despite the apparent similarity between the cadences to B major inmeasures 1718 and to E major in measures 5960, 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, measures5859 are connected through a tie, which makes the latter sound as acontinuation to the preceding events. The perceptually salient unfoldingF5A4G4E5 (measures 5862) also discourages perceiving the I6,occurring in the midst of the F5-A4 interval, as a real tonic. Finally, theprominence of the registrally isolated E2D2C2B2 (+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 5643 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 B5A5A5G5figure. Moreover, as illustrated in Example 4(c), graph (i),the Neapolitan sixth in measure 39 enables the permutationof this chromatic figure (BA[A]G) to form the top-voice framework of the entire second section, whereas thecounterpointing BCD 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, G5F5E5D5 (mea-sure 26) is enlarged in anticipation in the first section. Therelationship between the small and the large is reinforcedby a delayrelease pattern manifest in the rhythmic or-ganization on both scales. The associational network is fur-ther strengthened by the surface occurrences of theG5F5E5D5 figure in measures 1415 (summarizing thelarger figure) and in measures 3536 (marking the return tothe B major harmony after the BAAG enlargement).Finally, as shown in Example 4(d), graph (ii), theC4B3A3G3 line in the concluding section (measures

    5159) 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 118) 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 Pettys, 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 Pettys.

    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

    5

    5^? 3^?F5D5A4Arp.

    example 5(a). d: opening

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  • bachs 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 714 shifts the focus back to the vicinity of theoriginal 5 by leading the top voice to the syncopated B4(measures 1415). 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 1415 become a specialmoment by turning the implicit into explicit at the originallevel. In the ensuing cadence to III (F major), the AB con-nection is completed to form an enlargement of the themefigures top voice ABAGF as the framework of the en-tire section. The passing A is transferred to the bass of the Fmajor 63 chord in measure 16, another example of a local I6whose bass fulfills such a passing function.23

    ( )~ ~

    ~~( ) ( ) ( ) ( )

    ( )

    15

    7 10 15

    . . . becomes explicit.Implicit syncope . . .

    A

    A B A G F

    (B A G10 10 10 10

    F)10

    P

    53V

    64 IquasiF: II

    B A G F

    example 5(b). First section (measures 118)

    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 Pettys 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 enlargementthe syncopated Balso bears a special foreground relationship with the originalfigure. Moreover, additional features enhance the associationbetween the small and the large. The sequential passage inmeasures 714 subtly sustains the memory of the BAGFline in an inner voice.24 Furthermore, the sequential F5B4descent (measures 715) is anticipated at the start of the se-quence (measures 78), as is the accompanying D3G3 bassmotion (see brackets in Example 5[b]).

    The ABAGF 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 AGF motion by approaching A from above(CBA, measures 1838). 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 44461. 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 4749). Most remarkably, the 4 is madestand out through another explicit syncopethe second onein the piececonfirming the key role of this rhythmic devicefor the structure of d.

    ( ) ( ) ( )

    ( )( )

    7

    5^

    14 18 26 29 36 38 44 47 49

    5^ 4^ 3^ 2^ 1^

    (recap.gesture53(

    64 )

    53V

    64 I

    N

    VI 5 6V7IVIIII

    example 5(c). Overall sketch

    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 GB 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 (echappe 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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  • bachs 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 GFED or 8765. Asillustrated in Examples 6(a) and 6(b), the tetrachord occursfirst as a chromaticized form in the bass (measures 12), atypical passus duriusculus figure serving as the first counter-subject (even though the concomitant 65 upper-voice mo-tion is less typical).25 The second countersubject, given inthe right hand of measures 34, with its conspicuous ascentto 7 (F5), is more unusual. As indicated in Example 6(b),F5 points to a preceding implicit G5 as its point of origin.Consequently, the 8765 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 8765 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 120). The connection between this enlargement andthe foreground progression in measures 34 is reinforced bysimilar harmonic support IVIV for the 876 top-voice

    25 While the right-hand figure at the end of measure 1 may appear to ex-press a seventh chord (V65 ), there is no satisfactory resolution for theapparent seventh (C5), which implies that it has a more unusual antici-patory function.

    ( ) ( )

    38

    5^

    40 45

    4^

    50

    3^syncope!

    2^ 1^

    (recap. gesture)

    5 6V7 IVI 53V646 IIV75

    53

    64(= )

    example 5(d). Conclusion (measures 3852)

    ""

    V61

    P

    G F E DI ( 5 ) IV6 5 V86

    4

    753 I

    example 6(a). g: opening, underlying pattern

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  • 26 Schenkerians disagree on how rare 8-Urlinien are in Bach. In Visl2008, 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 54321 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 FA(F)C motion in a higher octave,whereas the descending scalar passages participate in the en-largement of the descending FEDC 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 FEDC 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 46 ina sequence of descending thirds. Adding to the remarkabledensity of interlevel connections, this pattern of descending

    27 Forte and Gilberts analysis (1982, 20607) shows the FEDC linebut not the FAC arpeggiation. (They indicate the 5 as establishedalready in measure 5 by the metrically weak C6.)

    ( ) ( )

    5

    8^. . . becomes explicitimplicit 8^ . . .

    ThemeCs. 2

    Cs. 1 Theme

    G F E P

    G

    F E DG F E

    F7^D

    G

    F E DG6V 43I ( ) IV7 VII ID

    G

    F E

    ( )

    ( )( )

    753

    VII)65IIIVII(Iexample 6(b). The establishment of 8

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  • Example 7(c) sketches the overall structure. After thetonicized dominant (measure 12), I43 (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 measures412. Through this transposition, the F5E5D5C5 pro-gression of the opening section becomes B4A4G4F4,which, in the absence of the ascending arpeggiation, assumestop-voice status and takes care of the Urlinie descent.

    bachs 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 BCA-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, A5F5D5B4, 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,135 (F5A5C6), which is elaborated by a passing 4 in alower register (B4). As the notation in Example 7(b) makesevident, the FAC arpeggiation is doubled at a lower octave,and the passing B connects registrally with these doublings.28

    The descending fourth (F4E5D5C5) represents an innervoice. The large descending-third pattern (A5F5D5B4),which brings about the transfer from the high 3 to the low 4,is formed by the combination of these voices.

    #

    ~

    7

    8^G

    9

    7^F

    11 14 15

    6^E

    17 20

    5^D

    22 23

    4^ 3^ 2^ 1^

    Cs. 1Th.Th.

    Cs. 2P

    (rchg.-over)

    Th.

    Cs. 1

    64

    853

    7

    I (V5

    65(c: IV VII I)

    6 ) IV8 7 V64 (IV6 65 ) 53 I

    ( ) ( ) ( )

    (I6)

    6(d: I IV V I)

    example 6(c). Overall sketch

    29 The I43 is emphasized by its basss extreme low register (C2). The I43does not occur as a surface harmony, but the reading aligns the strongbass C2 at the beginning of measure 25 with the upper voices of thefunctionally crucial I7 (V of IV) at the end of the measure.Comparable 43 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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  • 124 music theory spectrum 31 (2009)

    $ $ $ $ $ $

    *F

    *A

    *C(F)

    (summary)

    C

    DEF

    10

    DEF

    FEDC

    FA(F)C

    65 I)(C: V

    43

    5

    example 7(a). F: opening section, figure enlargement

    3

    P

    5^

    P

    7 9 12 14 5 105^

    P

    43

    6I5 6 II V

    I5 ( 6 II

    65 V )

    6

    C: II6 7

    II VI

    example 7(b). Opening section, structure

    4

    5^

    12 25 29 31 4 8 11 12 20 25 29 31

    4^ 3^ 2^ 1^ 5^ 4^ 3^ 2^ 1^

    I (V) I43 IV5 6 V I

    I5 6 II V I43 IV5 6 V I

    example 7(c). Overview

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  • bachs 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 135 arpeg-giation accompanied by IV, a fleeting return to the tonic

    as an applied dominant to IV, and a 432 motion abovethe prolonged IV; the prolongation of the IV also takes asimilar form in both cases (IVV65III6).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 thecomposers 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 135 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 IVIV7 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 measures23424 in a (Example 1[a]).

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

    I8 (V) 7

    IV5 (I) 6 V I

    French Suite VI,Allemande, opening

    4 12 25 26 29 31

    I8 (V) 7 IV5 (I) 6 V I

    F,

    3 8 14 15 21

    I8 (V) 7 IV5 ( V IIV5 ) IV7

    Prelude in E (overall structure

    WTC I ),

    overall structure

    example 7(d). Points of comparison

    30 For Schenkers 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, DAB, 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 DABfigure. 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 DABframework rather than DCB.35 Example 9(a) also showshow another important source of expressivity, the chromaticAA 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 G4G5 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. (Schenkers 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 GAGFE upper-voice figure, the larger of which hasbeen previously presented in Wintle 1986, Example 10. However, myanalysis differs in several respects from Wintles. A consquential detail

    ""

    a) Opening b) Overall sketch

    5 9

    5^

    14 17 18 19 23 26 28

    4^ 3^ 2^ 1^

    =G A G F E

    (smooth, scalar progression: laconic expression)

    F EC D C EF CDCIV7

    VII6I IV I

    51

    6I III5 IV V 43 I V I

    G A G F( E ) A G F E

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

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  • bachs 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 78, Example 9[b]), occurring after

    the opening thematic statements, implants the DAB fig-ure into a dominant prolongation (V). The diminishedfourth is given a pungent character by the way it serves toresolve the seventh between D4 and C5 (brought about bya bold use of counterpoint at the twelfth). The AA 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 DAB bassfigurewhich now occurs on two scalesproducing theouter-voice AA 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 V65 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.

    "

    "

    ( ) ( )* * * ~

    (A ) A

    * *

    ( )Underlyingscalarmodel

    Dim. 4th heightensdramaticism andexpressivity

    A BDD A BDI 6 II V65 I IV

    6 V

    example 9(a). b: opening

    7 A! ~

    AD B

    A ( )

    V 6 I

    A

    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 DAB figures. The small-scale

    D2A3B3 figure (measure 18) dramatically transforms thechord on D, which momentarily sounds like a D majorchord, into I6. The V65 of the larger D2A2B2 figure in turnprepares for the root-position tonic. The unusual temporalordera representative of I preceding the preparation for

    20

    % %

    ( ) ( )

    17

    Cs.

    Theme

    D A B

    (i)

    A B

    Theme

    Cs.

    D A B

    (ii)

    43

    64

    ( )

    A BDE

    E D A B

    65

    4

    2

    4^ 3^ 2^ 1^

    V 6542V

    6I

    V( 648

    53

    7)=V I

    D

    42 6

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

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  • bachs inventions: figuration, register, structure, and the clear way to develop inventions properly 129

    Isuggests 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 A3 and B3 of thesmall-scale D2A3B3 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 fromV42. But these E2 and D2 may in turn be understood as antici-patory upper voices in relation to the A2 and B2 of the even-tual V65I 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 Bachsimaginative and nuanced art of retransition and recapitula-tion. In this casein contrast to a and dthere 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

    1718) 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 121, BD(A)AB (IIIIVIIV65I),combines two central sources of expressivity, the DAB fig-ure and the chromatic AA 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 theVII in measure 14 and the V65 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 43 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 DAB figure forms the top voice of this cadence, Asubstituting 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 verybeginningin the theme or the countersubjectthe overalldesign of C is crucially affected by enlargements whose point

    36 The parentheses in Example 9(c) indicate that the anticipatory EDmotion 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 AB 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 AFE 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 AFEframework at three structural levels and in three registers, asindicated by letter names in Example 10(b).37

    ( ) ( )

    ~=

    3 15

    A F EIN

    65

    A5 without harmonic support VI offers harmonic support for A5

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

    ( ) ( )

    ( ) ( ) ( )

    (i)

    (ii)

    3 5 7 10 12 14 18 2117

    recap.

    D A A B

    recap.

    D

    D A B

    A B6

    I V I 65(D: IV 65

    5 V

    VIII V)

    III VII 5 6 65V I IVV= V42 I

    ( )

    2^ 1^3^4^5^

    2^ 1^3^4^5^

    3^4^ 3^4^

    example 9(d). Overall sketch

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  • bachs 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 A4F4E4 inner-voice figure but alsoby a C5B4C5 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

    ( ) ( )

    ( )( )

    7

    2^

    15 21 22

    1^3^

    C B C

    A F E

    I VI IV V I(CBC reproduces the originalcounterpoint of m. 1)

    P

    chr.

    ( )A F E

    A

    A F E

    A F E

    I V VI (I )6 IV I 6 V I

    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 17 and 715) andby a clearly contrasting concluding section (measures 1522), clarifiesthat the V in measure 7 is subordinate to a IVVI progression.

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  • 39 This model is similar to one of Neumeyers (1987) three-part Ursatzmodels.

    40 The 123 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 Visl 2008, in which I attempt toshow how the consideration of four types of analyticalcriteriain addition to design and register, these includemeter and rhetorical/gestural emphasishelps 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 triads 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 ABAGF 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 (A4D5F5) in d, whereas a123 Anstieg above the original 5 is discernible in e.40

    MTS3101_04 4/16/09 5:49 PM Page 132

  • While the 123 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 (A4B4A4G4F4),but the tones that should follow the high 3 (E5 and F5) failto occur. Conversely in e, the corresponding higher strandof voice-leading (G5F5G5) 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 Bachs tendency towards figure en-largement (see brackets in Example 11). The enlargement ofthe ABAGF figure in d has already been discussed, butthe GFG framework in e also reflects a detail in theopening thematic figure. While the figure starts from 5, 3 isactivated through the GFG 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

    bachs inventions: figuration, register, structure, and the clear way to develop inventions properly 133

    measures. However, the 123 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 measures35.

    && &&&

    ( )

    ( )( ) ( )

    ( )

    ( )

    7

    3^(i)

    (i)

    (ii)

    (ii)

    d

    e

    18 29 38 47

    3 7 11 13 15 20 22

    5^

    5^!(8(5(5 5)

    8)5)

    (5 5)

    V8 7IIII VI V I

    (not present)

    [ ] [ ] [ ] [ ][ ]

    cf. theme

    3^

    5^(8(5(5 5)

    8)5)

    (5 5)

    V8 7IIII VI V I

    3^!

    5 6

    (not present)

    cf. theme

    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 F5F4(measures 1519) and by the extended passing motion217, 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 eachothera lower tonic-chord tone is followed by a higheronetwo additional considerations pertain to the choicebetween them. The firs