Multidimensional Musical Objects In Mahler 7

MULTIDIMENSIONAL MUSICAL OBJECTS IN MAHLER’S SEVENTH SYMPHONY Jason Patterson, B.A., M.M.

Dissertation Prepared for the Degree of DOCTOR OF PHILOSOPHY

UNIVERSITY OF NORTH TEXAS May 2019

APPROVED: Timothy Jackson, Major Professor David Heetderks, Committee Member Peter Mondelli, Committee Member Felix Olschofka, Interim Director of Graduate Studies in the College of Music John W. Richmond, Dean of the College of Music Victor Prybutok, Dean of the Toulouse Graduate School

Patterson, Jason. Multidimensional Musical Objects in Mahler’s Seventh Symphony. Doctor of Philosophy (Music), May 2019, 139 pp., 50 figures, bibliography, 60 titles. Gustav Mahler’s Seventh Symphony seems to belie traditional notions of symphonic unity in that it progresses from E minor in the first movement to C major in the Finale. The repertoire of eighteenth and nineteenth century composers such as Haydn, Beethoven, and Brahms indicates that tonal holism is a significant factor for the symphonic genre. In order to reconcile Mahler’s adventurous key scheme, this dissertation explores a multidimensional harmonic model that expands upon other concepts like Robert Bailey’s double-tonic complex and transformation theory. A multidimensional musical object is a nexus of several interconnected chords that occupy the same functional space (tonic, dominant, or subdominant) and can be integrated into a Schenkerian reading. Mahler’s Seventh is governed by a threedimensional tonic object that encompasses the major and minor versions of C, E, and A-flat and the augmented triad that is formed between them. The nature of this multidimensional harmony allows unusual formal procedures to unfold, most notably in the first movement’s sonata form. To navigate this particular sonata design, I have incorporated my own analytical terminology, the identity narrative, to track the background harmonic events. The location of these events (identity schism, identity crisis, and identity reclamation) is critical to the entire structure of the Seventh.

Copyright 2019 By Jason Patterson

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ACKNOWLEDGEMENTS It is difficult to imagine this dissertation, or my grasp on music theory in general, without the guidance of my mentor, Timothy Jackson. His unique perspective on large-scale forms and Schenkerian analysis has been invaluable to me. He continuously challenged me to be a better theorist and to sharpen my knowledge of the repertoire. I truly cannot thank him enough for his years of instruction and inspiration. I am also greatly indebted to my committee members, Dr. David Heetderks and Dr. Peter Mondelli. Their feedback and questions helped me refine my model and analyses and strengthen my terminology. Fresh eyes on a years-long project is vital for its efficacy, and every chapter benefited from their new perspectives. The overall product is undoubtedly better and clearer because of their input. Finally, I want to thank my family and friends. Such an endeavor is profoundly easier with so many caring people at your side. And especially Laura, whose love and support has kept me sane through the entire process.

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TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ........................................................................................................... iii LIST OF FIGURES ....................................................................................................................... vi CHAPTER 1. INTRODUCTION ................................................................................................... 1 1.1

The Problem ............................................................................................................ 1

1.2

Literature Review.................................................................................................... 3

1.3

Multi-Movement Considerations .......................................................................... 15

CHAPTER 2. THE MULTIDIMENSIONAL MUSICAL OBJECT ........................................... 19 2.1

Introduction ........................................................................................................... 19

2.2

One-Dimensional Musical Objects ....................................................................... 19

2.3

Two-Dimensional Musical Objects ...................................................................... 23

2.4

Three-Dimensional Musical Objects .................................................................... 25

2.5

Early Examples of Three-Dimensional Musical Objects...................................... 29

CHAPTER 3. JUSTIFYING THE MULTIDIMENSIONAL MODEL ....................................... 37 3.1

Justification for a Three-Dimensional Tonic Object ............................................ 37

3.2

Enharmonicism and the Augmented Triad ........................................................... 41

3.3

The Three-Dimensional Dominant Object as Binding Element ........................... 45

3.4

Sonata Form and the Identity Narrative ................................................................ 47

3.5

Function Entanglement ......................................................................................... 67

CHAPTER 4. FIRST MOVEMENT ANALYSIS........................................................................ 69 4.1

Introduction ........................................................................................................... 69

4.2

Introduction Analysis ............................................................................................ 70

4.3

Exposition Analysis .............................................................................................. 84

4.4

Development Analysis .......................................................................................... 90

4.5

Recapitulation Analysis ........................................................................................ 97

4.6

Cubism ................................................................................................................ 105

CHAPTER 5. FINALE ANALYSIS .......................................................................................... 109 5.1

Introduction ......................................................................................................... 109

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5.2

Tonic Prolongation.............................................................................................. 109

5.3

Subdominant Prolongation.................................................................................. 114

5.4

Coda .................................................................................................................... 126

CHAPTER 6. CONCLUSION.................................................................................................... 127 6.1

Introduction ......................................................................................................... 127

6.2

The Multidimensional Musical Object ............................................................... 128

6.3

First Movement Conclusions .............................................................................. 130

6.4

Finale Conclusions .............................................................................................. 132

6.5

Future Projects and Final Thoughts .................................................................... 133

BIBLIOGRAPHY ....................................................................................................................... 135

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LIST OF FIGURES Page Figure 1.1: Schenker’s analysis of Chopin’s Op. 28, No. 2............................................................ 2 Figure 2.1: Two-Dimensional visual space metaphor .................................................................. 22 Figure 2.2: One- and two-dimensional musical objects................................................................ 22 Figure 2.3: Three musical objects and their dimensions............................................................... 25 Figure 2.4: Wormhole effect ......................................................................................................... 28 Figure 2.5: Three-dimensional musical object compressed into one-dimensional musical space. ....................................................................................................................................................... 29 Figure 2.6: Foreground of Schubert’s “Der Doppelgänger” ......................................................... 31 Figure 2.7: Middleground of Liszt’s Étude No. 3, Un Sospiro..................................................... 34 Figure 2.8: Middleground of Hiller’s Piano Concerto No. 1, Finale ............................................ 36 Figure 3.1: E major cadence embedded in C major cadence in Mahler’s Seventh Symphony, Finale, mm. 13-15 ......................................................................................................................... 40 Figure 3.2: Interjection of Af major into C major in Mahler’s Seventh Symphony, Finale, mm. 51-52 ............................................................................................................................................. 40 Figure 3.3: Emergence of C augmented triad in Mahler’s Seventh Symphony, Finale, mm. 588590................................................................................................................................................. 41 Figure 3.4: Middleground of Mahler’s Seventh Symphony, first movement, mm. 1-31 ............. 42 Figure 3.5: Superimposition of Af and Gs enharmonics in Mahler’s Seventh Symphony, first movement, mm. 19-20 .................................................................................................................. 43 Figure 3.6: Augmented triads as dominants, derived by Schoenberg in Theory of Harmony ...... 44 Figure 3.7: Mahler’s Seventh Symphony, first movement, mm. 114-117 ................................... 46 Figure 3.8: Deep middleground in Mahler’s Seventh Symphony, first movement, mm. 145-468 ....................................................................................................................................................... 47 Figure 3.9: Middleground of Haydn’s Symphony No. 93 in D major, first movement ............... 53 Figure 3.10: Middleground of Haydn’s Symphony No. 95 in C minor, first movement ............. 54 Figure 3.11: “Common sense” middleground of Schubert’s “Unfinished,” first movement ....... 57 vi

Figure 3.12: Alternative middleground of Schubert’s “Unfinished,” first movement ................. 59 Figure 3.13: (a) Deep middleground of Mahler’s Seventh Symphony, first movement, mm. 1-410 ....................................................................................................................................................... 64 Figure 4.1: Three-dimensional V, I, and IV objects ..................................................................... 69 Figure 4.2: Three-dimensional supertonic object ......................................................................... 70 Figure 4.3: Section 1, B cadence from Mahler’s Seventh Symphony, first movement, mm. 1-18 ....................................................................................................................................................... 72 Figure 4.4: B minor cadence in Mahler’s Seventh Symphony, first movement, mm. 15-19 ....... 74 Figure 4.5: Section 2, G major half cadence from Mahler’s Seventh Symphony, first movement, mm. 19-22 ..................................................................................................................................... 75 Figure 4.6: Section 2, Ef major ‘cadence’ from Mahler’s Seventh Symphony, first movement, mm. 23-27 ..................................................................................................................................... 76 Figure 4.7: Gs half-diminished and Af minor chords ................................................................... 78 Figure 4.8: Middleground, Mahler’s Seventh Symphony, first movement, mm. 1-50................. 80 Figure 4.9: Section 3 middleground and the ‘fourths’ motif from Mahler’s Seventh Symphony, first movement, mm. 32-49........................................................................................................... 82 Figure 4.10: Harmonic frequency ................................................................................................. 83 Figure 4.11: First group from Mahler’s Seventh Symphony, first movement, mm. 50-57 .......... 86 Figure 4.12: First group continued from Mahler’s Seventh Symphony, first movement, mm. 5864................................................................................................................................................... 87 Figure 4.13: Middleground, Mahler’s Seventh Symphony, first movement, mm. 50-134........... 88 Figure 4.14: Middleground, Mahler’s Seventh Symphony, first movement, mm. 135-245......... 92 Figure 4.15: Middleground, Mahler’s Seventh Symphony, first movement, mm. 247-317......... 93 Figure 4.16: Middleground, Mahler’s Seventh Symphony, first movement, mm. 328-410......... 96 Figure 4.17: Middleground, Mahler’s Seventh Symphony, first movement, mm. 413-478......... 99 Figure 4.18: Middleground, Mahler’s Seventh Symphony, first movement, mm. 478-508....... 102 Figure 4.19: Middleground, Mahler’s Seventh Symphony, first movement, mm. 510-515....... 103 Figure 4.20: Middleground, Mahler’s Seventh Symphony, mm. 466-512 ................................. 104

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Figure 5.1: Middleground, Mahler’s Seventh Symphony, Finale, mm. 1-70 ............................. 111 Figure 5.2: Middleground, Mahler’s Seventh Symphony, Finale, mm. 79-135 ......................... 111 Figure 5.3: Middleground, Mahler’s Seventh Symphony, Finale, mm. 136-218 ....................... 113 Figure 5.4: Middleground, Mahler’s Seventh Symphony, mm. 220-290 ................................... 116 Figure 5.5: Middleground, Mahler’s Seventh Symphony, mm. 291-351 ................................... 118 Figure 5.6: Middleground, Mahler’s Seventh Symphony, Finale, mm. 360-475 ....................... 120 Figure 5.7: Middleground, Mahler’s Seventh Symphony, Finale, mm. 476-522 ....................... 122 Figure 5.8: Middleground, Mahler’s Seventh Symphony, Finale, mm. 522-590 ....................... 125

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CHAPTER 1 INTRODUCTION 1.1

The Problem The topic of this dissertation grew out of a question: what is the key of Mahler’s Seventh

Symphony (1904-1905)? The Dover edition makes the simultaneously bold and yet indecisive claim that the symphony is in “B Minor / E Minor / C Major”; 1 it seemed that answering this question would be difficult. The latest critical edition does not attempt to assign a key for the whole work and instead leaves the title as “Symphonie Nr. 7,” which correlates with the original manuscript. 2 Of course, key designations on title pages appear to be largely a tradition of cataloging practice – a handy way to differentiate one symphony from another – and hardly a significant indicator of a symphony’s structure. Still, we must be aware of any potentially subliminal influence that title key-designations might impose on our analytical decisions. When a symphony is described as being in a key, then it would seem that an assumption is made that this multi-movement work exists within the confines of a single, global key. But is that the case? It is easy enough to understand that a modulation to a closely related key for a middle movement can be read in the context of the global key; however, more adventurous key regions could pose a challenge. The most difficult task is to account for multi-movement structures in which the first and last movements are in different keys. In these cases, the hierarchical authority of a single global key is at best challenged and at worst non-existent. A piece that exists within a unified, closed structure is less susceptible to global-key problems. If such a piece begins and ends in a different key, then a reasonable explanation is that

1

Gustav Mahler, Symphony No. 7 (New York: Dover Publications, Inc., 1992), iii.

2 Gustav Mahler, Symphonie Nr. 7, ed. Reinhold Kubik (Berlin: Boosey & Hawkes, 2012); my analyses and graphs are derived from this edition.

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it falls under the category of an auxiliary cadence. 3 For example, Chopin’s Prelude Op. 28, No. 2 begins on an E minor chord, which is understood as VI in the context of the first cadence on G major. However, neither E minor nor G major are the global tonic; the prelude ends on the actual global tonic of A minor. On a deeper level, the opening E minor chord is retrospectively understood as V of A minor and G major as III of V, which is how Heinrich Schenker reads the prelude (Fig. 1.1). 4 This reading is logical because each prolongation unfolds organically and structurally uninterrupted, and can be understood as functions in relation to the tonic; additionally, the order of the preludes in Op. 28 informs us that the tonic of Prelude No. 2 is undoubtedly A minor. 5 These analytical advantages do not extend to multi-movement works where different prolongations can be closed off from one another and there is no preordained, definitive tonic. The only real guideline by which one can gauge the tonal hierarchy in a multimovement work is the establishment of one primary key in the first movement and its definitive return in the Finale. When a multi-movement work falls outside of this design, then the analyst must rely on other less objective criteria to determine a tonal hierarchy. In these cases, a feasible conclusion is that the true tonic arrives in the finale and the other prolonged key of the first movement is part of a large-scale auxiliary cadence. Figure 1.1: Schenker’s analysis of Chopin’s Op. 28, No. 2

3

Heinrich Schenker, Free Composition, trans. Ernst Oster (New York: Longman, 1979), 88–89.

4

Schenker, Free Composition (Fig. 110.3).

5

The key of each prelude is not included in the titles in the original manuscript, only the number. However, the inclusion of the key in the title for each prelude is very common, which, given how a collection of preludes is organized, seems to be a safe conclusion.

2

However, some multi-movement structures that begin and end in different keys, such as Mahler’s Seventh Symphony, can be better understood as operating within the confines of a different tonal paradigm rather than relegated to an auxiliary cadence. My dissertation explores a new model that can reconcile multi-movement works that otherwise challenge a monotonal reading. The multidimensional musical object allows the analyst to expand the traditional tonic from a single triad into a nexus of interconnected triads. These additional tonal dimensions help to simplify the complicated harmonic language of late nineteenth-century composers by incorporating more chords within a single function – or a single Stufe for Schenkerian analysis. This model is informed by the music itself, as presented by composers like Mahler, and builds upon a number of other methodologies that have grappled with this species of musical structure.

1.2

Literature Review Some notable methodologies have been developed to address the issue of conflicting

tonic identities in multi-movement structures. The term progressive tonality was first proposed by Dika Newlin in her book Bruckner, Mahler, Schoenberg, although no real methodology is laid out. 6 Rather, it is merely a way to point out that some of Mahler’s musical structures appear to belie the traditional design of a single, global tonic (what Newlin refers to as concentric tonality) that bookends the work; that the hierarchical authority is at some point transferred from one key to another. Some of her applications of progressive tonality are in fact better understood as auxiliary cadences or otherwise within the framework of one global tonic. For example, she labels the first movement of Mahler’s Seventh Symphony as progressive since it begins in B minor yet ends in E minor/major. 7 However, this reading would assume that B minor is 6

Dika Newlin, Bruckner, Mahler, Schoenberg (New York: W. W. Norton & Company, Inc., 1978), 129.

7

Newlin, Bruckner, Mahler, Schoenberg, 186.

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hierarchically equal to E minor/major, when a more nuanced analysis would understand that the B minor prolongation ultimately functions as V of E. On the other hand, the applicability of progressive tonality to the entire symphonic structure is a more feasible proposition considering that it shifts from E minor/major to C major; E minor could serve as III of C in an auxiliary fashion, but the emergence of E major at the end of the movement begins to blur that diatonic relationship. Since Newlin does not qualify progressive tonality beyond a basic definition it is difficult to fully unpack its potential or specific meaning. It is also possible that her concept is not concerned with issues of hierarchy, and rather is simply categorical in nature. A number of other scholars have developed methodologies that attempt to address the phenomenon marked by Newlin’s term. Deborah Stein explores more deeply the concept under the moniker directional tonality. 8 She defines the process in the following passage: Directional tonality uses two different keys in the following way. One key functions as an opening tonality; and after the first key is clearly established as a tonic, a transformation occurs whereby the initial tonic becomes a nontonic function within a second tonality. The piece then concludes in the second key. The ultimate effect of directional tonality is twofold: first, the original tonality loses its identity as a tonal focus in deference to the second tonality; and second, the piece is heard as beginning and ending in two different keys. 9 This approach stands in stark contrast to Schenker’s concept of the auxiliary cadence, which retrospectively understands the opening key as a function of the structural tonic that closes the piece. Stein argues that an auxiliary cadence reading does not address the nuanced relationship between the opening and closing keys – nor the transformation from one to the other – in such pieces, particularly in works where key shifts are used to emphasize text. In other words, a hierarchical bias towards one key could impact the interpretation of the other. Rather than one 8

See Deborah Stein, Hugo Wolf’s Lieder and Extensions of Tonality (Ann Arbor, Michigan: UMI Research Press, 1985), 143–49. Stein attributes the term to Robert Bailey “in his lectures at Yale University and the Eastman School of Music,” 228.

9

Stein, Hugo Wolf’s Lieder, 143.

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superior tonic, “the overriding factor is the coexistence in directional tonality of two equally weighted [my emphasis] tonal centers within one musical work.” 10 In order to further differentiate directional tonality from a traditional auxiliary cadence reading, Stein also examines Chopin’s Op. 28, No. 2. Her reading emphasizes the tonally ambiguous beginning, which she posits can be understood as in either E minor or G major. Thus, the prelude has a tonal pairing between E minor/G major and A minor; that is, the piece has two, or possibly three, equally weighted and real tonics. However, as previously noted, the tonic designation of this prelude is very clearly A minor because of the very order of the preludes. There also is no text on which to base an argument whereby E minor or G major are poetically intended as equals to A minor. We must then ask what exactly is gained in a directional tonality scenario that would validate such an interpretation, which seems to be structural in nature. However, despite the claim of structural equality between the different keys, Stein’s conclusion is more appropriately phenomenological: [T]he difference between the Schenkerian reading and the directional tonal reading is that while both agree that the opening E minor can be understood retrospectively as a minor V, the directional tonal reading does not define the initial function of E minor as that of dominant. 11 In other words, the listener does not experience the opening E minor as a dominant function, which is a fair assessment in the given context of the piece. But experience alone does not alter the underlying structure or tonal hierarchy. The issue with Stein’s methodology is that its claims reach beyond its conclusions. She attempts to distinguish directional tonality as structurally distinct from the Schenkerian method, but her description comes across as vague and even contradictory. Consider the two following statements: “[In directional tonality] the opening

10

Stein, Hugo Wolf’s Lieder, 145.

11

Stein, Hugo Wolf’s Lieder, 149.

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ultimately yields to the closing tonality”; and “In Schenkerian analysis the opening is ultimately subsumed by the closing [tonality].” 12 The only difference between her definitions of directional tonality and an auxiliary cadence reading, then, is the interpretation of the words yield and subsumed. “Subsumed” would seem to accurately depict an auxiliary cadence reading where one tonic is hierarchically superior. “Yields,” on the other hand, does not, in my estimation, evoke structural equality in a tonal pairing; instead, it suggests that the opening tonal region is by default weaker than the closing tonic. Rather than argue the need for a new methodology, she has conflated directional tonality with an already well-established analytical tool. Directional tonality may shed some phenomenological insight into pieces with text, but its structural claims are redundant and needlessly confusing with the already well-established concept of the auxiliary cadence in Schenkerian analysis. Christopher Lewis finds the concept of progressive tonality to be problematic: [I]t is gravely misleading, since to distinguish between “progressive” and “concentric” pieces solely according to the beginning and ending keys is to imply that Mahler’s music alternates, apparently at random, between two tonal languages. It is internal syntactical relationships which define a musical language, and while progressive tonality requires a violation of certain of the rules of common practice, a violation of those rules need not necessarily produce a progressive background. 13 Lewis is concerned that such “progressive” designs may be better understood as auxiliary cadences, although he does not state that explicitly. He argues that, It is quite clear from the Schenkerian definition of common-practice tonality that the identity of beginning and ending keys arises from a syntactical imperative rather than from the composer’s choice. A common-practice piece does not just begin and end in the same key, is not simply “in” a given key, but expresses a single tonic triad. 14

12

Stein, Hugo Wolf’s Lieder, 145.

13

Christopher Lewis, Tonal Coherence in Mahler’s Ninth Symphony (Ann Arbor, Michigan: UMI Research Press, 1984), 2. It is possible that Lewis is likewise referring to directional tonality since he studied with Robert Bailey, but that point is never made clear.

14

Lewis, Tonal Coherence, 2.

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Syntax provides information through order and arrangement; thus, in one aspect of musical syntax, a great deal of information is revealed by the structural bookends of a piece. The inherent hierarchical authority of the tonic triad is made apparent because it is placed in these significant syntactical locations. In other words, tonic assignment is not random but a logical result of syntactic emphasis. For pieces in which the beginning and ending keys are different, then a hierarchical relationship should be determinable based on harmonic syntax. To return to Chopin’s Op. 28, No. 2 as an example, the progression from E minor to A minor as V to I is a more appropriate assessment within the common-practice tonal syntax than is I to IV – although an off-tonic ending is not out of the realm of possibility. However, Lewis argues that these hierarchical relationships collapse when the beginning key is disproportionately more weighted than the closing key. As an example, the “deceptive beginning” of the Finale of the Mahler Second Symphony occupies more than 90 percent of the movement, since the earliest point at which the Ursatz of Ef could be taken as beginning is m. 696 (of 764). It is true that the elements of the Ursatz are important because of function and not duration, and that the Ursatz itself is an abstraction, but an abstraction so far removed from one’s perception of the piece seems of limited value. 15 For these pieces he instead adopts Robert Bailey’s double-tonic complex model. 16 Lewis finds that this new methodology is necessary because music from the “post-Wagnerian tonal tradition” largely uses a new syntax that involves paired tonics. 17 The double-tonic complex model proposes that such a piece is structured around two equal tonics that are usually a third apart that often, but not always, manifest on the musical surface as a poly-sonority, sometimes even at important structural events. For example, Bailey’s analysis of the Prelude to Wagner’s Tristan 15

Lewis, Tonal Coherence, 3.

16

See Robert Bailey, “An Analytical Study of the Sketches and Drafts,” in Prelude and Transfiguration from Tristan and Isolde, ed. Robert Bailey (New York: W.W. Norton, 1985), 121–122. 17 Christopher Lewis, “Mirrors and Metaphors: On Schoenberg and Nineteenth – Century Tonality,” in Music at the Turn of Century, ed. Joseph Kerman (Berkeley: University of California Press, 1990): 21.

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und Isolde suggests that the piece is structured around a complex of the chromatic modes of A and C, and that this double-tonic complex manifests in the music as the sonority A-C-E-G – which can be understood as a superimposition of A minor and C major. 18 He states that, “this complex serves as the controlling tonic for the entire first act, which both begins (in the Prelude) and ends with the A/C complex.” 19 However, Bailey’s description and application of the model is brief and as a result is vulnerable to several concerns about the validity of such an approach. Matt BaileyShea in response raises the following questions: In what sense is the complex a “controlling tonic”? Is it a surface tonic sonority? Can it be prolonged like a traditional tonic? Does it function as the goal of directed linear and harmonic progressions? Bailey suggests that the answer is “yes,” but he presents the idea in a brief, ad hoc manner, leaving no guidelines for the general case. 20 These points are valid and are crucial elements for any model that claims to alter what can be a tonic. BaileyShea identifies three categories of double-tonic complex applications and ranks them according to their level of controversy: Category 1: Tonal Pairing A piece consistently vacillates between two keys, usually third related, often with dramatic/associative connotations. This is quite common in nineteenth-century music. Category 2: [Double Tonic Complex] (motivic) The piece exhibits tonal pairing as in category one, but with a crucial addition – the pairing of the two keys acts as an abstract motive, which manifests in a variety of idiosyncratic gestures: one tonic appearing in place of the other; ambiguous passages that could be interpreted in either key; striking, dissonant harmonies generated by the conflation of two different tonic triads, etc. This is somewhat rare in nineteenth-century music.

18

Bailey, “An Analytical Study,” 121–125.

19

Bailey, “An Analytical Study,” 122.

20 Matt BaileyShea, “The Hexatonic and the Double Tonic: Wolf’s ‘Christmas Rose,’” Journal of Music Theory 51, no. 2 (Fall 2007): 193.

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Category 3: [Double Tonic Complex] (structural) A piece exhibits the same features as category two, but in this case the conflation of tonic triads is not simply a motivic possibility – it operates as a prolonged tonic sonority, one that contains at least four constituent pitch classes. This is extremely rare in nineteenth-century music; arguably impossible. 21 In his own application of the double-tonic complex to Wolf’s Auf eine Christblume I and II, BaileyShea prefers the motivic emphasis of category two. He states that: This [motivic] interpretation does not weaken Bailey’s concept; on the contrary, it is precisely the opposite possibility – that double-tonic complexes operate as structural, “controlling tonics” in nineteenth-century music – that has most weakened the idea. By adopting a looser, motivic approach with regard to the double-tonic complex, we free it from the dead ends that it appears to have reached in prior research. 22 BaileyShea, perhaps because of Robert Bailey’s insufficient explanation of the model, finds the structural interpretation of the double-tonic complex to be misguided; that the current model is not strong enough to challenge the more traditional single-tonic background. In other words, extraordinary structural claims require extraordinary evidence. Christopher Lewis attempted to refine the methodology of the double-tonic complex in his analyses and laid out what he considered to be its most signal characteristics in a composition: 1. Juxtaposition of musical fragments implying the two tonics in succession or alternation. 2. Mixture of the two tonalities, exploiting ambiguous and common harmonic functions. 3. Use of a tonic sonority created by conflation of the two tonic triads. 4. Superposition of lines or textures in one key upon those in another. 5. Some combination of the above. 23 With these guidelines he is then substantially more able to utilize the double-tonic complex. He 21

BailyShea, “The Hexatonic and the Double Tonic,” 195.

22

BailyShea, “The Hexatonic and the Double Tonic,” 208–209.

23

Lewis, Tonal Coherence, 6.

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also expands the possible complex by incorporating the upper and lower thirds of both modes. For example, in his analysis of Mahler’s Ninth Symphony the principal tonic pitch of the first movement, D, is paired with B (from the major mode) and Bf (from the minor mode); and those tonal regions are further expanded by their lower thirds: Gs and Gn, and Gn and Gf, respectively. 24 The Finale’s tonic pitch, Df, is similarly derived as the minor upper-third of Bf. However, Lewis finds more convincing that the Df centered Finale is a result of a transposition of the “motto progression” that involves descending major thirds: D-Bf-Gf from the first movement, which becomes Df-A-F in the Finale. 25 As a result, Lewis seems to weaken the structural impetus of the double-tonic complex, and that the Df centered Finale is the consequence of other forces. Additionally, the expansion of the tonic complex to all possible third relationships, and not discerning any hierarchy between them, trivializes the tonic function – the complex becomes dangerously close to arbitrary. Graeme Downes in his dissertation on Mahler’s symphonies instead focuses exclusively on major third relationships. 26 His analytical model, axial tonality, relies more on the concept of progressive tonality than the double-tonic complex, although there are certainly some influences from the latter. In essence, pitches related by a major third are on an axis; members from the same axis share a number of voice-leading characteristics and, under the right circumstances, can substitute for one another in harmonic and structural functions. Altogether there are four axes within any tonal context: the subdominant axis, the tonic axis, the dominant axis, and the supertonic axis. For example, a piece in C major can have a clearly established tonic on the C major 24

Lewis, Tonal Coherence, 9.

25

Lewis, Tonal Coherence, 103–104.

26

Graeme Alexander Downes, “An Axial System of Tonality Applied to Progressive Tonality in the Works of Gustav Mahler and Nineteenth – Century Antecedents” (PhD diss., University of Otago, Dunedin, New Zealand, 1994).

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chord, but can easily shift the tonic authority to either E or Af – and either temporarily or permanently. In order to distinguish between the possible tonics, and maintain some sense of structural hierarchy, Downes labels the “true” tonic as I (C, in this example), the upper third tonic (E) as Iα, and the lower third tonic (Af) as Iβ. Accordingly, the functions of each tonic are labeled by the same method: G major as V, B major as Vα, and Ef major as Vβ. The fluidity between axis members is derived largely from the multifunctionality of the augmented triad. To continue the previous example, the sonority created by a combination of the dominant-axis members Ef-G-B is an enhanced dominant that could logically resolve to any tonic-axis member – it only requires a reinterpretation of the perceived root. In Downes’ model, there are two types of progressive tonality: progression within the same axis or progression to a different axis (trans-axial progression). Mahler’s Seventh Symphony would fall within the first category since the progression E minor to C major involves members from the same axis. The second category can be best understood as an auxiliary cadence, in which the initial tonic is retrospectively understood as a function of the real tonic that arrives at the structural close – but it is perhaps more flexible regarding the functional relationship between keys. For example, Mahler’s Fifth Symphony employs a trans-axial progression from Cs minor to D major; at the end of the symphony, and within the context of a D major tonic, Cs minor from the first movement is understood as Vα – a progression from the dominant axis (A-Cs-F) to the tonic axis. In both categories, the true tonic is found at the structural close of the finale; even when the progression involves members from the tonic axis, as in Mahler’s Seventh, only the final tonic has real hierarchical authority. To put it more simply, in Downes’ model E minor is not equal to C major in the Seventh Symphony, but rather is ultimately subservient to it. Unfortunately, Downes deliberately avoids any Schenkerian graphic

11

representations of his analyses which makes it difficult to estimate his methodology’s effectiveness. He instead provides general key schemes and their contextual functions for each movement, and the reader is given little evidence to confirm the reading. Although the topic is never mentioned in his dissertation, Downes’ model is uncannily similar in many aspects to transformational models, specifically the hexatonic cycle. The hexatonic cycle is a collection of six triads related by P (parallel) + L (leading tone) transformations, and, as a result, the three roots are a major third apart: C major, C minor, Af major, Af minor, E major, and E minor. 27 Of course, current transformational models are less concerned with function and instead focus on smooth, efficient voice leading in highly chromatic passages that might otherwise challenge traditional analytical techniques. For example, a cycle of chromatic mediants (a hexatonic cycle) may be difficult to reconcile in a tonal context, but the voice-leading mechanics are made comprehensible as a pattern of transformations. Downes, whether he was exposed to transformational ideas or not, picked up on these hexatonic relationships and unified them into his four tonal axes, upon which a function could be applied. A few years later in his article “As Wonderful as Star Clusters: Instruments for Gazing at Tonality in Schubert,” Richard Cohn proposed a methodology that resonates closely with Downes’ axial tonal model. Instead of an axis on which major-third related triads are placed, triads a major third from traditional functional roles – like the tonic – are considered to be in the same region. 28 Cohn does not suggest the same type of substitutability as does Downes, but rather that the regions offer non-diatonic ways in which diatonic triads can be prolonged. For example, in his analysis of Schubert’s Bf Major Piano Sonata, D. 960 the Bf tonic and F 27

Richard Cohn, Audacious Euphony: Chromaticism and the Triad’s Second Nature (New York: Oxford Press, 2012), 17–24. 28 Richard Cohn, “As Wonderful as Star Clusters: Instruments for Gazing at Tonality in Schubert,” 19th – Century Music 22, no. 3 (Spring 1999): 213–232.

12

dominant are both prolonged, in their respective sections, by keys a major third apart. 29 For Cohn, the tonal authority of Bf is never truly relinquished to its major-third counterparts; his primary concern is not necessarily function, but the semitonal system that guides these regions. [R]ecognizing consonant triads, authentic cadences, and monotonal sonata forms in Schubert’s music, we naturally interpret them in familiar tonal and diatonic terms. Encountering now syntactic combinations that resist adequate interpretation in those terms, we might surmise that the music is disjunct, purple, arbitrary, aimless, disunified, unstructured, incoherent, indeterminate, or coloristic. The “music itself” is now left behind as inaccessible to further analysis, its place occupied by an interchangeable set of hypostatized ascriptions, which might serve as a basis for speculation on the cultural features that these qualities reflect and that the music thereby inscribes. The approach developed in this article suggests that Schubert’s chromatic idiosyncracies are not arbitrary, aimless, or indeterminate by mere virtue of their irreconcilability to diatonic tonality. Some of them simply adhere to an alternative mode of determination…. 30 The alternative mode is the transformational model of the hexatonic cycle, which shows that these major-third related triads have a kind of kinship through shared pitches and semitonal voice leading. Cohn never goes so far as to say that each member of the hexatonic cycle is tonic or can be tonic, but rather that they operate as prolongational methods outside of the diatonic system. Another concept that is important for this discourse is Edward Laufer’s primary or referential sonority. For Laufer, post-tonal prolongation is possible in pieces in which a primary sonority, in place of a major or minor tonic triad, is emphasized through rhetorical or gestural compositional features. 31 In a 2003 interview with Stephen Slottow, Laufer provides some additional insights:

29

Cohn, “As Wonderful as Star Clusters,” 218–219.

30

Cohn, “As Wonderful as Star Clusters,” 232.

31

Edward Laufer, “An Approach to Linear Analysis of Some Early Twentieth – Century Compositions,” in A Composition as a Problem IV: Proceedings of the Fourth International Conference on Music Theory, ed. M. Khumal, 89–134 (Tallinn: Eesti Muusikaakadeemia, 2004), 133–134.

13

The thing is that one simply cannot have motives, or even complex motivic features just floating around on nothing…. [The primary sonority is] not going to be a major triad and it’s not going to be a minor triad, it’s going to be some other sort of sonority, and it is difficult sometimes. But if it’s too difficult, then it becomes something of an academic exercise. I think that if it doesn’t have a certain kind of directness, if it’s not in some way – I hate to use the word obvious, but that’s what I should say, perhaps – then it’s wrong. It’s not the primary sonority. It really has to stand out as such…. [The primary sonority] might be arpeggiated throughout the piece. Or it might move to such a sonority. Up to now I believe that it occurs at the ending of a piece. 32 Thus, the primary sonority facilitates the compositional roles typically filled by a tonic sonority: the establishment and development of motivic material, goal-oriented cadential or structural weight (particularly at the end of a piece), and prolongation at the middle and background levels. But the primary sonority must be apparent, or obvious as Laufer states it. Its hierarchical status should be a logical conclusion of a thorough analysis. All of the methodologies so far discussed have sought to explain music that appears to challenge the traditional monotonal model, especially those that begin and end in different keys. Each method brings something useful to the discourse: directional tonality argues that music that begins and ends in different keys should not necessarily be confined within the bounds of monotonality; the double-tonic complex suggests that two triads can have equal authority in a hierarchical model; axial tonality provides a way to sharpen the different types of progressive tonality and explores the substitutability of major-third related triads; transformational models and the hexatonic cycle further elucidate the mechanics of major-third related triads and offer a non-diatonic alternative to prolongation; the primary sonority expands what can be considered fundamental to a piece beyond a major or minor triad, and establishes some guidelines for post-

32 Stephen Slottow, “An Interview with Edward Laufer,” in Explorations in Schenkerian Analysis, ed. David Beach and Su Yin Mak, 328–348 (Rochester, NY: University of Rochester Press, 2016), 342.

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tonal prolongational techniques. Although I find certain aspects of each method to be useful, none of them are comprehensive enough to explain satisfactorily the tonal structure of Mahler’s Seventh Symphony and other works like it. The vocabulary of Schenkerian analysis is likewise too limited when it comes to musical works that challenge a monotonal reading. Outside of an auxiliary cadence reading, the analyst is forced to reconcile all harmonic and structural oddities, such as those that are so prevalent in the works of Mahler and other late-nineteenth-century compositions, within the confines of a single, global tonic triad. Instead, I propose a model that synthesizes these methodologies with the Schenkerian model: the multidimensional musical object.

1.3

Multi-Movement Considerations Before delving into my methodology, there is another issue that I must address: the

relevance of a global tonic to a multi-movement work. We find a possible answer in looking back to the suite, one of the earliest instrumental multi-movement genres. In his book The Courtly Consort Suite in German-Speaking Europe, 1650-1706, Michael Robertson unpacks the history of the suite and its true defining characteristics. The conclusion of chapter three provides the best summary: Seventeenth-century court suites were sequences of movements - mostly dances - which were united by a common key centre. With the exception of the bransle suite, the contents of these sequences varied widely and were subject to the whims of copyists or the performing musicians. It is unlikely that either scribes or members of the German Hofkapellen ever gave a moment’s thought to a ‘classical’ order. 33 Throughout the chapter he argues that the evidence discloses that the precise order of movements was not a defining factor for what constituted a suite; that, “the so-called ‘classical 33

Michael Robertson, The Courtly Consort Suite in German – Speaking Europe, 1650–1706 (Burlington, VT: Ashgate Publishing Company, 2009), 64.

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order’ was imposed upon the suite, presumably in an attempt to give it a readily identifiable sonata-like hierarchy and structure.” 34 Instead, the main identifiable characteristic of a suite that is confirmable is that each movement is united by key. This conclusion is corroborated through a contemporary musicologist of the seventeenth-century suite: The most important part of [Brossard’s] definition lies in what follows. He states that the movements of a suite are ‘tout cela compose sur le même Ton ou Mode’ (all composed in the same key or mode). Brossard emphasizes ‘le même Ton ou Mode’ by using the phrase twice in the same definition. And with the exception of Austria, and particularly the imperial court of Vienna, the same is true for nearly every suite written in the German lands. 35 The evidence suggests that what crystallizes the otherwise disparate movements of a suite – which in some cases could come from different sources and composers – into a coherent whole is the continuity of key. As one of the earliest instrumental multi-movement models, we can surmise that this genre-defining element of the suite influenced its compositional successors, like the symphony. The weight of coherence gradually shifted from total unity of key to other elements, such as the specific order of movements (the ‘classical order’ to which Robertson refers) and a greater sense of global thematic connection. This shift in the organic design freed composers from the necessity to relegate each movement within the confines of a single key. As a result, the interior movements, as in the early symphonies of Haydn, could be composed in closely related keys, most commonly the keys of the dominant and subdominant. But the exterior movements remained bound to the global key, like structural bookends that held the work together. Haydn explored the option of a modal shift on the global tonic in his Symphony No. 45, which begins in

34

Robertson, The Courtly Consort Suite, 45.

35

Robertson, The Courtly Consort Suite, 58; see also Brossard, [?] de, Dictionaire de Musique, contenant une explication des termes Grecs, Latins, Italiens, & Françios les plus usitez dans la Musique (Paris, 1703; facsm. F. Knuf (ed.), Hilversum, 1965).

16

Fs minor but ultimately concludes in Fs major. Beethoven famously employed this modal-shift design into his Fifth and Ninth Symphonies and on a much grander scale. It is difficult to find evidence that composers were keenly concerned that a multimovement work begin and end in the same key. The largest cache of evidence is the repertoire itself, in which the overwhelming practice is to begin and end in the same key; those modally fluid compositions were exceptional because they diverged from that tradition. Mahler’s symphonic oeuvre is likewise exceptional because it repeatedly belies that common practice. We must assume that Mahler was aware of those symphonic traditions and that his multi-movement designs were informed by its most salient characteristics. One could conclude that Mahler deliberately avoids or abandons tradition, but my analysis provides evidence to the contrary.

1.4

Glossary of Terms •

Dominant of opposition – A dominant that prevents a true structural close.

•

Functional entanglement – A compositional paradox in which two or more harmonic

functions (tonic, subdominant, or dominant) occupy the same space and obscure middleground or background structures. •

Identity crisis – The harmonic event in sonata form in which the oppositional

harmony confirms its separation from the tonic and becomes a background Stufe. •

Identity narrative – A series of three expected harmonic events in sonata form

(identity schism, identity crisis, and identity reclamation). •

Identity reclamation – The harmonic event in sonata form in which the tonic wrests

control of the background structure and regains its status as a background Stufe.

17

•

Identity schism – The harmonic event in sonata form in which the oppositional

harmony initiates a separation from the tonic. •

Multidimensional musical object (MMO) – A higher-dimensional sonority that is a

nexus of two or more chords. •

Oppositional harmony – A harmony that establishes itself as a background Stufe and

challenges the tonic’s status. •

Rotation – A means of composing-out a multidimensional music object in which it is

shifted from one side to another.

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CHAPTER 2 THE MULTIDIMENSIONAL MUSICAL OBJECT 2.1

Introduction The multidimensional musical object (MMO) can best be thought of as an expansion of

the triad, but not in the sense of a tertian extension like a 7th or 9th. Rather, a MMO is a more complex set of pitches that functions similarly to a triad as a compositional component. In the Schenkerian model, the triad is what undergoes all of the processes of diminution, prolongation, composing-out, etc.; the Ursatz is a temporal expression 36 of the tonic triad. My model suggests that a MMO can substitute for a triad in these traditional compositional roles. The MMO undergoes the processes of diminution, prolongation, composing-out, etc.; the Ursatz is a temporal expression of the tonic MMO. For all intents and purposes, my model provides new possibilities for Schenkerian analysis. Because the Schenkerian graphing method was derived around a traditional tonic and traditional conceptions of harmonic diminution, some adjustments were necessary to adapt it for multidimensional harmony. The unfolding sign is used when a multidimensional musical object is rotated (the concept of rotation is explored in section 2.3). In the case of three-dimensional musical objects, a “double unfolding” sign can be employed to show multiple rotations of the same object. The unfolding, or double unfolding, helps to clarify deeper contrapuntal structures in that each multidimensional musical object belongs to one Stufe.

2.2

One-Dimensional Musical Objects In order to understand the MMO properly, we must first reassess the triad within this new

36 I opt to use the term “temporal” in place of “horizontal,” what normally appears in Free Composition, so as to avoid any confusion with geometric terminology; see Schenker, Free Composition, 4.

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model. The Ursatz is a temporal expression of the tonic triad: the upper voice (Urlinie) is defined by the passing motion to 1 from either 3, 5, or 8; the lower voice (Bassbrechung) arpeggiates from the fundamental tone to its upper fifth and returns to the fundamental tone, or the “sacred triangle.” 37 Each point within the contrapuntal structure is derived from the tonic triad, either directly or indirectly – the 2 is not a direct member of the tonic triad, but supports a composingout of tonic triad’s upper fifth (V) and fills in the passing motion between 3 and 1. Although the tonic triad undergoes the various transformations into background, middleground, and foreground, the elements of its identity remain intact and unchanged – its constituent members realign vertically at the structural close. To invoke the geometrical metaphor, we could say that the identity of the tonic triad is one-dimensional; rather than a multidimensional musical object, it is a unidimensional musical object. In a mathematical sense, a dimension is a property of space – an extension in a given direction. An object in one dimension is measured by one set of coordinates; therefore, only one type of object is possible in one dimension: a line. We can imagine, then, that a major or minor triad is equal to a line. The coordinate of the first dimension of musical objects, or dimension X, defines the position of the fundamental tone; for the purposes of this model, the fifth and third, be it major or minor, are considered to be static in a one-dimensional musical object. A one-dimensional tonic built on C can generate one type of triad (major or minor), but cannot shift the position of its X coordinate without fundamentally changing its identity. In other words, if the X coordinate of a one-dimensional tonic object is C and it moves down a half step to B, then it is no longer the same object. A special explanation would be necessary for a piece that begins on C and ends on B – those events are fundamentally different.

37

Schenker, Free Composition, 15.

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To unpack this geometry metaphor fully – and to be able to explore the higherdimensional musical objects – we have to define the musical space, or what can be heard at any one moment in a piece of music. To be sure, all twelve pitches can sound at once either as a cluster or spread out over varying ranges, but any arrangement will sound dissonant. On the other hand, a major or minor triad is – at least in the tonal system – the model of consonance. In order to understand the higher-dimensional musical objects let us define the musical space as one-dimensional, where a one-dimensional musical object (i.e., a major or minor triad) sounds consonant. A higher-dimensional musical object can be heard in the one-dimensional musical space, but since it is losing at least one dimension in the process its true shape is distorted and it will sound dissonant. Figure 2.1 provides an analogy with visual dimensions that will help better to elucidate this phenomenon. The first shape is a two-dimensional square represented on a two-dimensional surface: the page. Each angle of the square appears correctly as 90° – there is no distortion because the number of dimensions of the shape matches the dimensions of the medium. The second shape, a three-dimensional cube, is distorted since an actual cube would be comprised completely of 90° angles. This cube, however, is compressed into only two dimensions and, as a result, some of the angles are acute or obtuse in order to give the illusion of depth – the dimension missing from the medium. The four-dimensional hypercube is even further distorted because this projection is missing two dimensions. 38 Every angle of the hypercube should also be 90°, but the majority of its angles are compromised in this two-dimensional shadow. We can think of the square as visually consonant, the cube as visually dissonant, and the hypercube as

38

A hypercube is a theoretical object that could exist in a four – dimensional space. It is the four – dimensional counterpart to the three – dimensional cube. In the same way that a cube has six squares for its sides, a hypercube has eight cubes for its sides. A three – dimensional “side” does not make sense in our three – dimensional world, hence it is only theoretical.

21

even more visually dissonant. In other words, the more dimensions an object contains outside the dimension limit of the medium, then the more dissonant it is perceived. To experience an object that contains more dimensions than the medium in which it is represented, we must rotate the object. The principle is essentially the same for higher-dimensional musical objects when confined within the one-dimensional musical space. Below I will further elaborate the process of “rotation” in musical space in the discussions of two- and three-dimensional musical objects. Figure 2.1: Two-Dimensional visual space metaphor

Figure 2.2: One- and two-dimensional musical objects

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2.3

Two-Dimensional Musical Objects The second dimension of a two-dimensional musical object, dimension Y, defines the

position of the third (Fig. 2.2). A two-dimensional musical object encompasses both the major and the minor third; they rest upon the Y axis and, depending on the orientation of the object, either one or both can appear on the musical surface. It is most common for the two sides to appear separately (e.g., C minor and then C major), but the entire object can be compressed onto the musical surface (a conflation, or superimposition, of C major and minor). This would, of course, sound dissonant, just like the example of the three-dimensional cube in Figure 2.1 that is squeezed into only two dimensions; we are able to perceive its true shape, but its geometrical balance is lost in the medium. Since a two-dimensional musical object does not consonantly fit onto the musical surface it is typically composed out through a rotation. Here the term rotation is used literally, in that the musical object is rotated around a specific axis and, as a result, a different side appears in the musical space. The concept of rotation is perhaps the biggest philosophical difference between my model and the transformation model. If a piece of music shifts its harmony from C major to C minor, then the transformation model would say that a P (parallel) transformation occurs – that the first harmony, C major, has actually transformed and become something different: C minor; that they are two separate entities. My model, on the other hand, suggests that C major and C minor can be two parts of the same object and, at the deepest level, would exist simultaneously, like the two sides of a coin. Therefore, a two-dimensional musical object is most often composed out through a Y rotation; a piece begins on one side of the two-dimensional musical object and then rotates around the X axis to allow the other side to be seen. In this way, the entire twodimensional musical object is experienced, but in one-dimensional slices.

23

That a piece can encompass the parallel major and minor modes of one fundamental tone is not a new concept; modal mixture is taught early on to all theory students. But in these cases, tonic assignment is typically one-dimensional in that hierarchical authority is granted to one mode over the other – typically the major mode is privileged. Others have argued that the distinction between modes is less fruitful, particularly in the late nineteenth century. For example, Robert Bailey suggests: The terms major and minor remain useful, of course, but only for the purpose of identifying the qualities of particular triads. When we want to identify the tonality of large sections, or that of whole pieces or movements, it is best simply to refer to the key by itself and to avoid specifying mode, precisely because the “chromatic” or mixed major-minor mode is so often utilized. By extension, the sense that a passage from a piece, or an entire movement, is in the major mode or in the minor mode is usually no more than an illusion, created by restricting the particular inflection of the tonic triad during the passage or movement in question to its major or minor form. 39 A prime – and early – example of this modal fluidity is Beethoven’s Fifth Symphony. The first movement is solidly in the minor mode of C and the Finale is solidly in the major mode of C. To specify either as the tonic lessens the significance of the other, but without the presence of both the symphony would lose its expressive impact. Instead, we can denote the Fifth as having a two-dimensional tonic object that begins on the minor side, rotates more freely in the third movement, and, once the object stops spinning, lands on the major side for the Finale. The frequency of the Y rotation in a composition increased as the nineteenth century progressed, and by Mahler’s time the spinning of a two-dimensional musical object reached frenzied speeds. 40 So much, in fact, that it began to lose its compositional effectiveness. Mahler, and other late nineteenth-century composers, had to seek a new dimension for composing out the tonic.

39

Bailey, “An Analytical Study,” 116.

40 For example, in the second movement of Mahler’s Seventh the tonal emphasis switches rapidly between C minor and C major, and at times they are even superimposed (m. 187).

24

2.4

Three-Dimensional Musical Objects The coordinate of the X dimension is the fundamental tone, and the coordinate of the Y

dimension is the third. Logically, then, in a three-dimensional musical object the coordinate of the Z dimension is the fifth (Fig. 2.3). As shown in Figure 2.3, the two Z coordinates – as generated from C – are G and Af (or Gs). In comparison to the transformation model, this rotation would be similar to the L (leading tone) transformation. If an L transformation is performed on a C minor triad, then G would shift to Af; an L transformation performed on C major would shift C to B because major and minor triads are treated as inversions of each other. And while my model accounts for these rotations, it also provides two additional rotations that do not occur in basic transformations: a Z rotation on C major shifts G to Gs, or C augmented; an X rotation on C minor shifts C to B, or Ef augmented. Of course, the interpretation of the root for an augmented triad can vary depending on the context. More discussion of augmented triads and this model follow below. Figure 2.3: Three musical objects and their dimensions

25

This third dimension has a significant effect on the behavior of the musical object in that, unlike the one- and two-dimensional musical objects, the interpretation of the fundamental tone can change based on the orientation of the object. For example, a Z rotation on C minor shifts G to Af and those pitches are now best interpreted as Af major. Another visual analogy would be helpful here. The cube in Figure 2.1 can be measured in height, length, and depth; if that cube were to be rotated so that it rests on a different side, then the interpretation of its height, length, and depth would change based on this new orientation. Likewise, the interpretation of the root, third, and fifth of a three-dimensional musical object can change depending on its orientation. Additionally, as briefly aforementioned, the third dimension opens up the possibility for an X rotation. The X rotation is a consequence of the Z dimension in that the interpretation of the root can now change, whereas that was not possible in the one- and two-dimensional musical objects. Altogether the triads of the three-dimensional musical object comprise the hexatonic cycle (C major, C minor, Af major, Af minor, E major, and E minor), along with two augmented triads: one a collection of the three fundamental tones (C, E, and Af; the primary members), and the other a collection of the three upper fifths (G, B, and Ef; the secondary members). For analytical purposes, I have separated the six pitches of a three-dimensional musical object into two categories: primary members, the pitch classes that can function as fundamental tones; and secondary members, the pitch classes that can function as upper fifths. To borrow Graeme Downes’ term, these augmented triads, when they appear in the musical space, are enhanced versions of their normal functions: tonic (primary members) and dominant (secondary members). 41 As a dominant function, the augmented triad works well since it is dissonant and has a heightened sense of instability that requires resolution. On the other hand, a tonic function

41

Downes, “An Axial System of Tonality,” 18–25.

26

requires stability and an augmented tonic triad can only exist fleetingly in that capacity. 42 Typically, the s5 is reinterpreted as a f6 and the augmented triad stabilizes back to a major triad through a 6-5 exchange. But the appearance of an augmented triad in a tonic capacity is significant in that it superimposes the three fundamental tones in the musical space. That augmented triads are dissonant is an issue for this model that must be explained. Since we have defined the musical surface as one-dimensional where one-dimensional musical objects are consonant, and augmented triads are dissonant on the musical surface, then it would be logical that augmented triads are not one-dimensional objects. However, they comprise two of the eight one-dimensional slices of a three-dimensional musical object. I propose that augmented triads are like wormholes: unstable, topological anomalies that link together different parts of spacetime, often separated by great distances. 43 Wormholes (hypothetically) exist in threedimensional space but behave differently in that they can seemingly violate the laws that govern normal three-dimensional spaces. Likewise, augmented triads exist in the same one-dimensional musical space as major and minor triads, but they behave differently and allow harmonic progressions to traverse great distances in pitch space. If we take the three-dimensional musical object built on the fundamental tones C, E, and Af, then its upper fifth, which is G, B, and Ef, is an enhanced dominant that can feasibly resolve to a chord of any of those fundamental tones. In this way, the musical space can traverse from C major to Af minor in a short amount of time. More closely aligned with the wormhole analogy, however, is the potential for an augmented triad to act like a portal that can lead to other three-dimensional musical objects (Fig. 2.4); the 42 However, Mahler does utilize the augmented triad as a point of arrival, particularly at the end of the Finale; see chapter 5. 43

For a recent discussion on wormholes, see Mauricio Cataldo, Luis Liempi, and Pablo Rodriguez, “Traversable Schwarzschild – like Wormholes,” The European Physical Journal C 77, no. 11 (2017): 1–9. They define wormholes as “hypothetical tunnels connecting two asymptotically flat universes, or two asymptotically flat portions of the same universe.”

27

three-dimensional tonic object shares a wall with the three-dimensional dominant and subdominant objects. As a result, the augmented triads allow the music to slip easily between the different functional spaces and opens up an entire universe of harmonic possibilities within a single composition. Figure 2.4: Wormhole effect

Another way to visualize the three-dimensional musical object would simply be to compress all of its constituent notes into the one-dimensional musical space (Fig. 2.5). In this arrangement we can see that the sonority is three sets of semitone pairs: B and C, Ef and En, and G and Af. This symmetry is what gives the three-dimensional musical object its special abilities to rotate because each pair of semitones can be interpreted on any axis. For example, C can represent the root of C major on the X axis; an X rotation would bring B to the surface and would represent the fifth of E minor – from this perspective the semitone pair is now on the Z axis. Or the semitone pair C and Cf can be reinterpreted on the Y axis as the major and minor third of Af, respectively. This fluidity of interpretation can have a significant impact on the

28

structural function of each pitch, and can even allow simultaneous or superimposed structures where certain pitches or chords have multifaceted functions. Ultimately, the different structures all serve to express the three-dimensional tonic object, since its one-dimensional projection, as seen in Figure 2.5, does not neatly fit into the confines of tonality. Figure 2.5: Three-dimensional musical object compressed into one-dimensional musical space

2.5

Early Examples of Three-Dimensional Musical Objects Although it has been implicit in the methodological discussion thus far, these musical

objects are spatial and exist outside of time – they are abstract materials that can be temporally sculpted into a composition. We can think of a composition as a musical spacetime where the spatial and temporal dimensions of music are combined. When we conceptualize a composition in this way, it allows a more nuanced reading than other analytical models can provide. Certain middleground and even background details can emerge through the lens of a multidimensional musical object that might otherwise go unnoticed in those methodologies that attempt to account for “progressive” and chromatic-mediant designs. It is easy to view the role of certain majorthird relationships as non-essential, when they may in fact represent an unfolding of deeper structures. These major-third relationships do not necessarily have to manifest as the bookends of a structure, as in a progressive-tonal composition; significant events can occur in the middle of a structure and still be relevant to a comprehensive background analysis. As an example of this latter point, I present a reading of Schubert’s “Der Doppelgänger” (1828). This piece is especially useful because there are many analyses in the literature to which 29

my reading can be compared. The prospect of revealing new information about a piece that has been so extensively covered would seem difficult; perhaps some small details could be refined, but the potential for any substantial observations is unlikely. Yet when Schubert’s setting is understood as operating within a three-dimensional tonic object, some striking features are revealed (Fig. 2.6). Most notably is the symbiotic relationship between B minor and Ds minor – both are a part of the three-dimensional tonic object. In David Bretherton’s article “Schenker, Cube and Schubert’s ‘Der Doppelgänger,’” he provides a comparison of multiple readings (sideby-side graphic reductions can be found on page 182). 44 Most of the analyses downplay the significance of the Ds minor passage (mm. 47-50) as a third-divider, or remove it completely from the middleground voice-leading; Cube, in his original reading, focuses on the As (V of Ds) as support for s4 (Es). This latter interpretation is closer to the reading I provide in that Es is a significant structural pitch, but not as part of the 5-line Urlinie as indicated by Cube. Rather, I perceive the Ds minor diversion as a rotation of the tonic object, and that Es is its structural 2. Schubert’s setting of the text justifies this hermeneutical choice because Ds minor clearly represents the Doppelgänger, which is another aspect, or form, of the narrator. The text indicates that they exist in different timelines. In the first verse, our narrator is reliving past events that occurred at a significant location: the house where his former beloved once lived. The trauma of these memories induces a dream-like state in which the narrator is unaware of the present. Schubert marks this passage with a vocal melody that likewise obsessively returns to the same location of the past, represented by Fs (3 of Ds minor).

44

See David Bretherton, “Schenker, Cube and Schubert’s ‘Der Doppelgänger,’” Music Analysis 34, no. 2 (July 2015): 175–199.

30

Figure 2.6: Foreground of Schubert’s “Der Doppelgänger”

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Not until the second verse that begins in m. 25 does the narrator break from this trance and return to the present, now marked with Dn (3 of B minor). The catalyst for this transition is that the narrator becomes cognizant of the other man. Once this other man is revealed to be his “own form” (“eigne Gestalt”), their timelines collide and unfold simultaneously in the third verse (mm. 43ff.). The Doppelgänger is the manifestation of the narrator’s past and the suffering that is associated with those events. One main issue discussed in Bretherton’s article is that most analyses of “Der Doppelgänger” take the initial vocal-line Fs as the Kopfton of B minor. Cube struggled to reconcile some consequences of a 5-line reading (the structural support of 4) and sought out Schenker’s advice. Schenker’s solution, which ultimately swayed Cube, was to designate the piece as a 3-line because of the thematic content in the opening B minor ostinato (I have marked this 3-2-1 in Fig. 2.6). However, I, like the other analysts discussed in the article, found the Fs too important to relegate to a status as cover tone. My solution is that it is a structural pitch, but for a different aspect of a three-dimensional tonic object: Ds minor. That Ds minor is used to represent the Doppelgänger means that this key also symbolizes the past of the narrator. At the very beginning of the vocal melody I have indicated in Figure 2.6 that the initial Fs is 3 of Ds minor despite its placement over the B minor ostinato because our protagonist is reminiscing about the past. When he mentally returns to the present (second verse, m. 25), the vocal line shifts to the Kopfton of B minor (Dn). Ultimately, there is a register transfer of the Kopfton up an octave in m. 53 and followed by the complete descent to 1 in m. 56. Beneath this descent, the Urlinie of Ds continues as an inner voice; for the narrator, it is a literal inner voice – the haunting, unspoken memories of his past. The Picardy third in m. 62 is thus the completion of the Ds 3-line and the first time in the song where the two timelines are fully united. Perhaps this

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union signifies an acceptance of the past, or simply indicates that the trauma is still apparent. How one reads this ending is dependent upon the opening B minor ostinato, which, like our protagonist, is presented as broken; in Figure 2.6 I have indicated that beats 1 and 3 are an unfolding of the tonic and beats 2 and 4 an unfolding of the dominant. The “completed” B major at the conclusion of the song is suggestive of some type of resolution, but the connotation is not necessarily positive. Liszt’s Étude No. 3, Un Sospiro from Three Concert Études (1845-49) is another example where a piece begins and ends in the same key, but the tonic space is prolonged by what could be considered a three-dimensional musical object (Fig. 2.7). The piece opens with the perceived tonic of Df major and then moves through a series of key changes before concluding in Df major. In m. 19, the key changes to three sharps and an arrival on A major occurs in m. 22. A few bars later in m. 28 the harmony shifts back to the flat side, and in mm. 31-32 we get a clear V-I in F major. Finally, in m. 42, a true fundamental shift in harmony occurs as we arrive on the dominant, but enharmonically spelled as Gs. The structure is interrupted on this dominant, which is prolonged until m. 56. Afterward, the Df major tonic returns in m. 57 with a modified version of the first theme, and the structural close is achieved in m. 66. The three-dimensional tonic object is briefly revisited in the penultimate measure, in which Df major arpeggiates to F major before concluding in the final measure back on Df major. In a traditional analysis, A major could be understood as fVI and F major as fVI of fVI, but I do not find these labels particularly informative in a functional sense. Here, Liszt employs these harmonic centers to expand the tonic space and delay the background shift to the dominant Stufe.

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Figure 2.7: Middleground of Liszt’s Étude No. 3, Un Sospiro

There are precedents for progressive designs that can be read as multidimensional musical objects. Two candidates for single movement designs are Chopin’s Ballade No. 2, Op. 38 (1836-1839) and Brahms’ String Quintet No. 1, Op. 88, ii (1882). Chopin’s Ballade begins in F major and concludes in A minor; the second movement of Brahms’ String Quintet begins in Csharp major/minor, but ultimately closes in A major. Another piece that is a remarkably early example of a multi-movement progressive design is Ferdinand Hiller’s Piano Concerto No. 1, Op. 5 (1829-1831). The concerto begins in F minor, proceeds to Df major in the second movement, and the Finale concludes in Af major; thus, the progressive design moves from F minor to Af major. It is apparent that Hiller intended this progressive design as the crux of the Finale. The rondo heavily emphasizes F minor, despite the structural close in Af major (Fig. 2.8). These two tonal areas are conflated in such a way that it is difficult to determine the middleground and background structures. One reading is that Hiller intended some type of expanded tonic object that encompasses both F minor and Af major. This dissertation is focused on three-dimensional musical objects that are constructed from major

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third relationships, but it is possible that a different species could instead be derived from minor thirds. In Figure 2.8 I show that the background structure of the initial Af tonic is directly connected to the background structure of F minor that begins in m. 120 and is later confirmed as a background Stufe in m. 150. The F major chord that arrives in m. 120 might be better understood as V of Bf minor that directly precedes it in m. 118; in fact, a prolongation of Bf minor in mm. 118-131, where the primary theme returns in Af major, is quite logical. Ultimately, I have chosen to privilege the prolongation of F because of the significant cadential arrival in m. 150. Moving forward, the structural upper voice, C, is reinterpreted from 3 of Af to 5 of F. A background 5-line descent begins in F minor in m. 202, although the 5-4-3 could easily be understood as 3-2-1 of Af major; the two Ursätze are synthesized together. I show F minor as the operative harmony because the music is headed toward a structural interruption on V of F in m. 255. In other words, the background structure actually shifts from an Af orientation to F. If Af and F can indeed be understood as two parts of a deeper, multidimensional tonic object, then mm. 1-255 can be reconciled as the first half of a divided structure (I-V||). After, the primary theme and Af major return in m. 270 and the structural close is achieved in m. 289. The parallel structures between Af and F are certainly an allusion to the I-III sonata design in the first movement, which further corroborates the meta-design of this concerto.

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Figure 2.8: Middleground of Hiller’s Piano Concerto No. 1, Finale

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CHAPTER 3 JUSTIFYING THE MULTIDIMENSIONAL MODEL 3.1

Justification for a Three-Dimensional Tonic Object Before I begin the step-by-step, detailed discussion of the first movement of Mahler’s

Seventh Symphony, I present my rationale for employing the three-dimensional tonic object in my analysis of Mahler’s Seventh Symphony; e.g., why I read E minor and C major in the exposition of the first movement as all tonic space. Traditional tonal practice would regard this progression as moving from tonic to submediant (I-VI), considering I and VI different chords with different harmonic functions. In the context of the first movement alone, that assessment would be appropriate. However, given that the first movement is just a single part of a larger tonal edifice, it is necessary to consider it within a holistic context since Mahler created a structure that contradicts the expected confirmation of a single tonic by the outer movements. To produce an accurate reading of the first movement, we must determine the ultimate role of E minor within the entire symphonic plan. My initial hypothesis is that there must be some significant relationship between the first and last movements. As discussed at the end of the Chapter 1, unification of key is paramount for suites: an early multi-movement structure. My presumption is that key unification remains an important element for later multi-movement structures, such as the symphony. For example, the overwhelming majority of Joseph Haydn’s symphonies begin and end in the same key, with some exceptions only when there is change of mode from the first to last movement; e.g., Symphony No. 45 begins in Fs minor and ends in Fs major. Mahler’s symphonic output, on the other hand, seems to belie this tradition. 45 In my estimation, there are two ways to approach

45

Symphony No. 2 begins in C minor and ends in Ef major; Symphony No. 4 begins in G major and ends in E major; Symphony No. 5 begins in Cs minor and ends in D major; Symphony No. 7 begins in B minor, moves to E

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these progressive symphonies: that Mahler has abandoned the ideals of symphonic unity or that Mahler is employing new methods of unification. I take the latter explanation as the impetus for my analysis, that the E minor tonic in the first movement and the C major tonic in the last movement are both different aspects of the same three-dimensional tonic object. Both chords fulfill the role of tonic in their respective movements, yet traditional tonal practice dictates that one must, by default, be subservient to the other. The logical choice would privilege C major since it is the tonic of the Finale, the goal towards which the entire symphony moves; considered in isolation, the tonic of the Finale is undoubtedly C major. However, we must reconcile its relationship with the E minor tonic of the first movement; is it merely a large-scale III-I progression? That is a potential reading, but one that I find to be unsatisfactory, if not completely unconvincing, because Mahler is at such pains to intricately connect E and C tonal areas in both outer movements. This interconnection of two putatively separate tonal centers suggests that they operate as a unity such that the appearance of a III-I progression is something of an illusion. The clearest evidence for a functional equivalence between E minor and C major – and indeed the entire three-dimensional tonic object – is found in the Finale. The first 78 measures encompass a motion from E minor (mm. 1-6) to C major (mm. 7-51), and finally to Af major (mm. 51-78). In this way, Mahler presents in direct succession all three primary members (E, C, and Af) of the three-dimensional tonic object. Furthermore, embedded in the first cadence in C major is a nested cadence in E major (Fig. 3.1). That E major and C major share a cadential space here is a further indication of their intimate relationship. Also telling is the way in which Af major arrives. In the middle of m. 51, Af major suddenly interjects into the C major chord

minor, and then ends in C major; Symphony No. 9 begins in D major and ends in Df major.

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(Fig. 3.2). The brief overlap of these two sonorities is a jolting experience, but is symptomatic of a three-dimensional musical object that abruptly rotates at the musical surface. Mahler revisits this moment in the penultimate measure of the Finale, although slightly altered. In this case, C major is interrupted by an augmented triad – which contains the three primary members of the three-dimensional tonic object, namely C-E-Gs; additionally, this interjecting sonority arrives on beat one of m. 589, instead of in the middle of a measure (Fig. 3.3). The downbeat placement of this augmented triad in the penultimate measure is a powerful foreground manifestation of the tonic object, and it aligns well with Laufer’s thoughts concerning the primary sonority and its tendency to arrive at the end of a piece. Its arrival is marked by a sudden change in texture and dynamic shift, and this rhetorical effect alerts us to the importance of this chord. Since the tonic object represents the most fundamental element in the Seventh Symphony, Mahler situates it here at the end of the work with as much finality as is possible for an augmented triad – the chord stabilizing on the C major triad only in the final measure. Prior to the end of the Finale, Mahler includes another important signal that E and C, as tonal centers, are interchangeable with both functioning as “tonic.” In mm. 573-576, the primary theme from the Finale (theme I), originally stated in C major, is presented in E major; then, in mm. 581-585, the primary theme from the first movement, originally stated in E minor, is presented in C major. That the two primary themes of the Seventh rotate between these two tonal centers is a further indication of a harmonic-functional equivalence between them. Shortly thereafter, as already observed, C and E are joined by Gs in the penultimate measure to complete the three-dimensional tonic object.

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Figure 3.1: E major cadence embedded in C major cadence in Mahler’s Seventh Symphony, Finale, mm. 13-15 46

Figure 3.2: Interjection of Af major into C major in Mahler’s Seventh Symphony, Finale, mm. 5152

46

All examples shown in concert pitch.

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Figure 3.3: Emergence of C augmented triad in Mahler’s Seventh Symphony, Finale, mm. 588-590

The third primary member of the three-dimensional tonic object, Af, makes another significant appearance in the first movement in mm. 27-30 (introduction). This passage in Af is significant because Mahler brings back the same material in m. 495, but in E minor, during the recapitulation – in fact, m. 495 is often read as the structural close. 47 Thus, to understand the role of E minor in m. 495, we must consider the function of Af minor in the introduction since Mahler presents them in the same way.

3.2

Enharmonicism and the Augmented Triad In my model, enharmonic pitches are often equivalent. For example, Af and Gs can both

be representative of the three-dimensional tonic object in Mahler’s Seventh Symphony. The manifestation of Af or Gs is usually due to foreground circumstances (e.g., clearer voice leading). Mahler often employs the flat side or the sharp side depending on the immediate

47

See John Williamson, “Mahler and Episodic Structure: The First Movement of the Seventh Symphony,” in The Seventh Symphony of Gustav Mahler: A Symposium, ed. James L. Zychowicz (Madison, WI: A – R Editions, 1990) 46; Henry – Louis de La Grange also locates the beginning of the coda at m. 495, Gustav Mahler, Volume 3: Vienna: Triumph and Disillusion (1904–1907) (New York: Oxford University Press, 1995), 856.

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context. In the introduction to the first movement, Mahler arrives in Af minor in m. 27; however, Af is a passing tone to a Gf 7th chord, and this chord means the dominant seventh of the B minor tonality; in other words, it should really be understood as an Fs 7th with a raised fifth (Fig. 3.4). That Mahler notates the previous passage in Af minor and not Gs minor is because it is immediately preceded by a passage in Ef major; thus Af is more appropriate for that circumstance. Figure 3.4: Middleground of Mahler’s Seventh Symphony, first movement, mm. 1-31

Often Mahler even superimposes enharmonic pitches, which, in my estimation, further validates my hypothesis regarding enharmonic equivalency in this piece. During the introduction of the first movement, a march theme is introduced in m. 19. Flutes 1 and 2 have the melody BAs-Gs, Ds-Cs-B; violins II have the same melody, but it is written as B-Bf-Af, Ef-Cs-B (Fig. 3.5). Neither instrument group has music in the previous measure that would justify alternative enharmonics for voice-leading purposes. Throughout the march theme there are contradictory

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enharmonics between simultaneous lines, although all instruments finally coalesce in Ef major at m. 25. Since ease of reading does not appear to be the impetus for this conflation of enharmonic notation, Mahler’s motivation here is likely to underscore the fluidity between the otherwise disparate tonal regions of Af and Gs (and their implied functions in relation to the tonic). Figure 3.5: Superimposition of Af and Gs enharmonics in Mahler’s Seventh Symphony, first movement, mm. 19-20

The power of enharmonics is most notable with augmented triads. That augmented triads can suggest several enharmonic spellings at once is what allows Mahler to quickly traverse the harmonic landscape (by means of “the wormhole effect” as discussed at the end of Chapter 2). Mahler fully exploits this potential in the Seventh, in that the kinship between E-Gs-Bs, C-E-Gs, and Af-C-E (and even Gs-Bs-D∗) is synthesized into a single harmony: the three-dimensional tonic object. The multiple harmonic possibilities of the augmented triad were well understood in Mahler’s time, as is documented by the treatises on harmony of the day. In his Theory of Harmony (dedicated “to the hallowed memory of Gustav Mahler”), Schoenberg discusses the augmented triad at length. At an important point in his discussion, he states that “the augmented triad is by virtue of its constitution, as indicated by its belonging to three keys, a vagrant chord like the diminished seventh.” 48 He goes on:

48 Arnold Schoenberg, Theory of Harmony, trans. Roy E. Carter (Berkeley: University of California Press, 1978), 243.

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Since in the resolutions to the major chords (III-VI of minor) the roots make the leap of a fourth, it is evident that they augmented triads may be used to produce [resolve to] a tonic, and that, to this end, they may be introduced artificially on the V of the major key in question, following the idea of the secondary dominants. They are most simply introduced through chromatic alteration upward of the fifth. …With the augmented triad it is not necessary to make a distinction between root position and inversions. It is indeed almost always reinterpreted, and, that being the case, the feeling of a six-four chord can hardly ever arise. To avoid complicated notation one will often use enharmonic change.… That the augmented triad resolves by the strong progression, the root progression a fourth upward, to a major key, even when it was derived from the minor key of the same name, favors its use for connection of major and minor. 49 To clarify, Schoenberg originally derives the augmented triad from harmonic minor, and specifically on the mediant chord (a). Since III-VI is a resolution, then so too is III+-VI (b). From there, Schoenberg simply substitutes the augmented triad into the role of the dominant in any context, be it the major or minor mode (c) (Fig. 3.6). Figure 3.6: Augmented triads as dominants, derived by Schoenberg in Theory of Harmony

Further, here we find several indications that the augmented triad is flexible in its spelling, its point of resolution, and its implication of major and minor. Additionally, Schoenberg’s placement of his discussion of the augmented triad in the chapter “At the Frontiers of Tonality” further suggests its potential for innovation within the tonal system. My model pushes the

49

Schoenberg, Theory of Harmony, 242–244.

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augmented triad from only a dominant function into the realm of other functions, such as tonic and subdominant.

3.3

The Three-Dimensional Dominant Object as Binding Element As further evidence that E minor and C major are two sides of the same three-

dimensional tonic object, let us examine more closely the relationship between the E minor first group and the C major second group in the exposition of the first movement. The initial clue that these two keys share the same tonic-functional role is the shared three-dimensional dominant object that binds them. A B pedal is prolonged in mm. 104-117, and within the context of an E minor first movement one might assume a simple V of E that resolves deceptively to VI (C major) in m. 118. However, the harmonic goal of this B pedal is the augmented triad in mm. 114117 (Fig. 3.7), which is most prominent in the trombones and bassoons (G-B-Ds). While this chord could be understood as an altered dominant of E minor (i.e., B-Ds-F∗), I believe Mahler deliberately conflates the dominant space of E minor (B-Ds-[Fs]) with the dominant space of C major (G-B-[D]). The conflation of these two dominants is further emphasized in m. 114 in which the second violins play the line Fs-G-B and the first violins play the line D-Ds against the augmented triad in the bassoons and trombones (the same melodic lines are repeated in m. 117 in the violas and first violins). Whereas the augmented triad suggests the conflation of both dominants, here there is a literal conflation of all five distinct pitches (B-Ds-Fs and G-B-D). Mahler’s explicit insertion of the three-dimensional dominant object between E minor and C major supports the hypothesis that both keys are simply different sides of the same threedimensional tonic object.

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Figure 3.7: Mahler’s Seventh Symphony, first movement, mm. 114-117

The augmented dominant, comprising the primary members of the three-dimensional dominant object, is composed out in both the introduction and the putative development section. In the introduction, the middleground prolongs the succession of harmonies B minor - G major Ef major in mm. 1-25 (Fig. 3.4). B minor has its own lengthy phrase (mm. 1-18), but G major and Ef major share an antecedent (mm. 19-22) and consequent (mm. 23-27) period. Measures 19-22 present a G major half cadence; the following phrase begins as though it will continue G major, but the harmony rotates towards an Ef major authentic cadence (although it resolves deceptively to iv (Af minor) in m. 27). A similar, but more expanded, composing out of the three-dimensional dominant object occurs throughout the development section: B (m. 145) - G (m. 228) - Ef (m. 258). This double unfolding of the dominant then moves to V/V in m. 315 and then to the emergence of the real second group in B major in m. 317 (Fig. 3.8). The arrival on B major is a climactic goal of the first movement. However, the B major second group is interrupted and broken off in m. 337; what follows is a reprise of the introductory material (m. 338) and the recapitulation (mm.

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394ff.). The second group is then recapitulated in m. 465, but now in G major and quickly cadences in m. 468. In this reading, the B major second group is the antecedent to the G major second group which functions as its consequent phrase. That Mahler reprises the second group in B major and G major, the respective dominants of E minor and C major, seems an especially telling indicator that these harmonic spaces are intimately connected. Mahler is certainly alluding to the I-VI progression that can be found in the sonata expositions of Beethoven and Schubert, but here the modulation is only apparent (mm. 50-118). The real harmonic progression is at a deeper level and later in the movement, namely from the tonic object (E minor and C major) to the dominant object (B major and G major). Figure 3.8: Deep middleground in Mahler’s Seventh Symphony, first movement, mm. 145-468

3.4

Sonata Form and the Identity Narrative First movements of symphonies are usually defined by their employment of, and

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interaction with, sonata form. The first movement of Mahler’s Seventh is no exception, and in fact Mahler’s manipulation of sonata form is of paramount importance for the analysis to be presented here. Before delving into the specifics of Mahler’s dialogue with sonata form, it is perhaps best to provide a cursory discussion of the form, its devices, and my own analytical terminology. A great deal of information regarding sonata form, its variants, deformations, and numerous categories can be found in James Hepokoski and Warren Darcy’s Elements of Sonata Theory; still, there is much to discuss because the form as a structural mold is highly malleable. 50 However, more central to my own focus is Schenker’s conception: “Only the prolongation of a division (interruption) gives rise to sonata form.” 51 And while Schenker did struggle to reconcile the inherent contradiction between an undivided background and a formal process that requires division, 52 his assertion that the real agency of sonata form lies in harmonic motion and not thematic groups is what I find most compelling. Schenker states that “the second theme, the subordinate theme, the lyrical theme, or the like… are in every respect inadequate terms and concepts which afford no insight into sonata form.” 53 However, current analytical approaches to sonata form often privilege the thematic elements at the neglect of the harmonic foundation. It is necessary to engage all dimensions of the sonata process before a comprehensive reading can be attained. My own three-fold summary of sonata form is informed by the types of major harmonic events with which Schenker grappled: (1) the tonic is established; (2) an opposing harmony then emerges and challenges the tonic’s hierarchical status; and (3) the tonic reestablishes its control. 50 James Hepokoski and Warren Darcy, Elements of Sonata Theory: Norms, Types, and Deformations in the Late – Eighteenth – Century Sonata (New York: Oxford University Press, 2006). 51

Schenker, Free Composition, 134.

52

Peter H. Smith, “Brahms and Schenker: A Mutual Response to Sonata Form,” Music Theory Spectrum 16, no. 1 (Spring 1994): 77–103.

53

Schenker, Free Composition, 135.

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Ideally for the tonic, the composition would be a space that allows the tonic to express itself fully and wield complete harmonic authority. But because the tonic is challenged by a competing harmony it necessarily needs some device through which it can reassert itself. One such device is the key feature of sonata form according to Schenker: the interruption, which provides the tonic an opportunity to restore its identity as the controlling sonority. With the “identity metaphor,” I supplement the conceptualization of sonata form with three identity-narrative events: identity schism, identity crisis, and identity reclamation. This identity narrative can help one to discern in a Schenkerian analysis the real background structure from those events that are better read at the middleground level. Additionally, it can help to explain when specific manipulations, or unusual procedures, occur in sonata form; in terms of my identity narrative, those manipulations can include the delay or exclusion of certain events. At least in major mode sonatas, the most typical harmony to oppose the tonic in the exposition is the dominant. In a minor mode sonata that role can be fulfilled by the major mediant (III) or, less commonly, the minor dominant (v); another harmonic option that developed later in the eighteenth century, and particularly in the nineteenth, is the major submediant (VI). 54 The identity schism is the point at which this oppositional harmony initiates a separation from the tonic and begins the process to assert itself as tonic; as shown below, this identity event often does not correlate with the second theme or the expositional “caesura.” 55 I have chosen the term “identity schism” because this event represents the initial rift in the fundamental harmony – the point at which the tonic begins to lose control. The schism begins to allow the opposing harmony the opportunity to be on the same level as the tonic and to threaten 54

Two examples are discussed below. This model is particularly relevant for Mahler’s Seventh since the putative exposition moves from E minor to C major; this harmonic progression, and what it means in the Seventh, will be further discussed in chapter 4. 55

Hepokoski and Darcy, Elements of Sonata Theory, 18.

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its structural priority. While the schism does not yet establish the opposing harmony as a real Stufe, it does lead to the necessary 2-1 descent in the upper voice with the supporting V-I progression in the new key. Once this new key area is confirmed through a cadential arrival it initiates the identity crisis; this identity event is marked by a time in which the normative relationships and hierarchy are upset, challenged, or, at the very least, obscured. There is a fundamental and important distinction between this event – i.e., the identity crisis – and what Hepokoski and Darcy call the essential expositional closure (EEC) in that in my model the cadence signaling the identity crisis can occur before or after the second theme. Therefore, my concept of the identity crisis is more analogous to the Kochian Quintcadenz or Terzcadenz (a V-I cadence that confirms the arrival of V or III as opposing harmony), which is the final punctuation of the Hauptperiode (first main period). Heinrich Christoph Koch’s eighteenthcentury conception of what would later be known as “sonata form” (coined ca. 1840 by Carl Czerny and A. B. Marx) was centrally concerned with the hierarchy of cadences rather than thematic groups. 56 One scenario for the culmination of the identity crisis is the moment of structural interruption on the dominant (2/V||). 57 This dominant, whether the opposing key in the exposition is V or not, represents the tonal subversion because it refuses to resolve directly to 1/I. I refer to this dominant at the interruption as the dominant of opposition. (Although it will usually coincide with the Schenkerian interrupting dominant, I want to distinguish situations in which the dominant of opposition does cadence to the tonic and produces a “failed” recapitulation.) At this point, the tonic must reassert itself as 3/I, revisit the thematic material, and establish the structural close (1/I) – the process of identity reclamation. Once again, this final For a discussion on Kochian terminology, see Veijo Murtomäki and Timothy L. Jackson, “‛Punctuation Form’ and Expressive Contents in the First Main Period of Selected G Minor Symphonie’s First Movements of the Classical Era—Kochian – Schenkerian Approaches,” Journal of Schenkerian Studies 11 (2018): 51–110. 56

57

Another possible scenario, in which there is no interruption, is discussed below regarding Beethoven’s Ninth.

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cadence may or may not align with what Hepokoski and Darcy call the essential structural closure (ESC). Each employment of sonata form can be evaluated through this identity narrative by locating the specific events and their placements relative to the exposition, development, and recapitulation spaces. To elucidate the discussion of the identity narrative, I explore two first movements from Haydn’s London Symphonies – one major and one minor. The first is Symphony No. 93 in D major (1791); a middleground sketch is provided in Figure 3.9. As expected for a major-mode sonata, the oppositional harmony is the dominant. Most significant for this example, however, is that the identity schism and crisis both occur before the second thematic group begins. The trajectory towards the tonicization of A major begins in m. 60 where D is converted into D-sharp as a chromatic passing-tone to E; as a result, the progression is retrospectively understood as IVsIV-V of A. An Anstieg in the upper voice likewise marks the drive towards A major. The 3 of A that arrives in m. 66, along with A in the bass, signals the identity schism. At this point, A major threatens to abandon its role as dominant and become its own tonic. That goal is achieved in m. 74 with a cadential confirmation of A major. 58 With this arrival, A is now considered a real Stufe on the same level as the opening tonic. In Figure 3.9, I differentiate the statuses of the A in m. 66 and the A in m. 74 with different stem lengths. From mm. 74-185, the identity crisis ensues with A, and not D, operating as the background Stufe. The dominant of opposition (VOpp.) is prolonged to the end of the development at m. 185. As a result of the interruption, the tonic is able to begin again. However, the tonic’s status as a real Stufe is not yet regained; it must traverse the material that was originally subverted by the dominant and reclaim it. In mm. 20458

Hepokoski and Darcy would qualify this arrival as a V:PAC MC—a third – level default option. That this V:PAC does not coincide with the EEC is not consequential for their model. They acknowledge that, from a Schenkerian perspective, the interpretation of the definitive arrival of the dominant (ZPAC) is often at the discretion of the analyst, and that their concern with the placement of the EEC is more rhetorical than structural. See Elements of Sonata Theory, 27, 147–149.

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205 the progression IV-sIV-V is reprised, but this time under the tonic’s control. The cadence in m. 218, which correlates with the cadence in m. 74, returns the tonic to the background structure. Finally, the recapitulation of the second group allows the tonic to confirm its status as a Stufe with a background cadence in m. 242, which signals the identity reclamation event. My second Haydn example is provided by Symphony No. 95 in C minor (1791); a middleground sketch is offered in Figure 3.10. Here the oppositional harmony is the major mediant (III) and the identity schism begins in m. 16. The identity crisis is delayed due to an interruption on V of Ef that is prolonged in mm. 21-28; measure 28 is where Hepokoski and Darcy would mark the “caesura.” 3 of III returns in m. 29 and initiates the second thematic group. Shortly after, Ef is cadentially confirmed as a real Stufe in m. 36 and finally signals the identity crisis. The development section drives toward the dominant of opposition; this transition to VOpp. is achieved through a transformation of III into an augmented-sixth chord (VI of V) in mm. 98-103. V, as a Stufe, arrives in m. 113 and is prolonged into the reprise of the first thematic group. This reading is feasible in that the reprise of the first group is not supported by a strong root-position tonic, as it is at the beginning of the movement. As a result, the C minor that begins in m. 120 is understood as a 6/4 upper-neighbor prolongation of the dominant of opposition. I read an interruption at the end of m. 128 because 3 of C is picked up by the bassoons in m. 129 along with the key change to C major. As in Symphony No. 93, the tonic’s reclamation occurs in the recapitulation of the second thematic group.

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Figure 3.9: Middleground of Haydn’s Symphony No. 93 in D major, first movement

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Figure 3.10: Middleground of Haydn’s Symphony No. 95 in C minor, first movement

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These Haydn symphonies provide eloquent examples of two main types of oppositionalharmony designs that are found in sonata form: I-V and I-III. I now explore the I-VI design in two other symphonies: Schubert’s “Unfinished” (1822) and Beethoven’s Ninth (1822-1824). These examples are especially relevant to our discussion because Mahler’s Seventh employs the I-VI design and the music of both Beethoven and Schubert was highly influential on Mahler as a composer. On these influences, Alfredo Casella observes: A further characteristic of Mahler’s art, which one notices at once, and whose effect is certainly no less captivating, is the pure goodness of this music. Through it Mahler can reach the most secret chord of our heart and often approaches Beethoven…. But there is also another musician whom he approaches in the same degree: Schubert. Of the latter we are reminded by the melodic invention, the melancholy grace of the many Ländler which run through his works, and finally by the touching naïveté of certain melodies. Beethoven and Schubert are Mahler’s two true teachers. 59 Mahler was very familiar with the works of these two composers and conducted the two aforementioned symphonies on many occasions. 60 He made orchestral revisions to Beethoven’s Ninth, although that practice was customary for Mahler’s time. 61 There is some evidence that Mahler made similar revisions to Schubert’s “Unfinished”; he certainly made a number of revisions to Schubert’s Ninth, as well as several suggested cuts in the first and last movements. 62 We can infer that Mahler’s intimate knowledge of these symphonies may have influenced his own compositional choices, particularly in how to approach a I-VI sonata. The I-VI design offers different possibilities – and also poses a new set of challenges – for the analyst. Indeed, I posit two interpretations of Schubert’s “Unfinished” that are vastly different 59

Alfred [sic] Casella, “Gustav Mahler et sa deuxième symphonie,” S.I.M., Revue musicale mensuelle VI, no. 4 (April 1910): 240–241, translated and quoted in Kurt Blaukopf and Herta Blaukopf, Mahler: His Life, Work and World (New York: Thames & Hudson Inc., 1991), 233. 60

Mahler conducted twelve performances of Beethoven’s Ninth and eight performances of Schubert’s “Unfinished”; Henry – Louis de La Grange, Gustav Mahler, Volume 4: A New Life Cut Short (1907–1911) (New York: Oxford University Press, 2008), 1593, 1607. 61

La Grange, Gustav Mahler, vol. 4, 390–394, 1534–1537.

62

La Grange, Gustav Mahler, vol. 4, 1538.

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from one another. The first could be considered a “common sense” reading – the result of a “traditional” Schenkerian analysis. As seen in Figure 3.11, the Stufen progression throughout the exposition and development is I-VI-IV-V. In this reading, G major (VI) is taken as the oppositional harmony and progresses toward the dominant of opposition that arrives at the end of the development in m. 202 (after a lengthy prolongation of IV). David Beach likewise reads the basic progression as I-VI-IV-V; however, Beach’s analysis differs in that he does not read an interruption at the end of the development section as I show in Figure 3.11. 63 Beach notes that Schubert “devoted the entire development section to the [introduction] motto theme” and, as a result, leads naturally into the recapitulation of the first group. He also reads that the Kopfton (3) does not emerge until the recapitulation of the second group, although it is supported by D major (III); therefore, an undivided structure is required. 64 Measure 218 presents the “double return” of the first thematic group and B minor, but the recomposition process alters the musical path in that the reprisal of the second thematic group is in the key of D major (III). The end of the second group is likewise recomposed so that it ends up on V7 in m. 279. A lower level interruption occurs and this V7 is understood as a back-relating dominant. After a measure of rest, the music returns on E minor (IV) in m. 281 – a recapitulation of the expositional material that begins in m. 63. The dominant Stufe arrives in m. 303, but as a 6/4; most importantly, the 6 is raised so that we have Ds (s3 of B major). B major is confirmed in m. 311, which coincides with the final G major cadence of the exposition in m. 93. However, the apotheosis is short lived and B minor resumes in mm. 328ff. of the coda.

63

David Beach, “Schubert’s ‘Unfinished’ Symphony: Analytical Observations,” in Explorations in Schenkerian Analysis, ed. David Beach and Su Yin Mak, 99–122 (Rochester, NY: University of Rochester Press, 2016), 111.

64

Beach, “Schubert’s ‘Unfinished’ Symphony,” 107.

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Figure 3.11: “Common sense” middleground of Schubert’s “Unfinished,” first movement

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Another, quite different reading of Schubert’s “Unfinished” is presented in Figure 3.12. Of the two interpretations, I prefer the second, which I shall designate my own reading. The first significant difference is the interpretation of G major, which is not presented as the oppositional harmony in this reading. My decision not to consider G major as a background Stufe is largely due to the return of B minor at the very end of the exposition, which begins as a unison B on beat two in m. 105 and unfolds to a full B minor triad in m. 108. Therefore, as shown in Figure 3.12, the G major section is reduced into a 5-6-5 motion (Fs-G-Fs) above the tonic. Rather than act as an oppositional harmony, then, G major is a harmonic ally that serves to prolong the tonic rather than deviate from it. In other words, in this reading I show that the exposition remains entirely under the tonic’s control. Here I take the dominant as the oppositional harmony, arguing that the identity schism occurs much later in the form, namely in m. 202, i.e., at nearly the end of the development. As in Beach’s reading, in my analytical model there is no interruption at the end of the development. To my ears, there is an undivided transition into the recapitulation of the first group in m. 218. The more controversial aspect of my reading, though, is that I do not take the B minor in m. 218 as a return of the tonic, but rather consider it an apparent tonic caught within a prolongation of the oppositional dominant harmony. The main evidence for this interpretation is that the final cadence for the recapitulation of the first group in m. 252 is on Fs and not B. In the previous reading, the Fs arrival is m. 252 is severely downgraded as the upper third of the following D major. But this arrival is far more significant in my estimation and signals a heightened moment in the identity narrative of this sonata because territory that was previously under the control of the tonic (mm. 9-38 of the exposition) is seized by an oppositional harmony. In the two Haydn examples, the oppositional harmony is established with new material – either the second group or the transition to the second group.

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Figure 3.12: Alternative middleground of Schubert’s “Unfinished,” first movement

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Schubert amplifies the drama by placing the identity schism and crisis around first group material and delaying the latter event into the recapitulation. As in the exposition, the recapitulation of the second group serves to prolong the operative background Stufe rather than deviate from it; D major functions as a 5-6-5 (Cs-D-Cs) upper-neighbor elaboration of the dominant harmony. Unlike Beach, I do ultimately read an interruption in m. 279 – albeit much later than in the “common sense” reading. The dramatic measure of rest naturally suggests this reading, although on which level the interruption occurs is debatable. Measures 281ff. are comparable to the “common sense” reading except that, because the tonic is not picked back up, the cadence on B major in m. 311 is auxiliary. This divided structure is unusual, but I believe it is a consequence of the delayed identity-narrative events. This interpretation is bolstered by the following consideration: that Schubert suggests a type of unity between the major-third pairs of B-G and Fs-D. The two-bar bass ostinato pattern that begins in m. 9 is the first indication. It is largely a repetition of B, but on the last eighth note of the second measure it drops to G; this pitch choice seems odd at first: Fs would be the more logical candidate as the upper fifth. A second indication of this pairing occurs on the first cadence in m. 20. This cadence is comprised of two events: 1) a perceived arrival on D (I-IV-VI, mm. 17-20), and 2) a half cadence in the tonic key (beat two of m. 20). The effect is a simultaneous arrival on both D and Fs. A traditional reading would surely privilege the half cadence as more structurally significant, but I posit that Schubert’s conflation of D and Fs into this arrival is an important clue for the structure of this movement. In the first phrase of the first group, Schubert pairs B and G in the tonic function and Fs and D in the dominant function. Another pairing of Fs and D occurs at the end of the development in mm. 202-209. Similar to the arrival in m. 20, here there is a cadence on D major (m. 204) and a sudden shift to Fs major (m.

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205). These events suggest to me that the major third pairs of B-G and Fs-D are used to prolong the same function (tonic and dominant, respectively) rather than present four different background Stufen. 65 Therefore, my reading (Fig. 3.12) more accurately represents the symbiotic relationship of the major-third pairs. All sonata forms previously examined have employed the divided structure as codified by Schenker. Through the interruption the identity narrative is able to progress towards the final event: the identity reclamation. However, sonata form can have an undivided structure, and in these circumstances some other device apart from the interruption can enable the identityreclamation process. In his dissertation, Benjamin Graf presents a reading of the first movement of Beethoven’s Ninth as an undivided structure. 66 The Stufen progression throughout the exposition and development is I-VI-IV-V, just as in the “common sense” reading of Schubert’s “Unfinished.” One would expect an interruption on this dominant at the end of the development, but the reprise begins on a 6/3 D major chord in m. 301, and later corrected to a 6/3 D minor chord in m. 315; the Fs, and later Fn, bass voice is accompanied by an upper-voice D. Graf reconciles the first-inversion D major (and then D minor) chord at the start of the reprise as the result of a massive voice exchange from the F/D at the beginning of the exposition (m. 17). This compositional device ultimately has the same effect as an interruption in that it resets the background structure and allows the tonic to reassert itself: everything caught within the voice exchange is subsumed within a tonic prolongation. As for the identity narrative, we can still read a schism (m. 80) and crisis (m. 150) when Bf major (VI) is confirmed; likewise, the drive toward

65

Richard Cohn has presented evidence for the significance of major – third relationships in the music of Schubert; see Cohn, “As Wonderful as Star Clusters” and Audacious Euphony.

66

Benjamin Graf, “An Analytical Study of Paradox and Structural Dualism in the Music of Ludwig van Beethoven,” (PhD diss., University of North Texas, 2016), 146–196, https://digital.library.unt.edu/ark:/67531/metadc849697/m2/1/high_res_d/GRAF – DISSERTATION – 2016.pdf.

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the dominant of opposition is apparent at the end of the development (m. 275ff.). However, because the large-scale voice exchange of the tonic takes precedence in the background structure, we must understand that the confirmation of an oppositional harmony as a real Stufe is only illusory. In other words, the other harmonies (VI, IV, and V) do not operate at the background level because the tonic never truly relinquishes control. Because of this circumstance, it is apparent that the identity narrative does not necessarily correlate with the background structure. Instead, it informs the analyst of potential background structures – sometimes the narrative is only a middleground phenomenon. The most important aspect of these Schubert and Beethoven examples is that they probably provided Mahler with potential models for sonata form, and the I-VI design in particular. In Figure 3.13 (a) and (b) I present my reading of the deep middleground of the first movement from Mahler’s Seventh. My analysis presented here should be regarded as an overview, with significant details to be discussed in subsequent chapters. The first group in E minor and the second group in C major seem, at the surface level of perception, to be in two different, contrasting keys. Indeed, Mahler appears to be following the model of a I-VI modulation in the exposition. However, as I have suggested, these two putatively distinct keys might be understood as just two different aspects of the same three-dimensional tonic object, so that no deep-level modulation really occurs. This strategy is analogous to my alternative reading of Schubert’s “Unfinished” (Fig. 3.12). Whereas in the Schubert example, VI is used to prolong the tonic Stufe, here VI is a part of the tonic Stufe (marked as an unfolding in Figure 3.13(a). In both cases, the real oppositional harmony is understood to arrive only later in the form, after the exposition, and on the dominant. Mahler marks this delayed arrival of the oppositional dominant by returning to the second group material at the end of the development in the dominant key of B

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major (mm. 317-337). In my analysis, this B major section is the real second group; it belongs in the exposition, but has been displaced by the rotation of the three-dimensional tonic object (E minor to C major). When compared to the B major second group, the C major presentation of the second group material in the exposition (mm. 118-144) feels harmonically static, incomplete, and unfulfilling; its cadential confirmation is marred with the addition of an As (m. 141), which transforms C major into an augmented sixth, leading to B as dominant. Thus, the C major section is used to enhance the motion towards the dominant in m. 145. This is where I place the identity schism and the remainder of the development articulates the Anstieg towards the Kopfton of the new key (as in Haydn’s Symphony No. 93 [Fig. 3.9]). Mahler further complicates the identity narrative in that he denies a cadential confirmation of B major; instead, an interruption (2/V of V) occurs in m. 337. Had a perfect cadence on B major supporting 1 in the top voice been achieved in m. 338, then this arrival would mark the identity crisis. Instead, m. 338 reverts to the introductory material and the return of the B minor + Gs sonority (although the Gs arrives one measure later and only in the first double basses). In my reading, this Gs, and the one at the beginning of the introduction, operates as part of the three-dimensional tonic object; for this movement, it is relegated to the upper voice as s3 at the deepest level, although it often functions as f4 in the middleground. A possibly contentious aspect of my reading is that the “double return” of the tonic and opening theme in m. 394 is really an apparent return to the tonic. 67 In terms of the identity narrative, then, the schism event is still ongoing since no new background (tonic) Stufe has been confirmed. As a result, the E minor asserted in m. 394 is caught within the ongoing attempt to establish the oppositional harmony, namely the dominant.

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The “false” return to E minor in m. 373 is discussed in Chapter 4.

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Figure 3.13: (a) Deep middleground of Mahler’s Seventh Symphony, first movement, mm. 1-410

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(b) Deep middleground of Mahler’s Seventh Symphony, first movement, mm. 413-543

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The recapitulation of the first group covers the same musical ground as in the exposition in that it first cadences on E and then on B; but because the structural and narrative context is shifted for the reprise, I have accorded structural weight to the B arrival (m. 410), while the putative tonic E is reinterpreted as IV of V (m. 393). I do not place the identity crisis at the B arrival in m. 410. Since the tonic had authority over both groups in the exposition, the dominant must secure both groups in the recapitulation to enact the identity crisis. This interpretation is similar to my reading of Haydn’s Symphony No. 93 (Fig. 3.9) in which the tonic first had to reclaim the material that was previously conceded to the dominant in the exposition – its background authority is not immediately regained at the start of the recapitulation. Here, in the Mahler example, the oppositional harmony must traverse both groups in order to establish itself at the background level. The confirmation of the dominant for the second group failed at the end of the development (m. 337), which necessitates the recapitulation of the second group in G major (m. 466). G major, which operates as part of the three-dimensional dominant object, is confirmed almost as soon as the second group begins (m. 468). It is here that I place the identity crisis and G as a background Stufe. From here, the music progresses toward the dominant of opposition, which is achieved in m. 512. This dominant of opposition should act like an interrupting dominant, but instead is forced to resolve for the structural close and subsequently complicates the identity narrative. Why I do not read the E minor in m. 495 as the structural close will be fully discussed in Chapter 4. Most important for the current discussion is that the identity narrative is left incomplete in this first movement. The reclamation event is not properly achieved because the structural close in m. 515 is corrupted by the dominant of opposition that precedes it. This sonata form, like Beethoven’s Ninth, is undivided. But unlike Beethoven’s Ninth, there is no background voice exchange that

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retrospectively redeems the tonic. In Mahler’s Seventh, both the interruption (m. 337) and voice exchange (mm. 486-512) devices are co-opted by the dominant. This design requires the remainder of the symphony, particularly the Finale, to complete the identity narrative and provide the tonic its reclamation event.

3.5

Function Entanglement Like Mahler’s use of sonata form, the harmonic language of the Seventh requires a

nuanced understanding because of its use of the three-dimensional tonic object. There are several sonorities throughout the symphony that present an analytical challenge, either because the collection of pitches is unclear within a tonal context or because the significance of putative seventh chords can easily be underestimated. For an example of the latter case, the opening chord of the Seventh Symphony may be cited: in isolation, this chord is a Gs half-diminished seventh in the first inversion; however, the key signature, the bass-voice B, and the confirmation of B minor throughout the slow introduction indicate that this sonority is better understood as B minor with an added Gs. I do not interpret this Gs as an added sixth or consider its inclusion merely coloristic. Instead, I propose that the initial sonority signals the conflation of different functional spaces: dominant and tonic. The dominant is represented by B minor (a part of the three-dimensional dominant object) and the tonic is represented by a partial Gs minor triad (GsB, a part of the three-dimensional tonic object). The implication of Gs minor superimposed upon B minor is not immediately apparent at the outset of the introduction but is later confirmed by the arrival of Af minor in m. 27. Thus, in hindsight, we can revalue the significance of the Gs in the opening sonority; that Mahler is suggesting a conflation of dominant and tonic spaces – a phenomenon that I call function entanglement.

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Function entanglement occurs when two or more different functional spaces are presented in a way in which the tonal hierarchy is obscured. For this discussion, function refers to Tonic, Subdominant, and Dominant relationships. In the previous example, the global functions of dominant and tonic are superimposed; that is, a conflation of primary members from the three-dimensional dominant and tonic objects. But at the local level that relationship is unclear, and, in fact, could be interpreted as inverted: B minor as three-dimensional tonic and Gs as three-dimensional subdominant. I find this reading compelling because it foreshadows the problematized sonata design employed in the first movement. The identity schism in the first movement, which unfolds for most of the symphony, begins when the dominant separates from the tonic (m. 145), but the event is delayed beyond the normal expositional space; due to the altered narrative, the recapitulation is marked by a hierarchical inversion in which the putative tonic (m. 393) is caught within a dominant prolongation, and is thus better understood as IV of V. When different functional spaces are entangled, these inverted relationships become possible. Mahler suggests these issues of entanglement through various sonorities that could be classified as polychords, such as the B minor + Gs minor chord at the beginning of the symphony. There are a few other prominent polychord-sonorities in the Seventh that fall into this category of function entanglement. Each case is discussed in the analytical chapters to follow.

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CHAPTER 4 FIRST MOVEMENT ANALYSIS 4.1

Introduction The analysis of the first movement, as well for the Finale, primarily focuses on the

aspects most relevant to the three-dimensional tonic object of the Seventh Symphony, and its corresponding dominant and subdominant objects (Fig. 4.1). One other object is possible (Fig. 4.2), but its function is more contextual in that it can be V/V or IV/IV. As a result, this threedimensional supertonic object acts like a link that connects all the objects into a circular model. As with any new analytical approach there is the danger of attempting to connect all aspects of a composition with that model, which can lead to weak or illogical conclusions. While my model does not provide answers to all possible questions about Mahler’s Seventh Symphony, it can facilitate significant insight into the fundamental ideas influencing the design of this complex work, and impacting its form, harmony, counterpoint, and motivic elements. Figure 4.1: Three-dimensional V, I, and IV objects

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Figure 4.2: Three-dimensional supertonic object

There are several key points in the following analysis. First, the compositional problem presented in the first movement is function entanglement between tonic and dominant; these two harmonic spaces often collide and are superimposed in a way that challenges any single viewpoint of the various structures. This problem manifests in a number of ways through the movement and is indeed one of its most significant elements. Second, the harmonic spaces are three-dimensional, namely the tonic and dominant. Third, the multidimensional tonic and dominant objects and the issues of function entanglement distort the process of sonata form: although a second group is presented in the exposition, the “real” second group does not arrive until the end of the development section; the recapitulation presents an apparent tonic that is caught within the dominant prolongation; the identity crisis does not occur until the recapitulation of the second group in G major (part of the three-dimensional dominant object); when the structural close is achieved late in the movement, it is corrupted by the dominant of opposition and represents a “failed” recapitulation.

4.2

Introduction Analysis Mahler suggests that the identity schism between tonic and dominant is problematic right

away with an introduction in B minor, which is dominant territory. As a result, the harmonic

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progression from the introduction to E minor in the exposition can be read as V-I in the tonic or I-IV in the dominant. However, the first sonority signals that the dominant space is precariously unstable. The first inversion Gs half-diminished seventh, as discussed at the end of Chapter 3, is best understood as a conflation of the global three-dimensional dominant (B minor) and tonic (Gs minor) objects: or function entanglement. One implication of this function entanglement is that the introduction has a significant role in the sonata design. The first narrative event is the oppositional harmony’s separation from the tonic, but a different reading of the introduction is that the tonic is attempting to separate from the oppositional harmony; that the symphony begins outside of the tonic’s control. In other words, when the tonic arrives at the beginning of the exposition in m. 50, we can understand it as extricated from the dominant and that it is a real tonic; or the extrication could be illusory, and the E minor that begins the exposition might be better understood as IV/V. This issue, the interpretation of the relationship between dominant and tonic, is central to the structure of the first movement. There is nothing apparent about the structural hierarchy when one considers the paradoxical nature of these harmonic relationships. Certain analytical decisions must be made and I have chosen to read the introduction as dominant and the exposition as tonic, but the obfuscation of this relationship seems deliberate by Mahler and must be understood in order to grasp the consequences of the first movement. Figure 4.3 provides a middleground assessment of the first phrase of the introduction, mm. 1-18. One important feature is the descending-fourth progression (B-As-Gs-Fs, mm. 1-11). Between As and Gs there is an ascending ninth in the bass, bifurcated by Ds in m. 9. The harmony here is B major, but the bass continues to rise and on beat four the harmony has rotated to G major – both chords are a part of the local three-dimensional tonic object. Globally, B major and G major serve as an unfolded dominant of Gs. Here the Gs sonority in m. 10 is the same

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from m. 1 – Gs half-diminished seventh – but now in root position and no longer interpreted as a B sonority. One consequence of function entanglement is the possibility of multiple interpretations. Essentially, the roles are reversed since Gs now functions as local tonic and B is subsumed into a dominant function. This paradox of function is further exaggerated by the instability of the diminished Gs sonority, which does not logically fit the role of a tonic. Figure 4.3: Section 1, B cadence from Mahler’s Seventh Symphony, first movement, mm. 1-18

In m. 11 the Fs in the bass marks the arrival of the local dominant, V of B. However, there are a number of complications with this dominant, which once again foreshadows the identity-schism problem that unfolds throughout the movement. Ironically, the sonority for the

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first half of the measure is the same Gs half-diminished seventh, but now in the third inversion. Of course, with Fs in the bass the B and D are appropriately read as 6/4 suspensions held over from the previous Gs chord in m. 10, and they do in fact resolve to As and Cs. Still, it seems significant that Mahler continues to employ this set of pitch classes, each time with a different meaning. The Gs, which occupies the highest voice, is more appropriately read as a passing tone between the Fs in m. 10 and the As in m. 12. Thus, in my reading, the As in m. 12 belongs to the Fs dominant in m. 11. I presented an analysis of this phrase in a paper in 2016 at the Indiana University Annual Symposium of Research in Music, and William Rothstein suggested that a new phrase begins in m. 12 after an interruption at the end of m. 11. He argued that the B minor chord and the tenor horn melody signal a return to the local tonic and initiates a consequent phrase. At the time I did not have a convincing rebuttal to justify my reading that the B minor is not a real tonic and that it is instead caught within a dominant prolongation. I argued that the upper voice continues to unfold a dominant harmony, beginning in m. 10 with Fs, to As in m. 12, and finally to Cs as 2 in m. 14. However, it is now clear to me that Rothstein’s suggestion highlights the phenomenon of function entanglement – that m. 12 exists in a conflation of (local) tonic and dominant spaces. As a result, m. 12 can be read as either a dominant or tonic function: the dominant continues to unfold toward the tonic arrival in m. 17, or the tonic arrives in m. 12 and continues to unfold, as a codetta, to m. 17. B minor, although globally a part of the dominant object, is locally entangled with its own dominant, which manifests as this B minor sonority against its own As leading tone. As a result of this function entanglement, the B minor cadence in mm. 15-17 is grotesquely distorted. The bass and upper voices seem to disagree where the dominant-to-tonic resolution occurs (Fig. 4.4). In m. 15 the bass and alto voices project a tonic function with B and

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Ds; the B is held as a pedal point and the Ds chromatically descends to B in m. 17. On the other hand, the upper voices, comprised of the violins and high woodwinds, firmly declare a dominant function in m. 15 as they arpeggiate from As down to Fs and then Cs. The violins concede to a tonic resolution in m. 16, but the high woodwinds continue to prolong the dominant arpeggiation, this time from As down to Fs and then Cn. Finally, in m. 17 every voice agrees on a tonic function, but almost immediately the basses descend a fourth to Fs as if to signal once more the entanglement between tonic and dominant. The high woodwinds likewise repeat the f2-1 resolution, echoed by the horns in the following measure. Figure 4.4: B minor cadence in Mahler’s Seventh Symphony, first movement, mm. 15-19

Beginning in m. 19, Mahler introduces a new march theme for the next section (Fig. 4.5). There are a number of interpretations that are possible in this section – mainly because of the rotating three-dimensional musical object – but I disclose the various levels that unfold. At the deepest level, this section continues the prolongation of the introduction’s local tonic: B minor. However, at the outset of m. 19 the harmony has rotated to G major in the first inversion through a 5-6 exchange. Directly following the G major chord is a harmonic progression (mm. 19-21) that would be difficult to assess in the traditional tonal paradigm, but when considered within the

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three-dimensional musical object model this series of chords can be understood as the rotation of a three-dimensional subdominant object. In other words, the Gs minor, E major, and C major chords are three different sides of the local subdominant object. That these three chords all prolong the same function is reflected in the underlying middleground counterpoint of a rising third progression: III-IV-V (B-C-D). Gs minor does not have to be reconciled as some type of s1 sonority; E major is not an altered VI chord. In my model, they are both subsumed within the three-dimensional subdominant object. What we hear with this rising third progression can best be understood as a four-measure (mm. 19-22) antecedent phrase that interrupts (3-2||) on the V7 of G in m. 22. Figure 4.5: Section 2, G major half cadence from Mahler’s Seventh Symphony, first movement, mm. 19-22

The march theme begins again in m. 23, but now on a root-position G major chord and at a quicker pace (Fig. 4.6). Measure 24 signals another rotation with the unfolding a V7 of Ef

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sonority that crystallizes on beat four. Ef is then clearly stated in m. 25 with a local 3/I followed by its own root-position 2/Vb9 on beat three. One would then expect a resolution to 1 of Ef, but in m. 26 only the basses move to Ef – the other voices maintain the Vb9. Here we have another instance of function entanglement that is reminiscent of what occurred in m. 15 at the B minor cadence. 1 of Ef does appear in the following measure, but over an Af minor chord and with a foreshadow of the exposition’s first theme. In a sense, this Ef major phrase is the consequent to the previous G major half cadence, or G: 3-2||Ef: 3-2-1. They are different key areas, but two parts of the local three-dimensional tonic object. Figure 4.6: Section 2, Ef major ‘cadence’ from Mahler’s Seventh Symphony, first movement, mm. 23-27

The Af minor chord is accompanied by a key change, seemingly to indicate that it is a new local tonic and that the Ef major chord is forced into a dominant role – an attempt to disentangle the function spaces. However, it is the wrong key signature: six flats, not seven. Is

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this wrong key signature merely an oversight by Mahler? My speculation is that this key signature is quite intentional, and that Mahler truly intended six flats to be read as Ef minor. The reason is that the introduction is governed by the local tonic B minor, but B minor is only one part of a three-dimensional musical object. Thus, Ef minor is simply a different side of the same object; the key change does not indicate a new function space, but a prolongation of the same function. Af minor is not tonic, but rather IV of Ef; in a deeper sense, Af is a part of the local three-dimensional subdominant object. However, on the global scale of the symphony, Af is a primary member of the three-dimensional tonic object. What Mahler presents here is a role reversal for the identity-schism narrative event: normally it is the dominant that attempts to break away from the tonic, but here the tonic is attempting to break away from the dominant. I emphasize this moment because Mahler reprises this section, transposed down a major third, in mm. 487-495, and the E minor that arrives in m. 495 is often read as the structural tonic close (1/I). If the Af minor arrival in the introduction is an apparent tonic that is caught within its own dominant, then the same arrival on E minor in the recapitulation may likewise be an apparent tonic. In that case, m. 495 cannot be the structural close. At the middleground level, this Af minor chord is a part of another descending fourth progression that initiates from the B that is prolonged in mm. 1-19, to Bf in m. 25, to Af in m. 27, and finally to Gf in m. 31. I read the Bf in m. 25 as the governing bass voice for two reasons: (1) the Ef chord at the beginning of m. 26 is a prolongation of the local three-dimensional tonic object that, for this section, began on B in the bass in m. 19 – Ef can be understood as the upperthird of B; and (2) the Vf9 of Ef chord is held over into m. 26 and directly precedes that Af minor chord. The Ef in the bass in m. 26 can be understood as an anticipatory upper-fifth of the following Af chord. A “traditional” Schenkerian reading would perhaps place more emphasis on

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this Ef in m. 26, or the Ef at the beginning of m. 25, as an applied dominant to the Af in m. 28, but in my estimation Mahler only wants to suggest the illusion of a cadence and that this is why most of the voices maintain the Bf dominant ninth chord. Since Bf is V of Ef, and Ef is a part of the local three-dimensional tonic object, then at a deeper level Bf is also V of B. In fact, this descending fourth progression is an enharmonic repetition of the descending fourth from the first section (B-As-Gs-Fs, mm. 1-15), but with altered harmonies. The As chord in m. 7 is V of B: FsAs-Dn (C∗) – as an augmented triad we could say that it is an enhanced dominant 68 that could potentially point to B, G, or Ef resolutions. In m. 25 it is altered into a Bf dominant – Fn substitutes for Fs. Figure 4.7: Gs half-diminished and Af minor chords

The Af minor chord in m. 27 is likewise similar to the Gs half-diminished sonority from m. 10, but is now a more stable reflection of the global tonic as if to signal that the separation between (global) tonic and dominant objects has begun (Fig. 4.7). Of course, the separation does not yet occur because this Af chord serves only a passing-tone function between B and its dominant (m. 31). Additionally, at the middleground level, the Ef in the upper voice can be understood as the major 3 of B that comes from the Dn (minor 3) in m. 1. The deeper structure of the first two sections is further connected in that the transferred Ursätzes of the B, G, and Ef

68 Here I am using Graeme Downes’ term “enhanced dominant” to invoke the augmented triad’s ability to resolve to three different tonal areas; see Downes, “An Axial System of Tonality,” 18.

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phrases can be conceptualized as extensions of one another (Fig. 4.8) – the 1 of B becomes 3 of G; and while the G phrase is interrupted and thus does not lead to 1, we can imagine the 3 of Ef as a pseudo resolution for the G phrase. As a result, the upper voice is largely a descending whole-tone scale apart from the initial Dn, which is ultimately corrected into Ds (Ef) in m. 27. More importantly, Mahler is providing a structural blueprint for how a three-dimensional musical object can be composed-out and that B, G, and Ef are all subsumed within the same function. The dominant that arrives in m. 31 further marks the synthesis of B, G, and Ef into a single function in that it consists of all three upper fifths (enharmonically): Fs, Dn, and Bf. At the end of m. 31, all of the instruments that continue into m. 32 have a breath mark, seemingly to indicate a division – a division not unlike the one found at the end of a development section (2/V||). Although the V in m. 31 does not contain 2, it does have Dn which could be interpreted as s2, or C∗. However, the upper voice moves from Ef to Ff, the 7th, or enharmonically 4 (En), and resolves back to 3 in m. 32 – what Schenker would call a delay, which only gives the illusion of an interruption. 69 Thus the middleground structure of the introduction is still incomplete.

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Schenker, Free Composition, 42.

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Figure 4.8: Middleground, Mahler’s Seventh Symphony, first movement, mm. 1-50

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The third and final section of the introduction (mm. 32-49) reprises the thematic material from the first, along with the B minor + Gs sonority. Here the trajectory is less obscured and moves quickly to II in m. 38 and V in m. 39 (Fig. 4.9). The V in m. 39 is initially an open fifth (Fs and Cs), but its dominant function is unquestionably implied despite the missing As. In m. 42, the dominant becomes more contorted by multiple levels of function entanglement. The basses maintain the Fs pedal, but the upper voices oscillate between E-G-B and Ds-Fs-B – the flutes and first violins elaborate Cn as an upper neighbor to B. E, G, and B could be understood as the 7th, 9th, and 11th of the V chord and the Ds and Fs could be understood as anticipations of the coming I chord in m. 43, but I read these pitches as superimpositions of other functions – i.e., namely IV and I. Mahler never gives us a real V of B, but rather this conglomerate sonority of V + I + IV. Despite the strange dominant, 1 over B major solidly emerges in mm. 43-44 and signals the end of the introduction. A short codetta follows in mm. 45-49 marked by a fourths motif in the trumpets (identified with slurs to show each trumpet). The trumpets enter one after another, each a fourth higher than the previous: B, E, and A, respectively. In my reading (Fig. 4.9), the notes of these fourths represent the paradigmatic circle-of-fifths harmonic progression: rising V-I-IV (B-E-A), then descending IV-I-V (A-E-B). We can understand tonic (I) like a taut string that is at rest; if we pull the string towards the dominant (V), it creates tension with potential energy; when released, the string will return to the center, but only briefly because the now-kinetic energy sends it towards the subdominant (IV). The subdominant mirrors the dominant in that its natural tendency is to return to the tonic (Fig. 4.10). In m. 47, the first trumpet replicates this motion: AE-B, or IV-I-V. This pattern of harmonic frequency could conceivably go on for a long time, but one would expect it eventually to come to rest back on tonic after all the energy is expended.

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Figure 4.9: Section 3 middleground and the ‘fourths’ motif from Mahler’s Seventh Symphony, first movement, mm. 32-49

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However, this expected pattern is suddenly interrupted by As (which seems to indicate a V/V function) in m. 48, and whose presence now suggests B as tonic instead of E. In this moment, we find a subtle implication of function entanglement between I and V (E and B), in that the relationship is obscured – or perhaps even completely reversed to IV and I (E and B). Figure 4.10: Harmonic frequency

The fourths motif slowly morphs the B major from m. 43 into a strange conglomerate sonority in m. 49, similar to the one found in mm. 41-42. The pitches are B, E, G, and As, and Mahler employs it as a quasi V of E – and E minor arrives directly after in m. 50, along with the exposition proper. In the context of E minor, the E and G could, again, be understood as anticipations of the coming I chord; the As – which had been absent from the previous V of B in m. 39 – a suspended leading tone. And at a deeper level these skewed surface voices could be separated and correctly realigned back into their respective functions, but I propose that this sonority has a far deeper meaning than some advanced counterpoint. I posit that this sonority is the culmination of function entanglement: that it truly represents, in the context of E minor, V/V + V + I; or in the context of B minor, V + I + IV. It signifies the compositional problem of the first movement; i.e., the first event of the identity narrative: the schism between I and V. Further, Mahler returns to this grotesque dominant for the structural close at the end of the recapitulation, which I discuss below.

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4.3

Exposition Analysis The primary theme emerges from the trumpets’ falling fourths motif, now heard in the

horns and cellos beginning in m. 50 (Fig. 4.11); an E minor chord is articulated by the high woodwinds and violins. That the horns and cellos are the lowest sounding instruments at this time is significant because the opening falling fourth from E to B seems to destabilize the arrival of E minor. On beat three of m. 50, and into m. 51, we have an E minor 6/4 chord, which is very similar to the dominant sonority from m. 49 – apart from the As. One could posit that the dominant, which was prolonged throughout the introduction, has not relinquished its control, and that tonic and dominant are suggested to be entangled at the start of the exposition. The theme continues, though, and moves back to E in m. 52, which then leaps down a fifth to A – later corrected up an octave in m. 54 to reflect better the rising fourths (Fig. 4.11, middleground). So far, our points of melodic arrival after the initial E have been B-E-A: a replication of the fourths motif. After the arrival of A in m. 53, the theme arpeggiates an A minor chord, but the upper woodwinds and violins continue to articulate E minor. Here we see another example of Mahler’s compositional technique that blends separate functions (I and IV) into each other. Ultimately, the A in m. 53 is an anticipation to the A in m. 54 as support for the fII harmony. 70 A local V-I cadence on E occurs in mm. 56-57, but the primary theme emphasizes 3 and does not lose its momentum. The consequent of the first theme, mm. 58-64 (Fig. 4.12), switches the harmonic emphasis from I to V, or E minor to B minor. This antecedent-consequent phrase structure that moves from I to V is a common design, but nonetheless highlights the hierarchical ambiguity in that E minor struggles to maintain its status as tonic and becomes subservient as IV of B. The 70 I want to emphasize that the Neapolitan chord fits into the three – dimensional subdominant (IV) object—it can also be conceptualized as IV of C.

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upper voice corroborates the shift of local tonicity as E takes on the role of 4 in mm. 60-61 and resolves to D, or 3 of B. 4 as upper neighbor to 3 is established in mm. 50-57 – in E minor – as a middleground motive, which makes the reinterpretation of the role of E within the context of B that much more significant. The theme returns for a second iteration in m. 65, but is delayed by one beat – a subtle change that is nearly undetectable due to the fast pace of the music. With this change, it is now B – and not E – that is placed on the strong downbeat and further confirms the complicated relationship that Mahler has established between tonic and dominant. After this point, the theme is altered into transition material and there is a rising third from the B in m. 66 to a prolongation of D that initiates in m. 70 (Fig. 4.13). This D, which is undoubtedly the upper third of B, ultimately moves to Ef in m. 77, and then enharmonically changed to Ds in m. 79. In m. 80 we encounter a key change to B major, accompanied by what sounds like new thematic material. Everything about these parameters suggest that we have entered the second group; however, this section of music is a red herring. The putative second group arrives in m. 118 in the key of C major (Fig. 4.13). Mahler incorporates this false second group to reiterate, once again, the entangled relationship between tonic and dominant. If we closely examine the thematic material found in mm. 79-98, we discover that it is a synthesis of the first theme and motivic elements from the introduction. In mm. 79-80 the violins reproduce the falling fourths that was originally associated with the trumpets at the end of the introduction and then subsumed into the first theme by the horns and cellos. Here, the fourths (Gs-Ds-As) are transposed in a way that suggest that they are upper thirds relative to B major, but we can understand them as a part of the larger three-dimensional dominant object – that is, the key of Ds (Ef).

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Figure 4.11: First group from Mahler’s Seventh Symphony, first movement, mm. 50-57

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Figure 4.12: First group continued from Mahler’s Seventh Symphony, first movement, mm. 58-64

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Figure 4.13: Middleground, Mahler’s Seventh Symphony, first movement, mm. 50-134

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Thus, Gs-Ds-As represent IV-I-V in Ds: the building blocks of the first theme (Fig. 4.11). This new presentation of the fourths motif is also reminiscent of the tenor horn’s melody in m. 2, both in rhythm and its outline of a descending minor seventh. There are also elements of this section that are suggestive of a Gesangsthema, particularly mm. 90-95. The melodic line shifts from the disorienting, leaping fourths to a melody that is lyrical and largely stepwise. This respite is brief, however, and the leaps return once again before a key change back to E minor in m. 99. While E is initially in the bass in m. 99 to coincide with the key change to E minor, the harmony here is A minor in the 6/4 position. I read the E as the unfolded upper fifth of A, which does appear, however briefly, at the end of m. 103. Immediately after, the bass moves to a B pedal in mm. 104-117, which suggests a larger prolongation of B from mm. 79-117. This extended B prolongation is ultimately a dominant function and a bridge between the first group and the putative second group. The most important attribute of this B prolongation is that it ultimately manifests an augmented triad: B-Ds-G in mm. 114-117, although there are moments of friction in which Dn is juxtaposed with Ds. Dn suggests G major, or a stabilized dominant of C – which is the key of the putative second group that arrives in m. 118. The dissonant contrast between Dn and Ds over the B pedal is perhaps the result of the deeper harmonic motion as the three-dimensional tonic object rotates from E minor to C major (C major is marked as unfolded from E in m. 50 in the deep middleground, Figure 3.13(a), to signify that it is a part of the threedimensional tonic object). That we perceive this moment as dissonant is a consequence of our limited, one-dimensional perception of musical sound. The putative second group, as one would expect, presents a character that is quite distinct from the first group, marked by the violins’ chromatically ascending lines and leaps to climactic pitches that are accented agogically on beat two. As a counterpoint, the horns are set with a

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chromatically descending line that is reminiscent of the horn part from the preceding B major section (specifically mm. 80-84); this similarity further suggests that that B major section is a thematic hybrid of the first and second groups, rather than an independent section. Harmonically, the C major second group is largely static; there is a progression of I-ii-viio7-I, but the bass voice, which amounts to a C-B-C lower neighbor, does not indicate a strong cadence. The absence of a root-position dominant renders this iteration of the second group as stagnant and undeveloped, and not at all the potent counter statement one would expect. Additionally, the thematic material feels rushed and not allowed the space to unfold fully, and its ending a contrapuntal web of dissonance that collapses into the development. That the second group presented here is unsatisfying is not initially apparent until it returns at the end of the development in the key of B major – one of the most transcendent moments of the symphony. There is a 3-2-1 descent in C major, but it only occurs after the second theme disintegrates in m. 133; thematically, the music returns to the march theme from the introduction. To further complicate the arrival on C major in m. 141, an As is present in the inner voice, which creates an augmented sixth. As stated in the sonata discussion at the end of Chapter 3, this augmented sixth serves to enhance the motion towards the dominant and the identity schism event in m. 145.

4.4

Development Analysis Although the B that arrives in m. 145 is a 6/4 chord (E and G in the upper voices), it

quickly moves toward a cadence in m. 160. However, I read G (3 of E) as a deeper background upper voice throughout the development, and that the Fs that accompanies the dominant prolongation is ultimately a middleground lower neighbor; in other words, the Fs that is present

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throughout the development is not a background 2. The upper voice is complicated by the mechanics of the three-dimensional dominant object since it contains both G and Fs, which means that the dominant space can prolong the minor scale degree 3 or 2 – and potentially both at the same time. As a result, the voice leading can at times appear to be paradoxical depending on which side of the three-dimensional dominant object has rotated to the musical surface. For example, the bass moves down from B, through a passing tone on A in m. 217, to G in m. 228; here the upper voice G corresponds better. But it would be understandable for one to prioritize Fs supported by B major in m. 160 as more structurally significant, and to read G as its upper neighbor. The section on G (mm. 228-247) contains no independent descent, but instead prolongs the important upper voices – namely G and Bf/Bn. In mm. 248-250, in my middleground graph (Fig. 4.15), I use an unfolding sign to connect D and Bf. Together they comprise the threedimensional V/V object and facilitate the rotation from G to Ef. While the Fs in m. 252 could be considered as a continuation of the three-dimensional V/V, here it acts as an incomplete upper neighbor to the Fn in m. 254. Here, the Fn prolongs the V/V function as part of a 4/3 Bf chord (V of Ef) – much like the A in m. 227 as part of a 4/3 D chord (V of G) – and acts as a passing tone between G and Ef. Mahler marks the arrival on Ef in m. 258 in two ways. The first is the trumpet fanfare that begins in m. 256 and reoccurs throughout the Ef section. Historically, the trumpet fanfare has been used to signal the arrival, or forthcoming arrival, of some important event or person 71.

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William Parker Melvin, "Instrument of Life and Death: The Symbolic Role of the Solo Trumpet in the First and Third Movements of Gustav Mahler's Symphony No. 5 in Cs Minor," (Ph.D. diss., University of Illinois, 1997): 52– 57.

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Figure 4.14: Middleground, Mahler’s Seventh Symphony, first movement, mm. 135-245

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Figure 4.15: Middleground, Mahler’s Seventh Symphony, first movement, mm. 247-317

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Here I believe Mahler employs the trumpet fanfare to announce the forthcoming B major reprise of the second group. The second way in which Mahler marks the Ef section is a sudden shift in the musical texture from polyphonic to homophonic. The chorale-like Ef section presents a stark contrast in mood compared to the previous one hundred measures of frenzied polyphony. Like the G section before it, there is no upper-voice descent to Ef and it instead prolongs the important upper-voice pitches G and Bf. Thus, the upper-voice stasis of the G and Ef sections further enforces that they are prolonging the dominant space rather than presenting a harmonic departure. In between the prolongation of Ef there is a brief detour back to G in mm. 266-297. I read this G as III of Ef, and not as hierarchically equal to the previous G section. However, this G section is much more fleshed out than the previous G section in that it has a full 3-line descent. My middleground also employs another double-unfolding symbol that connects G in m. 284 to Ef in m. 285 and Cf in m. 286. These three chords all represent the local three-dimensional tonic object. They are further connected by shared dominants in between, although each is only an open fifth (D-A, Bf-F, and Gf-Df); thus, the resolutions are comparable to deceptive motions, but a more nuanced understanding is that this harmonic progression represents a full rotation of the local three-dimensional tonic object. After the detour to G, the trumpet fanfare returns once again in m. 298 as well as the Ef chorale in m. 304. When the Ef chorale returns it is in the minor mode; thus, the upper-voice G has descended to Gf, which functions as an anticipation of the Fs that arrives with the B major second group in m. 317. The thematic material of both the Ef major and minor chorales is a slow version of the march theme from the introduction. In the Ef minor chorale the march theme rises

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from Ef-D-C (m. 304), to Gf-F-Ef (m. 306), then A-Gs-Fs (m. 310), and finally descends to GsFs-E (mm. 315-316) as the 7th of V of B. The second group in B major is one of the most climactic moments of the movement. Its arrival is marked by harp arpeggios and a pure homophonic texture in the woodwinds and strings. The harmonic trajectory toward a cadence is also clear; the bass rises fIII-IV-V in mm. 328-333 (Fig. 4.16); this harmonic progression facilitates a chromatic voice exchange from Fs/Dn in m. 328 to Ds/Fs in m. 333. Every aspect about this presentation of the second group appears fuller and more complete compared to the C major iteration (m. 118), which is harmonically stagnant and thematically disintegrates before its cadential arrival in m. 141. The B major presentation is how the second group should sound. Additionally, this is the first point in the movement where the dominant crystallizes into its own fully supported local tonic, unlike the introduction or the previous sections of the development. This evidence suggests that the B major section is the real expositional second group in terms of the standard sonata design; the dominant has fully separated from the tonic and now seeks to confirm itself as the oppositional harmony. However, it falls short of that goal in that its V-I cadence is interrupted at m. 338. Instead, the music collapses back to the B minor introduction – initially without Gs, but at the end of m. 339 the first double basses reprise the opening tenor horn melody: Fs-D-Gs. The interruption at m. 337 evokes, at a surface level, the structural interruption one would expect to find at the end of the development as described by Schenker; however, the interruption here is on V/V rather than V. Thus, while it provides the rhetorical effect of an interruption, it is a middleground phenomenon and does not initiate a background division.

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Figure 4.16: Middleground, Mahler’s Seventh Symphony, first movement, mm. 328-410

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Measure 338 and the return of the introductory material is sometimes read as the beginning of the recapitulation, or the introduction to the recapitulation. 72 I read it as a continuation of the oppositional dominant harmony as it seeks a cadential confirmation. That process is complicated, though, in that m. 338 reprises the B minor + Gs sonority, which signals that the dominant and tonic functions are still entangled – the identity schism still ongoing. A new 3-line descent in B begins with the reprise of the introductory material in m. 338, but now in the minor mode; however, the mode is rather ambivalent throughout this section (Ds appears frequently, and the key signature remains in B major). The descent is not completed until m. 410 – into the recapitulation. Nested within this B minor descent is the transition towards the “double return” of the first group and the putative tonic in m. 394. This transition first moves to A major in m. 365, which is then transformed, through a 5-6 exchange, into a first inversion F major chord in mm. 369-372. The fII (Fn) is one reason that I read the E major in m. 373 on such a low level – that the E major chord functions as a passing sonority between the A in m. 365 and the B in m. 393 (Fn-E-Ds in an inner voice). In other words, a IV-V progression towards the E arrival in m. 394. However, as stated, I read this E as a putative tonic and its role is discussed in the next section.

4.5

Recapitulation Analysis The recapitulation of the first group in m. 394 closely resembles the original first group

that began in m. 50. One adjustment is that it initiates on E major, but Gs is quickly reinterpreted as f4 and resolves to Gn in m. 397; additionally, the predominant is changed to II, rather than fII. 72

For example, see Constantin Floros, Gustav Mahler: The Symphonies, trans. Vernon and Jutta Wicker (Portland: Amadeus Press, 1993), 193; John Williamson, on the other hand, reads the recapitulation at m. 394, and that mm. 338–393 continue to prolong the dominant, “Mahler and Episodic Structure,” 46; another interpretation, given by Henry – Louis de La Grange, is that the recapitulation begins in m. 373, Gustav Mahler, vol. 3, 856.

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Otherwise, there are no significant foreground details to suggest that this iteration of E minor is different from the E minor at the beginning of the exposition. Yet the deeper structure and the identity narrative indicate that a true return to tonic in m. 394 is problematic. In my reading, I show that the identity schism is still unfolding and that an oppositional harmony, and the identity crisis, has not yet been established. To attribute the status of “tonic return” to the E minor in m. 394 would be incongruent with the necessary order of events that engender sonata form. Instead, this E minor is marked as IV of B; it is caught within the dominant prolongation. The following consequent phrase that begins in m. 403 is likewise comparable to the expositional version in mm. 57-64. Both versions of this consequent phrase change the harmonic trajectory from E to B. In the exposition, this progression is understood as I-V; however, the recapitulation, as it is caught within a dominant prolongation, reverses the hierarchical relationship of this progression to IV-I. Also comparable to the exposition is the move towards a prolongation of D at m. 417, which functions as III of B (Fig. 4.17). Afterward, the D prolongation unfolds to Bf in m. 425, and together they form part of a three-dimensional V/V object; the Bf acts as a lower neighbor to Bn in the bass. The motion back to B in the bass is accompanied by a key change to B major in m. 427. As in the shift to B major at m. 80 in the exposition, this section of music facilitates a transition to the second group. The usual normative process of the recapitulation is to transform the music of the transition in order to guide the arrival of the second group towards the tonic and continue the identity reclamation process. Rather than move to the tonic, the second group of the recapitulation is in the key of G major. Another unfolded three-dimensional V/V object facilitates the transition from B major to G major in mm. 436-449 (Fig. 4.17).

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Figure 4.17: Middleground, Mahler’s Seventh Symphony, first movement, mm. 413-478

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One fascinating aspect about the second group in G major at m. 465 is how quickly it moves to a cadential confirmation (Fig. 4.17). In fact, the climactic A on beat 2 in m. 466 is 2 of G, followed by 1 in m. 468 – four measures to reach a cadential close. This expedited arrival is even more significant when we consider that the first iteration of the second group in C major (m. 118) thematically dissolves before it completes a descent – and even then it terminates on an unstable resolution that contains an augmented sixth; the second group in B major (m. 317), while fully fleshed out in texture and harmony, is cut short of reaching its cadence, and is instead interrupted by the funeral march in B minor in m. 338. That G confirms its cadence so immediately after the second group is reprised suggests that it is connected to the B major second group. The cadence that was interrupted in m. 337 is, in a sense, resumed in m. 465, and together they form a complete structure – akin to an antecedent and consequent period. This reading is further supported by the lack of a strong 3 of G, although one can be implied from an inner voice in m. 466. Even with the implied 3, this cadence seems oddly balanced without the weight of some preceding build up, such as the previous B major second group. Mahler implies their connection in that immediately after the cadence on G, the music moves to B minor in m. 471, followed by a quasi-3-line progression D-C-B, or 3-f2-1. The bass moves from B to D in m. 474, which unfolds a local V of G through m. 477 (and over an anticipatory G pedal), and finally to G in m. 478. The cadence on G in m. 468 is where I mark the identity crisis event – that a real Stufe rival has arrived in the background structure. To be sure, I read G major not as III, but as part of the three-dimensional dominant object (marked as [= V] in Fig. 4.17). Because my reading connects the B major and G major second groups into one period, the E minor first group at m. 394 is necessarily caught within this dominant prolongation. The logical question to ask at this

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point is when do we get back to the real tonic Stufe, and what does that tonic mean for the sonata narrative? In terms of a standard sonata analysis, John Williamson reads the structural close at m. 495. 73 However, I find the placement of tonic at m. 495 to be problematic for reasons which I will disclose shortly; I read the real return of tonic (1/I) at m. 515 (Figs. 4.18 and 4.19). Thus, the dominant continues to be prolonged from m. 478, the conclusion of the G major second group, to m. 514. Once again, the putative E minor tonic at m. 495 is caught within a dominant prolongation. The way in which Mahler approaches E minor in m. 495 is paramount to a proper understanding of this movement’s structure. Measures 487-495 are nearly an exact reprisal of mm. 19-27 from the introduction but transposed down a major third (Fig. 4.18); measures 495ff. are more elaborate than mm. 27ff., but the same effect is achieved – and these sections employ the same foreshadowed variation of the first group (trombones in m. 27; trumpets and trombones in m. 495). The structure of the first two sections of the introduction is a descending-fourth progression in the bass: B-Bf-Af-Gf in mm. 1-31. For reasons that I disclosed in my discussion on the introduction, I read Af – which supports the first group variation – as a passing tone that is caught within a prolongation of B. Likewise, I read a descending-fourth progression in the bass from the second group on G (mm. 466-478) to Fs in m. 493, to E in m. 495, and finally to D in m. 507. This D chord in m. 507 does indeed present as a back-relating V9 of G: D-Fs-A-C-Ef; it is also agogically accented for three measures. Thus, G, and consequently the three-dimensional dominant object, is prolonged in mm. 478-508 (Fig. 4.18).

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Williamson, “Mahler and Episodic Structure,” 46; La Grange also puts the beginning of the coda at m. 495, Gustav Mahler, vol. 3, 856; Floros, on the other hand, puts the coda much later at m. 523, which is more aligned with my reading that puts the coda at m. 515, Gustav Mahler, 193.

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Figure 4.18: Middleground, Mahler’s Seventh Symphony, first movement, mm. 478-508

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Immediately after the V9 of G chord, an A minor chord (or IV of E) unfolds in mm. 510-511 and the dominant of opposition arrives in mm. 512-514. I read the A minor as a passing chord that facilitates a voice exchange from B over G at the end of the second group in m. 478 to G over B in m. 512 (Figs. 4.18 and 4.19). This voice exchange validates the shared functional space between G major and B major as different aspects of the three-dimensional dominant object. Figure 4.19: Middleground, Mahler’s Seventh Symphony, first movement, mm. 510-515

In my middleground sketch I have outlined four main voices (SATB) that connect this voice exchange (Fig. 4.20). Beginning in m. 478, the SATB arrangement is B, G, D, and G (marked with beams in the sketches) and moves toward G, E, B, and B in m. 512. The majority

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of the chord in mm. 513-514 is the same as the chord at the end of the introduction, m. 49: B-EG-As. In this case, the high strings and woodwinds unfold the B major triad in m. 514: Ds-Fs(C)-B. Figure 4.20: Middleground, Mahler’s Seventh Symphony, mm. 466-512

Buried within this dense texture, there is also the hint of a 3-line descent in B: Ds (horns, m. 512)-Dn (high strings and woodwinds, beginning of m. 513)-Cs (high strings and woodwinds, end of m. 512, picked up by horns in m. 514)-Cn (high strings and woodwinds, end of m. 514)-B (high strings and woodwinds, m. 515, picked up by horns in m. 516). In terms of rhetorical impact, this dominant chord has the necessary weight to signal the structural close – unlike the putative dominant in mm. 493-494, which is presented more as a fleeting thought (the chord suspended over B in m. 494 is actually V of B). Yet, despite the gravity and satisfaction of this cadence, the dominant in mm. 512-514 is destabilized by its internal mechanics that seem to

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suggest several functions simultaneously. In my discussion of the similar chord in m. 49, I suggested that B-E-G-As represents a conflation of harmonies relative to B; specifically, the chords IV + I + V (which is also representative of the fourths motive). The significance of this conflation is that it interprets E minor as IV of B, which is realized in the large-scale harmonic plan of the sonata form through the phenomenon of function entanglement. In mm. 512-514, the V of B is more pronounced, particularly with the 3-line descent against As in the cellos and trombones. I would suggest that this dominant chord can be understood in the same way as a cubist painting.

4.6

Cubism A few years after Mahler completed his Seventh Symphony in 1905, the cubist art

movement began to develop. One key component for the cubist style was the concept of simultaneous perspectives; that objects could be represented in multiple positions, in space or time, in one painting. This movement was influenced by theoretical explorations of the fourth spatial dimension (although this concept was sometimes conflated, incorrectly, with fourdimensional spacetime). Since four-dimensional objects cannot be perceived in the traditional way (as for a three-dimensional object), they must be represented in a way that compresses the number of dimensions. As a result, the subjects of cubist paintings appear distorted and even grotesque. The fascination with the fourth dimension, both the spatial and temporal types, had been on the rise since the late nineteenth century. In the very early twentieth century, the work of theoretical physicists such as Albert Einstein and Hermann Minkowski laid the mathematical foundation for four-dimensional spacetime. 74 The influence of the fourth dimension is apparent 74

Chiara Ambrosio, “Cubism and the Fourth Dimension,” Interdisciplinary Science Reviews 41, nos. 2–3 (June– September 2016): 202–221.

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in cubism, but its impact was felt throughout culture and society – these concepts were “in the air,” so to speak. Chiara Ambrosio, in their article “Cubism and the Fourth Dimension,” states: That a technical, mathematical notion was appropriated and used by artists at the beginning of the twentieth century is no longer an astonishing discovery, at least for art historians. The past decades have seen the emergence of a lively body of literature offering conjectures on how the concept of the fourth dimension, and the visual sources associated to it, participated in the reformulation of pictorial space that characterised Cubism in particular, as well as a range of subsequent avant-garde movements. 75 I believe it is highly plausible that Mahler was likewise influenced by the possibilities of the fourth dimension. In his essay “The Literary and Philosophical Worlds of Gustav Mahler,” Morten Solvik’s opening statement calls attention to links between musical and philosophical speculation in Mahler: Any thorough understanding of Gustav Mahler and his music must probe the complexities of his thoughts about life and existence. Mahler’s pursuit of these fundamental questions went far beyond idle speculation, haunting his personal reflections and informing his artistic project with a nearly obsessive quality. In significant ways, Mahler’s works represent a response to this existential inquiry, an extension of an overriding need to somehow fathom the universe. 76 One of Mahler’s life-long friends was the physicist Dr. Arnold Berliner. They first met in Hamburg in either 1891 or 1892 (their earliest extant correspondence is dated 9 June 1892). According to Bruno Walter, “Friends of his, professionally occupied with natural science, were hard pressed by his deeply penetrating questions. An eminent physicist whom he met frequently could not tell me enough about Mahler’s intuitive understanding of the ultimate theories of physics and about the logical keenness of his conclusions and counter-arguments.” 77 Berliner,

75

Ambrosio, “Cubism,” 203.

76

Morten Solvik, “The Literary and Philosophical Worlds of Gustav Mahler,” in The Cambridge Companion to Mahler, ed. Jeremy Barham (New York: Cambridge University Press, 2007), 21. 77

Bruno Walter, Theme and Variations, trans. James A. Galston (Westport, CT: Greenwood Press, 1981), 86. Morten Solvik speculates that the physicist in question was most likely Arnold Berliner.

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who later founded the Naturwissenschaften journal (the German counterpart to the British-based Nature) 78, was likely aware of the emerging interest in the fourth dimension. It seems feasible that a discussion of this topic could have occurred at some point between Mahler and Berliner. If we imagine the dominant in mm. 512-514 like a cubist painting, that it represents multiple harmonic dimensions of B in compression, then we can formulate a better sense of its meaning. That it conflates the horizontal progression V-I-IV (of B) into a verticality suggests that the dominant still maintains some amount of control in the hierarchical structure; the dominant and tonic remain caught in the phenomenon that I call function entanglement, and the structural close is marred as a result. For the sonata narrative, this dominant also poses a problem. The identity crisis is never resolved since I do not read a fundamental interruption in the structure of this sonata, or any other compositional device that would reset the background structure and allow the tonic to wrest control. Thus, the dominant in mm. 512-514 is what I refer to as the dominant of opposition. Typically, the dominant of opposition refuses to resolve to 1/I, which necessitates the structural reset and for the tonic to start again. That the dominant of opposition here – which is made even more grotesque and contrarian through its “cubism” treatment – is used for the structural close, denotes that the final event of the sonata narrative, identity reclamation, is not possible and the crisis still ongoing. Instead, the resolution, and the tonic’s reclamation of its identity, must occur outside the first movement. “Super-sonata form” is a concept that seeks to overcome the putative disconnect between movements in a structurally unified multi-movement work. In his book on Tchaikovsky’s Sixth Symphony, Timothy Jackson offers the following description:

78 Sven Thatje, “Dr. Arnold Berliner (1862–1942), Physicist and Founding Editor of Naturwissenschaften,” Naturwissenschaften 100 (2013): 1105–1107.

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In super-sonata form (sometimes called a “sonata-in-one”), the three spatial divisions of sonata form – exposition, development, and recapitulation – are superimposed upon the design of a unified – usually (but not always) continuous – four-movement macrosymphonic form. In this superposition, the first movement generally fills the exposition space containing the first and second groups of a normative sonata form, and the Finale is assigned the recapitulation space and encompasses the recapitulation of the first and second groups. 79 He goes on to explain that the middle movements typically will occupy the development space – or possibly augment the recapitulation space. Thus, the structural separation between movements can be understood as illusory. Jackson concludes that a narrative integration of the individual movements can reveal deeper, macro-structural underpinnings. For the purposes of this dissertation, the remainder of my analysis will focus on the Finale and its reclamation of the tonic’s identity.

79 Timothy Jackson, Tchaikovsky: Symphony No. 6 (Pathétique) (Cambridge: Cambridge University Press, 1999), 26.

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CHAPTER 5 FINALE ANALYSIS 5.1

Introduction One of the most salient features of the Finale is the vast number of key changes that

occur throughout the movement – as well as the keys that are curiously absent. If we can summarize the first movement as a problematization of the three-dimensional dominant object and its relationship to the tonic, then a corresponding summary of the Finale is that it is a problematization of the three-dimensional subdominant object and its relationship to the tonic. In other words, the first movement plus the Finale encompass, at a macro level, the essence of the V-I-IV fourths motive: V-I in the first movement, and I-IV in the Finale. The problematization of the first movement is that the tonic is often caught within a dominant prolongation, and is thus an apparent tonic that is better understood as IV/V. Likewise, in the Finale the tonic is often caught within a subdominant prolongation and is reinterpreted as V/IV. The main key of the three-dimensional subdominant object in the Finale is A major, which might seem strange in the context of a C major movement. However, when we consider the deepest tonic as three-dimensional, and that the first movement emphasized E as the local tonic, then the appearance of A major in the role of subdominant is logical (i.e., A is IV of E). A major also aligns exactly with the B-E-A fourths motive first presented by the trumpets at the end of the introduction to the first movement. Mahler further hints at this connection in that the Finale begins on E minor, mm. 1-6. My analysis largely reflects the expansion of the subdominant, which I read from mm. 219-550, or 56% of the movement.

5.2

Tonic Prolongation As discussed at the end of Chapter 4, the role of the Finale is recapitulatory and to

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reaffirm the tonic’s status. The final event of the sonata narrative, the identity reclamation, was not achieved by the end of the first movement. And as if to account for that lacuna, Mahler devotes the first 219 measures of the Finale to tonic space; 36% of the movement. In contrast, the tonic occupied only 23% of the entire first movement, with the remaining 77% largely under the control of the dominant. The beginning of the Finale also fully encompasses the threedimensional tonic object; after the introduction in E minor, the primary theme group (what Floros and Scherzinger refer to as the ritornello, although they both include the E minor introduction into this group) 80 is presented in C major, mm. 7-51 (the primary theme group consists of theme I in mm. 7-15, theme II in mm. 15-23, theme III in mm. 23-37, theme IV in mm. 38-44, and theme V in mm. 45-51); then, a secondary theme is presented in Af major, mm. 53-78 (fig. 5.1). The way in which Af major arrives is rather curious in that it interrupts C major in the middle of m. 51, and they overlap each other briefly. In my reading, this curiosity is explained as a sudden rotation of the three-dimensional tonic object. What follows is a lengthy prolongation of C major in mm. 79-218, in which there are several reprises from the primary theme group, as well as a tertiary theme in mm. 87-115 (figs. 5.2 and 5.3). There are some deviations from C major in this section, such as D major in mm. 106-119 and A minor in mm. 153-188, but these can both be understood as functions within a C prolongation, II and VI, respectively, and of a predominant nature. The A minor section is notable since it presents the secondary theme, which was originally in tonic space (Af major, mm. 53-78); A minor is further prolonged by its three-dimensional subdominant counterparts Df and F major in mm. 189-191 (marked with a double-unfolding sign in the middleground).

80 Floros, Gustav Mahler, 207; Martin Scherzinger, “The Finale of Mahler’s Seventh Symphony: A Deconstructive Reading,” Music Analysis 14, no. 1 (March 1995): 75.

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Figure 5.1: Middleground, Mahler’s Seventh Symphony, Finale, mm. 1-70

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Figure 5.2: Middleground, Mahler’s Seventh Symphony, Finale, mm. 79-135

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Figure 5.3: Middleground, Mahler’s Seventh Symphony, Finale, mm. 136-218

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This lengthy presentation of A minor seems to foreshadow the forthcoming entanglement between the tonic and subdominant spaces, which begins to unfold with the key change to A major in mm. 220ff. But before Mahler begins the process of entanglement, he confirms the tonic’s status multiple times. The primary theme group cadences on C major in mm. 15 and 23; nested within the first cadence, there is an internal cadence on E major, mm. 13-14: the 3-2-1 is in an inner voice in Horns 1 and 3, against a I-V-I in the bass voices. Additionally, Af major achieves its own cadential arrival in m. 70 within the secondary theme. There are two additional cadential arrivals on C major in the opening tonic prolongation: mm. 136 and 218 (there are also two half cadences in mm. 86 and 196). The length devoted to the tonic space as well as the number of times it is confirmed through cadential arrivals indicates that a resolution to the problem that was presented in the first movement, where entanglement with the dominant rendered the tonic structurally weak and left the identity crisis unresolved, is now achieved. In fact, one could argue that the dominant is almost underrepresented in the Finale. Within the opening tonic prolongation (mm. 1-219), the dominant only appears to facilitate a cadence; it does not receive its own prolongational sections, like the predominant functions II and VI. It is as though the dominant has now been tamed, unlike the grotesque dominant of opposition found at the structural close of the first movement (B-E-G-As-Fs-Ds-Cs/Cn).

5.3

Subdominant Prolongation However, the Finale cannot end here despite the resolution of the initial structural

problem. The Finale needs more weight and so Mahler implements a new problem, which is the entanglement of tonic and subdominant spaces as a mirror to the entanglement of dominant and

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tonic spaces in the first movement. This portion of the Finale can be understood as a coda at the macro, “super-sonata” level, since a motion to the subdominant within the coda space is a common design. The prolongation of the subdominant is a massive part of the Finale, and the mechanics through which it is prolonged are complex. Part of the complication is how to interpret the GfBf-D three-dimensional musical object that unfolds in mm. 307-447. As I mentioned at the beginning of my discussion in Chapter 4, this complex, at the global level, can be understood as V/V or IV/IV (Fig. 4.2). In the Finale, it is caught within a prolongation of the subdominant space, which suggests a IV/IV function. Also, that the Finale lacks a prolongation – or any emphasis at all – on V, further suggests that a V/V interpretation is not warranted. Below I will delineate exactly how this IV/IV space unfolds and fits within the subdominant prolongation. After the key change in m. 220 there is a brief prolongation of A major, but without any cadential confirmation (Fig. 5.4). Instead, the key shifts once again in m. 241 to Df major, which is an enharmonic reinterpretation of the bass voice Cs in mm. 239-240. Although Df is part of the three-dimensional subdominant object, in this context I believe its function is purely the upper third of A. A 3-2-1 descent in Df follows, which picks up the enharmonic 3 of A. The key returns to A major in m. 249 but a cadential motion is once again delayed, this time by a frantic chromatic passage that begins in m. 260. The goal of this passage is a first inversion Af dominant 7th chord in m. 267. My reading suggests that this chord is derived from a chromatic voiceexchange with the Fs chord at the end of m. 259: Cs/Fs to Gf/Cn. The arrival of the Af dominant 7th chord is accompanied by a key change to C major, and in the following measure (m. 268) the primary theme group material returns in C major.

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Figure 5.4: Middleground, Mahler’s Seventh Symphony, mm. 220-290

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However, I do not interpret this C major as a return to tonic; rather, this is the first instance in which the “tonic” is caught within the subdominant prolongation. Measures 267-290 expand the Af-C-E three-dimensional musical object, but here it ultimately functions as V of A. E major, the most literal, one-dimensional V of A, is even marked by its own cadence in m. 278. What follows is a rising-thirds progression; first to Af in m. 283 and then to C in m. 286. At first, this C major presents itself as a real tonic and even moves toward a cadential arrival marked with 2 of V in m. 290. But the cadence is deceptive in that m. 291 returns to A major; the upper-voice descent is actually 3-2-s1, which once again picks up 3 of A. This subversion is a particularly convincing bit of evidence to support a reading that the subdominant and tonic spaces are entangled, and that the tonic here is apparent and caught within a subdominant prolongation. The return to A major is heavily emphasized in that it superimposes themes I and III from the primary theme group and the opening timpani fanfare (mm. 1-6). This polyphonic presentation marks the significance of A major and its role in the global structure of the Finale. That several themes from the tonic space have been co-opted by the subdominant signals the hierarchical inversion and function entanglement. The status of A major is further confirmed by a cadential arrival in m. 303 (Fig. 5.5). Once A major is secured, it opens the harmonic floodgates and allows the music to explore some new, distant key areas from our original tonic. In m. 307 we arrive at the first new key area of Gf major along with a return of the secondary theme. The Gf section is notably long at 52 measures (mm. 307-359) and has three cadential arrivals: mm. 328, 342, and 351. It might seem logical to consider Gf within the context of the global key C major, but I believe we have to place it in relation to the threedimensional subdominant object which continues to make important structural appearances for the remainder of the Finale.

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Figure 5.5: Middleground, Mahler’s Seventh Symphony, mm. 291-351

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As a result, I read Gf as part of the three-dimensional IV/IV object (along with Bf and D); in other words, the Gf section facilitates a plagal extension of A major. The role of the IV/IV plagal extension can be understood, in the deepest sense, as a pedal 6/4: A5/3-6/4-5/3. A pedal point on A can be implied in the bass beneath the Gf section, and later the Bf and D sections as well; an A pedal is literally present in the D major section. In terms of figured bass at the local level, the Gf section moves the upper voices from 8/5/3 to f9/6/3 (Df is enharmonically understood as Cs, or 3). However, at the deepest level of this plagal extension, the Gf section prepares the 6 (Fs) of the 6/4. Perhaps this is the reason there are so many confirming cadences on Gf, as a way to emphasize its deeper structural purpose. The Bf major section (mm. 360-405) directly follows the Gf major section (Fig. 5.6). Here, theme I from the primary theme group is reprised and moves toward a cadential arrival. However, the V of Bf that is reached at m. 367 is interrupted with a rest at the end of the measure; in fact, the arrival on V is prepared with its own V/V and heavily suggests a half cadence. Bf reinitiates in m. 368 with a unison theme that was briefly originally presented in A minor in mm. 186-188. This Bf begins a lengthy rising third in the upper voice back to D in m. 404. In between is a passing tone on C that starts in m. 385 and is interrupted by another frantic chromatic passage that leads to a Df-Bn augmented sixth chord in m. 400, which finally resolves back to C over F in m. 403. Unlike the Gf major section, Bf never confirms a cadence. Instead, the upper voice continues to emphasize D, particularly with the rising third in mm. 368-404. Thus, the Bf section prepares the 4 of the 6/4 plagal extension. In terms of the local figured bass (in relation to A), the upper voices are f9/f6/4.

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Figure 5.6: Middleground, Mahler’s Seventh Symphony, Finale, mm. 360-475

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A section in C major follows with the tertiary theme in mm. 406-433, the unison theme in mm. 434-438, and theme III in mm. 439-442. This section is often read as a return to tonic; however, my reading suggests that this C is caught within the unfolding IV/IV. In the tenor voice, it facilitates a rising third Bf-C-D (mm. 404-447, middleground). The lower voice C initiates a rising third: III-IV-V of A. Along with a 3-2-1 descent, a cadence on A is achieved in m. 446. This cadence on A coincides with a key change to D major, and a D major chord does arrive in m. 447; however, this D major chord has one important distinction: it is in 6/4 position. A pedal A remains in the bass throughout the putative D major section (in the bassoons, tuba, and timpani in mm. 446-454, and picked up by the cellos and double basses in mm. 455-461). Thus, we have arrived at the proper A6/4 chord that began to unfold with the Gf major section in m. 307. At the end of this section in m. 461 the 6 (Fs) and 4 (D) move down to 5 (E) and 3 (Cs). I do not read these as resolutions; rather, I read them as anticipations of the Gs6/4 that arrives in m. 462. Indeed, there is a progression of descending 6/4 chords that repeat the same anticipation pattern: A6/4 (m. 447), Gs6/4 (m. 462), Gn6/4 (m. 476), Fs6/4 (m. 486), and Fn6/4 (m. 492). Once this final 6/4 chord is reached on Fn in m. 492, the pattern changes and there is no quasi-resolution of the 6 and 4 to an anticipation (Fig. 5.7). Instead, the upper voices chromatically ascend (mm. 499-505, middleground). One of the main reasons I read the 5/3s that occur in mm. 461, 472 (partial, 5 only), 485, and 491 as anticipations and not resolutions of the 6/4s is because the pattern freezes on the final F6/4 in m. 492; the 5/3 chords might be understood in terms of Riemann’s Scheinkonsonanzen, or apparent consonances, that default to deeper structures. 81

81 For a discussion of Scheinkonsonanzen, see Alexander Rehding, “Tonality between Rule and Repertory; Or, Riemann’s Functions—Beethoven’s Function,” Music Theory Spectrum 33, no. 2 (Fall 2011): 114.

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Figure 5.7: Middleground, Mahler’s Seventh Symphony, Finale, mm. 476-522

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In my estimation, this change signals that the 6/4 is more important and marks the deeper plagal extension design. Further, that F is marked as the next point of arrival is significant since it is a three-dimensional subdominant counterpart to A. Thematically, the music initially returns to theme I for the first part of the A6/4 in mm. 446-554. Then there is a reprisal of the first group from the first movement throughout the remainder of the descending 6/4 chords, mm. 455-499. The last counterpart of the three-dimensional subdominant object, Df, arrives in m. 506 after the bass descends F-E-D-Df. Here, the key change actually indicates Df major and it is presented as a 5/3 chord instead of a 6/4. Because Df major is in 5/3 position instead of 6/4, I have marked it as an unfolded anticipation to the 5/3 A major chord that begins in m. 531 (in the background structure, A is the operative Stufe). The first group (from the first movement) continues, but now polyphonically set against the tertiary theme. Harmonically, there is no motion from the sustained pedal on Df in mm. 506-516. Without any greater context, this Df might be interpreted as a Neapolitan, especially when one considers that the next bass pitch in m. 517 is G. However, this G does not function as V on the global level. It lacks the rhetorical power of a real dominant as it is presented with a sudden shift to piano, and the upper voice begins the third progression A-G-F. Everything about this measure suggests that its function lies elsewhere; I read it as II of F, and that the cadence on F occurs in m. 521. Although this F could be interpreted as part of the three-dimensional subdominant that has unfolded from A to Df, I read this F on a lower level as III of Df. After a brief stint in E minor in mm. 522-530, the music finally returns to A major in m. 531 (Fig. 5.8). This is where I interpret the resolution of the deep 6/4 plagal extension back to 5/3. At the end of m. 532 the music moves to V9 of A. The bass then arpeggiates from E in m. 532 to Af in m. 533 to C in m. 536 – almost a fully realized three-dimensional V/IV object. I say

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almost because the chord over C is really an Af dominant 7th. Regardless, this section represents V of A, which resolves to A minor in m. 539. Yet, every analysis that I have covered (Floros, La Grange, Downes, and Scherzinger) claim m. 539 as a return to C major. It is true that the upper voices return to C major and initiate theme I, but it is superimposed with the opening timpani fanfare transposed to A minor. The A minor arpeggiation at the bottom of the texture suggests to me that the subdominant prolongation is still ongoing, and that it is now vertically entangled with the tonic for the final presentation of the primary theme group. This insertion of A minor into the final reprise of the primary theme group signals the deeper significance of A and the three-dimensional subdominant space for the Finale’s structure. It also necessitates a recontextualization of the primary theme group’s design. In the original presentation (mm. 7ff.), an E major cadence was nested within the first cadential drive towards C major. Within that context we can assume its function is to facilitate a full unfolding of the threedimensional tonic object (E-C-Af). However, in this final presentation, the nested E major cadence (which returns in mm. 544-545) now suggests a dominant function with the preceding A minor timpani fanfare. Additionally, the first cadence of the primary theme group in this last presentation is deceptive; in m. 546 it is an Aadd6 chord, and not C major. In this sense, it seems Mahler deliberately designed two cadential arrivals in the primary theme group. While it may seem overly joyous, and even redundant, in the original presentation, in the final presentation the second cadence is necessary to achieve structural close on C major (m. 554) and finally disentangle the subdominant.

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Figure 5.8: Middleground, Mahler’s Seventh Symphony, Finale, mm. 522-590

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5.4

Coda With the structural close on C achieved, Mahler proceeds with the rest of the primary

theme group as well as a return of theme I in E major, mm. 573-576; then, a reprisal of the first movement’s primary theme in C major, mm. 581-585. Thus, E major and C major have swapped their primary themes. The coda space in the Finale allows a final recapitulation of the first movement material in order to address the problem that was originally posed: that the tonic’s status was challenged, and indeed in jeopardy. However, only two members of the threedimensional tonic object, E and C, have been reasserted. Gs/Af does make an appearance in the penultimate measure at the end of theme V, the final part of the primary theme group. It coincides with the original presentation when C major was suddenly interrupted by Af major in m. 51. In this case it is not Af major, but rather C augmented; thus, all members of the threedimensional tonic object are present: C-E-Gs (m. 589). In my estimation, this augmented triad represents the deepest structural hierarchy of the Seventh – it is the true tonic sonority from which all other diminutions have been generated.

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CHAPTER 6 CONCLUSION 6.1

Introduction The original intent of this dissertation was to confront the macro structure of Mahler’s

Seventh Symphony, particularly within the confines of a Schenkerian analysis. Since the Schenkerian model privileges tonal hierarchy, the Seventh serves as an excellent example to push that model, from the foreground to the deepest background, to its limits. Indeed, limitations were encountered that required new perspectives and analytical innovations. The most apparent issue when I began this project was the macro symphonic design in which the first and last movements seemed to disagree about the tonic. This initial problem led me to a number of other analytical models that offered some insight. At the macro level, progressive and directional tonality address the problem of a tonal design that seemingly indicates multiple tonic sonorities. However, these models lacked the structural rigor that I hoped to achieve with my analysis. In most cases, I found that the directional tonal readings fit more appropriately within an auxiliary cadence. However, in the case of the Seventh, a large-scale auxiliary cadence, from III to I, felt unsatisfying; I felt that E minor and C major were more strongly related. The argument that texted music could provide more fertile ground for a directional tonal reading – that poetic intentions could supersede a normal tonal hierarchy – seems more plausible. But since text is not pertinent to Mahler’s Seventh, directional tonality did not appear appropriate for my topic. On the other hand, Robert Bailey’s double-tonic complex and Graeme Downes’ axial tonality models offered more robust possibilities. In these models, the boundaries of the traditional tonic are expanded and able to encompass more liberal readings. For Bailey, a large-

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scale work could revolve around a pair of tonic chords, and even manifest in the music as a polychord. This conclusion resonated more clearly with Mahler’s Seventh and its progression from E minor to C major in the outer movements. Downes’ axial system expands a similar concept, but includes other functional spaces as well. These systems are well matched with the voice-leading mechanics of transformational models, particularly the hexatonic cycle, and can likewise reflect Edward Laufer’s concept of the primary sonority. My analytical model, the multidimensional musical object, was inspired by each of these methods and I was able to synthesize their best qualities.

6.2

The Multidimensional Musical Object The initial premise around which I began to design my model was to develop a way to

make E minor and C major hierarchically equal. I believe I achieved this goal with my multidimensional musical object, which became crucial to my analysis of Mahler’s Seventh. It provided a way to connect organically the different keys of the outer movements into the expression of a deeper, more fundamental element of the symphony. Additionally, this model indicated the relevance of Af/Gs, which originally was not apparent to me. Once I established the three-dimensional model (the octahedron that contains E, C, and Af triads), I began to work backwards with the assumption that two- and one-dimensional counterparts must be possible. My inferences led me to correlate modal mixture with the two-dimensional musical object, and pure diatonicism with the one-dimensional musical object. The complete model, with one-, two-, and three-dimensional musical objects, indicated the natural progression of harmonic techniques throughout tonal practice. Diatonicism was most common in the early eighteenth century; modal mixture became more apparent in the late-eighteenth century, and fully explored

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in the nineteenth century; music in the late-nineteenth century began to explore the relationships that are illuminated by my three-dimensional musical object – perhaps the term tonal mixture might be an appropriate label. In this context, modal mixture is a logical precedent to the types of tonal manipulations that are achieved with the three-dimensional musical object. For example, the effect of modal mixture is that two tonic chords are possible instead of one: the parallel major and minor. Even if only one is realized in the music, the other is at least implied through the mixed harmony. The example I provided in my methodology discussion was Beethoven’s Fifth Symphony, which fully realizes C minor and C major. One could choose to read the C major as an apotheosis that supersedes C minor, and that may indeed be the best reading. But I would argue that as the nineteenth century progressed the hierarchical distinction between the parallel modes becomes less defined, even to the point of dualism. This phenomenon is especially apparent in Mahler’s harmonic language, where in one phrase the mode can switch multiple times and even be superimposed. I would postulate that in the same way that the exhaustion of the diatonic system led to modal mixture, so too the exhaustion of modal mixture led to tonal mixture – an ever increasing need to expand the process of tonic prolongation and manipulation. The three-dimensional musical object expands the scope of a tonic from two to six chords. With such a larger pool of harmonic possibilities, it is easy for the tonic to be obscured, or completely misunderstood. Of course, my model suggests that each of those six chords is only part of the tonic, or, more precisely, one side of the three-dimensional tonic object. For a tonic of this type to be fully realized, it would require significantly more space to be composed out. It seems particularly apt for a multi-movement structure, like the symphony, where the outer movements can each unfold a different side of the three-dimensional tonic object.

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6.3

First Movement Conclusions The first movement is, in many ways, the most complicated of the Seventh. A great deal

of that complication is bound up in the movement’s dialogue with sonata form. The implications of a three-dimensional tonic object placed into the sonata process are already daunting, but Mahler further enhances the potential for chaos by manipulating formal expectations. This latter point required me to craft some additional analytical tools, such as the identity narrative. The implement is a series of harmonic events: identity schism, identity crisis, and identity reclamation. These events are then mapped onto the sonata form and help to indicate better the progression of background Stufe. In the case of the Seventh, the identity events are delayed beyond their typical formal boundaries. The result is that the first movement appears to complete the sonata process – one could identify the exposition, development, and recapitulation spaces – but the identity events unfold in a way that leaves the first movement unresolved. To better elucidate this reading, I coined the term dominant of opposition to refer to the dominant that typically precedes the interruption. In my reading, the dominant of opposition is a dominant that, for narrative purposes, refuses to resolve to the tonic. The interruption is then one possible device for the tonic to reassert itself over the dominant and reclaim its identity. For my analysis of the first movement, I do not interpret an interruption. Instead, the dominant of opposition, which manifests as a grotesque and dissonant B sonority, facilitates the structural close and signals that the identity crisis is left unresolved. Another conceptual tool that helped me to unpack the issues in the first movement is function entanglement. The essential premise is that two functional spacesfor the first movement, the tonic and dominantcan become entangled, both horizontally and vertically, in a way that obscures or inverts the normal hierarchy. In this way, I could disclose the areas of the

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movement where the tonic is merely apparent, and actually caught within a dominant prolongation. This tool also allowed me to better interpret some of the strange surface sonorities; for example, the first chord of the movement, the Gs half-diminished 7th in first inversion, is understood as B minor, from the dominant, as conflated with Gs, a primary member of the threedimensional tonic object. That the dominant and partial tonic are conflated reveals the fundamental problem of the first movement, which is largely made manifest through the identity schism and crisis narrative events. In my reading, embedded in the fourths motif, one of the main motivic elements of the first movement, is the tonic and dominant problem. I interpret each pitch of the fourths motif as representative of harmonic functions; thus, B-E-A is understood to represent V-I-IV. Harmonies that are separated by a fourth are context dependent in order for their functions to be determined. Thus, B to E could be understood as V-I or I-IV. It is this rather simple device that Mahler employs to present the tonal drama throughout the first movement. Each of these analytical devices helps to inform the unfolding three-dimensional musical objects that encompass the movement. The three-dimensional tonic object unfolds from the E minor first group to the putative C major second group. Then, the identity schism occurs and the three-dimensional dominant object unfolds through descending thirds: B-G-Ef-B. The final third descent marks the arrival of the real B major second group. Afterward is the recapitulation of the E minor first group, but this “tonic” is caught within the dominant prolongation. The dominant is picked back up by the recapitulation of the G major second group, which concludes the half cadence found at the end of the B major second group. Without the three-dimensional musical object, these function regions, as well as the sonata narrative, could easily be misunderstood.

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6.4

Finale Conclusions Traditionally one of the most criticized of all Mahler’s output, 82 a great deal of the Finale

is only made comprehensible after the structure of the first movement is clarified. Martin Scherzinger summarizes the qualities that have generally led to the confusion about the Finale: The lack of a clearly identifiable development section (customary for a movement of this size), together with the persistent cadencing (though not even the cadences necessarily coincide with the structural points of the movement), serves to undermine, rather than underscore, the overall logic of the finale. It seems impossible, then, to designate a sonata-like formal division. In fact, except for the passages beginning at b.249 and b.368, there are few passages that have a forward-pressing character at all. 83 The main criticisms, that the Finale lacks development and is overly diatonic (the persistent cadences), can be dealt with in the context of the identity narrative. At the end of the first movement the identity narrative is left unresolved with the identity crisis in process. That the identity narrative needs to conclude with the reclamation event is the impetus for the Finale’s design. The reason that the Finale lacks any significant development is because the first movement was heavily weighted with the development part of the narrative. The Finale’s ultimate purpose is to reassert the tonic’s identity, which is largely achieved through a series of emphatic, tonic-confirming cadences – and on all three primary members of the threedimensional tonic object: E-C-Af. After the lengthy tonic prolongation, the remainder of the Finale can be understood as occupying the coda space at the super-sonata level. In order for the Finale to be weighted properly, particularly when compared to the first movement, Mahler instigates a new problem: the entanglement between tonic and subdominant (I-IV). Apart from the subdominant as a

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See James L. Zychowicz, “Ein schlechter Jasager: Considerations on the Finale to Mahler’s Seventh Symphony,” in The Seventh Symphony of Gustav Mahler: A Symposium, ed. James L. Zychowicz, 98–106 (Madison, WI: A – R Editions, Inc., 1990). 83

Scherzinger, “The Finale of Mahler’s Seventh,” 76.

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typical harmonic diversion in the coda space, its presence in the Finale is also motivic in that it mirrors the V-I entanglement in the first movement. Thus, the first movement and the Finale together complete the ascending fourths motif: V-I-IV. The plagal design of the Finale is further emphasized in that the three-dimensional subdominant object is prolonged with its own plagal extension, or the three-dimensional IV/IV object (Gf-Bf-D). This particular aspect of my reading is perhaps the most adventurous, but I believe it is a more logical conclusion in the context of the first movement’s design. There the emphasis is on the three-dimensional dominant object, and its corresponding three-dimensional V/V object (Fs-As-D) is employed several times – in some cases through a double unfolding. The general harmonic plan for the first movement could be summarized as V/V-V-I, and for the Finale as I-IV-IV/IV.

6.5

Future Projects and Final Thoughts Now that I have fleshed out the multidimensional musical object as an analytical tool, I

plan to utilize it for other analyses. The most immediate case would be the middle movements of Mahler’s Seventh Symphony, and to discover how they inform the three-dimensional tonic object interpretation. But the most interesting endeavor would be the exploration of other latenineteenth-century composers, such as Wagner, Brahms, Liszt, Wolf, and Strauss, and to discover how relevant this model is in their music. In the music of these composers there are certainly examples of the types of major-third relationships that are so prevalent in Mahler’s music, and it would be significant to discover that any of them use a three-dimensional tonic object – or if their implementation of major-third relationships is less fundamental to the background structure. As for Mahler’s implementation of this harmonic language, it seems deliberate and

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appears at all structural levels. The usage of these three-dimensional musical objects pushes the tonal system into new territory and opens new structural possibilities. In my estimation, one of Mahler’s main compositional goals – in general, but specifically with the Seventh – is to illuminate the fragility of the tonic’s status in this late-nineteenth-century musical style. That the tonal system is about to reach a breaking point.

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