TRANSFORMATION, ANALYSIS, CRITICISM

484 TEMPORAL SPACE are now unknown. Eduard Hansick seems to have brought the poem in question to Brahms's attention (see Brahms/Herzogenberg, Briefwechse, 2: 135n); perhaps he ent Brahms a copy of Lingg's Gedichte as we. In any case, the transcription here foows the poem as found in Hermann Lingg, Gedichte. Dritte vermehrteaufage (Stuttgart and Augsburg: Cotta, 1857), 56. The poem is reproduced identicay in both the second and fourth editions (pubished by Cotta in 1855 and 1860, respectivey). 30. See Otto Friedrich Gruppe, Gedichte (Berin: Reimer, 1835), 55. PART VI TRANSFORMATION, ANALYSIS, CRITICISM T has been variousy noted that neo-riemannian theory emerged as a force to be reckoned with a t exacty the time when the project of musi c theory an d anaysis in the Ango-American academy had to parry a fundamenta critique of its aims and assumptions. This is hardy a coincidence. Certain positions of neo-riemannian theory can be seen as direct responses to the main points of criticism: tonai unity, the a-encompassing caims of anaysis, and utimatey the deep connections with the idea of the musica work. t is especiay these points that neo-riemannian theory has scaed back an d rethought, an d i t is these points, o ne might further add, that are most a t odds with (paeo-)riemann's own theoretica project. As a consequence, it is rare to find neo-riemannian theories being appied beginning to end in a p ieee of music. Yet some of the most powerfu insights can be gained through the interaction of neo-riemannian theories with other music-theoretica approaches. Indeed, i t is the very fexibiity of the approach that gives neo Riemannian anaysis its innovative strength. At the same time, a few fundamenta questions bave remained unanswered-or have received answers that are taiormade to specifìc situations. The questi o n of what kind of tonaity, if any, neo-riemannian theory represents has occasionay been raised. Simiary, the questi o n of repertoire-predominanty, 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 ead us into the wider aesthetica ream. In addressing how ideas that originated with Riemann may respond to contemporary anaytica chaenges, 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 pubished anaysis: Schubert's triadic but highy chromatic Gb-major Impromptu. Rings compares Riemann's own anaysis of the work with a neo Riemannian view inspired by the writings of Richard Cohn, assessing the differences in anaytica methodoogy and technoogy, and ocating those differences within the divergent ideoogies of the two approaches. Rings's centrai concern, however, is not with the anaytica technoogies themseves, but rather with the assumptions and vaues that underie the distinct anaytica perspectives. Rings focuses o n anaytica vaues with an eye toward synthesis: an enrichment of the neo Riemannian perspective through an engagement with the ethica an d methodoogica concerns of the paeo-riemannian approach. In the foowing chapter, Robert Cook performs a virtuosic hermeneutic anaysis of César Franck's Le chasseur maudit, which serves further as an extended and eegant refection on the potentia and imitations various anaytica frameworks. Cook situates his anaysis with respect to notions of chromatic music, in particuar the idea that chromaticism poses anaytica difficuties that Riemannian and neo Riemannian perspectives are particuary we suited to address. After considering the work from both functiona and inear perspectives and examining the conceptua probems that attend each, Cook iustrates how a contextua, neo-riemannian view can capture the work's saient gestures, an d offers a baance between a desire to understand the work as a refection of an ordery, conceptuay coherent reationa system an d the need to engage the aura! experience of the music. Danie Harrison coses this part of the book with a three-section essay, exporing certain interreated themes and questions centrai to the transformationa and neo-riemannian enterprise. Part one probematizes the natures of musica objects and reations within the transformationa wordview, 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 empoy more Kurthian anguage-not as sensuous but as energetic entities. Harrison deves further into the object/transformation dichotomy in the second section, defty exporing the structura an d functiona differences among dissonant and consonant trichords in a particuar nonatonic cyce. The essay, a fantasy o n a variety of specuative an d historica themes, expores how voiceeading, functiona, an d set-theoretica impications of the cyce might be profìtaby engaged by a transformationa perspective as a means to impart "sensuous distinctions" among otherwise indifferent transformations. The third section investigates the anaytica ramifications of the first two sections. Vaughan Wiiams' neo-m o da, triadic Fantasia on a Theme by Thomas Tais provides the soi in which these considerations can take root. CHAPTER 18 RIEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- STEVEN RINGS I The passage in exampe 18.1-from the coda of Schubert's Impromptu in Gi>-provides a usefu point of departure. Ex. 18.1. Schubert, Impromptu in GJ,, D. 899, no.3, mm. 78-82. 1:: n ~ h ~mrr=~mrttrtctri :::;;:;;;;,Mtfttrrtr~ C! 82 ~ "U" ~ '

488 TRANSFORMATION, ANALYSIS, CRITICISM RIEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 489 The tona 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 tri on C in measure 79 announces the imminent arrivai of the atter-we hear the G-minor chord coming before it sounds. (Such a menacing tri coud hardy be preparing us for G major.) Miracuousy, this forecasting does not essen the shock of the chord when i t actuay arrives. The G-minor chord of course admits of a tona interpretation: it is the minor Neapoitan, enharmonicay respeed. The chord nevertheess emanates a surpus of harmonic energy, overfowing the bounds of such a famiiar tona category. This surpus registers not ony sonicay, but aso in the notati an. Speed "correcty;' the chord woud be A*> minor; as a tonic, its key signature woud have 14 fats (that is, doube fats on every diatonie pitch). Schubert has aready begun from a point of fatward extremity: given his six-fat signature, any motion fatward wi exert pressure on the notation. 2 In the passage in question, the fatward pressure is so great that i t forces an enharmonic snap in the music, creating visua fissures on the page where the six-fat signature is canceed in measure 79 and then reinstated hafway through measure 80. The reinstatement coincides with a ffz augmented-sixth chord, which effortfuy haus the music back from its G-minor nadir, eading to a confirming cadentia 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 ast minute, under dominant energy. The descent in exampe 18.1, however, presses so far in the subdominant direction that it has the character of a tona crisis or trauma, the intensity of which registers visuay on the page, in the fissured notation. We can indeed hod the Gbtonic in our ears throughout the passage-thus retaining the minor-neapoitan hearing-but i t takes some effort to do so. If we isten whie ooking a t Schubert's fractured score-perhaps whie paying the piece-we may be encouraged to give up that effort atogether, opebing our ears to the chord's extratona surpus. How we respond to such a passage anayticay says much about what we vaue in music-and in musica anaysis (the two are not necessariy the same). Given its harmonic compexities, Schubert's passage provides an especiay fruitfu context for exporing some of the divergent vaues inherent in (echt-)riemannian and neo-riemannian approaches to harmonic anaysis. The various technica differences between Riemann's harmonic theory (in its many iterations) and neo-riemannian theory (in its many iterations) are, by now, reativey we known. 3 Less attention has been paid, however, to the theories' strikingy different attitudes toward the anaytica act itsef, incuding the different ways they seem to vaue music (in both senses: "cherish music" and "invest music with vaue")_ Such differences are, i t need hardy be said, products of the theories' highy distinct historica, ideoogica!, and cutura! moments. In what foows, I wi take an initia step toward mapping some of these divergences in vaue (and uncovering some unexpected points of contact), taking Schubert's Gb Impromptu as a point of reference. Section II compares a mode! neo-riemannian anaysis of the passage-based o n the work of Richard Cohn-to Riemann's own anaytica comments about the piece, which bookend his career, appearing first in the eary Musikaische Syntaxis (1877) and then in the sixth edition of te Handbuch der Harmonieehre (1917). Section III then expores the methodoogica and ethica contrasts between the two approaches in depth, tracing aspects of the inteectua and ideoogica! contexts in which they arose. The chapter concudes in section IV by considering some ways in which a technica rapprochement between the theories might open our ethica horizons, providing new ways in which we can vaue music through Riemann-inspired anaytica activity. II It seems safe to say tat te music in exampe 18.1 woud catch the ear of any neo-riemannian anayst, perhaps even providing the first point of anaytica entry into te p ieee. (O ne thinks h ere of the many anaytica forays into Parsifa that have begun no t a t the work's outset, but with the most chromaticay distorted version of te Grai motive, very near te end of act III.) Neo-Riemannians have often expored such passages by turning attention away from te traditiona categories of tona harmony and toward voice-eading efficiency, in an effort to detect pattern and reguarity where tere might otherwise appear to be tona strain or disorder. 4 By invoking enharmcinic equivaence, such approaches further sidestep enharmonic compexities such as tose discussed above. Exampe 18.2 sketches aspects of the Schubert passage from tis perspective. The grand staff at (b) shows a reduction of the passage. The singe staff at (a) extracts Kange from the music. 5 A key at the bottom of the exampe expains 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 reveas that every chord maintains at east one common tane with its predecessor; furthermore, motion by ic1 predominates. The annotations above staff (a) tay the resuts of the tota voice-eading between the chords. DVLS is Richard Cohn's "directed voice-eading sum:' 6 It measures the directed voice-eading motion between chords, distinguishing -between "up" an d "down:' 7 Thus, the first entry in the row, +2, indicates tota voice-eading of two semitones "up" from Gb+ to the B-: the two fied noteheads in te B- Kang indicate the two voices that have moved up by semitone from Gb+. The -2 that foows indicates tota voice-eading of two semitones "down" from B- to D+, as o ne voice descends by whoe tane. And so on. A cear pattern emerges: DVLS vaues aternate between +2 and -2 unti the Eb+ ( = D+) chord of measure 8o proceeds to the Gb+ of measure 81, yieding a DVLS vaue of o: h ere two voices move by semitone, but in opposi te directions, canceing each other out. This is the very moment a t which te ffz augmentedsixth chord wrenches the music back to a cadentia progression in Gb. The "wrenching" registers here in the contrary motion of DVLS = o, which ends the +2/-2 taispin.

490 TRANSFO RMAT ION, ANALYSIS, C RITICISJ\.1: R!EMANNIAN ANALYTICAL VALUES, PALEO- AND N EO - 491 Ex. 18.2. Harmonic reduction of Exampe 18.1, with voice-eading anaysis. (a) " v ) (b) " rv : DVLS: AVLS: G.+ 78 t +2 2 B- 79,! -2 2 D+ ~ +2 2 G- 80 '! - 2 2 o 2 Eih+ (=D+) q- jfz J,i,. -3 +3 G.+!)\,+ 81 ""'' ~. G.+ i 82 - Ex.18.3. (a) Weitzmann region incuding G~ +; (b) passes through this region in measures 78-81. (a) R Ii /I LP' "-11 / ~ \ ~ : 'PL N R..._ / N (b) R / :...,_ N I I r.t: ~: I ~ ".o /N Keyfor(a): o = common tones = motion by ic + = motion by ic2 DVLS = Directed voice-eading sum AVLS = Absoute voice-eading sum The row beow DVLS is abeed AVLS for "absoute voice-eading sum." This measurement takes no account of the direction of the voice-eading, instead measuring ony the absoute distance traversed in interva casses, registering what Joseph Straus cas the tota voice-eading "work" or "exertion" of the progression. 8 Again, there is a cear pattern: AVLS is 2 forai entri es unti! the cadentia osciation between G~+ and D~+, where it increments to 3. This reading notes a continuity in the progression from Eh+ to G~+, which traverses the same absoute voice-eading distance as ai of the preceding progressions. The wayward chromatic successions of the first p art of the phrase thus ai show AVLS = 2, whie the key-reaffirming cadentia tag in G~ projects AVLS = 3 The prevaence of 2s in the AVLS row suggests a particuar voice-eading space', which Cohn cas a "Weitzmann region:' Ai of the chords in such a region reate to one another by AVLS = 2. 9 Exampe 18.3(a) shows the Weitzmann region containing G+. 10 The soid, undirected edges circing the perimeter of the network indicate the transformations that reate adjacent triads within the system: N and R. The former is Cohn's transformationaabe for Weitzmann's nebenverwandt reation; 11 the atter is the famiiar neo-riemannian reative. Dashed edges indicate transformations between nonadjacent triads: Kiinge "two apart" o n the cyce are reated by PL or LP, and those apposite one another are reated by Lewin's SLIDE. These five transformations-n, R, LP, PL, and SLIDE-are the ony neo-riemannian transformations ( out of 24) for which AVLS = 2Y Ai of the Kiinge in exampe 18.2(a), with the soe exception of the dominant D~+, resi de in this Weitzmann region. Exampe 18.3(b) shows the passage's progression through the region, up to the G~+ cadentia six-four in measure 81. The progression begins with G~+ a t 12 o' dock an d proceeds cockwise aro un d the outer edge unti! i t reaches G- a t six o' dock. Aong the way, an LP arrow eads from B- to G-, indicating that the D+ chord that intervenes in the second haf of measure 79 pays a passing roe between the two harmonies o n the downbeats.. On the "return trip'; countercockwise from G-back to G~+, a simiar LP arrow eads from D+ to G~+; this is the "wrenching" LP motion associateci with the resoution of the augmented sixth to the cadentia six-four. (The return trip bypasses B- atogether.) After returning to G~+, the music eaves this Weitzmann region to engage in the confirming cadentia progression via AVLS = 3. Other passages in the Impromptu aso trace out significant portions of a singe Weitzmann region. Most notabe among these is the other highy "purpe" patch in the piece-the sojourn to C~ major and Eb major within the B section (measures 32-53). This is mapped in exampe 18.4, which presents the three Weitzmann regions containing G~+, C~+, and D ~ +, abeing them T, S, and D in a manner anaogous to Richard Cohn's abes for hexatonic systems in his anaysis of Schubert's B~ SonataY The exampe first shows the move to E~- and. its dominant a t the opening of the B section (measures 25-31); these harmonies stii resi de in the tonic region. 14 Then, at measure 32, there is a move to the subdominant region's 0+, via an interregiona L transform of E~-. The first "purpe" motion traced in the subdominant region is the aternation between 0+ and Fb- in measure 35. The dashed edges then ink these chords to the next!oca! tonic in the section, the E~+ that enters in measure 48, which eads to the transitiona A~- in measures 52-53. From here the music progresses to the dominant D ~ +, which is highighted within its network o n the right-hand side of the exampe. Notaby, no other node is "i t up" within the dominant network-in fact, none of the other Kange in the dominant region sound prominenty anywhere in the Impromptu.

492 TRANSFORMATION, ANALYSIS, CRITICISM_ RI EMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 493 Ex. 18.4. T, S, and D Weitzmann regions showing harmonic events in the B section. R-... /'N c:- This is indicative of the way in which the Impromptu thoroughy expores the subdominant si de of G~, but not its dominant side. 15 To be sure, there are severa! infeicities in the anaysis in exampe 18-4. For o ne, and most obviousy, the anaysis eaves out many harmonies within the B section that do not fit into its regions, most notaby a of the stormy diminished-seventhbased music in measures 40-45 and some!oca! dominants. Furthermore, it eaves the question of the reationship between voice-eading efficiency and harmonic function (T, D, and S) somewhat undertheorized, confating the two in a way that causes the distinction between chord and key to break down (e.g., by treating syntactica harmonies in the same way that i t treats tonicized harmonies). Simiary, the separation of the tonic, dominant, and subdominant triads into separate regions seems to do vioence to their!oca! syntactic connectedness a t the eve! of the phrase. These are famiiar probems in certain strands of neo-riemannian anaysis. Yet, despite these shortcomings, the anaysis provides a suggestive heuristic for tracing the piece's voice-eading activity, showing the ways in which i t navigates viaavls = 2 in its most ear-catching passages. This neo-riemannian reading-ony the beginning of a fuer anaysis-has proceeded as such readings often do, beginning with the most chromaticay exceptiona moment in the p ieee and moving outward from there to construct a broader interpretation of the movement. Cohn takes a simiar approach, for exampe, in his anaysis of Schubert's B~ Sonata, beginning by observing the hexatonic-poar reationship between the work's B~+ tonic an d its F#- secondary key area, then seeking out other hexatonic-poar progressions (such as iie one at the transition into the deveopment section), and utimatey constructing a hexatonic anaysis of the entire movement. The contrast with Riemann's own anaytica practice coud hardy be more stark. We can see it by consuting his anaysis of the G~ Impromptu, pubished in N ]Jusikaische Syntaxis in 1877. This is not ony Riemann's first pubished anaysis, but aso one of his ongest, at seven pages. Not unti the Beethoven piano sonata anayses from the end of his career woud he pubish further anayses of comparabe scope and technica detai. His primary concern is the Impromptu's phrase structure an d its demonstration of the principes of harmonic syntax that he deveops in the book, based on an arcane terminoogy deveoped from Oettingen. 16 Riemann's descriptions are often minutey detaied: he devotes an entire paragraph to the first phrase, and two paragraphs to measures 17-24. An d about the phrase in exampe 18.1 he says... amost nothing. The passage gets ony a very brief passing mention in his main prose, but i t is no t singed out; it is simpy isted as one of severa! progressions in the coda: "The fina! consoidation of the primary key through progressions [ Thesen] to C minor, G major, Ah minor, and G major has a whoy exceent effect." 17 Riemann is referring here to the music in measures 74-81; note that he anayzes the piece in G major. 18 The progression to A~ minor- the minor Neapoitan in measure 8o-is given no specia emphasis: i t is merey the goa of o ne of the four Thesen that Riemann mentions. Even more strikingy, Riemann cites these progressions as paying a roe in the "consoidation of the tonic" [Festigung der Haupttonaiti:it]. This is in vivid contrast to the comments a t the head of this chapter, in which I suggested that the music in exampe 18.1 can be heard to ead to a tonai crisis, creating a harmonic surpus that overfows the tonai frame. Riemann, by contrast, hears in these measures nothing more than a fina! confirmation of the goba tonic, a confirmation that, moreover, has an "exceent effect" [ vorziigicher Wirkung]. I t is hard to te exacty what aspects of the passage Riemann finds vorziigich, but whatever they are, they seem to have itte to do with any undermining of the tonai order. His anguage instead suggests a ceebration of the piece's confirmation of eternai ton~ aws-its exempary estabishment and reinforcement of a Haupttonaiti:it. Thus, the passage that received so much attention in the neo-riemannian account, serving as the starting point from which a other observations radiated, is itte more than a footnote for Riemann, a negigibe chromatic rippe on the surface of an exempary tonai masterwork. Indeed, Riemann frames his anaysis in just this way, describing the Impromptu as a mode! citizen of the tonai ream, a "formay rather ceary structured composition:' 19 He praises the piece's ordery construction: "The whoe ìs a masterpiece as regards not ony meodie form and metric structure, but especiay as regards the ordering of its progressions [Thesenordnung]. And over a of it reigns the tonaity of G major, the principa key." 20 The second sentence makes cear Riemann's firm commitment to monotonaity. (Modern readers wi be struck by the pre-echoes of Schenker and Schoenberg.) Ten years ater he voiced a simiar sentiment in a more generai context in his Systematische Moduationsehre: One is constanty struck by the controing force [ Getung] of the ma in tonic, even during the bodest and most wide-ranging moduations. When we find ourseves a t the end of the path, ooking back, we know that we have earned how to trace ever wider circes around the unshakeabe center. 21

494 TRANSFORMATION, ANALYSIS, CRITICISM RIEMANNIAN ANALYTICAL VALUES, PALEO- ANO NEO- 495 Though he is not specificay discussing the Impromptu here, this passage ceary appies to Riemann's understanding of the piece, in which a singe tonic not on!y contros the whoe, but does so in mode! fashion. The piece's harmonic excursions do not weaken the tonic, but instead contribute to its greater gory, concentricay expanding its dominion. (Again, the Schoenbergian resonance is striking.) There is ony o ne other passing reference to the minor Neapoitan of measure 8o in Riemann's extensive discussion of the piece. In a tabuar overview of the piece's harmonic progressions, he writes beneath the chord symbos for measures 79-80: "NB. Moduation to the antiogic antinomie third-key: g+- 0es." 22 The Oettingen-inspired terminoogy simpy means a progression from G major to the minor harmony whose dua root is a major third beow-that is, a progression from a G over-triad to an Eb under-triad (i.e., Ab minor). Again, it is not entirey cear just what we are to "note we" about the passage. The annotation coud be taken for an excamation of surprise, and perhaps admiration, at Schubert's harmonic audacity: "Note we: A remarkabe progression!" But the invocation of the arcane theoretica nomencature might aso suggest something quite different. We are to note not simpy a striking progression, but the fact that the music proceeds to the antiogic antinomie third-key. This yieds a very different sentiment: "Note we: My theory even has a name for this chord." Read in this atter sense, the statement seems to betray an anxiety, an attempt to contain the harmonic extravagance of the moment within the rationa bounds of the theory. 23 It suggests a desire to demonstrate that no part of the Impromptu eudes the theory's expanatory rea eh: a of its harmonic maneuvers are easiy contained and rationaized within the theory's bounds. Riemann himsef expicity thematizes the notion of spatiaized boundaries to harmonic possibiity in the book's cosing pages. Here h e suggests that h e has mapped out a spatiaized rearn of tonai order, comparing i t to a harmonic Garden of Eden: Thank God the combinations [ofharmonies] are inexhaustibe in number, and o ne cannot expore the area of harmony in its entirety by waking across i t step by step but ony by fying 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 eft us after the Fa; everybody may then fin d new si de paths for himsef eading to ever new perspectives on regions never entered before. 24 Schubert's progression woud seem to represent o ne of the exotic, "new si de patis" within this ream, off of the beaten track of the Hauptwege, but nevertheess admissibe. Yet, as Aexander Rehding has noted, Riemann's passage beies a profound worry: he presents the Garden of Eden as universa and transhistorica, but his anguage "impies a t the same time a premonition-conscious or not-of its actua, contingent nature... the whoe theory is buit o n a feeing of angst, a Spengerian feeing 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 ai, the invocations of the Garden an d the Fa vividy suggest tie possibiity ofharmonic sin. To sin against the tonai arder coud bring about permanent banishment from the Garden-that is, banishment from the ream of tonai order into the a tonai wids. Did Riemann have doubts as to whether Schubert's minor Neapoitan might represent just such a harmonic sin? He may have. For we find him preoccupi ed with tie chord forty years ater, when revising his Handbuch der Harmonieehre for its sixth edition. His foreword to this edition of the Harmonieehre contains the ast additions to the theory of functions, which he had first introduced in 1893's Vereinfachte Harmonieehre. Rehding observes that the theory of functions represented a way to contro! and corra the overy permissive possibiities for harmonic progression in some of Riemann's earier harmoriic theories, incuding that in Musikaische Syntaxis, thus better fortifying the boundary around the Garden of Eden. 26 It thus makes sense that he woud return to the Schubert chord to make sure its energy was contained within his new system. In the foreword to the Handbuch, Riemann adds symbos for direct third reations, as we as a symbo for the moda! Variante of any function-a v after a function symbo, which simpy switches the triad's mode. 27 The singe exampe Riemann adduces for the new symbo is the minor Neapoitan from Schubert's Impromptu, which he now anayzes as CJS : the variant of the eading-tone change of the minor subdominant. 28 Even with the new symbo, the chord ceary puts a strain on Riemann's functiona system, as it requires three aterations to the initia S function, which are made visibe in the three accretions to the S symbo: 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 mutipe aterations to a function symbo. 29 And indeed, his wariness is apparent here: he cas the chord "exceptiona" (an Ausnahmserscheinung) and otherwise uses the v symbo ony rarey in his ater anayses. Nevertheess, the sense of Riemann's symbo is qui te cear, an d i t is no t that far from the way in which the chord woud be anayzed in a modern American theory cassroom using Roman numeras. Exampe 18.5 compares a Roman-numera anaysis of the passage at (a) with a fu Riemannian-functiona reading at"(b) and the neo-riemannian reading at (c). In order to make the progression's correspondence to the tonai readings more egibe, I have adjusted the enharmonic speing. The Roman-numera interpretation constructs the chord on the downbeat of measure 8o as an ateration of a harmony bui! t o n the (owered) second scae degree, whie Riemann constructs it as a modification of subdominant function. There are important conceptua distinctions between the two theoretica concepts, 30 but overa, the readings at (a) and (b) are quite simiar: both trace a descent from tonic, via appied dominants, 31 to an exotic, "subdominant-side" harmony, before puing back to the dominant at the downbeat of measure 81. Both anayses 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-numera; (b) Riemannian (c) Neo-Riemannian. '" note vaues haved '.~ t ~.~ S= (a) G\.: I V '---iv\. V Ger"+6 '--- bi V-- 0 (b) T (D') os (D') 8" (JP,?') D (c) T SLIDE ~ u.;tzmann J regions augmented sixth chord shows considerabe interpretive exertion as we). The ne6- Riemannian anaysis, by contrast, simpy shows the first six chords unified by voiceeading motion of AVLS = 2, a unfoding within the tonic Weitzmann region. t is ony a t the resoution of the Gb+ cadentia six-four to Db+ that the toni c regio n is eft for the dominant one. The infeicities of the neo-riemannian account from a functiona perspective eap into view here: i t captures none of the phrase-eve functiona distinctions in the first six chords, and further is unabe to read the Gb+ cadentia sixfour as functionay distinct from the opening Gb+ tonic. The functionaabes for the Weitzmann regions shoud thus indeed be understood as a secondary overay-an informai tonaabe appied to a theoretica system whose essentiaogic is no t tona. Notaby, the Riemannian reading at (b) casts the progression as syntacticay normative, foowing the preferred paradigm for cadentia motion: T-S-D-(1). The subdominant regio n is expanded consideraby by modifications, but this does not obscure the phrase's overa syntactic sense. The anaysis further reveas a cear syntactica resembance to the opening progression of the piece (in measures 1-3), which Riemann woud aso anayze 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 tona crisis or trauma, a reading that ed to neo Riemannian exporation of its extratonaogic, is refashioned h ere as a mode tona phrase, one that vividy demonstrates the efficacy of Riemann's functiona principes in a chromatic context. The dangerous chord of measure So has now been fuy contained within the boundaries of Riemann's Garden of Eden. '----' D RIEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 497 III This returns us to the broader question of vaue. The contrasts are as obvious as they are vivid. Riemann anayticay constructs chromatic passages so that they show conformance to his tona theories, which he portrays as universaaws. 32 The neo-riemannian anayst, by contrast, constructs chromatic passages so that they appear tonay "disunified," and thus require nontona expanation. Riemann's theory thus paces a high vaue on order and conformance to putative universas of tona harmony, whie neo-riemannian theory, it woud seem, vaues crisis and disruption of that order. 33 We can better understand this sharp divergence if we briefy survey the inteectua and ideoogica! contexts that nurtured Riemann's theory, on the one hand, and neo-riemannian theories o n the other. Riemann's context has been masterfuy reconstructed by Aexander Rehding, so I wi merey summarize his argument here. Rehding characterizes Riemann as seeking to define a universa "dassicism" 34 that transcends history an d thus acts as a brake against further historica change in music, ceary demarcating the boundaries beyond which music shoud not progress. For Riemann, music theory had an ethica responsibiity to set imits for composers, acting as "a bastion against historica change:' 35 Appeas to the burgeoning natura sciences aowed Riemann to provide "hard" support for his daims of universaity, 36 whie institutiona an d pedagogica! factors payed a roe as we, as he sought to deveop ahatmonypedagogythatwoud dispace Roman-numera base d Webei-ian approaches, thus aowing him to infuence future musicians directy, instructing them in the aws and imits of musica possibiity. The resut was a conservative theory shot through with a "reentess normativity." 37 In short, Riemann sought, through this theory, to stem the tide ofhistorica change in music, which seemed to him (righty, i t turns out) to be periousy dose to transgressing the boundaries of the tona Garden of Eden. 38 By the time of the American reviva of interest in Riemann's theoretica ideas in the 1980s an d 1990s, that transgression had of course occurred ong ago. Indeed, the crossing of music over the atona threshod was o ne of the primary factors eading to the discipinary consoidation of music theory in the American academy: theorists had taken advantage of the chaenges posed by posttona musi c to argue for the institutiona necessity of music theory as a research discipine. Atonaity, which before had been a ooming threat to Riemann, to be resisted a t a costs, now enjoyed great institutiona priviege and prestige, especiay among theorists in the second haf of the twentieth century. That neo-riemannian theorists woud vaue tona crisis far differenty from Riemann shoud thus come as no surprise. If a given passage by, say, Wagner was perceived to veer periousy dose to tona incoherence, it coud now be embraced anayticay using the technoogies of atona theory, thus inheriting the institutiona vaues associateci with avant-garde atona musics through a sort of ethica transitive property. This is ony part of the story, however. In the new, institutionaized theory of the American academy, anaysis became an end in itsef-a means of engaging

498 TRANSFORMATION, ANALYSIS, CRITICISM deepy with individua! works via various technica hermeneutic genres, often with a iberai humanist focus on interpreting the teing compositiona idiosyncrasy. Such a practice carri es with i t a strong eement of Romantic ideoogy, with the individua! work vaued for its originaity and uniqueness, as a product of genius. Neo Riemannian theory ceary participates in this ideoogy, with its amost excusive focus o n the compositionay extraordinary. (The chromatic Grai motive is a cassic case.) This is in stark contrast to Riemann's own practice, in which anaysis serve& first and foremost to iustrate an d vaida te his theory. His anaytica emphasis is not on w ha t makes a work remarkabe or individuai, but o n the ways in which i t exempifies the normative, aw-ike aspects ofhis theory. As Rehding observes, regarding Riemann's anaysis of the "Wadstein" Sonata: [I]t seems that Riemann is not interested in the specia features of [the sonata's] opening. Rather-it woud appear-he pays down the particuarity of this opening in favor of its generai features. Whie we have come to appreciate the fìrst few bars of the "Wadstein" sonata as a paradigm of Beethoven's harmonic bodness, Riemann's anaysis of this passage is actuay a demonstration of its ordinariness. 39 Riemann's abiity to find the ordinary within the compositionay extraordinary extends to a aspects of his theory, from harmony, to rhythm, phrase structure, and form. For exampe, he says of the first movement of Beethoven's op. 130: "Correcty interpreted, the movement offers no cause to speak of disruption and formai difficuty, but instead ceary shows the norma framework of sonata form:' 40 This breathtakingy matter-of-fact assessment-turning one of Beethoven's most fissured movements into a sef-evident and unprobematic sonata-form-dissonates not ony with modernist (Romantic) theory and anaysis, which woud seek to expore the structura particuarities that make the movement unique, but aso with more recent criticai musicoogy. Danie Chua, for exampe, characterizes the same movement as nothing ess than a "direct assaut" on the istener: "the audience is simpy thrown into confusion by a disarticuated syntax, by a anguage so vioent and contradictorythat t o anayze the disunity is to be more obvious than 'post -structuraist:" 41 Obvious to us, perhaps, in our postmodern age, but not to Riemann, whose aesthetic system aowed no pace for disunity or disruption in a masterwork. Styes of criticai musicoogica thought such as Chua's are not irreevant to tis study. For it is not a coincidence that neo-riemannian theory roseto prominence. around the same time as the discipinary upheava caused by the New Musicoogy. The vaorization of disunity, crisis, fragmentation, an d heterogeneity in that iterature finds a curious an d distorted echo in neo-riemannian theory. Indeed, in his introductory artide t o the fo urna of Music Theory issue dedicated t o neo-riemannian theory, Richard Cohn expicity situates neo-riemannian anaytica approaches with respect to "an evoving post-structuraist criticai practice;' suggesting a community of purpose as regards caims of tona disunity. 42 There are further paraes. Neo-Riemannian theory focuses on the very iterature privieged by the New Musicoogists-nineteenth-century opera and concert music-turning its F.IEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 499 attention to many of the very same "probem" passages that woud capture the attention of the postmodern critic. The neo-riemannian iterature's emphasis on "de-centered" harmonic spaces aso resonates with such criticai practices (though, more cynicay, one might simpy recognize here a co-optation of jargon)y More substantivey, the emphasis on anaytica puraism in many transformationa approaches, with roots in Lewin's methodoogica writings, squares we with postmodern interpretive practices. Finay, certain recurring hermeneutic tropes in the neo-riemannian iterature-such as Cohn's exporations of the harmonic uncannf4-are ceary indebted to the New-Musicoogica spiri t. Neo-Riemannian theory thus appears to pu off a seemingy impossibe doube pay, benefìting from two sets of prestigious but conficting institutiona vaues a t once (both of which woud 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 cear, any commitment to the atter is trumped by more famiiar music-theoretica concerns: Both paradigms [neo-riemannian theory and post-structuraist music criticism] recognize the potentia for tonai disunity in musi c that uses cassica! harmonies, and accordingy resist shoehorning ai chromatic triadic music into the framework of diatonie tonaity. For the post-structuraist, the recognition of tonai disunity eads immediatey to an ascription of disunity tout court, and from there to a custer of cognate terms... : "unstructured," "incoherent," "indeterminate," "cooristic," disjunct;' "arbitrary," or "aimess." The recognition of tonai disunity coud instead ead to a question: "if this music is no t fuy coherent according to the principes of diatonie tonaity, by what other principes might i t cohere?" 45 Thus, despite nods to postmodern sensibiities, the most time-honored vaue of modernist music theory remains fìrmy intact: the demonstration of coherence through formaism. An d here we fin d common ground with Riemann himsef. For is the "coherence" of the neo-riemannian anayst reay that far removed from the "ogic" or "syntax" of Riemann? Despite some obvious differences in phiosophica underpinnings, both projects are underwritten by a drive toward systematization and ogica! rigor; a penchant for eegant, symmetrica theoretica structures; and a desire above ai to detect arder in compex music, containing harmonic extravagances in controed, rationa spaces. These vaues, it woud appear, are pan-riemannian. But neo-riemannian theory contains methodoogica tensions not present in Riemann. The high vaue it paces on both disruption and coherence eads to a pecuiar sort of hybridity or doube focus. Surey a prime reason for the success of neo-riemannian theory is the fact that it aows anaysts to dwe on the most remarkabe sounding passages in a chromatic work, those moments when the tona! fabric is stretched or torn. Buti t is no t the remarkabe so un d of those passages that is anayzed; i t is their coherence. One thus begins to wonder what the reationship is between the sound and the anaysis. Is the "coherence" that the method detects responsibe for what it is that makes these disorienting passages so auray captivating? Or are the two unreated? In other words, do we vaue the anaysis for the same reasons that we vaue the music?

500 TRANSFORMATION, ANALYSIS, CRITICISM; RIEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 501 This question coud surey be made of any stye of systematic music anaysis. But it has a specia urgency in neo-riemannian practice, as criticai responses to the theory attest. Chares Fisk, for exampe, states that Cohn's anaysis of Schubert's B~ Sonata runs the risk of "making even the most extraordinary progressions in Schubert seem ordinary-or a t east, in some respects, normative." 46 (The echoes of Rehding's interpretation of Riemann are striking, suggesting further pan-riemannian simiarities.) Fisk is concerned that Cohn's theory does not do justice to the sound of Schubert's music, making the auray arresting seem theoreticay commonpace. Cohn responds by invoking what amounts to a music-theoretic fact/ vaue distinction, arguing that theoretica categories do not necessariy correate with sonic affects in a simpe one-to-one fashion, even in traditiona theoryy Cohn's writing o n the uncanny effects of hexatonic-poar progressions is an eoquent testament to this. Nevertheess, Fisk's criticism is hard to dismiss. The Schubert passage in exampe 18.1 sounds extraordinary, but the anaysis of exampes 18.2-4 does not te us about that, instead reveaing order and pattern in its voice-eading. We are thus eft to wonder just what it is that this music does to us after it enters our ears, why it thris an d captivates us. Refecting o n this matter, we might come to vaue 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 aways framed in normative terms ("How shoud we hear music?") and that the answers were therefore tinged with a sense of prescriptive "ought" does no t diminish greaty the vaue 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 theoretica and rhetorica excesses and detours, some of his ideas hit the mark so successfuy 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 infuence is stii fet, no t ony in Germany's Musikhochschuen, but aso in many strands of Ango-American Roman-numerabased harmony, even in some Schenker pedagogy (however obfuscated the debt to Riemann may be). 48 In the concuding section of the chapter, I wi thus expore one way in which Riemann's functiona ideas can be reanimated in a transformationa context, thus shedding some ight on the remarkabe sonic effect of Schubert's passage, and narrowing the neo-riemannian fact/vaue gap, if ony sighty. IV Riemann's functions mode the reationship of harmonies to the tonic, either directy or via one of its two dominants. 49 One way to interpret his function symbos is thus no t as abes for chords, but as descriptions of the actions that isteners perform as they interpret sounding harmonies with respect to the tona center. In this understanding, to hear a chord as a subdominant is to perform the 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 istener reates it to the tonic. 50 I have esewhere referred to this action of directing awareness toward the tonic as "tona intention." 51 We can trace such Riemannian intentiona acts on various species of Tonnetz. Exampe 18.6 shows o ne such space that is usefu in exporing Schubert's Impromptu: a dua of the famiiar neo-riemannian Tonnetz that Michae Siciiano cas the "LRP map" and Douthett and Steinbach cai 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 eft. The network is rotated 90 from its usua presentation in neo Riemannian studies (and in historica Tonnetze), so that fifth-reated triads are on the vertica axis, capturing the famiiar metaphor of dominants residing "above" a specified tonic, and subdominants "beow." Dominant (D) and subdominant (S) arrows can be added to this vertica dimension as necessary, to indicate direct functiona reatio!ships, whie the P, L, and R dimensions aow for the modeing of Riemann's Paraee (R), Leittonwechse (L), and Variante (P) functiona modifications.53 If enharmonicay 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 doube border-has. a centrai position. Exampe 18.7(a) demonstrates the intentiona paths traced by two modified subdominants in the Impromptu: the pj,_ in measure 2, the second chord in the piece; and the Ab- ( = G-) minor Neapoitan of exampe 18.1, measure So. The reevant Kiinge are indicated with crosshatching, whie soid dark arrows show the intentiona paths traced by their Riemannian interpretations. The pj,_ chord is

502 T RANSFORMATION, ANALYSIS, CRITICISM RIEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 503 Ex. 18.7. (a) Intentiona paths traced by two different subdominants in the Impromptu; (b) Moda vectors within the functiona space. (a) (b) brightest ~ ' ' ' ' ' major ' 1! ~ ',Q minor "" ' ' ' ~ '... ' '... '' darkest ' ~ ~ interpreted as a subdominant eading-tone change, or S:,suggesting an intentiona path of LS back to the G~+ tonic, passing through the subdominant C>+ aong the way. Note that if the same Kang were interpreted as a tonic parae, T p, i t woud trace out a different intentiona path: directy back to the G~+ tonic via R. Riemann's different interpretations of the same sounding chord thus traverse different paths in the space, making cear that, in the present interpretation, the function describes not the chord, but the path whereby i t is reated back to the toni c. The minor Neapoitan, A.-, is anayzed foowing Riemann's interpretation of i t in the sixth edition of the Handbuch as os-v, that is, the Variante of the eading-tone change of the minor subdominant. This suggests a consideraby more compex intentiona path back to the tonic: PLPS. The initia P modes Riemann's Variante (v); that is, it traces our interpretation of the chord as a Variante of the "proper" minor subdominant eading-tone change, as we mentay interpret AA- as reated to Ai>+ via P. The remaining transformations indicate simiar interpretive activity, L corresponding to the eading-tone change (>) an d the second P to the moda ateration of the subdominant (o). I stated above that "i t takes some effort" t o "hod the G~ tonic in our ears throughout the passage-thus retaining the minor-neapoitan hearing." That effort registers in the PLPS intentiona path-specificay, in the concatenation of four interpretive moves it suggests. Whie AVLS in exampe 18.2 expressed the exertion invoved in voice-eading one chord to the next, the present system expresses iie exertion invoved in interpreting each chord tonay with respect to G~ via Riemannian categories. We can make vivid that exertion by imagining the intentiona arrows in 18.7(a) traveing through a substance or medium-a medium that offers a certain resistance, imiting how far we can trave in tracing a harmony back to a tonic. The resistance might cause us to give up atogether in our effort to interpret the AAchord with reation to G~+, thus breaking the chain of arrows departing from AAin r8.7(a). Or we may reach G~+, but ony weaky, our tonai intentions exhausted in the effort. Our sense of G~+ as tonic woud thus be consideraby attenuateci, its hod on the music in our ears now precarious. But the arrows are not ony a measure of exertion. They aso give us a way to think about the tonai quaity of a given harmonic function-its characteristic soni c affect (and effect). A subdominant sounds different from a dominant, after a, as do the many modifications of these harmonies from one another. These quaitative diffèrences are often described metaphoricay by reference to coor (dark harmonies, bright harmonies, and the ike). Such differences in sound are nota product of the raw acoustic signa, but a product of the way in which we reate a given sounding harmony to a toni c. (After a, the same chord can be a subdominant in o ne context and a tonic or dominant in another.) That is, the affect arises ony after the music "enters our ears;' setting our tona-interpretive activity in motion. The quaity can thus be understood to inhere in the path of arrows we trace from the sounding chord back to the tonic. Exampe 18.7(b) expores aspects of the resuting affective or cooristic regions in the space. As it shows, the "brightest;' sharpward regions are up and to the eft from the tonic, whie the "darkest," fatward regions are down an d to the right. These coordinates provi de a rough sense of the coor that wi accrue to a chord as i t is reated back to the tonic from a given region of the network. The farther in any given direction a chord resides, the more intensey wi it acquire that coor. The AJj,_ chord resides deep in the darkest quadrant of the space, tracking back to G~+ from the farthest fatward reaches and acquiring an extremey dark tinge in the process. It is as though with each interpretive/intentiona act-each darkened arrow-the harmony accrues a new ayer of coor, or better, of shading. (The word evokes Kurth, whose ideas are highy pertinent here.) These accretions of interpretive shading are what give ii e G-/ AA- chord its remarkabe chiaroscuro quaity; o ne thinks of the mutipe ayers of paint around the dark perimeter of a Rembrandt portrait. Again, though the harmonic coor seems to infuse the sounding medium

504 TRANSFORMATION, ANALYSIS, CRITICIS},r RIEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 505 itsef, it is nota property soey of the raw acoustic signa! (i.e., the minor triad). We can experience this by cearing our ears of the harmonic context of the Impromptu an d simpy paying a G-minor chord aone, hearing i t as a toni c. The effe et is o ne of stripping away the ayers of shading that are present when we hear the chord in the context of exampe 18.1, as though we have stripped away the many ayers of Rembrandt's deep browns, reveaing the bank canvas of the minor triad beneath. In this sense, the osv chord of measure So has not ess tona! character than the more traditiona tona harmonies in exampe 18.1, but more. Exampe 18.8 integrates the two chords from exampe 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. Soid arrows show the intentiona path back from each chord to the tonic. Exampe 18.8(a) anayzes the opening phrase up to the G~+ chord of measure 4; 18.8(b) anayzes the phrase in exampe 1s. 1 (measures 78-82). The opening phrase remains argey vertica within the space of 18.8(a), moving first down into the subdominant region, and then baancing this with a motion to the dominant. Note that this reading takes advantage of the toroida possibiities of the space, reinterpreting the Ai>- chord in a manner anaogous to Rameau's doube empoi. Exampe 18.8(b), by contrast, spreads out horizontayand chromaticay-across the network, dipping into its darkest corner. Note the intentiona interpretation of the E*>+ chord: It is heard not as the reative (or Riemannian Paraee) of the previous Ci>- chord (Riemannian osp). Instead, it is heard as the dominant of the upcoming AJ>-, consistent with the Riemannian anaysis in exampe 18.5(b). This is the chord that contains the "menacing bass tri 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 refected in the PD arrow chain that departs from E*>+ to the right and down, directing our attention toward the coming AJ>- via its dominant. The "menace" of the chord resides party 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 simiar T -S-D-T progression, via atered subdominants; this simiar trajectory is evident visuay on the two exampes, as the progressions move fìrst to the subdominant side, beow the tonic, then return to the tonic from above, "under dominant energy." Unike the progression in 18.8(a), however, the fip to the dominant si de in 18.8(b) does not occur through a Rameauian reinterpretation. Instead, there is simpy a snap from o ne extreme of the network to another, as the AJ>- 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 "effortfuy haus the music back from its G-minor [ = AJ>-] nadir." This reading vaues the music in exampe 18.1 by tracing something of a midde path between the Riemannian and neo-riemannian positions sketched above. As in neo-riemannian accounts, I have sought here to vaorize the remarkabe in tris passage; as in Riemann, I have cone so by situating the harmonies within a tonai Ex.18.8. (a) Anaysis of mm.r-4; (b) Anaysis of mm. 78-82. (a) (b) space. ButI have not invoked that space in order to contain the music-at east, that was not my intent. Instead, I wished to show the ways in which the Impromptu's harmonies are invested with quaitative intensity via their tona context an d the ways in which we as isteners participate in generating that quaitative intensity. The notso-impicit caim is that the tona! context is responsibe for the extraordinary sound: the atter resuts from the ways in which the musi c moves toward the outer, benighted reams of G~ major, or better, the ways in which we work to interpret its harmonies from those reams back to the tonic. This restores a sense of extremity-both harmonic and emotiona/expressive-to the music. 55 This extremity is aso captured by -

so6 TRANSFORMATION, ANALYSIS, CRITICISM:!iEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 507 the fragiity of the intentiona path from Ai>- back to G~+, refecting the danger We face of osing cognitive contact with the music's rationa center. The anaysis seeks to narrow the fact/vaue gap between the sounds we cherish and the anayses we construct. However successfu or unsuccessfu i t is in that effort it is cear that the gap is not cosed. No formai mode can capture ai aspects of ou; musica experience, even when we imit ourseves to o ne parameter, such as harmony. For mysef, I fìnd that the picture in exampe 18.8(b) turns what had been a fickering and contingent experience into something more fìxed and stabe, even overdetermined. In my prose I have sought to mitigate this, reinvesting the formai mode with some sense of fragiity. But this represents an intervention from outside the space of the formai theory, suggesting the continued persistence of the fact/vaue spit. The present approach has a signa meri t, however, in that it gives the "vaue" side of the equation specifìc hooks to attach to in the formai mode, aiowing our evanescent aura sensations to interact with the mode in suggestive ways. We are ucky in that our historica position aiows us, unike Riemann, to reish those moments in which chromatic works threaten to overfow the rationa bounds of our tonai theories, or in fact do overfow those bounds. As I suggested above, the reishing-indeed, vauing-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 ai costs one notes a continued reuctance to step over the next threshod, to reish that unruy part of musica experience that resists formai containment. Perhaps this is ony a matter of time, however-the neo-riemannian turn has introduced a new fexibiity into tonai anaytica thought, moving us toward a highy sautary methodoogica sef-awareness and interpretive puraism. This shift may utimatey ead us to reinquish coherence (and its impicit sense of rationa containment) as music theory's ethicaodestar, aowing us to empoy our anaytica methods freey in the exporation an d construction of manifod musica experiences, without feeing the need to caim comprehensiveness for any one of them. For surey the best way we can vaue music is to acknowedge that i t wi aways 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 aready traversed part of this path, from Gb major tò d minor and back. As Chares Fisk has noted, the BIO minor harmony in both passages recas the B-minor midde section in the previous Impromptu, in EJ.. Chares Fisk, Returning Cyces: Contexts far the Interpretation of Schubert's Impromptus and Last Sonatas (Berkeey: University of Caifornia Press, 2001), 118. For further discussion of the EJ, Impromptu, see my "Perspectives o n Tonaity and Transformation in Schubert's Impromptu in EJ., D. 899, no. 2:' ]ourna of Schenkerian Studies 2 (2007): 33-63; on the intertextua resonances of Schubert's B-minor harmonies, see 47 n. 26 in the atter artide. 2. O n notationa "pressure" forcing enharmonic shifts, see Danie Harrison, "Nonconformist Notions of Nineteenth-Century Enharmonicism: Music Anaysis 21.2 (Juy 2002): 140-142. O n the historica importance of the six-fat signature in Schubert's Impromptu, see Hugh MacDonad, "[Six-Fat Key Signature, 9!8],"19th-Century Music 11.3 (Spring 1988): 221-237. MacDonad cas the Impromptu "a breakthrough toward a new concept of the key" of Gb (p. 225). The Impromptu was fìrst pubished by Hasinger in 1857 in G major; Hugo Riemann seems to have based his discussions of the piece-about which, more beow-on Hasinger's (or a ater) G-major edition. 3 For an overview of some of the most saient technica differences between paeo- and neo-riemannian theories, see David Kopp, Chromatic Transformations in Nineteenth-Century Music (Cambridge: Cambridge University Press, 2002), 150-151. 4 For exampe, Richard Cohn states that neo-riemannian theory seeks to answer the question, "If this music is not fuy coherent according to the principes of diatonie tonaity, by what other principes might i t cohere?" Cohn, "Introduction to Neo Riemannian Theory: A Survey and a Historica Perspective: ]ourna of Music Theory 42.2 (Fa1998): 169. 5 I mean Kang here- and throughout this chapter-in the famiiar neo Riemannian sense of major an d minor triads (no t in Riemann's sense of a duaistic emanation of overtones and undertones from a singe pitch). The Kange in Exampe 18.2(a) orni t chorda sevenths and the one augmented sixth. The omissions woud need to be addressed in a broader transformationa anaysis, but they are no t consequentia here. For a transformationa mode! that integrates members of SC 3-11 and 4-27 see Juian 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: ]ourna of Music Thepry 42.2 (Fa 1998): 283-296. 7. The scare quotes make cear that DVLS vaues obtain in pitch-cass space, in which the concepts of"up" an d "down" are traditionay considered probematic. I have nevertheess retained those words in the text for their intuitive immediacy. I have aso repaced the directed pitch-cass intervas of Cohn's DVLS with positive and negative integers, for the same. reason. These numbers shoud be understood as substitutes for their mod-12 equivaents. (Recenty, Cifton Caender, Ian Quinn, an d Dmitri Tymoczko have recuperated the notions of"up" an d "down" in pitch-cass space; I do not, however, rey on their formaism here.) 8. Joseph Straus, "Uniformity, Baance, an d Smoothness in Atona! Voice Leading," Music Theory Spectrum 25.2 (2003) : 321-322; see especiay n. 39 AVLS is the same as Straus's "tota dispacement" and Cohn's "voice-eading effìciency" or VLE ("Square Dances: 284). 9 This ends a consistency of voice-eading distance to a Weitzmann regio n that is not present in a hexatonic cyce, in which AVLS vaues range from 1 to 3 On Weitzmann regions, see Richard Cohn, "Weitzmann's Regions, My Cyces, and Douthett's Dancing Cubes: Music Theory Spectrum 22.1 (Spring 2000): 89-103. See aso "Square Dances," 290-295. Cohn's Weitzmann regions arise from a transformationa interpretation of ideas in Car Friedrich Weitzmann's pamphet Der iibermafige Dreikang (Berin: T. Trautwein, 1853). Ai of the triads in a Weitzmann region share two tones with a singe augmented triad. Interestingy, severa augmented triads appear prominenty 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 word-especiay in moments of transition-is suggestive, but I woud not push the idea

508 TRANSFORMATION, ANALYSIS, CRITICISM RiEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 509 too hard: ai of these chords operate in ways far more traditiona than the Lisztian possibiities that Weitzmann had in mind. 10. In this an d ater networks, undirected edges are a shorthand fora symmetrica pair of arrows (or more cooquiay, doube-headed arrows). I have drawn the region in a hexagona forma t anaogous to Cohn's images of his hexatonic systems in "Maximay Smooth Cyces, Hexatonic Systems, an d the Anaysis of Late-Romantic Triadic Progressions," Music Anaysis 15.1 (1996): 9-40, and in "Weitzmann's Regions;' 95, ex. 7. This approach essentiay reverses Cohn's graph-theoretic priorities in "Weitzmann's Regions;' in which i t is the hexatonic cyces that are drawn cycicay, 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 exampe, C+~ F-. In more famiiar Ango-Anerican 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 reation is formay the same as Riemann's Seitenwechse and Oettingen's Wechse, which aso exchange triads that share the same dua! root. 12. Aong with the identity operation E, these fìve operations form a tidy diiedra group of order 6. O n SLIDE, see Davi d Lewin, Generaized Musica Intervas and Transformations, reprint ed. (NewYork: Oxford University Press, 2007), 178. 13. Richard Cohn, "As Wonderfu as Star Custers: Instruments for Gazing a t Tonaity in Schubert;' 19th-Century Music 22.3 (Spring 1999): 213-232. 14. Por Riemann, Eb-- in the key of Gb major coud act as either a tonic (T p ) or a subdominant (S). We wi expore the differences between Riemann's function theory and neo-riemannian regiona anayses such as that in exampe 18.4 beow. 15. Riemann himsef notes the piece's exporation of subdominant key areas a t the expense of dominant ones, using the fact as an argument for the duaistic equaity of the two dominants. Hugo Riemann, Musikaische Syntaxis: Grundrifi einer harmonischen Satzbidungsehre (Leipzig: Breitkopf und Harte, 1877), 71. 16. Michae Kevin Mooney provides a uci d and thorough overview of Riemann's Musikaische Syntaxis, incuding its Oettingen-inspired terminoogy an d the anaysis of the Schubert Impromptu, in his dissertation "The 'Tabe of Reations' and Music Psychoogy in Hugo Riemann's Harmonic Theory" (Ph.D. diss., Coumbia University, 1996), 162-175. 17. Riemann, Musikaische Syntaxis, p. 69. "Di e schieb!iche Festigung der Haupttonaitat durch di e Thesen von og, g, 0 es un d g ist von ganz vorziigicher Wirkung:' Riemann uses the word Th ese (a hodover from his earier Hauptmann-inspired work in "Musikaische Logik") throughout the book to refer to motions away from, or back to, the tonic, via its upper and ower dominants. I t comes to mean itte more than "progression;' and that is how I have transated it above. 18. As observed in n. 2, the Impromptu was fìrst pubished in 1857 in G major, ony 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 Harmonieehre, pubished in 1917. The voume of the Schubert Ate Gesamtausgabe incuding the present Impromptu in the correct key of Gb had been pubished in 1888, and the correct key was surey we known to Riemann by 1917. It is tempting to specuate that he retained the G-major version not ony for pedagogica! carity, but aso asi t does not exhibit the same notationa disruption as the Gb version does: there is no shift in key signature in measures 79-80 in the Hasinger edition, thus removing any notationa sign of tonai disruption an d visuay carifying the minor-neapoitan hearing of the harmony in measure So. 19. Riemann, Musikaische Syntaxis, 65. "eine forme recht iibersichtich gegiederte Komposition." 20. Ibid., 69. "Das Ganze ist ein Meisterstiick sowoh hinsichtich der Meodiebidung as der metrischen Struktur, besonders aber in Hinbick auf die Thesenordnung. Den bei weitem grtibten Thei beherrscht die Tonaitat von g+, di e Haupttonart." 21. Hugo Riemann, Systematische Moduationsehre (Hamburg: J, F. Richter, 1887), 202. "Immer wieder drangt sich uns die Getung der Haupttonaitat auch wahrend der kiihnsten und weitestaushoenden Moduation auf. Wenn wir daher n un am Schuss auf den Weg, den wir zuriickgeegt, zuriickbicken, erkennen wir, dass wir n un geernt haben, immer weitere Kreise um das unverriickbare Centrum zu beschreiben." 22. Riemann, Musikaische Syntaxis, 69. "NB. Ausweichung nach der antiogen antinomen Terztonart g - 0 es." 23. Riemann's anxiety about chromatic progressions such as this one is evident earier in the book, when h e twice stresses that o ne shoud treat su eh progressions (t o antiogicantinomic third chords, among others) with the greatest caution (Vorsicht). Ibid., 19-21. He observes (p. 21) that such progressions are best used ony a t the end of a piece, after the tonic has been estabished securey, as, presumaby, in Schubert's Impromptu. In his vauabe discussion of Riemann's anaysis of the Impromptu, Michae Kevin Mooney observes that the antiogic-antinomic Terzkang is one of the most distant harmonies from the toni c, when measured by a metric Riemann provides in his sighty ater Skizze einer neuen Methode der Harmonieehre of 1880. See Mooney, "The 'Tabe of Reations,"' 240-241. Notaby, if o ne were to take seriousy Schubert's enharmonic speing of the chord in its originai key-as a G-minor triad in Gb major-it woud represent the most distant harmony from the tonic on Riemann's scae, the Doppeterzwechsekang (gb+- 0 dq), 24. Ibid., 120, trans. Aexander Rehding, Hugo Riemann and the Birth of Modern Musica Thought (Cambridge: Cambridge University Press, 2003), 105. "Die Kombinationen sind Gott sei dank unerschtipfich an Zah un d man kann das Gebiet der Harmonik nicht Schritt fur Schritt abgehen, sondern nur iiberfiegen, aus der Vogeperspektive iiberschauen. Es geniigt a ber, die Hauptwege durch diesen herrichen Garten Eden, den uns der Himme nach dem Fae geassen, zu erkennen; jeder mag dann sebst weitere Seitenpfade zu immer neuen Durchbicken 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 duaist theory, in which major and minor thirds are not merey exchangeabe within a triad, but are instead generateci in opposed, duaist directions. Renate Imig, Systeme der Funktionsbezeichnung in den Harmonieehren sei t Hugo Riemann (Diissedorf: Geseschaft der Ftirderung der systematischen Musikwissenschaft, 1970), 51-52. 28. Hugo Riemann, Handbuch der Harmonieehre, 6th ed. (Leipzig: Breitkopf und Harte, 1917), xvii. 29. Rehding, Hugo Riemann, 58 n. 52 an d 168. 30. On the conceptua differences between Stufen and functions (for exampe, the difference between a IV Stufe and an S function), see David Lewin, "Music Theory, Phenomenoogy, an d Modes of Perception," Music Perception 3-4 (Summer 1986): 342-343; an d Brian Hyer, "Ton a Intuitions in Tristan und I sade" (Ph.D. diss., Yae University, 1989 ), 105. 31. The Ds in parentheses in the Riemannian reading indicate appied dominants of the foowing harmonies. Note that the preferred Riemannian reading of the German sixth is as an atered appied dominant. The sash through the D indicates that the root is omitted, whie the > symbo after an Arabi c numerai indicates chromatic owering.

510 TRANSFORMATION, ANALYSIS, CRITIC!SM RiEMANNIAN ANALYTICAL VALUES, PALEO- AND NEO- 511 32. O n universaity in Riemann's thought, see Rehding, Hugo Riemann, chapter 4. Riemann's anayticai practice cuts both ways, of course: he aso uses his theory to demonstrate the aeged vioiation of his universaiiaws in composers iike Berioz. 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 transformationa approaches to expiore specificay tonai characteristics of chromatic passages (asi have in my work). I base my discussion here on the infuentia species of neo-riemannian anaysis m ade popuar by Cohn, within which the vaiues outined above have been remarkabiy consistent. 34 Riemann's "universai" cassicism is of course simpyviennese cassicism (the roots of which he repeatediy traces back to the Mannheim symphonists, especiay Stamitz). On the nationaist 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 possibiity ernerges vividy a t the end of Riemann's history of nineteenthcentury rnusic, pubished 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) (Berin: 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 erautert von Hugo Riemann (Berin: Schiesinger, 1910), 129. "Verstandig interpretiert giebt der Satz keinerei Aniass, von Zerissenheit und schwerverstandiichern Aufbau zu sprechen, zeigt vieirnehr deutich das normae Geriist der Sonatenforrn." 41. Daniei Chua, The "Gaitzin" 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 discipinary response to the chaenge 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;' Journa of the American Musicoogica Society 57.2 (Fa2004): 285-323. 45. Cohn, "Introduction;' 169. 46. Chares Fisk, Cornrnent & Chronice, 19th-Century Music 23.3 (Spring 2000): 301. 47 Richard Cohn, Cornrnent & Chronice, 19th-Century Music 23.3 (Spring 2000): 303. 48. Aien Cadwaader 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. Aen Cadwaader and David Gagné, Anaysis oftona Music: A Schenkerian Approach, 2nd ed. (New York: Oxford University Press, 2006). 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, foowing Lewin ( Generaized Musica Intervas 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 Duaist 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, Tonaity and Transformation (New York: Oxford University Press, 2011). 52. Michaei Siciiano, "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," Journa of Music Theory 42.2 (Fa1998): 246-249. 53 I rnaintain the neo-riemannian transformationaiietters here (R, P, and L) for famiiarity, 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 underying graph is downgraded from "path consistent" to "universay reaizabie," 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 woud interpret as an appiied dominant; the quotes around the A>+ node in the exampie indicate that the aterations to the chord have significanty 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 infuentiai move-expiicity eiminated from his renewed, de-centered Tonnetz in "Reimag(in)ing Riemann;' fourna of Music Theory 39!1 (Spring 1995): 127-128.