484 TEMPORAL SPACE

are now unknown. Eduard Hanslick seems to have brought the poem in question to Brahms's attention (see Brahms/Herzogenberg, Briefwechsel, 2: 135n); perhaps he lent Brahms a copy of Lingg's Gedichte as well. In any case, the transcription here follows the poem as found in Hermann Lingg, Gedichte. Dritte vermehrteAuflage (Stuttgart and Augsburg: Cotta, 1857), 56. The poem is reproduced identically in both the second and fourth editions (published by Cotta in 1855 and 1860, respectively).

30. See Otto Friedrich Gruppe, Gedichte (Berlin: Reimer, 1835), 55.

PART VI

TRANSFORMATION,

ANALYSIS, CRITICISM

lT has been variously noted that neo-Riemannian theory emerged as a force to be reckoned with a t exactly the time when the project of musi c theory an d analysis in the Anglo-American academy had to parry a fundamental critique of its aims and assumptions. This is hardly a coincidence. Certain positions of neo-Riemannian theory can be seen as direct responses to the main points of criticism: tonai unity, the all-encompassing claims of analysis, and ultimately the deep connections with the idea of the musical work. lt is especially these points that neo-Riemannian the­ory has scaled back an d rethought, an d i t is these points, o ne might further add, that are most a t odds with (paleo- )Riemann's own theoretical project.

As a consequence, it is rare to find neo-Riemannian theories being applied beginning to end in a p ieee of music. Yet some of the most powerful insights can be gained through the interaction of neo-Riemannian theories with other music-the­oretical approaches. Indeed, i t is the very flexibility of the approach that gives neo­Riemannian analysis its innovative strength. At the same time, a few fundamental questions bave remained unanswered-or have received answers that are tailor­made to specifìc situations. The questi o n of what kind of tonality, if any, neo-Rie­mannian theory represents has occasionally been raised. Similarly, the questi o n of repertoire-predominantly, from Schubert to Strauss-is intriguing: do the triadic an d chromatic works for which neo-Riemannian theory works bes t form a coherent

486 TRANSFORMATION, ANALYSIS, CRITICIS:I.{

repertoire of their own? The essays in this section seek to address some of these questions that lead us into the wider aesthetical realm. In addressing how ideas that originated with Riemann may respond to contemporary analytical challenges, the essays in this section open up new paths an d offer suggestions for further work.

In the opening essay, Steven Rings considers a work that was the topic of Riemann's first published analysis: Schubert's triadic but highly chromatic Gb-major Impromptu. Rings compares Riemann's own analysis of the work with a neo­Riemannian view inspired by the writings of Richard Cohn, assessing the differ­ences in analytical methodology and technology, and locating those differences within the divergent ideologies of the two approaches. Rings's centrai concern, however, is not with the analytical technologies themselves, but rather with the assumptions and values that underlie the distinct analytical perspectives. Rings focuses o n analytical values with an eye toward synthesis: an enrichment of the neo­Riemannian perspective through an engagement with the ethical an d methodologi­cal concerns of the paleo-Riemannian approach.

In the following chapter, Robert Cook performs a virtuosic hermeneutic analy­sis of César Franck's Le chasseur maudit, which serves further as an extended and elegant reflection on the potential and limitations various analytical frameworks. Cook situates his analysis with respect to notions of chromatic music, in particular the idea that chromaticism poses analytical difficulties that Riemannian and neo­Riemannian perspectives are particularly well suited to address. After considering the work from both functional and linear perspectives and examining the concep­tual problems that attend each, Cook illustrates how a contextual, neo-Riemannian view can capture the work's salient gestures, an d offers a balance between a desire to understand the work as a reflection of an orderly, conceptually coherent relational system an d the need to engage the aura! experience of the music.

Daniel Harrison closes this part of the book with a three-section essay, explor­ing certain interrelated themes and questions centrai to the transformational and neo-Riemannian enterprise. Part one problematizes the natures of musical objects and relations within the transformational worldview, and asks what happens when we try to imagine ton es an d chords not as objects but as transformations, the prod­ucts of movement, or-to employ more Kurthian language-not as sensuous but as energetic entities. Harrison delves further into the object/transformation dichot­omy in the second section, deftly exploring the structural an d functional differences among dissonant and consonant trichords in a particular nonatonic cycle. The essay, a fantasy o n a variety of speculative an d historical themes, explores how voice­leading, functional, an d set-theoretical implications of the cycle might be profìtably engaged by a transformational perspective as a means to impart "sensuous distinc­tions" among otherwise indifferent transformations. The third section investigates the analytical ramifications of the first two sections. Vaughan Williams' neo-m o dal, triadic Fantasia on a Theme by Thomas Tallis provides the soil in which these con­

siderations can take root.

CHAPTER 18

RIEMANNIAN ANALYTICAL

VALUES, PALEO- AND NEO-

STEVEN RINGS

I

The passage in example 18.1-from the coda of Schubert's Impromptu in Gi>-provides a useful point of departure.

Ex. 18.1. Schubert, Impromptu in GJ,, D. 899, no.3, mm. 78-82.

n ~ h l •

1:: ~mrr=~mrttrtctri:::;;:;;;;·,Mtfttrrtr~l ~ "U" ~ '

82

C!

488 TRANSFORMATION, ANALYSIS, CRITICISM

The tonal bottom seems to drop out of the music here: in just aver two measures we progress from the tonic Gb major, thrm1gh B minor, to G minor. 1 A menacin~ bass trill on C in measure 79 announces the imminent arrivai of the latter-we hear the G-minor chord coming before it sounds. (Such a menacing trill could hardly be preparing us for G major.) Miraculously, this forecasting does not lessen the shock of the chord when i t actually arrives.

The G-minor chord of course admits of a tonal interpretation: it is the minor Neapolitan, enharmonically respelled. The chord nevertheless emanates a surplus of harmonic energy, overflowing the bounds of such a familiar tonal category. This surplus registers not only sonically, but also in the notati an. Spelled "correctly;' the chord would be Al*> minor; as a tonic, its key signature would have 14 flats (that is, double flats on every diatonie pitch). Schubert has already begun from a point of flatward extremity: given his six-flat signature, any motion flatward will exert pres­sure on the notation.2 In the passage in question, the flatward pressure is so great that i t forces an enharmonic snap in the music, creating visual fissures on the page where the six-flat signature is cancelled in measure 79 and then reinstated halfway through measure 80. The reinstatement coincides with a ffz augmented-sixth chord, which effortfully hauls the music back from its G-minor nadir, leading to a con­firming cadential progression in Gb.

The passage is a great intensification of a gesture Schubert has traced through­out the p ieee, beginning with the first phrase: a bass descent in thirds from the tonic into subdominant regions, with a return by ascent at the last minute, under domi­nant energy. The descent in example 18.1, however, presses so far in the subdomi­nant direction that it has the character of a tonal crisis or trauma, the intensity of which registers visually on the page, in the fissured notation. We can indeed hold the Gbtonic in our ears throughout the passage-thus retaining the minor-Neapolitan hearing-but i t takes some effort to do so. If we listen while looking a t Schubert's fractured score-perhaps while playing the piece-we may be encouraged to give up that effort altogether, opeBing our ears to the chord's extratonal surplus.

How we respond to such a passage analytically says much about what we value in music-and in musical analysis (the two are not necessarily the same). Given its harmonic complexities, Schubert's passage provides an especially fruitful context for exploring some of the divergent values inherent in (echt- )Riemannian and neo-Riemannian approaches to harmonic analysis. The various technical differ­ences between Riemann's harmonic theory (in its many iterations) and neo-Rie­mannian theory (in its many iterations) are, by now, relatively well known.3 Less attention has been paid, however, to the theories' strikingly different attitudes toward the analytical act itself, including the different ways they seem to value music (in both senses: "cherish music" and "invest music with value")_ Such differ­ences are, i t need hardly be said, products of the theories' highly distinct historical,

ideologica!, and cultura! moments. In what follows, I will take an initial step toward mapping some of these diver­

gences in value (and uncovering some unexpected points of contact), taking Schubert's Gb Impromptu as a point of reference. Section II compares a mode!

RIEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 489

neo-Riemannian analysis of the passage-based o n the work of Richard Cohn-to Riemann's own analytical comments about the piece, which bookend his career, appearing first in the early Musikalische Syntaxis (1877) and then in the sixth edition of tlle Handbuch der Harmonielehre (1917). Section III then explores the method­ological and ethical contrasts between the two approaches in depth, tracing aspects of the intellectual and ideologica! contexts in which they arose. The chapter con­cludes in section IV by considering some ways in which a technical rapprochement between the theories might open our ethical horizons, providing new ways in which we can value music through Riemann-inspired analytical activity.

II

It seems safe to say tllat tlle music in example 18.1 would catch the ear of any neo-Rie­mannian analyst, perhaps even providing the first point of analytical entry into tlle p ieee. (O ne thinks h ere of the many analytical forays into Parsifal that have begun no t a t the work's outset, but with the most chromatically distorted version of tlle Grail motive, very near tlle end of act III.) Neo-Riemannians have often explored such passages by turning attention away from tlle traditional categories of tonal harmony and toward voice-leading efficiency, in an effort to detect pattern and regularity where tllere might otherwise appear to be tonal strain or disorder.4 By invoking enharmcinic equivalence, such approaches further sidestep enharmonic complexities such as tllose discussed above. Example 18.2 sketches aspects of the Schubert passage from tllis perspective.

The grand staff at (b) shows a reduction of the passage. The single staff at (a) extracts Klange from the music.5 A key at the bottom of the example explains the noteheads in (a), which indicate whether the note in question is a common tane from the previous chord or has mòved by ic1 or ic2. A quick scan of the noteheads reveals that every chord maintains at least one common tane with its predecessor; furthermore, motion by ic1 predominates.

The annotations above staff (a) tally the results of the total voice-leading between the chords. DVLS is Richard Cohn's "directed voice-leading sum:'6 It measures the directed voice-leading motion between chords, distinguishing -between "up" an d "down:'7 Thus, the first entry in the row, +2, indicates total voice-leading of two semi­tones "up" from Gb+ to the B-: the two filled noteheads in tlle B- Klang indicate the two voices that have moved up by semitone from Gb+. The -2 that follows indicates total voice-leading of two semitones "down" from B- to D+, as o ne voice descends by whole tane. And so on. A clear pattern emerges: DVLS values alternate between +2 and -2 until the Elb+ ( = D+) chord of measure 8o proceeds to the Gb+ of measure 81, yielding a DVLS value of o: h ere two voices move by semitone, but in opposi te direc­tions, canceling each other out. This is the very moment a t which tlle ffz augmented­sixth chord wrenches the music back to a cadential progression in Gb. The "wrenching" registers here in the contrary motion of DVLS = o, which ends the +2/-2 tailspin.

490 TRANSFO RMAT ION, ANALYSIS, C RITICISJ\.1:

Ex. 18.2. Harmonic reduction of Example 18.1, with voice-leading analysis.

DVLS:

(a) AVLS:

" v

Gl.+

(b)

" 78

rv t

: )

Keyfor(a):

+2

2

B-

79

,!

-2

2

o = common tones • = motion by icl + = motion by ic2

D+

~

+2

2

G-

80

'!

- 2

2

Eih+

o 2

(=D+)

q-jfz J,i,.

-3 +3

Gl.+ !)\,+

81

""'' ~.

DVLS = Directed voice-leading sum AVLS = Absolute voice-leading sum

-

Gl.+

82

i

The row below DVLS is labeled AVLS for "absolute voice-leading sum." This measurement takes no account of the direction of the voice-leading, instead mea­suring only the absolute distance traversed in interval classes, registering what Joseph Straus calls the total voice-leading "work" or "exertion" of the progression.8

Again, there is a clear pattern: AVLS is 2 forali entri es unti! the cadential oscillation between G~+ and D~+, where it increments to 3. This reading notes a continuity in the progression from Elh+ to G~+, which traverses the same absolute voice-leading distance as ali of the preceding progressions. The wayward chromatic successions of the first p art of the phrase thus ali show AVLS = 2, while the key-reaffirming caden­tial tag in G~ projects AVLS = 3·

The prevalence of 2s in the AVLS row suggests a particular voice-leading space', which Cohn calls a "Weitzmann region:' Ali of the chords in such a region relate to one another by AVLS = 2.9 Example 18.3(a) shows the Weitzmann region containing G+.10

The solid, undirected edges circling the perimeter of the network indicate the transformations that relate adjacent triads within the system: N and R. The former is Cohn's transformationallabel for Weitzmann's nebenverwandt relation;11 the lat­ter is the familiar neo-Riemannian relative. Dashed edges indicate transformations between nonadjacent triads: Kliinge "two apart" o n the cycle are related by PL or LP, and those apposite one another are related by Lewin's SLIDE. These five transfor­mations-N, R, LP, PL, and SLIDE-are the only neo-Riemannian transformations ( out of 24) for which AVLS = 2Y

R!EMANNIAN ANALYTICA L VALUES, PALEO- AND N EO - 491

Ex.18.3. (a) Weitzmann region including G~+; (b) passes through this region in measures 78-81.

(a)

R Ili /Il

l l l l l l

l l l

N

LP' "-11 'PL / ~ l \

l ~ : l l

R......._ / N

(b)

R / :• ........,_ N Il Il l l l l

r.t:l l

~: l l l l l l l Il

~ " /N l .lo

Ali of the Kliinge in example 18.2(a), with the sole exception of the dominant D~+, resi de in this Weitzmann region. Example 18.3(b) shows the passage's progres­sion through the region, up to the G~+ cadential six-four in measure 81. The pro­gression begins with G~+ a t 12 o' dock an d proceeds clockwise aro un d the outer edge unti! i t reaches G- a t six o' dock. Along the way, an LP arrow leads from B- to G-, indicating that the D+ chord that intervenes in the second half of measure 79 plays a passing role between the two harmonies o n the downbeats .. On the "return trip'; counterclockwise from G-back to G~+, a similar LP arrow leads from D+ to G~+; this is the "wrenching" LP motion associateci with the resolution of the aug­mented sixth to the cadential six-four. (The return trip bypasses B- altogether.) After returning to G~+, the music leaves this Weitzmann region to engage in the confirming cadential progression via AVLS = 3.

Other passages in the Impromptu also trace out significant portions of a sin­gle Weitzmann region. Most notable among these is the other highly "purple" patch in the piece-the sojourn to C~ major and Eb major within the B section (measures 32-53). This is mapped in example 18.4, which presents the three Weitzmann regions containing G~+, C~+, and D~+, labeling them T, S, and D in a manner analogous to Richard Cohn's labels for hexatonic systems in his analysis of Schubert's B~ SonataY The example first shows the move to E~- and. its domi­nant a t the opening of the B section (measures 25-31); these harmonies stili resi de in the tonic region. 14 Then, at measure 32, there is a move to the subdominant region's 0+, via an interregional L transform of E~-. The first "purple" motion traced in the subdominant region is the alternation between 0+ and Fb- in mea­sure 35. The dashed edges then link these chords to the next !oca! tonic in the sec­tion, the E~+ that enters in measure 48, which leads to the transitional A~- in measures 52- 53. From here the music progresses to the dominant D~+, which is highlighted within its network o n the right-hand side of the example. Notably, no other node is "li t up" within the dominant network-in fact, none of the other Klange in the dominant region sound prominently anywhere in the Impromptu.

492 TRANSFORMATION, ANALYSIS, CRITICISM_

Ex. 18.4. T, S, and D Weitzmann regions showing harmonic events in the B section.

N

R-....._ _/'N c:-

This is indicative of the way in which the Impromptu thoroughly explores the subdominant si de of G~, but not its dominant side. 15

To be sure, there are severa! infelicities in the analysis in example 18-4. For o ne, and most obviously, the analysis leaves out many harmonies within the B section that do not fit into its regions, most notably all of the stormy diminished-seventh­based music in measures 40-45 and some !oca! dominants. Furthermore, it leaves the question of the relationship between voice-leading efficiency and harmonic function (T, D , and S) somewhat undertheorized, conflating the two in a way that causes the distinction between chord and key to break down (e.g., by treating syn­tactical harmonies in the same way that i t treats tonicized harmonies). Similarly, the separation of the tonic, dominant, and subdominant triads into separate regions seems to do violence to their !oca! syntactic connectedness a t the leve! of the phrase. These are familiar problems in certain strands of neo-Riemannian analysis. Yet, despite these shortcomings, the analysis provides a suggestive heuristic for tracing the piece's voice-leading activity, showing the ways in which i t navigates viaAVLS = 2

in its most ear-catching passages. This neo-Riemannian reading-only the beginning of a fuller analysis-has pro­

ceeded as such readings often do, beginning with the most chromatically exceptional moment in the p ieee and moving outward from there to construct a broader interpre­tation of the movement. Cohn takes a similar approach, for example, in his analysis of Schubert's B~ Sonata, beginning by observing the hexatonic-polar relationship between the work's B~+ tonic an d its F#- secondary key area, then seeking out other hexatonic-polar progressions (such as ilie one at the transition into the development section), and ultimately constructing a hexatonic analysis of the entire movement.

The contrast with Riemann's own analytical practice could hardly be more stark. We can see it by consulting his analysis of the G~ Impromptu, published in

RI EMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 493

]Jusikalische Syntaxis in 1877. This is not only Riemann's first published analysis, but also one of his longest, at seven pages. Not until the Beethoven piano sonata analyses from the end of his career would he publish further analyses of comparable scope and technical detail. His primary concern is the Impromptu's phrase struc­ture an d its demonstration of the principles of harmonic syntax that he develops in the book, based on an arcane terminology developed from Oettingen. 16 Riemann's descriptions are often minutely detailed: he devotes an entire paragraph to the first phrase, and two paragraphs to measures 17-24.

An d about the phrase in example 18.1 he says . .. almost nothing. The passage gets only a very brief passing mention in his main prose, but i t is no t singled out; it is simply listed as one of severa! progressions in the coda: "The fina! consolida­tion of the primary key through progressions [ Thesen] to C minor, G major, Ah minor, and G major has a wholly excellent effect." 17 Riemann is referring here to the music in measures 74-81; note that he analyzes the piece in G major. 18 The pro­gression to A~ minor- the minor Neapolitan in measure 8o-is given no special emphasis: i t is merely the goal of o ne of the four Thesen that Riemann mentions. Even more strikingly, Riemann cites these progressions as playing a role in the "consolidation of the tonic" [Festigung der Haupttonaliti:it]. This is in vivid con­trast to the comments a t the head of this chapter, in which I suggested that the music in example 18.1 can be heard to lead to a tonai crisis, creating a harmonic surplus that overflows the tonai frame. Riemann, by contrast, hears in these measures nothing more than a fina! confirmation of the global tonic, a confirma­tion that, moreover, has an "excellent effect" [ vorziiglicher Wirkung]. I t is hard to tell exactly what aspects of the passage Riemann finds vorziiglich, but whatever they are, they seem to have little to do with any undermining of the tonai order. His language instead suggests a celebration of the piece's confirmation of eternai ton~ laws-its exemplary establishment and reinforcement of a Haupttonaliti:it. Thus, the passage that received so much attention in the neo-Riemannian account, serv­ing as the starting point from which all other observations radiated, is little more than a footnote for Riemann, a negligible chromatic ripple on the surface of an exemplary tonai masterwork.

Indeed, Riemann frames his analysis in just this way, describing the Impromptu as a mode! citizen of the tonai realm, a "formally rather clearly structured composition:' 19 He praises the piece's orderly construction: "The whole ìs a master­piece as regards not only melodie form and metric structure, but especially as regards the ordering of its progressions [Thesenordnung]. And over all of it reigns the tonality of G major, the principal key."20 The second sentence makes clear Riemann's firm commitment to monotonality. (Modern readers will be struck by the pre-echoes of Schenker and Schoenberg.) Ten years later he voiced a similar sentiment in a more generai context in his Systematische Modulationslehre:

One is constantly struck by the controlling force [ Geltung] of the ma in tonic, even during the boldest and most wide-ranging modulations. When we find ourselves a t the end of the path, looking back, we know that we have learned how to trace ever wider circles around the unshakeable center.21

494 TRANSFORMATION, ANALYSIS, CRITICISM

Though he is not specifically discussing the Impromptu here, this passage clearly applies to Riemann's understanding of the piece, in which a single tonic not on!y controls the whole, but does so in mode! fashion. The piece's harmonic excursions do not weaken the tonic, but instead contribute to its greater glory, concentrically expanding its dominion. (Again, the Schoenbergian resonance is striking.)

There is only o ne other passing reference to the minor Neapolitan of measure 8o in Riemann's extensive discussion of the piece. In a tabular overview of the piece's harmonic progressions, he writes beneath the chord symbols for measures 79-80: "NB. Modulation to the antilogic antinomie third-key: g+- 0es."22 The Oettingen-inspired terminology simply means a progression from G major to the minor harmony whose dual root is a major third below-that is, a progression from a G over-triad to an Eb under-triad (i.e., Ab minor). Again, it is not entirely clear just what we are to "note well" about the passage. The annotation could be taken for an exclamation of surprise, and perhaps admiration, at Schubert's har­monic audacity: "Note well: A remarkable progression!" But the invocation of the arcane theoretical nomenclature might also suggest something quite different. We are to note not simply a striking progression, but the fact that the music proceeds to the antilogic antinomie third-key. This yields a very different sentiment: "Note well: My theory even has a name for this chord."

Read in this latter sense, the statement seems to betray an anxiety, an attempt to contain the harmonic extravagance of the moment within the rational bounds of the theory. 23 It suggests a desire to demonstrate that no part of the Impromptu eludes the theory's explanatory rea eh: all of its harmonic maneuvers are easily con­tained and rationalized within the theory's bounds. Riemann himself explicitly the­matizes the notion of spatialized boundaries to harmonic possibility in the book's closing pages. Here h e suggests that h e has mapped out a spatialized realrn of tonai order, comparing i t to a harmonic Garden of Eden:

Thank God the combinations [ofharmonies] are inexhaustible in number, and o ne cannot explore the area of harmony in its entirety by walking across i t step by step but only by flying over i t and surveying i t from a bird's-eye view. !t is sufficient, however, to recognize the chief paths through this magnificent Garden of Eden, which Heaven has left us after the Fall; everybody may then fin d new si de paths for himself leading to ever new perspectives on regions never entered before.24

Schubert's progression would seem to represent o ne of the exotic, "new si de patlis" within this realm, off of the beaten track of the Hauptwege, but nevertheless admis­sible. Yet, as Alexander Rehding has noted, Riemann's passage belies a profound worry: he presents the Garden of Eden as universal and transhistorical, but his language "implies a t the same time a premonition-conscious or not-of its actual, contingent nature ... the whole theory is built o n a feeling of angst, a Spenglerian feeling that the end of an age-the end of German music-is imminent."25 The harmonies on the borders of the Garden are thus fraught with perii, and perhaps temptation. After ali, the invocations of the Garden an d the Fall vividly suggest tlie

RIEMANNIAN ANALYTICAL VALUES, PALEO- ANO NEO- 495

possibility ofharmonic sin. To sin against the tonai arder could bring about perma­nent banishment from the Garden-that is, banishment from the realm of tonai order into the a tonai wilds.

Did Riemann have doubts as to whether Schubert's minor Neapolitan might represent just such a harmonic sin? He may have. For we find him preoccupi ed with tlie chord forty years later, when revising his Handbuch der Harmonielehre for its sixth edition. His foreword to this edition of the Harmonielehre contains the last additions to the theory of functions, which he had first introduced in 1893's Vereinfachte Harmonielehre. Rehding observes that the theory of functions repre­sented a way to contro! and corral the overly permissive possibilities for harmonic progression in some of Riemann's earlier harmoriic theories, including that in Musikalische Syntaxis, thus better fortifying the boundary around the Garden of Eden.26 It thus makes sense that he would return to the Schubert chord to make sure its energy was contained within his new system.

In the foreword to the Handbuch, Riemann adds symbols for direct third rela­tions, as well as a symbol for the moda! Variante of any function-a v after a func­tion symbol, which simply switches the triad's mode. 27 The single example Riemann adduces for the new symbol is the minor Neapolitan from Schubert's Impromptu, which he now analyzes as CJS•: the variant of the leading-tone change of the minor subdominant.28 Even with the new symbol, the chord clearly puts a strain on Riemann's functional system, as it requires three alterations to the initial S function, which are made visible in the three accretions to the S symbol:

1. Major Sto minor (0 );

2. Leading-tone change of that (> ); 3. Variant of that (v) .

As Rehding has noted, Riemann otherwise seemed wary about admitting multiple alterations to a function symbol.29 And indeed, his wariness is apparent here: he calls the chord "exceptional" (an Ausnahmserscheinung) and otherwise uses the v symbol only rarely in his later analyses.

Nevertheless, the sense of Riemann's symbol is qui te clear, an d i t is no t that far from the way in which the chord would be analyzed in a modern American theory classroom using Roman numerals. Example 18.5 compares a Roman-numeral anal­ysis of the passage at (a) with a full Riemannian-functional reading at"(b) and the neo-Riemannian reading at (c). In order to make the progression's correspondence to the tonai readings more legible, I have adjusted the enharmonic spelling.

The Roman-numeral interpretation constructs the chord on the downbeat of measure 8o as an alteration of a harmony bui! t o n the (lowered) second scale degree, while Riemann constructs it as a modification of subdominant function. There are important conceptual distinctions between the two theoretical concepts,30 but overall, the readings at (a) and (b) are quite similar: both trace a descent from tonic, via applied dominants,31 to an exotic, "subdominant-side" harmony, before pulling back to the dominant at the downbeat of measure 81. Both analyses show the greatest amount of cognitive work at the downbeat of measure 8o (though Riemann's

496 TRANSFORMATION, ANALYSIS, CRITICIS}.f

Ex. 18.5. Three harmonic readings of mm. 78-81: (a) Roman-numeral; (b) Riemannian· (c) Neo-Riemannian. '"

note values halved

' .~ t~ .~ S=l

(a) V V Ger"+6 G\.: I '---iv\. '--- bill V--

(b) T (D') oS (D') 08" (JP,?') D

(c)

SLIDE '----'

T D

~ u.;tzmann J regions

augmented sixth chord shows considerable interpretive exertion as well). The ne6-Riemannian analysis, by contrast, simply shows the first six chords unified by voice­leading motion of AVLS = 2 , all unfolding within the tonic Weitzmann region. lt is only a t the resolution of the Gb+ cadential six-four to Db+ that the toni c regio n is left for the dominant one. The infelicities of the neo-Riemannian account from a func­tional perspective leap into view here: i t captures none of the phrase-level functional distinctions in the first six chords, and further is unable to read the Gb+ cadential six­four as functionally distinct from the opening Gb+ tonic. The functionallabels for the Weitzmann regions should thus indeed be understood as a secondary overlay-an informai tonallabel applied to a theoretical system whose essentiallogic is no t tonal.

Notably, the Riemannian reading at (b) casts the progression as syntactically normative, following the preferred paradigm for cadential motion: T-S-D-(1). The subdominant regio n is expanded considerably by modifications, but this does not obscure the phrase's overall syntactic sense. The analysis further reveals a clear syntactical resemblance to the opening progression of the piece (in measures 1-3), which Riemann would also analyze as departing from a tonic, passing through modifications of S, and then arriving a t D. Thus, the passage that I read a t the open­ing of the chapter as representing a tonal crisis or trauma, a reading that led to neo­Riemannian exploration of its extratonallogic, is refashioned h ere as a model tonal phrase, one that vividly demonstrates the efficacy of Riemann's functional princi­ples in a chromatic context. The dangerous chord of measure So has now been fully contained within the boundaries of Riemann's Garden of Eden.

RIEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 497

III

This returns us to the broader question of value. The contrasts are as obvious as they are vivid. Riemann analytically constructs chromatic passages so that they show conformance to his tonal theories, which he portrays as universallaws.32 The neo-Riemannian analyst, by contrast, constructs chromatic passages so that they appear tonally "disunified," and thus require nontonal explanation. Riemann's the­ory thus places a high value on order and conformance to putative universals of tonal harmony, while neo-Riemannian theory, it would seem, values crisis and dis­ruption of that order. 33

We can better understand this sharp divergence if we briefly survey the intellec­tual and ideologica! contexts that nurtured Riemann's theory, on the one hand, and neo-Riemannian theories o n the other. Riemann's context has been masterfully recon­structed by Alexander Rehding, so I will merely summarize his argument here. Rehding characterizes Riemann as seeking to define a universal "dassicism"34 that transcends history an d thus acts as a brake against further historical change in music, clearly demarcating the boundaries beyond which music should not progress. For Riemann, music theory had an ethical responsibility to set limits for composers, act­ing as "a bastion against historical change:'35 Appeals to the burgeoning natural sci­ences allowed Riemann to provide "hard" support for his daims of universality,36

while institutional an d pedagogica! factors played a role as well, as he sought to develop ahatmonypedagogythatwould displace Roman-numeral base d Webei-ian approaches, thus allowing him to influence future musicians directly, instructing them in the laws and limits of musical possibility. The result was a conservative theory shot through with a "relentless normativity."37 In short, Riemann sought, through this theory, to stem the tide ofhistorical change in music, which seemed to him (rightly, i t turns out) to be perilously dose to transgressing the boundaries of the tonal Garden of Eden.38

By the time of the American revival of interest in Riemann's theoretical ideas in the 1980s an d 1990s, that transgression had of course occurred long ago. Indeed, the crossing of music over the atonal threshold was o ne of the primary factors leading to the disciplinary consolidation of music theory in the American academy: theo­rists had taken advantage of the challenges posed by posttonal musi c to argue for the institutional necessity of music theory as a research discipline. Atonality, which before had been a looming threat to Riemann, to be resisted a t all costs, now enjoyed great institutional privilege and prestige, especially among theorists in the second half of the twentieth century. That neo-Riemannian theorists would value tonal crisis far differently from Riemann should thus come as no surprise. If a given pas­sage by, say, Wagner was perceived to veer perilously dose to tonal incoherence, it could now be embraced analytically using the technologies of atonal theory, thus inheriting the institutional values associateci with avant-garde atonal musics through a sort of ethical transitive property.

This is only part of the story, however. In the new, institutionalized theory of the American academy, analysis became an end in itself-a means of engaging

498 TRANSFORMATION, ANALYSIS, CRITICISM

deeply with individua! works via various technical hermeneutic genres, often with a liberai humanist focus on interpreting the telling compositional idiosyncrasy. Such a practice carri es with i t a strong element of Romantic ideology, with the individua! work valued for its originality and uniqueness, as a product of genius. Neo­Riemannian theory clearly participates in this ideology, with its almost exclusive focus o n the compositionally extraordinary. (The chromatic Grail motive is a classic case.) This is in stark contrast to Riemann's own practice, in which analysis serve& first and foremost to illustrate an d valida te his theory. His analytical emphasis is not on w ha t makes a work remarkable or individuai, but o n the ways in which i t exem­plifies the normative, law-like aspects ofhis theory. As Rehding observes, regarding Riemann's analysis of the "Waldstein" Sonata:

[I]t seems that Riemann is not interested in the special features of [the sonata's] opening. Rather-it would appear-he plays down the particularity of this opening in favor of its generai features. While we have come to appreciate the fìrst few bars of the "Waldstein" sonata as a paradigm of Beethoven's harmonic boldness, Riemann's analysis of this passage is actually a demonstration of its ordinariness.39

Riemann's ability to find the ordinary within the compositionally extraordinary extends to all aspects of his theory, from harmony, to rhythm, phrase structure, and form. For example, he says of the first movement of Beethoven's op. 130: "Correctly interpreted, the movement offers no cause to speak of disruption and formai diffi­culty, but instead clearly shows the normal framework of sonata form:'40 This breath­takingly matter-of-fact assessment-turning one of Beethoven's most fissured movements into a self-evident and unproblematic sonata-form-dissonates not only with modernist (Romantic) theory and analysis, which would seek to explore the structural particularities that make the movement unique, but also with more recent criticai musicology. Daniel Chua, for example, characterizes the same move­ment as nothing less than a "direct assault" on the listener: "the audience is simply thrown into confusion by a disarticulated syntax, by a language so violent and con­tradictorythat t o analyze the disunity is to be more obvious than 'post -structuralist:"

41

Obvious to us, perhaps, in our postmodern age, but not to Riemann, whose aesthetic system allowed no place for disunity or disruption in a masterwork.

Styles of criticai musicological thought such as Chua's are not irrelevant to tllis study. For it is not a coincidence that neo-Riemannian theory roseto prominence. around the same time as the disciplinary upheaval caused by the New Musicology. The valorization of disunity, crisis, fragmentation, an d heterogeneity in that litera­ture finds a curious an d distorted echo in neo-Riemannian theory. Indeed, in his introductory artide t o the fournal of Music Theory issue dedicated t o neo-Rieman­nian theory, Richard Cohn explicitly situates neo-Riemannian analytical approaches with respect to "an evolving post-structuralist criticai practice;' suggesting a community of purpose as regards claims of tonal disunity.42 There are further parallels. Neo-Riemannian theory focuses on the very literature privileged by the New Musicologists-nineteenth-century opera and concert music-turning its

F.IEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 499

attention to many of the very same "problem" passages that would capture the attention of the postmodern critic. The neo-Riemannian literature's emphasis on "de-centered" harmonic spaces also resonates with such criticai practices (though, more cynically, one might simply recognize here a co-optation of jargon)Y More substantively, the emphasis on analytical pluralism in many transformational approaches, with roots in Lewin's methodological writings, squares well with post­modern interpretive practices. Finally, certain recurring hermeneutic tropes in the neo-Riemannian literature-such as Cohn's explorations of the harmonic uncan­nf4-are clearly indebted to the New-Musicological spiri t. Neo-Riemannian theory thus appears to pull off a seemingly impossible double play, benefìting from two sets of prestigious but conflicting institutional values a t once (both of which would be anathema to Riemann): those associated with the atona! avant-garde on the one hand, and with postmodern critica! practices on the other.

But, as Cohn makes clear, any commitment to the latter is trumped by more familiar music-theoretical concerns:

Both paradigms [neo-Riemannian theory and post-structuralist music criticism] recognize the potential for tonai disunity in musi c that uses classica! harmonies, and accordingly resist shoehorning ali chromatic triadic music into the framework of diatonie tonality. For the post-structuralist, the recognition of tonai disunity leads immediately to an ascription of disunity tout court, and from there to a cluster of cognate terms . .. : "unstructured," "incoherent," "indeterminate," "coloristic," disjunct;' "arbitrary," or "aimless." The recognition of tonai disunity could instead lead to a question: "if this music is no t fully coherent according to the principles of diatonie tonality, by what other principles might i t cohere?"45

Thus, despite nods to postmodern sensibilities, the most time-honored value of modernist music theory remains fìrmly intact: the demonstration of coherence through formalism. An d here we fin d common ground with Riemann himself. For is the "coherence" of the neo-Riemannian analyst really that far removed from the "logic" or "syntax" of Riemann? Despite some obvious differences in philosophical underpinnings, both projects are underwritten by a drive toward systematization and logica! rigor; a penchant for elegant, symmetrical theoretical structures; and a desire above ali to detect arder in complex music, containing harmonic extravagances in controlled, rational spaces. These values, it would appear, are pan-Riemannian.

But neo-Riemannian theory contains methodological tensions not present in Riemann. The high value it places on both disruption and coherence leads to a peculiar sort of hybridity or double focus. Surely a prime reason for the success of neo-Riemannian theory is the fact that it allows analysts to dwell on the most remarkable sounding passages in a chromatic work, those moments when the tona! fabric is stretched or torn. Buti t is no t the remarkable so un d of those passages that is analyzed; i t is their coherence. One thus begins to wonder what the relationship is between the sound and the analysis. Is the "coherence" that the method detects responsible for what it is that makes these disorienting passages so aurally captivat­ing? Or are the two unrelated? In other words, do we value the analysis for the same reasons that we value the music?

500 TRANSFORMATION, ANALYSIS, CRITICISM;

This question could surely be made of any style of systematic music analysis. But it has a special urgency in neo-Riemannian practice, as criticai responses to the theory attest. Charles Fisk, for example, states that Cohn's analysis of Schubert's B~ Sonata runs the risk of "making even the most extraordinary progressions in Schubert seem ordinary-or a t least, in some respects, normative."46 (The echoes of Rehding's interpretation of Riemann are striking, suggesting further pan-Rieman­nian similarities.) Fisk is concerned that Cohn's theory does not do justice to the sound of Schubert's music, making the aurally arresting seem theoretically com­monplace. Cohn responds by invoking what amounts to a music-theoretic fact/ value distinction, arguing that theoretical categories do not necessarily correlate with sonic affects in a simple one-to-one fashion, even in traditional theoryY Cohn's writing o n the uncanny effects of hexatonic-polar progressions is an eloquent testa­ment to this.

Nevertheless, Fisk's criticism is hard to dismiss. The Schubert passage in exam­ple 18.1 sounds extraordinary, but the analysis of examples 18.2-4 does not tell us about that, instead revealing order and pattern in its voice-leading. We are thus left to wonder just what it is that this music does to us after it enters our ears, why it thrills an d captivates us. Reflecting o n this matter, we might come to value Riemann's originai theory a bit more. Por Riemann throughout his career intended his theories to provide an answer to the question "Wie horen wir Musik?" That the question was always framed in normative terms ("How should we hear music?") and that the answers were therefore tinged with a sense of prescriptive "ought" does no t dimin­ish greatly the value of his approach in this regard. Riemann's focus was indeed on what happens to the musi c after i t enters our ears, an d despite his many theoretical and rhetorical excesses and detours, some of his ideas hit the mark so successfully that they remain with us, in some form or another, to this day. Chief among these is of course the idea of tonai function, whose influence is stili felt, no t only in Germany's Musikhochschulen, but also in many strands of Anglo-American Roman-numeral­based harmony, even in some Schenker pedagogy (however obfuscated the debt to Riemann may be) .48 In the concluding section of the chapter, I will thus explore one way in which Riemann's functional ideas can be reanimated in a transformational context, thus shedding some light on the remarkable sonic effect of Schubert's pas­sage, and narrowing the neo-Riemannian fact/value gap, if only slightly.

IV

Riemann's functions model the relationship of harmonies to the tonic, either directly or via one of its two dominants.49 One way to interpret his function symbols is thus no t as labels for chords, but as descriptions of the actions that lis­teners perform as they interpret sounding harmonies with respect to the tonal center. In this understanding, to hear a chord as a subdominant is to perform the

RIEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 501

Ex. 18.6. LRP ma p with G~+ tonic a t center.

~ =L --=P /=R

subdominant operation (S), directing awareness from the sounding chord to the tonic via S. The S-ness of the perception resides no t in the sounding harmony ( the raw acoustic signa!) but in the action whereby the listener relates it to the tonic.50 I have elsewhere referred to this action of directing awareness toward the tonic as "tonal intention."51

We can trace such Riemannian intentional acts on various species of Tonnetz. Example 18.6 shows o ne such space that is useful in exploring Schubert's Impromptu: a dual of the familiar neo-Riemannian Tonnetz that Michael Siciliano calls the "LRP map" and Douthett and Steinbach cali the "Chicken-Wire Torus."52 The edges rep­resent the three canonica! neo-Riemannian operators (P, L, and R), as shown by the key to the left. The network is rotated 90° from its usual presentation in neo­Riemannian studies (and in historical Tonnetze), so that fifth-related triads are on the vertical axis, capturing the familiar metaphor of dominants residing "above" a specified tonic, and subdominants "below." Dominant (D) and subdominant (S) arrows can be added to this vertical dimension as necessary, to indicate direct func­tional relatio!lships, while the P, L, and R dimensions allow for the modeling of Riemann's Parallele (R), Leittonwechsel (L), and Variante (P) functional modifica­tions.53 If enharmonically conformed, the network wraps around into a torus; it is arranged o n the page h ere so that Gb+-the tonic of Schubert's Impromptu, shown with a double border-has. a centrai position.

Example 18.7(a) demonstrates the intentional paths traced by two modified subdominants in the Impromptu: the pj,_ in measure 2, the second chord in the piece; and the Alb- ( = G-) minor Neapolitan of example 18.1, measure So. The rel­evant Kliinge are indicated with crosshatching, while solid dark arrows show the intentional paths traced by their Riemannian interpretations. The pj,_ chord is

502 T RANSFORMATION, ANALYSIS, CRITICISM

Ex. 18.7. (a) Intentional paths traced by two different subdominants in the Impromptu; (b) Modal vectors within the functional space.

(a)

(b)

brightest ~

' ' ' ' ' major

1! ~

' l ',Q

' "" '

minor

l ' ~ ' .. ' ... ' .... ' '

' darkest

~·~

interpreted as a subdominant leading-tone change, or S:,suggesting an intentional path of LS back to the G~+ tonic, passing through the subdominant Cl>+ along the way. Note that if the same Klang were interpreted as a tonic parallel, T p, i t would trace out a different intentional path: directly back to the G~+ tonic via R. Riemann's different interpretations of the same sounding chord thus traverse different paths in the space, making clear that, in the present interpretation, the function describes not the chord, but the path whereby i t is related back to the toni c.

The minor Neapolitan, All.-, is analyzed following Riemann's interpretation of i t in the sixth edition of the Handbuch as os-v, that is, the Variante of the leading-tone change of the minor subdominant. This suggests a considerably more complex intentional path back to the tonic: PLPS. The initial P models Riemann's Variante (v); that is, it traces our interpretation of the chord as a Variante of the "proper"

RIEMANNIAN ANALYTI CAL VALUES, PALEO- AND NEO- 503

minor subdominant leading-tone change, as we mentally interpret AA- as related to Ali>+ via P. The remaining transformations indicate similar interpretive activity, L corresponding to the leading-tone change (>) an d the second P to the modal altera­tion of the subdominant (o). I stated above that "i t takes some effort" t o "hold the G~ tonic in our ears throughout the passage-thus retaining the minor-Neapolitan hearing." That effort registers in the PLPS intentional path-specifically, in the con­catenation of four interpretive moves it suggests. While AVLS in example 18.2 expressed the exertion involved in voice-leading one chord to the next, the present system expresses ilie exertion involved in interpreting each chord tonally with respect to G~ via Riemannian categories.

We can make vivid that exertion by imagining the intentional arrows in 18.7(a) traveling through a substance or medium-a medium that offers a certain resis­tance, limiting how far we can travel in tracing a harmony back to a tonic. The resistance might cause us to give up altogether in our effort to interpret the AA­chord with relation to G~+, thus breaking the chain of arrows departing from AA­in r8.7(a). Or we may reach G~+, but only weakly, our tonai intentions exhausted in the effort. Our sense of G~+ as tonic would thus be considerably attenuateci, its hold on the music in our ears now precarious.

But the arrows are not only a measure of exertion. They also give us a way to think about the tonai quality of a given harmonic function-its characteristic soni c affect (and effect). A subdominant sounds different from a dominant, after all, as do the many modifications of these harmonies from one another. These qualitative diffèrences are often described metaphorically by reference to color (dark harmo­nies, bright harmonies, and the like). Such differences in sound are nota product of the raw acoustic signal, but a product of the way in which we relate a given sounding harmony to a toni c. (After all, the same chord can be a subdominant in o ne context and a tonic or dominant in another.) That is, the affect arises only after the music "enters our ears;' setting our tonal-interpretive activity in motion. The quality can thus be understood to inhere in the path of arrows we trace from the sounding chord back to the tonic.

Example 18.7(b) explores aspects of the resulting affective or coloristic regions in the space. As it shows, the "brightest;' sharpward regions are up and to the left from the tonic, while the "darkest," flatward regions are down an d to the right. These coordinates provi de a rough sense of the color that will accrue to a chord as i t is related back to the tonic from a given region of the network. The farther in any given direction a chord resides, the more intensely will it acquire that color. The AJj,_ chord resides deep in the darkest quadrant of the space, tracking back to G~+ from the farthest flatward reaches and acquiring an extremely dark tinge in the process. It is as though with each interpretive/intentional act-each darkened arrow-the harmony accrues a new layer of color, or better, of shading. (The word evokes Kurth, whose ideas are highly pertinent here.) These accretions of interpre­tive shading are what give ili e G-/ AA- chord its remarkable chiaroscuro quality; o ne thinks of the multiple layers of paint around the dark perimeter of a Rembrandt portrait. Again, though the harmonic color seems to infuse the sounding medium

504 TRANSFORMATION, ANALYSIS, CRITICIS},r

itself, it is nota property solely of the raw acoustic signa! (i.e., the minor triad). We can experience this by clearing our ears of the harmonic context of the Impromptu an d simply playing a G-minor chord alone, hearing i t as a toni c. The effe et is o ne of stripping away the layers of shading that are present when we hear the chord in the context of example 18.1, as though we have stripped away the many layers of Rembrandt's deep browns, revealing the blank canvas of the minor triad beneath. In this sense, the osv chord of measure So has not less tona! character than the more traditional tonal harmonies in example 18.1, but more.

Example 18.8 integrates the two chords from example 18.7(a) into their respec­tive progressions. Dashed arrows show the chord-to-chord progressions withiri. the passages; these are numbered to indicate their order. Solid arrows show the inten­tional path back from each chord to the tonic. Example 18.8(a) analyzes the opening phrase up to the G~+ chord of measure 4; 18.8(b) analyzes the phrase in example 1s.1 (measures 78-82). The opening phrase remains largely vertical within the space of 18.8(a), moving first down into the subdominant region, and then balancing this with a motion to the dominant. Note that this reading takes advantage of the toroi­dal possibilities of the space, reinterpreting the Ai>- chord in a manner analogous to Rameau's double emploi. Example 18.8(b), by contrast, spreads out horizontally­and chromatically-across the network, dipping into its darkest corner. Note the intentional interpretation of the El*>+ chord: It is heard not as the relative (or Riemannian Parallele) of the previous Ci>- chord (Riemannian oSp). Instead, it is heard as the dominant of the upcoming AlJ>-, consistent with the Riemannian anal­ysis in example 18.5(b). This is the chord that contains the "menacing bass trill on c;' which announces the imminent arriva! of the AA-/G- chord, which we can thus hear coming before i t sounds. The sense that we can "hear i t coming" is reflected in the PD arrow chain that departs from El*>+ to the right and down, directing our attention toward the coming AlJ>- via its dominant. The "menace" of the chord resides partly in the fact that it is pointing us further into the darkest regions of the space-further to the right an d downward.

As I noted above, both progressions trace a similar T -S-D-T progression, via altered subdominants; this similar trajectory is evident visually on the two exam­ples, as the progressions move fìrst to the subdominant side, below the tonic, then return to the tonic from above, "under dominant energy." Unlike the progression in 18.8(a), however, the flip to the dominant si de in 18.8(b) does not occur through a Rameauian reinterpretation. Instead, there is simply a snap from o ne extreme of the network to another, as the AlJ>- chord moves to ''Ai>+" via arrow 4.54 The snap from one edge of the space to the other coincides with Schubert's ffz dynamics and the second key-signature fissure, as the augmented-sixth "effortfully hauls the music back from its G-minor [ = AlJ>-] nadir."

This reading values the music in example 18.1 by tracing something of a middle path between the Riemannian and neo-Riemannian positions sketched above. As in neo-Riemannian accounts, I have sought here to valorize the remarkable in tlris passage; as in Riemann, I have clone so by situating the harmonies within a tonai

RIEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 505

Ex.18.8. (a) Analysis of mm.r-4; (b) Analysis of mm. 78-82.

(a) -l

(b)

space. ButI have not invoked that space in order to contain the music-at least, that was not my intent. Instead, I wished to show the ways in which the Impromptu's harmonies are invested with qualitative intensity via their tonal context an d the ways in which we as listeners participate in generating that qualitative intensity. The not­so-implicit claim is that the tona! context is responsible for the extraordinary sound: the latter results from the ways in which the musi c moves toward the outer, benighted realms of G~ major, or better, the ways in which we work to interpret its harmonies from those realms back to the tonic. This restores a sense of extremity-both har­monic and emotional/expressive-to the music.55 This extremity is also captured by

so6 TRANSFORMATION, ANALYSIS, CRITICISM:

the fragility of the intentional path from Ali>- back to G~+, reflecting the danger We

face of losing cognitive contact with the music's rational center.

The analysis seeks to narrow the fact/value gap between the sounds we cherish

and the analyses we construct. However successful or unsuccessful i t is in that effort

it is clear that the gap is not closed. No formai model can capture ali aspects of ou;

musical experience, even when we limit ourselves to o ne parameter, such as harmony.

For myself, I fìnd that the picture in example 18.8(b) turns what had been a flickering

and contingent experience into something more fìxed and stable, even overdeter­

mined. In my prose I have sought to mitigate this, reinvesting the formai model with

some sense of fragility. But this represents an intervention from outside the space of

the formai theory, suggesting the continued persistence of the fact/value split. The

present approach has a signal meri t, however, in that it gives the "value" side of the

equation specifìc hooks to attach to in the formai model, aliowing our evanescent

aural sensations to interact with the model in suggestive ways.

We are lucky in that our historical position aliows us, unlike Riemann, to relish

those moments in which chromatic works threaten to overflow the rational bounds of

our tonai theories, or in fact do overflow those bounds. As I suggested above, the rel­

ishing-indeed, valuing-of those moments seems to be o ne of the defìning traits of

the neo-Riemannian habitus. Yet, in the desire to detect coherence a t ali costs one

notes a continued reluctance to step over the next threshold, to relish that unruly part

of musical experience that resists formai containment. Perhaps this is only a matter of

time, however-the neo-Riemannian turn has introduced a new flexibility into tonai

analytical thought, moving us toward a highly salutary methodological self-awareness

and interpretive pluralism. This shift may ultimately lead us to relinquish coherence

(and its implicit sense of rational containment) as music theory's ethicallodestar,

allowing us to employ our analytical methods freely in the exploration an d construc­

tion of manifold musical experiences, without feeling the need to claim comprehen­

siveness for any one of them. For surely the best way we can value music is to

acknowledge that i t will always exceed the manicured gardens of our theories.

NOTES

1. The cited chords occur o n the downbeats of measures 78, 79, an d So. The passage expands a previous gesture in measures 74-77, which had already traversed part of this path, from Gb major tò d minor and back. As Charles Fisk has noted, the BIO minor harmony in both passages recalls the B-minor middle section in the previous Impromptu, in EJ.. Charles Fisk, Returning Cycles: Contexts far the Interpretation of Schubert's Impromptus and Last Sonatas (Berkeley: University of California Press, 2001), 118. For further discussion of the EJ, Impromptu, see my "Perspectives o n Tonality and Transformation in Schubert's Impromptu in EJ., D. 899, no. 2:' ]ournal of Schenkerian Studies 2 (2007): 33-63; on the intertextual resonances of Schubert's B-minor harmonies, see 47 n. 26 in the latter artide.

!liEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 507

2. O n notational "pressure" forcing enharmonic shifts, see Daniel Harrison, "Nonconformist Notions of Nineteenth-Century Enharmonicism:• Music Analysis 21.2 (July 2002): 140-142. O n the historical importance of the six-flat signature in Schubert's Impromptu, see Hugh MacDonald, "[Six-Flat Key Signature, 9!8],"19th-Century Music 11.3 (Spring 1988): 221-237. MacDonald calls the Impromptu "a breakthrough toward a new concept of the key" of Gb (p. 225) . The Impromptu was fìrst published by Haslinger in 1857 in G major; Hugo Riemann seems to have based his discussions of the piece-about which, more below-on Haslinger's (or a later) G-major edition.

3· For an overview of some of the most salient technical differences between paleo- and neo-Riemannian theories, see David Kopp, Chromatic Transformations in Nineteenth-Century Music (Cambridge: Cambridge University Press, 2002), 150-151.

4· For example, Richard Cohn states that neo-Riemannian theory seeks to answer the question, "If this music is not fully coherent according to the principles of diatonie tonality, by what other principles might i t cohere?" Cohn, "Introduction to Neo­Riemannian Theory: A Survey and a Historical Perspective:· ]ournal of Music Theory 42.2 (Fall1998): 169.

5· I mean Klang here- and throughout this chapter-in the familiar neo­Riemannian sense of major an d minor triads (no t in Riemann's sense of a dualistic emanation of overtones and undertones from a single pitch). The Klange in Example 18.2(a) orni t chordal sevenths and the one augmented sixth. The omissions would need to be addressed in a broader transformational analysis, but they are no t consequential here. For a transformational mode! that integrates members of SC 3-11 and 4-27 see Julian Hook, "Cross-Type Transformations and the Path-Consistency Condition," Music Theory Spectrum 29.1 (Spring 2007): 1-39.

6. Richard Cohn, "Square Dances with Cubes:· ]ournal of Music Thepry 42.2 (Fall 1998): 283-296.

7. The scare quotes make clear that DVLS values obtain in pitch-class space, in which the concepts of"up" an d "down" are traditionally considered problematic. I have nevertheless retained those words in the text for their intuitive immediacy. I have also replaced the directed pitch-class intervals of Cohn's DVLS with positive and negative integers, for the same. reason. These numbers should be understood as substitutes for their mod-12 equivalents. (Recently, Clifton Callender, Ian Quinn, an d Dmitri Tymoczko have recuperated the notions of"up" an d "down" in pitch-class space; I do not, however, rely on their formalism here. )

8. Joseph Straus, "Uniformity, Balance, an d Smoothness in Atona! Voice Leading," Music Theory Spectrum 25.2 (2003) : 321-322; see especially n. 39· AVLS is the same as Straus's "total displacement" and Cohn's "voice-leading effìciency" or VLE ("Square Dances:• 284).

9· This lends a consistency of voice-leading distance to a Weitzmann regio n that is not present in a hexatonic cycle, in which AVLS values range from 1 to 3· On Weitzmann regions, see Richard Cohn, "Weitzmann's Regions, My Cycles, and Douthett's Dancing Cubes:• Music Theory Spectrum 22.1 (Spring 2000): 89-103. See also "Square Dances," 290-295. Cohn's Weitzmann regions arise from a transformational interpretation of ideas in Carl Friedrich Weitzmann's pamphlet Der iibermaflige Dreiklang (Berlin: T. Trautwein, 1853). Ali of the triads in a Weitzmann region share two tones with a single augmented triad. Interestingly, several augmented triads appear prominently on the surface of Schubert's Impromptu-{Db, F, A~} in measures 4 an d 58; {Gb, Bb, D~} in measure 24; and {d, EJ., G~} in measure 73· The sense that the piece tends toward an augmented-triad sound world-especially in moments of transition-is suggestive, but I would not push the idea

508 TRANSFORMATION, ANALYSIS, CRITICISM

too hard: ali of these chords operate in ways far more traditional than the Lisztian possibilities that Weitzmann had in mind.

10. In this an d later networks, undirected edges are a shorthand fora symmetrical pair of arrows (or more colloquially, double-headed arrows). I have drawn the region in a hexagonal forma t analogous to Cohn's images of his hexatonic systems in "Maximally Smooth Cycles, Hexatonic Systems, an d the Analysis of Late-Romantic Triadic Progressions," Music Analysis 15.1 (1996): 9-40, and in "Weitzmann's Regions;' 95, ex. 7. This approach essentially reverses Cohn's graph-theoretic priorities in "Weitzmann's Regions;' in which i t is the hexatonic cycles that are drawn cyclically, with Weitzmann regions joining them via a mediating augmented triad.

11. "Weitzmann's Regions;' 92 and 98. N inverts a triad about its Riemannian (NB) root, for example, C+~ F-. In more familiar Anglo-Anlerican tonai terms, N maps a major triad to its minor subdominant (and back) or a minor triad to its major dominant (and back). Weitzmann's nebenverwandt relation is formally the same as Riemann's Seitenwechsel and Oettingen's Wechsel, which also exchange triads that share the same dua!

root. 12. Along with the identity operation E, these fìve operations form a tidy diliedral

group of order 6. O n SLIDE, see Davi d Lewin, Generalized Musical Intervals and Transformations, reprint ed. (NewYork: Oxford University Press, 2007), 178.

13. Richard Cohn, "As Wonderful as Star Clusters: Instruments for Gazing a t Tonality in Schubert;' 19th-Century Music 22.3 (Spring 1999): 213-232.

14. Por Riemann, Eb-- in the key of Gb major could act as either a tonic (T p ) or a

subdominant (S). We will explore the differences between Riemann's function theory and neo-Riemannian regional analyses such as that in example 18.4 below.

15. Riemann himself notes the piece's exploration of subdominant key areas a t the expense of dominant ones, using the fact as an argument for the dualistic equality of the two dominants. Hugo Riemann, Musikalische Syntaxis: Grundrifi einer harmonischen Satzbildungslehre (Leipzig: Breitkopf und Hartel, 1877), 71.

16. Michael Kevin Mooney provides a luci d and thorough overview of Riemann's Musikalische Syntaxis, including its Oettingen-inspired terminology an d the analysis of the Schubert Impromptu, in his dissertation "The 'Table of Relations' and Music Psychology in Hugo Riemann's Harmonic Theory" (Ph.D. diss., Columbia University, 1996), 162-175.

17. Riemann, Musikalische Syntaxis, p. 69. "Di e schlieB!iche Festigung der Haupttonalitat durch di e Thesen von og, g•, 0 es un d g• ist von ganz vorziiglicher Wirkung:' Riemann uses the word These (a holdover from his earlier Hauptmann-inspired work in "Musikalische Logik") throughout the book to refer to motions away from, or back to, the tonic, via its upper and lower dominants. I t comes to mean little more than "progression;' and that is how I have translated it above.

18. As observed in n. 2, the Impromptu was fìrst published in 1857 in G major, only twenty years before Riemann's book. I t is curious, however, that Riemann continued to referto the piece in its G-major version even in the sixth edition of his Handbuch der Harmonielehre, published in 1917. The volume of the Schubert Alte Gesamtausgabe including the present Impromptu in the correct key of Gb had been published in 1888, and the correct key was surely well known to Riemann by 1917. It is tempting to speculate that he retained the G-major version not only for pedagogica! clarity, but also asi t does not exhibit the same notational disruption as the Gb version does: there is no shift in key signature in measures 79-80 in the Haslinger edition, thus removing any notational sign of tonai disruption an d visually clarifying the minor-Neapolitan hearing of the harmony in

measure So.

RiEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO - 509

19. Riemann, Musikalische Syntaxis, 65. "eine formell recht iibersichtlich gegliederte Komposition."

20. Ibid., 69. "Das Ganze ist ein Meisterstiick sowohl hinsichtlich der Melodiebildung als der metrischen Struktur, besonders aber in Hinblick auf die Thesenordnung. Den bei weitem grtiBten Theil beherrscht die Tonalitat von g+, di e Haupttonart."

21. Hugo Riemann, Systematische Modulationslehre (Hamburg: J, F. Richter, 1887), 202. "Immer wieder drangt sich uns die Geltung der Haupttonalitat auch wahrend der kiihnsten und weitestausholenden Modulation auf. Wenn wir daher n un am Schluss auf den Weg, den wir zuriickgelegt, zuriickblicken, erkennen wir, dass wir n un gelernt haben, immer weitere Kreise um das unverriickbare Centrum zu beschreiben."

22. Riemann, Musikalische Syntaxis, 69. "NB. Ausweichung nach der antilogen antinomen Terztonart g•-0 es."

23. Riemann's anxiety about chromatic progressions such as this one is evident earlier in the book, when h e twice stresses that o ne should treat su eh progressions (t o antilogic­antinomic third chords, among others) with the greatest caution (Vorsicht). Ibid., 19-21. He observes (p. 21) that such progressions are best used only a t the end of a piece, after the tonic has been established securely, as, presumably, in Schubert's Impromptu. In his valuable discussion of Riemann's analysis of the Impromptu, Michael Kevin Mooney observes that the antilogic-antinomic Terzklang is one of the most distant harmonies from the toni c, when measured by a metric Riemann provides in his slightly later Skizze einer neuen Methode der Harmonielehre of 1880. See Mooney, "The 'Table of Relations,"' 240-241. Notably, if o ne were to take seriously Schubert's enharmonic spelling of the chord in its originai key-as a G-minor triad in Gb major-it would represent the most distant harmony from the tonic on Riemann's scale, the Doppelterzwechselklang (gb+-0 dq),

24. Ibid., 120, trans. Alexander Rehding, Hugo Riemann and the Birth of Modern Musical Thought (Cambridge: Cambridge University Press, 2003), 105. "Die Kombinationen sind Gott sei dank unerschtipflich an Zahl un d man kann das Gebiet der Harmonik nicht Schritt fur Schritt abgehen, sondern nur iiberfliegen, aus der Vogelperspektive iiberschauen. Es geniigt a ber, die Hauptwege durch diesen herrlichen Garten Eden, den uns der Himmel nach dem Falle gelassen, zu erkennen; jeder mag dann selbst weitere Seitenpfade zu immer neuen Durchblicken in nie betretene Reviere fìnden."

25. Rehding, Hugo Riemann, m-112. ' 26. Ibid. , 51-59, 105, an d 114. 27. Renate Imig observes that the Variante is in fact inconsistent with Riemann's

dualist theory, in which major and minor thirds are not merely exchangeable within a triad, but are instead generateci in opposed, dualist directions. Renate Imig, Systeme der Funktionsbezeichnung in den Harmonielehren sei t Hugo Riemann (Diisseldorf: Gesellschaft der Ftirderung der systematischen Musikwissenschaft, 1970), 51-52.

28. Hugo Riemann, Handbuch der Harmonielehre, 6th ed. (Leipzig: Breitkopf und Hartel, 1917), xvii.

29. Rehding, Hugo Riemann, 58 n. 52 an d 168. 30. On the conceptual differences between Stufen and functions (for example, the

difference between a IV Stufe and an S function), see David Lewin, "Music Theory, Phenomenology, an d Modes of Perception," Music Perception 3-4 (Summer 1986): 342-343; an d Brian Hyer, "Ton al Intuitions in Tristan und I salde" (Ph.D. diss., Yale University, 1989 ), 105.

31. The Ds in parentheses in the Riemannian reading indicate applied dominants of the following harmonies. Note that the preferred Riemannian reading of the German sixth is as an altered applied dominant. The slash through the D indicates that the root is omitted, while the > symbol after an Arabi c numerai indicates chromatic lowering.

510 TRANSFORMATION, ANALYSIS, CRITIC!SM

32. O n universality in Riemann's thought, see Rehding, Hugo Riemann, chapter 4. Riemann's analyticai practice cuts both ways, of course: he also uses his theory to demonstrate the alleged vioiation of his universaiiaws in composers iike Berlioz. See ibid 152-156. .,

33. I t is important not to monumentaiize neo-Riemannian theory as a singie practice-there are notabie instances of transformationai approaches to chromatic harmony that do no t fit this description. David Lewin, for exampie, never speaks of "disunity'' (or "unity," for that matter) in his harmonic-transformationai writings; both he and Davi d Kopp further empioy transformational approaches to expiore specifically tonai characteristics of chromatic passages (asI have in my work). I base my discussion here on the influential species of neo-Riemannian analysis m ade popular by Cohn, within which the vaiues outlined above have been remarkabiy consistent.

34· Riemann's "universai" classicism is of course simplyViennese classicism (the roots of which he repeatediy traces back to the Mannheim symphonists, especially Stamitz). On the nationalist motivations behind this project, most expiicit in the 189os, see Rehding, Hugo Riemann, chapter 4·

35· Ibid., no. 36. In addition to Rehding, see Scott Burnham, "Method and Motivation in Hugo

Riemann's History of Harmonic Theory;' Music Theory Spectrum 14.1 (Spring 1992): 1-14. 37· Rehding, Hugo Riemann, 9· 38. In addition to the Garden of Eden passage, the idea of spatiaiized boundaries to

cornpositionai possibility ernerges vividly a t the end of Riemann's history of nineteenth­century rnusic, published in 1901. After a negative assessrnent of Richard Strauss, he writes, "But o ne hopes that this tre n d [ toward prograrn musi c] has reached a boundary with Strauss, a t which i t rnust turn back." Hugo Riernann, Geschichte der Musik seit Beethoven (1800-1900) (Berlin: W. Spernann, 1901), 759. "Doch steht zu hoffen, daB diese Richtung mi t StrauB an einer Grenze angekornrnen ist, die zur Urnkehr zwingt."

39· Rehding, Hugo Riemann, 39. 40. Hugo Riernann, Beethovens Streichquartette erlautert von Hugo Riemann (Berlin:

Schiesinger, 1910), 129. "Verstandig interpretiert giebt der Satz keinerlei Aniass, von Zerissenheit und schwerverstandiichern Aufbau zu sprechen, zeigt vieirnehr deutlich das normale Geriist der Sonatenforrn."

41. Daniei Chua, The "Galitzin" Quartets of Beethoven (Princeton: Princeton University Press, 1995), 201.

42. Cohn, "Introduction;' 167. 43· The rise of neo-Riernannian theory can be read in one sense as a savvy

disciplinary response to the challenge of New Musicoiogy, in which the anaiyticai toois of a discredited high-rnodernist canon are turned toward new interpretive ends in the very repertory prized by criticai rnusicoiogists, with certain buzzwords retooied aiong the way.

44· Richard Cohn, "Uncanny Resernbiances: Harrnonic Signification in the Freudian Age;' Journal of the American Musicological Society 57.2 (Fall2004): 285-323.

45. Cohn, "Introduction;' 169. 46. Charles Fisk, Cornrnent & Chronicle, 19th-Century Music 23.3 (Spring 2000): 301. 47· Richard Cohn, Cornrnent & Chronicle, 19th-Century Music 23.3 (Spring 2000): 303. 48. Alien Cadwallader and David Gagné, for exarnpie, introduce the functionai

categories T, in t (for "interrnediate"), and D in their Schenker textbook without once mentioning Riernann by narne. Allen Cadwallader and David Gagné, Analysis ofTonal Music: A Schenkerian Approach, 2nd ed. (New York: Oxford University Press, 2006).

RiEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 511

49· As Riernann put i t in the fifth edition of his Musik-Lexikon: "Functions ... describe ... the various significances that chords possess, depending o n their position [ with respect] to the tonic." Transiated in Rehding, Hugo Riemann, 188. See aiso Kopp, Chromatic Transformations, 99.

50. This discussion raises a perenniai question in Riernannian exegetics: Are his functions iabeis for chords, reiations, or syntactic categories? The present study opts for the second choice, interpreting functions as symbois for reiationai paths from the sounding chord to the tonic, following Lewin ( Generalized Musical Intervals and Transformations, 177) and Hyer (''Tonai Intuitions;' 99-107). Cogent discussions of the function-as-chord versus function-as-category/reiation probiern rnay be found in Mooney, "The 'Tabie of Tonai Reiations,"' 102-108; Daniei Harrison, Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Account of Its Precedents (Chicago: University of Chicago Press, 1994), 266-276; and Rehding, Hugo Riemann, 61 and 78-79.

51. Steven Rings, Tonality and Transformation (New York: Oxford University Press, 2011) .

52. Michaei Siciliano, "Neo-Riernannian Transformations an d the Harmony of Franz Schubert" (Ph.D. diss., University of Chicago, 2002); Jack Douthett and Peter Steinbach, "Parsimonious Graphs: A Study in Parsirnony, Contextuai Transformations, an d Modes of Limited Transposition," Journal of Music Theory 42.2 (Fall1998): 246-249.

53· I rnaintain the neo-Riemannian transformationaiietters here (R, P, and L) for familiarity, even though they create some dissonance with Riemann's own terms. A formai note: once a D or S is added to the network, its underlying graph is downgraded from "path consistent" to "universally realizabie," per Juiian Hook's terminoiogy in "Cross-Type Transformations and the Path-Consistency Condition;' 29.

54· The iatter represents the German-sixth chord, which, as we noted,.Riemann would interpret as an appiied dominant; the quotes around the Al>+ node in the exampie indicate that the alterations to the chord have significantly obscured its triadic basis.

55· I t is hard to gain a sense of such extremity without a tonic center from which to measure such things, as in many neo-Riemannian "de-centered" spaces. The present approach thus restores the "distortions" created by a toni c that Brian Hyer-in an influentiai move-expiicitly eliminated from his renewed, de-centered Tonnetz in "Reimag(in)ing Riemann;' fournal of Music Theory 39!1 (Spring 1995): 127-128.