Ab-C-E Complex

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'The Ab-C-E Complex: The Origin and Function of Chromatic Major Third Collections in Nineteenth-Century Music

MATTHEW BRIBITZER-STULL

The Ab-C-E major-third constellation stands as a prototype for nineteenth-century composers' expressive and structural uses of chromatic major-third relations. After tracing the origins of the collection, this article presents a conglomeration of hierarchic and transformational analytic approaches to AL-C-E music by central European composers to demonstrate that recognition of the complex comprises a valuable added dimension to our structural and phenomenological hearings of 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.PE. Bach replied, "4nyone can and will assuredly do it who knows that E is the dominant ofa, and that a minor is very closely related to C major."1

A FASCINATION WITH TONAL RELATIONSHIPS based on major thirds has provided the motivation for in- quiries from C. P. E. Bach's day to the present.2 The

inspiration for the investigation herein is no different, but the premise-that a specific complex of sonorities can eluci- date major-third collections in central European music of the nineteenth century--introduces a new angle to this field of study. Succinctly put, this article suggests that the A?-C-E complex constitutes a romantic-era prototype- a benchmark for both structural and expressive trends in nineteenth-century music.3 A topic this rich necessarily in- vites numerous avenues of approach, but in the present context I restrict myself to three: first, how the AV-C-E complex most naturally demonstrates the emergence of major-third collections' expressive and structural functions from 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 might profitably approach examples of Ak-C-E music using a conglomeration of hierarchical and transformational thinking.

Earlier incarnations of this paper were delivered to the Music Theory Society of New York State (Columbia University, 2002) and to the Society for Music Theory (Columbus, 2002). At the MTSNYS meeting this was but one of three papers on the Ab-C-E complex; my discussions with Eric McKee and Charles Youmans, the authors of the other two papers,, were fundamental to shaping my thoughts on this topic. Additionally, I wish to thank the many scholars who shared with me examples of Ab-C-E; Michael Cherlin; David Damschroder; and the anonymous readers of this journal.

I 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, 93-115; Cinnamon 1992, 1-30; Todd 1996, 153-177; and Kopp 2002.

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

167

168 MUSIC THEORY SPECTRUM 28 (2006)

A number of studies in the last three decades address third relations in tonal music. Most either treat generic properties of these relations or focus on a specific work, genre (such as Lieder), or composer. By selecting music featuring Ab, C, and E, I am able to enjoy the benefits of a new vantage point. First, the music engaged by this approach cuts across genres and composers, featuring works composed throughout the "long" nineteenth century. (Many of these are listed in the appendix.) Second, major-third relations in general-and Ab-C-E, specifically-typify chromatic third relations in ways that other collections do not. And third, the consideration of works containing the complete cycle of thirds raises theoretic and analytic issues endemic to music that includes all three sonorities.

A recent study by David Kopp divides the eight possible third relations into three categories: diatonic (sharing two common tones), chromatic (sharing one common tone), and disjunct (sharing no common tones).4 (See Example 1.) Neo-Riemannian transformation labels explicitly show the common-tone relationship between sonorities, as each transformation indicates the motion of one pitch class between two triads.5 When A6, C, and E major or minor triads progress from one to another, they form eight possible root progressions whose tonal functions and directionality may bear extra-musical associations. (The move from I to WVI (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) These

L

(a) diatonic (C to e)

LP

(b) chromatic (C to E)

PLP (or LPL)

(c) disjunct (c to E)

Adapted from Kopp 2002, 10-11, Figs. 1.3-1.5

EXAMPLE I. Diatonic, chromatic, and disjunct major-third progressions.

eight root progressions are summarized in Example 2. Here Roman numerals and Neo-Riemannian operations are wed- ded in an attempt to place the parsimonious voice-leading transformations within a functionally tonal context. Four of these harmonic progressions, labeled with possible harmonic

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

5 For a fully formal exposition of the L and P operations see (among others) Hyer 1995. Despite their strengths, Kopp's M transformations are not used in this context since they, in effect, conflate two voice-leading transformations.

6 Just as individual key centers may have rich, extra-musical associations, so too may harmonic progressions between members of the Ab-C-E collection. The sense of progression from one key to another or of tonal motion between keys was crucial to Joseph Schalk's understanding of

musical association. See Wason 1997, 131. Hatten 1994, 44 goes so far as 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).

THE ORIGIN AND FUNCTION OF CHROMATIC MAJOR THIRD COLLECTIONS IN NINETEENTH-CENTURY MUSIC I69

C-E 2. C-At, I - III# I - kVI5 III - V# VIIP - V

LP PL

3. C - e 4. C-at, I - iii rare III - v

L PLP

5. c - E 6. c-Ab rare i- VI

vi - IV iv - rII

PLP L

7. c-e 8. c-at rare rare

PL LP

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

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 harmonic function. Does t~vi3 have a submediant function, due to its root? A dominant function due to its (respelled) leading tone? Both? Neither?7

Perhaps more than any other development in compositional technique, the increased application of chromatic third relations 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 triads whose roots were a major third apart.8 The reasons for this may include the following phenomena: first, major triads were preferred over minor simply due to the larger repertoire cast in major keys; second, chromatic-third relationships were preferred over diatonic relationships because they evoked a distinct sonic color, and they were preferred over disjunct relationships because they retained a common tone; and three, cycles of major-third-related triads were preferred over cycles of minor-third-related triads because each triad in the former shares one common tone with the others, un- like the complete minor-third cycle, which includes tritone root relationships (like c and f#, or eb and a) that are less directly intelligible.9

STRUCTURAL AND EXPRESSIVE UNDERPINNINGS

Chromatic major-third root relations are intrinsic to nineteenth-century central European music.10 A predilec- tion for these relations (more specifically, those including the complex of At, C, and E sonorities) is most obvious in the music of Beethoven, Schubert, Chopin, Wagner, Brahms, and Liszt."l Because the At-C-E complex was not invented by these composers, but rather emerged from earlier praxis, I begin with a consideration of how the tuning and compositional practices of the classic era contributed to the

7 Swinden 2005 opens his study of plural harmonic function in chromatic music with the "bvi3', chord from Wagner's "Tarnhelm" music. Swinden's article relies heavily on Harrison 1994 (especially 43-72). Both studies present a cogent scale-degree-based theory of harmonic function applicable to much nineteenth-century (and later) music.

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

9 See Krebs 1980, 117-18; Brown, Dempster, and Headlam 1997; and Kopp 2002, 217-18.

io See Kopp 2002, 151 and 213, and Hyer 1995, 130. ii 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 231-33 (Ex. 10); Berlioz's "Au cimitiere," mm. 9-15; and the opening of Act II of Rimsky-Korsakov's The Golden Cockerel.


emergence of Ab-C-E collections. In so doing, I examine both these collections' expressive (or coloristic) origins and their structural origins as notes, chords, and key areas within a tonal context.

Expressive origins and functions of the Ab-C-E complex. I first consider a suggestive idiosyncrasy of the eighteenth century-namely, its relative lack of works cast in the so- called "enharmonic" keys (B/CL, F#/G6, and C#/Db). The key choices of Haydn, Mozart, and Beethoven are represen- tative of the time; complete works in keys with five or more flats or sharps in the key signature are rare in both Haydn and Beethoven, and missing altogether in the music of Mozart.'2 (See Example 3.) The few exceptions that prove the rule fall into three categories: works whose overall tonic key includes five or more accidentals; interior movements; and extended sections within a single movement. These are illustrated in Examples 3(a), (b), and (c) respectively.13 Though slightly more common than their parallel major keys, minor-mode works in c#, f# and b are also rarer in classic-era music than those in f, c, and g minor, their counterparts on the flat side of the circle of fifths. Thus, it appears that it was not the diatonic collections alone that composers avoided, but rather that the sense of tonic, regardless of mode, influenced their key choices.

Ultimately, the rationale for eighteenth-century perform- ers' key preferences can be attributed to two related phenomena: C-centricity and temperament. In the classic era, the key of C major ranked as the most common; it was the key of the neophyte and of the amateur-the people's key. As Donald 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-acoustical frame of eighteenth-century musicians exhibited a clear preference for C major as the "default" tonality.15 This conception of C remained at least until Kurth's day, when the theorist 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 and point of departure of harmonic development in sharp and flat keys; the church modes already revolve around this center [sic]. Further, though -and this is by far more significant than the historical development- C major signifies again and again the origin and central starting point of musical sensibility for individual development, starting from the beginnings 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 distinguishes itself essentially from C major. The whole absolute character of a key, reflecting back to C major, is thus not given in the nature of music but rather in the particular course of [music] history and pedagogy.16

Though Kurth located the center of the church modes on C rather than D Dorian, his prose reflects a strong tradition in Western music theory--conflating a sense of key with a sense of location. Words like "middle" and "homeground" indicate tonality's spatial connotation. Thus, the distance one ventured from C could be measured metaphorically as the distance traveled from the commonplace toward the esoteric, a metaphor of alienation predicated upon keyboard intonation.17 The increasing intonational difficulties as one moved away from C were, in turn, a function of non-equal temperament.

While close approximations of equal temperament in Western Europe were used for fretted instruments as early as the sixteenth-century, true equal temperament on keyboard 12 C. P. E. Bach, for instance, also rarely ventured beyond key signatures

with four flats or sharps. See MacDonald 1988, 222. 13 Collections of pieces in all twenty-four major and minor keys, like

Bach's Well-Tempered Clavier and Chopin's preludes, are not cited here. Even in these contexts, however, composers seemed to favor certain enharmonic keys over others (like F# over G ). See MacDonald 1988, 222.

14 Tovey 1944, 61. 15 See Steblin 1981, 103-51, especially 105-6, 113-14, 117, 125, and 128. 16 Kurth 1923, 298, n. 1 (translated in Rothfarb 1991, 126, n. 18). 17 See the comments of Bruckner's disciple, Joseph Schalk, in Wason

1997, 130-31.

THE ORIGIN AND FUNCTION OF CHROMATIC MAJOR THIRD COLLECTIONS IN NINETEENTH-CENTURY MUSIC 171

Haydn Baryton Trio Hob. V: 5 B Haydn Minuet Hob. IX: 26 F# attributed Haydn Trio Sonata Hob. XV: 31 e6 "Jacob's Dream!" Haydn Divertimento Hob. XVI: 2c B Haydn Symphony Hob. 46 B

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

*0 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 D6 Haydn String Quartet op. 76 #5, Largo F# Haydn String Quartet op. 76 #6, Fantasia B Beethoven Sonata op. 26, "marcia funebre" ab Beethoven Sonata op. 27/2, Allegretto D6 Beethoven Sonata op. 57, Andante con moto D6 Beethoven Sonata op. 110, Arioso dolente ab Beethoven String Quartet op. 130, Presto bb Beethoven String Quartet op. 130, Andante con moto ma non troppo D6 Beethoven String Quartet op. 131, Adagio quasi um poco andante g# Beethoven String Quartet op. 135, Lento assai, cantante e tranquillo D6

(b) interior movements (sample listing)

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

(c) extended sections within movements (sample listing)

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


instruments was not universally accepted until 1917.18 Theorists and other musicians up through the nineteenth century espoused the virtues of equal temperament even though keyboard instruments of that century were almost universally tuned according to the principles of well- temperament, a tuning philosophy that made useable all the major and minor triads without sacrificing the "characters of the keys"-a set of extra-musical associations that arose, in part, from the meantone temperaments previously in use.19 By 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 producing the different qualities of meantone thirds that had, in turn, contributed to the establishment of the characters of the keys.20

Of the three contiguous major thirds within a given octave, only two (e.g., C-E and E-G# but not A6 (G#)-C) were intonationally suitable in meantone systems, thus leav- ing four major thirds as noticeably out-of-tune.21 While all twelve 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 C-E major third (as part of the common C major tonic) was to be among the most in tune of meantone thirds, then the smallest major thirds (i.e., most in tune) almost always included F-A, C-E, and/or G-B. The thirds belonging to major triads opposite these on the circle of fifths tended to be the largest and, consequently,

18 Jorgensen 1991, 4-7 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 other comments on the relationship between well-temperament, meantone tuning, and the characters of the keys.

20 Jorgensen 1991, 2. Though these intonational problems were a function of keyboard instruments, music for other instrumental forces written during this time period reflected the strong influence of keyboard thinking, probably because so many musicians used the keyboard during the act of composing.

21 Jorgensen 1991, 47 and 774.

the most out-of-tune.22 (See Example 4.) Hence, the sharp- side boundary interval of usable major thirds tended to be E-G#, the flat-side third, A6/(G#)-C.23 The three major thirds that lay outside these boundaries (B-D#, F#-A #, and Db-F) belonged to the tonic triads of the underused major keys.24

While there were more tonally-distant keys than Ab and E (speaking in terms of C-centricity), these two keys often marked the outer limits of acceptable intonation on unequally- tempered instruments-a boundary that has persisted into modern-day notation, as E and Ab still mark the edge of the enharmonic keys (D6/C#, G6/F#, and Ck/B). Like the dragon-infested waters that signaled the edge of terra incog- nito on the maps of early explorers, one can almost imagine the eighteenth-century circle of fifths breaking at this point. Venturing into this "musical beyond" during the age of ratio- nalism and enlightenment was rarely done, and then only

22 This is but one of countless meantone schemata. Tuning during these centuries 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 comprised more of a continuum than a hard-and-fast rule.

23 Notable exceptions did occur. For two, see Jorgensen 1991, Fig. 15-2, pg. 47; and Fig. 39-1, pg. 138.

24 Both tuning and compositional practice in the latter part of the eighteenth century reinforced the sense of the major key (and its tonic triad's 1-3 major third) as normative; minor keys were "marked" in the semiotic sense, shadowy reflections of their major-mode counterparts. (See Hatton 1994, 34-38, and Wheelock 1993, who uses the term "en-othered".) Since this markedness bore a reflexive relationship to the minor mode's greater degrees of chromaticism and tonal adventurousness, it is less easy to generalize about the intonational acceptability of minor keys simply 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 (b2, ti, ti, and 47f)- 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 fewer keys than their major-mode counterparts: b, f#, c#, g#/a6, d#/e6, and bb were all extremely rare in the classic era.


The largest thirds

4 enharmonically equivalent

,• .OS

Pythagorean thirds

/ C'E F"A" CbE& DbF GBb BD\

.*a the same size

S AC ---- the same size - EG' 4 flats and 4 sharps ~L a- a EbG equal temperament thirds A

. E -G

<-. - thesamesize -> AC' 0

3 flats and 3 sharps 4

B"D ------ the same size - >- DF' S . 2 flats and 2 sharps

•&,• the same size /

TheA CE GB thirds The smallest thirds

EXAMPLE 4. Major thirds in meantone temperaments. From Jorgenson 1991, 180, Fig. 51-1: Well-Tempered Tuning-- Vallotti's Theoretically-Correct Method.

with good reason. Hence, the edges of the known tonal world--Ab and E-could function as marked keys, destina- tion points that were far removed from the harmless tonal clarity of C. As such, these keys were often invested with rich associations and served as tonal settings for composers' most profound musical utterances-a habit that persisted even during the theoretical hegemony of equal temperament.

This leads us to a consideration of these keys as associative entities, markers of extra-musical significance. While "associative tonality," as it is referred to today, was most famously explored by Wagner in his Ring cycle, key symbolism, stemming from the aforementioned "characters of the keys," had a rich history long before the Wagnerian music

drama.25 Wagner's key associations were most often piece specific, but the associations comprising the characters of the keys infused all manner of works from the second half of the common practice era. C, lacking the artifice of black keys, was often used to represent light, truth, purity, and the common folk. D was the key of choice for triumphant and mili- tary music; Eb, for the heroic; F for the pastoral, and so forth.26 In addition to meantone intonation-largely a keyboard phenomenon-instrumental associations (e.g., trum- pets with D major, horns with Eb major, English horn with F major), tessitura, written notation, absolute pitch level, and prior compositional practice all added to the summary character of each key, even in compositions without a keyboard part.27

Because E and Ab were the most distant keys from C in common usage, their associations were among the most powerful. While these associations have never been fixed as to exact meaning, nor applicable to every work, there exists evidence of general expressive trends: Ab is linked to slumber, darkness, and death while E major is associated with transcendence, spirituality, and the sublime.28 Thus, we

25 For discussions of "associative tonality" see Bailey 1977, 48-61, and 1985, 113-46; McCreless 1982, 88-95, and 1983, 60-62; and Stein 1985, 43-44, and 141-87.

26 Schalk understood each key to have essential differences from the others; that is, he believed that music should not be treated as simply a transposable pattern (a misconception he laid at the feet of those who espoused equal temperament). See Wason 1997, 132-33.

27 Detailed descriptions of these key associations appear in tables com- piled by numerous eighteenth-century theorists. Since some of these tables 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 and theory on this issue.

28 E may have developed these associations since it is the dominant of vi in C major. The motion from I to vi as a spiritual symbol is discussed in McKee 2001. One might also conjecture that the upward arpeggiation of I-III#-V vs. the downward arpeggiation of I-WI-IV accounts for

Ry


might conceive of the eighteenth-century's E and Ab as positive and negative tonal-dramatic poles about a central C. The developing usage of E as the erotic key in the nineteenth century enriched this opposition between E and Ab by setting up an Eros-Thanatos antithesis.29

The Act II finale of Mozart's Coslfan tutte illustrates the eighteenth-century prototype of Ab and E serving as expressive boundaries about a central C. This finale, like most of Mozart's, is a conglomeration of independent tonal struc- tures, although C major is understood as the large-scale referential tonic. The opening C-major number is followed by a chorus in EL. The ensuing AL quartet features the main characters dwelling upon the virtues of wine for drowning sorrows in slumber or, in Guglielmo's case, death, should the wine be poisoned. By means of a chromatic 5-6 shift in mm. 199-200 there is a quick segue into the next scene, an active E-major ensemble piece in which Despina, disguised as a notary, reads the marriage contract. (See Example 5(a).) This moment is the action the four main characters have both feared and hoped for all along, a dramatic counterpoint to the preceding, reflective A6 reverie. Using a fascinating tonal gambit, Mozart then proceeds to make his way back to C major (at which point the truth is revealed and there is much rejoicing) via numbers cast in closely-related keys on both the flat and sharp sides of the circle of fifths. A summary of this tonal motion appears in Example 5(b). Before the final tonal-dramatic resolution can occur, Mozart illustrates, step- by-step, how far the tangled plot has come from the simple clarity of C major.30

Throughout the later common-practice period, Ab and E persisted as expressive tonal locales; increasingly, composers invoked their expressivity without reference to specific extra- musical associations.31 The same held true of the juxtaposi- tion of Ab, C, and E sonorities-sonorities whose harmonies had a profound impact on tonal structure.

Structural origins andfunctions of the Ab-C-E complex. When eighteenth-century composers featured two (or all three) of the members of the Ab-C-E complex in their works, these sonorities were usually related indirectly. In the excerpt from Cosi just examined, for instance, Ab and E as key areas are related only indirectly to the overarching tonic C via fifth cycles and to one another through the central C (as shown in Example 5(b)). However, the tenuous foreground link between the Ab and E triads provided by the 5-6 shift (mm. 199-200), produces the sound of a direct chromatic third relationship, a forerunner of the increasingly important role such relationships would play in romantic-era compositions.

The earliest strategies nineteenth-century composers used for incorporating direct chromatic third relationships into their music usually followed earlier diatonic models, providing coloristic alterations of them more than substan- tive changes to their structural functions.32 Thus, common surface- and middleground arpeggiation paradigms such as I-I6 (or iii)-V and I-vi (IV6)-IV (ii6) evolved into I-III#-V and I-LVI-iv6 (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. flat symbolism. Schalk noted that sharps "press upwards, to- wards light," while flats "strive toward the depths, into darkness." See Wason 1997, 130. Apparently, Riemann concurred: see Wason and Marvin 1992, 93, as well as the synopsis in Hatten 1994, 43.

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

30 Steptoe 1988,232-42 suggests that flat keys in Cosi represent 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 twilight realm 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 specificity of emotion in expressive music, see Kivy 1980, 46-49.

32 See Somer 1995, 219-27. 33 In Schenker's theory, ascending arpeggiations from tonic are also possi-

ble on the first order middleground, while descending arpeggiations operate on more surface levels. See Schenker 1979, Figs. 7b, 14/la-b,

Ry

Ry


5 66 3 63

Ab: I \VI E: 16 "7 V7 I

(a) modulation by chromatic 5-6 shift, mm. 199-203

Sharp Keys

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

developmental

E? EA ? A D E--------- (I-V) G ? (F---d/F)

Measures are numbered from the Flat Keys beginning of the finale.

(b) overall tonalplan

EXAMPLE 5. Act IIfinale of Mozart's Cosl fan tutte.

clude III# and 6VIb5.34 Such examples support the claim often made to undergraduates that modal mixture is essentially 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 or contrapuntal principles.

15/2b, 98/3a, 100/5, 108, 112, and 113/2 for examples. See also Beach 1997, and Kopp 2002, 109-12.

34 Krebs 1980, discusses oscillatory third progressions and circles of thirds involving tonic harmonies (94-121) and describes the same techniques prolonging non-tonic harmonies (84-94). Kielian-Gilbert 1990, 50- 52, uses terminology drawn from the definition of the transposition

operation described in Proctor 1978, 181-200, and describes these tonal itineraries in terms of their bass motion. Bass lines that articulate a series of the same interval (e.g., major thirds) may be directional (e.g., moving from C to Ab via E), circular (e.g., starting and ending on C with Ab and E by equal division of the octave), or axial (e.g., starting and ending on C with Ab and E providing upper and lower mediants).


Beethoven's "In questo tomba oscura" illustrates how this chromatic alteration functions. Example 6(a) shows that the Ab tonic is prolonged by a chain of descending major-third root progressions.35 Here we notice that the song's opening Ab Stufe is followed in m. 14 by a chromatic 5-6 shift to E, which in turn leads to a cadence on a C major triad in bar 19. This III Stufe is also labeled by its local function (V of vi) on the graph for two reasons. First, this chord makes reference to the diatonic vi Stufe that is replaced with the chromatic WVI, Fb, 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 relationship described by C. P. E. Bach at the opening of this article; it illustrates a common, indirect, and diatonic context for relating two of the three keys in the Ab-C-E complex. Rather than arising out of a direct chromatic relationship to Ab (as its III for instance), C's relationship to Ab can be heard indirectly, as the dominant of Ab's most closely-related key. Thus, both chromatic Stufen can be restored to a diatonic prototype without radically altering the middleground. This is shown in Example 6(b).

The ease with which the chromatic replaces the diatonic in such examples is perhaps predicated on the appearance of only two members of the A6-C-E collection. That is, one direct major-third relation is usually easy to accommodate within a tonal context that is still clearly controlled by a background tonic-dominant hegemony.36 The appearance of

0 3

4-6- 65 4

b"VI" V/vi IV Ab: I ,, V I

(a) graphic analysis

b dt oiro [IV6 Ab: I

(b) diatonicprototype

EXAMPLE 6. Beethoven, "In questa tomba oscura."

all three members of the collection does not necessitate the erosion of familiar structural functions, but if the three are directly related on the same level of tonal structure anything from the surface-level triad to the background Bassbrechung itself may be disrupted. The C-E-G# augmented triad, for instance, often arises because one of the three tones effects a

35 Proctor 1978, 178-79 analyzes the opening of this Lied as a bass arpeggiation of the augmented triad. The deep middleground here would look quite different if the Db and Eb quarter notes in m. 21 were taken to be bona fide harmonies, certainly a viable reading. Note that Schenkerian graphs throughout the present paper are middleground- oriented and thus lack foreground detail. Accidentals apply only to their 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 in

I-III#-V or I-bVIb5-V) or embellish V (V-III#-V or V-1VIs5-V). Numerous examples are cited in pages 24-59 and 73-84.


chromatic passing or neighboring motion. Even when all three are chord tones within an augmented dominant triad, the C-E-G# sonority may remain unquestionably dominant in function. But, when the augmented triad is not anchored by a diatonic Stufe its symmetry can threaten tonality altogether (as in Liszt's "Nuages gris"). Likewise, an E-major (III#) structural third divider between I and V on the middleground of a C major work (in the first movement of Beethoven's "Waldstein" Sonata, for instance) would hardly compromise the sense of tonal unity. An extension of this technique might feature a nested chromatic mediant relationship (III of III#) to invoke the third member of the complex (as in Wagner's Siegfried Idyll).37 When such chromatic third chains achieve independence from tonic and dominant, however, they may replace tonic prolongation with other structural functions as in the symmetrical division of the octave evinced by the Schlaffenakkorde of Wagner's Ring.38

Since the use of multiple major-third relationships requires great care to avoid disrupting the sense of tonality, it is unsurprising that nineteenth-century composers tended to rely on just a few strategies. In short, they anchored these third relations on tonic or dominant Stufen, thus prejudicing the tonal contexts in which an Ab-C-E collection could occur.39 Naturally, Ak, C, and E major were among the most common tonics for the incorporation of the complex. In

these keys, chromatic thirds could be strung from (or to) the tonic. This happens in Chopin's Polonaise, op. 53, in which the tonic Ab is prolonged first by its upper third, C (III), an arpeggiation both in m. 49 and again in m. 58 (functioning locally as V/vi), and later by its lower third, F6 (spelled as E), beginning in m. 81. (See Example 7.) This music comprises a concatenation of two separate oscillating progressions on different levels of structure-Ab-C-Ak and Ab-E-A .40 C major, a local expansion of A6, exists at a more foreground level than E, the tonic of the work's entire middle section. Despite this, C is emphasized both as the most obvious tonal departure from the tonic Ab within the first section and by virtue of its recurrence at the end of the retransition back into Ab (mm. 145-51) and in the final cadence (mm. 179-80).

When anchored by the dominant, two less obvious tonic contexts-a minor and f minor-predominate. In each key, one of the three members of the complex can function as III (the relative major) and another as V# (the functional dominant). The third member is often used to connect the two.41 The scherzo movement of Schubert's Sonata in a minor, op. 42, provides an illustration. (See Example 8.) Here, the expected modulation to the mediant during the first reprise of

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

Kurt. See Kurth 1923, 226-27 (translated in Rothfarb 1991, 133-34). 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, 111-16.

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. 1.46, v.2, 26) who illustrates this technique in Beethoven's piano concerto in E6 major ("Emperor"), iii, mm. 138-89 as a VI-IV-bIbs chain embellishing the motion from VI to V. See also, Beethoven's piano concerto in c minor, iii, mm. 138-220 (a VIP-IV-II succession

that leads eventually to V) and Schubert's Eb String Trio, i, mm. 434-52 (whose recapitulation features a biib-VIP-IV predominant chain).

40 Direct chains of thirds appear in the literature as well. See Schubert's Lied, "Fiille der Liebe" and the analysis in Krebs 1980, 110 (Fig. 11.37, v. 2, 49). Example 7 presents only the opening of the Ab-E-A6 progression. Interested readers may wish to consult Krebs for a graph of mm. 80 to the end. See Krebs 1980, Fig. 11.9 (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 Ab-C-E complex, perhaps because of its own relative scarcity in common-practice music. Note that the six keys mentioned (A6, c, E, a, f, and c#) together comprise Weitzmann's Region I, a grouping noted in Cohn 2000, 93, and further explored throughout his article.

Ry

Ry


A: I III ii V I

EXAMPLE 7. Analysis of the opening of Chopin's Polonaise, op. 53.

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 Ab in measure 43. Although this Ab is preceded by its own dominant and is followed by harmonies that pre- pare the arrival of the e-minor dominant (made major in bar 80 to set up the return of the opening material in a minor), these intervening sonorities do not prevent us from hearing a key succession of C-Ab-e /E. That is, Ab connects C and E by a descending major-third arpeggiation. Interestingly, surface-level references to the combination of A6, C, and E are also audible in the opening a-minor measures. Here E functions as a local dominant (passim) and Ab appears during the modulation to C major (m. 17).42

0 @@@@@@

a: i III V i ? 0

EXAMPLE 8. Analysis of the scherzo from Schubert's sonata, op. 42, iii.

ANALYTICAL ILLUSTRATIONS OF THE A--C-E COMPLEX

As chromatic-third usage evolved, nineteenth-century theory naturally developed alongside composition. Whether reactive or innovative, much of this work focused on A1- C-E 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- defined tonality specifically to model the A?-C-E collection.44 And Carl Friedrich Weitzmann both distinguished himself from his contemporaries and influenced Franz Liszt by his thorough treatment of the Ak-C-E augmented triad.45

42 Similar AL-C-E collections in a- and f-minor music occur in Brahms's Intermezzo op. 118, no. 4 (see Example 10); the prolongation of V in mm. 26-62 of the first movement of C. P. E Bach's Piano Sonata in f minor, H. 173; and the "dreamlike" Ab that intercedes between a back- related dominant, E, and motion to the mediant, C, in mm. 81-152 of Schubert's Allegro for piano, four hands, op. 144 ("Lebensstiirme"). Schmalfeldt 2002 describes some intriguing formal implications of the parenthetical Ab in the Schubert Allegro.

43 Riemann [1893], 165, stated that the third of a triad (Klang) can take on an independent significance just as the fifth of the tonic triad does. He even adapted a separate function symbol for chromatic mediants in the last edition of the Handbuch der Harmonielebre (1920) published during his lifetime. See Kopp 2002, 94. Other Al-C-E examples occur throughout Riemann's writings on third relations and tonality. See, for instance, Riemann 1882, 189, 1890, 38, and 1902-03, 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


Riemann and Weitzmann were accompanied by a host of others who used A6, C, and E as a prototypical collection to direct or explain the harmonic advances of the nineteenth century and the changing nature of tonality.46 Paraphrasing Kurth, one might go so far as to say that the symmetry of the Ab-C-E collection paved the way for the eventual dissolu- tion of functional tonality itself.47

This may be why chromatic-third relations continue to affect our modern-day conception of tonality.48 While relations by perfect fifths fit our existing theoretic and analytic approaches with few problems, chromatic thirds are another story. Some scholars have proposed that these chromatic relationships constitute another form of tonality, a sort of seconda prattica, distinct from the diatonic practice of the eighteenth-century.49 Others argue that chromatic-third relations, 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 been stretched to the breaking point by the predominance of chromatic-third relations (Liszt's Die Trauer-Gondel I, for one), determining which flavor of tonality governs a given work is perhaps less important than recognizing and articu- lating the ramifications created by the addition of chromatic thirds to a largely fifth-governed tradition.

As an example, consider the first movement of Beetho- ven's "Appassionata" sonata. Schenker's graph of the development section in Derfreie Satz indicates that, on the deep middleground, 3 of the Urlinie falls to 2 as an Ak (III) Stufe moves to a C Stufe (V) in f minor.51 Example 9 presents a slightly more extended middleground analysis, beginning at the end of the exposition and continuing through to the end of the development. The Schenkerian prolongation of an Ak Stufe supporting 3 is accompanied by the bubbles marking the appearance of A6, C, and E sonorities, and also by neo- Riemannian transformational symbols that illustrate how the Ab Stufe is prolonged by a series of P and L motions. Note that five of the six triads in Cohn's northern hexatonic collection are traversed.52 In chromatic-third chains of major triads, 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 the intervening minor triads are articulated. In such examples, like the "Appassionata" development, two common tones are retained by adjacent triads as the harmonic progression circles the northern hexatonic pole in a series of LP (or PL) cycles.

key of C major. See Weitzmann 1853 and the commentary in Todd 1996, 158-59.

46 These include Dehn 1840, 157; Kurth 1913, 124-28; Lobe 1861, 80; Rimsky-Korsakov 1895, 98, 102-103; Schwartz 1982, 70, n. 5, and 386-7; Weber 1846, 503; and Ziehn 1887, 8 and 119.

47 Taruskin 1985, 135-36, reproduces Rirnsky-Korsakov's "false progressions" by thirds from his harmony text, two of which feature major and minor triads built on AK, C, and E; McCreless 1983, 70-71, summa- rizes Kurth's belief that symmetrical, chromatic sequences were crucial forces in the destruction of tonality.

48 The prodigious body of scholarly literature on this topic aside, current music theory text books for undergraduates continue to present examples of Ab-C-E in "back-of-the-book" topics like augmented triads and enharmonic modulation. See, for instance, Laitz 2003, 645-46; Ottman 2000, 229; Roig-Francoli 2003, 830-31; and Kostka and Payne 1984, 383.

49 After Proctor 1978, this philosophy gained ground. Proponents include many authors in Kinderman and Krebs 1998.

50 These scholars support the applicability of Schenker's theory for nineteenth-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.

51 Schenker's analysis begins with the ab-minor sonority in m. 65. See Schenker 1979, Fig. 114.8.

52 See Cohn 1996, 17, Fig. 1. In Cohn's figure the Ab-C-E collection is given preferential placement at the north, the direction most commonly indicated on maps. It can be inferred from his remarks that Cohn made this choice consciously, due to the conventional primacy of C. Op. cit., 38, n. 34.

Ry

Ry

Ry

I80 MUSIC THEORY SPECTRUM 28 (2006)

( @ @@@@@@@@ o_?M??????' ?

f: i III VL Ab~'l?"H E He H ~ 4

EXAMPLE 9. Graphic analysis ofBeethoven's Sonata, op. 57 (W4ppassionata") with neo-Riemannian analysis.

While the Schenkerian graph in Example 9 shows the parsimonious voice leading of Beethoven's development section, the neo-Riemannian analysis, undergirded by an understanding of AK, C, and E triads as a group structure, highlights the skipped member of the northern hexatonic collection: C major. Its absence is audible because it disrupts the previous voice-leading transformation stream. One explanation for this omission is that C Major (as V) is required shortly at the retransition.53 But it may also suggest why the bass C remains active at the opening of the recapitulation, creating the sound of a tonic 6 underneath the return of the opening material.54

Just as a neo-Riemannian analytic vantage point may in- form a Schenkerian reading, as in the "Appassionata" analysis above, the converse is also true. The indeterminacy of directionality implied by "polar progressions"-motion across a hexatonic pole (PLP or LPL)-can be clarified by the prolongational context.55 While the distinction may seem academic, the two different labels-LPL and PLP-suggest two different hearings that imply a differentiation between clockwise and counterclockwise motion about a hexatonic pole, or-in linear rather than cyclic space-ascending and descending harmonic root motion. When considered as "up" vs. "down," the directionality of such harmonic progressions can play an integral role in a work's dramatic effect.

53 C refuses to relinquish its role in the development as dividing dominant. Rather, an unstable neighboring 6 elaborates the V Stufe when the primary 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 explanation for Beethoven's disruption of the paradigmatic sonata form retransitional tonal structure.

54 A motivic rationale for the idiosyncratic recapitulation that cites the transferal of the DB-C neighbor to the bass is also viable. See Smith 1995, 268-70, for an unconventional reading of this movement's tonal

structure that highlights these falling bass half-steps and suggests that the "apparent tonic" recapitulation grows out of a dominant prolongation at the opening.

55 Cohn's reading presupposes a lack of directionality. In examples from the literature, he cites direct motion between the hexatonic poles- motion that lacks common tones-rather than an incremental shift from one pole to the other is responsible for this progression's uncanny effect. See Cohn 2004 and Cohn 1996, 21-22.


This is the case in the middle section of Brahms's Intermezzo in f minor, op. 118, no. 4, in which a descending chain of major thirds prolongs the dominant. (See Example 10.) At first glance, it may seem immaterial whether the motion from Ab major to e minor in mm. 52 to 75 is marked PLP or LPL. But directionality is an important aspect of this work, whose tonal and dramatic structure is predicated upon contrast between the middle section and the sections that frame it. The energy of the opening section drives con- stantly toward the dominant. On the foreground and middleground levels, weak (or implied) tonics resolve to clear, emphasized dominants-comprising, in effect, a half cadence writ large. In the middle section, the illusion of motion within the dominant rather than motion to the dominant is of primary importance. The relaxed texture and feeling of descent created by the chain of thirds (PLP, or counter-clockwise about the northern hexatonic pole) and the upper-voice arpeggiation of C major contrast sharply with the opening section's ascent to the dominant. The opening of the middle section (mm. 52-67) firmly establishes the Ab Stufe, opening up another potential reading in which this III is part of a middleground I-III-V arpeggiation (relegat- ing the V that closes the opening section to a back-relating dominant). This reading, however, contradicts the strength of the many motions to V in first section and the sense of downward motion throughout the middle section. Thus, not only does the sense of counter-clockwise directionality in the Ab-e-C succession suggest one middleground reading over another, it also informs our choice of a PLP transformational model over an LPL model. This reading is also supported by foreground details (the cadence in ab in m. 67 and the prepa- ration of E in mm. 68-74) that point toward a tonal motion from Ab to ab to E, and then to the cadence in e (m. 75) that ultimately leads back to C.

The preceding analysis juxtaposes Schenkerian graphs and neo-Riemannian transformational symbols. At times the two analytic approaches complement one another, as in Example 10, in which the Schenkerian prolongation of C

Sv)

PL PLP L 0 ?

EXAMPLE IO. Analysis of mm. 52-83 of Brahms's Intermezzo, op. 118, no. 4.

(V) matches up with a counter-clockwise spin about the northern hexatonic pole. But, more often, this analytic juxta- position generates unavoidable conceptual frictions. These are patent in an excerpt from Wagner's Der Fliegende Holliinder.56

The progression in Example 11(a) occurs at the end of the Dutchman's Act I recitative and aria.57 The scale degrees above the score and the Roman numerals below it represent a Schenkerian hearing of a localized auxiliary cadence in which the Ab-major chord harmonizes an upper-voice passing tone.58 This reading presumes that Ab plays a more foreground melodic role than c and E. At odds with this reading are the neo-Riemannian symbols below the score. These imply that the structural dominant seventh chord on G is of

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

57 Measure numbers from Wagner's operas refer to the widely available Schirmer piano-vocal scores and are cited in the format: page/system/ measure.

58 For an overview of the Schenkerian auxiliary cadence, see Burstein 2005.


3 (PT) 2 1

auf! all!

auf! all!

Str. p espr. PP

'Cello

K.-dr c:

tII i LP L 9 @ ?

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

Section Recit. A B C Coda Measures 29/1/1 32/1/1 35/5/5 38/1/1 41/3/5 Key c ab c C-E-c Synopsis Dutchman Deathless Beseeches Longs for Crew welcomes

makes land wandering an angel the Day of death Judgment

(b) formal overview

EXAMPLE II. Dutchman's Act I recitative and aria.


less interest than the succession of triads: E, Ab, and c.59 In this second reading, the linear, root-motion cycle of ascending major thirds takes precedence over the hierarchical, Schenkerian reading. Since neo-Riemannian transformations are capable of incorporating the G dominant into the harmonic event stream, why omit it from the analysis? An investigation of the preceding music provides the answer. As Example 11(b) shows, the Dutchman's aria proper divides into three sections, cast in c, ab (with shifts to Ab major), and c respectively. The coda to the Dutchman's aria picks up the C major Picardy Third at the end of the third section and moves toward E for the ghostly crew's confirmation of the Dutchman's longing for death. E then passes through Ab to c, bringing an otherworldly quality to the number's close.60

In effect, then, we have two conflicting readings of this cadence. One favors a tonally-hierarchical view that models prolongation, while the other models a transformational event-stream.61 While Schenkerian analysis effectively represents tonal-prolongational structure, this structure is just one facet of musical construction and of musical experience. Associativity, referentiality, and salience are also important: even when Ab, C, and E are not adjacent on the same tonal level they are often marked by tonal, formal, rhetorical, referential, or associative processes, as in the Dutchman example, above. Stufen, significant cadential tonal centers, unexpected

or parenthetical tonal shifts, irregular formal units, and extra-musical connections can all draw the listener to a phenomenological awareness of AK, C, and E connections.62

Placing neo-Riemannian analytic symbols below a Schenkerian analysis shows where Ab, C, and E sonorities occur and suggests an abstract voice-leading connection between them.63 But the implications go far beyond merely la- beling an event stream. They point to a group structure-a connection between A1, C, and E-that is non-hierarchical in nature, though the members of the group may exist within a tonal hierarchy. While some analysts maintain these as dis- crete forms of tonality, this study suggests the possibility for the interpenetration of these two spaces-that the second practice of nineteenth-century tonality can exist within a diatonic background structure.

Brahms Piano Quartet in c minor, op. 60. My final analysis draws upon the E-major slow movement of Brahms's Piano Quartet to suggest that the relevance of the Ak-C-E complex in tonal contexts may range from the structural foreground to intra-movement connections, even within the same work. The G#-E-C major-third chain of the opening cello solo's first 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 abrogate a neo-Riemannian group structure. See Santa 2003, whose hybrid nonatonic/hexatonic group involves the dominants of the three tonics in question, in effect turning the model advocated in Krebs 1980 upside down.

60 There are also references to Ab-C-E in the recitative before the aria proper. See the foreground viio7 of c (29/4/2), viio7 of e (30/1/2), and resolution to Ab (30/2/1) in the recitative.

61 Samarotto 2003 sets forth an introductory model of interaction between tonal coherence (Schenkerian analysis) and transformational event streams (neo-Riemannian analysis).

62 Schachter 1987, 304-08, discusses the distinction (not made in Schenker's Free Composition) between "structural" keys and more foreground keys, all of which form part of the listener's "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 Ab-C-E complex as Cohn's northern hexatonic pole), as well as tonal and linear aspects of various works. See Cohn 1996, 33, in which he suggests the use of hexatonic thinking in conjunction with standard diatonic (sic) models such as Auskomponierung. See also Lewin 1986, 362 ff for examples illustrating the possibilities multiple perceptions have for multiple analytic approaches to the same passage, neither of which is "better" or "more valuable" than the others.


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

Later in the work, for instance, the retransition into the return of the opening material expands the G#-E-C major- third collection. (See Example 12.65) Two aspects of this passage are noteworthy. First, the descending augmented triad motive is itself stated three times on the musical surface, beginning on E, then C, and finally G#. The third iter- ation marks the return to the opening material played, this time, by the piano. Second, there is a discrepancy between the upper and lower voices. The structural upper line traces the chain of descending major thirds, E-C-G#, with the composing-out of the first third, E-C, transposed exactly to compose out C-G#. Thus, the upper line seems to model Proctor's "transposition operation" and suggests that Brahms has entered a fully-chromatic, equal-tempered tonal space.66 But the melodic pattern is not mirrored by the harmony. The g# marked in the neo-Riemannian analysis beneath the score is illusory, an implication that is never realized. For while the V? of c in m. 72 resolves as expected, the V# of g# in m. 77 does not. The deceptive bass motion to E coincides with the return of the opening material in a bait-and-switch tactic that reveals the passage to be a prolongation of the E Stufe by its lower third, C.67 While G# does not play a vital structural role in this prolongation, however, the neo-Riemannian analysis below the score points to its referential role in evok- ing the motivic descending third chain, a reference that enriches 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 we noted earlier, E major was often reserved for composers' most expressive music, a tonal marker for the spiritual and sublime. Given Brahms's allusions to remembrances of his feelings for Clara Schumann in this quartet, the expressive connotation is appropriate.68 But the tonal relationship between the E major Andante and the c minor of the other movements is odd. This unusual key relationship might be considered an isolated idiosyncrasy were it not for the number of other works in which it occurs. Despite the rarity of this third progression within a single Ursatz (i to # III#), the motion from tonic minor to the raised major mediant appears with surprising frequency between the movements of multi-movement works cast in c minor, and, as such, de- serves consideration as a further ramification of the Ab-C-E complex.69

Two things are immediately striking about this inter- movement tonal relationship. The first is the tonal contrast -the key of E represents a luminous and ethereal refuge from the surrounding c minor.70 But this tonal contrast seems to be predicated on the associativity of E, rather than absolute tonal distance, as E is not the most tonally distant major key from c minor; it lies five steps away on the circle of fifths, while A major lies six steps away-directly opposite c

64 Peter Smith argues that the G#-E-C augmented triad achieves a motivic and expressive importance that transcends its role in the structural hierarchy. See Smith 1994, 258-60, and 2005, 17-18.

65 Interested readers may wish to compare this graph with that in Smith 2005, 102.

66 Proctor 1978, 181-200. 67 This passage bears a striking resemblance to a similar retransitional

third chain with deceptive motion in Liszt's Orpheus, mm. 114-30.

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, 14-15, who notes that non-monotonal works often feature tonic keys related by third. See the keys of the movements in Beethoven, Piano Concerto no. 3; Brahms, Symphony no. 1 (c-E-Ab-c!); Grieg, Violin Sonata, op. 45; Rachmaninov, Piano Concerto no. 2; Schubert, Wanderer Fantasy (major mode variant: C-E-(c#)-Ab-C); and Liszt, Annees de Pdlerinage, "Premiere annie: Suisse" (multi-work variant) for other pieces that exhibit this tonal relationship. Note that many of these works feature A6-C-E collections on surface and middleground levels as well.

70 Smith 2005, passim (see, for instance 218) refers to the E-major Andante movement of Brahms's op. 60 as "solace" and a "dream."


5 6 c: V 4 15 6 V

motive on: E C G-

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

minor.71 Second, no other pair of keys with this unique relationship 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 with four or fewer accidentals requires enharmonic reinterpreta- tion in all cases but two-a possible reason for composers' avoidance of these pairs. And the one pairing apart from c minor and E major that doesn't require enharmonic reinter- pretation, f-minor works with A-major middle movements, apparently occurs with substantially less frequency than the c minor-E major relationship. Thus, in works like Brahms's op. 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 functional tonality had upon earlier music.72

Short of cataloging and counting all the tonal works from the late eighteenth century forward, it is impossible to con- tend that Ak-C-E collections are more common or more important than other major-third cycles. But the evidence for these keys' expressive significance and prevalence in romantic-era compositions suggest that they typify the chromatic-third relations that lie at the heart of nineteenth- century compositional practice. The composers who most

71 While intra-movement relationships of hexatonic poles are exceedingly rare in common-practice music, a few examples do exist. See, for instance, Mendelssohn's g-minor Piano Concerto, whose middle movement is in E major.

72 Smith holds that E is not only an expressive reprieve from the surrounding tragedy of c minor, but also bears motivic cross-references with the first movement. The Andante's E major can be heard as an out- growth of the pizzicato E's from the Allegro (Smith 2005, 101). Likewise, the foreignness of the C0 in the augmented triad highlights the tonal distance between E and the quartet's overarching c tonic (Smith 2005, 17).

i86 MUSIC THEORY SPECTRUM 28 (2006)

favored the use of the Ab-C-E complex span the final days of the classic era to the "progressive" New German school and its antipode, the "conservative," Johannes Brahms. Moreover, the examples cited above include chamber music and orchestral music, concerto and symphony, Lieder and opera, and musics both dramatic and absolute. From surface- level melodies to multi-movement connections, the Ab-C-E complex appears in almost every conceivable context, tran- scending the boundaries of genre, form, and tonal hierarchy.

More importantly, an awareness of its origins, tonal functions, and expressivity enriches our analytic practice. Recog- nition of these sonorities' group structure uncovers intersections and contradictions between tonal/hierarchical and phenomenological/referential hearings of music, impacting our understanding of musical form and musical meaning. That we can appreciate these intersections and contradictions in an artistic style period that embraced duality and ambiguity seems only fitting. And, despite the nineteenth- century's flowering of stylistic diversity fueled by individual expression, the AK-C-E complex allows us to trace a scarlet thread of sorts through the structural and expressive compositional practices of the romantic era.

APPENDIX: SOME Ab-C-E COMPLEX PIECES

Bach, C.P.E., Piano Sonata in f, H. 173, i, mm. 26-62 Beethoven, Piano Concerto no. 3 in c, op. 37, iii, mm.

182-265ff Beethoven, Piano Sonata in f, op. 57 ("Appassionata"), i,

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

i, mm. 138-89ff. Beethoven, Fidelio

Beethoven, String Quartet in e, op. 59, no. 2, i, mm. 209-21 Beethoven, String Quartet in E&, op. 127, ii Brahms, Piano Quartet in c, op. 60, mm. 70-78 Brahms, Symphony no. 1 in c, op. 68

Brahms, Concerto for Violin and Cello in a, op. 102, i, mm. 238-57ff

Brahms, Intermezzo, op. 118, no. 4 Brahms, Clarinet Sonata, op. 120, no. 1, ii, mm. 41-49 Chausson, Piano Trio, op. 3, ii, 139-48 Chopin, Rondo, op. 1, mm. 54-100 Chopin, Nouvelle Etude in A6, mm. 1-25 Chopin, Polonaise in AL, op. 53 Chopin, Mazurka in A6, op. 59, no. 2, esp. mm. 85-88 Chopin, Waltz in A6, op. 64, no. 3 Chopin, Piano Concerto no. 2 in f, op. 21, i, mm. 200-15 Debussy, "Soupir" from Troispoemes de Mallarme' Debussy, "Le jet d'eau" from Cinqpoemes de Baudelaire Elgar, Cello Concerto, op. 85, iv, mm. 197-255 Franck, Symphony in d minor, i, development Haydn, Sonata H. XVI: 52 in E6, i, development Haydn, Quartet, op. 76, no. 3 in C ("Emperor"), i Liszt, Annees des P6eerinage, "Premiere ann&e: Suisse" Liszt, Orpheus, C. 682, mm. 72-130 Liszt, "Blume und Duft," C. 698 Liszt, Eine Faust-Symphonie, C. 697b Liszt, Die Trauer-Gondell, C. 1279 Mahler, Symphony no. 2 in c ("Resurrection"), i, exposition Moussorgsky, Pictures at an Exhibition, "Limoges," mm. 16-18 Mozart, Cosifan tutte, Act II, Finale Mozart, Symphony No 39 in Eb, K. 543, iv, mm. 108-25 Prokofiev, Piano Sonata no. 7, op. 83, ii Rachmaninov, Piano Concerto no. 2 in c, op. 18, opening of

ii and iii Rimsky-Korsakov, The Golden Cockerel, Act II, opening Schubert, Piano Trio in Ek, D. 929, i, recapitulation Schubert, Piano Sonata in a, D. 845 (op. 42), i, mm. 1-80 Schubert, Allegro in a for Four Hands, D. 947 (op. 144,

"Lebensstfirme"), exposition Schubert, Wanderer Fantasy, op. 15 Schubert, Octet D. 803 (op. 166), vi, mm. 172-78 Schubert, Symphony no. 4 in c, D. 417 ("Tragic"), ii, mm.

83-109


Schubert, Symphony no. 6 in C, D. 589 ("Little"), iv, mm. 292-316

Schubert, Symphony no. 9 in C, D. 944 ("Great"), i Schubert, "Antigone und Oedip," D. 542 Schubert, "Der zurnenden Diana," D. 707 Schubert, "Fiille der Liebe," D. 854 Strauss, Richard, Horn Concerto no. 1 in EB, op.11, ii Strauss, Richard, Also Sprach Zarathustra, op. 30 Stravinsky, Piano Sonata Verdi, II Trovatore Wagner, Die Feen, "O ihr des busens Hochgeffihle" Wagner, Derfliegende Holldnder, "Die Frist ist um" Wagner, Die Walkiire, Magic Sleep music Wagner, Die Meistersinger von Niirnberg, Act III, "Wahn"

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

III; and final scene, among others Wolf, "Nimmersatte Liebe," no. 9 from Gedicbte von Eduard

M&rike

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