DOI: 10.1111/musa.12032 KENNETH M. SMITH THE TRANSFORMATIONAL ENERGETICS COHN, RINGS AND TYMOCZKO OF THE TONAL UNIVERSE: The Tonal Universe and the Hunt for the Harmonic Higgs Boson Richard Cohn’s Audacious Euphony, Steven Rings’s Tonality and Transformation and Dmitri Tymoczko’s A Geometry of Music are three new technologies for astronavigation, each offering new ways of charting the ‘universe in tones’.1 Since Cohn’s pre-millennial ‘As Wonderful as Star Clusters’ (1999), our ‘instruments for gazing at tonality’ have sharpened focus, exemplified by these three new alternative (or complementary) technologies for mapping the aweinspiring constellations of pitches and chords which form Western harmony. The wide scope of harmonic theory, as established by Hugo Riemann and, later, David Lewin, has extended far beyond the boundaries of neo-Riemannian (a term Cohn now shuns) and even neo-Lewinian space. That three volumes from leading North American theorists should appear in close succession is a godsend for students of music theory, who will by now have purchased all of them. Whilst the present article offers only miniature snapshots of each book for the purpose of contextualisation, it primarily elicits a more cosmological question which each book opens but inadequately addresses: what animates this tonal universe? However, before recalibrating our instruments for examining this issue, a brief (and unapologetically oblique) overview of each book. Audacious Euphony Cohn’s explorations of nineteenth-century harmonic space have been avidly followed by students across the globe, and with Audacious Euphony there now emerges a fully developed theory. His ‘neo-Riemannian’ zodiac consists of theorists from the last two centuries: Carl Friedrich Weitzmann (and his water bugs), Jack Douthett (and his cubes) and Benjamin Boretz (and his spiders) are perhaps the most prominent members. Cohn relishes the novel, the obscure and the Unheimlich, and while his feet aren’t always on the ground, his flights of gravity-defying fancy follow the spirits of composers as well as theorists. The analyses are derived from a loving engagement with the outré features of Romantic harmony; Riemannian dualism is defended but not to the hilt (Cohn 2012, pp. 37–9); and Cohn is wary of ideologies but works with them if they produce innovative readings which are unavailable to theorists that reject them out of Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA 1 2 KENNETH M. SMITH hand. His book is accessible: Cohn leaves behind his overly mathematical former life (Cohn 1997), offering instead a new range of voice-leading models with a wealth of chromatic examples to demonstrate his contention that Romantic harmony radically altered the triad’s coordinates within tonal space. Cohn’s most creative contributions are (a) his synthesis of Douthett with Boretz and (b) his ‘voice-leading zone’ – a new response to Riemann’s Funktionstheorie – which demonstrates the ‘up-shifting’ and ‘down-shifting’ of chords. However, while this latter records the stages that transforming configurations pass through, the technology shines only a dim light onto the question of why they transform in the first place. Nevertheless, voice-leading zones offer an incisive new tool, which will be evaluated presently. Tonality and Transformation Rings’s is a more meticulous book than Cohn’s, but it flies neither as fast nor as far. Its aims are clear and laudable: ‘to return to certain fundamental ideas from transformational and GIS [Generalized Interval System] theory, exploring their potential to illuminate familiar aspects of tonal phenomenology’ (Rings 2011, p. 1). He first wonderfully translates Lewin for the masses and then gently edges transformation theory onwards and in fact outwards, ‘prismatic[ally]’ (p. 8), preaching the integration of transformation theory with Schenkerian,2 psychological and energetic approaches. His own technology – the GIS space and its 84 ordered pairs of scale degrees and pitch classes – is used to map ‘dialogic’ hearings (p. 38). However, his ‘multiplicities’ are mostly binaries of just two interpretations which move in the right direction but don’t always go as far as they might. For William O’Hara, Rings’s fundamental principle comes from Husserl: the esthesic experience is privileged over the work’s immanent (or ‘neutral’) properties (O’Hara 2011, p. 9). Rings’s principle is the plurality of esthesic responses, although it is surprising that his GIS technology from Lewin is so distant from the other neo-Riemannian axis, the voice-leading parsimony of Brian Hyer (1995), Cohn and others. In fact, with the transformational aspects of ‘modelling’ the casual astronomer may have cause to wonder whether Rings has left his lens cap on; in Rings’s explication of Lewin we are often presented with mathematical complexities which yield extreme musical banality where we want profundity. Four pages of arithmetic and algebra lead to the algorithm ‘INT(C4, D4) = +1’, which translates as ‘[t]he interval from D4 to C4 is one diatonic step upwards’ (Rings 2011, p. 15). D is one note higher than C. A flippant, decontextualised illustration is more than unkind; obviously, Rings’s modelling is highly complex, and increased intricacy invites increasingly innovative insights. Nevertheless, many of his findings are available to the naked eye (and ear) and are not always as prismatic as we might like.3 The most Lewinian of the three, Rings comes closest to the spirit of ‘the transformational attitude’ (Rings 2011, p. 104), and he insists that this attitude has a vital force, that it has animating potential. This energetic attitude, together with the assumption that © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 3 chords become each other (or rather, we turn them into each other as listening transformers), begs to be problematised. Why exactly did we want C4 to become D4? A Geometry of Music A revisionist to the core, Tymoczko attempts an entire rewrite of music theory with a conscious balance between theory, composition, analysis, performance and pedagogy, striving to uncover concepts which ‘contemporary composers might find useful’ (Tymoczko 2011, p. 22). Armed with his five primary features of cross-cultural musical consistency – conjunct melodic motion, acoustic consonance, harmonic consistency, limited macroharmony and pitch centricity – his convincing chronicle proposes an extended common practice stretching from the dawn of Western counterpoint to the atonal music of the twentieth century which is characterised by continual shifts in the balance of the five features. This wends us through some cunning deconstructive twists and turns (for example, nineteenth-century Western chromaticism’s stasis signals a return to a medieval paradigm [p. 164]). Tymoczko offers insights into a broad repertoire: by turns adding fascinating new variations to the tale of the Tristan chord, on the one hand, and seamlessly drawing tighter attitudes to harmonic progression and scalar interaction from Bill Evans, John Coltrane and Jimi Hendrix, on the other. Together these suggest new ways of teaching which demonstrate underlying principles rather than hypostatised phenomena. Tacitly distanced from the neoRiemannians, Tymoczko proves himself to be fully neo-Lewinian in leaving behind the chord as ‘object’ in favour of an array of voice-leading models. Although his surface glistens with original 3-D geometries (downloadable software models user input), maths is kept to a minimum in this highly readable text.4 This sometimes comes at a cost: like Rings, Tymoczko’s modelling hinges on the ‘distance’ between pitches, but his formulae do not strike me as sophisticated in comparison with those of Fred Lerdahl (2001), who integrated tonal ‘tension’ (or ‘gravity’, as Rameau, who preferred Newtonian to Galilean metaphors, would have it).5 Tymoczko’s 3-D lattice shows voice-leading relations between familiar scales but registers only the neutralised pitch distances without factoring in the gravitational subtleties of Lerdahl, nor even those from his own more exhaustive attempts elsewhere.6 For me, the failure to account suitably for diatonic animation is the fundamental miscalculation of all three books and will be the focus of my critique. The God Particle? Contemporary with Peter Higgs’s theorisation of the ‘God particle’ (the sticky field which pulls particles together and animates matter), transformation theory set itself up as the animating force of the tonal universe. Its foundations were constructed on energetic principles. Lewin conceived transformations as Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 4 KENNETH M. SMITH ‘generative lusts’ (Lewin 2007, p. 341), and this image continues in the volumes discussed here: Cohn offers the usual metaphors of ‘energy, appellation, attraction, magnetism, charge, desire’ (2012, p. 151); and Rings talks of ‘centres of tonal attraction’(2011, p. 107), ‘energy’ and ‘urges’ (p. 133), sourcing this in Rameau’s désirer and souhaiter (p. 108). But also like the evasive Higgs boson particle, the energy which sparks the tonal universe has proved to be an elusive phenomenon, and our three music theorists sell us short when theorising why this energy circulates, or even speculating what this energy actually is. This theoretical black hole swallows light from many theorists, not just Cohn, Rings and Tymoczko. But, ironically, Riemann himself offered a ray of hope for understanding this phenomenon. From Schenker, Ernst Kurth and Leonard Meyer to Lerdahl, an animating force pulsates in the form of a tension-release pattern or in the ‘discharge’ paradigm offered in Daniel Harrison’s exegesis of that ‘other’ strain of Riemannian thought, Funktionstheorie (Harrison 1994). Like the Higgs boson, we theorise the existence of tonal tension only by observing its effect – its discharge – whereupon we impute a ‘function’ to it in relation to other elements in the system. Cohn gracefully distances himself from his earlier fraught application of harmonic Funktion’s animating potential to the neoRiemannian Schubertian world; Rings, sensitive to its elasticity, breezily flies it past us; Tymoczko, while pitching us theoretical curveballs left, right and centre, employs Riemann’s theory of the three Hauptfunktionen with surprising straightforwardness. Funktionstheorie is appropriated by all three, bringing with it a message of tonal discharge and animation, and when these authors short-change us, the main fault lies in their reluctance to integrate tonal discharge patterns fully into their post-Lewinian models. Another critical error of current transformation theory, one that has great bearing upon Riemann’s Funktionstheorie as our ‘animating principle’, is the marginalisation of minor-third chord relations (octatonic system) and overpromotion of major-third ones (hexatonic) or, worse still, the conflation of the two.7 Of the three authors, Tymoczko can be most thoroughly exculpated, but he is charged with over-segregating the two geometries, his tagline being, ‘[t]riads [major-third-related chords] are from Mars: [s]eventh chords [minorthird-related] are from Venus’ (Tymoczko 2011, pp. 97 and 220). He proposes that ‘four-note chords are particularly close to their minor-third and tritone transpositions’ (p. 97), whilst three-note chords are close to their major-third transpositions. And yet, for all of Tymoczko’s geometries, his admirable attempt to rein chromaticism in, reducing it to fundamentally diatonic principles, leads him to discuss ‘thirds’ in generic terms when discussing the simplest diatonic music, saving precision tools for complex chromatics. Applying Funktionstheorie in a more clearly defined way, major-third relations do become animators by allowing discharge between chords, and minor-third relations – which do not facilitate discharge – become significant harmonic channels nonetheless. In order to divine the deep links between Funktionstheorie and the problem of third relations, I take examples from each author’s work, teasing out alternative © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 5 Fig. 1 Cohn’s twelve voice-leading zones (Cohn 2012, p. 104). Reproduced by permission of Oxford University Press, USA readings which aim towards a different, and perhaps clearer, understanding of a harmonic animating force. Schubert’s Stars and Cohn’s Clusters Cohn’s sharpest tool is his ‘voice-leading zone’ – a new instrument for parsing Riemannian functions without the ‘implicit baggage’ of Funktionstheorie. It derives from Lewin’s upshifts and downshifts, that is, the upward or downward transformations of a chord’s pitches (Cohn 2012, p. 94). T4-related triads share voice-leading zones, 24 chords producing eight zones with augmented triads occupying the remaining spaces (see Fig. 1). The resulting clock face thereby resembles Douthett’s ‘cube dance’ (Douthett and Steinbach 1998, p. 254). Cohn asks the crucial question, ‘To what extent does the voice-leading dualism of down and up interact with the Riemannian dualism of subdominant and dominant?’(p. 124). His answer is basically ‘take it or leave it’, and he evades the question of whether these new zones coincide with or replace functional shifts, claiming that they simply have the ‘capacity to share a centre of balance’ Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 6 KENNETH M. SMITH (p. 103). At one point he rules that ‘[d]ownshifting and upshfiting may be interpreted in terms of plagal and authentic motions, respectively’ (p. 125). And this is true: (C–E–G) → (C–F–A) elevates us from Tonic (T) → Subdominant (S) through ‘upshift’ from zone 11 to 2 (modulo 12). Extending to the Dominant (D) from (C–F–A) to (D–G–B) at T2, we commute to zone 8. But is this 8 a.m. or p.m.? Is this an upshift or a downshift? (C–F–A) → (B–D–G) is equally dispersed in voice-leading zones so that, like the grand old Duke of York’s men, we are neither up nor down. A handful of borderline examples, in which upshifting and downshifting do coincide with transfers of function, are paradeigma which don’t persuade. The efficacy of this technology is evaluated when Cohn picks up where he left off in 1999. ‘As Wonderful as Star Clusters’ sketched a theory of hexatonic cycles as Riemannian functions in Schubert, commendably bridging the gulf between Funktionstheorie and voice-leading parsimony (Cohn 1999). Yet this was problematic: the four hexatonic cycles were (mis)aligned with the three Riemannian functions. Whilst not presenting a full U-turn, the present volume significantly changes course. Cohn compares his previous remarks to Ernő Lendvai’s ‘Procrustean’ (Cohn 2012, p. 125) axis system that promoted minorthird chord relations in Bartók, Verdi and Wagner. And yet Cohn rebuffs Lendvai at his peril. Cohn’s updated rendition of the first movement of Schubert’s Piano Sonata in B♭, D. 960, places his hexatonic reading in the ring with a new Wietzmannian interpretation. He follows the ‘downshift’ of voices through the expositional drift from zone 5 to zone 1, reversing to zone 7 only at the work’s core. His neutral layer of mod 12 integers produces a loose symmetry through the movement which takes the centre stage in Cohn’s play of functional readings. But voice-leading zones are fully contingent upon the hexatonic system, and Cohn does not sufficiently integrate the non-neo-Riemannian, but highly paleo-Riemannian, minor thirds. For Cohn, the triumph of the hexatonic is celebrated in this movement primarily because of the motion from B♭ major to G♭ major and back again in the exposition: a PL transformation around the Western hexatonic cycle. Yet, for this listener at least, the sudden flattened-sixth lift has a subdominant character that is more profound than a simple move to the flat side of the key spectrum (Ex. 1). My sensation of lift is not registered in Cohn’s analysis because the F–G♭ pitch raise (L: Leittonwechsel) is counterbalanced by the descent from D to D♭ (R), sharing voice-leading zone 5. Combined with the common hexatonic cycle, this confirms their shared T function, though Cohn hedges his bets, claiming, ‘[W]e are not stuck with the functional apparatus if we do not find added value in it’ (2012, p. 128). But this highlights a systemic problem of voice-leading neutrality. The rise in this instance is pitch F → G♭: the leading-note motion to the new region. Its contrary motion with the falling D → D♭ – whilst not the key-defining tritone resolution – also induces sensations of resolution from ♭6ˆ → 5ˆ (E♭♭ → D♭). Funktionstheorie is about the discharge of sensitised pitches into new function-defining units; these pitches contain strong charge, which © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 7 Ex. 1 Schubert, Sonata in B♭, D. 960/i, bars 1–23 Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 8 KENNETH M. SMITH Ex. 2 Schubert, Sonata in B♭, D. 960/i, bars 33–39 transformational neutrality kills. They are fully animated and animating. Kofi Agawu’s definition of tonality captures this quality in a way that none of the books under review quite manages: ‘a hierarchically organized system of pitch relations animated by semitonal desire’ (Agawu 2013). Transformation theory neutralises the ‘desire’ of these semitones, flattening them to 2-D (even if the image, as in Tymoczko, is rendered in 3-D). My impression of subdominant lift at bar 20 is strongly reinforced in bar 34 when G♭ major ‘transforms’ into a German sixth (which originates in the minor subdominant Phrygian move [Ex. 2]). The scare quotes denote the becoming of ♭VIGER because of the addition of E♮ (prepared in bar 2), rather than the alteration to existing pitches that would be involved in a ‘transformation’.8 For all of the transformational maths in these volumes, we do not deal particularly well with simple additions to the number of pitch classes, nor the integration of triads and tetrachords. Does the extra E♮ transform the chord itself or merely unlock its contextual meaning? Either way, German sixth chords bear S (or DP) function.9 We return to chord I (B♭ major) 6 after our flattened-sixth foray, but Schubert’s old chestnut is the ♭ VI GER → I 4 resolution, the F in the bass preserving the S → D effect. The expectation that 6 5 4 will resolve to 3 already shows that Funktionstheorie captures the sense of energy clearer than the voice-leading-zone clock face.’ © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 9 Ex. 3 Schubert, Sonata in B♭, D. 960/i, bars 44–49 A further formal curiosity is Schubert’s ‘second subject’s’ F♯ minor beginning, which Weitzmann would regard as D and Cohn as a downshift to 4.10 But again, Cohn’s analysis disregards the animations beneath. The F major of bar 44 resolves not to the expected B♭ major, but to a diminished seventh which substitutes B♭ for B♮. After recalibrating this symmetrical chord, Schubert orientates it towards C♯7 rather than B♭ major and from there lunges at F♯ minor (Ex. 3). Direct discharge between B♭ major and F♯ minor (the enharmonic parallel (henceforth ‘P’) of G♭ major) is mediated by the raw diminished seventh, which, rather than transform one chord to the other, redirects energy from B♭7 to C♯7. These kinships are not registered in Cohn’s voice-leading zones, but in Cohn’s version of Douthett’s ‘4-cube trio’ (or ‘power tower’) (Cohn 2012, p. 158) these minor-third- (rather than major-third-) related chords share the same region: 1, moving to 2 with the diminished chord, and back to 1. What is missing from this model, however closely it illustrates these tetrachordal relations, is the discharge factor: the F7 (bar 44) which precedes these nebulous T3 chords comes from a galaxy far away (region 5); this also requires us to switch between tetrachordal and triadic models. For all of Cohn’s dismissal of Procrustean old Lendvai, the key to understanding these progressions is, for me, the integration of hexatonic major-third relations (which actually discharge into each other), and octatonic minor-third relations (between functionally equivalent chords).This is the very dialectic which Lendvai tried to master. The functional affinity of T3 chords is profoundly paleo-Riemannian, in which ‘parallel’ (p: meaning ‘relative’ for Riemann) chords were deeply connected; as Joel Lester puts it, ‘Each secondary chord shares functions with one or more primary chords’ (1982, p. 195). Thus, (following Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 10 KENNETH M. SMITH Fig. 2 Schubert, Sonata in B♭, D. 960/i, bars 1–75: model of tonal functionality Fig. 2), in B♭ major, G♭ major is the chord for which the E♭ major S of bar 5 is the ‘Sp’ (P-related also in the neo-Riemannian sense; this is the relative of E♭’s minor mode). Schubert’s F♯ minor in bar 48 soon functions clearly as an A major substitute (both chords minor-third related, the former being the latter’s Sp), which, after discharging very briefly to D minor in bar 70 and subsequently E major (the parallel Tp of G major which would follow the D minor if the discharging cycle of fifths was continued), returns to A minor and then C major in bar 74 (the chord for which A minor is the Sp). This C major becomes a structural C7 downbeat to the full F major in bar 75. With localised discharges between, we have now heard the four T3 octatonic S nodes: E♭, G♭/F♯, A and C, creating a static environment of Sp relations in between T and D pillars (see again Fig. 2). Essentially, this recalls the promise of Lendvai that minor-third rather than major-third relations share functional categories. The progression macro-progression rendered this way is quite clear: T → S → D → T, etc., privileged by Riemann and later function theorists.11 We might risk modelling this with new tools. The following hypothetical map is designed to register the functional T → S → D → T rotation which lies at the heart of these chord progressions (Fig. 3). The central triangle represents the paradigm, wrapped in the cycle of fifths. Conceptually, it triangulates Weber’s ‘chart of the regions’ which Tymoczko reminds us was actually from F.G. Vial (Tymoczko 2011, p. 246) but was used latterly by Lerdahl (2001). The ‘cycle’ moves around the triangle, passing through T, S or D substitutions as it goes. Major and minor are collapsed into this monist model, P transformations being freely available and often tacit. Schubert’s journey could look like Fig. 4, with black lines representing the path taken and numbers representing the stations along the way. This further reconceptualises the recapitulation (Table 1), which consolidates many of these procedures. On the surface, Schubert’s recapitulation is characteristically straightforward; its commencement is auspicious enough to feel secure and there seems to be no danger of a relapse into developmental episodes. The G♭ major move is sufficiently assured, but Schubert anticipates Cohn’s ‘second subject’ by switching enharmonically to F♯ minor ahead of time (bar 239). He switches again to A major (bars 243–256), recalling material formerly © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 11 Fig. 3 Circularised model of table of Vial’s tonal relations in G♭ major and time-warping us back to the exposition to illuminate the minor-third ‘relative’ (R) connection. The subsequent discharge of A major to a full B♭ major root-position triad (bar 256) is noteworthy. Superficially, this S → T progression omits a D function, but closer inspection reveals C♯ of the A major triad discharging to the upper D while E discharges to F, offering hints of D minor at the crux. In any case, Schubert completes unfinished business at bar 266, when F♯ minor discharges more firmly than ever before into B minor for the recapitulation of Cohn’s ‘second subject’, now at the appropriate T5 level. Hearing the harmonic progression in this substitutional way, the ‘second subject’ which begins as S eventually migrates to the D. The B minor charge soon reaches to its equidistant minor-third partners: first to D7 (before a brief excursion into G minor), then to the F7 fermata which leads to the B♭ major closing theme, fully Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 12 KENNETH M. SMITH Fig. 4 Schubert, Sonata in B♭, D. 960/i: exposition Table 1 Schubert, Sonata in B♭, D. 960/i: comparison of exposition/recapitulation Exposition P1 P2 S1 S2 Recapitulation B♭ G♭ f♯ F T S D P1 P2 S1 S2 B♭ A/G♭/f♯ b B♭ T S D T recapitulated as T. Cohn’s 1999 hexatonic reading scores over mine in its encoding of the prototypical Schubertian Subdominant overshoot, but mine sheds new light upon Schubert’s directly transposed recapitulations, revealing D → T discharges to be transpositions of S → D. © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 13 Fig. 5 Brahms, Symphony No. 2/i, Op. 73, bars 102–118: cube dance model (Cohn 2012, p. 94). By permission of Oxford University Press, USA Cohn’s other analyses work likewise. His excerpt from Brahms’s Second Symphony progresses LRLR to produce a cycle of fifths, rerouted through R substitutions. This is transparent in Cohn’s cube dance, but the discharging cycle of fifths cannot connect up directly (Fig. 5). The wheel turns here only because of the major- and minor-third alternations, whose L transformations (with leading notes) resolve indirectly around the cycle of fifths. Cohn’s summary example is his most fruitful, setting into dialogue two different hearings of a passage from the third movement of Schubert’s C major Symphony; these hearings represent the book’s two dialectical antinomies. The chord progression, reproduced in my voice-leading graph, is quite simple (Ex. 4). Do we hear the passage as (a) diatonically tethered to C major? If so, how do we negotiate the enharmonic seam between the central D♭ major and C♯ minor? Or do we hear the ‘hexatonic alley’ (b) chromatically as a series of upshifts? My model suggests both are possible, because the upshifts (which don’t necessarily Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 14 KENNETH M. SMITH Ex. 4 Schubert, Symphony in C major, D. 944/iii, bars 101–142: voice-leading graph coincide with the circle of fifths) are heard as L progressions which discharge as temporary 7ˆ −8ˆ motions through chromatic representatives of the tonal functions, starting and ending on G major as D function. The only ‘substitutes’ for the primary major triads are the P and PR relations (oTP and oT, etc.) and the central D♭/C♯ minor. A diatonic T → S → D → T cycle rotates twice and the central chromatic drift is carefully directed back home (Fig. 6). This Schubert example helps us reconceptualise Ex. 5, the old neoRiemannian war horse which is explored by Kevin Mooney (1996), Harrison (1994), David Kopp (2002) and many more, and from whom Audacious Euphony derives its name (see Ex. 5). The passage comes from Riemann’s definition of ‘tonality’ from his Musik-Lexicon of 1909 (Cohn 2006, p. ix). As Avo Sõmer recently observed, ‘It provides an important late-nineteenth-century admission that a tonic preceded or surrounded only by chromatic mediant or submediant chords may still create a coherent tonal structure’ (1995, p. 215). Although Sõmer feels it ‘bizarre for omitting the dominant’ (p. 218), we might hear the missing G major as substituted by E major (its T3 relative). The potent driving force is the leading-note rise to the B pitch; the consonant bass leap of a fifth is replaced by another function-confirming jump – the major third. Matthew Bribitzer-Stull, examining ‘The A♭–C–E Complex’, recalls the famous anecdote about C.P.E Bach: ‘[w]hen asked “Who but yourself would dare go directly from C major to E major?” C.P.E. Bach replied, “Anyone can and will assuredly do it who knows that E is the dominant of A, and that A minor is very closely related to C major” ’ (Bribitzer-Stull 2006, p. 167). Ahead of time Bach pinpoints one of the tensions between neo-Riemannian transformation theory (where C major moves to E major because it transforms smoothly) and Funktionstheorie (where E major is a replacement for the dominant G major). Of course, A♭ major would fulfil an S function, just as an augmented sixth would do, the lower A♭ ( ♭6̂ ) imbuing a Phrygian quality. This progression reads: T → S(→ T) → D → T. Lendvai, we recall, stipulates that C major, A♭ major and E major best represent the T → S → D → T, rather than the more intuitive C major, F major and G major (1993, pp. 8–11). In fact, Alexander Rehding demonstrates that Riemann stops strikingly short of this exact interpretation (Rehding renders the more nuanced paleo-Riemannian functional interpretation as T → oSP → T → D(TP) → T) to avoid concatenations of Scheinkonsonanzen (Rehding 2011, pp. 167–90), which are disallowed in Riemann’s theory, although seemingly permitted in his analyses. © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 15 Fig. 6 Schubert, Symphony in C major, D. 944/iii, bars 101–142: model Ex. 5 The ‘audacious but effective and euphonious progression’ from ‘Tonalität’, in Riemann, Musik-Lexicon (1909) (Cohn 2012, p. ix) Chord progressions like these in Riemann’s little epitaph abound in the nineteenth-century literature and push right into the twentieth century. A short passage from Strauss’s Elektra demonstrates how these progressions overlap with borderline atonal repertoire and in fact highlights the ‘discharges’ Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 16 KENNETH M. SMITH Ex. 6 Richard Strauss, Elektra, §174a. © Copyright 1908, 1909 by Adolph Furstner. U.S. copyright renewed. Copyright assigned 1943 to Hawkes & Son (London) Ltd. (a Boosey & Hawkes company) for the world excluding Germany, Italy, Portugal and the former territories of the U.S.S.R. (excluding Estonia, Latvia, and Lithuania) and ‘substitutions’ as readily as any passage from the nineteenth century. In this extract from §174a (Ex. 6), the hero of the hour, Orestes, fortifies himself to exact revenge for the murder of his father. His vocal contour rises to match his new resolve to murder his mother and her lover. The harmony invites obvious neo-Riemannian interpretation, using as it does the same three chords as in Riemann’s example above. Whilst one chord transforms into the next, the discharge between them is palpable, leading notes guiding the progression. The G of the C minor triad leads to an A♭ major chord whose E♭ enharmonically doubles as D♯, squeezing to E major. Strauss’s nuances heighten the feeling of discharge: initially Orestes’ classic rising V → I bass voice fudges an E♭ major → A♭ major resolution. In the second bar, the A♭ becomes G♯ ‘ahead of time’ and resolves immediately to A♮. This A♮ sounds with the sustained E♭s below, forming a tritone pull (as D♯) towards E and G♯. A process of functionally rotational discharge occurs here, and, when Orestes reaffirms his resolve (‘The deed I will do!’), this repeats at T7: G minor → E♭ major → B major. A typically nineteenth-century passage of parsimonious neoRiemannian major-third cycles thus co-functions with a paleo-Riemannian T → S → D discharge of minor-third substitutions, procedures which change the analytical game of the opera and which could potentially alter the way we hear similar progressions in both the nineteenth- and early twentieth-century repertoires. © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 17 Fig. 7 Rings’s digraphs (Rings 2011, p. 118). By permission of Oxford University Press, USA Running Rings Around Romantic Harmony Of the three books, Rings takes tonal gravity most thoughtfully into account and asks the right question: ‘[c]an we reconcile energeticist metaphors with the transformational attitude?’ (Rings 2011, p. 104; italics in original). His hands are dirtier even than Cohn’s with metaphors of desire and lust. He refers to Rameau’s famous passage on the tonic’s powers of attraction: ‘[the tonic] is the chord that we find most agreeable, the one to which all of our desires tend, and beyond which we desire nothing more’ (p. 107).12 To validate his quest to trace the flows of ‘tonal energy’ with a Lewinian system of arrows, he appeals to Kurth: ‘[a]n arrow that proceeds forward in time urges the music forward to an anticipated tonic, with the forces of tonal attraction, or the flow of tonal energy; arrows that proceed backward in time, in contrast, check forward momentum, suggesting the pull of an earlier tonic as a force to be overcome’ (p. 104). From the opening chapter, he follows Huron in ‘organizing one’s aural sensations with respect to a single pitch class, the tonic’ (p. 3), and this becomes axiomatic (see p. 103). The direct magnetism which attracts (and sometimes repels) the tonic asserts itself in Rings’s transformation theory as ‘a special kind of transformational network that can impose an orientation on a given transformational space, directing all of its elements toward one central element ... . The directing of the listener’s attention toward a tonal center, which I call tonal intention, can be conceived as a special kind of transformational action’ (p. 5). The tonal centre which Riemann named the ‘middle point of our harmonic imagination’ appears on Rings’s ‘digraphs’ as a double-ringed node to which other nodes tend (p. 111 [Fig. 7]). Rings offers a new ‘Res’ function (although a RES function was also proposed in Bass 2007). Thus ‘ResC’: ‘[t]his transformation sends every element in S to the element C’ (Rings 2011, p. 25). For me, the factory for Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 18 KENNETH M. SMITH neo-Riemannian transformational terms has long since stopped manufacturing genuinely useful products; Riemann’s D function (or ‘DOM’), resurrected by Hyer (1995), is well-known to sit uncomfortably with transformation theory,13 and the same difficulties pertain also to Rings’s Res function. This is because we are trying to reconcile tension-resolution with the ‘transformational attitude’. The transformational attitude is too claustrophobic because very dissimilar chords, bound together in a knotty dialectical alterity, are reduced to a simplifying metamorphosis. In shifting focus from the ‘chord as object’ to something like Lyotard’s ‘libidinal band’ – a kind of self-regulating desire-producing Möbius strip, with no inside or outside – we leave too much behind. Do we really process chord progressions as recordable adjustments to a morphing sound object? Rings is not building bridges alone; Ramon Satyendra (2004) skilfully dissolves the difference between subject-oriented and object-oriented accounts by appealing to Lewin’s generalised conception of interval, but although he moves towards energising these experiences as something more than solipsistic manipulations to a hermetic harmonic self, he still fights shy of building tension into the GIS. In fact Rings is more radical than most, as we are encouraged to become active subjects; the music becomes something perhaps like Julia Kristeva’s transformational sujet-en-procès (Kristeva 1974, p. 37). Rings tellingly deliberates on ‘desire’ as access to an outer world. While reminding us that Hyer construes Rameau in sexual terms (Hyer 2002, p. 731), he invokes Riemann’s discussion of the tonic as ‘imaginary’ object: [t]he C major triad itself is part of the imagination as that Klang by which the significance of the G-major triad is determined as something deviating from it – the center of its imagination lies, so to speak, outside of it. That is to say, a moment of instability emerges, a desire to progress to the C-major triad, dissonance. So it is with the F major chord, and generally every chord of the key. (Rings 2011, p. 107)14 Rings also locates the performance of transformation (more than the establishment of mere function) in the listener’s mind: to hear a given harmony as a dominant is mentally to perform the dominant transformation, linking the sounding harmony to an understood tonic via D; it is our mental performance of D that invests the dominant with its special energetic charge, as we hear ‘through’ the sounding chord, so to speak, toward the tonic. (p. 105) Who or what is transforming – the listener, the composer, the analyst or the chords themselves? Rings thinks it’s the listener and cites Lewin’s question ‘I am at s, what characteristic transformation do I perform to arrive at t?’ (Lewin 2007, p. xxxi). But the celebrated Lewinian definition of transformation – ‘[s]omething one does to a Klang, to obtain another Klang’ (Lewin 2007, p. 177) – offers us © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 19 ample space for interpretative play. The interchange between the active, living listener and the passive, sonic object draws me rather to Riemann’s Funktionstheorie, which hypothesises imaginary active objects – the listener’s classification of chords into a network of metaphorical functional meanings. Rings retains from Lewin the dualistic notion that the D and S ‘drive the network’ (p. 101), and yet the network is replaced by much simplifying schemata in which every arrow points almost unswervingly to the tonic. Although we are persuaded that tonal energy cognitively emanates backwards as well as forwards, there is little scope for chords to radiate energy to non-tonics as root nodes in Rings’s digraphs do not tend to point away from themselves. This creates a simple see-saw between a weightless (often absent) tonic which swings one way and a grossly overweight body of S and D forces that swings the other. Needless to say, this uneven experience is no roller coaster, and the play of forces which push one to the other is a confusion/conflation of esthesic functional categories such as D and S with immanent dualistic Riemannian ones (P, L, R, and so on). Ever since Hyer’s ‘reimag(in)ing Riemann’, the use of S and D as transformations rather than functions has been problematic (Hyer 1995, p. 108). Hyer himself bemoans: Riemann never thought of disassociating the parallel, relative, and leittonwechsel from dominant-related harmonies, from functional tonics, dominants, and subdominants. There is, in other words, an enormous ontological difference between thinking of the dominant as a relation and thinking of it as an actual chord. Lewin argues that the desire to promote functional notation as a substitute for roman numerals prevented Riemann from working out the transformational potential of his harmonic theories. Riemann, that is, regarded the dominant as a label for a triad, rather than something one does to one triad in order to generate another (Lewin 1987, p. 177). Lewin’s own discursive rhetoric suggests a quasiHeideggerian distinction between ‘being’ and ‘doing’, where, for Riemann, the dominant is a ‘being’, something a triad is, rather than a ‘doing’, something one does to a triad. (1995, p. 128) Henry Klumpenhouwer similarly complains: The difference bothers me because the difference (and other similar differences in the system) is brought about by separating dualist chord models from dualist transformations (leittonwechsel, relative) and then using those transformations in conjunction with functions or relations (dominant, subdominant, submediant, mediant) derived from fundamental-bass conceptions of chord structure. (1994, p. 5) Klumpenhouwer locates the problem in Riemann’s later work, where his theory actually runs along neo-Rameauian lines, and Klumpenhouwer admits to his own purposeful misappropriation in returning to the Skizze einer neuen Methode der Harmonielehre of 1880 to confront Riemann with his earlier work in a new light. Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 20 KENNETH M. SMITH Riemann himself certainly under-theorised tonal tension and discharge. Kopp reproduces Riemann’s taxonomy of directed ‘root-interval’ progressions of thirds and fifths which were developed alongside the Harmonieschritte’s protofunctional concepts (2002, p. 70), but he demonstrates that, much later, Riemann still relied on the ‘root-interval’ as an explanation of tensile chord energy.15 Ian Cross nominates Kurth’s (1931) as the earliest psychological theory of tension/attraction as harmonic mobilisers (Cross and Woolhouse 2006, p. 763); Riemann was simply too early to admit a truly workable harmonic gravity into his system. Rings attempts to correct this but preserves Riemann’s conflation of transformational function and tonal function. A truism in Rings’s book is that musical events surge their meaning temporally backwards, as a corrective to the analytical predisposition to follow the forward flux of musical time. This may sound Adornian in conception (‘the structural listener hears the first measure only when hearing the last, which redeems it’)16 or even Lacanian (since, for Lacan, desire is retroactively constructed after points de capiton), but the analytical results are not always as admirable as their aspirations. Of Rings’s four analytical essays, the Adagio from Brahms’s String Quintet in G major, Op. 111, typifies this glitch. Training our telescopes onto the first two chords of the piece, we can identify another black hole between transformation theory and the music which it putatively models. For Rings, as for Cohn, the hole is caused by the overwhelming force of gravity – and again, gravity is the ‘overdetermined’ energy of Funktionstheorie. The irony of Rings’s parsing of the three-chord opening gambit of the Quintet is that, whilst on a formal level he stages the drama as a tripartite modular ‘rotation’ structure, sending us to Hepokoski and Darcy (2006) for reference, he constructs the seventeen-page analysis of the immediate harmonic surface as a binary exchange of just two chords rather than three. Rings’s toolbox is impressive: he has at his disposal sd/pc set networks, transformational networks, spatial networks, orientated networks and manifold makeshift digraphs to model the various hearings of the tonal functions at play in the two opening chords. Does the motto signal i → V in D minor, or iv → I in A major? Is the A major a ‘back relating dominant’, or is the D minor a ‘forward relating tonic’? (p. 206 [Fig. 8]). Rings offers two alternative ‘Riemannian’ accounts that ‘model the forward-relating and backward-relating kinetics’ (p. 206). Again he handles function as a transformational category, engaging mode-reversing subdominant (S*) and dominant (D*) transformations which ‘take’ a subdominant or dominant to a tonic of the reverse mode. He remarks, ‘There is thus a compelling (if abstract) retrograde-inversional relationship between the two hearings as energetic structures’ (p. 207). Energy is now rightly recorded in the analytical experience, but it is stifled by the refusal to let the chords speak as linear signifiers in an acutely Riemannian T → S → D chain.17 The problem stems from adherence to the dualistic Riemannian notion of the S and D as inversionally balanced around T, when the most profound source of energy is to be found in the rotation of these © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 21 Fig. 8 Brahms, String Quintet Op. 111/ii: Rings’s two hearings of the opening two chords (Rings 2011, p. 205). By permission of Oxford University Press, USA Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 22 KENNETH M. SMITH functions and the ambiguity is to be found through their retroactive functional (not transformational) interpretation. For me, the three chords which warrant such close inspection tender a relatively simplistic ‘modulation’: D minor → substitute E7 → A major. We may call this I → II → V progression ‘To → S(DP) → D’ or ‘So → D → T’, but the passage is structurally little more precarious than the opening of the Beethoven’s Waldstein Sonata, which Lerdahl scrutinised in energetic terms: T → S → D (2001, pp. 202–7). Rings explains the progression’s modal adjustments by invoking the Gypsy scale (broken into two tetrachords which lever each other across a tonic fulcrum), which acts as a catch-all scalar collection – an appropriate tactic, but one that fails to account for the more Romantic energetics of the appoggiatura A on the second beat of bar 1 which wants to descend to G♯, or the B♮ in the alto that Rings, in his warmest moment, imbues with a ‘subdominant tinge tilting it towards A as tonic’ (2011, p. 211), reinforced by the G♯. This is heard as S via Rameau’s double emploi between G major and E minor (P transformation notwithstanding), relating backwards to the tonic. Rings discloses the ‘massive dominant’ pedal on D in the fifth variation but demonstrates that Brahms keeps us guessing about the key of the motto until the final bars. (Spoiler alert: the movement is in D minor.) The Ringsian listener of these variations continually fluctuates between T and D hearings, just as he or she shuttles back and forth between the two chords. But the three chords, for us, can each bear three hearings (as S, D or T), and rather than a binary alternation in which the trapped listener is condemned to alternate left and right, the threechord model sets the harmonies and their energies in the ever-diminishing spiral of a T → S → D → T vortex. We might cast Rings as Caronte in L’Orfeo, condemned to sail back and forth across the Styx, when he could be the Flying Dutchman and navigate the entire globe (an equally eternal sentence, but a prettier night sky). In the spirit of Ringsian temporal twists, the listening subject’s functional interpretations push anticlockwise around the T ← S ← D ← T circle (remember Adorno and Lacan). The issue, Rings reminds us, is not as trivial as ‘[i]s the piece in A major or D minor?’; the problem of the missing chord resonates higher up. Rings takes on board the ambiguity immanent to the work and allows it to emanate into the listening experience, but his feedback loop is constricted. The energy of transformation is a feedback loop between the listener and the work, but a central dimension is missing: the circular nature of chords’ energies themselves. In my view, Brahms runs rings around transformation theory. So too does Wagner. One celebrated splendour of Tonnetz navigation is that it discards the tonic, but the price we pay for such liberation is an impoverished theory of voice-leading distances. As mentioned in my precis, for all of the energy which Rings focuses on the tonic, his technology for measuring voice-leading distance is lifeless, and its algorithms pay little attention to tonal attraction. Rings’s qualia (which do of course involve fixity of key) prove equally troublesome in both the diatonic and the chromatic versions of the other © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 23 Fig. 9 Wagner, Parsifal, diatonic ‘Grail’ motive: Rings’s GIS model (2011, p. 85); my annotations. By permission of Oxford University Press, USA neo-Riemannian warhorse: Parsifal’s ‘Grail Motive’. Adding to Cohn (2006), David Clampitt (1998), Lerdahl (2001) and Lewin (1984), Rings proposes a tonal qualia reading, which requires a fixed key in a GIS-based system. This works perfectly for the diatonic ‘Dresden Amen’, and few could quarrel with Rings’s model (Fig. 9), nor, I imagine, with my functional labels added beneath. In standard Riemannian nomenclature we register a simple T → Tp → S → Sp → D → T string, but Rings’s effort to capture chromatic adaptation suffers from the exchange of three distinct keys (E♭ major, D major and D♭ major) with only abstract subliminal modulations to secure each. Rings’s graph is thus compromised by layer upon layer of key strata. Each triad is laid out in the GIS with its root at the base and the governing tonic dictating the qualia. The modelling thus shows a concatenation of abstract operations which celebrates a passage that ‘thrums with energy’ (Rings 2011, p. 88) but ignores some basic activity. Like Cohn’s hexachordal chains, each chord contains the leading note of the next, and, parsimonious though each exchange may be, these sensitised pitches have heightened energy efficiency; leading notes resolve powerfully from one to the other rather than transform in an equally weighted GIS. An alternative graph might dispense with the qualia altogether and connect the leading note to root motions. I retain Rings’s original in light shading in Fig. 10 with my superimposed black boxes which highlight leading note → local ‘tonic’ resolutions in each chord change. The T → S → D → T paradigm described earlier (Lendvai’s Procrustean bed) pertains here also, but the functional labels or ‘anchoring points’ remain ever elusive (S → D → T → S or D → T → S → D?). Lewin (1984, pp. 345–6) found a ii → (V) → I progression at the example’s close; Clampitt (1998, p. 330), however, describes the pivotal E♭ minor triad as D functioned in A♭ major, but Sp functioned in D♭ major. Clampitt’s concern is a hierarchic levelling of functionality, but we are also in a Derridean open-ended chromatic universe, in which concepts are never fixed but in constant motion through plays of différance. This concept may sound like the chain of rightward arrows which in one of Rings’s examples moves towards a conventional cadential progression as a Schoenbergian monotonal ‘conveyor belt’ (Rings 2011, p. 105). It runs the obvious risk of mechanising some very supple modulatory twists and turns, but Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 24 KENNETH M. SMITH Fig. 10 Wagner, Parsifal, chromatic ‘Grail’ motive: Rings’s GIS model (2011, p. 85); my annotations. By permission of Oxford University Press, USA the performance of Janus-like différances between sounds and concepts is in fact a liberating compositional tool and a suggestive (if inconclusive) way of conceptualising nineteenth-century animated harmonic progression. For this reader, Rings’s book is to be applauded for establishing (a) the plurality of models and the dialectics between them, (b) the integration of ‘energy’ into the GIS, (c) the retroactivity of his hearing and (d) problematising subject-object relations. The answer to the book’s fundamental question – ‘Can we reconcile energeticist metaphors with the transformational attitude?’ (p. 104) – is ‘yes, but the book is only a step closer’. While Ringsian transformation theory shuttles between subject and object positions, between retroactivity and forward motion, between energy and transformation, between one hearing and another, it doesn’t reconcile fully. A Brief History of Tymoczko Tymoczko’s new historical account of theory traces energetic lines from S and D to T (Fig. 11), across various repertoires; in permitting any left-to-right moves (only those prescribed are backwardly plausible), he produces a ‘harmonic cycle’ of T → S → D → T (Tymoczko 2011, p. 227), like Rings. Also like Rings, he shows that all roads lead to Rome, although the rationale behind the tonic attractor is based on quantitative frequency analysis; he relegates discussion of ‘absent tonics’ to a mere footnote (p. 179). He subscribes to this T → S → D → T progression as one of the fundamental grammars of tonal music: © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 25 Fig. 11 Tymoczko’s permissible progressions in the major mode (2011, p. 227, his Fig. 7.1.5). By permission of Oxford University Press, USA Chords seem to obey specifically harmonic laws ... . This is one of the features of Western music that is most suggestively language-like. For just as in English, the subject normally precedes the verb, which in turn precedes the object, so too does Western music seem to have a harmonic ‘grammar’ according to which subdominant chords precede dominants that in turn precede tonics. (p. 226) His diatonic rightward motion also honours falling-third bass progressions: ‘falling thirds are more fundamental than falling fifths’ (p. 228). In his diatonic model, prototypical minor-third substitutions preserve function, while their major-third equivalents tend to discharge it. I/vi implies functionally equivalent R-related T chords, moving to IV/ii as functionally new Ss, progressing to the ambiguous viio leading to V(/iii?) as Ds. Another model of early chord progression is the ‘down a third, up a step’ sequence, as shown in Ex. 7. The reharmonisation of this Bach fugue subject yields a perfect cycle of fifths until the final notch (D♭7 → E♭ major). The only changes (apart from various P transformations) are the B♭7/Go trade and the F/A♭7 exchange, both fluent functionpreserving minor-third relatives. Like the cycle of fifths, T2 progressions functionally upshift, as in the Waldstein’s C major → D major, which retroactively becomes T → S. Under certain conditions, Tymoczko’s ‘down a third, up a step’ sequences can also usher in functional exchanges (the first Go → E♭ minor → F minor would be S → D → T). Tymoczko’s historically considerate account of scale geometrics recognises the gravitational quality of leading-note adjustments, caused by fourthprogressions in the itinerant cycle of fifths (F becomes the F♯ leading note in a C → G modulation). Following the tonal neutrality characteristic of medieval music, increasing tension eventually produced ‘harmonic cycles’ which required chromatic cadential outlets (2011, p. 210). Tymoczko prioritises T → S → D → T alongside ‘modulation’ (p. 214) but confesses that progressions such as the tritone substitution (p. 216; his example is from the ‘Chopin’ movement of Schumann’s Carnaval, Op. 9) obfuscate the need for modulation. In this way Classical geometry was transfigured as Romantic. Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 26 KENNETH M. SMITH Ex. 7 Bach, WTC II, Fugue in A♭, bars 6–7 and 13–14 (Tymoczko 2011, p. 243). By permission of Oxford University Press, USA After considering the rise and fall of the ‘chord as object’, Tymoczko introduces the ‘major-third system’ and the ‘minor-third system’ (2011, p. 280). But his Mars and Venus come perilously close to collision in his reading of a progression from Schubert’s D major Sonata, D. 850, in which, following a D major tonic, a hexatonic progression (D → F major → C♯ major → A7) ‘leaves traditional tonality in favour of the major-third system’. The final chord’s seventh is considered ‘inessential’ and ‘decorates the more basic triadic relationship’ (p. 280). A (as a triad) is considered close to F (purely based on proximate voice-leading parsimony), and for Tymoczko the C♯ is the alien triad. I would contend that the seventh is essential, even if it causes Mars and Venus to collide. The basic progression for me is T → S → D → T, with F (as the RP of D major) discharging to C♯ via enharmonic leading-note resolution; C♯ likewise discharges to A (via the leading note G♯), whose seventh secures a full discharge to the true T of D major. And Tymoczko almost slips into Cohn’s black hole: ‘since major-third-related triads can be connected by efficient voice-leading, they can substitute for one another without much disrupting the music’s contrapuntal or harmonic fabric’ (p. 283). But, as proposed in my extended reading of Cohn’s Schubert analysis, hexatonically related triads contain each other’s leading notes and can therefore discharge into each other and alter function; octatonically related chords have no leading notes and can only store up functional energy.18 Tymoczko finds his two polarised ‘systems’ exchanging spaces in Chopin’s E major Prelude, Op. 28 No. 9, bars 5–8 (2011, p. 218). His B major (chord 2 of Ex. 8) is a major-third relative of G major but is also a parallel minor third D[ominant] of E, so that the B major provides a deceptive embedded cadential motion to G major with its F♯ leading note. The seventh of G major appears in the bass, which (a) discharges to the following C major and (b) breaks the © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 27 Ex. 8 Chopin, E major Prelude, Op. 28 No. 9, bars 5–8: Tymoczko’s interpretation (2011, p. 218). By permission of Oxford University Press, USA ‘Mars’/‘Venus’ segregation (pp. 97 and 220). For Tymoczko, the subsequent bars outline the minor-third system with the dominants (or dominant substitutes). This is an octatonic succession of functionally synonymous chords. The B♭ minor (which Tymoczko’s analysis glosses over) now leads to a mock E♭ major discharge as a T11 retrograded parody of the earlier E major → B major progression. Tymoczko’s assertion that nineteenth-century music ‘abandons functional norms in favour of direct chromatic voice-leading between triads and seventh chords’ (p. 217) is thus only partially correct. The functional tectonics here move T → S → D → T again with a full mixture of tetrachords and triads. Despite my protestation, Tymoczko is nonetheless correct: octatonically related triads have sevenths because sevenths are available in the collection; hexatonically related triads are denied sevenths. Geometrically, this makes perfect sense. Musically, it doesn’t always work, because these planets are in a diatonic solar system and are still subject to the same gravitational forces. Adrian Childs, the first neo-Riemannian to take account of seventh chords sufficiently, claims that ‘the composers whose works seem best suited for neo-Riemannian analysis rarely limited their harmonic vocabulary to simple triads’ (1998, p. 181). The irony is that octatonic seventh chords are full of ‘drive’ but the absence of leading notes prevents them from discharging, and thus they remain static; hexatonic chords can’t have sevenths but can nonetheless discharge into each other as leading notes. Both are gravitationally bound, needing each other to become animated. An indicator of the inability of Tymoczko’s theory of tonal geometrics to integrate triads as objects of tetrad resolution is an innocuous example for which Tymoczko doesn’t offer a mathematical model, nor even a functional hearing. Rather, he provides an austere voice-leading reduction which demonstrates the ‘resolutions’ of the Tristan chord in Brahms’s Op. 76 No. 4, ‘a teasing piece that begins on a dominant seventh chord that resolves only at the end’ (2011, p. 276). His reading of the opening has a comparatively simple premise: a Tristan chord Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 28 KENNETH M. SMITH Ex. 9 Brahms, Intermezzo, Op. 76 No. 4, bars 1–3 Fig. 12 Brahms, Intermezzo, Op. 76 No. 4, opening chords: a Ringsian digraph (F–G♯–B–E♭) is nestled between instances of an F7 tetrachord (Ex. 9), which refracts and later ‘resolves’ to A♭ major. The progression is relatively straightforward, but a geometric interpretation is complex. Because both sonorities contain tense tritones, Tymoczko adopts the well-chosen vocabulary of ‘resolution’ and ‘discharge’. However, the language is misleading. Since when did a ‘root position’ Tristan chord on F want to ‘resolve’ to another F major–rooted seventh? When, for that matter, did it want to resolve to A♭ major? We could look instead to Rings. His technology might model the opening progression as both a ResV (in which the Tristan chord resolves to V7) and a ResI (with a resolution to the Tristan chord as ‘I’ [Fig. 12]). Flaws immediately materialise. The D in Brahms’s score, which Tymoczko ignores, transforms the chord into a full B♮o7 chord, replacing F7’s desired B♭ with a prohibitive B♮. Rings’s technology copes with standard abstract resolutions but collapses in situations when (a) we wish to express that the resolution given was not the desired one; (b) the tonic centre is veiled; (c) we want to outlay each pitch’s multivalent resolution tendencies (which they always have); (d) chord x ‘resolves’ to y, which then ‘resolves’ back © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 29 Fig. 13 Brahms, Intermezzo, Op. 76 No. 4, bars 1–3: my model in Tymoczko’s 3-D four-note chromatic space to x, thus requiring two asymmetrical profiles of tension and release; and (e) the tonic resolves away from itself. These five issues render my Ringsian modelling in Fig. 12 faulty. Chromatic though the voice leading is, no leading notes signal discharge between chords, nor do they develop in the second ‘resolution’ to A♭ major (the A♭ major also contains a delicate F♯, omitted from Tymoczko’s reduction, that assembles a German sixth). F7 and A♭7 major chords are minorthird relatives and share two pitches each with the Fø7. The only Tymoczkan technology to model the sequence is his three-dimensional four-note chromatic space, which can demonstrate the two ways of parsimoniously connecting Fø7 with the piece’s ‘resolutions’.19 These are unfilled particles in my Fig. 13. Tymoczko’s space proves that (a) the chords are two parsimonious steps apart, Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 30 KENNETH M. SMITH Fig. 14 Douthett’s 4-cube trio, modified by Cohn (2012, p. 158). By permission of Oxford University Press, USA and (b) a single octatonic collection is employed. However, the A♭ major chord is purely triadic (the F♯ pitch occurring late in the game) and therefore alienated. My model tells us nothing about how these chords dubiously ‘discharge’ into each other. Douthett’s ‘4-Cube Trio’ is appropriated by Cohn (Fig. 14) and, coincidentally, offers identical information in 2-D (although I have not modelled the intermezzo here) with the addition of voice-leading zones. One thing which both graphs illuminate is that, while all genera of octatonic chords have connectable chromatic tones, no discharge is possible because the four excluded pitches are the leading notes (both the major 7ˆ −8ˆ leading note and the minor ♭6ˆ −5ˆ ) of each. The diminished sevenths are portals into discrete octatonic worlds, and discharge occurs only as the gateway closes. Although 2-D, Cohn’s © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 31 Table 2 Formal rotation of TSDT A B A′ (Recap) F7 (B♭) → g D → Tp e♭/G♭ → F7 So → D F7 (B♭) → B♭ (D) → T more compact graph is more navigable because of its cyclic format, which embraces functional rotation. What Cohn’s fresh voice-leading zones (now modelling tetrachords) add is an understanding that some motions relate octatonically (3,4,5–7,8,9–11,0,1), while 2, 4 and 10 are resolving agents. In an octatonic matrix the dominant seventh is like an orbiting space rocket which requires a huge thrust in order to escape the gravitational field of one planet and circle another. But Cohn’s new voice-leading cardinals always run against the now anticlockwise cycle of fifths: upshifting does not equate to functional D → T discharge. Douthett’s and Tymoczko’s claustrophobic models register D → T ‘transformations’ only after four theoretical moves. Is it possible that a model of nineteenth-century harmony, even specific aspects of it such as those modelled by ‘neo-Riemannians’, cannot integrate these obvious releases of tension, and can register kinships based only on minute mechanical adjustments? Tymoczko’s demonstration of the resolutions of the Tristan chord is startlingly reminiscent of Lendvai’s analysis of Tristan, which picks out its four T3-related resolutions (Lendvai 1993, p. 82). Brahms’s miniature yields only two of these, but it still pays to avail ourselves of Lendvai’s hospitality once more while analysing Brahms’s form. The form is transparent, for all its chromatic harmony: a Baroque rounded binary (Table 2 and Fig. 15). Typically A partitions into two clear themes, the first based on B♭ (D), the second deceptively resolving to G minor. A G♭ pedal grounds B while E♭ minor harmonies animate the opening chromatic progression. The bass briefly rises to B♭ before descending again to G♭, indicating B♭ major through a Phrygian cadence on F7 which signals the recapitulation. A′ corrects the R move in A, now fixing itself to T (no longer G minor but B♭). In this version of the T → S → D → T model, the dominant (F7) is the primary contact point. Relative substitutes are innocuous and paleoRiemannian; the only unusual elements are: (a) the Weitzmannian N progression, which makes the S section E♭ minor, and (b) the G♭ major (upper relative of this E♭ minor) as S representative. That this becomes the ‘flattened sixth’ through Phrygian motion fully commissions the G♭ major as the S envoy. Tymoczkan models closely represent the harmonic progressions of parts of this piece, but to account for discharge motion we must burst these particular theoretical bubbles and mix tetrads and triads together in a substitution system. I have tried to demonstrate that Tymoczko’s ‘triads are from Mars; seventh chords are from Venus’ leads to a false analytical segregation rather than a genuinely musical one when we factor in tension and resolution; a simple V7 → I Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 32 KENNETH M. SMITH Fig. 15 Formal rotation of TSDT progression – the paradigmatic ‘discharge’ motion – connects the two planets in the most obvious way. But on reflection, it seems that Tymoczko’s two ‘planets’ are actually two different functions or forces: one force releases tension while the other stores it. Ironically, while we might imagine the discharger to be the seventh chord and the storer to be the triad; the opposite is true because of the association between triads and hexatonics and between seventh chords and octatonics, the former allowing discharge, the latter arousing the desire for but prohibiting it. This theory is latent in Tymoczko and Cohn, but only Rings really attempts to model the crucial X factor: the discharge motion. In order to stage a true dialogue, we must gaze once more into the night sky, bringing our three theorists into a synthetic analysis. For me, the high-resolution lenses for gazing at nineteenth-century night sky are (a) Cohn’s ‘voice-leading zones’, (b) the Ringsian digraphs and (c) the Tymoczkan geometries of three/four-note space, © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 33 and it is these very different lenses which we take with us into the same observatory, together with the Romantic-cum-Modernist composer referred to earlier – Richard Strauss. Strauss’s Twilight Zone: Funktion as Energetic Particle Tymoczko claims that, along with Max Reger, Strauss phased out the T → S → D → T progression (2011, p. 181). To disprove this, I will bring our three theorists together to analyse a whole piece (quite a rare phenomenon in these rather atomistic volumes). Cohn takes a convincing hexatonic snapshot of ‘Spring’, the second of Richard Strauss’s Four Last Songs (1949) (Cohn 2012, pp. 78–81); but I will analyse the fourth song, ‘Twilight’, in its entirety. Its weary tempo and lush orchestration make every chord change a special event; we mull over the voice leading of each individual pitch. Some passages celebrate minorthird progression (rarely presented as seventh chords), others venerate the major third (not exclusively triadic in nature); but behind this hazy afterglow remains a faint T → S → D → T, ever straying from E♭ major through substitution, but always returning to its E♭ major evening mantra: ‘Ist das etwa der Tod?’ (‘Could this be death?’), Eichendorff’s poetry asks. It is immediately apparent that the chord progressions in the song resist some of the approaches from our three authors critiqued in this article. Cohn’s voice-leading zones are displayed above the stave in Ex. 10. Except for an initial rotation around zones 1, 5, 8 and 10, I discern no pattern save a general impression of balance between upshift and downshift (as pieces obeying the law of averages should). The many L and P transformations suggest hexatonic inquiry with progressions like B♭ major → G♭ major → D major motion in bars 33–39 (Ex. 11). But this is a faint star. The DOM ‘transformations’ are generally V7 → I progressions whose tetrachord dulls the triadic lustre of the transformation networks and further weaken DOM’s plausibility as a category of cognition. We could call them ResI à la Rings, but we would still conflate transformation with discharge. Besides, other transformations make nonsensical strings: (1) the simple T2 is NPRP or RPRL, (2) T11 is LN/PS and (3) T6 is RPRP or NPSP. These transformations – too tenuous to be ‘maximally smooth’ – are mostly marked ‘?’ on the transformational strand of Ex. 10. Tymoczko’s three-note and four-note chromatic models are problematic here because there is no association between triads/seventh chords and major/minor systems in Strauss (seventh chords only occur when discharging in a clear DOM ‘transformation’). Strauss’s progressions, although for the most part clearly third-related, traverse too many geometric modules to be of value. On Tymoczko’s website, an animation of Chopin’s E minor Prelude, Op. 28 No. 4, superbly models the dense network of seventh chords as they move deftly through four-note space. But as triads surface, a miscellaneous green dot appears outside of his grid, indicating that the model cannot process them as it stands.20 This approach is unhelpful, and a similar thing is happening here. Given that Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 34 KENNETH M. SMITH Ex. 10 Richard Strauss, ‘Twilight’ (‘Im Abendrot’), Four Last Songs, Op. posth.,: reduction Strauss’s song is broadly triadic (in contradistinction to Chopin), the extract reproduced as Ex. 11 maps well onto three-note chromatic space, and we might ‘ignore’ sevenths when they arise as auxiliary in function. Fig. 16 demonstrates the symmetry of this purple patch, the E♭ major and C♯ major appearing as discharge outlets for the central hexatonic triads. Voice-leading stages between chords can be counted readily on this graph, and V and I lie three steps apart. But is this useful? We could have learned this from Cohn’s 2-D book, and the Tymoczkan 3-D models are occasionally a clumsy way of rendering progressions.21 If the aim of the geometric models really is to ‘clearly illustrate musical principles that can otherwise be quite difficult to grasp’ (p. 64), then we sometimes fall short with these cluttered 3-D diagrams rendered on Oxford University Press’s woefully 2-D paper. Rings’s GIS transposition network graphs look different. Like the star that collapses catastrophically just before it becomes a spectacular supernova, his digraph proves most insightful when it breaks apart. Hexatonics (B♭ major → G♭ major → D major) model efficiently, and the grid admirably assumes a listener who accesses his or her database of previously heard chord progressions and matches patterns which take paths of least diatonic resistance. The digraph © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 35 Ex. 11 Strauss, ‘Twilight’, bars 33–39. ©Copyright 1950 by Boosey & Co Ltd. Complete edition 1959. B&H 16925. Reproduced with permission (Fig. 17) shows the hexatonic L shapes, assuming ‘ghost’ discharges (B♭ major → E♭ minor). I fully approve of these hypothetical, imaginary steps, but these are bypassed with direct-discharge leading note → temporary tonic motion (overlaid with my boxes and arrows). Thus the G♭ major → D major discharge renders a ghost B minor as local T and the existing D as Submediant; direct leading-note discharge would suggest D major as a new T, and the GIS’s qualia would need recalibrating. This would yield a descending chromatic cascade of T(e, -1) keys as in Rings’s version of Parsifal. (NB: chromatic descents rotate T → S → D → T in my model of tonal function.) This issue pertains also to the bottom rung of F♯ minor that C♯ major posits as firm T. Shuttling backwards produces two ‘ghost’ B minor triads, but the retroactively working GIS suggests that we were in F♯ already from the previous G♭, also retroactively reaffirming B♭ major → G♭ major as D → T. Retroactivity makes us constantly shunt our functions backwards. I approve of this GIS because it assumes an extremely diatonic hearing, but it fixes us to specific tonics, whereas chromatic music pulls me rather to function. The model is too diatonic for its own good. Rings’s GIS Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 36 KENNETH M. SMITH Fig. 16 (a) My Tymoczkan model of three-note chromatic space; (b) Strauss, ‘Twilight’, bars 33–39: model technology breaks down when it needs to model the retroactivity and dialectic of hearings so fundamental to his aesthetic of chromatic listening, but problematic to his graphical technique. Crucial to the piece, therefore, seems to be the perpetual T → S → D → T rotation of Ex. 10, whose functions are inscribed below it. Occasionally something like the Funktionstheorie equivalent to a neighbour-note motion will occur, where a chord will leave a functional category and return to it – usually to the same chord. This happens only twice in the piece, which otherwise rotates functions no fewer than eleven times. Each and every hexatonic transformation discharges leading notes (F → G♭/C♯ → D), pushing around the circle, and in fact this usually occurs because a hexatonic cycle is being rotated ‘downwards’ (i.e. by downwards T8 transposition rather than T4). Things change slightly in the orchestral postlude, when the sudden global freeze of D, F, A♭ and C♭ octatonics closes the song’s universal oneness with the chilling question ‘Ist dies etwa der Tod?’. Here the A♭ minor completes an octatonic cycle of all four minor-third-related triads (not seventh chords, © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 37 Fig. 17 Strauss, ‘Twilight’, bars 33–39: a Ringsian GIS model Tymoczko!). Tymoczko would doubtless approve of the juxtaposition of the minor-third system and the major-third system as two separate planets, but the bigger picture emphasises their functional difference: these octatonics are nondischarging (lifeless? objective?) and confirm for us the fully discharging (living? subjective?) hexatonics which coincided with rotations of tonal function. After approaching E♭ major obliquely (a D minor chord: PSP), the C♭ major ♭ VI ‘Tod’ erupts as the only proper reversal of the T → S → D → T paradigm. Here, and throughout the song, the wandering harmony searches for the true dominant (B♭ major) but finally discharges to the E♭ major tonic as Strauss returns home. Back to Earth And so do we. My contention is that we cannot understand the spaces that chords occupy unless we explore their inherent tensions and integrate models of how they relate and how they move (or transform) with why they move, why we want them to move and what transformational possibilities are on offer at any given moment. The reason I critique these astronomical (or geometrical) books with cosmological questions is that all three authors talk about energy in some form or other and betray an aspiration to model the paths in which it circulates in different musics. Rings talks about harmonic intention, Tymoczko draws harmonic DNA strands and Cohn even uses Freudian terminology at times, with all of its hydraulic associations. It is here that the books sell us short. My own approach pertains only to certain strains of diatonic/borderline chromatic music Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 38 KENNETH M. SMITH (see Smith 2010a and 2010b) and is one attempt to tie up two of these books’ loose ends: (a) the misapplied theory of tonal function and (b) the misalignment of major and minor thirds. My account of tonal energetics is hardly comprehensive but may move us one step closer to finding the ever-elusive harmonic God particle. Make no mistake: all three books are exceptional texts. My reading of them, crooked though it is and bending towards specific problems which I find, in no way attempts to seal off the debates of transformational theory. The discipline as represented by Cohn, Rings and Tymoczko (among other practitioners) signifies a dialectical spirit of inquiry that seeks renewal and reinvigoration. My hope is that my peculiar curveballs may spark one of those hits (like in Superman films) which are launched through the stratosphere into deep space and return to earth (even if they knock out the pitcher) after a fascinating interplanetary journey. Transformation theory can handle it either way; it has been expanding now for a century or so, and the project of mapping the universe of tones was already fairly vast. With these new texts, and our responses to them, it just got bigger. NOTES 1. Richard Cohn, Audacious Euphony: Chromaticism and the Consonant Triad’s Second Nature (Oxford and New York: Oxford University Press, 2012). xviii + 256 pp. £22.99 (hb). ISBN 019977269X. Steven Rings, Tonality and Transformation (Oxford and New York: Oxford University Press, 2011). xi + 256 pp. £22.99 (hb). ISBN 019538427X. Dmitri Tymoczko, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice (Oxford and New York: Oxford University Press, 2011). xviii + 432 pp. £25.99 (hb). ISBN 0195336674. The term ‘Universe in Tones’ was coined by Ives as the subtitle for his unfinished ‘Universe Symphony’. 2. See Ramirez (2013) for a recent survey of this issue. 3. All three theorists strike a balance between theory, analysis and musicianship. The formula is: (1) introduce a theoretical system; (2) use it to model a piece; (3) tell us something new; (4) instruct us to play in a new way until we hear the analytical claim; and (5) confirm (or exaggerate at times) the impact of stage 1 on stages 3–4. Rings offers valuable insights on specific works, but while these showcase his gifts as a brilliant analyst as well as a calculator, the musical pay-off is not always contingent upon the maths. 4. Tymoczko’s software is free, downloadable from www.dmitri.mycpanel .princeton.edu. 5. See Christensen (1993), pp. 185–9. 6. Tymoczko’s voice-leading distance strips diatonic gravity from Lerdahl’s model, claiming, for example, that ‘F major and A7 are harmonically quite © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd Music Analysis, ••/•• (2014) THE TRANSFORMATIONAL ENERGETICS OF THE TONAL UNIVERSE 39 close’, which only accounts for parsimony, achievable through semitone manipulations. One could compare this less favourably with Tymoczko’s ‘Three Conceptions of Musical Distance’ (2009, pp. 258–73). The Geometry utilises only the ‘no crossings principle’, while exploring both Euclidean and non-Euclidian methods of reaching a cardinality of voice-leading size in a substantial appendix. The issue is consequential to Tymoczko’s Möbius of dyadic space. 7. The conflation pertains to others in the discipline: Kopp’s exegesis of neo-Riemannian study proposed a whole new concept of third-related chord interactions, and he even invented the M function to add to the R, L, P, D, S (the D-1), H, N and S list (Kopp 2002). But even here, major and minor thirds are given distinctive treatment only as upper and lower chromatic mediants around a tonic, and thus are stirred into Kopp’s pot to melt beyond recognition. 8. For alternative discourse on ‘becoming’ and musical form see Schmalfeldt (2011). 9. Marion Guck called the relationship ‘P’ (1978, p. 33); Tymoczko classifies it as ‘PD’ (pre-dominant) (2011, p. 270); Charles Smith suggests a general auxiliary-dominant function: ‘it is in fact these [augmented sixth] chords that so clearly proclaim the dominant function of their successors, since this particular succession of sonorities directly defines the second chord as a dominant’ (1986, p. 97). For reasons that will become clear, I classify these dominant preparation (DP) chords as various species of S. 10. Cohn (2012) ontologically under-complicates the second subject; Webster demonstrates that this is a ‘three-key exposition’, fitting an historical tradition from Schubert to Brahms. See Webster (1978), p. 26. 11. It is something of a cliché that Riemann advocated the T → S → D → T progression above all. Kopp explains that Riemann’s 1902 composition manual discusses its strengths over the weaker T → D → S → T, still permitting it for his students but making them aware of obvious cadential weakening. ‘The lesson is merely that T → S → D → T works and sounds better; the purpose of the discussion is chiefly to discourage the student composer from writing the progression from D to S’ (Kopp 1995, p. 10). 12. Translated by Rings, from Rameau (1737), pp. 27–8. 13. Julian Hook put this most articulately: ‘[w]hat is to be made of the fact that, applied to a major triad, D has the same effect as R followed by L, but applied to a minor triad, D has the same effect as L followed by R? Is D superfluous?’(2002, pp. 57–8). However, Krumhansl’s empirical studies suggest that the D transformation is heard very differently to LR: ‘these results established the independent psychological status of the D transforMusic Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd 40 KENNETH M. SMITH mation, indicating that combinations of R and L transformations cannot substitute for the D transformation’ (1998, p. 272). 14. Translated by Rings, from Riemann (1882), p. 188. 15. Root motion by a fifth drove Rameau’s system, as Scott Burnham, comparing the two theorists, reminds us: ‘Rameau understands such motion to be a logical deduction from the properties of the fifth as they were themselves deduced from the principe sonore’ (Burnham 1992, p. 7). 16. Cited in Hoeckner (2002), p. 86. 17. See Hyer (1994) for a discussion of Lacanian signifiers in the T → S → D paradigm. Although I would contend that Hyer’s Lacanian/Derridean interpretation lacks some of the subtlety of Lacan’s understanding of signifier (which ultimately only signifies itself to another signifier), his approach marks an excellent step forward for an understanding of the signification processes involved in Funktionstheorie. For an alternative approach to the same problem see Smith (2010a) and (2010b). 18. I disagree with Rings (2011, p. 128) and Eytan Agmon (1995, p. 201), who classify iii as dually related to I and V. I would concede that in certain contexts, iii can stand in place of I (and thus 7̂ replace 8̂ ); but 7̂ is so strongly D functioned that it saturates the whole chord’s character and wants to discharge upwards to 8̂ in the vast majority of cases, all other things being equal. Minimum perturbation can have maximum effect where certain sensitised pitches are concerned: E minor → C major is a functional shift in the majority of cases. 19. A potential way of connecting octatonically related sevenths is via a French sixth, but this does not pertain between minor-third-related sevenths. 20. Available for free at http://dmitri.tymoczko.com. 21. See for example the complex model of 3-D geometry, on which Tymoczko traces the route through Schubert sequences from the Quartett-Satz, D. 703, bars 173ff. Here, a simple fifth jumps G to D and then a minor third from D to F, creating unsightly leaps across his hexatonic cuboid string. The only complexities in the graph are parallel relations, which Cohn’s 2-D apparatus would express well (Tymoczko 2011, p. 284). A more productive case is the geometrically hypothesised rules behind Chopin’s improvisation technique, as embodied in the Mazurka Op. 68 No. 4 (pp. 285–9), where Tymoczcko’s cubic model is unique and matches the piece exactly. 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Smith, Kenneth, 2010a: ‘Skryabin’s Revolving Harmonies, Lacanian Desire and Riemannian Funktionstheorie’, Twentieth Century Music, 7/ii, pp. 167–94. ______, 2010b: ‘Lacan, Zemlinsky, and Der Zwerg: Mirror, Metaphor, and Fantasy’, Perspectives of New Music, 48/ii, pp. 78–113. Sõmer, Avo, 1995: ‘Chromatic Third-Relations and Tonal Structure in the Songs of Debussy’, Music Theory Spectrum, 17/ii, pp. 215–41. Tymoczko, Dmitri, 2009: ‘Three Conceptions of Musical Distance’, in Elaine Chew, Adrian Childs and Ching-Hua Chuan (eds), Mathematics and Computation in Music (Heidelberg: Springer), pp. 258–73. ______, 2011: A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice (Oxford and New York: Oxford University Press). Webster, James, 1978: ‘Schubert’s Sonata Form and Brahms’s First Maturity’, 19th-Century Music, 2/i, pp. 18–35. NOTE ON CONTRIBUTOR KENNETH SMITH is Lecturer in Music at the University of Liverpool. His book Skryabin, Philosophy and the Music of Desire was published by Ashgate in 2013. He completed his PhD at Durham University in 2009 and has published articles on harmonic theory in Skryabin, Zemlinsky and Ives. He is events officer and vice president of the Society for Music Analysis. Music Analysis, ••/•• (2014) © 2014 The Author. Music Analysis © 2014 John Wiley & Sons Ltd