The Role of Invertible Counterpoint within Schenkerian Theory ...

The Role of Invertible Counterpoint within Schenkerian Theory

by

Peter Jocelyn Franck

Submitted in Partial Fulfillment

of the

Requirements for the Degree

Doctor of Philosophy

Supervised by

Professor Matthew Brown

Department of Music Theory Eastman School of Music

University of Rochester Rochester, New York

2007

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

To my father,Jurgen Peter Franck (1930 - 2002)


Curriculum Vitae

Peter Jocelyn Franck was bom in Edmonton, Alberta, Canada, on October 15,

1969. He attended Grant McEwan Community College from 1989 to 1992 and

graduated with a Diploma in Performance (Jazz Guitar). He then attended McGill

University from 1992 - 1995 and obtained a Bachelor of Music in Music Theory,

graduating with distinction. He returned to McGill during the period 1999 - 2001

and graduated with a Masters of Arts in Music Theory. He came to the Eastman

School of Music of the University of Rochester in the Fall of 2001 and began doctoral

studies in Music Theory. He pursued his research in invertible counterpoint and

Schenkerian theory under the direction of Professor Matthew Brown, beginning in

2005. He received the Raymond N. Ball Dissertation Year Fellowship in 2006.


Acknowledgments

The completion of this dissertation would not have been possible without the

peerless support of my dissertation advisor, Matthew Brown, who met with me (what

seems like) countless times at the coffee shop next to Eastman over a period of more

than two years. Through our weekly meetings, he taught me how to think about

music, how to transform my fledgling ideas into compelling arguments, and the value

of resorting to first principles. It was Matthew who made me realize the potential of

researching invertible counterpoint. To that end, I must also acknowledge Peter

Schubert at McGill, under whose tutelage I first gained an appreciation for the study

of counterpoint. I would also like to acknowledge the input of the other members of

my dissertation committee, William Marvin and Patrick Macey, who provided

valuable editorial suggestions and comments and helped me to create a more polished

document. Of course, I must acknowledge the input and support of my wife, Leila,

who helped to proofread through drafts of the dissertation and provide simple

solutions to seemingly insurmountable editorial problems. More significantly, she

provided the moral support and understanding necessary for overcoming the

difficulties of writing a dissertation. Finally, I could not have completed this

dissertation without the financial support provided by the Raymond N. Ball

Dissertation Year Fellowship (Summer 2006 - Spring 2007), which allowed me to

write unfettered by the constraints of teaching duties.



by


Abstract

This dissertation examines the role that invertible counterpoint plays within

Schenkerian theory. The work is divided into five chapters. Chapter 1 establishes

invertible counterpoint as an essential component of polyphonic composition within

the Western tradition, from parallel organum to the music of the common practice

period. As part of this tradition, three factors necessary for invertible counterpoint

are identified—ways of vertically stacking tones, rules dictating relative motion

between voices, and preexistent material for writing new compositions. Changing

aspects of these factors are discussed, as expressed through the development of

musical style and compositional thought. Included within this development are the

changing definitions of invertible counterpoint, including those by Zarlino, Cerone,

Marpurg, Albrechtsberger, and Kimberger. These definitions are juxtaposed against

Schenker’s, whose opinion of this compositional technique is mixed. Despite his

ambivalence towards this practice, his predecessors demonstrated properties of

invertible counterpoint—including contrary/oblique motion between consonances and

triadically-based counterpoint—that intersect with Schenker’s ideas on tonality.

Some of these ideas include his notion of combined linear progressions and intervals


between harmonic tones, all of which are subject to inversion. Chapter 2 redefines

invertible counterpoint at the twelfth and shows how it occurs within fugal

composition in particular and tonality in general. Analyses of J. S. Bach’s Fugue in C

Minor from the Well-Tempered Clavier, Book I show that counterpoint at the twelfth

exists at deep levels of structure, not just at the surface. Chapter 3 applies the same

tack to invertible counterpoint at the tenth but also considers how this technique

engages specific voice-leading transformations, in this case, reaching-over. Bach’s

Fugues in B^ Major from the Well-Tempered Clavier, Books I and II serve as

analytical test-pieces. Chapter 4 continues in the same manner with invertible

counterpoint at the octave but also explains how this technique engages the voice-

leading transformations of register transfer and voice-exchange. The explanations are

illustrated with analytical excerpts from the Fuga of Bach’s Sonata for Solo Violin in

G Minor (BWV 1001). Chapter 5 projects further lines of research, including one

that considers the correlation between invertible counterpoint and musical form.


vii

Table of Contents

Volume 1: Text

Curriculum Vitae iii

Acknowledgements iv

Abstract v

Table of Contents vii

List of Tables in Volume 2 x

List of Examples in Volume 2 xi

Chapter 1 What is Invertible Counterpoint? 1

1.1. Introduction 1

1.2. Intervallic Spaces of Invertible Counterpoint 22

1.2.1. Triadic Spaces: The Twelfth,Tenth, and Octave 22

1.3. Invertible Counterpoint and the Elements Which ItInverts 32

1.3.1. Counterpoint at the Octave 37

1.3.2. The Seven Intervals of Counterpoint 38

1.3.3. Counterpoint at the Octave, Tenth, and Twelfth 41

1.3.4. The Problem Imposed by the Inversion Table: A Solution Provided by Cerone 43

1.4. Schenker and Invertible Counterpoint 46

1.4.1. Schenker’s Discussion of Combined Linear Progressions 47

1.4.2. Schenker’s Notes on Invertible Counterpoint Contained within the Oster Collection 59


viii

1.4.3. Intervallic Inversion 69

1.5. Conclusion 74

Chapter 2 Why Use Counterpoint at the Twelfth? 77

2.1. Introduction 77

2.2. Definitions/Problems of Counterpoint at the Twelfth 78

2.3. Compositional Applications of Counterpointat the Twelfth 84

2.4. The Fugue Subject and Counterpoint at the Twelfth 93

2.5. Counterpoint at the Twelfth: Background orForeground? 102

2.6. Conclusion 107

Chapter 3 Why Use Invertible Counterpoint at the Tenth? 111


3.2. Definition/Problems of Counterpoint at the Tenth 113

3.3. Compositional Applications of Counterpoint at the Tenth 122

3.3.1. Descending Thirds at the Surface 124

3.3.2. Descending Thirds at the Middleground 125

3.3.3. Illustrations of Counterpoint at the Tenth 127

3.4. Schenkerian Transformations andCounterpoint at the Tenth 131

3.5. Conclusion 142

Chapter 4 Why Use Counterpoint at the Octave? 145


4.2. Definitions/Problems of Counterpoint at the Octave 151


ix

4.3. Compositional Applications of Counterpointat the Octave 153

4.4. Schenkerian Transformations andCounterpoint at the Octave 158

4.5. Analytical Applications within the Fuga(BW V 1001) 165

4.5.1. Octave-progressions 173

4.5.2. Register Transfer 174

4.5.3. Voice-exchange 178

4.6. Conclusion 179

Chapter 5 Conclusion 184

Volume 2: Examples

Examples 193

Bibliography 322


X

List o f Tables in Volume 2

Table Title Page

Table 1.1a Treatises that discuss invertible counterpoint from1555 to 1773. 214

Table 1.1b Treatises that discuss invertible counterpoint from1774 to the present. 216

Table 1.2 Categorization of combined linear progressions inDer freie Satz. 229

Table 1.3 Dated items from File 83 in the Oster Collection. 238


xi

Example

Example 1.1

Example 1.2

Example 1.3

Example 1.4

Example 1.5

Example 1.6

Example 1.7

Example 1.7 contd.

Example 1.8

Example 1.9

Example 1.1

List o f Examples in Volume 2

Title

Parallel organum as found in the Musica enchiriadis.

In exitu Israel, as set by Binchois.

Alma Redemptoris Mater/Ave Regina coelorum, by Josquin.

Magnificat Tertii Toni: Et misericordia, by Palestrina.

“Fantazia upon One Note,” by Purcell.

Diagram showing the change in the three factors necessary for creating invertible counterpoint, from the Musica enchiriadis, to the time of Tinctoris, to the time of Schenker.

Zarlino’s example of invertible counterpoint at the twelfth. Adapted from Marco and Palisca trans., The Art o f Counterpoint, Exx. 113a and 113b.

Zarlino’s example of invertible counterpoint at the twelfth. Adapted from Marco and Palisca trans., The Art o f Counterpoint, Exx. 113a and 113b.

Diagram of invertible counterpoint at the twelfth as it is applied to modes.

Model of counterpoint at the twelfth in three voices; adapted from Marco and Palisca trans., The Art o f Counterpoint, Ex. 150a and 150b.

Zarlino’s example of counterpoint using contrary motion with parts inverted. Adapted from Marco and Palisca trans.,The Art o f Counterpoint, Exx. 116 and 117.

Page

193

194

195

196

197

198

199

200

201

202

203


xii

Example 1.10, contd.

Example 1.11

Example 1.12


Example 1.13

Example 1.14


Example 1.15


Example 1.16

Example 1.17

Example 1.18

Zarlino’s example of counterpoint using contrary motion with parts inverted. Adapted from Marco and Palisca trans., The Art o f Counterpoint, Exx. 116 and 117.

Model of invertible counterpoint in contrary motion.

Zarlino’s example of invertible counterpoint in contrary motion in three voices. Adapted from Marco and Palisca trans., The Art o f Counterpoint, Exx.l51a and 151b.

Zarlino’s example of invertible counterpoint in contrary motion in three voices. Adapted from Marco and Palisca trans., The Art o f Counterpoint, Exx. 15la and 151b.

The possible combinations of adding either a tenth above the bass or a tenth below the soprano.

Zarlino’s example of three-voice double counterpoint at the twelfth. Adapted from Marco and Palisca trans., The Art o f Counterpoint, Exx. 150a and 150b.

Zarlino’s example of three-voice double counterpoint at the twelfth. Adapted from Marco and Palisca trans., The Art o f Counterpoint, Exx. 150a and 150b.

Zarlino’s example of counterpoint at the tenth. Adapted from Marco and Palisca trans., The Art o f Counterpoint, Ex. 114.

Zarlino’s example of counterpoint at the tenth. Adapted from Marco and Palisca trans., The Art o f Counterpoint, Ex. 114.

Inversion table of invertible counterpoint at the octave.

Illustration of counterpoint at the octave by Rodio (1609).

Illustration of counterpoint at the ninth by Marpurg, 1753, Volume I, Table LVI, Fig. 8.

204

205

206

207

208

209

210

211

212

213

218

219


X lll

Example 1.19 Inversion tables for counterpoint at the ninth,eleventh, thirteenth, and fourteenth. 220

Example 1.20 Kimberger’s two-part example used to illustratecounterpoint at the octave, tenth, and twelfth. 221

Example 1.21 Kimberger’s example of counterpoint at the twelfth. 222

Example 1.22 Kimberger’s example of counterpoint at the tenth. 223

Example 1.23 Kimberger’s examples of counterpoint at the octave. 224

Example 1.24 Kimberger’s examples of doubling in thirds. 225

Example 1.25 Kimberger’s example of doubling in thirds, in fourparts. 226

Example 1.26 Illustrations of counterpoint at the octave, twelfth,and tenth; adapted from Cerone, El melopeo y maestro, 1613. 227

Example 1.27 Schenker’s Figure 1 from Derfreie Satz. 228

Example 1.28 Reproduction of Fig. 95, a, 4, from Der freie Satz. 230

Example 1.29 Fig. 98, Nr. 3c of Der freie Satz. From C. P. E. Bach,Generalbass IX/2, § 1 a. 231

Example 1.30 Fig. 99, Nr. la, b, and c from Der freie Satz. 232

Example 1.31 5-line Schenkerian Ursatz with intervals betweenouter voices. 233

Example 1.32 3-line Schenkerian Ursatz. 234

Example 1.33 3-line Schenkerian Ursatz with annotations showingharmonic-tone inversion. 235

Example 1.34 Alto and Bafibrechung of 5-line Ursatz. 236

Example 1.35 Tenor and Bafibrechung of 5-line Ursatz. 237


xiv

Example 1.36 File 83, Item 478 from the Oster Collection. 239


Example 1.38 File 83, Item 460 of the Oster Collection. 241

Example 1.39 Schenker’s acknowledgement of disallowing parallelconsonances under counterpoint at the tenth. 242

Example 1.40 File 83, Item 445 of the Oster Collection. 243

Example 1.41 File 83, Item 454, first page, of the Oster Collection. 244

Example 1.41, File 83, Item 454, second page, of the Ostercontd. Collection. 245

Example 1.42 File 83, Item 447 (second page) from the OsterCollection. 246



Example 1.45 List of melodic intervals (without inversion)according to Kontrapunkt I. 249

Example 1.46 The list of melodic intervals (under inversion)according to Kontrapunkt I. 250

Example 1.47 The list of vertical intervals. 251

Example 1.48 The list of harmonic tones. 252

Example 2.1 Inversion chart illustrating invertible counterpoint atthe twelfth. 253

Example 2.2 Counterpoint at the twelfth, from mm. 5-6 and mm.17-18 of Bach’s Fugue in C Minor (BWV 847). 254

Example 2.3 Annotated version of Example 2.2. 255

Example 2.4 Pattern of consonances occurring under invertiblecounterpoint at the twelfth. 256


XV

Example 2.5 Harmonic-tone inversion at the twelfth. 257

Example 2.6 The Ursatz. 258

Example 2.7 Unfolding. 259

Example 2.8 Voice-exchange. 260

Example 2.9 Form chart of Bach’s Fugue in C Minor (BWV 847). 261

Example 2.10 Revised form chart of Bach’s Fugue in C Minor(BWV 847). 262

Example 2.11 Invertible relationship between the initiatingforeground harmonies of Episodes one and four from Bach’s Fugue in C Minor (BWV 847). 263

p 1 '“) 1 O A A A A A

example z .iz Melodic stereotype outlining 5 - 6 - 5 - 4 - 3 , from Handel’s Fugue in C Minor, from Six Grandes Fugues. 264

Example 2.13 Subject/answer Paradigm 1, Category 1. 265

Example 2.14 Revision of subject/answer paradigm. 266

Example 2.15 Revised paradigm in mm. 1 -5 of Bach’s Fugue in CMinor (BWV 847). 267

Example 2.16 Neighbor note around the third of the dominant in theepisode, mm. 5-7, of Bach’s Fugue in C minor (BWV 847). 268

Example 2.17 Analysis of mm. 5-6 of Bach’s Fugue in C Minor(BWV 847). 269

Example 2.18 Analysis of mm. 5-7 of Bach’s Fugue in C Minor(BWV 847). 270

Example 2.19 Harmonic-tone inversion at the twelfth. 271

Example 2.20 Deep-middleground of Bach’s Fugue in C Minor(BWV 847). 272


xvi

Example 2.21

Example 2.22

Example 2.23

Example 2.24

Example 2.25

Example 3.1

Example 3.2

Example 3.3

Example 3.4

Example 3.5

Example 3.6

Example 3.7

Example 3.8

Example 3.9

Example 3.10

Analysis of the exposition of Bach’s Fugue in C Minor (BWV 847).

Voice-leading summary of the episode from mm. 17- 20 of Bach’s Fugue in C Minor (BWV 847).

Comparison of episodes from mm. 5-7 and mm. 17- 18 of Bach’s Fugue in C Minor (BWV 847).

The fourth-progression at the deep-middleground of Bach’s Fugue in C minor (BWV 847).

Middleground sketch of Bach’s Fugue in C Minor (BWV 847).

Counterpoint at the tenth.

Inversion table for counterpoint at the tenth.

Counterpoint at the tenth. Creation of three-part example from two-voice counterpoint. Adapted from Alfred Mann, The Study o f Fugue (New York: Norton, 1958), 116.

Extrapolation of Piston’s notion of counterpoint at the tenth.

Diagram of counterpoint at the tenth.

Inversion tables for counterpoint at the tenth with sample illustration.

Counterpoint at the tenth at the background.

Descending thirds sequence. Based on Aldwell and Schachter, 2003.

Descending thirds and counterpoint at the tenth.

Background and deep-middleground of the Fugue in B^ Major (BWV 890). Use of contrary and oblique motion.

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287


xvii

Example

Example

Example

Example

Example

Example

Example

Example

Example

Example

Example

Example

Example

Example

Example

3.11 Bach’s Fugue in Major (BWV 866). Demonstration of counterpoint at the tenth between parallel sections.

3.12 Excerpts from Bach’s Fugue in B^ Major (BWV 890) that are related by counterpoint at the tenth.

3.13 Subject of Bach’s Fugue in B^ Major (BWV 866), mm. 1-5.

3.14 Subject of Bach’s Fugue in B^ Major (BWV 890), mm. 1-5.

3.15 Model of reaching-over.

3.16 Schenker’s illustration of reaching over.

3.17 Middleground sketches of B Maj or Fugue (BWV 866).

3.18 Form chart of Bach’s Fugue in B^ Major (BWV 890).

3.19 Middleground of Bach’s Fugue in B Major (BWV 890).

3.20 Measures 32-36 and mm. 40-44 of Bach’s Fugue in B^ Major (BWV 890).

3.21 Parallel entries from Bach’s Fugue in B Major (BWV 890), mm. 47-51 and mm. 54-58.

3.22 Measures 78-86 of Bach’s Fugue in B^ Major (BWV 890).

3.23 Exposition of Bach’s Fugue in B^ Major (BWV 890), mm. 1-32.

3.24 Measures 32-78 of Bach’s Fugue in B^ Major (BWV 890).

3.25 Graphs of mm. 78-93 of Bach’s Fugue in B^ Major (BWV 890).

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302


xviii

Example 3.26 Measures 1-5 from Bach’s Fugue in Major (BWV 890); superimposition of CS2; mm. 5-9 from Bach’s Fugue in B^ Major (BWV 890); superimposition of CS2. 303

Example 3.27 Scale-degree functions of CS2 in Bach’s Fugue in B^Major (BWV 890). 304

Example 4.1 Inversion tables for counterpoint at the octave. 305

Example 4.2 Illustrations of counterpoint at the octave within theFuga. 306

Example 4.3 Example 4.3. Invertible counterpoint at the octaveoccurring at structural cadences. 307

Example 4.4 8-line Ursatz in G minor. 308

Example 4.5 Ascending register transfer. 309

Example 4.6 Register transfer in the Fuga. 310

Example 4.7 Voice-exchange; model of register transfer andvoice-exchange. 311

Example 4.8 Form chart of the Fuga. 312

Example 4.9 Exposition of the Fuga, mm. 1-14. 313

Example 4.10 Octave descents in the tonic and dominant within theFuga, mm. 14-24. 314

Example 4.11 Octave descent from 3 to 3, within the Fuga, mm.73-80. 315

Example 4.12 Voice-leading summary of the Fuga, mm. 24-42. 316

Example 4.13 Invertible cadence at mm. 52-55. 317

Example 4.14 Invertible cadence moving through falling fifths. 318

Example 4.15 Voice-leading summary of the Fuga, mm. 14-19. 319


Example 4.16 Register transfer and voice-exchange.

Example 4.17 Kellner’s alternate cadence in the Fuga, mm. 34-35.


1

Chapter 1: What is Invertible Counterpoint?

1.1. Introduction

This dissertation examines the role that invertible counterpoint plays within

Schenkerian theory. Invertible counterpoint is the practice of exchanging registral

positions of melodies, enabling one melody to appear above and below another.

Schenkerian theory is a theory that explains tonal music as the product of applying

recursive transformations to contrapuntal prototypes at hierarchical levels of

structure. Invertible counterpoint abounds within tonal music, a repertory whose

structure Schenkerian theory is designed to elucidate. Based upon this observation,

one could infer that invertible counterpoint informs aspects of Schenkerian theory,

and vice versa. But as we shall see, Heinrich Schenker (1868 - 1935) expressed

ambivalence towards invertible counterpoint, thus putting into doubt the relationship

between this contrapuntal device and the theory that he developed.

As a way of erasing this doubt and strengthening the ties between invertible

counterpoint and Schenkerian theory, this chapter focuses on the following five

issues. First, it establishes the significance of invertible counterpoint as a

compositional device and how it has changed and developed throughout the history of

tonal music. Second, the chapter examines which intervals are used to transpose and

invert the registral positions of melodies when using invertible counterpoint. Third, it


2

discusses what elements are being exchanged when melodies are inverted with each

other. Fourth, this chapter identifies aspects of invertible counterpoint, both explicit

and implicit, that Schenker integrated into his published and unpublished writings.

Fifth, it establishes the framework for the following chapters of the dissertation,

which will explicate the relationships between invertible counterpoint and the

following theoretical concepts: Schenker’s prototype of tonality (the Ursatz), musical

form, and voice-leading transformations.

There can be no doubt that invertible counterpoint has played a large role

within the history of musical composition. This contrapuntal technique has informed

compositional practice from the origins of polyphony in parallel organum in the ninth

century, to the advent of twelve-tone techniques in the twentieth century. That

invertible counterpoint has appeared in compositions exhibiting wholly different

systems of pitch-organization, and belonging to disparate historical periods, illustrates

how fundamental this practice has been within musical composition.

Within the history of tonal compositional practice—I include, for our

immediate purposes, music in the Western tradition written between the ninth to early

twentieth centuries, and that which is based around a final, tonic, or tonal center;

post-tonal repertories are excluded—the technique of writing invertible counterpoint

was dependent upon three primary factors of composition: the first was a list of ways

that tones could be stacked vertically; the second was a rule that dictated the way that

vertically stacked tones could be approached horizontally; the third was a corpus of

melodies that could be borrowed and used as the basis for new compositions. The


3

first factor deals generally with the concepts of consonance, dissonance, and triadic

membership; the second factor describes the concept of relative motion between

concurrent melodies; the third factor addresses the use of a cantus firmus as the

melodic foundation of a composition. With respect to this third factor, composers

would weave this preexistent material into compositions by placing it above and

below newly-composed melodies, each of which was referred to as a counterpoint. A

small set of intervals—these included the twelfth, tenth, and octave—was used to

transpose, and thus, invert the registral position of the cantus firmus with the

counterpoint(s), leading to three types of invertible counterpoint: namely, invertible

counterpoint at the twelfth, tenth, and octave. (Henceforth, I shall refer to these

techniques in a slightly shorter form as follows: “counterpoint at the twelfth”,

“counterpoint at the tenth”, and “counterpoint at the octave”.) The specific nature of

the three factors used to create invertible counterpoint—vertical stacking, relative

motion, and cantus firmus treatment—changed over time as did musical styles

throughout the history of music, but the use of invertible counterpoint itself has

maintained a continuous foothold upon compositional practice ever since the

beginning of polyphony.

Indeed, it is the beginning of polyphony itself, as expressed through the genre

of parallel organum, that heralded the practice of invertible counterpoint. A

discussion of parallel organum is found within the Musica enchiriadis (latter half of

the ninth century); the explanation offered within this treatise addresses the three

factors used to create invertible counterpoint, as discussed above. First, parallel


4

organum featured a cantus firmus as its highest voice (referred to as the principal

voice). Second, a counterpoint (referred to as the organal voice) would appear below

the cantus firmus at the interval of either a fourth or fifth (both intervals at this time

were considered to be consonant). Third, the organal voice would follow the

principal voice in parallel motion. With these factors set in place, octave-doublings

of either the principal or organal voice could expand the original two-part texture to

three or more voices (see Example 1.1, where both the principal and organal voices

are doubled); in so doing, the original arrangement of the principal and organal voices

becomes inverted, as expressed by the anonymous author(s) of the Musica

enchiriadis:

If a man and a boy sing simultaneously as organal voices, the two make a diapason [octave] with each other. However, the higher [voice], which is the boy’s, is heard a fifth above, and the lower [voice], which is the man’s, is heard a fourth below that middle voice which they contain between each other and to which both correspond as organum. Thus the symphonies bind themselves together by means of a mutual relationship, so that any tone that lies a fourth from another on one side looks at the other an octave away from the distance of a fifth.1

The author(s) of this passage describe(s), implicitly, the practice of counterpoint at

the octave, since it is through the transposition of an octave by which the principal

1 Musica enchiriadis and Scolica enchiriadis, trans. Raymond Erickson, ed. Claude V. Palisca (New Haven: Yale University Press, 1995), 23.


5

and organal voices exchange places.2 The application of invertible counterpoint

continued from this point forward throughout the development of polyphonic music,

however, changes in musical style resulted from concomitant changes within the three

factors necessary for creating invertible counterpoint.

One of these changes was the way that tones could be stacked vertically. As

shown previously in the illustration of parallel organum (Example 1.1), the fourth was

considered to be a consonance. By the beginning of the fifteenth century, ideas

concerning consonance and dissonance had changed considerably. Example 1.2

shows a case in point, the three-voice motet, In exitu Israel, set by Binchois (c. 1400-5 t

- 1460) in the mostly homorhythmic texture offawcbourdon. The music by this

composer and his contemporaries4 was praised by Johannis Tinctoris (c. 1435 —

1511), a theorist whose writings established the rules of consonance and dissonance

that would characterize much of the contrapuntal treatment within music from the

Renaissance period. Specifically, Tinctoris classified consonances and dissonances

in Books 1 and 2, respectively, of his Liber de arte contrapuncti (1477); he gave a list

of 22 consonances, ranging from the unison to the triple octave. The fourth

(diatessaron) was listed among these consonances; however, he quickly dismissed its

viability as a consonance since it was heard as an “intolerable discord,” at least by

2 For a discussion of the authorship of this treatise, see Erickson trans., Musica enchiriadis and Scolica enchiriadis, xxii-xxiii; Nancy Phillips, “Musica and Scolica Enchiriadis: The Literary, Theoretical, and Musical Sources” (Ph.D. diss., New York University, 1984), 396- 410.3 This example is taken from The Sacred Music o f Gilles Binchois, ed. Philip Kaye (Oxford: Oxford University Press, 1992), 203.4 These include Ockeghem, Regis, Busnois, Caron, Faugues, Dunstaple, and Dufay. See Johannes Tinctoris: Opera Theoretica, ed. Albert Seay, in Corpus Scriptorum de Musica, vol. 22 (American Institute o f Musicology, 1975), 12.


6

“educated ears.”5 The only way that a fourth could enter into a polyphonic, note-

against-note setting is between the upper voices of a three-voice texture, where the

bottom voices create the intervals of either a fifth or third: i.e., a fourth could not

appear above the lowest voice in note-against-note counterpoint. To illustrate this

concept, he used fauxbourdon, the same style depicted in the music of Example 1.2.6

Referring back to Binchois’s motet, we can see that parallel fourths occur between the

upper voices, and that the intervals created by the lower voices are restricted, for the

most part, to thirds, and at the cadences, fifths.7 The relative motion between the

upper voices is the same as that of parallel organum (refer back to Example 1.1), but

changes in the way that the distinction between consonance and dissonance were

understood in the mid-fifteenth century reflect the change in musical style that

occurred during this time—it was one factor, among the three listed at the beginning

of this chapter, that would alter the future treatment of invertible counterpoint.

Another of these three factors necessary for creating invertible counterpoint,

relative motion between melodies, had also changed by the fourteenth and fifteenth

centuries through the inclusion of a new rule, namely the prohibition of parallel

perfect consonances. As can be seen in the illustration of parallel organum in

Example 1.1, parallel fifths and octaves were created through the doubling of voices.

5 Tinctoris, Liber de arte contrapuncti [1477] 1975, 26. “Simpliciter tamen concordantia non est, immo per se emissa apud aures eruditas, quae, ut inquit Cicero, discrepantem concentum audire non possunt, intolerabiliter discordat.”6 Tinctoris, Liber de arte contrapuncti [1477] 1975, 27.7 For more on the status o f the fourth in the fourteenth century, see Richard L. Crocker, “Discant, Counterpoint, and Harmony,” Journal of the American Musicological Society 15/1 (1962): 5-8.


7

One theorist from the early fifteenth century,8 Prosdocimo de’ Beldomandi (d. 1428),

strictly forbade the use of parallel perfect consonances in his Contrapunctus (1412)

for the reason that “one voice would sing the same as the other...which is not the

purpose of counterpoint... its purpose is that what is sung by one voice be different

from what is pronounced by the other, and that this be done through concords that are

good and properly ordered.”9 His instructions addressed the melodic nature of both

voices set in counterpoint, dictating that both should express different melodic

motion, i.e., both voices should move horizontally by different, specific intervals (a

condition not possible when the voices are separated by a perfect consonance and

move to another of the same type in parallel motion). This contrapuntal behavior is

demonstrated in Binchois’s motet in Example 1.2, e.g., in m. 2, when the tenor moves

up a whole tone from C to D, the superius moves up a semitone from A to bK To be

sure, this motet is premised on parallel motion, since it is an example offauxbourdon;

however, the only intervals expressed between voice-pairs are thirds (between the

lower voices), sixths (between the outer voices), and fourths (between the upper

voices)—no parallel perfect consonances occur between any voice-pairs. Thus,

Binchois illustrated how the prohibition of parallel perfect consonances affected the

contrapuntal treatment in his motet. And this prohibition also impacted the way that

8 There are many who state this as a rule: Franchinus Gaffurius (1451-1522), Practica musicae (Milan: Guillermus Le Signerre, 1496); trans. Clement A. Miller (Dallas: American Institute o f Musicology, 1968), 125; Pietro Aaron, Toscanello in musica (Venice, 1539), Chapter XIIII; trans. Peter Bergquist, Toscanello in Music, Book II, Chapters I - XXXVI (Colorado Springs: Colorado College Music Press, 1970), 23-24.9 Prosdocimo de’ Beldomandi, Contrapunctus (Montagnana: MS, 1412); trans. and ed. Jan Herlinger (Lincoln and London: University o f Nebraska Press, 1984), 63.


8

invertible counterpoint was written, since it limited the palette of consonances that

could be used above and below the cantus firmus.

But it was the cantus firmus itself, as the third in the triumvirate of factors

used to create invertible counterpoint, that underwent the greatest change throughout

the development of polyphonic music. The most significant of these changes was the

placement and role of the cantus firmus within the polyphonic fabric. Within the

context of parallel organum, the cantus firmus occupied the uppermost voice (that is,

prior to doubling voices); however, it gravitated towards the lower end of the texture,

specifically in the tenor, within music dating from the twelfth century onward.10

Indeed, the tenor often occupied the lowest melodic strand, most notably within

motets consisting of two or more voices. But even the tenor would thread its way

into different registral positions as musical styles changed, and this was certainly the

case in motets from the fifteenth century. Binchois’s motet, shown earlier in

Example 1.2, is a good case in point, since it shows the cantus firmus-canying voice

in the middle of the texture and surrounded by counterpoints in the superius and

tenor.11 In this way, Binchois’s motet expresses the concept of invertible

counterpoint: the superius, occurring above the cantus firmus, may be understood as a

10 For a general overview of cantus firmus treatment, see M. Jennifer Bloxam, “Cantus Firmus,” in The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie, 2nd edition (London: Macmillan, 2001), 5:67-74. Also see Edgar H. Sparks, Cantus Firmus in Mass and Motet, 1420-1520 (Berkeley: University o f California Press, 1963).11 Although the tenor is the lowest voice in the edition shown in Example 1.2 (Kaye ed., The Sacred Music o f Gilles Binchois), some scholars view the middle voice as the cantus flrmus- bearing voice, albeit transposed to the upper fifth. See J. Marix, ed., Les Musicians de la Gourde Bourgone auXV6 Siecle (1420-1467) (Paris: Editions de L’Oiseau-Lyre, 1937), 196- 208; Manfred F. Bukofzer, “Fauxbourdon Revisited,” The Musical Quarterly 38/1 (1952):42; Brian Trowell, “Faburden and Fauxbourdon,” Musica Disciplina 13 (1959): 75-76.


9

transposition of the tenor, occurring below the cantus firmus. To be sure, the cantus

firmus would occupy other positions within the sonic fabric of compositions during

the course of the fifteenth century. For example, John Dunstaple (ca. 1390 - 1453)

placed the cantus firmus into the highest voice in his three-voice setting of the hymn,

Ave Maris stella,12 and most four-voice masses and motets featured the cantus firmus

in the tenor, the second voice from the bottom. In the latter half of the fifteenth

century, composers would use imitation, canon, and multiple cantus fiirmi as a way of

weaving the borrowed material into other voices of the musical texture. Example 1.3

shows an illustration of some of these techniques in Josquin’s motet, Alma

Redemptoris Mater/Ave Regina coelorum, where the incipit in the superius/altus pair

is inverted by the tenor/bassus pair in mm. 9ff. Ironically, as the cantus firmus began

to saturate the polyphonic fabric through imitation and canon, its role as a structural

pillar of composition began to wane.

Imitation (and to a lesser degree, canon), therefore, began to supplant the

cantus firmus as a structural determinant of musical composition. To be sure, cantus

firmus settings still continued throughout the works of such composers as Lassus

(1530 or 1532 - 1594) and Palestrina (1525 or 1526 - 1594); however, imitation was

beginning to be a structuring technique that would influence the development of

composition for many years to come. Indeed, composers such as Palestrina relied

heavily on imitation, and with it, invertible counterpoint, as compositional

determinants. Example 1.4 shows the opening measures of his Magnificat Tertii

12 See Musica Britannica: A National Collection o f Music, vol. 8: John Dunstable: Complete Works, ed. Manfred F. Bukofzer (London: Stainer and Bell, 1953), 95.


10

Toni: Et misericordiaP Each voice enters with the same melodic incipit as first

stated by the altus; however, the cantus enters at the fourth above in m. 3 and the

tenor enters at a fifth below in m. 5. The melodies shared between the altus and

cantus in mm. 3-4 are inverted and passed along to the tenor and cantus as the tenor

makes its entrance—specifically, this produces counterpoint at the twelfth.14

Although the imitation does not last for very long, the appearance of the incipit

throughout the three voices demonstrates the use of imitation and invertible

counterpoint as a way of structuring a composition. Theorists such as Zarlino (1517 —

1590) and Vicentino (1511 - ca. 1576) attempted to explain such contrapuntal

techniques within their respective treatises, Le Istitutione Harmoniche (1558) and

L ’antica musica ridotta alia modernaprattica (1555). For example, Zarlino

distinguished between different sorts of imitative techniques, such as imitationi and

fuga,15 which elucidated the types of melodic intervals at which a subject could be

imitated, and he also wrote one of the earliest exegeses on double or invertible

counterpoint.16 The use of imitation and invertible counterpoint, however, was not

13 This example is taken from Gustave Frederic Soderlund and Samuel H. Scott, Examples of Gregorian Chant and Other Sacred Music of the 16th Century (New York: Appleton-Century- Crofts, 1971), 59.14 Example 2.1 provides an illustration o f an inversion table for counterpoint at the twelfth, which articulates the change in intervals that takes place as voices are inverted at the twelfth. This type o f invertible counterpoint will be discussed in more detail in Chapter 2.15 Gioseffo Zarlino, Le Istitutioni Harmoniche (Venice, 1558), Chapter 51, 212; Chapter 52, 212; translated as The Art of Counterpoint: Part Three ofhe Istitutioni Harmoniche, 1558, trans. Guy A. Marco and Claude V. Palisca (New Haven: Yale University Press, 1968), 126; 135.16 Essentially, “double counterpoint” is the same as “invertible counterpoint,” with the former being a specific case o f the latter. Some theorists, however, distinguish between double counterpoint—adding a counterpoint to an already existing contrapuntal framework—and invertible counterpoint, which specifically defines a registral exchange o f voices. See Peter


11

restricted to music from the Renaissance. Indeed, the contrapuntal techniques

demonstrated by Palestrina and as explained by Zarlino would, unwittingly, diminish

the structural role of the cantus firmus within musical composition and would serve

as exemplars to countless theorists and composers from the sixteenth century onward.

Following the examples established by Palestrina and Zarlino, the techniques

of imitation and invertible counterpoint continued to flourish into the seventeenth

century and weave their way into the fabric of tonal music. This is not to say that

cantus firmus techniques were completely abandoned by composers in this century.

For example, Samuel Scheidt composed keyboard pieces around cantus firmi in his

Tabulatura nova (1624) and Henry Purcell (1658 or 1659 - 1695) wrote different

instrumental settings of In nomine, an English genre based on a cantus firmus

segment taken from Taverner’s mass, Gloria tibi Trinitas. But imitation and

invertible counterpoint also played a large role within many compositions from the

latter part of the seventeenth century, including those that employed cantus firmi.

Example 1.5, which shows the opening measures of Purcell’s “Fantazia upon One

Note,” provides a good illustration. Here, the part second from the bottom

(presumably, the “tenor”) contains a constructed cantus firmus (one designed by the

composer himself, not borrowed from external sources) consisting of a single note, C.

Meanwhile, the remaining parts engage in motivic play occurring above and below

the cantus firmus, e.g., the stepwise ascending fifth from F to C in the bass in mm. 1-

2 is imitated at the upper fifth, C to G, in the part third from the bottom. This is also

Schubert, Modal Counterpoint, Renaissance Style (New York: Oxford University Press, 1999), 174- 175.


12

an example of counterpoint at the twelfth. Although his “Fantazia” employs a cantus

firmus, it differs greatly from those used by composers from previous centuries. For

example, it does not end with a stepwise descent to the tonic (or final), nor does it

contain, for that matter, any stepwise motion at all. But Purcell’s “Fantazia”

demonstrates, indeed, the explicit use of imitation and invertible counterpoint,

techniques that would continue to prevail throughout tonal compositions and be

written about in treatises elucidating music from the common practice period.

Although the cantus firmus began to take a back seat to imitation and

invertible counterpoint in tonal music, it took over the driver’s position as a

pedagogical aid to counterpoint instruction in the late seventeenth and early

eighteenth centuries, especially within Gradus ad Parnassum (1725) of Johann

Joseph Fux (1660 - 1741). Even though this treatise was written during the time of J.

S. Bach and Telemann, Fux looked backward to Palestrina as his source of

inspiration. His veneration of Palestrina is expressed through the following words

from Gradus ad Parnassum: “By Aloysius, the master, I refer to Palestrina, the

celebrated light of music... to whom I owe everything that I know of this art, and

whose memory I shall never cease to cherish with a feeling of deepest reverence.”17

Instead of using the cantus firmus as a borrowed melody to be used for new

compositions, he used it as a foundation upon which to demonstrate principles of

counterpoint, such as the passing tone, cambiata, and suspension, introducing each

17 Johann Joseph Fux, Gradus ad Parnassum (Vienna: Van Ghelen, 1725), third page of preface; trans. and ed. Alfred Mann, The Study of Counterpoint (New York: W.W. Norton, 1965), 18.


13

through a graduated set of species. He also included sections on writing imitation,

invertible counterpoint, and fugue, for by this time, these techniques had become

essential components of the composer’s tool-kit. The cantus firmus figured into

almost all of these exercises, not as preexistent melodic material, but as a guide to

acquiring compositional technique. Some of the preeminent composers of the

common practice period benefited from this technique: indeed, Haydn, Mozart, and

Beethoven—to name but a few—all reaped the benefits of this pedagogical

approach.18 But the compositions that they wrote—these pieces were the products of

following Fux’s pedagogical plan—did not result in cantus firmus based pieces. On

the contrary, most of the output from the First Viennese School was devoid of any

cantus prius factus. As the cantus firmus established a new role as a pedagogical aid

within counterpoint treatises, including those by Albrechtsberger (1790), Cherubini

(1835), and Bellermann (1862), its absence from tonal composition would need to be

replaced with a new type of “preexistent material,” one that would direct the creation

of compositions that used imitation and invertible counterpoint.

One theorist who recognized the lacuna left open by the departure of the

cantus firmus from tonal composition was Heinrich Schenker. He inherited the

Fuxian species-approach to counterpoint instruction, including the use of cantus firmi,

as demonstrated in his two counterpoint manuals, Kontrapuntkt 7(1910) and

Kontrapunkt 7/(1922). But by the end of the second of these volumes, he removed

the cantus firmus for the purpose of creating a link between species-counterpoint

18For more on Fux’s influence, see Mann’s introduction to The Study of Counterpoint, xi-xv.


14

exercises and the “free composition” of master composers writing in the tonal idiom.

With respect to the removal of the cantus firmus, he wrote,

...a voice moving in whole-notes [the cantus firmus] can also justifiably be imagined to be absent from a combined-species setting, even through it is true that by the nature of combined-species exercises the counterpoints were able to arise only through the mediation of that voice. It will therefore be shown to be possible to infer, from the course of two or several counterpoints of the second, third, fourth, or fifth species (in any combination), a voice in whole-notes that binds together and explains the others. One need only assume the voice in whole-notes to be comparable to a surrogate for the scale degrees of free composition in order to understand the value of such an elision [omission] as a bridge to free composition.19

Even though the “voice moving in whole-notes” could be deleted from the musical

texture, he made it clear that there is an imagined melody that guides the direction of

the voice-leading and harmonies that occur within free composition. To this end, he

wrote, “It is possible to add conceptually, without any doubt, another voice which by

itself provides the first explanation of the voice leading and the foundational

concepts, and which completes, clarifies, and supports the harmonies.”20 Schenker,

thus, took on the challenge to discover a new sort of cantus firmus, one that could

account for the contrapuntal behavior, including that of invertible counterpoint, that

occurred within the context of tonal music.

19 Heinrich Schenker, Neue Musikalische Theorien und Phantasien, zweiter Band: Kontrapunkt, zweiter Teil, zweiter Halbband: Drei- und Mehrstimmiger Satz, Ubergange zum freien Satz (Vienna and Leipzig: Universal-Edition A. G., 1922); trans. John Rothgeb and JUrgen Thym, ed. John Rothgeb, Counterpoint, Book II (Ann Arbor: Musicalia Press, 2001), 176.20 Rothgeb and Thym trans., Counterpoint II, 270.


15

In an effort to fill in the gaps left open by the departure of the cantus firmus,

Schenker envisioned a new, multi-leveled paradigm that could explain the general

nature of tonal composition. To bring this paradigm to fruition, it was necessary for

him to reinvent received notions of consonance and dissonance treatment and the rule

prohibiting parallel perfect consonances, along with creating a replacement for the

cantus firmus. Inadvertently, he was redefining the three factors necessary for

creating invertible counterpoint, identified earlier. For the first factor (consonance

and dissonance), he recognized that one must acknowledge the triad as the primary

harmonic unit of composition, rather than intervals alone. In this light, he wrote that

“[T]he nature of three-voice polyphony itself...imposes first of all the demand...that

each individual tone of the cantus firmus become wherever possible a constituent of a

0 1complete triad, 5/3 or 6/3.” This triadic view of counterpoint changed the

orientation of vertical stacking from intervals to harmonic tones. Within free

composition, the triad assumed a more abstract guise as a Stufe, a harmonic unit that

exists at different levels of structure and is able to undergo a finite list of

transformations (to be discussed shortly). For the second factor, he recognized that

tones occurring at different levels of derivation could come into contact with each

other at the surface, and thus create the illusion of parallel perfect consonances. In

this respect, he wrote that “[parallel] successions are justified by the voice-leading in

the middleground and background from which they originate, where 8—8 and 5—5

21 Rothgeb and Thym trans., Counterpoint II, 4.


16

are nonexistent.”22 His solution to the third factor was more abstract and complex

than those for the previous two. For this he posited a contrapuntal prototype, the

Ursatz, through which one could apply voice-leading transformations

(Stimmfuhrungsverwandlungen) at subsequent voice-leading levels

(Stimmfuhrungsschichten). The application of transformations would lead,

ultimately, to the foreground of a tonal work. Schenker, therefore, revised the three

factors necessary for creating invertible counterpoint within the tonal idiom; most

importantly, the Ursatz provided a powerful replacement for the now-departed cantus

firmus.

In his aims to develop the Ursatz, Schenker relied primarily on the

counterpoint instruction of Fux (as stated above) and the thoroughbass instruction of

C. P. E. Bach (1714-1788). Indeed, he championed the teachings of both as keys to

understanding “organic coherence.” He wrote:

The following instructional plan provides a truly practical understanding of this concept [organic coherence]. It is the only plan which corresponds exactly to the history and development of the masterworks, and so is the only feasible sequence: instruction in strict counterpoint (according to Fux-Schenker), in thorough-bass (according to J. S. and C. P. E. Bach), and in free composition (Schenker).24

22 Heinrich Schenker, Neue musikalische Theorien und Phantasien: Dritter Band: Der freie Satz (Vienna: Universal-Edition, 1935), §162; trans. and ed. Ernst Oster, Free Composition (New York: Longman, 1979), §162, 57-58.23 Schenker, Der jreie Satz, §45.24 Oster trans., Free Composition., xxi-xxii.


17

25Although he acknowledged the teachings of Fux and C. P. E. Bach as exemplary, he

recognized that both contained drawbacks. On the one hand, he criticized Fux for

concentrating solely on vocal music, excluding instrumental music entirely from his

purview:

By elevating voice leading to the rank of a binding theory of composition—specifically, one based exclusively on a purely vocal foundation—he [Fux] unfortunately closed the door to instrumental music from the outset. He was thus unable to show the most important aspect: the fact that all voice leading remains in the final analysis one and the same, even if it appears in a new guise in instrumental music because of the different circumstances present there.26

On the other hand, he admonished C. P. E. Bach for discussing foreground

prolongations without relating them to an understood, compositional model:

...the thoroughbass theory of Bach was faulty because, unfortunately, problems are shown there not in their origin but in an already advanced state. Thoroughbass shows us prolongations of archetypes (Urformen), without first having familiarized the reader with the latter in any way.27

Schenker summarized the flaws of both with respect to his notion of a hierarchical

compositional paradigm, one that would evolve into the Ursatz in his later writings.

25 Heinrich Schenker, Kontrapunkt: Erster Halbband: Cantus firmus und zweistimmiger Satz (Stuttgart und Berlin: Cotta, 1910), xxvii; trans. John Rothgeb and Jurgen Thym, ed. John Rothgeb, Counterpoint, Book I, (Ann Arbor: Muscalia Press, 2001), xxvii.26 Rothgeb and Thym trans., Counterpoint I, xxvii-xxviii.27 Ibid., xxviii.


18

He wrote, “in Fux’s theory, what is missing is the ‘future,’ so to speak—the

prolongations—, while [C. P. E.] Bach’s theory lacks the ‘past’—the archetypes

(Urformeri) ? 28 Both Fux and Bach provided theories of voice-leading; however,

neither provided a multi-leveled model that could explain the structure of tonal music

composed without a cantus firmus. Schenker, therefore, applied and expanded their

theories within his germinal ideas presaging the Ursatz, a compositional model that

could elucidate the nature of tonal compositions, including those that engaged the

technique of invertible counterpoint.

Although Schenker provided a way of understanding the contrapuntal

environment inhabited by tonal compositions—his was a theory that could elucidate

the rich and varied techniques of composers ranging from Bach to Brahms—he

discounted, ironically, invertible counterpoint as a viable subject of counterpoint

instruction. He wrote:

...the concept of so-called double counterpoint at the tenth or twelfth can have no validity. Double counterpoint therefore takes its place in the ranks of such fallacious concepts as the ecclesiastical modes, sequences, and the usual explanation of consecutive fifths and

29octaves.

This admission by Schenker is especially striking, since he looked backward to Fux

and C. P. E. Bach as exemplars, both of whom regarded invertible counterpoint as a

viable compositional technique. For instance, Fux wrote specifically about

28 Ibid., xxviii.29 Oster trans., Free Composition, §222, 78.


19

counterpoint at the octave, tenth and twelfth and C. P. E. Bach had written a small

essay on the technique o f writing invertible counterpoint “without a knowledge o f the

rules.”30 Although he characterized invertible counterpoint as a theoretical trifle in

Der freie Satz, this attitude did not hold throughout his oeuvre o f theoretical writings.

For instance, he cast a more favorable light upon invertible counterpoint in

Kontrapunkt I, where he excluded it from the sphere o f voice-leading, and classified

it, along with fugue and canon, as procedures “which properly have their place in the

•5 1

theory o f [musical] form,” the topic o f which provided the capstone to Der freie

32 * •Satz. His attitude towards invertible counterpoint, therefore, was mixed. Indeed, he

made reference to and incorporated principles o f invertible counterpoint throughout

the majority o f his published writings. In addition to this, his personal papers contain

copied-out exercises o f invertible counterpoint from other theorists, including those

o f Fux, Albrechtsberger, Cherubini, and Bellermann. He may have regarded

invertible counterpoint as a “fallacious concept” within the pages o f Der freie Satz,

but his reasons for holding this v iew are complicated and in need o f explanation. Just

as Schenker felt obliged to revise the contrapuntal theories o f Fux to accommodate

the behavior o f tonal music, so must we revise Schenker’s notion o f invertible

counterpoint as an integral component o f tonal composition.

30 See Fux, Gradus ad Parnassum, 174-217; C. P. E. Bach, “Einfall, einen doppelten Contrapunct in der Octave von sechs Tacten zu machen, ohne die Regeln davon zu wissen,” in F. W. Marpurg, Historisch-kritische Beytrdge zur Aufriahme der Musik (Berlin: J. J. Schiitzens Witwe, G. A. Lange, 1757), 3:167-181; trans. Eugene E. Helm, “Six Random Measures ofC . P. E. B a c h Journal o f Music Theory 10/1 (1966): 139-151.31 Rothgeb and Thym trans., Counterpoint I, 16.32 See Schenker, Der freie Satz, §322.


20

These changes to Schenker’s understanding of invertible counterpoint go hand

in hand with those that occurred to the three factors necessary for creating invertible

counterpoint itself: namely, the ways that tones can be stacked vertically, the rules

dictating how vertically stacked tones may be approached simultaneously in the

horizontal direction, and a corpus of preexistent melodies that may be used as the

basis for creating new compositions. Example 1.6 diagrams the development of these

three factors: Example 1.6a shows the general template of the three factors, each of

which is connected with double-headed arrows, demonstrating the effect that each has

on the other. Example 1.6b shows the parallel organum of the Musica enchiriadis,

where parallel fifths are allowed; Example 1.6c shows the changes in contrapuntal

rules in effect during the time of Tinctoris, specifically, the reevaluation of the fourth

as a dissonance and the prohibition of perfect parallel consonances; Example 1.6d

illustrates the revisions to contrapuntal theory as found in the writings of Schenker,

including the change in contrapuntal environment from intervals to triads and Stufen,

and the replacement of the cantus firmus with the Ursatz and the application of

transformational levels.

And it is the Ursatz which is of primary concern in this chapter, since it is the

conceptual engine that runs the Schenkerian project: voice-leading transformations as

applied to voice-leading levels flow out from this contrapuntal motor. The Ursatz

would be Schenker’s way of elucidating the structure of tonal music, a repertory that

would part ways with the cantus firmus as its underlying melodic foundation. But it

is this same conceptual engine, the Ursatz, that drives invertible counterpoint


21

composed within the tonal idiom. And herein lays the paradox: Schenker used voice-

leading transformations as a way of deriving tonal pieces, but he criticized the

technique of invertible counterpoint, which resulted from the application of voice-

leading transformations. To reconcile this paradox, we must reevaluate the received

notions of invertible counterpoint and the Ursatz to show that both concepts, in

essence, work hand in hand.

In efforts to understand how invertible counterpoint engages with Schenkerian

theory, we must first gain a clear definition of invertible counterpoint itself. For just

as the three factors used to create invertible counterpoint changed throughout the

history of music, so did the definitions used to describe it. To be sure, invertible

counterpoint is the practice of exchanging registral positions of melodies, enabling

one melody to appear above and below another, but this provokes the following four

issues, each of which will be discussed in turn. First, what determines which

intervals are used to transpose and invert the registral positions of melodies? As

stated at the beginning of this chapter, the twelfth, tenth, and octave are the usual

intervals used for transposition, but theorists such as Zarlino only addressed

counterpoint at the twelfth and tenth. Second, what elements are being exchanged

when melodies are inverted? Third, what aspects of invertible counterpoint did

Schenker—knowingly or unknowingly—integrate into his theory of tonality? Fourth,

what further relationships can be made between Schenker’s theories and invertible

counterpoint?


22

1.2. Intervallic Spaces of Invertible Counterpoint

According to Zarlino, two primary conditions must be satisfied when writing

invertible counterpoint. First, both voices participating in the inversion must

maintain their melodic character. That is, one must be able to recognize melodies as

being the same after they have been transposed via particular intervals. Second, and

relevant to our current discussion here, voices can only be transposed by intervals that

exist within the triad (this condition was only made tacitly by Zarlino). In other

words, melodies may be transposed by only a twelfth (or fifth), tenth (or third), or

octave.

1.2.1. Triadic Spaces: The Twelfth, Tenth, and Octave

Zarlino gave two types of intervals in which invertible counterpoint may be

written: the twelfth and tenth. In the case of the twelfth, he wrote that one could

produce this type of counterpoint in the following way:

raising the lower voice by an octave and dropping the upper by a twelfth...From this it may be seen that the counterpoint of the inversion is much different from the principal, and its harmony is very different also. This is called double counterpoint at the twelfth. One might also place the upper part an octave lower in the bass and raise the lower part a twelfth. The inversion would not be altered; so it would be

-3 -3

superfluous to discuss the procedure.

33 Marco and Palisca trans., The Art o f Counterpoint, 161-62.


23

He thus provided two ways in which to create invertible counterpoint at the twelfth.

Both ways involved transposing one part by a twelfth and the other by an octave in

opposite directions, respectively. Although he gave two options, he gave a musical

illustration of only the first one (refer to Example 1.7). Here, the inversion (on the

second page of the example) was obtained by transposing the upper voice of the

principal down a fifth (or twelfth) while moving the lower voice of the principal up

an octave.34

We may ask ourselves the following question: why did Zarlino illustrate only

downward transposition of a fifth? To answer this, we must consider two things.

First, he specified that after voices had exchanged places, both would retain the same

movements. In this case, we may understand “movement” as the melodic, note-to-

note progressions occurring in both melodies. Referring back to Example 1.7, the

lower voice of the principal (the original combination) retains the same melodic

movements within the inversion since it is displaced by an octave, thus reproducing

the same set of pitch classes. Likewise, the upper part of the principal also retains the

same melodic movements within the inversion after being transposed by a fifth.

Second, downward transposition of a fifth kept the inverted upper melody (now in the

lower voice of the inversion) within the same mode, albeit transposed to a g-

34 Zarlino used the downward transposition o f a fifth in his illustration— rather than a twelfth, as per his instructions— which, when coupled with an upward transposition o f octave in the other part, produced an inversion with voices separated by a twelfth. Thus, we may infer from this observation that Zarlino was more interested in the product o f inversion at the twelfth (voices separated by a twelfth), rather than the process o f inversion at the twelfth (transposing one voice o f a pair o f voices up or down a twelfth).


24

final.35 In the case of Example 1.7, the upper melody of the principal is built on the

second, or Hypodorian mode; the final of this mode is D and its range (or ambitus)

consists of a fourth from a to d1 and a fifth from d1 to a1.36 As a complement to this

upper part, the lower part of the principal is built on the first, or Dorian mode; its final

1 37is also D but its range consists of a fifth from d to a and a fourth from a to d . Both

the first and second modes are built from the first species of fourth and fifth; a first-

species fourth contains the intervallic sequence tone-semitone-tone and a first-species

-> o

fifth contains the intervallic sequence tone-semitone-tone-tone. Despite the change

in final, the second mode transposed to g1 fills out an octave from d to d1, an octave

lower than the range articulated by the first mode melody in the upper part of the

inversion. This is shown in Example 1.8, which diagrams the arrangement of modes

in the principal and the inversion. Upward transposition would have placed the lower

melody into the wrong mode (ninth, or Aeolian) and, thus, would not have preserved

the melodic movements of the principal’s lower melody. Constrained by the limits of

35 Gioseffo Zarlino, Le Istitutione Harmoniche, Book IV, Chapter 17, 319; trans. Vered Cohen, On the Modes (New Haven: Yale University Press, 1983), 52. Zarlino discusses the transposition o f modes as a practical device: “These transpositions are useful and extremely necessary to every expert organist who accompanies choral music, and to players o f other sorts of instruments as well, in order to accommodate the sounds o f the instruments to the voices, which sometimes cannot ascend or descend as much as is dictated by the proper places o f the modes on the said instrument.”

See Cohen trans., On the Modes, 58.37 Ibid., 54.38 Ibid., 41-42; 58. For more on species o f fourth and fifth within modes, see Peter Schubert, Modal Counterpoint, Renaissance Style (New York: Oxford University Press, 1999), 10. Theorists grouped modes together that contained the same species o f fourth and fifth; the mode that featured the fifth on the bottom was referred to as the “authentic” mode while the one with the fourth on the bottom was referred to as either the “plagal” or “collateral” mode. Christoph Bernhard takes this up later in the mid-seventeenth century in his Tractatus compositionis augmentatus under the rubric o f the “association o f modes.” See Walter Hilse, “The Treatises of Christoph Bernhard,” in Music Forum III (1973): 133.


25

the modal system, Zarlino was forced into using a downward transposition of a

fifth.39

Zarlino’s quest for preserving melodic content via transposition of a twelfth

had significant harmonic consequences for counterpoint using three voices or more.

Transposition of a twelfth not only situated tones around a different final, it also

mapped them to different harmonic tones of the triad. For example, transposing a

triadic fifth downward by a twelfth transfers that tone onto a triadic root. This

operation can also be reversed (an upward transposition moves a tone from a triadic

root to a triadic fifth) as shown in Example 1.9. Here, the first three sonorities

correspond to those found within a hypothetical principal combination, the next three

to its inversion (the filled-in notes refer to the transposed melody). The bass of the

principal occupies the triadic root while the soprano of the inversion articulates the

triadic fifth. He made it clear that “in the principal the bass must never form a sixth

39 Of course, he could have incorporated a sharp into the key signature o f the upward transposition as he did a flat in the actual, downward transposition. Sharp key signatures, however, were rare in the sixteenth century. Indeed, flat signatures had been in use since the eleventh century. As well, partial key signatures— Example 1.7 is one such instance— had been in existence at least since the thirteenth century through to the sixteenth century. It was common for lower parts to carry a flat in the signature whilst none occurred in the highest. The reasons for partial key signatures, however, are varied. For more on key signatures, including partial key signatures, see the following: Margaret Bent: ‘Musica Ficta’, Grove Music Online ed. L. Macy (Accessed 20 September 2006), <http://www.grovemusic.com>: Willi Apel, The Notation of Polyphonic Music: 900-1600, 5th ed. (Cambridge, MA: The Mediaeval Academy of America, 1953), 102; Richard H. Hoppin, “Partial Signatures and Musica Ficta in Some Early 15th-Century Sources,” Journal of the American Musicological Society 6/3 (1953): 197-215; Edward E. Lowinsky, “Conflicting Views on Conflicting Signatures,” Journal o f the American Musicological Society 7/3 (1954): 181-204; Edward L. Kottick, “Flats, Modality, and Musica Ficta in some Early Renaissance Chansons,” Journal of Music Theory 12/2 (1968): 264-274.


http://www.grovemusic.com

26

with another part,”40 which is why the intervals above the bass (in both the principal

and inversion) are restricted to 8, 5, and 3. His preference for these consonances was

elucidated in his introduction to three-voice composition. He wrote:

Where such consonances are heard, the harmony is truly perfect. Now these consonances that offer diversity to the ear are the fifth and third or their compounds.41

Zarlino showed, albeit implicitly, that the limits of the triad—represented by the root

and the fifth—governed the nature of invertible counterpoint at the twelfth. Triadic

limits would have profound ramifications for his other demonstrations of invertible

counterpoint.

One of the ramifications was a different sort of invertible counterpoint where,

in Zarlino’s words, “the reply proceeds in contrary motion after the interchange of

parts, lower for upper and vice versa.”42 In two-voice counterpoint, the resulting

inversion produces the same intervals between parts as those found in the principal

(see Example 1.10). Expanded to three voices, triadic roots exchange places with

triadic fifths while triadic thirds remain invariant. Example 1.11 gives an abstract

schematic of this type of invertible counterpoint. Here, different note-heads signify

correspondences between the notes found in the principal and those within the

inversion. For example, a melody occupying the fifth of an E minor triad in the

40 Marco and Palisca trans., The Art of Counterpoint, 205-208.41 Ibid., 186.42 Ibid., 159.


27

principal will occupy the root of an F major triad in the inversion. This manner of

invertible counterpoint allows for more intervallic variety—Zarlino allowed the use

of sixths43—because more than one voice is being transposed among triadic members.

In Example 1.9 only one voice (the bass) is transposed by a twelfth from the principal

to the inversion (now the soprano). Three-voice invertible counterpoint in contrary

motion, however, entails that two voices—those that occupy the root and fifth,

respectively—exchange places with each other. The illustration in Example 1.9—

adapted to accommodate contrary motion between principal and inversion—

therefore, may be rewritten with 6/3 triads. As a means of demonstrating this type of

three-voice counterpoint, Zarlino provided a sample composition (shown in Example

1.12). The outer voices exchange places from principal to inversion; the upper voice

in the principal ends on a final of E and the lower voice in the inversion ends on a

final of C. The most obvious demonstration of triadic roots exchanging with triadic

fifths can be seen in the final cadence. Here, the lower voice of the principal ends on

the root (A), but when it occupies the upper voice of the inversion, it ends on the fifth

(G). Zarlino, therefore, showed how the exchange of triadic roots and fifths play a

pivotal role within compositions using invertible counterpoint, even ones that use

contrary motion.

Another ramification of the triadic principle that Zarlino implicitly showed

was the process of doubling-, adding a third voice that moves in parallel tenths or

thirds with another voice of an extant two-voice composition. He wrote,

43 Marco and Palisca trans., The Art of Counterpoint, 208.


2 8

To the bass of the principal and of the...inversion^] we may add a part a tenth above. Similarly we may add to the treble of the...inversion a part a seventeenth below. We may, finally, move the lower voice up an octave and add a part a tenth beneath the treble.44

His directions came without illustrations, yet they proved to be instructive later in the

treatise when he discussed writing invertible counterpoint at the twelfth in three

voices. Doubling was achieved in two ways. First, he made sure to use only contrary

or oblique motion in his two-voice principal combination; this ruled out the creation

of any parallel octaves or fifths with the addition of the third voice. Second, all

consonant rhythmic coincidences between both parts of the principal combination

were limited to the intervals of either an octave, fifth, or third (and their respective

compounds). By doing this, adding a tenth above the bass (or a tenth below the

treble) resulted in either 5/3, 8/3, or 3/3 sonorities (this alludes to our model of

counterpoint at the twelfth shown earlier in Example 1.9). The results of doubling are

shown in Example 1.13, which diagrams the addition of a third voice (depicted with

filled-in notes) either a tenth above the bass or a tenth below the soprano to each of

the three intervals referred to above. The harmonic constraints outlined in Example

1.13 were of paramount importance in his section on three-voice invertible

counterpoint using three independent melodies. Although his discussion did not

address doubling (specifically) as a compositional device, he still restricted his

harmonic palette within his principal composition to only those three sonorities

shown in Example 1.13. The illustration he provided, shown in Example 1.14,

44 Ibid., 169-170.


29

demonstrated counterpoint at the twelfth. Here, the inversion was produced by

transposing the lower voice upward by a fifth while moving the upper voices down

by an octave.45 All consonant vertical combinations within the principal—as

predicted—consist of only 8/3, 5/3, or 3/3 sonorities. As with any example of

counterpoint at the twelfth, he instructed that “the bass must never form a sixth with

another part,” but he conceded that “the other two parts [the upper voices] may form

sixths.”46 Fewer restrictions were possible between the upper voices since their

respective relationship to each other within the inversion was held invariant.

Therefore, sixths occurring between the upper voices in the principal become sixths

appearing above the bass in the inversion. Nevertheless, Zarlino modeled his

composition on sonorities expressing the intervals of the octave, fifth, and third—in

other words, the three primary intervals of the triad.

Since his examples of invertible counterpoint expressed intervals within the

triad, Zarlino included a discussion of counterpoint at the tenth. But his

experimentation with this sort of composition was not as successful as it was with

counterpoint at the twelfth. He wrote:

Then there is the inversion formed by raising the lower voice an octave and dropping the upper voice a tenth. It is known as double counterpoint at the tenth...The upper voice may also be lowered an octave, and the lower voice raised a tenth. This is the method I prefer,

45 He was able to use upward transposition since his bass melody was in the first mode (outlining an octave from d to d1); the resulting soprano in the inversion moved the final from d to a (at least in this melody), however, it preserved its modal identity— it was still in first mode—and it also outlined an octave from a1 to a2, or the cofinal o f the mode built on d.46 Marco and Palisca trans., The Art of Counterpoint, 205-208.


30

because the harmony is thus best maintained within the limits of the mode. On the other hand the principal counterpoint does not turn out so well as the inversion.47

He provided an example of his first option (the one that he preferred less), as shown

in Example 1.15. According to Zarlino’s principles, the inversion fails in two ways.

First, it ends on an imperfect consonance, D-F. In his words, “[I]f the last chord is

not an octave or unison, it would be easy to mistake the mode by assuming the top or

bottom note of the chord to be the final.”48 Thus, the final cadence of the inversion

produces modal ambiguity. Second—and related to the first problem—the transposed

melody of the inversion does not retain its modal affiliation to the principal. The

inversion’s lower melody is created by transposing the upper melody of the principal,

occurring in the sixth (Hypolydian) mode, to a melody occurring in the second mode

(Hypodorian), via downward transposition of a tenth. The inversion, therefore, not

only mixes finals—F in the upper melody with D in the lower melody—but also

disintegrates the authentic-plagal kinship. The severance of modal ties explains why

he preferred the other method, where the inversion results from the upward

transposition of tenth. By doing this, he implied that the resulting inversion would

consist of an authentic-plagal pair. But, he lamented, this inversion comes at a cost:

the principal “does not turn out so well” because it shares the same faults with the

inversion shown in Example 1.15, i.e., unmatched modes built on different finals.

For him, either the principal or the inversion suffered, depending on which method

47 Ibid., 162.48 Marco and Palisca trans., The Art o f Counterpoint, 85.


31

one chose to follow. Thus, he did not prefer counterpoint at the tenth because it

prevented perfect consonances appearing at final cadences and it broke the bonds

between authentic-plagal pairs.

Although he offered many ways of utilizing invertible counterpoint, Zarlino

restricted the intervallic spaces or transpositions in which he composed his invertible

compositions to those occurring within the triad: specifically, the twelfth (or fifth),

tenth (or third), and—as a participant in the previous two types of invertible

counterpoint—octave (or unison). He preferred counterpoint at the twelfth because it

preserved melodic intervals as they were transposed from the principal to the

inversion. Further, counterpoint at the twelfth allowed for both the principal and the

inversion to end on perfect consonances—an impossibility under counterpoint at the

tenth. His second “mode” of counterpoint—the one using contrary motion in the

inversion—also relied on the transposition of a fifth, since the bass-as-root in the

principal inverted into the soprano-as-fifth in the inversion. The harmonic spaces

within the triad became an important, albeit, tacit component of his compositional

strategy, since he restricted the use of three-voice sonorities—either in the principal

or inversion, but not both—to complete or incomplete root-position triads. Zarlino,

therefore, handed us the intervallic space that was to dominate tonal music of later

generations: the triad.


32

1.3. Invertible Counterpoint and the Elements Which It Inverts

Although we were able to use the triad as a yard stick against which to

measure Zarlino’s use of invertible counterpoint, there is no evidence that he or his

contemporaries did the same. We were able to surmise that the properties of the

triad—especially the relationship between the root and fifth—influenced the

technique of invertible counterpoint, but Zarlino explained this style of composition

only as a way of exchanging registers of voices or parts. He made no mention of

“triads,” “chords,” or “harmonic tones.” What his illustrations did do, however, was

give a set of directions—transpose and exchange voices via particular intervals—and

show the compositional results. The instructions that he gave, unfortunately, did not

explain how or why they worked. Composers of his time could, with enough

tinkering, devise new pieces as per his instructions. But his method focused more on

the process of invertible counterpoint—voices trade places given a set of

conditions—rather than its explanation. As a result, he showed only the end-product

of this compositional technique, rather than its inner workings. He could point out

the exchanged voices, but not the musical materials—be they intervals, pitch-classes,

or harmonic tones—of which they were composed. Due to this lacuna of knowledge,

Zarlino could not convey what elements were being exchanged when one performed

the operation of invertible counterpoint.


33

This gap in knowledge exists to this day, apart from a few exceptions.49 Most

contemporary textbooks define invertible counterpoint as a process that exchanges

registral positions of performing forces and inverts the intervals between them.

Richard S. Parks provides a case in point, answering the question of how to write two

parts that are invertible at the octave:

The answer is that the pitch relationships that are consonant in one instance may become dissonant when the voices—and therefore the intervals formed between them—are inverted; thus, consonant intervals that are potentially dissonant (when inverted) must be treated as dissonances.50

Intervals, however, are simply distances between elements, be they pitches, pitch-

classes, harmonic tones, etc. What is missing from his definition, therefore, are the

elements that are conveyed by each of the voices. To be fair, Parks explains the

nature of these elements in a footnote: “The pitch classes of each voice thus remain

the same but the octave registers change...”51 But the concept of “pitch-class”

denotes a musical substance devoid of tonal identity, orientation, or meaning. If we

49 One of these is Peter Schubert and Christoph Neidhofer, Baroque Counterpoint (Upper Saddle River, NJ: Pearson Prentice Hall, 2006), which considers harmonic factors as well as the exchange o f voices within invertible counterpoint. See pp. 279-285.50 Richard S. Parks, Eighteenth-Century Counterpoint and Tonal Structure (Englewood Cliffs, NJ: Prentice-Hall, 1984), 204. At the time o f writing this chapter, the most current definition o f invertible counterpoint, as transmitted by William Drabkin on Grove Music Online, states that “[T]he underlying principle o f invertible counterpoint is the inversion of intervals with respect to some fixed interval.” In “Invertible Counterpoint,” Grove Music Online, ed. L Macy (Accessed 3 October 2006) http://www.grovemusic.com.51 Parks, Eighteenth-Century Counterpoint, 204.



34

are to interpret invertible counterpoint from a tonal perspective, then we must

understand the elements of exchange as tonal entities.

Nevertheless, there is a long history of counterpoint treatises that explain

invertible counterpoint through the agency of the inversion table. An illustration of

such a table is shown in Example 1.16; the numbers represent intervals between two

parts. The top row indicates intervals as they occur within a contrapuntal

combination; the bottom row illustrates the intervals that result when the voices from

this combination exchange registral positions. The numbers in bold type represent,

according to Parks, “the consonant intervals that are potentially dissonant.”

Unfortunately, like Zarlino’s invertible compositions, these tables only document the

resultant effects of invertible counterpoint rather than providing an explanation of

how it is done. But before dismissing the inversion table, we must first understand its

place within compositional history. We will, therefore, examine some earlier

examples of inversion tables as a way of explaining their role within the story of

invertible counterpoint.

The inversion table, curiously enough, did not appear alongside Zarlino’s

discussion of invertible counterpoint.52 It was not until later in the seventeenth

century that inversion tables appeared in counterpoint treatises. One of the earliest

52 Paul Walker credits Zarlino as one o f the first theorists to discuss invertible counterpoint within a treatise. See Theories of Fugue from the Age ofJosquin to the Age of Bach (Rochester: University o f Rochester Press, 2000), 204-205. An earlier discussion of invertible counterpoint—albeit, one that does not go into nearly as much detail as Zarlino’s— exists in Nicola Vicentino’s L ’antica musica ridotta alia modernaprattica (Rome: Antonio Barre, 1555); reprinted in facsimile, ed. Edward E. Lowinsky (Basel: Barenreiter, 1959), 91 (verso); trans. Maria Rika Maniates, Ancient Music Adapted to Modern Practice (New Haven: Yale University Press, 1996), 287-290.


35

examples is found within J. A. Herbst’s translation of Giovanni Chiodino’s Arte

Prattica & Poetica (1653).53 Here, Herbst laid out the tables for counterpoint at the

octave, tenth, and twelfth (in that order) in a vertical presentation.54 Thus, not only

did he show inversion tables, he also included a discussion of counterpoint at the

octave—an interval of inversion that was missing from Zarlino’s treatise.55 The

inversion table, thus, became a new way of transmitting ideas pertaining to invertible

counterpoint.

Indeed, the use of inversion tables continued from this point onward: Berardi

copied Herbst’s vertical formatting in his Documenti armonici (1687) and Fux

presented the same tables in the (now) more familiar horizontal presentation in

Gradus ad Parnassian. The tradition of using tables took a left-tum with Marpurg’s

Abhandlung von der Fuge (1753-1754), which included not only tables illustrating

counterpoint at the octave, tenth, and twelfth, but also ones for counterpoint at the

ninth, eleventh, thirteenth, and fourteenth. Treatises that followed this lead include

Cours de contre-point et de fugue (1835) by Cherubini, Double Counterpoint and

Canon (1891) by Prout, Der strenge Satz (1929) by Bussler, and La Polifonia

53 Giovanni Chiodino, Arte Prattica & Poetica', trans. Johann-Andreas Herbst (Frankfurt, 1653).54 Ibid., 39,41-42.55 As well, Herbst’s examples were set for three or more voices; two voices were set in counterpoint to each other while another or more voices would “follow” one o f these voices (or both) in upper or lower parallel thirds. The added third part is reminiscent of Zarlino’s discussion of expanding a two-voice model to a composition with three voices, but the prevalence o f this technique in Herbst’s treatise illustrates doubling in thirds as an essential component o f this type o f counterpoint. The practice o f doubling in thirds would continue in the treatises o f Bernhard and Fux, among others. See Marco and Palisca trans., The Art of Counterpoint, 169-170; Hilse, “The Treatises o f Christoph Bernhard,” 169-173; Alfred Mann, The Study of Fugue (New York: Norton, 1965), 124-125.


36

nell ’Arte Moderna (1931) by de Sanctis. The tradition of including tables for

counterpoint only at the octave, tenth, and twelfth continued as well throughout

treatises, such as Grundliche Anweisung zur Composition (1790) by Albrechtsberger,

Traite du Contrepoint et de la Fugue (1846) by Fetis, Lehrbuch des einfachen und

doppelten Kontrapunkts (1897) by Richter, Lerbuch des einfachen, doppelten, drei-

und vierfachen Contrapunkts (1884) by Jadassohn, Kontrapunkt (vokalpolyfoni)

(1930) by Jeppesen, and Counterpoint (1947) by Piston. These two groups of

treatises correspond to different schools of thought concerning the technique of

invertible counterpoint, but they do not tell the entire story. There still existed many

treatises that did not include any tables at all but only explained invertible

counterpoint through examples. Some of these treatises include Esemplare o sia

saggio fondamentale pratico di contrappunto sopra il canto fermo (1774) by Martini,

Cours Complet d ’Harmonie et de Composition (1806) by Momigny, Kurzgefafites

Handwdrterbuch der Musik fur praktische Tonkunstler undfur Dilettanten (1807) by

Koch, and the Musik-Lexikon: Theorie und Geschichte der Musik (1882) by Riemann.

All of these treatises and others are documented in Table 1.1a (earlier

treatises) and Table 1.1b (later treatises). A glance at both shows that inversion tables

became more prevalent within treatises as time passed by. The earlier treatises,

however, show a less unified picture of counterpoint instruction: only eight of the

twenty-three treatises listed in Table 1.1a contain inversion tables—the remainder

teaches only by example. This variation in counterpoint instruction indicates that

there were (and still are) different ways of teaching the skill of invertible


37

counterpoint. More significantly, these differing methods of instruction—especially

the variation in inversion tables—highlight a general disagreement among theorists as

to what musical elements are being inverted. We will now investigate some of these

divergent perspectives.

1.3.1. Counterpoint at the Octave

As we have already seen, counterpoint at the octave was not discussed as

much in earlier treatises as it is today. But judging by the treatises listed in Table

1.1a, counterpoint at the octave started entering treatises in the early seventeenth

century, roughly fifty years after Zarlino and Yicentino wrote some of the first

discussions of invertible counterpoint. One of these treatises was Regole di Mvsica

by Rocco Rodio from 1609.56 His explanation of counterpoint at the octave was short

but instructive. Indeed, he showed that counterpoint at the octave grew out of the

same principles as those which Zarlino used to demonstrate counterpoint at the

twelfth: transpose melodies such that they maintain the same melodic movements. In

the case of counterpoint at the twelfth, transposition by twelfth preserved the melodic

intervals of a melody as it moved from the principal to the inverted combinations.

Rodio applied the same tack to counterpoint at the octave; however, he applied

transposition of a twelfth to both voices of a two-part example (see Example 1.17).

56 Rodio Rocco, Regole di Mvsica (Naples, 1609; reprint, Bologna: Fomi, 1981).


38

His instructions dictated that the upper voice be moved down a fifth while the lower

voice be moved up a fourth. As with Zarlino’s examples of counterpoint at the

twelfth, the lower melody preserves its melodic intervallic structure as it is transposed

up a fourth in the inversion. The downward transposition by fifth of the upper voice

to the lower voice of the inversion is not altogether successful, since there should be a

in the signature. Nevertheless, the transpositions by fifth reflect the continuation

of a compositional practice; he simply adapted the principles of this particular

practice to create a new type of invertible counterpoint. To be sure, he provided an

example of counterpoint at the octave using octave transposition—this is only after he

had first illustrated counterpoint at the tenth and twelfth—but it is curious that he did

not present such an illustration for his first example. We may infer, therefore, that

counterpoint at the octave—much like counterpoint at the twelfth—preserves melodic

integrity of lines, rather than their particular pitch-classes. This integrity breaks

down, however, with the inclusion of other intervals of counterpoint.

1.3.2. The Seven Intervals o f Counterpoint

In his treatise on fugue, Marpurg wrote that “just as there are seven sizes of

interval in each scale, so there are seven fundamental types of double counterpoint.”57

57 Friedrich Wilhem Marpurg, Abhandlung von der Fuge, vol. 1 (Berlin: Haude and Spener, 1753), 161-162. My translation: “Da es nun sieben vergleichen Intervallen in dem Umfange


39

His list of seven brands of counterpoint included both simple and compound

intervals, e.g., there is counterpoint at the second or ninth. One of his examples of

counterpoint at the ninth is shown in Example 1.18. He ingeniously fitted this type

of counterpoint into a triadic mold, but to do so, it was necessary to use suspensions.

Indeed, each of his illustrations included suspensions in the combination that is to be

inverted. The nature of these suspensions, however, changed from the original

combination (Hauptcomposition) to its inverted form. For instance, m. 3 of Example

1.18 shows a 7 - “6” suspension occurring between the two upper voices; when this is

58inverted, it becomes a more dubious 3 - “4” suspension (or is it a passing tone?). A

more pronounced problem occurs at the end of the inverted combination, where the

final note of the inversion in the bass has been changed from D—were this to adhere

strictly to transposition of a ninth—to F, so as to accommodate the final 5/3 sonority.

His modification of the inverted melody, slight though it may be, illuminates a

glaring problem in this type of counterpoint: there is only one pair of consonances—

one for the original combination and one for its inverted form—that may be inverted

at the ninth. In this case, the pair consists of two of the same interval, a perfect fifth

(refer to Example 1.19). Indeed, this property also applies to counterpoint at the

eleventh, where only the interval of a sixth may be used as a consonance in both the

original and inverted combinations.

einer jeden Tonleiter giebt: so kommen sieben Hauptgattungen des doppelten Contrapuncts heraus.”58 The suspended tone within each o f these suspensions is measured against an implied half- note, here diminished into eighth-notes.


40

Counterpoint at the thirteenth and fourteenth is somewhat easier, since they

each include two invertible consonances. (That is, there are only two discrete

consonances within each row of their respective inversion tables, disregarding octave-

equivalent intervals.) In the case of the thirteenth, the sixth and octave invert into

each other; at the fourteenth, it is the fifth and tenth. As we will see, each of the four

dissonant intervals of counterpoint lacks the consonant possibilities afforded by

counterpoint at the octave, tenth, and twelfth. The significance of this property for

Marpurg was that he was forced to modify his counterpoint to create a consonant,

triadic harmony. In so doing, he sacrificed the integrity of his melody at the cadence

of his example—the locus of formal expression of every tonal composition. Just as

there is a dearth of consonant possibilities under these dissonant intervals of

counterpoint, so there is a scarcity of melodic potential. Prout, who followed

Marpurg’s example, had this to say about writing counterpoint in the “rarer

intervals”:

There are, however, as will be seen directly, such difficulties connected with all these [rarer intervals], as to render them practically useless, except incidentally...It would be useless to give the student any rules for writing such counterpoints.59

59 Ebenezer Prout, Double Counterpoint and Canon (London: Augener, 1891), 105.


41

1.3.3. Counterpoint at the Octave, Tenth, and Twelfth

Perhaps as a reaction to Marpurg’s inclusive approach to invertible

counterpoint, Albrechtsberger put the brakes on this type of compositional theory. He

wrote:

Remark—Ancient masters also mention counterpoints in the second or ninth—in the fourth or eleventh—in the sixth or thirteenth—and in the seventh or fourteenth. Themes may certainly be invented, which might occasionally be capable of a two-part transposition, in the proportion of the above-mentioned intervals; but they would rarely produce a flowing, harmoniously satisfactory melody—they would consist of free, unprepared dissonants [sic], which are prohibited in strict style, and in free style would still continue harsh and uncouth.As we can gain no essential advantage from them, and can obtain a better result in a more perfect manner, by means of the three counterpoints just explained, these useless artifices are banished from our systems, and modem teachers seldom touch upon their possible existence.60

His comment directly focused on the melodic difficulties that one may encounter

under intervals other than the octave, tenth, or twelfth. But are there other reasons for

some theorists favoring only these three intervals of counterpoint?

Johann Philipp Kimberger (bap. 1721 - 1783), a contemporary of

Albrechtsberger’s, gave a clear reason in his Gedanken iiber die verschiedenen

Lehrarten als Vorhereitung zur Gugenkenntniss (1782).61 He wrote: “In order to

60 J. G. Albrechtsberger, Guide to Composition, trans. Sabilla Novello (London: Novello, Ewer, and Co., 1855), 213. See also Sammtliche Schriften iiber Generalbafi, Harmonie- Lehre, und Tonsetzkunst, ed. Ignaz Ritter von Seyfried (Leipzig, 1837; reprint, Kassel: Barenreiter-Verlag, 1975), third volume, 70.61 Johann Philipp Kimberger, Gedanken iiber die verschiedenen Lehrarten als Vorbereitung zur Gugenkenntniss (Berlin, 1782); trans. Richard B. Nelson and Donald R. Boomgaarden,


42

construct a fugue with double counterpoint which is exact, adequate, and sufficient,

one must know the three types of double counterpoint, which have their basis in the

harmonic triad, as: at the octave, at the tenth, and at the twelfth.” Thus for him,

invertible counterpoint was a contrapuntal expression of the primary intervals of the

triad. To demonstrate this principle, he constructed short illustrations of counterpoint

at the octave, tenth, and twelfth, all of which were based on a single two-part model,

as shown in Example 1.20. He first showed counterpoint at the twelfth by

transposing the upper melody of the model down a twelfth (see Example 1.21), and

then he showed counterpoint at the tenth by transposing the upper voice of the model

down a tenth (see Example 1.22). For counterpoint at the octave, he adapted his two-

part model to a four-voice setting, as shown in Examples 1.23a and 1.23b. (The

soprano and tenor are inverted to become the alto and soprano, respectively.) The

triadic context which he espoused becomes clear in these latter two examples; the

tones of the original two-part model (occurring here as b and b1) occupy the chordal

fifth of the opening sonority (hence the reason why these tones appear as the upper

voices in Example 1.23a and its inversion at the octave in Example 1.23b). His final

examples illustrated the technique of doubling in thirds, as had been shown by many

theorists before him; see Example 1.24. What had been an harmonic fifth in the tenor

of the opening sonority of Example 1.23 a now becomes an harmonic root as the bass

(e1) of Example 1.24; this occurs as fall-out from counterpoint at the twelfth (note

“Kimberger’s Thought’s on the Different Methods o f Teaching Composition as Preparation for Understanding Fugu e ” Journal of Music Theory 30/1 (1986): 71-94.62 Nelson and Boomgaarden trans., “Kimberger’s Thought’s, 80.


43

that this bass is the same as that of Example 1.20). Likewise, the harmonic fifth in

the soprano of the same opening chord of Example 1.24 (b1) is doubled a third below

with the chordal third (g#1); in a similar manner, this is the result of performing

counterpoint at the tenth. Since he based his contrapuntal practice on the structure of

the triad, he was able to flesh out the three-voice model in Example 1.24 to four

voices, as shown in Example 1.25. The lower voice, which occupied mostly roots in

Example 1.24, simply moves in parallel motion with chordal thirds placed above it in

Example 1.25. Kimberger’s examples were short in duration, but far reaching in their

consequences. They illustrated that counterpoint at the octave, tenth, and twelfth—

and the transpositions attendant to each—were related to the harmonic tones of the

triad.

1.3.4. The Problem Imposed by the Inversion Table: A Solution Provided by Cerone

Ironically, it was Pietro Cerone (1566 - 1625) who solved the very problems

of the inversion table. Instead of focusing on the difficulties of counterpoint at the

octave, tenth, and twelfth, he documented their compositional possibilities. Of course

at this time, the inversion table had not yet appeared in full form within counterpoint

treatises. Nevertheless, his illustrations of invertible counterpoint share many

features with the inversion tables that we know today. His examples concentrated on

two of the three factors necessary for invertible counterpoint, referred to earlier:


44

consonance/dissonance treatment and relative motion between voices (see Example

1.26). Relating to the first factor, he limited the intervallic relationship between

voices exclusively to consonances. As seen in Example 1.26, each example uses only

the following intervals: 3, 5, 6, or 8 (along with their compounds). Concerning the

second factor, he restricted the relationship between voices to contrary and oblique

motion. Each illustration in Example 1.26 shows two melodies creating either a

diverging wedge on the left-hand side or—after voices have been inverted—a

converging wedge on the right-hand side. His restriction of using only consonances

between voices moving in parallel or oblique motion resulted in contrapuntal patterns

occurring in the three types of invertible counterpoint: 8 - 6 - 3 inverts into 1 - 3 - 6

under counterpoint at the octave; 1 2 - 1 0 - 8 inverts into 1 - 3 - 5 under counterpoint

at the twelfth; 1 0 - 8 - 6 inverts into 1 - 3 - 5 under counterpoint at the tenth. These

intervallic patterns are consonances that occur in sequential order as the voices

progress, together, from one note to the next. For instance, the first three vertical

intervals occurring in the illustration of counterpoint at the octave in the upper left-

hand side of Example 1.26 produce the pattern 1 - 3 - 6; the next three intervals

replicate this pattern (albeit with compounds), 8 - 1 0 - 1 3 . After the voices have

been inverted (shown in the upper right-hand comer of Example 1.26), the pattern for

the first three vertical intervals reads as 15 - 13 - 10 (or 8 - 6 - 3). Cerone, therefore,

provided a solution to the inversion table by focusing on a particular compositional

possibility, that of intervallic patterns of consonances.


45

Through the agency of the intervallic pattern, Cerone highlighted three

properties of invertible counterpoint. First, he showed that strict invertible

counterpoint involves note-against-note settings of melodies. Such strict treatment

reflects the historical connection between counterpoint and discant: the practice of

note-against-note polyphony in music from the Medieval period.63 Second, he

demonstrated that strict invertible counterpoint can only use consonances. The

intervallic patterns represent ordered sequences of consonances that may be used for

the three intervals of counterpoint. Third, he showed that contrary motion (and to a

lesser extent oblique motion) between voices provides a contrapuntal paradigm that is

applicable to the three types of invertible counterpoint. Strict adherence to this type

of contrapuntal patterning ensures the avoidance of parallel perfect intervals.64 By

restricting his purview to strict consonance treatment and contrary/oblique motion,

Cerone was able to navigate the universe of invertible counterpoint with ease and

elegance.

Cerone’s illustrations did not demonstrate, however, the third factor relating to

invertible counterpoint, that of cantus firmus treatment. Although his illustrations

highlighted the role that consonance/dissonance treatment and relative motion play

63 See Janet Knapp, “Discant,” in The Harvard Dictionary of Music, ed. Don Michael Randel, 4th ed. (Cambridge, MA: The Belknap Press o f Harvard University Press, 2003), 243-245.64 Girolamo Diruta (1609), for example, employed similar note-against-note settings as illustrations of moving from one consonance to the next. See The Transylvanian (II Transilvano), Vol. II, trans. and ed. Murray C. Bradshaw and Edward J. Soehnlein (Hemyville: Institute for Mediasval Music, [1609] 1984), 47. Similar to this is an example by Angelo Berardi, which, referring to Diruta’s, shows how to move from one consonance to the next using contrary motion. See Arved Marita Larsen, “Angelo Berardi (1636-1694) as Theorist: A Seventeenth-Century View of Counterpoint,” Ph.D. diss., The Catholic University o f America, 1979, 234.


46

within counterpoint at the octave, tenth, and twelfth, they failed to show how they

should be applied to a cantus firmus. Indeed, he did not specify any meaningful,

horizontal span that should occur in both voices as they are counterpointed against

each other. By “meaningful,” I mean the melodic expression of harmonic intervals as

found within tonal music, including the third, fourth, fifth, sixth, and octave. As a

case in point, Cerone’s example of counterpoint at the octave (Example 1.26)

juxtaposes two spans of a ninth against each other: D up to E in the upper voice and D

down to C in the lower voice; these same spans are used in his illustration of

counterpoint at the tenth, shown at the bottom of Example 1.26. The absence of a

cantus firmus and the lack of specificity of span within his examples illustrate one

shortcoming of his explanation of invertible counterpoint. The two factors of strict

consonance/dissonance treatment and contrary/oblique motion would need to be

grounded in a paradigm that would uniquely express the content of tonal music. One

paradigm that fits this description is Schenker’s prototype of tonality, the Ursatz.

1.4. Schenker and Invertible Counterpoint

Schenker’s attitude towards invertible counterpoint was mixed, to say the

least. As demonstrated earlier, he dismissed this technique in Der freie Satz, but he

did not maintain this opinion throughout his published and unpublished writings.


47

This section will focus on three aspects that address his ambivalence towards

invertible counterpoint. First, we will reappraise the concept of combined linear

progressions, which he posited as a replacement for invertible counterpoint. Second,

we will focus on his unpublished notes within the Oster Collection, which contains a

sizable number of papers devoted to invertible counterpoint. Third, we will show

how the concept of inversion—and by extension, invertible counterpoint—fueled

Schenker’s understanding of consonant intervals and triadic construction.

1.4.1. Schenker’s Discussion o f Combined Linear Progressions

Schenker summarized the nature of the Ursatz in Der freie Satz. He wrote,

“the descending fundamental line and the melodically rising bass constitute the first

example of two linear progressions in contrary motion; this motion, regulated

according to strict counterpoint, indicates the path for what follows.”65 Here, a linear

progression (Zug) is a unidirectional, stepwise melodic line that connects two

harmonic tones of a single Stufe or two different Stufen. What followed from this

prototype were voice-leading transformations—such as the neighbor note, reaching-

over, and register transfer—applied at successive levels and which led the way to the

foreground of a tonal composition. He diagramed this process in his Figure 1 of Der

freie Satz, which provided an abstract picture of the Ursatz (see Example 1.27). The

two linear progressions—the fundamental line (Urlinie) and bass arpeggiation

65 Oster trans., Free Composition, §67, 32.


48

(Bafibrechung)66—are located at the top of the diagram and exist at the background

(.Hintergrund); directly beneath these progressions are the transformational levels

(Verwandlungsschichten) occurring at the middleground (Mittelgrund); the bottom of

the diagram illustrates “tonality” (Tonalitdt) occurring at the foreground

(Vordergrund). The linear progression, itself, was also a voice-leading

transformation;67 counterpointing two linear progressions against each other, such as

occurs at the background within the Ursatz as described by Schenker, could take

place at successive transformational levels and even provide a context for writing


Indeed, Schenker included an entire section on combining two or more linear

progressions (Von der Verbindung zweier oder meherer Ztige) within the part of Der

freie Satz devoted to the foreground.68 Subheadings within this section correspond to

the types of relative motion that may occur between two (or more) voices; these

include parallel, oblique, contrary, and mixed motion. In a similar manner as setting

a cantus firmus, he demanded that one of the linear progressions, within a

combination exhibiting any of the four types of relative motion, occur at an earlier

transformational level than the others. He clarified this point at the beginning of this

section:

66 The Bafibrechung is considered a linear progression, since it can be filled in with passing tones. For more on this, see Schenker, Der freie Satz, §§ 53 - 54, 54.6 7

68Schenker, Der freie Satz, §§113-114, 203-229. Ibid., §§221-229.


49

With the combination of two or more linear progressions, it is indispensable to fix at the background, middleground, and foreground, which of the linear progressions comes up as the leader. In comparison with the leading linear progression, the others are valid only as counterpoints, be they in outer or inner voices moving in parallel, oblique, or contrary motion. Indeed, after the leading linear progression has been firmly posited in either the lower or upper voice, the counterpoints are to be understood only as upper thirds, tenths, sixths or lower thirds, tenths, or sixths.69

He followed in the footsteps of his forbears by positing a melody that could receive

further development through the addition of contrapuntal voices. In this way,

Schenker’s conception of combined linear progressions—and specifically, the

structurally prior nature of the leading linear progression—provided a suitable, tonal

environment in which invertible counterpoint could flourish, much as it did within the

era of vocal polyphony.

But invertible counterpoint, ironically, was considered by Schenker to be

anathema to his notion of combined linear progressions. Just as he introduced

transformational levels as a surrogate for the cantus firmus, so he also explained

combined linear progressions as a replacement for invertible counterpoint (see the

introduction of this chapter). He chose to concentrate, therefore, on the derivation of

combined linear progressions via transformation levels, rather than the technique of

69 Ibid., §221. My translation. “Bei der Verbindung zweier oder mehrerer Ziige ist es unerlaBlich, aus Hinter-, Mittel- und Vordergrund zu bestimmen, welchem der Ziige die Fiihrung zukommt: Gegeniiber dem fuhrenden Zug gelten die anderen nur las Kontrapunkte, mogen sie in gerader, Seiten- oder Gegenbewegung, im AuBen- oder Innensatz verlaufen; je nachdem der fiihrende Zug bei der Unter- oder Oberstimme festgestellt worden ist, sind die Kontrapunkte nun als Oberterzen, -Dezimen, -Sexten oder Unterterzen, -Dezimen, -Sexten zu verstehen.”


50

invertible counterpoint. For him, the Ursatz became the engine that powered the

direction of tonal music, not the practice of inverting voices. Level by level,

transformations could be applied, including the combination of linear progressions,

until the foreground of an infinite number of tonal compositions had been reached. In

this way, invertible counterpoint had become a vestige of a bygone era that was in

need of excision from the body of compositional theory. For Schenker, invertible

counterpoint, at least judging by his comment as quoted in the introduction to this

chapter, had become superfluous.

As a counterforce to invertible counterpoint, Schenker used the notion of

combined linear progressions in Der freie Satz. Along with cataloguing these

combinations according to relative motion, other criteria were used for categorization,

including length of linear progression, size of intervals for parallel and oblique

combinations, and relative lengths of progressions for contrary combinations. The

length of each linear progression within a combination could span the distance of

either a third, fourth, fifth, sixth, or octave.70 For combinations moving in parallel

and oblique motion, a further distinction was made between those moving in parallel

thirds and those moving in parallel sixths. (It should be noted that oblique motion,

here, is treated as two linear progressions moving in, primarily, parallel motion

71against a stationary pedal.) For combinations moving in contrary motion, a further

distinction was made between those containing linear progressions of equal length

70 There is one exception: Schenker refers to one example o f a sixth-progression against a seventh progression, Der freie Satz, §228.71 See Schenker, Der freie Satz, §226.


51

and those of unequal length. These descriptive criteria for combined linear

progressions and the lengths discussed specifically in Der freie Satz are summarized

in Table 1.2. The table shows that he did not discuss each length for every type of

relative motion; the only category that did receive complete consideration was

parallel progressions moving in thirds. The data from Table 1.2 summarize the way

that combined linear progressions are categorized; however, they do not specifically

convey how the leading linear progression is determined.

As mentioned earlier, the leading linear progression occurs at earlier levels of

derivation than the counterpoints that are added to it. Schenker made this point clear

in his remarks at the beginning of the section on combined linear progressions.

Example 1.28 provides a good case in point, which reproduces his Fig. 95, a, Nr. 4,

from Der freie Satz. Here, the progression from a2 to f2 constitutes the leading

progression, spanning a third, while the bass voice moves with it in parallel tenths.

His summary of this combination of linear progressions is shown on the right-hand

side of the illustration in parentheses, which demonstrates that the lower voice exists

as a leap from A to d1 at an earlier level of derivation. Closer to the foreground,

however, the lower voice moves in parallel motion with the leading progression

(shown at the left-hand side of the illustration). This explains how the lower voice

may begin its linear progression on f1, even though this tone resides outside of the

governing A major triad (this is acknowledged in Example 1.28 via the asterisk

beneath the lower voice). The example demonstrates that the added counterpoints

may undergo changes, through subsequent levels, that even take them outside of the


52

governing harmony. The leading progression, however, must be constrained by the

criteria owing to all linear progressions; Schenker addressed this issue when he wrote

that “in a progression in parallel thirds, one must consider whether it remains within

the same chord or progresses to another chord, within the dimensions and space of the

11linear progressions.” More importantly, the illustration of Example 1.28

demonstrates that the leading third-progression in the upper voice provides the

melodic foundation upon which counterpoints are to be added at levels closer to the

foreground.

The leading linear progression, however, was not always identified in

Schenker’s graphs within the section on combined linear progressions in Der freie

Satz. This is the case in his discussion of progressions in contrary motion. The

significance of these combinations was highlighted by his opening comment in this

section, where he wrote the following:

Already, the forms of the Ursatz show linear progressions in contrary motion. The setting in contrary motion in the foreground depends, in the first place, upon the adjustment in the number of tones; thereafter, [it depends] in the particular case upon the ascertainable requirements of a particular composing-out.73

72 Schenker, Der freie Satz, §224. My translation. “Bei einem Terzen-(Dezimen-)Satz kommt es darauf an, ob er im selfben Klang bleibt oder zu einem anderen Klang fortgeht, femer auf AusmaB und Weite der Ziige...”73 Ibid., §227. My translation. “Schon die Ursatzformen zeigen Ziige in Gegenbewegung...Der Satz der Gegenbewegung im Vordergrund hangt in erster Reihe vom Ausgliech in der Zahl der Tone, sodann von dem im einzelnen Fall feststellbaren Bediirfnis einer bestimmten Auskomponierung ab.”


53

Despite the significance of contrary motion, he did not designate one progression to

be structurally prior to the other. A case in point is shown in Example 1.29, which

reproduces Fig. 98, Nr. 3c from Der freie Satz. Two linear progressions, a fourth-

progression from E - A and a fifth-progression from E - A, are juxtaposed against

each other; neither progression, however, assumes structural primacy. The upper

voice is distinguished as a string of thirds occurring above the fourth progression, not

as a progression on equal footing with the fourth- and fifth-spans identified above.74

Schenker’s ambivalence about designating one as the leader may originate from his

notion of the Ursatz, where, in his words, “the combination of fundamental line and

bass arpeggiation constitutes a unity.”15 In this light, he went on to say that “neither

the fundamental line nor the bass arpeggiation can stand alone.”76 Thus for Schenker,

contrary-motion combinations consist of two linear progressions occurring at the

same level of derivation.

Schenker’s opinion of contrary progressions, however, changed abruptly in

the following section devoted to combinations of progressions in mixed motion. His

most striking example of this phenomenon is shown in Example 1.30, which

reproduces his Fig. 99, Nr. la, b, and c from Der freie Satz. Here, he presented three

permutations of a four-part example from J. S Bach’s Fugue in C minor, unfinished

(BWV 906). Both parallel and contrary motion are illustrated in this example. He

presented this illustration as a way of downplaying the technique of invertible

74 Ibid., §228.75 Oster trans., Free Composition, §3, 11.76 Ibid.


54

counterpoint, while emphasizing the virtue of the leading linear progression. And

even though contrary motion occurs between the fifth- and fourth-progressions in

Example 1.30, he singled out one of these as the leading progression. He wrote,

We see the combination of a fifth-progression in the bass with a fourth-progression in contrary motion in the tenor, with upper thirds above the fourth-progression in the alto and upper tenths above the bass in the soprano. The fifth-progression of the bass alone leads, including the upper tenths that belong to it. This score diminishes in importance what the teachings of double counterpoint would otherwise show.77

That the leader is the fifth-progression, and not the fourth-progression, is of no

surprise. As we will see, he regarded the fourth as a derivative of the fifth.

Nevertheless, he presented this illustration as a counterexample to invertible

counterpoint. For him, the leading linear progression was consistent with his notion

of voice-leading transformations applied at successive structural levels, all emanating

from the Ursatz', invertible counterpoint—at least according to Schenker—could not

be afforded such a rich and multi-leveled interpretation.

Invertible counterpoint, however, may reside at deeper levels of structure than

Schenker had initially realized. With the concept of combined linear progressions, he

established that tonal counterpoint must involve melodies that move between

77 Schenker, Der freie Satz, §229. My translation. “Wir sehen die Verbindung eines Quintzuges beim Basse mit einem Quartzug in Gegenbewegung beim Tenor, mit Oberterzen zum Quartzug beim Alt und Oberdezimen zum Basse beim Sopran. Fiihrend ist allein der Quintzug des Basses einschlieBlich der zu ihm gehorenden Oberdezimen. Gegeniiber dieser Wertung tritt zuriick, was die Lehre vom doppelten Kontrapunkt sonst vermittelt.”


55

harmonic tones of Stufen. Matthew Brown has recently articulated this concept; he

wrote,

Indeed, Schenker went so far as to claim that “all melody is the composing-out of sonorities.” Since Stufen can only be expressed by major, minor, and diminished triads, it follows that melodic motion must always occur between triadic intervals, such as thirds, fourths,

7 8fifths, sixths, and octaves.”

Triadic intervals, therefore, are formed between harmonic tones of Stufen and linear

progressions are melodic spans within these intervals. These types of melodic spans

find their ultimate expression within the linear progressions of the Ursatz, where

contrary motion between the upper voices and the Bafibrechung creates strings of

intervallic patterns. Indeed, these patterns are the same as those shown earlier in

Cerone’s illustrations of invertible counterpoint. As a case in point, consider the 5-

line Ursatz shown in Example 1.31. Here, the numerals at the bottom of the

illustration, 1 2 - 1 0 - 8 - 5 , correspond to the intervals created between the outer

voices. They also correspond directly to the intervallic patterning found in Cerone’s

illustration of counterpoint at the twelfth, shown earlier. One difference between

Example 1.31 and that of Cerone’s is that the spans traversed by the linear

progressions in the present example are delimited by harmonic tones of Stufen, I - V

- 1, occurring at the background; the upper and lower voices of Cerone’s illustration,

however, do not traverse any identifiable span. That the intervals between the outer

78 Matthew Brown, Explaining Tonality: Schenkerian Theory and Beyond (Rochester, NY: University o f Rochester Press, 2005), 45-46.


56

voices of the 5-line Ursatz correspond to those that occur under counterpoint at the

twelfth suggests that the potential for invertible counterpoint may reside at levels

other than the surface.

Indeed, invertible counterpoint between harmonic tones of Stufen resides at

the background level; see Example 1.32, which reproduces a 3-line Schenkerian

Ursatz (Fig. 9 from Der freie Satz). Here, the Urlinie moves through 3 - 2 - 1, the

A A A

Bafibrechung through I - V - 1, and the inner voice through 1 -7 - 1. In addition to

this, the vertical disposition of harmonic tones in the upper voices is reordered with

the progression from one Stufe to the next. For example, the opening tonic Stufe

features its third in the soprano, whereas the following dominant Stufe features its

fifth in the soprano. A similar alternation between harmonic tones occurs in the inner

voice, which moves from the root of the tonic Stufe to the third of the dominant Stufe.

The succession of harmonic tones of these Stufen is illustrated in Example 1.33. The

annotations above the staff show how the disposition of harmonic tones changes

along with the progression from I to Y: the third appears above the root within the

tonic Stufe, while the third appears below the fifth of the dominant Stufe. The

crossing arrows suggest inversion of voices: this is not the case. Each voice—

soprano, alto, and bass—maintains its registral position; however, the harmonic tones

undergo inversion between adjacent Stufen. In an abstract way, the illustration in

Example 1.33 demonstrates how the harmonic tones undergo counterpoint at the

twelfth: the root of one Stufe moves up to the fifth of another Stufe and in so doing,

inverts the relative position of these harmonic tones to the thirds of both respective


57

Stufen. This process is analogous to transposing one melody up a fifth (or a twelfth)

while transposing another down an octave. The result of this harmonic-tone

inversion, as shown in Example 1.33, is that one vertical third, C-E, inverts into

another, B-D; this corresponds to thirds inverting to tenths (or vice versa) under

counterpoint at the twelfth. Significantly, this particular illustration of counterpoint at

the twelfth shows that invertible counterpoint is not just a technique available at the

surface, but one that resides at deep levels of structure.

Such a deep-level perspective can also be extended to counterpoint at the tenth

and octave. As shown previously with counterpoint at the twelfth in the 5-line Ursatz

in Example 1.31, intervallic patterning corresponding to counterpoint at the tenth and

octave occurs in the inner voices within this same prototype. Example 1.34 shows

how the alto reproduces the sequence found in Cerone’s illustration of counterpoint at

the tenth. Example 1.35 applies the same tack to the tenor voice, which, when set

against the bass, mirrors the set of intervals shown in Cerone’s example of

counterpoint at the octave. Thus, the three upper voices and the Bafibrechung (as

shown in Examples 1.31, 1.34, and 1.35) replicate the intervallic patterning as that

found under counterpoint at the twelfth, tenth, and octave. In this way, the potential

for invertible counterpoint is built into the intervallic structure of the Ursatz. This is

not to say that the Urlinie and Bafibrechung can be inverted at deep levels of

structure: they cannot. But the juxtaposition of the upper and lower voices of the


58

Ursatz at the deep-middleground sets up a contrapuntal template that can be

replicated, through recursion, at later levels of derivation.79

All three intervallic sequences produced between the three upper voices and

Bafibrechung articulate contrapuntal patterns created between harmonic tones,

fulfilling Schenker’s demand that three-voice counterpoint express complete, triadic

harmonies wherever possible.80 He alluded to the polyphony of harmonic tones

within a triadic environment when he wrote that “the harmonic degrees [Stufen] are

inextricably bound up with counterpoint.”81 Part of this counterpoint is the harmonic

tones of Stufen, which continually change disposition from one Stufe to the next and

also create intervallic patterns with the Bafibrechung corresponding to counterpoint at

the twelfth, tenth, and octave. Combined linear progressions also play a role in this

counterpoint, since they provide melodic connections between the harmonic tones of

Stufen at later levels of derivation. In this way, the potential for invertible

counterpoint is inextricably bound up with the Ursatz.

79 For more on recursion and how it applies to Schenkerian theory, see Brown, Explaining Tonality, 70.80 Some of these harmonic tones exist at later levels o f derivation.81 Oster trans., Free Composition, §79, 36.


59

1.4.2. Schenker’s Notes on Invertible Counterpoint Contained within the Oster Collection

Schenker’s most explicit references to invertible counterpoint are illustrated

through his notes contained within the Oster Collection.82 Significantly, all of these

notes are contained within File 83 of the collection, entitled “Studies for a Theory of

Form.” The file consists of eleven folders, the last of which contains the notes

mentioned above, as well as those on fugue. Indeed, the second item of Folder 11 is a

title-page bearing the following words: “Dopp. Kp....Fuge als Satz.” (“double

counterpoint...fugue as composition”).83 The time-frame in which the notes from File

83 were written includes the years 1911 to 1923, which dates these materials to the

period between the publications of Kontrapunkt 7(1910) and Kontrapunkt 7/(1922),

respectively. See Table 1.3, which lists the dated items from File 83. Curiously,

none of the notes from Folder 11 (Items 443 - 534) are dated, thus making it

impossible to establish their exact origin.

The contents of Folder 11 include annotated examples from counterpoint

JM

treatises, annotated transcriptions of fugues (primarily authored by Fux), and fugues

82 The Oster Collection: Papers of Heinrich Schenker: A Finding List, ed. Robert Kosovsky (New York: New York Public Library, 1990).83 Ibid., File 83, Item 444.84 These treatises include the following: Johann Georg Albrechtsberger, Grundliche Anweisung zur Composition (Leipzig: Breitkopf: 1790); Heinrich Bellermann, Der Contrapunkt; oder, Anleitung zur Stimmfiihrung in der musikalischen Composition (Berlin: Julius Springer, 1862); L. Cherubini, Cours de contre-point et de fugue (Paris: Maurice Schlesinger, 1835); Schenker’s notes in the Oster Collection, however, refer to example numbers that are found in the German translation, Theorie des Contrapunktes und der Fuge, trans. Franz Stoepel (Leipzig: Kristner, 1835) and the English translation, A Treatise on Counterpoint & Fugue, trans. Chowden Clarke (London: Novello, 1854); Johann Joseph Fux, Gradvs ad Pamassvm, oder Anfuhrung Musikalischen Composition auf eine neue, gewisse, und bishero noch niemahls in so deutlicher Ordung an das Licht gebrachte Art uasgearbeitet


60

written by Schenker himself. We can relate his studies to the three factors necessary

for creating invertible counterpoint, listed at the beginning of this chapter. First, he

highlighted the intervals (consonances and dissonances) and harmonic tones that

could be used within invertible counterpoint and explained how they changed under

inversion. Second, he demonstrated the types of relative motion that could be used

between voices under the different intervals of counterpoint. Third, he studied and

composed fugues as a way of learning tonal composition without using a cantus

firmus.

As a way of showing what could be used for vertical stacking within tonal

composition, Schenker focused on consonances and dissonances, and by extension,

harmonic tones and non-harmonic tones. With respect to consonances, he identified

those that would transform into other consonances under the three intervals of

counterpoint in File 83, Item 478 of the Oster Collection (see Example 1.36). In the

middle of the page of this sketch, we see how he summarized the types of

consonances that could be used: 3, 6, and 8 could be used for counterpoint at the

octave and tenth; 3,5, and 8 for counterpoint at the twelfth; and 3 and 8 could be used

for all three intervals of counterpoint. The consonant intervals of 3, 5, 6, and 8,

therefore, could be used as consonances within a composition, depending on which

interval of counterpoint was being employed. The recognition of these consonant

intervals was made explicit through his sketches, since he often wrote them above the

von Johann Joseph Fux, trans. Lorenz Mizler (Leipzig, 1742); Gustav Nottebohm,Beethoven’s Studien: Erster Band: Beethoven’s Unterricht bei J. Haydn, Albrechtsberger und Salieri (Leipzig und Winterthur: Verlag von J. Rieter-Biedermann, 1873).


61

staff, especially at the beginning of measures. File 83, Item 446 provides an

illustration of this practice (see Example 1.37), which shows demonstrations of

counterpoint at the tenth taken from Mizler’s translation of Fux (identified from now

on as “Fux-Mizler”), all based on Fux’s d-dorian cantus firm usP The first exercise

in the middle of the page, Table XXXI, Fig. 9 from Fux-Mizler, illustrates how he

singled out the consonances, 5, 6, 8, or 10, at the beginning of each measure. He did

the same with the next example, Table XXXIII, Fig. 1 from Fux-Mizler, where the

counterpoint from the former exercise has been transposed down a tenth in the latter.

He thus showed which intervals retain their consonant character after they have been

inverted. Translated into triadic parlance, Schenker showed, albeit indirectly, that

harmonic tones maintain their triadic status under invertible counterpoint.

Schenker clarified the distinction between harmonic tones and non-harmonic

tones, slightly, in his sketches of suspensions within the context of invertible

counterpoint. Indeed, many of the illustrations within this folder of the Oster

Collection include examples of suspensions and how they behave after voices have

been inverted. Some of his notes, such as those found in File 83, Item 460, are based

on examples taken from Albrechtsberger’s Grundliche Anweisung zur Composition

(1790), pp. 326-350, which deals with counterpoint at the twelfth (see Example 1.38).

Within this section of the Anweisung, Albrechtsberger established seven rules that

one should follow when writing counterpoint at the twelfth; Schenker focused on the

85 These illustrations are taken from table XXXI, Fig. 9 and XXXIII, Figs. 1, 2, 4, and 5 in the separate book o f tables in Fux-Mizler (1742).


62

second of these rules, since it addressed the issue of sixths transforming into sevenths.

The second rule reads as follows:

Second rule: one is not permitted to skip to a major or minor sixth, because in the inversion, this is understood as poor treatment of a major or minor seventh. N. B. The augmented sixth, however, can be approached through a skip quite well, because in the inversion, it becomes the diminished seventh. Within harmonic settings, one is allowed to have two [sixths] of different sizes follow one another; one can also create a suspension with the lower voice. Two sixths, however, are not allowed to follow one another if the first is major and

o r

the second is augmented.

Schenker summarized this second rule by writing “6 = 7” on his sketch. The

examples in the Anweisung featured arabic numerals for every interval created

between a pair of voices, whereas Schenker highlighted, primarily, sixths and

sevenths. He was clearly interested in the behavior of these intervals within

suspensions, since he skipped over examples in this section of the Anweisung that did

not contain them. These intervals were of interest, since the sixth was considered

consonant; the seventh, dissonant. Schenker alluded to the fact in his notes, however,

that both the sixth and seventh, as suspensions, are non-harmonic tones.

86 Albrechtsberger, Griindliche Anweisung zur Composition, 326. My translation. “Zweyte Regel: DaB man eine kleine oder groBe Sexte nicht sprungweise anbringen darf; weil daraus in den Versekungen eine frey angeschlagene groBe, oder kleine Septime entstunde. NB. Die iibermaBige Sexte aber thut zum freyen Satze sprungweise gut; weil in den Versezungen die verminderte Septime daraus wird... Stufenweise darf man die andem zwo anbringen; auch wenn die Unterstimme eine ligatur damit macht. Zwo Sexten aber diirfen nicht auf einander folgen, wenn nicht die erste groB, und die zweyte iibermaBig ist.”


63

We can see this principle at play in the illustration marked with a “P” in

Example 1.38, the second measure (marked with a “6” beneath it) of which features a

suspended A in the lower voice resolving to G. The corresponding measure in the

inversion (located directly beneath the previous illustration and marked with a “7”

beneath it) features a suspended E in the upper voice resolving to D. Although the

suspension in the former illustration is technically consonant, both suspensions

feature non-harmonic tones resolving to harmonic tones. In the first illustration, A is

a non-harmonic tone that resolves to the root, G. In the second illustration, E is a

non-harmonic tone that resolves to the root, D (or possibly the third of a B diminished

triad). He showed that dissonant suspensions—ones that employed suspended non

harmonic tones resolving to harmonic tones—preserved their behavior under

invertible counterpoint. More globally, Schenker showed that both harmonic tones

and non-harmonic tones remain invariant under invertible counterpoint.

Along with demonstrating which tones could be vertically stacked under

invertible counterpoint, Schenker also demonstrated the types of relative motion that

are necessary for invertible counterpoint. In the case of counterpoint at the tenth, he

consulted Fux-Mizler as a reference (refer back to Example 1.37). Near the top of

this sketch, he demonstrated that consonances must never occur in parallel motion by

showing how they are transformed under inversion. Example 1.39 transcribes this

part of the sketch, illustrating that two consonances must never occur in direct

succession. Based on this evidence, he realized that parallel motion could not occur

at all under counterpoint at the tenth. In support of this claim, he paraphrased the


64

instruction from Fux-Mizler that one must use only contrary or oblique motion in this

type of invertible counterpoint.87 (In reference to Fux-Mizler, Schenker often cited

the page and line numbers; in this case, he cited p. 145, lines 19 - 20.) In addition to

this, he sometimes annotated musical examples with arrows, so as to accentuate the

contrary motion between voices. The musical illustration in File 83, Item 445 is a

good case in point; see Example 1.40, bottom of the page. Here, he used two arrows

to show how the downbeat of each measure is approached in contrary motion (that is,

if the counterpoint is not tied over the barline). As did many theorists before him,

Schenker recognized the importance of contrary motion when writing counterpoint at

the tenth.

Schenker also recognized the importance of contrary motion for the purpose

of doubling in thirds. The musical examples found in File 83, Item 454 (second

page), illustrate this procedure; see Example 1.41. All of the illustrations are taken

from Albrechtsberger’s Anweisung (287-290), which are contained in a section of the

treatise devoted to counterpoint at the octave (277-296). The top of the second page

of Item 454 shows the contrapuntal model (Hauptsatz); however, Schenker added

annotations not found in the original: he used arrows to designate contrary motion

between voices at every downbeat, and he only designated intervals between voices,

via arabic numerals, at the beginnings of measures (unlike Albrechtsberger, who

indicated [almost] every interval occurring between voices).88 The same manner of

indicating downbeat consonances continues into the next illustration, which shows

87 Fux-Mizler, Gradus ad Parnussum, 145.88 Albrechtsberger, Grundliche Anweisung zur Composition, 287.


65

the voices of the model being inverted at the octave. We can see that Schenker was

highlighting Albrechtsberger’s instruction to only use the third, sixth, and octave in

metrically strong positions, which was paraphrased at the bottom of the first page of

Item 454 (contained within a drawn, partially-closed box set next to an asterisk). The

instruction reads as follows in the Anweisung: “In the model of a two-part

contrapuntal setting of this kind, only the third, sixth, and octave should appear,

alternately, in strongly accented positions.”89 The remaining illustrations, as copied

from the Anweisung, demonstrate the concept of doubling by adding upper thirds to

either or both of the voices of the model. His copies of these doublings emphasize

how this procedure is done: e.g., he underlined “decima acute” at (a) and (b) and he

used upward arrows to indicate that upper thirds, not lower thirds, are added to either

voices of the model. This procedure results in three-voice and four-voice settings,

depending on the number of voices doubled. Variations in this procedure occur, such

as inverting the doubled upper voices from parallel thirds to sixths (second from the

bottom of page two, Item 454),90 and adding two voices a third lower to the pair of

voices of the model (bottom of page two, Item 454), which Albrechtsberger

acknowledged as being the same as counterpoint at the tenth.91 Schenker thus

demonstrated that contrary motion occurring between two voices of a contrapuntal

model was a necessary condition for the purpose of doubling in thirds.

89 Ibid., 287. My translation. “Wenn in der Hauptcomposition eines zweystimmigen Contrapunctischen Satzes von dieser Gattung nur die Terz, die Sexte, und die Octave immer wechselsweise als Streichnoten erscheinen...”90 Discussed in Albrechtsberger, GriXndliche Anweisung zur Composition, 289.91 Albrechtsberger, GriXndliche Anweisung zur Composition, 290.


6 6

In addition to these sketches, Schenker also took notes concerning fugal

composition. As a point of departure for his study, he focused primarily on Fux’s

fugues in the Gradus. Fux had written a number of fugues based on the same subject

in D minor, and Schenker had copied and studied a number of them. One of these is a

four-voice fugue, catalogued as File 83, Item 447 (second page), as shown in

Example 1.42.92 Although this fugue originally appeared in the Gradus, other

authors, whom Schenker had consulted, included it within their teachings on

composition, including Bellermann (1862) and Nottebohm (1873). These latter

sources are of interest here, since both used this fugue as a way of demonstrating the

cadential structure of a typical three-voice fugue. For instance, Bellermann described

its three-part cadential structure:

In a three-voice fugue, the cadences should be set up in the best way, so that the first one occurs in the dominant of the key, the second one in the mediant, and finally, the third one (the close of the entire fugue) in the home key or tonic...These three cadences on the dominant, mediant, and tonic should be maintained in every fugue with three Durchfiihrungen, the first of which theorists call the “exposition” of the fugue, and the second the “repercussio” or “counterexposition” [Wiederschlag] .93

9 2 •Fux-Mizler, Gradus as pamassum, Table XXXV, Fig. 1.93 Heinrich Bellermann, Der Contrapunkt, vierte auflage (Berlin: Verlag von Julius Springer, 1901), 326. My translation. “In der zweistimmigen Fuge werden die SchluBfalle am besten so eingerichtet, daB man das erste Mai eine Cadenz auf der Quinte der Tonart macht, das zweite Mai auf der Terz und schlieBlich das dritte Mai (zum SchluB der ganzen Fuge) auf dem Hauptton oder der Tonica. [...] Durch jene drei Cadenzen auf Quinte, Terz und Tonica erhalt jede Fuge drei Durchfuhrungen, von dene die erste von einigen Theoretikem auch die Expositio der Fuge und die zweite die Repercussio oder der Wiederschlag genannt wird.”


67

Nottebohm also recognized this tripartite structure.94 Although Bellermann had some

reservations about Fux’s contrapuntal treatment within this fugue, he nevertheless

recommended its cadential structure as one of its redeeming features.95 Schenker,

too, acknowledged the three-part structure of the fugue (refer to Example 1.42) by

designating each section (Durchfiihrung) with a roman numeral, much in the same

manner that Bellermann had done with other fugues.96 For example, the four entries

of the D minor fugue each has a “I” next to it in mm. 1, 2, 4, and 6, respectively.

After this point, the roman numerals “II” and “III” designate the entries occurring

after the cadences in the dominant minor (m. 12) and the mediant (m. 17),

respectively. In addition to acknowledging the formal structure of this fugue, he also

highlighted the parallel thirds occurring either above or below the subject, an explicit

use of invertible counterpoint. For example, he wrote “dec. gravis” beneath mm. 4-5

in reference to the parallel thirds occurring beneath the subject and he wrote “10 ac.”

above mm. 8-9 in reference to the parallel sixths (inverted thirds) occurring above the

subject. Schenker thus recognized how invertible counterpoint, as demonstrated by

the parallel thirds set against the subject, interacted with form, as expressed through

the three sets of entries.

Schenker, however, did not limit his study of fugue to analyses: he also made

discoveries through his own compositions. Two fugues, composed by him and based

94 Nottebohm, Beethoven’s Studien, 71.95 Bellermann, Der Contrapunkt, 395. This specific cadential structure, however, did not apply to all modes, according to Bellermann. For Phrygian, the first cadence must occur on either VI or IV (p. 331), whereas in Mixolydian, the second cadence must occur on either II or IV (pp. 332-333).96 For example, see Bellermann, Der Contrapunkt, 390-392.


6 8

on the same subject, are contained within File 83, Items 510 and 525 (see Examples

1.43 and 1.44). Both fugues are in C major; the one in Example 1.43 is for two

voices and the other in Example 1.44 is for three. The two-voice fugue in Example

1.43 follows the tripartite model set out by Bellermann and Nottebohm, referred to

earlier: the first set of entries leads to a cadence in the dominant in m. 13; the second

set to a cadence in the mediant in m. 25; the third set to the final cadence in the tonic

in m. 38. Moreover, there are instances of invertible counterpoint peppered

throughout this short composition. To be sure, this fugue is not an invertible tour-de-

force; nevertheless, it implements some of the contrapuntal techniques demonstrated

in his earlier sketches within the Oster Collection. For example, the subject in the

soprano voice is accompanied by the alto in parallel thirds in mm. 7-8, though when

the voices invert in mm. 16-17, they proceed in parallel sixths. Likewise, the subject

in the soprano in m. 19 is placed above a 2-3 suspension; however, when the subject

moves to the alto voice in m. 28, it is placed beneath a 7-6 suspension. Schenker’s

two-voice fugue thus demonstrated the three-part fugal design as prescribed by his

forbears and illustrated how invertible counterpoint fit into this form.

Schenker’s three-voice fugue in Example 1.44 also follows a tripartite plan;

however, the specific nature of this plan deviates from that set out by Bellermann and

Nottebohm. One notable change is the first cadence in m. 18, which occurs in the

submediant (A minor), and not in the dominant as prescribed by Bellermann and

Nottebohm. In addition, the entries are distinguished not by their ordinal appearance

(as in his copy of Fux’s fugue in Example 1.42), but by their affiliation with either the


69

tonic or dominant harmonies. In short, the subject is designated with a “I”, while the

answer with a “V.” The three-voice fugue also demonstrates the technique of

invertible counterpoint: e.g., the answer in the upper voice is accompanied by the

middle voice in parallel thirds in mm. 7-8, while the subject in the lower voice is

accompanied by the upper voice in parallel tenths in mm. 31-33. Such treatment is an

example of counterpoint at the twelfth. The change in formal design illustrated in this

fugue may have been a way of proving that form was not merely a concatenation of

cadences, but rather a more abstract idea existing at deeper levels of structure,

namely, the transformational levels of the Ursatz. For him, the linear expression of

harmonic intervals—illustrated in this fugue via the subject and answer—was an

integral transformation that could take place at successive structural levels. As we

will see, Schenker’s concept of intervals, both harmonic and melodic, was intimately

connected to invertible counterpoint.

1.4.3. Intervallic Inversion

Schenker applied the concept of inversion, and by extension, invertible

counterpoint, to his notion of melodic and harmonic intervals. For example, he wrote

in Kontrapunkt I that “the [melodic] fourth is not found in the overtone series and

thus is to be regarded, both at the outset and always, merely as the inversion o f the


70

fifth .”97 In this light, some intervals were primary, while others (such as the fourth)

were considered to be derivative in nature. With this in mind, he laid out the range of

available melodic intervals into two groups: consonances and dissonances (please see

Example 1.45). The example clearly shows only three consonances (the octave, fifth,

and third) and a single dissonance (the second). As a counterpart to these intervals,

he listed their inversions as shown in Example 1.46. The example shows only two

inverted consonances (the sixth and fourth; the unison is left out) and a single

inverted dissonance (the seventh). To conclude his classification of the melodic

intervals, he identified two aspects owing to every interval with terminology

borrowed from Harmonielehre. He wrote that every interval is classified according

to the following criteria:

1. the aspect of size, and2. the aspect of development [represented by the ascending

direction]OSversus inversion [represented by the descending direction].

Thus for him, the primary melodic intervals bear an identical resemblance to the

harmonic intervals of the triad (this alludes to the concept of linear progressions). He

made this resemblance explicit when he wrote in Kontrapunkt /, “Nature as it is

manifested in the overtone series represents itself not only in the vertical direction

(the harmonic principles within the triad) but also in the horizontal direction (melodic

97 Rothgeb and Thym trans., Counterpoint I, §13, 79.98 Ibid., 78.


71

succession).”99 For Schenker, the overtone series was merely an organicist metaphor

of tonality that inspired his notion of the Ursatz; the triad—and the intervals

contained within in it—however, was a representative of this metaphor that could be

applied at successive levels of derivation.

Following his explanation of melodic intervals, Schenker also explained

harmonic intervals as originating from the triad. For example, he wrote that “...the

fourth, as a boundary-interval of the harmonic triad, is inferior to the fifth as original

boundary-interval in that it is gained only through the artifice of inversion, therefore

by a secondary method.”100 His comment not only demonstrated the derivative nature

of the fourth, but also illustrated his rationale for expunging the fourth from the list of

harmonic consonances—inversion. In light of this, he continued to say that “as an

inverted boundary-interval, it calls attention immediately to the fact that it lacks the

perfection of the fifth... .”101 For him, the fifth represented the harmonic limit of the

triad. In his words, “the fifth, although a smaller interval than the octave, is

nevertheless, by virtue of the overtone series, the boundary interval of the harmonic

102triad.” Based on his observations, we can establish a list of the available vertical

intervals as shown in Example 1.47. Here, the left-hand column illustrates the

vertical consonances (the same as those shown in Example 1.45) and the right-hand

99 Rothgeb and Thym trans., Counterpoint 1, §10, 78.100 Ibid., 112.101 Ibid., 112.102 Ibid., 79. Also see Heinrich Schenker, Harmonielehre (Stuttgart: Cotta, 1906), §11, 39; trans. Elisabeth Mann Borgese, ed. Oswald Jonas, Harmony (Chicago: University o f Chicago Press, 1954) §11, 26: “it should be clear now what is meant by my observation that it may be a wonderful, strange and inexplicably mysterious fact, but a fact, nevertheless, that the ear can penetrate only up to the fifth division.”


72

column shows inversions (only the unison and sixth are available). His distinction

between the octave and unison is unclear. On the one hand, the unison occupied the

first position in his hierarchy of perfect consonances: 1, 8, 5.103 On the other hand, he

wrote, “[I]t is the overtone series that affirms that the octave is the most perfect

interval, since it manifests identity of pitch-class [Ton] coupled with differentiation

only in pitch [H dhe\”m In addition, he later wrote that “consonance adds to a tone

only that which the tone carries within its own bosom by nature in the form of the

overtone series, regardless of whether the particular interval is an octave, fifth, third,

or only the artificial inversions of those [unison, fourth, sixth]!”105 Based on this last

admission, we distinguish the unison as an inversion of the octave within Example

1.47.106 The right-hand column also shows the sixth as a product of inversion. He

wrote that the sixth “is not even ratified by the overtone series itself, but is to be

regarded only as the inversion of the third.”107 He later did some back-pedaling to

explain why he did not discount the sixth as a vertical consonance due to its

derivative origin (as he did the fourth); first, the sixth does not produce a boundary-

interval, second, the sixth is “less sensitive” than the perfect fourth.108 Whether or

103 Rothgeb and Thym trans., Counterpoint I, 124.104 Ibid., 124.105 Ibid., 265.106 Though it might be the other way around. He also wrote that “[T]hus from the most perfect identity o f pitch-class and specific pitch, as represented by the unison, the path leads to the offspring o f the overtone series: to the octave, which, alongside differentiation of pitch, still repeats the pitch-class...” See Rothgeb and Thym trans., Counterpoint I, 125.107 Ibid., 125.108 Ibid., 125-126; 113. The vertical fourth causes problems for Schenker since it can represent either an inverted fifth, a suspension, or an accented passing tone. The sixth, however, does not create this kind of ambiguity (according to Schenker).


73

not we take stock in Schenker’s explanations, the sixth occupied a secondary role

with respect to the third as a product of inversion.

Thus, Schenker incorporated inversion into his understanding of melodic and

harmonic intervals, both consonant and dissonant. For him, inversion was a

necessary component of intervals, since he was determined to show that all

consonance grew out of the primary intervals of the harmonic triad: the octave, fifth,

and third. His notion of intervals, therefore, referred to the spaces existing between

the harmonic tones of the triad as listed in Example 1.48. By implementing this

notion, he was able to link all consonances, inverted or otherwise, directly back to the

well-spring that delivered them: the triad. We may or may not appraise the overtone

series as a viable explanation for consonance and dissonance, however, for Schenker,

it was a powerful icon that linked melody, harmony, and form.

That the intervals within the triad formed the basis of Schenker’s intervallic

thinking also presumed that harmonic tones could be inverted so as to produce a

change in their vertical disposition. For example, he applied harmonic-tone inversion

to his understanding of the triadic third. In this light, he wrote, “[A]s an inversion of

the third..., the sixth from the outset contradicts the roothood-tendency of the lowest

tone... .”109 In the case of the inverted sixth, he made it clear that it negated the

existence of a root in the lowest voice. Thus, he was not only discussing intervals

and their necessary inversions, he was also demonstrating that intervals were formed

109 Rothgeb and Thym trans., Counterpoint II, 9.


74

between harmonic tones, which themselves could be inverted. Therefore, the triad

was an integral component of invertible counterpoint. His position on the presence of

triads within three-part note-against-note counterpoint was made clear in the opening

pages of Kontrapunkt II, when he stated that three-voice counterpoint necessarily

demands for each note of the cantus firmus to be a harmonic tone within complete

triads, wherever possible.110 Thus for Schenker, intervals within three-voice

counterpoint articulated the distance between harmonic tones of the triad.

1.5. Conclusion

This chapter has helped to establish the role of invertible counterpoint within

the history of tonal composition and, more specifically, Schenker’s theory of tonality.

As demonstrated in the introduction to this chapter, the three factors necessary for

writing invertible counterpoint—these include vertical stacking, relative motion, and

preexistent melodic material—underwent many changes throughout the history of

music. By the time that Schenker was developing his theories of tonal music, triads

had become the basic building blocks of tonal composition, exercises in species

counterpoint dictated how relative motion between voices was to be observed, and the

110 Rothgeb and Thym trans., Counterpoint II, 4. This is obviously borrowed from Fux, who wrote “[HJere it is to be observed first o f all that the harmonic triad should be employed in every measure if there is no special reason against it.” Mann trans., The Study of Counterpoint, 71.


75

cantus firmus had become merely a pedagogical tool for budding composition

students. As mentioned earlier in this chapter, Schenker felt it incumbent upon

himself to upgrade these three factors so that they would be better suited to the

complexity of tonal compositions. Ironically, he also felt it necessary to abandon the

practice of invertible counterpoint altogether. But as I have shown, his opinion of

invertible counterpoint was not consistent throughout his published and unpublished

writings.

Based on these observations, I posit the following questions, which will be

answered in the following chapters of this dissertation. First, is invertible

counterpoint really embedded within Schenker’s Ursatzl My preliminary look at the

intervallic patterns created between the bass and the upper voices would suggest that

the relationship between invertible counterpoint and his notion of tonality is more

than coincidental; a more in-depth study of this relationship, however, is required. As

a way of going into more detail, I will look at Schenkerian Ursatze that begin on each

of the three Kopftonen: 3 , 5 , and 8. Second, if invertible counterpoint is embedded

within the Ursatz, then which particular musical forms express this property? As a

case in point, I will consider fugal form; this is a natural choice, since Schenker

himself studied fugue as a formal expression of invertible counterpoint. Here, the

interplay between fugal subjects and countersubjects is an instance of surface-level

invertible counterpoint; my investigations, however, will determine if the

compositional surface is really just a manifestation of properties that exist at earlier

levels of derivation. My study of fugal form will not just limit its scope to fugal


76

expositions, but will also explore other formal areas—the fugal episode is one such

location—where invertible counterpoint exerts its influence. Third, are there voice-

leading transformations that correspond to the properties of invertible counterpoint?

If the Ursatz shares properties with that of invertible counterpoint, then

transformations—these include, but are not limited to, reaching over, register transfer,

and voice-exchange—also share this provenance as they are applied at later levels of

derivation. Finally, I will conclude with ideas for further research that examine the

interaction of tonality with invertible counterpoint.


Chapter 2: Why Use Counterpoint at the Twelfth?

77

2.1. Introduction

In discussing invertible counterpoint, it is useful to start with counterpoint at

the twelfth, since Zarlino discussed this as his first example of this contrapuntal

technique, and the previous chapter demonstrated the potential of this technique as

occurring within a 3-line Schenkerian Ursatz at the background. As a way of

countering Schenker’s dismissal of counterpoint at the twelfth in Der freie Satz, we

will focus on the following four issues. First, we will review Zarlino’s definition of

invertible counterpoint at the twelfth and contrast it with problems of this technique

as identified by contemporary theorists. As an example of counterpoint at the twelfth,

we will examine a classic demonstration of this device within Bach’s Fugue in C

Minor from the Well-Tempered Clavier, Book I (BWV 847). Second, since this

example occurs within a fugal episode, we will discuss episodes as one possible

compositional application of counterpoint at the twelfth. In this light, we will review

some standard definitions of fugal episodes from other theorists. To place this

episode into a larger context, we will demonstrate how it functions within the overall

form of the fugue. Third, since the episode is used to link up recurring statements of

the fugal subject and answer, we will examine the structure of the subject itself.


78

Fourth and finally, we will investigate how invertible counterpoint at the twelfth

informs tonal organization at deep levels of structure, not just the surface.

2.2. Definitions/Problems of Counterpoint at the Twelfth

First, we must define invertible counterpoint at the twelfth. To do this, let’s

consult one of the first known definitions, as given by Zarlino:

Here a composition is so ingeniously designed that it may be sung with the parts interchanged...the reply is obtained...by exchanging registers while retaining the same movements...if the reply is sung at the twelfth there is no change in intervals.1

Zarlino’s focus on movement provides a concrete, melodic foundation upon which to

base invertible counterpoint.

More recent theorists, however, treat this practice as compositional walking

on eggshells. For instance, Piston warns us of impending difficulties:

But the circumstance that a sixth inverts to a seventh presents an obstacle to flexibility in melodic writing.2

1 Marco and Palisca trans., The Art o f Counterpoint, 159.2 Walter Piston, Counterpoint (New York: W. W. Norton, 1947), 178.


79

Tovey anticipates this cautionary advice:

...but it is in Double Counterpoint in the twelfth, which has the important property of changing sixths into sevenths.3

Gauldin chimes in,

Here the problem interval is the sixth, which inverts into a seventh.Therefore all sixths must be treated as though they were originallydissonant in nature.4

Gauldin uses an inversion chart to show how invertible counterpoint produces

an exchange of intervals (Example 2.1). The numbers represent intervals between a

pair of voices. The top row corresponds to intervals between voices within a

particular combination; the bottom row represents the change in intervals that occurs

between voices when this combination has been inverted at the twelfth. One can

verify the presence of counterpoint at the twelfth if the intervals between voices of

corresponding invertible passages add up to thirteen. This is evidenced in the chart of

Example 2.1, where a sum of thirteen results by adding together each of the vertically

aligned numbers. Moreover, each vertical pair of numbers corresponds to a pair of

operations by which to transpose voices up and down, respectively, of a particular

combination to its inversion at the twelfth. For example, one could transpose a lower

3 Donald Francis Tovey, J. S. Bach, Forty-Eight Preludes and Fugues, vol. 1 (London: The Associated Board of the Royal Schools o f Music, 1924), 29.4 Robert Gauldin, Eighteenth-Century Counterpoint (Englewood Cliffs, NJ: Prentice Hall, 1988), 187.


80

voice up a fifth and an upper voice down an octave to produce the desired result.

Again, both members of each pair of operations used to invert voices at the twelfth

can be added together to produce a sum of thirteen. This test for invertibility,

however, is more of a “spell-check” for invertible counterpoint, not a definition of

what it is.

As an illustration of counterpoint at the twelfth, many theorists have focused

on its use within J. S. Bach’s Fugue in C Minor from the Well-Tempered Clavier,

Book I (BWV 847). Piston was included among these theorists; as shown in Example

2.2, he isolated two passages from this fugue, mm. 5-7, and its inversion at mm. 17-

18.5 Example 2.3 applies the technology of the inversion table, shown earlier in

Example 2.1, as a way of demonstrating how the parallel passages of this fugue are

related to each other via counterpoint at the twelfth. Here, the soprano moves down

an eleventh and the bass up by a second. By doing the math, 1 1 + 2 = 1 3 , one can

confirm that these parallel passages are related to each other by counterpoint at the

twelfth. This test for inversion, however, is a heuristic for identifying the presence of

counterpoint at the twelfth, not a definition of what it is or method of how to use it.

Our reconstituted definition of invertible counterpoint at the twelfth reads as

follows: an exchange of voices that impedes flexibility in melodic writing, whereby

problematic sixths turn into dissonant sevenths. This definition is obviously

deficient, since it only identifies problematic mechanics of surface-level invertible

counterpoint at the twelfth. Instead of emphasizing problems, it is more beneficial to

5 Piston, Counterpoint, 178.


81

concentrate on possibilities. Counterpoint at the twelfth may invert particular

consonances into dissonances—this is true of any interval of counterpoint. But

focusing on consonances that remain consonant under inversion, however, provides a

more fruitful mode of inquiry of counterpoint at the twelfth.

As shown in the previous chapter, Cerone illustrated the possibilities of

counterpoint at the twelfth by showing only the intervals that remain consonant under

inversion (shown earlier in Example 1.26). Here, the consonances of 12, 10, and 8

invert into 1, 3, and 5, respectively. Both parts move primarily by step—save for

skips between F-A and B^-D—and the parts move in contrary motion, preventing any

possibility of creating parallel fifths. The intervals shown in Cerone’s example group

into three pairs of consonances: 12/1, 10/3, and 8/5; the member of each pair inverts

into the other. His example also provides a more “musical” illustration of the

possibilities of counterpoint at the twelfth than that shown in ordinary inversion

charts (such as in Example 2.1). Instead of mixing consonant with dissonant intervals

into a numerically ordered chart, Cerone’s example gives patterns of consonances—a

contrapuntal template—that one may use to fashion a musical exercise. Example 2.4

illustrates these patterns by grouping the consonant pairs into a table. Each row

consists of a pattern, moving left to right or vice versa, e.g., 1 2 - 1 0 - 8 or 1 - 3 - 5

are two such patterns. As demonstrated in the previous chapter, the wedge-shaped

patterns of Cerone’s illustrations are similar to those found in Schenkerian Ursatze,

where the Urlinien and the ascending Bafibrechungen converge toward the dominant

(see Example 1.31). One must make a slight modification to this patterning,


8 2

however, since the outer voices of an Ursatz comprise a counterpoint of harmonic

tones, not mere intervals. Thus, the possibilities of Cerone’s simple solution to

counterpoint at the twelfth have far reaching implications with respect to the structure

of tonality.

Within a tonal environment, counterpoint at the twelfth involves an inversion

of harmonic tones, not just a simple exchange of musical parts or intervals. Prout

alluded to this triadic foundation of invertible counterpoint when speaking about the

episodes from Bach’s Fugue in C Minor: “Notice...sixths in the pattern, the inversions

of which...become fundamental sevenths.”6 His recognition of “fundamental

sevenths” presumes the presence of triads and Stufen. Schenker also recognized that

sixths invert into sevenths; however, he made it clear, through his illustrations of

inverted suspensions in the Oster collection, that both of these intervals—within the

context of counterpoint at the twelfth—represent non-harmonic tones, however much

the sixth is consonant with the bass.7 In short, non-harmonic tones remain invariant

under inversion at the twelfth. Likewise, harmonic tones also remain invariant under

inversion at the twelfth; their type may change, such as roots inverting to fifths;

however, their status as harmonic tones will be preserved under counterpoint at the

twelfth. In this light, we can revise the chart of Example 2.4 to reflect better how

counterpoint at the twelfth necessitates an inversion of harmonic tones; refer to

Example 2.5. The table at Example 2.5a shows how the intervals have been replaced

6 Prout, Double Counterpoint and Canon (London: Augener, 1891), 80.7 Matthew Brown addresses the issue o f consonant non-harmonic tones and dissonant harmonic-tones in Explaining Tonality, 51-52.


83

with harmonic tones. This revised table demonstrates how harmonic tones are

transposed under counterpoint at the twelfth: e.g., a root occurring in one voice will

be transposed to a fifth when that voice undergoes inversion at the twelfth with

another voice. This interpretation of counterpoint at the twelfth is illustrated further

in Example 2.5b. Here, the example integrates information from the tables in

Examples 2.4 and 2.5a: the top of the example shows the consonant intervals and

their inversions; harmonic-tone inversion is listed beneath the intervals; finally,

harmonic-tone inversion is illustrated in musical notation. Based on this revised

interpretation, our new definition of invertible counterpoint at the twelfth reads as

follows: an inversion of voices that coincides with the inversion of harmonic tones,

whereby triadic fifths invert into triadic roots or vice versa. (Triadic thirds remain

invariant.)

These preliminary solutions focus on the possibilities of counterpoint at the

twelfth, rather than its problems. The solution of harmonic-tone inversion shows how

to write invertible counterpoint at the twelfth within a triadic environment. It fails to

show, however, how counterpoint at the twelfth functions within tonal spans or

complete pieces. Thus, this leads to our first question: where should one write

invertible counterpoint at the twelfth within a composition?


84

2.3. Compositional Applications of Counterpoint at the Twelfth

Most manuals of counterpoint instruction discuss invertible counterpoint as a

technique that one uses without giving explicit instructions as to where it should

occur within a composition. Indeed, Prout mentions this technical emphasis in the

introduction to his book Double Counterpoint and Canon: “It must be understood that

this is simply the preparatory technical work to the free Double Counterpoint used in

actual composition.”8 More recently, William Renwick devotes an entire chapter to

invertible counterpoint in his book on analyzing fugue. The analyses from this

chapter, however, “do not consider the compositional context within which invertible

counterpoint operates, but merely illustrate the way in which it functions at

foreground and high middleground levels of structure.”9 Just as counterpoint manuals

include preparatory sections on species counterpoint exercises, so they also present a

nuts-and-bolts approach to the mechanics of invertible counterpoint. Despite this

emphasis on contrapuntal technique, most discussions of invertible counterpoint

highlight its pivotal role within fugal composition. Prout, for example, identifies it as

one of the “main ingredients” of fugue, and he goes on:

But its utility is by no means restricted to this branch of composition.It frequently plays an important part in large instrumental works, such

gProut, Double Counterpoint and Canon, 5.

9 William Renwick, Analyzing Fugue: A Schenkerian Approach (Stuyvesant, NY: Pendragon Press, 1995), 107.


85

as symphonies and sonatas, and is even to be met with in vocal music which is not fugal.10

Within the context of fugue, Prout claims that invertible counterpoint is necessary

when using countersubjects in the exposition, since the countersubjects must appear

both above and below the subject and answer throughout the piece.11 In many fugal

expositions, however, counterpoint at the octave comprises the primary relationship

between subject and countersubject, though as Prout mentions, counterpoint at the

tenth or twelfth can also occur.12 Also within the context of fugue, he writes that one

finds invertible counterpoint within episodes, where “some of the best episodes are

made by repetition of an earlier episode with inversion o f parts. This gives the

requisite variety, and at the same time preserves the artistic unity.”13 Renwick also

values invertible counterpoint at the octave. For him, invertible counterpoint at the

octave relates closely to the concepts of strict counterpoint and structural levels as

understood from a Schenkerian perspective, whereas invertible counterpoint at the

tenth and twelfth “at best bear[s] a loose connection”14 with such concepts. Prout

integrates invertible counterpoint at the octave into fugal expositions and parallel

episodes, though as he makes clear, invertible counterpoint applies to a wide variety

of compositional genres and forms, thus provoking further research. Renwick injects

invertible counterpoint at the octave into Schenkerian theory—an important

10 Prout, Double Counterpoint and Canon, 58.11 Ebenezer Prout, Fugue (London: Augener, 1891), 3.12 Ibid., 72.13 Ibid., 103.14 Renwick, Analyzing Fugue, 107.


86

contribution to the study of tonal music—but disregards invertible counterpoint at the

tenth and twelfth as integral concepts and, more importantly, fails to address how

invertible counterpoint relates to large-scale form. The insights of both Prout and

Renwick open up avenues of inquiry in which to investigate how invertible

counterpoint at the twelfth relates to issues of form and tenets of Schenkerian theory.

Thus, even contemporary approaches to invertible counterpoint, ones that

engage Schenkerian methodology, neglect to discuss its formal function and continue

to identify it as a technique occurring closer to the surface, rather than as a property

of the background. Attempts to reconcile Schenkerian theory with the concepts of

invertible counterpoint remain incomplete for four reasons. First, theorists disagree

about the very nature of the Ursatz itself—the foundation of Schenkerian theory.

Some invoke the diagram of the Ursatz from Free Composition, shown in Example

2.6, suggesting that tonality grows out of two-part counterpoint.15 This illustration,

however, is an editorial distortion of Schenker’s original conception (shown earlier in

Example 1.27); this original model includes the background, the transformational

levels of the middleground, and the foreground.16 Second, voice-leading

transformations—ones that can engage multiple voices, and thus, invertible

15 David Neumeyer, “The Three-Part Ursatz,” In Theory Only 10/1-2 (1987): 3. “The Ursatz in Schenkerian theory is two-part counterpoint; the joining o f a diatonic descending upper voice with a diatonic ascending bass (Schenker 1935, 10-11).” Renwick also resorts to this same illustration of the Ursatz in Analyzing Fugue, 81.16 Example 2.6 is from Oster’s translation in Free Composition, based largely on Jonas’s edition o f Der freie Satz (Vienna: Universal Edition, 1956). For the original illustration, see the first edition in Schenker, Der freie Satz, 1 (volume of examples). Gregory Proctor and Herbert Lee Riggins investigate the transformational levels o f the Ursatz in “Levels and Reordering o f Chapters in Schenker’s Free Composition,” Music Theory Spectrum 10 (1988): 102-26.


87

counterpoint—necessarily require inner voices, as shown in Schenker’s illustration of

an Ursatz at the background starting from 3 (refer back to Example 1.32).

Unfolding and voice-exchange are two such transformations. Unfolding is a melodic

connection made between an upper and inner voice of a single harmony or of multiple

harmonies.17 In one example, given here as Example 2.7, Schenker shows “how the

unfolding figure is counterpointed by itself and thus creates a two-part setting...the

two voices which are spread out in the upper or inner parts are not to be taken for a

fourth-progression, since their origin lies only in the step of a second in the upper

voice.”18 Thus, he implies that the upper voice must appear below the inner voice,

though he does not express this explicitly. Voice-exchange also engages invertible

counterpoint (see Example 2.8). The example shows exchanges of pitch-classes, e.g.,

and D invert with E and F#, respectively, in the first illustration of Bach’s Fifth

Brandendburg Concerto. But his illustrations do not clarify, precisely, what is being

exchanged—is it voices, pitch-classes, harmonic tones, or all of the above?19 Third,

integration of invertible counterpoint with Schenkerian principles must necessarily

include all three intervals of counterpoint (at the twelfth, tenth, and octave), since all

relate to intervals of the harmonic triad. Fourth, theorists, in general, have not made

any explicit connection between invertible counterpoint, tonality, and form (as

understood in the Schenkerian sense). Indeed, Charles J. Smith acknowledges this

lacuna of knowledge in his expansive essay on form:

17 Schenker, Der freie Satz, §140.18 Oster trans., Free Composition, §141, 50.19 Neumeyer engages invertible counterpoint in his reconstructions o f the Ursatz in “The Ascending Urlinie,” Journal o f Music Theory 31/2 (1987): 282.


88

Another type of form left relatively unexplored by Schenker can be termed ‘ritomello’ form—the pattern displayed most clearly in large Baroque concerto movements, but also in a surprising number of solo instrumental and vocal pieces, including many of Bach’s fugues. This pattern can be most efficiently described as a journey through a series of keys, each of which is more or less equally weighted, and (almost) all of which are articulated by a recurring refrain. Since the number and interrelationships of these keys cannot be standardised, it is unclear whether ‘ritomello form’ should be regarded as a ‘form’ in the same sense as sonata or rondo form—but if it is, structural models for ritomello-form pieces will no doubt look quite different from those examined here [in this article].20

Thus, the lack of research concerning the role of invertible counterpoint at the twelfth

within formal function—specifically as it occurs within fugue—motivates our first

inquiry.

To establish a firmer connection between invertible counterpoint and form, we

examine the role of counterpoint at the twelfth within fugal episodes as one viable

application. The investigation employs the parallel, invertible episodes from Bach’s

Fugue in C Minor from the Well-Tempered Clavier, Book 1 as a specific test case

(refer back to Example 2.2). The study proceeds in three stages: first, we examine

standard definitions of episodes as understood by theorists; second, we locate the

episodes from the Fugue in C Minor within a musical form; third, we relate the

formal context of these episodes to the larger tonal scheme of the fugue, illustrating

how they ameliorate the transition from one stmctural station in the Ursatz to the

next.

20 Charles J. Smith, “Musical Form and Fundamental Structure: An Investigation of Schenker’s Formenlehre,” Music Analysis 15/2-3 (1996): 272.


89

Theorists understand episodes, primarily, in two ways: first, as a linking

section between fugal entries, and second, as a source of variety and contrast from

these entries. The former interpretation emphasizes structure and form; the latter

focuses on melodic and motivic relationships shared between entry and episode.

Robert Gauldin neatly summarizes the dual-purpose of episodes:

The episodes, which join the subject re-entries, are especially interesting. The composer’s craft and imagination are tested in these passages. They are often sequential and may be based on previous material or introduce new motivic elements. Their basic function is that of transitional modulation, linking the various keys of the recurring subject. Indeed, the most likely place for cadences within the fugue is at the conclusion of an episode.21

Here, he identifies the episode’s formal and structural purpose—spinning a thread

between entries—and he alludes to it as a locus of heightened interest, where the

composer displays her technical prowess by creating unity through motivic

integration of previously heard material, and variety through the introduction of new

motivic elements. From a contemporary, Schenkerian perspective, Renwick

emphasizes the first interpretation of an episode; he writes that an “episode is a

connecting or contrasting passage between subject statements in fugue, thus [an]

episode is a component of form and structure.”22 The second interpretation of fugue,

conveyed earlier by Prout, describes episodes as a means of creating variety while

maintaining unity. Specifically, one may inject variety into a composition through

21 Gauldin, Eighteenth-Century Counterpoint, 219.22 Renwick, Analyzing Fugue, 139.


90

invertible counterpoint, by repeating entire episodes—preservation of unity—while

exchanging parts. Tovey makes a similar observation with respect to Bach’s Fugue in

C Minor, who writes that “its Episodes are derived one from another and

recapitulated symmetrically.”23 Indeed, Gauldin expresses this symmetry through his

form chart of the Fugue in C Minor, shown in Example 2.9. The chart clearly shows

the alignment of Episodes one and four and their invertible relationship with one

another, specifically, creating invertible counterpoint at the twelfth. The arrows

signify, first, their movement towards a subject entry, and second, their connection

between the foreground keys of V and I. Interestingly, the deployment of invertible

episodes—identified here as a means of creating variety and maintaining unity—also

becomes an integral component of the form. Schenker, however, ignored the motivic

content of episodes, and indeed, the very notion of episodes as a formal category;

instead, he emphasized contrapuntal connections between the dominant and tonic:

Preparations for the tonic triad of the next entry would first have to be made, namely the transformation of V^3 into V^3—a return modulation

in respect to the foreground or a chromaticization of the third of V in respect to the basic tonality. (Conventional fugal theory would call this passage an episode.)24

23 Tovey, Forty-Eight Preludes and Fugues, vol. 1, 29.24 Schenker, “Das Organische der Fuge: aufgezeigt an der I. C-Moll-Fuge aus dem Wohltemperierten Klavier von Joh. Seb. Bach,” in Das Meisterwerk in der Musik: Ein Jarbuch, Band //(Munich: Drei Masken Verlag, 1926), 67; trans. Hedi Siegel, “The Organic Nature o f Fugue as Demonstrated in the C Minor Fugue from Bach’s Well-Tempered Clavier, Book I,” in The Masterwork in Music: A Yearbook, vol. II (1926), ed. William Drabkin (Cambridge: Cambridge University Press, 1996), 37.


91

Episodes, thus, structurally link the answer to the subject, facilitating a smooth

transition from the dominant to the tonic area, creating a sense of return in thematic

and tonal domains. The composer’s craft ensures that the episodes often incorporate

previously heard material while integrating new motivic elements. Moreover,

recycled materials—the episode en bloc—may appear in parallel episodes of the

fugue, creating formal symmetry.

One can summarize the observations of Gauldin, Renwick, Tovey, and

Schenker into a form chart of the Fugue in C Minor illustrating the role of the

invertible episodes (see Example 2.10). The chart emphasizes three things. First, it

highlights the foreground harmonic progressions occurring in both episodes; both

express local harmonies of III and VI, first in the tonic, then in the dominant. Second,

the chart shows the modal shift of the dominant mediated by an intervening neighbor-

note, thus avoiding a direct chromatic succession over the dominant. Third, the chart

emphasizes the symmetrical position of the invertible episodes: both follow the

answer and precede the subject; one in the tonic, the other in the dominant. The chart

resembles and owes a debt to Gauldin’s, though the revised chart contains two subtle

differences. For example, each row of Example 2.10 begins with a statement of the

subject, beginning on I, III, and V (within the tonic), respectively; this symmetrical

arrangement carries through each row, aligning thematic and tonal elements.

Moreover, the revised chart illuminates the deep-middleground structure of the fugue,

whereby the entries initiating each row occupy a structural position within the Ursatz,

namely, I, III, and V, respectively, of the divided ascending Bafibrechung. Thus, the


92

invertible episodes become a hub of tonal activity, mediating the transition between

pivotal moments within the tonal and formal structure. Globally, Example 2.10

demonstrates how thematic and tonal events support each other: i.e., subject entries

melodically signal the progression through the Ursatz. In Schenker’s words, “this

unified composing-out guarantees the organic life of the fugue and even reinforces, in

-yethe background tonality, the particular feature of the three-part structure.”

All definitions of the episodes emphasize a sense of return, whether in a

formal, tonal, or contrapuntal sense. Formally, the episodes link re-entries of the

subject, first near the end of the exposition, and later near the end of the piece.

Tonally, they connect keys, in a foreground sense, of recurring statements of the

subject; these episodes unite the keys of the dominant with the tonic. Contrapuntally,

neighboring motion necessitates a return from an earlier departure; motion around the

third of the dominant avoids creating a direct chromatic succession from V-minor to -

major, and allows for a return to C minor for the final entry of the exposition and for

another near the close of the fugue. The invertible nature of the parallel episodes, one

and four, reflects the tonal areas that they occupy, at least in a local sense. Episode

one occurs within the tonic; Episode four in the dominant. Our new definition of

invertible counterpoint at the twelfth—an inversion of harmonic tones, specifically,

the fifth with the root—elucidates the tonal connection between the episodes, both of

which begin with foreground mediant harmonies in their respective keys, tonic and

dominant. Episode one begins with the third and root of an E^-major harmony;

25 Siegel trans., “The Organic Nature o f Fugue,” 32.


93

Episode four begins with the fifth and third of a B^-major harmony. Example 2.11

summarizes this relationship. (Later, we will show how this foreground instance of

invertible counterpoint at the twelfth relates to deep-levels of structure.) Thus,

invertible counterpoint at the twelfth, within the context of fugal episodes, plays

many different roles, prompting further questions. For example, if invertible episodes

link the answer to the subject, does the structure of these episodes depend on that of

the subject?

2.4. The Fugue Subject and Counterpoint at the Twelfth

Prout remarked on the general nature of fugue subjects,

It would be possible to write a fugue of some kind—or, to speak more correctly, a piece in fugal form, on almost any subject that might be selected; but it is by no means every melody that is adapted for fugal treatment; and it is no more possible to make a really good fugue on a bad subject than it would be to make a really good coat out of rotten cloth.26

In support of his claims, contemporary theorists recognize particular melodic patterns

that recur within subjects, weaving their way throughout a vast repertory of fugues.

For instance, Gauldin recognizes that “from the vast range of linear and rhythmic

26 Prout, Fugue, 6.


94

variety of fugue subjects, several striking melodic idioms recur like a ‘scarlet thread’

A A A A A A A

throughout the literature.” He identifies one such thread, 5 6 5 4 3 ( 2 1), as a

melodic pattern that provides the framework for many subjects from the late Baroque

period. Example 2.12 reproduces one of his illustrations of this pattern, from

Handel’s Fugue in C Minor, from Six Grande Fugues?* Here, the boxed notes, G -

A ^ - G - F - E K outline the structural tones of the subject. Within the context of C

minor, or 6 serves as a neighbor to 5 before it descends a third to 3. Similarly,

Renwick uses scale-degree content to classify subject- and answer-pattems (labeled

as “Paradigms”), each of which are supported by particular harmonic progressions

(labeled as “Categories”).29 Example 2.13 shows his subject and answer Paradigm 1,

Category 1, which he uses for Bach’s Fugue in C Minor; we will first focus on the

subject. Like Handel’s subject, this one expresses a third-progression from 5 to 3,

confirming Gauldin’s previous observation, and is supported with a harmonic

progression of I - V - 1 (Renwick’s Category 1). Schenker, too, recognized that “the

descending third-progression g1 - f1 - e^1 determines the content of the subject.”30

The beginning and ending of the subject take on structural significance; according to

Renwick, “together, they define the limits of its tonal expression and largely

determine the tonal structure of the answer.”31

27 Gauldin, Eighteenth-Century Counterpoint, 212.28 Ibid., 212.2 9 For an overview o f these paradigms and categories, see Renwick, Analyzing Fugue, 24-27.30 Siegel trans., “The Organic Nature o f Fugue,” 34.31 Renwick, Analyzing Fugue, 21.


95

The answer, therefore, must express an equivalent beginning and ending as

that of the subject, though it must occur in the dominant. Prout acknowledged the

relationship between subject and answer in a similar fashion:

The answer is the transposition of the subject into the key of the perfect fourth or fifth above or below the key of the subject...The answer is frequently an exact transposition of the subject...At other times the answer is a modified transposition of the subject, alterations being necessitated by the form of the subject itself.

Here, an exact transposition of the subject corresponds to a “real answer,” where each

interval of the subject is preserved under transposition within the answer. We can

apply this concept by simply transposing the subject of Example 2.13 (G - F - E) to

the key of the dominant, as shown in Example 2.14. Here, we recast Renwick’s

subject paradigm in the dominant, creating an answer that expresses 5 - 4 - 3 in the

key of G major. Renwick, however, appeals to the notion of tonal answer, where

subtle intervallic changes allow for the characteristic skip from 5 to 1 in the answer

to reply to that between 1 and 5 in the subject; such alterations affect his answer

paradigm (refer back to Example 2.13). For him, “the basic answer paradigm, 8 8

7 , embodies a tonal alteration: the descending third becomes a descending second.”

The alteration from a third to a second alludes to Prout’s observation, above, that the

answer may be a “modified transposition” based upon the form of the subject itself;

32 Prout, Fugue, 2-3.33 Renwick, Analyzing Fugue, 27.


96

he implied, however, that this modification deviates from a transpositional norm.

Transposed to Schenkerian parlance, the interval of a second is problematic, since it

does not constitute a linear progression. For the answer to be in the dominant, even in

a local sense, it must project, according to Renwick, “a simple linear basis that

expresses the main harmonies involved.”34 Linear progressions express harmonies,

which according to Schenker, “must be at least the size of a third.”35 Therefore, the

answer is understood as a tonal transposition of the subject—in essence, a real

answer—filling out the span of a third within the local key of the dominant.

We can apply the revised subject/answer paradigm of Example 2.14 to the

surface of the fugue in mm. 1-5, as shown in Example 2.15. Here, the subject

outlines a third-progression from G to E^ in mm. 1-3 and the answer outlines a third-

progression from D to in mm. 5-7. Both the subject and answer are preceded by

upper neighbors, ^6, in their respective foreground keys. Example 2.15 demonstrates

the transpositional relationship between subject and answer, but fails to show the

“tonal alteration” of the answer occurring at the surface. This alteration, however,

results from an elision between the end o f the subject and the beginning of the

answer, acting as a bridge to the key of the dominant; the pivot chord of Example

2.15, where I of C minor becomes IV of G minor, dramatizes this elision in m. 3.

The E^ on beat one of m. 3 closes off the subject in the middle voice just as the

34 Ibid., 26.35 Oster trans., Free Composition, §205, 74.36 Schenker speaks about this elision in Siegel trans., “The Organic Nature of Fugue,” 36.


97

answer begins in the top voice on G. Thus, the beginning and ending of the answer—

shown here as a real answer at an earlier level of derivation, yet becoming a tonal

answer at the surface—preserves the tonal integrity of the exposition.

Just as the end of the subject overlaps with the beginning of the answer, so

does the end of the answer with the beginning of the first invertible episode in mm. 5-

7. Since the answer ends in the dominant, the episode, according to Gauldin, “links

the tonal motion to tonic in preparation for the next subject.”37 As diagramed earlier

in Example 2.10, this tonal motion comprises a neighbor-note around the third of the

dominant: - C - prevents a direct chromatic succession from occurring over the

dominant (this was alluded to earlier by Schenker). Example 2.16 shows the large-

scale neighbor motion of the episode. Here, the upright beam illustrates the tones in

question: B^ in m. 5, C and B in m. 6. A fourth-progression from G-C, shown with a

downward beam, amplifies the neighboring motion and comprises the foreground

content of the episode. The progression from G-C utilizes the lowered forms of 6 and

7 (A1, and B^) since B can only appear after C (the neighbor note) has been

introduced; after this point, C can descend to B in preparation for the final subject-

entry of the exposition in m. 7. The elision created between the end of the answer

and the beginning of the episode in m. 5 motivates the neighboring motion and the

fourth-progression that lead the way from the dominant to the tonic.

37 Gauldin, Eighteenth-Century Counterpoint, 214.


98

Schenker, though, provided different readings of this episode within analyses

found in Meisterwerk II and Der freie Satz. Although he observed in the former

analysis that “[T]he chromatic step b^1 - b^1.. .ought to be achieved via a diatonic

progression,”38 neighbor motion was not the only way to achieve it. His solution was

to rely solely on the fourth-progression from G-C, as shown in Level b of Example

A A I

2.17. This version of the fourth-progression utilizes the raised forms of 6 and 7 (A^

and B^), thus providing the requisite diatonic approach to B . But the chromatic step,

B^ - B , that he wished to avoid exists at the deepest level of his analysis as shown in

Level a of Example 2.17. The subsequent levels that he provided (ones closer to the

surface), therefore, were meant to correct this error of voice-leading existing at a

deeper level of structure. Instead of correcting this error at the deepest level (as does

the analysis shown previously in Example 2.16), Schenker instead chose to allow

later levels of derivation to smooth out the path between B^ and B .

Schenker incorporated this same episode into Der freie Satz as a way of

illustrating motion from an inner voice (Untergreifen).39 See Example 2.18. This

analysis also emphasized the necessity of avoiding the direct chromatic succession

between B^ and B , highlighted above the staff with asterisks. As with his previous

reading, this analysis also makes use of a linear progression that begins on G, but the

similarities between these two readings begin to wane after this point. His later

38 Siegel trans., “The Organic Nature o f Fugue,” 38.39 Schenker, Der freie Satz, §233, 135.


99

reading implies that is approached diatonically from C above. This deviates from

his earlier reading (Example 2.17), which demonstrated that is approached

diatonically from below. This shift in analytical perspective impacts the nature of

the linear progression that begins on G. Although his analysis from Der freie Satz

articulates an arpeggiation from G to B via the slur (see Example 2.18), it also

illustrates the stepwise ascent from G to C that occurs “within” this arpeggiation. The

fourth-progression from G-C, in this later analysis, moves through lowered forms of

6 and 7 (A^ and B^), since B can only be introduced until after the neighboring C

arrives in m. 6. This reading of the fourth-progression G-C bears out in the graph of

Example 2.18: the dotted quarter-notes, save for the G that begins the progression,

occur at an earlier level of derivation; the eighth-notes occur at a later level of

derivation. Based on this interpretation of Schenker’s graph from Der freie Satz, a

fourth-progression running through G - A ^ - B ^ - C provides the approach to the

neighbor note C.

The analysis that I presented earlier (Example 2.16) incorporates elements of

and is commensurate with Schenker’s later interpretation from Der freie Satz (as just

shown in Example 2.18): the neighboring C, existing at an earlier derivational level,

provides the necessary diatonic step to B^; the fourth-progression provides a stepwise

approach to the neighboring C, utilizing lowered forms of 6 and 7 (raised forms of

these scale degrees would negate the need for neighbor motion to introduce B^). The


100

scale-degree content of the fourth-progression— 5 - 6 - 7 - 8 in the tonic—neatly

threads together the dominant conclusion of the answer to the tonic beginning of the

final subject-entry of the exposition. Later, the same linear progression appears in the

parallel episode in invertible counterpoint at the twelfth in mm. 17-20, first

transposed to the dominant, and then back to the tonic. The fourth-progression from

5 up to 1 not only binds together the primary components of the exposition (answer

to subject), but it also weaves a web throughout the entire fugue.

Thus, the invertible episode forms part of the rhetoric of the exposition, a

formal unit articulating the thematic content of each part and establishing the

relationship between tonic and dominant. The exposition presents the tonal

“arguments” of the fugue—subject in the tonic and answer in the dominant—before

progressing to new stages within the form.40 With the tonal arguments set in place,

the episode of the exposition mediates the transition between answer and subject with

the fourth-progression from 5 to 1. Linking together the essential components of the

exposition, answer with subject, this episode completes a compositional block to be

used later in the fugue (refer back to Example 2.10, which shows the exposition

occupying nearly the entire first row in the form chart). Later, the parallel invertible

episode of mm. 17-18 completes part of another compositional block, again linking

40 Warren Kirkendale associates imitation pieces, fugal and otherwise, with the presentation of “arguments,” which are revealed one by one within an exordium. See “Ciceronians Versus Aristotelians on the Ricercar as Exordium, from Bembo to Bach,” Journal of the American Musicological Society 32/1 (1979): 25. The analogy o f fugue to debating was anticipated by Thomas Mace (c. 1613 - 1709), who regarded fugue as a discourse meant to augment a given proposition; see Gregory G. Butler, “Fugue and Rhetoric,” Journal o f Music Theory 21/1 (1977): 63-64.


101

answer to subject, but this time through 5 to 1 in the dominant (D - G); this is

followed immediately by 5 to 1 in the tonic (G - C) in mm. 18-20. This scale-degree

pattern begins on the third of the local mediant harmonies occurring at the foreground

(shown earlier in Example 2.11); in Episode 1, the pattern occurs above the added

counterpoint that begins on the root; in Episode 4, the inversion at the twelfth forces

the pattern to occur below the added counterpoint beginning on the fifth. This

confluence of formal, tonal, and contrapuntal events within fugal composition appeals

to Schenker’s understanding of the development of tonality and form. He alluded to

the growth of tonality out of fugal expositions, stating that “the fugue became the first

unified form of larger dimensions. The fifth-relation between the first three entries

(subject, answer, subject) provided the form with direction and stability.”41 For him,

the three entries translated into an expression of tonal rhetoric, where the exposition

served as a metaphor of tonality. For Schenker, this metaphor threaded its way into

deeper levels, weaving together the melodic strands of the Urlinie, divided

Bafibrechung, and inner voices into a rich, contrapuntal, and tonal fabric.

41 Oster trans., Free Composition, 143, §322.


102

2.5. Counterpoint at the Twelfth: Background or Foreground?

This tonal fabric, however, starts to unravel, when we consider received

notions of invertible counterpoint and its role within large-scale tonal structure.

Renwick, for instance, observes that “invertible counterpoint often becomes

secondary to the unfolding of larger tonal areas as expressed primarily through the

motions of the upper voice and bass.”42 Within this secondary sphere, he

distinguishes between “invertible counterpoint at the octave, which is closely tied

with concepts of strict counterpoint and structural levels, and invertible counterpoint

at the tenth and twelfth, which at best bears a loose connection with such

principles.. .Indeed, it is Schenker’s middleground that enables us to understand the

true structural basis of invertible counterpoint in the tonal era as a counterpoint of

linear progressions rather than simply a series of invertible intervals.”43 A

counterpoint of linear progressions necessarily involves a counterpoint of harmonic

tones, because linear progressions provide uni-directional, stepwise connections

between harmonic tones within a single Stufe or between two Stufen.44 And since we

now understand invertible counterpoint at the twelfth as an exchange of harmonic

tones—root with fifth—we may surmise that linear progressions within the Ursatz

have the potential to express counterpoint at the twelfth. Thus, what Renwick

42 Renwick, Analyzing Fugue, 107.43 Ibid., 108.44 Oster trans., Free Composition, §204, 74. Schenker distinguishes between genuine linear progressions and illusoiy linear progressions which do not express a “harmonic relationship between the point o f departure and the final tone.” For more on linear progressions, see §§113-124,43-46 and §§203-220, 73-78.


103

identifies as only a “loose connection” between structural levels and counterpoint at

the twelfth may actually constitute a binding relationship that extends from

background to foreground.

A simple demonstration of counterpoint at the twelfth occurring between two

voices is illustrated in Example 2.19. The left-hand side of the illustration represents

one combination of voices, while the right-hand side represents its inversion at the

twelfth. The center of the diagram shows the three harmonic tones of the triad, with

the arrows representing how the root may be transposed to the fifth or vice versa. It is

important to note that the alto and tenor maintain their vertical disposition with

respect to each other as the inversion takes place. That is, the fifth occurring in the

alto in the left-hand combination is transposed down to the root occurring in the tenor

in the right-hand combination. Thus, it is the harmonic tones that undergo inversion,

not the voices themselves. As demonstrated in the previous chapter, this

interpretation of counterpoint at the twelfth exists, in potential, at the background (see

Example 1.33). Harmonic tone inversion thus illustrates a tight connection between a

prototype of tonality—the Ursatz at the background level—and counterpoint at the

twelfth. Such a binding connection exerts its influence at later levels of derivation.

We can demonstrate that harmonic-tone inversion at the twelfth occurs in the

middleground of the Fugue in C Minor, as shown in Example 2.20. The annotations

above the staff refer to the exchange of harmonic tones occurring in the inner voices

as the harmony progresses from the tonic to the mediant. Here, the root of I inverts to

the fifth of III. The voices, themselves, do not invert—only the harmonic tones.


104

Counterpoint at the twelfth, therefore, occurs between I and III, not I and V, as shown

in the 3-line Ursatz from Chapter 1 (Example 1.33). Nevertheless, the essential

concept remains the same: harmonic-tone inversion occurs, in these cases, between

adjacent Stufen. That they are different Stufen is irrelevant; that the contrapuntal

principle is obeyed, however, is paramount. Here, the transition from background to

deep-middleground involves a preservation of rules—in this case, the inversion of

harmonic tones between adjacent Stufen—not of specific objects. Thus, the fabric of

tonality, the Ursatz, necessarily threads together the motion of the outer and inner

voices with the mechanics of invertible counterpoint at the twelfth.

It is these mechanics of invertible counterpoint, however, that appeal to

opponents of Schenkerian methodology. Laurence Dreyfus, for example, suggests

that foreground invertible counterpoint, not the background, serves as the “primary

motor behind the deepest structure of the piece.”45 We address his comment by

showing how foreground invertible counterpoint within the two episodes of the Fugue

in C Minor relates to that occurring within the Ursatz. First, let us review the role of

the episode within the exposition (see Example 2.21). The episode links together the

answer with the subject (marked with “A” and “S,” respectively); large-scale

neighbor motion o f B ^ - C - B ^ allows the dominant to change from minor to major

(indicated by the lower beam), and the fourth-progression from G - C (indicated by

an inner beam) fills out the approach to the neighboring C. After this, the final

subject-entry occurs in the bass, supported by a cadential 6/4, and closes off the

45 Laurence Dreyfus, Bach and the Patterns of Invention (Cambridge: Harvard University Press, 1996), 178.


105

exposition. Second, we can show how the neighbor motion and the fourth-

progression occur in the parallel episode in mm. 17-20, summarized in Example 2.22.

Here, the fourth-progression occurs in the bass from D to a neighboring G and

implies a resolution to V in G minor. Resolution, however, never arrives (indicated

by the dotted line). Instead, a progression from V to I, this time in the tonic, supports

passing motion in the upper voice from D to C. In addition, the move back to C

minor involves the fourth-progression from G to C, or 5 - 6 - 7 - 8 , a “scarlet

thread” that runs through both invertible episodes.

Next, we can use this thread to build up both episodes, level by level, as

shown within the boxed areas in Example 2.23. Levels a, b, and c refer to mm. 5-7,

Levels d, e, and f to mm. 17-18. Level a depicts the fourth-progression in the alto

leading the way from G to the neighboring C; Level b shows the tenor following the

leader in parallel thirds; Level c shows the bass providing triadic support to the inner

voices, which reveals their harmonic-tone membership at the foreground level; in this

sense, the episode begins with the third and fifth of an E^-major harmony (compare

with Example 2.11).46 Level d shows the fourth-progression from D to G; Level e

shows the parallel counterpoint in tenths, this time occurring above the fourth-

progression; Level f completes the triads in the tenor. In mm. 17-18, the parallel

tenths comprise the fifth and third of the framing foreground harmonies (B^-major

and E^-major). The first half of Episode 4 (shown in the box) clearly relates to the

46 This relationship between the inner voices holds, more or less, throughout the duration of the exposition.


106

exposition-episode via foreground counterpoint at the twelfth (shown at the bottom of

Example 2.23). This relationship ties together the deep-middleground and foreground

levels, which both display harmonic-tone inversion. The thirds of the I- and III -Stiffen

occurring at the deep-middleground, shown earlier in Example 2.20, provide a goal

for the fourth-progression at the foreground, which moves through 5 - 6 - 7 - 8 and

binds together the fugal entries at the surface into a unified contrapuntal tapestry.

A A A A

The fourth-progression through 5 - 6 - 7 - 8 , a structural element of the

parallel episodes, also weaves its way into later levels of the middleground.

Referring back to Example 2.20, the deep-middleground demonstrates counterpoint at

the twelfth occurring between the harmonic tones of the inner voices of I and III. The

alto creates 5-6 motion over III, - C, thus establishing a deep-middleground

neighboring motion, B^ - C - B^, reminiscent of that occurring in the exposition (refer

back to Example 2.21). Also like the exposition, the approach to the neighboring C in

m. 20 within the deep-middleground involves a fourth-progression from G-C, as

shown in Example 2.24. Here, the fourth-progression links up subject-entries on III

and V^, respectively; the connection between entries at this level is suggestive of that

between the foreground harmonies of III and in the exposition, beginning with

Episode 1 in m. 5, and concluding with the final subject-entry on the dominant in m.

7. Based on this analysis, the deep-middleground of the entire piece exhibits many

affinities with that of the exposition. Indeed, the structure of the exposition parallels

the form of the entire piece, as shown earlier in Example 2.10. We can further


107

express this parallelism by combining later levels of the middleground with the

formal scheme of the fugue, shown in Example 2.25. Here, the three systems

correspond to the three parts of the fugue, as diagramed in Example 2.10. The graph

of Example 2.25 shows the versatility of the fourth-progression within the invertible

episodes. In mm. 5-7, it prolongs the ioriic-Stufe as part of the exposition; in mm. 17-

20, it provides the contrapuntal link between III and V. The many correspondences

between the exposition (mm. 1-9) and the entries of mm. 11-22 suggest that the later

section is a recomposition of the earlier one. Thus, the foundation of the invertible

episodes—the fourth progression, 5 - 6 - 7 - 8, which is used to link up harmonic

tones derived from the background —also plays a large role in later levels of the

middleground.

2.6. Conclusion

We conclude by reviewing the issues covered in this chapter. First, invertible

counterpoint at the twelfth is defined as an exchange of harmonic tones; specifically,

a root lying beneath the third of one harmony in a particular passage may be

transposed to the fifth lying above the third of a corresponding harmony in a parallel

passage (the reverse process may also occur). Second, we may use counterpoint at

the twelfth within fugal episodes, which serve to link up subject- and answer-entries


108

and facilitate the return from dominant harmony to tonic harmony. Third, the

structure of fugal episodes is dependent upon that of the subject, which is a linear

progression that expresses the tonic Stufe\ the structure of the subject, in turn,

constrains that of the answer. Fourth, the presence of harmonic-tone inversion at the

background suggests that counterpoint at the twelfth is not just a technique available

only at the surface, but one that resides at deep levels of structure.

It is this pervasiveness of invertible counterpoint, at both early and later levels

of derivation, that appeals most to Schenker’s program of creating “organic

coherence” and “organic unity.”47 Although he relegated invertible counterpoint to

the sphere of musical form, it is the latter that grew out of contrapuntal processes. In

support of this perspective, he wrote the following:

In the music of the early contrapuntal epoch, including even Palestrina, the basic voice-leading events, such as passing tones or neighboring notes, had not yet come to fruition, like flower in bud. Who would have suspected, at that time, that these phenomena, through the process of diminution, were to become form-generative and would

• • 4 8give rise to entire sections and large forms!

Thus, form exists as a large-scale expression of contrapuntal procedures.

Counterpoint, in the background sense, involves the juxtaposition of the Urlinie,

Bafibrechung, and inner voices—i.e., the Ursatz. Since form grows out of deep-level

contrapuntal procedures found within the Ursatz—e.g., the passing tones of the

47 Oster trans., Free Composition, §301, 128.48 Ibid., §301, 128.


109

Urlinie—he insisted that form itself exists at early levels of derivation. In Schenker’s

words,

All forms appear in the ultimate foreground; but all of them have their origin in, and derive from, the background...Previously...I have repeatedly referred to form as the ultimate manifestation of that structural coherence which grows out of [the] background, middleground, and foreground.49

Fugue, ultimately, expressed structural coherence for Schenker. Invertible

counterpoint, a procedure related to fugue, branched out, not from voice-leading, but

from the theory of form, according to him.50 But since form grew out of the

contrapuntal procedures found within the Ursatz, so did invertible counterpoint, thus

tracing its roots back to its contrapuntal origins. As the tones of the Urlinie descend

against the divided Bafibrechung, harmonic tones invert with the progression from

one essential harmony to the next. As demonstrated earlier, this invertible

relationship extends as far as the background. Thus, counterpoint at the twelfth is an

essential component of the background, the same location that stores the seeds of

tonality and form. Such an organic connection between tonality, form, and

counterpoint addresses concerns posed by opponents of Schenkerian methodology,

such as Dreyfus, who complains that “one overriding problem that the graph [or any

graph] cannot conceal is the relationship between Schenker’s large-scale reading of

49 Ibid., § 306, 130.50 Rothgeb and Thym trans., Counterpoint I, 16.


110

the piece and the obvious demands of foreground invertible counterpoint.”51 Indeed,

the background and deep-middleground—the large-scale reading—exist “like flower

in bud,” only to branch out later at the foreground, and thus establish coherence and

unity between the properties of tonality, form, and invertible counterpoint.

Invertible counterpoint plays an integral role within a tonal composition,

especially fugue. Fux recognized its importance, as he made clear through his alter

ego, Aloysius:

“[Let’s] use the time to proceed to fugues with several themes. Since this cannot be done without a knowledge of double counterpoint, we shall have to discuss this first... You will soon realize how exceedingly useful this contrapuntal device is in any kind of composition, especially in a fugue where several themes are to be combined.”52

As the student of Aloysius, Josephus enthusiastically thanked his teacher for

imparting such valuable advice: “I am amazed at this device of counterpoint, and I am

most anxious to learn how it is to be applied.”53 As we have discovered throughout

this chapter, invertible counterpoint, especially at the twelfth, has many

compositional applications, as it is a fundamental component of tonality.

51 Dreyfus, Bach and the Patterns of Invention, 175.52 Mann trans., The Study of Fugue, 107-108.53 Ibid., 112.


I l l

Chapter 3: Why use Invertible Counterpoint at the Tenth?

3.1. Introduction

Having discussed counterpoint at the twelfth, we now turn our attention to

counterpoint at the tenth. We begin with a quote from Kimberger and an illustration

as shown in Example 3.1:

Knowledge of double counterpoint greatly facilitates this variety of harmony...If the bass of the first measure were to be repeated in the second measure, this passage would lose all its beauty. It is notable that each bass note is taken a third lower with the repetition of the same notes in the upper voices. This transposition is based on counterpoint at the tenth. It must be kept in mind that even when a melody repeats the same note several times in succession, the harmony may be varied sufficiently, since the same note can now be the third, now the sixth, fifth, or octave of the same or another root, and thus new harmonies always occur.1

Kimberger did not extrapolate any further concerning the connection between the

music of Example 3.1 and the technique of counterpoint at the tenth. Nevertheless,

he used this type of invertible counterpoint as a means of providing variety to

melodies that contain repeated melodic tones. For Kimberger, invertible counterpoint

at the tenth created variety by placing the same note into different harmonic contexts.

1 Johann Philipp Kimberger, Die Kunst des reinen Satzes in der Musik, Vol. 1 (Berlin and Konigsberg, 1776-1779), 147; trans. David Beach and Jurgen Thym, The Art of Strict Musical Composition, trans. (New Haven: Yale University Press, 1982), 163.


112

Other theorists, however, treated counterpoint at the tenth as a necessary evil

that one must endure. Ebenezer Prout remarked,

It will be seen that double counterpoint in the tenth requires so much to be avoided that its rules may be compared to the laws of the Decalogue, nearly all of which begin with words, ‘THOU SHALT NOT.’ Consequently, this kind of counterpoint is far less frequently met with, and much less useful than that in the octave. It is nevertheless important that the student should be able to work it and he will find its practice very beneficial.2

His advice concerning counterpoint at the tenth is not unique. Past theorists have

remarked on the difficulty of this brand of counterpoint, including Zarlino, Morley,

Kennan, and Gauldin.3 Suffice it to say, these theorists agree that invertible

counterpoint at the tenth presents a number of difficult obstacles that one must

overcome.

Two of Prout’s observations given above, however, are at odds with each

other. On the one hand, counterpoint at the tenth is difficult to do. Because of this

difficulty, it is seldom used. On the other hand, counterpoint at the tenth is a

necessary skill that all composition students should acquire, because—according to

Prout—it will prove to be useful in further compositional endeavors. If counterpoint

at the tenth is used so infrequently (due to its difficult nature), then why would a

2 Prout, Double Counterpoint and Canon, 26.3 Marco and Palisca trans., The Art of Counterpoint, 162-63; Thomas Morley, A Plain and Easy Introduction to Practical Music (London: Peter Short, 1597); ed. R. Alec Harman (New York: Norton, 1952), 193; Kent Kennan, Counterpoint: Based on Eighteenth-Century Practice (Upper Saddle River, NJ: Prentice-Hall, 1999), 118; Gauldin, Eighteenth-Century Counterpoint, 186.


113

student gain proficiency at utilizing this technique? The inconsistency between

Prout’s two observations—infrequency vs. usefulness—elicits the three following

questions. First, how do we define counterpoint at the tenth, especially as it pertains

to tonal composition? Second, where should one use this technique within the

context of tonal compositions? Third, which Schenkerian voice-leading

transformations engage counterpoint at the tenth? Answers to the latter two questions

will involve analyses of Bach’s Fugues in Major, from the Well-Tempered

Clavier, Books I and II (BWV 866 and BWV 890, respectively).

3.2. Definition/Problems of Counterpoint at the Tenth

Similar to the previous chapter, we will first examine Zarlino’s definition of

counterpoint at the tenth. He wrote,

Then there is the inversion formed by raising the lower voice an octave and dropping the upper voice a tenth. It is known as double counterpoint at the tenth...The upper voice may also be lowered an octave, and the lower voice raised a tenth.4

4 Marco and Palisca trans., The Art o f Counterpoint, 162.


114

His definition identified, clearly, the necessary procedure for creating this type of

invertible counterpoint: one voice moves an octave in one direction, while the other

moves a tenth in the opposite direction.

And as with counterpoint at the twelfth, one can use simple arithmetic to

calculate how intervals are transformed when voices invert at the tenth. In this vein,

Gauldin wrote the following:

During the inversional process the original soprano becomes the lower part, being held in the same octave, while the other voice is moved a tenth higher (or 1 + 10 = 11, the ‘magic’ number for this type of double counterpoint).5

Of course, he is referring to the inversion table for counterpoint at the tenth, as shown

in Example 3.2. As with any inversion table, the numerals represent intervals

between voices and/or intervals by which to transpose voices to create an inversion.

After an inversion has taken place, the intervals between the voices change from

those occurring in the top row to their corresponding counterparts in the bottom row

(or vice versa). Gauldin here uses the table to explain how voices are transposed to

create a corresponding inversion at the tenth. By looking at the table in Example 3.2,

one can see that the corresponding transpositions, 1 and 10, are vertically aligned and

add up to 11. One can arrive at the same sum by adding up any of the aligned pairs of

numerals.

5 Gauldin, Eighteenth-Century Counterpoint, 186-87.


115

Not only does inversion at the tenth produce corresponding intervals that add

up to eleven, it also swaps perfect consonances for imperfect ones. Knud Jeppesen

articulated this intriguing result of counterpoint at the tenth:

Thus all consonances retain their consonant character, just as all dissonances remain dissonant upon inversion. But the perfect consonances become imperfect, the imperfect perfect...”6

Therefore, counterpoint at the tenth maintains generic consonances while exchanging

specific consonances: perfect with imperfect. This unique property influences the

type of relative motion that must be observed when utilizing this kind of invertible

counterpoint.

With these preliminary definitions out of the way, we now turn to an example

that demonstrates the technique of counterpoint at the tenth. See Example 3.3, which

provides an illustration adapted from Fux’s Gradus ad Parnassum.1 An original

combination of voices is shown in Example 3.3a, which consists of a cantus firmus

and a counterpoint lying above it. The inversion at the tenth is achieved in two

different ways. First, the counterpoint is transposed down a tenth while the cantus

firmus remains at the same pitch level, as shown in Example 3.3b. Second, the cantus

firmus is transposed up a tenth while the counterpoint is moved down an octave, as

6 Knud Jeppesen, Kontrapunkt (vokalpolyfoni) (Copenhagen: Wilhelm Hansen/Musik-Forlag, 1930), 228; trans., Glen Haydon, Counterpoint: The Polyphonic Vocal Style of the Sixteenth Century (New York: Prentice-Hall, 1939), 282.7 Mann trans., The Study of Fugue, 116.


116

shown in Example 3.3c. Both of these applications are combined, together, to

produce a three-voice example, as illustrated in Example 3.3d (this alludes to the

process of doubling in thirds, as discussed in Chapter 1). Here, the original cantus

firmus of Examples 3.3a and 3.3b is placed down an octave into the bass voice (see

the arrows); its transposition a third higher from Example 3.3c is placed into the

soprano voice in Example 3.3d (doubling in thirds); and the counterpoint, displaced

down an octave in Example 3.3c, is placed into the middle voice in Example 3.3d.

By applying the procedure of counterpoint at the tenth to a two-voice example, Fux

created a three-voice example that is able to express triadic harmonies.

But to produce a successful inversion at the tenth that is free of errors, it is

required that one follow a strict rule of relative motion. Fux articulated this rule

within his dialogue between Aloysius and Josephus:

Josephus'. How amazing is this technique of counterpoint, in which transcribing one part makes a third part arise. But could any two-part setting be extended to three parts in this species [double counterpoint at the tenth]?Aloysius'. Yes, if one observes carefully the rules of this species and the fact that each downbeat should be approached by either contrary or oblique motion, as in the preceding examples.8

Here, Fux recognized the necessary relative motion that one must use when writing

counterpoint at the tenth: contrary and oblique motion. The strict adherence to this

type of relative motion becomes more obvious when we reexamine Jeppesen’s earlier

8 Mann trans., The Study of Fugue, 117.


117

observations concerning counterpoint at the tenth: since perfect consonances invert

into imperfect consonances (and vice versa), parallel motion becomes impossible.

Indeed, Fux clearly recognized the ramifications of employing parallel motion as

demonstrated in the inversion table. He wrote:

From the following table of numbers it may be seen which intervals are not to be used... You will realize that two thirds or two tenths must not be used successively in direct motion, for by inversion the latter would result in two unisons, the former in two octaves. Similarly, the use of two sixths is forbidden, since their inversion at the tenth would produce two fifths.9

He explained, via the inversion table, that parallel motion between inverted parts will

ultimately lead to parallel perfect consonances, an act prohibited by the laws of strict

counterpoint. The use of contrary and oblique motion—a rule of contrapuntal

motion—is required when writing counterpoint at the tenth.10

This rule of relative motion results from Jeppesen’s observation, referred to

earlier, that perfect and imperfect consonances invert into each other under

counterpoint at the tenth. But why is this so? As one answer, Piston stated that

“inversion at the tenth changes the position of the whole melody in the scale, so that

its melodic intervals undergo slight changes.”11 His comment dictates that a melody,

9 Mann trans., The Study of Fugue, 115.10This rule, o f course, has been recognized by a number of theorists from past to present, including Zarlino, Gauldin, and Schubert and Neidhofer, to name but a few. Marco and Palisca trans., The Art of Counterpoint, 162-63; Gauldin, Eighteenth-Century Counterpoint, 186; Schubert and Neidhofer, Baroque Counterpoint, 277.11 Piston, Counterpoint, 176.


118

consisting of generic pitch-intervals, moves up or down a diatonic scale by the

interval of a third, thus altering the specific pitch-intervals of the melody. Example

3.4 illustrates this scenario. A C-major scale extends from bottom to top and a box

surrounds a hypothetical original melody, whose chronological order is indicated by

arrows between pitches. The diagonal arrow represents an upward transposition by a

third (or a tenth) of the melody. The transposed melody retains its generic intervals

(three diatonic steps) while altering its specific intervals (tone-tone changes to

semitone-tone). We can imagine the original melody as being placed below C in the

soprano voice (see the bottom of Example 3.4). The intervals between the original

combination consist of 8 - 7 - 6: the initial interval is perfect (indicated with “P”), the

last interval is imperfect (indicated with “I”), and the middle interval (representing a

passing tone) is dissonant. We can then imagine the inversion (also shown at the

bottom of Example 3.4), which illustrates the upward transposition of the melody by

a third and a downward transposition of the upper voice by an octave. The intervals

resulting from the exchange equal 3 - 4 - 5 . Now the initial interval is imperfect, the

last interval perfect, and the middle interval dissonant. A complete reversal has

occurred between consonance-types. This bears out Jeppesen’s observation

concerning how consonances change under counterpoint at the tenth and also why it

occurs according to Piston’s understanding of this procedure.

Piston’s notion of moving up and down by the interval of a tenth is accurate

enough, but the space in which these movements take place is not. As discussed in

the previous chapter, invertible counterpoint involves harmonic tones of Stufen. In


119

this sense, we must shift our attention from scales to triads.12 Instead of moving a

melody up or down a scale, one can transpose a melody up or down so that it begins

on different harmonic tones of a triad (Schubert and Neidhofer 2006 take a similar

tack).13 See Example 3.5. Here, transpositions by a tenth occur between adjacent

members of a triad: root, third, and fifth. The double-headed arrows signify

transposition up or down between harmonic tones. The bottom of the example

reproduces the illustration from Example 3.4, this time changing the orientation from

intervals to harmonic tones. The root and third in the melody of the original

combination are transposed upward so that they begin on the third and fifth in the

inverted combination (upward transposition of a tenth such as this has prompted

recommendations from past theorists to use only the root and third [or unison and

third] in the original contrapuntal combination when writing counterpoint at the

tenth).14 The illustration at the top of Example 3.5 summarizes the results of

counterpoint at the tenth: roots and fifths invert to thirds, and thirds invert to either

roots or fifths. This updates Jeppesen’s comment concerning the exchange of perfect

with imperfect intervals under counterpoint at the tenth; perfect intervals are now

replaced by roots and fifths, imperfect intervals by the chordal third.

As in the previous chapter, we can focus on the possibilities of counterpoint at

the tenth, rather than focusing on its problems or difficulties as shown in the inversion

12 See Matthew Brown’s discussion on the relationship between Schenkerian theory and scales in Explaining Tonality, 146-151.13 Schubert and Neidhofer, Baroque Counterpoint, 285.14 For instance, see Bernhard’s comments in Walter Hilse, “The Treatises o f Christoph Bernhard,” The Music Forum 3 (1973): 175; Schubert and Neidhofer, Baroque Counterpoint, 279 and 293.


120

table. First, we can illustrate how consonances remain invariant under counterpoint at

the tenth, as shown in Example 3.6a. Here, the consonances of one row will invert to

those of the other, e.g., 3 will invert to 8 as voices are inverted at the tenth. This table

also summarizes Cerone’s interpretation of counterpoint at the tenth, discussed in

Chapter 1 (refer to Example 1.26), where two voices moving in contrary and oblique

motion and proceeding by step, are counterpointed against each other. The relative

motion demonstrated in his illustration produces the intervallic patterns shown in

Example 3.6a: 1 - 3 - 5 and 1 0 - 8 - 6 . These patterns may be reversed and/or

rotated; however, the adjacent ordering of intervals must remain intact. Second, the

table of Example 3.6a can be revised so that it reflects the inversion at the tenth of

harmonic tones; see Example 3.6b. Here, the intervals have been replaced with

harmonic tones. Similar to the last chapter, this revised table demonstrates how

harmonic tones are transposed under counterpoint at the tenth: e.g., a root occurring

in one voice will be transposed to a third when that voice undergoes inversion at the

tenth with another voice. This interpretation of counterpoint at the tenth is illustrated

in the Example 3.6c. Here, the example integrates information from the tables in

Examples 3.6a and 3.6b: the top of the example shows the consonant intervals and

their inversions; harmonic-tone inversion is listed beneath the intervals; finally,

harmonic-tone inversion is illustrated in music notation. One can see that the three

consonant intervals available under counterpoint at the tenth, as demonstrated so long

ago by Cerone, correspond to the inversion of the three harmonic tones of the triad:


121

the root, third, and fifth. The possibilities of inverting harmonic tones at the tenth are

many, however, voices must move against each other in contrary or oblique motion.

Of course, this strict contrapuntal treatment of harmonic tones alludes to the

Ursatz, where the outer voices move against each other in contrary motion. Indeed,

one can demonstrate the principles of counterpoint at the tenth as occurring within the

Ursatz. See Example 3.7, which depicts a 3-line Schenkerian Ursatz; the voices that

take part in counterpoint at the tenth are the alto and bass. The arrows between and

above the staves indicate the inversion of harmonic tones, not of voices. First,

• 9 1contrary motion occurs between the two voices: as c moves down to b , the bass

moves up from C to g; as b1 moves up to c2, the bass moves back down to C. Second,

the root of the first I-Stufe, occurring in both the alto and bass, exchanges with the

third and root of the V-Stufe in the same respective voices. Likewise, the third and

root of the V-Stufe, occurring in the alto and bass, respectively, exchange with the

root of the final I -Stufe in these same respective voices. Example 3.7 also

demonstrates the rule of thumb of only using the root and third in the original

contrapuntal combination.15 Taking this tack, one can add a third voice above the alto

that alternates between the third and fifth of the l-Stufe and V-Stufe, respectively; this

alludes to the practice of doubling in thirds, discussed in Chapter 1. By doing this,

the first two tones of the Urlinie—e2 and d2—result as parallel thirds above the alto

voice. Thus, one can explain the relationship between the soprano, alto, and bass

voices of the Ursatz as an illustration of counterpoint at the tenth.

15 See Schubert and Neidhofer, Baroque Counterpoint, 25.


122

To summarize, counterpoint at the tenth must utilize contrary and oblique

motion between parts that are to be inverted. This is because harmonic roots and

fifths invert into thirds and vice versa. These types of relative motion—contrary and

oblique—also exist within the Ursatz, thus establishing a structural connection

between the workings of invertible counterpoint at the tenth and Schenker’s

foundation of tonality. This structural connection pervades tonal composition—

especially fugues—from background to foreground. We now turn our attention to

compositional contexts in which to use counterpoint at the tenth.

3.3. Compositional Applications of Counterpoint at the Tenth

For the sake of good diversity of harmony one can never recommend enough to young composers that they study the four-part works of Handel, Bach, and Graun with persistent diligence. But before they are able to do this with desirable success, they must be familiar with double counterpoint.16

Kimberger’s insistence, above, that composers be fluent in the practice of

double counterpoint at the tenth—and the twelfth and octave—becomes even more

profound as we discover its harmonic potential. Prout would have us believe that

counterpoint at the tenth rarely occurs in music (see the quoted passage at the

16 Beach and Thym trans., The Art of Strict Composition, 164.


123

beginning of this chapter), yet Kimberger’s example of counterpoint at the tenth

(previously shown in Example 3.1) illustrated a simple progression of descending

thirds in the bass. We need only consult a few theory textbooks to realize that this

progression occurs with moderate (if not regular) frequency within the tonal idiom.17

This progression gives us a useful harmonic context in which to apply the technique

of counterpoint at the tenth. This section of the chapter will explore the

compositional possibilities of this harmonic pattern in three different ways. First, we

will study this pattern as a progression residing at the surface. Second, we will

reevaluate this progression as a deep-middleground pattern. Third, we will consider

how this deep-middleground pattern may affect tonal events closer to the surface as it

occurs within the Major fugues (BWV 866 and BWV 890); specifically, we will

identify foreground instances of counterpoint at the tenth engaging the subjects of

both fugues.

17 One cannot create a comprehensive list, but here is a random sampling within the pedagogical literature: Edward Aldwell and Carl Schachter, Harmony and Voice-Leading (Belmont, CA: Wadsworth Group/Thomson Learning, 2003), 276-78; Allen Cadwallader and David Gagne, Analysis of Tonal Music: A Schenkerian Approach (New York: Oxford University Press, 1998), 94-97; Stefan Kostka and Dorothy Payne, Tonal Harmony, with an Introduction to Twentieth-Century Music, 4th ed. (Boston: McGraw-Hill, 2000), 107; Steven G. Laitz, The Complete Musician An Integrated Approach to Tonal Theory, Analysis, and Listening (New York: Oxford University Press, 2003), 336-337.


124

3.3.1. Descending Thirds at the Surface

The first part of this chapter has shown that counterpoint at the tenth consists

of two components: 1) contrary and oblique motion and 2) inversion of harmonic

tones: thirds with roots/fifths and vice versa. Application of both components will

shed light on the tonal patterns that are available under this type of invertible

counterpoint.

First, contrary and oblique motion control the types of tonal patterns that one

may encounter under counterpoint at the tenth. As a demonstration, refer to Example

3.8, which builds upon an example from Aldwell and Schachter.18 Here, the music

resembles the descending thirds sequence shown earlier in Kimberger’s illustration of

counterpoint at the tenth (in Example 3.1), but continues the pattern through one more

scale-step. The bass moves through descending thirds (B^ - G - E^ - C) as do the

harmonies (I - VI - IV - II). The annotations above the staff illuminate the inversion

of harmonic tones occurring between the soprano and bass. There are two types of

inversion. First, root movement by ascending fifth—e.g., between major and F

major—involves the root and third of the first harmony inverting with the doubled

root in the second harmony. Second, root movement by ascending second—e.g.,

between F major and G minor—involves the doubled root inverting with the root and

third. Continuation of this inversional pattern produces moderately disjunct voice

18 Aldwell and Schachter, Harmony and Voice Leading, 278.


125

leading at the surface, but this is not necessarily required.19 In contrast, we can

compress the descending thirds pattern into a register of smaller compass, as shown in

Example 3.9. Despite the attenuation of register, harmonic-tone inversion still

occurs, though between different pairs of voices. For example, the first exchange

occurs between the alto and bass (from I to VI), the second between the tenor and

bass (from VI to IV). To summarize, the descending-third pattern at the surface

encapsulates the two necessary components of counterpoint at the tenth: contrary and

oblique motion and harmonic-tone inversion of thirds with fifths or roots. This

pattern will play a role in our understanding of the Fugues in Major (BWV 866

and BWV 890), to be discussed later.

3.3.2. Descending Thirds at the Middleground

Descending thirds can also be used at the middleground as a way of

structuring a tonal composition. For example, the illustration of Example 3.9 could

be interpreted in such a way.20 Indeed, Carl Schachter has shown in his analysis of

the Fugue in B^ Major (BWV 866) that descending thirds through I, VI, and IV—

referred to as the “bass arpeggio”—can exist at a very deep level of the

19 For example, see Cadwallader’s and Gagne’s analysis o f the opening measures o f the last movement o f Beethoven’s Piano Sonata, Op. 79, which demonstrates a similar voice leading pattern. Analysis of Tonal Music, 94-97.

Schenker illustrates the use o f descending thirds in the following figures from Der freie Satz: 30b, 37, 40.3, 154.3, and 155 (not an exhaustive list).


126

middleground.21 We may also understand descending thirds as resulting from the

rules of counterpoint at the tenth.

We can apply the constraints of contrary and oblique motion—the necessary

relative motion for counterpoint at the tenth—to a tonal prototype, shown in Example

3.10 (this models Bach’s Fugue in Major [BWV 890]). Level a shows a 3-line

Schenkerian Ursatz. Level b illustrates a repetition of the opening tonic Stufe and a

filling-in of the Bafibrechung through 16. The boxed inner voices in the latter

illustration, F/bK highlight a potential interval for unfolding.22 Level c realizes this

potential: the span o f a fourth from F to B^ now occurs in the horizontal direction and

is filled in with passing tones, thus creating a linear progression. Level d illustrates

triads built upon each of the tones of the ascending fourth-progression and applies the

rules of contrary and oblique motion. The resulting harmonizations are triads built

upon I, VI, IV, and II. The rule of contrary and oblique motion remains in effect

throughout the duration of the descending thirds from I - II (this is reminiscent of the

descending-thirds progression shown earlier in Example 3.8). The illustrations of

Example 3.10 show that rules of relative motion necessary for counterpoint at the

tenth—contrary and oblique motion—can also interact with Schenkerian

transformations such as unfolding and linear progressions. More globally, these rules

21 Carl Schachter, “Bach’s Fugue in Major, Well-Tempered Clavier, Book I, No. XXI,”Music Forum 3 (1973): 242.22 ♦Schachter follows a similar tack in his analysis, where the inner voice that forms a thirdwith the upper voice “becomes activated and forms part o f the melodic line.” See “Bach’s Fugue in B Major,” 241.


127

of relative motion transform a tonal prototype, level by level, into a template for tonal

composition—one that specifically uses counterpoint at the tenth.

3.3.3. Illustrations o f Counterpoint at the Tenth

Concerning the study of fugue, Carl Schachter has remarked that “at one time

or another, we have all learned...to construct Frankensteinian fugues, robbing one

23graveyard for the subject, another for the counter subject, a third for the episodes.”

This comment suggests that the corpus of studied fugues, as understood by teacher

and student alike, exists as dismembered compositional parts, knitted together willy-

nilly into freakish compositional monstrosities. Writing fugues—understood here as

an assembly of disparate parts—may help the fledgling composer acquire local

contrapuntal techniques but hinder the aspiring analyst’s understanding of large-scale

tonal strategy. In his words, “a pedagogical help in teaching the student to write

fugues becomes a stumbling block to the understanding of fugues by the great

composers.”24 Schenker, too, understood the gap between composer and analyst. For

him, it is the composer’s task that “leads him from a background Ursatz through

23 Schachter, “Bach’s Fugue in Major,” 263.24 Ibid,” 263.


128

prolongations and diminutions to a foreground setting.”25 But when the analyst

confronts a piece and attempts to break it down into levels, it is, in his words, “from

the perspective of the observer, not that of the composer—that is, they [structural

26levels of the piece] are presented...in the direction from foreground to background.”

For Schenker, people initially encounter an individual piece through the surface, but

they can only learn about its large-scale structure by viewing it from the background.

Thus, the only way that the analyst can make sense of the fragments scattered across

the foreground of a fugue—what Schachter refers to as “disjecta membra'’'’21—is to

incorporate them, as would a composer, into the living body of the background.

As an example, let us look at instances of foreground counterpoint at the tenth

in both of the Fugues in Major (BWV 866 and BWV 890). Example 3.11 shows

excerpts from the first fugue, all of which contain a subject and two countersubjects

(referred to as CS1 and CS2, respectively). The left-hand side of the example shows

the two countersubjects converging on a unison, indicated by the number “1.” For

example, CS2 appears in mm. 9-13 as D-B^-D in sixteenth notes counterpointed

above CS1 with F-D in eighth notes; both countersubjects converge on unisons B^

and C as the entire progression moves up a scale-step. The same strategy is taken in

mm. 13-17 in the key of F major and mm. 26-30 in the key of C minor. Things

25 Heinrich Schenker, “Forsetzung der Urlinie-Betrachtungen,” in Das Meisterwerk in der Musik: Ein Jahrbuch [1925] (Munich: Drei Masken Verlag, 1925), 188; trans. John Rothgeb, “Further Considerations o f the Urlinie: I,” in The Masterwork in Music: A Yearbook (1925), Volume I, ed. William Drabkin (Cambridge: Cambridge University Press, 1994), 104-105.26 Rothgeb trans., “Further Considerations o f the Urlinie: I,” 105.27 Schachter, “Bach’s Fugue in B Major,” 240.


129

change, however, in mm. 22-26, which is in the key of G minor (see the right-hand

side of Example 3.11). Here, the two countersubjects invert and converge on a tenth.

The switch from unisons to tenths resulting from the inversion of countersubjects

confirms that counterpoint at the tenth has been performed.

Likewise, we find similar instances of this type of invertible counterpoint in

the second Fugue in Major (BWV 890), as shown in Example 3.12. Five excerpts

are shown, each of which isolate two particular intervals that occur between the

subject and CS2 (here, CS2 is the voice that moves in dotted half-notes). The first

two excerpts at the top of the example (in F major and B^ major, respectively) are

related through counterpoint at the tenth: the fifths B^/F and C/G in mm. 34-35 invert

to thirds G/B^ and A/C in mm. 42-43. Here, the relative position between the subject

and CS2 stays the same, but the interval between them does not. In this regard, Prout

has remarked, “[I]t sometimes happens, in double counterpoint other than the octave,

that the two voices will be in the same relative position to one another, but the

counterpoint will be at a different interval”; Schubert and Neidhofer refer to this as

“uninverted double counterpoint.”28 Nevertheless, inversion between both parts does

occur in the next two passages (in G minor and E^ major, respectively), inverting the

thirds from the previous section into octaves G/G and A/A in mm. 49-50, and E^/E^

and F/F in mm. 56-57. The subject and CS2 invert back to their original disposition

in the final passage (moving from F major to G minor) in mm. 78-86, inverting the

28 Prout, Double Counterpoint and Canon, 3; Schubert and Neidhofer, Baroque Counterpoint, 275.


130

octaves from the G minor and major sections into tenths. Furthermore, this last

section extends the groupings of parallel intervals from two to three iterations.

These isolated passages are easy enough to identify and explain, yet they pose

a problem. How can instances of counterpoint at the tenth employ parallel motion?

The excerpts from both fugues illustrate parallel voice leading: unisons and tenths in

the first fugue; fifths, tenths, and octaves in the second fugue. We answer this

question by studying the fugue subjects from both pieces, since they participate in all

of the instances of counterpoint at the tenth identified above. The technique used in

both subjects—a Schenkerian transformation—occurs at both foreground and

middleground levels of structure.

Concerning the subject from the first Fugue in Major (BWV 866),

Schachter writes that the ascent to the Kopfton occurs as a line that “rises in three

overlapping segments (Ubergreifstechnik) rather than in a single, unbroken curve.”29

Example 3.13 illustrates what he means. The beamed line through 1 - 2 - 3 results

from the outer voices playing a game of leap-frog: the soprano descends into an inner

voice by step (indicated by the straight arrow) as the bass ascends to the soprano by

skip (indicated by the curved arrow). The pattern continues in this fashion and

concludes as the bass leaps up to D in m. 4. Similarly, the subject from the second

Fugue in B^ Major (BWV 890) progresses in much the same way (see Example 3.14).

This time, 1 - 2 - 3 appear at the bottom of the texture, but the bass skips up to

29 Schachter, “Bach’s Fugue in Major,” 248.


131

implied tones A and in the upper register (the identity of these tones becomes

clearer as voices are added). As in the first fugue, the game of leap-frog continues

until D arrives, this time in the bass in m. 5.

Our preliminary glance at both fugue subjects shows that the stepwise ascent

to 3 is a polyphonic process, rather than a progression in a single voice. This

knowledge changes our understanding of the apparent instances of parallel voice-

leading in both fugues (refer to Example 3.15). Here, the two upper voices move

down as the lowest voice skips up. Significantly, any two ascending consecutive

intervals of the same type can never be represented by the same pair of voices when

using counterpoint at the tenth, thus eliminating the chance of creating parallels. This

constant leap-frog motion alludes to Kimberger’s preference for “good diversity of

harmony.” It also fulfils his requirement that students “be familiar with double

counterpoint,” since the continual turning over of voices necessarily demands that

each be invertible, one with the other.

3.4. Schenkerian Transformations and Counterpoint at the Tenth

Example 3.15 shows us that one can create the effect of parallel voice-leading

without committing voice-leading errors. Schenker explained a transformation—

reaching-over (Uebergreifen)—that corresponds to the schematic of this example. He


132

defined it in the following way: “When a group of at least two descending tones is

used to place an inner voice into a higher register...”30 In Der freie Satz, he claimed

that reaching-over takes effect at the first level of the middleground, where this

transformation is used as a means of approaching the Kopfton. Execution of this

transformation occurs in one of two ways. First, in Schenker’s words, “[T]he first

tone of the top voice descends one or more steps, whereupon an inner voice crosses

above, in order to establish the new pitch of the top voice.” Second, “[T]he

superimposed inner-voice tone introduces a tone of the upper voice from above and

so resembles an upper neighboring note.” He gives illustrations of both scenarios,

but none is as precise as the one he used in Meisterwerk I in the section entitled

• • 'X'X“Elucidations.” See Example 3.16, which reproduces his illustration. Here, the

interlocking slurs between both voices clearly show the leap-frog motion referred to

earlier. Reaching-over creates the effect of parallel thirds without any true parallel

motion occurring between both voices. From a more global perspective, one may use

reaching-over to arrive at the Kopfton. In Schenker’s words,

30 Oster trans., Free Composition, 47, § 129.31 Ibid., 48, §134.32 Ibid., 49, §134.33 Heinrich Schenker, “Erlauterungen,” in Das Meisterwerk in der Musik: Ein Jahrbuch [1925] (Munich: Drei Masken Verlag, 1925), 204; trans. Ian Bent, “Elucidations,” The Masterwork in Music: A Yearbook (1925), Volume I, ed. William Drabkin (Cambridge: Cambridge University Press, 1994), 113. Matthew Brown referred me to this source, personal communication. Also see Matthew Brown’s comment concerning reaching-over in Explaining Tonality, 237.


133

In the service of the primary tone [Kopfton], a reaching-over can occur with the effect of—

1. a neighboring note...2. a linear progression which has the sense of an initial ascent...[or]3. an arpeggiation....34

It is these effects of reaching-over that interest us here, since they appear with regular

frequency in both fugues in Major. With this in mind, we will divide this last

section into two parts. First, we look at how reaching-over occurs at the

middleground. Second, we will show how this influences later levels of derivation,

where reaching-over engages properties of counterpoint at the tenth.

We noted earlier in Example 3.13 how the subject from the first Fugue in

Major (BWV 866) uses reaching-over to move through 1 - 2 - 3. We may also

understand this example as an illustration of a preliminary ascent or Anstieg at the

middleground. Referring to Schenker’s description above, this constant leaping-over

of voices creates the sense of an initial ascent. The sense that he alludes to is the

apparent linear progression in the upper voice (amply demonstrated by now as being

a polyphonic transformation). We can illustrate the role of this ascent with a

middleground graph, shown in Example 3.17. Level a shows the reachings-over in

mm. 1-5 as happening between the tenor and soprano voices. In fact, the ascent to the

Kopfton resembles, quite clearly, the schematic outlined earlier in Example 3.15.

Notice also in Level a the reachings-over that occur on tones other than the Kopfton.

34 Oster trans., Free Composition, 48, §132. Emphasis added.


134

Although Schenker restricts this transformation to the acquisition of the Kopfton, he

also writes that “in the presentation of later structural levels I shall show how the

reaching-over technique can relate to tones other than the primary tone.” The

1 2 • application to other tones can be seen in mm. 17-19, where c moves up to c , and in

mm. 19-26, where d1 reaches up to e^2. Level b fills out some of the spans from the

previous level through reaching-over, two of which I will point out. First, a series of

reachings-over facilitate a register-transfer of in mm. 13-19. This confirms

Schenker’s statement that “the purpose of reaching-over is either to confirm the

original pitch-level or to gain another.”36 Second, another series of reachings-over in

mm. 41-45 creates a linear ascent to the passing tone d2, which fills in the third-span

between e^2 in m. 37 and 2 in m. 47. Significantly, these two instances of reaching-

over incorporate the answer in the former and the subject in the latter, both of which

create the “sense” of 1 - 2 - 3 in the dominant and tonic, respectively.

The preliminary ascent within the second Fugue in Major (BWV 890) also

uses reaching-over; however, the Kopfton does not appear within the first statement

of the subject. Rather, it arrives in m. 13, following the presentation of the subject,

answer, and linking material in the previous measures. Example 3.18 illustrates a

form-chart of the fugue, identifying subject-answer entries with their attendant

countersubjects, along with the key-scheme of the entire piece. Two instances of

35 Oster trans., Free Composition, 48, §134. Schenker refers only to foreground levels, though he does not discount later levels o f the middleground.36 Oster trans., Free Composition, 47, §129.


135

reaching-over appear at the middleground, shown in Example 3.19. First, the alto f1

of the opening tonic Stufe reaches up an octave, descends by step to the e^2 neighbor-

note in m. 12, and then falls to the Kopfton in m. 13. Second, the tenor (b^) of m. 37

reaches up to d2 as part of the soprano in m. 44.37 This latter gesture serves to regain

the Kopfton, since the fugal exposition in mm. 1-32 drives the upper line into the

inner voice, traversing a sixth from d2 to f1. The resumption of the Kopfton, however,

appears above VI in m. 47—the first step of the descending-thirds bass line through

VI, IV, and II that governs this piece at the deep-middleground (refer back to

Example 3.10, which also serves as a reading of this piece at an earlier level). Now

that we have examined how reaching-over occurs at middleground levels of structure,

we will next examine how it occurs at the foreground within the second Fugue in

Major (BWV 890), specifically, within the sections that are related to each other via

counterpoint at the tenth.

Details of the first pair of invertible passages are illustrated through graphs

shown in Example 3.20. Level a shows 1 - 2 - 3 of the subject in beams, arrows

signifying the pathways that voices will take, and boxed tones indicating the dotted

half-notes of CS2; Level b shows reaching-over and register transfers of voices; Level

c adds diminutions, including suspensions and passing motion between voices. Refer

to Level a. The first passage (mm. 32-36) is in F major and the second (mm. 40-44)

in B^ major, but both express similar harmonic progressions: the first moves through I

37 The bass transfers from d to d2.


136

- IV - V - 1 and the second through I - IV - V - III. Level b of mm. 32-36 shows the

tenor reaching up to the soprano as the alto passes into the tenor; mm. 40-44 of the

same level shows the alto transferring to the bass, the bass transferring to the soprano,

and the soprano descending into the alto voice. (In mm. 32-36, the game of leap-frog

takes place between the soprano, alto, and tenor voices; in mm. 40-44, however, it

occurs in the bass, soprano, and alto voices [respectively] as voices are inverted.

Thus the bass of this latter section must be understood as a soprano voice in the bass

register—not as a true bass.) Level c adds suspensions and passing motion between

voices (identified with brackets). For instance, the alto passes to the soprano through

c'-d'-e^1 in m. 32; e^1 ties over into the next measure, creating a 4-3 suspension

against the B^-major harmony. The bracketed motives migrate to the bass register in

the parallel passage in mm. 40-44. The subject and CS2 maintain their relative

position with each other in both passages; however, they invert in the next pair of

passages, to be discussed next.

The second pair of passages, mm. 47-51 and mm. 54-58, is shown in Example

3.21, Level a. Here, CS2 appears in the soprano voice (shown in boxes). The setting

of 1 - 2 - 3 in mm. 47-51 differs with others since it begins with tonic, rather than

subdominant harmony (compare with Example 3.20). Level b (of both passages)

shows the bass transferring to the alto, the tenor passing into the bass, and the alto

moving into the tenor. (The final harmony of mm. 54-58, a VII°7/II, deviates from

the voice-leading plan: here, the bass transfers to the soprano d^2 as the soprano


137

moves into the alto b^2.) Level c adds suspensions and passing motion between inner

voices (shown with brackets). For example, a register transfer moves the bass G of

m. 49 into the alto within the same measure; g1 is then tied over into the next

measure, creating a 7-6 suspension over the bass A. As in the previous example, the

bracketed motives migrate from the upper to lower register in the parallel passage in

mm. 54-58.38

Finally, let us consider the passage from mm. 78-86, shown in Example 3.22.

The passage begins in F major and ends with an authentic cadence in G minor. See

Level a. The section begins in F major in m. 78; however, by its conclusion, the tonal

center has moved to major in m. 82. Moreover, the tail-end of the subject from

mm. 80-82 sequences down a third in mm. 83-85, initiating a move to G minor.

Countersubject two—previously restricted to two dotted-half notes—expands to three

dotted half-notes, highlighted with boxes in Level a. Significantly, the expansion of

CS2 alters the harmonic expression of the subject-entry; most entries end on I,

whereas this entry ends on V. Indeed, the forward motion through 3 - 4 - 5 in CS2

TQanticipates a similar drive to the structural cadence in m. 92. As a means of

preventing parallel motion with CS2 in the bass, the upper voices invert with each

38 The change from 7-6 to 2-3 suspensions reflects the presence of counterpoint at the octave between these two passages.39 David Ledbetter remarks on the change o f CS2 in this section; he identifies it as a “common formula” used to unify the complexity o f this passage. See Bach’s Well-Tempered Clavier: The 48 Preludes and Fugues (New Haven: Yale University Press, 2002), 322. In this regard, the “common formula” is reminiscent o f William Caplin’s “expanded cadential progression,” used to achieve cadential closure in the latter half of Classic-style phrases. See Classical Form: A Theory of Formal Functions for the Instrumental Music o f Haydn, Mozart, and Beethoven {Oxford: Oxford University Press, 1998), 19-20.


138

other in the following manner: the tenor reaches over to the soprano as the soprano

and alto move by step into the alto and tenor, respectively. Despite the general

maintenance of the voice-leading pattern, the material in the tenor and soprano inverts

at the octave from mm. 80-82 to mm. 83-85. This can be seen in Level b, where 9-8

suspensions first appearing in the tenor in mm. 81-82 then migrate to the soprano in

mm. 84-85. The third-motive, which participates in these suspensions, is shown with

brackets (as in the previous examples) in Level c. And like the 9-8 suspensions in

this passage, the third motive migrates from the tenor to the soprano voice from mm.

80-81 to mm. 83-84. More globally, the contrapuntal action of the upper voices in

mm. 80-86 recapitulates the diagram of reaching-over shown earlier in Example 3.15.

This observation emphasizes the prevalence of this Schenkerian transformation, since

it produces the effect of parallel motion between the subject and CS2 without

breaking the rules of relative motion for counterpoint at the tenth, a deeply-embedded

property at the middleground.

We conclude our analysis by contrasting two fundamental forces at play

throughout this fugue: descending and ascending motion. We will assign the role of

“descending motion” to the neighbor-note. With respect to the first level of the

middleground, Schenker writes that “only the upper neighboring note is possible.”40

And later he writes that “the neighboring note of the fundamental line thus belongs to

the primary tones 3 or 5 .”41 His observations bear out in middleground levels of the

second major fugue (refer to Example 3.23). The Kopfton arrives via a

40 Oster trans., Free Composition, 42, §106.41 Ibid., 42, §107.


139

neighboring e^2 in m. 13. Likewise, the resumption of 3 occurs in mm. 44-47,

preceded by an upper neighbor in m. 43 (shown in Example 3.24). Indeed, the

neighboring in the upper voice controls a large swath of music from mm. 54-78

(see Level a in Example 3.24) and delays the descent of the Urlinie, which only

resumes at mm. 89-93 (see Example 3.25).42 We must also acknowledge the upper

neighbor’s silent partner in descending motion, namely, reaching-over. Both work

hand-in-hand in approaching the Kopfton in m. 13 and m. 47: a skip from the alto f1

to the soprano f2 in the former (see Example 3.23), and a leap from the tenor b^ to the

soprano f2 in the latter (see Example 3.24). Reaching-over also exerts its influence at

a later level. For example, Level b of Example 3.23 shows the alto f1 surging up to

the soprano f2 in the last ten measures of the exposition, mm. 22-32, followed by an

octave descent f2-f '. And Level b in Example 3.24 illustrates the large-scale upper-

neighbor of mm. 54-88 arriving through reaching-over from the tenor e^1 to the

soprano e^2 in mm. 52-54. Most directly, the role of descending motion, as played by

both upper neighbor and reaching-over, asserts its authority in the fugue subject itself.

Shown earlier in Example 3.14, the first two measures detail upper neighbors to b^1

and f1, while mm. 3-5 show reaching-over from the lower to upper voice.

As a counterforce to the weight of descending motion, ascending motion also

plays a role. The deepest form of upward motion is, of course, preliminary ascent,

42 See Schenker, Der freie Satz, § 109 and §111.


140

which in Schenker’s words “represents a delaying at the very outset of the piece.”43

We can observe this delay in Level a of Example 3.23: the ascent to the Kopfton

occurs over thirteen measures.44 The most direct expression of ascending motion,

however, lies with CS2, for its dotted half-notes exist as the only ascending motion

that occurs within a single voice. The treatment of CS2 was analyzed closely in

Examples 3.20-3.22; we can apply this same treatment to the opening subject in mm.

1-5, shown in Example 3.26a. The example illustrates the tacit role of CS2 moving

through 1 - 2 - 3 as it would appear in the upper voice; meanwhile, the lower voices

continue their undulating game of leap-frog. The same procedure, albeit transposed

to the dominant, occurs in the answer in mm. 5-9 (see Example 3.26b). The answer

also achieves a higher register, f2, as a means of creating stepwise motion to the

deeper-level neighbor e^2 in m. 12 (shown earlier in Level a of Example 3.23).

Ascending motion, thus, occurs within both preliminary ascent and reaching-over,

two voice-leading transformations that interact with counterpoint at the tenth.

As a corollary to the ascending motion of CS2 and its interaction with

counterpoint at the tenth, we can demonstrate how it migrates to different scale

degrees separated by a third (these different instances of CS2 occur in different local

key areas). We can summarize the different scale degree functions of CS2 in a table,

as shown in Example 3.27. The rows correspond to the passages that use

counterpoint at the tenth. After the exposition in mm. 1-32 has concluded, the scale

43 Oster trans., Free Composition, 46, §124.44 Interestingly enough, it takes another thirteen measures (mm. 32-44) to regain the Kopfton after the cadence to the dominant in m. 32 (see Example 3.24, Level a).


141

degrees in the following passages—measuring from the first member of each

instance—move up in thirds from one section to the next: 4 moves to 6 in the

second row; 6 moves up to 1 from the second to the third row; finally, 1 moves up

to 3 from the third to the fourth row. The move up to 1 in major is especially

important, since this scale degree represents the upper neighbor to the Kopfton at

earlier levels (refer back to Levels a and b in Example 3.24). The changes of scale

degree also correspond to changes in harmonic function. For example, the scale

degrees in the second row express pre-dominant to dominant functions (albeit with

different scale degrees); this changes, however, to tonic function in G minor in the

third row. Finally, the scale degrees of the fourth row express tonic, pre-dominant,

and dominant functions: the expansion of CS2 necessarily corresponds to a similar

expansion of harmonic function as it appears in the bass. The ascending motion

through 3 - 4 - 5 in mm. 80-82 and mm. 83-85 presages the final cadence in mm.

92-93, which contracts the previous three-measure units into a one-measure gesture

that leads to the structural dominant. The group of scale degrees and harmonic

functions shown in Example 3.27 illustrates that different formal areas of the fugue

express different tonal behaviors. Moving CS2 around different scale degrees

through the interval of a third or tenth is a way of tapping into these different

behaviors. The multiple settings of CS2 echo Kimberger’s sentiment, presented at

the beginning of this chapter, that invertible counterpoint provides the necessary

means for creating harmonic variety.


3.5. Conclusion

To conclude, we will recount the answers to our three questions posed at the

beginning of this chapter. First, counterpoint at the tenth is a tonal, compositional

technique of setting two parts together, provided that one uses only contrary or

oblique motion between parts and that (triadic) transpositions of a tenth occur

between adjacent tones of the triad: i.e., harmonic thirds transpose to harmonic roots

or fifths, and vice versa. Second, we may utilize this compositional technique within

descending third progressions since this tonal pattern obeys the two conditions

outlined above. This pattern can exist at the surface or at deep levels of the

middleground. Specifically, we may employ counterpoint at the tenth as a means of

creating countersubjects within fugues. Interestingly, the descending-thirds

progression occurs at the deep-middleground of both fugues discussed in this chapter.

Third, Schenkerian transformations, especially reaching-over (and register transfer),

participate with the properties of counterpoint at the tenth, since they ensure that

parallel motion never occurs between the same two pairs of voices. Globally, such

transformations occur at early and later levels, thus integrating apparently isolated

foreground instances of counterpoint at the tenth into deep levels of tonal structure.

With our questions answered, we may finally address issues raised by Prout,

Kimberger, and Schachter. First, we may summarily dismiss Prout’s contention that

“counterpoint at the tenth requires so much to be avoided.”45 This is not true. He

45 See footnote 2.


143

focused on what one should not do. This chapter, however, emphasizes what one can

do; that is, employ contrary and oblique motion, and transpose triadic thirds to

roots/fifths and vice versa. Moreover, we may also, with caution, discount his charge

that counterpoint at the tenth occurs with little frequency; the plethora of tonal

compositions that employ descending-third patterns suggests otherwise.46 Second,

we address Kimberger’s first tenet of four-part writing: “that the harmony have

appropriate variety and diversity along with good continuity.”47 The analyses of the

second Fugue in Major (BWV 890) show the flexibility with which the second

countersubject occupies different harmonic tones and scale degrees throughout the

course of the fugue. Ultimately for Kimberger, it was invertible counterpoint that

provided the key to such variety and diversity. Finally, we confirm Schachter’s

observation that any meaningful study of fugue “goes beyond the identification of

elements and proceeds to an understanding of their function within the whole.”48 His

comments ring true, though we may add that we should study fugue as would-be

composers—not as those who fret over simple machinations of foreground

counterpoint—but as those who work from background to foreground. Instead of

presenting what he calls “timid little monsters”49 purloined from history’s tomb, we

46 This latter observation is offered as a “hunch” o f where to find more instances of counterpoint at the tenth.47 Beach and Thym trans., The Art of Strict Musical Composition, 161.48 Schachter, Bach’s Fugue in B Major,” 263.49 Ibid.,” 263.


144

may progress from the background, where, in Schenker’s words, “the gradual growth

of the voice-leading prolongations, [are] all pre-determined in the womb.”50

50 Heinrich Schenker, “J. S. Bach: Zwolf kleine Praludien Nr. 5,” in Der Tonwille, Funftes Heft [1923] (Vienna: Albert J. Gutmann, 1923), 8; trans. Joseph Dubiel, “Bach’s Little Prelude No. 5 in D minor, BWV 926,” in Der Tonwille, Vol. 1, ed. William Drabkin, (Cambridge: Cambridge University Press, 2004), 180.


145

Chapter 4: Why Use Counterpoint at the Octave?

4.1. Introduction

Having discussed counterpoint at the twelfth and tenth, we now turn our

attention to counterpoint at the octave. There can be no doubt that counterpoint at the

octave played an important role within compositional practice of the Baroque period.

Its presence was felt especially in imitation pieces, where opening statements often

featured the same thematic material in different voices. Such pieces included

inventions and fugues, where subjects and countersubjects could trade places as upper

and lower voices. According to Marpurg, restating thematic material in this manner

was a form of repetition.1 But the nature of this repetition was not exact; according to

present-day theorists Peter Schubert and Christoph Neidhofer, “we use invertible

counterpoint primarily to vary thematic presentation.”2 Whether or not we agree that

counterpoint at the octave was a form of repetition or variation, we can agree that it

was a fundamental component of the Baroque composer’s tool-kit. Indeed, J. S.

Bach incorporated counterpoint at the octave into his Clavier-Biichlein for Wilhelm

Friedemann Bach (circa 1720), which included a set of fifteen two-part Preambuluma

(or “Preambles”) in order of increasing difficulty;3 these pedagogical pieces were

1 See Mann, The Study of Fugue, 142.2 Schubert and Neidhofer, Baroque Counterpoint, 275.3 See Gauldin, Eighteenth-Century Counterpoint, 122.


146

later reordered and revised as the Two-Part Inventions in 1723. Elwood Derr has

written that their original ordering as Preambles provided a series of compositional

models of increasing complexity, ones where “longer segments of invertible

counterpoint begin to accrue to each piece.”4 Indeed, we have evidence that

composers from the Bach circle, such as Kimberger, followed a gradual, step-by-step

acquisition of compositional skills, ones that ultimately led to the writing of fugues.5

One of these skills, to be sure, was the ability to compose pieces that employed

counterpoint at the octave.

Within the more general context of tonality, counterpoint at the octave finds

expression in the cadence, where each of the three upper voices— 5 - 4 - 3 , 3 - 2 -

i , and 1 - 7 - 1 —may appear as the highest sounding part of the texture while the

bass carries the characteristic leap from 5 to 1 (this collection of scale-degree

patterns is the same as Renwick’s voice-leading complex, referred to in Chapter 2).

For example, another member of the Bach circle, Johann Peter Kellner (1705 - 1772),

wrote that “one can...invert these clausulas [the three upper parts] under each other,

so that they can (almost all) be placed either over or under each other;”6 however, he

4 Ellwood Derr, “The Two-Part Inventions: Bach’s Composers’ Vademecum [sic]," Music Theory Spectrum 3 (1981): 40.5 Derr, “Bach’s Composers’ Vademecum,” 47.6 Johann Peter Kellner, Treulicher Unterricht im General-Bafi (Hamburg: Herold, 1737; reprint, New York: Georg Olms Verlag, 1979), 23. My translation. “Man kan aber die Clausuln unter einander verwechseln, daB bald diese bald jene unten oder oben stehet; doch die einzige Bass-Clausul will ihren Siz unten alleine behalten.”


147

t y

remarked that the bass must maintain its position as the lowest sounding voice. The

o

cadence—what Edward Lowinsky refers to as the “cradle of tonality” —plays a

structural role within tonal composition, since it articulates formal boundaries and

defines structural closure. Schenker, too, recognized the formal properties of the

cadence. He wrote in his Hamonielehre,

If we now consider that...the return to the tonic coincides with the formal conclusion...and that it thus signifies a return to the harmonic point of departure, we see that the motion has reached its goal: form as well as harmony have closed their circle.9

His notion of the Ursatz also employed a “cadential formula” of 2 - 1 over V - 1

which achieved triadic completeness of the dominant, “as required by the

arrangement for closing in three-part strict counterpoint.”10 A fundamental

component of this closure is the uniting of both leading tones (i.e., 2 and 7),

occurring as the upper and inner voices, respectively, over the dominant. For

Schenker, the cadence expressed properties of counterpoint, harmony, tonality, and

form. To this group we may add counterpoint at the octave, since it too is a particular

property of cadential motion. Thus by association, counterpoint at the octave not

7 See also Georg Muffat, An Essay on Thoroughbass (1699; ed Hellmut Federhofer, Tubingen: American Institute o f Musicology, 1961), 104. Citation refers to the edited version.8 Edward E. Lowinsky, Tonality and Atonality in Sixteenth-Century Music (Berkeley and Los Angeles: University o f California Press, 1961), 4.9 Borgese trans., Harmony, § 119, 217.10 Oster trans., Free Composition, 16, §23.


148

only plays a tacit role within cadences, it also exists as a silent partner within tonal

structure and formal expression.

Schenker’s opinion of counterpoint at the octave, however, was mixed. As we

have already seen in Chapter 1, his attitude towards counterpoint at the tenth and

twelfth was dismissive. Both negative and positive references to counterpoint at the

octave can be found within his published writings. On the one hand, he dismissed

this concept in favor of combined linear progressions, as discussed in Chapter 1: with

the establishment of a leading linear progression at an earlier level of derivation, one

could then add parallel counterpoints as either upper thirds or lower sixths. He

focused on the leading progression because of its proximity to earlier levels of

derivation; the counterpoints, however much they established counterpoint at the

octave by appearing above and then below the leading progression, were added at

later structural levels, thus assuming a subordinate role within the tonal hierarchy. In

this respect, he diminished the relevance of counterpoint at the octave.11

On the other hand, Schenker regarded counterpoint at the octave as a valuable

compositional resource, one that could express motivic details. In this sense, he even

bracketed instances of invertible counterpoint in his analysis of Bach’s Short Prelude

No. 7 in E Minor (BWV 941) in Meisterwerk I as a way of highlighting the motivic

interplay between the upper and inner voices. Regarding his analysis, he wrote the

following:

11 See Schenker, Der freie Satz, §229.


149

an inversion turns the sixths into tenths (or thirds), as is shown by the brackets, and the line is advanced upward to c2. What was originally the inner voice now becomes the treble; it moves in sixths with the new series of thirds below, and even retains—this is important—the decorating suspensions...12

He went on to say that “the continuing chain of suspensions originates in the

technique of reaching-over.” His comments are illuminating for three reasons. First,

he demonstrated that counterpoint at the octave is a viable compositional technique.

Second, he showed that this technique could be used in tandem with a voice-leading

transformation, in this case, reaching-over (discussed in the previous chapter of this

dissertation). Third—and most importantly—he showed that counterpoint at the

octave created imitative textures, ones where imitation in the upper voice became the

Anstieg, another deep-level voice-leading transformation at the middleground (one

that he had yet to fully formulate within his theoretical writings). Thus, Schenker’s

opinion of counterpoint at the octave was not consistent, but his positive references

towards this technique illuminated a world of compositional possibilities.

The possibilities of combining the properties of counterpoint at the octave

with those of tonality, however, tend to dissolve when we consider Schenker’s later

writings in Der freie Satz. By this time, he had branded invertible counterpoint as a

frivolous mechanism that could not possibly engage deep levels of tonal structure. As

demonstrated above, however, his earlier writings in Meisterwerk I revealed a more

12 Heinrich Schenker, “Joh. S. Bach: Zwolf kleine Praludien, Nr. 7,” in Das Meisterwerk in der Musik: Ein Jahrbuch [1925] (Munich: Drei Masken Verlag, 1925), 110; trans. Hedi Siegel, “Bach: Twelve Short Preludes, No. 7 [BWV 941],” in The Masterwork in Music /, 58- 59.


150

sympathetic attitude towards integrating invertible counterpoint with specific voice-

leading transformations. He continued to develop these transformations—including

preliminary ascent, register transfer, and voice-exchange—in Der freie Satz, by

explaining them as fundamental components of tonal composition, and tonality in

general. But if counterpoint at the octave is also a fundamental element of tonal

composition—especially that from the Baroque period—then this creates the

following paradox: Schenker claimed that counterpoint at the octave participated in

specific voice-leading transformations, yet also dismissed counterpoint at the octave

as being “insignificant.” To address this paradox, we will focus on the following

three issues. First, we will define counterpoint at the octave, especially as it pertains

to tonal composition. Second, we will discuss compositional applications of

counterpoint at the octave. Third, we will explore Schenkerian voice-leading

transformations that interact with counterpoint at the octave. As a way of

demonstrating the principles learned in this chapter, we will utilize analytical

illustrations based on the Fuga from Bach’s Sonata for Solo Violin in G Minor (BWV

1001).


151

4.2. Definitions/Problems of Counterpoint at the Octave

Texts that discuss invertible counterpoint invariably begin with counterpoint

at the octave, because in the words of Kent Kennan, it is “by all odds the most

frequent and natural form of invertible counterpoint.”13 His opinion suggests that

counterpoint at the octave owes its naturalness to the preservation of melodies once

inversion has taken place. Indeed, some theorists analogize the ease of this technique

to the concept of receiving financial rewards. For instance, Henry Martin writes that

“one gets ‘music for free,’” and Peter Schubert writes that “it is economical, since

you get two combinations for the price of one.”14 But despite these endorsements,

even something allegedly natural as counterpoint at the octave bears a hidden cost.

Kennan explains the nature of this fee:

There is only one small difficulty that is likely to arise. If an essential interval of a perfect 5th, generally between the first and fifth scale degrees, appears in the original, the inversion of it will be a perfect 4th. That interval is classed as a dissonance in two-voice counterpoint of this style, and is normally unusable as an essential interval—at least on the beat.15

His comment identifies the problem of counterpoint at the octave: a consonant fifth

inverts to a dissonant fourth. In addition, he only considers the inversion of intervals

13 Kennan, Counterpoint: Based on Eighteenth-Century Practice, 115.14 Henry Martin, Counterpoint: A Species Approach Based on Schenker’s Counterpoint (Lanham, MD: Scarecrow Press, 2005), 98; Schubert, Modal Counterpoint, Renaissance Style, 165.15 Kennan, Counterpoint: Based on Eighteenth-Century Practice, 116.


152

as they occur within counterpoint at the octave. But as we have demonstrated earlier

in Chapters 2 and 3, invertible counterpoint within tonal music involves the inversion

of harmonic tones. Thus, to define counterpoint at the octave as it occurs within tonal

compositions, we must focus on the possibilities of counterpoint at the octave, not its

problems, and then explain how harmonic tones are inverted.

To do this, we resort to the same methodology as applied to counterpoint at

the twelfth and tenth, as discussed in Chapters 2 and 3, respectively. As in these

previous chapters, we can adapt Cerone’s illustrations of invertible counterpoint,

shown earlier in Example 1.26, where two voices are counterpointed against each

other in contrary and oblique motion. The relative motion demonstrated in his

illustration of counterpoint at the octave produces the following intervallic patterns of

consonances, as shown in Example 4.1a: 1 - 3 - 6 and 8 - 6 - 3 . The intervals of

both rows of the table invert into each other. As with the intervallic patterns for

counterpoint at the twelfth and tenth, the patterns for counterpoint at the octave may

be reversed and/or rotated, provided that the adjacent ordering of intervals remain

intact. We can also revise the table of Example 4.1a so that it reflects the inversion at

the octave of harmonic tones; see Example 4.1b. Similar to the last two chapters, this

revised table demonstrates how harmonic tones are transposed under counterpoint at

the octave. The chart shows that harmonic tones undergo no change with this type of

invertible counterpoint: i.e., each harmonic tone preserves its identity. And this is as


153

it should be, since each harmonic tone moves by an octave as voices are inverted.16

This principle is demonstrated in Example 4.1c, which integrates the information of

Examples 4.1a and 4.1b: the top of the example shows the consonant intervals and

their inversions; beneath the intervals, harmonic-tone inversion is illustrated; at the

bottom, harmonic-tone inversion is depicted in musical notation. Significantly, each

harmonic tone—the root, third, and fifth—is able to undergo octave transposition and

invert positions with another harmonic tone. Even the fifth may be transposed by an

octave, provided that it does not appear in the bass. Thus, counterpoint at the octave,

within tonal environments, is the octave-transposition of harmonic tones, provided

that they create the intervals of an octave, third, or sixth between inverted parts.

4.3. Compositional Applications of Counterpoint at the Octave

Now that we have a general definition of counterpoint at the octave, we will

locate some instances of this technique in the Fuga from Bach’s Sonata for Solo

Violin in G Minor (BWV 1001), as shown in Examples 4.2a and 4.2b. Here, different

settings of the subject, answer, and countersubject are illustrated. Example 4.2a

depicts the answer (designated with the letter “A”) beginning on G in m. 2 set below

a countersubject, consisting of a string of descending fifths: B^ - E^ - A - D. The

16 Peter Schubert and Christoph Neidhofer recognize the transposition o f harmonic tones under counterpoint at the octave. See Baroque Counterpoint, 275.


154

numbers beneath the score in m. 2 demonstrate the intervals created between the

answer and countersubject: 6 - 2 - 6 . Inversion of this contrapuntal model occurs in

mm. 26-27, albeit in a slightly modified form. Here, the countersubject does not

retain its original contour; however, it still follows the path of descending fifths: A -

D - G. Nevertheless, the pattern of intervals created in this instance, 3 - 7 - 3 ,

represents the inversion of intervals shown earlier in Example 4.2a. Melodic

fragments of both the subject and countersubject occur in the passage directly after

the exposition, which spans mm. 1-14; Example 4.2b details some of these

occurrences from mm. 17-20. The example shows the subject set beneath the

countersubject in m. 17, albeit both are in slightly truncated form; this setting

translates into the intervallic pattern 2 - 6 . In the following measure, this

contrapuntal model is inverted, thus producing the intervallic pattern 7 - 3 . The

alternation between these intervallic patterns occurs in mm. 19-20 as the subject and

countersubject invert positions with each other.

From a slightly deeper perspective, we can observe how counterpoint at the

octave plays a role within cadences of the Fuga. Refer to Example 4.3, which

juxtaposes mm. 32.5 - 42 in the dominant with mm. 85 - 87 in the tonic. The

beginnings and endings of these corresponding sections are practically identical;

however, this identity is not maintained throughout their entirety. For example, the

later section in the tonic is short and swift, comprising three measures, whereas the

earlier section in the dominant comprises eleven measures. Nevertheless,

counterpoint at the octave appears at their respective beginnings, each of which


155

articulates the arrival of the dominant with the same motivic gesture. In mm. 32.5,

the three-note motive, consisting of two sixteenths and an eighth (B^ - G# - A),

appears in the soprano above the leading-tone (C#) in the alto; in m. 85, the same

motive (E^ - C - D) appears in the alto below the leading-tone (F#) in the soprano.

Both sections culminate with sixteenth-note passage work articulating the moment of

cadential closure in m. 42 and m. 87, respectively. Differences in surface-details

begin to emerge after the statement of the three-note motive (referred to earlier) in

mm. 32.5 - 33 and m. 85, respectively , but the presence of counterpoint at the

octave, working at deeper levels, still persists. For example, the leading-tone (F#) in

the soprano is placed above 2 in the alto in m. 86 (see the boxed areas in Example

4.3); this arrangement inverts in mm. 35-36, placing the leading-tone (C#) in the alto

below 2 in the soprano. Moreover, movement to V4/2 in m. 36 subverts closure to

the local tonic of D minor. A root-position tonic does not arrive until seven measures

later in m. 42 (the analogous place to the cadential closure in m. 87), which is

preceded in mm. 38-41 by what Russell Stinson calls a “tension-generating pedal

point.”17 We may ascribe this generation of tension to another case of counterpoint at

the octave occurring over this pedal, also shown in Example 4.3. Here, the parallel

thirds of mm. 38-39 invert into parallel sixths in mm. 40-41; only after this inversion

takes place does a firm cadence in the dominant take place in m. 42. As referred to

earlier in this chapter, cadences are comprised of scale-degree patterns that are

17 Russell Stinson, “J. P. Kellner’s Copy of Bach’s Sonatas and Partitas for Violin Solo,” Early Music 13/2 (1985): 207.


156

invertible at the octave. The particular instances, shown here, of counterpoint at the

octave within cadences, are no exception to this rule.

These are not the only cases of invertible counterpoint within this piece, but

suffice it to say foreground counterpoint at the octave plays a considerable role in the

Fuga. From a more general perspective, counterpoint at the octave also plays a role

within the structure of tonal cadences. Within this vein, we can also apply the

properties of counterpoint at the octave to Schenkerian Ursatze. As a case in point,

we will examine the Ursatz starting from 8, since this prototype applies to the Fuga

in particular. But before we examine the invertible properties of this prototype, we

will first discuss its general nature.

The Urlinie starting from 8 is one of but three options for Ursatze occurring

at the background.18 The Urlinie—either starting on 3, 5, or 8—appears above a

counterpointing Bafibrechung of I - V - 1, which creates unique contrapuntal

relationships between the outer voices, depending on the upper line’s point of origin,

i.e., the Kopfton. In the cases of Urlinien beginning on 5 or 8, passing tones exist

between the Kopfton and 2. Such tones create dissonances with the tonic Stufe,

18 *See Schenker, Der freie Satz, §41 - §44. For further discussion of 8 as a viable, andessential, Kopfton option, see the following: David Beach, “The Fundamental Line fromScale Degree 8: Criteria for Evaluation,” Journal o f Music Theory 32/2 (1988):271-294;Steve Larson, “Questions about the Ursatz: A Response to Neumeyer,” In Theory Only 10/4(1987): 11-31; David Neumeyer, “The Urlinie from 8 as a Middleground Phenomenon,” InTheory Only 9/6 (1987): 3-24; David Neumeyer, “The Ascending Urlinie,” Journal o f MusicTheory 31/2 (1987): 275-303; David Neumeyer, “The Three-Part Ursatz,” In Theory Only10/1-2 (1987): 3-29; David Neumeyer, “Reply to Larson,” In Theory Only 10/4 (1987): 33-37; David Smyth, “Schenker’s Octave Lines Reconsidered,” Journal of Music Theory 43/1(1999): 101-133.


157

though Schenker made it clear that “tones of the fundamental line may sometimes

appear as dissonant passing tones.”19 He referred to the dissonant zone between

supported Urlinie-tones as a Leerlauf,20 translated often as “unsupported stretch.”

For example, 4 is a dissonant passing tone between 5 and 3. His preliminary

solution to bridging this dissonant gap involved creating apparent subdivisions, thus

providing consonant support to originally dissonant tones. For example, one may

A /\ ~ |

place V beneath 7 or 5 in the case of octave-lines. This experiment fails, however,

9 9since in his words, “the unsupported stretch has not been eliminated.” This is

because the Urlinie possesses a “forward compulsion toward 1,” one that “reaffirms

the indivisibility of the progression... [of the Urlinie]..An the fundamental structure, no

23matter how much the appearance of a subdivision might seem to deny it.” In efforts

to solve the problem, he recommended that “the tasks of the middleground and

foreground are to eliminate the unsupported stretch”24 by providing consonant

support to originally dissonant passing tones. Through the accretion of structural

levels, each tone of the octave-line receives consonant support, thus making it a

viable, compositional template.

19 Oster trans., Free Composition, 32, §69.20 See Schenker, Derfreie Satz, §36.21 Ibid., §42.22 Oster trans., Free Composition, 20, §42.23 Ibid., 20, §38. Schenker’s notion of the Crime's indivisibility is made abundantly clear earlier on p. 12, §6: “No matter what upper voices, structural divisions, form, and the like the middleground or foreground may bring, nothing can contradict the basic indivisibility o f the fundamental line.”24 Ibid., 20, §39.


158

With respect to 8-line Ursatze, we will investigate how they interact with

counterpoint at the octave. As shown previously, counterpoint at the octave simply

involves octave transpositions of harmonic tones of Stufen, provided that the

harmonic fifth never occurs in the bass. We will start with an Ursatz that moves

through the octave in G minor, which is the key of the Fuga (Example 4.4). (This

illustration preserves the one-flat signature of the Fuga.) The example shows an

• • 8 7octave descent in the upper voice G-G with a bass arpeggiation I - V ' - 1. The

annotations above the staff show harmonic-tone inversion occurring between the root

and third of the tonic and dominant Stufen. The root appears above the third in the

tonic; it then appears below the third in the dominant. As well, the inversion takes

places between two different pairs of voices: soprano and tenor in the tonic, alto and

tenor in the dominant. Also notice that the fifth appears below the root in the tonic

and then appears above the root in the dominant. From this very global perspective,

counterpoint at the octave plays a role within this 8-line Ursatz.

4.4. Schenkerian Transformations and Counterpoint at the Octave

The instances of counterpoint at the octave in the cadences at mm. 35-36 and

mm. 86-87 of the Fuga suggest that there must be a Schenkerian transformation that

allows this inversion to take place. Historical notions of cadences confirm this first


159

observation. Cadences were understood in the fifteenth- and sixteenth-centuries as

being composed of two parts, a tenor and a discant. The former typically carries 2 -

1 and the latter 7 - 1 ,25 Either voice may be placed over the other, creating the

harmonic interval-succession major sixth to octave or minor third to unison,

depending upon vertical placement. The discant-tenor model carried into theoretical

thinking in the seventeenth- and eighteenth-centuries, fleshing out the two-voice

framework into a four-voice complex. An alto 5 - 4 - 3 and a bass 1 - 5 - 1 were

A A A A A A

added, along with the expansion of the discant to 1 - 7 - 1 and the tenor to 3 - 2 - 1.

Finally, Schenker adopted the discant-tenor model to one that expresses leading-tone

A A

tendencies, where 2 is the descending leading tone and 7 the ascending leading

26tone. He acknowledged that both should appear together at cadences in two-voice

counterpoint, either expressing the interval succession 6 - 8 or 3 - 1; however, he did

not distinguish them as inversions of each other.27 Both leading-tones then became

the backbone of his discussion of three-voice cadences, which allowed for the

inclusion of complete triads built on the structural dominant.28 As in his discussion of

two-voice counterpoint, three-voice cadences may place either leading tone above the

other, but inversion was never introduced as a guiding principle. Nevertheless,

25 For more on this, see Richard Crocker, “Discant, Counterpoint, and Harmony,” Journal of the American Musicological Society 15/1 (1962): 1-21, and Elisabeth Schwind and Michael Polth, “Klausel und Kadenz,” in Die Musik in Geschichte und Gegenwart, ed. LudwigFinscher (Barenreiter: Kassel, 1996), 5: 266-267.26 Rothgeb and Thym trans., Counterpoint I, 102.27 Ibid., 171.28 Rothgeb and Thym trans., Counterpoint II, 45-47.


160

Schenker integrated prior notions of cadence-construction—including its invertible

properties—into his theory of tonality.

One such transformation that engages aspects of counterpoint at the octave,

including at cadences, is register transfer, which can occur in ascending or descending

directions. For Schenker,

“ascending register transfer means a raising to a higher octave. The octave interval is the basis of this concept.”29

He went on to say that “register transfer is applicable to fundamental-line tones or

their neighboring notes, and to a single tone or succession of tones of an inner voice.”

Thus, ascending register transfer entails an upward octave-displacement of tones of

the Urlinie and those of the inner voice. This transformation works nicely with

counterpoint at the octave, since both involve octave-transpositions of harmonic tones

(see Example 4.5, which illustrates register transfer). Here, we see the G minor

octave-line prototype shown earlier in Example 4.4, albeit with a register-transfer at

III5'6: the alto G surges up an octave, placing the inner voice above the final 3 - 2 - 1

descent in the Urlinie.30 Now the interval between the harmonic third and fifth of the

dominant Stufe is a sixth, whereas before it was a third (compare Example 4.4 to

Example 4.5). This illustration of register transfer is also significant, since it relates

29 Oster trans., Free Composition, 51, §147. The interval o f an octave, however, does not solely mean octave-displacement o f particular tones, e.g., neighbor-notes may appear in octave-displaced registers, without actually transferring the note being neighbored. See Fig. 107.30 See Schenker, Der freie Satz, Fig. 46/2 for a similar illustration.


161

directly to the cadence from mm. 86-87 of the Fuga, shown earlier in Example 4.3.

We can add another level of derivation, shown in Example 4.6, which illustrates

A A A A A |

instances of register transfer acting upon 6 , 5, 4 , and 3 in the Urlinie. Most

^ • 1 • 2 *notable is 5, which occurs in three successively higher registers: d in m. 24, d in m.

A

38, and d in m. 42. The Urlinie submerges into the tenor at 4 in m. 55, only to later

shoot up into the two-line register in m. 58. These octave-displacements necessarily

create invertible relationships between the three upper voices: i.e., the tones of the

Urlinie occur as either the soprano, alto, or tenor throughout the course of the piece.

In addition, register transfer brings attention to structural tones: e.g., 3 in m. 58

signals the division in the ascending bass arpeggiation at III. Register transfer also

accentuates different registers throughout the range of a particular instrument. Thus,

counterpoint at the octave—as articulated by register transfer—finds expression

through formal articulation and instrumental technique. Schenker eloquently

summarized the confluence of these and other compositional techniques that may

result from register-transfer:

A desire to exploit the brilliance of an instrument, the establishment of a relationship between registers, the general necessity of creating new content, the accentuation of formal divisions—all these lead to an ascending register transfer at the later levels.32

31 Justin London and Ronald Rodman discuss registral changes in the Urlinie in “Musical Genre and Schenkerian Analysis,” Journal o f Music Theory 42/1 (1998): 115-116.32 Oster trans., Free Composition, 85, §238.


162

One may also engage counterpoint at the octave through voice-exchange,

another transformation discussed by Schenker. He described this transformation in

the following way:

The necessity for exchanging voices arises only at the later levels.Here the diminutions have expanded considerably, and so it frequently becomes necessary to bring the demands of the diminutions and the requirements of the voice-leading into agreement with one another.33

With respect to level, voice-exchange (according to Schenker) only occurs at the

foreground—it is not used to create large-scale voice leading structures.34 According

to his description, voice-exchange is a foreground transformation that responds to

earlier levels of derivation. He did not clarify, however, what is being exchanged.

The term “voice-exchange” is clear enough: it implies that voices trade places, e.g.

soprano changes with bass. But the term does not precisely express the true nature of

this transformation as described in Der freie Satz. Here, Schenker’s explanations and

illustrations showed that voices trade content—they do not trade places. He

demonstrated three ways in which to use this transformation. First, outer voices may

exchange content that allows them to stay within the same harmony.35 Second, the

33 Oster trans., Free Composition, 84, §236.34 Those who would disagree with this viewpoint include Carl Schachter. See “The First Movement o f Brahms’s Second Symphony: The Opening Theme and Its Consequences,” Music Analysis 2/1 (1983): 63-68. For material related to Schachter’s argument, see Schenker, Der freie Satz, §250.35 For example Allen Cadwallader and David Gagne write that “perhaps the most characteristic type o f voice exchange occurs. ..where the soprano and bass exchange two tones within the same chord.” Analysis of Tonal Music, 136.


163

inner voice may exchange content with the upper voice. Third, the bass may

exchange content with the upper voice if the register becomes too low. In all cases,

voices maintain their registral disposition while exchanging content.

To illustrate this concept, we turn to the pedal-point in mm. 38-42 of the Fuga

(refer back to Example 4.6). Here, 5 transfers from the alto to the soprano over a D-

minor Stufe. At a later level shown in Example 4.7a, a voice-exchange allows the

soprano and alto to exchange content: the crossing arrows demonstrate how the alto

melody D - C# - D continues into the soprano as the soprano melody F - E - D

continues into the alto. Example 4.7b shows how arpeggiation and passing tones fill

in the gap between f2 and c#3, thus creating a new upper voice. This later level

further confirms that voice-exchange does not trade essential voices, per se, but rather

essential content.

Register transfer and voice-exchange, therefore, engage counterpoint at the

octave, though in different ways. First, register transfer alters the registral order of

essential voices through octave-transposition (Example 4.7c). Here, a tonal model

consisting of I - V - 1 and a linear descent from 3 in the upper voice is shown; the

registral order of the upper voices from top to bottom (soprano, alto, and bass) is

indicated above the staff via the letters “S,” “A,” and “T,” respectively. Register

transfer is applied to the alto voice in Example 4.7d: this transformation moves the

alto voice above the soprano (this is reflected in the letters above the staff, where A

appears above S over the V-Stufe). Second, voice-exchange at later levels provides a

means of navigating through octave-displaced voices inherited from previous levels:


164

i.e., voice-exchange alters the registral order of essential content. Voice-exchange is

applied to the content of the soprano and alto voices in Example 4.7e; here, the upper

two voices exchange their respective content over the V-Stufe while the registral

order of all voices remains intact. The arrows, in this case, indicate that two

melodies, E - D - C and C - B - C (i.e., content) move from one voice to another.

(The passing tones in parentheses in Example 4.7e merely fill in the space left open

by the register transfer that occurred at the previous level in Example 4.7d.) Based

upon the nature of these two transformations, voices (ironically) maintain their

registral disposition more strictly as one approaches the foreground.

Thus, one can view counterpoint at the octave and octave-displacement

through a Schenkerian lens, filtering harmonic tones through the transformations of

register transfer and voice-exchange. Octave-displacement behaves like an icon

throughout the Fuga, articulating section beginnings and endings. Not only does it

influence the way we look at the cadences in the dominant and tonic (previously

discussed), it also enhances our vision of other cadences within the piece—

specifically those that involve the tones of the Urlinie. Next, we will turn our

attention towards the octave, not only as a part of transformations at the

middleground and foreground, but also as an overarching compositional principle

within the Fuga.


165

4.5. Analytical Applications within the Fuga (BWV 1001)

We have already shown that the Fuga results from an 8-line Schenkerian

Ursatz. The octave, however, also exerts its influence at the surface. Indeed, Joel

Lester has remarked in his recent book on Bach’s music for solo violin that the octave

plays a role in the exposition of the Fuga. He explains that “the descending bass

octave scale in the violin [g1 - g] in mm. 2-5...helps propel the music toward the

episodic sixteenths, and that same bass octave descent in mm. 11-14 helps round off

the opening section of the Fuga with particular clarity.” He also writes that

components of octave-scales in the bass appear at the surface in other movements

from the same sonata, e.g., the descent from G - D in mm. 2-4 of the opening

Adagio?1 The octave also influences the structure of other pieces by Bach, including

many of the other works for solo violin and ‘cello. Schenker analyzed many of these

pieces as octave-lines, including the Sarabande from the C Major Suite for

Violoncello, the Largo from the C Major Sonata for Solo Violin, and the Preludio

from the E Major Partita for Solo Violin.38 David Beach, in his recent monograph on

Bach’s Partitas and Suites, observes,

36 Joel Lester, Bach’s Works for Solo Violin: Style, Structure, Performance (Oxford: Oxford University Press, 1999), 81.37 Lester, Bach’s Works for Solo Violin, 72.38 Heinrich Schenker, “Joh. Seb. Bach: Suite III C-Dur fur Violoncello-Solo, Sarabande,” in Das Meisterwerk in Der Musik: Ein Jarbuch, Band //(Munich: Drei Masken Verlag, 1926), 97-104; trans. Hedi Siegel, “The Sarabande o f J. S, Bach’s Suite No. 3 for Unaccompanied Violoncello [BWV 1009],” Music Forum 2 (1970): 274-282; “Joh. S. Bach: Sechs Sonaten fur Violine: Sonata III Largo,” in Das Meisterwerk in Der Musik: Ein Jarbuch [1925] (Munich: Drei Masken Verlag, 1925); 61-73; trans. John Rothgeb, “The Largo o f Bach’s SonataNo. 3 for Solo Violin [BWV 1005],” Music Forum 4 (1976): 63-73; “Joh. S. Bach:


166

The prominent role of the octave as a large-scale organizing device...in Bach’s works points to an interesting question: why do we encounter structural octaves in Bach’s music, or for that matter, in late baroque music in general, but not in compositions of the classical period? Perhaps the answer to this question lies in the differing nature of the forms employed. For example, the tonal motion of the typical baroque binary form is continuous, a condition favorable to the large-scale unfolding of octave lines, while classical binary forms (rounded binary

IQ

and sonata form) involve interruption.

Thus, the octave has cropped up in many prior analyses of Bach’s music, but why is

this so?

Most composers from the eighteenth century would have been familiar with

the regie de I ’octave, a thorough-bass technique used to accompany the ascending

and descending scale on the clavecin, guitar, or theorbo. Although it is difficult to

ascribe authorship of the regie to any one person, most attribute Francois Campion

(ca. 1685 - 1747) as the first to coin this term in his treatise of 1716.40 The influence

of the regie was widespread; in this light, Thomas Christensen writes, “[I]n virtually

every eighteenth-century thorough-bass and composition treatise one finds a series of

scale harmonizations figured above all 24 ascending and descending major and minor

scales.”41 J. S. Bach was not immune to the techniques endowed by the regie, for we

Sechs Sonaten fur Violine: Partita III (E-Dur) Praludio,” in Das Meisterwerk in Der Musik [1925], 75-98; trans. John Rothgeb, “The Prelude o f Bach’s Partita No. 3 for Solo Violin [BWV 1006],” in The Masterwork in Music, ed. William Drabkin, (Cambridge: Cambridge University Press, 1994), 39-53.39 David Beach, Aspects of Unity in J. S. Bach’s Partitas and Suites: An Analytical Study, Eastman Studies in Music (Rochester: University o f Rochester Press, 2005), 84, footnote 21.40 Francis Campion, Traite d ’Accompagnement et de Composition selon la regie des octaves de musique (Paris, 1716). See Thomas Christensen, “The Regie de VOctave in Thorough- Bass Theory and Practice,” Acta Musicologica 64 (1992): 91.41 Christensen, “The Regie de I ’Octave in Thorough-Bass Theory and Practice,” 91.


167

find this technique resonating throughout his music and the few pedagogical sources

that survive today. For example, authors such as William Renwick and Joel Lester

have commented on the descending C major scale that initiates Bach’s Prelude in C

Major from the Well-Tempered Clavier, Book I.42 Bach also used descending and

ascending octaves to demonstrate different sequential techniques in his Vorschriften

und Grundsatze zum vierstimmigen Spielen des General-Bass oder Accompagnement

(1738),43 a treatise that had been greatly influenced by the Musicalische Handleitung

(1721) of Friedrich Erhardt Niedt (bapt. 1674 - 1708). For example, a descending

octave was used to demonstrate how to play parallel sixth-chords in four parts at the

keyboard.44 The exercise included descending scales on the tonic, dominant, and

submediant, followed by a da capo sign, and concluding with a cadence on the tonic.

This latter detail suggests that realization of octave-scales was not just a manner of

accompaniment, but a technique of composition. Indeed, Christensen distinguishes

between opposing attitudes toward the regie. First, German composers typically

regarded it as a “compositional prescription”45 for realizing thorough-bass, one that

emphasizes rules of counterpoint and harmony, rather than instrumental technique.

Second, Italian composers regarded it as a type of keyboard pedagogy (expressed in

the form of partimenti), one acquired through rote memorization.46 Whichever way it

42 Renwick, Analyzing Fugue, 14; Joel Lester, Compositional Theory in the Eighteenth Century (Cambridge: Harvard University Press, 1992), 74.43 See the translation, J. S. Bach, Precepts and Principles for Playing the Thorough-Bass or Accompanying in Four Parts, trans. Pamela L. Poulin (Oxford: Clarendon Press, 1994). See especially p. xviii, which discusses the connection between Bach and Niedt.44 Poulin trans., Precepts and Principles, 46.45 Christensen, “The Regie de FOctave in Thorough-Bass Theory and Practice,” 117.46 Christensen, “The Regie d e l’Octave in Thorough-Bass Theory and Practice,” 114.


168

was applied, there can be no doubt that the regie de I ’octave was a concept that was

ingrained into the minds of eighteenth-century composers and performers alike.

Absent from these compositional and pedagogical perspectives is any

discussion of how the regie relates to invertible counterpoint, specifically at the

octave. For example, how does one treat octave-scales that appear in the soprano?

This question is never dealt with specifically, since the regie was meant to simplify

the practice of unfigured bass—a rule-laden procedure47—not as a solution for

invertible counterpoint. Despite this lack of explanation in contemporaneous

treatises, we know that Bach employed octave-scales in both the bass and the soprano

within his compositions. For example, Renwick notes that his two-part Invention in

G major is based on a descending scale in the soprano 48 Chorale harmonization

presents another obvious compositional context where the soprano must be

harmonized: e.g., Bach’s setting of “O Ewigkeit, du Donnerwort,” whose opening

phrase features an ascending F major scale.49 Nevertheless, the octave—whether it

appeared in the bass as part of the regie de I ’octave or the soprano as part of a chorale

melody—was used as a basis for performance and composition during the eighteenth

century. It is no surprise, then, that we find many examples of it in Bach’s music,

including his fugues.

From a Schenkerian perspective, however, Renwick claims that most fugues

exhibit 5 as the Kopfton, though some, such as Bach’s Fugue in B^ Major from the

47 See Lester, Compositional Theory in the Eighteenth Century, 69-72.48 Renwick, Analyzing Fugue, 15; this occurs in mm. 1-9.49 See chorale #26 in Johann Sebastian Bach, 371 Harmonized Chorales and 69 Chorale Melodies with Figured Bass, ed. Albert Riemenschneider (New York: Schirmer, 1941), 7.


169

Well-Tempered Clavier, Book I (BWV 866), progress from 3. Despite the

abundance of 5-line fugues, he makes his position clear: “There are no a priori

restrictions as to which background structure a fugue must follow, and any of the

fundamental line forms is possible with any subject type.”50 Indeed, as we have seen

already, the Fuga from the solo violin sonata has an Urlinie that begins from 8.

Although this prototype is allegedly not as typical for fugues as the other two, the

fugue-subject itself is very typical: it expresses 5 - 4 - 3 (this is the same paradigm

that was used for the subject of the Fugue in C Minor [BWV 847], discussed in

Chapter 2). The real answer that follows expresses the same scale degrees in the

subdominant, instead of the expected dominant. Thus, a conventional subject is

paired with an unconventional answer. But why is this so? A tonal answer would

place its final note on F, thus allowing for motion to the dominant. This type of

answer would also necessitate numerous repetitions of G (4 in D minor), which, as

Lester remarks, “would be intolerably dull.”51 As an antidote to this ennui, he writes,

“Bach’s solution is to abandon altogether a fugal answer in the dominant and instead

place the answer in the subdominant key...so that the subject’s and answer’s repeated

notes express the key-defining 5-1 (D-G).” The subdominant answer may provide

respite from impending dullness. The brevity of the subject and the fairly expansive

length of the piece may support this reading. Difficulty ensues, however, when we

base theoretical assertions on aesthetic criteria such as “ennui” and “excitement.”

50 Renwick, Analyzing Fugue, 207.51 Lester, Bach’s Works for Solo Violin, 67.52 Ibid., 67.


170

Thus, we must ground our theoretical observations upon the foundation of harmony

and voice leading. In short, we must view the exposition from a Schenkerian

perspective.

We begin our investigation by placing the exposition within the overall form

of the Fuga as shown in Example 4.8. The entire piece divides, essentially, into two

halves. We will first focus on the second half, since it contains the structural cadence.

Closure to the tonic occurs in m. 87; passagework in sixteenth-notes follows in mm.

87-94, bringing the piece to its conclusion. The second half begins at m. 58 on the

lll-Stufe, which corresponds to the division in the Bafibrechung at the deep-

middleground (shown earlier in Example 4.5). The first half of the piece begins with

the four entries of the exposition in mm. 1-14; after the close of the exposition,

motion towards the dominant occurs in mm. 14-42 and is articulated by three

cadences at m. 24, m. 38, and m. 42. After confirmation of the dominant, a move

towards the subdominant begins at m. 42 and is confirmed with a perfect authentic

cadence in m. 55. A short imitative section follows in mm. 55-57, setting up the

arrival of III (or major) in m. 58. Many correspondences exist between the two

halves of the fugue: e.g., the subject entries at m. 28 and m. 74, the use of IV6 as an

approach to the dominant in m. 32 and m. 84.5, the double-neighbor motive (marked

with “DN”) at mm. 32.5-33 and mm. 85-85.5, and especially the passagework

sections in mm. 42-55 and mm. 87-94. Such regular recurrences of material at the

surface also play a role at deeper levels of structure, where similar voice-leading


171

strategies are deployed throughout the course of the fugue. We will now investigate

those strategies that occur within the exposition.

The exposition expresses similar tonal details as that of the entire piece

(please see Example 4.9, Level a). Within mm. 1-14, the upper voice traverses the

span of an octave from g - g . Level a shows the upper voice moving from 8 - 3 in

mm. 1-5 as the bass undergoes a register-transfer from g1 to g. The Bafibrechung

/ A

divides at I in m. 13, supporting an implied 3 in the upper voice. The dominant

arrives in the same measure and supports 2 in the upper voice; this is followed by a

PAC in m. 14, which closes off the exposition. Level a shares similarities with the

deep-middleground of the entire piece shown in Example 4.5. These similarities

include the octave-line descent and the division at 3; however, a dividing dominant

V7 occurs in mm. 6-11, and I6 harmonizes 3 instead of III. Despite the subtle

differences, the similarities make a convincing argument for the exposition as a

compositional model for other sections of the piece. From the model, one can extract

particular linear spans and techniques that are used elsewhere in the piece, such as

octave-progressions, sixth-progressions, register transfer, and voice-exchange. As an

illustration, we will show how register transfer works within the exposition.

Two instances of register transfer control much of the music that occurs

within the exposition; refer to Level b of Example 4.9. The first instance of register-

transfer occurs in m. 5: 3 makes the octave-leap between b^1 and b^2 (indicated with

a bracket on the example); the bass and tenor follow suit (the bass, however, moves


172

up two octaves), and the alto, g1, passes into the tenor, e^1, in mm. 5-6. The second

instance occurs in m. 11, where the tenor c2 (the dissonant seventh of the dominant)

transfers its resolution to the soprano b^2, thus resuming the progression of the

Urlinie. Both instances of register transfer discussed here initiate descending linear

progressions, such as the fifth-progression from a -d in mm. 7-9 and the octave-

progression b ^ -b ^ 1 in mm. 11-13.53

The sixth-progression from g2 to b^1 engages the entry-scheme of the

exposition shown in Level c. (Entries are indicated with beams; “S” refers to the

subject and “A” refers to the answer.) The order of entries occurs in the following

manner: subject - answer - answer - subject. Such a pattern deviates from the usual

alternation of subject and answer in a four-voice fugue. This is so for three reasons.

First, the framing subjects at the beginning and ending of the entry-scheme allow for

a tonally closed expression of thematic material; concluding with an answer would

steer the harmony towards the subdominant. Second, the immediate repetition of the

answer in m. 3 highlights g2 as the Kopfton. The third entry does not juxtapose

different tonal material with that of the second; rather, it contrasts different registers

between the soprano and tenor voices. Third, the last two entries—the answer and

subject in mm. 3-5—present the sixth-progression from g2 - b^1, a foreground version

of the same progression that exists at the deep-middleground (Example 4.5). The

53 The transfer from b 2 to includes the chromatic inflection from a2-a^2 in mm. 11-12, alluding to the same inflection that occurs m. 7 in the score. See Lester’s discussion o f the juxtaposition o f A/A^ within the Fuga and how this contributes to creating progressive levels of intensity throughout the piece in Bach’s Works for Solo Violin, 68-69.


173

subject and answer, therefore, appear in two different formats: first, in imitation

between different voices, e.g., the opening gesture of mm. 1-3, and second, as a linear

progression within a single voice, i.e., the sixth-progression in mm. 3-5. Both

formats appear throughout the course of the fugue. Thus, the subject and answer exist

as a compositional unit, one that has properties of harmony, voice-leading, and form.

4.5.1. Octave-progressions

Octave-progressions guide the structure of the Fuga from early to later levels

of derivation. As we have seen, the descending octave controls the composition all

the way to the background, as it does the foreground pattern of entries in the

exposition. The descending octave, therefore, serves as a compositional model in

general, as part of the Ursatz, and in particular as part of the exposition. Three

instances of octave-progressions, apart from the exposition, should be mentioned.

Refer to Example 4.10. First, a descending octave from G-G controls mm. 14-19.

Indeed, this span resembles the one in the exposition, since it divides the descending

progression at b^2 (harmonized with G minor; compare m. 18 with m. 5).

Significantly, a change in register of the descending line emphasizes the division at

b^2 in m. 18. Second, another octave-progression occurs in mm. 20-24, this time from

D-D, as the Urlinie makes its descent through 7 - 6 - 5. As a parallelism with


174

previous instances, the descending progression subdivides into a sixth—D to F in

mm. 20-22—and a third—F to D in mm. 22-24. One difference, however, is the

change to a lower register for the last two tones of the progression e1 - d1 in m. 24,

placing the soprano into the tenor voice. As well, the D-D span features the subject in

the bass in m. 20 (beginning on a1); this contrasts with mm. 14-19, where the subject

occurs only in the upper voices. The migration of the subject to the bass creates a

string of parallel tenths in mm. 20-22: c3/a' - b^ /g1 - a2/f* - g2/e' - f2/d1. Compare

this to the parallel sixths in mm. 15-16: d3/f2 - c3/e^2- b^/d2. Third, an octave-

progression from b^2 to b^1 in mm. 74-80 (shown in Example 4.11) returns 3 to its

proper register. Interestingly, similar treatment of this scale degree occurs at the end

of the exposition in mm. 11-13 (shown in Example 4.9, Level b). Thus, the

descending octave-progression—a model established in the exposition—is applied to

each of the harmonic tones of the tonic triad: G, B^, and D.

4.5.2. Register Transfer

Register transfer is another technique that structures the Fuga from early to

later levels (refer back to Example 4.6, which shows displaced Urlinie-tones

throughout the course of the piece). This transformation is used in two ways; first, as

a means of displacing tones of the Urlinie at earlier levels; second, as a way of


175

creating imitative textures at later levels. We will now discuss the first application.

Refer to Example 4.12, which summarizes mm. 24-42. Here, the example illustrates

the most pronounced instances of displacing tones of the Urlinie, where 5 appears in

three different registers—d1, d2, and d3. The three registers occur at m. 24, m. 38, and

A 54m. 42, where 5 occupies the tenor, alto, and soprano, respectively. Another

instance of register transfer—separate from the ones that act upon 5 —is also shown.

Here, the tenor a1 of m. 35 transfers to the soprano register in m. 36; both measures

occur within the dominant Stufe of D minor. After the transfer in m. 36, the soprano

moves through a passing seventh over the dominant to 3 over the tonic in mm. 37-38.

This move achieves cadential closure, but simultaneously it precludes the

middleground 5 from appearing in the highest sounding part. Perfect closure arrives

in m. 42 as D breaks into the three-line register over the tonic pedal-point in mm. 38-

42. Complicit with the register transfer of A in mm. 35-38 is the use of imitation,

which occurs between the tenor and soprano voices. We now turn our attention to

later levels of derivation which feature imitation as a component of register transfer.

We have already shown that counterpoint at the octave allows for imitation to

occur at the surface. Register transfer plays a role in these imitative schemes since it

54 Carl Schachter shows that the entire Urlinie may exist as an inner voice at deep levels, in “The Prelude from Bach’s Suite No. 4 for Violoncello Solo: The Submerged Urlinie,” Current Musicology 56 (1994): 54-71 (see especially pp. 60-61 and pp. 68-69); the same author also shows that the Urlinie may begin as an upper voice but later enter the inner voice as it descends to the tonic, in “The Triad as Place and Action,” Music Theory Spectrum 17/2 (1995): 149-169 (see especially pp. 151-153). Eric Wen writes that the Urlinie may occupy the bass voice exclusively in “Bass Line Articulations of the Urlinie,” in Schenker Studies 2, ed. Carl Schachter and Hedi Siegel (Cambridge: Cambridge University Press, 1999), 276- 297.


176

displaces particular voices above or below other voices by an octave. We have also

shown that the three upper voices of a cadence are completely invertible above the

bass, allowing for imitation to occur at structural closes of phrases. This is exactly

what occurs at mm. 52-55, preceding the cadence to C minor (refer to Example'4.13).

The illustration at Example 4.13a reproduces a typical cadence I - V7 - 1, where the

three upper voices articulate the following scale degree patterns: 5 - 4 - 3 , 3 - 2 -

A A A A h

1, and 1 - 7 - 1. We amend this model in Example 4.13b by preceding it with V ,

keeping 5 in the bass until the end of the cadence, and adding a 4-3 suspension over

V. Next, we carve out the subject from the line carrying 5 - 4 - 3, shown with

beams at Example 4.13c. As well, we use register transfer to place the alto into the

soprano voice by moving b1 to c3. This octave-displacement allows the new soprano

voice in m. 53 to connect more smoothly with the imitation of the subject in mm. 53-

54. Other register transfers occur in these latter two measures. For example, the bass

g moves to g1 in m. 53 and continues through g1 - a^1- a^1- g1 as part of the alto

voice in m. 54: this melody is a metrical compression of the bass line of mm. 47-52.

Also, the soprano in m. 54 transfers down to the next register from e^2 to an implied

f1; this suggests a continued line through e^’ - d1 - c1, which secures C minor as a

local tonic and 4 of the Urlinie.

We can also apply the invertible cadence model to other sections, such as mm.

24-28 and mm. 55-58. Instead of imitating the subject at the same pitch-class level

(as just discussed with the cadence in mm. 52-55), we can apply the rhetoric of


177

subject and answer, where the latter lies a fifth below the former. Immediately after

the cadence to C minor in m. 55, the cadential model sequences through falling fifths

as a way of moving to the local key of major (Example 4.14). The illustration at

Example 4.14a shows two interlocking instances of the model (the second instance is

classified as an imitation); as with the previous example, the subject is represented

with beams. Here, the cadential model is altered to accommodate the interlocking

subjects: the subject retains its distinctive character with 5 - 4 - 3 while the other

two voices carry 2 - 1 and 7 - 1, respectively. Two iterations of the subject

facilitate the transition to major; however, the final goal of this sequence is not

approached through a root position dominant. To rectify the cadential approach to

major, repetition of the imitation transfers the subject F - E^ - D from the bass to the

soprano in m. 57 (illustrated in Example 4.14b); this is accompanied by a

reharmonization of the subject with VI7 - II7 - V7 - 1. In addition, since the subject

ends on the third of the local tonic in the soprano, a final register transfer displaces

the alto b^1 into the highest sounding voice as b^2 in m. 58, thus perfecting the

cadence in B^ major and initiating 3 in the Urlinie. (A similar descending fifths

technique is used in mm. 24-28.)

Finally, imitation and descending fifths occur in mm. 14-19 after the structural

cadence to G minor in the exposition (see Example 4.15). Here, the subject (shown at

Level b) weaves its way through the descending octave G-G (shown at Level a).

Register transfer at Level c positions the inner voices above the beginning of the


178

octave-progression in the soprano; now the imitation occurs alternately between the

(new) soprano and tenor voices. Another transfer places the tenor d2 into the soprano

in m. 17, thus preserving the soprano/tenor alternation of subject-entries.55 A final

transfer concludes the passage in m. 19, where the tenor c2 moves into the soprano

and comes to rest on b^2. This final gesture brings attention to BK a structural tone at

many different levels of derivation.

4.5.3. Voice-exchange

As we saw at the beginning of the chapter, voice-exchange smoothed out the

gaps left open by register transfer during the pedal-point over D in mm. 38-42, the

final cadential gesture in the dominant. The initial cadential gesture to the dominant

in mm. 22-24 is no different (refer to Example 4.16). A cadential model is shown at

Example 4.16a, illustrating 7 - 6 - 5 (of G minor) in the soprano and harmonies I -

8 7 a / \

II6/5 - V ' - 1. Level b displaces 6 and 5 through register transfer, leaving a gap

between the soprano and alto voices in mm. 23-24. Level c invokes voice-exchange,

dictating that the tenor will continue the content from the soprano; meanwhile, the

soprano continues the content of the alto while the alto continues that of the tenor.

Level d uses passing tones to fill in most of the holes left open by register transfer.

55 Contrast this with Level b, where the subject cycles through the soprano, alto, and tenor, respectively.


179

The leading tone to D minor now appears in the soprano in m. 24 as 6 descends to 5.

Here, the appearance of the leading tone in the soprano suggests that it might

substitute for e2 resolving to d2. But e1 resolving to d2 clearly appears at the surface

in m. 24, negating any need for a substitute. More significantly, the application of

register transfer and voice-exchange elegantly captures the invertible properties of the

cadence.56

4.6. Conclusion

In this chapter, we have focused our energy on counterpoint at the octave and

how it interacts with the Schenkerian transformations of register transfer and voice-

exchange. Instead of only concentrating on surface-level invertible counterpoint—the

object of inquiry for most textbooks—this investigation has directed our attention to

deeper-level compositional concerns. In short, we have examined how counterpoint

at the octave influences overall compositional organization. As a window into this

compositional world, we have addressed the three issues raised at the beginning of

this chapter. First, we defined counterpoint at the octave as the octave-transposition

56 In regard to substitution, Schenker writes that “such a substitution is generally combined with an interruption, an unfolding, or an ascending register transfer. However, it is easily recognizable as a substitution because the counterpointing bass arpeggiation clearly indicates the actual tone o f the fundamental line, even though it is hidden.” Oster trans., Free Composition, 51, §154. Clearly, substitution requires that a replacement tone stand in for a structural tone that is literally not there at the surface.


180

of harmonic tones, provided that they create the intervals of an octave, third, or sixth

between inverted parts. Second, we located the presence of counterpoint at the octave

as far as the background, where harmonic tones invert positions, namely, over the

tonic and dominant Stufen. At later levels, this inversion translates into invertible

cadences within local foreground keys. Third, we identified register transfer and

voice-exchange as Schenkerian transformations that neatly combine our definition of

counterpoint at the octave with the invertible properties of the deep-middleground (in

general) and the cadence (in particular). Finally, we used Bach’s Fuga (BWV 1001)

as a way to illustrate applications and consequences of these transformations.

The analytical illustrations from the Fuga may not only illuminate the inner

workings of counterpoint at the octave, they may also shed light on Bach’s

compositional method. As a glimpse into this process, Russell Stinson has written on

the body of Bach’s manuscripts that survived within the Bach circle, especially those

copied by J. P. Kellner.57 Although an autograph exists of the Sonatas and Partitas

for Solo Violin (dated at 1720 in Bach’s own hand and now kept at the

Staatsbibliothek Preussischer Kulturbesitz [Berlin], P 967), a fair copy also exists infO

Kellner’s hand from 1726. Kellner’s copy omitted mm. 35-39 of the Fuga—seven

full measures—including the invertible counterpoint over the pedal-point on D. To

make up for this deletion, he inserted material not found within the autograph. This

new material consisted of a cadence to D minor on the downbeat of m. 35 in his

57 Stinson, “J. P. Kellner’s Copy of Bach’s Sonatas and Partitas for Violin Solo”; Russell Stinson, The Bach Manuscripts o f Johann Peter Kellner and His Circle (Durham: Duke University Press, 1990).58 Stinson, The Bach Manuscripts, 55.


181

version of the Fuga, shown in Example 4.17 (the cadence immediately preceded the

passagework section in mm. 42ff in the autograph). Stinson makes note of the

parallel sixths in Kellner’s cadence, since “they are in abundance throughout the

movement...and they lead to a perfectly effective cadence (complete with parallel

sixth [s/c]) on the downbeat of m. 35.”59 There can be no doubt that Kellner brings a

close to the D minor section in his version of the Fuga, however, its effectiveness

may be in question for three reasons. First, the lower voice F on the downbeat of m.

35 challenges the very existence of an authentic cadence; compare this with m. 42 of

the autograph, which firmly positions the open string D at the beginning of the

measure. Second, the soprano of the cadential tonic (D) lies an octave lower than that

found in the autograph; Kellner’s stemming confirms this observation by applying

upward stems to the structural descent from a and downward stems to the inner

voices c#2 - d3.60 Third, repositioning the upper voice into the lower octave at the

cadence prevents the upper voice from making the long, precipitous descent to 4 in

mm. 42-55. To be sure, one could invoke register transfer at the down beat of m. 35

of Kellner’s copy, thus repositioning the soprano into the three-line register. But the

solution by Bach was to incorporate register transfer, voice-exchange, and

imitation—indispensable components of invertible counterpoint—as a way of

convincingly attaining the higher register for 5. Stinson himself has noted that

59 Ibid., 65.60 The illustration in Example 4.17 preserves the stemming found in Kellner’s copy, reproduced in Stinson, “J. P. Kellner’s Copy of Bach’s Sonatas and Partitas for Violin Solo”, pp. 206-207, however, Stinson resorts to modem notational practices in The Bach Manuscripts, 66, thus obscuring these slight but revealing details.


182

Kellner’s copy may have been a first draft of the Fuga, since Bach would often add

measures to his revised pieces.61 As part of this revision process, Bach may have

added mm. 35-41 in the autograph as a way of smoothing out the connection between

the cadence in D minor in m. 38 and the passagework section in mm. 42-55. We can

only speculate that Kellner would have approved of Bach’s revisions since invertible

counterpoint was used in the cadence as a means of attaining the higher register. And

we know from the beginning of this essay that Kellner was well aware of the

invertible properties of cadences.

And we can rest assured that Schenker, too, would have approved of Bach’s

invertible solutions, since they engaged the very transformations that comprised the

bedrock of Schenker’s theory of tonality. The application of register transfer and

voice-exchange examined in this chapter forced voices and content to exchange

places, thus compelling analysts to deal with the properties of invertible counterpoint.

The application of these transformations in the Fuga is complex, but this need not be

the case. Indeed, we showed that Schenker illustrated the use of counterpoint at the

octave in the opening measures of Bach’s Short Prelude No. 7 in E Minor, BWV

941—a piece written for children. His discussion emphasized the germinal third-

progression from e1 to g1 in the upper voice that opens the prelude, though he made

clear, “[T]wo other voices take part in this linear movement,” especially an inner

voice that follows in parallel sixths.62 He boldly claimed, “[T]his ascending third-

progression is the seed from which the prelude emerges; once this seed is sown,... the

61 Stinson, “J. P. Kellner’s Copy o f Bach’s Sonatas and Partitas for Violin Solo,” 201.62 Siegel trans., “Bach: Twelve Short Preludes, No. 7 [BWV 941],” 58.


183

entire harvest is determined.”63 His comments suggest that counterpoint at the octave

is part of this bounty, since it is the inversion of linear progressions that motivates

“the general ascending thrust of the upper voices.”64 He aimed to show that the

simple initiating third-progression became the life-force that flows through the veins

of the prelude, branching out into different voices through the aid of invertible

counterpoint. He remarked on the richness of tonal, contrapuntal, and harmonic

relationships that manifested themselves within this children’s piece:

This is the gift—a world of profundity—that Bach gave to beginners, to children. What a wealth of relationships! All of the individual aspects of the voice-leading, the many parts necessitated by the whole, continually offer each other their best features. Just as water seeks its own level according to the principle of communicating vessels, each separate attribute of the voice-leading does not pertain to itself alone; it exists not only for itself but contributes to all the others.65

This is the gift that Schenker gave to us: that counterpoint at the octave is inextricably

bound up with tonal composition and tonality in general. Just as Johann Sebastian

Bach gave his Preambles to Wilhelm Friedemann as a way of integrating invertible

counterpoint into composition instruction, so did Schenker offer his analytical and

theoretical insights to us as a way of incorporating invertible counterpoint into our

understanding of tonality.

63 Siegel trans., “Bach: Twelve Short Preludes, No. 7 [BWV 941],” 58.64 Ibid., 59.65 Ibid., 61.


184

Chapter 5: Conclusion

This dissertation has examined invertible counterpoint as an essential

component of Schenkerian theory, and by extension, tonal voice-leading and

harmony. Chapter 1 surveyed the concept of invertible counterpoint as it was

demonstrated through compositional practice, and conveyed through theory treatises

from Zarlino, to Schenker, and to the present day. Chapter 2 addressed counterpoint

at the twelfth and its role within fugal composition and tonality. Chapter 3 discussed

counterpoint at the tenth within fugal composition and tonality, but also showed how

this type of invertible counterpoint intersects with the transformation of reaching

over. Chapter 4 illustrated the connections between counterpoint at the octave and

fugal composition and tonality, but also demonstrated the relevance of this type of

invertible counterpoint to the transformations of register transfer and voice-exchange.

Chapter 1 identified three factors necessary for writing invertible counterpoint

within music ranging from the advent of polyphony to the common-practice period: a

list of ways to vertically stack tones, rules dictating the relative motion between

voices, and a corpus of preexistent material for writing new compositions. These

three factors went through many changes over the course of history, thus reflecting

the concomitant changes in musical style. Just as these factors changed, so did

theorists’ understanding of invertible counterpoint. For example, Marpurg dictated

that there are seven intervals of counterpoint, not three. But as demonstrated by


185

Cerone before him, counterpoint at the octave, tenth, and twelfth are the only ways of

setting two voices together, such that they move (primarily) by step, in contrary

and/or oblique motion, and only create consonances. By doing this, Cerone revealed

the three-member intervallic sequences that could only result from these three types

of invertible counterpoint, and no other. Translated to tonal parlance, Kimberger

related the three intervals of invertible counterpoint to the structure of the harmonic

triad. Grounding invertible counterpoint within a triadic framework served as a way,

for us, of linking this compositional practice to Schenker’s theories of tonality: a

conception of tonal composition that considers the interaction of both contrapuntal

and harmonic factors. But Schenker’s attitude towards invertible counterpoint proved

to be varied and mixed, as seen in his published writings and unpublished notes.

Despite some of his negative assertions, his analytical essays detailing the interaction

between voice-leading transformations and invertible counterpoint, and his copious

notes on invertible counterpoint within the Oster collection, provided the necessary

impetus for further investigation. The following chapters used analyses as a means of

reappraising Schenker’s theories and their interdependence with the properties of


Chapter 2 examined counterpoint at the twelfth and its role within fugal

composition in particular and tonality in general. As a case in point, the chapter

studied the parallel invertible episodes from Bach’s Fugue in C Minor from the Well-

Tempered Clavier, Book I (BWV 847). Both episodes play formal and tonal roles

within the fugue: they link the answer to the subject and they occur between structural


186

locations within the deep-middleground (I - III for the first episode; III - V for the

second episode). With the establishment of this deep connection, I considered the

episodes as compositional fallout from the invertible properties of the Ursatz itself.

Here, harmonic tones invert positions between adjacent Stufen. In this case, the root

appears below the third of the I-Stufe, while the fifth appears above the third of the

lll-Stufe. It is the presence of deep-level invertible counterpoint between harmonic

tones of Stufen that guided this analysis and the others in the remaining chapters.

Chapter 3 examined counterpoint at the tenth and considered its role within

fugal composition and tonality, similar to the direction taken in the previous chapter.

As with counterpoint at the twelfth, counterpoint at the tenth involves an inversion of

harmonic tones. In this case, thirds of Stufen exchange places with roots or fifths of

other Stufen, and vice-versa. This contrapuntal property bears out at deep-levels of

the middleground, primarily within harmonic patterns of descending thirds. Chapter

3, however, extended its purview to the study of counterpoint at the tenth and its

relationship to voice-leading transformations. From this perspective, I looked at

reaching-over as one such transformation that engages counterpoint at the tenth. The

contrapuntal constraints owing to this type of invertible counterpoint—that is, the use

of contrary or oblique motion—intersected neatly with the constantly shifting voice-

pairs that participate in reaching-over. Due to this connection between invertible

counterpoint and voice-leading, the analyses in the chapter demonstrated two fugues

that express reaching-over as an integral part of their respective fugue-subjects and

counterpoint at the tenth as a feature of their respective countersubjects. The focus on


187

voice-leading transformations and their connection to invertible counterpoint

continued into the next chapter.

Chapter 4 addressed the relationship between counterpoint at the octave to

fugal composition and tonality, but also considered its connection to the voice-leading

transformations of register transfer and voice-exchange. As with the other intervals

of counterpoint, counterpoint at the octave involves the inversion of harmonic tones.

In this specific case, harmonic tones invert positions with others, provided that each

maintains its specific identity. In other words, harmonic tones are displaced via

octave transposition. This definition worked nicely with register transfer, which is

the process of displacing tones, including those of the Urlinie (or their respective

neighbor notes) and of the inner voices, upward or downward by an octave. By doing

this, counterpoint at the octave was achieved by transposing harmonic tones by

octave transposition. The registral rearrangement of harmonic tones—especially at

earlier levels of derivation—provided a structural basis for instances of counterpoint

at the octave. At deep levels, registral reordering of harmonic tones established a

method of explaining relationships between structural cadences that used invertible

counterpoint. At later levels, voice-exchange was a means of making linear

connections between harmonic tones that had been inverted via register transfer.

Although this dissertation uncovered many connections between invertible

counterpoint and Schenkerian theory, many questions are left unanswered. I include

three topics that must be left for further research.


188

First, do different intervals of counterpoint correspond to particular types of

voice-leading transformations? For example, more investigation is required to

understand the relationship between reaching-over in both B^-major fugue-subjects

(discussed in Chapter 3) with counterpoint at the tenth. We may ask the following

question: does it necessarily follow for fugue-subjects employing reaching-over to

occur within fugues that employ counterpoint at the tenth? Many different methods

could be employed. One tack would involve analyzing a number of different fugue

subjects', determining those that are suitable candidates for reaching-over, and testing

the positive matches for the presence of counterpoint at the tenth throughout the

course of each fugue. Such a study could be statistical in nature, allowing one to

make generalizations about the structure of a large number of fugue-subjects and their

suitability for counterpoint at the tenth. The purpose of conducting an investigation

on a large scale such as this would be to verify a hypothesis: that fugue-subjects

incorporating reaching-over often employ counterpoint at the tenth as a compositional

device.2 The challenges at this point in time include the conversion of surface-level•5

instances of reaching-over into an accessible format for automated databases.

1 A place to start analyzing fugue-subjects would be Andre Gedalge, Traite de la Fugue (Paris: Enoch et Co., 1901), 357-371; trans. and ed. Ferdinand Davis, Treatise on Fugue (Norman: University o f Oklahoma Press, 1965), 296-312.2 In this vein, Nicholas Cook writes that the processes used to group large repertories into databases “can become a means o f generating or verifying musicological hypotheses.” See “Computational and Comparative Musicology,” in Empirical Musicology: Aims, Methods, Prospects, ed. Erick Clarke and Nicholas Cook (Oxford: Oxford University Press, 2004), 1113 The database that I am considering here is the Humdrum Toolkit developed by David Huron. For more on this, see Cook, “Computational and Comparative Musicology,” 113- 123.


189

Second, how does invertible counterpoint intersect with Schenker’s notion of

the leading linear progression? We addressed the concept of combined linear

progressions in Chapter 1 as a replacement for the cantus firmus in tonal composition.

Here, he established specific lengths of progressions and the types of relative motion

that could be exhibited between them. Most importantly, he made it clear that the

leading linear progression was established at earlier levels of derivation than the

counterpoints that would be added to it. What he did not make clear, however, were

his criteria for creating the leading linear progression in the first place. William

Rothstein, who posits one criterion, claims that the leading progression horizontalizes

a harmonic interval of the governing harmony.4 But his claim can be refuted, since

leading progressions may involve more than one harmony.5 If the Ursatz includes the

inversion of harmonic tones, then leading linear progressions that navigate through it

must intersect with these properties. With the establishment of a deep-middleground

that inverts harmonic tones, a leading progression would comprise the next

middleground level of derivation. The challenge for further research is to determine a

“firing order” of linear progressions occurring within the outer and inner voices. One

solution would be to create a rhetorical model that assigns leadership based upon

priority of voices (with the highest given to the upper voice) and opportunities for

passing motion between harmonic tones and voices. Both criteria fall in line with

Schenker’s notion of the Ursatz. For instance, he wrote,

4 William Rothstein, “Articles on Schenker and Schenkerian Theory in The New Grove Dictionary of Music and Musicians, 2nd edition, ed. Stanley Sadie,” Journal of Music Theory 45/1 (2001): 204-227.5 Schenker, Der freie Satz, §224.


190

The traversal of the fundamental line is the most basic o f all passing- motion\ it is the necessity (derived from strict counterpoint) of continuing in the same direction which creates coherence, and, indeed, makes this traversal the beginning of all coherence in a musical composition.6

The creation of a leading linear progression would, in a sense, establish a tonal cantus

jirmus that would weave its way through later middleground levels. The

establishment of such a rhetorical model would not supplant the technique of

invertible counterpoint—as Schenker would have liked, at least according to his

opinion within Der freie Satz—but rather enhance it as a formal application of

compositional procedures.7

Third—and most significantly—does invertible counterpoint play a role in

musical forms other than fugue? For example, can one incorporate invertible

counterpoint into a description of binary form? As a case in point, I consider J. S.

Bach’s Six French Suites (BWV 812-817). Many of the movements within these

suites utilize invertible counterpoint as a way of articulating formal sections,

including the Sarabande and Menuet I from the First Suite in D Minor (BWV 812),

the Sarabande from the Third Suite in B Minor (BWV 814), and the Gavotte from the

Sixth Suite in E Major (BWV 817).8 But contemporary notions of musical form only

address thematic and harmonic criteria, not contrapuntal treatment. For instance,

cadences punctuate reprise-endings within typical Baroque-style binary forms.

6 Oster trans., Free Composition, §5, 12.71 am thinking here o f composing against a cantus Jirmus.8 Matthew Brown referred me to this last example, personal communication.


191

Formal classification depends upon the nature of such cadences, e.g., a perfect

authentic cadence in the tonic at the end of the first reprise denotes a “sectional”

binary form.9 Likewise, thematic design influences formal classification. For

example, thematic material that occurs at the opening of the first reprise and the end

of the second reprise is often designated as a “rounded” binary form.10 This formal

description applies to the Gavotte from Bach’s Sixth Suite in E Major; however, it

does not describe the inversion of voices that takes place with the return of opening

material occurring at the end of the second reprise. New formal categories, therefore,

are needed to explain how invertible counterpoint intersects with our current notions

of formal classification. A survey of many binary movements—by Bach and his

contemporaries, such as Handel and Telemann—would be necessary to paint a more

complete picture of the compositional possibilities within this small, but dynamic

formal design. Compositional schemas describing different contrapuntal strategies—

for example, inverting voices at the beginning of the second reprise—will provide a

means of grouping pieces into formal categories. The creation of new formal designs

will enhance our understanding of binary forms and help us to better explain the

9 For an explanation of sectional binary form, see Steven G. Laitz, The Complete Musician: An Integrated Approach to Tonal Theory, Analysis, and Listening (New York: Oxford University Press, 2003), 416.10 For a more detailed explanation o f rounded binary form, see Laitz, The Complete Musician, 419-420. Contemporary theorists, however, do not agree whether or not this form is in two or three parts. On the one hand, supporters o f the binary interpretation focus on the symmetrical halves and the bipartite tonal design. On the other hand, others emphasize the tripartite repetition scheme and the concept o f recapitulation. For more on this debate, see William Caplin, Classical Form: A Theory of Formal Functions for the Instrumental Music of Haydn, Mozart, and Beethoven (Oxford: Oxford University Press, 1998), 71-72.


192

compositional process of tonal music from the Baroque period and how it intersects

with the properties of invertible counterpoint.




by


Volume 2: Examples


Example 1.1. Parallel organum as found in the Musica enchiriadis.

OR.

PR.OR.

PR.

Tu pa - tris s e m - p i - ter - nus es fi - li - us.

Tu pa - tris sem - pi - ter - nus es fi - li - us.M. JL m. M. ML IT'

mTu pa - tris sem - pi - ter - nus es fi - li - us.

-0------- 0..... »......»-

OR. _*----9---•----- •--- #------- m-----m--- 0-- m 0---- 0—Sit gio - ri - a do - mi -n i in sa e -cu - la.

• 0 «►___ m__ •_m m 0----0—PR.OR.

Sit glo - ri - a do - mi - ni in sae - cu - la.

PR.

Sit glo - ri - a do - mi - ni in s a e -c u - la.IOR.

PR.OR.

PR.

0 '9---W"" W" W 9 m--9--- “ 9 ' 0-------9—Le- ta - bi - tur do- m i- nus in o - p e- ri - bus su - is.

• 0

L e -ta - bi - tur d o -m i-n u s in o - p e - r i - bus su - is.

L e - t a - b i - t u r d o -m i-n u s in o - p e -r i - bus s u - i s .

Erickson trans., Musica enchiriadis and Scolica enchiriadis, 22 and 24.


194

Example 1.2. In exitu Israel, as set by Binchois.

Ir« c\ - i - tu Is • ra - el de E - £vp - to

• uuxs j«i - - cub vie pe - pu - jo bar - b.i

; - -tr™:*- -tjt..xr...re.

_) =

fa - -.la es; ju ■■ de - a1 .......r

sati - cti •• fi - ca - ti • <5 e jus---Hf-f--- f---- f---- ..it..ft—it ..-..f-j*— ...- ..|-p f .....

d . ri}£ - - ' -

fa - eta e.-i ju ■de - a san • cii • ii • ca ti - o e ........I V ji.is‘ rO'

..\ ::;=...........i?--- i 1 tn'mimr i

Kaye ed., The Sacred Music o f Gilles Binchois (Oxford: Oxford University Press,1992), 203.


195

Example 1.3. Alma Redemptoris Mater/Ave Regina coelorum, by Josquin.

Soprano

Redempto

Altus

Re

Tenor

Bassus

7

ter.

Alt.

lo -

Bass.

demp - to - risma Re -A1


Reproduced with

permission

of the copyright owner.

Further reproduction prohibited

without permission.

Example 1.4. Magnificat Tertii Toni: Et misericordia, by Palestrina.

I J | J J J J If J. J^| |» J |» J~D | J J | J J p p -j-ij f f p~ '||Cantus

Altus

Et mi - s e - r i - c o r - d i - a e ju s a_______ p ro -g e - m - e in_ pro-

f ] i u N j jEt mi - s e - r i - c o r - d i - a e jus, et______ m i - s e - r i - c o r-d i • e - ju s a pro - ge - m - e

j i j j j j I? j. j jTenor

Et mi - s e - r i - c o r - d i - a e ju s a pro -

VOOs

197

Example 1.5. “Fantazia upon One Note,” mm. 1-13, by Purcell.

_o_*


198

Example 1.6. Diagram showing the change in the three factors necessary for creating invertible counterpoint, from the Musica enchiriadis, to the time of Tinctoris, to the

time of Schenker.

a. templaterelative motion

vertical stacking cantus firmus

b. Musica enchiriadisparallel

P8, P5, P4 cantus firmus

c. Tinctorisno parallel perfect

consonances

cantus firmustriads

d. Schenker no parallel perfect consonances at the same level

of derivation

harmonic tones; Stufen

-> Ursatz',transformational levels


Example 1.7. Zarlino’s example o f invertible counterpoint at the twelfth.Adapted from Marco and Palisca trans., The Art o f Counterpoint, Exx. 113a and

113b.

Upper part of the principal

& J L - - j J J J J Lr ^ > --------- J -e- -J

------------------1©-----J t ----------------------------- r r j

—©--------- p---- p----

Lower part

5

J- j- j ^ -------a.-e--------

r f fffl r r o—P—r-f- r r r ff- - ii9 r jy •............. i r r 1 r r I

9

t r f - ire. j -

XL

..._.._..ET..........C<.......................1 &

—P

-<©------------------<©------------------

-r :----------6 --------* --------r

hr r F py * o j—----------75--------©----------------—f©----------P----------------------------= 1=1 11 1 ' r

13

4.j j j - f = # W ^ = ,-a-e» -------------------k ° ■ J• —

7 .r. f - E 1“ r r 1r r r r r f -

17

i ■■ 1-e--- <J-- sJ—-©----4:--- —©-- Hull

>>: f r f - r r r r p r ^ f JOILIy r 1 - - - - - - - -L 1 r> r r i■i r


200

Example 1.7, contd. Zarlino’s example o f invertible counterpoint at the twelfth.Adapted from Marco and Palisca trans., The Art o f Counterpoint, Exx. 113a and

113b.

Upper part o f the inversionAn i =if f - — J- r [ r r f f r r » r

-----------S-----&----

r r ° --- p—------- p-----1—*—L ow er part

5

j------1— _l-------------------------

#p - i r r J J U - J - J — T A A . p r r r i

----------e----------p-----

a v 8- - r r r f ~ [ f r r r r f z r r f - ^ J f f . - g ....:...

9

- J ........H r J J J j.-tJ----u---f f f r fs j- j 2- |0 |# ~

-------- ------^ ---------------f9 -------

13

p J j ° a a p J Jrr- - - - - - - - - - - - Is ---- ...........1* ..f ..

r p— - - - - - - - - - - - - - f 1- - - - - - - - - - - -

*

l A r r r r r r = H..f.r r r n17

Hlefr~w

n n


201

Example 1.8. Diagram of invertible counterpoint at the twelfth as it is applied to modes.

Principal Inversion

mode 2 (d) mode 1 (d)

oXT

~W_ o _XT

O

mode 1 (d) mode 2 (g)


202

Example 1.9 Model of counterpoint at the twelfth in three voices; adapted from Marco and Palisca trans., The Art o f Counterpoint, Ex. 150a and 150b.

principal: melody in bass inversion: melody in soprano

- / ------------------« ------------ -ft------------ ft*J------------ -ft--------------

*r-H------------ H------------

------------------- -0 ------------- -©--------------------------- —*-------------

8 5 3 5 3 83 3 3 3 3 3


203

Example 1.10. Zarlino’s example o f counterpoint usingcontrary motion with parts inverted. Adapted from Marco and Palisca trans.,

The Art o f Counterpoint, Exx. 116 and 117.

Upper part of the principal

1 J 1 1T O * 1--1- ^

-z -----e--------e-------------------------------

* * -----m &—&—

os------------e»-------- ^ 0

J - j d J J

-e ---------1.-------------

Lower part5

\ f , 1 I , i i j I| p J J 1 1.... frJd & _ .

4; : ° °-*-»---------- o ---------- - e » ---------- u -------------- . » --------------° -------=

^Ti J' JJ[lC o

U j. JJT ,--- 11---

5 »

- e --- n--L jJ-e>---------


204

Example 1.10, contd. Zarlino’s example o f counterpoint usingcontrary motion with parts inverted. Adapted from Marco and Palisca trans.,

The Art o f Counterpoint, Exx. 116 and 117.

Upper part o f the inversion

TT-O-------1» (T O O-e-

Lower part

- n ---------- e*---------- « ----------n -------------- ^ --------------

p---- ------ ------ 1--------------- —n — ~T~ f P m------P~^—7?— — J -m -P ..... ---- ------------- -------------

V f — p— I— — U— |----------- W -g- f .I...I-J

o<> ~rr -o- ~rr ~TT

TT

73

-e - -HeH-

-HeH-


205

Example 1.11. Model of invertible counterpoint in contrary motion; notes of the same shape map onto each other under inversion.

Principal

Inversion


206

Example 1.12. Zarlino’s example of invertible counterpoint in contrary motion in three voices. Adapted from Marco and Palisca trans.,

The Art o f Counterpoint, Exx.l51a and 151b.

Cantus o f the principal

Tenor

r r ....TT

Bass & &TT _QI TT

0 i i j j J , , |- J - M - i - r ..... 1 J j J f l

§ J J "

y r 0 ■ — ■■■

- s - — — p —

- J — p — J — P -

------------ U J —

-(S’------ J ------ 'J

^ ------<5------- a

J J r~ r f H nvfl) f &

W ? f o --------- - rj rj

- t - 1 . f M

- C l--------------» ------ V

# • p . a

- e -------------- n -----------y

V - r i

m l — p H -------------------------~\* 7 — ^

i— < 9

rJ —m^ a - a U 9 ■ " a l - e

< d

- e— s i p M e H ------------- U

iTT -(S'—<S>- -HeH-


Example 1.12, contd. Zarlino’s example of invertible counterpoint in contrary motion in three voices. Adapted from Marco and Palisca trans.,

The Art o f Counterpoint, Exx. 151a and 151b.

Soprano o f the inversion✓ A r i l l I i/ *» .^ ............... ■=?-... -------- *>• .. p

Tenor6V ( *

----------- m &----&---- ----------------- j J

2 = ^ — I f X- p p~_ 5 p ...

B ass

u ------------ 1------1----- M— 1— — 1—

4V-----_------ _-------J*—:----------------- *> o - rjy

5

fiff- F j o _ fj rj fj r - r f n 0

*): p ----P -

.=f =f 1

-f®—r>—I®--- \~

_A1------s--d_

1 J p - J j j r f p J 1 p—i t —r ----

J J-------

f ^ r

.. .jl-J..-Hir i t 1 ^

r, / rJ rJy--------- ©---LI—J-J -J. m rJJ J * 9---- ----------------1—j_\—&—&—^---- *

9

a . . 1O'" J « fJ ? -p' ---------=1% 1

p g ^ — *-

^ — —

=73—73--- S^-9----

- « — \

y —

4): J J j— |—

r r 1 r1 i j j p 1

t r f t 1

h h ..................... F=F=t=

—Iinm— j

1MI-----1f rJ & rJy - - - - - - - - 1 .J 1? ^ -----J J ^ J ^ 9—&—


208

Example 1.13. The possible combinations of adding either a tenth above the bass or a tenth below the soprano; the filled-in note depicts the added voice.

tenth above bass tenth below soprano

octavei i---------------------------

fifth third octave fifth third

3.3


209

Example 1.14. Zarlino’s example of three-voice double counterpoint at the twelfth. Adapted from Marco and Palisca trans., The Art o f Counterpoint, Exx. 150a and

150b.

. Soprano o f principal

TT(1 •-------O'T enor

3E

B ass£-a-

iCJ& — & ■

TT

= 4 = ... J ^ z = z z q$ 9 — -r. — a --------------

---- 0 -----------------

r r r r-

- 0 -

£

tot

—HQH—

I J e t


210

Example 1.14, contd. Zarlino’s example of three-voice double counterpoint at the twelfth. Adapted from Marco and Palisca trans., The Art o f Counterpoint, Exx. 150a

and 150b.

Soprano of the inversion

f\ Tenor l = = ^ L j 1

1 1 1 I 1 1 j j

L d

(n) ^•J o- a Bass

b : n - - 1

Lu----g---cl---19 * ^— rj-----

5

|-e* * ° ^—G

—hTT

_J—J— --- J_T.f T3. O l | J

11 JI+-H

<•»:--------

t—6-<9------si ■'«=\ j j j w * ^yd

^ u---- 5k .J ------ -- <>

9

y - 1 , 1 H— — ------------ < = ------------4

F = F ^ k >

— s L o —

h > : ... - . J :----------------z r ------------------:-------------

d * m ~ -----------------------

........................... ■ m - -

----------------- 6

y °

V

- c l -------- J --------- s i --------1 ~ | 1 | -1------------- e -----------------------L

13

^ J --------<e

r LI

L- n •

_

— 1J J °

----------------- 1

----------------- j--------<s

*): f r M

?-------- s -------- ^ — * m~--- CJ.------ ©------------1

t

IMI

1 r— - — *_=----------- • __ -B---- — ------------^------


211

Example 1.15. Zarlino’s example of counterpoint at the tenth. Adapted from Marco and Palisca trans., The Art o f Counterpoint, Ex. 114.

Upper part o f the principalo op pT>

Lower part

mrrCT

TT TT..~~p & g)

PP¥P PP TT

- e -

• j —j p

^ jA |=> ° —

_ ti i -----------------s —

r r 6 =m ? r - - - - - -T J

r r * 0 rj -p r r r i ^ .17

m pmnw&— &

rJ a TT 30C


212

Example 1.15, contd. Zarlino’s example of counterpoint at the tenth. Adapted from Marco and Palisca trans., The Art o f Counterpoint, Ex. 114.

Upper part of the inversionts &■

¥TT

n TT

Lower part t> &— &

\JL p r r |° p -rr .-----* n rj

§ r 1 r — r — F-—e ------------0 —&—

,J- r r r• 1* o V m J p ,

h N-----rj/ ----------------

1 r ° r r £>..........

o

¥

tSs—©■

7J

: -

H

o <

::

............... .......o ■■■■

lO

P j * J J g ) ^

-p—

“ f T T — F l— —— &—

-s>-----n---------

- i r " J

17

4 JJ r r j ^ n ~rJ 0 - J t

-Den-


213

Example 1.16. Inversion table of invertible counterpoint at the octave.

1 2 3 4 5 6 7 8 8 7 6 5 4 3 2 1


214

Table 1.1a. Treatises that discuss invertible counterpoint from 1555 to 1773

Intervals of counterpoint

Source 8ve* 10th 12th Tables

C. Angleria, 1622 X X X

C. P. E. Bach, 1757 X

A. Berardi, 1687 X X X X

C. Bernhard, 1657-64 X X X

G. M. Bononcini, 1673 X X X X

S. Cerreto, 1601

P. Cerone, 1613 X X X (X )

G. Chiodino, 1653 X X X X

J. F. Daube, 1773 X X X X

G. Diruta, 1593; 1612

J. J. Fux, 1725 X X X X

V. Lusitano, 1561


215

Table 1.1a, contd.


Source 8ve 10th 12th Tables

F. W. Marpurg, 1753; 1754 X X X X

T. Morley, 1597 X X

R. Rocco, 1609 X X X

T. Sancta Maria, 1565

J. P. Sweelinck, 1640; 1647; 1670 X X

J.Theile, 1670s- 1680s X X X

R. M. O. Tigrini, 1588 X

N. Vicentino, 1555 X X

J. G. Walther, 1708 X X X X

J. G. Walther, 1732

G. Zarlino, 1558 X X

* Numbers (including ordinals) correspond to counterpoint at the octave, tenth, and twelfth, respectively.

t Inversion table


216

Table 1.1b. Treatises that discuss invertible counterpoint from 1774 to the present________________________________________________________


Source 8ve 10th 12th Tabl

J. G. Albrechtsberger, 1790 X X X X

J. G. Albrechtsberger, 1837 X X X X

H. Bellermann, 1862 X X X X

T. Benjamin, 2003 X X X X

L. Bussler, 1877 X X X X

L. Cherubini, 1835 X X X X

F. Davis and D. Lybbert, 1969 X X X X

F-J. Fetis, 1824 X X X X

R. Gauldin, 1988 X X X X

A. Gedalge, 1901 X

S. Jadassohn, 1884 X X X X

K. Jeppesen, 1930 X X X X

K. Kennan, 1999 X X X X

J. P. Kimberger, 1776-79 X X X X

H. C. Koch, 1807 X X X


217

Table 1.1b, contd.


Source 8 ye 10th 12th TableA. F. C. Kollmann, 1796 X X X X

A. F. C. Kollmann, 1799 X X X

A. Lavignac, 1930 X X X X

H. Martin, 2005 X X X X

J-J. de Momigny, 1806 X

G. B. Martini, 1774 X X X

R. 0 . Morris, 1922 X X X X

H. Owen, 1992 X X X (X)

W. Piston, 1947 X X X X

E. Prout, 1891 X X X X

E. F. Richter, 1872 X X X X

H. Riemann, 1882 X X X

H. Riemann, 1888 X X X X

C. de Sanctis, 1934 X X X X

A. Schoenberg, 1963

P. Schubert, 1999 X X X X

G. J. Vogler, 1776; 1817


218

Example 1.17. Illustration of counterpoint at the octave by Rodio (1609).

Canto

g ----- -© ------------- O

TT----- e -------------- -6 ------p----- _ © ------------- o _o_

Tenor

The tenor is presented a fourth higher:

m = m

4>: _----- j ----- -&----- ------

.O' .......... - o ----------- J u ..

j r r H- G J —m o l—o --------U

The soprano is presented a fifth lower:


219

Example 1.18. Illustration of counterpoint at the ninth by Marpurg, 1753, Volume I, Table LVI, Fig. 8.

Hauptcomposition:

Inversion of Hauptcomposition:


220

Example 1.19. Inversion tables for counterpoint at the ninth, eleventh,thirteenth, and fourteenth.

counterpoint at the ninth:

1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1

counterpoint at the eleventh:

I 2 3 4 5 6 7 8 9 10 11II 10 9 8 7 6 5 4 3 2 1

counterpoint at the thirteenth:

1 2 3 4 5 6 7 8 9 10 11 12 1313 12 11 10 9 8 7 6 5 4 3 2 1

counterpoint at the fourteenth:

1 2 3 4 5 6 7 8 9 10 11 12 13 1414 13 12 11 10 9 8 7 6 5 4 3 2 1


221

Example 1.20. Kimberger’s two-part example used to illustrate counterpoint at theoctave, tenth, and twelfth.

\t r 1t o t o lt o t o t o


Example 1.21. Kimberger’s example of counterpoint at the twelfth.

------ r -&--------- 7?---- n

P r r p

—&---------

/ \ -1---


Example 1.22. Kimberger’s example of counterpoint at the tenth.

m m cj S J ■d- • —&----------

—&— -p—f r ------


Example 1.23. Kimberger’s exampleso f counterpoint at the octave.

\#f J | i , j . — i -J J J¥ = r fL:r J

r r r,\ j

r rJ J j

r rr-j— —

rj

^ * r 1i ^ r 1r"rrr r 1 v.rftr - J j=j=4=p i= ^¥ = = = ? V,

|= =tfp=0 r ~

------

------

r_

““J

J_ “UL 5 ■


Example 1.24. Kimberger’s examples o f doubling in thirds.

r T - H » J , ) —

■ § * - * ? - H — * t =

. J : , f = £ = ■


226

Example 1.25. Kimberger’s example o f doubling in thirds, in four parts.

4* r - t - H 9=4|V=t f -J—


Reproduced with

permission



without permission.

Example 1.26. Illustrations of counterpoint at the octave, twelfth, and tenth; adapted from Cerone, El melopeo y maestro, 1613.

......; ...... O---- O---- O---- O-------- o -----° ----

1 3 6 8 o ---- o ----

10 13 15 +

- e ---- n —

counterpoint at the octave

% o t,------.............— — — O-------p,------

«J------------------15 13 10 8

-------- 0----6 3r»

0 ^1 +

XL ^4>:-----------------,,— 11 ®t — O-------o----

i_o_

5bo

8<>

10 12counterpoint at the twelfth 12

m

10

o

5XL

1.O .

5 6bo CL

8 10 O — r r -

12 13 15 17 counterpoint at the tenth

-/-------e—n—..... ..... 1 XI---0----------------------

17 15 13 12 10 8-*-*—o—

6 5..«.................

3 1 «•> ■©•

-e—x>—o o7-------e—11— 227

228

Example 1.27. Schenker’s Figure 1 from Der freie Satz.

H inter grund: Urs&tz, Diatonic

Urlinie

: Brechung durch die Oberquint

Mittelgrund:Verwandlungsschichten

Vordergrund:Tonalitat


229

Table 1.2. Categorization of combined linear progressions in Per freie Satz*

Relative Motion Parallel Oblique Contrary

Length! 3rds 6ths 3rds 6ths Equal Unequal3 x x x x4 x x x5 x x6 x x8 x3/3* x8/8 x3/4 x4/5 x5/8 x6/3 x6/7 x

* Mixed progressions are not listed.t Lengths discussed by Schenker in Der freie Satz.t Both values correspond to the lengths of both progressions in contrarymotion.


230

Example 1.28. Reproduction of Fig. 95, a, 4, from Der freie Satz.

(leads)

J i — *■ — [ = di~1*L

— ]

'S r(lower lOths)A --------------------D

r 1

is

p.

— i


Example 1.29. Fig. 98, Nr. 3c of Der freie Satz. From C. P. E. Bach, Generalbass IX/2, §1 a.

( 5 = Z u g )


232

Example 1.30. Fig. 99, Nr. la , b, and c from Der freie Satz.

m m . 7-8

l a ) (u p p e r lO ths)

m m . 16-17

J - ub )

(leads)

(4 -p rg .)m m . 12-13

c )

b e(u p p e r 3 rd s)

(leads)

(4 -p rg .) (lo w er 6 ths)(5 -p rg .) (5 -p rg .)

j r f l J . m(5 -p rg .) ( lo w er 6 th s) w ritten 8va


233

Example 1.31. 5-line Schenkerian Ursatz with intervals between outer voices.

£ 1o 0

I - 16 - V - I

intervallic pattern ------------ ► 12 10 8 5


234

Example 1.32. 3-line Schenkerian Ursatz.

o- f L r f t m

----- - - - &(1) (2)

I V I


Example 1.33. 3-line Schenkerian Ursatz with annotations showing harmonic-tone inversion.

soprano: third fifththfalto root third

A3

A2

A1

(1) (2)V


Example 1.34. Alto and Bafibrechung o f 5-line Ursatz.

I - 16 - V - I

intervallic pattern ► 10 8 6 5 3


Example 1.35. Tenor and Bafibrechung o f 5-line Ursatz.

I 16 - V I

intervallic pattern 6 - 6 3 8*


238

Table 1.3. Dated items from File 83 in the Oster Collection

Year Item Numbers1908 300*, 308*, 412*19091910 484t1911 112*, 309, 365,366, 3671912 81,82, 122, 123, 124, 244, 2451913 217, 280, 286, 338, 342, 344, 346, 349, 350, 355, 356, 3611914 335, 336, 337, 341, 351, 357, 358, 359, 360, 363, 364, 36819151916 140, 148, 151, 154, 247t191719181919 1441920192119221923 254*

* Dates written on published materials (newspapers, periodicals, etc.) and official documents.

t Date unclear due to being on opposite side of page.

$ Dates of published materials (newspapers, periodicals, etc.).


239

Example 1.36. File 83, Item 478 from the Oster Collection.

hrn / 2 LWX'/LyK ^r,

6 , y / . j J t , n -U+juS+y j$ *»> * (^ f . £ u ' t —s ,

d - i . ^ A t u C

4n,4+*>J ^ € ^ . - I^JL j-J-, , 7 - <?>^./^.j '/n ' / t ^ **

-< -/*

a ,* . I(Jbycw • . j

1/ irp r ir j i *» // ^ 3 - *Xr j

* ) 1--------- ^ X- , ______ ^ tu J J u ^ j ^ . HusCXt- I

v ' ^ ' ’2 /^ '^ w . 1V »t y )^hV , *A. *jt!U *** ~ 1

\riyLr-ty*~-C' \Xr ^5v^~ , jf I


240


f , l*.' iA*r t(uU -

X

■4*. Ai . 414 J* e u* {fa/*, ■/ z i * s~ f 7 t ? /o

/O 1 <f 7 S r v i i /92cÂt .• \

<V-*Vz j & /W &*}*i4. = j***' 8 ) .

+/ , * •, | { J Aj ., ¥ & ■ * ! , S')S/M M U O V » fa- t)

V ^ 4 *X » / / 9 r /r.?ai

/<}■ f f a & / M f a f a / * > : ^ g' / ; / /f>, <£«• 4» <f7■An**, mmi/t #t*4^*n. W w *~*vL, ..

- 'I*' IJI^PI. ‘i w ? S

^ ' iJuJm* *&** tAM*/ *«•*< I

X v 7 :', f /

V vJBb,*?mA* «**•}/

I : Kcchi7mr<i

- f f i f w W r t 'rf M 1 ? ?

* '*■ ,« - 4* ■Tffvt k a ;

cj lX*K. m oil* yn,. -i/lH* / *^ <4- c.yf t. y*}iti~y AAijf't*..

»»-V l U n . VT » W '4v>/ M sfe'jH * **•3. 5/. fa**-, fa'*-/ J**- &k-, *C/ *“*• /fa++*

i -j i j H j I j ✓C J jV

i/*] /f**Sf.44A Hi4.t''-/

<*~ *5 WiX'?


241

Example 1.38. File 83, Item 460 o f the Oster Collection.

tJ . 'U , / 2 s i ■ (~n'£/rs o }

. / * 3 V S' / J. f f f e / / / z ./ z r / o p f p f f - y * -2- t

’ •%* j1)!t H f—^ 0^ -Ou y * d d -fr f. 4 /x U *km.

,y *jf j ^ cj ani ^ <Cx' £t *s&~ */£. JhrA t,

± itr -

fe^-lol.j j ,.. .0 WtJ

t 2ttr~7tfr( a i t . f r . ) T F ' J '

f) the;

r r j* ^ ?>■7 ”A .


242

Example 1.39. Schenker’s acknowledgement of disallowing parallel consonancesunder counterpoint at the tenth.

‘zwei Terzem = zwei 8 10 = zwei 16 5”


243

Example 1.40. File 83, Item 445 o f the Oster Collection.

v r f

( / I f - /VVJJ > (n sX t* + . : t /G tq rfttk fk . * ■ 9/y> tX u / y / ^ w ^ / K ^ j i n

- 2 s*y

^ : / / *}>u**k. $ t A n "*** vî>

&*0*n-> MhAr//~rJ>*u yfaA* . (?r- &" /V /. /h*4 ‘*A°^ *-+»*' fi**/*' *6**t .

} / ^ -¥ ^ L /L (tu r f. ^ > p < U M / * » ' ’> « » *

vi*- ,

/ /*r*Zy/>~y W X 2 / */ * ' ‘k ' t y , t //, & .J

>rn- ><C«^«o *4+ ’ **'&•» *L*

/ > j J / ' f y f f c / ' fr fo /r - itj

*4r wut 0tiLJy( ?(/¥/

. . — • • - - — .\ (A + ^ {A m t X X X /." S J ~ . * v -C ^ /4 L 1 U + 1 J jL & r /fa p t* * ) * * * £ ? !

I " *. ,. w X~n6u/t*>VLy jI „ / , ^ - w ^ /

* I •*»'. «*»/*»«■ £-.•'•■/. 2, i I« M < » < / ' //L s& m .'/tA U f/ * f& v ix Y fr/r* p € a 4 , % * /. ^ W i y » ’i y

i f'gp&r/ / f 4 ^ ^ / ■*» ^«<in> , •jri/i T/ hCk

" - f i * * ' - t * . , * * *<*-■*4.


244

Example 1.41. File 83, Item 454, first page, o f the Oster Collection.

v r

Tfr3 * U r f r f f i ■ 'v

i{fî—j H =• ^

F ' jjr * f r T-

ttAiw- >Xk£;

r w

r r

T' 1 J ‘ v

ii t ±

j* r

g stiV/-

**/■ »

J~Vj 9 - i j/ #Ti-/ ^ 7 - ? (7T-

j if y . *4 ,

/ w \JS4'3r .

tjffiu * * * . S*dî r k ttt# /T*Zr 6 t +- ' P « » •* » ■ ’

y # ^vk * *? -mt~.

, c

<■#**&- ^kJâ. - ^

^ j n U J U . r . n j . / i f , f y . ^


245

Example 1.41, contd. File 83, Item 454, second page, o f the Oster Collection.

a n . .

2 = 2 5 V

t i i i i n U i t i l

!■_ t i l r ~ .

? n Z j J - u / i ' L Jjj

^ 4 tit. te■ JL

mtA*‘ 4})

3 CV/4 ^*\fit ^ A wfcwwi»f vf* A*. 4jfc. *4*-> I

o^- " a ifj

*.& t )Ar%u»> fcv r

/*£»»»»>, »*/C4./ *4**% r+€€ fi*4f>*'r/ MJ > -£*tC£. «v dcf,

" a / f r+v4lllfa» 4yA J'.fy.A i

jM -r* fin, /h ■ f \ . Kf*^g»*si •**- ^>.i 6*C <yX6U.

%***/$ m »y<V

^ ^ k4yv~~ A£ < /C ^. /%*ht*XaA ma*t <Ar>^f r ’. ,/ ,


246

k -

Example 1.42. File 83, Item 447 (second page) from the Oster Collection.

1 X ■>~T

V r **=E= » = i

>* -------

Jfe-j. - I" 'I--o—

r ::

^ a t*" t r r - f - r rJ 4 i 4j I r r ^ —

P r ‘ •Ut- —

10


247


/»

J 1±=KtajSfPM r r

- k r -

f e

553= i f

i f t

= f

;P? j 3 J J-.

t i 14# W

f f e S f e i = t t =

p f p fe n Effi^ 321


248


-----L -J..... . j , IQ -r- - - = f =j ®—

4 —

-6----t w H

-V...

4 4

4

4r~------L - r 4

* Jrf t

- r - 4 )4A

j .tL._TT

P— f—

t 4 M. . . . . . . .

1- - r -J — ■J— 4

j «• 374 ? ?L ........

' t4 ) ...1 : 9 . .

liftrtj *? t

.5 - ..

co’ ;

— - Ft TV-

i v ^* rJ \

- i j .r r

»-------? x X 4 4 14-

] ? ^ 4I U ?

T 4 i4 J

_r-+-4—

* i r 4

^..... .'c$>

_

... . j> f

. i 4

A - 4

t + t r -

1

,

. ..-_1... 1....

-F- . .. ......

4 r4 r\ O

- -Od...

A.....

_e---^L1! n - r t

0

----I f

6

st? ~

f f— t— -0------,r ^ t i

C_- "": f 1 \

’ — r

\r *~

JT ? -----J— 4 —X=t--- V - «r-- \y- -a------------&— ------- .


Example 1.45. List of melodic intervals (without inversion) according to Kontrapunkt I.

consonances:

the octave, the fifth, and the third

dissonances:

the second


250

Example 1.46. The list of melodic intervals (under inversion) according to Kontrapunkt I.

consonances: dissonances:

the fourth (as inversion of the fifth), and the seventh (as inversion of the second)the sixth (as inversion of the third)


251

Example 1.47. The list o f vertical intervals.

consonances inverted consonances

the octave, the unison, (as inversion of the octave)the fifth,the third the sixth (as inversion of the third)


Example 1.48. The list o f harmonic tones.

harmonic tones

the root, the fifth, the third


253

Example 2.1. Inversion chart illustrating invertible counterpoint at the twelfth.

3 2 ! 1 2 3 4 5 6 7 8 9 10 11 12: 13 14

14 13 i 12 11 10 9 8 7 6 5 4 3 2 1 : 2 3

Gauldin, Eighteenth-Century Counterpoint, 187.


254

Example 2.2. Counterpoint at the twelfth, from mm. 5-6 and mm. 17-18 of Bach’sFugue in C Minor (BWV 847).

Piston, Counterpoint, Ex. 273.


255

Example 2.3. Annotated version o f Example 2.2.

11 Invemon

mm. 5-6

mm. 17-18

1 1 + 2 = 1 3 (invertible counterpoint at the twelfth)


Example 2.4. Pattern of consonances occurring under invertible counterpoint at the twelfth.

interval of inversion intervallic patterns

twelfth 1 3 5

12 10 8


257

Example 2.5. Harmonic-tone inversion at the twelfth.

interval of inversion harmonic tone patterns

twelfth root third fifth

fifth third root

b.

Intervals: 12

Harmonic tones: fifth

1

root ro o t-^ 1 ^ root

12

10

t h i r c k ^ rro o t^ ^ k .

|T2l

3 8

fifth rootthird rootX

5

fifthroot

12o-

■» 0 0 8-

~Tfr


Example 2.6. The Ursatz.

Background:fundamental structure

I V

Oster trans., Free Composition, Fig. 1.


Example 2.7. Unfolding.

ZJ Cj' U ' t f10 - 10, 6 - 6 6 - 6, 10-10

IS IS6 - 6, 10 - 10 10 - 10, 6 - 6

Oster trans., Free Composition, Fig. 43 (b) 4 and 5; (c) 4 and 5.


*<«i W

B«5 “iic?V

V- a

I

ino

a>OX)

30 X <u10)oo

>oo(N<D

./:e.’T lffl9

.£PEsT©•*>«»

• *»>*

01

£©©

£

C!jXw

fII E;

fci v ;>»

U<utoo

jSJ

USO


Reproduced

with perm

ission of the

copyright ow

ner. Further

reproduction prohibited

without

permission.

Example 2.9. Form chart of Bach’s Fugue in C Minor (BWV 847).

S = subject A = answer CSl = countersubject 1 CS2 * countersubject 2 Ep. = episode Circled numbers ~ measure numbers Boxed Roman numerals = tonal areas

®s

m

CSlA

0

© Ep. 1 ©CSlCS2S

0

® Ep. 2

incomp, cycle of Sths

®sCS2CSl

m

® _ .Ep. 3(based on inv. offig. in Ep. 2)

® C S 1 ® Ep. 4 @ Ep. 5s (beat on ext. CSl (beat on ext.CS 2 ofE p . 1; CS2 of Ep. 2;

dbl. cpt. dbL at 8va)at 12th)

complete cycle0 □ of 5ths

JLd D ® (ext.) < m

CS2 sCSlS

m......... ............ 0

Gauldin, Eighteenth-Century Counterpoint, Fig. 17-1, 224.N>On

Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

Example 2.10. Revised form chart of Bach’s Fugue in C Minor (BWV 847).

mm. 1-3 mm. 9-11mm. 3-5 mm. 7-9mm. 5-7

Ep. 2desc. fifths

CSlCS2

Ep. 1CSl

v—illi - I I I

counterpoin t at 12'

mm. 25- 26.5mm. 11-13 mm. 13-15 mm. 15-17 mm. 20-21 mm. 22-25mm. 17-20

Ep. 3 CSl Ep. 5desc. fifths

V - p e d a lEp. 4CS2CSl

CSlCS2CS2

fvTl i — III — VI

I I I

1

mm. 26.5-28 mm. 28-29 mm. 29-31

CS2CSl

L i n k

262

263

Example 2.11. Invertible relationship between the initiating foreground harmonies of Episodes one and four from Bach’s Fugue in C Minor (BWV 847).

fifththird

V ° I9 -

* = § = ------------------------ y -- M - n

i: III

m. 5

v: III

m. 17


264

Example 2.12. Melodic stereotype outlining 5 - 6 - 5 - 4 - 3, from Handel’s Fuguein C Minor, from Six Grandes Fugues.

m

Gauldin, Eigtheenth-Century Counterpoint, Ex. 17-2 A, 212.


Example 2.13. Subject/answer Paradigm 1, Category 1.

A A AParadigm 1 8 8 7

I V V

Renwick, Analyzing Fugue, 26.


Example 2.14. Revision o f subject/answer paradigm.

subject answerr

$

A A A5 4 3V: 5

A4

A3

i v 1 V V /V V


E

^r00

PQShOo

u_co3003P-Ht/5

XOCOPQ<4Hoin

S B

_3

B _op”3

CO

3Oh03<DV

•

<UOh

in<NJhuiXM

" O '

E

osoEO


Exam

ple

2.16

. N

eigh

bor

note

arou

nd

the

third

of the

do

min

ant

in the

ep

isod

e,

mm

. 5-

7, o

f B

ach’

s Fu

gue

in C

min

or (

BWV

847)

.

268

VO

c ; :

f :

/ . :•

1::3

a

/ . : •

I "rI ^

aa

<oo

S-iOCau


269

Example 2.17. Analysis o f mm. 5-6 o f Bach’s Fugue in C Minor (BW V 847).

cmoil V

Schenker, “Das Organische der Fuge,” Figure 8.


270

Example 2.18. Analysis o f mm. 5-7 o f Bach’s Fugue in C Minor (BW V 847).

5 . <♦ .

. j r . -p f iT f r ^ W r fW ic minor: vk3 L------------------------------- _____----------------- 3------ I

Schenker, Der freie Satz, Figure 102, 5.


271

Example 2.19. Harmonic-tone inversion at the twelfth.

fifth] alto

thirdj tenor

root

<c=

<c=

i alto 4 i 1

3 tenor 4 .........11


Example 2.20. Deep-middleground of Bach’s Fugue in C Minor (BWV 847).

© @ © @ @

alto: third — ______ ___p. fifthtenor: root — third

A A A A A

o n u s

5 — 6 .8 7


273

Example 2.21. Analysis o f the exposition o f Bach’s Fugue in C minor (BW V 847).

(D (D ® ©

A A

exposition


Exam

ple

2.22

. V

oice

-lead

ing

sum

mar

y of

the

episo

de

from

mm

. 17

-20

of B

ach’

s Fug

ue

in C

Min

or (

BWV

847)

.

274

<oo-a

a

oo

S z f l <oo

HH

a


Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

Example 2.23. Comparison of episodes from mm. 5-7 and mm. 17-18 of Bach’s Fugue in C Minor (BWV 847).

mm.: 5 6 7 8 9 17 18 19 20

0

-6— 4+ f>____A -I 1

parallel tenthsparallel thirds0

parallel sixths 6— 4+ i

third thirdroot

alto: fifth bass:thirdalto: third

tenor:rootthirdroot

thirdroot

rootthird0

6-----4I---- 4—2

1tinvertible counterpoint at the twelfth

275

276

Example 2.24. The fourth-progression at the deep-middleground of Bach’s Fugue in C minor (BWV 847).

mm.: 3 10 11 20 25 29S

4th-prg.

— 7


Exam

ple

2.25

. M

iddl

egro

und

sketc

h of

Bac

h’s F

ugue

in

C M

inor

(BW

V 84

7).

277

<N


Example 3.1. Counterpoint at the tenth.

A - n - i /

? D m —

1 i r i —

Beach and Thym trans., The Art o f Strict Musical Composition, 163.


Example 3.2. Inversion table for counterpoint at the tenth.

10

10 9


Reproduced with

permission



without permission.

Example 3.3. Counterpoint at the tenth. Creation of three-part example from two-voice counterpoint. Adapted from AlfredMann, The Study o f Fugue (New York: Norton, 1958), 116.

a.

■ f - :p f r~ - Holt

o L IM I

IICUT

280

281

Example 3.4. Extrapolation o f Piston’s notion o f counterpoint at the tenth.

C

B

A

G

F

3(10)EtDt

C

B

A

GtFtE

D

C


282

Example 3.5. Diagram o f counterpoint at the tenth.

fifth

10

third

root

third fifth

XT

root third


283

Example 3.6. Inversion tables for counterpoint at the tenth with sample illustration.

a.


tenth 1 3 5

10 8 6

b.


tenth root third fifth

third root third

c.

Intervals:

Harmonic tones:

1 10ro o t-^^ r third ro o t^ ^ k root

$10

thirdroot

rootroot

fifthroot

_Q_i

rootthird


284

Example 3.7. Counterpoint at the tenth at the background.

alto: rootbass: root

V


285

Example 3.8. Descending thirds sequence. Based on Aldwell and Schachter, 2003.

lOi lOt lOi lOt lOl lOt

soprano: third —root —third .-root —third —root -thirdyC yC yc ycbass: | root rootj rooK root , root ^ ro o t7! | root |

III I


286

Example 3.9. Descending thirds and counterpoint at the tenth.

root, third

root root - ^ ^ ^ t h i r d root- r third

V


Exam

ple

3.10

. Ba

ckgr

ound

and

de

ep-m

iddl

egro

und

of the

Fu

gue

in m

ajor

(BW

V 89

0).

Use

of co

ntra

ry

and

obliq

ue

mot

ion.

287

«N

c3


Reproduced with

permission



without permission.

Example 3.11. Bach’s Fugue in Major (BWV 866). Demonstration of counterpoint at the tenth between parallel sections.

10 li 12 13

A t f t — H

l_

!i i_ — i#-

J - ........— J J 5 f# i

Bb maj.10

13 14 15 16 17

-hjBgae■ La i- t - J - r LCCn^"1..r t ...

C min.

22 23 24 25 26

10 10

>

G min.

288

Exam

ple

3.12

. Ex

cerp

ts fro

m B

ach’

s Fu

gue

in M

ajor

(BW

V 89

0) t

hat

are

relat

ed

by co

unte

rpoi

nt a

t the

te

nth.

289

o

LU


F m

aj.—

*■ G

min

.

290

Example 3.13. Subject of Bach’s Fugue in Major (BWV 866),mm. 1-5.

I .. 1].Ml p '5— 1

b-iH ~ =jtr LLy [gTsffl L£a


291

Example 3.14. Subject o f Bach’s Fugue in Major (BW V 890), mm.1-5.

J g j i . a , ----- -----4 '— [ r

t v ; 1,3................................

..*.. |»..m..9 ..- .. 4 ---------------- ^r ▼

! i ii

h

?

h / - y t

A

3


292

Example 3.15. M odel o f reaching-over.

h ighA

▼

lo w


293

Example 3.16. Schenker’s illustration o f reaching over.

nOcf..(durch iibtrgcttfen von Stimmvn)

Schenker, “Erlauterungen,” in Das Meisterwerk in der Musik: Ein Jahrbuch [1925] (Munich: Drei Masken Yerlag, 1925), 204.


Exam

ple

3.17

. M

iddl

egro

und

sketc

hes

of M

ajor

Fugu

e (B

WV

866)

.

00

£?Q.-a

294

£?CL

X>

00uCL

■a

id cn

O O 'sO T f

>

>

VO

>


Reproduced with

permission



without permission.

e x p o . ->

Example 3.18. Form chart of Bach’s Fugue in Major (BWV 890).

mm. 1-5

S

m

mm. 5-9

A

m

mm. 9-13

Ep. 1

mm. 13-17

CS1s

mm. 17-20

link to V

ctpt. at the 8ve

mm. 21-25

CS1s

* n n

mm. 25-32

cadencephrase

PAC

C Dmm. 32-36

CS3S

CS2

n n «-

mm. 36-40

link to I

ctpt. at the 10th

mm. 40-44

SCS2CS3

mm. 44-47

link to vi

mm. 47-51

CS2CS3

S

1 vi 1 +

mm. 51-54

link to IV

ctpt. at the 8ve

mm. 55-58

CS1s

mm. 58-63

link to ii

mm. 63-67

SCS3CS2

D O

mm. 67-78

link

PAC

L i <v i | i i - > V

mm. 78-82 S

CS3 (extended)

CS2

m

mm. 86-93

PAC

m

t oVO

296

Example 3.19. Middleground of Bach’s Fugue in Major (BWV 890).

exposition1

® @ (2 8 ) (3 2 ) @ ( 4 4 ) @ @ @ @ @ @ @ @

A A A A A

3 3 3 2 1

8 — 7V

8-7 V I1— 6I vi IV ii


Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

Example 3.20. Measures 32-36 and mm. 40-44 of Bach’s Fugue in Major (BWV 890).

(40)

0

V6/5F major: I IV V

E

V6/5 IV

Bl’ major: I

0

42 -3

47 2 - 3 ------- 7

III6I 6

IV V

04 -3IVV6/5 4 - 3

0

42 -3

42 -3

III6IV V

297

Reproduced with

permission



without permission.

Example 3.21. Parallel entries from Bach’s Fugue in Major (BWY 890), mm. 47-51 and mm. 54-58.

48) (49

-- ■____ !!2:V ^ " r /

^ = = =f .......... .

3, , 1 + \_/ Zj

------»....... ■j.... ■■■■■— w, .......= Hr*7!.--- .......... ^

G minor: i--13j6 Eb major: I-

..o .. Vll / 11

0

• • O / • •vii / u

6—1.7 4—5 2—3 v ii°/ ii

7 - i>67- 8 2 -

298

Reproduced with

permission



without permission.

Example 3.22. Measures 78-86 of Bach’s Fugue in Major (BWV 890).

@ ( 7 9 ) (80) (8?) @ @ ( 8 4 ) @ @

0

F major: IBt> major: G minor:

9 -9 - v i9 — \>9-I V 6F major: I ii 9 —Bl> major: G minor:

0vi9 -9 - 8 - 7 i>9 -ii 9 —F major: I

Bb major: G minor:

299

Exam

ple

3.23

. Ex

posit

ion

of B

ach’

s Fug

ue

in M

ajor

(BW

V 89

0),

mm

. 1-

32.

©

©

©

©

os{X


Fmaj

or:

ii

Exam

ple

3.24

. M

easu

res

32-7

8 of

Bac

h’s

Fugu

e in

Majo

r (B

WV

890)

.

301

<N

<


Exam

ple

3.25

. Gr

aphs

of

mm

. 78

-93

of B

ach’

s Fu

gue

in M

ajor

(BW

V 89

0).

302

oc>

00>

CQ


G m

inor

: j

II Bb

maj

or:

303

Example 3.26a. Measures 1-5 from Bach’s Fugue in Major (BWV 890);superimposition of CS2.

® ® @ ©CS2

7 - 6 7— 6I ---------------- 1 V 4 I

3

Example 3.26b. Measures 5-9 from Bach’s Fugue in B Major (BWV 890);superimposition of CS2.

Bb Major: I ' 6 5F Major: IV V4/3-----► IV V I


304

Ex. 3.27. Scale-degree functions o f CS2 in Bach’s Fugue in Major (BW V 890).

measures S D ’s fo r local harmonicCS2 keys function

mm. 1-5 1 - 2 - 3 B^ Imm. 5-4 F V T

mm. 34-35 4 - 5 F Vmm. 42-43 6 - 7 B^ I P D - D

mm. 49-50 i - 2 G min. VI Tmm. 56-57 E^ maj. IV P D - D

mm. 80-82 3 - 4 - 5 B^ I T - P Dmm. 83-85 G min. VI T - P D - D


305

Example 4.1. Inversion tables for counterpoint at the octave.

a.


octave 1 3 6

8 6 3

b.


octave root third fifth

root third fifth

c.

Intervals: Harmonic tones:

1rootroot X

8 3 6 6root th ird s . _».root thirdroot root— f"'* third fifth-

fifth "7— third[8]

& r ■<>


306

Example 4.2. Illustrations o f counterpoint at the octave within the Fuga.

a.CS

6 — 2 6

r

c s3 -----7--------3


Reproduced with

permission



without permission.

Example 4.3. Invertible counterpoint at the octave occurring at structural cadences.

sixthsthirds

D minor: 7

G minor:

o

308

Example 4.4. 8-line Ursatz in G minor.

third

fifth root

third x rootroot third

A

7A

6A

4

- r i

8 -----7V I


309

Example 4.5. Ascending register transfer.

A A A A A A

8 7 6 5 4 3

I J p , g ---------------------------® ^ -------------------o ----------------------------------------------- — £

A

2« ------------------------

A

1

^ ° ® — m 0 -------------------------------------------------------------------------------------------------------------------------------------------------------* f ♦ V'

V ' L------------- ---------------------------------------------------

S>------------............................................................................. .. ............. - .... ----------------------------

5 - 6 8 ----- 7I ---------------------------------------------------III-------------- V I


310

Example 4.6. Register transfer in the Fuga.

® (22) (24) (36) (38) @ @ ® @ @ @

8-75 - 6! ----------------------------------------------------------------------III------------V—I


311

Example 4.7. Voice-exchange,

(a). (b).

D minor: I I I --------------------------------------- I

Example 4.7, contd. Model of register transfer and voice-exchange.

upper voices:

I V I I V I I V

(c). tonal model (d). register transfer; (e). voice-exchange;(inverting voices) (inverting content)


Reproduced with

permission



without permission.

Example 4.8. Form chart of the Fuga.

exposition

m m. 1-2

S

□

mm. 2-3

CSA

mm. 3-4

A

mm. 4-5

S

mm. 5-14 14 - 24 - 28 31

S ‘D7’

32 32.5 33 35.5

DN DN

iv6 V7 6/4----------4/2

38 42

i — iIAC PAC□

42 47 55

16th note passagework V/1V

PAC

□

55 57

imitationV7/IV

m. 58

III

m. 64

- IPAC

64 - 66 - 74 76 80 82.5 84.5 85 85.5

S ‘D7’ S S DN DN

III V i6 iv6 V7— 6/4

87

□I

PAC□

87 91 94

16th note passageworki V i

PAC

□

Exam

ple

4.9.

Ex

posit

ion

of the

Fu

ga,

313


314


Exam

ple

4.11

. Oc

tave

desc

ent

from

3 to

3, w

ithin

the

Fu

ga,

mm

. 73

-80.

315

O00

D-

©

©

©


316

Example 4.12. Voice-leading summary o f the Fuga, mm. 24-42.

(24) (28) © ( 3 2 ) © @ © @A A

I: 5 5

V: 5

7iv V7t

v s ---4—#

61 l l

::&6 \/811 5 4 — #


Exam

ple

4.13

. In

verti

ble

cade

nce

at m

m.

52-5

5.

317

ilfci o o >

i n

y

| |5

11=


Reproduced with

permission



without permission.

Example 4.14. Invertible cadence moving through falling fifths.

Ill: 1imitation imitation repeatedmodelmodel imitation

4 4

oo

Exam

ple

4.15

. V

oice

-lead

ing

sum

mar

y of

the

Fuga

, m

m.

14-1

9.

319

©

CL

Cl I

>

> >

>

o o n

O n T fr

>

1 03 |£O


320

Example 4.16. Register transfer and voice-exchange.

[a~j model

registertransfer

0 voice-exchange

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission

321

Example 4.17. Kellner’s alternate cadence in the Fuga, mm. 34-35.

:


322

Bibliography

Albrechtsberger, Johann Georg. Grundliche Anweisung zur Composition. Leipzig: Johann Gottlob Immanuel Breitkopf, 1790.

. Sammtliche Schriften iiber Generalbafi, Harmonie-Lehre, und Tonsetzkunst.Ed. Ignaz Ritter von Seyfried. Leipzig, 1837. Reprint, Kassel: Barenreiter- Verlag, 1975.

. Guide to Composition. Trans. Sabilla Novello. London: Novello, Ewer, andCo., 1855.

Aldwell, Edward and Carl Schachter. Harmony and Voice Leading. 3rd ed. Belmont, CA: Wadsworth Group/Thomson Learning, 2003.

Angleria, Camillo. La regola del contraponto e della mvsical compositione. Milan, 1622. Reprint, Bologna: Amaldo Fomi, 1983.

Apel, Willi. The Notation o f Polyphonic Music: 900-1600. 5th ed. Cambridge, MA: The Mediaeval Academy of America, 1953.

Aaron, Pietro. Toscanello in musica. Venice, 1539. Trans. Peter Bergquist. Colorado Springs: Colorado College Music Press, 1970.

Bach, C. P. E. “Einfall, einen doppelten Contrapunct in der Octave von sechs Tacten zu machen, ohne die Regeln davon zu wissen.” In F. W. Marpurg. Historisch- kritische Beytrage zur Aufnahme der Musik. Berlin: J. J. Schiitzens Witwe, G. A. Lange, 1757, 3:167-181. Trans. Eugene E. Helm. “Six Random Measures of C. P. E. Bach.” Journal o f Music Theory 10/1 (1966): 139-151.

Bach, J. S. 371 Harmonized Chorales and 69 Chorale Melodies with Figured Bass. Ed. Albert Riemenschneider. New York: Schirmer, 1941.

. Precepts and Principles for Playing the Thorough-Bass or Accompanying inFour Parts. Trans. Pamela L. Poulin. Oxford: Clarendon Press, 1994.

Beach, David. “The Fundamental Line from Scale Degree 8: Criteria for Evaluation.” Journal o f Music Theory 32/2 (1988): 271-294.

. Aspects o f Unity in J. S. Bach’s Partitas and Suites: An Analytical Study.Eastman Studies in Music. Rochester: University of Rochester Press, 2005.


323

Beldomandi, Prosdocimo de’. Contrapunctus. Montagnana: MS, 1412. Trans, and ed. Jan Herlinger. Lincoln: University of Nebraska Press, 1984.

Bellermann, Heinrich. Der Contrapunkt. Berlin: Julius Springer, 1901.

Benjamin, Thomas. The Craft o f Tonal Counterpoint. New York: Routledge, 2003.

Bent, Margaret. “Musica Ficta.” Grove Music Online. Ed. L. Macy. Accessed 20 September 2006. <http://www.grovemusic.com>.

Berardi, Angelo. Documenti armonici. Bologna, 1687.

Bloxam, M. Jennifer. “Cantus Firmus.” In The New Grove Dictionary o f Music and Musicians. Ed. Stanley Sadie. 2nd edition, 5: 67-74. London: Macmillan, 2001 .

Bononcini, Giovanni Maria. Mvsico prattico che breuemente dimostra II modo di giungere alia perfetta... Bologna, 1673.

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

Bukofzer, Manfred F. “Fauxbourdon Revisited.” The Musical Quarterly 38/1 (1952): 22-47.

, ed. Musica Britannica: A National Collection o f Music, vol. 8: JohnDunstaple: Complete works. London: Stainer and Bell, 1953.

Bussler, Ludwig. Der strenge Satz. Berlin: C. Habel, 1877.

Butler, Gregory G. “Fugue and Rhetoric.” Journal o f Music Theory 21/1 (1977): 49- 109.

Cadwallader, Allen and David Gagne. Analysis o f Tonal Music: A Schenkerian Approach. New York: Oxford University Press, 1998.

Campion, Francois. Traite d ’Accompagnement et de Composition selon la regie des octaves de musique. Paris, 1716. Reprint, Geneva: Minkoff, 1976.

Caplin, William E. Classical Form: A Theory o f Formal Functions for theInstrumental Music o f Haydn, Mozart, and Beethoven. Oxford: Oxford University Press, 1998.



324

Cerreto, Scipione. Della prattica musica vocale, et strumentale. Bologna, 1601. Reprint, Bologna: Amoldo Fomi, 2003.

Cerone, Pietro. El melopeoy maestro. Naples, 1613.

Cherubini. L. Cours de contre-point et de fugue. Paris: Maurice Schlesinger, 1835. Trans. Franz Stoepel. Theorie des Contrapunktes und der Fuge. Leipzig: Kristner, 1835. Trans. J. A. Hamilton. A Course o f Counterpoint and Fugue. London: Cocks and Co., 1837. Trans. Chowden Clarke. A Treatise on Counterpoint & Fugue. London: Novello, 1854.

Chiodino, Giovanni. Arte Prattica & Poetica. Trans. Johann-Andreas Herbst. Frankfurt, 1653.

Christensen, Thomas. “The Regie de I ’Octave in Thorough-Bass Theory and Practice.” Acta Musicologica 64 (1992): 91-117.

Cook, Nicholas. “Computational and Comparative Musicology.” In EmpiricalMusicology: Aims, Methods, Prospects. Ed. Erick Clarke and Nicholas Cook, 103-126. Oxford: Oxford University Press, 2004.

Crocker, Richard. “Discant, Counterpoint, and Harmony.” Journal o f the American Musicological Society 15/1 (1962): 1-21.

Daube, J. F. Der musikalische Dilettant: eine Abhandlung der Komposition. Vienna: von Trattner, 1773.

Davis, Ferdinand and Donald Lybbert. The Essentials o f Counterpoint. Norman: The University of Oklahoma Press, 1969.

Derr, Ellwood. “The Two-Part Inventions: Bach’s Composers’ Vademecum.” Music Theory Spectrum 3 (1981): 26-48.

Diruta, Girolamo. II Transilvano. Bologna, 1612; 1622. Reprint, Bologna: Fomi, 1969. Trans, and ed. Murray C. Bradshaw and Edward J. Soehnlein. The Transylvanian (II Transilvano), Vol. II. Henryville: Institute for Mediaeval Music, [1609] 1984.

Drabkin, William. “Invertible Counterpoint.” Grove Music Online. Ed. L. Macy. Accessed 3 October 2006. <http:/7www.grovemusic,com>.

Dreyfus, Laurence. Bach and the Patterns o f Invention. Cambridge: Harvard University Press, 1996.


325

Erickson, Raymond, trans. Musica enchiriadis and Scolica enchiriadis. Ed. Claude V. Palisca. New Haven: Yale University Press, 1995.

Fetis, Fran9ois-Joseph. Traite du Contrepoint et de la Fugue. Paris: Charles Michel Ozu [au magisin de musique du Conservatoire], 1824. New and revised edition, Paris: E. Troupenas et cie., 1846.

Fux, Johann Joseph. Gradus ad Parnassum. Vienna: Van Ghelen, 1725. Reprint, New York: Broude Brothers, 1966.

. The Study o f Counterpoint. Trans. Alfred Mann. New York: Norton, 1965.

Gaffurius, Franchinus. Practica musicae. Milan: Guillermus Le Signerre, 1496.Trans. Clement A. Miller. Dallas: American Institute of Musicology, 1968.

Gauldin, Robert. A Practical Approach to Eighteenth-Century Counterpoint. Englewood Cliffs, NJ: Prentice Hall, 1988.

Gedalge, Andre. Traite de la Fugue: Ire Partie: De la Fugue d ’Ecole. Paris: Enoch et Co., 1901. Trans, and ed. Ferdinand Davis. Treatise on Fugue. Norman: University of Oklahoma Press, 1965.

Hilse, Walter. “The Treatises of Christoph Bernhard.” The Music Forum 3 (1973): 1- 196.

Hoppin, Richard H. “Partial Signatures and Musica Ficta in Some Early 15th-Century Sources.” Journal o f the American Musicological Society 6/3 (1953): 197- 215.

Jadassohn, S. Lerbuch des einfachen, doppelten, drei- und vierfachen Contrapunkts. Leipzig: Breitkopf & Hartel, 1884. Trans. T. Baker. A Manual o f Single, Double, Triple and Quadruple Counterpoint. New York: G. Schirmer, 1902.

Jeppesen, Knud. Kontrapunkt (vokalpolyfoni). Copenhagen: Wilhelm Hansen/Musik- Forlag, 1930. Trans. Glen Haydon. Counterpoint: The Polyphonic Vocal Style o f the Sixteenth Century. New York: Prentice-Hall, 1939.

Kaye, Philip, ed. The Sacred Music o f Gilles Binchois. Oxford: Oxford University Press, 1992.

Kellner, Johann Peter. Treulicher Unterricht im General-Bafi. Hamburg: Herold,1737. Reprint, New York: Georg Olms Verlag, 1979.


326

Kennan, Kent. Counterpoint: Based on Eighteenth-Century Practice. Upper Saddle River, NJ: Prentice-Hall, 1999.

Kirkendale, Warren. “Ciceronians Versus Aristotelians on the Ricercar as Exordium, from Bembo to Bach.” Journal o f the American Musicological Society 32/1 (1979): 1-44.

Kimberger, Johann Philipp. Die Kunst des reinen Satzes in der Musik. Berlin and Konigsberg, 1776-1779. Reprint. Hildesheim: Georg Olms Verlagsbuchhandlung, 1968. Book I trans. David Beach and Jurgen Thym.The Art o f Strict Musical Composition. New Haven: Yale University Press,1982.

--------- . Gedanken uber die verschiedenen Lehrarten als Vorbereitung zurGugenkenntniss. Berlin, 1782. Trans. Richard B. Nelson and Donald R. Boomgaarden. “Kimberger’s Thought’s on the Different Methods of Teaching Composition as Preparation for Understanding Fugue.” Journal o f Music Theory 30/1 (1986): 71-94.

Koch, Heinrich Christoph. Kurzgefafites Handworterbuch der Musik fur praktische Tonkunstler undfur Dilettanten. Leipzig: Joh. Friedr. Hartknoch, 1807.

Kollmann, Augustus Frederic Christoph. Essay on Musical Harmony. London, 1796.

. An Essay on Practical Musical Composition. London, 1799.

Kosovsky, Robert. The Oster Collection: Papers o f Heinrich Schenker: A Finding List. New York: New York Public Library, 1990.

Kostka, Stefan and Dorothy Payne. Tonal Harmony, with an Introduction to Twentieth-Century Music. 4th ed. Boston: McGraw-Hill, 2000.

Knapp, Janet. “Discant.” In The Harvard Dictionary o f Music, ed. Don Michael Randel. 4th ed., 243-245. Cambridge, MA: The Belknap Press of Harvard University Press, 2003.

Kottick, Edward L. “Flats, Modality, and Musica Ficta in Some Early Renaissance Chansons.” Journal o f Music Theory 12/2 (1968): 264-274.

Laitz, Steven G. The Complete Musician: An Integrated Approach to Tonal Theory, Analysis, and Listening. New York: Oxford University Press, 2003.


327

Larsen, Arved Marita. “Angelo Berardi (1636-1694) as Theorist: A Seventeenth- Century View of Counterpoint.” Ph.D. diss., The Catholic University of America, 1979.

Larson, Steve. “Questions about the Ursatz: A Response to Neumeyer.” In Theory Only 10/4 (1987): 11-31.

Lavignac, Albert. Encyclopedia de la Musique et Dictionaire du Conservatoire. 2nd part. Paris: Librairie Delgrave, 1930.

Ledbetter, David. Bach’s Well-Tempered Clavier: The 48 Preludes and Fugues. New Haven: Yale University Press, 2002.

Lester, Joel. Bach’s Works for Solo Violin: Style, Structure, Performance. Oxford: Oxford University Press, 1999.

. Compositional Theory in the Eighteenth Century. Cambridge: HarvardUniversity Press, 1992.

London, Justin and Ronald Rodman. “Musical Genre and Schenkerian Analysis.” Journal o f Music Theory 42/1 (1998): 101-124.

Lowinsky, Edward E. “Conflicting Views on Conflicting Signatures.” Journal o f the American Musicological Society 173 (1954): 181-204.

. Tonality and Atonality in Sixteenth-Century Music. Berkeley and LosAngeles: University of California Press, 1961.

Lusitano, Vincenzo. Introduttione facilissima et novissima. 1561. Reprint, Bologna: Liberia Musicale Italiana, 1989.

Mann, Alfred. The Study o f Fugue. New York: W. W. Norton, 1965.

Marpurg, Friedrich Wilhelm. Abhandlung von der Fuge. 2 vols. Berlin: Haude and Spener, 1753 - 1754. Reprint, Hildesheim: Georg Olms, 1970.

. Historisch-kritische Beytrage zur Aufnahme der Musik. Vol. 3. Berlin: J. J.Schutzens Witwe, G. A. Lange, 1757.

Marix, J., ed. Les Musicians de la Cour de Bourgone au X V Siecle (1420-1467). Paris: Editions de L’Oiseau-Lyre, 1937.

Martin, Henry. Counterpoint: A Species Approach Based on Schenker’s Counterpoint. Lanham, MD: Scarecrow Press, 2005.


328

Martini, da F. Giambattista. Esemplare o sia saggio fondamentale pratico dicontrappunto sopra il canto fermo. 1774. Reprint, Ridgewood, NJ: Gregg Press, 1965.

Momigny, Jerome-Joseph de. Cours Complet d ’Harmonie et de Composition. Paris: the author and Bailleul, An XI [1802-3] - 1806.

Morley, Thomas. A Plain and Easy Introduction to Practical Music. London: Peter Short, 1597; ed. R. Alec Harman. New York: Norton, 1952.

Morris, R. O. Contrapuntal Technique in the Sixteenth Century. Oxford: Clarendon Press, 1922.

Muffat, Georg. An Essay on Thoroughbass. 1699. Ed. Hellmut Federhofer. Tubingen: American Institute of Musicology, 1961.

Neumeyer, David. “The Ascending Urlinie.” Journal o f Music Theory 31/2 (1987): 275-303.

:--------. “Reply to Larson.” In Theory Only 10/4 (1987): 33-37.

. “The Three-Part Ursatz.” In Theory Only 10/1-2 (1987): 3-29.

. “The Urlinie from 8 as a Middleground Phenomenon.” In Theory Only 9/6(1987): 3-24.

Owen, Harold. Modal and Tonal Counterpoint: From Josquin to Stravinsky. New York: Schirmer, 1992.

Parks, Richard S. Eighteenth-Century Counterpoint and Tonal Structure. Englewood Cliffs, NJ: Prentice-Hall, 1984.

Phillips, Nancy. “Musica and Scolica Enchiriadis'. The Literary, Theoretical, and Musical Sources.” Ph.D. diss., New York University, 1984.

Piston, Walter. Counterpoint. New York: Norton, 1947.

Proctor, Gregory and Herbert Lee Riggins. “Levels and Reordering of Chapters in Schenker’s Free Composition.” Music Theory Spectrum 10 (1988): 102-26.

Prout, Ebenezer. Double Counterpoint and Canon. London: Augener, 1891.

. Fugue. London: Augener, 1891.


329

Renwick, William. Analyzing Fugue: A Schenkerian Approach. Stuyvesant, NY: Pendragon Press, 1994.

Richter, Ernst Friedrich. Lehrbuch des einfachen und doppelten Kontrapunkts. Leipzig: Breitkopf & Hartel, 1872.

Riemann, Hugo. Musik-Lexikon: Theorie und Geschichte der Musik. Leipzig: Bibliographisches Institut, 1882.

--------- . Lehrbuch des einfachen, doppelten und imitierenden Kontrapunkts. Leipzig:Breitkopf & Hartel, 1888.

Rodio, Rocco. Regole di Mvsica. 1609. Reprint, Bologna: Fomi, 1981.

Rothstein, William. “Articles on Schenker and Schenkerian Theory in The New Grove Dictionary o f Music and Musicians, 2nd edition, ed. Stanley Sadie.” Journal o f Music Theory 45/1 (2001): 204-227.

Sancta Maria, Thomas de. Libro llamado arte de taher fantasia. 1565. Reprint, Heppenheim: Gregg Publishers, 1972.

Sanctis, C. de. La Polifonia nellArte Moderna. Milan: G. Ricordi, 1934.

Schachter, Carl. “Bach’s Fugue in B^ Major, Well-Tempered Clavier, Book I, No. XXI.” The Music Forum 3 (1973): 239-267.

. “The First Movement of Brahms’s Second Symphony: The Opening Themeand Its Consequences.” Music Analysis 2/1 (1983): 55-68.

. “The Prelude from Bach’s Suite No. 4 for Violoncello Solo: The SubmergedUrlinie.” Current Musicology 56 (1994): 54-71.

—. “The Triad as Place and Action.” Music Theory Spectrum 17/2 (1995): 149-169.

Schenker, Heinrich. Neue Musikalische Theorien und Phantasien.Vol. 1: Harmonielehre. Stuttgart: Cotta, 1906. Trans. Elisabeth Mann Borgese, ed. Oswald Jonas. Harmony. Chicago: University of Chicago Press, 1954.Vol.. 2, Part 1: Kontrapunkt: Erster Halbband: Cantus Firmus und Zweistimmiger Satz. Vienna and Leipzig: Universal Edition, 1910. Trans. John Rothgeb and Jurgen Thym, ed. John Rothgeb. Counterpoint. Book I. Ann Arbor: Musicalia Press, 2001.


330

Part 2: Kontrapunkt, zweiter Teil, zweiter Halbband: Drei- und Mehrstimmiger Satz, Tibergange zum freien Satz. Vienna and Leipzig: Universal-Edition A. G., 1922. Trans. John Rothgeb and Jurgen Thym, ed. John Rothgeb. Counterpoint. Book II. Ann Arbor: Musicalia Press, 2001 .

Vol. 3. Derfreie Satz. Vienna: Universal Edition, 1935. Ed. Oswald Jonas. Vienna: Universal Edition, 1956. Trans, and ed. Ernst Oster. Free Composition. New York: Longman, 1979.

-. “J. S. Bach: Zwolf kleine Praludien Nr. 5.” In Der Tonwille, Fiinftes Heft [1923]. Vienna: Albert J. Gutmann, 1923, 8-9. Trans. Joseph Dubiel. “Bach’s Little Prelude No. 5 in D minor, BWV 926.” In Der Tonwille: Pamphlets in Witness o f the Immutable Laws o f Music, Offered to a New Generation o f Youth, Vol. I, ed. William Drabkin, 180-181. Cambridge: Cambridge University Press, 2004.

-. “Erlauterungen.” In Das Meisterwerk in der Musik: Ein Jahrbuch [1925]. Munich: Drei Masken Verlag, 1925, 201-205. Trans. Ian Bent. “Elucidations.” In The Masterwork in Music: A Yearbook (1925), Volume I, ed. William Drabkin, 112-114. Cambridge: Cambridge University Press, 1994.

-. “Forsetzung der Urlinie-Betrachtungen.” In Das Meisterwerk in der Musik: Ein Jahrbuch [1925]. Munich: Drei Masken Verlag, 1925,185-200. Trans. John Rothgeb. “Further Considerations of the Urlinie: I,” in The Masterwork in Music I, ed. William Drabkin, 104-111. Cambridge: Cambridge University Press, 1994.

-. “Joh. S. Bach: Sechs Sonaten fur Violine: Partita III (E-Dur) Praludio.” In Das Meisterwerk in Der Musik [1925]. Munich: Drei Masken Verlag, 1925, 75-98. Trans. John Rothgeb. “The Prelude of Bach’s Partita No. 3 for Solo Violin [BWV 1006].” In The Masterwork in Music I, ed. William Drabkin, 39-53. Cambridge: Cambridge University Press, 1994.

-. “Joh. S. Bach: Zwolf kleine Praludien, Nr. 7.” In Das Meisterwerk in der Musik: Ein Jahrbuch [1925]. Munich: Drei Masken Verlag, 1925, 107-113. Trans. Hedi Siegel. “Bach: Twelve Short Preludes, No. 7 [BWV 941].” In The Masterwork in Music I, ed. William Drabkin, 58-61. Cambridge: Cambridge University Press, 1994.

-. “The Largo of Bach’s Sonata No. 3 for Solo Violin [BWV 1005].” Trans. John Rothgeb. Music Forum 4 (1976): 63-73.

-. “Das Organische der Fuge: aufgezeigt an der I. C-Moll-Fuge aus dem Wohltemperierten Klavier von Joh. Seb. Bach.” In Das Meisterwerk in der


331

Musik: Ein Jarbuch, Band II. Munich: Drei Masken Verlag, 1926, 55-95. Trans. Hedi Siegel. “The Organic Nature of Fugue as Demonstrated in the C Minor Fugue from Bach’s Well-Tempered Clavier, Book I.” In The Masterwork in Music: A Yearbook, vol. II (1926), ed. William Drabkin, 31-54. Cambridge: Cambridge University Press, 1996.

. “Joh. Seb. Bach: Suite III C-Dur fur Violoncello-Solo, Sarabande.” In DasMeisterwerk in Der Musik: Ein Jarbuch, Band II. Munich: Drei Masken Verlag, 1926, 97-104. Trans. Hedi Siegel, “The Sarabande of J. S. Bach’s Suite No. 3 for Unaccompanied Violoncello [BWV 1009].” Music Forum 2 (1970): 274-282.

. “Bach’s Little Prelude No. 4 in D Major, BWV 925.” In Der Tonwille:Pamphlets in Witness o f the Immutable Laws o f Music, Offered to a New Generation o f Youth. Vol. I: Issues 1-5 (1922-1923), trans. Joseph Dubiel, ed. William Drabkin, 177-179. Oxford: Oxford University Press, 2004.

Schoenberg, Arnold. Preliminary Exercises in Counterpoint. Ed. Leonard Stein. London: Faber and Faber, 1963.

Schubert, Peter. Modal Counterpoint, Renaissance Style. New York: Oxford University Press, 1999.

Schubert, Peter and Christoph Neidhofer. Baroque Counterpoint. Upper Saddle River, NJ: Pearson Prentice Hall, 2006.

Schwind, Elisabeth and Michael Polth. “Klausel und Kadenz.” In Die Musik inGeschichte und Gegenwart. Ed. Ludwig Finscher, 5: 256-282. Barenreiter: Kassel, 1996.

Smith, Charles J. “Musical Form and Fundamental Structure: An Investigation of Schenker’s FormenlehreC Music Analysis 15/2-3 (1996): 191-297.

Smyth, David. “Schenker’s Octave Lines Reconsidered.” Journal o f Music Theory 43/1 (1999): 101-133.

Soderlund, Gustave Frederic and Samuel H. Scott. Examples o f Gregorian Chant and Other Sacred Music o f the 16th Century. New York: Appleton-Century-Crofts, 1971.

Sparks, Edgar H. Cantus Firmus in Mass and Motet, 1420-1520. Berkeley: University of California Press, 1963.


332

Stinson, Russell. The Bach Manuscripts o f Johann Peter Kellner and His Circle. Durham: Duke University Press, 1990.

. “J. P. Kellner’s Copy of Bach’s Sonatas and Partitas for Violin Solo.” EarlyMusic 13/2(1985): 199-211.

Sulzer, Johann Georg. Allgemeine Theorie der schonen Ktinste. Leipzig, 1771 and 1774.

Sweelinck, Jan Pieterszn. Compositions-Regeln. In Werken. Vol. X, ed. Hermann Gehrmann. 1901. Reprint, Westmead: Gregg, 1968.

Tigrini, R. M. O. II Compendio della Musica ne quale Brevemente...Venice, 1588.

Tinctoris, Johannes. Opera Theoretica. Ed. Albert Seay, in Corpus Scriptorum de Musica, vol. 22. American Institute of Musicology, 1975.

Tovey, Donald Francis. J. S. Bach, Forty-Eight Preludes and Fugues. Vol. I.London: The Associated Board of the Royal Schools of Music, 1924.

Trowell, Brian. “Faburden and Fauxbourdon.” Musica Disciplina 13 (1959): 43-78.

Vicentino, Nicola. L ’antica musica ridotta alia moderna prattica. Rome: Antonio Barre, 1555. Reprint, ed. Edward E. Lowinsky. Basel: Barenreiter, 1959. Trans. Maria Rika Maniates. Ancient Music Adapted to Modern Practice.New Haven: Yale University Press, 1996.

Vogler, Georg Joseph. Tonwissenschaft und Tonseztkunst. Mannheim, 1776.

. System fur den Fugenbau als Einleitung zur harmonischer Gesang-Verhindungs-Lehre. Offenbach am Main, 1817.

Walker, Paul. Theories o f Fugue from the Age ofJosquin to the Age o f Bach. Rochester: University of Rochester Press, 2000.

Walther, Johann Gottfried. Musikalishes Lexikon oder musikalische Bibliothek. 1732. Reprint, ed. Richard Schaal. Kassel: Barenreiter-Verlag, 1953.

. Praecepta der composition. 1708. Ed. Peter Benary. Leipzig: Breitkopf &Hartel, 1955.

Wen, Eric. “Bass Line Articulations of the UrlinieC In Schenker Studies 2, ed. CarlSchachter and Hedi Siegel, 276-297. Cambridge: Cambridge University Press, 1999.


333

Zarlino, Gioseffo. Le Istitutioni Harmoniche. Venice, 1558. Book III trans. Guy A. Marco and Claude V. Palisca. The Art o f Counterpoint: Part Three o f Le Istitutioni Harmoniche, 1558. New Haven: Yale University Press, 1968. Book IV trans. Vered Cohen. On the Modes. New Haven: Yale University Press,1983.


Date post:	01-Feb-2023
Category:	Documents
Upload:	khangminh22
View:	1 times
Download:	0 times

The Role of Invertible Counterpoint within Schenkerian Theory ...

Documents