The Death and Resurrection of Function

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THE DEATH AND RESURRECTION OF FUNCTION

A Dissertation

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

John Gabriel Miller, B.A., M.C.M., M.A.

*****

The Ohio State University 2008

Doctoral Examination Committee: Approved by Dr. Gregory Proctor, Advisor

Dr. Graeme Boone ______

Dr. Lora Gingerich Dobos Advisor Graduate Program in Music

John Gabriel Miller

2008

ABSTRACT

Function is one of those words that everyone understands, yet everyone understands a little differently. Although the impact and pervasiveness of function in tonal theory today is undeniable, a single, unambiguous definition of the term has yet to be agreed upon. So many theorists—Daniel Harrison, Joel Lester, Eytan Agmon, Charles

Smith, William Caplin, and Gregory Proctor, to name a few—have so many different nuanced understandings of function that it is nearly impossible for conversations on the subject to be completely understood by all parties. This is because function comprises at least four distinct aspects, which, when all called by the same name, function , create ambiguity, confusion, and contradiction.

Part I of the dissertation first illuminates this ambiguity in the term function by giving a historical basis for four different aspects of function, three of which are traced to

Riemann, and one of which is traced all the way back to Rameau. A solution to the problem of ambiguity is then proposed: the elimination of the term function . In place of function , four new terms—behavior , kinship , province , and quality —are invoked, each uniquely corresponding to one of the four aspects of function identified. The meanings of these new terms are elucidated by such harmonic topics as secondary dominants and six-four paradigms. A notation system is developed for behavior , in particular, which is

used in conjunction with two standard systems of harmonic analysis to form a Three-fold

System of Analysis that yields deeper explanations of harmony characteristics.

Part II of the dissertation reveals how my theory of behavior leads to new explanations for chromatic harmonies. A definition of tonicization is proposed based on behavior paradigms. The models suggest that tonicization is a better explanation than is mixture for many chromatic notes and harmonies that are typically explained with mixture. Further, so-called linear or voice-leading chords (such as augmented-sixth chords, common-tone diminished seventh chords, chromatic mediants, etc.) are examined through the lens of the theory of behavior. Finally, I discuss ramifications of the theory of behavior for musics beyond the realm of classical diatonic tonality, including 19 th -

century chromatic tonality, jazz, and popular music.

Part III of the dissertation traces the history the four aspects of function from

Rameau to Riemann. This account includes such figures as Rameau, Béthizy, Daube,

Kirnberger, Koch, Vogler, Momigny, Weber, Fétis, Sechter, Hauptmann, Helmholtz, and

Riemann.

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ACKNOWLEDGMENTS

I would like to thank my advisor, Gregory Proctor, for inspiring this study, carefully considering my ideas, and helping me to refine them. Thanks to my committee members, Lora Gingerich Dobos and Graeme Boone, for time and effort in helping me achieve clarity. Thanks also to Burdette Green for his helpful suggestions. I am indebted to Blake Henson, David Tomasacci, and Ben Williams for their enthusiastic feedback.

Mom and Dad, thank you for everything. To Samantha, thank you for your support and encouragement. And last, but certainly most, I would like to give thanks to God for all good gifts.

VITA

November 25, 1977...... Born – Winter Haven, Florida

2000...... B.A. Music, Florida State University

2001-2002...... Director of Bands, Wildwood High School Wildwood, Florida

2003...... M.C.M., Church Music, Lee University

2005-2008...... Graduate Teaching Associate, The Ohio State University

2005...... M.A., Music Theory, The Ohio State University

PUBLICATIONS

Miller, Gabriel. “The Death and Resurrection of Function.” The Ohio State Online Music Journal 1, no. 1 (Spring 2008): 21 pages.

FIELDS OF STUDY

Major Field: Music

TABLE OF CONTENTS

Page Abstract...... ii Acknowledgments...... iv Vita...... v List of Tables...... ix List of Figures...... x List of Examples...... xi

Chapters:

Introduction...... 1

Part I. The Death of Function: A Solution to the Problem of Ambiguity...... 4

1. Historical Foundations: Sketching the Many Faces of Function...... 5

1.1 Riemann and Function...... 7 1.1.1 Daniel Harrison...... 12 1.1.2 Joel Lester and Eytan Agmon...... 15 1.1.3 Charles Smith and William Caplin...... 18 1.2 Rameau and Function...... 20 1.2.1 Gregory Proctor...... 25 1.3 Summary...... 30

2. A New Approach to Function Theory...... 32

2.1 Preliminary Assumptions...... 35 2.2 Characteristics of the Dominant Triad and Seventh Chord...... 36 2.3 Characteristics of the Tonic and Subdominant Harmonies...... 37 2.4 Definitions...... 38 2.4.1 Kinship...... 40 2.4.2 Province...... 42 2.4.3 Quality...... 43 2.4.4 Behavior...... 44

2.5 Reexamining the problem of V 7/V with the new terms...... 45 2.6 Behavior and Province...... 46 2.6.1 Behavior complementing Province: Six-four paradigms...... 47 2.6.2 Behavior contradicting Province...... 49 2.7 Notation...... 50 2.8 Analysis: Beethoven, Piano Sonata no. 9 , Op. 14, no. 1, mm. 1-16...... 55 2.9 Diatonic triad motion explained by the theory of behavior...... 57 2.10 Summary...... 58

Part II. The Resurrection of Function: Practical Applications for the Theory of Behavior...... 60

3. Tonicization Versus Mixture: The First Explanatory Power of the Theory of Behavior is Revealed...... 61

3.1 Tonicization codified with the theory of behavior...... 62 3.2 Downward pointing leading tones...... 66 3.3 Tonicization (d-t and s-t)...... 67 3.3.1 Combining d-t and s-t tonicization...... 69 3.3.2 Incomplete tonicization...... 70 3.4 Quasi-tonicization...... 75 3.5 Pseudo-tonicization...... 82 3.6 Microtonicization...... 84 3.7 Analysis: Beethoven, Waldstein Sonata , I, mm. 1-35...... 86 3.8 Chromatic notes in classical diatonic tonality not explained by tonicization...... 92 3.9 Alternatives to tonicization...... 93

4. The Theory of Behavior as an Explanation for Chromatic Harmonies...... 96

4.1 The Death and Resurrection of Roots...... 98 4.2 Distinguishing seventh chords from added-sixth chords...... 100 4.3 S, D, and their characteristic dissonances...... 103 4.4 Root motion down by step...... 104 4.5 The diminished seventh chord...... 105 4.6 Local behavior versus kinship...... 105 4.7 The augmented-sixth chord...... 106 4.8 Romantic resolutions of the augmented-sixth chord...... 108 4.9 The common-tone diminished seventh chord...... 110 4.10 Determining whether or not chords may be explained with behavior...... 111

vii

4.11 The real linear chords...... 112 4.12 Summary...... 116

5. Behavior in Chromatic Music, Jazz, and Popular Music...... 117

5.1 Behavior in chromatic music...... 117 5.2 Behavior in jazz...... 123 5.2.1 The subdominant seventh chord...... 125 5.2.2 Behavior and the tritone substitution...... 129 5.3 Province, behavior, and harmonic structures in popular music...... 133

Part III. The Birth of Function: Historical Precursors of Behavior, Kinship, Province, and Quality...... 140

6. The Lineage of Behavior, Kinship, Province, and Quality from Rameau to Riemann...... 141

6.1 Jean-Philippe Rameau...... 141 6.2 Jean-Laurent de Béthizy...... 144 6.3 Johann Friedrich Daube...... 146 6.4 Johann Phillip Kirnberger...... 147 6.5 Heinrich Christoph Koch...... 148 6.6 Georg Joseph Vogler...... 149 6.7 Jérôme-Joseph de Momigny...... 151 6.8 Gottfried Weber...... 155 6.9 François-Joseph Fétis...... 156 6.10 Simon Sechter...... 157 6.11 Moritz Hauptmann...... 157 6.12 Hermann von Helmholtz...... 160 6.13 Hugo Riemann...... 160

Glossary...... 162

Bibliography...... 166

viii

LIST OF TABLES

Table Page

1.1 Harrison’s Bases, Agents, and Associates...... 13

2.1 Diatonic triads grouped by kinship...... 41

2.2 Chromatic harmonies grouped by kinship...... 41

2.3 Diatonic harmonies grouped according to behavioral paradigms...... 58

3.1 Position-finding with a hypothetical secondary dominant...... 72

3.2 Position-finding with a hypothetical secondary subdominant (iv/x)...... 73

LIST OF FIGURES

Figure Page

1.1 Secondary triads and the Diminished Seventh Chord in Harrison’s Model...... 14

1.2 Joel Lester’s Function Model...... 16

1.3 Eytan Agmon’s Function Model...... 18

LIST OF EXAMPLES

Example Page

1.1 I-IV 6 with 7-6 suspension. S comprises suspension and resolution...... 9

1.2 Six-four paradigms functioning according to F2...... 10

1.3 Perfect and Irregular Cadences in Rameau’s Treatise on Harmony ...... 23

1.4 Proctor’s voice-leading paradigms...... 25

1.5 Functional voice-leading paradigms featuring neighbor motion...... 26

1.6 Functional voice-leading paradigms featuring passing motion...... 26

1.7 Functional paradigms for seventh chords and added-sixth chords...... 27

1.8 Root movements by step as functional voice-leading paradigms...... 28

1.9 Nesting notation in a progression featuring the cadential 6-4 chord...... 29

1.10 Haydn, Piano Sonata H. XVI: 37 , III, mm. 9-12...... 30

2.1 The functional ambiguity of the secondary dominant...... 33

2.2 I-IV-V(7) -I, representing the prototypical progression TSDT...... 36

2.3 Cadential six-four paradigms labeled with nesting notation...... 47

2.4 Behavior contradicting province...... 50

2.5 Bach, “Prelude No. 1,” Das Wohltemperierte Clavier , Book I, mm. 1-6...... 53

2.6 Beethoven, Piano Sonata no. 9 , Op. 14, no. 1, mm. 1-16...... 55

2.7 Province analysis expanded to account for quasi-Schenkerian levels...... 56

3.1 Chromatic notes explained by tonicization...... 63

3.2 Bach, “Gott lebet noch,” mm. 1-2...... 64

3.3 Bach, Invention 1 , mm. 7-15...... 65

3.4 6ˆ and 7ˆ as leading tones...... 67

3.5 Chromatic notes explained by s-t tonicization...... 68

3.6 Schubert, “Im Dorfe” from Winterreise , mm. 44-47...... 69

3.7 vii o7 /x as combination of d-t and s-t...... 70

3.8 Mozart, Piano Sonata in F Major , K. 280, mm. 73-83...... 74

3.9 Tonicization, Quasi-tonicization, and Pseudo-tonicization...... 77

3.10 Chromatic notes resulting from quasi-tonicization...... 79

3.11 Quasi-tonicization (IV/IV) in Mozart, Sonata K. 545, II, mm. 68-74...... 81

3.12 The Neapolitan as Quasi-tonicization (IV/VI) in Chopin, Prelude 20 ...... 82

3.13 Pseudo-tonicization...... 83

3.14 “Rocky Top Tennessee,” Words and Music by Boudleaux and Felice Bryant...... 83

3.15 “Unchained Melody,” Words by Hy Zaret, Music by Alex North...... 84

3.16 Beethoven, Piano Sonata Op. 26 , III, mm. 23-30...... 86

3.17 Beethoven, Piano Sonata in C Major, Op. 53, Waldstein , I...... 88

3.18 Brahms, Symphony no. 3 , Mvt. I, mm. 1-7...... 94

4.1 Combination of d-t and s-t paradigms...... 97

4.2 Perfect, tonal, and behavioral roots...... 100

4.3 Seventh chord and added-sixth chord models...... 101

xii

4.4 Resolutions of Chromatic Chords...... 102

4.5 Voice-leading paradigms for d and s including characteristic dissonances...... 103

4.6 Voice-leading paradigms for d-t, s-t, and the diminished seventh chord...... 105

4.7 A 6 – V progression...... 108

4.8 Gr 6/I – I progression...... 109

4.9 CT +6 – I progression...... 110

o 4 4.10 CT 2 redefined as a common-tone added-sixth chord...... 111

4.11 Richard Wagner, Der Ring des Nibelungen ...... 113

4.12 Chromatic mediant as triad or fourths chord...... 115

4.13 CT o7 redefined as a harmony that does not exhibit behavior...... 116

5.l Schoenberg, "Schenk Mir Deinen Goldenen Kamm," Op. 2, no. 2, mm. 1-6.... 118

5.2 “Autumn Leaves,” mm. 1-8. Words by Johnny Mercer, Music by Joseph Kosma...... 123

5.3 Diatonic seventh chords in the major and minor systems...... 124

5.4 Blues form in C...... 126

5.5 Behavior redefines F 7 as F add#6 ...... 126

5.6 CT o7 -I progression and F 7-C7 progression...... 127

5.7 C Blues changes with characteristic scales...... 128

5.8 ♭VI as tritone substitution for ii...... 130

5.9 Voice-leading comparison of A ♭7-G7 and Gr 6-V in C...... 132

5.10 ♭II as decoration of V...... 132

5.11 "Message in a Bottle," Words and Music by Sting...... 134

xiii

5.12 “Saturday in the Park,” Horn line in Introduction. Words and Music by Robert Lamm...... 136

5.13 “Feelin’ the Same Way,” Words and Music by Lee Alexander...... 137

5.14 "You've Got a Friend," Words and Music by Carole King...... 138

6.1 Momigny’s voice-leading rule 1...... 152

6.2 Momigny’s voice-leading rule 2...... 153

6.3 Momigny’s voice-leading rule 3...... 154

6.4 Momigny’s voice-leading rule 4...... 154

6.5 Chromatic alteration of ii 7...... 156

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INTRODUCTION

We have learned much from Hugo Riemann’s theory of harmonic function. The concept has persisted to virtually every music theory textbook in print today. And although the impact and pervasiveness of function in contemporary tonal theory is undeniable, a single, unambiguous definition of the term has been impossible to settle upon, even from the beginning. So many theorists have so many different nuanced understandings of function that it is nearly impossible for conversations on the subject to be completely understood by all parties. In many cases, when it comes to conversations about function, we talk past each other rather than to each other.

I propose a solution to this problem: the elimination of the term function . An alternative system of terms and notation is proposed, one which demands one-to-one correspondence of terms and definitions, and therefore yields clearer descriptions of many of the meanings of function. Function theory can only reach its full potential as an explanation for harmonic characteristics through the death of the term function ; function

theory may then be resurrected with more clarity and power through the careful

articulation of its many meanings.

Part I of this dissertation, “The Death of Function,” explains how the term

function is and always has been ambiguous, and proposes a solution to this ambiguity.

Chapter 1 lays the historical foundation for the refined theory, from Riemann’s concept of function to Rameau’s, as well as contemporary advances on each. Chapter 2 identifies specific problems with the various meanings of function, and solves these problems by proposing four terms—behavior , kinship , province , and quality —to replace function in describing the various characteristics of the primary harmonies. An analytic notation is developed for behavior , in particular, which is used in conjunction with two standard types of harmonic analysis to form a Three-fold System of Analysis that yields a broader range of explanations for harmony characteristics.

Part II, “The Resurrection of Function,” reveals how the proposed theory of behavior leads to new explanations for chromaticism, as well as jazz and popular music.

In Chapter 3 a definition of tonicization is proposed based on paradigms associated with the theory of behavior. The models suggest that tonicization is a better explanation than is mixture for many chromatic notes and harmonies that are typically explained through mixture. Chapter 4 examines so-called linear or voice-leading chords (such as augmented-sixth chords, common-tone diminished seventh chords, chromatic mediants, etc.) through the lens of the theory of behavior. Three problems and three solutions arise from this examination. First, some harmonies (e. g., augmented-sixth chords) are so chromatic that it is difficult to ascertain what their roots are, or even if they have roots.

The behavior of seventh chords and added-sixth chords (regardless of chromatic inflection, or even clef!) is proposed as a way to consistently determine roots for these harmonies. Second, some harmonies (e. g., diminished seventh chord) are composed of members with opposite behaviors; that is, both subdominant and dominant

microbehaviors can be found in a single harmony. In these cases, it is proposed that the behavior of the root should serve as the behavior of the harmony as a whole. Third, the difference among three categories of chords—harmonies, linear chords, and harmonies that are also linear chords—is often unclear. I propose that a harmony is a structure that has a behavioral root , and that a harmony is a linear chord if it cannot be explained by the theory of behavior (that is, if its root does not move by step or by fourth). Chapter 5 examines ramifications of the theory of behavior for musics beyond classical diatonic tonality, including highly chromatic music, jazz, and popular music. It is suggested that even where other aspects of function fail (e. g., kinship and province), behavior may still yield relevant information about harmonies. Behavior is useful wherever root movements by fourth or step occur, such as the circle-of-fifths dominated harmonic syntax of jazz.

Part III, “The Birth of Function,” which comprises Chapter 6, traces the history of the four aspects of function defined in this dissertation—behavior, kinship, province, and quality—from Rameau to Riemann. This examination includes Rameau, Béthizy, Daube,

Kirnberger, Koch, Vogler, Momigny, Weber, Fétis, Sechter, Hauptmann, Helmholtz, and

Riemann.

PART I

THE DEATH OF FUNCTION:

A SOLUTION TO THE PROBLEM OF AMBIGUITY

CHAPTER 1

HISTORICAL FOUNDATIONS:

SKETCHING THE MANY FACES OF FUNCTION

Hugo Riemann was never quite clear himself about what a harmonic function is, and his confusion inspired many subsequent authors to attempt clarifications and refinements that unfortunately, in too many cases, trapped the idea further in a sticky web of ambiguity. 1

Function is one of those words that everyone understands, and yet everyone understands it a little differently. As described in the quote above, even Riemann himself seems to color the concept of function with several shades of meaning. Contemporary function theorists have rightly expanded on function theory in various ways, each emphasizing one aspect or another about function. The strands of function theory are so varied now that only the most generic of ideas behind each is held in common; namely, that the three pillars of harmony—tonic, dominant, and subdominant—have primacy over all other harmonies. That is rather vague.

It gets worse. Function theory must not only contend with the various interpretations of Riemann and his successors, but also with the (arguably) “real”

1 Daniel Harrison, Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Account of Its Precedents (Chicago: The University of Chicago Press, 1994), 37.

inventor of function, Rameau. After all, it was Rameau who redefined the subdominant and first attempted to place it on equal footing with the dominant, second in status only to tonic. 2 The characteristics of the three pillars of harmony that interested Rameau were

different from those that interested Riemann. Whereas Riemann was more concerned

with the essence of tonic, dominant, and subdominant, and how secondary and chromatic

harmonies can be understood as transformations of those essences; Rameau seemed to be

more concerned with the behavior of the three pillars of harmony and their interaction

with one another. Thus, Rameau developed an entirely different brand of function

theory 150 years before the term was even coined. This Ramellian version of function

theory has been usurped by the Riemannian tradition, and largely ignored or forgotten.

This is unfortunate. The maximum potential for the explanatory significance of

harmonic function can only be reached by accessing both traditions. The nuances of each

tradition only give more avenues for explanatory power. It is fitting, then, to revisit each

of the many faces of function, showing how each may contribute to a greater

understanding of harmony. That is the goal of this chapter. The first portion of the

chapter contains a summary of Riemann’s concept of harmonic function, as well as

several of his successors’ interpretations. The second portion summarizes Rameau’s

view on the three pillars of harmony, and clarifies this view further through Gregory

Proctor’s voice-leading paradigms and nesting notation.

2 Jean-Philippe Rameau, Nouveau système de musique théorique (Paris: Ballard, 1726), 38; translated in Glenn B. Chandler, “Rameau’s Nouveau système de musique théorique : An Annotated Translation with Commentary” (Ph.D. diss., Indiana University, 1975), 270.

1.1 Riemann and Function

Identifying one clear and consistent meaning of harmonic function in the writings of Hugo Riemann is impossible. Instead, it is convenient to view Riemannian function as comprising three aspects. 3

First Aspect of Function (F1) : Function involves the grouping together of harmonies that share scale degrees. Harmonies that are not primary triads may be derived from or associated with one (or more) of the primary triads.

Second Aspect of Function (F2) : (a) Function implies that the three pillars of harmony maintain a sphere of authority or field of activity over portions of harmonic progressions, and that (b) These spheres of authority or fields of activity tend to be organized according to the prototypical progression, TSDT.

Third Aspect of Function (F3) : Function may be transferred to different scale degrees.

The first aspect of the Riemannian concept of function (F1) is that non-primary

triads may be grouped with I, IV, or V into the categories T, S, or D. Secondary triads

may be derived from primary triads through the parallel and leading-tone changes. 4 For

3 William Mickelsen’s summary of the “basic features of Riemann’s mature harmonic system” includes four points, two of which parallel what I have termed F1 and F2. These are that “Chords other than the three primary harmonies are mixtures of notes from these chords and thus may be comprehended as representing two or even three of the primary chords” (F1) and “harmonic function (and tonality) basically involves the movement away from the tonic to chords having dominant or subdominant significance and back to the tonic chord” (F2). Mickelsen rightly does not include F3 as one of the basic features because it is tangential to the other two primary aspects. I include it here not to give it theoretical weight equal to F1 and F2, but rather to separate it from these other aspects of function in an effort to erase ambiguity. William Mickelsen, Hugo Riemann’s Theory of Harmony and History of Music, Book III (Lincoln: University of Nebraska Press, 1977), 5.

example, the parallel and leading-tone transformations of IV (ii and vi respectively) may all function as S. However, vi may also function as T, since it is the parallel transformation of I. Chromatic alterations of ii, IV, and vi may also have S function, as is the case with the augmented-sixth and Neapolitan chords. Thus, any chord, chromatic or diatonic, that shares at least two scale degrees with IV may function as S. According to

Riemann, “logical meaning [of scale degrees and therefore of secondary triads and chromatic chords] depends on the degree of relationship with one or another primary triad.” 5

The second aspect of harmonic function (F2) implies that the three pillars of harmony consistently relate to one another according to the prototypical progression

TSDT. 6 Whereas F1 is an abstract concept that suggests that, for example, certain harmonies are subdominant-like regardless of where they appear or how they are ordered in a passage of music, F2 is less abstract in that it suggests that harmonies that are subdominant-like tend to occur before cadential dominant-like harmonies in harmonic progressions.

F2 entails more than simply a prototypical ordering of the three pillars of harmony. It also suggests that the pillars maintain a “sphere of authority or field of

4 Hugo Riemann, Harmony Simplified or the Theory of the Tonal Functions of Chords , trans. H. W. Bewerunge (London: Augener Ltd., [1895]), 69-106.

5 Hugo Riemann, “Musikalische Logik: Ein Beitrag zur Theorie der Musik,” Neue Zeitschrift für Musik 68 (1872): 279ff; translated in Harrison, 268.

6 Riemann, Harmony Simplified , 45. Riemann also posits oToDoSoT as the minor prototypical progression. In this dissertation, TSDT is considered the complete prototypical progression, while TDT, TST (plagal), TSD (half), and TD (half) are also viable prototypical progressions.

activity” over portions of harmonic progression. That is, T, S, and D each maintain the highest status over their particular section of harmonic progression; within that section, all harmonies are subject to the authority of that particular pillar of harmony. The possibilities for the expansion of these T, S, and D windows of harmony include but are not limited to: non-harmonic tones, including suspensions; six-four paradigms; and expansions through harmonic inversion.

Riemann’s notation for suspensions reveals his belief that the suspension is subject to the “sphere of authority” of whichever (primary) harmony it embellishes.

Example 1.1, taken from Harmony Simplified , reveals that S function applies to the suspension as well as the resolution, thus expanding the subdominant harmonic window to include two sonorities.

Example 1.1 I-IV 6 with 7-6 suspension. S comprises suspension and resolution. a

a Riemann, Harmony Simplified , 109; Ex. 126 a.

If single suspensions may expand harmonic windows, so too may double suspensions, including the cadential six-four, which Riemann clearly views as a double

6 - 5 neighbor to V, given his notation, D 4 - 3 . In other words, the dominant maintains a sphere of authority over the cadential six-four chord in this paradigm (Example 1.2). This is a significant feature of function theory given that the cadential six-four chord is nominally tonic. According to this version of function theory (as opposed to F1), local chord significance within a progression trumps the global status of a particular harmonic identity.

The other six-four paradigms are also good examples of F2 in action. In Example

1.2, the arpeggiating six-four of m. 1, the passing six-four of m. 2, and the neighboring six-four of m. 4 are all subject to their respective spheres of authority. In the first three

6 7 measures, a single harmony, I 4 , functions (F2) as tonic, subdominant, and dominant.

6 6 6 6 6 - 5 6 C: I I 4 I IV I 4 IV D 4 - 3 I IV 4 I T ------S ------D ------T ------

Example 1.2 Six-four paradigms functioning according to F2

7 Section 2.6.1 reveals more about this progression.

The third aspect of function (F3) is that function may be transferred to other scale degrees. It is interesting that Riemann notates V/V with dominant function ( DD ), since it shares two scale degrees with IV, and could therefore function as S according to F1. 8 In using the notation DD instead of S, Riemann shows that he understands dominant function not only as the essence of dominantness (F1), and the order and placement of

dominantness (before tonic in TSDT), but also the quality and behavior of dominantness

in relation to other chords (F3). In some cases, this dominant quality and dominant

behavior may occur even when the root of the harmony is not 5ˆ (e.g., secondary dominants).

So far three different Riemannian concepts have been examined, all of which share a single term, function . Contemporary theorists who have advanced these

Riemmanian concepts of function predictably tend to favor one aspect over the others.

The following review summarizes some important contributions to F1 theory by Daniel

Harrison, Joel Lester, and Eytan Agmon; as well as contributions to F2 theory by Charles

Smith and William Caplin. Once again, F3 is tangential to F1 and F2, and is treated as such by proponents of both F1 and F2.

8 Riemann, Harmony Simplified , 101. Riemann’s insistence on D instead of S (and thus, F3 over F1) is surely a tribute to Moritz Hauptmann, whose theory had perhaps the strongest direct influence on Riemann. D

1.1.1 Daniel Harrison

Harmonic function essentially results from the judgment that certain chords and tonal combinations sound and behave alike, even though these individuals might not be analyzed into equivalent harmonic classes... 9

Daniel Harrison believes function to be more about similarities between primary and secondary harmonies (or in other words, the harmonic “attitude” of a chord, or how much a harmony is like T, S, or D) than the idea of place holder within a progression, a sphere of authority, or the expansion of a harmonic window.10 That is, Harrison

emphasizes F1. His terminology for functional scale degrees—bases, agents, and

associates—clarifies this aspect of function. Table 1.1 shows how each scale degree

relates to each harmonic function. Bases correspond to the roots of the primary triads,

agents correspond to the thirds of primary triads, and associates correspond to the fifths;

however, it is not necessary that these scale degrees appear as members of a primary triad.

In a supertonic triad, for example, 4ˆ functions as a subdominant base, although it is not

the root of the chord. For a scale degree to be considered a base, it must either appear in

the lowest voice, or be accompanied by its agent. Thus, the supertonic triad functions as

S because it has more S-functioning chord members than the other primary triads ( 4ˆ and

6ˆ have S function, while 2ˆ has D function).

9 Harrison, 37.

10 Ibid., Harrison uses the words harmonic “attitude” to describe how secondary and chromatic chords function as T, S, or D.

Associate 1ˆ 5ˆ 2ˆ

Agent 6ˆ 3ˆ 7ˆ

Base 4ˆ 1ˆ 5ˆ

Subdominant Tonic Dominant

Table 1.1 Harrison’s Bases, Agents, and Associates. b

b Harrison, 45.

All agents are entirely dedicated to the function in question. This is made clear by

Table 1.1 in which 6ˆ , 3ˆ , and 7ˆ appear only once, each belonging to only one primary triad. In addition, agents communicate the modal character of each harmonic function.

That is, 6ˆ , 3ˆ , and 7ˆ are the possible agents of mixture. Bases and agents make up the bulk of functional power. Associates have little functional power, and “are entirely dependent on the presence of agents or bases for what little functional power they have.” 11

With this terminology, Harrison clarifies the first of Riemann’s interpretations of function; it is now easy to see how secondary triads can function as T, S, or D. Figure

1.1 shows how the three secondary triads, along with the diminished seventh chord, are given functional value in Harrison’s system. The submediant triad has two chord members with S function, and two with T function. Whichever function appears in the

11 Ibid., 55. Other theorists, such as Gregory Proctor, take exception to this devaluation of the associate, pointing out that in some cases (e. g., 4-3 suspensions) the associate is the only indicator of function. The reader may see section 1.2.1 for further discussion.

bass is more likely to be perceived as stronger. Similarly, the mediant triad has two chord members with tonic function and two with dominant function. If 3ˆ is in the bass, the chord will more likely have T function, whereas if 5ˆ or 7ˆ is in the bass, the chord will likely be interpreted with D function. As previously stated, the supertonic triad has S function since its two most meaningful chord members, the base and agent, have S function, and only the associate has D function. The diminished seventh chord is intriguing in that it has two chord members uniquely functioning as D and two chord members uniquely functioning as S. This helps explain the diminished seventh chord’s strong pull toward tonic, since the two members with D function are pushing up to tonic while the two members with S function are pulling down to tonic. (Further discussion of

Harrison’s approach may be found in Chapter 4.)

a. Submediant triad b. Mediant triad

S T

6 1 3 3 5 7

T D c. Supertonic triad d. Leading-tone seventh chord

D D

2 4 6 7 2 4 6

S S

Figure 1.1 Secondary triads and the Diminished Seventh Chord in Harrison’s Model. c

c Harrison, 60-65.

1.1.2 Joel Lester and Eytan Agmon

Like Harrison, Joel Lester and Eytan Agmon emphasize F1 over F2. Lester implies F1 when he writes of function, “Each secondary chord shares functions with one or more primary chords.” 12 He makes further remarks on the function of each secondary

chord specifically. Regarding the leading tone chord, “The VII 6 chord, like V, contains scale steps 2 and 7 and therefore is distant from I in a harmonic sense.” 13 Regarding the supertonic chord, “Most usages of the II chord arise through its resemblance to only one primary chord—IV.” 14 Regarding the submediant chord, “[VI] may be functionally ambiguous [since] the primary triads with which VI is associated are I and IV.” 15

Regarding the mediant chord, “usages of III arise from its position between I and V [but]

the tonic and dominant functions that unite in the III chord share no usages.” 16

Lester creates a model showing the relationships between secondary and primary triads, shown below in Figure 1.2. Adjacent triads share two common tones. The two triads adjacent to IV on either side may function as S; the two triads adjacent to I on either side may function as T; the two triads adjacent to V on either side may function as

12 Joel Lester, Harmony in Tonal Music , vol. 1, Diatonic Practices (New York: Alfred A. Knopf, 1982), 195.

13 Ibid., 24.

14 Ibid., 196.

15 Ibid., 232.

16 Ibid., 251.

Subdominant Tonic Dominant Modal dominant of the dominant

Figure 1.2 Joel Lester’s Function Model. d

d Lester, 251.

Although Lester tends to emphasize F1 over F2, he also makes reference to the

functionality of progression on several occasions. In his model, for example, he notates

functional progressions (i. e., circle-of-fifths motion) by drawing arrows connecting

dominant-tonic related harmonies: iii-vi, vi-ii, ii-V. Elsewhere Lester notes of the three

primary triads:

We may conclude that tonic, dominant, and subdominant chords can move from one to another in progressions directed toward a harmonic goal in any combination except dominant to subdominant:

Subdominant

Tonic Tonic 17

Dominant

Eytan Agmon follows Lester in emphasizing F1 over F2, even describing function

in similar ways and depicting it with similar models. He notes:

17 Ibid., 22.

The hallmarks of functionalism are: (1) the characterization of individual chords as tonic (T), subdominant (S), and dominant (D) in function; and (2) the notion that the so-called primary triads I, IV, and V somehow embody the essence of each of these functional categories.18

For Agmon, functions (TSD) are categories of harmonies; the primary triads are prototypes for each category. Although Agmon is interested in the functional value of

triads and not scale degrees as is Harrison, the end result of both theories is similar, with

function implying how similar or dissimilar a secondary triad is to a primary triad.

Agmon’s model of function, shown in Figure 1.3, wraps Lester’s linear model around

end-to-end to create a circular model for function theory. Here, I is the prototype for T,

which includes the maximally similar triads III and VI; IV is the prototype for S, which

includes the maximally similar triads VI and II (and the minimally similar triad VII); and

V is the prototype for D, which includes the maximally similar triads III and VII (and the

minimally similar triad II).

18 Eytan Agmon, “Functional Harmony Revisited: A Prototype-Theoretic Approach,” Music Theory Spectrum 17, no. 2 (1995): 197.

I III VI TONIC SUBDOMINANT DOMINANT

V IV

VII II

Figure 1.3 Eytan Agmon’s Function Model. e

e Agmon, 201.

Thus, Harrison, Agmon, and Lester conceive of function as the abstract

relationship of secondary triads to prototypical primary triads (F1). They concede that

ordering through time (F2) is a valid theoretical concept, though it should be kept

separate from function theory. In other words, F1 and F2 are both valid theories, but

should not necessarily be linked. Agmon explains:

The term “theory of harmonic functions” is stripped of any chord- progressional connotations, and thus has no standing with regard to how the three functional categories may follow each other in time. … The TSDT paradigm remains a part of harmonic theory in general – that is, the interaction of the theory of functions with a theory of chord progression. 19

1.1.3 Charles Smith and William Caplin

Whereas Harrison, Lester, and Agmon develop a concept of function that most

closely resembles F1 (i. e., the similarity in essence between secondary chords and

19 Ibid., 202, 204.

primary chords), other contemporary theorists emphasize the notion that function implies a prototypical progression, an ordering of harmonies, and a series of place holders within that progression (F2). In his article, “The Functional Extravagance of Chromatic

Chords,” Charles Smith does just that. This becomes clear when his explanation of function below is contrasted with that of Agmon above:

The paradigmatic functional progression is the following succession of chords: tonic [to dominant preparation(s)] to dominant(s) to tonic. …. No chord should be labeled as functional unless it participates in such a functional progression, or some segment thereof. 20

Here it is clear that Smith disagrees with Agmon that function and progression should remain separate concepts. For Smith, function necessitates progression. Even though

Smith is happy to marry function to progression, he also conveys the idea that F1 is a legitimate part of function theory. He sums up F1 in two sentences: “Any chord containing a leading tone is a dominant, if it has any clear function at all. All other functioning chords, except for the tonic triad, are normally considered dominant preparations.” 21

In his article, “Tonal Function and Metrical Accent: A Historical Perspective,”

William Caplin draws attention to the importance of order among the pillars of harmony

20 Charles Smith, “The Functional Extravagance of Chromatic Chords,” Music Theory Spectrum 8 (Spring 1986): 111. Italics mine.

21 Ibid., 110.

in harmonic progressions (F2). 22 He writes, “By functional harmonic progressions, I am

referring primarily to the motion within a given tonal region between tonic and dominant

harmonies (and occasionally, even tonic and subdominant).” 23 Caplin notes examples

from several theorists from Rameau to Riemann for which metrical accent is cited as a

determining factor for status between harmonic functions. Specifically, Riemann implies

the TSDT prototype when he writes, “The tonic chord ‘impresses itself upon us’ and we

‘desire its return,’” as well as, “In the case of all two-chord progressions, the metrical

accentuation decides which chord is to be considered the actual tonic.” 24

1.2 Rameau and Function

Long before Riemann proposed harmonic function as an explanation of tonal music and the many offshoots of this theory began to take shape, Jean-Philippe Rameau had already done much work in function theory, though he never used the term. Rameau redefined the subdominant, and was the first to juxtapose it against the dominant as rivals for second place status compared with tonic. 25 But the aspects of the three pillars of

harmony that Rameau emphasized were different from those aspects Riemann developed.

22 William Caplin, “Tonal Function and Metrical Accent: A Historical Perspective,” Music Theory Spectrum 5 (Spring 1983): 1-14.

23 Ibid., 1.

24 Ibid., 10-11; translated from Hugo Riemann, Musikalische Syntaxis (Leipzig, 1877): 79.

25 Rameau, Nouveau système , 38; trans. in Chandler, 270.

Whereas Riemann gave primacy to the essence of T, S, and D, Rameau gave primacy to

the relationships among T, S, and D. In particular, he was interested in the relationships

of D to T and S to T, relationships he called perfect cadence and irregular cadence,

respectively. 26

Cadence , for Rameau, did not have the same connotation the word does today, that is, the harmonic close of a phrase. Instead, cadence for Rameau referred to any move from one harmony to the next. Similarly, the term dominant was not necessarily

associated with 5ˆ , but rather was applied to any harmony, the root of which then moves

up by fourth. In other words, dominant was always a local concept. If the dominant in

question were the real dominant (i. e., the one built on 5ˆ ), it was called dominant-tonic .27

So a perfect cadence is not translated into modern parlance as authentic cadence, but

rather as any adjacent pair of harmonies, the roots of which are separated by an ascending

fourth. Likewise, an irregular cadence is not limited to plagal cadences, but to any

adjacent pair of harmonies, the roots of which are separated by a descending fourth.

Rameau also proposed characteristic dissonances for dominant and subdominant

harmonies. A dominant harmony (i. e., a harmony whose root moves up by fourth)

carries with it a seventh (even if it does not appear on the surface of the music), which

pulls downward by step. A subdominant harmony (i. e., a harmony whose root moves

26 Jean-Philippe Rameau, Treatise on Harmony , trans. with an introduction by Phillip Gossett (New York: Dover Publications, 1971), 63-70, 73-82.

27 Ibid., 237.

down by fourth) carries with it a sixth (even if it does not appear on the surface of the music), which pushes upward by step. 28

Rameau writes, “Whereas the perfect cadence ends with a progression from the dominant to the tonic note, the [irregular cadence] ends on the contrary with a progression from the tonic note to its dominant, or from the fourth note to the tonic.” 29

The fact that Rameau combines the T-D move with the S-T move into a single category called irregular cadence shows that he understands the term cadence as a local event, and not necessarily at the end of a phrase. More importantly for our purposes, it hints that

Rameau’s concept of harmonic function is less dependent on the identity of T, S, and D and more dependent on the root and voice-leading relationships among the three primary triads, and how similar root and voice-leading moves may occur on other scale degrees.

Thus, in addition to the three aspects of the Riemannian concept of function, a fourth aspect of function can be found in the work of Rameau.

Fourth Aspect of Function (F4) : Function connotes a relationship between two adjacent harmonies whose roots lie a fourth apart. If the roots are separated by an ascending fourth (from lower to higher status), D-T is implied; if the roots are separated by a descending fourth (from lower to higher status), S-T is implied.

28 Ibid., 65, 75, 235, 241.

29 Ibid., 73.

Example 1.3 Perfect and Irregular Cadences in Rameau’s Treatise on Harmony .f

f Rameau, Treatise , 80.

Examples can be found throughout the Treatise of cadences (global and local) involving

root movements by fourth. One of these is shown as Example 1.3. Here, perfect

cadences (D-T) are found not only on 5ˆ -1ˆ , but also on 6ˆ - 2ˆ . Irregular cadences occur on 4ˆ -1ˆ , but also on 2ˆ - 6ˆ . Rameau writes of this example, "In the perfect cadence the

octaves [A and C] … compel the sevenths [B and D] to descend. In the irregular cadence … the fifths [E and G] … compel the sixths [F and H] to ascend." 30

Thus, a seventh chord is distinguished from an added-sixth chord by how it is

approached or how it resolves, and not by some element of its essential make up. This

leads Rameau to another important concept in the history of function theory, the double

6 7 employment of the dissonance. In the typical progression, I-ii 5 -V -I, the second harmony may be either a seventh chord (ii 7) or an added-sixth chord (IV add6 ). Because Rameau favors root moves by fourth over root moves by step, he prefers to think of the chord as

IV add6 as it is approached from I, so that the root move is up a fourth. If the harmony had

resolved back to I, it would have remained IV add6 and its dissonance, the sixth, would resolve up to the third of the tonic harmony. However, since it resolves to V, the

6 harmony transforms itself into ii 5 so that its root may progress to V by a fourth. The dissonant seventh now descends by step to the third of the dominant harmony. 31

30 Ibid., 80.

31 Rameau, Génération harmonique, ou Traité de musique théorique et pratique (Paris: Prault, 1737), 115; trans. in Deborah Hayes, “Rameau’s Theory of Harmonic Generation: An Annotated Translation and Commentary of Génération harmonique by Jean-Philippe Rameau” (Ph.D. diss., Stanford University, 1968), 140. It should be noted that Rameau’s double employment of the dissonance may be considered a predecessor of Riemann’s F1 and F2. First, the idea that a single harmony may have more than one add6 6 6 7 meaning (IV or ii 5 ) shows that Rameau already understands F and Dmin as related 6 in some way (F1). Second, that he chooses I-ii 5 -V-I as his prototypical progression shows that he already understands that music may progress according to a TSDT scheme (F2). The Ramellian concept of distinguishing seventh chords from added-sixth chords is explored further in Chapter 4. Rameau’s contributions to the development of function theory are explored further in Chapter 6.

1.2.1 Gregory Proctor

Gregory Proctor follows Rameau in conceiving of function as related to a chord’s

behavior rather than its essence. He situates local function in the context of voice-leading

and root-movement possibilities. Example 1.4 shows the voice-leading moves (and root

moves) for triads. 32

a. Neighbor aspect

b. Passing aspect

Example 1.4 Proctor’s voice-leading paradigms

These models (Example 1.4a) show that roots may move up or down by third, up or down by fourth, or up or down by step. The corresponding voice-leading moves are: one note displaced by step, two notes displaced by step, and three notes displaced by step, respectively. Passing motion may only result from root movements by step or by fourth

(Example 1.4b). Of the paradigms, the most important are those in which the root moves

32 Gregory Proctor, “Harmonic Function and Voice Leading,” unpublished paper, 21-28. Examples 1.4-1.10, and all discussion related to these examples, are synthesized versions of this paper and personal communication between Gregory Proctor and the author.

by fourth, and two notes are displaced by step; these yield Rameau’s perfect and irregular cadences.

Example 1.5 shows functional voice-leading paradigms. In this example, the t-d-t model features lower-neighbor motion whereas the t-s-t model features upper-neighbor motion. We could say, then, that D is a lower-neighbor/lower-status element with respect to T, and that S is an upper-neighbor/lower-status element with respect to T.

t d t t s t

Example 1.5 Functional voice-leading paradigms featuring neighbor motion

Alternative voice-leading moves may be achieved through passing motion. Example 1.6 shows how the lower-status element of each voice-leading paradigm may result from passing motion instead of neighbor motion. D may result from passing motion from 1ˆ to

3ˆ ; S may result from passing motion from 3ˆ to 5ˆ . We can, therefore, revise the earlier

statement regarding the status of D and S with respect to T to read: D is a lower-neighbor

(or 1-3 passing)/lower-status element with respect to T; S is an upper-neighbor (or 3-5 passing)/lower-status element with respect to T.

t d t t d t t s t t s t

Example 1.6 Functional voice-leading paradigms featuring passing motion

Thus far the voice-leading paradigms have comprised only triads. When

characteristic dissonances are added to these triads, new voice-leading possibilities

emerge. Example 1.7 shows the primary voice-leading moves for the dominant seventh

and subdominant added-sixth chords. 33

t d t t s t

Example 1.7 Functional paradigms for seventh chords and added-sixth chords

If seventh chords and added-sixth chords are treated as the complete version of D

and S respectively, then the triadic paradigms from Example 1.5 may be seen as subsets

of the complete version. Furthermore, another triadic subset emerges that takes on either

D or S characteristics. This subset is the one found in the last two paradigms of Example

1.4a, in which the root moves down or up by step and all three voices are displaced by

step. In Example 1.8 these two moves are added to the list of functional root moves.

33 I only wish to emphasize the “natural” voice-leading possibilities for these chords, and have therefore not included unnatural resolutions of the dissonances (i. e., sevenths moving up or sixths moving down) as does Proctor. In order for dissonances to move contrary to their natural resolutions, they must be overpowered by a superior musical force. Parallel (sixth or third) motion, for example, is one musical force that overrides the tendency of the seventh to pull down. The distinction between seventh chords and added-sixth chords based on their resolution does not necessitate key signatures, accidentals, or even clefs, which leads Proctor to attribute “S-like” and “D-like” status to chords that are highly chromatic. This point will be developed further in Chapter 4. Gregory Proctor, Real and Apparent Simplicity in Musical Explanation, Occam’s Razor and Music-theoretic Wormholes , Keynote Address for Music Theory Midwest, Bloomington, IN, 13 May 2006.

Thus, Proctor’s paradigms account for like items (i. e., ii with IV and V with vii o) not by way of their shared pitches/scale degrees (as in Harrison, Agmon, and Lester) but rather in view of their similar behaviors.

t d t t s t

Example 1.8 Root movements by step as functional voice-leading paradigms

Proctor follows Rameau in conceiving of function as local behavior of the three pillars of harmony, but he also finds usefulness in the Riemannian concept of function as a sphere of authority of T or S or D over a section of harmonic progression (F2). He uses a nesting notation that combines the two concepts.

A simple example of the descriptive power of nesting notation is the cadential six- four paradigm. Example 1.9 shows a progression featuring a cadential six-four chord.

We have already seen that Riemann understands the cadential six-four chord as under the sphere of authority of the dominant. Since that is so, the dominant may be treated as a local tonic within the boundaries of its sphere of authority. Using Proctor’s voice-leading

6 paradigms then, compared with V, I 4 is an upper-neighbor/lower-status element. It

therefore fits the S-T model. When V is treated as a local tonic (which is always the case

within its sphere of authority), the cadential six-four chord is the subdominant of the

dominant. The nesting notation clearly shows this, with S-T being shown on a local level, yet under the sphere of authority (notated by the curly bracket) of the global dominant. 34

S T

T S D T

Example 1.9 Nesting notation in a progression featuring the cadential 6-4 chord

Nesting notation can perhaps be best exemplified by a progression that Proctor has found to be characteristic of the Classical period. It is often found, among other places, in rounded binary forms just before the return of the opening. Example 1.10 shows such a progression.

34 Signification of nested levels with the curly bracket may be traced back to Felix Salzer, Structural Hearing: Tonal Coherence in Music (NewYork: Dover Publications, 1982), first published 1952.

9 11

D T S T D Example 1.10 Haydn, Piano Sonata H. XVI: 37 , III, mm. 9-12

Although a Roman-numeral analysis of the progression shows only alternation

6 between dominant and tonic (e. g., V 5 -i-V in mm. 11-12), consideration of status

distinctions reveals that the first dominant is subordinate to the tonic that follows it,

which in turn is subordinate to the dominant that follows it, which in turn is subordinate

to the tonic that follows it. Thus, the tonic in m. 11 is actually functioning as the

subdominant of the dominant. Likewise the dominant in m. 11 is functioning as the

dominant of the subdominant of the dominant.

1.3 Summary

This account of the history of function theory has shown that the term function is

and always has been ambiguous. Riemann’s concept of function included at least three

distinct aspects: function as similarity in essence among harmonies as exemplified by two

or more shared scale degrees, function as place holder and sphere of authority or field of

activity within a prototypical harmonic progression, and function as transferable to local

harmonic levels. Riemann’s successors rightly expanded and clarified the theory of function in various ways, but without ever solving the problem of ambiguity. A case has been made for the inclusion of a fourth aspect of function theory, courtesy of Rameau, for which function involves the motion and resolution of harmonies more than their essence.

This fourth aspect of function loads the term with even more ambiguity. All four aspects reveal important information about harmonic characteristics, and all four are therefore worthy of usage as explanatory tools for tonal harmony. However, if all four aspects go by only one name, function, ambiguity remains, and function theory can never reach its

maximum potential for explanation. This problem must be solved.

CHAPTER 2

A NEW APPROACH TO FUNCTION THEORY

Function is a suggestive term which is still inspiring creative work in theory after over a century of use. But careful definition and elaboration is crucial. …. The term carries so many associations for us that it is difficult not to read some of them into the historical subject, thereby occluding perception of subtle yet important differences from our own views. Even if we know exactly what we mean, there is no guarantee that our reader will accurately grasp our meaning when we use the term without scrupulous qualification, since there are so many acceptable interpretations of the concept. Furthermore, the widespread and casual use of the term nowadays has diminished its descriptive power. It may prove helpful to investigate our assumptions and more clearly articulate and differentiate the myriad concepts which function has come to represent for us. 35

The historical account of function presented in Chapter 1 revealed examples of

the differing aspects of function described in this quote from David Kopp. Function now

has (and always has had) so many interpretations that it is impossible to know precisely

what people mean when they say function . Some of the meanings of function are

complementary, but some are contradictory. What is most confusing is when two or

more of these meanings are implied concurrently. Consider the following scenario as an

example.

35 David Kopp, “On the Function of Function,” Music Theory Online 1, no. 3 (May 1995): [15].

4 7 C: I V 3 /V V I T S D T

Example 2.1 The functional ambiguity of the secondary dominant

The four-voice progression in Example 2.1 adorns a chalkboard in a freshman theory classroom. The teacher defines secondary dominant, and then attempts to attribute functional significance to the harmony …

Teacher: Notice how the secondary dominant is the same as a ii 7 chord , except the F is sharp instead of natural. In other words, according to our TSDT model , it still functions as S. 36

Student: I kind of get it, but can you explain why it’s called secondary dominant again?

Teacher: Sure, the secondary dominant functions as a dominant because, as opposed to ii 7, V 7/V has a leading tone . It is the dominant of the dominant.

Student: Wait, I thought you said it functions as S. Does it function as S or D?

Most likely the teacher will uncomfortably make some sort of excuse for the confusion,

such as “… function in this sense of the word, but … function in this other sense of the

36 Riemann’s notation, S for subdominant function, has been modified to P or PD for pre-dominant function in some schools of thought. This modification shows a bias toward F2 over the other aspects of function. This dissertation, therefore, uses S exclusively.

word,” and may even deny that the teacher has made an error. An error has occurred, nevertheless, whether one of carelessness or ignorance. All three aspects of Riemannian function, at least two of which are contradictory, have converged on the same harmony; yet the teacher only has one word to describe all three aspects. The teacher comments on

F1 by saying that the secondary dominant is “the same as ii 7” except for the F#. The teacher notes that the harmony maintains the same place in the “TSDT model” as ii 7, and therefore evokes F2 in the harmony. But since the harmony “has a leading tone,” it is also functional in the F3 sense. Unfortunately for the teacher and the student, the harmony is D according to F3, while it is S according to F1 and F2. Furthermore, the teacher fails to mention at all the Ramellian concept of function, F4, which helps explain why the harmony is D. The harmony “functions” as a dominant in that its root moves up

by fourth to a higher-status harmony.

In order to solve the problem of ambiguity in function theory and the confusion to

which it leads, it is first necessary to clarify the language we use to describe function.

This means we must be sure that every significant aspect of function carries its own

unique term. That is the goal of this chapter. Four aspects of function are separated, and

each is given a unique name. By rejecting the ambiguous term function in favor of these

new precise terms, all of these important aspects of function will become clearer and

more meaningful.

2.1 Preliminary Assumptions

Eight definitions must be accepted before the new theory of function may be

developed.37

Definitions 1. The tonic ≡ 1ˆ 2. Tonic triad ≡ the diatonic triad built on 1ˆ 3. The dominant ≡ 5ˆ 4. Dominant triad ≡ the diatonic triad built on 5ˆ 5. Dominant seventh chord ≡ the diatonic seventh chord built on 5ˆ 6. The subdominant ≡ 4ˆ 7. Subdominant triad ≡ the diatonic triad built on 4ˆ 8. Subdominant added-sixth chord ≡ the diatonic added-sixth chord built on 4ˆ

I understand that these are familiar terms with definitions intuitive for most readers.

However, I also know that these distinctions are not always maintained, nor does

everyone define these terms the same way. For instance, the dominant triad is commonly referred to as merely the dominant . This is so common, in fact, that I know of at least

one colleague who understands the dominant not as a scale degree, but as a chord, and therefore fails to make a distinction between the chord and the scale degree. Another example of failure to maintain these distinctions comes from the jazz community, where the term dominant seventh chord is commonly used to describe any Mm 7 chord,

regardless of the scale degree of its root. For these people (I have been among them), the

term dominant seventh chord is defined by quality only, and not a relationship to tonic.

37 These definitions are adapted from Definition 1 of “Dominant,” in Don Michael Randel, ed., The New Harvard Dictionary of Music (Cambridge: The Belknap Press of Harvard University Press, 1986), 236.

Maintaining these distinctions in the following discussion will erase confusion and allow us to distinguish between the essence and the characteristics of tonicness, dominantness, and subdominantness.

Another axiom that must be accepted before the new theory can be developed is that the progression TSDT may serve as a harmonic prototype for a substantial portion of the Western-music canon. If it is assumed that this music follows this prototype, then the simple diatonic version of the progression shown in Example 2.2 may be dissected to reveal several unique characteristics of tonicness, dominantness, and subdominantness.

G: I IV V(7) I T S D T

Example 2.2 I-IV-V(7) -I, representing the prototypical progression TSDT

2.2 Characteristics of the Dominant Triad and Seventh Chord

Several characteristics may be noted about the dominant harmony in Example 2.2:

(1) It is composed of 5ˆ , 7ˆ , 2ˆ , and possibly 4ˆ ; (2) It maintains a sphere of authority in

the position immediately before the final tonic in the progression; (3) It is a Mm seventh

chord (any note of which may not appear on the surface of the music, in particular the

seventh); and (4) It is the lower-status element of a lower-adjacency voice-leading

paradigm (that is, its root moves up by fourth to a harmony of higher status). Often,

when any of these characteristics are found in a harmony other than the dominant, that harmony is said to “function as dominant.” For example: (1) One might say vii o functions as a dominant because it shares at least two scale degrees with the dominant harmony; (2) One might say a cadential six-four functions as a dominant because it is under the sphere of authority of the dominant harmony; (3) One might say that V 7/V

functions as a dominant because it is Mm; (4) One might say that vi 7 moving to ii (or

even vii o moving to I, which involves a root move up a step as opposed to a root move up

a fourth) functions as a dominant because it is the lower-status element of a lower-

adjacency voice-leading paradigm. Each of these characteristics of dominant harmony

has a corresponding functional connotation that must bear its own unique name if the

meanings of function are to gain clarity and power.

2.3 Characteristics of the Tonic and Subdominant Harmonies

In the same way that certain characteristics of the dominant harmony are found in

other harmonies, characteristics of the tonic and subdominant harmonies may also be

found in other harmonies. When this is so, one might say that those harmonies function

as tonic or subdominant. For examples: (1) One might say that vi functions as tonic or

subdominant because it shares at least two scale degrees with the tonic harmony and with

the subdominant harmony; (2) One might say that a passing six-four chord functions as

6 6 6 6 tonic (I-V 4 -I ) or subdominant (IV-I 4 -IV ) because it is under the sphere of authority of

the tonic or the subdominant; (3) One might say that iv add6 functions as subdominant

because it is a minor added-sixth chord; 38 (4) One might say that the cadential six-four moving to V functions as (local) subdominant moving to (local) tonic because it is the lower-status element of an upper-adjacency voice-leading paradigm. Once again, each functional characteristic of tonic and subdominant must bear its own unique name if the meanings of function are to gain clarity and power.

2.4 Definitions

Since function has been shown to have several aspects, some of which are contradictory, and since the establishment of new unique names for each aspect of function will clarify specific meanings for each of these aspects, let us eliminate the ambiguity of function theory by eliminating the term function and replacing it with four new terms: (1) kinship, (2) province, (3) quality, and (4) behavior.

Dominant

1. A harmony has Dominant Identity if and only if it is built on the dominant. That is, dominant triads and dominant seventh chords (as well as dominant ninth, eleventh, thirteenth chords, and even dominant added- sixth chords) have dominant identity. No other harmonies have dominant identity.

2. A harmony has Dominant Quality if and only if it is Mm (the seventh might not appear on the surface).

3. A harmony has Dominant Kinship if and only if it shares at least two scale degrees with the dominant triad.

38 The minor added-sixth chord as subdominant quality is treated more extensively in section 2.4.3.

4. A harmony exhibits Dominant Behavior if it is the lower-status element of a lower-neighbor (or 1-3 passing) voice-leading paradigm. Such a paradigm will feature root motion by ascending fourth or step from lower to higher status (d-t) or root motion by descending fourth or step from higher to lower status (t-d) or both.

5. A harmony (or group of harmonies) belongs to the Dominant Province if it is governed by the sphere of authority of the dominant. The dominant province usually occupies the position immediately before the final tonic province in a prototypical harmonic progression, TSDT.

Subdominant

6. A harmony has Subdominant Identity if it is built on the subdominant. That is, subdominant triads and subdominant added-sixth chords (as well as subdominant seventh chords) have subdominant identity. Harmonies not built on the subdominant do not have subdominant identity.

7. A harmony has Subdominant Quality if and only if it is a minor added- sixth chord (the sixth might not appear on the surface).

8. A harmony has Subdominant Kinship if it shares at least two scale degrees with the subdominant triad.

9. A harmony exhibits Subdominant Behavior if it is the lower-status element of an upper-neighbor (or 3-5 passing) voice-leading paradigm. Such a paradigm will feature root motion by descending fourth from lower to higher status (s-t) or root motion by ascending fourth from higher to lower status (t-s) or both.

10. A harmony (or group of harmonies) belongs to the Subdominant Province if it is governed by the sphere of authority of the subdominant. The subdominant province usually occupies the position immediately after the initial tonic province in a prototypical harmonic progression, TSDT.

Tonic

11. A harmony has Tonic Identity if and only if it is built on the tonic.

12. A harmony has Tonic Kinship if and only if it shares at least two scale degrees with the tonic triad. A harmony may have both tonic kinship and dominant kinship exclusively, or it may have both tonic kinship and subdominant kinship exclusively.

13. A harmony exhibits Tonic Behavior if it is the higher-status element of an upper-neighbor or lower-neighbor (or 1-3 passing or 3-5 passing) voice-leading paradigm. Such a paradigm will feature root motion by ascending fourth from higher to lower status (t-s), root motion by descending fourth from lower to higher status (s-t), root motion by descending fourth or step from higher to lower status (t-d), or root motion by ascending fourth or step from lower to higher status (d-t).

14. A harmony (or group of harmonies) belongs to the Tonic Province if it is governed by the sphere of authority of the tonic. The tonic province usually occupies the first and last positions in a prototypical harmonic progression, TSDT.

2.4.1 Kinship

Kinship yields an array of harmonies categorized together based on their similarity in essence. Harmonies that share two or more scale degrees with one of the pillars of harmony are said to have kinship with that pillar. In this dissertation kinship has thus far been known as F1, one of the three aspects of Riemannian function, and the one emphasized and expanded by Harrison, Lester, and Agmon. A substantial, but not comprehensive list of harmonies grouped by kinship is given in Tables 2.1 and 2.2.

T S D I, i IV, iv V, v vi, VI ii, ii o vii o, VII iii, III vi, VI iii, III

Table 2.1 Diatonic triads grouped by kinship

T S D V/IV, vii o/IV V/V, vii o/V vii o7 Gr 6 It 6, Fr 6, Gr 6, Sw 6 Gr 6/I N6

Table 2.2 Chromatic harmonies grouped by kinship

It is important to remember that kinship has no direct relation to ordering in

harmonic progressions. Neither does it have any direct connection to status between

pillars of harmony. That is not to say that kinship does not tend to coincide with, for

example, province, where order and status are of primary importance. Be that as it may,

a harmony such as the Neapolitan has subdominant and only subdominant kinship

regardless of how well or poorly it fits into a province or behavior scheme.

It is also important to note that kinship, unlike behavior and quality, is unable to be transferred to other scale degrees. Whereas V 7 (moving to I), for example, has dominant kinship, quality, and behavior; V 7/V (moving to V) only has dominant behavior

and quality. That is, whereas fi -sol is a local ti -do move in behavioral terms, it may only

be fi -sol in terms of kinship.

2.4.2 Province

Province is defined as a “sphere or field of activity or authority.” 39 I have used

these phrases, “sphere of authority” and “field of activity,” when referring to the second

of the Riemannian aspects of function (F2). Province implies two characteristics: first,

that tonic, dominant, and subdominant maintain spheres of authority over portions of

harmonic progressions, and second, that these spheres of authority are arranged according

to the prototypical progression, TSDT. As opposed to kinship, which is more abstract

and does not even require a real musical setting for its existence (e. g., ii has S kinship

regardless of whether ii even appears in a particular passage of music), province requires

context within a tonal progression. 40 Order and status are vital components of province.

For example, a tonic chord does not necessarily belong to a tonic province. To do so, it must be the first or last element of a tonal progression, and maintain the highest status among harmonies in that portion of progression. The cadential six-four chord, on the other hand, is a tonic chord that fits neither of these criteria. It is not the first or last element, and it is the lower-status element within its province, subordinate to the dominant.

39 Webster’s New Universal Unabridged Dictionary , rev. ed. (New York: Barnes & Noble, 2003), 1556.

40 That is not to say that province and kinship are not related. On the contrary, province presupposes kinship. For example, vii o may wholly comprise the dominant province precisely because it has dominant kinship. The converse, however, is not necessarily true. Kinship does not presuppose province. For example, the cadential six- four chord necessarily does not belong to a tonic province, even though it has tonic kinship.

2.4.3 Quality

When diatonic seventh chords are built on all scale degrees in the major mode, the

Mm 7 appears only once, and is unique to 5ˆ . It is called the dominant seventh chord.

Since Mm quality is unique to the dominant in the diatonic system, we can say that Mm 7 chords have dominant quality , even if they are not built on the dominant (i. e., if they are

secondary dominants created through the process of tonicization). 41

In the same way, when diatonic added-sixth chords are built on all scale degrees in the minor mode, the minor (triad with major) added-sixth appears only once, and is unique to 4ˆ . Since m add6 quality is unique to the subdominant in the diatonic system, we can say that m add6 chords have subdominant quality , even if they are not built on the subdominant.

One could suggest that IV add6 should also have subdominant quality, but this fails on two points. First, there would then be two qualities known by the same name. Second, if IV add6 is a subdominant quality, then v 7 would have to be included as a dominant

quality. Most readers will intuit the problems with identifying v 7 as dominant quality; it

destroys the whole reason for having the term quality in the first place. In the same way

then, IV add6 may not be considered to have subdominant quality. Instead, v 7 and V 7 are both said to have dominant identity , meaning that each is a chord whose root is 5ˆ ; IV add6

and iv add6 have subdominant identity , meaning that each is a chord whose root is 4ˆ .

41 As previously noted, many jazz musicians refer to any Mm seventh chord as a dominant seventh chord. According to our list of eight accepted definitions, the dominant seventh chord may only be the one built on 5ˆ . Thus, it is clearer to refer to any Mm seventh chord (whether built on 5ˆ or not) as a dominant-quality seventh chord.

It might also be suggested that the half-diminished seventh chord (ii Ø7 ) be a

subdominant quality. After all it is identical to iv add6 in essence (i. e., they share all four

notes). However, this suggestion shows a bias toward seventh chords over added-sixth

chords and ignores Rameau’s concept of double employment. The half-diminished

seventh chord (as S) will move to V, whereas the iv add6 will move to (or has moved from)

tonic. Thus, subdominant quality suggests a move directly to or from tonic (or perhaps

local tonic) just as dominant quality does, and not a move to dominant. As shown in the

next chapter, this dualistic view of quality, seventh and added-sixth chords, and behavior,

leads to some clear and simple explanations of chromatic notes.

2.4.4 Behavior

Rameau’s concept of function (F4) is roughly equivalent to behavior.42 Behavior

involves the application of functional properties to local levels. Two chords behave as

tonic and dominant (or tonic and subdominant) regardless of their identity (i. e., what

scale degree they are built on) if there occurs between the two chords a root move of a

fourth or a step. 43 Whichever of the chords has higher status is considered the local tonic

42 The reader may recall that in some examples Rameau draws no distinctions between T-S and D-T, and thus only views function as voice-leading and root motion events without status distinctions. This differs from the definition of behavior in that behavior implies both voice-leading/root-motion paradigms and status distinctions.

43 In Sections 2.2 and 2.3, root moves by step were grouped with root moves by fourth to represent a category now known as behavior. Root moves by step were included in this category because the voice-leading and status distinctions between root moves by step and root moves by fourth are similar, as supported by any number of tonal progressions where root moves by step substitute for root moves by fourth (e. g., vii o-I instead of V-I, and I-ii instead of I-IV).

(i. e., behaves as tonic). Depending on the type of root move, the other chord behaves as dominant or subdominant.

The cadential six-four paradigm again serves as a clear and simple example.

Between the two harmonies, I and V, V has the higher status, and is therefore the local tonic. The move from I to V is a root move down a fourth to a higher-status element.

Therefore I behaves as the subdominant of V.

In the progression ii 7-V, ii 7 exhibits dominant behavior since its root moves up a fourth to a higher-status element. Even though the chord does not have dominant quality

(i. e., V 7/V), it behaves as a dominant.

2.5 Reexamining the problem of V 7/V with the new terms

At the beginning of the chapter, a hypothetical problem (Example 2.1) was posed that exemplified the confusion that results from the many-to-one mapping of aspects of function to the term function . The definitions above solve this problem completely. The

teacher may say that the harmony (V 7/V) has dominant quality since it is Mm. It also

Conspicuously absent from this grouping are root moves by third (diatonic or chromatic). I have not included root moves by third as a behavior for two reasons. First, root moves by third are found far less frequently as substitutes for root moves by fourth (e. g., iii-I rarely assumes the role of cadence, and I-vi is usually heard as a tonic expansion). Second, root moves by third result in harmonies that are akin to each other. By limiting behavior moves to root moves by fourth or step, a distinction regarding root moves is drawn between kinship and behavior. It should be noted that a gray area clouds this last distinction. Usually root moves by third do evoke kinship, but sometimes they may evoke behavior instead. That is, sometimes what is changed (one note displaced by step) is more significant that what stays the same (two scale degrees in common). Therefore, even though in this dissertation I will continue to adhere to the rule that behavior involves root moves by step or by fourth, the reader is encouraged to contemplate possible exceptions to this rule: behavior-like relationships in root moves by third.

exhibits dominant behavior since its root moves up by fourth to a harmony of higher

status. It has subdominant kinship since it shares three scale degrees with IV. It also continues to occupy the position between T and D in the prototypical progression TSDT, and therefore belongs to the subdominant province .

2.6 Behavior and Province

Identity, quality, and kinship remain constant regardless of the context of a

harmony within a progression given a single key. Behavior and province, however, are

variable. A single harmony may exhibit two different behaviors given two different

contexts; the same harmony may belong to two different provinces given two different

progressions. There are, therefore, more analytical decisions to be made regarding

behavior and province than regarding identity, quality, and kinship. The interaction of

behavior and province within a progression is at the heart of functional significance.

The relationship between behavior and province can be complementary or

contradictory. When a behavior paradigm is wholly contained within a province,

harmonic relationships can be easily seen and heard on two levels. When behavior

paradigms cross into more than one province, as in the example of the secondary

dominant above, relationships may become tenuous. In both situations, distinguishing

between behavior and province (as opposed to function and function ) brings clarity.

2.6.1 Behavior complementing Province: Six-four paradigms

A few musical examples are in order to show how a harmony may exhibit different behaviors and belong to different provinces. Perhaps the simplest examples are the four six-four paradigms: cadential, neighboring, passing, and arpeggiating. For the sake of simplicity, all four six-four paradigms are shown in the progression in Example

2.3. The arpeggiating six-four chord embellishes the tonic harmony in m. 1; the passing six-four chord embellishes the IV chord in m. 2; the cadential six-four chord embellishes the V chord in m. 3; and the neighbor six-four chord embellishes the tonic harmony in m.

4. Proctor’s nesting notation reflects the behavior and province ramifications of the progression. Each curly bracket points to a province, while behavioral paradigms are shown within each curly bracket.

6 6 6 6 6 6 ID C: I I 4 I IV I 4 IV I 4 V I IV 4 I T T T T D T S T T S T

T S D T

Example 2.3 Cadential six-four paradigms labeled with nesting notation

The simplest of the four paradigms occurs in the first tonic province, where the arpeggiating six-four chord (as well as I 6) simply expands the province through inversion.

6 The I 4 chord exhibits tonic behavior by default since it does not move to a new harmony.

It also belongs to the tonic province.

6 Within the subdominant province, however, the same harmony, I 4 , is a passing

6 6 chord that exhibits dominant behavior. The IV and IV that surround I 4 have higher

status due to the fact that S maintains a sphere of authority over this section of the

6 progression. Since IV is the local tonic (i. e., highest-status element), I 4 behaves as a local dominant. It is a 1-3 passing/lower-status harmony. Its root is approached by

6 descending fourth, and moves forward by ascending fourth. In other words, in m. 2, I 4 is the dominant of the subdominant. Although it has tonic identity , it exhibits dominant behavior , and belongs to the subdominant province .

6 A third appearance of I 4 occurs within the dominant province, which comprises m.

3. The same harmony that belonged to the tonic province in m. 1 and to the subdominant

province in m. 2 now belongs to the dominant province. Since the dominant maintains a

sphere of authority over m. 3, it is the element of highest status, and is therefore treated as

a local tonic. The cadential six-four chord, then, is an upper-neighbor harmony to V. As

an upper-neighbor/lower-status harmony, the cadential six-four chord exhibits

subdominant behavior. Its root also moves down by fourth to a root of higher status.

6 Thus, in m. 3, I 4 has tonic identity , but exhibits subdominant behavior and belongs to the

6 dominant province . In the first three measures, a single harmony, I 4 , exhibits each of the three behaviors and belongs to each of the three provinces.

The final tonic province, which comprises m. 4, is embellished by the neighbor

6 six-four paradigm. In this case, IV 4 belongs to the tonic province even though it has subdominant identity. It also behaves as a subdominant, since it is an upper-neighbor harmony to the higher status element of tonic.

2.6.2 Behavior contradicting Province

In Example 2.2, nesting notation accurately accounted for behavior and province because the boundaries of each behavioral paradigm exactly corresponded to the boundaries of each province. Many times, however, this is not the case. It is possible for behavior paradigms to cross over province boundaries. Such a progression is shown in

Example 2.4. In this example, a paradigm consistent with dominant behavior exists from

V7/V to V. V is the higher-status harmony, and its root is approached by ascending

fourth. Thus, V 7/V exhibits dominant behavior with respect to V. However, province

analysis would find that V 7/V is S, while V is D. 44 Thus, the nesting notation, while

explanatory of harmonic behavior within a single province, is unable to sort out the

confusion that results from harmonic behavior that reaches into more than one province.

It is fitting then, to develop a system of notation that accounts for such crossovers.

44 An alternative reading may feature a TDT province scheme where the D comprises a d-t behavior paradigm for which V/V would belong to the D province, and not to the S province. In this case, behavior and province are once again complementary, not contradictory.

T ? T

Example 2.4 Behavior contradicting province

2.7 Notation

Identity, quality, and province are all described by current and widespread

notational systems. Identity is notated with the Roman-numeral system, where tonic

identity is notated with I or i, subdominant is notated with IV or iv, and dominant is

notated with V, V 7, v, or v 7. Dominant quality is represented in jazz-pop nomenclature by the superscript 7, and in the Roman-numeral system by the label V 7/x. It is now common to see province notated with capital letters T, S (or PD), and D underneath a

Roman-numeral analysis, often with horizontal lines stretching to the boundary of each province. 45

Riemann developed a notation for kinship via the parallel and leading-tone

transformations. The subscript p denotes the parallel transformation, while the

45 Two recent textbooks may serve to illustrate this claim. The reader may see, for example, Steven Laitz, The Complete Musician: An Integrated Approach to Tonal Theory, Analysis, and Listening , 2d ed. (New York: Oxford University Press, 2008), 246- 54; and Jane Piper-Clendinning and Elizabeth West-Marvin, The Musicians Guide to Theory and Analysis (New York: W. W. Norton & Co., 2005), 270.

strikethrough < denotes the leading-tone transformation. For example, the parallel transformation of tonic is T p, and the leading-tone transformation of tonic is T< .

Behavior has two notations, but neither is sufficient to comprehensively describe

it. In the Roman-numeral system, dominant behavior on scale degrees other than the

dominant is notated with the slash, as in V/x or vii o/x. This notation fails to account for

subdominant and tonic behavior, however (although in principle it could: e. g., IV/IV.

The reader may refer to Section 3.4 for more on this subject.). It also ignores behavior in

paradigms such as the passing six-four and cadential six-four. In other words, the

notation only accounts for behavior associated with tonicization, and not for diatonic

harmonies and non-tonicization related chromatic harmonies.

The other notation for behavior, nesting notation, does account for these other

types of behavior, but still fails on two levels. First, it does not account for behavior that

reaches into more than one province (as was the case in Example 2.4). Second, the

notation for behavior (i. e., capital T, S, and D) is the same as that of province, which can

lead to ambiguity and confusion, in much the same way that describing four different

aspects of function with a single term leads to ambiguity and confusion.

It is with these strengths and weaknesses of current notational systems in mind

that I propose the following Three-fold System of Analysis.

The first line of analysis, which is a low-order analysis where decisions in

labeling are based only on the key of the piece/section, is called identification . Harmonic identification simply involves labeling individual harmonies and their inversions,

regardless of their contexts (behavior, province, etc.). The harmonies may be notated with the Roman-numeral system, although other systems are also acceptable.

*Aside*

I prefer to identify harmonies with a notation that does not mistake figured bass for inversion. Figured bass was originally used as a guide to performance, not a tool for analysis. It has since been linked to the Roman-numeral system as a way of indicating the inversion of a harmony. Unfortunately, figured bass does not indicate inversion; it only indicates intervals above the bass. Consequently, the standard Roman-numeral system, with its mistaken use of figured bass to indicate inversion, actually thwarts real 6 analysis by eliminating the option of the added-sixth chord. For example, the figure 5 does not indicate a seventh chord in first inversion. It indicates a harmony arranged with a third, fifth, and sixth above a bass note. By analysis we might find that this chord is a seventh chord in first inversion or an added-sixth chord in root position. If we want to identify it as an added-sixth chord (let’s say our chord is F-A-C-D in the key of C), then 6 we cannot call it IV 5 because the notation’s bias toward seventh chords would lead us to 6 spell that chord A-C-E-F. We cannot call it ii 5 because we believe the root to be F. It is not so much that the figured bass does not provide us with a way to notate added-sixth 6 chords, the notation 5 does that just fine. It is the fact that we have perverted figured bass notation to mean inversion, and that we have biased ourselves toward seventh chords to the exclusion of added-sixth chords, that leaves us no consistent notation for inversion in a world where seventh chords and added-sixth chords are both legitimate. (As an illustration of my point, imagine the opposite. If we were to take figured bass symbols to represent inversions of added-sixth chords instead of seventh chords, we would be identifying a root position G 7 chord in the key of C as vii o7 : an added-sixth chord in third inversion. I am not sure that what we actually do is much less ludicrous.) The Roman-numeral system may be refined so as to more clearly account for added-sixth chords and harmonic inversion. First, figured bass symbols may be replaced with the Arabic numbers from jazz-pop notation. These numbers—2 through 13— indicate intervals above the root, not the bass. Second, the Roman-numeral system may also borrow from jazz-pop notation its symbol for inversion, the forward slash. Inversions may be notated with slashes followed by the number of the chord member that occurs in the bass. For example, V in first inversion would be written V/3, and pronounced “five over three.” Third, invoking the slash as a notation for inversion leaves the applied chord without a notation in the modified Roman-numeral system. We may borrow from mathematics its notation for the word of : the parentheses, as in f(x) . An applied chord such as “five of five in first inversion” would then be V(V)/3. We even get the satisfaction of invoking a mathematical notation for function to become a musical notation for function !

There are two advantages to this modified Roman-numeral system. First, pedagogically speaking, learning jazz-pop notation, which usually occurs subsequent to learning the Roman-numeral system for university-trained musicians, becomes much easier. Second, and more importantly, the modified Roman-numeral system recognizes the added-sixth chord. For virtually all music from Bach to Beethoven, superscript Arabic figures would be limited to 7 and 6! Furthermore, the modified notation allows analysts to make decisions about whether a four-note harmony is a seventh chord or an added-sixth chord based on its context within a progression. Such an analysis may look like the one in Example 2.5. Specifically, the modified notation allows the analyst to account for Rameau’s double employment of the dissonance in m. 2, where the harmony is initially an added-sixth chord as it is approached from I, and then becomes a seventh chord as it progresses to V.

ID C: I IV 6/5 becomes ii 7/7 V 7/3

I vi/3 V 7(V)/7

Example 2.5 Bach, “Prelude No. 1,” Das Wohltemperierte Clavier , Book I, mm. 1-6

I do not mean to imply that figured bass symbols are inherently bad. On the contrary, figured bass is an excellent pedagogical tool for keyboard, counterpoint, and voice-leading. Figured bass is even successful in an analytic notation system in one respect: its ability to depict melodic motion, in particular suspensions and suspension-like 6 - 5 harmonies such as the cadential six-four chord ( 4 - 3 ). Fortunately, my Three-fold System of Analysis accounts for these motions without having to rely on figured bass. If the analyst wishes, figured bass symbols may still be marked above the notes, in much the same way as common notations for suspensions. 46

46 The modified notation described above builds on that of Carolyn Alchin, who notates inversions by placing the number of the chord member occurring in the bass

The second line of analysis is behavior analysis . On this line, two or more harmonies are grouped together and analyzed on the basis of status and voice leading.

The highest-status element in a grouping is notated with a lower case t, which indicates

local tonic. Lower-status elements are notated with lower case d or s to indicate local dominant or subdominant. The harmonies are grouped together with a horizontal square bracket (as opposed to the curly bracket in nesting notation that necessitates that all elements in the behavior paradigm belong to the same province). A grouping in behavior analysis may reach into more than one province, but the behavior group is always self- contained. The lower case t, s, and d in the behavior analysis model will differentiate that notation from the upper case T, S, and D that will signify province . In the behavior

level of analysis, chromatic harmonies may be distinguished from diatonic harmonies by

slashes. An upward-pointing slash (/) through a d or s will designate a raised chromatic

note; a downward-pointing (back)slash (\) will designate a lowered chromatic note.

The third line of analysis is province analysis. Provinces are notated with capital

T, S, and D, as well as with horizontal lines that extend out to the boundary of the province. The prototypical progression for province analysis is TSDT, although incomplete versions may exists, including TSD (half-cadence), TST (plagal cadence), and TDT. 47

I 6 under the Roman numeral, for example, 5 instead of I 4 . Carolyn A. Alchin, Applied Harmony (Los Angeles: Carolyn Alchin, 1917), 28.

47 Only behavior and province (not quality and kinship) receive individual lines of analysis in the Three-fold System of Analysis. This is because quality and kinship remain the same regardless of context within a progression, and therefore do not warrant

2.8 Analysis: Beethoven, Piano Sonata no. 9 , Op. 14, no. 1, mm. 1-16.

Several of the concepts presented in this chapter are found in the opening of the

Beethoven piano sonata in Example 2.6. The Three-fold System of Analysis accounts for

harmonic identity, behavior, and province in the excerpt. Behavior paradigms may be

found that are wholly contained within a province; other behavior paradigms cross into

more than one province.

o7 o 4 6 6 4 o7 ID e: i VI vii /V vii 3 i V 4 i V 3 i V vii /V V Beh. d t t d t d t s t t d t Prov. T ------S D ------T ------D ------

o7 o 4 6 6 4 6 7 ID. i VI vii /V vii 3 i V 4 i V 3 i i 4 V i Beh. d t t d t s t Prov. T ------S D ------T ------D ------T

Example 2.6 Beethoven, Piano Sonata no. 9 , Op. 14, no. 1, mm. 1-16

their own lines of analysis. Kinship is not notated at all in the system; quality may continue to be notated with slashes in the Roman-numeral system (as well as slashes in the behavior system). For that matter, the Roman-numeral system may be expanded to include iv (add6) /x as an indication of subdominant quality.

A few points may be made about the analytical choices above. First, because of behavior and province notation, identity notation may revert back to its original role. That is to say, for example, the cadential six-four chord may be notated as a tonic, since its status is described with behavior and province notation. When the analyst notes that the chord belongs to the dominant province, and is therefore the subdominant of the dominant,

6 - 5 identifying the chord as V 4 - 3 becomes redundant. Second, province notation should

draw attention to the structural level of harmony once removed from the foreground.

This level may be realized as the prototype TSDT or it may be one of the other

prototypes (TDT, TST, TSD, TD). Province analysis is compatible with a quasi-

Schenkerian levels analysis, which could show, for example, that the opening TSDT

move is on a more foreground level than the TD background of the first eight measures.

For the opening of the Beethoven sonata above, such an expansion of the notation would

look something like Example 2.7.

m.1 m.5 m.9 m.12 Prov. T S D T D T S D T D T T ------D || T ------D T

Example 2.7 Province analysis expanded to account for quasi-Schenkerian levels

Such a refinement of the basic notation for province is welcome if the analyst wishes to

account for structural levels in addition to function (i. e., behavior and province).

However, the most foreground province analysis, shown in Example 2.6, is sufficient if

the analyst does not wish to account for structural levels. 48 Third, behavior analysis may

be as detailed as the analyst wishes. Behavior analysis should at least account for any harmony whose behavior contradicts its province. This would include six-four chords, for which the boundary of the behavior paradigm is completely within the boundary of the province. Examples of these paradigms can be found in mm. 5, 7-8, 13, and 15 (of

Example 2.6). It would also include behavior paradigms that cross over province boundaries, such as the example of the secondary dominant that began this chapter.

Other examples found in the excerpt above include mm. 3-4, 11-12, for which the behavior d-t contradicts the province S-D. In those places where behavior and province

(and perhaps identity) are identical, it is a task of busy work to notate behavior. For example, a phrase that ends V-i, such as the one in mm. 15-16 above, need not be labeled d-t in the behavior analysis, since it is already labeled V-i in the identification and D-T in the province analysis.

2.9 Diatonic triad motion explained by the theory of behavior

There are four types of two-harmony behavioral paradigms: d-t, t-d, s-t, and t-s.

Several common diatonic progressions can be grouped into these four models. Table 2.3 provides a generous but not comprehensive list.

48 When a behavior paradigm occurs wholly within a late middleground province, the behavior paradigm itself is a foreground level with respect to the late middleground structural level. That is to say, Schenkerian levels are also implied when behavior complements province.

d-t t-d s-t t-s V-I, vii o-I, V-vi I-V, I-vii o IV-I I-IV, I-ii ii-V, IV-V

Table 2.3 Diatonic harmonies grouped according to behavioral paradigms

The second row of harmonic paradigms, which only appears in the d-t column, represents an important footnote to behavior. A d-t behavior may appear as an S to D identity or kinship. This raises the question of a possible s-d behavior. Embracing such a behavior, however, actually renders behavior impotent! To legitimize s-d behavior ignores one of the axia of behavior: that there be a status distinction between the two harmonies in question. By ensuring that all behavior paradigms contain a t, we ensure that there is a higher-status harmony (t) and lower-status harmony (s or d).

2.10 Summary

Chapter 1 described a fundamental flaw in modern function theory: it has at least four different aspects, which, at a minimum cause confusion, and may at times even be contradictory. Chapter 2 proposed a solution to this problem: “The Death of Function,” that is, the elimination of the term function in favor of four replacement terms—behavior, kinship, quality, and province—that each correspond to one of the four aspects of function. Replacing function with the four new terms erases the problem of ambiguity and makes the meanings of function clearer and more powerful. Harmonic analysis, for example, may be enhanced through the use of a Three-fold System of Analysis that accounts for identity, behavior, and province.

Other ways that function theory may be expanded through the new concepts and terminology include the application of the theory of behavior to chromatic music, jazz, and popular music. We now turn to these new applications in Part II, noting how “The

Death of Function” makes possible “The Resurrection of Function,” in which function theory reaches its full potential for explanation.

PART II

THE RESURRECTION OF FUNCTION:

PRACTICAL APPLICATIONS FOR THE THEORY OF BEHAVIOR

CHAPTER 3

TONICIZATION VERSUS MIXTURE:

THE FIRST EXPLANATORY POWER OF

THE THEORY OF BEHAVIOR IS REVEALED

Tonicization and mixture are complementary explanations for most of

chromaticism. 49 In the major mode, for example, tonicization accounts for raised chromatic notes on 1ˆ , 2ˆ , 4ˆ , and 5ˆ ; while simple mixture accounts for lowered

chromatic notes on scale degrees 3ˆ , 6ˆ , and 7ˆ . But tonicization and mixture may also be contradictory explanations for many chromatic notes. For example, in the case of V 7/IV, tonicization is the explanation for ♭ 7ˆ , a note often explained by mixture. Since

tonicization and mixture compete for explanatory significance of chromaticism in cases

such as this, it is sometimes difficult to tell which theory is a better explanation of a

49 This idea stems from the Schenker/Salzer tradition, and may be most succinctly summarized by Proctor, who writes, “There are but two ways … in which chromatic notes may arise: mixture and tonicization.” Gregory Proctor, “Technical Bases of Nineteenth-century Chromatic Tonality: A Study in Chromaticism” (Ph.D. diss., Princeton University, 1977), 43. Proctor’s condition on this assertion is that the music to which the statement applies is governed by what he calls classical diatonic tonality . It should be noted that this dissertation—particularly Chapters 1-4—also primarily concerns classical diatonic tonality. Chapter 5 reveals that my theory of behavior also has some ramifications for other styles of music, including nineteenth-century chromatic tonality, jazz, and popular music.

particular event. I submit that this difficulty is more widespread than most people realize, primarily because most chromatic notes usually explained with mixture are better explained with different forms of tonicization. 50 Perhaps more significantly, I know of

no set of criteria by which to judge whether a chromatic note is a product of mixture or

tonicization (or something else). Arriving at a decision between tonicization and mixture

seems to be some combination of unwritten rules and the discretion of the analyst.

Fortunately, the theory of behavior leads to a clear and simple definition of tonicization

as well as a series of clear and simple heuristics for determining when tonicization is the

correct explanation of a particular chromatic event.

3.1 Tonicization codified with the theory of behavior

The most common form of tonicization may be defined as a chromatic event

represented by a paradigm featuring an upward pointing leading tone supported by

dominant behavior . The complete version of the tonicizing chord is the secondary dominant seventh chord, V 7/x. Two subsets of this chord are also agents for tonicization:

V/x and vii o/x. All three forms of the tonicizing chord conform to the definition above; they all have an upward pointing leading tone, and their roots move up by fourth or by step.

50 This is not a new idea. Proctor’s dissertation features several examples where chromatic notes usually explained by mixture are instead explained by tonicization (e. g., the Neapolitan complex); Proctor, “Technical Bases,” 97-114. Furthermore, Proctor now believes most of the examples in his dissertation for which he invoked mixture are actually products of tonicization. What is unique about my position are the specific criteria (or paradigms) by which to judge chromatic notes as tonicization based on the proposed theory of behavior.

Example 3.1 shows all of the accidentals generated by these tonicizing chords in

the major system (read left to right) and in the minor system (read right to left). The

diminished triad (vii o in major or ii o in minor) is ineligible for tonicization since its diminished fifth makes it unable to sustain its own key.

C: V7 V7/ii V7/iii V7/IV V7/V V7/vi V7/III V7/iv V7/v V7/VI V7/VII V7 :a

Example 3.1 Chromatic notes explained by tonicization

Four chromatic leading tones are shown in the example, each appearing as the lowest note of the voicing, and resolving by half step to the local tonic ( 7ˆ -8ˆ ). As the “active ingredients” of the tonicization, the leading tones are direct results of toncization. They yield #1ˆ , # 2ˆ , # 4ˆ , and # 5ˆ in major; and # 7ˆ , # 6ˆ , # 4ˆ , and # 3ˆ in minor. Two other chromatic notes are found in the example. These are not active ingredients, but rather occur incidentally as part of the harmonic field (i. e., secondary key) generated by the tonicization. Thus, these notes are indirect results of tonicization. The F# as part of

V7/iii (or V 7/v in minor) is not a leading tone as is the F# in V 7/V (V 7/VII), but rather it is

a note incidentally generated by the tonicized key of E minor. Likewise, the B ♭ is not

the leading tone in V 7/IV (V 7/VI), but rather indirectly results from the F-major field

invoked by the tonicization.

Example 3.2 reveals how the simplest of tonicizations, V 7/V to V, may be understood as a leading tone, B ♮, supported by dominant behavior.

4 ID. F: I V 3 /V V Beh. d t Prov. T ----- S D

Example 3.2 Bach, “Gott lebet noch,” mm. 1-2

Although there are only two indirect results of tonicization in the paradigms of

Example 3.1, other chromatic notes may result indirectly as parts of scales generated by tonicizing chords. For example, the V 7/iii shown in Example 3.1 generates an E-minor universe that may include C# (as part of melodic minor) in addition to the chromatic notes—D# and F#—that are part of the secondary dominant harmony.

Example 3.3 reveals how indirect results of tonicization may result from scalar representations of tonicizing harmonies in addition to the tonicizing harmonies themselves. In mm. 10-11, D minor is tonicized. The C-sharps are the leading tones of this tonicized harmony, and are therefore direct results of the tonicization of D minor.

The B-flats are not the leading tones of the tonicized harmony (i. e., not part of the

harmony V 7/ii), and are therefore indirect results of the tonicization of D minor.

Similarly, in mm. 12-14 A minor is tonicized. The G-sharps are the leading tones of this tonicized harmony, and are therefore direct results of the tonicization of A minor. The F- sharps are not leading tones of the tonicized harmony (i. e., not part of the harmony

V7/vi), and are therefore indirect results of the tonicization of A minor.

Example 3.3 Bach, Invention 1 , mm. 7-15

3.2 Downward pointing leading tones

Currently, tonicization is not widely accepted as an explanation of lowered chromatic notes, as it is with raised chromatic notes. Instead, mixture is often invoked for that purpose. Lowered 3 rd , 6 th , and 7 th scale degrees in major are virtually always

explained as borrowed from the parallel minor mode, while the lowered 2 nd scale degree

is often accounted for by mixture with the parallel Phrygian mode. Mixture, however, is

not the only explanation for these notes. Theorists’ failure to place the downward

pointing leading tone (local ♭ 6ˆ ) on equal (or at least near equal) footing with the upward pointing leading tone (local # 7ˆ ) has resulted in the omission of a completely different

brand of tonicization theory.

The great divide of the diatonic scale falls between 6ˆ and 7ˆ . Scale degree 7 is

tied to the tonic while 6ˆ is tied to the dominant. Example 3.4 represents this on a staff. 51

The raised 7ˆ is native to major and creates a stronger pull toward tonic than the lowered

7ˆ in minor. The lowered 6ˆ is native to minor and creates a stronger pull toward the

dominant than the raised 6ˆ native to major. When 7ˆ is raised in minor to strengthen the

move from dominant to tonic, it is a lower adjacency resolving upward to a higher-status

element. In other words it is an upward-pointing leading tone supported by dominant

behavior. This is the well known form of tonicization codified in Section 3.1. Similarly,

when 6ˆ is lowered in major to strengthen the move from subdominant to tonic, it is an

51 This representation of the mirror-image tendencies of 6ˆ and 7ˆ is rooted in the dualism of Riemann. For further discussion, the reader may refer to Harrison, 25-28, 254-265.

upper adjacency resolving downward to a higher-status element. This downward pointing leading tone supported by subdominant behavior is also a form of tonicization.

7ˆ 1ˆ 2ˆ 3ˆ 4ˆ 5ˆ 6ˆ

Example 3.4 6ˆ and 7ˆ as leading tones

3.3 Tonicization (d-t and s-t)

Tonicization was defined in Section 3.1 as a paradigm featuring an upward pointing leading tone supported by dominant behavior. However, a downward pointing leading tone supported by subdominant behavior is also a form of tonicization. Thus, henceforth the paradigm defined by upward pointing leading tones supported by dominant behavior will be referred to as d-t tonicization . Similarly, the paradigm defined

by downward pointing leading tones supported by subdominant behavior will be referred

to as s-t tonicization . Tonicization of I (i) will be called trivial tonicization .

The complete version of the s-t tonicizing chord is the minor added-sixth chord, iv add6 /x. Two subsets of this chord are also agents for tonicization: iv/x and ii o/x. All

three forms of the tonicizing chord conform to the definition above; they all have a

downward pointing leading tone, and their roots move down by fourth.

Example 3.5 shows all of the accidentals generated by s-t tonicizing chords in the

major system (read left to right) and in the minor system (read right to left).

C: iv add6 iv add6 /ii iv add6 /iii iv add6 /IV iv add6 /V iv add6 /vi iv add6 /III iv add6 /iv iv add6 /v iv add6 /VI iv add6 /VII iv add6 :a

Example 3.5 Chromatic notes explained by s-t tonicization

Four chromatic leading tones are shown in the example, each appearing as the highest note of the voicing, and resolving by half step to the local dominant ( 6ˆ -5ˆ ). As the active ingredients of the tonicization, the leading tones are direct results of toncization. They

yield ♭ 6ˆ , ♭ 7ˆ , ♭ 2ˆ , and ♭ 3ˆ in major; and ♭ 5ˆ , ♭ 4ˆ ,♭ 2ˆ , and ♭1ˆ in minor. Two other chromatic notes are found in the example. These are not active ingredients, but rather occur incidentally as part of the harmonic field (i. e., secondary key) generated by the tonicization. Thus, these notes are indirect results of tonicization. The B ♭ as part of

iv add6 /VI in minor (or iv add6 /IV in major) is not a leading tone as is the B ♭ in iv add6 /ii

(iv add6 /iv) but rather a secondary chromatic note that is brought along by the field generated by the tonicization. Likewise, the F# is not the leading tone in iv add6 /v

(iv add6 /iii), but rather indirectly results from the E-minor field invoked by the tonicization.

One fairly common example of s-t tonicization is the minor cadential six-four chord in the major mode. As described in Chapters 1-2, the cadential six-four chord belongs to the dominant province, and behaves as subdominant of the dominant. Thus, it and the dominant together form an s-t behavior paradigm. The minor version of this chord includes the downward pointing leading tone, ♭ 6ˆ of the dominant. Since the

downward pointing leading tone is supported by subdominant behavior, by definition, an s-t tonicization is generated. Example 3.6 shows an s-t tonicizing minor cadential six- four chord in Schubert’s “Im Dorfe” from Winterreise . The F ♮ is a downward pointing leading tone, ♭ 6ˆ of V; it is supported by the subdominant behavior, i-V.

Beh. s t Prov. D ------T

Example 3.6 Schubert, “Im Dorfe” from Winterreise , mm. 44-47

3.3.1 Combining d-t and s-t tonicization

When the paradigms for d-t and s-t tonicization are combined, and all common

tones eliminated, the secondary diminished seventh chord emerges. This harmony,

vii o7 /x, has three notes from V 7/x and three notes from iv add6 /x, with two notes shared by both. Therefore, the fully diminished seventh chord is equally dominant and subdominant.

t s t 5ˆ ♭ 6ˆ 5ˆ

1ˆ # 7ˆ 1ˆ t d t

Example 3.7 vii o7 /x as combination of d-t and s-t

3.3.2 Incomplete tonicization

The paradigms given above serve as clear and simple heuristics for determining whether a chromatic note is a product of tonicization. If a tonicizing chord (V 7/x or

iv add6 /x) and its object of tonicization (x) lie in proximity to each other, tonicization is the explanation. There are situations, however, for which the object chord of a seemingly tonicizing chord is never reached. In cases such as this, the tonicization is incomplete , in that the passage lacks its local tonic. 52 The question to be answered, then, is “When there is a harmony that seems to be a tonicizing chord, but the object of tonicization is not present, how will I know whether or not it is an example of incomplete tonicization?”

Richmond Browne’s theory of position-finding helps us here. Specifically,

Browne notes that rare (diatonic) intervals help determine keys. For example, given a tritone, only one other diatonic note need be added for a unique key to be generated. 53

52 This is what Proctor calls “tonicization in the service of secondary chords in the key.” I take everything in this section of his dissertation to be true. I only add here a series of paradigms by which one can determine whether or not a chromatic chord is an incomplete tonicization. Proctor, “Technical Bases,” 71.

Since the process of tonicization is a process of generating a (secondary) key, this principle is directly applicable to the problem of incomplete tonicization. In d-t tonicization, a secondary key is generated by the third (leading tone) and seventh of V 7/x,

along with one other note (i. e., any other note of the V 7/x or the x). Thus, within the

V7/x or vii o/x there is enough information to establish a secondary key, and therefore establish tonicization. In the case of V/x, however, there is no tritone present; thus, several keys may be suggested by the harmony. In order for the key of x to be uniquely suggested, one (or more) additional notes needs to be present.

Table 3.1 shows an A-major triad that may potentially be a secondary dominant triad. Several notes are added to the triad to create diatonic (or harmonic minor) sets, some unique, some not unique. The process reveals that any of three notes is sufficient to uniquely suggest a secondary key. The hypothetical ♭ 6ˆ and ♭ 3ˆ , either of which suggest

D minor, and the hypothetical 4ˆ , which suggests D major/minor, are all sufficient to imply that the A triad is a secondary dominant. All other notes (even taken all together) are either inconclusive, meaning they do not uniquely suggest a key, or they conclusively and uniquely suggest a key for which A is not V. For example, the presence of a G# negates the possibility that A is V.

With Browne’s position-finding in mind, we can establish the following rule for determining whether or not a chromatic note is a product of incomplete d-t tonicization:

When a possible tonicizing chord, V/x, lacks its object chord, x, the chord is still

53 Richmond Browne, “Tonal Implications of the Diatonic Set,” In Theory Only 5, nos. 6-7 (July-August 1981): 3-21.

explained by incomplete d-t tonicization if it is adjacent to a harmony containing its local

4ˆ , ♭ 6ˆ , or ♭ 3ˆ . Generally speaking, this means that chords with S kinship in the

secondary key may combine with (local) V to uniquely suggest the secondary key.

Triad out of context: A C# E Add any or all of the following: B D F# Result: Key of D or Key of A?

Triad out of context: A C# E Add G#, then B, D, and F# follow: G# (B) (D) (F#) Result: Key of A; A is not a sec. dominant

Triad out of context: A C# E Add G ( 4ˆ ), then B( ♭), D, F(#) follow: G (B)( ♭) (D) (F) Result: G uniquely suggests D major/minor

Triad out of context: A C# E Add B ♭ (♭ 6ˆ ), then D, F, G follow: (G) B♭ (D) (F) Result: B ♭ uniquely suggests D minor

Triad out of context: A C# E

Add F (♭ 3ˆ ), then G, B ♭, D follow: (G) (B ♭) (D) F Result: F uniquely suggests D minor

Table 3.1 Position-finding with a hypothetical secondary dominant

The same argument may be made for s-t tonicization. Given a possible iv/x, two possible notes uniquely suggest a key: local 2ˆ and local # 7ˆ . Thus, when a possible tonicizing chord, iv/x, lacks its object chord, x, the chord is still explained by incomplete

s-t tonicization if it is adjacent to a harmony containing its local 2ˆ or # 7ˆ . Generally

speaking, this means that chords with D kinship in the secondary key may combine with

(local) iv to uniquely suggest the secondary key.

Triad out of context: A C E Add any or all of the following: G B D Result: Key of e or Key of a?

Triad out of context: A C E Add F, then G, B, and D follow: (G) (B) (D) F Result: Key of a; a is not iv/e

Triad out of context: A C E Add F# ( 2ˆ ), then G, B, and D follow: G (B) (D) F# Result: F# uniquely suggests e minor

Triad out of context: A C E Add D# ( # 7ˆ ), then G, B, and F# follow: (G) (B) D# (F#) Result: D# uniquely suggests e minor

Table 3.2 Position-finding with a hypothetical secondary subdominant (iv/x)

Incomplete tonicization is often found in the development sections of Classical sonata forms. Mozart’s Piano Sonata no. 6 , K. 280 (Example 3.8) is one such example.

The A-major triad in m. 78, 79, and 80 may be a tonicizing chord, but as no D harmony is

ever reached, it is difficult to specify how such a tonicization might take place. There are

hints in the first system. There is a D-minor harmony in m. 76. However, since it is a

passing six-four chord, it does not yet establish itself as the tonic. Furthermore, this D-

minor harmony is not a direct object chord of the tonicizing chord, A major, in m. 78ff.

The echoing C# that is paired with E does directly point to D, also hinting that D is the object of tonicization in this passage. Nevertheless, this tonicization is never made explicit; there is no V-I in D minor.

During the dominant prolongation in m. 78, however, incomplete tonicization accounts for the chromatic note C# as part of a D-minor harmonic field. The B♭ in m. 78, in conjunction with the A-major triad, uniquely suggests D minor, as is described in the fourth hypothetical situation in Table 3.1. Measure 80 is a microcosm of the tonicization features of the entire passage. The A-major triad alone is not enough to suggest tonicization, but the addition of B ♭ uniquely suggests the key of D minor. Since the

object chord, D minor, is never reached, the tonicization is incomplete.

Example 3.8 Mozart, Piano Sonata in F Major , K. 280, mm. 73-83

3.4 Quasi-tonicization

Paradigms for tonicization feature V 7/x and iv add6 /x along with their triadic subsets. Sometimes, however, d-t and s-t paradigms feature tonicizing chords without their respective qualities; that is, without leading tones. Thus, chromatic notes may arise from d-t paradigms featuring v 7/x, or from s-t paradigms featuring IV add6 /x. When v 7/x is

applied to (diatonic) minor triads or IV add6 /x is applied to (diatonic) major triads, the

result is quasi-tonicization . When v 7/x is applied to (diatonic) major triads or IV add6 /x is

applied to (diatonic) minor triads, the result is pseudo-tonicization (See Section 3.5).

Example 3.9 shows how quasi-tonicization and pseudo-tonicization are derived from

tonicization. Examples 3.9a and 3.9b show the chromatic notes generated by d-t

tonicization and s-t tonicization respectively. (These are exact copies of Examples 3.1

and 3.5.) Examples 3.9c and 3.9d show the tonicization paradigms from Examples 3.9a

and 3.9b without their leading tones. Example 3.9e shows how quasi-tonicization

paradigms are a subset of the paradigms in Examples 3.9c and 3.9d. Specifically, quasi-

tonicization results when v 7/x is applied to diatonic minor triads (i. e., the second, third,

and sixth paradigms of Example 3.9c) or when IV add6 /x is applied to diatonic major triads

(i. e., the first, fourth, and fifth paradigms of Example 3.9d). In other words, quasi-

tonicization requires the quality of the tonicizing harmony (without its dissonance) to

match the quality of the tonicized harmony. Example 3.9f shows how pseudo-

tonicization paradigms are a subset of the paradigms in Examples 3.9c and 3.9d.

Specifically, pseudo-tonicization results when v 7/x is applied to diatonic major triads (i. e., the first, fourth, and fifth paradigms of Example 3.9c) or when IV add6 /x is applied to

diatonic minor triads (i. e., the second, third, and sixth paradigms of Example 3.9d). In other words, pseudo-tonicization requires the quality of the tonicizing harmony (without its dissonance) to oppose the quality of the tonicized harmony. Quasi-tonicization paradigms and pseudo-tonicization paradigms are complements, which when combined, result in the complete set of paradigms shown in Examples 3.9c and 3.9d.

a. d-t tonicization

C: V7 V7/ii V7/iii V7/IV V7/V V7/vi V7/III V7/iv V7/v V7/VI V7/VII V7 :a

b. s-t tonicization

C: iv add6 iv add6 /ii iv add6 /iii ivadd6 /IV iv add6 /V iv add6 /vi iv add6 /III iv add6 /iv iv add6 /v iv add6 /VI iv add6 /VII iv add6 :a

c. d-t tonicization paradigms without leading tones

C: v 7 v7/ii v7/iii v 7/IV v 7/V v7/vi v 7/III v 7/iv v 7/v v 7/VI v 7/VII v 7 :a

d. s-t tonicization paradigms without leading tones

C: IV add6 IV add6 /ii IV add6 /iii IV add6 /IV IV add6 /V IV add6 /vi IV add6 /III IV add6 /iv IV add6 /v IV add6 /VI IV add6 /VII IV add6 :a

Example 3.9 Tonicization, Quasi-tonicization, and Pseudo-tonicization

Continued

Example 3.9 (Continued)

e. Quasi-tonicization (d-t from Example 3.9c and s-t from Example 3.9d)

C: v 7/ii v7/iii v 7/vi v 7/iv v 7/v v 7 :a

C: IV add6 IV add6 /IV IV add6 /V IV add6 /III IV add6 /VI IV add6 /VII :a

f. Pseudo-tonicization (d-t from Example 3.9c and s-t from Example 3.9d)

C: v 7 v 7/IV v 7/V v 7/III v 7/VI v 7/VII :a

C: IV add6 /ii IV add6 /iii IV add6 /vi IV add6 /iv IV add6 /v IV add6 :a

The complete version of the d-t quasi-tonicizing chord is v 7/x. Two subsets of

this chord are also agents for quasi-tonicization: v/x and ♭VII/x. The complete version of the s-t tonicizing chord is IV add6 /x. Two subsets of this chord are also agents for quasi-

tonicization: IV/x and ii/x. Example 3.10 shows all of the accidentals generated by quasi-

tonicization in the major system (read left to right) and in the minor system (read right to

left). The first line shows the chromatic note generated by d-t quasi-tonicization; the

second line shows the chromatic note generated by s-t quasi-tonicization.

C: v 7/ii v7/iii v 7/vi v 7/iv v 7/v v 7 :a

C: IV add6 IV add6 /IV IV add6 /V IV add6 /III IV add6 /VI IV add6 /VII :a

Example 3.10 Chromatic notes resulting from quasi-tonicization

Examples of quasi-tonicization seem to favor the s-t variety over the d-t variety.

In the major system, the chromatic note generated by quasi-tonicization, ♭ 7ˆ , comes from

IV (add6)/IV, often less clearly labeled as ♭VII. In the minor system, quasi-tonicization

yields ♭ 2ˆ by way of IV (add6)/VI, usually referred to as the Neapolitan.

The end of the second movement of Mozart’s Piano Sonata, K. 545, features an example of quasi-tonicization in the major mode (Example 3.11). The F-major harmony in m. 69 is startling, since it clearly and suddenly departs from the tonic universe. Once the progression reaches m. 70, however, we can hear that the province goal of C (S) is reached, and that the harmonies in m. 69 (F and G 7) were propelling us toward this goal.

Thus, the chromatic note, F, is generated by the invocation of the key of C, and is part of

the harmony IV/IV. This is an s-t quasi-tonicization, since in order to be tonicization, the

downward-pointing leading tone of the secondary key, A ♭ ( ♭ 6ˆ -5ˆ in C), must be present.

In other words, iv/IV would be tonicization, but IV/IV is quasi-tonicization. The F, which persists through the V 7/IV chord, then becomes an indirect result of the d-t

tonicization from G 7-C. In this way, both IV/IV and V 7/IV are progressing toward their

common object chord, IV. This is noted in the two behavior analyses. 54 If, by contrast,

IV/IV was shown to progress to V 7/IV directly, then a d-t behavior paradigm would be invoked, and the quasi-tonicization would not be revealed.

54 This is the first time in this dissertation that two behavior analyses have overlapped. Showing them separately simply emphasizes that the object of s is t (not d), as is the object of d. The analysis could be condensed into one line, except that the analysis s-d-t implies a behavior paradigm s-d, which does not exist. (This is because one of the preconditions for behavior is that there must be a status distinction between the constituent harmonies. Since there is no status distinction between s and d, no paradigm may consist of those two elements only. There must be at least one t, which signifies highest-status element, in every behavior paradigm. Thus, an s-d move is, in behavior terms, always either d-t or t-s.) Separating the analysis into two lines maintains this distinction and shows clearly that t is the behavioral object chord of s.

6 6 o 6 o7 ID. G: I IV /IV V 5 /IV IV vii 5 vii /V Beh. d t s t Prov. T ------S ------

6 7 7 7 ID. G: I 4 V I V I V I Beh. s t Prov. D ------T ------

Example 3.11 Quasi-tonicization (IV/IV) in Mozart, Sonata K. 545, II, mm. 68-74

The Neapolitan is often a product of quasi-tonicization in minor, in the same way

that ♭VII may be a product of quasi-tonicization in major. Chopin’s Prelude 20 features an example of this (Example 3.12). The Neapolitan in m. 2 is really the subdominant of

VI, which immediately precedes it. Thus, a chromatic note is generated in conjunction with subdominant behavior. However, the chromatic note is not the downward pointing leading tone native to that s-t model (i. e., F ♭). Thus, quasi-tonicization rather than

tonicization is the explanation.

ID. c: i iv 7 V7 i VI IV/VI V 7/VI VI Beh. t s

Example 3.12 The Neapolitan as Quasi-tonicization (IV/VI) in Chopin, Prelude 20

3.5 Pseudo-tonicization

Sometimes a leading tone is present within the key (i. e., diatonic), yet is

subverted by a chromatic note, thereby creating a whole-step rather than half-step voice

leading. This is called pseudo-tonicization . Pseudo-tonicization features v 7/x and

IV add6 /x, as does quasi-tonicization, but whereas these qualities are applied to object

chords of matching quality in quasi-tonicization (i. e., v 7/x is applied to minor triads and

IV add6 /x is applied to major triads), they are applied to chords of opposite quality in

pseudo-tonicization (i. e., v 7/x is applied to major triads and IV add6 /x is applied to minor

triads).

Example 3.13 shows all of the accidentals generated by pseudo-tonicization in the

major system (read left to right) and in the minor system (read right to left).

C: v 7 v 7/IV v 7/V v 7/III v 7/VI v 7/VII :a

C: IV add6 /ii IV add6 /iii IV add6 /vi IV add6 /iv IV add6/v IV add6 :a

Example 3.13 Pseudo-tonicization

A trivial example of d-t pseudo-tonicization (i. e., pseudo-tonicization of I) is

found in the cadence of “Rocky Top Tennessee.” Example 3.14 shows a three-fold

analysis of the cadence. The cadence, ♭VII-I, which accompanies the word “Tennessee,” features a root move up a step, as well as a lower-adjacency harmony resolving up to a harmony of higher status. It therefore exhibits dominant behavior. However, the element that defines tonic, the leading tone, is omitted in favor of ♭ 7ˆ . Scale degree 1 is pseudo- tonicized.

C: I ♭VII I I ♭VII I t d t t d t

Example 3.14 “Rocky Top Tennessee,” Words and Music by Boudleaux and Felice Bryant

A non-trivial example of pseudo-tonicization occurs in the bridge of “Unchained

Melody.” Example 3.15 shows a three-fold analysis of the beginning of the bridge.

Dominant behavior occurs from E ♭ to F. This dominant behavior is not accompanied by leading tone motion, however, but by a whole step. Scale degree 4 is pseudo-tonicized.

No leading tone

ID: C: IV V IV ♭III IV V I Beh: t d t Prov. PD ------D T

Example 3.15 “Unchained Melody,” Words by Hy Zaret, Music by Alex North

3.6 Microtonicization

Sometimes a chromatic leading tone appears without the support of its tonicizing

harmony (V 7/x, iv add6 /x, or their subsets). Thus, tonicization is applied to a single note rather than a whole harmony (or group of harmonies). This is called microtonicization .55

55 Proctor’s discussion of microtonicization, including the chromatic chords explained by microtonicization, is thorough and accurate in nearly all respects. My brief summary here only differs from Proctor in that: (1) Given the existence of the downward pointing leading tone, Proctor’s statement, “In general, virtually all … chromatic neighbor or passing notes turn out … to be instances of either microtonicization or mixture,” may be revised to read “In general, virtually all chromatic neighbor or passing notes turn out to be instances of either upward or downward microtonicization;” (2) Similarly, the augmented-sixth chord features microtonicization exclusively, as opposed to part microtonicization and part mixture; (3) The b 2ˆ in the Neapolitan is a

Examples of microtonicization include chromatic passing or neighbor notes, the augmented triad, the Neapolitan, the augmented-sixth chord, and the common-tone diminished seventh chord.

Among these, the Neapolitan is interesting in that it may also be explained as a quasi-tonicization, IV/VI in minor, as we have seen above. There are situations, however, in which the Neapolitan is found divorced from its context with VI. When VI is not in proximity to N, N may not be considered a quasi-tonicization (i. e., IV/VI). In the case of

N (not IV/VI), as shown in Example 3.16, a downward pointing leading tone resolves downward, but it is not supported by its characteristic harmony and behavior (IV/VI moving to or from VI). The downward pointing chromatic note, B º, is not supported by

an s-behaving harmony (i. e., it is not a local ♭ 6ˆ moving to local 5ˆ ). Instead, B º is a leading tone that points downward to the A ♭ tonic in m. 29. It is therefore tonicizing a single note, A ♭, rather than tonicizing an entire harmony, A ♭-minor. This is therefore a case of microtonicization.

microtonicization and not a quasi-tonicization in cases where it is not supported by its characteristic harmony (i. e., IV/VI in minor). The reader may refer to Proctor, “Technical Bases,” 68-130 for a complete discussion of microtonicization, including relevant examples.

6 N V 5 i

Example 3.16 Beethoven, Piano Sonata Op. 26 , III, mm. 23-30

3.7 Analysis: Beethoven, Waldstein Sonata , I, mm. 1-35.

This chapter has clarified and codified the concept of tonicization in a variety of ways. First, tonicization was defined as a paradigm in which certain notes are supported by certain behaviors. Depending on these notes and behaviors, tonicization may be d-t or s-t, and may also qualify as tonicization, quasi-tonicization, or pseudo-tonicization.

Second, criteria were established that allowed tonicization to be invoked, even without the presence of the secondary tonic, thus creating incomplete toniciztion. Third, when a leading tone is not supported by one of its characteristic tonicizing harmonies, but rather tonicizes only a single note, it is microtonicization. With all of these codifications of tonicization in mind, it would be revealing to see how they may all work together in a single analysis. The first theme and transition of the first movement of Beethoven’s

Waldstein Sonata will reveal many of the types of tonicization described above.

The opening of the Waldstein Sonata exemplifies s-t quasi-tonicization. Although

the C-major triad in mm. 1-2 is tonic, it may be retrospectively heard to be lower in status

than the G-major triad in mm. 3-4, and therefore, the subdominant of the dominant. 56 It passes through the dominant of the dominant on its way to the object chord, G. This model, s through d to t, is restated in mm. 5-8, this time with F as the object chord. The three-fold analysis of the reduction in Example 3.17b shows that the subdominant behavior within the subdominant province results in an identity of IV/IV, and the chromatic note B b as a result of quasi-tonicization. As the B b persists through the V 7/IV,

it becomes the indirect result of the d-t tonicization of F. In the first phrase, the F# is the

leading tone of the tonicizing harmony, and is therefore the direct result of the d-t

tonicization of G. The other chromatic note in the first phrase, the C# in m. 3, is simply a

chromatic neighbor, a form of microtonicization.

56 Godfrey Winham’s reading of this passage is that the first articulation of the harmony represents the tonic, whereas the second articulation (or voicing) represents the subdominant of the dominant. Indeed, every time there is a harmony with dual meaning in the movement, Beethoven supplies two distinct voicings of the harmony. Godfrey Winham, personal communication, as reported by Gregory Proctor.

a. Score, mm. 1-35.

Example 3.17 Beethoven, Piano Sonata in C Major, Op. 53, Waldstein , I

Continued

Example 3.17 (Continued)

b. Reduction, mm. 1-8.

ID. C: IV/V V 7/V V IV/IV V 7/IV IV 6 iv 6 Beh. d t d t s t s t Prov. (T) D ------S ------

c. Middleground reduction, mm. 1-14.

Beh. t s

In m. 8, IV gives way to iv as the downward pointing leading tone trivially tonicizes I. This tonicization is not immediately apparent from the score, however, since the object of tonicization, C major, is not in proximity to the s-t tonicizing harmony, F minor. The tonicization, which occurs at a middleground level, can be more clearly seen in the reduction of mm. 1-14 shown in Example 3.17c. Here the F-minor chord features a downward pointing leading tone supported by subdominant behavior, thus tonicizing C in

the middleground. The B-flat in m. 8 and the E-flats in mm. 9-11 are scalar indirect results of the tonicization of C. That is, they are brought along with the A ♭ from the invoked field of C minor.

In mm. 12-13, an s-t paradigm accompanies a downward pointing leading tone, creating s-t tonicization of the dominant harmony. Much like the minor cadential six- four chord from Schubert’s “Im Dorfe,” the E ♭ points downward to D ( ♭ 6ˆ of V) as the minor tonic resolves to the higher-status dominant.

At m. 15 the harmonic structure echoes the beginning of the piece, but after four bars, there is a tonicization of A minor instead of F. Thus, the G# in m. 19 is a direct result of d-t tonicization. The D-sharps in mm. 21-22 are microtonicizations, chromatic neighbors to E not supported by their characteristic tonicizing harmony. The A# in m. 22 is also a microtonicization, not as a chromatic neighbor, but rather as a part of the augmented-sixth chord that resolves to V/iii in m. 23.

By m. 23, the tonicization of E minor is made explicit by the alternation of B 7 and second-inversion E-minor chords. This dominant prolongation, which extends from m.

23 all the way to m. 34, features chromatic notes generated in a variety of ways. First, the D-sharps that occur as part of the B 7 harmony are the leading tones of the tonicized

harmony, and are thus direct results of d-t tonicization. The F-sharps that accompany the

D-sharps in this passage are also part of the tonicizing harmony, B 7, but are not leading tones, and therefore, are indirect results of d-t tonicization. The A-sharps in mm. 24, 26,

27 and the G sharps in mm. 24, 26 are all microtonicizations, the former as chromatic neighbors to B, the latter as chromatic passing tones to A.

From mm. 29-34, the B 7 dominant harmony is prolonged as the scalar indirect

results of tonicization shift from implying E minor to implying E major. The gaps of the

B7 harmony are filled with C#, E, G#, and A# in mm. 31-34, which result from the tonicized key of E major. Measure 35, the beginning of the second theme, is set in the key of E major, the culmination of the dominant prolongation and scalar indirect results of tonicization present since m. 23.

3.8 Chromatic notes in classical diatonic tonality not explained by tonicization

We now have four types of tonicization—tonicization (complete or incomplete), quasi-tonicization, pseudo-tonicization, and microtonicization—that serve as explanations for chromatic notes in classical diatonic tonality. Further, we have several paradigms by which to judge whether a particular harmony is explained by one of these types of tonicization. A good many chromatic notes hitherto explained by mixture are better explained by one of these four types of tonicization. Indeed, the only chromatic chords in classical diatonic tonality that remain to be explained involve root movements by third, in other words, the so-called chromatic mediants (and doubly chromatic mediants).

Felix Salzer explains these harmonies with three different kinds of mixture. In the major system, ♭VI and ♭III may be explained by simple mixture; VI and III may be explained by secondary mixture; ♭vi and ♭iii may be explained by double mixture. In the

minor system, #vi and #iii may be explained with simple mixture; vi and iii may be

explained with secondary mixture, and #VI and #III may be explained with double mixture. 57

Whether or not mixture is used as the explanation of these harmonies, it is

interesting to note that only six harmonies in major and six harmonies in minor require an

explanation other than tonicization. The explanatory value of tonicization increases sharply when two important theoretical entities are taken into consideration: the downward pointing leading tone, and behavior .

3.9 Alternatives to tonicization

Even if the criteria for tonicization are met, there may be occasions when one still prefers mixture or some other explanation over tonicization. These occasions do not diminish the importance of the theory of tonicization laid out above. Rather, one analyst may prefer one theory (mixture) over another (tonicization) on a given occasion.

One such example may be found in the opening of Brahms’ Third Symphony ,

shown in reduced form in Example 3.18. In keeping with Brahms’ F-A-F motto, the

common-tone diminished seventh chord in m. 2 is spelled with A ♭ instead of G#. If it were spelled with G#, it and the B would both be microtonicizations of the tonic chord members. The A ♭, however, has no half-step resolution (to G), and is therefore not an

57 Felix Salzer, Structural Hearing: Tonal Coherence in Music (New York: Dover, 1982), 178-181. I am not suggesting that mixture is the best explanation for these harmonies, but rather only that these harmonies cannot be explained by tonicization since tonicization requires root moves by fourth or by step (in the case of tonicization, quasi- tonicization, and pseudo-tonicization) or a half-step resolution of the chromatic note in the proper direction (in the case of microtonicization).

example of microtonicization … or is it? The rest of the phrase reveals a progression from F through F minor, D ♭ Major, and the same common-tone diminished seventh

chord to the goal harmony of C in m. 7. As the goal of the phrase, C major can be

viewed as a structural harmony, and thus, an object of microtonicization (and tonicization

in the case of the iv in m. 4) in the middleground. According to this view, all of the

chromatic chords of the phrase, which all contain A♭, are pointing toward the G in the C-

major triad. Thus, the F-minor triad is explained by s-t tonicization; the D ♭ triad is explained by microtonicization; and the common-tone diminished seventh chords are explained by microtonicization.

Example 3.18 Brahms, Symphony no. 3 , Mvt. I, mm. 1-7

I concede that although the tonicization explanation is plausible, it is complex.

The analyst may wish to opt for the simpler explanation of mixture, whereby the

invocation of A ♭ is simply a toggle between F major and F minor. 58 This explanation

may even be more in keeping with Brahms’ FAF motto, which seems to be overly

complicated by the tonicization explanation given above.

Even if an alternative to tonicization is preferred in a passage in which

tonicization could be invoked, the breadth of explanatory power for tonicization

presented in this chapter remains substantial. Further, the theory of behavior, as well as

that of province, is revealed to produce explanatory power that has not been previously

available from the weighted down term function .

58 This explanation is preferred by Proctor; Proctor, “Technical Bases,” 94-5.

CHAPTER 4

THE THEORY OF BEHAVIOR AS AN EXPLANATION FOR

CHROMATIC HARMONIES

The theory of behavior presented in Chapter 2 is a continuation of the Ramellian

tradition that emphasizes root movement and voice leading over kinship and province. It

is based primarily on Gregory Proctor’s voice-leading paradigms for triadic motion,

which reveal that dominant behavior is a lower-neighbor (or 1-3 passing)/lower-status

phenomenon, while subdominant behavior is an upper-neighbor (or 3-5 passing)/lower-

status phenomenon. In four-note harmonies, however, voice leading often includes a

combination of upper-neighbor and lower-neighbor motion. This leads to ambiguity

when determining the behavior of the harmony as a whole. This is the case with the

diminished seventh chord (secondary or not) shown in Chapter 3, where three notes are

subsets of V 7/x and three notes are subsets of iv add6 /x, with two notes shared by both. 59

Example 4.1 demonstrates this ambiguity: (a) the dominant, lower-neighbor model and (b)

59 Much has been written about the dual function of the diminished seventh chord. I have already cited Harrison’s account in Chapter 1, for example, where he notes that 7ˆ and 2ˆ are the agent and associate of dominant respectively, while 4ˆ and 6ˆ are the base and agent of subdominant respectively. My discussion here differs from Harrison’s in that Harrison is discussing dual kinship whereas I am discussing dual behavior. A more complete discussion of the difference between the two is found below.

the subdominant, upper-neighbor model are combined to make (c) a seventh chord with root resolution up by step.

a. b. c.

t d t t s t t d/s t

Example 4.1 Combination of d-t and s-t paradigms

The diminished seventh chord is, thus, an example of a single harmony with

multiple behaviors. Each voice has its own microbehavior . The lowest voice has a

dominant microbehavior; the highest voice has a subdominant microbehavior; the two

inner voices may have either dominant or subdominant microbehavior, since they may

move by step in either direction. When a harmony has voices with opposite

microbehaviors, it may not be immediately evident which behavior to assign to the

harmony as a whole. This problem is solved by the following rule: in cases where both

dominant and subdominant microbehaviors are present in a single harmony, the behavior

of the harmony as a whole will be consistent with the behavior of the root . Unfortunately, it is sometimes not that simple to ascertain what the root of a harmony is. Indeed, the concept of root, much like the concept of function, has different shades of meaning that must be distinguished from one another if the term root is to achieve clarity. Furthermore,

the legitimacy of the added-sixth chord, as well as the distortion of heavy chromaticism

makes the task of root-finding even more difficult. Thus, in order to determine the

behavior of harmonies with opposing microbehaviors (by establishing the microbehavior

of the root), first a consistent definition of root must be established, and second, criteria for judging when and if a chord has a root must be established.

4.1 The Death and Resurrection of Roots

Depending on which theorist is asked, a harmonic root may have up to three increasingly lax definitions. In the strictest of definitions, a root may only be the lower note of a perfect fifth or the higher note of a perfect fourth. This “rooted interval” may or may not be accompanied by a major/minor third, or major/minor seventh, sixth, ninth, etc.

This type of root may be called a perfect root .

Only six of seven diatonic triads have perfect roots, since one has a diminished

fifth. Given our history and practice of harmonic inversion, even with the diminished

triad (and augmented triad), many theorists have adopted the term root to apply also to the lower note of a diminished (or augmented) fifth or the higher note of an augmented

(or diminished) fourth. Under this definition, if a third is to be interpolated, it should be a minor third in the case of the diminished fifth and a major third in the case of the augmented fifth. Furthermore, if a seventh is added, it should be a major or minor third above the fifth. In other words, rooted harmonies are those harmonies for which each third is major or minor, and can therefore not be enharmonically confused with a step or a fourth. This type of root may be called a tonal root .

An even looser definition of root would allow for diminished and augmented thirds. Under this version of root, accidentals and even clefs may be ignored. If the chord may be reduced to a triad, seventh chord, or added-sixth chord based on structure

alone (i. e., the chord has two adjacent generic thirds and a generic fourth, with or without a dissonance filling the space of the fourth), the root is the lowest note of such a structure when it is stacked in thirds (or thirds with an added sixth). It is this third type of root that I am most concerned with in this dissertation, since, in combination with the theory of behavior, it yields some explanatory value for certain chromatic chords. Thus, this type of root may be called a behavioral root . I will continue to refer to the behavioral root simply as root ; all occasions of the word root in this dissertation should

be understood as behavioral roots unless otherwise noted. 60

60 Not included in this discussion are other well known but somewhat counterintuitive definitions of roots, such as those of Paul Hindemith and Hugo Riemann. Hindemith’s concept of root is similar to the perfect root in that the root must be the lower note of a perfect fifth, but whereas in a B Ø7 , for example, no perfect root exists since there is a diminished fifth, Hindemith finds the root to be D, since it is the lower note of a perfect fifth (with A). In other words, the root, for Hindemith, need not be the lowest note of a tertian (or added-sixth chord) structure (i. e., behavioral root). Paul Hindemith, The Craft of Musical Composition , Vol. 1, Theoretical Part , trans. Arthur Mendel (New York: Associated Music Publishers, Inc., 1945), 68-73, 121-131. Riemann’s concept of root is founded in his rigid adherence to dualism. For a major triad, the root is the lower note of the fifth, but for a minor triad, the root is the higher note of the fifth. This is understandably rejected immediately by many contemporary theorists. However, it may be that those who dismiss Riemann’s dualistic theory are really only dismissing the use of the term root for the context in which Riemann wishes to invoke it, and are thereby overlooking an entire system that yields good theoretical fruit. Surely Riemann also knew that the lowest note of a minor triad in root position is the primary note in terms of aural significance. Perhaps if instead of root Riemann had invoked the phrase note of maximum stability , readers would stay with him a little longer. After all, the G in a C-minor triad is more a note of stability than the B in a B-diminished triad, for example. Furthermore, if we dismiss the notion that the major tonal system is based in some way on the overtone series (which we should), then we should have no problem divorcing a downward generation of the minor system from the fictional undertone series. Downward generation of the minor system yields a complete mirror image of the major system (generated upward), and therefore leads to many consistent explanations of harmonic events between the two systems. One such consistent explanation is the one given for tonicization in Chapter 3 of this dissertation. Riemann, Harmony Simplified , 1-11.

Example 4.2 shows three increasingly chromatic triads. The root of the first is perfect, tonal, and behavioral; the root of the second is only tonal and behavioral; the root of the third is only behavioral.

Perfect Root: G Tonal Root: G Behavioral Root: G#

Example 4.2 Perfect, tonal, and behavioral roots

4.2 Distinguishing seventh chords from added-sixth chords

The most basic harmony of the common-practice period is the triad, which is the most even division of the octave by three diatonic notes. That is, the triad is the only three-note structure that divides the diatonic octave without creating an adjacency.

Instead, the triad leaves two small gaps filled by one scale step and one large gap filled by two scale steps. When a chordal dissonance is added to a triad, as in Example 4.3, it must occupy one of the two scale steps filling the large gap. Another small gap and a dissonance result. The dissonance forms an adjacency with the triad on the opposite side of the small gap. Thus, the dissonance may occur adjacent to the root and a third away from the fifth, or it may occur adjacent to the fifth and a third away from the root. 61

When the dissonance is adjacent to the root, the structure is called a seventh chord.

When the dissonance is adjacent to the fifth, the structure is called an added-sixth chord .

61 Rameau, Treatise , 80.

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Seventh chord: Added-sixth chord: Dissonance points down Dissonance points up

Example 4.3 Seventh chord and added-sixth chord models

Given the existence of inversion in common-practice music, there is no way to distinguish an isolated seventh chord from an isolated added-sixth chord. A seventh chord in first inversion and an added-sixth chord in root position appear identical. It is how the two chords behave that determines whether they are seventh chords or added- sixth chords. There are two clues for determining whether a structure is a seventh chord or an added-sixth chord: the motion of the root and the motion of the dissonance. Given a four-note harmony: (1) if its supposed root moves up by fourth or up by step, and its supposed seventh moves down by step, the structure is a seventh chord; (2) if its supposed root moves down by fourth and its supposed added-sixth moves up by step, the structure is an added-sixth chord .62

62 The word supposed here takes on its modern English meaning, as in, “supposing we treat note x as the root” and is not meant to evoke the Ramellian “subposed” root. Proctor follows Rameau in distinguishing between seventh chords and added- sixth chords, noting that certain resolutions of the questioned structure are “S-like” while others are “D-like.” He cites several examples of chromatic chords that feature these resolutions. Proctor, Real and Apparent Simplicity .

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This leads to an alternative explanation (as added-sixth chords) of many chords typically understood as seventh chords. Since classical analytic notation systems have no way of accounting for the added-sixth chord, we have abandoned it altogether in favor of strict tertian harmony. We identify all of these structures as seventh chords simply because that is the only label we have. Furthermore, in the case of certain chromatic harmonies, accidentals bias us not only toward seventh chord identification rather than added-sixth chord identification, but also away from seventh chord identification altogether, in favor of so-called “linear” identification. For example, if such a structure is modified by accidentals so that one of the thirds is diminished, we no longer accept that it may be a triad or seventh chord. We tend to discount the importance of behavioral roots.

We instead refer to it as a “linear” chord or a “voice-leading” chord. While it is true that these kinds of chords (most notably augmented-sixth chords) do invite us to emphasize their linear and voice-leading aspects, this is no reason to deny that the chords also

a. S-like resolutions

b. D-like resolutions

Example 4.4 Resolutions of Chromatic Chords

Much of the current chapter is based on this idea. My goal here is to situate the example above in the context of behavior. Two points about these harmonies are central to my exposition of the example. First, each pair generates a behavior paradigm. Second, each questioned harmony may be categorized as seventh chord or added-sixth chord, based on the motion of its supposed root and supposed dissonance.

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resolve as seventh and added-sixth chords. The essence of seventh chords and added- sixth chords lies in their structure and their behavior; it is in no way based on accidentals, key signatures, or even clefs.

4.3 S, D, and their characteristic dissonances

Before examining the question of how chromatic chords can be explained with the

theory of behavior, we must first return to the voice-leading paradigms of s and d, and

determine the role of characteristic dissonances in behavior. Example 4.5 shows the

voice leading paradigms d-t and s-t with the characteristic dissonances added. In the d-t

model, 2ˆ may move by step in either direction (to 1ˆ or 3ˆ ), while 7ˆ moves only to 1ˆ and

4ˆ moves only to 3ˆ . Although 4ˆ appears to have two stepwise moves available to it,

Rameau explains that the minor dissonance is propelled downward by the static 5 th scale degree (i. e., the root). 63

t d t t s t

5ˆ 6ˆ (to 5ˆ ) 4ˆ (to 3ˆ ) 4ˆ (to 3ˆ or 5ˆ ) 2ˆ (to 1ˆ or 3ˆ ) 2ˆ (to 3ˆ ) 7ˆ (to 1ˆ ) 1ˆ

Example 4.5 Voice-leading paradigms for d and s including characteristic dissonances

63 Rameau, Treatise , 80.

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In the s-t model, 4ˆ may move up or down by step in either direction (to 3ˆ or 5ˆ ), while 6ˆ

moves only to 5ˆ and 2ˆ moves only to 3ˆ . The characteristic dissonance of the subdominant, the added sixth, is pushed upward by the static dominant. Thus, the move from 2ˆ to 1ˆ may only be accomplished in the s-t model through the subversion of the upward tendency of the dissonance by a superior musical force.

4.4 Root motion down by step

It is interesting to note that 2ˆ -1ˆ is a d-t move even though I-ii, for example, is

usually considered a t-s move. Regardless, when transferred to a local level, local 2ˆ moving to local 1ˆ as part of a move from a lower- to a higher-status harmony, must be

considered a d-t microbehavior. If this motion occurs between roots, according to the

rule presented earlier in this chapter—in cases where both dominant and subdominant

microbehaviors are present in a single harmony, the behavior of the harmony as a whole

will be consistent with the behavior of the root—the behavior of the complete harmonies

must also be considered to be dominant. Since root movement down by step to a higher-

status element may only be a d-t behavior, we can now add more diatonic progressions to

our list of behaviors from Table 2.3. Now ii-I and vi-V, as well as their chromatic

alterations will be understood to exhibit dominant behavior. 64

64 The vi-V progression again raises the notion of a possible s-d behavior. This was denied in Chapter 2 on the grounds that behavior, by definition, requires a difference in status. Since there is no consistent status distinction between s and d, a behavior paradigm involving the two would be vague. Behavior paradigms, therefore, must include a t, which signifies the higher-status element.

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4.5 The diminished seventh chord

Our brief excursion into roots and seventh chords/added-sixth chords has made it possible to now complete a discussion of the diminished seventh chord shown at the beginning of the chapter. Example 4.6 shows that 7ˆ has d microbehavior, 6ˆ has s microbehavior, and 2ˆ and 4ˆ may behave as either s or d. Since the harmony has multiple microbehaviors, the root is the determining factor for the behavior of the harmony as a whole. Since the root moves up by step to a higher-status harmony, the diminished seventh chord exhibits dominant behavior.

a. b. c.

t d t t s t t d t

Example 4.6 Voice-leading paradigms for d-t, s-t, and the diminished seventh chord

4.6 Local behavior versus kinship

In Chapter 1 a summary of Daniel Harrison’s version of function theory revealed an explanation of what I have termed kinship based on scale degrees. Different scale degrees “function” as base, agent, or associate of one of the primary triads, regardless of where they fall in the harmonic progression. Harrison’s explanation for the vii o, then, is

similar to mine (Example 4.6). For Harrison, 2ˆ and 7ˆ are dominant (have dominant kinship) whereas 4ˆ and 6ˆ are subdominant (have subdominant kinship); in my

explanation, 2ˆ and 7ˆ are dominant (have dominant behavior) whereas 4ˆ and 6ˆ are

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subdominant (have subdominant behavior). The major difference between the explanations, however, is that when the function is transferred to the dominant, my model based on behavior remains the same, whereas Harrison’s model based on kinship is completely changed. Now, for Harrison, 4ˆ and 6ˆ are subdominant (have subdominant kinship) and 1ˆ and 3ˆ are tonic (have tonic kinship); in my model, V is now treated as local tonic, and # 4ˆ becomes the new (local) 7ˆ , etc. It is where # 4ˆ goes (to 5ˆ , local 7ˆ -1ˆ ) that makes it dominant, not what it is ( # 4ˆ ) that makes it subdominant. From this point forward, it should be understood that all behavioral models must have a local tonic, and that scale degree microbehaviors, which are based on Example 4.5, are local phenomena.

When a local scale degree is different from its global scale degree, it will be notated with an L, as in L 7ˆ .

4.7 The augmented-sixth chord

The augmented-sixth chord is often explained as a linear chromatic chord or a voice-leading chord, and therefore a chord that is not a harmony. While it is true that the voice-leading aspects of the chord are significant with respect to its behavior, it is also true that the augmented-sixth chord is a harmony with functional value, that is, kinship and behavior.

Example 4.7 shows an augmented-sixth chord in its Classical resolution (to V).

The behavior of each voice is diagrammed to the right. The behavior of each of the chord members is as follows: F# is L 7ˆ (to L 1ˆ ) and behaves solely as d, A ♭ is L 2ˆ moving to

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L1ˆ and therefore behaves solely as d, C is L 4ˆ moving to L 3ˆ and may behave as s or d,

D is L 5ˆ and behaves solely as d, E ♭ is L 6ˆ (to L 5ˆ ) and behaves solely as s. Here, then,

is a chord whose members have both s and d behavior. The behavior of the entire chord

is thus based on the movement of the root. But what is the root? If it is A ♭, then the chord is an added-sixth chord, and we would expect a root move down a fourth with s behavior. The A ♭ moves down a step, however, creating the dominant move from 2ˆ to

1ˆ . F# is the logical root choice, then, in the case of the Italian and German versions, since it results in a triad and seventh chord respectively. In the seventh-chord model, we expect d behavior with the root moving up a fourth or up a step. The F# does move up a step and the 7ˆ -1ˆ move has already been classified as d behavior. In the French version,

D is the root, since the seventh chord may be built on it, and its root movement (D-G) has d behavior. In all three versions of the augmented-sixth chord, then, the root moves up by step or up by fourth. The augmented-sixth chord in its Classical resolution thus follows the d-t paradigm. 65

65 The legitimacy of behavioral roots with repect to augmented-sixth chords may be found in the notation systems and/or writings of several prominent theorists. Notable among modern theorists is Piston [Walter Piston, Harmony (New York: W. W. Norton & Company, Inc., 1941), 279], but traces of the concept may be seen in Fétis and Vogler, and even as far back as Béthizy and Rameau. For more on the lineage of behavioral roots in theorists from Rameau to Riemann, including pertinent bibliographic information, see Chapter 6.

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a. Italian b. French c. German

L 7ˆ -L1ˆ (d-t) L5ˆ -L5ˆ (d-t) L 6ˆ -L5ˆ (s-t) L 4ˆ -L3ˆ (d-t or s-t) L 2ˆ -L1ˆ (d-t) Root: F#, L 7ˆ -L1ˆ (d-t) Root: D, L5ˆ -L1ˆ (d-t) Root: F#, L 7ˆ -L1ˆ (d-t)

Example 4.7 A 6 – V progression

4.8 Romantic resolutions of the augmented-sixth chord

When the augmented-sixth chord resolves to V, as shown above, it follows the d-t behavioral model, but also the S-D provincial model. When the augmented-sixth chord resolves directly to I, as it often does in the Romantic period, it follows different models.

Example 4.8 shows the Gr 6/I resolving to I. Here the active ingredients of the

augmented-sixth chord are ra and ti instead of le and fi as was the case in the Classical

resolution. For all intents and purposes, however, the behavior of the chord is exactly the

same as the Classical resolution. B is the root of the chord, and the d-t behavioral model

still holds. All of the notes and voice leadings in this model are identical to the last,

which is why the behavior is the same. What has changed is that the provincial model

has shifted so that the object chord falls in the tonic province where it had previously

fallen in the dominant province. The augmented-sixth chord thus shifts into the dominant

province.

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Beh. d t Prov. D T

Example 4.8 Gr 6/I – I progression

Example 4.9 shows another Romantic resolution of the augmented-sixth chord.

In this model, the original augmented-sixth chord (the one with le and fi as active ingredients) resolves to I. Le and fi still resolve to sol, as in the Classical model, but the other chord members resolve to do and mi instead of ti and re . In this model, F# which is

now L 4ˆ (and 4ˆ ) , moves up to 5ˆ , a move that solely belongs to s behavior. A ♭ is 6ˆ (to

5ˆ ) which belongs solely to s. D is 2ˆ moving to 3ˆ which may be s or d. In this case, F# is again the root. The root movement is down by fourth (F# to C), which is typical of s behavior. The D# now behaves as the characteristic dissonance (the added sixth) and moves up to mi. If a German sixth (with E ♭ instead of D#) resolves in this way, I submit that the E ♭ is really an enharmonic D#, since the dissonance in this resolution must be the upward pointing 6 th and not the downward pointing 7 th . In other words, German augmented-sixth chords may only resolve to V, whereas Swiss augmented-sixth chords may only resolve to I; any spelling to the contrary is a misspelling (unless mixture is invoked as an alternative explanation).

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L 6ˆ -L5ˆ (s-t) L 4ˆ -L3ˆ (d-t or s-t) L 2ˆ -L3ˆ (s-t) L 1ˆ -L1ˆ (s-t) Root: F#, L 4ˆ -L1ˆ (s-t)

Example 4.9 CT +6 – I progression

4.9 The common-tone diminished-seventh chord

Example 4.10 shows the common-tone version of the diminished-seventh chord in third inversion. This Romantic resolution of the diminished-seventh chord follows a different behavioral pattern from that of the Classical resolution. C is L 1ˆ (and 1ˆ ) and

belongs solely to s, D# is 2ˆ moving to 3ˆ and may belong to either s or d, F# is 4ˆ moving to 5ˆ and belongs solely to s, and A is 6ˆ (to 5ˆ ) and belongs solely to s. This harmony is

o 4 often identified as CT 2 which implies that the root is D#. However, a root of D# means the root movement is down a step, a dominant behavior. Furthermore, using 2ˆ as the

root goes against the subdominant nature of this chord. If accidentals are ignored, the

chord in question is Rameau’s chord of double emploi . If the chord moves on to V, then the root changes to 2ˆ , but the move from I requires that the root be 4ˆ . Since in this case,

the resolution is back to 1ˆ , the root did not change to 2ˆ . When the accidentals are

reattached, nothing changes about the behavior of the chord. It is an added-sixth chord

built on 4ˆ . It just so happens that 4ˆ is chromatically raised, as is 2ˆ . Therefore, this is

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really not a diminished-seventh chord at all, but rather, a diminished added-sixth chord!

F# is the root; the root moves down a fourth, and the harmony follows the s-t model of behavior.

L 6ˆ -L5ˆ (s-t) L 4ˆ -L3ˆ (d-t or s-t) L 2ˆ -L3ˆ (s-t) L1ˆ -L1ˆ (s-t) Root: F#, L 4ˆ -L1ˆ (s-t)

o 4 Example 4.10 CT 2 redefined as a common-tone added-sixth chord

4.10 Determining whether or not chords may be explained with behavior

There are four functional, that is, behavioral root movements: up by fourth, down by fourth, up by step, and down by step. The two types of root movement that are not behavioral are: up and down by third. For a chord (that is, a simultaneity) to be

explained with the theory of behavior, it must: (1) be a triad, seventh chord, or added-

sixth chord (in other words, it must be a harmony), and (2) exhibit root movements by

fourth or by step. It follows that chords that cannot be explained by the theory of behavior are either not harmonies, or have roots that move by third.

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4.11 The real linear chords

Augmented-sixth and diminished-seventh chords in Classical or Romantic resolutions are often referred to as linear or voice-leading chords, thus diminishing their harmonic implications. It is true that they are linear, but they are also functional harmonies, whether seventh chords or added-sixth chords, exhibiting subdominant or dominant behavior. There are chords, however, that do not act according to the theory of behavior. They are either not harmonies, or their roots move by third. These are the real linear chords.

The first type of linear chord results from a chord that is not a harmony, that is, not a triad, seventh chord, or added-sixth chord. In some cases, chords can feign a harmonic structure (through notation) without being a harmony. Kevin Swinden has found one such example in a motive by Wagner. 66 Example 4.11a shows the

"Tarnhelm" motive from Der Ring des Nibelungen . The motive is composed of alternating G#-minor and E-minor harmonies. Swinden asserts that the E-minor triad is not really a triad at all, but is rather a pair of chromatic neighbors to the root and fifth of the higher-status G#-minor triad. Example 4.11b shows this spelling in a voice-leading reduction. The so-called E-minor triad is not a harmony in this interpretation, but rather a fourths chord. The upper neighbor to D#, E, exhibits subdominant microbehavior, as the lower neighbor to G#, F X, exhibits dominant microbehavior. Regardless of Wagner’s intentions, spelling the chord as an E-minor triad under this interpretation would simply be a notational choice for easy part-reading.

66 Kevin J. Swinden, “When Functions Collide: Aspects of Plural Function in Chromatic Music,” Music Theory Spectrum 27, no. 2 (Fall 2005): 249-82.

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Nevertheless, the chord is spelled E minor, and the fact that the chord is supported by an E in the bass gives support to the reading of the chord as an E-minor triad.

Conversely, the motion of the progression—the alternation of harmony and neighbor notes rather than direction toward a goal—gives support to the reading of the chord as a linear chord. Either way, the chord does not exhibit behavior. Either it is not a harmony

(as in Swinden's reading), or it is a triad whose root moves by third.

a. "Tarnhelm" motive.

b. Alternate spelling of "Tarnhelm" motive with conflicting microbehaviors.

Example 4.11 Richard Wagner, Der Ring des Nibelungen

The second type of linear chord results from a harmony that does not exhibit behavior, that is, that does not feature root moves by fourth or by step. The degree to which a chromatic mediant like the one above is understood as a legitimate triad or as a fourths chord with parsimonious voice leading is surely partially dependent on the chord's context in its progression, and more specifically, the bass line. Example 4.12a

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shows a chromatic mediant relationship like the one in Example 4.11b, except with a major triad as the higher-status element. The A ♭ is exhibiting subdominant microbehavior as the B is exhibiting dominant microbehavior, in much the same way as the diminished seventh chord. When this voice-leading move is situated within the context of Chopin's Nocturne in C Minor , Op. 48 (Example 4.12b), however, the chord is more likely to be understood as an E-major triad. In this excerpt, as opposed to the

"Tarnhelm" motive, the chord is situated within a province-guided progression. The bass moves from C to E through F to G. Thus, the E-major harmony is filling the space between tonic and dominant rather than merely decorating tonic as in Example 4.11.

Regardless, C major and E major do not generate a behavior paradigm, since there is root movement by third. Thus, E major is still understood as a linear chord. 67 But, as opposed to the decorative fourths chord in the Wagner excerpt, this non-behavioral chord is at least a harmony.

67 E major does move by step to F 6, and therefore participates in a behavior paradigm with it.

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a. Chromatic mediant spelled as fourths chord with conflicting microbehaviors.

b. Chopin, Nocturne in C minor, Op. 48.

Example 4.12 Chromatic mediant as triad or fourths chord

Another example of a harmony that does not exhibit behavior is the common-tone

diminished seventh chord in root position, shown in Example 4.13a as an expansion of a

G-major triad (G#-B-D-F). In this chord, the third and fifth are the common tones. Since

the root bifurcates into upper- and lower-neighbor motion, a more accurate spelling of the

chord might be A ♭-B-D-F, shown in Example 4.13b. This incidentally creates a B o7 (or

Do add6 ) instead of G# o7 ; however, the chord behaves neither of these ways. If the root is

B, it moves down a third; if it is D, it moves up a fourth. The harmony is therefore linear since it does not exhibit behavior.

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a. No root move b. Root move by third

Example 4.13 CT o7 redefined as a harmony that does not exhibit behavior

4.12 Summary

Chapter 4 has shown how the theory of behavior can generate heuristics for

distinguishing between harmonies and non-harmonies, distinguishing between seventh

chords and added-sixth chords, and determining the behavior of harmonies that exhibit

multiple microbehaviors. First, any chord is considered a harmony if it has a behavioral

root. Second, seventh chords may be distinguished from added-sixth chords by the

behavior of the root and dissonance. Third, if a harmony exhibits multiple

microbehaviors; as is the case with such chromatic chords as the diminished seventh

chord, added-sixth chord, common-tone diminished seventh chord, and common-tone

augmented-sixth chord; the behavior of the harmony as a whole will be consistent with

that of the behavioral root. Fourth, truly linear chords are those chords that either do not

form harmonies (i. e., triads, seventh chords, added-sixth chords) or do not exhibit

behavior (i. e., roots move by third).

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CHAPTER 5

BEHAVIOR IN CHROMATIC MUSIC, JAZZ, AND POPULAR MUSIC

The theory of behavior is not style-dependent. Although the concepts and examples presented so far have emphasized music from the common-practice period, the theory of behavior is an abstraction that applies to any music that features certain combinations of root movements and status distinctions. Thus, behavior does carry explanatory power for some chromatic music, as well as jazz and popular music.

5.1 Behavior in chromatic music

Criteria for the application of behavior are less constrained than the criteria for the application of province. Consequently, the explanatory power of behavior extends to more music than that of province. This includes some highly chromatic music for which

TSDT schemes have no bearing, yet root movements by step and by fourth still flourish.

Thus, behavior is the aspect of function that provides the “functional” link between highly chromatic music and the music of the Baroque and Classical periods.

Example 5.1, taken from the beginning of Schoenberg’s "Schenk Mir Deinen

Goldenen Kamm," Op. 2., no. 2, exemplifies music for which province fails but behavior remains valid.

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a. Score

b. Metric Reduction showing voice leading, root movement, and status distinctions. 1 2 3 4 3 2 1 1 2 3 2 1 1 2 1 (3) 2

Example 5.l Schoenberg, "Schenk Mir Deinen Goldenen Kamm," Op. 2, no. 2, mm. 1-6

Continued

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Example 5.1 (Continued)

c. Metric Reduction: close and root position; without key, clef, and accidentals.

Beh. t d d t d t d t d t t s

Example 5.1b shows a metric reduction that features behavioral roots in the bass clef,

voice leading in the treble clef, and status distinctions above the harmonies. The first

problem that becomes apparent in this example is: how are status distinctions to be

determined in music that does not feature provinces? In order to answer this question it

would be beneficial to first codify the intuitive preference rules for status distinction in

music with provinces, and then to create preference rules for status distinction in music

without provinces. 68

68 The term “preference rules” is taken from Fred Lerdahl and Ray Jackendoff. The authors explain that preference rules “establish [flexible] decisions about structure, but relative preferences among a number of logically possible analyses.” They further admonish that “the term preference rule should not mislead the reader into thinking that preference rules model conscious preferences. Much of musical understanding is unconscious and hence seems automatic. A ‘preferred’ analysis represents how a musical passage is coherent to a listener.” Fred Lerdahl and Ray Jackendoff, A Generative Theory of Tonal Music : (Cambridge: The MIT Press, 1983), 42, 336. Whereas preference rules are given for grouping, meter, time-spans, and prolongational reductions in Lerdahl and Jackendoff, the preference rules given in this chapter concern status distinctions among harmonies in various musical styles.

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Preference Rules for Status distinction in music featuring provinces

1. Where a set of status distinctions is contained within a province (e. g., six-four paradigms), attribute higher status to the harmony of the province. 2. Attribute higher status to harmonies in progress toward a cadence (i. e., S-D-T = 3-2-1). 3. Attribute lower status to harmonies progressing away from a phrase beginning (e. g., T-S = 1-2). 4. Attribute higher status to tonicized harmonies over tonicizing harmonies.

Even for music that does not feature provinces, determination of status distinctions remains fairly intuitive, although often more difficult. The following preference rules are offered as ways to supplement musical intuition regarding this task.

Preference rules for determining status distinctions in music not featuring provinces.

1. Attribute higher status to harmonies in progress toward a cadence or phrase ending, in the same spirit as the S-D-T province phrase ending (3-2-1). 2. Attribute lower status to harmonies progressing away from a phrase beginning, in the same spirit as the T-S province phrase beginning (1- 2). 3. Attribute higher status to harmonies with extreme registers.

With these preference rules, the decisions about status distinctions made in Example 5.1b become clearer. In m.1, increasingly lower status is attributed to harmonies progressing away from the beginning of the phrase (1-2-3). In mm. 2-3, increasingly higher status is attributed to harmonies approaching a cadence, as well as moving toward the local

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extreme of the register (F2 in m. 3). Likewise in mm. 5-6, increasingly higher status is attributed to harmonies as they approach the cadence.

Even with these preference rules, however, some decisions concerning status distinctions remain difficult to make. For example, the move in mm. 4-5 from G ♭7 to A 7

is ambiguous in terms of status. Some may hear the G ♭7 as a sort of ornamentation of

B♭min7 on the way to A 7, and therefore attribute higher status to A 7. Others may hear the

G♭7 as enharmonically equivalent to tonic, thus encapsulating the cadence F#-A-C#-F#,

and therefore be inclined to attribute lower status to the A 7. These fringe arguments about the gray areas of analysis are of little significance here. Rather, it is significant to focus on areas where there are clear status distinctions, as well as clear behavioral root movements, and therefore, clear examples of behavior in highly chromatic music.

Example 5.1c translates the combination of status distinction and root movement shown in Example 5.1b into behavior analysis. It is important to note that whereas clef, key, and accidentals are necessary to determine status distinctions, once those distinctions are made they can be translated into behavior without clef, key, and accidentals.

Example 5.1c shows the harmonies in root position so the behavior can be easily visualized. In m. 1, higher status was given to F# minor (in Ex. 5.1b), and since its root moves down by fourth, it behaves as t-d. Measures 2-3 feature increasingly higher status accompanied by a series of root moves up a fourth, therefore each pair of harmonies exhibits d-t behavior. Similarly, the cadence in mm. 5-6 features a root move up a fourth from lower to higher status, and is therefore a d-t behavior.

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Several harmonic moves in the example that do not exhibit behavior are worth

noting. The first is the move from C# o7 to C minor in m. 1. Example 5.1b shows that

there is a status distinction between the two, however, there is no stepwise root

movement, only chromatic root movement. Since only one of the two requirements for

behavior is met (status distinction, but not a root movement by step or by fourth), the

harmonies do not exhibit behavior. A similar situation occurs in m. 3, where there occur

two adjacent harmonies with root C followed by two adjacent harmonies with root F.

Although there is a status distinction between each of the chords, there is no root

movement, and thus no behavior. The final example, the move from B ♭min7 to G ♭7 in m.

4, is only different from the others in that the root moves by third. As established in

Chapter 4, this is not a behavioral move, even though there is a status distinction between the chords.

This behavior analysis of the opening of “Schenk Mir Deinen Goldenen Kamm” is not meant to deny the preponderance of so-called “linearity” in the music. It is not even meant to somehow assert that behavior is a more powerful explanation for this music than, say, parsimonious voice leading, or tonnetz , or quasi-Schenkerian graphs. It

is, rather, meant to demonstrate two things about some examples from highly chromatic

tonality: (1) Linear analysis is not the only way to approach this music; behavior can

offer additional explanatory information about harmonies and harmonic movement, and

(2) This particular aspect of harmony and harmonic movement (behavior) remains

unchanged from Classical diatonic tonality to highly chromatic tonality and beyond; this

fact establishes a unique link across style boundaries with regard to “function.”

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5.2 Behavior in jazz

The propensity of jazz to feature circle-of-fifths progressions makes it simultaneously as conducive to behavior as it is non-conducive to province. Much like the sequence in Classical music, where the rule of phrase structure guided by province breaks down, perpetual modulation by circle-of-fifths progression in jazz cannot be explained with province. Instead, behavior explains every move in a progression such as the one from “Autumn Leaves” shown in Example 5.2.

d t t s d t d t d t d t

Example 5.2 “Autumn Leaves,” mm. 1-8. Words by Johnny Mercer, Music by Joseph Kosma

All root moves are up a fourth, and therefore behavior is either d-t or t-s depending on which element has the higher status. This status is determined by a variety of context clues including chord qualities and hypermetric placement. For example, the first four chords are all diatonic in the key of B ♭; therefore, B ♭Maj7 has the highest status

during the first four bars. Since E ♭Maj7 has lower status, the move from B ♭Maj7 to E ♭Maj7

is t-s and not d-t.

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Quality is an essential component of the circle-of-fifths aspect of jazz harmony.

With the exception of dominant quality, which may be substituted for any harmony in a circle-of-fifths progression, quality helps determine local scale degrees. Example 5.3 shows the diatonic seventh chords (with some leading tones) in the major and minor systems. The example illustrates that in the major system, major seventh chords are native to tonic (and also subdominant), minor seventh chords are native to supertonic

(and also mediant and submediant), and dominant quality seventh chords are native to dominant. Thus, when a jazz musician sees a set of changes Cm 7-F7-B♭Maj7 , it is assumed that the local key is B ♭, and by implication, Cm 7 is a ii 7 chord, and F 7 is a V 7

chord. This so-called “ii-V-I” does not lose its identity even if the global key were, say,

G minor. The ii o7 -V7-i in minor at the end of the phrase in Example 5.2 is characterized by the half-diminished seventh chord, followed by the dominant-quality seventh chord

(with possible alterations) and the minor seventh chord (with minor or major seventh).

MM mm mm MM Mm mm dd

I ii iii IV V vi vii o

mm (mM) dm MM mm Mm MM dd

i ii o III iv V VI vii o

Example 5.3 Diatonic seventh chords in the major and minor systems

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In language of jazz, therefore, quality influences preference rules for determining status distinctions in a way that it does not for Classical diatonic tonality.

Preference rules for status distinctions in jazz given root movement up a fourth.

1. Attribute higher status to any harmony following a dominant-quality seventh chord (local V-I). 2. Attribute higher status to a dominant-quality seventh chord following a minor or half-diminished seventh chord (local ii-V). 3. Attribute higher status to a major seventh chord followed by another major seventh chord (local I-IV).

5.2.1 The subdominant seventh chord

It was suggested in Chapter 4 that behavior should be the test to determine whether a four-note structure is a seventh chord or an added-sixth chord. Sometimes, however, this test contradicts well established notions of harmony. In the blues form, for example, the move from the tonic major-minor seventh chord to the subdominant major- minor seventh chord and back is a clear move from higher status to lower and back to higher. This status move is accompanied by a root move up and back down a fourth.

Thus, subdominant behavior is strongly suggested. However, subdominant behavior implies that the dissonance be an added-sixth, and that it point upward. The dissonance in this case does resolve up, but it is spelled as a seventh.

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Example 5.4 Blues form in C

L3ˆ - L 2ˆ - L 3ˆ (t-s-t) L1ˆ -L1ˆ -L1ˆ (t-s-t) L 7ˆ - L 6ˆ -L 7ˆ L5ˆ - L 4ˆ -L5ˆ Root: L 1ˆ -L 4ˆ - L 1ˆ (t-s-t)

Example 5.5 Behavior redefines F 7 as F add#6

If the dissonance is determined to be an added-sixth, then the behavior of the F 7 chord is the same as that of the common-tone diminished seventh chord in Example 4.9.

Example 5.6 shows the two paradigms for side-by-side comparison. Two differences are evident: the root of the CT o7 has an accidental, and the third of the F 7 chord resolves up to the seventh (B ♭) in addition to the fifth.

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C: CT o7 I

Example 5.6 CT o7 -I progression and F 7-C7 progression

A decision must be made: should the structure be labeled an added-sixth chord,

thereby reaffirming behavior as the test for distinguishing seventh chords from added-

sixth chords, or should it be labeled a seventh chord, thus repudiating the behavior test?

The answer lies not in the harmony itself, but in what scale potentially accompanies the

harmony. Chord-scale relationships are a central component of jazz improvisation and

jazz pedagogy. The concept implies that a given chord suggests at least one harmonically

compatible scale. Example 5.7 shows the scales associated with the chords in a basic C

blues. Where dominant-quality seventh chords occur, the mixolydian scale may be

employed. This scale is simply a consequence of filling the gaps of the dominant-quality

seventh chords with diatonic passing tones. When the scale is completed in this way, the

dissonance in the harmony must be the seventh since the diatonic passing tone fills the

space of the sixth. This suggests that the chord in question is a seventh chord, and

therefore, that behavior fails.

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Example 5.7 C Blues changes with characteristic scales

The example above indicates that, although the behavior test for distinguishing

between seventh chords and added-sixth chords is beneficial when dealing with pairs of

harmonies, scalar representations of those harmonies may also clarify the difference by

indicating which gap is filled by passing motion. Thus, a second test should be

considered. 69

The scale degree test for determining whether a dissonance is a seventh or added-sixth:

Given a four-note structure with known root, the dissonance is a seventh if a passing note occupies a scale degree adjacent to the dissonance and the fifth; the dissonance is an added-sixth if a passing note occupies a scale degree adjacent to the dissonance and the root.

69 The scale degree test has its foundation in Proctor’s discussion of the opening of Schubert’s String Quintet , Op. 163, where he supports the conclusion that the common-tone diminished seventh chord is spelled with a (mixture-based) E-flat rather than a (tonicization-based) D-sharp by pointing to the “composing out of the minor third [C to D to E-flat] in the first violin.” Proctor, “Technical Bases,” 95.

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If the scale degree test contradicts the behavior test, then the dissonance does not act according to the definition of behavior. However, if the root movement and status distinctions are compatible with the definition of behavior, it seems reasonable to allow that the harmony still exhibits behavior. Whereas the behavior test is also a way of determining the root of a four-note structure, and is therefore a more general test, the scale degree test requires a known root, and is therefore essentially a test of enharmonicism and correct spelling. 70

5.2.2 Behavior and the tritone substitution

The concept behind the tritone substitution is that a circle-of-fifths progression may be varied by substituting descending half-step root motion for descending fifth root motion. Thus, the root of the substituted harmony lays a tritone away from the original harmony. Example 5.8 shows a circle-of-fifths progression varied to include a tritone substitution for ii 7.

70 This discussion of whether the dissonance is an added-sixth, and therefore a voice-leading move , or a seventh, and therefore a chromatic slide is reminiscent of the discussion of chromatic mediants in Chapter 4. In that instance, it was determined that a chromatic mediant may be understood as two stepwise voice-leading moves , which generate a fourths chord, as opposed to one stepwise move and one chromatic slide , which generates the triad. This distinction between moves and slides is addressed in various ways in the so-called Neo-Riemannian literature. A few germane articles are listed below. I simply add the scale degree test here as an alternative to several solutions to the problem of move versus slide . Richard Bass, “Enharmonic Position Finding and the Resolution of Seventh Chords in Chromatic Music,” Music Theory Spectrum 29, no. 1 (Spring 2007): 73-100; Adrian P. Childs, “Moving Beyond Neo-Riemannian Triads: Exploring a Transformational Model for Seventh Chords,” Journal of Music Theory 42, no. 2 (Fall 1998), 181-93; Kevin Swinden, “When Functions Collide: Aspects of Plural Function in Chromatic Music,” Music Theory Spectrum 27, no. 2 (Fall 2005): 249-82.

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Example 5.8 ♭VI as tritone substitution for ii

Behavior analysis reveals much about these two progressions. First, A ♭7 is a seventh chord with root A ♭. This is so because the harmony passes both the behavior

test and scale degree test for determining whether a dissonance is a seventh or added

sixth. According to the behavior test, it is the lower-status element of a voice-leading

paradigm, its root moves down by step (d-t) and its seventh resolves down by step (d-t).

The scale degree test confirms this, since the accompanying scale, A ♭ mixolydian,

reveals G ♭ as a harmonic tone (seventh) and F as a passing tone (sixth).

Second, A ♭7 exhibits the same behavior as the harmony it replaces, Dmin 7. Both harmonies behave as dominants to G 7, the former with its root move down by step and

seventh move down by step, and the latter with its root move up a fourth and seventh

move down a step.

Third, the jazz spelling and resolution of A ♭7 in this progression is different from its enharmonic Classical counterpart, the German augmented-sixth chord. Whereas Gr 6

is a seventh chord with root F#, A ♭7 is a seventh chord with root A ♭. Both are spelled

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correctly. The seventh of A ♭7 pulls down (to the seventh of G 7!), whereas the root and

local leading tone of Gr 6 pushes up. Example 5.9 shows this distinction in voice

leading. 71

Fourth, if the harmony is spelled as A ♭7, and not G# 7, then there exists no behavior between the Am 7 and the A ♭7. Although A ♭7 substitutes for D, which would be

the object of dominant behavior from Am 7, A ♭7 is nominally the same root as Am 7, and therefore no behavioral root move exists even though there is a status distinction. This is similar to one of the problems noted in m. 2 of the analysis of “Schenk Mir Deinen

Goldenen Kamm,” where between two types of C harmonies (C# Ø7 and C min) there was a status distinction but no root move, and therefore, no behavior. It seems odd that there should be no behavior between the two harmonies, particularly considering the status distinction. Perhaps a way around this problem is to adapt Rameau’s concept of double employment to render a sort of double employment of the enharmonic spelling. That is, as the tritone substitution is approached from Am 7, it is a G# 7; as it resolves to G 7, it

inaudibly transforms into A ♭7. In this reading, Am7 exhibits dominant behavior toward

G# 7 just as A ♭7 exhibits dominant behavior toward G 7. It is lower in status, its root moves down by step, and its seventh resolves down by step.

71 The Gr 6 may, of course, also resolve to a dominant seventh chord. As described in Chapter 1, sevenths point down and sixths point up, though they do not always resolve to where they point. In this case, the scale degree test again helps illuminate the distinction between A ♭7 and Gr 6.

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Example 5.9 Voice-leading comparison of A ♭7-G7 and Gr 6-V in C

Sometimes the tritone substitution is not actually a substitute for, but is rather a

decoration of the original harmony. In Example 5.10 the dominant harmony is decorated

by its tritone partner. The two harmonies share enharmonically equivalent thirds and

sevenths, and thus both have dominant kinship. Either harmony would be considered to

exhibit dominant behavior with respect to C Maj7 ; both have lower status and sevenths that resolve down by step, the root of G 7 moves up by fourth, and the root of D ♭7 moves

down by step. There is no behavior between G 7 and D ♭7, however. Although there is a

root move by fourth, there is no status distinction. Thus, the opposite problem arises

from that of the Am 7-A♭7 move in Example 5.9. Instead of a paradigm with a status

distinction but no root move, there is now a root move with no status distinction. Both

must occur to denote behavior.

Example 5.10 ♭II as decoration of V

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5.3 Province, behavior, and harmonic structures in popular music

One of the primary differences between popular-music harmony and common- practice harmony is that the former need not be guided by provinces. That is, harmonic progressions are not required to follow the TSDT scheme, nor is the dominant nor subdominant guaranteed higher status than so-called secondary harmonies. Although some popular music lacks any association with provinces, many songs (or portions of songs) do feature provinces. Example 5.11 shows a lead sheet of a portion of “Message in a Bottle” by The Police. Whereas the harmonic progression of the verse and parts of the refrain (those sections marked A, C, and D) does not feature provinces, the beginning of the refrain (marked B) does.

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Example 5.11 "Message in a Bottle," Words and Music by Sting

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The verse is constructed around a four-chord repeating progression: i-VI-VII-iv.

There is no dominant harmony at all, but rather a sort of plagal cadence upon each restatement. The refrain begins with the most prototypical of province progressions: I-

IV-V-I in A major. This is quickly replaced by an alternation between F# minor and D major, a progression in no way associated with province (or behavior). The final phrases of the refrain also follow the model of alternation of third-related harmonies: C# minor and A major. Thus, in a single composition, portions of music may be guided by province while others are not. As we have seen in jazz and chromatic music, whether or not province is featured in a composition, behavior may nevertheless be exhibited where there is root movement by fourth or by step.

One way behavior is illuminated in popular music (as well as jazz) is with the so- called guide tone . The guide tone is a line of pitches in stepwise motion, often generated by alternating thirds and sevenths among fifth-related harmonies. More simply, a guide tone is a voice-leading line. Not only is the resulting line a string of good voice leadings, it often provides a countermelody that can serve as a hook in popular music. For example, in the introduction to “Saturday in the Park” by Chicago (Example 5.12), the guide-tone melody in the horn section serves as the focal point of the introduction.

Although this excerpt does not feature provinces, it does feature behavior exclusively, with three consecutive d-t moves. The horn line makes the dominant-behavior voice leading explicit by beginning on the seventh of the first harmony, the moving to the third, then the seventh, and finally resting on the third of the tonic.

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d t d t d t

Example 5.12 “Saturday in the Park,” Horn line in Introduction. Words and Music by Robert Lamm

So far this study has shown that popular music may or may not feature provinces or behavior. Even though there is some popular music without either, it is undeniable that popular-music listeners are intuitively familiar with both province and behavior.

That is, when provinces and behavior are found in popular music, it does not sound abnormal. Since provinces and behavior are commonplace, variation or abandonment of these features is common in the perpetual search for the “new sound” in popular music.

One change has already been shown: the abandonment of provinces and behavior in

“Message in a Bottle.” Another change in harmonic language is found in the harmonic structures themselves. That is, in popular music, the term harmony encompasses a wider

range of structures than just triad, seventh chord, and added-sixth chord.

One example of a popular-music harmony is the one featuring a second or added

ninth. This color tone may replace (as a second) or obscure (as an added ninth) the third.

The chord is most often built on the tonic or subdominant.

Another variation from tertian harmony in popular music is the sus4 chord. This

chord occurs most often on the dominant, and, unlike a 4-3 suspension, is not prepared

and does not resolve within the dominant province, but rather, is held as a common tone

with tonic in the successive tonic harmony.

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“Feelin’ the Same Way,” from Norah Jones’ album, Come Away With Me

(Example 5.13), features all three of these harmonies. The tonic B ♭(add9) concludes the verse and leads into the F sus4 , the dominant harmony that begins the refrain. Passing

motion in the bass generates the E ♭(add9) on the way to vi 7.

Example 5.13 “Feelin’ the Same Way,” Words and Music by Lee Alexander

In each harmony— B♭(add9) , E ♭(add9) , and F sus4 —it is Harrison’s “agent” that has been rejected in favor of a coloring dissonance. The listener still intuits the chords as primary harmonies, I, IV, and V, since the consonant version of these harmonies is still an established part of the harmonic language of popular music. The base and associate of each harmony are available for the listener to grab onto while processing these dissonant modifications. The sus2 (or add9) and sus4 replacements (or obscurings) for the agent generate new tonic, subdominant, and dominant harmonies that have kinship with

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themselves (i. e., they each share two scale degrees with their corresponding primary triad).

Another frequently used harmony in popular music is the subdominant triad over dominant bass note, which is frequently substituted for V at cadences. This harmony, like the sus4, is a type of suspension without preparation or resolution. Also like the sus4, this harmonic modification is a way of rejecting the now predictable and cliché agent.

“You’ve Got a Friend,” by Carole King, features both types of dominant sus chords. The sus4 (this time with added seventh) leads into the refrain; the subdominant triad over dominant bass, D ♭/E ♭, occurs at each of the cadences in the refrain.

Example 5.14 "You've Got a Friend," Words and Music by Carole King

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The subdominant triad over dominant bass note is an obvious case of conflicting microbehaviors. All of the members of the subdominant triad are present and behave as local subdominant; however, the bass, even as the lone representative of the dominant triad, seems to overpower the functional force of the subdominant triad. Its position in the bass, as well as its position in the progression, before tonic, seem to support the reading of this chord as a dominant-behaving chord.

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PART III

THE BIRTH OF FUNCTION:

HISTORICAL PRECURSORS TO

BEHAVIOR, KINSHIP, QUALITY, AND PROVINCE

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CHAPTER 6

THE LINEAGE OF

BEHAVIOR, KINSHIP, QUALITY, AND PROVINCE

FROM RAMEAU TO RIEMANN

If the birth of function is appropriately attributed to Riemann, then its gestation period reaches back at least to Rameau. The following survey traces the lineage of the four aspects of function through germane theorists from Rameau to Riemann.

6.1 Jean-Philippe Rameau

Rameau contributed so much to the concept of function that it is difficult at best

to sort out all of the connections. For example, none of the four aspects of function—

behavior, kinship, province, quality—are possible without such Ramellian concepts as

the fundamental bass and harmonic inversion. There are, however, some Ramellian ideas

with direct connection to function theory. The genesis of behavior is in Rameau’s understanding of the dominant as a voice-leading/root-moving relationship, and not necessarily as a harmony built on 5ˆ . His separation of this concept of dominante from

the dominante-tonique reveals a distinction in quality . His understanding of the leading-

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tone harmony (vii o) as derived from the dominant harmony points toward kinship . And his concept of double employment is at once a predecessor of both behavior and kinship .

The discussion of Rameau in Chapter 1 revealed that his understanding of tonic, dominant, and subdominant was more about relationships among structures (that is, added-sixth chords and seventh chords resolving to triads) than magnetic attraction toward a global tonic (i. e., province and/or kinship). His explanation of the irregular cadence as “from the tonic note to its dominant, or from the fourth note to the tonic” exemplifies his conception of cadence as a behavior (e. g., root motion down a fourth),

and not as identities. 72 Example 1.3 demonstrated that perfect cadences may occur on 5ˆ -

1ˆ , but also on 6ˆ - 2ˆ . Likewise, irregular cadences may be found on 4ˆ -1ˆ , but also on 2ˆ -

6ˆ . Again, it is the behavior (i. e., root motion) that determines these cadences, not their

identities. Dahlhaus succinctly describes the concept this way: “Rameau defines the dominante and sousdominante not primarily in terms of tonality, but in terms of compositional technique – as chordal types requiring a specific resolution of dissonance and a corresponding progression of the fundamental bass.” 73

Since dominant refers to a behavior rather than an identity for Rameau, he coins a

special term to specify the dominant before the tonic, dominante-tonique .74 This distinction is one of quality . The dominant-tonic is a dominant with a leading tone. In other words, it has both dominant behavior and dominant quality. The dominant-tonic

72 Rameau, Treatise , 73.

73 Carl Dalhaus, Studies on the Origin of Harmonic Tonality , trans. Robert Gjerdingen (Princeton: Princeton University Press, 1990), 26.

74 Rameau, Treatise , 83.

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may then also be used to describe secondary dominants, and distinguish them from diatonic dominant behaviors.

Rameau does not accept tonal roots, and therefore must find an alternative

explanation for the leading-tone harmony. His solution is that the leading-tone harmony

is really a dominant harmony without the root. 75 By deriving the leading-tone harmony

this way, Rameau implies kinship between V 7 and vii o. The leading-tone triad is a subset

of the dominant seventh chord, and therefore shares at least two scale degrees with it.

One of the crowning achievements of Rameau’s theory is his concept of the

double employment of the dissonance. This concept points toward behavior and kinship

at once. First, that the same chord (e. g., F-A-C-D in the key of C) may have two

different meanings is at the heart of Riemannian function theory. As F-A-C-D is

approached from the tonic C triad, it is acting as a IV add6 ; as F-A-C-D resolves to the

6 dominant G triad, the added-sixth chord is transformed into a seventh chord, ii 5 . Thus,

6 add6 ii 5 is akin to IV in that they share all four scale degrees. Second, the reason why this transformation is necessary is so that the approach and resolution of the chord (F-A-C-D) may both conform to the rules of behavior . As a IV add6 , the chord is approached from a

6 higher-status element with root motion up a fourth (t-s); as a ii 5 , the chord resolves to a

higher-status element with root motion up a fourth (d-t).

Rameau’s revolutionary theories, thus, help begin to define three of the four

aspects of function: behavior, quality, and kinship. And these concepts continue to be

75 Ibid., 48-52, 93-95.

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shaped until their explication in Riemann’s work, and beyond, beginning with one of

Rameau’s contemporaries, Jean-Laurent de Béthizy.

6.2 Jean-Laurent de Béthizy

As a contemporary of Rameau, Béthizy’s work is based largely on that of the master, but he also clarifies and supplements several of Rameau’s explanations.

Béthizy’s concepts of implied (or reputed) tonics and augmented-sixth chords add new dimensions to the concept of behavior . His understanding of the cadential six-four chord can be seen as a precursor to province .

Béthizy expands on Rameau’s concept of dominant as a local behavioral structure

(and not as a chord built on 5ˆ ) by suggesting the possibility of a local tonic. He writes:

One frequently encounters in the best pieces, passages where there is no key and yet are quite agreeable. The fundamental bass in these passages is at times a succession of seventh chords [ dominantes simples ], and at other times a succession of notes that I call reputed tonics [ censées toniques ] because they bear the perfect chord just as tonics do, without having the character of the latter. 76

76 Jean-Laurent de Béthizy, Exposition de la théorie et de la pratique de la musique, suivant les nouvelles decouvertes (Paris: Lambert, 1754), 140; translated in A. Louise Hall Earhart, “The Musical Theories of Jean-Laurent de Béthizy and their Relationship to those of Rameau and d’Alembert” (Ph.D. diss., The Ohio State University, 1985), 293. As shown in Béthizy’s corresponding musical example, these areas where there is no key may be understood to mean sequences. Lester translates censées toniques as “implied tonics;” Joel Lester, Compositional Theory in the Eighteenth Century (Cambridge: Harvard University Press, 1992), 207.

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Rameau had defined dominante as distinct from dominant-tonique , but had not made the same distinction between local and global tonics. This is the first time such a distinction has been made. Thus, Béthizy moves harmonic theory one step closer to the concept of behavior, and with it, voice-leading paradigms, for which (local) tonic simply means higher status.

Rameau fails to comment on augmented-sixth chords in his early writings (i. e., his work published before Béthizy’s treatise of 1754), thus creating one of the few opportunities for one of his contemporaries to take the lead in theorizing about these chords. Béthizy takes advantage of this opportunity, writing that “this extraordinary chord comes from no direct chord and has no fundamental bass,” but also that “if [for example, F-A-B-D#] had a fundamental bass, the direct chord would be B-D#-F-A, and the fundamental bass would be B.” 77 Although Béthizy dismisses the idea of behavioral

root, he nevertheless is inclined to arrange the augmented-sixth chord as a seventh chord

with a quasi-root that resolves up by fourth to a higher status harmony, V. Thus, he

posits a hypothetical dominant behavior for the French sixth. Rameau, on the other hand,

is bound by the notion that every harmony must have a perfect root. Earhart correctly

points out that since is the case, either the augmented-sixth chord has a root, or it is not a

harmony. 78 Remarkably, Rameau chooses the former! In his Code de musique pratique ,

published after Béthizy’s treatise, he shows B as the fundamental bass of the French sixth

in A minor. 79 With this, Rameau first accepts the legitimacy of behavioral roots.

77 Béthizy, 205; translated in Earhart, 404.

78 Earhart, 409.

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In addition to propelling the concept of behavior, Béthizy betrays a sympathetic

point of view toward province in his analysis of the cadential six-four chord. Of the

cadential six-four chord in C major, he writes, “The G fundamental [!] and dominante- tonique thus bear the notes C-E, a fourth and a sixth.” He also refers to the chord as a

“species of suspension.” 80 By understanding the cadential six-four and V as both having

5ˆ as fundamental, Béthizy is hinting at an expansion of the dominant province.

6.3 Johann Friedrich Daube

Daube’s system, perhaps more than any other, looks back toward Rameau and

forward toward Riemann. First, he posits three Hauptakkorde , the triad, the added-sixth chord (which is native to 4ˆ ), and the seventh chord (which is native to 5ˆ ). 81 Thus, tonic, subdominant, and dominant receive the highest status in the system, while “iii and vi are ignored as if they are not legitimate chords.” 82 The leading-tone triad is an incomplete

dominant seventh chord in Daube’s system, and therefore has dominant kinship. 83 The most interesting derivation of secondary harmonies, however, is the one on 2ˆ . “The

79 Jean-Philippe Rameau, Code de musique practique, ou Méthodes pour apprendre la musique … avec de nouvelles réflexions sur la principe sonore (Paris: L’Imprimerie Royale, 1760), 56; musical example p. 4.

80 Béthizy, 205-6; translated in Earhart, 394.

81 Johann Friedrich Daube, General-Bass in drey Accorden, gegründet in den Regeln der alt- und neuen Autoren (Leipzig, 1756), 14.

82 William Mickelsen, “Hugo Riemann’s History of Harmonic Theory with a Translation of Harmonielehre ” (Ph.D. diss., Indiana University, 1970), 85.

83 Ibid., 83; Daube’s discussion of dominant harmonies is found in Daube, 17-20.

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seventh chord on the second scale degree is not a fundamental chord at all, but an inversion of the chord on the fourth degree.” 84 With this statement, Daube is willing to take Rameau’s concept of double employment further than Rameau himself. In Nouveau système , Rameau only allows the added-sixth chord in root position and first inversion. 85

Daube now allows it in third inversion, identical in structure to a root position ii 7!

6.4 Johann Phillip Kirnberger

Kirnberger is perhaps best known for his distinguishing between essential and non-essential dissonances, but it is his distinction between consonant and dissonant six- four chords that hints at province. For Kirnberger, the cadential six-four chord is dissonant, a product of two suspended tones that must resolve to the chord tones of the dominant. It is, thus, an expansion of the dominant province. All other six-four chords (i. e., passing, neighbor, and arpeggiating) are consonant, products of inversion of stable harmonies. 86 There is no hint that these six-four chords, too, may be expansions of provinces.

Joyce Mekeel describes how the consonant and dissonant six-four chords are actually derivations of the essential dissonances (i. e., dissonant harmonic tones, such as

84 Daube, 17; translated in Mickelson, “Translation of Harmonielehre ” (Ph.D. diss.), 83.

85 Chandler, 351.

86 Johann Phillip Kirnberger, The Art of Strict Musical Composition , trans. David Beach (New Haven: Yale University Press, 1982), 71. A parallel discussion of six-four chords may be found in Johann Phillip Kirnberger, Grundsätze des Generalbasses als erste Linien zur Composition (Berlin, J. J. Hummel, 1781), 67-69.

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the chordal seventh) and unessential dissonances (i. e., dissonant non-harmonic tones, such as the suspension).

The consonant six-four chord, like any essential dissonance, can occur either on a weak or on a strong beat. It may be taken without preparation and any note may be doubled. … The dissonant six-four chord, like any unessential dissonance, can occur only on the strong beat – as a suspension or as an appoggiatura. It must be treated as a dissonance with both fourth and sixth prepared and resolved. In the former instance the six-four chord represents a triad in second inversion; in the latter it is a triad with the fifth and third delayed by suspension or appoggiatura. 87

6.5 Heinrich Christoph Koch

Koch follows Kirnberger in distinguishing between essential and non-essential dissonance, as well as consonant and dissonant six-four chords. He writes:

The [consonant] 6-4 chord must always be consonant because it arises from the inversion of a consonant chord. If on the other hand the sixth and fourth appear together in the harmony as suspended tones in such a way as to delay the fifth and third of the chord, they appear as dissonances and as such belong to a special dissonant chord in the same way as the fourth which singly delays the third. 88

He goes one step further, however, positing essential and non-essential triads.

The tonic, dominant, and subdominant triads are the essential triads, whereas the

87 Joyce Mekeel, “The Harmonic Theories of Kirnberger and Marpurg,” Journal of Music Theory 4, no. 2 (November 1960): 180.

88 Heinrich Christoph Koch, Handbuch bey dem Studium der Harmonie (Frankfurt, 1811), 71; translated in Mickelsen, “Translation of Harmonielehre ” (Ph.D. diss.), 366. The reader may also refer to Shirlaw, 330.

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supertonic, mediant, and submediant triads are unessential. The leading-tone triad, as in so many previous explanations, is a rootless dominant seventh chord. Thus, Koch codifies Rameau’s three most important harmonies as essential, while grouping and labeling as unessential those less important triads. This may also be understood as a precursor to kinship in that once the distinction is made between essential and unessential triads, the stage is set for unessential triads to be understood as transformations of essential triads. 89

6.6 Georg Joseph Vogler

Vogler is best known for his attribution of Roman numerals (I, II, III, IV, V, VI,

VII) to harmonies built on diatonic scale degrees. While this notation alone does not directly connect to any of the four aspects of function, the underlying principle of rigid adherence to chord building in thirds, and hence, a less strict definition of roots, does point toward behavior. Rameau’s fundamental bass is synonymous with what I have called “perfect root.” There is no “tonal root” (e. g., 7ˆ in vii o) for Rameau, since the

fundamental of the leading-tone harmony is the implied dominant. And although

Rameau seems to accept the behavioral root in the case of the French sixth, this choice

must be seen as the lesser of two evils in Rameau’s attempt to solve a contradiction in his

theory: that a harmony must have a fundamental (i. e., perfect root), and that the French

sixth should be understood as a harmony. Vogler’s concept of harmonic generation and

his notation of it renders Rameau’s concerns moot. Vogler unabashedly accepts both

89 Shirlaw, 330; Mickelsen, “Translation of Harmonielehre ” (Ph.D. diss.), 364.

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tonal and behavioral roots, in the process inching one step closer to theory of behavior and behavioral paradigms.

By notating the leading-tone triad/seventh chord as VII, Vogler rejects Rameau’s view that the harmony has an implied subposed root. He, therefore, accepts the leading tone, with its diminished fifth, as a legitimate (tonal) root. Although this downplays the kinship between the leading-tone harmony and the dominant harmony, it opens up new opportunities for behavior comparisons between the two. For example, if these two harmonies both tend to approach tonic at cadences, then the concept of d-t cadence may be expanded to include not only root motion up a fourth, but also root motion up a step.

In conjunction with the diminished triad built on the leading tone, Vogler also recognizes the diminished triad built on # 4ˆ . Although he labels the chord #IV, and not with the modern secondary “slash,” he nevertheless makes note of the dominant behavior of the harmony. As Grave and Grave write, “the progression of raised IV to V yields closure on the fifth scale degree just as VII-I produces a cadence on the tonic.” 90

Bernstein gives further insight to the dominant behavior Vogler is evoking by referring to

the #IV as “analogously cadential to the dominant as the leading-tone chord is to the

tonic.” 91

90 Floyd K. Grave and Margaret G. Grave, In Praise of Harmony: The Teachings of Abbé Georg Joseph Vogler (Lincoln: University of Nebraska Press, 1987), 24.

91 David W. Bernstein, “Nineteenth-century Harmonic Theory: The Austro- German Legacy,” In The Cambridge History of Western Music Theory , ed. Thomas Christensen (Cambridge: Cambridge University Press, 2002), 780. Vogler’s musical examples of these cadences may be found in Georg Joseph Vogler, Handbuch zur Harmonielehre und für den Generalbass, nach den Grundsätzen der Mannheimer Tonschule, zum Behuf der öffentlichen Vorlesungen im Orchestrations-Saale auf der k. k.

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In addition to legitimizing tonal roots in the form of leading-tone and secondary leading-tone triads, Vogler also legitimizes behavioral roots when he discusses augmented-sixth chords. He labels the chords with altered Roman numerals: #IV for It 6,

II for Fr 6 and #IV for Gr 6.92 Thus he views these as altered chords as presented in

Chapter 4. The It 6 is a triad whose (behavioral) root (# 4ˆ ) moves up by step to a higher-

status element (V); the Fr 6 is a seventh chord whose root ( 2ˆ ) moves up a fourth to a

higher-status element (V); the Gr 6 is a seventh chord whose root (# 4ˆ ) moves up a fourth

to a higher-status element (V). Not only does Vogler accept behavioral roots with this

explanation, he expands the dominant behavior paradigm to include root motions up by

step in addition to those up by fourth.

6.7 Jérôme-Joseph de Momigny

Momigny anticipates behavior, and with it, Proctor’s voice-leading paradigms,

when he presents his four rules of voice leading.

“Rule 1: The natural resolution of the notes of an antecedent chord should occur

on the notes of the consequent chord with which they make contact.” 93 Example 6.1

shows that this rule simply means that voices should move by step. Proctor’s voice-

Karl-Ferdinandeischen Universität zu Prag (With accompanying set of music examples.), (Prague, 1802), Table II, Figures 12-13.

92 Vogler, It 6-V labeled #IV-V in Table II, Figure 15; Fr 6-V labeled II-V in Table II, Figure 16; Gr 6-V labeled #IV-V in Table VII, Figure 5. 93 Jérôme-Joseph de Momigny, Cours complet d’harmonie et de composition (Paris, 1806), 58-59; translated in Glenn Gerald Caldwell, “Harmonic Tonality in the Theories of Jérôme-Joseph Momigny” (Ph.D. diss., The Ohio State University, 1993), 57.

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leading paradigms reflect this, only allowing stepwise motion (or common tones) between each of the three possible root movements (i. e., by step, by third, or by fourth).

Example 6.1 Momigny’s voice-leading rule 1. a

a Glenn Gerald Caldwell, “Harmonic Tonality in the Theories of Jérôme-Joseph Momigny,” Ph.D. diss., The Ohio State University, 1993, 57.

“Rule 2: When the note of the antecedent chord lies between two notes of the consequent, it preferably moves by a semitone interval because it is more attracted by the distance of a semitone than a whole-step. But if the antecedent note is a whole step from one or the other, the principle of variety or unity decides or gives one freedom of choice.” 94 Example 6.2 reveals that Rule 2 is a variant of Proctor’s voice-leading paradigms that show that voice-leading may take a neighbor-motion path or a passing- motion path.

94 Momigny, 58-59; Caldwell, 58.

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Example 6.2 Momigny’s voice-leading rule 2. b

b Ibid., 58.

“Rule 3: When two voices double the same pitch in the antecedent chord, either in unison or in an octave relationship, and this tone goes to a single consequent note, then only one voice resolves ….” Example 6.3 shows that this rule, to some extent, foreshadows Proctor’s notion that voice leading and root motion are separate ideas.95

95 Momigny, 58-59; Caldwell, 58.

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Example 6.3 Momigny’s voice-leading rule 3. c

c Ibid., 59.

“Rule 4: When a tone is common to both chords of a cadence, the voice sounding this tone has no movement to make ….” 96 Example 6.4 shows that Rule 4 echoes the spirit of Rule 3 (except the root move is by third rather than by fourth) that voice leading is an abstract notion of movement by step or common tones.

Example 6.4 Momigny’s voice-leading rule 4. d

d Ibid., 60.

According to Caldwell, “One of the most radical attributes of Momigny’s tonal

theory is the idea that a key contains chromatic and enharmonic components as naturally

96 Momigny, 58-59; Caldwell, 59.

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as it contains the seven diatonic scale degrees.” This is what Momigny calls the Grand

Musical System. 97 For example, F# may be a chromatic component of the C-major system instead of a temporary invocation of the diatonic G-major system. While this

Grand Musical System fails to be compatible with upward and downward tonicization as generators of chromatic notes, it does lead to a unique understanding of kinship. For example, if F# is native to C major as a chromatic alteration of F, rather than explained as the invoked leading tone of G (i. e., tonicization), then D-F-A-C and D-F#-A-C are akin to each other in a way that V 7/V and ii 7 are not. In other words, shared scale degrees, and

therefore kinship , are a product of Momigny’s system, while tonicization is not. 98

6.8 Gottfried Weber

Weber’s contribution to the concept of function is one of notation. He modified

Vogler’s Roman-numeral system to account for Koch’s essential and unessential triads by representing harmonies ii, iii, and vi in the lower case. Today this is seen as a designation of major versus minor triads, but Weber’s description indicates he is rather viewing the dichotomy as one of significance within the key, at least in the major mode. 99

97 Ibid., 140.

98 Ibid., 138.

99 In his presentation of the Roman numerals in the major mode, Weber calls the “essential” chords of the key the tonic triad, the dominant triad, the dominant seventh chord, and the subdominant triad. Later, however, in the presentation of Roman numerals in the minor mode, the i and iv chords are lower case, indicating that the notation does indeed favor the major/minor dichotomy over the essential/non-essential dichotomy, even though Weber’s description favors the latter. Gottfried Weber, The

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Thus, the exaltation of I, IV, and V above ii, iii, vi, and vii o, renders Weber’s notation a

predecessor of Riemannian functions, be they kinship oriented or province oriented.

6.9 François-Joseph Fétis

Fétis follows Momigny in conceiving of certain chromatic chords as chromatic

alterations of diatonic chords. In other words, Fétis allows for behavioral roots. In the

process, he understands chromatic chords as having kinship with their diatonic

counterparts. Example 6.5 shows a progression that Fétis explains as “the second chord

[representing] a chromatically altered form of the harmony f-a-c-d.” 100

Example 6.5 Chromatic alteration of ii 7.e

e François-Joseph Fétis, Traité complet de la théorie et de la pratique de l’harmonie (Paris: Brandus et Cie, 1853), 91.

Theory of Musical Composition , Vol. 1, trans. with notes by James F. Warner, ed. John Bishop (London: Robert Cocks and Co., 1851), 258, 287.

100 Shirlaw, 351; Shirlaw incorrectly comments, “If Fétis holds that the chromatic alteration of the first chord does not change the harmonic meaning and significance of the chord, he is plainly in error.” Shirlaw’s discussion is taken from François-Joseph Fétis, Traité complet de la théorie et de la pratique de l’harmonie (Paris: Brandus et Cie, 1853), 91.

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6.10 Simon Sechter

Sechter is another theorist who simultaneously builds on Rameau while

foreshadowing Riemann. He is sympathetic to Rameau’s prioritization of root movement

by fourth, and therefore also open to reinterpretation of harmonic meaning, in a way

similar to the double employment of the dissonance. This reinterpretation of harmonic

meaning, however, leads Sechter to tenets of kinship that no one had previously claimed.

Dahlhaus explains:

Sechter interprets the progression I-ii as vi 7-ii. The fifth of degree I is thus ‘really’ a seventh. … Sechter’s reductive method thus assumes a distinction fundamental to a theory of functions: the differentiation between appearance and significance, between what is presented and what is represented.” 101

By conceiving of I as an incomplete vi 7, Sechter incorrectly prizes local root movement by fourth over global movement away from and back to tonic. However, he correctly implies that harmonies may have functional significance (kinship) beyond their identity.

6.11 Moritz Hauptmann

Hauptmann is best known for situating his theory of harmony (and meter) within the context of the Hegelian dialectic: thesis, antithesis, synthesis. Concerning function,

“S represents unity, D duality, and T union,” which he shows with the notation I-III-II

101 Dahlhaus, 37; Dahlhaus’ discussion concerns Simon Sechter, Die Grundsätze der musikalischen Komposition , Part III, Vom drei- und zweistimmigen Satze; Rhythmische Entwürfe; Vom strengen Satze, mit kurzen Andeutungen des freien Satzes; Vom doppelten Contrapunkte (Leipzig, 1854), 96.

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(STD). 102 And although this conception of the pillars of harmony did influence Riemann, it does not directly translate into kinship, province, or behavior. The Hegelian model,

I-III-II, which is translated into pitch-class notation as FaCeGbD (for the C-major system), does not mean kinship for Hauptmann. Instead of referring to aCe as ‘like CeG’ because they share two scale degrees—a kinship relationship that is clearly discernable from the notation—Hauptmann notes that in the move from CeG to aCe, “C changes meaning and becomes third, while e changes meaning and becomes fifth.” 103 Further,

I-III-II does not lead directly to province, since the abstract positioning of T, S, and D does not concern an ordering through time, TSDT. The union of subdominant and dominant in tonic is not meant to evoke cadence, but rather the duality of the upper and lower fifth finding union in the tonic. Finally, Hauptmann’s Hegelian model does not translate into behavior. The fact that tonic becomes “a dominant to its subdominant and a subdominant to its dominant,” as found in the model I-III-II, is not meant to evoke the concept of behavior, for which a chord with tonic identity may be the dominant of the

6 subdominant (e. g., a passing I 4 chord in the subdominant province) or the subdominant of the dominant (e. g., a cadential six-four chord in the dominant province). 104 Yet, even

102 Mickelsen, “Translation of Harmonielehre ” (Ph.D. diss.), 105-6.

103 Shirlaw, 362; Shirlaw’s discussion concerns Moritz Hauptmann, The Nature of Harmony and Metre , trans. and ed., W. E. Heathcote (New York: Novello, Ewer, and Co., 1888), 45.

104 Hauptmann, 13. Further discussion may be found in Dahlhaus, 42 and Harrison, 218-25.

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though Hauptmann is not evoking kinship, province, or behavior when he posits the model I-III-II, these connections are easily derivable from the model.

Hauptmann does have some ideas that directly translate into behavior and kinship.

He writes:

A triad can pass (1) Into another triad lying next [to] it, i.e. joined to it by two common notes; e.g. … C-e-G [to] C-e-a [or] C-e-G [to] b-e-G. (2) Into another joined to it by one common note; [e.g.] C-e-G [to] C-F-a [or] C-e-G [to] b-D-G. (3) Into one wholly separate; [e.g.] C-e-G [to] a-D-F [or] C-e-G [to] D-F-b. 105

These rules are virtually identical to Proctor’s voice-leading paradigms, except that

Proctor’s paradigms are generalized so as to not require clefs (nor exact pitch classes).

Thus, Hauptmann evokes behavior by positing all root moves and voice-leading possibilities for triads.

Hauptmann also evokes kinship with his model, FaCeGbD, which becomes the real basis of Harrison’s base, agent, and associate. Hauptmann’s “triadic unities” are

Harrison’s bases; Hauptmann’s “opposed determinations” are Harrison’s associates;

Hauptmann’s “unity of unions” is Harrison’s agent. 106 The model FaCeGbD also shows

105 Hauptmann, 59-60.

106 Harrison, 232-3

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that Hauptmann conceives of “tonal relationships in a key [as] every note is a fifth or a third above 1ˆ , 4ˆ , or 5ˆ .” 107 This is clearly a precursor to kinship.

6.12 Hermann von Helmholtz

Regarding kinship, Helmholz is willing to go where Hauptmann is not. In fact, he

essentially defines the term: “When two chords have two tones in common they are more

closely related than when they only have one tone in common. Thus [c-e-g] and [a-c-e]

are more closely related than [c-e-g] and [g-b-d].”108

6.13 Hugo Riemann

As shown in Chapter 1, Riemannian function implies three aspects: kinship, province, and quality. Kinship is reflected by Riemann’s categories T p, S p, D p, T,< S,<

all of which refer to secondary harmonies as transformations of primary harmonies. 109

Riemann elaborates, “The most fundamental idea of all harmonic theory is that dissonant

chords are only transformations of harmonic chords.” 110 Province is made explicit in the

Riemannian prototypical progression, TSDT. 111 Province is further shown to imply a

107 Ibid., 233.

108 Hermann L. F. Helmholtz, On the Sensations of Tone as a Physiological Basis for the Theory of Music , trans. Alexander J. Ellis, 6 th ed. (New York: Peter Smith, 1948), 296.

109 Riemann, Harmony Simplified , 69-106.

110 Mickelsen, “Translation of Harmonielehre ” (Ph.D. diss.), 367.

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6 - 5 “sphere of authority” by the Riemannian notation D 4 - 3 , which depicts the tonic chord as

being under the sphere of authority of the dominant province. Quality (and to a certain

extent, behavior) is implied by the Riemannian notations DD and SS , which reveal that Riemann wishes to transfer functional attributes to secondary levels. 112

111 Riemann, Harmony Simplified , 45.

112 Ibid., 101.

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GLOSSARY

1-3 Passing Paradigm A voice-leading paradigm created by Gregory Proctor featuring passing motion from the root to the third (or from the third to the root) of the local tonic. Such a motion generates a t-d-t behavior paradigm. See also lower-adjacency paradigm .

3-5 Passing Paradigm A voice-leading paradigm created by Gregory Proctor featuring passing motion from the third to the fifth (or from the fifth to the third) of the local tonic. Such a motion generates a t-s-t behavior paradigm. See also upper-adjacency paradigm .

Behavior The aspect of function concerned with the relative status and voice-leading of adjacent harmonies. There are three behaviors: dominant, which appears as a lower- adjacency/lower-status element with respect to local tonic; subdominant, which appears as an upper-adjacency/lower-status element with respect to local tonic; and tonic, which appears as a higher-status element with respect to either local dominant or local subdominant.

Behavioral Root The root of any tertian or added-sixth harmony, regardless of the specific quality of its intervals d-t Tonicization A chromatic event represented by a paradigm featuring an upward pointing leading tone supported by dominant behavior

F1 The aspect of function that involves the grouping together of harmonies that share scale degrees. According to F1, harmonies that are not primary triads may be derived from or associated with one (or more) of the primary triads. See also kinship .

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F2 The aspect of function that implies that (1) the three pillars of harmony maintain a sphere of authority or field of activity over portions of harmonic progressions, and that (2) these spheres of authority or fields of activity tend to be organized according to the prototypical progression, TSDT. See also province .

F3 The aspect of function that allows for the transfer of functional properties to scale degrees other than tonic, subdominant, and dominant. See also quality and behavior .

F4 The aspect of function favored by Rameau, which connotes a relationship between two adjacent harmonies whose roots lie a fourth apart. If the roots are separated by an ascending fourth (from lower to higher status), D-T is implied; if the roots are separated by a descending fourth (from lower to higher status), S-T is implied. See also behavior .

Identity The property of a harmony that is defined by what scale degree the harmony is built on. For example, a harmony built on the dominant has dominant identity, regardless of its quality or number or types of extensions. V 13(#11) and v 7 are examples of harmonies with dominant identity.

Incomplete Tonicization A tonicization paradigm in which the object of tonicization does not appear on the surface of the music.

Kinship The aspect of function that involves the grouping together of harmonies that share at least two scale degrees with one of the primary harmonies. See also F1 .

Lower-adjacency Paradigm A voice-leading paradigm created by Gregory Proctor for which voices of the lower- status element are adjacent to and below voices of the higher-status element. Such a voice leading may take the form of passing motion or neighbor motion, and results in a t-d, d-t, or t-d-t behavior paradigm. See also 1-3 passing and lower-neighbor paradigm .

Lower-neighbor Paradigm A voice-leading paradigm created by Gregory Proctor for which voices of the lower- status element are lower neighbors to voices of the higher-status element. Such a voice leading results in a t-d, d-t, or t-d-t behavior paradigm. See also lower-adjacency paradigm .

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Microbehavior The behavior of a single voice, apart from its context within a harmony. Dominant microbehaviors include: local 5ˆ , 7ˆ (to 1ˆ ), 2ˆ (to 1ˆ or 3ˆ ), and 4ˆ (to 3ˆ ). Subdominant microbehaviors include: local 1ˆ , 6ˆ (to 5ˆ ), 4ˆ (to 3ˆ or 5ˆ ), and 2ˆ (to 3ˆ ).

Microtonicization A chromatic event represented by a leading tone without the support of its tonicizing harmony (V 7/x, iv add6 /x, or their subsets). This form of tonicization is applied to a single note rather than a whole harmony (or group of harmonies).

Perfect Root The lower note of a perfect fifth or the upper note of a perfect fourth

Province The aspect of function that implies that (1) the three pillars of harmony maintain a sphere of authority or field of activity over portions of harmonic progressions, and that (2) these spheres of authority or fields of activity tend to be organized according to the prototypical progression, TSDT. See also F2 .

Pseudo-tonicization A chromatic event represented by a tonicization paradigm in which no leading tone is present, and for which the quality of the tonicizing harmony is opposite the quality of the tonicized harmony (e.g., IV (add6) /v or v (7) /IV)

Quality The aspect of function concerned with the relative position of intervals within a harmony. There are only two functional qualities: dominant, which is the major-minor seventh chord (the seventh might not appear on the surface), and subdominant, which is the minor added-sixth chord (the sixth might not appear on the surface).

Quasi-tonicization A chromatic event represented by a tonicization paradigm in which no leading tone is present, and for which tonicizing harmonies share the same quality as the tonicized harmony (e.g., IV (add6) /IV or v (7) /v)

s-t Tonicization A chromatic event represented by a paradigm featuring a downward pointing leading tone supported by subdominant behavior

Three-fold System of Analysis A system consisting of three distinct analytic goals: identification, behavior analysis, and province analysis

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Tonal Root The root of a harmony whose base triad is major, minor, augmented, or diminished

Trivial Tonicization A chromatic event featuring a tonicization paradigm for which the object of tonicization is I (or i). Examples include V-i and iv-I.

Upper-adjacency Paradigm A voice-leading paradigm created by Gregory Proctor for which voices of the lower- status element are adjacent to and above voices of the higher-status element. Such a voice leading may take the form of passing motion or neighbor motion, and results in a t-s, s-t, or t-s-t behavior paradigm. See also 3-5 passing paradigm and upper-neighbor paradigm .

Upper-neighbor Paradigm A voice-leading paradigm created by Gregory Proctor for which voices of the lower- status element are upper neighbors to voices of the higher-status element. Such a voice leading results in a t-s, s-t, or t-s-t behavior paradigm. See also upper-adjacency paradigm .

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