direccionalidad s xix.pdf

8/10/2019 direccionalidad s XIX.pdf

http://slidepdf.com/reader/full/direccionalidad-s-xixpdf 1/25

Society for usic Theory

The A♭–C–E Complex: The Origin and Function of Chromatic Major Third Collections inNineteenth-Century MusicAuthor(s): MATTHEW BRIBITZER-STULLSource: Music Theory Spectrum, Vol. 28, No. 2 (Fall 2006), pp. 167-190Published by: on behalf of the Society for Music Theory

Stable URL: http://www.jstor.org/stable/10.1525/mts.2006.28.2.167 .Accessed: 09/09/2014 03:00

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

http://www.jstor.org/action/showPublisher?publisherCode=smt

http://www.jstor.org/stable/10.1525/mts.2006.28.2.167?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


http://www.jstor.org/stable/10.1525/mts.2006.28.2.167?origin=JSTOR-pdf

http://www.jstor.org/action/showPublisher?publisherCode=smt



167

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

matthew bribitzer-stull

The A –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 thecollection, this article presents a conglomeration of hierarchic and transformational analytic ap-proaches to A –C–E 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. Bach’s day to the present.2 The

inspiration for the investigation herein is no different, butthe 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 fieldof 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 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: first, how the A –C–E 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 mightprofitably approach examples of A –C–E 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 –C–E 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 –C–E; 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, 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.

This content downloaded from 193.144.2.35 on Tue, 9 Sep 2014 03:00:14 AMAll use subject to JSTOR Terms and Conditions






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 specific work,genre (such as Lieder ), or composer. By selecting music fea-turing A , C, and E, I am able to enjoy the benefits 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 ingeneral—and A –C–E, specifically—typify chromatic thirdrelations in ways that other collections do not. And third, theconsideration of works containing the complete 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 root

progressions whose tonal functions and directionality may bear extra-musical associations. (The move from I to VI(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, Kopp’s M transformations arenot used in this context since they, in effect, conflate two voice-leadingtransformations.

6 Just as individual key centers may have rich, extra-musical associations,so too may harmonic progressions between members of the A –C–Ecollection.The sense of progression from one key to another or of tonalmotion between keys was crucial to Joseph Schalk’s 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, 10–11, Figs. 1.3–1.5

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

Š Ł Ł Ð ÐŁ Ł C

L

e

Š Ł ÐÐ ÐÐŁ ²C

LP

E

Š ÐÐÐ− ÐÐÐ²¦c

PLP (orLPL)

E







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 vi3 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: first, 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 tritone

root 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 specifically, those including thecomplex of A , C, and E sonorities) is most obvious in the

music of Beethoven, Schubert, Chopin, Wagner, Brahms,and Liszt.11 Because the A –C–E 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, 117–18; Brown, Dempster, and Headlam 1997; and

Kopp 2002, 217–18.10 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 231–33 (Ex. 10); Berlioz ’s “ Au cimitière,”mm. 9–15; 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 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 harmonicfunction applicable to much nineteenth-century (and later) music.

1. C – E 2. C – A

I – III

I – VI5

III – V

VII53

– V LP PL

3. C – e 4. C – a I – iii rare III – v

L PLP

5. c – E 6. c – A

rare i – VI vi – IV iv – II

PLP L

7. c – e 8. c – a

rare rare PL LP

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







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

Expressive origins and functions of the A –C–E complex. I

first 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 fi ve or moreflats 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 tonic

key includes fi 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 flat side of the circle of fifths. Thus, it appearsthat it was not the diatonic collections alone that composersavoided, 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 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 amateur—the people’s 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 the “default” tonality.15 This con-ception 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 andpoint of departure of harmonic development in sharp and flat keys; thechurch 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 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,reflecting 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 reflects a strong traditionin Western music theory —conflating a sense of key with asense of location. Words like “middle” and “homeground”indicate 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 ficulties 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 flats or sharps. See MacDonald 1988, 222.

13 Collections of pieces in all twenty-four major and minor keys, likeBach’s Well-Tempered Clavier and Chopin’s 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, 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.








Haydn Baryton Trio Hob. V: 5 BHaydn Minuet Hob. IX: 26 F attributedHaydn Trio Sonata Hob. XV: 31 e “ Jacob’s 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)

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 D

Haydn String Quartet op. 76 #5, Largo F Haydn String Quartet op. 76 #6, Fantasia BBeethoven Sonata op. 26, “marcia funebre” a

Beethoven Sonata op. 27/2, Allegretto D

Beethoven Sonata op. 57, Andante con moto D

Beethoven Sonata op. 110, Arioso dolente a

Beethoven String Quartet op. 130, Presto b

Beethoven String Quartet op. 130, Andante con moto ma non troppo D

Beethoven String Quartet op. 131, Adagio quasi um poco andante g

Beethoven String Quartet op. 135, Lento assai, cantante e tranquillo D

(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 ––> 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 minuet’s parallel key and has fi 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.”







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 sacrificing the “characters of the keys”—a set of extra-musical associations that arose, inpart, 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 producingthe 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 oc-tave, only two (e.g., C–E and E–G 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 C–E major third (as partof 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/orG–B. The thirds belonging to major triads opposite these onthe circle of fifths 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 beE–G, the flat-side third, A /(G)–C.23 The three majorthirds that lay outside these boundaries (B–D , F – A , andD –F) belonged to the tonic triads of the underused major

keys.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 instruments—a 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 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


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 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 reflected the strong influence 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. 15–2,

pg. 47; and Fig. 39–1, 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 tonictriad’s 1̂–3̂ major third) as normative; minor keys were “marked” in thesemiotic 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 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 ( 2̂, 3̂, 6̂, and 7̂)—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.







with good reason. Hence, the edges of the known tonal world— A and E—could 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 composers’most profound musical utterances—a 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 significance. While“associative tonality,” as it is referred to today, was mostfamously explored by Wagner in his Ring cycle, key symbol-ism, stemming from the aforementioned “characters of thekeys,” had a rich history long before the Wagnerian music

drama.25 Wagner’s key associations were most often piecespecific, but the associations comprising the characters of thekeys infused all manner of works from the second half of thecommon practice era. C, lacking the artifice 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 intonation—largely a key-board phenomenon—instrumental 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 fi xed asto exact meaning, nor applicable to every work, there existsevidence of general expressive trends: A is linked to slum-ber, darkness, and death while E major is associated withtranscendence, spirituality, and the sublime.28 Thus, we


25 For discussions of “associative tonality ” see Bailey 1977, 48–61, and1985, 113–46; McCreless 1982, 88–95, and 1983, 60–62; and Stein1985, 43–44, and 141–87.

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, 132–33.

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 I–III–V vs. the downward arpeggiation of I–VI–IV accounts for

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

Vallotti’s Theoretically-Correct Method.





http://www.jstor.org/action/showImage?doi=10.1525/mts.2006.28.2.167&iName=master.img-000.jpg&w=260&h=248



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 A

by setting up an Eros-Thanatos antithesis.29

The Act II finale of Mozart’s Così fan tutte illustrates theeighteenth-century prototype of A and E serving as expres-sive 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 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 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 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, reflective 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 much

rejoicing) via numbers cast in closely-related keys on boththe flat and sharp sides of the circle of fifths. A summary of this tonal motion appears in Example 5(b). Before the finaltonal-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 specific extra-musical associations.31 The same held true of the juxtaposi-tion of A , C, and E sonorities—sonorities whose harmonies

had a profound impact on tonal structure.

Structural origins and functions of the A –C–E complex. Wheneighteenth-century composers featured two (or all three) of the members of the A –C–E complex in their works, thesesonorities were usually related indirectly. In the excerpt fromCosí just examined, for instance, A and E as key areas are re-lated only indirectly to the overarching tonic C via fifth cy-cles and to one another through the central C (as shown in

Example 5(b)). However, the tenuous foreground link be-tween the A and E triads provided by the 5–6 shift (mm.199–200), produces the sound of a direct 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 asI–I6 (or iii)–V and I– vi (IV 6)–IV (ii6) evolved into I–III

–V

and I– VI–iv (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 andMarvin 1992, 93, as well as the synopsis in Hatten 1994, 43.

29 Wagner uses E as the erotic key in Tannhäuser . 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 Così 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 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 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 arpeggiationsoperate on more surface levels. See Schenker 1979, Figs. 7b, 14/1a–b,







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, 181–200, 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, 109–12.

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

Š Ý ÐÐ

ÐÐ−−−

ÐÐÐÐ

−−−−

ÐÐÐÐ

²¦² ¦

ÐÐÐÐ²

Ð

ÐÐÐ²²

ÐÐÐÐ²!

−−

( )

( )

=

53 63

A−:E:

I −VII6 ii7 V 7 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

C E A E A D E B (I–V) G C

(F---d/F)

Flat Keys

(b) overall tonal plan

example 5. Act II finale of Mozart’s Così fan tutte.

Measures are numbered from the

beginning of the finale.







Beethoven’s “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 song’s opening A 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 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 vi Stufe 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 A –C–E complex. Ratherthan arising out of a direct chromatic relationship to A (asits III for instance), C’s relationship to A can be heard in-directly, as the dominant of A ’s most closely-related key. Thus, both chromatic Stufen can 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 –C–E 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 background Bassbrechung itself may be disrupted. The C–E–G 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, 178–79 analyzes the opening of this Lied 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 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 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 inI–III

–V or I– VI

5–V) or embellish V (V –III–V or V – VI

5–V).Numerous examples are cited in pages 24–59 and 73–84.

Š Ý

−−−−

−−−− ð ð ð ð

Ł Ł Ł Ł

¦²¦ ¦

Ł

Ł Ł Ł Ł Ł Ł ¦¦

Ł

Ł Ł Ł −

ð ð ð ð

ð ð ð ð

− ð ð ð −

ð ð ð ð

! ¦−564

53 6

453

8 7

A−: A−

5̂

1

E

14

N

C

19 22

4̂

31

3̂

32

2̂ 1̂

33

I

−“VI” V/vi

“III¦”IV

V I↓

Š Ý

−−−−

−−−− ð ð ð ð

Ł Ł Ł Ł

Ł Ł Ł

Ł

Ł Ł

Ł

Ł Ł Ł ! [ ]

A−:

5̂

or

N

6

I

vi [IV 6]







chromatic passing or neighboring motion. Even when allthree are chord tones within an augmented dominant triad,the C–E–G sonority may remain unquestionably dominantin function. But, when the augmented triad is not anchoredby a diatonic Stufe its symmetry can threaten tonality alto-

gether (as in Liszt ’s “Nuages gris”). Likewise, an E-major(III

) structural third divider between I and V on the mid-

dleground 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 thistechnique might feature a nested chromatic mediant rela-tionship (III

of III

) to invoke the third member of the

complex (as in Wagner’s Siegfried Idyll ).37 When such chro-matic third chains achieve independence from tonic and

dominant, however, they may replace tonic prolongation with other structural functions as in the symmetrical divisionof the octave evinced by the Schlaffenakkorde of Wagner’sRing .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 dominant Stufen, thus prejudicing

the tonal contexts in which an A –C–E 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 Chopin’s Polonaise, op. 53, in whichthe tonic A is prolonged first 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 first section andby virtue of its recurrence at the end of the retransitionback into A (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 domi-nant). 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 ex-pected modulation to the mediant during the first reprise of


that leads eventually to V) and Schubert ’s E String Trio, i, mm.434–52 (whose recapitulation features a ii

–VI

–IV predominant

chain).40 Direct chains of thirds appear in the literature as well. See Schubert’s

Lied , “F ülle der Liebe” and the analysis in Krebs 1980, 110 (Fig. II.37, v. 2, 49). Example 7 presents only the opening of the A –E– 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 A –C–E 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

Weitzmann’s 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 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. I.46, v.2, 26) who illustrates this technique in Beethoven’s pianoconcerto in E major (“Emperor”), i ii , mm. 138–89 as a VI–IV – II

5

chain embellishing the motion from VI to V. See also, Beethoven’spiano concerto in c minor, iii, mm. 138–220 (a VI

–IV – II succession







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 –c–e complex

As chromatic-third usage evolved, nineteenth-century theory naturally developed alongside composition. Whetherreactive or innovative, much of this work focused on A–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 collec-tion.44 And Carl Friedrich Weitzmann both distinguishedhimself from his contemporaries and influenced Franz Lisztby his thorough treatment of the A –C–E augmented triad.45


42 Similar A –C–E collections in a- and f-minor music occur in Brahms’s

Intermezzo op. 118, no. 4 (see Example 10); the prolongation of Vin mm. 26–62 of the first movement of C. P. E Bach’s 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. 81–152 of Schubert’s Allegro for piano, four hands, op. 144 (“Lebensstürme”).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 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 Harmonielehre (1920) publishedduring his lifetime. See Kopp 2002, 94. Other A–C–E examples occurthroughout Riemann’s writings on third relations and tonality. See, forinstance, 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

Š Ý

−−−−

−−−− ð

Ł Ł ð

Ł

Ł Ł Ł ¦

Ł Ł Ł Ł −

Ł

Ł Ł Ł

Ł

Ł Ł Ł ¦

Ł

Ł Ł Ł

Ł

Ł Ł Ł ¦

Ł

Ł Ł Ł

Ł Ł Ł Ł

Ł

Ł Ł Ł

Ł Ł Ł Ł

¦²¦ ¦

! A−:

A−

5̂

1 49 53 57

C

58 63 64 65

E

−6̂81

IIII¦ ii V I −VI−5

example 7. Analysis of the opening of Chopin’s Polonaise,

op. 53.

Š

Ý ð

Ł Ł ð

Ł

Ł Ł ð

ð

Ł Ł ð

Ł

Ł Ł Ł −−−

Ł

Ł Ł Ł

−

−− Ł Ł Ł −−−

Ł Ł Ł Ł

²²² ¦

Ł Ł Ł Ł

²²¦

Ł

Ł Ł Ł

¦

¦¦

ð

Ł Ł ð ²

ð Ł Ł ð

¦!a:

5̂

1

4̂

C

3̂

A−

43 58 66 67

E

2̂

80

1̂

97

i III V ² i

example 8. Analysis of the scherzo from Schubert’s sonata, op.42, iii.







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 –C–E 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 perfect fifths fit 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 the

eighteenth-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 (Liszt ’s Die Trauer-Gondel I, forone), 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 chromaticthirds to a largely fifth-governed tradition.

As an example, consider the first movement of Beetho- ven’s “ Appassionata” sonata. Schenker’s graph of the develop-ment section in Der freie Satz indicates that, on the deepmiddleground, 3̂ of the Urlinie falls to 2̂ as 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 end

of the development. The Schenkerian prolongation of an A Stufe supporting 3̂ 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 fi ve of the six triads in Cohn’s 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 Schenker’s analysis begins with the a -minor sonority in m. 65. SeeSchenker 1979, Fig. 114.8. 52 See Cohn 1996, 17, Fig. 1. In Cohn’s figure the A –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 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, 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, and386–7; Weber 1846, 503; and Ziehn 1887, 8 and 119.

47 Taruskin 1985, 135–36, 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, 70–71, summa-rizes Kurth’s 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 –C–E in “back-of-the-book ” topics like augmented triadsand enharmonic modulation. See, for instance, Laitz 2003, 645–46;Ottman 2000,229; Roig-Francolí 2003, 830–31; 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 Schenker ’s 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.







While the Schenkerian graph in Example 9 shows theparsimonious voice leading of Beethoven’s 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 tonic 6

4 underneath the return of theopening material.54

Just as a neo-Riemannian analytic vantage point may in-form a Schenkerian reading, as in the “ Appassionata” analysisabove, the converse is also true. The indeterminacy of direc-tionality implied by “polar progressions”—motion across ahexatonic pole (PLP or LPL)—can be clarified by theprolongational context.55 While the distinction may seemacademic, the two different labels—LPL and PLP—suggesttwo different hearings that imply a differentiation between

clockwise and counterclockwise motion about a hexatonicpole, or—in linear rather than cyclic space—ascending anddescending harmonic root motion. When considered as “up” 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 6

4 elaborates the V Stufe 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 Beethoven’s 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, 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 prolonga-tion 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 shiftfrom one pole to the other is responsible for this progression ’s uncanny effect. See Cohn 2004 and Cohn 1996, 21–22.

Š

Ý

−−−−

−−−− ð

Ł Ł ð

ð Ł Ł ð

Ł Ł Ł Ł

²¦² ²

Ł Ł Ł Ł

¦¦¦ ²

Ł Ł Ł Ł

¦¦¦ ¦

Ł Ł Ł Ł −¦

Ł Ł Ł Ł

Ł Ł Ł −

Ł Ł Ł Ł

−−

Ł Ł Ł Ł −¦ ¦

ð

Ł Ł Ł ¦¦Ł

Ł Ł ð

!á

64f:

5̂

1

A−

3̂

35

P L“a−”

51

PE

67

LPe

79

Lc

83

A−

87 93 97

2̂

132

5̂

i III V ¦

example 9. Graphic analysis of Beethoven’s Sonata, op. 57 (“Appassionata”) with neo-Riemannian analysis.








Š

Ý

Š Ý

−−−

−−−

−−−

−−−

Ð²

Ð²

Ð

ÿ

¦

\

Ł

Ł

ÐÐÐÐ

¦²¦\

¼

¼

½

½

ÿ

ÿ

ÐÐÐÐ−−

ÿ

ÿ

ÐÐ

Ðð ý

¦¦ Ł

ÿ

ÿ

ð

ð ð ð \\ ð ð

½

ð

$

%

!

all!

auf!

all!

auf!

’Cello

c:

E

3̂

LP

Str.

(PT)

A−

2̂

L

1̂

K.-dr

c

¦III¦² 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 C–E–cSynopsis 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.








62 Schachter 1987, 304–08, discusses the distinction (not made inSchenker’s Free Composition) between “structural” keys and more fore-ground 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 A –C–E complex as Cohn’s northern hexatonic pole), as well as tonaland linear aspects of various works. See Cohn 1996, 33, in which hesuggests the use of hexatonic thinking in 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 “better” or “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 the

harmonic event stream, why omit it from the analysis? Aninvestigation of the preceding music provides the answer. AsExample 11(b) shows, the Dutchman’s aria proper dividesinto three sections, cast in c, a (with shifts to A major),and c respectively. The coda to the Dutchman’s aria picks upthe C major Picardy Third at the end of the third sectionand moves toward E for the ghostly crew ’s confirmation of the Dutchman’s longing for death. E then passes through A to c, bringing an otherworldly quality to the number’s

close.60

In effect, then, we have two conflicting 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 the Dutchman example,above. Stufen, significant 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 sonorities

occur 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 structure—aconnection between A , C, and E—that 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 spaces—that the sec-ond 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 PianoQuartet to suggest that the relevance of the A –C–E com-plex in tonal contexts may range from the structural fore-ground to intra-movement connections, even within thesame work. The G–E–C major-third chain of the openingcello 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 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 A –C–E 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).


8 8 ( 6)







64 Peter Smith argues that the G–E–C augmented triad achieves a mo-tivic and expressive importance that transcends its role in the structuralhierarchy. See Smith 1994, 258–60, and 2005, 17–18.

65 Interested readers may wish to compare this graph with that in Smith2005, 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 keysrelated by third. See the keys of the movements in Beethoven, PianoConcerto no. 3; Brahms, Symphony no. 1 (c–E– A –c!); Grieg, ViolinSonata, op. 45; Rachmaninov, Piano Concerto no. 2; Schubert,Wanderer Fantasy (major mode variant: C–E–(c )– A –C); and Liszt,

Années de Pélerinage, “Première année: Suisse” (multi-work variant)for other pieces that exhibit this tonal relationship. Note that many of these works feature A –C–E collections on surface and middlegroundlevels 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.”

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 G –E–C 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, and finally 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, E–C–G , with thecomposing-out of the first third, E–C, transposed exactly tocompose out C–G . Thus, the upper line seems to model

Proctor’s “transposition operation” and suggests that Brahmshas entered a fully-chromatic, equal-tempered tonal space.66

But 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. 77

does 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 E Stufe

by its lower third, C.67

While G does not play a vital struc-tural role in this prolongation, however, the neo-Riemanniananalysis below the score points to its referential 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 composers’most expressive music, a tonal marker for the spiritual andsublime. Given Brahms’s 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 progression within a single Ursatz (i to III

), the

motion from tonic minor to the raised major mediant ap-pears 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 A –C–Ecomplex.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 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 fi ve steps away on the circle of

fifths, while A major lies six steps away —directly opposite c


the origin and function of chromatic major third collections in nineteenth century music 185







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 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 C in the augmented triad highlightsthe tonal distance between E and the quartet’s 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, Mendelssohn’s 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 two—a possible reason for composers’avoidance of these pairs. And the one pairing apart from cminor 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 cminor–E major relationship. Thus, in works like Brahms’sop. 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 –C–E collections are more common or moreimportant than other major-third cycles. But the evidencefor these keys’ expressive significance 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

Š

Ý

²²²²

²²²² Ł

Ł Ł Ł ¦

Ł

Ł Ł Ł

¦

¦¦

Ł

Ł Ł Ł

−

−¦

Ł

Ł Ł Ł

¦

−¦−

Ł

Ł Ł Ł ¦¦

Ł

Ł Ł

−

¦ ¦

Ł

Ł Ł Ł

¦

¦¦−

Ł

Ł Ł Ł

¦

¦¦

Ł − Ł − Ł ¦ Ł

Ł Ł Ł

¦

−¦ ¦

Ł

Ł Ł Ł

−

²¦

Ł

Ł Ł Ł

−

¦²

Ł

Ł Ł Ł

−

¦

Ł

Ł Ł Ł

¦

²²

Ł

Ł Ł ²

Ł

Ł Ł Ł

²

¦ Ł Ł Ł ×²

Ł

Ł Ł

!

( )

5¦ 5×5

e

Augmented triadmotiveon: E

L

6

C

c:

Pc

L

C

6

G²

g²:P

g²E:

LE

G²

3̂

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.








favored the use of the A –C–E complex span the final daysof the classic era to the “progressive” New German schooland its antipode, the “conservative,” Johannes Brahms.Moreover, the examples cited above include chamber musicand orchestral music, concerto and symphony, Lieder and

opera, and musics both dramatic and absolute. From surface-level melodies to multi-movement connections, the A–C–Ecomplex 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 sonorities’ group 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 only fitting. And, despite the nineteenth-century ’s flowering of stylistic diversity fueled by individualexpression, the A –C–E complex allows us to trace a scarletthread of sorts through the structural and expressive compo-sitional practices of the romantic era.

appendix: some a–c–e complex pieces

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

182–265 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. 138–89 ff.Beethoven, FidelioBeethoven, String Quartet in e, op. 59, no. 2, i, mm. 209–21Beethoven, String Quartet in E, op. 127, iiBrahms, Piano Quartet in c, op. 60, mm. 70–78Brahms, Symphony no. 1 in c, op. 68

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

Brahms, Intermezzo, op. 118, no. 4Brahms, Clarinet Sonata, op. 120, no. 1, ii, mm. 41–49Chausson, Piano Trio, op. 3, ii, 139–48

Chopin, Rondo, op. 1, mm. 54–100Chopin, Nouvelle Etude in A ,mm.1–25Chopin, Polonaise in A ,op.53Chopin, Mazurka in A , op. 59, no. 2, esp. mm. 85–88Chopin, Waltz in A ,op.64,no.3Chopin, Piano Concerto no. 2 in f, op. 21, i, mm. 200–15Debussy, “Soupir” from Trois poè mes de Mallarmé Debussy, “Le jet d’eau” from Cinq poè mes de Baudelaire Elgar, Cello Concerto, op. 85, iv, mm. 197–255

Franck, Symphony in d minor, i, developmentHaydn, Sonata H. XVI: 52 in E, i, developmentHaydn, Quartet, op. 76, no. 3 in C (“Emperor”), iLiszt, Anné es des P é lerinage , “Première année: Suisse”Liszt, Orpheus , C. 682,mm. 72–130Liszt, “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.16–18Mozart, Cos ì fan tutte , Act II, FinaleMozart, Symphony No 39 in E, K. 543, iv, mm. 108–25Prokofiev, 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. 1–80

Schubert, Allegro in a for Four Hands, D. 947 (op. 144,“Lebensstürme”), expositionSchubert, Wanderer Fantasy, op. 15Schubert, Octet D. 803 (op. 166), vi, mm. 172–78Schubert, 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”), iSchubert, “ Antigone und Oedip,” D. 542Schubert, “Der zurnenden Diana,” D. 707

Schubert, “F ülle 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 ühle” Wagner, Der fl iegende Holl ä nder, “Die Frist ist um” Wagner, Die Walküre , Magic Sleep music Wagner, Die Meistersinger von N ürnberg , Act III, “ Wahn”

monologue and Dream theme Wagner, Siegfried , Act III, Siegfried and Brünnhilde 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 Gedichte von Eduard

M ö rike

references

Anson-Cartwright, Mark. 1996. “Chord as Motive: The Augmented-Triad Matrix in Wagner’s Siegfried Idyll .” Music Analysis 15: 57–71.

Bailey, Robert. 1977. “ The Structure of the Ring and itsEvolution.” 19th-Century Music 1:48–61.

———. 1985. “ An Analytical Study of the Sketches andDrafts.” In Richard Wagner: Prelude and Trans fi guration

from Tristan und Isolde. Edited by Robert Bailey. New

York: W. W. Norton and Company, 113–46.Beach, David. 1997. “ The Submediant as Third Divider: ItsRepresentation at Different Structural Levels.” In Music Theory in Concept and Practice . Edited by James Baker,David Beach, and Jonathan Bernard. Rochester: Univer-sity of Rochester Press, 309–35.

Bribitzer-Stull, Matthew. 2006. “ The End of Die Feen and Wagner’s Beginnings: Multiple Approaches to an Early Example of Double-Tonic Complex, Associative Theme,and Wagnerian Form.” Music Analysis , forthcoming.

Brown, Matthew. 1986. “ The Diatonic and the Chromatic

in Schenker’s Theory of Harmonic Relations.” Journal of Music Theory 30:1–33.

Brown, Matthew, Douglas Dempster, and Dave Headlam.1997. “ The IV ( V) Hypothesis: Testing the Limits of Schenker’s Theory of Tonality.” Music Theory Spectrum 19:155–83.

Burnham, Scott. 1994. “Mozart ’s felix culpa: Cos ì fan tutte and the Irony of Beauty.” Musical Quarterly 78: 77–98.

Burstein, Poundie. 2005. “Unraveling Schenker’s Concept of

the Auxiliary Cadence.” Music Theory Spectrum 27.2:159–85.Cinnamon, Howard. 1984. “ Third-Relations as Structural

Elements in Book II of Liszt ’s Anné es de Pelerinage and Three Later Works.” Ph.D. dissertation, University of Michigan.

———. 1992. “ Third-Related Harmonies as Elements of Contrapuntal Prolongation in Some Works by Franz Liszt.” In Theory Only 12: 1–30.

Cohn, Richard. 1996. “Maximally Smooth Cycles, Hexa-tonic Systems, and the Analysis of Late-Romantic Triadic Progressions.” Music Analysis 15:9–40.

———. 2000. “ Weitzmann’s Regions, My Cycles, andDouthett’s Dancing Cubes.” Music Theory Spectrum 21.1:89–103.

———. 2004. “Uncanny Resemblances: Tonal Significationin the Freudian Age.” Journal of the American Musicological Society 57: 285–323.

Dehn, S.W. 1840. Theoretische – Praktische Harmonielehre mit angef ü gten Generalbassbeispielen. Berlin: Wilhelm Thome.Drabkin, William. 1982. “Characters, Key Relations and

Tonal Structure in Il Trovatore .” Music Analysis 1: 143–53.Gilliam, Bryan. 1991. Richard Strauss ’ s Elektra . Clarendon

Press: Oxford.

the origin and function of chromatic major third collections in nineteenth century music 187








Harrison, Daniel. 1994. Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Account of Its Precedents . Chicago: Chicago University Press.

Hatten, Robert. 1994. Musical Meaning in Beethoven.Bloomington: Indiana University Press.

Hyer, Brian. 1995. “Reimag(in)ing Riemann.” Journal of Music Theory 39: 101–38.

Irving, Howard, and Lee Riggins. 1988. “ Advanced Uses of Mode Mixture in Haydn’s Late Instrumental Works.”Canadian University Music Review 9: 104–35.

Jorgensen, Owen. 1991. Tuning: Containing the Perfectionof Eighteenth-Century Temperament, the Lost Art of

Nineteenth-Century Temperament, and the Science of Equal Temperament Complete with Instructions for Aural and

Electronic Tuning . East Lansing: Michigan StateUniversity.Kielian-Gilbert, Marianne. 1990. “ The Functional Diffe-

rentiation of Harmonic and Transpositional Patternsin Liszt’s Consolation No. 4.” 19 th-Century Music 14: 48–59.

Kinderman, William and Harald Krebs, eds. 1996. The Second Practice of Nineteenth-Century Tonality . Lincoln:University of Nebraska Press.

Kivy, Peter. 1980. The Corded Shell: Re fl ections on Musical Expression. Princeton: Princeton University Press.Kopp, David. 2002. Chromatic Transformations in Nineteenth-

Century Music . Cambridge: Cambridge University Press.Kostka, Stephan and Dorothy Payne. 1984. Tonal Harmony .

New York: Knopf.Kramer, Richard. 1985. “ The New Modulation of the 1770s:

C. P. E. Bach in Theory, Criticism, and Practice.” Journal of the American Musicological Society 38: 551–92.

Krebs, Harald. 1980. “ Third Relation and Dominant in Late18th- and Early 19th-Century Music.” Ph.D. dissertation, Yale University.

———. 1981. “ Alternatives to Monotonality in Early Nineteenth-Century Music.” Journal of Music Theory 25:1–16.

Kurth, Ernst. 1913. Die Voraussetzungen der TheoretischenHarmonik und der Tonalen Darstellungssysteme . Bern: Akademische Buchhandlung von Max Dreschel.

———. 1923. Romantische Harmonik und ihre Krise inWagners “ Tristan.” Berlin: Max Hesses Verlag.

Laitz, Steven. 2003. The Complete Musician. New York:Oxford University Press.

Lewin, David. 1986. “Music Theory, Phenomenology, andModes of Perception.” Music Perception 3: 327–92.

Lobe, J. C. [1861]. Vereinfachte Harmonielehre . Leipzig:C. F. W. Siegel.

MacDonald, Hugh. 1988. “G 9/8.” 19th-Century Music 11:221–37.

MacDonald, Malcolm. 1990. Brahms . London: J. M. Dent

and Sons, Ltd.McCreless, Patrick Philip. 1982. Wagner ’ s Siegfried: Its Drama,History, and Music . Ann Arbor: UMI Research Press.

———. 1983. “Ernst Kurth and the Analysis of the Chro-matic Music of the Late Nineteenth Century.” Music Theory Spectrum 5: 56–75.

McKee, Eric. 2001. “Hope and Salvation in the Instru-mental Music of Beethoven.” Paper read at the BeethovenEaster Festival, Krak ó w.

Neumeyer, David. 1982. “Organic Structure and the SongCycle: Another Look at Schumann’s Dichterliebe .” Music Theory Spectrum 4: 92–105.

———. 1997. “Synthesis and Association, Structure, andDesign in Multi-Movement Compositions.” In Music Theory in Concept and Practice . Edited by James Baker,David Beach, and Jonathan Bernard. Rochester: Uni- versity of Rochester Press, 197–216.

Ottman, Robert W. 2000. Advanced Harmony: Theory and

Practice , fifth edition. Upper Saddle River: Prentice Hall.Proctor, Gregory. 1978. “ Technical Bases of Nineteenth-Century Chromatic Tonality: A Study in Chromaticism.”Ph.D. dissertation, Princeton University.

Riemann, Hugo. 1882. “Die Natur der Harmonik.” Samm-lung musikalischer Vortr ä ge 4.40: 157–90.

( )








———. 1890. Katechismus der Harmonie- (und Modulations)lehre . Leipzig: Max Hesse.

———. [1893]. Harmony Simpli fi ed or The Theory of theTonal Functions of Chords . Translated by H. Bewerunge.London: Augener & Co.

———. 1902–03. Gro ß e Kompositionslehre . Berlin and Stutt-gart: W. Spemann.

———. 1922. “ Tonalität.” In Musik-Lexikon. Edited by Alfred Einstein. Berlin: Max Heffes Verlag.

Rimsky-Korsakov, Nicolai. 1895. Praktisches Lehrbuch der Harmonie . Leipzig: M. P. Belaieff.

Roig-Francolí, Miguel. 2003. Harmony in Context . New York: McGraw-Hill.

Rothfarb, Lee A. 1991. Ernst Kurth: Selected Writings . Cam-

bridge: Cambridge University Press.Samarotto, Frank. 2003. “ Treading the Limits of TonalCoherence: Transformation vs. Prolongation in Selected Works by Brahms.” Paper read at the annual meeting of the Society for Music Theory, Madison.

Santa, Matthew. 2003. “Nonatonic Systems and the Parsi-monious Interpretation of Dominant-Tonic Progres-sions.” Theory and Practice 28: 1–28.

Schachter, Carl. 1987. “ Analysis by Key.” Music Analysis 6.3:

289–318.Schenker, Heinrich. 1979. New Musical Theories and Phantasies, Part Three: Free Composition. Edited and Translated by Ernst Oster. New York: Longman.

Schmalfeldt, Janet. 2002. “Music that Turns Inward: New Roles for Interior Movements and Secondary Themesin the Early Nineteenth Century.” Paper read at theannual meeting of the Society for Music Theory,Columbus.

Schwartz, Richard Isadore. 1982. “ An Annotated Translationof Harmonielehre of Rudolph Louis and Ludwig Thuille.”Ph.D. dissertation, Washington University.

Smith, Peter H. 1994. “Liquidation, Augmentation, andBrahms’s Recapitulatory Overlaps.” 19th-Century Music 18.3: 237–61.

———. 1995. “Structural Tonic or Apparent Tonic?: Para-metric Conflict, Temporal Perspective, and a Continuumof Articulative Possibilities.” Journal of Music Theory 39.2:245–83.

———. 2005. Expressive Forms in Brahms ’ s Instrumental

Music . Indiana University Press.Somer, Avo. 1995. “Chromatic Third Relations and Tonal

Structure in the Songs of Debussy.” Music Theory Spectrum 17: 215–41.

Steblin, Rita Katherine. 1981. “Key Characteristics in the18th and Early 19th Centuries: A Historical Approach.”Ph.D. dissertation, University of Illinois, Urbana-Champaign.

Stein, Deborah. 1985. Hugo Wolf ’ s Lieder and Extensions of

Tonality . Ann Arbor: UMI Research Press.Steptoe, Andrew. 1988. The Mozart-DaPonte Operas: The Cultural and Musical Background to Le Nozze di Figaro,Don Giovanni, and Così fan tutte. Oxford: ClarendonPress, 1988.

Swinden, Kevin. 2005. “ When Functions Collide: Aspects of Plural Function in Chromatic Music.” Music Theory Spectrum 27.2: 249–82.

Taruskin, Richard. 1985. “Chernomor to Kashchei: Har-

monic Sorcery, or, Stravinsky ’s ‘ Angle’.” Journal of the American Musicological Society 38: 72–142. Todd, R. Larry. 1988. “ The ‘Unwelcome Guest’ Regaled:

Franz Liszt and the Augmented Triad.” 19th-Century Music 12: 93–115.

———. 1996. “Franz Liszt, Carl Friedrich Weitzmann,and the Augmented Triad.” In The Second Practice of

Nineteenth-Century Tonality . Edited by William Kinder-man and Harald Krebs. Lincoln: University of Nebraska,

153–77. Tovey, Donald Francis. 1944. Musical Articles from the Encyclopedia Britannica . Oxford University Press:London.

Wason, Robert. 1997. “ Joseph Schalk and the Theory of Harmony at the End of the Nineteenth Century.” In

9








Bruckner Studies . Edited by Timothy Jackson and PaulHawkshaw. Cambridge: Cambridge University Press,122–39.

Wason, Robert and Elizabeth Marvin. 1992. “Riemann’sIdeen zu einer ‘Lehre von den Tonvorstellungen’: An

Annotated Translation.” Journal of Music Theory 36:69–117.

Weber, Gottfried. 1846. Theory of Musical Composition, Vol.2. Translated by James F. Warner. Boston: Wilkins,Carter, and Co.

Weitzmann, Carl Friedrich. 1853. Der Ü bermä ssige Dreiklang . Ber lin: T. Trautweinschen Buch undMusikhandlung.

Wheelock, Gretchen. 1993. “Schwarze Gredel and the

Engendered Minor Mode in Mozart ’s Operas.” In Musicology and Difference . Edited by Ruth Solie. Berkeley:University of California Press, 201–24.

Ziehn, Bernhard. 1887. Harmonie- und Modulationslehre .Berlin: R. Sulzer.

Music Theory Spectrum, Vol. 28, Issue 2, pp. 167–190, ISSN 0195-6167,

electronic ISSN 1533-8339. © 2006 by The Society for Music Theory. All rights reserved. Please direct all requests for permission to photo-copy or reproduce article content through the University of CaliforniaPress’s Rights and Permissions website, at http://www.ucpress.edu/

journals/rights.htm.





direccionalidad s xix.pdf

Documents