sign in sign up
<<
Home , Chromatic mediant, Diatonic and chromatic

University of Calgary PRISM: University of Calgary's Digital Repository

Graduate Studies The Vault: Electronic Theses and Dissertations

2018-04-25 Trans-Diatonic Theory and the Harmonic Analysis of Nineteenth-Century de Andrade Negri, Alexandre Jorge Jr de Andrade Negri, A. J. (2018). Trans-Diatonic Theory and the Harmonic Analysis of Nineteenth-Century Music (Unpublished master's thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/31872 http://hdl.handle.net/1880/106586 master thesis

University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY

Trans-Diatonic Theory and the Harmonic Analysis of Nineteenth-Century Music

by

Alexandre Jorge de Andrade Negri Júnior

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF ARTS

GRADUATE PROGRAM IN MUSIC

CALGARY, ALBERTA

APRIL, 2018

© Alexandre Jorge de Andrade Negri Júnior 2018 ii

Abstract

A hallmark of nineteenth-century music is the conscious and systematic exploration of ‘non- normative’ harmonic relations especially by composers such as Liszt, Brahms, Wagner, and others. Non-normative in this context is taken to refer to any harmonic relation that fulfills the following two conditions: (1) at least one, but normally both chords (or harmonic regions) are chromatic to each other and/or to their context; and (2) the intervallic relationship between their roots is derived from a symmetrical division of the (i.e., they must be either a , a whole-tone, a , a , or a tritone apart). By their nature, diatonic-based analytical systems do not have the necessary tools or flexibility to explain Romantic non- normative harmonic relations, as defined above. This thesis proposes a new approach to analysis of later nineteenth-century . In chapter 1, this thesis considers four modern theories of harmony that were developed to address the issues raised above. The theories examined are by Gregory Proctor (1978), Harald Krebs (1980), David Kopp (2002), and Richard Cohn (1996, 2000, 2012). Because all of these theories are ultimately rooted in an adaptation of conventional diatonic analytical practices or concentrate upon very limited aspects of non-normative harmonic relations (mainly, chromatic relations), they are found to be inadequate to deal with characteristically difficult passages of nineteenth-century harmony. In chapter 2, this thesis proposes a new theory of harmony, entitled ‘trans-diatonic theory.’ The key concept of trans-diatonic theory is the ‘diptych principle,’ which fuses parallel keys (and chords) into a new, greater unit. The diptych principle generates a trans-, which provides richer, more complex harmonic relations that, in turn, allows for a more logical understanding of non-normative harmonic relations. The final chapter shows the practical advantages of the trans-diatonic system through the analysis of a variety of harmonically challenging passages from different works. These analyses demonstrate the use and effectiveness of the trans-diatonic theory.

iii

Preface

This thesis is original, unpublished, independent work by the author, Alexandre Negri.

iv

Acknowledgements

My first debt is to my advisor, Dr. Kenneth DeLong, whose never-ending support and steady guidance have been of the utmost importance to me during these last two years. Without his invaluable advices, this thesis would not have been possible. To my exam committee members,

Drs. Joelle Welling, Allan Bell, Leonard Manzara and Adam Bell, my deepest gratitude for their engagement with my work and invaluable insights and contributions. My heartfelt thanks to the music faculty as well, specially to Drs. Friedemann Sallis and Laurie Radford for their comments on earlier stages of my research. To my friends and colleagues, Dr. J. Orlando Alves and Lauro

Pecktor, my sincere thanks for their candid observations during various stages of my work. My sincere gratitude to the School of Creative and Performing Arts staff members, especially to

Alison Schmal, for their constant help and support.

I would like to extend my gratitude to my dear friends, Júlio and Andrea, Paulo and Ana

Amélia, and Giselle whose sincere friendship and sense of community have made the absence and distance from home less daunting. To my dear wife, Taciana, to whom this work is dedicated, there are no words to express my deepest regard and admiration for her unconditional support and motivation.

The greatest gratitude to my family. To my grandmother, Martha, for the first lessons in music and theory. To my sister, Camila, for always believing in me. To my parents, Alexandre and Fátima, for the precious gift of life and unwavering, enthusiastic support. None of this would have been possible without them.

v

For Taciana

vi

Table of Contents Abstract...... ii Preface...... iii Acknowledgments...... iv Dedication...... v Table of Contents...... vi List of Tables...... vii List of Figures...... viii List of Musical Examples...... ix Introduction: The Need for a New Harmonic System...... 1 1. Chapter 1: Modern Theories of Harmony...... 8 1.1 Gregory Proctor...... 8 1.2 Harald Krebs...... 11 1.3 David Kopp...... 13 1.4 Richard Cohn...... 20 2. Chapter 2: The Trans-Diatonic Theory...... 35 2.1 The Diptych Principle...... 35 2.1.1 Towards Mode Fusion...... 36 2.1.2 The Trans-Diatonic Scale...... 38 2.2 Orders of Complexity and Tonnetzing...... 46 2.2.1 Third Order of Complexity...... 48 2.2.2 Second Order of Complexity...... 54 2.2.3 First Order of Complexity...... 57 2.3 Trans-Diatonic Transformational System...... 60 3. Chapter 3: Trans-Diatonic Analyses...... 67 3.1 Franz Schubert...... 67 3.1.1 Mass in Eb Major, Sanctus, mm. 1-13...... 67 3.1.2 Piano Sonata in D Major, I, mm. 1-16...... 69 3.1.3 Die Zauberharfe, Overture, mm. 1-47...... 71 3.1.4 Die Sterne...... 72 3.1.5 String Quartet in G Major, I, three excerpts...... 73 3.2 Johannes Brahms, Ein deutsches , II, mm. 261-273...... 77 3.3 Franz Liszt...... 79 3.3.1 Polonaise I from Die legend vom heiligen Stanislaus, two excerpts...... 79 3.3.2 Lélio Fantasy, mm. 195-198...... 82 3.4 Cesar Franck, Quintet for Piano and Strings, I, mm. 90-107...... 83 3.5 ...... 84 3.5.1 Tarnhelm progression...... 84 3.5.2 Parsifal...... 85 3.5.2.1 Act III, mm. 1098-1100...... 85 3.5.2.2 Act I, mm. 241-257...... 86 3.5.2.3 Overture, mm. 44-55...... 87 4. Final Remarks...... 90 5. Bibliography...... 93

vii

List of Tables

Table 1.1: Kopp’s mediant relations in C major...... 17 Table 1.2: Cohn’s unified model transformations...... 32

viii

List of Figures

Fig. I.1: Symmetric divisions of the octave...... 4 Fig. I.2: Cohn’s “Five notations for an equal division.” ...... 5 Fig. 1.1: Proctor’s ...... 9 Fig. 1.2: Relation between the diatonic and trans-diatonic systems...... 10 Fig. 1.3: Kopp’s functional demonstration of mode-preserving third relations...... 15 Fig. 1.4: Kopp’s scheme on Schubert’s Die Sterne...... 18 Fig. 1.5: Kopp’s full set of common-tone relations from C...... 20 Fig. 1.6: Cohn’s hyper-hexatonic system...... 21 Fig. 1.7: Cohn’s union of Weitzmann water bugs and his hexatonic groups...... 27 Fig. 1.8: Cohn’s graph of SSD-relations among and between the twenty-four Klänge and the four augmented triads (Cube Dance) ...... 28 Fig. 1.9: Cube Dance Schubert’s Overture to Die Zauberharfe, mm.1-47...... 29 Fig. 1.10: A proposed model for the hypothetical hyper-octatonic system...... 31 Fig. 2.1: Hauptmann’s minor-...... 39 Fig. 2.2: Schenker’s ‘The Six Products of Combination.’ ...... 40 Fig. 2.3: Marra’s ‘tonal forms.’ ...... 41 Fig. 2.4: Fusion of the scales into the trans-diatonic scale...... 43 Fig. 2.5: Trans-diatonicism as the missing link between diatonicism and ...... 44 Fig. 2.6: Trans-diatonic triadic collection...... 46 Fig. 2.7: Steps towards the trans-diatonic Tonnetz of the third order of complexity...... 48 Fig. 2.8: Minimal expansion of the trans-diatonic Tonnetz of the third order of complexity...... 51 Fig. 2.9: Non-redundant trans-diatonic Tonnetz of the third order of complexity...... 52 Fig. 2.10: Diptych Tonnetz of the third order of complexity...... 54 Fig. 2.11: Normative diptych pair resolutions of a ...... 55 Fig. 2.12: Normative resolutions of an ...... 56 Fig. 2.13: Diptych Tonnetz of the second order of complexity...... 57 Fig. 2.14: Five Tonnetze of the first order of complexity...... 58 Fig. 2.15: Layered Tonnetz of the first order of complexity...... 59 Fig. 2.16: Diptych Tonnetz and its underlying axial structure...... 62 Fig. 2.17: Trans-diatonic transformations...... 63 Fig. 2.18: Trans-diatonic transformational cycles...... 64 Fig. 2.19: Trans-diatonic transformations in music notation...... 66 Fig. 3.1: Schubert, Die Zauberharfe, Overture, mm. 1-47...... 72 Fig. 3.2: Modulation path of Schubert’s Die Sterne...... 73

Fig. 3.3: Harmonic path of Ex. 3.6...... 77 Fig. 3.4: Trans-diatonic bass outline of Ex. 3.7...... 78 Fig. 3.5: Diptych Tonnetz of the second order of complexity of Ex. 3.7...... 79

Fig. 3.6: Overall harmonic plan of Ex. 3.9...... 81 Fig. 3.7: Expanded Tonnetz of the second order of complexity of Ex. 3.10...... 83 Fig. 3.8: Harmonic scheme to Franck’s Quintet for Piano and Strings, I, mm.90-107...... 84 Fig. 3.9: Overall harmonic plan of Ex. 3.11...... 85 Fig. 3.10: Diptych Tonnetz of Ex. 3.14...... 88

ix

List of Musical Examples

Ex. 1.1: Schubert, Piano Sonata in D Major, I, mm. 1-16...... 11 Ex. 1.2: Liszt, Polonaise I from Die Legende vom heiligen Stanislaus, mm. 98-103...... 22 Ex. 1.2: Continuation, mm. 104-110...... 23 Ex. 1.3: Franck, Quintet for Piano and Strings, I, mm. 90-93...... 24 Ex. 1.3: Continuation, mm. 94-101...... 25 Ex. 1.3: Continuation, mm. 102-106...... 26 Ex. 2.1: Liszt, Polonaise I from Die Legende vom heiligen Stanislaus, mm. 1-20...... 45 Ex. 3.1: Schubert, Mass in Eb Major, Sanctus, mm. 1-13...... 69 Ex. 3.2: Tymoczko’s analysis of the opening of Schubert’s Piano Sonata in D Major...... 70 Ex. 3.3: Schubert, Piano Sonata in D Major, I, mm. 1-16...... 71 Ex. 3.4: Schubert, String Quartet in G Major, I, mm. 1-14...... 74 Ex. 3.5: Schubert, String Quartet in G Major, I, mm. 15-24...... 75 Ex. 3.6: Schubert, String Quartet in G Major, I, mm. 54-63...... 76 Ex. 3.7: Brahms, Ein deutsches Requiem, II, mm. 261-267...... 77 Ex. 3.7: Continuation, mm. 268-273...... 78 Ex. 3.8: Liszt, Polonaise I from Die Legende vom heiligen Stanislaus, mm. 20-36...... 80 Ex. 3.9: Liszt, Polonaise I from Die Legende vom heiligen Stanislaus, mm. 98-110...... 81 Ex. 3.10: Liszt, Lélio Fantasy, mm. 195-198...... 82 Ex. 3.11: Wagner, Tarnhelm progression...... 85 Ex. 3.12: Wagner, Parsifal, act III, mm.1098-1100...... 85 Ex. 3.13: Wagner, Parsifal, act I, mm. 241-251...... 86 Ex. 3.13: Continuation, mm. 252-257...... 87 Ex. 3.14: Wagner, Overture to Parsifal, mm. 44-48...... 87 Ex. 3.14: Continuation, mm. 49-55...... 88 Introduction The Need for a New Harmonic System

Everywhere, Romanticism exploits the ability to hear one and the same phenomenon in two or more ways; it is fond of this coexistence and its indefiniteness...1 Ernst Kurth

Despite the rekindled interest in nineteenth-century harmonic practices seen in the past few decades, there is at present still no analytical model in place that is at the same time systematic and flexible enough to engage straightforwardly with challenging aspects of

Romantic harmony. Nineteenth-century harmonic practices are often treated as an extension of that of the previous century. A consequence is that characteristically Romantic harmonic procedures are seen as ‘non-normative’ when they do not conform to Classical expectations.2

A favorite non-normative harmonic relation, widely employed throughout the nineteenth- century, is the , or chromatic third relation, on both the foreground and middle ground structural levels. Indeed, almost the entirety of the current literature on Romantic harmony concentrates its efforts towards ‘normalizing’ chromatic third relations. Herald Krebs, taking a Schenkerian stand, examines “the degree of connection between third-related triads

(mediant and ) and the dominant within [large scale music structures] in the late eighteenth and early nineteenth centuries.”3 David Kopp elevates mode-preserving to a functional status, alongside Tonic, Dominant, and . Richard Cohn erects an entire

1 Lee Rothfarb, Ernst Kurth: Selected Writings, (New York: Cambridge University Press, 2006), 134. 2 By non-normative I mean any harmonic relation that in a greater or smaller degree undermines the diatonic Tonic-Dominant axis imperative. Unless otherwise stated, ‘harmonic relation’ will always refer to the relation between two triads or harmonic regions. For lack of a better term, ‘normative’ and ‘non- normative’ will be used throughout this essay in a provisory fashion. 3 Harald M. Krebs, “Third Relation and Dominant in Late 18th- and Early 19th-Century Music” (Ph.D. diss., Yale University, 1980), (abstract) i. 2 analytical system that privileges major third harmonic relations as well as minimal voice leading displacements, and so forth.

However, the remaining non-normative harmonic relations, such as the ones based on the semitone, the whole-tone, and the tritone intervallic relationship, are yet to receive the same attention as their mediant cousins. In other words, there has been an exaggerated theoretical concentration in recent scholarship on third related harmonic relations to the detriment of the remaining non-normative relations. The purpose of this research is to propose a new harmonic model in which nineteenth-century ‘normative’ and ‘non-normative’ harmonic practices are equally and non-hierarchically contemplated in a manner that is systematic, yet flexible enough to deal with highly chromatic music.

In chapter 1, I briefly consider some of the most recent developments of Romantic harmony theories and concentrate on Kopp and Cohn’s systems. In chapter 2, in face of the shortcomings of the theories explained in chapter 1, I lay out my own theory of Romantic harmony, which I call ‘trans-diatonic.’ As will be elaborated in chapter 2, the trans-diatonic theory is based on a refined concept of mode mixture – the diptych principle – and has the advantage of being able to contemplate all triadic non-normative harmonic relations in conjunction with all normative ones. In chapter 3, I demonstrate the validity of my theory with analyses of several passages from the literature, as well as a trans-diatonic version of the analyses discussed in previous chapters. Finally, I offer an overview of this essay and my final considerations. In what follows, I will briefly discuss some of the properties of non-normative harmonic relations, consider the limitations of the traditional diatonic system in dealing with them and, by doing so, justify the need for a new harmonic system. 3

All non-normative harmonic relations are chromatic, from a diatonic perspective.

However, chromaticism by itself is not enough to render a harmonic relation ipso facto non- normative. As it is often the case, chromatic relations act as a local enforcer of diatonic syntax or as a stylized coloring, or effect, such as a secondary Dominant or a minor subdominant within a major mode context, respectively. Thus, within a well-established diatonic context, a chromatic event can be easily understood as subordinate to it. Therefore, in order to become non-normative de facto, i.e., diatonically incompatible, a harmonic relation must fulfill the following two conditions: first, as already discussed above, at least one, but normally both chords (or harmonic regions) must be chromatic to their context and/or to one another; and second, and most importantly, the intervallic relation between their roots must be derived from a symmetrical division of the octave; in other words, they must be either a semitone, a whole-tone, a minor third, a major third, or a tritone apart – anything but a fifth.

Consider a modulation from C minor, as tonic, to Eb major.4 Although it fulfills the second condition stated above, the relation between C minor and Eb major is not chromatic. They are actually each other’s key relative, which amounts to one of the highest degrees of diatonicity there is. Now, consider a further modulation from Eb major to Gb major. This is indeed a non- normative harmonic relation: it is chromatic and a ‘symmetrical interval’ (minor third) apart. A diatonically driven approach would be to quickly resort to some degree of modal mixture or

4 Throughout this essay, I will strive to be the most efficient possible regarding chord labelling and Roman numeral assignment. As I see it, efficiency means to use the least number of symbols or characters to convey the most information possible. Hence, rather than saying C major or C minor, or CM or Cm, or C+ or C-, or any combination thereof, my analyses will use an uppercase letter to indicate major and a lowercase letter to indicate minor mode chords and keys – thus, ‘C’ and ‘c’ respectively. Diminished chords will display a lowercase notation with a superscript ‘o’ symbol and Augmented chords will present an uppercase notation with a superscript ‘+’ sign, e.g., ‘co’ and C+. The same notions apply to Roman numerals symbols and to harmonic regions. However, while writing, I will abide to the traditional ‘C- major’ or ‘C-minor’ terms to indicate chords, and ‘C major’ or ‘C minor’ to indicate keys. 4

substitution to account for this relationship. However, this pseudo-explanation fails to

satisfactorily account for the macro-relationship between the initial C minor and Gb major. Going

even further, consider now a modulation from the enharmonically equivalent F# major to A

major. Once again, this is a non-normative relation easily dismissed in its most superficial,

immediate level as a chromatic mediant relation, yet not contemplated in its larger, overall

harmonic logic. Note that as one consistently continues down the road of symmetrically-based

intervallic relations, it gets increasingly difficult to bend diatonicity to encompass the resulting

harmonic relations, which raises the question: are we in a diatonic environment? Apparently not.

Then, why press it for diatonically oriented analyses?5

By consistently dividing an octave by symmetrical intervals, one arrives at one of the five

structures depicted in Fig. I.1 below. The division by semitone renders a chromatic scale (Fig.

I.1a); by whole-tone, a whole-tone scale (Fig. I.1b); the division by minor third outlines a fully

diminished (Fig. I.1c); by major third, an augmented triad (Fig. I.1d); and by

tritone, the symmetric partition of the octave in two equal parts (Fig. I.1e).

a. b. c. d. e. bœnœbœnœœ #œœ œœ œ œ & œ#œœbœnœœ#œœ œ œœ#œ#œ œbœ#œ œ œ#œ œ#œ

Fig. I.1: Symmetric divisions of the octave. ∑ ∑ ∑ ∑ ∑ ∑ ∑ & Clearly, the resulting harmonic relations achieved among chords whose roots are based {on? the intervallic∑ relations∑ depicted above∑ defy diatonicity.∑ According∑ to Cohn,∑ “[t]he limitations∑ [of diatonically based models] leap most rapidly into view in the familiar case of triadic

∑ ∑ ∑ ∑ ∑ ∑ ∑ 5 I &will address this issue shortly. Incidentally, this was not a mere hypothetical scenario. The same modulation path is taken by Schubert in the introduction to his overture to Die Zauberharfe. To be sure, Schubert? reiterates the same process one last time, modulating from A major to C major, thus finishing {the Andante ∑introduction with∑ an overall∑ non-diatonic ∑C minor to C ∑major arch. In∑ chapter 1, I will∑ critique Cohn’s analysis of this passage and, in chapter 3, I will offer my own.

{D(')} {R(')} {D} {D'} {R} {R'} {X} œ bœ œ bœ bœ œ œ #œ b bœ n #œ & ˙ b˙ œ œ œ œ b œ bœ b œ # œ D d D' d' R r r' R' X x & ∑ ∑ ∑ ∑ ∑ ∑ {? ∑ ∑ ∑ ∑ ∑ ∑ {L(')} {W(')} {S(')} {L} {L'} {W} {W'} {S} {S'} bœ bœ & œ #œ b œ œ œ œ bœ œ œ œ œ #œ bœ bœ b œ bœ #œ #œ l L L' l' w W W' w' S s s' S' 5 & ∑ ∑ ∑ ∑ ∑ ∑ progressions whose roots divide the octave in equal parts. […] The standard term ‘equal division of{ the? octave’∑ captures intuitions∑ about two∑ types of equivalence∑ [shown∑ in Fig. I.2 below]:∑ between the individual segments of the span, and between its two boundary events. Classical methods of analysis can capture either of these intuitions individually, but not both simultaneously.” 6

a. b. c. d. e. ˙ ˙ b˙ ˙ ˙ ˙ b˙ b˙ ˙ ˙ #˙ n˙ ˙ #˙ ˙ n˙ ‹˙ #˙ ˙ n˙ & #˙ n˙b ˙n#˙ #˙ n˙b ˙b ˙ #˙ n˙## ˙ ˙ #˙‹‹˙ #˙ ˙ #‹‹˙‹ ˙##˙ ˙ {? ˙ ˙ ˙ ˙ ˙ ˙ ˙ #˙ ‹˙ #˙ b˙ n˙ b˙ b˙ #˙ ˙ #˙ ˙ #˙ ˙ E: I ¼VI ¼IV I I ¼VI ¼IV ¼¼II I ¼VI III¾ I I ¾V III¾ I ¾VII¾ ¾V III¾ I

Fig. I.2: Cohn’s “Five notations for an equal division.”7

Cohn’s examples sound all the same, however, their respectively Roman numeral analyses differ greatly among each other. Cohn arguments that one may either choose to consistently& ∑ depict the∑ major∑ third intervallic∑ relation∑ ship∑ between∑ adjacent∑ chords∑ (Fig. I.2b∑ . and e.{), ?or to stablish∑ both∑ the first∑ and final∑ chord ∑as identical∑ and adjust∑ one’∑s Roman∑ numeral∑ analysis accordingly (Fig. I.2a., c., and d.). He adds that, “[t]he problem cannot be laid solely at the door of the fundamental bass tradition. Equal divisions are equally paradoxical from a

Schenkerian/linear perspective, in part because they erode the fundamental distinction between consonance and dissonance. […] The conceptual problem springs from the same source as the notational problem: the culprit, in both cases, is the diatonic scale.”8 I would argue, however, that the ‘culprit’ is not the diatonic scale itself, but rather the widespread need to ‘force’ clearly non- diatonic phenomena into a diatonic syntax. Additionally, since they all sound the same, we do

6 Richard Cohn, “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions.” Music Analysis 15, no. 1 (1996): 9. 7 Cohn, “Maximally Smooth Cycles,” 10. 8 Idem, 11. 6 not have an actual musical problem, merely a notational one. The ‘tyranny of the eye’ is a strong force to be overcome by ear-oriented (i.e., musical) analysis.

Cohn further comments that “[c]lassical analytical methods, whether oriented towards basse fondamentale or towards Auskomponierung, follow the notational system in assuming the diatonic determinacy of events. Yet the relationship between the constituents of a symmetrical division and the diatonic Stufen is fundamentally indeterminate. A symmetrical division of the chromatic twelve cannot also be a symmetrical division of the diatonic seven without engaging with some sleight-of-hand.”9 Indeed, there is no scenario in which the harmonic relations derived from the symmetrical divisions of an octave can coherently coexist within the asymmetry of the diatonic system – they are utterly incompatible.

And yet, the standard approach towards Romantic harmony, in which non-normative relations are widely employed, is to force the music to fit diatonic models of analysis. On that note, Kopp writes that “[m]ost of our prevailing analytic models and methods, predicated on eighteenth-century practice, have traditionally explained chromatic music as the elaboration of diatonic structures. The music’s frequent lack of conformity with these models has often been interpreted as a sign of weakness or inferiority in the music itself rather than due to any inappropriateness of the model.”10

Indeed, there are many elements in common between eighteenth- and nineteenth-century music: the blatantly obvious would be the use of triads as basic harmonic units, that are connected to one another by common principles of voice-leading. However, the manner in which chord progressions are treated differs considerably between the Classical and the Romantic era.

9 Idem. 10 David Kopp, Chromatic Transformations in Nineteenth-Century Music (New York: Cambridge University Press, 2002), 1. 7

Arguing this same point, Cohn writes that “[t]he central position of [major and minor triads] constitutes the strongest bond of continuity that connects eighteenth-century music to the progressive music of the last quarter of the nineteenth-century, and the strongest motivation for the prevailing practice of adapting analytical approaches to late from models forged for music of the Classical era. But the diatonic indeterminacy of triadic progressions in late Romantic music often limits the efficacy of these diatonically based models.”11

Having contextualized the inefficacy and inappropriateness of Classical models of harmonic analysis, the need for an actual Romantic analytical system of harmony becomes self- evident. It is the purpose of this research to propose such a model. As I will argue in chapter 2 and demonstrate in chapter 3, the trans-diatonic theory of harmony possesses the necessary conceptual flexibility to be able to straightforwardly deal with normative (diatonic) and non- normative (non-diatonic) harmonic relations at the same time, without resorting to ‘corrupt diatonicity’ explanations or general chromaticism categorizations.

11 Cohn, “Maximally Smooth Cycles,” 9. Chapter 1 Modern Theories of Harmony: As discussed in the introduction, issues with harmonic analysis of nineteenth-century music spring mainly from the inability of the diatonic system, and diatonic-based models, to deal with harmonic relations based on the symmetric divisions of the octave (see Fig. I.1). Facing the limitations of diatonic-based systems, a plethora of theoretical system and harmonic models was put forward in older and current literature.1 From the multitude of perspectives, I have chosen four modern theories of harmony that will enable me to further develop the issue of non- normative harmonic relations as well as to set the stage for my own theory in the following chapter. To be sure, the theories discussed in this chapter do not engage with the full spectrum of non-normative harmonic relations resorting to general, broad considerations on chromaticism and/or concentrating their attention on specific, isolated issues, such as chromatic thirds relations.

1.1 Gregory Proctor

In face of the non-normativeness of some nineteenth-century harmonic procedures,

Gregory Proctor argues for two distinct types of : one being the ‘classical diatonic tonality’ and the other ‘19th-century chromatic tonality.’ On the one hand, classical diatonic tonality has its harmonic relations governed by the diatonic scale and eventual chromatic perturbations are seen as the result of the combination of different diatonic collections. Proctor adds that “Schenker’s formulations of mixture and tonicization [can] adequately account for such traditional chromatic constructs as: the diminished seventh chords, Neapolitan chords,

1 For a comprehensive, historically oriented examination of theories of harmony focusing on harmonic third relations, see Kopp, Chromatic Transformations, chapters 3 to 5. 9 chords, and augmented triads.”2 On the other hand, 19th-century chromatic tonality is harmonically based not in any particular diatonic scale, but rather in the ‘equally- tempered twelve pitch-class collection’ provided by the chromatic scale. Conversely, in chromatic tonality, diatonic events are “construed to be a special derivative of [the] underlying chromatic scale.” Fig. 1.1 below schematically depicts how Proctor’s tonalities interact. They can be both seen as independent, but interactive systems in which overlapping elements from one are always perceived as disturbances on the other.

Fig. 1.1: Proctor’s diatonic and chromatic tonalities.

The notion of two distinct tonalities in Proctor has a similar origin as my previous formulation of normative and non-normative, i.e., diatonic and non-diatonic, harmonic relations.3

However, there is a crucial distinction between Proctor’s proposition and mine. Whereas Proctor sees both systems as quasi-independent from each other, with elements from one having a cursory presence and influence on the other, I see the existence and raise of non-normative harmonic relations as a symptom of the end of classical diatonicity and the establishment of a new system – the trans-diatonic system – in which all harmonic relations are to be re-signified by its new logic. By comparing Fig. 1.1 above with Fig. 1.2 below one can have a clear idea of how

Proctor’s approach differs from mine.

2 Gregory M. Proctor, “Technical Bases of Nineteenth-Century Chromatic Tonality: A Study in Chromaticism” (Ph.D. diss., Princeton University, 1978), iv. 3 Note the tacit bias in my nomenclature. It assumes diatonicity as the norm and any departure from it as a deviation from that norm. The theory I propose in the next chapter bridges the gap between what might be considered normative and non-normative from a diatonic perspective by considering all triadic harmonic relations as equally valid under the new trans-diatonic system. 10

Fig. 1.2: Relation between the diatonic and trans-diatonic systems.

Proctor’s notion of separateness between both systems, which is later developed by Cohn under the banner of ‘double syntax’ (see section 1.4 below), invariably favors a narrow view of the harmonic phenomena. The somewhat artificial alternation between systems might account for very localized events, but it lacks an overall arch-like understanding of harmony. I believe that was what prompted Charles Smith to write: “Unfortunately, such an approach […] suggests the separateness of chromatic and diatonic tonalities, since it treats them as virtually distinct languages. I cannot believe that any chromatic master conceived of his musical terrain as so partitioned; I hear no grind of gears as one area is left and the other entered.”4 I agree with Smith in this point. Passages such as the one presented in Ex. 1.1 below, for instance, do not seem to shift between conflicting logics, but rather to flow effortlessly towards their proper closure. In

Proctor’s logic, there would be a ‘suspension’ of diatonicity over the blackened notes at the second system, that would be quickly resumed for the . As I see it, the case of the suspension of diatonicity is moot for the passage was never diatonic, nor chromatic, to begin with but rather trans-diatonic. This passage will be further discussed, and analyzed in chapter 3.

4 Charles J. Smith, “The Functional Extravagance of Chromatic Chords,” Spectrum 8, (1986): 109. 11

Ex. 1.1: Schubert, Piano Sonata in D Major, I, mm. 1-16.

1.2 Harald Krebs

Whereas Proctor considers the issue of non-normative harmonic relation in a broad, systemic sense, Harald Krebs concentrates upon the features of third-based harmonic relations in the music from the late eighteenth- and early nineteenth-century. To be sure, by not considering the music from mid- and late-nineteenth century, many issues of non-normativeness simply do not appear. However, the systematic categorization of third-related triads developed by Krebs does offer a useful frame of reference that might be applied to later music. He divides third- relations in three categories: first, third-related triads entirely subordinate to the tonic and, therefore, within the tonic-dominant axis; second, third-related triads temporarily independent from the tonic-dominant axis; and, finally, third-related triads permanently detached from the tonic-dominant axis.

Within the corpus of musical examples studied, Krebs asserts that “[a]nalysis of numerous passages from the works of Haydn, Mozart and Beethoven reveal that it is typical in their works for large-scale mediant and submediant triads to be employed within the large-scale progressions I-III-V, V-III-I, V-III-V, and VI-V, and thus, to be used in connection with the 12 dominant [i.e., subordinate to the tonic-dominant axis].”5 Before addressing third relations temporarily, and permanently independent from the tonic-dominant axis, Krebs defines the two processes of tonic embellishment that facilitated the detachment, “namely the oscillatory third- progression (a motion from one triad to another a third removed, followed by the return of the first triad) and the circular third-progression (a progression involving bass motion by equivalent thirds, arriving back at its starting point).”6

The context in which such operations are used will determine the degree of detachment of the harmonic progressions from the tonic-dominant axis – whether temporarily or permanently.

In the case of temporary independence from the tonic-dominant axis, the oscillatory and circular embellishment of the tonic “are connected to the dominant at an earlier or later point of the work.

Furthermore, the dominant triad usually returns before the final tonic of the work, the work as a whole thus being based on the tonic-dominant axis.”7 It is in the works of Schubert and Chopin that Krebs finds examples of third relations that are independent from the tonic-dominant axis. In the pieces of this category, Krebs points out that “the permanent disassociation of a third-related triad from the dominant of an initial apparent tonic results not only in the weakening of that tonic, but in its dissolution and replacement by a new third-related tonic.”8

Although the division into three categories based on the degree of non-diatonicity of third-related harmonic relations provides a useful frame of reference, Krebs’ proposal ignores entirely the remaining non-normative harmonic relations and is biased towards the Schenkerian view of an underlying diatonic structure a priori. Therefore, Krebs’ considerations can be perceived as operating within the traditional manner of inquire, that is, pressing forward diatonic

5 Harald M. Krebs, “Third Relation and Dominant in Late 18th- and Early 19th-Century Music” (Ph.D. diss., Yale University, 1980), abstract i-ii. 6 Krebs, “Third Relation and Dominant,” abstract ii. 7 Idem. 8 Krebs, “Third Relation and Dominant,” 125. 13 based models to explain harmonic procedures that clearly undermine classical diatonicity and foreshadow a new system. It was indeed a hallmark of classical diatonicism to flirt with ‘what lies beyond,’ but without ever trespassing its diatonic fences – this metaphor can be applied to both Proctor and Krebs’ views on diatonic harmony – however, once the threshold of ‘beyond diatonicity’ is not only crossed, but dwelt upon, one ought to reassess the premises by which analyses are made.

1.3 David Kopp

David Kopp concentrates his theoretical cosiderations on the chromatic mediant, or chromatic third, harmonic relations. In that regard, it can be seen as a development of Krebs’s approach to analysis. A particular feature of Kopp’s theory is the elevation of mode-preserving mediants to a functional status alongside of, and independent from, the traditional Tonic,

Dominant and Submediant. He argues that “[t]he preferred approach in the past century has been to treat chromatic mediants as derivative entities. They have been characterized as alterations or combinations of other, more basic progressions; as ornamental, secondary voice-leading events; or as incipiently degenerate coloristic phenomena, contributing to the eventual breakdown of the tonal system.”9 Feeling that “these explanations fall short,” Kopp examines, classifies, and justifies his new functional category.

Before addressing Kopp’s functionally charged mediants, it is important to explain his

‘common-tone tonality’ concept. Drawing from Proctor’s ideas (see section 1.1), Kopp argues that the chromatic harmonic space in which dwells most of the nineteenth-century harmonic practices is not entirely chromatic, nor is it diatonic, but rather something in between. The crucial aspect of such harmonic space is “the requirement of a in any direct chromatic

9 Kopp, Chromatic Transformations, 3. 14 relationship. [He adds that] common-tone relationships constituted the first group of chromatic relations between triads and keys to become fully normalized in nineteenth-century harmonic practice.”10 To Kopp common-tone relationships can be classified in three ways: first, by interval of motion; second, by absence or presence of mode change; and, third, by the number of common-tones. Most common-tone relations are readily explained by traditional harmony, such as the diatonic and parallel relations. Chromatic third relations, however, “have always proven contentious for theory. Throughout the nineteenth and twentieth centuries, theorists have disagreed on their nature, their origin, their role within the harmonic system, and their value. [To

Kopp, however,] they serve as a cornerstone of common-tone tonality.”11

By identifying common traits among the traditional harmonic functions (i.e., Tonic,

Dominant, and Subdominant) and mode-preserving third-relations, Kopp considers the latter as a functional harmonic relation as well. According to him, features like “the presence of a linear semitone, root motion by a consonant interval, connection to the tonic, and the existence of a common-tone,”12 are enough to justify his claim that mode-preserving third relations are functional. Fig. 1.3 below depicts Kopp’s example in which he illustrates the features just mentioned. Note how example C (chromatic mediant relation to Tonic) shares similar attributes with examples A and B (Dominant to Tonic, and Subdominant to Tonic, respectively), as shown below by factors 1 to 4.

10 Kopp, Chromatic Transformations, 2. 11 Idem, 3. 12 Idem, 7-8. 15 Common-tone tonality 7

Figure 1.1 Contributing factors of identity, similarity, and difference in three common-tone progressions to the tonic 1: characteristic interval, different in each case 2: single common tone 3: descending diatonic semitone 4: root motion by consonant interval A: from dominant seventh B: from subdominant C: from major-third mediant 1: leading-tone resolution 1: descending whole tone 1: chromatic semitone 2: common tone 5ˆ 2: common tone 1ˆ 2: common tone 1ˆ 3: semitone descends to 3,ˆ 3: semitone descends to 3ˆ 3: semitone descends to 5ˆ resolving tritone 4: root descends by fourth 4: root ascends by third 4: root descends by fifth (ascends by fifth) Fig. 1.3: Kopp’s functional demonstration of mode-preserving third relations.

The pointsthe raised relationship by Kopp of dominant-seventh to justify the elevation chord to of tonic mode triad,-preserving shown in mediant Figure 1.1a.s relations Several elements contribute to the overall effect of the progression. Some of the most basic are: the progression by diatonic semitone of leading tone to tonic (linear); the to functional status are indeed true: there is a common-tone; the root does move by a consonant root relation of a perfect fifth descending to the tonic (harmonic); the resolution of the tritone created by adding a dissonant seventh to the triad (contrapuntal); interval; and soand forth. the H presenceowever, of I abelieve common a functional tone (linear/contrapuntal). approach to nineteenth The absence-centur ofy any music of to these factors would deprive the relationship of some or much of its accustomed be misguided insense. its inception13 Likewise,. This other issue progressions will be further also derive developed their substance in the fromnext achapter, combination but for of linear, harmonic, and contrapuntal factors. Figure 1.1b shows a progression from now I would pointsubdominant out that harmonic triad to tonic progressions triad. It shares throughout features with nineteenth Figure 1.1a,-century although practice somes were are differently realized: the descending diatonic semitone still leads to 3;ˆ the root increasingly lessrelation regarded of a fifthin terms now ascends;of their thefunctiona commonlity tone, but is approached1ˆ rather than with5;ˆ and regards the leading- to their tone and tritone resolutions are gone. Figure 1.1c shows a commonly encountered actual harmonicchromatic relationship third. relation.Additionally, It, too, there contains is the a descending crude issue diatonic of how semitone, a chord although sounds in a from a chromatic pitch; a root relation of a major third leading to the tonic; and ˆ given context –a there common is a specific tone, 1. sound Other associated details of voice with leading the functionally also vary from-charged progression progression to progression, but the factors traced above in combination give these progressions the from Dominantbulk or Subdo of theirminant defining to Tonic, strength inand all their major sense of and functionality: minor inflections the presence and of a combinations, that13 While mode the presence-preserving of the tritone mediant may not bes relations essential to this (or relationship, any mediant it contributes relationship) a fair share of its cadential simply do and key-defining sense. Some mid-nineteenth-century theorists clearly considered the dominant seventh the fundamental chord of opposition to the tonic, far superior to the plain triad on the fifth scale degree, as related not share. Despitebelow the infact chapter that 3. ‘mode-preserving third relations’ can be accomplished by drastically different chords, too distinct to be awarded the same functional classification, such as:

A-major, Ab-major, E-major and Eb-major (having C major as Tonic), Kopp’s mediant function

(M) also lacks the modal flexibility offered by the traditional, diatonic functions. 16

Kopp is also concerned with the appropriate wording one should use in dealing with third relations. For Kopp, the terms ‘mediant’ and ‘submediant’ are not the best to refer to the chords whose roots are on 3 and 6, respectively. He claims, “the term ‘mediant’ conveys the impression of a more primary form, in contrast to the term ‘submediant,’ whose prefix sub- connotes an inferior quality.”13 Alternatively, he proposes the usage of ‘upper mediant’ and ‘lower mediant,’ as more appropriate, neutral substitutes to mediant and submediant, respectively.14

Kopp then classifies third relations according to the number of common-tones with the tonic. To any given tonic there will be eight possible third-related chords which Kopp divides into three categories: (1) relative mediants – the lower relative mediant (LRM) and the upper relative mediant (URM), with two common-tones; (2) chromatic mediants – the lower flat mediant (LFM), the lower sharp mediant (LSM), the upper flat mediant (UFM), and the upper sharp mediant (USM), with one common-tone; and (3) disjunct mediants – the lower disjunct mediant (LDM) and the upper disjunct mediant (UPM), with no common-tones with tonic.15

Table 1.1 below summarizes Kopp’s mediants and shows what chord would they be in C major and the intervallic relation between their root and that of the tonic.

13 Kopp, Chromatic Transformations, 11. 14 For an opposite view, which I favor, see Francis Donald Tovey, “Tonality in Schubert,” in The Main Stream of Music and Other Essays (New York: Oxford, 1949), 134. “The subdominant should be thought of not as the note below the dominant, but as an anti-dominant having the opposite effect to the dominant, and lying a fifth below the tonic as the dominant lies a fifth above. The meaning of the term submediant then becomes clear: it is a third between tonic and subdominant, as the mediant is a third between tonic and dominant.” Italics mine. Additionally, I do not believe the terms ‘upper’ and ‘lower’ advocated by Kopp to be of a more neutral nature as he claims, but rather to reinforce the distinction and apparent sense of hierarchy between the mediant and submediant. 15 Kopp, Chromatic Transformations, 12. 17

LRM m3 down a UFM m3 up Eb URM M3 up e USM M3 up E LFM M3 down Ab LDM M3 down bb (sic.) LSM m3 down A UDM m3 up eb Table 1.1: Kopp’s mediant relations in C major.16

I raise three issues with Kopp’s third relation classification system. First, the prefixes

‘upper’ and ‘lower,’ besides not conveying the intended neutrality, do not make clear the quality of the intervallic relation between the mediants and the tonic, i.e., if they are a major or minor third apart. Second, the ‘flat’ and ‘sharp’ classifications are inconsistent. Whereas he calls the upper and lower flat mediant based on the alteration of their root, the upper and lower sharp mediant are so called with reference to their thirds. There is also a degree of counter- intuitiveness to this flat/sharp classification. Consider Db as tonic: its lower sharp mediant would the Bb and its upper sharp mediant would be F – with no actual sharp in sight. Finally, even with its three categories and eight different acronyms, it fails to clearly convey a crucial information – that of the mode of the goal chord.

One piece analysed by Kopp to demonstrate how such concepts would be applicable is

Schubert’s Die Sterne. The crucial aspect here, observed by Kopp, are the modulations from the tonic Eb major to C major, Cb major, and G major in stanzas one and four, two, and three, respectively. He writes that this piece “reads like a deliberate experiment in creating functional chromatic mediant relations at the level the phrase.”17 Fig. 1.4 below summarises these relations.

Note that, under Kopp’s requirement of common-tone with tonic and same mode retention, all

LSM, LFM, and USM would be equally functionally charged, in other words, such different

16 The LDM should have been Ab-minor and not Bb-minor as shown in Kopp’s Chromatic Transformations Figure 1.6 b and c. 17 Kopp, Chromatic Transformations, 23. 18 chords as C-major, Cb-major, and G-major would all perform the very same function of mediant.

I consider this24 piece in chapter Chromatic 3. transformations in nineteenth-century music

a) first and fourth stanzas

b) second stanza

c) third stanza Figure 2.2 Chromatic mediants in Die Sterne (black notes vocal part; slurs common= tones) = Fig. 1.4: Kopp’s scheme on Schubert’s Die Sterne.18

Havingor aspresented an alteration his ofideas some on other third chord. relations The only and pitches having outside considered the triad how appearing the issue of third in the passage are the lower neighbors B and D, which imply C’s dominant, serving only to strengthen the impression of C as a momentarily stable harmony. The move relations wasback historically to tonic Edealt♭ at m.with 38 by is achievedtheorists by from the directRameau juxtaposition to Hauptmann, of root positionfrom Riemann to C major and an E ♭ major triad in first inversion. The use of first inversion here Schenker andinstead beyond, of rootKopp position proposes does what not result he calls from a any modified inherent chromatic mechanical transformation difficulty in returning to the tonic directly from a chromatic mediant. Rather, this I6 is used system.19 He forstablishes purely musical six categories reasons: a of root-position transformation tonic, triad they would are: the be farprime, too definite the mediant, an the arrival so soon here at the beginning of the phrase. Schubert does not employ I in relative, the plainroot position-fifth, the until fifth the-change, very last momentand the (m.slide. 46); In he the deliberately prime category avoids it, we not have only the at m. 38 but again at m. 42, preserving a measure of tension not fully resolved until identity (I) andthe parallel end of the (P) phrase. transformations This, of course, – the conveys latter thechanges feeling the of mode one unified of a chord musical and the phrase setting the fourth line of text. It also gives a sense of continuity between it and the previous phrase by keeping harmony consistently open between the C of m. 38 and the E ♭ major chord of m. 46. Thus the two harmonic areas are heard in direct relation to each other both before and after the chromatic mediant 18 Kopp, Chromatic Transformations, 24. 19 The ‘modified’ attribute accounts for the fact that it is similar to the transformational system proposed by David Lewin, but not entirely equal. The main modification presented by Kopp’s system would be the tolerance for more than one voice leading displacement per voice at the same time, as well as the possibility for such displacements to exceed a single semitone. 19 former preserves all the pitches of a chord. Unique to Kopp’s system, the mediant transformations can be M (root motion of a major third) or m (root motion of a minor third). The default direction of any transformation in Kopp’s system is downwards; upward root motion is indicated as M-1 and m-1. The relative transformations R/r are mode-altering transformation by minor thirds. In this case, the uppercase transformation indicates upward direction, and lowercase, downward direction.20 Plain-fifth transformations D/D-1 stand for mode-preserving dominant-type relation by root interval of a fifth; and fifth-change transformations F/F-1 are defined as mode-altering relations also by root interval of a fifth. Finally, the slide transformation

S, representing the most distant common-tone relation possible, accounts for the C-major to

C#/Db-minor transformation, in other words, a chromatic semitone higher with mode change.21

Kopp does not consider a S-1 transformation due to the lack of common-tones. Figure 1.5 below schematically summarizes the ambitus of Kopp’s transformational system. Note that the transformation from C-major to E-major, given it moves a major third upwards, should read M-1; and the one from C-major to Eb-major, a minor third upwards, should read m-1. Fig. 1.5 shows the opposite, however. Note also that the disjunct mediants, the UDM (Eb-minor) and the LDM

(Ab-minor), due to the lack of common-tones, are missing from the scheme, which brings attention to the inefficiency of Kopp’s M transformations and mediant classification system.

Furthermore, every non-common-tone progression is only identifiable as the result of a compound transformational process, despite their direct succession in an actual piece.

20 Note that, apart from its counter-intuitiveness and inconsistence, the mode of the goal chord is not specified either by the mediants, or the relative transformations. 21 Kopp, Chromatic Transformations, 165-76. 20

Fig. 1.5: Kopp’s full set of common-tone relations from C.22

1.4 Richard Cohn

Richard Cohn’s theory considers the harmonic relations achieved by the symmetric division of the octave exclusively by major thirds intervals and solely involving consonant triads. Like

Kopp’s theory, Cohn’s system may also be seen as a later development of Proctor and Krebs’ theories. At the outset of his theory, Cohn argues that “consonant triads have two sets of unique properties that are apparently independent of each other. One set may be characterised in terms of acoustics, and is the primary basis of the syntactic routines of diatonic tonality. […] The other set concerns the voice-leading potential of motion between triads […] and is the basis of many of the syntactic routines of chromatic music.”23 These two sets of properties are essential to Cohn’s theory, as he sees it: the first set stablishes major and minor chords as ‘optimal acoustic object’ independent either of acoustical or dualistic origins and explains the absence of other chord types, such as diminished, augmented, seventh chords, etc.; the second, contrapuntally connects consonant triads by means of what Cohn calls single semitonal displacement (SSD).

By applying maximally parsimonious voice leading procedures to any consonant triad, one will invariable obtain a harmonic cycle comprised of six members, which Cohn calls a

22 Kopp, Chromatic Transformations, 174. 23 Cohn, “Maximally Smooth Cycles,” 12-13. 21 hexatonic cycle. Let us take C-major as a starting point. There are only two possible consonant triads achievable by a single semitonal displacement: C-minor, by flattening the third (3 to b3) and E-minor, by lowering the tonic to the leading tone (1 to 7). This implies that the members of a hexatonic cycle are linked through alternating parallel (P) and Leittonwechsel (L) transformations (C-major to C-minor, and C-major to E-minor, respectively). Thus, having C- major as starting point, and alternating P and L transformations, one obtains the following hexatonic cycle: C-major 24 – CRICHARD-minor – Ab-major COHN – Ab-minor – E-major – E-minor – (C-major). In order to encompass all twentyintersecting-four stable triads, Cohn ovals undergoes in this which process three theymore times, are enclosed portray the four hexatonic collections of pitch-classes, labelled Ho(pc) to H3(pc), each of which includes two T4-cycles. The arrows from centre to periphery show the affiliations between hexatonic collections and hexatonic systems (cf. Table 2). each time a semitone higher Neighbouring. By hexatonicconsidering systems a(thosell four connected cycles directly) within share athree single pcs, system, Cohn obtains while the pc content of opposite systems is complementary with respect to the what he calls the hyper twelve-pc-hexatonic aggregate.35 system, presented below as Fig. 1.6. Fig. 5 The hyper-hexatonic system

C+

E- C- Ho (triad) E+ A6+ At,+

E?+ C + Ho (p c) G- EG Et F- G, C#-B C, E, Abl H3 H3 Hi oHi (triad) (pc) (pc) (triad) G+' B+ D3 F B B CB, F, A F+ A+ H2 B- (pc) A-

F#- D- H2 (triad) F#+ B6+ Bt Fig. 1.6: Cohn’s hyper-hexatonic system.24

The outer cycles C onBlackwell the hyper Publishers-hexatonic Ltd. 1996 system Music are Analysis, each a15/i hexatonic (1996) cycle whose

This content downloaded from 136.159.134.219 on Thu, 28 Sep 2017 23:07:02 UTC members are triads. The inner cycles depictAll use subject the to http://about.jstor.org/terms pitch class collection of the outer cycles. As can

24 Cohn, “Maximally Smooth Cycles,” 24. 22 be seen in Fig. 1.6, each hexatonic cycle is independent, having in common only three pitch classes with an adjacent cycle. Despite the straight lines connecting consecutive groups, they are nonetheless harmonically isolated. The lack of connection among groups, and consequently among triads, is evident in the analyses offered by Cohn. Ex. 1.2 below depicts Cohn’s analysis of measures 98 to 110 of Liszt’s Polonaise I from Die Legende vom heiligen Stanislaus. He first identifies some of the chords. Thus, we have E-major in m. 98, and C-minor in m. 99. Measures

100 and 101 repeat the same harmonic relation. From the hyper-hexatonic system above, one notes that E-major and C-minor belong to the same hexatonic group, the H0. The following four measures, however, (presenting F#-major and D-minor) are assigned to a different group, the H2.

The pattern of Cohn’s analysis is set. He will first identify the chord in question and will then assign it to its respective hexatonic group from his hyper-hexatonic system. In this Liszt example, one can note the awkward shifting from one hexatonic groups to another and how cumbersome the whole analysis seems to be when comparing its ‘tonal cadence’ in the last two measures with previous measures.

AAddaaggiioo qq.=.=4488 hexatonic 0 j ## ## 12 hexatonic 0 bœj œ™ j & ## ##812 œ™ œ™ œ œ œ œ œ œ nœ™ nnœ bœ œ™™ œ œj #œ™ œ™ œ œ œ œ œ œ & 8 œ™ œ™ œ œ œ œ œ œ nnœœ™™ bb œ œ œ™™ œ œ #œ™ œ™ œ œ œ œ œ œ œœ™™ œœ™™ œœ œœ œœ œœ œœ œœ nnnœœ™™ œ œ œ #œ™ œ™ œ œ œ œ œ œ nnœ œ™™ j ??## ###1122 œ™ œ™ œ œ œ œ œ œ bœ™ nnœ œ œ™™ œ œj nœ™ œ™ œ œ œ œ œ œ {{ ## #88 œœ™™ œœ™™ œœ œœ œœ œœ œœ œœ nnbœœ™™ nœ œ œ™™ œ œ nœ™ œ™ œ œ œ œ œ œ œœ™™ œœ™™ œœ œœ œœ œœ œœ œœ bbœœ™™ nœ J œ œ nœ™ œ™ œ œ œ œ œ œ E+ C_ E+ C_ # ## jj hheexxaattoonniicc 22 nœ œ™ j ### ## ™ nnœœ bbœœ œœ™™ œœ jj ™ ™ ™ nœ nœ œ™ œ j && nnœœ™™ b œ œœ œœ™™ œ œœ ##œœ™™ œœ™™ œ œ œ œ œ œ nnœ™ n œ œ œ™ œ œ nnœœ™™ bœœ œ œ œœ œœ ##œœ™™ œœ™™ œ œ œ œ œ œ nnœ™ œ J œ œ nœ™ nœ œ™™ œj nœ œ™ j ??## ## bœ™ nnœœ nœœ œœ™™ œœ œj œœ™™ œœ™™ œ œ œ œ œ œ nœ™ œ œ œ™ œ œj {{ ## ##nnbœœ™™ nœœ œœ œœ™™ œœ œœ ##œœ™™ œœ™™ œ œ œ œ œ œ n œ™ œ œ œ™ œ œ bbœœ™™ nœ JJ œ œœ œœ™™ œœ™™ œ œ œ œ œ œ nœ™ œ J œ œ

Ex. 1.2: Liszt, Polonaise I from Die Legende vom heiligen Stanislaus, mm. 98-103. 25

# ™™ hexatonic 0 j ### ### nnœ nnœ œ œ j nœ™ ˙™ œ #œ && # ##œœ™™ œœ™™ œœ œœ œœ œœ œœ œœ nnœœ™™ nn œ œ œ™™ œ œ bnœ™ ˙™ nœœ #œ #œœ™™ œœ™™ œœ œœ œœ œœ œœ œœ nnnœœ™™ œ J œ œ bnœ™ ˙™ nœœ™™#œ #œ œ œ œ œ œ œ œ nœ nnœ œ™™ œj 25 # ™™ ™™ ™™ œ œ œ™™ œ j bœ™ ˙™ nœ™™ Cohn,{?? “Maximally#########œœ™™ œœ Smooth™™ œœ œœCycles,”œœ œœ œ29.œ œ œ nnnnœœ™™ œ œ œ™™ œ œ b œ™ ˙™ nnœ™™ { ##œœ™™ œœ™™ œœ œœ œœ œœ œœ œœ nnœœ™™ œ J œ œ bœ™ ˙™ nœ™™ A¼+ E__

ttonall cadence hexatonic 0 ## ## hheex™xaattoonni™icc 22 jj hœe™xato˙n™ic 0 ™ ™ œ™ #œ ™ œ™ œ™ œ™ œ œ œ œ œ œ ## ## nnœœ™ ˙˙™ ##œœ ##œœ œ™™ ˙™™ ##œœ™™ œ ™ œ™ #œ˙™™ œ™ œ™ œ œ œ œ œ œ && nnnœœ™™ ˙˙™™ ## œœ™#™#œœ nnœœ™™ ˙˙™™ ####œœ™™ œ˙™™ œ™™ #œ ™™ œ™ œ œ œ œ œ œ œ œ ##œœ ™ ™ bœ™ ˙™ ™ nnœœ™™ ˙˙™™ œ™ ˙™ ˙™™ œ™ œ™ œ œ œ œ œ œ ??## ## nnbœœ™™ ˙˙™™ ##œœ™™ nnœœ™™ ˙˙™™ ##œœ™™ ˙™™ ˙™ œ™ œ™ œ œ œ œ œ œ {{ ## ##bbœœ™™ ˙˙™™ œœ™™ nnœœ™™ ˙˙™™ œœ™™ ˙™™ ˙™™ œ œ œ œ œ œ œ œ CC++ GG¾¾__ Adagio q.=48 Adagiohe qx.a=t4o8nic 0 j ####12 nœ bœ œ™ œ j & 8 hœex™ atonœic™ 0 œ œ œ œ œ œ nœ™ œj œ™ œ #œ™ œ™ œ œ œ œ œ œ # ##12 œ™ œ™ œ œ œ œ œ œ n œ™ b œ bœ œ™ œ œj œ™ œ™ œ œ œ œ œ œ & # 8 œ™™ œ™™ œ œ œ œ œ œ nnœ™ nœ œ œ™™ œ œ ##œ™™ œœ™™ œœ œœ œœ œœ œœ œœ ?# # 12 œ™ œ™ œ œ œ œ œ œ n œ™ bnœ nœ œ™ œ œj œ™ œ™ œ œ œ œ œ œ { # # 8 œ™™ œ™™ œ œ œ œ œ œ nnbœ™ œ œ œ™™ œ œ #nœ™™ œœ™™ œœ œœ œœ œœ œœ œœ ™ ™ ™ nœ nJœ œ œ j ™ ™ ?# ##12 œ™ œ™ œ œ œ œ œ œ bbœ™ nœ œ œ™ œ œ nnœ ™ œœ™ œœ œœ œœ œœ œœ œœ { # 8 Eœ+ ™ œ™ œ œ œ œ œ œ nCœ_ ™ œ œ™ œ œ œ™ œ™ œ œ œ œ œ œ œ™ œ™ œ œ œ œ œ œ bœ™ nœ J œ œ nœ™ œ™ œ œ œ œ œ œ E+ C_ j hexatonic 2 ™ #### nœ bœ œ™ œ j nœ nœ œ œ j & nœ™ œj œ™ œ #hœex™ atonœic™ 2 œ œ œ œ œ œ nœ™ n œ œ œ™ œ œ # ## n œ™ b œ bœ œ™ œ œj #œ™ œ™ œ œ œ œ œ œ nnœ™ nœœ nJœ œ™ œœ œj & # nœ™™ nœ œ œ™ œ œ #œ™ œ™ œ œ œ œ œ œ nœ™ n œ nœœ œœ™™ œ œ ?# # n œ™ bnœ nœ œ™ œ œj œ™ œ™ œ œ œ œ œ œ nnœœ™™ œœ œJ œ™ œœ œœj { # #nnbœ™™ œ œ œ™ œ œ #œ™ œ™ œ œ œ œ œ œ nnœœ™ nœœ œœ™™ œ œœ bœ™ nœ nœJ œ™ œ œj œ™ œ™ œ œ œ œ œ œ nœ™™ œœ Jœ œ™ œœ œj {?####nbœ™ nœ œ œ™ œ œ #œ™ œ™ œ œ œ œ œ œ nnœ™ œ œ œ™ œ œ bœ™ nœ J œ œ œ™ œ™ œ œ œ œ œ œ nœ™ œ J œ œ 23 # ™ hexatonic 0 j ## # nœ nœ œ œ j nœ™ ˙™ œ #œ & #œ™ œ™ œ œ œ œ œ œ nœ™ n œ œ œ™ œ œ b hœe™xaton˙ic™ 0 n œ j # ###œ™ œ™ œ œ œ œ œ œ nnœ™ nœ nJœ œ™ œ œj nœ™™ ˙™™ œ ™#œ & # #œ™ œ™ œ œ œ œ œ œ nœ™ n œ nœ œ™™ œ œ nœ™ ˙™ œ #œ ?# # #œ™™ œ™™ œ œ œ œ œ œ nnœ™ œ œJ œ™ œ œj bbnœœ™™ ˙˙™™ nœœ™œ™#œ { # ##œ ™ œ™ œ œ œ œ œ œ nnœ™ nœ œ™™ œ œ b œ™ ˙™ n œ™ #œ™™ œ™™ œ œ œ œ œ œ nœ™ œ Jœ œ™ œ œj bbœœ™™ ˙˙™™ nœ™™ {?#####œ™ œ™ œ œ œ œ œ œ nnœ™ œ œ œ™ œ œ b œ™ ˙™ nnœ™ #œ™ œ™ œ œ œ œ œ œ nœ™ œ J œ œ Ab¼œ+ ™ ˙™ Enœ_ ™ A¼+ E_ tonal cadence # hexatonic 2 j hexatonic 0 ™ œ™ œ™ œ œ œ œ œ œ ## # nœ™ ˙™ #œ #œ œ™ ˙™ #œ™ tœona™l cœad™ en#cœe ™ œ ™ ™ & hexatonic 2 nhœex™ ato˙ni™c 0 # œ™ ˙ ˙™ œ™ œ™ œ œ œ œ œ œ # # nnœ™ ˙™ ##œœ™#œj œ™ ˙™ #œ™ œ ™ œ™ #œ ™ œ™™ œ™ œ™ œ œ œ œ œ œ # # nœ™ ˙™ #œ #œ œ™ ˙™ #œ™ œ ™ œ™ #œ˙™ œ™™ œ™™ œ œ œ œ œ œ & nnbœ™™ ˙™™ # œœ™ ™#œ nnœ™ ˙™ ##œ™ ˙œ˙™ ™ œ™ #˙œ™ ™ œ™ œœ™ œœ™ œœ œœ œœ œœ œœ œœ ?# ##n œ™ ˙™ ##œ™ nœ™ ˙™ #œ™ ˙™ œ™™ œ™™ œ œ œ œ œ œ { # bœ™™ ˙™™ œ™ n œ™ ˙™ œ™ ˙™™ ˙™™ œ™ œ™ œ œ œ œ œ œ ?# ##n œ™ ˙™ #œ™ nœ™ ˙™ #œ™ ˙™ ˙ œ™ œ™ œ œ œ œ œ œ { # bœ™ ˙™ nCœ+ ™ ˙™ Gœ¾_ ™ ˙™ ˙™ œ™ Ex.C+ 1.2: ContinuationG¾_ , mm. 104-110.

Another example is Cohn’s analysis of the first movement of Franck’s Quintet for Piano and Strings presented below as Ex. 1.3. In this analysis, the cumbersomeness of having such close systems is more evident than in the previous Liszt analysis (Ex. 1.2). Note how Cohn shifts from one hexatonic group to another with no regard to the overall harmonic direction. New to this analysis is the presence of an ‘extra’ group, the octatonic. According to him, while Liszt shifts among hexatonic cycles with ease, “Frank is at pains to smooth the transitions between successive iterations of the hexatonic systems.”26 I would argue that the Frank’s ‘pain’ as perceived by Cohn relates to the passage’s resistance to a hexatonic analysis, for the music seems to flow effortlessly from one hard-to-explain chord to another. The remainder of Ex. 1.3 presents more issues still. The ‘octatonic’ assignment of a single beat in m. 99 (third system, second measure) is another example of how inadequate Cohn’s theory really is to deal with such repertoire. Finally, measures 102 and 103 pose the biggest problems to Cohn’s analysis (fourth system, first two measures). His a priori exclusion of augmented and diminished chords at the outset of his theory constitutes a problem now. His ‘hexatonic 1/octatonic’ analysis highlights the failure of the hexatonic system in dealing with passages such as this. To use Cohn’s own

26 Idem, 28. 26 RICHARD COHN

In Ex. 6, the first two24 transpositions are centred around C# major and E major respectively. The final two transpositions, around G major and B% words, major, his overall are approach durationally and assessment compressed. of both pieces The make T3 transpositionsapparent the “illusion tour that theto four hexatonic systems. Each transposition shares half of its pcs with the previous describe iteration, is to explain.” and27 half It is notwith enough the tosubsequent allocate that iteration.or the other chordThe sequencein predetermine, concludes with the reappearance of the initial system, represented by an A major triad rather artificial groups and assign them random names – the overall harmonic logic and relationships than the C# major triad that our tonal ears demand. must be accounted for. I offer my own analysis of both Liszt and Frank’s excerpts in chapter 3. Ex. 6 Franck, Quintet for Piano and Strings, first movement, bars 90-106

hexatonic 1 octatonic

Ai I L

A I~ 2, , ,. K? . ,

15? --p i Ti- 4z... ..-

AK. ix L K -u- m octatonic

tenero ma on passion Ex. 1.3: Franck, Quintet for Piano and Strings, I, mm. 90-93.28 :-am ^ Ihexatonichexatonic 0 p[J _ . 0....,_. piu ocatni

A I Piu

piu

piu

A i tlt t

27 Richard Cohn, Audacious Euphony: Chromaticism and the Triad’s Second Nature, (New York: Oxford Press, 2012), 11. 28 Cohn, “Maximally Smooth Cycles,” 26-27.

C Blackwell Publishers Ltd. 1996 Music Analysis, 15/i (1996)

This content downloaded from 136.159.123.178 on Mon, 05 Feb 2018 04:03:19 UTC All use subject to http://about.jstor.org/terms 26 RICHARD COHN

In Ex. 6, the first two transpositions are centred around C# major and E major respectively. The final two transpositions, around G major and B% major, are durationally compressed. The T3 transpositions tour the four hexatonic systems. Each transposition shares half of its pcs with the previous iteration, and half with the subsequent iteration. The sequence concludes with the reappearance of the initial system, represented by an A major triad rather than the C# major triad that our tonal ears demand.

Ex. 6 Franck, Quintet for Piano and Strings, first movement, bars 90-106

hexatonic 1 octatonic

Ai I L

A I~ 2, , ,. K? . ,

15? --p i Ti- 4z... ..-

AK. ix L K -u- m octatonic

tenero ma on passion 25 :-am ^ Ihexatonichexatonic 0 p[J _ . 0....,_. piu ocatni

A I Piu

MAXIMALLY SMOOTH CYCLES, HEXATONIC SYSTEMS 27 piu

piu Although the passage is dominated by the piano, the strings are twice summoned to mark a transition between hexatonic systems. The first transition connects C# major with E major through two , the first of which A isi tlt interpretable t as iiu7 of C# minor, the second of which acts as a German sixth of E major, resolving directly to the new tonic in . Although these references to Classical syntax are surely a component of the transition's effectiveness, there are important group-theoretic connections as

Ex. 6 (cont.)

C Blackwell Publishers Ltd. 1996 Music Analysis, 15/i (1996) hexatonic 3 octatonic hexatonic 2 octatonic _ __ _ I I I

This content downloaded from 136.159.123.178 on Mon, 05 Feb 2018 04:03:19 UTC All use subject to http://about.jstor.org/terms

Ex. 1.3: Continuation, mm. 94-101.

k<.'69 1 FIX- 4.-_ ifr octatonic hexatonic 1/ octatonic -- ,pasi. .? I IT,| I Y LL Y lop,'.IF i 9I k&.39 I- I..I kh ' I I [ w"

A u, I----

q:,., 6L 1

Music Analysis, 15/i (1996) C Blackwell Publishers Ltd. 1996

This content downloaded from 136.159.123.178 on Mon, 05 Feb 2018 04:03:19 UTC All use subject to http://about.jstor.org/terms MAXIMALLY SMOOTH CYCLES, HEXATONIC SYSTEMS 27

Although the passage is dominated by the piano, the strings are twice summoned to mark a transition between hexatonic systems. The first transition connects C# major with E major through two tetrachords, the first of which is interpretable as iiu7 of C# minor, the second of which acts as a German sixth of E major, resolving directly to the new tonic in root position. Although these references to Classical syntax are surely a component of the transition's effectiveness, there are important group-theoretic connections as

Ex. 6 (cont.)

hexatonic 3 octatonic hexatonic 2 octatonic _ __ _ I I I

26 k<.'69 1 FIX- 4.-_ ifr octatonic hexatonic 1/ octatonic -- ,pasi. .? I IT,| I Y LL Y lop,'.IF i 9I k&.39 I- I..I kh ' I I [ w"

A u, I----

q:,., 6L 1

Ex. 1.3: Continuation, mm. 102-106. 29

In order to promote a smoother connection between hexatonic cycles, Cohn draws upon Music Analysis, 15/i (1996) C Blackwell Publishers Ltd. 1996 some findings by the theorist Carl Friedrich Weitzmann in his monograph Die Übermässige

Dreiklang (The Augmented Triad), from 1853. According to Weitzman, “[t]he closest relatives This content downloaded from 136.159.123.178 on Mon, 05 Feb 2018 04:03:19 UTC of an augmented triad are thus the Allmajor use triadssubject [built] to http://about.jstor.org/terms on its bass tone, third, and fifth, [plus] the minor triads to whose roots each of [the augmented triad’s] three voices forms the leading tone.”30

In other words, Weitzmann sees the augmented triad as having been generated by, but also generating, six different chords: a major chord for each note of the augmented chord as root and their relatives (see Fig. 1.7 below). Note how the augmented triad is the central node of each so-called ‘water bug’ and that it is reciprocally connected to its resolutions/origins, called

Weitzmann Regions by Cohn. Noting that half of one ‘water-bug’ combined with the consecutive half of an adjacent ‘water bug’ makes up a hexatonic cycle, Cohn proposes the union

29 Cohn, “Maximally Smooth Cycles,” 26-27. 30 Carl Friedrich Weitzmann, “Two Monographs by Carl Friedrich Weitzmann, I: The Augmented Triad.” Translated with an introduction by Janna Saslaw. Theory and Practice 29 (2004): 189. 27 of the two systems into a single model. Fig. 1.7 below shows the overlapping of Weitzmann

‘water bugs’ and Cohn’s hexatonic systems connected by an intermediary augmented triad. CHAPTER 5 A Unif ed Model 85 !

Fig. 1.7: Figure Cohn’s 5.2. Fourunion Weitzmann of Weitzmann water bugs waterin union bugs with four and hexatonic his hexatonic pools. groups.31

As a further elaborationwith melodic direction: of Fig. clockwise 1.7 above, and counterclockwise Cohn proposes motion denote a model upshif ingbased on Douthett and and downshif ing, respectively. Since Cube Dance includes the four hexatonic32 cycles and the four Weitzmann Steinbach’s ‘cube dance,’regions shownas contiguous in Fig.subgraphs, 1.8 itbelow. models all ofCohn the passages argues that arethat internal the “tocyclic components of them, as studied in chapters 2 and 4, respectively. But it also supplies, for the f rst time, a way to model progressions that move between regions, which is the usual the hexatonic graph [Fig.case. To1.6 get and a preliminary 1.7], when sense combined of how Cube Dancewith modelsits two a compositionflanking augmented triad, that crosses boundaries between regions, consider the Adagio opening of the gives rise to a cubic superOverture graph. to Schubert’s The Die union Zauberharfe of these from 1821.four Schubert cubes later forms used thea connected same graph composition as the Overture to Rosamunde , under which name it is normally per- formed today. T e score is available at Web score 5.4 . Af er an eight-measure presented here as [Fig.fanfare, 1.8 ].”the33 oboe Altho soundsugh an antecedentit did not phrase seem in c tominor, be followedan issue by afor conse- earlier versions of his quent in its relative major. Af er an E major cadence, Schubert pulsates on this chord and then drops G to G and pulsates further on e minor. T e bass now 34 theory and analysis, slips Cohn down add throughs that D (forming the cube a transient dance augmented, “for thetriad) first to D ,time supporting [supplies] a a way to 6 cadential 4 that resolves in classical fashion to G major. Af er a tonally closed period in G major, the same sequence of events occurs twice in transposition: 35 model progressions thatpulsations move on Gbetween major and [hexatonic] f minor, transient regions, motion through which FAC is, cadencethe usual in case.” A major; pulsations on A major and a minor, transient motion through ACE, 6 cadential 4 in C major. In the event, the latter initiates a deceptive motion to A major, and a standing-on-the-dominant, before proceeding with a C major Allegro.

31 Cohn, Audacious Euphonies, 85. 32 See Jack Douthett and Peter Steinbach, “Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transformations,” Journal of Music Theory 42, no. 2 (1998):

254. 05-Cohn-05.indd 85 10/13/2011 12:34:29 PM 33 Richard Cohn, “Weitzmann regions, My Cycles, and Douthett’s Dancing Cubes,” Music Theory Spectrum 22, no. 1 (2000): 96. 34 Cohn earlier analysis from 1996 (the Liszt and Frank discussed above) was not concerned at all about the lack of connection between hexatonic cycles. It a later moment (2000, 2012), he seems to have noticed the problem and make some adjustments by adding the Weitzmann regions to further a better connection between hexatonic cycles. 35 Cohn, Audacious Euphony, 85. 96 Music Theory Spectrum

Example 8. Voice-leading of the Klange in a hexatonic system Example 9. Cubic graph of SSD-relations among and between the six

A A Klange of the Northern cycle and its two flanking augmented triads b o ? . _.-. o JJ o0.. ,o Lo oM--- {C, E, A} o_o o hooo Io #,, c- A6

0) - 0 -, o I 0I 0 I0 0,r 0 0 o 0 0 1 1 1 0 1 1 o 0 0 1 I values. The two remaining combinations are <0,0,0>, the set of all lowered pitch /iclasses AK- and <1,1,1>, the set of all raised pitch {B, E;, G} classes. These combinations represent the two augmented triads that, in Example 5, "flank" the Northern cycle. Example 9 presents the cubic graph implied by the above rela- 28 tions. Six of the Example 10. Graph graph's of SSD-relations among and between theeight twenty- vertices represent the Klange of the Northern cycle, four Klange and andthe four augmented triadsthe (Cube Dance) remaining two represent the augmented triads that flank that cycle. Half of the cube's twelve edges con- nect Kldnge directly {C, E, Ab} to each other; these edges are drawn in bold in Example 9. The subgraph formed by these six edges is a com- ponent of the hexatonic graph (Example 7), where it represents the Northern hexatonic system. The remaining six edges join one of the Klange to one of the flanking augmented triads. The discon- nected subgraph formed by these six edges is also a subgraph of the Weitzmann graph (Example 6).

Each of the {B,other ES, G} three {C,, F,A} cyclic components of the hexatonic graph, when combined with its two flanking augmented triads, gives rise to a cubic supergraph in the same manner. The union of these four cubes forms a connected graph presented here as Example 10, which is a version of the figure that Douthett calls Cube Dance.'9 Cube Dance originated as an elaboration of my

'9Example 10 reverses and rotates the elements of the figure presented in Douthett and Steinbach, "Parsimonious Graphs," 254, in order to preserve graphic consistency with {D, FJ, Bb} Example 6, but the two graphs are formally identical. Fig. 1.8: Cohn’s graph of SSD-relations among and between the twenty-four Klänge and the four augmented triads (Cube Dance).36

This content downloaded from 136.159.123.191 on Tue, 06 Feb 2018 04:00:58 UTC All useAs subject a practical to http://about.jstor.org/terms example of how Cube Dance analysis might take place, Cohn analyses the

introduction of the Overture to Schubert’s Die Zauberharfe. The harmonic scheme of this

passage can be found in chapter 3 (section 3.1.3), along with my considerations. According to

Cohn, this excerpt features a cyclic move on his Cube Dance graph shown here as Fig. 1.9

below. In this analysis, the arrival points in the Cube Dance are supposed to represent harmonic

regions, not chords. As one can see, the piece starts in C minor and makes its first modulation to

Eb major, i.e., it modulates to its relative major and bypasses entirely the ‘connecting’ augmented

chord (Eb-augmented). There is a shift to the parallel minor (Eb-minor) and a swift, one-beat long

augmented chord (Gb-augmented) makes the transition to Gb major. This same harmonic pattern

is carried out two more times. First, from Gb/F# major, passing through F#-minor and A-

augmented chords en route to A major; and secondly, from A major through A-minor and C-

augmented towards C major. The double headed arrow between C-major and Ab-major makes

reference to the deceptive cadence in measure 33 and its gradual return to C major.

36 Cohn, Weitzmann Regions, 96. 29 CHAPTER 5 A Unif ed Model 87 !

37 Fig. 1.9: Cube Figure Dance 5.4. Schubert, Schubert’s Overture Overture to Die Zauberharfe to Die , openingZauberharfe measures,, mm.1portrayed-47. on Cube Dance. To have chosen this particular Schubert excerpt to demonstrate how hexatonic cycles and

Weitzmann regions interact in the Cube Dance was, I believe, a poor choice from Cohn’s part.

Not only does it not showcase Cohn’s hexatonic cycles, nor Weitzmann regions features prominently, it actually undermines them. Note how Schubert directly makes the modulation from C minor and Eb major – a very ‘normal’ modulation (tonic to Relative), I might add – with no regard to the single semitonal voice leading displacement ‘rule.’ Cohn’s graph is also not clear in the meaning of the elements it depicts. Do they represent harmonic regions or individual

Figure 5.5. Schubert, Overture to Die Zauberharfe , portrayed on the Tonnetz . triads? Are these equivalent for Cohn? From the beginning, one can assume he means harmonic regions, for there is much harmonic activity within the tonic region of C minor en route to the next region of Eb 05-Cohn-05.inddmajor that 87 is not portrayed in his analysis. However, Eb-minor10/13/2011 and 12:34:31 G PMb- augmented are not at harmonic regions but chords en route to Gb major. The same happens with the modulation to A major, and to C major. Therefore, the actual path of harmonic modulations of this passage is: C minor – Eb major – Gb major – A major – C major, i.e., a modulation path based on minor thirds intervals, not majors thirds.

37 Cohn, Audacious Euphony, 87. 30

This issue opens up space for a hypothetical experimentation. The intervallic relation between regions in this Schubert passage is of minor thirds, as said before. Conversely, Cohn’s system is based on major thirds harmonic relations. To limit one’s theory to a single intervallic relation seems doomed to failure right from its inception. However, for the sake of argumentation I will briefly consider how a system exclusively based on minor thirds might take place. Fig. 1.10 below depicts what I call, after Cohn, the hyper-octatonic system. Like Cohn’s hyper-hexatonic system (Fig. 1.6), the outer cycles of the hyper-octatonic system are each an octatonic cycle and its members are consonant triads. The inner blocks represent the pitch class collection of each cycle. Note that each octatonic cycle shares with its clockwise consecutive cycle half of its pitch class collection. Differently from Cohn’s system, the connection between octatonic cycles, as well as the inner intervallic outline achieved by the roots of each cycle, is articulated by a fully , rather than an augmented triad. Another difference concerns the voice leading procedures to complete an octatonic cycle. Whereas in a hexatonic cycle all voice leading is based in a single semitonal displacement, what Cohn refers to as ‘maximally smooth cycles,’ in an octatonic cycle the voice leading displacement alternates between a semitone and a whole-tone, what I call ‘near maximally smooth cycles.’

Take Oct0 for instance, from C-major to C-minor it is needed but a single semitone shift on the third: ê to êb.38 From C-minor to Eb-major, however, the root of the former must descend a whole tone to the fifth of the latter: ĉ to b̂ b, and so on. Going back to the Schubert example, one can easily see how its inherent harmonic logic is better explained by my hypothetical octatonic system than Cohn’s tortuous and inconsistent path through his Cube Dance. Note how the modulatory path of this Schubert passage fits perfectly with my Oct0 below. The opposite,

38 In this study, a lowercase letter with an circumflex (e.g., ĉ), like the scale degree notation (1, 2, 3, etc.), indicates pitch. 31 however, is just as true. In passages based more closely on major third harmonic relations, such as the Liszt and Frank’s examples above, my octatonic system would have to be heavily adjusted to try to explain such progressions. This all to say that a system based on a single intervallic harmonic relation will certainly never be able to properly explain music whose harmonic relations are based on several intervallic relations.39

Fig. 1.10: A proposed model for the hypothetical hyper-octatonic system.

A final feature of Cohn’s theory would be its harmonic transformation dimension. As discussed above, a hexatonic cycle can be achieved by the alternation of P and L transformations. Cohn proposes an extra transformation to classify progressions between triads diametrically opposed to one another in a hexatonic cycle, such as C-major to Ab-minor, C-minor to E-major, etc. He calls this transformation hexatonic (H). Weitzmann regions are also bound by specific transformations. By alternating Nebenverwandt (N) and relative (R) transformations,

39 Based on observations made by Benjamin Boretz on the harmonic properties and implications of fully diminished seventh chords, Cohn briefly considers an octatonic space in which more complex chords, such as half-diminished and seventh, operate. (See Cohn, Audacious Euphony, 152-156.) However, Cohn does not seem to systematically apply octatonic principles to his hexatonic system, nor does he reconsider the overemphatic hole major third harmonic relationships have in his model. Therefore, my hyper- octatonic system, as briefly described above, stands more as a parody and critique to Cohn’s system than as a viable alternative of a harmonic model. 32 one obtains a complete Weitzmann cycle.40 The transformation between diametrically opposed chords on a Weitzmann cycles is called Slide (S). Table 1.2 below summarises the harmonic transformations of Cohn’s theory.

Hexatonic Weitzmann Transformation P L H R N S From C to c e ab a f db Table 1.2: Cohn’s unified model transformations.

As can be seen from Table 1.2 above, the range of harmonic possibilities achieved by

Cohn’s six transformations is very restricted. To be sure, his transformations can be combined among themselves in order to arrive at chords not listed above. A progression from C-major to

Ab-major, for instance, wold be classified in Cohn’s transformational terms by the compound

PL, i.e., first C-major suffers a P transformation to C-minor, which then undergoes a L transformation to Ab-major. Even if C-major moves directly to Ab-major, Cohn’s system would assume an intervening C-minor in between. Plain diatonic progressions are even harder to be achieved by this transformational system. Imagine C-major to D-minor, i.e., a common I to ii in a cadential context, for instance. A possible voice leading path would be: C-major to E-minor, E- minor to G-major, G-major to G-minor, G-minor to Bb-major, and Bb-major to D-minor. In transformational terms this path would result in the overly cumbersome LRPRL symbol.

Another possibility, more direct but still too cumbersome for such a simple progression, would be from C-major to A-minor, A-minor to A-major, and A-major to D-minor. This time the symbol would be RPN. Regardless of the path chosen, the transformational symbol does not bear any resemblance with the direct musical reality of the progression. To not have an unary transformation for an actual unary progression is a clear weakness of Cohn’s system.

40 According to Cohn: “Although the triads of a Weitzmann region have no natural cyclic ordering on the basis of voice leading proximity, they do fall quasi naturally into a cycle on the basis of their historical origins in classical syntactic routines.” (Audacious Euphony, 61). 33

The fixation with optimal, i.e., minimal, voice leading procedures, I believe, does more harm than good to Cohn’s theory. On top of resulting in cumbersome transformations as the ones mentioned above, it excludes from the picture normative, diatonic progressions, such as V-I or I-

IV, etc., which are, needless to say, extremely common throughout the repertoire. In other words,

I would argue an exaggerated weight has been given to the contrapuntal features of hard-to- explain progressions at the expense of more harmonic oriented considerations. On that note,

Dmitri Tymoczko asks: “couldn’t it be that composers chose these [third-related] chord progressions for other reasons, applying the efficient voice leading only as an afterthought? [The harmonic relations] may be as important as their contrapuntal relations. To be sure, [passages may] exhibit efficient voice leading – but then again, so does most other tonal music. Perhaps voice leading is a secondary matter, subservient to deeper harmonic forces.” 41

Imbedded throughout Cohn’s theory and analytical practices is the principle of double syntax.42 As mentioned above, Cohn’s hexatonic system is based exclusively on major third harmonic relationships. This choice might have given Cohn the degree of systematization, uniformity and symmetrical rendering that he felt was necessary to develop his theory, but it has also severely crippled the overall reach and applicability of his model. Double syntax, then, is the strategy devised by Cohn to cope with the limitations of his model. In the examples cited here, one can see how it works. The final two measures of the Liszt passage (Ex 1.2), for example, operate entirely within a diatonic logic (a V7-I cadence). At that moment, Cohn suspends the hexatonic analysis, or else, shifts the ‘syntax,’ and assigns a general ‘tonal cadence’ to the progression. In the Frank example (Ex. 1.3) there is a disturbing number of syntax changes

41 Dmitri Tymoczko, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice, (New York: Oxford University Press, 2011), 303. Italics mine. 42 See the discussion on Proctor’s classical diatonic tonality and 19th-century chromatic tonality above (section 1.1). 34 between hexatonic groups and octatonic swift progressions. Although not as evident from the graph, the Schubert passage (Ex. 1.9) is filled with syntactical shifts, as well. By connecting harmonic evets of different natures (i.e., chords to harmonic regions), Cohn concocts a graph well-suited to his a priori assumptions, even though it does not necessarily reflect the true harmonic nature of the passage.

It goes without saying that the theories and systems considered in this chapter are much richer and more complex than it is possible to be addressed here. My discussion of them concentrated in very specific issues and served the bigger purpose of briefly contextualizing the current state of non-normative (but really chromatic third-relations) theoretical considerations. In the next chapter, in response to the issues raised thus far, I offer my own theory of harmony. Chapter 2 The Trans-Diatonic Theory

Having considered the limitations of the diatonic system and diatonic-based analytical models to coherently explain nineteenth-century harmonic procedures and having found that the examined modern theories of harmony, that were specifically designed to address some of the issues raised in the Introduction, do not offer models reliable enough to consistently deal with

Romantic harmony in its entirety, I now offer my own analytical model in the hopes that it may provide the modern analyst with a set of tools and concepts that are at the same time systematic and flexible enough to properly engage with nineteenth-century harmonic language.

2.1 The Diptych Principle

The core concept of the trans-diatonic theory is what I call the diptych principle. A diptych is a term normally used to refer to a painting comprised of two hinged, smaller paintings.

Usually, the individual paintings can stand on their own, but only together do they achieve their complete meaningfulness. Thus, a diptych can be said to be a larger, composite unity articulated by two subsidiary, smaller unities. The diptych principle works the same way in trans-diatonic theory. Simply put, the diptych principle sees homonymous keys (e.g., C major and C minor) not as independent keys per se, but rather as complementary, and fully integrated equals under a new trans-diatonic super-key called diptych C, or trans-diatonic C. It is a quite simple construct, but when coherently developed and thoroughly applied it provides a logical system of relationships that is ideal to analyse ‘non-normative’ nineteenth-century harmonic procedures. Moreover, the diptych principle is not a concept artificially invented ex nihilo, but rather entirely born from 36 observation and deducted from the literature, as will be demonstrated and develop throughout this chapter.1

Naturally, mode mixture plays a significant role here. However, differently from the theories and approaches that will be discussed in the next section, the trans-diatonic theory takes mode mixture as a mean to an end, rather than an end in itself. To be sure, mode mixture is not an uncommon phenomenon in music. One could easily trace back its usage to pre-tonal era in which musica ficta was used to ‘adjust’ a given mode. The increasing appearance of borrowed scale degrees from other modes, especially in structural points, was instrumental to the establishment of what we call today the major and minor modes/scales. The exchange of selected scale degrees between opposite mode keys entails several harmonic implications. The tierce de

Picardie, fully-diminished seventh chords, the , the augmented , augmented chords, and even the dominant major chord in minor keys are all the result of mode mixture procedures.

2.1.1 Towards Mode Fusion

According to Proctor, Heinrich Schenker was one of the first theoreticians to systematically consider notions of mode mixture. Schenker asserts that “any composition moves in a major-minor system. A composition in C, for example, should be understood as in C major-

1 The diptych principle is by no means to be confused with tonal pairing or double-tonic complex. While the latter, envisioned by Robert Bailey and developed by several other theorists, applies to “situations where two keys simultaneously occupy the highest position in a tonal hierarchy” (Krebs 1996, 17) usually combining a major Tonic and its minor relative, or vice-versa; the trans-diatonic diptych principle inextricably fuses parallel keys and, by doing so, achieves a new super-key with an increased, and more complex harmonic pool. The consequences of carrying out such concept will be explored here. Furthermore, it must be said that neither Hugo Riemann’s dualism, in which the minor mode is the mirrored inversion of the major, nor Daniel Harrison’s new dualism, in which major and minor are still seen as separable, feature here. To use Harrison’s own analogy, trans-diatonic theory sees homonymous keys not as the Adam and Eve of harmonic function, but as the original couple, an inextricable unity comprised by two hierarchically identical members. 37 minor (C !"#$%); for a pure C major, without any C minor ingredient, or, vice versa, a pure C !&'$% minor, without any C major component, hardly ever occurs in reality.”2 According to Kopp,

Gottfried Weber considered the relation between parallel major and minor to be the closest one:

“The similarity [between parallel modes] is … so great, that it almost ceases to be a mere identity, and well-neigh passes into an absolute identity.”3 Donald Tovey also considers the blending of major and minor modes. While addressing Schubert’s String Quintet he talks of a

“half minor and half major”4 context and declares beforehand that “the first basis […] of wider key-relations is that major and minor keys on the same tonic are identical.”5

Walter Piston writes that “the modal implications of the chord progressions have less basic significance than their tonal implications. Major and minor modes are not as distinct in usage as their two scales would seem to indicate […]. Fluctuation between major and minor has always been common.”6 Along the same lines, Proctor says “mixture simply allows the interchange of scale degrees between different parallel diatonic scale degrees of opposite mode, without alteration of the function of the mixed scale degree.”7 In a more systematic manner,

Edward Aldwell, Carl Schachter and Allen Cadwallader assert three degrees of mixture: (1) the simple mixture borrows an element from the parallel mode (e.g., F-minor in C major); (2) the secondary mixture alters the quality of a triad that belongs to the key without borrowing it from the parallel mode (e.g., E-major in C major); and (3) the double mixture combines the previous

2 Heinrich Schenker, Harmony (Chicago: University of Chicago Press, 1954), 86. Italics mine. 3 Kopp, Chromatic Transformations, 42, n. 30. Italics mine. 4 Donald F. Tovey, “Tonality in Schubert,” in The Main Stream of Music and other essays (New York: Oxford, 1949), 151. Italics mine. 5 Tovey, “Tonality in Schubert,” 147. Italics mine. 6 Walter Piston, Harmony (New York: W. W. Norton, 1978), 57. Italics mine. 7 Proctor, “Technical Bases of Nineteenth-Century Chromatic Tonality,” 44. 38 two types by applying secondary mixture to a triad achieved through simple mixture (e.g., Ab- minor in C major).8 Patrick McCreless, based in the findings of Robert Bailey, writes that:

in the late nineteenth-century we have moved from a tonal universe in which there are twenty- four diatonic major and minor keys to one in which there are twelve keys with interchangeable mode. Although a given passage can be, of course, diatonically in either the major or minor mode, it may also incorporate elements of both. More importantly, the modes are functionally equivalent: a tonic performs the same role, regardless of modal choice. What we have here is essentially Schenker’s principle of mixture, plus the explicit acknowledgment that neither mode needs to be designated as taking precedence over the other.9 All these writers agree as to the ubiquity of mode mixture, although they do not seem to systematically elaborate this notion towards Romantic harmonic analysis. What I propose to do with the trans-diatonic theory is just that. Note that throughout the citations above terms like major-minor, half major half minor, !"#$%, etc., were very common. They rightly convey the !&'$% meaning of mixture between major and minor modes. The trans-diatonic theory however takes this notion a step further with its diptych principle. Consequently, a diptych key is more than just the combinations of some degrees from parallel keys, more than major/minor – the whole being larger than its constituent parts – as will be demonstrated below. For clarity sake, I propose the use of the term mode fusion, rather than mode mixture, to convey the special conditions afforded by the diptych principle.

2.1.2 The Trans-Diatonic Scale

In response to the increasing interchangeability between major and minor, several theorists proposed new scalar collections as the result of what I call mode fusion. Moritz

Hauptmann, for instance, concerned with the disparate number of major and minor scales, proposed what he called a ‘minor-major’ key. To be sure, “Hauptmann’s formulation of minor-

8 Aldwell Edward, Carl Schachter and Allen Cadwallader, Harmony and Voice Leading (Boston: Schirmer, 2011), 590. 9 Patrick McCreless, “Ernst Kurth and the Analysis of the Chromatic Music of the Late Nineteenth Century,” Music Theory Spectrum 5 (1983): 60. Italics mine. Andante mesto q = 88 3 œ ˙ œ ?#4 œ œ™ œ œ œ ˙ œ ˙™ #œ Œ œ œ™ œ œnœ ˙ #œ ˙ #˙ œ

f 39 ?#3 ˙ œ œ ˙ œ { 4 œ œ™ œ œ œ ˙ œ ˙™ #œ Œ œ œ™ œ œ10nœ ˙ #œ #˙ major [key] explicitly rejects the idea of modal mixture” however, the fact remains that its “‘ 10 >˙™ œ ™ scale?# is born outŒ of#œ the ˙same #processœ Œ #œesn ˙of™ modalœ Œmixture#œ ˙.™ Essentially,œ Œ Œ the∑ minor-major∑ scale can

be thought of as a majordim scale. with a lowered 6, or conversely, as puts Harrison, “consisting of a p >˙™ œ ˙™ ?# #œ n˙™ œ 11 ˙™ œ œ {harmonic minor Œscale#œ fitted withŒ maj#œ or mode’s 3Œ.”#œ Given that theŒ Œharmonic∑ #minorœ Œ Œ scale is itself #œ œ :“; a secondary entity derived from the natural minor, I favor the derivation of the minor-major scale 21 ?# n from the primary major scale.∑ To Hauptmann,c however, this scale is not∑ the product of mode & {mixtu?# re, as said before, but∑ an attempt towardsc modal isomorphism between∑ the major and minorn systems. Fig. 2.1 below depicts Hauptmann’s minor-major scale.

bw w w & w w nw w w

Fig. 2.1: Hauptmann’s minor-major scale. 31 & Schenker∑ ∑ systematically∑ ∑ considers∑ several∑ combinatory∑ ∑ possibi∑lities between∑ ∑the pitch∑ {collection? ∑ of the∑ major ∑and minor∑ scale,∑ depicted∑ below∑ in Fig.∑ 2.2. Having∑ the∑ major∑ scale as∑ upper and the minor as lower limits, Schenker conducts a series of combinations. The first series, 43 from& top ∑to bottom,∑ is “the so-∑called melodic∑ minor∑ scale.” The∑ second∑ series is the∑ same as∑ the minor-major scale proposed by Hauptmann above. The third, the Mixolydian mode. The fourth is {? ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ “the so-called harmonic .” The fifth, the Dorian mode. And “the sixth series, finally,

is5 2clearly the product of a combination.”12 & ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ {? ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

10 Daniel Harrison, Harmonic Functions in Chromatic Music: A Renewed Dualist Theory and an Account of Its Precedents (Chicago: University of Chicago Press, 1994), 231. 11 Idem. 12 Schenker, Harmony, 87-93. 4 MATTHEW RILEY 40 Ex. 2. Schenker on `the six products of combination'

3

6

7

3, 6

3, 7

6, 7

3, 6, 7

Fig. 2.2: Schenker’s ‘The Six Products of Combination.’

James Marra, after!1906), identifying Schenker issues in claimed four slightly that distinct all versions compositions of the chromatic should be regarded as inhabiting a `major-minor' system. !He explicitly distinguished this from scale, proposes what heHauptmann's calls the ‘tonal chromatic minor-major scale,’ shownmode.) below A piece as Fig. in2.3. C According would thusto be in C major/ ^ ^ ^ Marra, minor, with any amount of mixture permissible on 3, 6 and 7. Schenker lists the various mixed scales !Ex. 2), including the major scale with [6,^ but sees The tonal chromaticthem scale as self-sufficient can claim advantages only in as the a functional weakest sense. model after They the inhabit a space between suggested scalespure [Allen major Forte’s !top)melodic and chromatic pure minor scale; William !bottom), Drabkin’s and areintervallic evidently not to be taken as chromatic scale;independent Roger Session’s ontological synthetic chromatic entities scale; that and could Ebenezer serve, Prout’s even conceptually, as the harmonic chromaticbasis scale for]. wholeFirst, the pieces requisite or indeedtop-down even hierarchical for extended structure ispassages. The plentiful captured. […] Second, modality is determined in terms of the structure of the tonic triad and the relationshipillustrations of its elements that he to draws neighboring from chro thematic eighteenth- and diatonic and tones. nineteenth-century litera- […] Third, it suggeststure show that within many a chromatictonally functional alterations chromatic taking scale placerare intervals in quick succession, with no […] can act as positionsingle markers. scalar13 formation remaining in place for long. In particular, the major scale with [6,^ although it `offers the artist the possibility of rich colouring', is ultimately to be understood as a local deformation of major or minor, depend- ing on context.6 From this perspective, there is no need to posit anything so cumbersome as a separate mode to account for the formation in Ex. 1. It simply represents the of a chromatic alteration applied to the underlying E[ major collection. 13 James Marra, “The TonalThe Chromatic difference Scale as a between Model for Functional the viewpoints Chromaticism,” of Hauptmann Music and Schenker stems Perception: An Interdisciplinaryfrom theirJournal differing 4, no. 1 (1986): accounts 76-77. of the nature of tonality itself. Hauptmann proceeds from the starting-point of `things directly known', which for him are

ß Blackwell Publishing Ltd. 2005 Music Analysis, 23/i !2004) 76 James Marra

Further, it does not distinguish between the use of the diatonic de- gree in minor and the chromatic raised sixth scale degree in major. Figure 3 displays two chromatic scale forms which will be referred to as "tonal chromatic" scales4 and which can serve as a model for functional chromaticism. The "grammar" of these scales is represented in the follow- ing nonformal list of constraints:

1. The scale will be linearly ordered either as an unidirectional ascent or descent. 2. It will be delimited by tonic scale degrees, one octave apart. 3. All 12 tones of the equal tempered system must be included. 4. A diatonic scale must be a proper subset of the tonal chro- matic scale. 5. 1,3, and 5 may not be chromatically altered. 6. All chromatic degrees must be represented as either ascending or descending semitonal neighbor of 1, 3, or 5; except #6/t 7. 7. #6 and \>1 in major are chromatic neighbor tones to 7 and 6, respectively.

The tonal chromatic scale can claim advantages as a functional model over other suggested scales. First, the requisite top-down hierarchical struc- ture is captured. The tonic triad is melodically and harmonically prolonged with respect to (1) diatonic scale degrees and (2) more surface chromatic transformations of those degrees. Second, modality is determined in terms of the structure of the tonic triad and the relationship of its elements to neighboring chromatic and diatonic tones. It will be recalled that the me- lodic chromatic scale clouded these functional relationships. Third, it sug- gests that within a tonally functional chromatic scale rare intervals (and here the "rare" denotes not interval classes mod 12 but functional intervals 41

Fig. 2.3: Marra’s ‘tonal chromatic scale forms.’14 Fig. 3. Tonal chromatic scale forms. However, none of the scale systems considered above present the features or tools I deem necessary 4. As to far analyze as this nineteenth author- centuryis aware, harmonic similar procedures. scales were Therefore, first proposed in what follows, by Arnold I will Franchetti and Joseph Mulready in The dissolution of architectonic tonality (unpublished, no date or propose place of my writing own scale provided).. As said before, by applying the diptych principle to parallel keys one would obtain a new diptych super-key. This super-key is naturally equipped with its own scale which I call the trans-diatonic scale. Figure 2.4 below illustrates the modal fusion process between the Thisscales content of both downloaded major and from minor 136.159.123.20 modes into the on transTue, -27diatonic Feb 2018 scale 17:51:51. The term UTC ‘trans’ All use subject to http://about.jstor.org/terms in trans-diatonic scale alludes to the fact not only of its transversality within major and minor modes, but also to its transcendence to both of them.

To fuse parallels keys is not as smooth a task as it would appear at first. There is an issue here that must be addressed before proceeding. On the one hand, the major scale is a well- stablished entity and poses no problems whatsoever. On the other, the minor mode traditionally comprises not one but three distinct scales – the natural, the harmonic, and the melodic (which has two different versions, one upwards and the other downwards) – which poses an issue, albeit easily fixed. This situation was what led Hauptmann to formulate his minor-major key, discussed above.

14 Idem, 76. 42

According to Harrison, there are apparently two solutions to solve this situation and to achieve isomorphism between major and minor systems. The first method, inaugurated by

Hauptmann “encouraged a broadening of the major system by proposing hybrid scale forms analogous to those of minor […] in order to bring about parity between major and minor.”15 The second method, favored by Hugo Riemann, instead of creating new major scales would select one minor scale from the already existent ones to be the minor scale par excellence – naturally, the natural minor. Of the two methods to achieve modal isomorphism, Harrison prefers, as do I, the latter one for the following three reasons: (1) the ideal scenario of having one essential scale for both modes; (2) the a priori unaltered, diatonic structure of both scales; and (3) the clear harmonic differentiation between the major scale and its minor counterpart. Although such distinctions are important, I would argue that perhaps Harrison overemphasises this matter. Once one realizes that the harmonic and melodic minor scales are themselves ‘impure,’ altered versions of the natural minor, clearly achieved by musica ficta (mode mixture) procedures, the choosing of the one ‘true’ minor scale is no longer an issue.

Accordingly, in Fig. 2.4 one can see how the trans-diatonic scale is the result of the complete fusion between major and minor. Notice the simultaneous coexistence of scale degrees b3 and n3, b6 and n6, and b7 and n7. In the trans-diatonic scale, the typical mode-defining scale degrees share their place and strength with that of the opposite mode. Thus, this new scale is neither truly diatonic nor fully chromatic – it is trans-diatonic.

15 Harrison, Harmonic Functions in Chromatic Music, 23. 43

& w w w w Major & w w w w w w w w Major w w w w w Minor {& w w bw bw w Minor {& w w bw w bw bw w w bw w

w Trans- & w w b˙ n˙ b˙ n˙ w dTiraatnons-ic & w w b˙ n˙ w w b˙ n˙ b˙ n˙ diatonic w w b˙ n˙ Fig. 2.4: Fusion of the major and minor scales into the trans-diatonic scale.

Incidentally, Proctor arrives at this same scale in a discussion considering the chromatic œ œ œ œ œ wœ œwœ & œ œ œ wœ œwœ œw œ œ scaleMa jasor a productwœ of modew mixture.œ Hew writes:œ “The mixtureœ of parallelwœ majorw and minorw scales & œ œ wœ wœ w Major wI wii iii IV V vi vii° I produces the compositeI imajor/minori iii collectionIV [equivalentV to the transvi -diatonicvœii° scale].I œ[…] We Minor œ œ bœ œ bwœ {& œ bœ bœ bœ bwœ bbœw bœw bœ Minor bœ œ b w w œ œ w can attempt{& toœ completew b œthisw chromatbœ ic scale bbyœ applyingbwœ mixtureb ofwœ major andbw minor with bœ œ b w w wi wii° ¼III iv v ¼VI ¼VII i traditional churchi modes.”ii° 16 Proctor¼III then arguesiv that mixturev with¼V theI Phrygian¼VII mode wouldi Trans- ∑ 2 Ú Ú Ú Ú Ú Ú provide the& b2 and mixture21 with the parallel Lydian would yield the #4. Noticeably, Proctor dTiraatnons-ic ∑ 1 Ú Ú Ú Ú Ú Ú diatonic & seems to be more concerned in filling out the ‘missing’ degrees (b2 and #4) to form a complete

chromatic scale, the basis of his nineteenth-century chromatic tonality (see section 1.1), and does bœ nœ bœ nœ œ œ bœ nœ Trans- bœ nœ bœ nœ œ œ bœ nœ œ œ œ œ bwœ nwœ w w & œ œ bœ nœ œ œ œ œ bwœ nwœ w w bbœ˙ bnœ˙ nbœ˙ nnœ˙ œ œ bœ nœ notdTiraat norecognizens-ic thebwœ transnwœ -diatonicw w b scale˙ b ˙in its˙ own˙ œ right.œ œ œ œ œ œ œ bwœ nwœ w w & œ œ bœ nœ bœ nœ bœ nœ bwœ nwœ bwœ nwœ b ˙ b ˙ n ˙ n ˙ diatonic bwœi nwœI wii° wii b ¼˙IIIb ¼˙III= i˙ii° i˙ii iv IV v V ¼VI ¼VI= vi° vi ¼VII vii° i I Similarly,i HarrisonI ii° i iwrite¼IsII that¼III = “theiii° i ichromatici iv IV scalev V can ¼beVI thought¼VI= vi° ofvi as¼V aI Icompletevii° i I

mixture of major and minor, along with other elements ( above 1 and 4). This Trans- Ú Ú Ú Ú Ú Ú diatonic & particularTrans- mixture doesÚ have a specialÚ name butÚ not necessarilyÚ special tonalÚ properties, insofarÚ diatonic & as it can be heard within a major-minor context and, thus, in a sphere where harmonic function Ú Ú Ú Ú Ú Ú 17& operates.Major ” Here again,Ú the trans-Údiatonic scale Úis considered Úonly en route Úto the chromaticÚ one, Major & andM idoesnor { not& receive Úthe attention orÚ status I deemÚ necessary. TheÚ urge to addÚ scale degreesÚ b2 Minor {& Ú Ú Ú Ú Ú Ú and #4 to the trans-diatonic scale can be seen as the culmination of a historic macro-process that

16 Proctor, “Technical Bases of Nineteenth-Century Chromatic Tonality,” 54. 17 Harrison, Harmonic Functions in Chromatic Music, 35-36, note 23. 44 began well before the tonal era and was entirely driven by musica ficta procedures. As said before, musica ficta-led adjustments to some modes were eventually responsible for the narrowing down from the church modes to the major and minor diatonic modes and, finally, to the chromatic scale. Being the result of major and minor fusion, the trans-diatonic scale, or trans- diatonicism, is thus the ‘missing link’ between Classic diatonicism and full chromaticism, being therefore ideal for dealing with Romantic harmony, as will be demonstrated. Fig. 2.5 below illustrates this process.

Fig. 2.5: Trans-diatonicism as the missing link between diatonicism and chromaticism.

Considering the trans-diatonic scale as an entity in its own right, Ex. 2.1 below shows an interesting instance of its utilization. In the introduction to his Polonaise I from Die Legende vom heiligen Stanislaus, Liszt employs an almost complete trans-diatonic scale. The only missing scale degree is #6, i.e., the sixth degree from its major diptych element, E major. However, n6 is emphasised throughout the passage, rendering its overall ‘minorness.’ Although the diptych principle fuses both major and minor keys into a super-key, it however does not trump their modal qualities. In other words, the pieces can still be in major or minor keys or be heard as 45 such, but nonetheless operate under trans-diatonic harmonically expanded premises. Thus, when referring to a clearly trans-diatonic piece or passage that favors its major-mode side, the term

‘trans-diatonic major’ should be employed; when it favors minor, ‘trans-diatonic minor.’ Liszt’s introduction to Polonaise I is an example of a trans-diatonic minor passage. However, even though the differentiation between trans-diatonic major and minor is a possibility, it may generally be better, and more direct, to generally refer to such passages as simply ‘trans- diatonic,’ bearing in mind the coexistence of major and minor harmonic features under the auspices of the diptych principle. Traditional approaches to the passage below would probably engage the n7 and #7 appearances as deriving from natural and harmonic minor scale, respectively. The #3 in m. 13 would probably be simply ignored or quickly dismissed as a generic chromatic elaboration. Conversely, by using the trans-diatonic system, all scale degrees from this passage are explainable and accounted for.

Andante mesto q = 88 3 œ ˙ œ ?#4 œ œ™ œ œ œ ˙ œ ˙™ #œ Œ œ œ™ œ œnœ ˙ #œ ˙ #˙ œ f ?#3 ˙ œ œ ˙ œ { 4 œ œ™ œ œ œ ˙ œ ˙™ #œ Œ œ œ™ œ œnœ ˙ #œ #˙ “‘

10 >˙™ œ ™ ?# Œ #œ ˙ #œ Œ #œ n˙™ œ Œ #œ ˙™ œ Œ Œ ∑ ∑

dim. p >˙™ œ ˙™ {?# Œ #œ #œ Œ #œ n˙™ œ Œ #œ ˙™ œ Œ Œ ∑ #œ Œ Œ œ #œ œ :“;

Ex. 2.1: Liszt, Polonaise I from Die Legende vom heiligen Stanislaus, mm. 1-20.

Now I explore the harmonic possibilities furthered by the trans-diatonic scale. In Fig. 2.6 21 ?# ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ below I build one triad upon each scale degree using only the pitches indigenous to that scale. {?# ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

35 ?# ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

{?# ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

48 ?# ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

{?# ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ œ œ œ œ wœ 46 Trans- & œ œ œ œ wœ wœ w diatonic wœ wœ w w I ii iii IV V vi vii° I Whereas the& major and minor scale offer seven triadic possibilitiesw w each, thew trans-diatonicw Major w w w w presents eighteen.° Clearly the whole of the trans-diatonic scale is greaterbœ thanœ the sumb ofœ its Trans- œ œ œ bwœ w & œ bœ bœ bwœ bwœ b w w dMiaitnoonric{& bwœ w b w w bw bw constituent parts. The extraw four chordsbw are onlyw possible due to the coexistence of scale degrees wi ii° III iv v VI VII i + o + b3, n3, b6, n6, b7, and n7 mediated by the diptych principle.œ The extrabœ chords œare: bIII , bnœiii , bVI , Trans- bœ nœ bœ nœ b wœ b˙nw n˙ ww Voice && bœ wœ b˙ bnwœ˙ w w w w b˙ n˙ dandiato nviico, as canw wbe seen wbelow. As the seven triads from major and minor are diatonic to their i ii° III= iv V VI vii° i respective key, so are the eighteen chords trans-diatonic to a diptych key. œ œ nœ œ nœ bœ nwœ bwœ Voice ¢& œ nœ œ nwœ w n w bwœ w b w œ œ œ œ wœ wœ & œi œii œIII= wœIV Vwœ wvi° vii° i Major wœ w w I ii iii IV V vi vii° I 2 œ œ MTirnaonrs- & ∑ œ 1 œ bœ Ú œ bwœ diatonic{& œ bœ bœ bœ bwœ b w bw bwœ wœ b w w i ii° ¼III iv v ¼VI ¼VII i œ œ œ œ Trans- 2 œ œ bœ nœ bœ nœ œ œ bwœ nwœ Trans- && œ ∑ œ b1œ nœ Úbœ nœ bœ nÚœ bœ nœ Úbwœ nwœ b ˙œÚ b ˙œ n ˙œ n ˙œÚbw nw Ú ddiaitaotonnicic bwœ nwœ wœ wœ b ˙b ˙n ˙n ˙ w w i I ii° ii ¼III ¼III= ½iii° ½iii iv IV v V ¼VI ¼VI= ½vi° ½vi ¼VII ½vii° i I

Fig. 2.6: Trans-diatonic triadic collection.18 œ œ bœ nœ bœ nœ œ œ bwœ nwœ Tran2s.2- Orders& œ ofœ Complexitybœ nœ b andœ n œTonnetzingbœ nœ bœ nœ bwœ nwœ b ˙œ b ˙œ n ˙œ n ˙œ bw nw diatonic bwœ nwœ wœ wœ b ˙ b ˙ ˙ ˙ w w In its originali I formulation,ii° ii ¼III the¼III =diptychiii° iii principleiv IV v actsV upon¼VI parallel¼VI= vi° keysvi ¼byVII renderingvii° i Ithem

equivalent. However, by not limiting its application to keys alone but rather allowing it to be Trans- Ú Ú Ú Ú Ú Ú dapplicableiatonic & to parallel chords as well, one obtains a fuller, richer theory of harmony with far

reaching consequences. Although the effect of the diptych principle either upon parallel keys or

parallel chords is theÚ same, i.e., theÚ rendering ofÚ modal equivalencyÚ betweenÚ major and minorÚ Major & members, it works fundamentally on different levels or orders of complexity. Minor {& Ú Ú Ú Ú Ú Ú

18 In trans-diatonic theory, Roman numerals do not bare any functional significance. They merely indicate the scale degree whereupon a chord is built, its quality (if major, minor, augmented or diminished), and inversion. 47

There are three orders of complexity in which harmonic activity can operate. The first one, the simplest, is limited to the pitch level of activity. It does not necessarily convey a harmonic sense, but it can certainly imply it. Ex. 2.1 above presented a passage from Liszt’s

Polonaise I which operates entirely within the first order of complexity. The second order of complexity is the one reserved for the chord level of harmonic activity and it concentrates on the relationship among them. The trans-diatonic scale in Fig. 2.6 depicted the trans-diatonic triadic collection whereby its members ought to interact to one another on this order of complexity. The third order of complexity, the highest one, deals with harmonic regions and how they might relate to a central key on a macro-level.19 Although one might swiftly shift from one order of complexity to another, a clear sense on which order one’s analysis is operating is paramount to its clarity. A city is not the same as a province inasmuch that a chord is not the same as a harmonic region. The analysis offered by Cohn of the introduction to the Overture of Schubert’s

Die Zauberharfen (see section 1.4), as discussed before (section 1.4), lacks such clarity. He constantly relates events of the second order with that of the third, in other words, he considers chords and harmonic regions as operating in the same level, and shifts indiscriminately between them.

From a trans-diatonic perspective every event, independent of its order of complexity, must be directly relatable to a given centrality or point of reference. In other words, every relationship – be it between pitches, chords, or harmonic regions or keys – can be immediately assessed without intermediary agents. That way, instead of imagining a C-minor and an Ab-

19 Dyads can be assigned to either the first order of complexity or the second – it will mostly depend on the context and on the analyst’s intended purpose. More complex chords, such as seventh chords, 9th, and so on, as well as dyads, will not be systematically contemplated in this study. To be clear, seventh chords do appear in the examples and analysis presented here, (the most common being Dominant seventh chords), however they are treated in light of their triadic structure. 48 major between an actual C-major to Ab-minor progression, with trans-diatonic theory one can deal with such a progression as it really is – a direct, unmediated relationship. For convenience sake, I begin by addressing the third order of complexity (harmonic regions) and its spatial representation or Tonnetz, to then move on to the second and first orders respectively.

2.2.1 Third Order of Complexity

The third order of complexity is dedicated to events on the macro-level of harmonic regions. A harmonic region is a principal or secondary key (i.e., a key within a key) to which events from lesser orders of complexity are subordinate. As a rule, a harmonic region must always be stable in order to support less stable, or unstable activities from subordinate orders of complexity. In other words, a harmonic region/key must be consonant and based on a single triad or diptych pair. In what follows I build, step by step, the Tonnetz of the third order complexity and consider some of its features.

a. b. c. c Eb Eb

F C G f c g f c g

a C F C G

a

ab eb bb d. e. Ab Eb Bb Ab Eb Bb

f c g f c g

F C G F C G

d a e d a e

D A E Fig. 2.7: Steps towards the trans-diatonic Tonnetz of the third order of complexity. 49

Fig. 2.7a., above, depicts the main harmonic regions of a diatonic major Tonic – the

Dominant, the Subdominant, the relative and the parallel. Fig. 2.7b. presents the same relations for a diatonic minor tonic. The next figure, Fig. 2.7c., overlaps the two previous figures by suggesting the diptych union of the major and minor tonics (the dashed frame involving C and c). Note the consequent diptych pairing of the major and minor Dominants and as well. At any given time, a composer may choose to modulate to one harmonic region out of twenty-three. Consequently, any Tonnetz of any order of complexity will only be complete if it encompasses all possibilities. Thus, the Tonnetz depicted in Fig. 2.7c must be expanded. The manner in which such expansion is to be carried on was already established by the harmonic relations offered by the previous Tonnetze, i.e., perfect fifth relation horizontally (Dominant and

Subdominant) and minor third and homonymous relation vertically (relative and parallel).

With that in mind, Fig. 2.7d. carries out a discreet first stage of expansion towards a complete third order of complexity Tonnetz. Fig. 2.7d. provides an useful frame of reference, for all of its members correspond ipsis litteris to all the stable chords from the trans-diatonic scale – the only ones capable of becoming a harmonic region (see Fig. 2.6). Although it looks symmetrical, Fig. 2.7d. presents two levels of asymmetry – one functional and the other structural. Note how a move sideways or diagonally from any region to another necessarily implies a change of function, whereas moving vertically does not necessarily do so. Note also that the trans-diatonic diptych principle is in effect only in the middle layer, leaving the outer regions unattended. By applying the diptych principle to the outer regions, Fig. 2.7e. corrects the structural asymmetry of Fig. 2.7d. The functional asymmetry, however, stays in place for now.

In order to differentiate trans-diatonic regions proper, i.e., those predicted by the trans- diatonic scale (Fig. 2.6 and Fig. 2.7d.) from those achieved by the secondary application of the 50 diptych principle (the outer regions in Fig. 2.7e.), I propose the use of two different categories of classification called soft- and hard-regions. Thus, the mediants regions predicted by the trans- diatonic scale (A minor, Eb major, E minor, and Ab major) are called soft mediants; whereas their parallels, achieved through the secondary application of the diptych principle (A major, Eb minor, E major, and Ab minor), are called hard mediants. The same holds true to the regions a whole-tone removed from the tonic: D minor and Bb major are called soft whole-tone regions and their diptych pairs (D major and Bb minor) are referred to as hard whole-tone regions.

Going back to Fig. 2.7e., one notices that the Tonnetz is not yet complete for some regions are still missing. In trans-diatonic theory terms, the harmonic regions based on the semitone above, the semitone below and a tritone removed from the Tonic (Db major, Db minor,

B major, B minor, Gb/F# major, and Gb/F# minor), are the de facto chromatic regions. To clarify, the seemingly chromaticism up to Fig. 2.7e. is not actually chromatic, but rather trans-diatonic to the system and entirely subsumed under an expanded notion of diatonicism, i.e., trans- diatonicism. In order to encompass the missing chromatic regions to the Tonnetz, further expansion must be carried out, always respecting the intervallic relations already established. As soon as all the remaining chromatic regions are contemplated, the Tonnetz is complete and no longer in need of further expansion. Fig. 2.8 below depicts such Tonnetz. 51

Bbb Fb Cb Gb Db Ab Eb

gb db ab eb bb f c

Gb Db Ab Eb Bb F C

eb bb f c g d a

Eb Bb F C G D A

c g d a e b f#

C G D A E B F#

a e b f# c# g# d#

Fig. 2.8: Minimal expansion of the trans-diatonic Tonnetz of the third order of complexity.

To be sure, Fig. 2.8 above presents a high degree of redundancy, both in literal repetition of some regions and in enharmonic equivalency of others, and although Tonnetze in general tend to exist in ad infinitum terms, there is no need to condone unnecessary reiteration here. Fig. 2.9a. below eliminates such repetitions and presents a more elegant trans-diatonic Tonnetz of the third Bbb Fb Cb Gb order of complexity. Fig. 2.9b. and the list below name all regions within a trans-diatonic logic, that is, always relatinggb them directdb ly toab the toniebc diptych.

Gb Dd Ab Eb Bb

eb bb f c g

F C G D A

d a e b f#

A E B F#

f# c# g# d#

SSS SS StRtr tRtr tRtr tRtr ssta sta ta tr

SStA StA tA tR Rd

sss ss s t d

S T D DD DDD

Sr Tr Ta dTa ddta

TR TA DTA DDTA dd ddd TrTR dTrTR TrTR TrTR

Gb

db ab eb bb

Dd Ab Eb Bb

f c g

F C G

d a e b

D A E B

f# Bbb Fb Cb Gb Db Ab Eb

gb db ab eb bb f c

Gb Db Ab Eb Bb F C

eb bb f c g d a

Eb Bb F C G D A

c g d a e b f#

C G D A E B F#

a e b f# c# g# d#

Bbb Fb Cb Gb

gb db ab eb

Gb Dd Ab Eb Bb

eb bb f c g

F C G D A

d a e b f#

A E B F#

f# c# g# d#

SSS SS StRtr tRtr tRtr tRtr ssta sta ta tr

SStA StA tA tR Rd

sss ss s t d

S T D DD DDD

Sr Tr Ta dTa ddta

TR TA DTA DDTA dd ddd TrTR dTrTR TrTR TrTR

52

Gb X a. b. db ab eb bb n ta tr sb

Db Ab Eb Bb N tA tR SB

f c g s t d

F C G S T D

d a e b sp Tr Ta an

D A E B SP TR TA AN

# f x

T Tonic tR tonic Relative SB Sub-tonic t tonic tr tonic relative sb sub-tonic D Dominant Ta Tonic anti-relative N Neapolitan d dominant TA Tonic Anti-relative n neapolitan S Subdominant tA tonic Anti-relative an anti-Neapolitan s subdominant ta tonic anti-relative AN Anti-Neapolitan Tr Tonic relative sp super-tonic X X-region TR Tonic Relative SP Super-tonic x x-region Fig. 2.9: Non-redundant trans-diatonic Tonnetz of the third order of complexity.

A few comments on the naming of the regions are due. Trans-diatonic theory strives for

an efficient notation system and nomenclature method. Thus, the region symbols and names are

as succinct and convey the most information possible. Thus, the uppercase ‘T’ represents the

‘major Tonic’, and the lowercase ‘t,’ the ‘minor tonic’. The same is true throughout. As said

before, every trans-diatonic event must be relatable with a central point of reference – in this

case, the Tonic diptych. Therefore, every harmonic region name makes clear its direct relation to

it. Take A minor, for example: it relates directly to C major as its minor relative, a minor third

below. Consequently, the symbol ‘Tr’ indicates such relation – the lowercase ‘r’ indicates it is

the minor relative of the major Tonic, as indicated by the uppercase ‘T.’ The relative relationship 53 is a special one in the sense its direction varies according to the mode of the starting region. The relative region to a major region will be found a minor third below it; conversely, the relative to a minor region will be found a minor third above it. The relative region to C minor, for instance, will not be found a minor third below it, but rather a minor third above it. The symbol ‘tR’ indicates this relation, i.e., the Relative major of the minor tonic – Eb major.

Conversely, the anti-relative relation will operate under the same premises of the relative relation, but in the opposite direction and always a major third apart. Thus, ‘Ta’ means the minor region (lowercase ‘a’) a major third above T, otherwise known as the e-minor region; ‘tA’ means the major region a major third below t, or the Ab major region. The remaining harmonic regions follow the same logic: the super-tonic regions are a whole-tone above T, and the sub-tonic regions, a whole-tone below. The Neapolitan regions and its antipode, the anti-Neapolitan regions, are respectively a semitone above and a semitone below the Tonic. The furthest regions from the Tonic, which I call X-regions, are a tritone away from T in either direction.

It is time to introduce a trans-diatonic notational novelty. As mentioned before, the diptych principle is the founding concept of the theory. Thus, to refer to a diptych pair in a simple, direct way without having to make reference to its major and/or minor members, I propose the usage of curved brackets to indicate its trans-diatonicity. Therefore ‘{C}’ stands for, depending on the order of complexity, the key, the harmonic region, or the chord of diptych C.

Needless to say, neither the curved bracket notation, nor the diptych principle, apply to the first order of complexity (pitch level). By applying this notational resource to the Tonnetz presented above one corrects the functional asymmetry mentioned earlier and achieves what I call the diptych Tonnetz of the third order of complexity (Fig. 2.10 below). As will be seen in the next chapter, there are advantages in maintaining both the non-redundant, or expanded Tonnetz (Fig. i I

54

2.9) and the diptych Tonnetz below – it will mostly depend on what features of the piece’s harmonic behaviour one would like to highlight.

{Gb} {X} {bV}

{Db} {Ab} {Eb} {Bb} {N} {tA} {tR} {SB} {bII}{bVI}{bIII}{bVII}

{F} {C} {G} {S} {T} {D} {IV} {I} {V}

{D} {A} {E} {B} {SP} {TR} {TA} {AN} {II} {VI} {III} {VII}

{F#} {X} {#IV}

{T} Tonic diptych region {tA} tonic Anti-relative diptych region {D} Dominant “” {SP} Super-tonic “” {S} Subdominant “” {SB} Sub-tonic “” {TR} Tonic Relative “” {N} Neapolitan “” {tR} tonic Relative “” {AN} Anti-Neapolitan “” {TA} Tonic Anti-relative “” {X} X-region “” Fig. 2.10: Diptych Tonnetz of the third order of complexity.

2.2.2 Second Order of Complexity

The second order of complexity deals with harmonic events at the chord level and therefore, unlike the third order, must accommodate not only stable members (i.e., major and minor triads) but also unstable ones, such as diminished and augmented chords. The diptych

Tonnetz of the third order presented in Fig. 2.10 will serve as a frame of reference for this order of complexity. However, what were once seen as harmonic regions, must now be seen as triads.

The curved bracket, or diptych pair notation will be preserved.

For general purposes, dissonant chords will always be considered en route harmonic entities, drawing their meaning not from within themselves but from their expected and/or actual resolution. Thus, since the positioning of the consonant triads are already given from the Tonnetz of the third order, the issue with the Tonnetz of the second order of complexity is essentially 55 where to place the dissonant chords in relation to its already stablished consonant members.

Examining the trans-diatonic scale (Fig. 2.6), one can see six dissonant chords – four diminished

(iio, niiio, nvio, and nviio), and two augmented (bIII+, and bVI+). Since the meaningfulness of the dissonant chord is derived from their resolutions, they will be the ones to determine the positioning of diminished and augmented triads on the Tonnetz.

Regarding diminished triads, consider bo: the standard procedure is to resolve the b̂ -f̂ tritone into ĉ-ê or some variation thereof (ĉ-êb or ĉ#-ê).20 Such resolutions would render {C} or

{A}. However, there are still several other possible resolutions of bo. One might, for instance, not properly resolve the b̂ -f̂ tritone. By fixing the f̂, one could ‘resolve’ it to {F}; by retaining the d̂ , one could resolve to {D}. Many other resolutions are possible, such as: Ab major, Bb major, B minor, Eb major, their diptych pairs and more. However, the first eight resolutions considered here (depicted in Fig. 2.11 below) can be safely deemed to be the most common ones, and consequently, normative resolutions. This same reasoning can be extrapolated to all remaining diminished triads and it will be instrumental to their proper placing in the Tonnetz of the second order of complexity.

Fig. 2.11: Normative diptych pair resolutions of a diminished triad.

20 Although diminish chords commonly appear as fully- or half-diminished seventh chords, i.e., as tetrads, not as triads, this study will consider them, for convenience and consistence purposes, in light of their triadic structure. In that way, the treatment of diminish seventh chords is similar to that of inasmuch as their triadic structure is enough to render clear their role and place in the Tonnetz. 56

The notion that dissonant triads draw their meaning from their resolution is particularly evident concerning augmented triads. Its structural intervallic symmetry renders its inversions aurally irrelevant. Only by assessing its resolution can an augmented chord be truly identified.

Fig. 2.12 below depicts three distinct instances in which the same augmented triad might occur and how it can be defined by its resolution. Note that in every instance the augmented chord is placed between two diptych pairs a minor third apart from each other, and that it acts as a dissonant, contrapuntal link between them in either direction. This characteristic behaviour of the augmented triad is enough to assign its place in the Tonnetz.

Fig. 2.12: Normative resolutions of an augmented triad.

Taking into account the peculiarities of diminished and augmented triads, Fig. 2.13 below presents the diptych Tonnetz of the second order of complexity. Notice its overall resemblance with the diptych Tonnetz of the third order and how each chord can be related to one another.

The stable chords maintain their placing from the Tonnetz of the third order of complexity, while the unstable chords are positioned in relation to their resolutions. Thus, any augmented chord is vertically placed between two diptych pairs; and any diminished triad is equidistantly positioned among its normative diptych pairs resolutions. Naturally, this Tonnetz, as any other, must not be taken as an absolute entity with a fix set of constraining rules, however, it should nonetheless be considered a referential framework whereby the ideal relationships among all triads can be more clearly assessed. 57

{Gb} fo Gb+ {Db} {Ab} {Eb} {Bb} co Db+ go Ab+ do Eb+ ao Bb+ {F} {C} {G} eo F+ bo C+ f#o G+ {D} {A} {E} {B} c#o D+ g#o A+ d#o E+ a#o B+ {F#} F#+

Fig. 2.13: Diptych Tonnetz of the second order of complexity.

2.2.3 First Order of Complexity

The first order of complexity supports events on the pitch level of activities. For the sake of continuity, the intervallic relationships established by the third, and maintained in the second order of complexity, will be preserved here as well. Since it depicts actual pitches, the Tonnetz of the first order is especially suitable for graphically dealing with voice leading issues. Fig. 2.14 below presents five first order Tonnetze: Fig. 2.14a. shows a ‘clean’ Tonnetz. Note that each member is written in a lowercase letter with a circumflex to indicate that it is a pitch. Fig. 2.14b. and 2.14c. show a major and a minor triad, respectively. Major and minor triads will always be depicted as right triangles. In the major chord, the right angle will be on the top right corner; in the minor, on the lower left. Diminished and augmented triads, depicted in Fig. 2.14d. and 2.14e. respectively, will always appear as a straight line: either vertically, in the case of a diminished chord; or in a top to bottom diagonal, for augmented triads. 58

a. b. ĝb ĝb

d4b âb êb bb4 d4b âb êb bb4

f#4 ĉ ĝ f#4 ĉ ĝ

d̂ â ê b̂ d̂ â ê b̂

f#4 f#4

c. d. e. ĝb ĝb ĝb

d4b âb êb bb4 d4b âb êb bb4 d4b âb êb bb4

f#4 ĉ ĝ f#4 ĉ ĝ f#4 ĉ ĝ

d̂ â ê b̂ d̂ â ê b̂ d̂ â ê b̂

f#4 f#4 f#4

Fig. 2.14: Five Tonnetze of the first order of complexity. 59

In terms of voice-leading approaches to triadic displacement, however, one might better benefit from a more abstract spatial representation of pitch space, as presented in Fig. 2.15 below. By anticipating three hierarchical levels of pitch inter-relationship within a triad (the root, the third, and the fifth), and by laying out each level through a chromatic continuum, the alternative layered Tonnetz of the first order of complexity might be better suited to deal with triadic voice leading procedures than the previous one. As an example of how voice leading could be traced in the layered Tonnetz, Fig. 2.15 below depicts all four possible triads with d̂ as root, i.e., D-diminished, -minor, -major, and -augmented. Note how each triad is represented below: for the diminished chord, we have two left braches linking d̂ to f̂, and f̂ to âb or, in other words, a left-left branching pattern. For the , we have a left-right pattern; the major triad presents a right-left pater; and the augmented chord a right-right pattern.

Fig. 2.15: Layered Tonnetz of the first order of complexity.

Even though the first order of complexity is a logical consequence of the reasoning developed thus far, its usefulness and applicability is limited. Whereas the Tonnetze of the third and second orders of complexity can serve as a frame of reference for long, harmonically complex passages or even entire pieces, the Tonnetze of the first order quickly become cumbersome if one tries to painstakingly follow every single voice leading displacement. Also, I believe that due to the lack of a harmonic analytical system specifically tailored to account for

‘hard-to-explain’ progressions, too much emphasis is given to voice-leading procedures in current scholarship in detriment of more pressing issues, such as the actual harmonic relations in 60 play. Voice-leading is par excellence the composer’s domain – he or she is the one who should pay close attention to its intricacies and appropriateness, leaving little room for the analyst to dwell upon, apart from spotting common-tones and contrapuntal smoothness. “After all, couldn’t be that composers chose [hard-to-explain] chord progressions for other reasons, applying the efficient voice leading only as an afterthought?”21 Therefore, I propose the first order of complexity to be regarded mainly as a logical, theoretical construct without much further application in actual analyses, yet didactically helpful and applicable to small, localized contexts.

2.3 Trans-Diatonic Transformational System

Trans-diatonic harmony is beyond function. Harmonic experimentation has evolved to such a degree in the course of the nineteenth-century that to abide to diatonic ideals of harmonic function – tonic, dominant, and subdominant – is as inefficient as it is constrictive for harmonic analyses. Equally ineffective are the torturous adaptations of diatonic models discussed in the previous chapter, such as Kopp’s proposition of a new functional category – the mediant – or

Daniel Harrison’s attribution of functional value to each individual pitch member of a chord.22

The sooner one realizes Romantic music established, developed, and explored its own harmonic universe, its own Gestalt, and was not merely an extension of Classic-period practice, the better.

Thus, trans-diatonic theory offers a larger, overall encompassing concept of harmonic relationship instead of the limited, diatonically oriented harmonic function construct. For every function implies a harmonic relationship, but not all harmonic relationships are functional.

A harmonic relationship between two chords or regions, i.e., between two events from the second or third order of complexity, respectively, can be identified by taking into

21 Dmitri Tymoczko, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice, (New York: Oxford University Press, 2011), 302. 22 See Harrison, Harmonic Function in Chromatic Music, 15-42. According to him, “harmonic function resides in the [individual] scale degrees that make up chords” (Harrison 1994, 42). 61 consideration their position on their respective Tonnetz, and the intervallic distance between their roots. Each relationship implies a specific transformation. The entirety of all transformations constitutes the trans-diatonic transformational system. In this study, transformations will only involve major and minor chords or regions, i.e., consonant events from the second and third order of complexity. Dissonant chords are seen as transformational aids justifiable by their resolution (or lack thereof). Therefore, the diptych Tonnetz of the third order (Fig. 2.10) will be taken as reference here and, unless otherwise stated, will serve as the Tonnetz for the second order of complexity as well. In other words, the Tonnetze presented below can depict harmonic regions and stable chords at the same time. Transformations on the first order will not be contemplated.

There are five axes of harmonic transformation coexisting within a trans-diatonic

Tonnetz. Each axis is representative of a specific intervallic cycle and occupies a specific position on the Tonnetz: the horizontal X-axis embodies the cycle of fifths; the vertical Y-axis, the cycle of minor thirds; the top-down diagonal Z-axis encompasses the cycle of major thirds; the bottom-up diagonal W-axis, the or whole-tone cycle; and the 30o tilted diagonal S-axis go through the minor second or chromatic cycle. Needless to say, they all operate within a diptych universe in which parallel major and minor events are seen as equivalent. Therefore, the parallel P transformation from systems discussed in the previous chapter, are actually seen as a parallel non-transformation P occurring freely between diptych pairs. Fig. 2.16 below overlaps the trans-diatonic diptych Tonnetz with its inherent axial structure. Note how every member relates directly to the Tonic diptych {T}. i I

62

{Gb} {X} {bV}

{Db} {Ab} {Eb} {Bb} {N} {tA} {tR} {SB} {bII}{bVI}{bIII}{bVII}

{F} {C} {G} {S} {T} {D} {IV} {I} {V}

{D} {A} {E} {B} {ST} {TR} {TA} {AN} {II} {VI} {III} {VII}

{F#} {X} {#IV}

Fig. 2.16: Diptych Tonnetz and its underlying axial structure.

Since all regions and chords are directly relatable to {T}, trans-diatonic transformations have the advantage of being all unary, in other words, there are no compound transformations between any chords or regions. To be sure, although the transformation system works more naturally dealing with events on the second order of complexity, it is not restricted to this order.

Some analyses that take into account macro-harmonic arches might benefit from a trans-diatonic transformational approach on the third order of complexity as well. Fig. 2.17 below depicts both the expanded and the diptych Tonnetze with the transformational symbols beside the goal chord or region. Note that there are two non-transformations in trans-diatonic theory: that of the parallel (P), for it is merely the application of the diptych principle, as mentioned before, and that of identity (I), for it connects two equal chords. Fig. 2.18 depicts a cyclic representation of each trans-diatonic axis separately. Each cycle presents a unique intervallic pattern derived from its intervallic relationship with {T}. Note how R and X transformations coexist in the same axis/cycle. 63

D perf. 5th up G X Tritone - Gb/F# d “” up g x “” - gb/f# W’ major 2nd down Bb D’ “” down F l major 3rd up e w' “” down bb d' “” down f L “” up E S minor 2nd up Db R minor 3rd up Eb L’ “” down Ab s “” up db r “” up eb l' “” down ab s' “” down b r' “” down a w major 2nd up d S’ “” down B R’ “” down A W “” up D Fig. 2.17: Trans-diatonic transformations. 64

Fig. 2.18: Trans-diatonic transformational cycles.

65

For convenience sake, in the case of the expanded Tonnetz all transformations departed from C-major. It could just as easily have departed from c for the transformations would have been exactly the same. Note that between C-major and C-minor there is only a straight line, i.e., not an arrow as in every other case. As said before, this makes reference to the effect of the diptych principle and to the non-transformational, non-directional status of P. The list on Fig.

2.17 summarizes all trans-diatonic transformations. The prime (’) after a transformational symbol indicates its downwards, counter-clockwise move in the cycle of each axis as shown in

Fig. 2.18. The dominant transformations (D, d, D’ and d’) operate on the X-axis and, as all trans- diatonic notation, bare an uppercase letter if the goal chord or region is major and a lowercase letter if it is minor. D’ and d’ are antipodes to D and d respectively. The diptych notation {D} makes reference to both D and d and {D’} to D’ and d’. The notation {D(’)} encompasses all four of them. 23 The relative ({R} and {R’}) and x or tritone ({X}) transformations occur along the Y- axis. There is no {X’} since the tritone is equally far from {T} on both directions. The

Leittonwechsel transformations ({L} and {L’} or simply {L(’)}) make up the Z-axis. The W-axis comprises the whole-tone transformations ({W(’)}). Finally, the slide transformations ({S(’)}) are set onto the S-axis. Fig. 2.19 below depicts in all possible transformations applied to {T}. Again, notice how every chord (and consequently, every harmonic region) can be achieved by a single trans-diatonic transformation, without the need of imaginary, intervening chords – just as it happens in the music.

23 They read: D – Dominant transformation; d – dominant minor transformation; D’ – Dominant prime transformation; d’ – dominant minor prime transformation; {D} – Dominant diptych transformation; {D’} – Dominant diptych prime transformation; {D(‘)} – Dominant omnibus transformation. The same logic applies to all transformations. 66

{D(')} {R(')} {D} {D'} {R} {R'} {X} œ bœ œ bœ bœ œ œ #œ b bœ n #œ & ˙ b˙ œ œ œ œ b œ bœ b œ # œ D d D' d' R r r' R' X x

{L(')} {W(')} {S(')} {L} {L'} {W} {W'} {S} {S'} bœ bœ & œ #œ b œ œ œ œ bœ œ œ œ œ #œ bœ bœ b œ bœ #œ #œ l L L' l' w W W' w' S s s' S'

Fig. 2.19: Trans-diatonic transformations in music notation.

As can be noticed from Fig. 2.18, all five trans-diatonic transformational axes form closed cycles based on a single trans-diatonic transformation and thus on a single interval. Such one-transformational cycles are referred to by their axis’s name (X-cycle, Y-cycles, etc.), or by

& ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ its constituent trans-diatonic transformation as needed, e.g., D-cycle, d’-cycle, {R}-cycle, and so on. As will be seen in the next chapter, multi-transformational cyclic gestures are a common trait of Romantic harmony. Cohn’s Weitzmann cycle, for instance, is an example of a ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ transformational & cycle involving two different transformations. Thus, this cycle is referred to as two-transformational R/d’-cycle.24 Transformational combinations are virtually endless and were able to provide a degree of systematic variety that was greatly explored by nineteenth-century composers. & Now,∑ I turn to the∑ practical∑ application∑ of the trans∑ -diatonic theory.∑ In the∑ following chapter, I will demonstrate through a series of analyses of selected passages of the repertoire how one can use the trans-diatonic theory to better understand the nineteenth-century harmonic syntax. & ∑ ∑ ∑ ∑ ∑ ∑

24 Even though Cohn perceives three transformations within that cycle (his R, N, and S), it is however primarily achieved by two transformations: R (a minor third up to a major chord) and d’ (the trans- diatonic equivalent to Cohn’s Nebenverwandt – a perfect fifth down from a major to a minor chord). Cohn’s S (here the equivalent of a trans-diatonic s – C-major to Db-minor) and other transformations such as L’, L, X, etc., are certainly possible within the cycle once it is formed, but they are not structural to the actual forming of the cycle itself. Chapter 3 Trans-Diatonic Analyses

In the previous chapter, I have laid out the principles and concepts governing trans- diatonic theory. Now, in this chapter, I demonstrate how trans-diatonic-based analyses might take place. The pieces chosen to be analyzed here present specific issues that are, in a general sense, ‘hard-to-explain’ with diatonic-based analytical models, but particularly suited to trans- diatonic inquires. Since the main purpose of this chapter is to demonstrate the validity of the trans-diatonic theory, I have decided not to concentrate on any particular composer, or genre, but rather to apply the theory to a number of excerpts by various composers throughout the nineteenth-century. Additionally, the analyzes presented here do not encompass entire compositions, or movements, but specifically ‘problematic’ passages in which the inadequacy of diatonic-based analytical models is most evident. Furthermore, like any analytical exercise, the analyses presented here are the result of my personal interpretations and, thus, must not be taken as absolute explications, but as subjective understandings.

3.1 Franz Schubert

Although this is not a study solely devoted to Schubert’s harmonic language, his output is among the first to thoroughly explore non-normative harmonic procedures, especially those involving chromatic third relations, to the point that I consider him to be the symbolic first trans- diatonic composer. Accordingly, a large number of the examples examined here are by him.

3.1.1 Mass in Eb Major, Sanctus, mm. 1-13

As can be seen from Example 3.1 below, the opening gesture of this Sanctus does not subscribe to the Tonic-Dominant diatonic logic. However, it fits perfectly within the trans- diatonic Z-axis (major third cycle). Beginning with a Eb-major chord, it undergoes three l’ 68 transformations ending up in the diptych pair of the initial chord – Eb-minor. What we have here is nothing more than an elaboration of {T} through the Z-axis (mm. 1 to 7). The remaining measures of the phrase make even clearer the trans-diatonicity of this passage. How else to coherently explain that 2 out of 3 tonic chords are presented in the ‘wrong’ mode (Eb-minor) at this point of the piece? What about the presence of the Cb-major chord in m. 7? Indeed, they are the result of mode mixture procedures, however, they cannot be comfortably explained within a diatonic logic. Conversely, to trans-diatonic harmony, they are all readily available. After arriving at the Eb-minor in m. 7, Schubert alters slightly his {T} prolongation through the Z-axis with a L’ transformation to Cb-major, instead of another l’ transformation; he then inverts the transformational direction with l back to Eb-minor. The phrase ends with a half cadence in Bb- major.

The second phrase begins with a much shorter prolongation of the Tonic, but this time along the Y-axis: Eb-major to C-minor. The trans-diatonicity of the passage is further enriched, and confirmed, by the presence of Eb-minor, Cb-major and Ab-minor chords which are not indigenous to the key of Eb major stricto sensu. Note also the presence of a neat diptych symmetry: the first transformation of the passage (i.e., l’ from Eb-minor to B/Cb-minor) is the symmetrical opposite of the one that occurs after the Z-cycle is first completed (m.7); but this time it goes from Eb-minor to Cb-major, completing the diptych pair of the members of the first progression (Eb-major and Cb-minor, notated as B-minor). 69

Ex. 3.1: Schubert, Mass in Eb Major, Sanctus, mm. 1-13.

3.1.2 Piano Sonata in D Major, I, mm. 1-16

This passage was mentioned in chapter 1 (section 1, Ex. 1.1), as a general critique of double syntax approaches. Below I comment upon Dmitri Tymoczko’s analysis (Ex. 3.2) of that same passage and offer my own as well (Ex. 3.3). The main issue with this excerpt is highlighted under a ‘major third system’ bracket in Tymoczko’s example below. Tymoczko’s approach to this issue clearly subscribes to a double syntax modus operandi, i.e., any time diatonic logic is weakened or absent (as is the case), he suspends the working premise of diatonicity and applies a highly localized system to solve that particular spot. The biggest problem I see with this ‘clip- on/clip-off’ approach is the tendency towards a reduction ad absurdum situation in which any single harmonic event can be analyzed in its own right, regardless of its contextual situation.

Turning to Ex. 3.2, it is evident that Tymoczko ‘solves the problem’ by focusing on highly localized harmonic events. In an opening 16 measures-long phrase, he resorts to three modulations – from D major to D minor, then to F major, and back to D major. The 70

‘problematic’ spot is analyzed in F major, in which the g#o6 chord is enharmonically analyzed as viio7/V followed by F-major (as I) in second inversion. He then ‘modulates’ back to D major for the final V7-I progression, leaving the C#-major chord unanalyzed. He argues that “having reached F major (the relative major of the parallel minor), Schubert then leaves traditional tonality in favor of the major-third system: a pair of efficient chromatic voice leadings connecting F major to C# major to A7, which in turn leads back to the tonic [and the traditional tonality].”1 That uninhibited, swift transition between different systems, as Tymoczko puts it, seems to be a palliative measure, a highly localized gambit employed to avoid the real problem rather than actually solve it. One might ask what is there to prevent the whimsical usage of modulations at will in order to make analysis easier or more convenient? Chromaticism 281

Figure 8.4.1 A reduction of the opening of Schubert’s D major Piano Sonata, D. 850, Op. 53.

Ex. 3.2: Tymoczko’s analysis of the opening of Schubert’s Piano Sonata in D Major.2

Now, compare my analysis below with Tymoczko’s. Note how the macro-logic of the Figure 8.4.2 passage is made clearer. The dashed arrows show macro-transformations while the solid ones A geometrical representation of the voice leadings 1 Dmitri Tymoczko, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice,(F, A, C)®(E , (New York: Oxford University Press, 2011), 280-1. G , C ) and (Es , 2 Tymoczko, A Geometry of Music, 281. Gs, Gs, C )®(E,s G,s A, Cs ),s which exploit sthe near major-third symmetry of the triad.

leadings connecting F major to C major to A7, which in turn leads back to the tonic. We can think of this passage as exploitings the fundamental geometry of three-note chord space, in which the major triads F, C , and A are all adjacent (Figure 3.10.2b). From this point of view the seventh is inessential—ans embellishment that merely decorates the more basic triadic relationship. (An alternative interpretation, shown in Figure 8.4.2, uses the pitch-class circle and retains the seventh.12) No matter how we conceptualize it, the progression represents a dazzling, gratuitous, and over-the-top transition from F major to A7—“gratuitous” since F major and A7 are harmonically quite close, and do not require the intermediation of a C chord.13 There is, in fact, an interesting precedent for these sortss of major-third juxtaposi- tions: in both baroque and classical music, one occasionally fi nds major-third-related triads across phrase boundaries (Figure 8.4.3). Typically, the earlier phrase ends with a half cadence on the dominant of the relative minor, while the next phrase begins

12 Note that the progression from C to A7 uses the standard German-sixth resolution in retrograde. 13 A classical composer would have sno problem moving directly from F major to A7, the dominant of the relative minor. 71 show local progressions. There is no need for a modulation here – the passage is entirely in {D}.

The portion where Tymoczko changed to a ‘major third system,’ is understood in trans-diatonic terms as a normative elaboration on the Z-axis of the Dominant, i.e., a prolongation, or expansion of Dominant zone, but entirely within {T}. As one can see below the g#o6 chord opens up ‘Dominant space’ which is used by Schubert as an opportunity to deflect to other members of the Dominant’s Z-axis, in this case F-major, followed by C#-major, to then arrive at the

Dominant proper, A-major.

Ex. 3.3: Schubert, Piano Sonata in D Major, I, mm. 1-16.

3.1.3 Die Zauberharfe, Overture, mm. 1-47

This passage illustrates the importance of a clear differentiation between events from distinct orders of complexity. Cohn’s analysis of this excerpt (section 1.4) considered events from the second order of complexity (chords) in the same level of those from the third order

(harmonic regions). The result was a tendentious analysis, as discussed previously. Turning to the passage itself, it is evident that its interesting harmonic feature is the ‘path of modulation’ 72 taken by Schubert. As can be seen from Fig. 3.1 below, Schubert elaborates the initial tonic within its Y-axis. Although the introduction ends on a HC (measure 32) in C minor, what follows is in the major Tonic which can be understood as the harmonic goal of the overture.

30 (33) 1 16 19 n˙ bœ ˙ ? ˙ b˙ b˙ ∑

{C}: t tR X TR T

Fig. 3.1: Schubert, Die Zauberharfe, Overture, mm. 1-47.

3.1.4 Die Sterne 3 ? 8 Kopp’s analysis of this∑ piece was considered in chapter∑ 1 (section2 1.3). Like the previous∑ example, this is an analysis of the third level of complexity, i.e., of the harmonic regions of

Schubert’s Die Sterne. Note how easily the diptych Tonnetz below depicts the harmonic features 90 94 98 100 102 106 107 of this lied. Instead of using cumbersome terms, suchb˙ as ‘lowered˙ sharp mediant,’bœ ‘lowered flat ? #˙ ˙ nœ ˙ bœ œ n#œ b˙ mediant,’ or ‘upper sharpnœ mediant,’ I advocate for understanding the underlying˙ logic guiding b˙ {Db}: I ½III X ¾VI Gel6/6 V V7/V V7... the modulations.(Db=C#) The first aspect to note is that all modulations in this piece are oscillatory, inasmuch that they always return to Eb major (note the double headed arrow in Fig. 3.2 below). 7 The binding ?harmonic∑ logic of the∑ passage is∑ made evident∑ by examining∑ Fig. 3.2∑ – what we∑ ∑ have here is a complete elaboration of the {T}’s Z-axis, framed by two elaborations over the Y- 15 axis. The number? by ∑the arrows indicate∑ their∑ order and ∑respective stanza.∑ ∑ ∑ ∑

23 ? ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

31 ? ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

39 ? ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ 73

Fig. 3.2: Modulation path of Schubert’s Die Sterne.

3.1.5 String Quartet in G Major, I, three excerpts

Schubert’s last string quartet is a remarkable piece in which the diptych principle itself is deliberately explored. Below I present three excerpts from its first movement in which the modal flexibility afforded by the diptych principle can be seen not only as a coloristic effect, but also as structurally relevant. The first example (Ex. 3.4) depicts the opening gesture from measure 1 to

14. As can be seen below, the diptych elaboration of {T} is Schubert’s first concern, with a long, sustained G-major (mm. 1-2) followed by an energetic G-minor (m. 3). The remainder of the opening bars veers the harmony towards the Dominant. Note that what was supposed to be a regular 4-measure opening gesture is extended one extra bar, with a further emphasis on the

Dominant. The next five bars (mm. 6-10) rearticulate the same motivic pattern of the previous bars, but on the level of the Dominant. Note how bars 1-5 and 6-10 receive the same Roman numeral analyses. The last four bars, retake the emphasis on the Dominant proper and conclude the opening gesture of the quartet with a suspenseful HC (m. 14). 74 Allegro molto moderato . ° # ™ . œ ™ . ™ ˙™ ˙ œ œ 3 A˙llegro ˙moltoœ mœod™beœrat‰o™ œ #œ œ . Œ ‰™bœ Œ I & 4 ™ . . œ œ œ . ° # ™ . œœ œ. ™ . ™ R ˙™ ˙ œ œ 3 ˙ ˙ œ œ ™bœœ ‰™ œ. #œ. œ . Œ ‰™bœ œ Œ Violin I &# 4 œ œ œ œ R ˙™ ˙ œnœ. 3 ˙™ ˙ œbœ. ™ œœ ‰™ . bœ™ Œ ‰™ r Œ Violin II & 4 ™ œ . œ œ œ # bœ œ . œ™#œ œ ˙™ ˙ œnœ. 3 ˙™ ˙ œbœ. ™ œ ‰™ . bœ™ . . Œ ‰™ r Œ ™ . Violin II & 4 ™ . b™ œœ ..™ œ.œ . ™ œ œ ˙ ˙ œ œ #3 ˙ ˙ œ œ ™ bœœ ™ œ.œ. œ . œœ. ™#œ. œ ™ œ Viola B 4 ™ ‰ . œ Œ ‰#œ Œ . ™ . bœœ . ™ œ. . ™œ R ˙™ ˙ œ œ B#3 ˙ ˙ œ œ ™ œ ‰™ œ. œ œ Œ ‰™#œ œ Œ Viola 4 . ™ œ ™ œ. œ ?#3 œ ™ ™ r bœ bœ R . Violoncello 4 ∑ Œ Œ ‰ J ™ Œ ‰ bœ œ Œ ∑ ∑ Œ Œ ‰ œ ¢ . œ bœ™ J ?#3 ∑ Œ Œ ‰ œ ™ Œ ‰™ r bœ œ Œ ∑ ∑ Œ Œ ‰ œ. Violoncello ¢ {G4}: I J ™ i œ vii°b4"/œV i6 Iw6 V vii°7/V V V ({D}): I J {G}: I i vii°4"/V i6 Iw6 V vii°7/V V V ({D}): I

8 œ . . . ° #nœ œ™#œ œ™ . nœ ™ ™ ™ U™ œ ‰™ œ. œ#œ Œ ‰™ R œ Œ b˙ ˙ b˙ ˙ &8 œ . . . ° #nœœ œ™#œ œ™ . nœ ™ ™ ™ U™ œ ‰™ œ. œ#œ Œ ‰™ R œ Œ b˙ ˙ b˙ U˙ &# œœ nœ. ™ . œ ‰™ œ. œ œ™ œ Œ ‰™ œ #œ Œ n˙™ ˙™ ˙™ ˙™ & nœ . . #œ. R b˙™ ˙U™ # œ nœ™ .œ. œ & œ ‰™ .œ. œ. ™nœ œ.œ™™#œ œ Œ ‰™ #œ Œ n˙™ ˙™ ˙™ ˙™ #nnœœ œ . .. #œR œ œ#œ b˙™ r ˙™ U B ‰ ≈ . nœ. . œ œ Œ ‰™ R Œ ‰™ R Œ ∑ ‰™ Œ ∑ œœ . œ™ œ™ œ œ #nœ œ œ. œ ™#œ œ ™ œ#œ ™ r U B œ ‰ ≈ nœ™ Œ ‰ R Œ ‰ R Œ ∑ ‰ œ œ Œ ∑ ?# œ ™ nœ bœ œ ™ œ#œ ™ rU Œ ‰ R nœ™ Œ ∑ ∑ ‰ R Œ ∑ ‰ œ ¢ œ bœ œ œ ˙U ?# Œ ‰™ nœ Œ ∑ ∑ ‰™ R#œ Œ ∑ ‰™ r ¢({D}): iœ vii°4"/RV i6 Iw6 V vii°7/V V ({G}): vii°7 V6% vii°7 V6% œ ˙

({D}): i vii°4"/VEx.i6 3.4Iw6 V: Schubert,vii° 7/StringV V ( {QuartetG}): vii°7 in G MajorV6%, I, mm. 1v-ii14°7 . V6%

15 ° # The .next. n œ.examp˙ le (Ex. 3.5. ) shows. the music that immediatelyn˙ followsœ ™theœ previous & Œ ‰™œnœ™ œ‰ Œ ‰™œnœ. ™nœ ˙ œ‰ Œ ‰™bœ œ™œ œ œ œ œ œ J œ œ œ œ 15 . J J . . . example,° # i.e.,™œ. measures. ™nœ ˙ œ 15 to 24.™. Thenœ. interesting˙ aspect™ of thatœ n˙ passageœ œliesœ notœ™ onlyœ œ in œits bassœ & Œ ‰ nœ J ‰ Œ ‰ œnœ. ™ œ‰ Œ ‰ bœ œ™œ œ J œ # æ æ æ J æ . . . æ æ æ æ æ æ æ æ line, but˙ specially™ in˙æ the™ trans˙æ™-diatonic harmonicæ™ æ™ relations exploredæ æ æ by Schubert.æ æ æ Asæ canæ beœ seen & æ ˙ ˙ œ ˙ ˙ œ œ ˙ ˙ œ #æ ™ æ æ æ æ æ æ æ æ æ æ æ æ & ˙ ˙™ ˙™ ˙æ™ ˙æ™ œæ ˙æ ˙æ œæ œæ ˙æ ˙æ œæ œ below,# theæ passage presents what is commonly æreferred to asæ a ‘lamentæ bass’æ – a descendingæ B æ æ æ æ b˙™ nœ ˙ ˙æ™ œ ˙ ˙æ™ œ ˙æ™ ˙æ™ n˙æ™ ˙æ™ æ ææ ææ chromaticB# line normally ‘filling in’ the space betweenæ 1 and æ5.˙ Whatæ we haveæ ˙ here, æhowever, is æ æ æ æ b˙™ nœ æ ˙™ œ æ ˙™ œ ?# æ˙™ æ˙™ næ˙™ ˙™ æ æ no ordinaryæ™ .æ Dueæ to the harmonicæ contentæ built overæ æ eachæ bass ænote,æ I understandæ ∑ it as ¢ ˙ ˙™ n˙™ ˙æ™ b˙æ™ œæ æ ˙æ™ œæ æ ˙æ™ ?# æ æ æ æ æ ˙ ˙ ∑ ¢{G}: I æ™ V6æ ½VIæI IVæ6 ¼VIæ I6$æIVæ Væ I6$æIVæ Væ I a normative˙ descending˙™ transn˙-™diatonic scale˙™ fromb˙™ 1 to 4. Noteœæ ˙theæ two˙æ™ non-œdiatonicæ ˙æ ˙ æchords™ in {G}: I V6 ½VII IV6 ¼VI I6$ IV V I6$ IV V I measures 17 and 19. To interpret the F-major and the Eb-major as the result of general mode Allegro molto moderato œ ˙™ ˙ œ œ. ° #3 ˙™ ˙ œ œ. ™ ™ œ™#œ œ. ™ . ™ Violin I & 4 ™bœ ‰ . . œ œ Œ ‰ bœ œ Œ œ œ. R ™ . #3 ˙™ ˙ œbœ. ™ œ ‰™ bœ™ Œ ‰™ r Œ ˙ ˙ œnœ Violin II & 4 ™ œ . œ œ œ bœ œ. . œ. ™#œ œ . . ™ . ˙™ ˙ œ œ. bœ œ. ™ œ œ. ™ ˙ ˙ œ œ B#3 ™ œ ‰™ œ. Œ ‰™#œ œ Œ Viola 4 œ œ. œ R . bœ™ ?#3 ∑ Œ Œ ‰ œ ™ Œ ‰™ r bœ œ Œ ∑ ∑ Œ Œ ‰ œ. Violoncello ¢ 4 J œ bœ J {G}: I i vii°4"/V i6 Iw6 V vii°7/V V V ({D}): I

8 œ . . ° œ™#œ. œ™ . U #nœ ‰™ . œ#œ Œ ‰™nœ œ Œ b˙™ ˙™ b˙™ ˙™ & œ œ R œ . ™ U 75 # œ ™ .nœ œ. ™ œ #œ n˙™ ˙™ ˙™ ˙™ & nœ ‰ œ œ. ™#œ œ Œ ‰ R Œ b˙™ . . ˙™ mixture œprocedure. œ.s™n isœ toœ. misunderstand™ the binding trans-diatonic logic of the passage. Like the #nœ œ . #œ œ œ#œ r U B œ ‰ ≈ œ œ Œ ‰™ R Œ ‰™ R Œ ∑ ‰™ œ Œ ∑ previousœ example, what we have here is a strikingly original elaboration of {T}œ through its X- nœ™ bœ œ œ U axis.? I#n otherŒ words,‰™ nœ in the initialŒ measure∑s of the piece∑ Schubert‰™ R #wasœ Œ able to take∑ the ‰most™ r ¢ œ R œ ˙ diatonic({D}): ideterminatevii°4" /routineV i6 Iw6 V (i.e., prolongationvii°7/V V ({G }of): v theii°7 Tonic-DominantV6% axis)vii°7 and stripV6% it of its diatonicity by resorting to trans-diatonic maneuvers.

°15 # . . . ™ œ Œ ‰™œnœ. ™nœ ˙ œ‰ Œ ‰™œ.nœ. ™nœ ˙ œ‰ Œ ‰™bœ œ™ œ n˙ œ œ œ œ œ J œ œ œ œ & J J . . œ. # ™ æ æ æ æ æ æ æ æ æ æ æ æ & ˙ ˙™ ˙™ ˙æ™ ˙æ™ œæ ˙æ ˙æ œæ œæ ˙æ ˙æ œæ œ æ # æ æ æ æ æ B æ æ æ æ b˙™ nœ ˙ ˙æ™ œ ˙ ˙æ™ œ ˙æ™ ˙æ™ n˙æ™ ˙æ™ æ æ ?# æ æ æ ¢ ˙æ™ æ æ æ æ æ æ æ æ æ æ ∑ ˙™ n˙™ ˙™ b˙™ œæ ˙æ ˙æ™ œæ ˙æ ˙æ™ {G}: I V6 ½VII IV6 ¼VI I6$ IV V I6$ IV V I

Ex. 3.5: Schubert, String Quartet in G Major, I, mm. 15-24.

Yet another remarkable example of trans-diatonic explorations can be found in this same movement, only thirty bars after the music of Ex. 3.5 above. Ex. 3.6 below presents a modulation from {T} to the Anti-Neapolitan Major region {AN}, i.e., from {G} to {F#} – a highly unusual destination for a first modulation. The manner in which Schubert achieves the {AN} region is also very interesting. Note how he sequentially moves upwards from {G} up to {F#}, along the

X-axis (cycle of perfect fifths), always articulating the diptych pair of each member. Upon arriving at the desired destination, Schubert reinforces it with emphatic alternations between

Tonic and Dominant chords (now in {F#}). The brackets highlight the sequence pattern. Note how a diatonic-based analysis would be at pains to explicate the relation between G major and F# major, whereas in the trans-diatonic system they are nothing but normative. 76

To be sure, by considering linear procedures (sequences, voice-leading efficiency, etc.) of passages such as this and others, the how a composer achieves that or the other chord or harmonic region is rendered clearer. However, the relationship among individual chords and the macro-harmonic structure – the what – remains unconsidered. Fig. 3.3 below graphically depicts the harmonic path of Ex. 3.6. The solid arrows show the connection between diptychs through an ascending sequence in the X-axis (cycle of fifths). The dashed arrow highlights the overall, macro-harmonic Anti-Neapolitan drive of the passage that is not evident if one only considers its sequential features. It is easy to see how the composer arrived where he has, but what is the relationship2 between the departureœ point and the final destination?œ 2 54 ˙ œ œ ™ œ œ ° # œ œ #˙ œ nœ œ™ œ #˙ œ nœ °5&4 ˙˙ œœbœœ œ™™ œ Œ œ œ Œ # ˙ œ bœ œ Œ #˙ œ nœ œ™ Œ #˙ œ nœ & œ™ œ œ # ˙ œ œ œ ™ œ ™ #œ & ‰ R œ ˙ œ œ #œ ‰ #Rœ œ ˙ œ œ # ˙ œ œ œ ‰™ œ ‰™ R ˙ œ œ & R ˙ œ œ #˙œ œ œ œ™ # œ ˙ œ nœ œ™ œ #˙ œ nœ œ™ œ B œ Œ Œ #œ Œ ˙ œ œ œ™ #œ Œ # œ ˙ œ nœ œ™ œ #˙ œ nœ œ™ œ B œ Œ Œ #œ Œ ˙ œ œ #œ Œ ˙ œ œ œ r œ ?# Œ Œ ˙ œ œ ‰™ #œ ˙ œ œ ‰™ #œ œ ¢?# œ œ rœ œ R Œ Œ ‰™ #œ ‰™ #œ œ ¢{G}: Iœ i V v II œ ii ½VI ½vi ½III R ½iii {G}: I i V v II ii ½VI ½vi ½III ½iii

59 #œ#œ œ™ œ ° # œ™ #œ œ #œ œ™ œ™#œ œ™#œ œ™#œ#œ™ œ œ œ 59 #œ™ œ œ Œ ‰ œ ™#œ #œ#œ ‰ Œ °& # œ™ œ™#œ #œœ J #œœ™ œ™#œ œ™#œ#œ™ Jœ #œ™#œ œ#œ Œ ‰ #œ ™#œ #œ œ ‰ Œ & œ™#œ #œ J #œ œ #œ™ J # œ # #œ ™ ™ j & #œ Œ ‰ #œ #œ Œ ‰ œ #œ œ™#œ œ œ #œ™ œ#œ™ œ œ™ œ #œ ‰##œœ Œ # ™ #Rœ #œ Jœ #œ™#œ œ j #œ & #œ Œ ‰ R Œ #œ‰ J œ™#œ œ#œ™ œ œ™ œ #œ ‰ œ Œ # #˙ œ ™ œ™#œ #œ ™ ™ ™ #œ B ##˙˙ œœ œ #œ œ #œ#œ œ #œ Œ ‰ œ #œ œ œ œ œ œ #œ ‰ #œœ Œ B## ˙ œ œ #œ™ œ œ™ œ #œ Œ ‰ Jœ #œ™ œ œ™ œ œ™ œ #Jœ ‰ œ Œ ™ ™#œ#œ J ™ ™ J œ ?# ˙ œ#œ œ #œ œ #œ œ Œ ‰ j #œ œ #œ œ œ œ ‰ Œ ¢ ™ ™#œ#œ #œ œ™ ™ ™ J œ ?# ˙ œ#œ œ #œ œ #œ œ Œ ‰ j #œ œ #œ œ œ œ ‰ Œ ¢ ¾VII #œ œ™ J œ {F#}: I V4£ I V4£ I V4£ I V4£ I ¾VII {F#}: I Ex.V4£ 3.6I: Schubert, StringV4£ QuartetI in G MajorV4£ , I,I mm. 54-63. V4£ I

35 ° # 5 b 3&5 ∑ 2 ∑ b b ° # 5 b & ∑ 2 ∑ b b # 5 b & ∑ 2 ∑ b b # 5 b & ∑ 2 ∑ b b # 5 b B ∑ 2 ∑ b b # 5 B ∑ 2 ∑ bbb ?# 5 ∑ 2 ∑ bbb ¢?# 5 b ¢ ∑ 2 ∑ b b 37 ˙ ? ˙ ˙ 37 bb ˙ ˙ ? b ˙ ˙ ˙ bbb ˙ ˙ 38 ? 38 bb ∑ ? b bbb ∑ 77

Fig. 3.3: Harmonic path of Ex. 3.6.

3.2 Johannes Brahms, Ein deutsches Requiem, II, mm. 261-273

Ex. 3.7 below shows a common procedure of nineteenth-century harmony not yet discussed thus far – the drive to the tritone ({X}). The harmonic weight of the chord built a tritone apart from {T} is by itself strong enough to be seen as the harmonic climax of the passage. In this case, however, x (m. 266) is further emphasized by its applied-Dominant, V6/x

(m. 265-6). As can be seen below, I highlight what I consider the main harmonic events by connecting them with a solid arrow with the appropriate trans-diatonic transformational sign.

Secondary harmonic events are connected by a broken line, and their transformational symbols are between parenthesis.

Ex. 3.7: Brahms, Ein deutsches Requiem, II, mm. 261-267. 78

Ex. 3.7: Continuation, mm. 268-273.

After achieving the tritonic apex (m. 266), Brahms turns his attention towards the

Dominant. There is an underlying harmonic symmetry in place here, for the manner in which the

Dominant is achieved is mirrored by that in which the minor tonic is elaborated – x acting as a fulcrum. In Fig. 3.4 below I offer the same information conveyed by the arrows above in a schematic notation. Note the structural weight given to x, placing the passage well outside the diatonic system. The note values represent the fundamentals. However, the symmetry I mentioned before might be more clearly depicted in a diptych Tonnetz of the second order of complexity, shown as2 Fig. 3.5 below. The solid arrows connect the truly structural . ˙ By bringing to the fold ?the final chord of the tonic elaboration (i.e., the dotted arrows to and from 3 bb Ó ∑ 1 Ú {Eb}) one can better see the neat symmetrical design of the passage.

˙ bœ œ œ b˙ ?bb œnœn˙ ˙ Ú Ú Ú Ú Ú Ú {Bb}: i x V I

° Figb . 3.4: Trans-diatonic bass outline of Ex. 3.7. &b Ú Ú Ú Ú Ú Ú Ú Ú Ú

? b ¢ b Ú Ú Ú Ú Ú Ú Ú Ú Ú

b &b Ú Ú Ú Ú Ú Ú Ú Ú Ú ? { bb Ú Ú Ú Ú Ú Ú Ú Ú Ú

° b &b Ú Ú Ú Ú Ú Ú Ú Ú Ú

? b Ú Ú Ú Ú Ú Ú Ú Ú Ú ¢ b

b &b Ú Ú Ú Ú Ú Ú Ú Ú Ú

{?bb Ú Ú Ú Ú Ú Ú Ú Ú Ú

° b &b Ú Ú Ú Ú Ú Ú Ú Ú Ú

? b ¢ b Ú Ú Ú Ú Ú Ú Ú Ú Ú

b &b Ú Ú Ú Ú Ú Ú Ú Ú Ú

{?bb Ú Ú Ú Ú Ú Ú Ú Ú Ú 79

Fig. 3.5: Diptych Tonnetz of the second order of complexity of Ex. 3.7.

3.3 Franz Liszt

3.3.1 Polonaise I from Die legend vom heiligen Stanislaus, two excerpts

In the previous chapter, I considered the introduction to this piece in the context of a trans-diatonic scale (see Ex. 2.1). Now I present the music that immediately follows it. Ex. 3.8 below shows trans-diatonic harmonic relations in both the second and third orders of complexity.

In the second order, the progression moving back and forth between E-minor and C-minor defies a clear diatonic understanding. In trans-diatonic terms, however, the relation is normatively seen as the interaction between tonic and a hard mediant, in this case the tonic anti-relative (ta, see

Fig. 2.9). The trans-diatonicity of the passage is further enriched by the articulation of #III6, enharmonically written as Ab-major (m. 28-9). The remainder of the passages articulates a modulation to {G} (m. 36) as analyzed below. The relation between {E}-minor and {G}-minor regions3 (i.e., in the third order of complexity) is also diatonically resistant, but yet another case of a trans-diatonic hard mediant relation (in this case, G minor is the tonic relative (tr) of {E}).

3 They read: ‘E diptych minor,’ and ‘G diptych minor.’ 80

20 bœ. œ. œ. œ. œ. bœ. œ. . 20 œ. œ. œ. œ. œ. œ. œ. œ. bœ. œ. œ. œ. œ. bœ. œ. œ. œ œ œ œ œ œ œ œ 2?0 # œ. œ. œ. œ. œ. œ. œ. œ. bœ œ œ œ œ bœ œ œ. œ œ œ œ œ œ œ œ ?# ‰ Jœ œ œ œ œ ‰ Jœ ‰ Jœ ‰ Jœ ‰ Jœ œ œ œ œ ‰ Jœ ‰ Jœ ‰ Jœ ‰ Jœ œ œ œ œ ‰ Jœ ‰ Jœ ‰ Jœ ?# ‰ J ‰ J ‰ J ‰ J ‰ J ‰ J ‰ J ‰ J ‰ J ‰ J ‰ J ‰ J p p p ?# j œ j j œ {?# œ™ #œj œ œ œ™#œ œ nœ œ™ œj œ œ œ™ œ œ #œ œ™ #œj œ œ œ™#œ œ nœ {?# œ™ œ œ œ™ œ œ™ œ œ œ™ œ œ œ™ œ œ œ™ œ { œ™ ##œ œ œ œ™##œ œ nnœ œ™™ œ œ œ œ™™ œ œ ##œ œ™ ##œ œ œ œ™##œ œ nnœ œ œ œ œ œ™ œ œ œ™ œ œ œ œ œ œ œ™ #œ œ œ œ™#œ œ nœ œ™ œ œ œ œ™ œ œ #œ œ™ #œ œ œ œ™#œ œ nœ {E}: i ½vi œ œ i {E}: i ½vi i {E}: i ½vi i

26 26 bœ œ œ œ œ bœ œ œ bœ œ œ œ œ bœ œ œ b œ œ œ œ œ 2?6 # bœ œ œ œ œ bœ œ œ bbœ œ œ œ œ bbœ œ œ bnœ œ œ œ œ ?# ‰ Jœ œ œ œ œ ‰ Jœ ‰ Jœ ‰ Jœ ‰b Jœ œ œ œ œ ‰b Jœ ‰ Jœ ‰ Jœ ‰ nJœ œ œ œ œ ?# ‰ J ‰ J ‰ J ‰ J ‰ J ‰ J ‰ J ‰ J ‰ J ?# j j {?# œ™ œj œ œ œ™ œ œ bœ œ™ œj œ œ œ™ œ œ bœ œ™ œ œ bœ {?# œ™ œj œ œ™ œ œ bœ œ™ œj œ œ™ œ œ bœ œ™ œ œ bœ { œ™™ œ œ œ œ™™ œ œ bœ œ™™ œ œ œ œ™™ œ œ bœ œ™™ œ œ bœ œ œ œ œ bœ œ œ œ œ bœ œ œ bœ œ™™ œ œ œ œ™™ œ œ bœ œ™™ œ œ œ œ™™ œ œ bœ œ™™ œ œ bœ œ œ œ œ œ œ œ œ œ œ œ œ {G}: œ œ œ ½vi ¾III66 {G}: ii°66 ½vi ¾III6 {G}: ii°6

31 bœ- 31 bœ- œ ˙ œ ˙ œ 3?1 # bœ- œ œ ˙ œ œ œ œ ˙ œ ?# Œ œ Œ Œ œ œ ˙ œ œ œ œ ˙ œ bb ?# Œ Œ Œ œ bb p ?# ˙ œ œ pœ™ {?# ˙ œ œ œ ˙ œ ˙ œ bb œ™ {?# ˙ œ œ œ œ ˙ œ œ œ ˙ œ bb œ™™ { ˙ œ œ œ œ ˙ œ œ œ œ ˙ œ b œ™ ˙ œ œ œ œ œ œ œ™ vii°77 i vii°7 Ex. 3.8: Liszt, Polonaise I from Die Legende vom heiligen Stanislaus, mm. 20i -36. 53 5?3 ?bb ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ?bbThe next∑ excerpt∑ was already∑ considered∑ in light of∑ Cohn’s analysis∑ on chapter∑ 1 (see∑ section? 1.4). Now I offer my own analysis. Instead of assigning chords to ad hoc groups, as does {?bb ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ {?b { bb ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ Cohn with his hexatonic cycles (Ex. 1.2), a trans-diatonic analysis of this passages strives to highlight the binding harmonic logic of the entire passage as a whole. The interesting aspect of

Ex. 3.9 is the prolongation of {T} through an almost complete, underlying W-axis (whole-tone cycle). The transformations between adjacent chords, however, may mask the true harmonic structure of the passage by locally embellishing each step of the W-axis with transformations along the Z-axis (l’ and L transformations) signaled by the brackets. Fig. 3.6 summarizes the overall harmonic frame of the passage. As can be seen by examining both the Ex. 3.9 and Fig.

3.6, the complete whole-tone prolongation of {T} is interrupted on the last possible moment by a 81

S’ transformation, from C-major (nVI) to B-major (V). The shift to the Dominant breaks the overall whole-tone logic of the passage and veers it back to the X-axis, to which an authentic cadence to E-major follows (mm. 109-110). Note how this approach offers a more straightforward understanding of the overall harmonic logic of the passage, without the necessity of changing between (hexatonic) systems at every local ‘perturbation.’

Ex. 3.9: Liszt, Polonaise I from Die Legende vom heiligen Stanislaus, mm. 98-110.

Fig. 3.6: Overall harmonic plan of Ex. 3.9. 82

3.3.2 Lélio Fantasy, mm. 195-198

The interesting feature of this passage is the concomitant prolongation of two parallel Y- axes (minor third cycle). If one considers the immediate progression between chords, it will soon become evident the alternation of S’ and l transformations, shown in Ex. 3.10 with solid arrows.

If maintained long enough this transformational pattern (a two-transformational S’/l-cycle) completes a cycle within eight steps. It is exactly what Liszt does: from the Bb-minor in m. 195 to the next in m. 198, he systematically employs these two transformations until the S’/l-cycle is completed. There is however, an underlying, larger logic behind this excerpt: if one, starting on the first Bb-minor chord, only considers every other chord, one will soon notice how they go along the Y-axis. The same thing is true if one begins with the second chord and skips every other chord: it also goes through a Y-axis parallel to the first one. Fig. 3.7 below shows the transformations of this passages in the context of an expanded trans-diatonic Tonnetz of the second order of complexity. The central, circled Bb-minor is the starting point. The solid arrows indicate the actual , the curved lines highlight the r transformations along the two parallels Y-axes. The broken arrow lines compensate for the limits of a two-dimensional representation by acting as connectors of the top and bottom of the Tonnetz.

Ex. 3.10: Liszt, Lélio Fantasy, mm. 195-198. 83

Fig. 3.7: Expanded Tonnetz of the second order of complexity of Ex. 3.10.

3.4 Cesar Franck, Quintet for Piano and Strings, I, mm. 90-107

Cohn’s analysis of this passage was considered in the first chapter. The main problem perceived in Cohn’s approach was perhaps his excessive reliance on double syntax shifts (section

1.4, Ex. 1.2). This is a remarkable passage of trans-diatonic harmonic elaboration. Fig. 3.8 below schematically depicts how I understand its harmonic logic. The passage begins after a stable cadence in Db-major (enharmonically notated as C#-major). The Db/C#-major is then elaborated throughout its Y-axis, albeit not completing the cycle, but outlining a fully diminished chord

(mm. 90-100). Each term from the Y-axis is locally elaborated in their respective Z-axis (shown below in black note heads). This pattern is maintained up to the last term of the Y-axis prolongation (m.100). The Bb-major is not elaborated in its Z-axis but, in its Y-axis, briefly down 30 (33) 1 16 19 n˙ bœ ˙ ? ˙ b˙ b˙ ∑ 84 {C}: t tR X TR T to its relative and back up. The German-Sixth in m. 102, by using the sounding Y-axis elaboration3 both shifts the harmonic focus of the passage towards the Dominant and re-signifies ? 8 the presiding passage∑ as an anticipatory elaboration∑ of itself (shown2 below by the∑ arrowed bracket). The passage ends relatively stable in a HC in m. 107.

90 94 98 100 102 106 107 ? ˙ b˙ œ ˙ bœ #˙ nœ ˙ nœ bœ n#œ ˙ b˙ b˙ {Db}: I ½III X ¾VI Gel6/6 V V7/V V7... (Db=C#)

Fig. 3.8: Harmonic scheme to Franck’s Quintet for Piano and Strings, I, mm.90-107. 7 ?3.5 Richard∑ Wagner∑ ∑ ∑ ∑ ∑ ∑ ∑ Without a doubt, Wagner’s harmonic language is among the ones that still poses one of the biggest15 challenges for the modern theorist. My intent in analyzing the following passages ? ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ here is not to engage with the current analytical literature, but rather to demonstrate the validity of a 2trans3 -diatonic approach to such repertoire. Therefore, like the analyses above, I will not ? ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ engage with expressive meanings, or dramatic situations that in a greater or smaller degree might have influenced the composer’s harmonic decisions. The intent here is to show the applicability 31 ? of the trans-diatonic∑ mod∑el to nineteenth∑ -century∑ harmonic∑ contexts.∑ With that in∑ mind, I have∑ chosen the Tarnhelm progression, and three passages from Parsifal. 39 ?3.5.1 Tarnhelm∑ progression∑ ∑ ∑ ∑ ∑ ∑ ∑

What we have here is a tonally closed {G#} passage (Ex. 3.11). The characteristic i-nvi progression (mm. 1-3), albeit memorable, does not carry structural weight during the first four bars of the passage, the nvi being then an elaboration of the tonic in its Z-axis. The structural relevance of nvi changes in the latter part of the excerpt, however. Its new metric position, longer duration, and alternation with its Dominant (albeit with empty fifths), is enough to render it 3 262 ? ∑ ∑ ∑ ∑ ∑ ∑ ∑

& ∑ ∑ ∑ ∑ ∑ ∑ ∑ {? ∑ ∑ ∑ ∑ ∑ ∑ ∑

269 5 ? ∑ ∑ ∑ ∑ ∑ 2 ∑ ∑

85 5 & ∑ ∑ ∑ ∑ ∑ 2 ∑ ∑ structurally significant (mm. 5-6). Due to the local importance of nvi, the cadence in bar 4 is ? 5 reinterpreted{ in hindsight∑ as a HC ∑tonizicing {E},∑ after which∑ G#-minor is retaken∑ 2and confirmed∑ ∑ as tonic in the last two bars (mm. 7-8). Fig. 3.9 below summarizes my analysis. ™ ™ ˙™ œ ˙™ œ ˙ œ™ œ w n˙™ œ n˙™ œ #˙ œ œ w # 4 ˙™ nœ #˙™ nœ #˙ nœ™ œ w ˙™ œ ˙™ œ ˙ œ œ w ?## # 4 œ œ œ™ œ w ˙™ œ ˙™ œ ˙ œ ‹œ # J ˙™ ˙™ #˙

{G#}: i ½vi i ½vi i ½vi ½III ½vi ½III ½vi ½III i i6$ V i ({E}:) i i V V V ¾iii Ex. 3.11: Wagner, Tarnhelm progression.

? #˙ n˙ ˙ ˙ nœ ™ œ w n˙™ #˙œ n˙™ œ #˙ ∑ ˙™ œ ˙™ œ ˙ œ™ œ w ˙™ œ ˙™ œ ˙ œ œ w ?# ## 4 ˙™ nœ #˙™ nœ #˙ nœ™ œ w ˙™ œ ˙™ œ ˙ œ ‹œ w 9 # #4{G#}: i ½vi œ wi Vœ œ #i˙ #### J b ˙™ ˙™ 4 & # ∑ Fig. 3.9: Overall∑ harmonicb b plan of Ex. ∑3.11. ∑ 4 {G#}: i ½vi i ½vi i ½vi ½III ½vi ½III ½vi ½III i i6$ V i ({E}:) i i ?#3.5.2# Parsifal V b V V ¾iii 4 { # ## ∑ ∑ b b ∑ ∑ 4 3.5.2.1 Act III, mm. 1098-1100

Like the opening of the Sanctus in Schubert’s Mass (Ex. 3.1), this passage, a trans-

diatonic9 elaboration of the Grail leitmotiv, also presents an elaboration of {T} in its Z-axis (mm. # # 278 j b bœ ˙ b# 4## ? ∑ ™ œ ˙∑ œb b œ bbœ ∑œ b˙ ∑ 4 &b b4 ˙ #nœ™ ∑œ n˙ bbœ bœ œ œ∑ 4 ∑ 1-2/2). Unlike˙ Schubert’snœ™ example, however, once having completed the Z-cycle, Wagner œ ˙ b˙ b˙ ?#{E#b}: b 4 {modulates# ## I the piece∑ to¼vi its sub-tonic regions,½III ∑ {Db}, confirmingb b ii ¼III it∑ with an authentic cadence∑ . This4 i {Db}: IV V I short example and brief comment stands as an example of how an seemingly impossible

16 progression,b in diatonic terms, can be effortlessly engaged by trans-diatonic theory. &b b ∑ ∑ ∑ j bœ ˙ b 4 ™ œ ˙ œ œ bbœ œ b˙ &b b4 ˙ nœ™ œ n˙ bbœ bœ œ œ ˙ #nœ™ œ ˙ b˙ b˙ {Eb}: I ¼vi ½III ii ¼III i {Db}: IV V I Ex. 3.12: Wagner, Parsifal, act III, mm.1098-1100.

16 b &b b ∑ ∑ ∑

19 b # 4 &b b ∑ ∑ 4

19 b # 4 &b b ∑ ∑ 4 86

3.5.2.2 Act I, mm. 241-257

The next example depicts a longer passage in which enharmonic equivalencies might be

an issue. The E-minor chord in m. 241 is achieved from D-major, as an evaded cadence ({G}: V-

vi). However, for my purposes, it will be considered as the starting point of a trans-diatonically

closed passage, as can be seen by the final cadence in mm. 256-257. Without having established

the tonic, Wagner shifts the focus towards {C}. The following measures are analyzed having

{C} as reference, but are nonetheless subordinate to the macro, albeit weak {E}. From m. 247

onwards, the focus gradually returns to {E} up to the point in which it is confirmed with a

perfect authentic cadence (mm. 256-257). Regarding enharmonic equivalence, note measures

245 and 246. The Ab-minor could have been analyzed as the enharmonic G#-minor, i.e., #iii of

{E};2 and the Eb-major in m. 246, could have been analyzed as the D#-major, or #VII of {E}. This

is an important aspect of nineteenth-century music and, therefore, enharmonic equivalency must 241 œ™ #>œ œ™ bœ™ j bœ j 2 ?#4 J œ ‰ Œ Ó Œ œ œ œ ‰ œ œ œ Œ bœ bœ bœ bœ bœ be freely4 embracedJ in trans-diatonic analyzes. J J J J J Treu' und Glück? Ihr nährt sie nicht, sie naht euch nie, nichts hat sie mit euch ge- 241 > œ™ #œœ œ˙™ bœ™ j bœ ™ j ?#4 œJ œ ‰ Œœ œœ Ó˙ Œ˙ œ b˙ œ œ˙ ‰ œ nb˙ œ bœ˙ Œ b b˙ bœ bœ bœ ™ bœ bœ { 4 Œ JŒ ˙ J ˙ J ˙ bb˙ J J Jœ™ ‰ œ œ ˙ ˙ n˙ ˙ bœ™ Treu' und Glück? Ihr nährt sie nicht, sie naht euch nie, nichts hat sie mit euch ge- {E}: i {C}: i˙ V iv i ¼vi Np/¼™III ?#4 œ œ œœ ˙ ˙ b˙ ˙ nb˙ b˙ b b˙ bœ™ { 4 Œ Œ ˙ ˙ ˙ bb˙ œ™ ‰ œ œ ˙ ˙ n˙ ˙ bœ™

{E}: i {C}: i V iv i ¼vi Np/¼III 3 nœ 246 ˙ ˙ 3œ œ œ œ ?# bœ ŒŒ #œ ‰ #œ œ œ ‰ œ œ™ œ ‰ œ nœ™ œ œ J J J J #œ ŒÓ Ó Œ‰œ J J R R J J R J J J main; dochw, ann's in Ge-fahr der Hil-fe gilt, der Ei-fer führt sie schier durch die Luft, die 246 3œ nœ ˙ w ˙ ˙™ œ 3œ œ œ j ?# bœ˙™ ŒŒ #œ ‰ #œ œ œ n‰œ œ œw™ œ ‰ œ nœ˙™™ œ J # Jœ J J #œw ŒÓ Óœ Œ‰œ { b ˙™ JŒ nJ˙™ R R # œ J Jw R J ˙™ J œœ # w #œ ‰ŒÓ J b˙™ ™ #w œ main; dochw, nan˙n's in Ge-fahr der Hwil-fe gilt, der Ei-fer führt sie schier durch die Luft, J die {E}: j ¼III V nœ ½wVI G˙el™6 œ #Vw/V ¾œvi°6 ?# b˙™ Œ ™ # œ w n˙™ #œœ œ ‰ŒÓ { b ˙™ n˙ Ex. 3.13: Wagner,w Parsifal,˙ act™ I, mm. 241-251œ . ##w #œ b˙ n˙™ w Jœ ¼III {E}: V ½VI Gel6 V/V ¾vi°6 252 œ œ™ œ 3 ˙ œ ?# œ J œ œ #œ œ Œ Œ‰ œ œ œ Œ nœ ‰ ‰ œ J œ #œ J#œ œ œ J J J J J J J œ œ ˙ J J J J J J J nie euch dann zum Danke ruft. Ich wähne, ist dies Schaden, so tät' er euch gut ge raten. 252 j nœ œ œ œ™ œ œœ œ œ3 ˙ œœ ?# œ J œ œ#œ œ Œ Œ‰ œ œ œ Œ nœ ‰n‰œ J #œ J#œ œ œœ { Jœ JŒ J Œ #œ ‰ Œ ˙™ J Jœ J ŒœÓ œ ˙Ó JŒ œJ‰ Ó J J #œŒ JŒ J J J ˙™ œ J J nie euch dann zum Danke ruft. Ich wähne, ist dies Schaden, so tät' er euch gut ge raten. nivœ Vj7 ½VI Npœ6 V i ?# œ œœ nœ œ œ { œ Œ Œ #œ ‰ Œ ˙™ œ ŒÓ Ó Œ œ ‰ Ó #œŒ Œ J ˙™ œ J iv V7 ½VI Np6 V i

258 ?# ∑ ∑ ∑ ∑ n

258 ?# n & ∑ ∑ ∑ ∑ ?# n { # ∑ ∑ ∑ ∑ n & ∑ ∑ ∑ ∑ {?# ∑ ∑ ∑ ∑ n 2

241 œ™ #>œ œ™ ™ ?#4 J œ ‰ Œ Ó Œ œ œ œ ‰ œ bœ j Œ bœ bœ bœ bœ bœ j 4 J J J œ œ J J J bœ Treu' und Glück? Ihr nährt sie nicht, sie naht euch nie, nichts hat sie mit euch ge- ˙ ™ ?#4 œ œ œœ ˙ ˙ b˙ ˙ nb˙ b˙ b b˙ bœ™ { 4 Œ Œ ˙ ˙ ˙ bb˙ œ™ ‰ œ œ ˙ ˙ n˙ ˙ bœ™

{E}: i {C}: i V iv i ¼vi Np/¼III

246 3œ nœ ˙ ˙ œ 3œ œ œ ?# bœ ŒŒ #œ ‰ #œ œ œ ‰ œ œ™ œ ‰ œ nœ™ œ J J J J #œ ŒÓ Ó Œ‰œ J J R R J J R J J J main; dochw, ann's in Ge-fahr der Hil-fe gilt, der Ei-fer führt sie schier durch die Luft, j die nœ w ˙™ œ #w œ ?# b˙™ Œ ™ # œ w n˙™ #œœ œ ‰ŒÓ { b ˙™ n˙ w ˙™ œ ##w #œ b˙ n˙™ w Jœ ¼III {E}: V ½VI Gel6 V/V ¾vi°6

87

252 œ œ™ œ 3 ˙ œ ?# œ J œ œ #œ œ Œ Œ‰ œ œ œ Œ nœ ‰ ‰ œ J œ #œ J#œ œ œ J J J J J J J œ œ ˙ J J J J J J J nie euch dann zum Danke ruft. Ich wähne, ist dies Schaden, so tät' er euch gut ge raten. nœ j œ ?# œ œœ nœ œ œ { œ Œ Œ #œ ‰ Œ ˙™ œ ŒÓ Ó Œ œ ‰ Ó #œŒ Œ J ˙™ œ J iv V7 ½VI Np6 V i

Ex. 3.13: Continuation, mm. 252-257.

3.5.2.3 Overture, mm. 44-55

The final passage considered here presents a non-normative elaboration of the normative

X-2axis58 , or else, a trans-diatonic elaboration of a common diatonic routine. The overall harmonic ?# ∑ ∑ ∑ ∑ n path of this excerpt is the modulation from Tonic to dominant (from Ab major to Eb minor).

Rather# than going straight to the dominant, Wagner first moves up through the Y-axis, using n & ∑ ∑ ∑ ∑ only its major constituents (Cb-major and Ebb/D-major); then, once he achieves the point of {?# ∑ ∑ ∑ ∑ n furthest remove from Tonic, i.e., the Ebb/D-major chord a tritone apart, he returns to the tonic minor (Ab-minor) which is reinterpreted as iv of {Eb} and as such, a modulatory fulcrum to the new region, to which follows a confirmatory perfect authentic cadence to Eb-minor. Note that by the time we arrive at the harmonic apex with X (m. 51), Wagner has a completed a two- transformational D/L’ semi-cycle (Ab-major – Eb-major – Cb-major – Gb-major – D-major). Fig.

3.10 shows the underlying harmonic logic of Ex. 3.14 in the diptych Tonnetz.

Ex. 3.14: Wagner, Overture to Parsifal, mm. 44-48. 88

Ex. 3.14: Continuation, mm. 49-55.

Fig. 3.10: Diptych Tonnetz of Ex. 3.14.

The analyses provided here should not be taken as absolute renderings of trans-diatonic- based analysis. As in any analysis, the subjective and interpretative dispositions of the analyst play a significant role. The idea here, however, was to demonstrate the practical applicability and validity of trans-diatonic concepts as developed in the previous chapter. I tried to provide different ways in which to render a trans-diatonic analysis throughout. From non-functional 89

Roman numeral analysis to transformational analysis with arrows and brackets on the score, from Tonnetze graphs to summarized harmonic outlines, and so on. None of them are ideal. They are better or worse suited to a given specific situation and ought to be employed at the discretion of the analyst, having in mind what features are important and how one might better present them. In what follows I offer a brief overview of this essay and my final remarks. Final Remarks

It was demonstrated that diatonic-based analytical models are unequipped to coherently deal with harmonic relations based on symmetrical divisions of the octave, widely employed in nineteenth-century music. To be sure, the resistance to symmetrical harmonic relations alone does not constitute a problem a priori – for the diatonic system can indeed account for some subsidiary, local symmetric/chromatic harmonic relations. However, as seen in the previous chapter, a considerable number of Romantic harmonic routines consistently, and intentionally, crosses the threshold of ‘non-normativeness’ and, as such, poses an unsurmountable challenge to diatonicism as a whole. Therefore, with the demise of diatonicism by nineteenth-century harmonic practices, the need for a new harmonic system is evident.

In chapter 1, I considered four theories of harmony that were formulated in response to the limits of diatonicism. Proctor’s system established two distinct tonalities: diatonic tonality, and chromatic tonality – the former based on the diatonic scale, the latter, on the chromatic.

Strange events, i.e., chromatic elements in diatonic tonality and vice versa, were seen as perturbations of one tonality into the other. Krebs’ Schenkerian approach extended diatonicism to encompass harmonic third relations in the Ursatz, albeit making the distinction among the degrees of subordination of such harmonic relations to an overall diatonicity (i.e. harmonic third relations within, temporarily, and permanently independent of the Tonic-Dominant axis). Kopp, like Krebs, also concentrated upon chromatic third relations. The main features of Kopp’s theory are the assignment of functional status to mode-preserving mediants and a modified transformational system based on common-tone preservation. Finally, Cohn articulates what he calls the hexatonic hyper-system around the harmonic possibilities afforded by stable chords a 91 major third apart and linked by minimal voice leading displacement. In face of the limitations of these systems, as discussed in chapter 1, I offered my own theory in chapter 2.

The trans-diatonic theory, as elaborated in the second chapter, is based on the fundamental concept of the ‘diptych principle,’ which fuses parallel keys (and chords) into a new, greater unit. The trans-diatonic scale is a natural, logical consequence of the mode fusion carried out by the diptych principle. The orders of complexity and their respective Tonnetze refine the theory by precisely indicating in which level of harmonic activity one is operating – if in that of the harmonic region, chord, or pitch level – thus, rendering a clearer analysis. The trans-diatonic transformation system displays the full potential of the thorough application of the diptych principle. Its axial structure welcomes all symmetrical division of the octave, as well as the diatonic asymmetrical cycle of fifths (the X-axis). Additionally, all events are possible to be directly related to a given centrality, which allows for a greater flexibility and precision when dealing with the Romantic harmonic universe. The favoring of harmonic relations over harmonic functions allows for an analytical system more closely related with the musical phenomena itself, without the need either to imagine intervening chords where there are none or to base analysis in inconvenient substitutions processes.

As it was demonstrated with the analyses in the previous chapter, trans-diatonic theory offers a reliable set of concepts and principles that are specifically tailored, and appropriate to nineteenth-century harmonic practices. Differently than the theories discussed in chapter 1, trans- diatonic theory is an empirical model, which bears the advantage of being able to deal with the musical reality as it is. There are at least three features that distinguish trans-diatonic theory from the theories discussed earlier: (1) trans-diatonic theory does not assume an ‘ideal’ scenario of only consonant triads; (2) it does not imagine non-existent chords between existent ones for the 92 sake of an arbitrary necessity of common-tone preservation or minimal voice-leading displacement; and finally (3), it does not need to retreat to double-syntax explanations of highly localized harmonic events, for it is able to coherently engage with the overall, binding harmonic logic of a given passage, or piece.

However, I do not claim trans-diatonic theory to be the panacea of nineteenth-century harmonic analysis. As any other theory or system, it has its particular applications, and limitations. Further developments of the trans-diatonic theory ought to consider chords more complex than a triad, such as 7th-, 9th-, and so on, accounting for their qualitative variants

(major 7th, minor 7th, diminished 7th, etc.) as well. Additionally, and of a more speculative nature, another further step in trans-diatonic theory developments ought to be the granting of transformational privileges to dissonant chords as well as the increasing of their intervallic complexity. Parallel to further developments of the theory itself, I also plan to explore the historical circumstances that fostered trans-diatonic harmonic exploration and perhaps trace back its embryonic inception to earlier periods. Regardless of the future improvements I envision, I believe the trans-diatonic theory, as portrayed in this essay, to be a valuable tool towards a better understanding of the harmonic universe of the ever-fascinating repertoire of nineteenth-century music.

Bibliography

Agawu, Kofi. “How We Got out of Analysis, and How to Get Back in Again.” Music Analysis 23, no. 2/3 (2004): 267-286.

———. Music as Discourse: Semiotics Adventures in Romantic Music. New York: Oxford University Press, 2009.

Aldwell, Edward and Carl Schachter with Allen Cadwallader. Harmony and Voice Leading. New York: Schirmer, 2011.

BaileyShea, Matt. “The Hexatonic and the Double Tonic: Wolf’s ‘Christmas’ Rose.’” Journal of Music Theory 51, no. 2 (2007): 187-210.

Beach, David. “Schenkerian Theory.” Music Theory Spectrum 11, no. 1 (1989): 3-14.

———. “The Functions of the Sixth-Four Chord in Tonal Music.” Journal of Music Theory 11, no. 1 (1967): 2-31.

Beach, David W. “The Origins of Harmonic Analysis.” Journal of Music Theory 18, no. 2 (1974): 274-306.

Black, Brian. “The Function of Harmonic Motives in Schubert’s Sonata Forms.” Intégral 23 (2009): 1-63.

Botstein, Leon. “Music and Ideology: Thoughts on Bruckner.” The Musical Quarterly 80, no 1 (1996): 1-11.

Bribitzer-Stull, Matthew. “The Ab-C-E Complex: The Origin and Function of Chromatic Major Third Collections in Nineteenth-Century Music.” Music Theory Spectrum 28, no. 2 (2006): 167-190.

Brown, Matthew. “The Diatonic and Chromatic in Schenker’s ‘Theory of Harmonic Relations.’” Journal of Music Theory 30, no. 1 (1986): 1-33.

———. Explaining Tonality: Schenkerian Theory and Beyond. Rochester: University of Rochester Press, 2005.

Burnham, Scott. “Thresholds Between, Worlds Apart.” Music Analysis 33, no. 2 (2014): 156- 167.

Caplin, William. Classical Form: A Theory of Formal Functions for the Instrumental Music of Haydn, Mozart and Beethoven. New York: Oxford University Press, 1998.

———. Analyzing Classical Form: An Approach for the Classroom. New York: Oxford University Press, 2013. 94

———. “Recent Research in the History of Theory: The Eighteenth and Nineteenth Centuries.” Music Theory Spectrum 11, no. 1 (1989): 29-34.

Caplin, Willian, and Nathan J. Martin. “The ‘Continuous Exposition’ and the Concept of Subordinate Theme.” Music Analysis 35, no. 1 (2016): 4-43.

Carver, Anthony F. “Bruckner and the Phrygian Mode.” Music & Letters 86, no. 1 (2005): 74- 99.

Clark, Suzannah. Analyzing Schubert. New York: Cambridge University Press, 2011.

———. “Schenker’s Mysterious Five.” 19th-Century Music 23, no. 1 (1999): 84-102.

Cohn, Richard. “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late- Romantic Triadic Progression.” Music Analysis 15, no. 1 (1996): 9-40.

———. “Introduction to Neo-Riemannian Theory: A Survey and a Historical Perspective.” Journal of Music Theory 42, no. 2 (1998): 167-180.

———. “As wonderful as Star Clusters: Instruments for Gazing at Tonality in Schubert.” 19th- Century Music 22, no. 3 (1999): 213-232.

———. “Weitzmann regions, My Cycles, and Douthett’s Dancing Cubes.” Music Theory Spectrum 22, no. 1 (2000): 89-103.

———. “Uncanny Resemblances: Tonal Signification in the Freudian Age.” Journal of the American Musicological Society 57, no. 2 (2004): 285-324.

———. Audacious Euphony: Chromaticism and the Triad’s Second Nature. New York: Oxford University Press, 2012.

Dahlhaus, Carl. Studies on the Origin of Harmonic Tonality, trans. Robert Gjerdingen. Princeton: Princeton University Press, 1990.

Ferris, Joan. “The Evolution of Rameau’s ‘Harmonic Theories.’” Journal of Music Theory 3, no. 2 (1959): 231-256.

Fétis, François-Joseph. Esquisse de L’Histoire de L’Harmonie, trans. Mary I. Arlin. Stuyvesant: Pendragon Press, 1939.

Fisk, Charles. Returning Cycles: Contexts for the Interpretation of Schubert’s and Last Sonatas. Berkeley and Los Angeles: University of California Press, 2001.

———. “Nineteenth-Century Music? The Case of Rachmaninov.”19th-Century Music 31, no 3, (2008): 245-265. 95

———. “Chopin’s ‘Duets’ – and Mine.” 19th-Century Music 35, no. 3 (2012): 182-203.

———. “Performance, Analysis and Musical Imagining.” College Music Symposium 36 (1996): 59-72.

———. “Performance, Analysis and Imagining Part II: Schumann’s ‘Kreisleriana,’ No. 2.” College Music Symposium 37 (1997): 95-108.

Frisch, Walter. Brahms: The Four Symphonies. New York: Schirmer Books, 1996.

———. Brahms and the Principle of Developing Variation. Berkley: University of California Press, 1990.

Forte, Allen and Steven E. Gilbert. Introduction to Schenkerian Analysis. New York: W. W. Norton, 1982.

Gollin, Edwars, and Alexander Rehding, eds. The Oxford Handbook of Neo-Riemannian Music Theories. New York: Oxford, 2011.

Harrison, Daniel. Harmonic Function in Chromatic Music. Chicago: University of Chicago Press, 1994.

———. “Nonconformist Notions of Nineteenth-Century Enharmonicism.” Music Analysis 21, no. 2 (2002): 115-160.

Hauptmann, Moritz. The Nature of Harmony and Metre. New York: Da Capo Press, [1853] 1991.

Hawkshaw, Paul. “The Bruckner Problem Revisited.” 19th-Century Music 21, no. 1 (1997): 96- 107.

Hebert, Trevor. Music in Words: A Guide to Researching and Writing about Music. New York: Oxford, 2009.

Hepokoski, James and Warren Darcy. “The Medial Caesura and Its Role in the eighteenth- Century Sonata Exposition.” Music Theory Spectrum 19, no. 2 (1997): 115-154.

———. Elements of Sonata Theory: Norms, Types, and Deformations in the Late-Eighteenth- Century Sonata. New York: Oxford University Press, 2006.

Hepokoski, James. “Sonata Theory, Secondary Themes and Continuous Expositions: Dialogues with Form-Functional Theory.” Music Analysis 35, no. 1 (2016): 44 -74.

Horton, Julian. “Recent Developments in Bruckner Scholarship.” Music & Letters 85, no. 1 (2004): 83-94. 96

———. “Stasis and Continuity in Schubert’s String Quintet: Responses to Nathan Martin, Steven Vande Moortele, Scott Burnham and John Koslovsky.” Music Analysis 33, no. 2 (2014): 194-213.

Hsu, Dolores M. “Ernst Kurth and His Concept of Music as Motion.” Journal of Music Theory 10, no.1 (1966): 2-17.

Jackson, Timothy. “Bruckner’s Metrical Number.” 19th-Century Music 14, no. 2 (1990): 101- 131.

———. “Bruckner’s ‘Oktaven.’” Music & Letters 78, no. 3 (1997): 391-409.

Kerman, Joseph. “How We Got into Analysis, and How to Get Out.” Critical Inquiry 7, no. 2 (1980): 311-331.

———. Write All This Down: Essays on Music. Berkley: University of California Press, 1994.

Kinderman, William, and Harald Krebs, eds. The Second Practice of Nineteenth-Century Tonality. Lincoln: University of Nebraska Press, 1996.

Kopp, David. Chromatic Transformations in Nineteenth-Century Music. New York: Cambridge University Press, 2002.

Koslovsky, John. “Timeless Reflections: Form, Cadence and Tonal Structure in the Scherzo and Finale of Schubert’s String Quintet.” Music Analysis 33, no. 2 (2014): 168-193.

Kraus, Joseph C. “Tonal Plan and Narrative Plot in Tchaikovsky’s Symphony No. 5 in E Minor.” Music Theory Spectrum 13, no. 1 (1991): 21-47.

Krebs, Harald M. “Third Relation and Dominant in Late 18th- and Early 19th-Century Music.” Ph.D. diss., Yale University, 1980.

———. “Returning Cycles: Contexts for the Interpretation of Schubert’s Impromptus and Last Sonatas.” Music Theory Spectrum 25, no. 2 (2003): 388-400.

La Rue, Jan. Guidelines for Style Analysis: A Comprehensive Outline of Basic Principles for the Analysis of Musical Style. New York: W. W. Norton, 1970.

Lerdahl, Fred and Ray Jackendoff. A Generative Theory of Tonal Music. Cambridge: MIT Press, 1983.

Lerdahl, Fred. Tonal Pitch Space. New York: Oxford University Press, 2001.

Lewin, David. “A Formal Theory of Generalized Tonal Functions.” Journal of Music Theory 26, no. 1 (1982): 23-60. 97

———. “Amforta’s Prayer to Titurel and the Roll of D in ‘Parsifal’: The Tonal Spaces of the Drama and the Enharmonic Cb/B..” 19th-Century Music 7, no. 3 (1984): 336-349.

———. “Cohn Functions.” Journal of Music Theory 40, no. 2 (1996): 181-216.

Lewis, Christopher. “Into the Foothills: New Directions in Nineteenth-Century Analysis.” Music Theory Spectrum 11, no. 1 (1989): 15-23.

Marra, James. “The Tonal Chromatic Scale as a Model for Functional Chromaticism” Music Perception: An Interdisciplinary Journal 4, no. 1 (1986): 69-84.

Marston, Nicholas. “Schubert’s Homecoming.” Journal of the Royal Music Association 125, no. 2 (2000): 248-270.

Martin, Nathan John, and Steven Vande Moortele. “Formal Functions and Retrospective Reinterpretation in the First Movement of Schubert’s String Quintet.” Music Analysis 33, no. 2 (2014): 130-155.

Mayrhofer, Robert. Der Kunstklang: Das Problem der Durdiatonik. Leipzig and Vienna: University Edition, 1910.

McCreless, Patrick. “Ernst Kurth and the Analysis of Chromatic Music of the Late Nineteenth Century.” Music Theory Spectrum 5, (1983): 56-75.

———. “Syntagmatics and Paradigmatics: Some Implications for the Analysis of Chromaticism in Tonal Music.” Music Theory Spectrum 13, no. 2 (1991): 147-178.

Morgan, Robert P. “Symmetrical Form and Common-Practice Tonality.” Music Theory Spectrum 20, no. 1 (1998): 1-47.

Proctor, Gregory. “Technical Bases of Nineteenth-Century Chromatic Tonality: A Study in Chromaticism.” Ph.D. diss., Princeton University, 1978.

Rehding, Alexander. “Rousseau, Rameau, and Enharmonic Furies in the French Enlightenment.” Journal of Music Theory 49, no. 1 (2005): 141-180.

———. “Tonality between Rule and Repertoire: Or, Riemann’s Functions – Beethoven’s Function.” Music Theory Spectrum 33, no. 2 (2011): 109-123.

Richards, Mark. “Beethoven and the Obscured Medial Caesura: A Study in the Transformation of Style.” Music Theory Spectrum 35, no. 2 (2013): 166-193.

Rings, Steven. Tonality and Transformations. New York: Oxford, 2011.

98

Rosen, Charles. The Classical Style: Haydn, Mozart, Beethoven. New York: W. W. Norton, 1997.

———. Sonata Forms. New York: W. W. Norton, 1988.

———. The Romantic Generation. Cambridge, Mass.: Harvard Press, 1995.

Rothfarb, Lee. Ernst Kurth: Selected Writings. New York: Cambridge University Press, 2006.

Salzer, Felix. Structural Hearing: Tonal Coherence in Music. New York: Dover, 1962.

Satyendra, Ramon. “An informal Introduction to Some Formal Concepts from Lewin’s Transformational Theory.” Journal of Music Theory 48, no. 1 (2004): 99-141.

Schenker, Heinrich. Harmony, trans. Elisabeth Mann Borgese. Chicago: University of Chicago Press, 1954.

Siegel, Hedi. Schenker Studies. Cambridge: Cambridge University Press, 1990.

Smith, Charles J. “The Functional Extravagance of Chromatic Chords.” Music Theory Spectrum 8, (1986): 94-139.

Smith, Kenneth M. “The Transformational Energetics of the Tonal Universe: Cohn, Rings, Tymoczko.” Music Analysis 33, no. 2 (2014): 214-256.

Smith, Leland. Handbook of Harmonic Analysis. Palo Alto: San Andreas Press, 1979.

Smith, Peter H. “Brahms’ Motivic Harmonies and Contemporary Tonal Theory: Three Cases Studies from the Chamber Music.” Music Analysis 28, no. 1 (2009): 63-110.

———. “Tonal Paring and Monotonality in Instrumental Forms of Beethoven, Schubert, Schumann, and Brahms.” Music Theory Spectrum 35, no. 1 (2013): 77-102.

Somer, Avo. “Chromatic Third-Relations and Tonal Structure in the Songs of Debussy.” Music Theory Spectrum 17, no.2 (1995): 215-241.

———. “Musical Syntax in the Sonatas of Debussy: Phrase Structure and Formal Function.” Music Theory Spectrum 27, no. 1 (2005): 67-96.

Tymoczko, Dmitri. A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. New York: Oxford University Press, 2011.

Tovey, Donald F. “Tonality in Schubert.” In The Main Stream of Music and Other Essays, 134- 159. New York: Oxford University Press, 1949.

Tuttle, Marshal. Musical Structure in Wagnerian . Lewiston: Edwin Mellen Press, 2000.