bribitzer ab c e major third collections in nineteenth-century music

8/12/2019 BRIBITZER Ab C E Major Third Collections in Nineteenth-Century Music

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167

T he A CE Complex:The Origin and Function of Chromatic Major Third Collections in Nineteenth-Century Music

matthew bribitzer-stull

The A CE major-third constellation stands as a prototype for nineteenth-century composersexpressive and structural uses of chromatic major-third relations. After tracing the origins of thecollection, this article presents a conglomeration of hierarchic and transformational analytic ap-proaches to A CE music by central European composers to demonstrate that recognition of thecomplex comprises a valuable added dimension to our structural and phenomenological hearingsof romantic-era music.

Keywords: Chromaticism, Schenkerian Analysis, Neo-Riemannian Analysis, Third Relations,19th-Century Music

When asked Who but yourself would dare go directly from C major to E major? C.P.E. Bach replied, Anyone can and will assuredly do it whoknows that E is the dominant of a, and that a minor is very closely related toC major.1

Afascination with tonal relationships based onmajor thirds has provided the motivation for in-quiries from C. P. E. Bachs day to the present.2 The

inspiration for the investigation herein is no different, butthe premisethat a specic complex of sonorities can eluci-date major-third collections in central European music of the nineteenth centuryintroduces a new angle to this eldof study. Succinctly put, this article suggests that the

A CE complex constitutes a romantic-era prototypea benchmark for both structural and expressive trends innineteenth-century music.3 A topic this rich necessarily in-

vites numerous avenues of approach, but in the present con-text I restrict myself to three: rst, how the ACE com-plex most naturally demonstrates the emergence of major-third collections expressive and structural functionsfrom classic-era compositional and tuning practices; second,how tonal music theory copes with some problems posed by chromatic major-third collections; and third, how one mightprotably approach examples of ACE music using a con-glomeration of hierarchical and transformational thinking.

Schenker[1906] Earlier incarnations of this paper were delivered to the Music Theory Society of New York State (Columbia University, 2002) and to theSociety for Music Theory (Columbus, 2002). At the MTSNYS meet-ing this was but one of three papers on the A CE complex; my dis-cussions with Eric McKee and Charles Youmans, the authors of theother two papers, , were fundamental to shaping my thoughts on this

topic. Additionally, I wish to thank the many scholars who shared withme examples of A CE; Michael Cherlin; David Damschroder; andthe anonymous readers of this journal.

1 Kramer 1985, 552; cited in Irving and Riggins 1988, 106.2 In recent years, the topic has received much attention. See for instance,

Krebs 1980; Cinnamon 1984; Todd, 1988, 93115; Cinnamon 1992,130; Todd 1996, 153177; and Kopp 2002.

3 Throughout this paper, upper-case letters signify major keys and triads while lower-case letters signify minor keys and triads.


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A number of studies in the last three decades addressthird relations in tonal music. Most either treat genericproperties of these relations or focus on a specic work,genre (such asLieder ), or composer. By selecting music fea-turing A , C, and E, I am able to enjoy the benets of a new

vantage point. First, the music engaged by this approach cutsacross genres and composers, featuring works composedthroughout the long nineteenth century. (Many of theseare listed in the appendix.) Second,major-third relations ingeneraland A CE, specicallytypify chromatic thirdrelations in ways that other collections do not. And third, theconsideration of works containing thecomplete cycle of thirdsraises theoretic and analytic issues endemic to music that in-cludes all three sonorities.

A recent study by David Kopp divides the eight possiblethird relations into three categories: diatonic (sharing twocommon tones), chromatic (sharing one common tone), anddisjunct (sharing no common tones).4 (See Example 1.)Neo-Riemannian transformation labels explicitly show thecommon-tone relationship between sonorities, as each trans-formation indicates the motion of one pitch class betweentwo triads.5 When A , C, and E major or minor triadsprogress from one to another, they form eight possible rootprogressions whose tonal functions and directionality may bear extra-musical associations. (The move from I toVI(PL), for instance, relies not only upon the use of mixture,but also upon the falling root motion to evoke the dream-

world state so often associated with this progression.6) Theseeight root progressions are summarized in Example 2. HereRoman numerals and Neo-Riemannian operations are wed-ded in an attempt to place the parsimonious voice-leadingtransformations within a functionally tonal context. Four of these harmonic progressions, labeled with possible harmonic

168 music theory spectrum 28 (2006 )

4 The uncanny nature of disjunct (hexatonic polar) progressions istreated at length in Cohn 2004.

5 For a fully formal exposition of the L and P operations see (among oth-

ers) Hyer 1995. Despite their strengths, Kopps M transformations arenot used in this context since they, in effect, conate two voice-leadingtransformations.

6 Just as individual key centers may have rich, extra-musical associations,so too may harmonic progressionsbetweenmembers of the A CEcollection.The sense of progression from one key to another or of tonalmotion between keys was crucial to Joseph Schalks understanding of

musical association. See Wason 1997, 131. Hatten 1994, 44 goes so faras to imply that associations based on relationships between keys are of greater analytic value than absolute key characteristics, a position pro-pounded earlier by Donald Francis Tovey ( 1944, 61).

(a) diatonic (C to e)

(b) chromatic (C to E)

(c) disjunct (c to E)

Adapted from Kopp 2002, 1011, Figs. 1.31.5

example 1. Diatonic, chromatic, and disjunct major-third progressions.

C

Le

C

LPE

c

PLP (orLPL)E


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interpretations, occur with relative frequency in common-practice music. The remaining four progressions are rela-tively rare, perhaps due in part to the lack of clear harmonicfunction. Does vi 3 have a submediant function, due to itsroot? A dominant function due to its (respelled) leadingtone? Both? Neither?7

Perhaps more than any other development in composi-tional technique, the increased application of chromatic thirdrelations distinguished the harmonic practice of the nine-

teenth century from that of the eighteenth. Even a cursory survey of the literature strongly suggests that nineteenth-century composers favored progressions featuring major tri-ads whose roots were a major third apart.8 The reasons forthis may include the following phenomena:rst, major triads were preferred over minor simply due to the larger repertoirecast in major keys; second, chromatic-third relationships were preferred over diatonic relationships because they evoked a distinct sonic color, and they were preferred overdisjunct relationships because they retained a common tone;and three, cycles of major-third-related triads were preferredover cycles of minor-third-related triads because each triadin the former shares one common tone with the others, un-like the complete minor-third cycle, which includes tritoneroot relationships (like c and f , or e and a) that are less di-rectly intelligible.9

structural and expressive underpinnings

Chromatic major-third root relations are intrinsic tonineteenth-century central European music.10 A predilec-tion for these relations (more specically, those including thecomplex of A, C, and E sonorities) is most obvious in themusic of Beethoven, Schubert, Chopin, Wagner, Brahms,and Liszt.11 Because the A CE complex was not inventedby these composers, but rather emerged from earlier praxis,I begin with a consideration of how the tuning and com-positional practices of the classic era contributed to the

the origin and function of chromatic major third collections in nineteenth-century music 169

8 Somer 1995, 216 notes that the most frequent chromatic third relationsearlier in the nineteenth century involve major triads.

9 See Krebs 1980, 11718; Brown, Dempster, and Headlam 1997; and

Kopp 2002, 21718.10 See Kopp 2002, 151 and 213, and Hyer 1995, 130.11 Examples of third relations from the music of Verdi, Debussy, and

Rimsky-Korsakov, among others, are also copious. See, for instance,Somer 1995, 227 (Ex. 5) and 23133 (Ex. 10); Berlioz s Au cimitire,mm. 915; and the opening of Act II of Rimsky-Korsakov s The GoldenCockerel .

7 Swinden 2005 opens his study of plural harmonic function in chro-matic music with the vi 3 chord from Wagners Tarnhelm music.Swindens article relies heavily on Harrison 1994 (especially 4372).Both studies present a cogent scale-degree-based theory of harmonicfunction applicable to much nineteenth-century (and later) music.

1. C E 2. C AI III I VI 5III V VII 53 V

LP PL

3. C e 4. C aI iii rare III v

L PLP

5. c E 6. c Arare i VI

vi IV iv II

PLP L

7. c e 8. c arare rare

PL LP

example 2. Some tonal contexts for root motions by major third.


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emergence of A CE collections. In so doing, I examineboth these collections expressive (or coloristic) origins andtheir structuralorigins as notes, chords, and key areas withina tonal context.

Expressive origins and functions of the A CE complex.Irst consider a suggestive idiosyncrasy of the eighteenthcentury namely, its relative lack of works cast in the so-called enharmonic keys (B/C , F /G , and C /D ). Thekey choices of Haydn, Mozart, and Beethoven are represen-tative of the time; complete works in keys with ve or moreats or sharps in the key signature are rare in both Haydnand Beethoven, and missing altogether in the music of Mozart.12 (See Example 3.) The few exceptions that provethe rule fall into three categories: works whose overall tonickey includes ve or more accidentals; interior movements;and extended sections within a single movement. These areillustrated in Examples 3(a), (b), and (c) respectively.13 Though slightly more common than their parallel majorkeys, minor-mode works in c, f and b are also rarer inclassic-era music than those in f, c, and g minor, their coun-terparts on the at side of the circle offths. Thus, it appearsthat it was not the diatonic collections alone that composersavoided, but rather that the sense of tonic,regardless of mode ,inuenced their key choices.

Ultimately, the rationale for eighteenth-century perform-erskey preferences can be attributed to two related phenom-ena: C-centricity and temperament. In the classic era, thekey of C major ranked as the most common; it was the key of the neophyte and of the amateurthe peoples key. AsDonald Francis Tovey put it:. . . nobody can name a key

without being aware of its distance from C major.14 Thus,the notation, physical instruments, and psycho-acousticalframe of eighteenth-century musicians exhibited a clearpreference for C major as thedefault tonality.15 This con-ception of C remained at least until Kurths day, when thetheorist wrote:C major is perceived as the middle and foundation for two reasons.First, in the historical sense the C major region is the homeground andpoint of departure of harmonic development in sharp andat keys; thechurch modes already revolve around this center [sic ]. Further, thoughand this is by far more signicant than the historical developmentC major signies again and again the origin and central starting pointof musical sensibility for individual development, starting from the be-ginnings of musical training. This position establishes itself and deter-

mines not only the character of C major itself but all other keys as well. The effect of E major, for example, depends on the way it distinguishesitself essentially from C major. The whole absolute character of a key,reecting back to C major, is thus not given in the nature of music butrather in the particular course of [music] history and pedagogy.16

Though Kurth located the center of the church modes onC rather than D Dorian, his prose reects a strong traditionin Western music theory conating a sense of key with asense of location. Words likemiddle and homegroundindicate tonality s spatial connotation. Thus, the distanceone ventured from C could be measured metaphorically asthe distance traveled from the commonplace toward theesoteric, a metaphor of alienation predicated upon keyboardintonation.17 The increasing intonational dif culties as onemoved away from C were, in turn, a function of non-equaltemperament.

While close approximations of equal temperament in Western Europe were used for fretted instruments as early asthe sixteenth-century, true equal temperament on keyboard


12 C. P. E. Bach, for instance, also rarely ventured beyond key signatures with four ats or sharps. See MacDonald 1988, 222.13 Collections of pieces in all twenty-four major and minor keys, like

Bachs Well-Tempered Clavier and Chopins preludes, are not cited here.Even in these contexts, however, composers seemed to favor certainenharmonic keys over others (like F over G ). See MacDonald 1988,222.

14 Tovey 1944, 61.15 See Steblin 1981, 10351, especially 1056, 11314,117, 125, and 128.16 Kurth 1923,298, n. 1 (translated in Rothfarb 1991,126, n. 18).17 See the comments of Bruckners disciple, Joseph Schalk, in Wason

1997, 13031.


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Haydn Baryton Trio Hob. V: 5 BHaydn Minuet Hob. IX: 26 F attributedHaydn Trio Sonata Hob. XV: 31 e Jacobs Dream!Haydn Divertimento Hob. XVI: 2c BHaydn Symphony Hob. 46 B

5 out of more than 1,500 compositions (including attributed works and folksong arrangements)

Mozart0 out of more than 600 compositions

Beethoven Sonata op. 78 F 1 out of more than 200 compositions

(a) complete works

Haydn Sonata Hob. XVI: 46, Adagio DHaydn String Quartet op. 76 #5,Largo F Haydn String Quartet op. 76 #6, Fantasia BBeethoven Sonata op. 26,marcia funebre aBeethoven Sonata op. 27/2, Allegretto DBeethoven Sonata op. 57, Andante con moto DBeethoven Sonata op. 110, Arioso dolente aBeethoven String Quartet op. 130,Presto bBeethoven String Quartet op. 130, Andante con moto ma non troppo DBeethoven String Quartet op. 131, Adagio quasi um poco andante gBeethoven String Quartet op. 135,Lento assai, cantante e tranquillo D

(b) interior movements (sample listing)

Haydn Symphony H. 45Farewell,ending F (Picardy third)Haydn String Quartet op. 64 #2, ending B > (Picardy third)Beethoven Fantasia op. 77 g> BBeethoven Sonata op. 106, Adagio F parallelBeethoven String Quartet op. 131, Allegro C (Picardy third)Both Haydn and Beethoven wrote many minuet/trio pairs in which the trio is in the minuets parallel key and has ve or more accidentals in thekey signature.

(c) extended sections within movements (sample listing)

example 3. Works by Haydn, Mozart, and Beethoven cast in enharmonic keys.


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instruments was not universally accepted until 1917.18 Theorists and other musicians up through the nineteenthcentury espoused the virtues of equal temperament eventhough keyboard instruments of that century were almostuniversally tuned according to the principles of well-temperament, a tuning philosophy that made useable all themajor and minor triads without sacricing the characters of the keysa set of extra-musical associations that arose, inpart, from the meantone temperaments previously in use.19By the eighteenth century meantone tuning had been aban-doned, largely due to its intonational problems. It was these very problems, however, that were responsible for producingthe different qualities of meantone thirds that had, in turn,contributed to the establishment of the characters of thekeys.20

Of the three contiguous major thirds within a given oc-tave, only two (e.g., CE and EG but not A (G )C) were intonationally suitable in meantone systems, thus leav-ing four major thirds as noticeably out-of-tune.21 While alltwelve major thirds were used in eighteenth-century music,those that were most out-of-tune were not usually part of the stable tonic sonority. If the CE major third (as partof the common C major tonic) was to be among the most intune of meantone thirds, then the smallest major thirds (i.e.,most in tune) almost always included F A, CE, and/orGB. The thirds belonging to major triads opposite these onthe circle offths tended to be the largest and, consequently,

the most out-of-tune.22 (See Example 4.) Hence, the sharp-side boundary interval of usable major thirds tended to beEG , the at-side third, A /(G )C.23 The three majorthirds that lay outside these boundaries (BD , F A , andD F) belonged to the tonic triads of the underused majorkeys.24

While there were more tonally-distant keys than A andE (speaking in terms of C-centricity), these two keys oftenmarked the outer limits of acceptable intonation on unequally-tempered instrumentsa boundary that has persisted intomodern-day notation, as E and A still mark the edge of theenharmonic keys (D /C , G /F , and C /B). Like thedragon-infested waters that signaled the edge ofterra incog-nitoon the maps of early explorers, one can almost imaginethe eighteenth-century circle offths breaking at this point.Venturing into thismusical beyondduring the age of ratio-nalism and enlightenment was rarely done, and then only


18 Jorgensen 1991, 47 and 45.19 In 1721 well-temperament began to surpass meantone temperament in

usage. See Jorgensen 1991, 714, as well as xxi, 48, and 715 for othercomments on the relationship between well-temperament, meantone

tuning, and the characters of the keys.20 Jorgensen 1991, 2. Though these intonational problems were a functionof keyboard instruments, music for other instrumental forces writtenduring this time period reected the strong inuence of keyboardthinking, probably because so many musicians used the keyboard dur-ing the act of composing.

21 Jorgensen 1991, 47 and 774.

22 This is but one of countless meantone schemata. Tuning during thesecenturies belonged more to the realm of art than to science. Since many subtle variations of both meantone and well-tempered tunings prolifer-ated, intonation and the concomitant characters of the keys comprisedmore of a continuum than a hard-and-fast rule.

23 Notable exceptions did occur. For two, see Jorgensen 1991, Fig. 152,pg. 47; and Fig. 391, pg. 138.

24 Both tuning and compositional practice in the latter part of the eigh-teenth century reinforced the sense of the major key (and its tonictriads 13 major third) as normative; minor keys weremarked in thesemiotic sense,shadowy reections of their major-mode counterparts. (SeeHatton 1994, 3438, and Wheelock 1993, who uses the termen-othered.)Since this markedness bore a reexive relationship to the minor modesgreater degrees of chromaticism and tonal adventurousness, it is lesseasy to generalize about the intonational acceptability of minor keyssimply because intonational purity was compromised by the nature of

the minor mode itself, which had to admit to augmented seconds,augmented sixths, and altered scale degrees (2, 3, 6, and 7)intonational miscreants that were much less common in the well-ordered world of the relative major mode. The result was that minor-mode works in the eighteenth century were restricted to even fewerkeys than their major-mode counterparts: b, f , c , g /a , d /e , and b were all extremely rare in the classic era.


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with good reason. Hence, the edges of the known tonal world A and Ecould function as marked keys, destina-tion points that were far removed from the harmless tonalclarity of C. As such, these keys were often invested withrich associations and served as tonal settings for composersmost profound musical utterancesa habit that persistedeven during the theoretical hegemony of equal temperament.

This leads us to a consideration of these keys as associa-tive entities, markers of extra-musical signicance. Whileassociative tonality, as it is referred to today, was mostfamously explored by Wagner in hisRing cycle, key symbol-ism, stemming from the aforementionedcharacters of thekeys, had a rich history long before the Wagnerian music

drama.25 Wagners key associations were most often piecespecic, but the associations comprising the characters of thekeys infused all manner of works from the second half of thecommon practice era. C, lacking the artice of black keys, was often used to represent light, truth, purity, and the com-mon folk. D was the key of choice for triumphant and mili-tary music; E , for the heroic; F for the pastoral, and soforth.26 In addition to meantone intonationlargely a key-board phenomenoninstrumental associations (e.g., trum-pets with D major, horns with E major, English horn withF major), tessitura, written notation, absolute pitch level, andprior compositional practice all added to the summary char-acter of each key, even in compositions without a keyboardpart.27

Because E and A were the most distant keys from C incommon usage, their associations were among the mostpowerful. While these associations have never been xed asto exact meaning, nor applicable to every work, there existsevidence of general expressive trends: Ais linked to slum-ber, darkness, and death while E major is associated withtranscendence, spirituality, and the sublime.28 Thus, we


25 For discussions ofassociative tonality see Bailey 1977, 4861, and1985, 11346; McCreless 1982, 8895, and 1983, 6062; and Stein1985, 4344, and 14187.

26 Schalk understood each key to have essential differences from the oth-ers; that is, he believed that music should not be treated as simply atransposable pattern (a misconception he laid at the feet of those whoespoused equal temperament). See Wason 1997,13233.

27 Detailed descriptions of these key associations appear in tables com-piled by numerous eighteenth-century theorists. Since some of thesetables also end upon reaching the keys with four accidentals (e.g., those

of Vogler and Knecht in Steblin 1981, 133), it is tempting to hypothe-size on the chicken-and-egg relationship between composition andtheory on this issue.

28 E may have developed these associations since it is the dominant of viin C major.The motion from I to vi as a spiritual symbol is discussed inMcKee 2001. One might also conjecture that the upward arpeggiationof IIII V vs. the downward arpeggiation of I VIIV accounts for

example 4. Major thirds in meantone temperaments. From Jorgenson 1991, 180, Fig. 51-1: Well-Tempered Tuning

Vallottis Theoretically-Correct Method.


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might conceive of the eighteenth-century s E and A aspositive and negative tonal-dramatic poles about a central C. The developing usage of E as the erotic key in the nine-teenth century enriched this opposition between E and Aby setting up an Eros-Thanatos antithesis.29

The Act II nale of Mozarts Cos fan tutte illustrates theeighteenth-century prototype of A and E serving as expres-sive boundaries about a central C. Thisnale, like most of Mozarts, is a conglomeration of independent tonal struc-tures, although C major is understood as the large-scale ref-erential tonic. The opening C-major number is followed by achorus in E . The ensuing A quartet features the maincharacters dwelling upon the virtues of wine for drowningsorrows in slumber or, in Guglielmos case, death, should the wine be poisoned. By means of a chromatic 56 shift in mm.199200 there is a quick segue into the next scene, an activeE-major ensemble piece in which Despina, disguised as anotary, reads the marriage contract. (See Example 5(a).) Thismoment is the action the four main characters have bothfeared and hoped for all along, a dramatic counterpoint tothe preceding, reective A reverie. Using a fascinating tonalgambit, Mozart then proceeds to make his way back to Cmajor (at which point the truth is revealed and there is muchrejoicing) via numbers cast in closely-related keys on boththe at and sharp sides of the circle offths. A summary of this tonal motion appears in Example 5(b). Before thenaltonal-dramatic resolution can occur, Mozart illustrates, step-by-step, how far the tangled plot has come from the simpleclarity of C major.30

Throughout the later common-practice period, A and Epersisted as expressive tonal locales; increasingly, composersinvoked their expressivity without reference to specic extra-musical associations.31 The same held true of the juxtaposi-tion of A , C, and E sonoritiessonorities whose harmonieshad a profound impact on tonal structure.

Structural origins and functions of the A CE complex. Wheneighteenth-century composers featured two (or all three) of the members of the A CE complex in their works, thesesonorities were usually related indirectly. In the excerpt fromCos just examined, for instance, A and E as key areas are re-lated only indirectly to the overarching tonic C viafth cy-cles and to one another through the central C (as shown inExample 5(b)). However, the tenuous foreground link be-tween the A and E triads provided by the 56 shift (mm.199200), produces the sound of adirect chromatic third re-lationship, a forerunner of the increasingly important rolesuch relationships would play in romantic-era compositions.

The earliest strategies nineteenth-century composersused for incorporating direct chromatic third relationshipsinto their music usually followed earlier diatonic models,providing coloristic alterations of them more than substan-tive changes to their structural functions.32 Thus, commonsurface- and middleground arpeggiation paradigms such asII6 (or iii)V and I vi (IV 6)IV (ii6) evolved into IIII V and I VIiv (iio6) respectively.33 Likewise, diatonic oscilla-tion patterns expanding tonic with iii and/or vi came to in-


these positive and negative associations. Finally, one must not overlook sharp vs. at symbolism. Schalk noted that sharpspress upwards, to-

wards light, while ats strive toward the depths, into darkness. See Wason 1997, 130. Apparently, Riemann concurred: see Wason andMarvin 1992, 93, as well as the synopsis in Hatten 1994, 43.

29 Wagner uses E as the erotic key inTannhuser . See also Gilliam 1991,68 for a discussion of Strauss, E major, and the erotic.

30 Steptoe 1988, 23242 suggests thatat keys inCosrepresent falseness;keys near C, neutrality; and sharp keys, sincerity. The whole opera is

thus organized around a central, neutral C major. Burnham 1994,98, n.35, citing E-major music in this opera, states:In its exotic twilightrealm at the far edge of the tonal world of Mozartian opera, E major

may well stand for the phoenix that is this opera. 31 For more on the degree of specicity of emotion in expressive music,see Kivy 1980,4649.

32 See Somer 1995, 21927. 33 In Schenkers theory, ascending arpeggiations from tonic are also possi-

ble on the rst order middleground, while descending arpeggiationsoperate on more surface levels. See Schenker 1979, Figs. 7b, 14/1ab,


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clude III and VI 5.34 Such examples support the claimoften made to undergraduates that modal mixture is essen-tially a coloristic device that inserts chromatic alterations

into one or more voices of the tonal structure without requir-ing a shift in understanding of fundamental harmonic orcontrapuntal principles.


operation described in Proctor 1978, 181200, and describes thesetonal itineraries in terms of their bass motion. Bass lines that articulatea series of the same interval (e.g., major thirds) may be directional (e.g.,moving from C to A via E), circular (e.g., starting and ending on C with A and E by equal division of the octave), or axial (e.g., startingand ending on C with A and E providing upper and lower mediants).

15/2b, 98/3a, 100/5, 108, 112, and 113/2 for examples. See also Beach1997, and Kopp 2002, 10912.

34 Krebs 1980, discusses oscillatory third progressions and circles of thirdsinvolving tonic harmonies (94121) and describes the same techniquesprolonging non-tonic harmonies (8494). Kielian-Gilbert 1990, 5052, uses terminology drawn from the denition of the transposition

!

( )

( )

=

53 63 A:

E:I VI

I6 ii7 V 7 I

(a) modulation by chromatic 56 shift, mm. 199203

Sharp Keys

mm. 1 66 149 208 280 291 310 372 483 539 576

developmental

C E A E A D E B (IV) G C

(F---d/F)

Flat Keys

(b) overall tonal plan

example 5. Act II nale of Mozarts Cos fan tutte.

Measures are numbered from thebeginning of the nale.


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Beethovens In questo tomba oscura illustrates how thischromatic alteration functions. Example 6(a) shows that the A tonic is prolonged by a chain of descending major-thirdroot progressions.35 Here we notice that the songs opening A Stufe is followed in m. 14 by a chromatic 56 shift to E, which in turn leads to a cadence on a C major triad in bar19. This III Stufe is also labeled by its local function (V of vi) on the graph for two reasons. First, this chord makes ref-erence to the diatonic viStufe that is replaced with the chro-matic VI, F , enharmonically respelled as E major. Second,hearing this C chord arising in some sense from an unarticu-lated f-minor Stufe is an example of the exact tonal relation-ship described by C. P. E. Bach at the opening of this article;it illustrates a common, indirect, and diatonic context for re-lating two of the three keys in the ACE complex. Ratherthan arising out of a direct chromatic relationship to A(asits III for instance), Cs relationship to A can be heard in-directly, as the dominant of As most closely-related key. Thus, both chromatic Stufencan be restored to a diatonicprototype without radically altering the middleground. Thisis shown in Example 6(b).

The ease with which the chromatic replaces the diatonicin such examples is perhaps predicated on the appearance of

only two members of the A CE collection. That is, onedirect major-third relation is usually easy to accommodate within a tonal context that is still clearly controlled by abackground tonic-dominant hegemony.36 The appearance of


all three members of the collection does not necessitate theerosion of familiar structural functions, but if the three aredirectly related on the same level of tonal structure anythingfrom the surface-level triad to the backgroundBassbrechung itself may be disrupted. The CEG augmented triad, forinstance, often arises because one of the three tones effects a

(a) graphic analysis

(b) diatonic prototype

example 6. Beethoven, In questa tomba oscura. 35 Proctor 1978, 17879 analyzes the opening of thisLied as a bass arpeg-giation of the augmented triad. The deep middleground here wouldlook quite different if the D and E quarter notes in m. 21 were takento be bona de harmonies, certainly a viable reading. Note that

Schenkerian graphs throughout the present paper are middleground-oriented and thus lack foreground detail. Accidentals apply only totheir immediate context and do not carry throughout.

36 Krebs 1980 argues that in Haydn, Mozart, and Beethoven, thirds(often chromatic) above and below tonic either lead directly to V (as inIIII V or I VI 5V) or embellish V (V III V or V VI 5V).Numerous examples are cited in pages 2459 and 7384.

! 5 64 53 64 538 7 A:

A

51

E

14

N

C

19 22431

332

2 133

IVI V/vi

III IV

V I

! [ ] A:

5

or

N

6

I

vi [IV 6]


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chromatic passing or neighboring motion. Even when allthree are chord tones within an augmented dominant triad,the CEG sonority may remain unquestionably dominantin function. But, when the augmented triad is not anchoredby a diatonicStufe its symmetry can threaten tonality alto-gether (as in Liszts Nuages gris). Likewise, an E-major(III ) structural third divider between I and V on the mid-dleground of a C major work (in therst movement of Beethovens Waldstein Sonata, for instance) would hardly compromise the sense of tonal unity. An extension of thistechnique might feature a nested chromatic mediant rela-tionship (III of III ) to invoke the third member of thecomplex (as in Wagners Siegfried Idyll ).37 When such chro-matic third chains achieve independence from tonic anddominant, however, they may replace tonic prolongation with other structural functions as in the symmetrical divisionof the octave evinced by theSchlaffenakkorde of WagnersRing .38

Since the use of multiple major-third relationships re-quires great care to avoid disrupting the sense of tonality, it isunsurprising that nineteenth-century composers tended torely on just a few strategies. In short, they anchored thesethird relations on tonic or dominantStufen, thus prejudicingthe tonal contexts in which an A CE collection couldoccur.39 Naturally, A , C, and E major were among the mostcommon tonics for the incorporation of the complex. In

these keys, chromatic thirds could be strung from (or to) thetonic. This happens in Chopins Polonaise, op. 53, in whichthe tonic A is prolongedrst by its upper third, C (III ), anarpeggiation both in m. 49 and again in m. 58 (functioninglocally as V/vi), and later by its lower third, F (spelled as E),beginning in m. 81. (See Example 7.) This music comprisesa concatenation of two separate oscillating progressions ondifferent levels of structure A C A and A E A .40 Cmajor, a local expansion of A, exists at a more foregroundlevel than E, the tonic of the work s entire middle section.Despite this, C is emphasized both as the most obvious tonaldeparture from the tonic A within the rst section andby virtue of its recurrence at the end of the retransitionback into A (mm. 14551) and in the nal cadence (mm.17980).

When anchored by the dominant, two less obvious toniccontextsa minor and f minorpredominate. In each key,one of the three members of the complex can function as III(the relative major) and another as V (the functional domi-nant). The third member is often used to connect the two.41 The scherzo movement of Schuberts Sonata in a minor, op.42, provides an illustration. (See Example 8.) Here, the ex-pected modulation to the mediant during therst reprise of


that leads eventually to V) and Schuberts E String Trio, i, mm.43452 (whose recapitulation features a ii VI IV predominantchain).

40 Direct chains of thirds appear in the literature as well. See SchubertsLied , F lle der Liebe and the analysis in Krebs 1980, 110 (Fig. II.37, v. 2, 49). Example 7 presents only the opening of the AE A pro-gression. Interested readers may wish to consult Krebs for a graph of mm. 80 to the end. See Krebs 1980, Fig. II.9 (v. 2, 34), which links

both chromatic Stufen to V.41 The other minor key capable of containing these tonal relationship, c,does not provide many examples of the ACE complex, perhaps be-cause of its own relative scarcity in common-practice music. Note thatthe six keys mentioned (A , c, E, a, f, and c ) together comprise Weitzmanns Region I, a grouping noted in Cohn 2000, 93, and furtherexplored throughout his article.

37 See Anson-Cartwright 1996, 60, Ex. 3. 38 This excerpt wasrst described as a chain of chromatic thirds by Ernst

Kurt. See Kurth 1923, 22627 (translated in Rothfarb 1991, 13334).More recently, Brian Hyer demonstrated the manner in which neo-Riemannian L and P transformations control both the harmonic and

melodic structure of the Magic Sleep music. See Hyer 1995, 11116. 39 Another strategy was to include the three sonorities in a chain of pre-dominants that ultimately lead to the dominant. See Krebs 1980, 60(Fig. I.46, v.2, 26) who illustrates this technique in Beethovens pianoconcerto in E major (Emperor), iii , mm. 13889 as a VIIV II 5

chain embellishing the motion from VI to V. See also, Beethovenspiano concerto in c minor, iii, mm. 138220 (a VI IV II succession


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the scherzo eventually leads to the structural dominant in m.80. These two key areas, C and E, are connected by a toni-cization of A in measure 43. Although this A is precededby its own dominant and is followed by harmonies that pre-pare the arrival of the e-minor dominant (made major in bar80 to set up the return of the opening material in a minor),these intervening sonorities do not prevent us from hearing akey succession of C A e /E. That is, A connects C and Eby a descending major-third arpeggiation. Interestingly,surface-level references to the combination of A, C, and Eare also audible in the opening a-minor measures. Here Efunctions as a local dominant ( passim) and A appears dur-ing the modulation to C major (m. 17).42

analytical illustrations of the a ce complex

As chromatic-third usage evolved, nineteenth-century theory naturally developed alongside composition. Whetherreactive or innovative, much of this work focused on ACE collections. Hugo Riemann, for instance, eventually came to believe that chromatic third relations were percepti-ble as direct harmonic progressions,43 and, at one point, re-dened tonality specically to model the A CE collec-tion.44 And Carl Friedrich Weitzmann both distinguishedhimself from his contemporaries and inuenced Franz Lisztby his thorough treatment of the ACE augmented triad.45


42 Similar A CE collections in a- and f-minor music occur in Brahmss

Intermezzo op. 118, no. 4 (see Example 10); the prolongation of Vin mm. 2662 of the rst movement of C. P. E Bachs Piano Sonata in f minor, H. 173; and the dreamlike A that intercedes between a back-related dominant, E, and motion to the mediant, C, in mm. 81152 of Schuberts Allegrofor piano, four hands, op. 144 (Lebensstrme).Schmalfeldt 2002 describes some intriguing formal implications of theparenthetical A in the Schubert Allegro.

43 Riemann [1893], 165, stated that the third of a triad (Klang ) can takeon an independent signicance just as thefth of the tonic triad does.He even adapted a separate function symbol for chromatic mediants in

the last edition of the Handbuch der Harmonielehre (1920) publishedduring his lifetime. See Kopp 2002, 94. Other ACE examples occurthroughout Riemanns writings on third relations and tonality. See, forinstance, Riemann 1882, 189, 1890, 38, and 190203, 76.

44 See Riemann 1922, 1304.45 The continuing force of C-centricity led Weitzmann to choose the col-

lection as his augmented triad prototype, deriving it from the default

! A:

A

51 49 53 57

C

58 63 64 65

E

681

I III ii V I VI5

example 7. Analysis of the opening of Chopins Polonaise,op. 53.

!a:

51

4

C

3

A

43 58 66 67

E

280

197

i III V i

example 8. Analysis of the scherzo from Schuberts sonata, op.42, iii.


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Riemann and Weitzmann were accompanied by a host of others who used A , C, and E as a prototypical collection todirect or explain the harmonic advances of the nineteenthcentury and the changing nature of tonality.46 ParaphrasingKurth, one might go so far as to say that the symmetry of the A CE collection paved the way for the eventual dissolu-tion of functional tonality itself.47

This may be why chromatic-third relations continue toaffect our modern-day conception of tonality.48 While rela-tions by perfectfths t our existing theoretic and analyticapproaches with few problems, chromatic thirds are anotherstory. Some scholars have proposed that these chromaticrelationships constitute another form of tonality, a sort of seconda prattica , distinct from the diatonic practice of theeighteenth-century.49 Others argue that chromatic-third re-lations, rather than replacing a still-viable tonal tradition,simply added another dimension to it.50 While there are ob-

vious examples in which functional monotonality has beenstretched to the breaking point by the predominance of chromatic-third relations (Liszts Die Trauer-Gondel I, forone), determining whichavor of tonality governs a given work is perhaps less important than recognizing and articu-lating the ramications created by the addition of chromaticthirds to a largelyfth-governed tradition.

As an example, consider therst movement of Beetho- vens Appassionatasonata. Schenkers graph of the develop-ment section in Der freie Satzindicates that, on the deepmiddleground, 3of the Urlinie falls to 2as an A (III) Stufe moves to a C Stufe (V) in f minor.51 Example 9 presents aslightly more extended middleground analysis, beginning atthe end of the exposition and continuing through to the endof the development. The Schenkerian prolongation of an AStufe supporting 3 is accompanied by the bubbles markingthe appearance of A , C, and E sonorities, and also by neo-Riemannian transformational symbols that illustrate how the A Stufe is prolonged by a series of P and L motions. Notethat ve of the six triads in Cohns northern hexatonic col-lection are traversed.52 In chromatic-third chains of majortriads, the third of one triad becomes the root of the next(ascending thirds) or vice versa (descending thirds). The

smoothest voice leading, however, is maintained when theintervening minor triads are articulated. In such examples,like the Appassionata development, two common tones areretained by adjacent triads as the harmonic progressioncircles the northern hexatonic pole in a series of LP (or PL)cycles.


51 Schenkers analysis begins with the a-minor sonority in m. 65. SeeSchenker 1979, Fig. 114.8. 52 See Cohn 1996, 17, Fig. 1. In Cohns gure the A CE collection is

given preferential placement at the north, the direction most commonly indicated on maps. It can be inferred from his remarks that Cohn madethis choice consciously, due to the conventional primacy of C. Op. cit.,38, n.34.

key of C major. See Weitzmann 1853 and the commentary in Todd1996, 15859.

46 These include Dehn 1840, 157; Kurth 1913, 12428; Lobe 1861, 80;Rimsky-Korsakov 1895, 98, 102103; Schwartz 1982, 70, n. 5, and3867; Weber 1846, 503; and Ziehn 1887,8 and 119.

47 Taruskin 1985, 13536, reproduces Rimsky-Korsakov s false progres-sions by thirds from his harmony text, two of which feature major andminor triads built on A , C, and E; McCreless 1983, 7071, summa-rizes Kurths belief that symmetrical, chromatic sequences were crucialforces in the destruction of tonality.

48 The prodigious body of scholarly literature on this topic aside, currentmusic theory text books for undergraduates continue to present exam-ples of A CE in back-of-the-book topics like augmented triadsand enharmonic modulation. See, for instance, Laitz 2003, 64546;Ottman 2000,229; Roig-Francol 2003, 83031; and Kostka and Payne

1984, 383.49 After Proctor 1978, this philosophy gained ground. Proponents includemany authors in Kinderman and Krebs 1998.

50 These scholars support the applicability of Schenkers theory fornineteenth-century music, arguing that his analytic method is fully chromatic, lacking only the ability to model direct tritone relationships.See Brown 1986 and Brown, Dempster, and Headlam 1997.


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While the Schenkerian graph in Example 9 shows theparsimonious voice leading of Beethovens development sec-tion, the neo-Riemannian analysis, undergirded by an un-derstanding of A , C, and E triads as a group structure,highlights the skipped member of the northern hexatoniccollection: C major. Its absence is audible because it disruptsthe previous voice-leading transformation stream. One ex-planation for this omission is that C Major (as V) is required

shortly at the retransition.53

But it may also suggest why thebass C remains active at the opening of the recapitulation,creating the sound of a tonic64 underneath the return of theopening material.54

Just as a neo-Riemannian analytic vantage point may in-form a Schenkerian reading, as in the Appassionataanalysisabove, the converse is also true. The indeterminacy of direc-tionality implied bypolar progressionsmotion across ahexatonic pole (PLP or LPL)can be claried by theprolongational context.55 While the distinction may seemacademic, the two different labelsLPL and PLPsuggesttwo different hearings that imply a differentiation between

clockwise and counterclockwise motion about a hexatonicpole, orin linear rather than cyclic spaceascending anddescending harmonic root motion. When considered asup vs.down, the directionality of such harmonic progressionscan play an integral role in a work s dramatic effect.


53 C refuses to relinquish its role in the development as dividing domi-nant.Rather, an unstable neighboring 64 elaborates the VStufe when theprimary material appears, only giving way to a root-position tonic later

in the recapitulation. Thus, the inclusion of neo-Riemannian third-centric analysis with the Schenkerian graph provides one explanationfor Beethovens disruption of the paradigmatic sonata form retransi-tional tonal structure.

54 A motivic rationale for the idiosyncratic recapitulation that cites thetransferal of the D C neighbor to the bass is also viable. See Smith1995, 26870, for an unconventional reading of this movements tonal

structure that highlights these falling bass half-steps and suggests that

the apparent tonic recapitulation grows out of a dominant prolonga-tion at the opening. 55 Cohns reading presupposes a lack of directionality. In examples from

the literature, he cites direct motion between the hexatonic polesmotion that lacks common tonesrather than an incremental shiftfrom one pole to the other is responsible for this progressions uncanny effect. See Cohn 2004 and Cohn 1996, 2122.

!

64f:

5

1

A

3

35

P La

51

PE

67

LPe

79

Lc

83

A

87 93 97

2

132

5

i III V

example 9. Graphic analysis of Beethovens Sonata, op. 57 (Appassionata) with neo-Riemannian analysis.


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This is the case in the middle section of BrahmssIntermezzo in f minor, op. 118, no. 4, in which a descendingchain of major thirds prolongs the dominant. (See Example10.) At rst glance, it may seem immaterial whether the mo-tion from A major to e minor in mm. 52 to 75 is markedPLP or LPL. But directionality is an important aspect of this work, whose tonal and dramatic structure is predicatedupon contrast between the middle section and the sectionsthat frame it. The energy of the opening section drives con-stantly toward the dominant. On the foreground and mid-dleground levels, weak (or implied) tonics resolve to clear,emphasized dominantscomprising, in effect, a half ca-dence writ large. In the middle section, the illusion of mo-tion withinthe dominant rather than motion to the domi-nant is of primary importance. The relaxed texture andfeeling of descent created by the chain of thirds (PLP, orcounter-clockwise about the northern hexatonic pole) andthe upper-voice arpeggiation of C major contrast sharply with the opening sections ascent to the dominant. The open-ing of the middle section (mm. 5267) rmly establishes the A Stufe , opening up another potential reading in which thisIII is part of a middleground IIIIV arpeggiation (relegat-ing the V that closes the opening section to a back-relating

dominant). This reading, however, contradicts the strengthof the many motions to V inrst section and the sense of downward motion throughout the middle section. Thus, notonly does the sense of counter-clockwise directionality in the A eC succession suggest one middleground reading overanother, it also informs our choice of a PLP transformationalmodel over an LPL model. This reading is also supported by foreground details (the cadence in a in m. 67 and the prepa-ration of E in mm. 6874) that point toward a tonal motion

from A to a to E, and then to the cadence in e (m. 75) thatultimately leads back to C. The preceding analysis juxtaposes Schenkerian graphs

and neo-Riemannian transformational symbols. At times thetwo analytic approaches complement one another, as inExample 10, in which the Schenkerian prolongation of C

(V) matches up with a counter-clockwise spin about thenorthern hexatonic pole. But, more often, this analytic juxta-position generates unavoidable conceptual frictions. Theseare patent in an excerpt from Wagners Der Fliegende Hollnder .56

The progression in Example 11(a) occurs at the end of the Dutchmans Act I recitative and aria.57 The scale degreesabove the score and the Roman numerals below it represent a

Schenkerian hearing of a localized auxiliary cadence in whichthe A -major chord harmonizes an upper-voice passingtone.58 This reading presumes that A plays a more fore-ground melodic role than c and E. At odds with this readingare the neo-Riemannian symbols below the score. Theseimply that the structural dominant seventh chord on G is of


56 For another discussion of the benets and contradictions generated by juxtaposing Schenkerian and neo-Riemannian thought, see Bribitzer-Stull 2006.

57 Measure numbers from Wagners operas refer to the widely availableSchirmer piano-vocal scores and are cited in the format: page/system/measure.

58 For an overview of the Schenkerian auxiliary cadence, see Burstein2005.

!f:5

1

CPL

A

52

PLPe

75

LC

83

i V

example 10. Analysis of mm. 5283 of Brahmss Intermezzoop. 118, no. 4.


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\

\

\\

$

%

!

all!auf!

all!auf!

Cello

c:

E

3

LP

Str.

(PT)

A

2

L

1

K.-dr

c

III V i

espr.

(a) Ghost-ship cadence, Hd/42/3/1

Section Recit. A B C CodaMeasures 29/1/1 32/1/1 35/5/5 38/1/1 41/3/5Key c a c CEcSynopsis Dutchman Deathless Beseeches Longs for Crew welcomes

makes land wandering an angel the Day of death Judgment

(b) formal overview

example 11. Dutchman s Act I recitative and aria.


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62 Schachter 1987, 30408, discusses the distinction (not made inSchenkers Free Composition) between structural keys and more fore-ground keys, all of which form part of the listeners moment-by-moment experience.

63 As Cohn suggests, neo-Riemannian analysis can be used in conjunction with Schenkerian analysis to understand both group structure (the A CE complex as Cohns northern hexatonic pole), as well as tonaland linear aspects of various works. See Cohn 1996, 33, in which hesuggests the use of hexatonic thinkingin conjunction with standard dia-tonic (sic) models such as Auskomponierung . See also Lewin 1986, 362 ff . for examples illustrating the possibilities multiple perceptions havefor multiple analytic approaches to the same passage, neither of whichis betteror more valuable than the others.

less interest than the succession of triads: E, A, and c.59 Inthis second reading, the linear, root-motion cycle of ascend-ing major thirds takes precedence over the hierarchical,Schenkerian reading. Since neo-Riemannian transforma-tions are capable of incorporating the G dominant into theharmonic event stream, why omit it from the analysis? Aninvestigation of the preceding music provides the answer. AsExample 11(b) shows, the Dutchmans aria proper dividesinto three sections, cast in c, a (with shifts to A major),and c respectively. The coda to the Dutchmans aria picks upthe C major Picardy Third at the end of the third sectionand moves toward E for the ghostly crew s conrmation of the Dutchmans longing for death. E then passes through A to c, bringing an otherworldly quality to the numbers

close.60

In effect, then, we have two conicting readings of thiscadence. One favors a tonally-hierarchical view that modelsprolongation, while the other models a transformationalevent-stream.61 While Schenkerian analysis effectively rep-resents tonal-prolongational structure, this structure is justone facet of musical construction and of musical experience. Associativity, referentiality, and salience are also important:even when A , C, and E are not adjacent on the same tonal

level they are often marked by tonal, formal, rhetorical, refer-ential, or associative processes, as in theDutchmanexample,above.Stufen, signicant cadential tonal centers, unexpected

or parenthetical tonal shifts, irregular formal units, andextra-musical connections can all draw the listener to a phe-nomenological awareness of A, C, and E connections.62

Placing neo-Riemannian analytic symbols below aSchenkerian analysis shows where A, C, and E sonoritiesoccur and suggests an abstract voice-leading connection be-tween them.63 But the implications go far beyond merely la-beling an event stream. They point to a group structureaconnection between A , C, and Ethat is non-hierarchicalin nature, though the members of the group may exist withina tonal hierarchy. While some analysts maintain these as dis-crete forms of tonality, this study suggests the possibilityfor the interpenetration of these two spacesthat the sec-ond practice of nineteenth-century tonality can exist within

a diatonic background structure.Brahms Piano Quartet in c minor, op. 60.My nal analysis

draws upon the E-major slow movement of Brahmss PianoQuartet to suggest that the relevance of the ACE com-plex in tonal contexts may range from the structural fore-ground to intra-movement connections, even within thesame work. The G EC major-third chain of the openingcello solos rst notes comprises the initial statement. While these notes are naturally divided by the harmonic

progression from I to iv, their linear appearance together

59 The presence of a functional dominant does not necessarily abrogatea neo-Riemannian group structure. See Santa 2003, whose hybridnonatonic/hexatonic group involves the dominants of the three tonicsin question, in effect turning the model advocated in Krebs 1980 upsidedown.

60 There are also references to ACE in the recitative before the ariaproper. See the foreground viio7 of c (29/4/2), viio7 of e (30/1/2), andresolution to A (30/2/1) in the recitative.

61 Samarotto 2003 sets forth an introductory model of interaction be-tween tonal coherence (Schenkerian analysis) and transformationalevent streams (neo-Riemannian analysis).


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64 Peter Smith argues that the G EC augmented triad achieves a mo-tivic and expressive importance that transcends its role in the structuralhierarchy. See Smith 1994, 25860, and 2005, 1718.

65 Interested readers may wish to compare this graph with that in Smith2005, 102.

66 Proctor 1978, 181200.67 This passage bears a striking resemblance to a similar retransitional

third chain with deceptive motion in Liszts Orpheus , mm. 11430.

68 See MacDonald 1990, 225 and Smith 2005, 24 and 227.69 For discussions of tonal relationships between the movements in multi-

movement works see Neumeyer 1982 and 1997. See also Krebs 1981,1415, who notes that non-monotonal works often feature tonic keysrelated by third. See the keys of the movements in Beethoven, PianoConcerto no. 3; Brahms, Symphony no. 1 (cE A c!); Grieg, ViolinSonata, op. 45; Rachmaninov, Piano Concerto no. 2; Schubert,Wanderer Fantasy(major mode variant: CE(c ) A C); and Liszt, Annes de Plerinage, Premire anne: Suisse (multi-work variant)for other pieces that exhibit this tonal relationship. Note that many of these works feature A CE collections on surface and middlegroundlevels as well.

70 Smith 2005, passim (see, for instance 218) refers to the E-major Andante movement of Brahmss op.60 assolaceand a dream.

here constitutes a striking motivic gesture that recursthroughout the piece, on both surface and deeper levels.64

Later in the work, for instance, the retransition into thereturn of the opening material expands the GEC major-third collection. (See Example 12.65) Two aspects of this

passage are noteworthy. First, the descending augmentedtriad motive is itself stated three times on the musical sur-face, beginning on E, then C, andnally G . The third iter-ation marks the return to the opening material played, thistime, by the piano. Second, there is a discrepancy betweenthe upper and lower voices. The structural upper line tracesthe chain of descending major thirds, ECG , with thecomposing-out of therst third, EC, transposed exactly tocompose out CG . Thus, the upper line seems to model

Proctors transposition operation and suggests that Brahmshas entered a fully-chromatic, equal-tempered tonal space.66But the melodic pattern is not mirrored by the harmony. Theg marked in the neo-Riemannian analysis beneath the scoreis illusory, an implication that is never realized. For while theV of c in m. 72 resolves as expected, the V of g in m. 77does not. The deceptive bass motion to E coincides with thereturn of the opening material in a bait-and-switch tacticthat reveals the passage to be a prolongation of the EStufe

by its lower third, C.67

While G does not play a vitalstruc-tural role in this prolongation, however, the neo-Riemanniananalysis below the score points to itsreferential role in evok-ing the motivic descending third chain, a reference thatenriches our hearing.

The previous use of falling thirds throughout the Andante , in part, lends this movement its tender, contempla-

tive aspect. Arguably, so does its tonic key of E major. As wenoted earlier, E major was often reserved for composersmost expressive music, a tonal marker for the spiritual andsublime. Given Brahmss allusions to remembrances of hisfeelings for Clara Schumann in this quartet, the expressive

connotation is appropriate.68 But the tonal relationship be-tween the E major Andante and the c minor of the othermovements is odd. This unusual key relationship might beconsidered an isolated idiosyncrasy were it not for the num-ber of other works in which it occurs. Despite the rarity of this third progressionwithin a singleUrsatz (i to III ), themotion from tonic minor to the raised major mediant ap-pears with surprising frequencybetweenthe movements of multi-movement works cast in c minor, and, as such, de-

serves consideration as a further ramication of the A CEcomplex.69 Two things are immediately striking about this inter-

movement tonal relationship. Therst is the tonal contrastthe key of E represents a luminous and ethereal refugefrom the surrounding c minor.70 But this tonal contrastseems to be predicated on the associativity of E, rather thanabsolute tonal distance, as E is not the most tonally distantmajor key from c minor; it lies ve steps away on the circle of

fths, while A major lies six steps away directly opposite c


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72 Smith holds that E is not only an expressive reprieve from the sur-rounding tragedy of c minor, but also bears motivic cross-references with the rst movement.The Andante s E major can be heard as an out-growth of the pizzicato Es from the Allegro (Smith 2005, 101).Likewise, the foreignness of the C in the augmented triad highlightsthe tonal distance between E and the quartets overarching c tonic(Smith 2005, 17).

71 While intra-movement relationships of hexatonic poles are exceedinglyrare in common-practice music, a few examples do exist. See, for in-stance, Mendelssohns g-minor Piano Concerto, whose middle move-ment is in E major.

minor.71 Second, no other pair of keys with this unique rela-tionship seems to occur with any frequency. (How many multi-movement works in d minor, for instance, have slow movements in F major?) Narrowing the pairs that exhibit

this tonal relationship to the accepted classic-era keys withfour or fewer accidentals requires enharmonic reinterpreta-tion in all cases but twoa possible reason for composersavoidance of these pairs. And the one pairing apart from cminor and E major that doesnt require enharmonic reinter-pretation, f-minor works with A-major middle movements,apparently occurs with substantially less frequency than the cminorE major relationship. Thus, in works like Brahmssop. 60 Quartet, we can hear tenuous connections back to the

previous century, faint echoes of the enormous impact tun-

ing systems, key associations, and the conventions of func-tional tonality had upon earlier music.72

* * *

Short of cataloging and counting all the tonal works fromthe late eighteenth century forward, it is impossible to con-tend that A CE collections are more common or moreimportant than other major-third cycles. But the evidencefor these keys expressive signicance and prevalence inromantic-era compositions suggest that they typify thechromatic-third relations that lie at the heart of nineteenth-century compositional practice. The composers who most

!

( )

5 55

e

Augmented triadmotiveon: E

L

6

C

c:

Pc

L

C

6

G

g:P

gE:

LE

G

3

V 64 i5 V 64 VI

I

example 12. The A C E complex in Brahms Quartet, op. 60, Andante: Graphic analysis of the retransition to the opening material,mm. 70 78.


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favored the use of the ACE complex span thenal daysof the classic era to theprogressive New German schooland its antipode, the conservative, Johannes Brahms.Moreover, the examples cited above include chamber musicand orchestral music, concerto and symphony,Liederand

opera, and musics both dramatic and absolute. From surface-level melodies to multi-movement connections, the ACEcomplex appears in almost every conceivable context, tran-scending the boundaries of genre, form, and tonal hierarchy.

More importantly, an awareness of its origins, tonal func-tions, and expressivity enriches our analytic practice. Recog-nition of these sonoritiesgroup structure uncovers intersec-tions and contradictions between tonal/hierarchical andphenomenological/referential hearings of music, impacting

our understanding of musical form and musical meaning. That we can appreciate these intersections and contradic-tions in an artistic style period that embraced duality andambiguity seems onlytting. And, despite the nineteenth-century s owering of stylistic diversity fueled by individualexpression, the A CE complex allows us to trace a scarletthread of sorts through the structural and expressive compo-sitional practices of the romantic era.

appendix: some a ce complex pieces

Bach, C.P.E., Piano Sonata in f, H. 173, i, mm. 2662Beethoven, Piano Concerto no. 3 in c, op. 37, iii, mm.

182265 ff.Beethoven, Piano Sonata in f, op. 57 ( Appassionata), i,

developmentBeethoven,In questa tomba oscura, WOO 133Beethoven, Piano Concerto no. 5 in E, op. 73 (Emperor),

i, mm. 13889 ff.Beethoven,FidelioBeethoven, String Quartet in e, op. 59, no. 2, i, mm. 20921Beethoven, String Quartet in E , op. 127, iiBrahms, Piano Quartet in c, op. 60, mm. 7078Brahms, Symphony no. 1 in c, op. 68

Brahms, Concerto for Violin and Cello in a, op. 102, i, mm.23857 ff.

Brahms, Intermezzo, op. 118, no. 4Brahms, Clarinet Sonata, op. 120, no. 1, ii, mm. 4149Chausson, Piano Trio, op. 3, ii, 13948

Chopin, Rondo, op. 1, mm. 54100Chopin, Nouvelle Etude in A ,mm.125Chopin, Polonaise in A ,op.53Chopin, Mazurka in A , op. 59, no. 2, esp. mm. 8588Chopin, Waltz in A ,op.64,no.3Chopin, Piano Concerto no. 2 in f, op. 21, i, mm. 20015Debussy,Soupir from Trois po mes de Mallarm Debussy,Le jet deaufrom Cinq po mes de Baudelaire Elgar, Cello Concerto, op. 85, iv, mm. 197255

Franck, Symphony in d minor, i, developmentHaydn, Sonata H. XVI: 52 in E , i, developmentHaydn, Quartet, op. 76, no. 3 in C (Emperor), iLiszt, Ann es des P lerinage , Premire anne: SuisseLiszt, Orpheus , C. 682,mm. 72130Liszt, Blume und Duft,C. 698Liszt, Eine Faust-Symphonie , C. 697bLiszt, Die Trauer-Gondel I , C. 1279Mahler, Symphony no. 2 in c (Resurrection), i, exposition

Moussorgsky,Picturesat an Exhibition, Limoges,mm.1618Mozart, Cos fan tutte , Act II, FinaleMozart, Symphony No 39 in E , K. 543, iv, mm. 10825Prokoev, Piano Sonata no. 7, op. 83, iiRachmaninov, Piano Concerto no. 2 in c, op. 18, opening of

ii and iiiRimsky-Korsakov,The Golden Cockerel , Act II, openingSchubert, Piano Trio in E , D. 929, i, recapitulationSchubert, Piano Sonata in a, D. 845 (op. 42), i, mm. 180

Schubert, Allegro in a for Four Hands, D. 947 (op. 144,Lebensstrme), expositionSchubert,Wanderer Fantasy, op. 15Schubert, Octet D. 803 (op. 166), vi, mm. 17278Schubert, Symphony no. 4 in c, D. 417 ( Tragic), ii, mm.

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Schubert, Symphony no. 6 in C, D. 589 (Little), iv, mm.292316

Schubert, Symphony no. 9 in C, D. 944 (Great), iSchubert, Antigone und Oedip,D. 542Schubert,Der zurnenden Diana,D. 707

Schubert,F lle der Liebe,D. 854Strauss, Richard, Horn Concerto no. 1 in E , op.11, iiStrauss, Richard, Also Sprach Zarathustra ,op.30Stravinsky, Piano SonataVerdi, Il Trovatore Wagner, Die Feen, O ihr des busens Hochgef hle Wagner, Der iegende Holl nder,Die Frist ist um Wagner, Die Walkre , Magic Sleep music Wagner, Die Meistersinger von N rnberg , Act III, Wahn

monologue and Dream theme Wagner, Siegfried , Act III, Siegfried and Brnnhilde love duet Wagner, Siegfried Idyll Wagner, Parsifal , Prelude to Act I; transition music in Act

III; and nal scene, among others Wolf, Nimmersatte Liebe, no. 9 from Gedichte von Eduard

M rike

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C ontributors

matthew bribitzer-stull is Assistant Professor of Music Theory at the University of Minnesota School of Music. Heis also contributing co-editor of New Millennium Wagner Studies: Essays in Music and Culture.

arnie cox is Associate Professor at the Oberlin CollegeConservatory of Music.

marion a. guck , the Louise Cuyler Collegiate Professor of

Music at the University of Michigan, is Visiting Professor of Music at Columbia University for 200607.

david huron is Professor of Music at the Ohio StateUniversity.

henry klumpenhouwer is the Associate Chair of theMusic Department at the University of Alberta.

judy l ochhead is Professor and Chair of the Departmentof Music at Stony Brook University.

yonatan malin is Assistant Professor of Music at WesleyanUniversity.

robert morris is Professor of Composition at the EastmanSchool of Music, University of Rochester.

ann ommenis currently pursuing a Ph.D. in musicology atthe Ohio State University.

chitravina n. ravikiran is an internationally known mas-ter musician, composer, and pedagogue of South Indianmusic.

david smyth is Professor of Music Theory at LouisianaState University. He is currently working on a book aboutStravinskys sketches.

alastair williams is Reader in Music at Keele University,

UK. He authored Constructing Musicologyand also wrote thechapter on modernism since 1975 for theCambridge History of Twentieth-Century Music.

bribitzer ab c e major third collections in nineteenth-century music

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