Aspects of Plural Function in Chromatic Music

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When Functions Collide: Aspects of Plural Function in Chromatic MusicAuthor(s): Kevin J. SwindenSource: Music Theory Spectrum, Vol. 27, No. 2 (Autumn, 2005), pp. 249-282Published by: University of California Press on behalf of the Society for Music TheoryStable URL: http://www.jstor.org/stable/4499838

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When Functions Collide:Aspectsof Plural Functionin ChromaticMusic

KEVIN J. SWINDEN

Daniel Harrison's 1994 study, Harmonic Function in Chromatic Music, questions the traditional

mapping of chords onto function, and instead suggests that scale steps embody the source of har-

monic function. His reformulation creates a new one-to-one mapping of scale steps onto harmonic

function, which may be, at times, problematic. This article examines aspects of Harrison's theoryand advances differentmechanism orthe evaluation f harmonic unctionbasedon the Tonnetz.It examinesa particularet of chromatic armonies hat displayplural unction,which maybe or-

ganized according o agenusandspeciesmode of classification.

INTRODUCTION

The study of nineteenth century harmony abounds with

rich chords and striking relationships. Sadly, many of these

elude clear understanding. Our existing theories, though ele-

gant and expressive, have not penetrated the moments in

chromatic music about which we most care. The best-knowntheories give a tidy and systematic picture of diatonic har-

mony, but often finesse so many of these wonderful chords as

exceptions to their harmonic principles, or simply fail to ex-

plore fully the functional effect of these chords. Such treat-

ment, however, highlights shortcomings of theories that can-

not explain such configurations within their systems of rules

and structures. We would be well served to search for a mode

of understanding that values these moments for what they

are, rather than for being warped versions of something else.

Consider Wagner's "Tarnhelm" motive from scene three

of Das Rheingold, shown in Example 1(a). The repetition of

the G#-minor triad invites us to hear this chord as a tempo-

rary tonic, alternating with an E-minor triad whose relation-

ship to this tonic is far from clear. The chromatic-third rela-

tionship between G4 minor and E minor at once suspends

tonality, and yet the passage somehow holds together. At

the end of the passage cited, the phrase comes to rest on

B major, recasting E minor as the new tonic of the modulat-

ing phrase.1What is so spectacular about the alternation of E minor

and G# minor? Such an uncommon progression may wellattract attention, but if we want to know how it works, the

answer lies in the particular contrapuntal and functional

association between the chromatically-related G#-minor and

E-minor triads.

The essential counterpoint that governs the G# minor/E minor alternation is fairly straightforward. The upper

I The passagecited followsa fermataovera restthat was precededby ahalf-cadence n C major.The mysteryof the "Tarnhelm" otive is cer-

tainly ntensifiedbythe relationship f the G#-minor riadfollowing soclosely on the heels of V of C major,and a fullerstudyof the passagewould need to consider he harmonicrelationshipn this largercontext.Forpresentpurposes, amconcernedwith thevery ocal tonicityof G#minor within the key scheme of descending major thirds (from C,

throughG#minor, o E at the end of the passagecited).

249



250 MUSIC THEORY SPECTRUM 27 (2005)

(a) Wagner,Der Ring desNibelungen, "Tarnhelm"motive.

(b) Wagner,Der Ring desNibelungen, "Tarnhelm" oice leading.

EXAMPLE I

voice, 5, is embellished with an upper neighbor, ?b; the lower

voice begins on G# (i) and moves in chromatic contrary mo-

tion through a lower neighbor G?.2 Against this counter-

point, a stationary B? stabilizes the progression. Finally, E? isadded to the bass, presumably to add color and emphasis,thus harmonizing the first inversion triad with its own root.

While this explanation delineates the contrapuntal associa-

tion of the two chords, it does not adequately address their

functional relationship.3 If the E-minor triad were merely a

linear phenomenon embellishing G# minor, there would be

no need to add an emphasizing bass note E? to the Klang.There is something more to be said about this progression.

Functionally, there are competing elements within the

progression. If one overlooks the E? in the bass for a mo-ment, the counterpoint might suggest that G? is a clever

disguise for Fx, the leading tone. In this situation, the con-

trapuntal motion would be governed by a tonic 1 and 5 ex-

panding outward to a Dominant-functioning 7 and b6. The

BTwould be an inner-voice pedal tone, stabilizing the func-

tion of the passage in the orbit of G# minor, as shown in

Example 1(b). This interpretation grows out of the condition

already agreed upon that E minor is a chord that is subordi-

nate to-and in some way prolongs-G# minor. With re-

spect to the presumed Tonic, 7 and kb are dissonant neigh-bors. But there is that slippery k6 in the bass. A lesser

composer might well have left us this hypothetical version

with 7 in the bass, or perhaps have put 5 in the bass to

emphasize the Dominant function of the middle chord.

Wagner, instead, created an ambiguity by placing E? in the

bass, leaving a plagal bass line to support an otherwise au-

thentic counterpoint, and changing the effect of the har-

mony altogether.

What I find fascinating is the existence of an entire fam-ily of chords in the literature that have common elements

and functional behaviors, lending themselves to a genus-

and-species mode of organization. In this paper, I provide a

way of thinking that groups many of these chords into a

larger network of relations. This essay explores collisions of

2 As modal mixture is a fundamental principle of chromatic harmony,

throughout the article all scale-step numbers and Roman numerals are

based on a fixed notation in relation to the tonic pitch, regardless of

modality. That is, in relation to A major-minor, F and an F-major triadare notated as k6 and VI respectively.The modality of a local key area

may be inferred by the quality of the tonic chord (I or i) should the

reader wish to take special note of instances of mixture.

3 Other authors have considered this passage from the perspective of

transformational theories (Lewin 1992) and mediant relations in the

context of a theory based on common-tone voice leading (Kopp 2002,

182f). Harmony texts typically describe this relationship as a chro-

matic mediant relation or "Coloristic Chord Succession" (Kostka andPayne 1995) without further comment on the harmonic function that

may be present in the progression. Indeed, Kostka and Payne 1995,

439-40 go so far as to advise the student to abandon any attempt at in-

terpretive analysis altogether in favor of simply labeling the chords with

a letter-name root and chord quality.



ASPECTS OF PLURAL FUNCTION IN CHROMATIC MUSIC 251

Subdominant and Dominant functional elements. Part I de-

velops a theoretical apparatus; Part II considers specific in-

stances of this phenomenon in various guises.

I. THEORIES OF HARMONIC FUNCTION AND PROLONGATION

Many theorists have struggled with the difference be-

tween diatonic and chromatic tonality, with varying degreesof success. Most Schenkerians address music of the late

nineteenth century with little apology, while Schenker's de-

tractors cite his own inability to deal adequately with the

repertoire.4 My initial position is that Schenker's contribu-

tions are more than valuable-indeed, it is essential to ad-

dress his contributions regarding the interaction of harmony

and counterpoint. By the same token, however, Schenker's

system inadequately addresses the slippery issues of har-

monic function in the music of the late-nineteenth century.To rehearse these arguments at this late date would serve lit-

tle use. Further, under the nouveau regime of functional

analysis initiated by Harrison 1994, a critique of Schenker

on these grounds becomes inappropriate. However, I believe

it is neither necessary nor appropriate to discard many of

Schenker's most important contributions. My intent is not to

revise, rewrite, or extend Schenker'stheory,5but only to in-

vestigate harmonic function in chromatic music with sensi-

tivity to his ideas.

Some authorspropose that the nineteenth century repre-

sents a second harmonic practice,governed not so much bytraditional rules of diatonic harmonyand counterpoint, but

by interactions and amalgamsof scales (including the possi-

bility that a chromatic scale might serve the role once occu-

pied by diatonic scales), symmetric divisions of the octave,models of directional tonality, or dual-tonic structures.6These bipartitedivisions imply that chromatic music cannotbe considered an extension of diatonic, but rather, must be

wholly separate. By extension, theories and methodologies

appropriate o the earlier firstpracticeneed to be retooled or

replacedto address this new way of composing.Theories of harmonic function, on the other hand, hold

that diatonic and chromatic harmony may display differentidioms and customs, but are bound together by a commonfunctional basis. Some of the most important work in thisareahas come from theoriesof CharlesJ. Smith (Smith 1981and 1986) and Daniel Harrison (Harrison 1994 and 1995),both of whom rely to some extent on Riemannian notionsinformed by a host of other historicalcontemporaries. Both

4 Daniel Harrison writes: "Schenker's failure to deal with Reger's op. 81

is emblematic of a general failure to understand the harmonic structures

and procedures of chromatic music, or at least to understand them with

the same sensitivity that can be brought to the analysis of common-

practice and atonal musics. [...] We cannot repay this debt with

Schenker'scoin or with coins stamped from his bullion; Schenker'sown

experience is warning enough that his currency is not convertible"

(Harrison 1994, 5). In a similar spirit, David Kopp writes: "Most of our

prevailing analytic models and methods, predicated on eighteenth-century practice, have traditionally explained chromatic music as the

elaboration of diatonic structures.The music's frequent lack of confor-

mity with these models has often been interpreted as a sign of weakness

or inferiority in the music itself, rather than due to any inappropriate-ness of the model" (Kopp 2002, 1).

5 Blasius 1996 providesan excellentbackgroundo understandingwhysuch a course s ill advised.

6 GregoryProctorwrites:"there s not a single commonpractice'extend-

ing from the earlyseventeenthcenturythroughthe end of the nine-teenth century.Rather, he era can be divided nto two large,overlap-ping style systems,hereinreferred o as:classicaldiatonictonality,andnineteenthcenturychromatictonality."Proctor1978, iii). The essen-tial feature hat distinguisheshese two tonal languages s the underly-ing scale on which each is based: he diatoniclanguage s groundedin

the traditionalmajor-minor ystem,while the chromatic/enharmoniclanguageis based on the 12-note, equal-temperedchromatic scale.Other representativexamplesof these approaches o the nineteenth

centuryas a second practicemay be found, among other places, in

Benjamin1975, Krebs1981 and 1991, Stein 1985, Kindermanand

Krebs,1996.




Hugo Riemann and Arnold Schoenberg believed that the

differentiation between diatonic and chromatic music was a

matter of style, rather than substance;neither recognized a

strict division between the two styles. Unfortunately,neither

Schoenbergnor Riemann

developedan

analyticmethodol-

ogy sophisticated enough to present detailed analytical pic-tures of both vertical and linear dimensions. Without a sys-tem that allows the linear representation of harmony, any

analytical theory is impractical,and ignores what is perhapsSchenker'smost significantcontribution.7

In an effort to capture the powerful syntactic model of

a theory of harmonic function while also capturing the

linear-harmonic musical aspects as afforded by a Stufen-theorie,I adopt a Riemannian theoretical model: a harmonic-

function/bass-line system following lines developed byCharles J. Smith, and further informed by Harrison'sre-

newal of harmonic function. The analytic system holds to

the tenet that just as surfacevoice-leading events may pro-

long a harmony (in the Schenkeriansense of the term), lin-

ear events may also be brought into the service of harmonic

function. The Smith/Harrison distinction regards he nature

of harmonic function itself. Smith's function is more akin to

Schenker's harmony--it is a property of a chord, part ofwhat a chord is.Harrison's unction is an action-it is some-

thing that a chord does.Thus we are not speakingof func-

tional prolongation,per se, but rather,of functional persis-

tence, despite interveningvoice

leading activity.Thus,the

analyticnotation employed is designed to drawattention to

functionalconnections between chords and acrosspassages.The analytic notation used modifies Smith's approach

(Smith 1981 and 1986). It is a hierarchicalnotationdesignedto privilege the prolongation of harmonic function rather

than a specific harmony per se. At the layer closest to the

musical surface open noteheads indicate chord tones;filled-in noteheads indicate non-chord tones; and barlines

separate discrete harmonies (often trivial harmonies).The

structuralouter-voice counterpoint is identified with stems,while slursrelate the voice leading of non-chord tones to the

chord tones they embellish. Subsequent layers re-interpretharmonies as embellishing chords that prolong or connect

regionsgoverned by a particularharmonic function,showingthe interactionof harmonyand counterpoint.On deeper ay-ers, barlinesseparatefunctional regions rather than chords;the deeper one goes into the hierarchy, the closer one

approachesa Schenkerianbackground,showing a very large

expanseof Tonic

prolongation.While a more orthodox

Schenkerian notation could be used for this purpose, it

would not sufficiently highlight the aspects of harmonic

function that I discuss. In the interest of keeping the exam-

ples as compact as possible, I have compressedthe analytic

layers closest to the foreground to functional symbols ap-

plied to the music itself, and presented a reasonablemiddle-

groundlinearanalysis o supportthe discussion.

DETERMINING HARMONIC FUNCTION

Small-scaleandlarge-scaleharmonicstructuresareclassi-

fied by harmonicfunction and bass line context. Categoriesof harmonicfunction identify chords in terms of their rela-

tionships within familiar diatonic contexts, and providethe

7As an

analytical system,it is on this

pointwhere Harrison 1994

mayfall short. Harrison's three methods of analysis (segmental, linking, and

accumulative) generalize functional moments and points of functional

discharge according to segmentations based on the perception of an

imaginative listener. What Harrison gains in analytic flexibility, he sac-

rifices in analytic rigor and the more scientific notions of experimental

repeatability. Harrison never claims to provide a sophisticated system of

analysis; rather, he values the simplicity of his analytic notation

(Harrison 1994, 127). As a replacement for traditional models of coun-

terpoint, Harrison's new counterpoint weighs the functional dischargeof several individual melodic lines; these lines don't so much interact as

competefor functional presence. From the outset, he denies both the ap-plicability of Schenker's system to this music as well as attempts at revi-

sionism (see note 1, above). For arguments that critique Harrison from

a Schenkerian perspective, see Whittall 1995. Despite this shortfall,

however, Harrison casts a gauntlet that no scholar of harmonic function

can afford to ignore.



ASPECTS OF PLURALFUNCTION IN CHROMATIC MUSIC 253

basis for a harmonic syntax.8Principally,authenticprogres-sions involve directed motion from Tonic (T) to Dominant

(D) regions and typically return to Tonic; these categoriesshould require ittle defense at this point. The third signifi-cant

category betraysthe fact that harmonic function cannot

be defined by pitch-class identity alone, even in the simplestdiatonic contexts. While some authorspreferto consider all

IV chords as Dominant-Preparation (DP), others prefer he

label Subdominant (S). The former betrays a bias for au-

thentic models of harmonic progression, relegating plagalmodels as distinctly secondary;this seems to follow Classical

diatonicpractice,and is closely alignedwith Schenkerianas-

sumptions.The latter implies that plagal progressions ie on

equal ground with authentic systems as understood in a

Riemannian dualist model.10In this study,context shall dif-ferentiate DP function from S function; when speaking in

general terms about the sonorities involved, I shall use the

term "Subdominant."

In addition to reckoning the relationshipof a Klangto a

tonic pitch, the second principal functional determinant s a

chord'scontext. If we privilege the status of bass scale steps,we can categorizechords accordingto their linearconfigura-

tions.11 Following Harrison's bold lead (Harrison 1994), Ishall abandon Roman numeral

designationsfor all but the

most straightforwardharmonies.12The occasional Roman

numerals that are mentioned are provided only to aid the

8 All functionalcategoriesarecapitalized,o distinguishhe termDomi-

nant (the functionalcategory) rom dominant(the V triad),andTonic

(the category) romtonic (the chord),etc. Otherfunctionaldescriptorsthat do not share heir name with a chord are likewisecapitalizedor

the sakeof consistency.9 The termDominant-Preparationas firstused in Forte1979; he term

Predominantwould serveequallywell in this regard.io This position s well defended n Harrison1994.

ii The importanceof bass scalestepswhen reckoningharmonic unction

is too often undervaluedn treatiseson functionalharmony,with Smith

1981 and 1986 andHarrison1994 providingnotableexceptions.12 Smith 2003 alsopresents his challenge,but does offeranalternative.

reader; hey areneithermeant to convey functional meaning,nor to suggest a privileged status for the indicated root. In-

escapably,Roman numerals remain the lingua ranca of our

discipline, even if through them we retain biases that mayneed to be discarded when

considering highlychromatic

music.

Defining a single functional affiliation for each of the

diatonic chords is often problematic, enough so to promptHarrison to abandon the effort altogether. In a traditional

conception of harmonic function, we needed to rely on

definitions of harmonicfunction ascribed o a predefined,"ac-

ceptable"pitch-classcollection (Akkord).n contrast,Harrison

thoroughlyworks out a theory in which the individual scale

steps themselves bear the burden of expressingfunctionality

(Harrison 1994). This theory is most promising, and indeedrenews the notion of harmonic function in a powerfulway.For

readersnot well versed with Harrison'stheory, a brief synop-sis is provided n the appendixto this article.13

Because harmonic function resides in the constituents of

chords, we are given a tool with which to deconstruct any

sonority (Klang) and to reckon its harmonic function care-

fully. The argument is predicated on the proposition that

each primaryscale step (with its own orbit of governance)stands in a

uniquerelation to its Tonic.

More recently,how-

ever, conclusions in Harrison 2002 seem to challenge this

position. Harrison2002 builds a case to support the "uncon-

formed Tonnetz"as a model for key relations-a Tonnetzthat is not conceived as a closed system in equal tempera-ment (as on a torus), but rather one that may extend infi-

nitely in every direction, conceived in just intonation.14

13 Rifkin 2004 applied Harrison's theory in a promising way. Rifkin heeds

the call of Burkhart 1978, that a recurrence of a structural motive must

be presented in a similar or parallel functional context, but reckons thatfunction according to Harrison's theory. In this manner, Rifkin presentsa compelling theory of motivic/functional analysis in the music of

Sergei Prokofiev.

14 It is widely accepted that the elements of the Tonnetz may stand for

pitch relations or key relations. Pragmatically, of course, one does not




Harrison 2002 concludes that when a composertakes ajour-

ney through keys on the grid, let us say,moves from one C

on the grid to a different C on the grid, these two Cs are not

thesame.That is to say, he composerhas ended in a different

place; enharmonically equivalent pitcheson the

grid,or in-

deed anyrecurrencesof the same pitch class,do not have the

same meaning. In a way,Harrison broadens the scope of di-

rectional tonality; now, a piece may begin and end in the

same key, yet still have directionalproperties.Harrison 2002 tacitly contains the correction to what I

see as the basic error of Harrison 1994. On the unconformed

Tonnetz(Example 2, centered on a C majortonic triad), the

4 primitive for S function is not thesame4 that extends the

Dominant triad to a seventh chord.These two degrees stand

in a different relation to the originalTonic, contraryto thesuggestion in Harrison 1994, which asserts that there is onlya single 4 in relation to the Tonic, and that it is a primitive of

S function. The unconformed Tonnetzreconciles the percep-tual difficulty I have with Harrison's formulation of har-

monic function. Sometimes 4 expressesSubdominant func-

tion, and sometimes it expresses Dominant function. The

same point can be made about 2: sometimes 2 is an amplifierfor the Dominant and sometimes 2 comes about through an

extension in the Subdominant direction. Likewise the sub-

mediants: to my ear,k6 is perceived differentlywhen it ap-

pears in iv and when it appearsas the seventh of vii .This begs a new question. How do we know which4 we

arelooking at on the score?Allow me to presenta readingof

the major-minor system on the unconformed Tonnetzthat

remains faithful to a dualist mode of theorizing. For ease of

reading, I shall work with the C major-minor complex. In

Example 3, the shaded arearepresents he composite scale of

the major-minor system as generatedby the primarytriads.

In this figure, motions toward the West represent moves

towardthe Subdominant side (T to S, D to T) and motions

towardthe East representmoves toward the Dominant side

(S to T, T to D). The shaded area (the major-minorsystem)is sufficient to account for the three primarytriads and the

functionallyslippery mediants and submediants as contigu-ous lozenges on the Tonnetz. Each triad quality has a dis-

tinctive shape:an upward pointing triangle is a major triad

and a downward pointing triangle is a minor triad. Less

common, mixed structures are observed on the diagonals(Ak-C-E, A-C-EB, Ek-G-B, and E-G-Bk). The main

East-West axis contains the bases and associates of the three

primaryfunctions, and in the offset rows to its North and

South, we find the functional agents.This region forms the

main provinceof the key.The rows that are twice removed

from the main East-West axis contain the elements of moredistant modulations, and the strong tendency for a key to

dissolve.The pitches located in these regions point to Tonics

on a different East-West axisthan our home key,even if they

point to a different location of the same tonic (a differentC).Likewise, should one stray too far East or West, one might

easily get caught in another tonal gravity.However,the flexi-

bility along the main East-West axis is somewhat greater,

allowingfor D ofD and S of S regions to maintain an affilia-

tion(if

moreremote)

to theoriginal key.As essential seventh chords became fundamental har-

monies, we may look to this Tonnetz to stretch the basic

major-minor system. The first stretching element would be

to consider the seventh of V7. Projectedfrom the dominant

triad,we stretch the system to include the F found on the

Dominant side of C. If we were to mirror this action on the

Tonnetz,we would stretch the system Westward, invokingthe Riemannianpractice of generating a minor triad down-

wardfrom its dual root (its fifth, in traditionalterms).Thus,

we gain access to a supertonic triad that is fully on theSubdominantside of Tonic and a leading tone triad that is

fully on the Dominant side of Tonic. Since nineteenth cen-

tury harmonic practice typically permits dominant ninths

and leading tone sevenths, it is easy to allow for a hyper-

meander into the realm of theoretic key signatures and expect that the

music shall not be adjusted for sensibility's sake.




F C G D A E B, Fl C# GI

Ab E6 B6 C Gt DI A E B

B EG G 6 D 6 A 6 E B

EXAMPLE 2. Unconformedonnetzand4.

extension of the basic chromaticcomplex to include the A

and Ak lozenges toward the East. A balancingWestward-

extension towardB and B1 is also possible; strictlyfrom a

theoreticalpoint of view, there is nothing that would pro-hibit this move,and it seems to sustain an equivalent ustifi-cation as a duality. t must be noted, however, hat a hyper-extension in either direction results in the leading tone or

subtonic seventh chord. The harmonicfunction of the for-mer is at times slippery;that of the latter has a nebulous

function that rarelyhas an unmediated associationwith the

home key. It is often read as either Dominant of lIII or

Subdominantof IV.Example4 outlinesthis largerregion.A comment is warrantedregarding he functional status

of the hyper-extended lementsjust proposed.Notably, hese

pitches are located far from the originalTonic on the Ton-

netz.Significantly,wo of them they are found in the parallelrow, two steps removedfrom the main East-West axis. As

suggested, the axes that lie two steps above and below themain tonal axis of the key bring scale steps that may be

drawnto the gravityof another tonic. Specifically,hese two

axesaboveand below representmoves toward the sharpside

of the key and the flat side of the key respectively.As

Harrisonpoints out, a move toward the sharpside is a gen-eralizationof Dominant motion, and a move toward the flat

side, of Subdominant(Harrison 1994, 27). Thus, althoughthe hyper-extended7 is achievedby a Westward(Subdomi-

nant) extension,it crossesa Northern tonal field boundary.The newly acquired7 is once againDominant, but a Domi-

nant caught in the tonal gravityof a different 1-the one

found to its Southwest.The parallelargumentalso standsforthe hyper-extendedb6that extends into the flat-side axis,and its recoveryof Subdominantfunction.Therefore,paceHarrisonand Erpf,the vii' chordbecomesthe firstdiatonic

harmony hat is functionallymixedby nature.

Adjacentto the outlined area n Example4, we can rec-

ognize some basicchromaticchordsthat includeF# (stretch-

ing furthertowardD ofD function)and the Neapolitande-

gree, 62, extending furthertoward the West. If we stretch

furtherWestward,we reach the realmof S ofS.

These traditional diatonic harmonies of the complex

major-minor system form the experientialbasis for an un-

derstandingof harmonicfunction in more complexKliinge.I suggest that our understandingof harmonic function is

based on pitch-class groupings, according to two guiding




F1 c G D 1 F C G

Ab E6 b B Ct G A E B

E 6 ,

EXAMPLE 3. UnconformedTonnetz and major-minorsystem.

principles.The firstprinciple s functionalparsimony.Given

a choice, it is more natural to group together pitch classes

in closergeographicproximity.Thus, in a D-F-A complex,functionalparsimonywould dictate that the D results from

stretching the system toward the Subdominant side. It

would be decidedlyunparsimoniouso take the F-A from the

Subdominantside,and to pickup the D from the Dominantside of our home Tonic. One would need strong contextual

justification to locate a D-F-A complex completelyn the

Dominant side of Tonic, where it might be considered a

modallyunusualvariant of D ofD function.15 Alternatively,in a Dominant-sided D-F-A complex, the A might be

perceivedas a non-chordalpitch class againsta more stable

harmony--in which case the argumentfor an independentfunctionbegins to dissolve.(I shall return o this in the dis-cussion of linearchordsand Example34.) The secondprin-ciple is a left-privileged nterpretation,which allows the left-most element on the Tonnetz o overshadow he harmonic

function of elements on the central East-West axis, where

the contextis appropriate.When notes are added to a Klangin an Eastwarddirection, t is easyto imaginethese as added

factors to a tertian sonority: fifths, sevenths and ninths.

Added notes in a Westwarddirectioncould representaddedsixths (which might not disruptthe prevailingfunction of

the Klang),or new bases,which redefine he originalnotes asagentsand associates.Such a redefinition s bound to changethe functionalorientationof the sonority.While the A-C-E

complex might sustainTonic function by virtue of the pow-erful C-E Tonic elements, the left-privilege suggests that

this complex maylean towardSubdominant unction.

The topographyof the Tonnetzprovides a structure n

which to rethink the primitivesof harmonic unction.How-

ever,ourreckoningof harmonic unctionis incompletewith-out consideringourexpectationsof tonal syntaxand the lin-

earpatternsof the bassline.The paradigmaticauthentic harmonic progression is a

successionof chords T-DP-D-T (with DP optional).Any

contiguous segment of the paradigmis considered an au-thentic harmonicprogression;any element that disruptsthe

15 This argument assumes sympathy with the discussion of the minor

Dominant in Harrison 1994,passim, especially 53-54.




C O cl D

G G ES F

V

EXAMPLE. Unconformed

Tonnetz and extendedmajor-minorsystem.

paradigmis considerednon-functionalwith respect to the

progression.The paradigmatic lagalprogressions a succes-

sion of chordsT-S-T.16A categoryof linear chords s essential to an investigation

of harmonic unction.These chordsare often functional,but

they do not participate n typical paradigms.Examplesin-

cludepassing

orneighbor

chords(forexample,

apassingIV6chord between V and V6), chords that result from linear

processes(suchas the 5-6 motion above tonic thatgeneratesa vi6chord without a strongsense of Dominant-Preparationor Subdominant function), and many of the standard

4-chord techniques and sequences.The notion of a linearchord allows us to assess chordsby their voice-leadingand

metriccontexts,where a blindapplicationof functionaccord-

ing to identityis unmusicalandunconvincing.Linear chords

suggest a hierarchy, espite any sense of harmonic function

discharged n the process.A roster of typical bass lines, cataloguedby their linear

shape, is a useful accompaniment o the abstract unctional

paradigm.Such standardbass-lineprogressionsarecommon

among harmony textbooks, although they are normallyfound amidst discussions

pertainingto individual

chords,rather than organized according to chords that share the

same function and are built over the same bass scale step.17In this way, characteristicbass-line patternsand functional

progressionsbecome inextricablyntertwined n this system,each informingand guiding the functionalinterpretationof

the other.

16 In laternineteenthcenturypractice,Harrison ollowsRiemannn pos-tulatingthe possibilityof a dualparadigm,T-D-S-T. As this is the

most problematic aradigm, shall leave t unconsideredor the time

being.While I am sympathetico the model,its considerations not

germaneo mypurpose t the moment.

17 Forexample,AldwellandSchachter 003 address he behavior f IV,

ii6, IV7, and ii as they move to variouskinds of cadentialand non-

cadentialDominantchords,but these discussionsarewell dispersedthroughouthe text.Certainly, hen thesediscussions reconcatenated

accordingo bass-linemotion, here s a danger hat somesubtlety e-

gardingeachparticularDP(4) chordmightbe lost,but this is at the

expenseof the clearpresentationndimportancef bass-line ohesionin functionalharmony.




COLLIDING FUNCTIONS

Progressions are classified according to their bass lines.

Within an unembellished harmonic paradigm, nine

Dominant-Preparation to Dominant possibilities and sixDominant to Tonic possibilities comprise the common au-

thentic progressions.While there are only a few common

Classic era bass lines for plagal progressions, nineteenth-

century practice greatly expanded the possibilities. These

bass-line rosters, given in Examples 5 and 6, can be con-

firmed in most standard theory texts by concatenating the

discussions of appropriatechords and their typical resolu-

tions. Exceptions to these practices may constitute "marked"

musical events, but certainlyare not so prevalentas to invali-

date the theory as a whole.18Examples 5(a) and 5(b) present rosters of authentic bass

lines according to their functional disposition; Example 6

presents the comparableroster for the nine plagal bass lines

found in nineteenth-century music. Using these figures,we

see bass lines that may representmore than one type of pro-

gression (authentic or plagal). Any bass lines found in both

authentic and plagal contexts are considered functionally

ambiguous, and therefore should be eliminated from a new

list of bass lines thatcharacterize

neparticular

functional

disposition. For example, since a 4-3 bass line might reason-

ably support either D-T or S-T progressions,4-3 is elimi-

nated as a characterizingbass line. On the other hand, it is

reasonable to declare that a 4-I bass line is emblematic of a

plagal progression, since the only typical context for a 4-1bass line is S-T. Similarly,bass lines that could reasonably

arpeggiate a single harmonic function (such as 4-2, which

can potentially support either DP-D or DP-DP) and bass

lines that might commonly support T-D or T-S are also

eliminated, since they do not characterizeany single func-tional disposition. Rosters that summarize the possiblecommon-practice motions of T-D or T-S are not terribly

valuable,because the motion from Tonic is generally unre-

stricted. Example 7 catalogs the bass lines that are elimi-

natedon the groundsthat they do not characterizea particu-lar functional disposition; Examples 8 and 9 provide the

resultinglists of

characterizing authentic and plagal basslines respectively.

Examples8(a) and 8(b) tell us that a 5-i bass line is, in it-

self, strongly suggestive of authentic D-T harmony:there isno legitimate plagal succession with such a bass line.19Like-

wise, Example 9 shows that a bass line that moves from 4 toIstrongly ndicatesplagal harmonyfor the same reason. It is

not surprising hat the resultingrosters nclude only motions

from the paradigmatic S and D agents and bases. When

Dominant or Subdominant agents are placed in the lowest

voice and move to the Tonic Base, that condition alone issufficient to supporta strong paradigmatic unctional articu-

lation. Having postulated the tonal focusing power of bass

lines, we must still consider the implications of the uppervoices.

Any single upper-voicepitch maybe a constituent of sev-

eral different diatonic triads or seventh chords-an obvious

point with an important ramification.Since any pitch may

imply one of severalpossible chords, a single pitch within a

Klangcannot

representa

singlechord. The unconformed

Tonnetzdiscussed above adds depth to this observation.In a

major-minortonal context, there may be different attitudes

available or many different scale steps.That is to say,there

are two distinct kinds of 4 in this system: one is found due

West of the Tonic (a strong Subdominant location) and

the other is an off-axis extension of the Dominant in the

stretched system. The principles of functional parsimony

suggestedabovearguethat we should look at the companionscale steps in the Klang to help us determine which 4 we

have beforeus.

Example4, in its extended version,shows us that we have

only one 1 and one 5 in the system.If one of these is the left-

18 The term "marked" s used in this context following Hatten 1994. 19 See also Harrison 1994, 48, Example 2.1, which confirms this finding.




DP D

2

4 -

2

4

7•i - 7

D - T

2 - i

- (b)3

A

4 - (~~3(b)

EXAMPLE 5. Roster

of.authentic

bass ines.

S - T

i - i2 - 1

((03

(-)b ) 3

EXAMPLE 6. Rosterofplagalbass ines.

most element of the Klang, then it will likely subsume the

function of the Klangto its purpose.If we find ourselves at a

different 1, then the music has undertaken a noteworthyjourneyto a new place.20There is also only one pairof Tonic

agents (3, b3),and two pairs of Subdominantagents (6, b6)

Bass Line T-D T-S DP-D D-T S-T Other

i-i x x

1i- x x2-i x x

()3-i X X T-T

(6)3-2 x x x

4_7 X D-D

4-4 x S-S

4-(2X X

4-2 X D-D

EXAMPLE 7. Non-characterizing ass ines.

DP D

4 5

(06)57

(a)

D - T

-i

(b)

EXAMPLE . Characterizing bass lines: authentic.

S - T

4 - i

(0)3

EXAMPLE 9. Characterizing bass lines.:plagal.

and Dominant agents (7, b). Regardingthe two Subdomi-

nant and Dominant pairs, one is found close to the Tonic

whereas the other is in the hyper-extended tonal field. In0 Cf Harrison 2002.




either case, invoking the more remote agent will requirean

extraordinary circumstance.21The viif chord is striking in

that it invokes a remote agent, attesting to the slippery na-

ture of this symmetricalharmonyand accountingfor its abil-

ityto

slipso

easilyinto a new tonal field. This observation

supports Harrison'sposition regarding the devotion of the

functional agents to a single tonic, even if it turnsout to be a

different manifestation of the same tonic pitch. RegardingDominant agency,the last case suggested (a Dominant with

?7agency) is ratherextraordinary.n most cases,left-privilege

suggests that the Klangwhich containsk7will function as ei-

ther T or D of S. In a more extreme case, the context might

suggest that k7 s found off the Western edge of the major-minor complex, and that ?7 reaches toward S of S function.

However, two versions of 2 and 4 can be alignedwith eitherstrong S or weak D. The closest chromatic scale steps to the

major-minor tonal field are#4, available n close proximityas

the first step towardD ofD function,andv2,

vailable n close

proximity on the subdominant side.

II. SUBDOMINANT-DOMINANT COLLISIONS

The criteria above constitute a new instrument for ob-

servingharmonic function. When

evaluatingharmonic

function, two elements must be observed-the bass line, and

the component scale steps of the chord in their relation to

Tonic on the Tonnetz.Characterizingbass-line patterns arethe primary determinants of harmonic function. In tradi-

tional harmonic language, the function of the upper voices

(the sonority) typically agreeswith the function of the bass-

line pattern. Occasionally,however,these two elements con-

tradict each other. For example, the bass line 1-4-i charac-

terizes the plagal paradigm T-S-T; however, if the chordabove 4 contains a leading tone that resolves correctly to

tonic with the change of bass (i.e., it behavesas an authentic

leading tone),then

there is an inherent contradiction be-tween the presence and resolution of the leading tone (the

agent of Dominant function) and the motion of the bass

(one of the bass lines that characterizesa plagalparadigm).If such phenomenawere rare and confined to the musical

surface, t would suffice to comment on the curious effect of

the progression,and to explain the processes that brought it

about. Such contradictoryprogressions appearwith surpris-

ingly regularity in the music of the nineteenth century.Given that they appearso frequentlyon both the surfaceand

deeper levels of structure, t makes sense to investigate thefamilyof chordsthat contain collisions of SubdominantandDominant functions.22

21 One might also locate 6 due East of the Dominant associate 2 (three

fifths to the East of Tonic). If that 6 is used, then we are getting re-mote, but we are still along the main East-West axis, and are perhaps

engaged in "Dominant Accumulation?" (D of D of D) as per Harrison

1994, 153 f Note that the remote version of b6 is further removed in

the Southern direction, and is much more prone to behave as Sub-

dominant of a new tonal center.

22 A comment regarding Harrison 1994 and his discussion of Mixed

Function (a concept he credits to Riemann and Erpf) is necessary at

this point to clarify the difference between our discussions. Harrison

uses this term to describe chords of inherently mixed function: "Since

secondary triads contain scale degrees associated with different func-

tions, theyare

functionallymixed

structures,able to communicate more

than one function"(Harrison 1994, 60). That is to say,since 2 is associ-

ated with D harmony,while 4 and 6 are associated with DP harmony, a

supertonic triad is functionally mixed. Of course, it is quite a different

matter to claim that a chord has the capabilityto express more than one

function (Harrison, after Erpf) versus the claim that a chord expressestwo functions simultaneously,as I shall argue below. In the applicationof Harrison's version of mixed function, conflicting and ambiguousfunctions are weighed to give a harmony a three-dimensional func-

tional vector. My reformulation of function-finding obviates the major-

ity of diatonic chords that have this property, with the exception of

viio7. The kind of functional mixture I speak of here is the very particu-lar mixture found when a characterizing functional bass line conflicts

with upper voices that characterize a different function. In Harrison's

use of the term, mixed function chords are common diatonic phenom-ena. While the KldngeI explore are, by comparison, less common theyare nevertheless important and worthy of consideration.




There aretwo distinct ways for Subdominant and Domi-

nant functions to collide. The first type is defined as a chord

whose bass scale step participates n a context that character-

izes a Dominant-Preparation or Subdominant chord, but

which also contains the Dominantagent

in anupper-voice.These chords shall be designatedby their primary basscon-

text) function with a superscript D' indicating the element

of Dominant function in the upper voice. Thus, the two

paradigmatic contexts for this first type are T-SD-T and

T-DpD-D-T. As with any structuralparadigm,the basic

forms of these possible progressions may be decorated

through various non-functional chords or successions of

chords that share the same function. Example 10 presentsthe basic structural contexts that contain SD and DPD

chords. It is noteworthy that the mixed-function chordsin these two paradigmshave markedly different effects. In

the first instance, the Dominant element in the SD chord

appearsbetween two manifestations of Tonic; in the second

instance, the Dominant element in the DPD chord antici-

pates the arrival of the stronger, authentic Dominant.

Nevertheless,even though the two contextsrepresentdiffer-

ent manifestations of a chord containing a functionalcolli-sion, the collision is the common element that allows us todrawa

similaritybetween them.

The second type of collision occurs when a bass scalestep

contextuallycharacterizesDominant function,but its essen-

tial harmonic character s imbued with Subdominantfunc-

tion, accordingto the function-finding techniquesdiscussed

above.This type shall be indicated in the same manner as

above,with roles reversed:Ds. A Ds chordcannot,by defini-

tion, exist in the context of a plagal succession-it mayonly

appearover a characteristicallyauthentic bass.On the other

hand, in the context of an authentic progression,a Ds chord

may stand in place of an expected Dominant chord in thesame fashion as a SD chord.The theoretic contexts of pro-

gressions involving Ds chords are given in Example 11. We

shall return to Ds structures ater,afterexploringthe SDand

DPD categoriesin greater depth.

SDchordsnplagalparadigms

In theory,the basic form of the plagal succession T-SD-T

may appear over any characteristically plagal bass line, as

defined in Example 9, above.

Interesting examples of SD progressions appear over(b)6-1 or 4-1 bass lines, where the leading tone is occasion-

ally spelled enharmonically as bi.23 Note that while Kbis

proximate to the Tonic on the Tonnetz, t is twice removed

from the main East-West axis in relation to the Tonic. An

authentic (hypothetical)bi seriously challenges the nature of

a tonal system, perhaps in a way that cannot be sustained.

Most importantly,spelling such a pitch as b1 and treating it

contrapuntallyas such arevery different things. In every case

I'vefound, b1appears

o bespelled

as such to form a tertian

sonority that is easy to read, yet it is contrapuntallytreated

as a leading tone. (Refer to Example 1, where b1 (G?) was

treated as a leading tone (Fx) in the key of G# minor.)

Therefore, on functional grounds, I find it more compellingto challenge the spelling of the note and to understand it as

the Dominant agent, thinly disguised, thus occupying a dif-

ferent (disjunct)location on the Tonnetz.In these situations,

Ki s a proxyfor 7.

Because the SD chord classification is inherently ambigu-

ous in a traditionalsense, an obvious choice for the label andnotation of each chord is likewise unclear. SD chords could

be reasonablyshown as altered, or non-standard Dominant

chords, or conversely,as altered Subdominant sonorities.24

23 While I do not discount he possibility hat the characterizing ass line

(b)6-(b)3 although admittedlythe weakest of the characterizingbass

lines) may support uchstructures, haveyet to find convincingexam-

plesto demonstratehispattern.

24 Havingattempted o reduce all chordswith colliding functionsto ex-pressionsof singleRomannumeralswith appropriate lterations, havediscovered hat there s simplynot a singletidysymbolthatwill satisfyall casesand conditions.I am grateful o Daniel Harrison for warningme (both in his book, and in reading an early draft of this paper)against etishizingRomannumeralsymbologyat the expenseof a clear




Plagal Bass

T-SD-T

T-S-SD-T

AuthenticBass

T-DPD-D-T

T-DP-DPD-D-T

EXAMPLE IO. Possible contextsfor SD andDPD chords.

Authentic Bass

T-DS-T

T-DS-D-T

T-D-DS-T

T-DP-DS-T

EXAMPLEII.

Possible ontextsfor

Dschords.

As the purpose of this article is to explicate the functional

behavior of certain Kldnge,the analytic label for SD chords

simply appendsthe bass scale step to the functional designa-tion. (While this symbol does not discriminate between all

of the possible sonorities for the category, he accompanyingdiscussion shall make those differencesclear.)

Ahealthy

roster of SD chordsmay

be derivedby placingfamiliar diatonic and chromaticvii' or vii' chords over either

a bass 4 or (6)6. Example 12 providessuch a roster of possi-ble

SD(,)and SD(b6)chordswith their scale steps;it also in-

cludes misleading Roman numerals that might be recog-nized through common enharmonic spelling. The most

common enharmonicspelling substitutes 1 for 7, such as the

G? in the 6vi chord of Example 1. 7 by proxy produces a

comfortable triadic spelling, despite the linear behavior of G

as a lower neighbor, Fx. The last column on Example 12 is

presented to draw attention to this equivalency in the con-

text of a Roman numeral interpretation. By definition, all SD

chords are assumed to resolve to a T(i). In an authentic par-

adigm, this roster would also suffice for DPD chords resolv-

ingto

D(5)chords. This list is not

exhaustive,but it

providesinsight into the nature of the chords and their possible

spellings in the literature.

SD(4) and SD(6)

SD(4) chords appear as embellishments of plagal cadences

in some liturgical pieces such as the traditional hymn-tune

Caithness, given as Example 13. Although the SD(4) chord

sounds innocuous due to its embellishing context in a

T(i)-S(4)-SD(4)-T(i) succession, its features are nonetheless

evident. First, the function of the succession is unaffected bythe decoration because it is governed, at the deepest level, bya diatonic plagal cadence; that is, its resolution denies the

Dominant function often afforded to an unqualified vii4 in

favor of recognizing the chord's Subdominant potential.

Second, the insertion of the leading tone before the resolu-

tion to tonic strengthens and directs the progression more

forcefully than the unadorned plagal close. For these reasons,

vii3j behaves as SD(4) rather than D(4). On the Tonnetz, this

manifestation of vii' is achieved in thehyper-extended

field

to the North-West of Tonic. The chord is principally S(4),which has been stretched to include a Dominant agent.

An example of a similar sonority in a non-liturgicalcontext is found in Robert Schumann's Kreisleriana, no. 5,

shown in Example 14.25 Here, the leading tone, F#, embell-

representation f harmonic unction. It is certainlypossibleto examinethese harmoniesas Roman numeralswith complex figuredbass ap-pendages,but suchsymbolstend to obscureandcomplicate he discus-sion of harmonic unction.

25 This piece is also discussed in Agmon 1995, 210, Aldwell and Schachter

1989, 392 and 2003, 418. Agmon exhibits the passage to support his

case for the weak Subdominant function potential of VII7. However,

the strength of Agmon's argument is undermined bythe

privileged sta-tus given to chordal roots in his theory. It must be stressed that it is

only in its second inversion (over the bass scale degree 4) that anySubdominant function may be realized. Although the first edition of

Aldwell and Schachter 1989 does not acknowledge the implied mixed

function of this progression, the 2003 edition makes this explicit.




Genus Quality Structure RN(?)

dmd 4 2vii..

dm1,no 3rd 4 vii"4

sD)

(4 ) ... .... ..................................

d 4 b vii1, (iv??)

dd7 b 2 ^4 vii112

d6 b 4 viil, (iv6?)

Fr 3 4 1 #2i IVo0 6)

sD4) dm+6

,30 4-b (26ivL,

ivo6A"gm4 a 5 7 i V+4dm # 4v"4

sD(,7) m• 2 V vi

Mmi b8 2 b3II,

EXAMPLE 12. RosterofSD chords.

ishes a plagal progression, resulting in a SD(4) chord that is

approachedfrom its diatonic subdominant and resolves to

a root-position Tonic chord. While Aldwell and Schachter

(2003) point out that such treatment of a diminished-

seventh sonority built on the subdominantis not rare,I am

especially interested in the germinal propertiesof this pro-

gression,and in the fact that many more complex examples

might be related back to this simplercontext in a larger ax-

onomy of chromatic chords that arenot so easily explained.

Example 15 (Schubert's Phantasie,op. 15, mm. 28-29)provides an interesting context for a SD(44) tructure.The

passage is situated between an opening section in C majorand a structural cadence in the dominant;it can be read as

the pivot region between these two keys. In measure 28,

Oia

E: 16 'V6 (vii/IV) IV sD(4) I

(D)

E: T(i) S(6/4) T(i)

EXAMPLE 13. Caithness.

Schubert embellishes a V3 chord in an approach to I6 in C

that is expanded with 6/10 voice-exchange between the

outer voices, moving to a G-major dominant triad. The fig-ure repeatsbefore continuing to a cadence in the key of the

dominant. The pivot harmony is thus the C-major triad in

m. 29: Tonic becomes IV of G. Themodulating dominant,however,is a SD(4) chord in the key of G. Rather than mov-

ing to a traditionalroot-position C-major triad in the 6/10

voice-exchange, Schubert includes inner-voice motion that

transforms he C-major triadinto a secondaryvii4j of G thatresolves to G over a S-T (4-1) bass line relative to the new

key.This is a decidedlyweak modulatingdominant that eas-

ily allows the ear to remain anchored to the key of C for a

repetition of the two-measure figure,before continuing and

confirming the new key.A noteworthy feature of this exam-

ple is the inclusion of the secondary 6 in the SD(4) chordratherthan b6,which is by far more common.

The examples to this point have looked at SD(44)chords

that function as embellishmentsof diatonic S-T progressions.

Examples of the SD(4) are more convincing when they are




sehr ebhaft

,6 a ,I-lG: iv SD(4)

G: S(4) (D) T(i)

EXAMPLE 14. Schumann,Kriesleriana,op. 16, no. 5, mm. 51-53.

freed from a decorative role. An instance of an independentSD chord resolving directly to a root-position tonic triad is

found in the opening to Brahms's Academic Festival Overture,

op. 80, given as Example 16. Here, the leading tone B? is su-

perimposedover a

repeating1-4-1 bass line in C minor, cre-

ating a conflict between a plagal bass and an authentically re-

solving leading tone in the top voice-precisely the categorySD as defined.

A SD(4) chord of a different variety may be found in the

final measures of Richard Strauss's Till Eulenspiegels lustigeStreiche, op. 28. The chord in the second measure of Ex-

ample 17 (sometimes called the "Till Sixth," Bb-Db-E?-G#)is an augmented sixth chord whose characteristic augmentedsixth (Bk-G#) resolves to 3 (A?) of the following F major

tonic triad, over a structural 4 to 1 bass-a sonority that hasstrong Dominant associations (by virtue of the leading tone,

E0) resolving to tonic. The Till Sixth thus combines the Sub-

dominant and Dominant elements necessary to qualify as a

SD(4) chord. Two measures later, Strauss uses a similar chord

4 6117 3V3,6"

[V]G: IV6 SD I

C: D(2) T(3/1i) [V]G: S(6/4) T(i)

EXAMPLE15. Schubert,Phantasie, op. 15, mm. 28-29.

with two small differences. As the upper voice arpeggiates

through the tonic triad over the final four measures, another

SD(4) supports 5 in the soprano, with 6 appearing in an inner

voice rather thanb6

as used in the earlier TillSixth,

above. In

this case, the resulting SD(4) chord might appear to be an

embellished third-inversion V+9resolving to tonic over a 4 to

i bass line, but such a Roman numeral analysis seems grossly

misleading. The final embellishing chord of the piece is a

plagally-resolving German augmented-sixth chord.

Hypothetically, Strauss could have spelled the Till Sixth

by proxy (with bi and kb), and used a contiguous segment of

the Tonnetz to do so (see Example 18). But this spellingwould have obscured the linear behavior of the chord, and

hence its function. The representation in Example 18 showsthe functional behavior of the individual elements of this

chord and demonstrates its resolution to Tonic from both

the Subdominant and Dominant sides. On this figure, the

nature of the functional discharge is explicit. The 4-1 motion




,r ?

pp sepre e sotto voce

SD(4) i SD(4)

EXAMPLE I6. Brahms,Academic Festival Overture, op. 80, mm.

1-2.

is due East, representing base-to-base, S to T motion. b6-5 is

S to T, agent-to-associate motion; the 7-1 motion is exactlysymmetrical to b6-5 with a Dominant agent to Tonic base

discharge. The last motion, #2-3 accompanies 7-i in parallel

motion, and thus has a latent D-T of 3 functional discharge;

upon the arrival of 3, it is subsumed by the Tonic base, and

thus #2-3 is a kind of synchronized (as opposed to successive)dominant accumulation, reconciled in the strength of the

1-3-5 Tonic Klang. In short, S and D functions are equallybalanced in the discharge to T.

The

openingseven measures of the

Adagioof Bruckner's

Ninth Symphony present a passage that defies traditional

Roman numeral analysis, but which can be illuminated func-

tionally. The passage is given as Example 19(a). The move-

ment is in the key of E, and the opening phrase ends on an E-

major triad; as no other clear position-finding elements are

available, this information will suffice to provide a starting-

point for the analysis. This accepted, the opening unharmo-

nized Bhfunctions as 5 of E, whose promise to move to tonic

is fulfilled by a substitute Tonic chord with the E# in the bass

of m. 2. The upper voices of this chord, however, complicatesuch a simplistic opening gambit. A page from Harrison

1994 supports this interpretation: the chord built on E# con-

tains the bass and agent of the Tonic triad, displacing the

associate by the two semitones on either side of it, A# and

C#. Certainly, he 5-i bass line provides sufficient groundingfor this interpretation.But it is what follows that is especially

intriguing.The E# chord initiates an elaborate tonicization of A,

shown inExample

19(b). Theleading

tone of A isdelayedthrough a stepwise chromatic ascent in the upper voices,

against which the bass moves to D# (4 of A). D# is harmo-

nized with a transposition of the Till Sixth chord before

leaping down a fourth to A#. The 5-4-1 bass-line motive,

together with the SD(4')chord, is then cleverly used as a

structural motive to tonicize A. Immediately following, this

same motive is used to effect a tonicization of D major,which arriveson the downbeat of m. 5.

When the bass leaps to A# on the third beat of m. 3,

Bruckner resolves the SD(4) chord deceptively by flatting thefifth of the goal chord (the A majortriad),which then serves

as a chromatic dominant of D (V65), and initiates the subse-

quent tonicization. (See Example 19(c).) The chromatic

dominant passes through a bass G# on the way to G#, fillingin the 5-4 motivic bass line with a passing #4 that supportsan enharmonically respelledviiW/V.4 then supports another

SD(4), similar to the "Till Sixth" but for the substitution of

6 for b6.Thus, the bass line and the functional motive used

to tonicize A in m. 3 is elided into an embellished version of

the same progression(adding a passing chord and slightly al-

tering the color of the SD(4)), which tonicizes the key of D.

The remainderof the passageis much clearer;Bruckner uses

the D majortriad as a diatonic pivot chord, from which he

moves to a half-cadence in the key of A, closing the phraseon the first E major triad of the movement in the famous

quotation of the Dresden Amen.

In summary,plagally resolving SD(4) chords seem to be

found in only a few specific contexts. Not surprisingly, the

most common approachchord to a plagally-resolving SD(4is either a diatonic Subdominant chord with 4 in the bass

(SD(4)), or a move directly from tonic. Approaching SD(4)chords from S(6) is uncommon; an approachfrom an altered

Dominant (as in Bruckner'sSymphony no. 9 "Adagio") is




i 6F 44n

ISD(4) I D(4) I

Gr

T-,T(i)

EXAMPLE 17. R. Strauss, TillEulenspiegels lustige Streiche,op.28, last six measures.

clearly extraordinary.Only occasionallydoes

6appearas part

of the SD(4) chord at all; there is a clearpreferencefor b6 n

this context. When 6 is found, the chord resolvesto a major-mode tonic.

Plagal bass:SD( 6)-T(J)

While 6 in a major key may theoretically support a SD(6)

structure,I haveyet to find examplesthat areconvincing.On

the other hand, SD(I6) chords areamong the most common

manifestations of SDstructures n the literature.

In the minor-modeprogression,T(i)-SD( 6)-T(i), b isoften held as a common tone, while I and 5 expand outward

to 7 and L6respectively, creating an enharmonically spelled6vi chord.This voice-leading paradigm s precisely llustrated

in Example 1, the "Tarnhelm"motive fromWagner'sRing. It

is in this context that the chosen notation "SD" sidesteps the

problems encountered when forcing a Roman numeral to fit

the circumstance. The symbol Mvi undamentally misrepre-sents the linear behavior of the leading tone, whether or not

it is spelled with the common enharmonic notation of bi.Likewise, the linear behavior of #2 is belied by the common

spelling as b6.26However misleading the context may appear,the fact remains that bk is, for all purposes, a functional lead-

ing tone, despite the plagal bass line. The spelling of #2 as g3

26 The interchangeablepellingof#2andb3 s commonplacen both chro-

maticDP and chromaticD chords,dependingon the modalityof the

tonic that follows.For example,when a Gr5resolves o a majormode

tonic (or cadential-6) b3 is usuallynotated with #2, and yet alternate

symbolsarerarelyproposed or such substitutions.




* ** A0

EXAMPLE I8. "Till Sixth"resolutionon the Tonnetz.

(a) Bruckner,Symphonyno. 9, iii, A4dagio,mm. 1-7 (score).

EXAMPLE 19




Ti

A: D() sDA: D(5) SDk(T)

(b) Bruckner,Symphonyno. 9, iii, Adagio, "mm. 2-3.

Ai

ITI

A: "T(1)"

D: D(5) (P2) sD(4) T(i)

(c)Bruckner,ymphonyo.9, iii, Aldagio,mm. -5.

EXAMPLE19. [continued]

is then simply a maneuverto show the chord in the music as

a tertian triad, avoiding the harmonic doubly-augmentedsecond from b1to #2,preferring nstead the majorthird,bito

b3.Furthermore,there are instances when 4 is added to this

basic sonority (as we shall see below) and the same unusual

spellings appear.

A passagefrom Franck'sPiano Quintet in F minor,shownin Example 20, demonstrates the same SD( 6) chord as

Wagner's"Tarnhelm"motive,but one that resolves to a majormode tonic. In this passage,Franckalso simplifiesthe spellingof the chord to appearas a 6vi triad in the music.The linear

analysis respells the appropriate notes to demonstrate their

linear, rather than their purely harmonic behavior.

In the Overture-Fantasy to Romeo andJuliet, mm. 32-36,

Tchaikovsky provides a similar context for the SD('6) chord

as above, but includes 4 in a T()-SD( )-T(i) progression.The passage, omitting the Harp part, which arpeggiates the

framing tonic chords, is given as Example 21. In this exam-

ple, the leading tone E# is part of an arpeggiation in the

Flute part, clarifying its chord-tone status; 5 moves to a

neighbor note b6 while K3 is retained as a common tone

through the progression. All this appears over a 14-6-1 bass

line. While the upper-voice 7-b3 diminished fourth (in the

Flute) appears anomalous with respect to normal Dominant-

to-Tonic voice leading, the authentic resolution of the lead-

ing tone (7-1) is present in the English Horn and FrenchHorn II.

A remarkable manifestation of a SD(I6) formation ap-

pears in the famous aria "Nessun dorma" in the third act of

Puccini's Turandot.27As shown in Example 22, Puccini pre-sents a context in G major, where the bass note b6 (Ek) sup-

ports a V' triad with an added seventh. In the first presenta-

tion, 5 (D#) is held across the upper voice of the progression,

stabilizing the progression. In subsequent presentations, the

voice moves to the seventh, 4, which resolves to 3 in the fol-

lowing tonic chord, while the accompaniment parts maintain

5 as an essential element of the harmony. Thus, acknowledg-

ing the well-known difficulty of finding a satisfactoryRoman numeral for what might appear to be a Dominant-

ninth chord in fourth inversion, this instantly recognizablechord is better left with the generalized analysis of SD(I6).

Furthermore, it would clearly be a stretch of the imaginationto think that the bass tone b6 is any kind of ninth. As in the

previous examples, this bass tone is operating independently

of the whole sonority, although in this case, the situation ismore extreme.

27 I am gratefultoJohn Cuciureanfor bringing this exampleto my attention.




Allegro

C#: I sD6) I

C#: I

EXAMPLE 20. Franck,Piano Quintet(F minor), i, mm.90-93.

Authenticparadigms: DPD (4-D(5)When placed in an authentic context, where DPD chords

resolve to traditional Dominant chords, the nature of the

leading tone in the mixed-function chord subtly changes.Rather than bearing the entire weight of the functional dis-

charge from Dominant to Tonic, the leading tone in this

context anticipates the Dominant proper, blurring the func-

tional boundary between the intermediate Dominant-

Preparation chord and the Dominant that follows. The bass

line 4 to 5 is clearly laden with the functional rhetoric of

DP-to-D, but Pablo de Sarasate complicates this progres-sion in his Zigeunerweisen for Violin and Piano, op. 20, given

as Example 23. In this example, the leading tone, B?, arrives

prematurely above the bass 4 in m. 14; thus, the resulting

progression intensifies the Dominant-Preparation chord

with Dominant function and thereby anticipates the Domi-

nant of the passage atop a DP bass. Strikingly, in this con-

text, the arrival of the cadential- provides the resolution of 7in the DPD chord, but this "resolution" is also an anticipationof the authentic resolution in the ultimate tonic.

Example 24 shows a similar progression in Schubert's

Moment Musical, D. 78, no. 3.28 In mm. 34-38, Schubert

28 This example is also used in Agmon 1995, 210, and Aldwell and

Schachter1989, 534 and2003, 572.Again,in Agmon's analysis t is not

the VII Stufe hatdefines he subdominantfunction,but rather he bass

line. Schubert'suxtaposition f a chordthat contains a functional ead-

ing tone againsta 4 to 5 bass line creates he functionaltension. Also,the earliereditionsof Aldwell and Schachterdo not acknowledgethe

mixed functionalpotential

of thischord,

but in the 2003edition,

the

authorshavereinterpretedhe progressiono be a diminished-_chord

thatstands n for a ii56chord, he chord-tonesof which (not presentin

the music)are"elided"nto the neighbortone embellishments.How-

ever, heseanalysesaddress he voiceleading originsof the passage,and

do not purport o clarify ts harmonic function. If the leading tone is




Fl.I, II

1 1

Cl. I, II

Eng. Hn.

61:,

6. |

..sn. I, II

and Cb.

F: i sD(,)

i

ii

F: i

EXAMPLE 21. Tchaikovsky, Romeo and Juliet, Fantasy-Overture, mm. 28-33.

presents a bass line that proceeds from i-4_#4-5-i, clearly

implying a plain functional paradigm [T(i)-DP(4i)-D(5)-T(i)], into which a secondary dominant to V has been in-

serted. However, instead of the traditional DP(4) chord that

would be expected in m. 35, Schubert prematurely intro-

duces Dominant function into the chord by superimposingthe leading tone, E?, on the bass 4. The result is a DPD(4)chord that is prolonged through a non-functional, passing

secondary dominant of V on the way to V7 in m. 37.29

Authenticass. PD(B6)-D(5)It is apparent from the literature that when SD( 6) chords

are used plagally, they are not normally embellished; rather,

they normally embellish Tonic chords. This is not the case

when DPD( 6) chords are used in an authentic paradigm.A common manifestation of DPD('6) is the viio4 chord.

By definition, a Dominant functioning chord with b6 in the

bass cannot resolve functionally to tonic-b6 in the bass

must resolve down to 5, over which afunctional Tonic chord

is extraordinary. Rather, viio4 normally dissolves into thedominant as an embellishing chord, functioning instead in a

weak pre-Dominant role, while already containing the lead-

ing tone. An oft-cited example of this progression is found at

the beginning of the development section in Beethoven's

"Pathetique" sonata, op. 13, given as Example 25.

In this example, Beethoven uses the fully-diminished sev-

enth chord as an enharmonic pivot between the keys of

G minor and E minor, where viio4 in G is reinterpreted as

viio, in E, which subsequently resolves to a V7 chord of E.

V7 of E is then prolonged until the Allegro molto e con brio.

Although the notation viio4 correctly identifies the pitchcontent of the harmony, it does little to engage the subtle

element of DP function in the progression. However one

chooses to label the sonority, DPD(b6) remains a helpfulmode of classification. Furthermore, this notation acknowl-

edges that the pivot undergoes more than an enharmonic

indeed standing in for 1, does it bear the function of i (not present) or

does it retain the function of the leading tone that is actually heard?

Aldwell and Schachter's analysis suggests that we attribute supertonic

function to the chord while the leading tone is, in concept at least, a

dissonant neighbor. My analysis claims that both functions are presentand are in conflict, intensifying our experience of the moment. For

more on "elision" n Schenkerian analytic practice, see Laufer 1997 and

Cadwallader and Gagne 1998.

29 Laufer 1997, 218, note 9 shows an example from Mozart's Piano

Concerto, K. 481, that is exactly parallel to this circumstance. He ar-

gues that the origin of the voice leading is found in the Schenkerian

concept of "Elision." Through elision, Laufer demonstrates that the

DPD(4) chord has its origin in a diatonic DP(4), with which the com-

poser has elided a neighbor-tone 7.




Andante ostenuto

THE PRINCE p

Nes-sun dorma ... Nes-sun dor-

ma ...

i4X

G: ISD(I) I SD(

G: I

EXAMPLE 22. Puccini, Turandot,Act IIL/1, "Nessundorma."

reinterpretation. t also undergoesa subtle functional trans-

formation,where the Dominant functioningchordin the old

key mutates into a chord with Dominant-Preparation unc-

tion in the bass. This new interpretationof an old problemadds to our understanding of this aspect of many enhar-

monic modulations by recasting the functional behaviorof

the pivot chord in the context of the new key,where it pre-

paresthe modulating dominant proper.

An example where a DPD(I6) governs a deeperlevel

ofstructure s found in measures1-4 of Liszt'sAnnetes epdeeri-

nageII, S. 191, no. 2, "I1penseroso,"given in Example 26.

Here, Liszt prolongs a DPD(6) chord by arpeggiatingdown to a DPD(4) structurebefore arrivingon a V7 chord

on the final beat of m. 3. Liszt spells 7 as bi through the

mixed-function chords, but restores the correct spelling of

the leading tone with the arrival ofV09; while he uses C (biin the key of C# minor) in the upper voices of both the

DPD(6) and the DPD(4) chords, he enharmonically rein-

terprets C? as B# when the Dominant proper arrives.

While the "Vorspiel"to Wagner's Tristan und Isolde is one

of tonal music's most discussed passages, a mixed-function

interpretationhas been

implicitin

analytic writings for quitesome time. The opening to the "Vorspiel" is shown in Ex-

ample 27. Many analysts have commented on the G#-B

voice-exchange between the soprano and tenor in the open-

ing measures, which suggests a sense of shared harmonic




Lento

1:

ll.

r a l l

f trispassion

porall.

C: iDpD5(4)

V4 i

C: i8-7

EXAMPLE 23. Sarasate, Zigeunerweisen, op.20, mm. 12-15.

function between the Tristanchord and its resolution.30The

bass line from ?6 to 5 of the Tristan progression(one of the

characterizing bass lines associatedwith DP to D function)is coupled with the prolonged leading tone (in this case,

through a voice-exchange), which gives the Tristan succes-sion so much of its character.The addition of the innervoice

#4 o 4 between these two harmonies is an additional com-

plication. The tendency of #4 o move to 5 (D of D to D) is

thwarted; #4 instead moves to the weaker Dominant factor,4. Represented on the Tonnetz, he resolution of the Tristanchord has a remarkableelegance.The 2/7 voice-exchangerep-resents

the D functionalpersistence.b6to 5 is a functionaldis-charge from the Subdominant agent to the Dominant base;

this motion preciselymirrorsthe expected resolution of theDominant-of-the-Dominant agent to the Dominant base (#4

to 5). Insteadof this resolution, he polarityof the Dominant-of-the-Dominant agent changes from majorto minor (#4 o

84),at which point 4 becomes a factor of the Dominant.Unlike the other DPD chords discussed thus far,the Tristanchord is not a tidy chromatic Dominant placed atop a bassline that suggestsDP function, owing to the presenceof #4.

However, I believe that the association of this chord with

other similarDPD(b6) chordsis a useful correlation.

FUNCTIONALCOLLISIONS

OF TYPEDs

Authenticbass:Ds (5)-T()

DS(5) chordsin generalareclosely relatedboth to Domi-nant chords with added dissonance, and to dominant pedal

30 This interpretation of the Tristan progression was explored in Smith

1986, 136-39 and later in Harrison 1994, 156-57.




Allegro moderato

I

F: i DPD(4) (v/ V4-3

F: T(i) D(5) T(i)

EXAMPLE4. Schubert,Moment Musical, D. 78, no. 3, mm. 34-38.

TI

,•T-fp f

2" i' . . . f"[• "1 1 "1 r

-I I _

G: D(4) T(63) D(4)

E: DpD(6) D(5) T(i)

EXAMPLE 25. Beethoven, Sonata, op. 13, "Pathetique,", mm. 134-37.



274 MUSIC THEORY SPECTRUM27 (2005)

C#: iDPD(6/) V95 9-8

C#: T() D(5) T(i)

EXAMPLE 26. Liszt, Annees depe'lerinageII, S. 191, no. 2, "Ilpenseroso,"mm. 1-4.

tones. From the former, a DS(5) chord is distinct in that it

often refers to Dominant Elevenths and Thirteenths, the

so-called "tall-chords,"31 whose upper, dissonant elements

behave as real chord tones rather than unresolved embellish-

ments. Dominant pedal tones, on the other hand, typically

support Subdominant or Tonic chords in a non-functional

context-that is, they resolve to the diatonic dominant be-

fore the change of the bass. A DS(5) chord is different from

these because it bears the entire Dominant function of the

passage, where there is no resolution of the upper-voiceSubdominant elements.

As with SD chords, the notation of Ds chords is similarly

fraught with problems. Thus, rather than contriving a com-plex Roman numeral to account for the many possible

configurations, the notation will be simplified to a straight-

g2

A: iDPD(6g)

V7

A: T(i) D(5)

EXAMPLE27. Wagner,Tristan undlsolde, "Vorspiel,mm. 1-3.1 Kostka and Payne 1995 use the term "Tall chord" in this manner.




forward account of the chord's function. From a practical

perspective, DS(5) chords may be thought of as completeSubdominant chords built atop 5 in the bass, normally

placedin a texturewhere the two functionalhalves,and their

behaviors, are clearly separate. A roster of DS(5) chords

(such as ii7 built atop 5, IV7 built atop5, etc.) is easily imag-ined, and an enumeration of all possible permutationsyieldslittle direct benefit.

The opening of the first movement of Schumann'sFan-

tasy, op. 17, given as Example 28, contains a striking DS(formation that is subsequently undermined at a deeperstructurallayer. In what is apparentlya conscious effort to

blur together Subdominant and Dominant function,Schumannsuperimposes a ii7 chord over 5 in the bass.The

resolution of theDS(5)

toD(5)

across mm. 7-9 blurs the

harmonic function, as the right hand resolves to Dominant

harmonywhile the left hand continues to outline the super-tonic triad in m. 7. On a deeper level, the Ds(5) chord is re-

alized as non-chord-tone motion, embellishing a structural

V7 chord. In this case, the nature of the prolongation of

these non-chord-tones preserves a surface-level indepen-dence of the harmony.

In the opening to the "Rigaudon" rom Le Tombeaude

Couperin Example 29), Ravel blurs the Dominant function

in the second measureby superimposingan arpeggiationof a

complete ii9 chord over an authentic bass. Unlike the

Schumann example above, Ravel does not decorate a dia-

tonic Dominant with a DS(5) chord;rather, heDS(5)

chord

stands alone and resolves directly to Tonic in aDP(4i/2)-

Ds()-T(i) progression.In the first few measures of Debussy's "Prelude"o the

SuiteBergamasque, iven as Example 30, Debussy strongly

emphasizes a ii7 chord built atop a bass5 for all but the final

sixteenth note of m. 2, which introduces the leading toneand so fleetingly turns the harmonytoward an explicit,tradi-

tional Dominant. Once again, the surface-layer analysisholds the DS(5) as a recognizable and significant aspect of

the composition; the second layerrevealsDebussy'selaborate

suspension. The metric context and the emphasis of the

arpeggiated ii7 (Gm7) chord reinforce this interpretationacross the sixteenth-note figuration.The metric emphasis of

the G minor-seventh chord suggests that the real upper-voice voice leading prolongs a G througha downward arpeg-

giation from G5 to G4, resolving into the suspended ninthabove the Tonic chord in m. 3. At the deepest level, the

Dominant chord remainsprivilegedby convention.

Authenticbass:T(i)-DS(7)-T(i)

From a theoretical perspective, the neighbor bass 1-7-iholds some of the most intriguingmixed-function situations.

This bass line is also the most problematicfrom a notational

perspectiveand the most sensitive to the

modalityof the

Tonic chords.

The examples that follow all hinge upon the spelling of a

Dominant chord that is altered with #2. A vii` chord inher-

ently presents a conflict between the bass note, which is em-

blematic of Dominant function, with the upper-voice scale-

steps#2andb6.These scale steps aredisjunct on the Tonnetz,and thus naturallydivide the chord's allegiance to a Tonic.

When #2 resolvesto 3 in a majormode Tonic, there is a clear

sense that it functions as #2; when #2 resolves to a minor-

mode Tonic, it is normally spelled as b3,and, more impor-tantly,sounds ike b3resolvingby common tone.

Chopin presents a #2alterationof a viiOchord in his Bbminor Sonata, op. 35.32As Example 31 shows, the openingmeasures tonicize F, as the dominant of Bb minor, implyinga secondary viiO/V,from which an implied 2 of F moves

through #2 on the way to 3 of F (V of Bb) in an extended

structuralanacrusis.In this context, the chromatic scale-step#2 clearlymoves to 3 n a Dominant-to-Tonic impulse; imag-

iningthis resolution as K3 o 3 is not

productive,and seems

to cloud the issue more than it clarifies.

32 This interpretationmaybe found n Smith1986, 122-24.




Durchaus antastischund eidenschaftlich orzutragen

EXAMPLE 28. Schumann, Fantasy, op. 17, mm. 1-9.

ASPECTS OF PLURAL FUNCTION IN CHROMATIC MUSIC



277

C:P(/) D> T(i)

C:IV7

iilV13

EXAMPLE 29. Ravel, Le Tombeaude Couperin,"Rigaudon,"mm. 1-2.

Example 32 presents a contrast to Example 31. The

opening measuresof "Salome'sTanz,"from Strauss'sSalome,embellish the tonic with a surface-level plagal succession

i-bV16-i,which establishes the key and mood of the dance.

After the first phrasecloses with a half-cadence,the secondphrasebegins in the same fashion, but as the expectedWVIchord embellishes tonic in the uppervoices, the bass moves

to the lower neighborbi,apparentlysubstitutingbvi6for WVI6.However,we have seen this respellingbefore, in the context

of SDchords,where the pair Kiand0i are proxyspellings of

pitches that have a linear function of 7 and #2 respectively.When appearing n the bass like this, the secondphrasethus

begins with an authenticstatement to contrast the opening

plagalgesture, resultingin a

Ds(7)chord.The essential

pla-gal aspect of the progressionis embodied in the "resolution"

of #2 to b3 in Tonic, exhibiting the linear behavior of a

common-tone b3alongside the Subdominant ~6 esolvingto

5. Carefulanalysesof this behaviormust separate he appar-

M o d e r a t o

F: I 7 19

-

Ti D(5)?F: T() Ds(5) T()

EXAMPLE 30. Debussy, Suite Bergamasque, "Prelude,"mm. 1-3.

ent vertical spelling and acknowledgment of a triadic sonor-

ity from the linear behavior and clear authentic functional

discharge that characterizes the sense of double-functionafforded this striking progression.

Rimsky-Korsakov, in the second movement of the

CapriccioEspagnol, op. 34 (see Example 33) uses Ds chords

rather than straightforward Dominants in both the initial

compound basic idea and the continuation phrase of the

second variation. The DS(7) chord in m. 43 is identical to

the one found in Example 32, above. The passage concludes

with a strikingly rich cadence that superimposes a Nea-

politantriad above a bass 5 in m. 47. While this cadential

dominant might also be interpreted as an altered V7 (with a

lowered 2) the texture and spacing of the chord suggests a

DS(5) interpretation, where the Neapolitan chord is heard

superimposed over the lower-voice Dominant harmony.

MUSIC THEORY SPECTRUM



278 27 (2005)

Grave Doppiomovimento

f-

L

Bb: (viio/V) V6-5F: vii' DSO() "I"

Bb: [DS(7)/V] V7

EXAMPLE 31. Chopin, Sonata in Bbminor,op.35, i, mm. 1-6.

CONCLUDING REMARKS

The Kldnge I have discussed in this article are related in

that they all involve a collision of Subdominant and Domi-nant functional elements. While in practice bifrequently ap-

pears instead of 7, the linear nature of the progression is evi-

dent. It is so strong, in fact, that the authentic functional

discharge embodied in 7-1 must be acknowledged as the

guiding element of the progression, granting some degree of

Dominant function to the chord so that the result is a true

functional hybrid.Scholars of the nineteenth century speak often of the

Romanticpenchant

to blur boundaries in art and in music.

The Kldnge I have discussed blur the boundaries of harmonic

function on the musical surface. This blurring, rather than

creating ambiguity, suggests a conflict that draws our atten-

tion to functionally hybrid chords. I have chosen my exam-

ples in an effort to explain the "big moments"-the most

interesting and striking events in the compositions at hand.

Hatten 1994 develops a theory of markedness for the in-

terpretation of music.33 There is tremendous opportunity toassimilate the present study with Hatten's interpretive strate-

gies. In this context, plural-functioned harmonies constitute

one half of the equipollent opposition (Plural-function vs.

Singular-function) where Plural-functionality is clearly the

marked term in the opposition. I argue that the presence of a

functional collision in a harmony constitutes a stylistic type,within which the variants I've explored above represent a

range of musical tokens. This approach can lead us to make

33 This theoryis adapted rom the field of linguisticsemiotics;Hattencreditsthe work of Michael Shapiro(1968) as his source for these

ideas.




m.7

m.073

Lm

C6: i 6VI6 6i 1vi6? i

DS(7)

c : T(i) T(i)

at~ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

~ ~ _ ___ __ __ ___ __ __ __ ___ __ ___4____L

EXAMPLE32. R. Strauss, Salome, "Salome'sTanz," mm. 1-3, 7-9.

Andante con motodolce r--- 3 >

r---3

D

-4o

D: i Ds(7) iII Cad6 Ds(5)F: I vi

D: T() DS T(i)

EXAMPLE 33. Rimsky-Korsakov, Capriccioespagnol, op.34, ii, mm. 41-48.




informed interpretive decisions where these harmonies are

found.

But there is yet muchwork to be done; this studyis only a

first step toward a fuller understanding of these fascinatingharmonies. Furtheravenues for study might attempt to draw

intertextual connections between plural-functioned har-monies; to define a tradition for their use; to make general-izations about their location; and to come to grips with

issues of stylistic evolution. It is my hope that this investiga-tion might provea usefulpoint of departure or such studies.

APPENDIX:

A BRIEF SYNOPSIS OF DANIEL HARRISON S THEORY

Harrison's approach is a mature dualist formulation

whose nineteenth-century origin may be traced to Haupt-mann.

Usinga dialectic

apparatus,34 Hauptmannwas con-

cerned not with building a dualist tonal system, but rather

with using oppositional structures to explain the tonal sys-tem as he found it. Helmholtz countered this theory with a

scathing critique supported by the natural sciences-a disci-

pline so much in vogue at the time that his critique carried

a great deal of weight. To counter Helmholtz's argument,

Ottingen invoked Hauptmann's Having/Being dialectic in

the context of the overtone series, and constructed a natural

basis for the minor mode (thus for harmonic dualism itself)

in his explanation of Tonicity/Phonicity. Ottingen's rationalein the natural sciences was sufficient for Riemann to devise

his system of harmonic function and transformation. Al-

though he misread Ottingen's theory when he proposed his

infamous "undertone series," Riemann later revised his argu-ment to find its basis in the mind rather than in nature, and

thus realigned his theories more closely with Hauptmann's

original formulation. Harrison's dualism is likewise uncon-

cerned with providing a natural basis for the diatonic system;

simply,Harrison finds the

explanatory powerof

oppositionalpairs to be a useful entry into chromatic harmony. Since this

kind of dualism offers substantial explanatory power, it is

therefore a valid vehicle for musical analysis.The primitive of Harrison's dual system is the simple op-

position of major and minor, from which he generates a dual

network of scale degrees. Onto this network, Harrison graftsa second postulate, that of the "three-termed dualism,"

which places a neutral Tonic in between the marked ex-

tremes of Subdominant and Dominant. From these central

34 See Klumpenhouwer002, 459-62, for discussionabout he degree o

whichauthenticallyHegelianthoughtcan be ascribed o Hauptmann's

theoryasopposed o Hauptmann'smereuse of dialecticlanguagewith-

out a deepunderstandingf trueHegeliandialectics.




postulates, Harrison then explores the harmonic function

that can be inferred from each individual scale-degree in the

tonal system, and accordingly, in scale-degree assemblies.

In relation to a Tonic, the basic functional dualities, S and

D, are represented by 4 and 5 in particular, but also by the

primary triads projected from 4 and 5. When placed in thelowest voice, either 4 or 5 normally suffices to convey a sense

of its respective function and to subsume the remainder of

the Klang to its purpose. When placed in an upper voice, 5

requires the support of its functional agent, 7. 4 is not bur-

dened by this constraint, as it is not found in either of the

other primary triads (Harrison 1994, 46

ff.).

Functional

agents (the thirds of the primary triads) "are entirely dedi-

cated to the function in question" (Harrison 1994, 49) and

thusoperate unconditionally.

Functional bases have the

power they do in large part because they imply the presenceof their agents. Functional associates (the fifths of the pri-

mary triads) are weak functional signifiers; they are able to

amplify their comrades, but unable to bear functional expres-sion alone.

Any Klang may be disassembled and may represent a

functionally mixed structure; witness Harrison's disassemblyof the supertonic triad. Since the function of the Klang as a

whole is dependent on the functional status of its con-

stituents, the supertonic triad is functionally mixed: 4 and 6are base and agent of S function; 2 expresses a weak D func-

tion. Thus, there resides in the supertonic triad a latent

Dominant potential. Compare the two progressions in

Example 34. The first progression provides a context where

the latent Dominant potential of the supertonic triad is

apparent-the supertonic triad is subsumed into the func-

tion of the Dominant. The metric context and shared bass

scale step suggest that the soprano 2 is an arpeggiationwithin the dominant

harmony;6 is a dissonant

upper neigh-bor to 5. In the second progression, however, the metric con-

text suggests that the soprano 2 has a greater degree of inde-

pendence from the leading tone that follows. The leapingbass supports a compelling change of harmonic function.

4.i A.

n- I

T D T T DP D T

EXAMPLE 34. Modelprogressions involving DP(2)?-D.

In Harrison's theory, a similar argument is advanced for

the dominant seventh chord. In V7, the seventh, 4, provides a

strong element of S function, and thus empowers this har-

mony with S potential, especially if 4 is found in the bass.

Context or competing functional elements coexisting in a

Klang do not deny the potential of the functional primitivesin the system. For Harrison (following Erpf), functional

mixture is an abstract property of a chord. In musical con-texts, these elements are reconciled as the scale steps com-

pete for functional presence.

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