Bartok 5th String Quartet

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8613806

Bates, Karen Anne

THE FIFTH STRING QUAR TET OF BELA BARTOK: AN ANALYSIS BASED

ON THE THEORIES OF ERNO LENDVAI

The University of Arizona Ph.D. 1986

UniversityMicrofilms

International 300 N. Zeeb Road, Ann Arbor, Ml 48106

Copyright 1986

by

Bates, Karen Anne

All Rights Reserved

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UniversityMicrofilms

International






THE FIFTH STRING QUARTET OF BELA BARTOK:

AN ANALYSIS BASED ON THE

THEORIES OF ERNO LENDVAI

by

Karen Anne Bates

A Di ss er ta tio n Submitted to the Fa culty of the

SCHOOL OF MUSIC

In Partial Fulfillment of the Requirements

For the Degree of

DOCTOR OF PHILOSOPHY

WITH A MAJOR IN MUSIC THEORY

In the Graduate College

THE UNIVERSITY OF ARIZONA

1 9 S 6

Copyright 1986 Karen Anne Bates




THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE

As members of the Final Examination Committee, we certify that we have read

the dissertation prepared by ______ Karen Anne Bates _________________________

entitled ______ The Fifth String Quartet of B&la Bartdk: _____________________

______ An Analysis Based on the Theories of Ernfl I.endvai

and recommend that it be accepted as fulfilling the dissertation requirement

for the Degree of Doctor of Philosophy

Date

Date

Date

Date

Date

Final approval and acceptance of this dissertation is contingent upon the

candidate's submission of the final copy of the dissertation to the Graduate

College.

I hereby certify that I have read this dissertation prepared under my

direction and recommend that it be accepted as fulfilling the dissertation

requirement.

( j-

Lssertation Director




STATEMENT BY AUTHOR

This dis se rta tion has been submitted in pa rt ia l fu lfi llm e n t

of requirements for an advanced degree at The University of Arizona

and i s deposited in the Un ive rs ity Libr ary to be made av ai lab le to

borrowers under rules of the Library.

Brief quotations from this dissertation are allowable without

special permission , provided t ha t accurate acknowledgement of source

is made. Requests fo r permission fo r extended quo tat ion from or

reproduction of this manuscript in whole or in part may be granted

by the copyright holder.

SIGNED




ACKNOWLEDGEMENTS

I wish to acknowledge, with g ra ti tu de , Dr. Edward W. Murphy

for patiently answering questions and always setting an example to

foll ow in the purs uit o f le arn ing . To my paren ts, Welland J. and

Els ie F. Bates and si s te r , Dehlia Bates L oz ier , I say thank you fo r

the love and support a ll these years.

A specia l thank you to Dana Liv ings ton Co llins fo r reading

the drafts, Veronica Engel for hours of proofreading, Todd Seelye

for his assistance in editing and surviving the birth of a Ph.D.,

and John M. K is sl er fo r h is constan t ass istance and encouragement

over the years.

i i i




TABLE OF CONTENTS

LIST OF FIGURES................................................................................................................... v i i

ABSTRACT................................................................................................................................... xxi

INTRODUCTION

NEED FORSTUDY........................................................................................................... 1

PURPOSE OFTHE STUDY................................................................................. 1

METHOD OF ANALYSIS................................................................................................ 1

METHOD OF PRESENTATION....................................................................................... 2

BACKGROUND..................................................................................................................2

CHAPTER ONE

LENDVAI'S THEORETICAL PRINCIPLES

Fibonacci Series ........................................................................................... 6

Closed System.................................................................................................. 8Pentatony................................................................................................ .10

Mipentatony ......................................................................................... 12Six four Structure ............................................................................ 13Phrygian................................................................................... 14

Symmetry........................................................................................................... 15Harmony............................................................................................................. 16

Relative Solmization ........................................................................ 16Ma jo r Mi norSubmi no r ................................................................................. 24

Fully Diminished Seventh Chords........................................................ 26Hypermajor and Hyperminor Chords...................................................... 28

Movement Between Chords.......................................................................... 32Substitute Chords....................................................................................... 37

Function ........................................................................................................... 41Axis System.................................................................................................... 44

Functional Derivation .............................................................................. 45Pentatonic Origin .............................. 50

Harmonic Origins ......................................................................................... 54Axis Scales .................................................................................................... 61

1:2 Model................................................................................................ 611:5 Model................................................................................................ 641:3 Model................................................................................................ 66

iv

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Authentic Cadences.................................................................................... 68Dominant Poles as Roots................................................................. 72

Tonic Poles as Roots....................................................................... 73Alpha Harmonies........................................................................................... 74

Alpha Chord........................................................................................... 75Beta Chord............................................................................................. 80Gamma Chord........................................................................................... 82Delta Chord........................................................................................... 86Epsilon Chord.......................................................................................88

Fibonacci Origins ofAlpha Harmonies............................................... 90Eq uid istan t Scales and Harmonies...................................................... 92

CHAPTER TWO

THE QUARTET............................................................................................................. 97

CHAPTER THREE

MOVEMENT ONE ANALYSIS.....................................................................................100

CHAPTER FOUR

MOVEMENT TWO ANALYSIS................................. ..................................................131

CHAPTER FIVE

MOVEMENT THREE ANALYSIS................................................................................ 146

CHAPTER SIX

MOVEMENT FOUR ANALYSIS.................................................................................. 183

CHAPTER SEVEN

MOVEMENT FIVE ANALYSIS.................................................................................. 200

CHAPTER EIGHT

SUMMARY AND CONCLUSION

Fibonacci Series andPentatony.'....................................................... 220Function........................................................................................................ 222

Axis System.................................................................................................. 223

Alpha harmonies......................................................................................... 224Equidistant Scales and Harmonies.......................................... 226The Quartet.................................................................................................. 227Movement One,.............................................................................................. 227

Movement Two.......................................................................................... .228

Movement Three................. ••....................................................................... 229

produced with permission of the copyright owner. Furthe r reproduction prohibited without permission.



vi

Movement Four .............................................................................................230

Movement Five .............................................................................................230

Conclusion................................................................................................... 231

LIST OFREFERENCES.............................................................................................................. 235

SELECTED BIBLIOGRAPHY............................................................................................236




LIST OF FIGURES

CHAPTER ONE

FIGURE 1 Golden Section ......................................................................... 7

2 Fibonacci Labels of H alf s tep Gra dation s ................. 8

3 Closed System............................................................................ 9

4 Chromatic Sc ale ......................................................................11

5 In te rv a ll i c Chain of Mipentatony...............................12

6 Solmization Labeling ofMipentatony...........................13

7 Si x four S tr ucture ................................................................13

8 Folksong..................................................................................... 14

9 Overlay of Phrygian and Pentatony...............................15

10 Symmetry..................................................................................... 15

11 Closed Cir cle of Fi ft h s .....................................................16

12 L adimi Tr iad ........................................................................ 17

13 'Ma' Major ................................................................................. 18

14 Domaso T r ia d ........................................................................ 19

15 Re lat ive Chords....................................., .............................19

16 Movement 3, Scherzo I : m. 54 ...................................19

17 Po lar Chords............................................................................ 20

18 Movement 3, Scherzo I : mm. 19 20 ..............................20

19 De riva tio n of 'Ma' Maj or .................................................. 21

20 Combination Chord..................................................................22

vi i




vi i i

CHAPTER ONE (Continued)

FIGURE 21 A lt e ra ti on of Combination Chord to

Produce Polar Chords.......................................................... 23

22 Types of Major and Minor Chords...................................23

23 De riv at ion of Subminor Chord......................................... 24

24 Movement 3, Scherzo I : m. 3 ......................................... 25

25 Movement 3, Scherzo I I : mm. 757 6 .............................25

26 MajorMinorSubminor ...........................................................25

27 Majorminor 7 Becomes Fu ll y Diminished Seventh .26

28 Movement 1: mm. 159 1 60.................................................. 26

29 Resolu tions of a F u lly DiminishedSeventh................27

30 Movement 3, Scherzo I I : m. 17......................................27

31 Replacement Chords............................................................... 28

32 Thir d Tower and Pa rti tio ni ng o fF i f t h s ...................... 29

33 Movement 4: m.63................................................................. 29

34 Movement 3 , Scherzo I : mm. 545 9................■.............30

35 Hyperminor .................................................................................31

36 Movement 3, Scherzo I : m. 19........................................31

37 Movement 3, Scherzo I I : m. 2 ............................ 31

38 Movement 3, Scherzo I : m. 55 ........................................32

39 Movement 3, Scherzo I I : m. 9.........................................

32

40 Movement 3, Scherzo I I : m. 16......................................33

41 Movement 3, Scherzo I I : m. 79.......................................33

42 Movement 3, Scherzo I I : mm. 35 3 6 ..............................34

43 Positive Dir ec tion ................................................................ 35

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,ix


FIGURE 44 Movement 3, Scherzo I : m. 3 ......................................... 35

45 Movement 3, Scherzo I : mm. 545 5 .............................. 35

46 Negative D ire ct io n ................................................................ 36

47 Movement 1: rn. 57 ................................................................ 36

48 Movement 3, Scherzo I I : m. 76 ..................................... 37

49 Movement 3, Scherzo I : m.56 .......................................... 38

50 Hypermajor Produced by Su bs titute Chords..............38

51 Movement Between Sub st itut e Chords............................ 38

52 Movement 3, Scherzo I : mm. 7 8 ...................................39

53 Chord Placed Between Subs ti tu te Chords..................39

54 Movement Between Chords ...................................................40

55 Inward Closing of Chords................................................. 41

56 Outward Re lat ion sh ip of Chords................................... 42

57 Common Notes Between Polar Chords.............................42

58 Chart of Chords Having the Same Function .............. 43

59 Symmetrical Basis of Funct ion ...................................... 44

60 Primary Tr ia ds ....................................................................... 45

61 Secondary T ri ads .................................................................. 46

62 Addit ion of Tr iad s Above Primary Chords................ 46

63 Completion of Circ les o f Minor Th ird s .................... 47

64 Axis System.............................................................................. 48

65 Axis System Based on Bb................................................... 48

66 Ind ivid ual Axes..................................................................... 49

67 Symmetry of Pentatony Around 'R e ' .............................. 50





FIGURE 68 Diagram of Symmetry Placed on a S t a f f .....................50

69 Join ing of Two Pe nt ato nie s ............................................. 51

70 Movement 2: mm. 3536 ..................................................... 51

71 Linking Pen tato nies ............................................................ 51

72 Tonic and Dominant Axes by Join ingPentatonies.52

73 Dominant and Subdominant Axis by

Joining Pentatonies ............................................................ 52

74 Axis System..............................................................................53

75 In te rv a ll ic Relat ionships in a Mipentatony------ 53

76 Numeric Re lat ion ships Within WovenPentatonies.54

77 Joining of P ar al le l and Re lativ e Keys.................... 54

78 Completion of Circ le Using Re lativ e

and Parallel Keys................................................................. 55

79 Comparison of the Two Methods of D eri va ti on ------ 55

80 Three Ind ivid ua l Axes................................... ...................56

81 Axis System as Derived by Rel at iv e

and Parallel Keys................................................................. 57

82 Main and Side Branches of the Axis System.............58

83 Poles and Counterpoles of the Axis System............. 59

84 Po lar Exchange....................................................................... 60

85 Dominant Pole Mm7 Chords................................................. 61

86 1:2 Model .................................................................................. 61

87 Movement 1: mm. 4243 ..................................................... 62

88 Movement 1: m. 103............................................................ 62





FIGURE 89 1:2 Model Constructed from Po lar Chords.................63

90 1:2 Model Constructed from Axis System..................63

91 1:5 Model Constructed from Po les ...............................64

92 Movement 5: mm. 351356................................................65

93 Movement 5: mm. 490492 ................................................65

94 1:5 Model Cons tructed from Axis System..................66

95 1:3 Model................................................................................. 66

96 Ambiguity of To na lit y in 1:3 Model.......................... 67

97 Movement 1: m. 35............................................................. 67

98 Movement 1: mm. 130131 ................................................67

99 1:3 Model Constructed from Axis System..................68

100 Dominant and Tonic Pole s ................................................68

101 Example of a 'C la s s ic a l' Cadence...............................69

102 Movement 3, Scherzo I I : mm. 919 2 ..........................69

103 Example of a Modal Dominant Cadence........................70

104 Movement 2: mm. 9 1 0 .......................................................70

105 Example of a Phrygian Dominant Cadence..................71

106 Movement 1: mm. 1314 .................................................... 71

107 Example of a 'Romantic' Dominant Cadence............. 71

108 Mm7 Chords on Dominant Poles.......................................72

109 Examples of Dominant to Tonic Cadences

Using Varying Qualities of Dominant Chord.............72

110 Examples of Dominant to Tonic Cadences

Using Varying Qualities of Tonic Chords.................73




x i i


FIGURE 111 Inve rted Alpha Chord........................................................... 76

112 Examples of Alpha C h o r d s ., , , , , , . . ...............................76

113 Two Note Alpha Chord........................................................... 78

.114 Six Note Alpha Chord........................................................... 78

115 Three Layer Alpha Chord.....................................................79

116 Movement 5: mm. 686691 .................................................. 80

117 Examples of Beta Chords.................................................. 81

118 Movement.1: m. 56 ........................................................... ^.82

119 Movement 1: m. 135 .............................................................82

120 Symmetry of Gamma Chord.................................................... 83

121 Succession of Gamma Chords..............................................84

122 Examples of Gamma Chords .................................................. 84

123 Movement 3, Scherzo I : m. 32 .......................................85

124 Movement 4: m. 39............................................................... 85

125 Movement 5: mm. 173181 .................................... 86

126 Examples of Delta Chords.................................................. 87

127 Movement 1: m. 48 ................................................................88

128 Movement 4: m. 16 ............................................................... 88

129 Examples of Epsilon Chords ..............................................89

130 Movement 1: mm. 1415 .......................................................90

131 In te rv a ll ic Relationships ofAlpha Chords................91

132 Omega Scale ...............................................................................92

133 Movement 1: m. 73............................................................... 93

134 Fu ll y Diminished Seventh Chord.....................................93




x i i i


FIGURE 135 Sequence of Per fe ct Four ths........................... 94

136 Movement 3, Scherzo I: mm. 4548 ............................... 94

137 Movement 1: m. 81 ............................................................... 94

138 Movement 4: m. 37 ................................................................ 95

139 Movement 5: m. 497..............................................................95

140 Augmented Tri ads ................................................................... 95

CHAPTER TWO (The Quartet) .

FIGURE 141 Formal St ructur e of the Quar te t ................................... 98

142 Axis System of the Quart et ..............................................99

143 Tonal Str uc tur e of the Quartet ...................................... 99

CHAPTER THREE (Movement One)

FIGURE 144 Formal Str uctu re of Movement One............................... 100

145 Tonal Center ofMovement One........................................ 101

146 Measures 1 3 ........................................................................ 101

147 Bb Mi pen ta tony .............. ..... ............................................. 102

148Transformation of Pentaton ies .................................... 102

149 Measures 4 5 ......................................................................... 102

150 Measure 8 ............................................................................... 103

151 Measures 9 1 0 ....................................................................... 103

152 Measures 13 14 ..................................................................... 104

153 Measures 14 20 ..................................................................... 105

154 Diagram of Measures 14 1 6 ............................................. 106




xiv

CHAPTER THREE--Moverrient One (Continued)

FIGURE 155 G Epsilon Chord.................................................................... 106

156 Juxta po siti on of Pole andCounterpole...................... 106

157 Ab Mi pe nta tony .................................................................... 107

158 Cb and F Counter poles....................................................... 107

159 Measure 16 ............................................................................... 108

160 Ab and D Count erpo les....................................................... 108

161 Dominant and SubdominantPoles...................................... 109

162 Measures 17 20 ....................................................................... 109

163 Measures 21 23 .................................................................... 110

164 Inve rted C# Alpha Chord, Measure 21 ...................... I l l

165 Closed System Diagram on 'C ' ..................................... 112

166 Measures 25 29 .................................................................... 113

167 Closed System Based on 'F # 1....................................... 114

168 Measure 36 ............................................................................. 114

169 Measures 37 3 9 ......................................................... 115

170 Measures 44 4 7 .................................................................... 116

171 Measures 445 8 .................................................................... 117

172 Measures 45 48 .................................................................... 119

173 Measures 565 8 .................................................................... 120

174 Phrygian Cadence in Measure 58 .................................. 121

175 Measures 59 62 ..................................................................... 121

176 Measures 63 69 ..................................................................... 122

177 Diagram of E and A# Pe nta tonies ................................ 123

178 Chord of the Fourth , Measure 69 ................................ 123

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



XV

CHAPTER THREEMovement One (Continued)

FIGURE 179 Measures 737 5 ................................................................... 124

180 Measures 869 0 ................................................................... 124

181 Measures 104111 and AxisSystem...............................125

182 Measures 12612 8 ...............................................................126

183 Measures 16 016 5...............................................................128

184 Measures 177181 and Axis System.............................129

185 Measures 20 921 8 ................................................................130

CHAPTER FOUR (Movement Two)

FIGURE 186 Formal Struc tu re of Movement Two.............................131

187 Tonal Axis fo r Movement Two.......................................132

188 Measures 1 4 ........................................................................132

189 Measures 5 9 ........................................................................133

190 Poles and Counterpoles ofMeasures 59 ..................133

191 1:2 Model of Measures 5 9 ............................................134

192 Measures 9 10 ......................................................................134

193 Measures 102 5 ....................................................................135

194 A Mi pe nta ton y....................................................................136

195 Measures 464 9 ....................................................................136

196 Measures 1 9 ..............................................................., . . . .137

197 Measures 263 0 ....................................................................138

198 Measure 27, Motive 'X ' .................................................. 139

199 Measures 313 4 ....................................................................140

200 Measures 35 4 1 ....................................................................141




xvi

CHAPTER FOURMovement Two (Continued)

FIGURE 201 In te rli nk ed Pent aton ies .................................................. 142

202 Measures 4041 ..................................................................... 143

203 Diagram of Measures 404 1 ..............................................143

204 Measures 434 6 ..................................................................... 144

205 Measures 52 56 ..................................................................... 145

CHAPTER FIVE (Movement Three)

FIGURE 206 Formal Str uctu re of Movement Three.......................... 147

207 Tonal Axis fo r Movement Three.....................................147

208 Measures 1 6 ..........................................................................148

209 Measures 7 8 ..........................................................................149

210 Diagram of Movement Between Chords

in Measures 7 8 .................................................................. 149

211 Measures 5 7 ..........................................................................150

212 Comparison of G# Mi pentatony and

EM, c#m and a#dm7............................................................150

213 Measures 13 and 16 .............................................................151

214 Measure 14.............................................................................151

215 Diagram of Measure 14 Progress ion.............................152

216 Measure 18.............................................................................152

217 Diagram of Measure 18 Progress ion............................153

218 Measure 17 .............................................................................153

219 Measure 19 .............................................................................154

220 Measures 141 9 ..................................................................... 155




xv i i

CHAPTER FIVE—Movement Three (Continued)

FIGURE 221 Measure 2728 ........................................................................ 156

222 Measure 29 ................................................................ 156

223 Diagram of Measure 29 Pr og ress ion.............................157

224 Measure 29 (V io la L in e ) ................................................... 157

225 Diagram of Measure 29 Vio laLine Progression..158

226 Continuation of Figure 20Progression ................... .158

227 Measure 30 ............................................................................. 159

228 Measures .4547 .................................................................... 159i

229 (A) Secti on Tonal Ce nt er ............................................... 160

230 Measure 52 ............................................................................. 160

231 Measures 54 5 9 .................................................................... 161

232 Measures 64 6 6 .................................................................... 162

233 Tonal Center o f the T r io .............................................. 163

234 Measure 1 and So lmiza tionSy l lables ......................... 163

235 Measure 15 ............................................................................. 164

236 Measures 21 22 .................................................................... 164

237 Measure 24 and So lmizati onS y ll ab le s .......................165

238 Measures 41 43 .................................................................... 166

239 Measure 44 ............................................................................. 167

240 Measure 52 ............................................................................. 168

241 Golden Secti on Diagram................................................... 168

242 Measures 44 6 5 .................................................................... 169

243 Pitch C olle ctio n of Climax Area ................................ 171




xvi i i

CHAPTER FIVE—Movement Three (Continued)

FIGURE 244 E Mipen ta to ny .................................................................... 171

245 Measure 1 ................................................................................. 172

246 Measure 2 ................................................................................. 173

247 Measures .1 9 28 .................................................................... 173

248 Tonic Axis : EGA#C#..................................................... 174

249 Measures 19 25, V io lin 1 ................................................. 175

250 Measures 19 22 , Ce llo ....................................................... 175

251 Measures .1922, Vio lin I I and V io la ........................ 176

252 Tonic Axis: C#EA#......................................................... 176

253 Re la tiv e and Modall y Related Chords of emm7...177

254 Diagram of Polar Exchange............................................ 177

255 Measures 2328.....................................................................178

256 Diagram of Vio la Progres sion ...................................... 179

257 Measures 44 and 51 ............................................................ 180

258 Measure 49 ................................................................... 180

259 Measures 58 6 6 ....................................................................... 181

260 Measures 91 9 2 ................................ 182

CHAPTER SIX (Movement Four)

FIGURE 261 Formal Str uctu re of Movement Four ........................... 183

262 Tonal Axis of Movement Four ................... 184

263 Movement 2: Measures 1 4 ............................................... 185

264 Movement 4: Measures 1 5 ............................................... 185

265 Measures 11 8, C ello L ine ............................................... 186




x i x

CHAPTER SIXMovement Four (Continued)

FIGURE 266 Measure 16 , G De lta Chord.............................................. 187

267 Measures 23 32 ...................................................................... 187

268 A Mi pen tatony Diagram.................................................... 188

269 Measures 29 and 35 ............................................................. 188

270 Diagram of Measures 29 and 35 ...................................... 189

271 Measure 37 ................................................................................ 189

272 Measures 39 4 0 ...................................................................... 189

273 Measure 42, Motive X . . ..................................................... 190

274 Measures 44 4 5 ...................................................................... 190

275 Measures 4 3 4 4 ., .................................................................. 191

276 Measures 4445 and Diagram........................................ . .192

277 Measures 60 6 3 ...................................................................... 193

278 Measure 63 ............................................................................... 193

279 Measures 67 6 8 ...................................................................... 194

280 Measures 65 and 676 8 ....................................................... 195

281 Diagram of Measures 65 and6768 ................................. 196

282 Measures 899 1 ...................................................................... 197

283 Measures 100101 ..................................................... 197

284 Measures 81 101 .................................................................... 198

CHAPTER SEVEN (Movement Five)

FIGURE 285 Formal St ru ct ur e of Movement F iv e ............................. 200

286 Tonal Center fo r Movement Fi ve .................. 201

287 Measures 172181 .................................................................. 202

roduced with permission of the copyright owner. Further reproduction p rohibited without permission.



XX

CHAPTER SEVENMovement Five (Continued)

FIGURE 288 Measures 184188............................................................ 203

289 Measures 184188 Diagram.....................................203

290 Measures 196200...................................................... 204

291 Measures 275291...................................................... 205

292 Measures 334345 ......................................................206

293 Re lat ive Solmization of Measures 334 34 5.207

294 Measures 351 356...................................................... 208

295 Measures ,35 93 64 ..................................................................208

296 Measures 457, 473475 and 47 6 47 7 ............................ 209

297 Measure 469............................................................................ 210

298 Measure 461............................................................................ 210

299 Measures 465 , 467and 471.....................................210

•300 Measures 490 49 2,501503 and 509511 ............211

301 Measures 497500...................................................... 212

302 Measures 507512 ............................................ 212

303 Measures 624635 ...................................................... 213

304 Measures 6516 60 .......................................................214

305 Measures 6736 85 .......................................................215

306 Measures 6866 91 .......................................................216

307 Measures 6926 98 .......................................................217

308 Measures 6997 20 ........................ ......................................218

309 Measures 8258 28 ..................................................................219






xx i i

whose roots are a tr it o n e ap ar t. Through the use of polar exchange,

i t is possible to s h if t the tonal center by six key signatu res, yet

never a lt e r the function of the two po lar ly re late d chords.

The analysis portion of this paper is designed to give a

s tr u c tu ra l, tonal and harmonic overview of each movement, giving

pa rtic ula r atten tion to three areas: pentatony; re la tiv e , modally

re la te d and su bs tit ut e chord harmonies; alpha harmonies. These areas

assume varying degrees of importance depending on the particular

movement.

The theories of Lendvai are too new and untried to place

them into any kind of perspective at th is time. Lendvai's own wri ting s

are concerned more with a few specific pieces of Bartok's works which

conform n ea tly to golden sec tion p rin ci pl es , cl ea r cut use of models

(1 :2 , 1 :3 , 1 :5 ), or alpha harmonies. His wr itin gs avoid those portions

of Bartok's music which defy explanation using this methodology.




INTRODUCTION

NEED FOR STUDY

The F if th St rin g Quarte t of Bela Bartok has long been neglected

by music the or ists because of the d if fi c u lt ie s i t presents when ana-

lyzed by tra d iti o n a l methods of analysis . Although freq uen tly cited

fo r i ts palin dromic st ru ctu re* no auth or has undertaken a thorough9

harmonic an aly sis . A no nt rad itio na l approach is ther efor e necessary

to f i l l this void le f t by t rad i t iona l harmonic analysis .

PURPOSE OF THE STUDY

The purpose of this dissertation is to present the nontradi-

tio na l th eo re tic al techniques of Erno Lendvai and introduce the a p p li-

cation o f these techniques in a deta iled analysis o f the F ift h String

Quartet of Bela Bartok.

METHOD OF ANALYSIS

The method of a na lys is used in t h is paper is based on the theo -

re ti c a l/a n a ly ti c a l techniques of Erno Lendvai. Lendvai (b. 1925), a

Hungarian m usico logis t, has spent the la s t t h ir ty years analyzing and

cod ifying the compositional techniques of Bela Bartok. Un til re ce nt ly ,

*Paul G r if fi th s , Barto k, The Master Musicians Series (London: J.M. Dent & Sons, L td ., 1984).

George P er le , "The St rin g Qua rtets o f Bela Bartok, " A Musical Offe rin g" Essays in Honor of Martin Bernste in (New York: PendragonPress, 1977). ^

Halsey Steven, The Life and Music of Bela Bartok (New York:

Oxford University Press, 1964).

1




the m aj or ity of Lendv ai's w ritin gs were in Hungarian and lim ite d to

discussions of the golden sec tion . In 1983, his wri tin gs were compiled / / 9

in a sin gle 762 page volume, The Workshop of Bartok and Kodaly .

The f i rs t port ion of th is d isser ta tion is a d is t i l la t io n of

Lendvai's the or ie s, showing the or ig ins and components. I t also

introduces the application of these theories to the Fifth String

Qua rtet. This condensation of the the or et ic al techniques contains a

minimum of Lendvai's h igh ly sub jectiv e approach. I t is the au tho r's

int en t to present only the most objec tive parts of the theory.

METHOD OF PRESENTATION

Lendvai's theoretical principles and introduction of the appli-

cation of these theories to the Fifth String Quartet are shown in the

f i r s t part of the dis se rta tio n. The remaining portion of the paper

is a detailed analysis of the Quartet based on these theoretical

pr inc ip les .

St ructu ra l and tonal chart s fo r the Quartet and each movement

are included in the analys is section. I t is not the inte nt of thi s

paper to pursue compositional techniques not addressed within Lendvai's

theoretical framework, such as canon, stretto and inversion.

BACKGROUND

The F if th Str ing Qu artet of Bela Bartok, w rit te n in 1934 in

less than a month, was commissioned by the Elisabeth SpragueCoolidge

Êrno Lendvai, The Workshop of Bartok and Kodaly (Budapest:

Editio Musica, 1983).




Foundation wi th it s premiere performance by the Kolisch Quartet in

Washington D.C. Ap ri l 8 , 1935. I t was not premiered in his homeland

u n ti l March 3 , 1936 when i t was performed in Budapest by the New

Hungarian String Quartet.

The five year period between the Piano Concerto No. 2 (written

in 1930/32) and the Fifth String Quartet was a time of frustration

and p o li ti c a l upheaval in Bartbk 's l i f e . His compositional output

decreased during these five years as his attention became focussed

on European p o li ti c s and d if fi c u lt ie s in ge ttin g his music performed.

In 1931, Bartok was honored with both the medal of the Legion

d'honnear and the Corvin Medal (a high Hungarian award before 1 945),

though he did not at tend the ceremony which was to honor him. Per-

formances of the Wooden Prince, Bluebeard's Castle and the Budapest

premiere of The Miraculous Mandarin had been planned, but, like other

performances scheduled during this same period, never realized.

By 1932, his mood was melancholy:

I have no contra cts f o r concerts at a ll th is season," he

wrote. " I f thi s continu es, wi thin eighteen months I shall have

to move to a tworoom f l a t and economize to the highes t degree

.. .T he most famous conductors show no in te re s t a t a l l in my

works...As long as they are playing all kinds of trash and not a single work lik e th is , i t is a waste of energy to attempt any

promotion of my orch estra l wo rks ... 3

oGyorgy Kroo, A Guide to Bartok , T ra ns la ted by Ruth Pataki and Maria Ste in er , tr an sla tio n revised by Elisabe th West (By the

aut ho r, 1974) [O rig ina l t i t l e : Bart6k Kalauz. Budapest: Zenemukiado,

1971], p. 173.




A statement w ritte n in August 1933 is also pa rt ic u la rly re le -

vant: " . . . I too have los t al l desire to make any kind of an off erin g

to audiences which show the minimum possible interest in my works.

The p o li ti c a l sit u at io n in Europe was worsening. Stron gly

opposed to the onslaught of the Third Reich, Bartok's letters are

sa rca stic and rep ud iato ry. An example of his at tit u d e would be the

l a t t e r sent in response to a request in April 1934 fo r proof of his

aryan orig in s. "I would notdream of sending ce rt if ic a te s of baptism

5to Germany, not even i f I hadthe documents in my hand..."

Through a ll of th is , Bartok continued serving at the Budapest

Academy of Music. In autumn o f 1934, he was f i n a l ly re lie ve d of

his teaching duties and commissioned by the Hungarian Academy of

Sciences to prepare his collection of folksongs for publication.

This folksong co lle ct io n , begun in 1913, contained some 13,000 en tri es .

After the performance of the Second Piano Concerto (23 January

19 33 ), Bartok never p layed in Germany and in 1937 he denied r ig hts

to broadcast his music in both Germany and I t a ly . In 1939, he shipped

his papers and manuscripts to London. A ft e r his mother's death in

December of 1939, h is l a s t t i e to Budapest was broken and Bartok

se rio us ly considered emigration to the United Sta tes. By the end

of 1940, Bartok and his wi fe had l e f t Hungary and emigrated t o America.

From 1930 to 1935, Bar tok composed various minor works, in -

cluding the Szekely Songs for male choir, a work which preceded his

4 Ibid.

5Ib id . , p. 174.




Two and Th re e Par t Choruses composed in 1935. There were also orches-

tr a l pie ces , and the vi o lin and piano pieces fo r young people. 193133

were the years of transcription (Transylvanian Dances, Hungarian

Pic tu re s, Hungarian Peasant Songs and Hungarian Fol kso ngs ). The

Fo rty Four Duos and Mikrokosmos were also w ri tt e n during t h is time.

These works which so intimately involved the use of folksong and

the folksong idiom prepared the way for the Fifth String Quartet.

Reception of the F ifth String Quartet was enth usias tic.

Antal Mo lnar "the inventor of a new sty le can also create its f ir s t

great cla ss ic." Referring to the "more relaxed , f i l t e re d and classic

nature" of the Quartet at the Hungarian premiere (March 3, 1936),

Sandor Jem nitz ~" the calm ease of clas si c sages." Ernst Krenek,

the Au stria n Composer, spoke of "the marvellous balance of s p ir it

and material."®

Structural ly, the Fifth String Quartet is modelled after

the Fourth Str in g Quartet and the Second Piano Concerto . Five move-

ments are arranged in pal indromic o rde r. Movement th re e, a scherzo,

is at the center of the bridge str uc tu re . Movement four re fle c ts

the motives of movement two and movement fi v e re fl ec ts the motives

of movement one. The overall stru cture of the Quartet is A B C B1 A '.

A fte r the F ift h String Qu artet, Bartok abandoned the in tr ic a te

arc hite ctu re of a palindrome. The only exceptions are its lim ited

use in the Concerto for Orchestra (1943/rev. 1945) and Piano Concerto

No. 3 (1945).

6 Ib id . , p. 175.

produced with permission of the copyright owner. Further reproduction p rohibited without permission.



CHAPTER ONE

LENDVAI'S THEORETICAL PRINCIPLES

Noted Bartok scholar Erno Lendvai , over a t h ir t y year span,

has developed a system of theoretical principles, in an attempt to

describe and c od ify the harmonic language of Bela Barto k's music.

Lendvai considers the music of Bartok to have descended from that

of Verdi and Wagner. Because of th is d ir e c t link age between la te

romantic music and Bartok, i t is important to understand ce rta in

aspects of romant ic harmonic language as Lendvai perceives them to

applyto the music of Bartok. This sec tion w il l address these aspects

of the romantic harmonic language as they app ly to Lendvai 's th eo re ti -

cal concepts. The major concern of th is se ct io n, however, is to

as si st in the understanding of how Lendvai's t he or ies may be applie d

to Bartok's F ifth String Quartet. This portion w il l present a dis-

cussion of the theory's derivations, various components, and intro-

duce its application to the Fifth String Quartet.

Fibonacci Series

At the base of Lendvai's the ory is the Fibonacci Series.

By using the numbers th at are generated by th is Se rie s, Lendvai

constructs the pentatonic system which lies at the heart of his theory.

To show this important link between the Fibonacci Series and Lendvai's

theor y i t w il l be necessary to discuss the Fibonacci Series and how

i t applies to pentatony.

6




The Fibonacci Series is an additive series of positive integers

in which each number is the sum of the preceding two in te ge rs . The

series begins wit h the in teg er 1 , which, when added to i t s e l f , produces

2. By adding 1 and 2 to ge th er , the next number in the se ries, 3,

is generated. By continuing th is process, the follow ing series can

be constructed: 1,2 ,3,5 ,8,1 3,2 1,3 4,5 5,8 9,1 44 . . .

The Fibonacci Series i s si g ni fic an t fo r being the "simplest

golden section series which can be expressed in whole numbers."

(Lendvai 1983, p. 45) The golden section is a geometrical pro por tion ,

o ri g in a ll y ca lle d the "golden rule" by Leonardo da Vi nc i. (Lendvai

1983, p. 33) This p ar ti cu la r proportion re fe rs to a geometric whole

divided into two unequal parts and is considered by Lendvai to be the

most a es th e ti c al ly pleasing d ivis ion o f a whole. The golden section

is obtained when the proportion of the whole to the larger part corres-

ponds with the prop ortion of the lar ge r pa rt to the smaller pa rt.

In o ther words, i f the whole distance is equal to the value of 1,

the la rg er sect ion (shown as X in the diagram below) w i ll be approx-

imate ly 0.618 [X = (tpT - 1) / 2] (rounded off from the irrational

number 0.6 1 80 33 9. ..) and the smaller section (1 X ), 0.382. This

is i l lustrated in Figure 1.

X ■ l-X ■ . 0.618 . 0.382 .

1:X = X:(1-X)

x =rr 1T~

X = 0.618

1-X = 0.382

Figure 1: Golden Section




By taking any number from the Fibonacci Series and multiplying

i t by 0.6 18 , one w il l a rr iv e a t, or near, the number preceding it

in the series . For example, i f from the series 1 ,2 ,3 ,5 ,8 ,1 3 ,2 1 .. .

the number 13 is chosen and multiplied by 0.618, the product 8.0

is obta ined. By s im il a r process: 5 x 0.618 = 3.09 and 144 x 0.618

= 88 .99. Each number then has two func tio ns . I t is the la rg er number

when in relation to the previous integer, and is the smaller number

when in re la tio n to the int eg er which follows i t .

In addition to the golden section, Lendvai uses Fibonacci

numbers to label sp ec ifi c h al f st ep gradations of the twelve tone

system based on the integ ers o f the numeric ser ie s. This is shown

in Figure 2.

1 = m2 (minor second)

2 = M2 (major second)

3 = m3 (minor th i rd )

5 = P4 (perfect fourth)

8 = m6 (minor s ix th )

Figure 2: Fibonacci Labels of Ha lfs tep Gradations

Closed System

Lendvai uses Fibonacci numbers to create what he calls a

'closed sys tem'. A closed system, as he defines i t , is any sequentia l

chain of in ter va ls which returns to its s tart ing pitc h. The chromatic

1 2 3 5 8




scale may be considered a closed system as its sequential chain of

half steps (minor seconds) returns the system to its starting pitch.

A p a rt ic u la r closed system which Lendvai considers im portant

is a sequential chain of 3 + 3 + 2 (h a lf s te p s ). This sequential

chain of a minor th ir d ( 3 ) , minor th ir d (3 ) , major second (2) re turns

to its starting pitch when repeated three times, thus forming a closed

system. The in t e r v a ll ic stru ctu re is shown in Figure 3.

Figure 3: Closed System

Further discussion of the closed system may be seen in the

analysis of the second movement using Lendvai's theoretical techniques.

me




10

Pentatony

Lendvai makes use of a particular system of labeling notes with

solmization syllables based on the method established by Kodaly.

In this method, which employs a movable do, each major scale is made

up of d o re m ifa s o la t i d o, and each minor scale consists of l a t i

doremi fa so la .

For example, in the key of C major or it s r e la tiv e minor,

'a ' minor, the foll ow ing notes are always associated with these sy ll a -

bles:

c = do

d = re

e = mi

f = fa

g = so

a = la

b = t i

Altered scale degrees are described as follows:

do raised 1/2 step is di

fa raised 1/2 step is fi

so raised 1/2 step is si, etc.

mi lowered 1/2 step is ma

ti lowered 1/2 step is ta, etc.

The diagram below shows the chromatic scale fo r C ma jor /

'a ' minor. In the key of C major, c# would be di and f# would be




11

f i , eb would be ma, b^ would be ta .

do di re ma mi fa f i so si la ta t i do

Figure 4: Chromatic Scale

This method of movable do is known as re la t iv e so lmiza tio n.

Relative solmization also gives each note/syllable a distinct character

and meaning. (Lendvai 1983, p. 93) Lendvai's choice of meaning

is purel y sub jecti ve onhis pa rt . References made to meaning in

thi s paper w il l be in sin gle quotation marks to set them apart as

Lendva i's choice of words. This w il l become important in the section

dealing with harmony.

Lendvai discusses thre e important concepts in re la tio ns hi p

to pentatony. These are: mipentatony , si x fo ur str uc tur e and Phry-

gian influences.

Lendvai link s many of his theoretical perceptions together

with the concept o f pen tatony , a systemwith sources in tra di tio na l

Eastern folk music which has melodic, as opposed to harmonic, origins.

Thepen tatonic sca le is made up of do, re , mi , so and l a degrees.

Using each of these sy lla bl es as the primary note of importance,

as a 'keynote', five pentatonic scales can be constructed:

do pentatony: do re mi so la ( e .g . c d e g a )

re pentatony: re mi so la do (e .g . d e g a c )




12

mi pentatony: mi so la do re (e .g . e g a c d )

so pentatony: so la do re mi (e .g . g a c d e )

la pentatony: la do re mi so (e .g . a c d e g)

Any of the pentatonic scales, including mipentatony, can

be generated on any pitch; they are not limited to 'C'.

Lendvai gives greatest importance to the one form of pentatony

which he feels represents the basic scale of Bartok's pentatonic

st yle , the mipentatonic scale.

Mipentatony

He considers mipenta tony to a lso be the musical f ru it io n

of the Fibonacci Se ries . Mipentaton y is derived using the Fibonacci

in te ge rs 2 and 3 in a re pea tin g sequence: 2 + 3 + 2 + 3 + 2 t o cre ate

a chain of in te rv a ll i c relatio nsh ips. For example, egac d as

seen in Figure 5.

Figure 5: In te rv a ll ic Chain of Mipentatony

Solmization syllables are used to label this intervall ic

chain . Lendvai does not consider the syl la b le 'do' as a ton ic in

his pen tato nic system, as in Western theo ry , ins tea d, mi is the most

important note in mipentatony as re would be in repentatony, so

in sopentatony, etc.




Mipentatony is a particular arrangement of intervals and

solmizatio n lab eli ng . This is shown in Figure 6.

Lendvai defines one of the primary distinguishing scalar

ch ara cte risti cs of mipentatony as the descending minor th ird : somi.

Since mi is the most impor tant note and is used to id e n ti fy and labe l

this pa rti cu la r pentatony, the minor th ird int erv al between so and

mi i s oft en the fac to r which determines a pentatony. Lendvai also

bases this on the frequency of the somi minor third interval in

the tr a d iti o n al eastern European fo lk idiom which influen ced the

writ ings of Bartok.

Sixfour Structure

Unl ike fun cti on al harmony which tr ea ts a tr ia d in second

inversio n as an unstable emb ellishing chord, pentatony accepts this

chord as h iera rc hic al ly equivalent to the root posit ion tr i a d . While

common pra ct ic e harmony tr e a ts the s ix f o u r as a dissonance which

requires res olu tion , the modally derived sixf ou r is a consonant

chord, often the resolution of a preceding dissonance.

Figure 6: Solm ization Labeling of Mipentatony

mi la do (« )

Figure 7: Six fo ur Structure




14

The k eynote of the mi system s ix f o u r shown above is c.

The pe rfec t fourth int erv al from (c f ) is the consonant in te rv al .

On the other hand, the p erfect f i f t h , ty p ifi e d by scale degree mi

to t i in th e penta tonic system, is considered a hig hly dissonant

in te rv a l. The fo llo wi ng example is a folksong from Lend vai's Workshop,

p. 12 showing a melody in 6/4 structure:

J. I|J. J J i l j pl a m i ( J )

Figure 8: Folksong

Having the P4 as an important structural interval establishes

the descending l am i plag al cadence as the eq ui va lent of the common

pra ctic e sodo au then tic cadence. This lami cadence is drawn d ir e c tl y

from the six fo u r struc ture . Melodies of mipentatony auto matically

assume a sixfour structure by the inherent properties of pentatony,

hence the s t a b il it y o f a lami cadence.

Phrygian

According to Lendvai, mipentatony can be considered a subset

of the phryg ian sca le on E. The fi gure below shows an ov er lay of

the mi pe nta ton y and phrygian sca le. Notes in parentheses are commonly

used within the pentatonic system, although not part of the skeletal

pentatonic forma tion. These notes co ntr ibu te to the phrygian co lor ing ,

esp ecia lly the fami ha lfs tep con figuratio n which normally denotes

a phrygian scale.




15

e f g a b c d e

mi (f a ) so la ( t i ) do re mi

Figure 9: Overlay of Phrygian and Pentatony

Fur ther examples of phrygian influ enc es w i ll be seen in the

discussion of dominant cadences.

Symmetry

Lendvai determines symmetry by arranging the other notes

of the pentatony around a cen tral pi tc h. This pit ch is always re

in a mipentatony as the other notes can be arranged in te rv al 1ic a ll y

around i t to produce a symmetrical co nf igur at io n.

so »■ do | re | mi «■ la

'■F4 “ M2 " M2 '“ " P4~‘

Figure 10: Symmetry

This symmetry is p art of Lendvai's ra tio n al iz a tio n of the

lami cadence, in particular, being the equivalent to the sodo cadence

of Western theo ry. His explanation is sub jec tiv e and is one of the

components of his the ory which must be accepted a t face value . The

symmetry of do and mi in r e la ti o n to the c en tra l re w il l be impor tant

in the discussion of Lendvai's theo ry of harmony. This prop erty

of symmetry w i ll be discussed in subsequent sect ions .




16

Harmony

Relative Solmization

The chromatic language of the romantic p erio d is considered

by Lendvaito be a 'closed system', based on the cir c le of fi ft h s .

As i l lu st ra te d below, the cir cl e of f i ft h s is a selfconta ined system

which always returns to it s st ar tin g po int. Enharmonic res pe llin g

of keys is necessary to form th is kind of closed system. Because

of th is , i t is "impossible to speak of f ixed points or 'progress',"

(Lendvai 1983, p. 44)

Figure 11: Closed Ci rc le of Fi fth s

For th is reason, Lendvai in si st s the l a te works of Verdi

and Wagner remain impregnable to traditional analytical approaches.

Lendvai, therefore, discards the figured bass of the diatonic system,

suggesting that romantic music is best approached by a system which

compares any given chord to the chord fo llow in g i t . He compares

these chord re la tio ns hi ps by determ ining, what he c a ll s , th e ir modal




17

ten sion. Modal tension is based on how fa r the next chord is 'l o g i-

c a l ly removed' from the preceding chord . (See "Movement between

chords" p. 32) 'Meaning' is su bj ec tiv ely assigned to a chord based

on this modal tension.

According to Lendvai, by using Kodaly's didactic method of

re la tiv e solmization i t is possible to make la te romantic music,

which he feels defies traditional analytical approaches, comprehensi-

ble . Re lat ive sol miz atio n, as mentioned be fo re, i s a system whereby

solmization labels for individual notes are altered to reflect not only

chromatic a lte ra tio n s, but also to associate with those labels d if fe r-

ent charac ters and meanings. Each syl la b le may be al te re d . Although

i t is th eo re tic al ly possible to do th is , Lendvai l im its such a lte ra -

tions to specific syllables which he feels to be more important than

othe rs, namely do to d i , mi to ma, fa to f i and t i to ta .

Using the process of re la tiv e solm izatio n, i t is possible

to realize the character distinctions Lendvai makes between chords.

I f C major is considered to be a domiso chord and ' a' minor (the

re la tiv e minor of C major) a ladomi chord, then by rais ing do to

d i , A major is a la di m i chord. Relat ive solmization distinguishes

between the character of a domiso triad and that of a ladimi

t r i a d .

. r r r 3= f l 3 3 = =

do mi so la do mi

pdLS----------- L

la di mi

Figure 12: Ladimi Tria d




18

Both are major tr ia d s , but because of the raised d i, the

la di m i is said to have a 'b rig h te r' chara cter. Also, by rais ing

d i, a s h if t from C major (no sharps) to A major (3 sharps) has resu lted

in a key sig na ture change of three sharps. (Lendvai uses the terms

chord and key in terc han gea bly when the chord repres ents the t on ic

of the key, so i t is possible to speak of 'chords' sh ift in g in key

s ignature ) .

This canbe contrasted with the alt er in g of the other impor-

tan t degree, me. Again, C major (no f la ts ) is a domiso major tr ia d .

I f , instead of rai sin g do, mi is lowered to ma (and t i lowered to

ta ) Eb major (3 fl a ts ) is produced (a masota major tr ia d or 'ma'

major) .Ma chords are said to have a 'dar ke r' tone than e it h e r domi

so or ladomi chords. A key signatu re change of three fl a t s is

always implied.

According to Lendvai, the same process holds true for minor

chords. An 'a ' minor chord (la d o m i) is transformed in to c minor

by lowe ring mi to ma (do maso ). Th ere fore, the domaso minor tr ia d

is considered to have a 'darker' tone than the ladomi triad.

do mi so ma so ta

Figure 13: 'Ma' Major




19

rJTJ ,ft77^=

la do mi do ma so

am cm

Figure 14: Domaso Tri ad

The chords in Figures. 15 and 16 are re la t iv e chords (keys)

bearing the same key signature.

do mi so la do mi

CM am

Figure 15: Re lat ive Chords

CMM7 = major

amm7 = re la ti v e minor

Figure 16: Movement 3, Scherzo I : m. 54




20

Since raising do to di results in a key change of three sharps,

and the lowering of mi to ma changes the key by three f l a t s , i t th er e-

fo re follows t h a t a simultaneous a lt e ri n g of do to di and mi to ma

w il l re su lt in a key change of six key signa tures .

Figure 17 diagrams the a lt e ra ti o n of do to di which produces

a domaso tr i a d , and the a lt e ra ti o n of mi to ma which produces a

la d im i tr ia d . Figure 18 is a musical example of the procedure.

do mi so do ma so

la do mi la di mi

Figure 17: Po la r Chords

do ma so la di mi

Figure 18: Movement 3, Scherzo I: mm. 1920




21

As seen in F igure 17, by the a lt e ra ti o n o f do to di and mi

to, ma, c minor and A major are derived from C major and 'a ' minor.

These two chords (cm and AM) have a d if fe re nce o f si x key signatures

(three f la ts of c minor and three sharps of A ma jor). This re la tio n -

ship of six key signatures difference is cal led 'po lar ' or 'po lar

te nsi on '. This concept assumes greated importance in the discussion

of the axis system.

The examples so fa r have shown on ly the a lt e r in g of one or

two notes to construct a di ffe rin g q ua lity of chord. Solmization

can also r e fl e c t a g rea ter change in chord stru ctu re by al te rin g

an important note and implying the alteration of the other chord

members ac co rd ingl y. This is diagrammed in Fig ure 19.

i t

Ab = do

— es

AbM

I S —- . —CbM = 'ma ma jo r'

Figure 19: Der ivatio n of 'Ma' Major

In Figure 19, Ab is 'd o ', since c is mi. By a lt er in g mi

to ma, the char ac te r of Ab major has been changed and is now Cb major

or 'ma m ajo r'. The a lt e ra tio n of mi to ma could have ju s t as ea si ly

yie lde d the chord ab mino r, however, the poi nt here was not to a lt e r




22

an exis ting chord, but rather to move to an e n ti re ly d if fe re n t one

in the course of such a lt e ra ti o n s . Sh ift in g from Ab major to ab

minor is perceived as a modal change, whereas Ab major moving to

Cb major could be seen as a type of modulation or transformation.

Yet another aspect of re la t iv e solm ization can be seen below.

Rel at iv e chords may be combined, then the new combination chord a l -

tere d. For example, i f the re la tiv e keys of C major and 'a' minor

are added tog eth er, an 'a ' minorminor seventh chord re su lt s . This

is shown in Figure 20.

.Q— s p

CM + am = amm7

Figure 20: Combination Chord

By the simultaneous altering of do to di and mi to ma, the

po lar chords o f A majorminor 7 and Eb majo rminor 7 are produced.

I t is important to note tha t chords whose roots are a tr it o n e apa rt

au tom ati cal ly assume a pol ar rela tio ns hi p to one another. This is di a -

grammed in Figure 21.

roduced with permission of the copyright owner. Further reproduction prohibited witho ut permission.



23

ma

ll''n \ud i a r a n 7 A M m 7 E b M m 7

Polar Chords

Figu re 21: A lt er at io n o f Combination Chord to Produce Pol ar Chords

To summarize, Lendvai uses relative solmization to form three

types each of major and minor chords and th e ir ass ociated dark or

l ight characters:

1 ;

Majo r chords la di mi do mi so ma so ta

Mino r chords f i la di la do mi do ma so

Light ........................................................... Dark

Figure 22: types o f Major and Minor Chords

I t is possib le to generate these major and minor chords,

and pol ar r el at ed chords by the mani pula tion o f di and ma (the use

of re la ti v e so lmizatio n to deriv e chords). However, Lendvai does

not discuss this alteration to derive new key signatures as a method

fo r modulation. The manipu lation of chords is fo r the purpose of

assuming new 'ch ara ct er s' to r e f le c t the meaning of the music. I t

is th is rel at io ns hi p between the various characters which creates




24

the tension between chords. Lendvai makes tr a d it io n a l, func tional

harmony subservient to the 'te ns io n' of the music. (Lendvai 1983,

pp. 97122)

In addition to the standard distinction made between major

and minor, romantic harmony inc ludes a th ir d chord of importance,

the h alf di mi nis hed seventh chord. Since Lendvai considers the music

of Bartok to be di re c tl y link ed to the harmonic language of Verdi

and Wagner, i t is important to understand ce rtai n aspects of the

romantic harmonic language as i t applie s to Bar tok.

Lendvai derives the halfdiminished seventh, which he calls

the 'subminor', by combining the minor tr ia d (lad om i) and it s re la -

t iv e minor chord (f i la d i) b u ilt a minor th ird below the ( ladomi)

minor tr ia d (e .g . CMamf#m). The subminor is considered the re la -

ti v e chord of a minor tr ia d in the same way the minor is the re la ti v e

of a major triad.

MajorMi norSubmi nor

CM

major

am + f#m f#dm7

subminor re l a t i v e

minor

re l a t i v e

minor of

mi nor

Figure 23: De riv at ion of Subminor Chord




25

u "imT!

c#mm7 = minor

a#dm7 = subminor

Figure 24: Movement 3 , Scherzo I : m. 3

'm ino r *■ subminor minor major

Figure 25: Movement 3, Scherzo I I : mm. 7576

When placed next to one anot her , the re la tio ns hi p between

these three chordsmajor, minor and subminorbecomes apparent:

CM amm7 f#dm7

Figure 26: MajorMinorSubminor

Because the subminor is based on f i , a tr it o n e away from

do, the major and subminor chords stand in polar relationship to

one ano ther . Majo r, minor and subminor chords are a ll the re s u lt

of relat ive solmization techniques.




26

Fully Diminished Seventh Chords

Fully diminished sevenths are considered, not as incomplete

dominant ninths, a traditional concept in functional harmony, but

as alter ed majorminor sevenths. Domisota is alte red by rais ing

do to d i , making i t di m is o ta , a fu ll y diminished seventh chord.

The sy lla b le do is the root of the chord. Fu lly diminished seventh

chords are a result of relative solmization techniques.

Mm7 •> °7

do *• di

Figure 27: Majo rminor 7 Becomes Fu ll y Diminished Seventh

BbMm7

d o ------------------------ > di

Figure 28: Movement 1: mm. 159160

Because o f the unique p rop erty o f symmetry in a f u l l y dim in-

ished seventh, any of the notes can, in turn, be altered, resolving

to eith er the origin al domisota, or yet a d iffe re nt majorminor

seventh. This is diagrammed in Figure 29.

A -e




c # ° 7 A M m 7 F #M m 7 E b M m 7 C M m 7

Figu re 29: Resolutions of a F u ll y Diminished Seventh

I »Jl>

1

bb a

9 ♦ 9e *• e

c# c#

Fig ure 30: Movement 3, Scherzo I I : m. 17

Fu lly diminished sevenths are fr eq ue nt ly used as d ir ec t re-

placement chords fo r th e ir majorminor cou nte rpa rts: (Example from

Lendvai 1983, p. 163)






29

Hypermajor

Figure 32: Third Tower and P ar tit io ni n g of Fifths

I t is sig nific an t th at a series of perfect f i ft h s can be

pa rtitio ne d from this tower: the per fect f if t h i nte rva l being con-

sidered highly dissonant in qu alit y. The use of this f i ft h p a rt i-

tioning is seen in the passage shown in Figure 33.

Figure 33: Movement 4: m. 63

The four lowest notes of this third tower (CEGB) are what

form the hypermajor chord, a majormajor seventh chord quality.

It s charac ter i s one of 's ol em ni ty '. Below are examples of hypermajor

chords from a passage in the th ird movement.




30

y |S:

m..ujzrmn

Figure 34: Movement 3, Scherzo I : mm. 5459






Movement Between Chords

In actual musical composition, movement between major, minor

and subminor chords is seen on two lev e ls . The f i r s t of these le ve ls

is the simplest, the relat ive key relat ionship.

1. major to re la tiv e minor (e .g . C major to 'a' minor)


2. minor to re la ti v e subminor (e .g . 'a ' minor to f#dm7)

amm7 = re la ti v e minor

CMM7 = major

dmm7 = minor

bdm7 = subminor

minor subminor

Figure 39: Movement 3, Scherzo I I : m. 9




33

3. major to re la tiv e subminor (e .g . C major to f#dm7)

BMm7 = majo r

e#dm7 = subminor

Fig ure 40: Movement 3, Scherzo I I : m. 16

or the reverse of any of the above.

The next l ev el is one of modal change.

1. major to pa ra lle l minor (e.g. C major to cm)

2. minor to p a ra ll e l subminor (e .g . cm to cdm7)

Negative direction:

ddm7 = subminor

dmM7 = minor (hyperminor)


1°I •-

subminor minor

major subminor




34

3. major to pa ra lle l subminor (e .g . C major to cdm7)

Negative direction:

W &

i n ■ " T

*

m

¥ m

ddm7 = subminor

DM = major

cJ 1 bvA

subminor major


or the reverse of any of the above.

Lendvai discusses progression and retrogression in terms

of 'po si tiv e' and 'negativ e' dire ction s of chord motion. Positive

direct ion is:

1. a major chord always moves to i ts re la ti v e minor,

2. a minor chord always moves to i ts p a ra ll e l m ajor.

This can be seen in Figure 43. C# major moves to it s r e la t iv e

minor, a# minor (bb min or) . The re la ti v e minor, in tu rn , moves to

it s p ar al le l majo r, Bb major. By always moving in th is p a rt ic u la r

sequence, direction is positive.




35

C#M

I

a#m )BbM

\|r gm------------ >GM

I em............... »EM

]r c#m >(C#M)

Figure 43: Po sitive Direction

c#mm7 = minor

a#dm7 = subminor

minor subminor


L i r r r l l i CMM7

amm7

major minor


major

minor




36

The reverse of this gives the negative direction:

1. a minor chord always moves to i t s r e la ti v e major,

2. a major chord always moves to i ts pa ra ll e l minor.

This i s seen in the diagram below. The A major chord moves

to it s p ar all el minor, 'a ' minor. The p ar al lel minor, in turn moves

to i ts re lat iv e major , C major . A cont inuation of this pa rt icu lar

sequence w il l re su lt in a di re ct io n which is considered neg ative.

AM

TGbM >f#m

TEbM êb,

tCM >cm

t(AM) >am

Figure 46: Negative Di rec tion

major minor

amm7 = minor

FMM7 = major





37

Substitute Chords

The concept of substitute chords within a modal system is

d iff e re n t from tha t of a functional system. Instead of using 'a'

minor to substitutefo r C ma jor , as seen in Western harmony, modal

theory sub stitute s e minor fo r C major. They are sub stitu te chords

because they share two chord tones in common. This is s im il a r to

Western harmony, but here i t is the th ird and f i f t h of the orig ina l

chord which becomesthe root and th ird of the su bst itute chord.

Western harmony uses the roo t and th ir d of the o rig in al chord as

the thi rd and f if t h of the substitute chord. By Lendvai's d e fin it io n ,

the major triad is always substituted by a minor triad a major third

high er, (See Figu re 48) and the minor tri a d by a major tr ia d a major

third lower (See Figure 49).

1

major minor su bstitute

FMM7 = major

amm7 = minor substitute





minor major sub stitute

amm7 = minor

FMM7 = major substitute


An alternative method of constructing a hypermajor chord

is by combining the or igin al chord with it s su bst itut e (C major +

e minor).

AbM

Hyper'major

Hyper

major

Figure 50: Hypermajor Produced by Su bs ti tu te Chords

Substitute chords often move directly from one to another

as shown below.

|PPg#m >■ EM

Figure 51: Movement Between Sub st itut e Chords

roduced with permission of the copyrigh t owner. Furthe r reproduction prohibited without permission.



39

M A v v i1


In the above example g#m moves d i re c t ly to EM. But the y

can have a chord between them. By ra is in g the root g# to ' a 1, BMm7

is constructed, which in turn resolves to EM, the substitute chord

of g#m. __i ________________

g#m BMm7 EM

I ___________ I

Substitute chords

Figure 53: Chord Placed Between Su bs ti tu te Chords

Movement from a major to it s minor sub st itu te chord is pe r-

ceived as positive, whereas the opposite, minor to its major substitute

is neg ati ve. Thus, the movement can be e it h e r posi ti ve or negat ive

depending on the direction:

1. in dire ctio n of p a ra lle l major or minor

Po sitiv e: to pa ra lle l major (e.g . c minor to C major)

Negative: to pa ra lle l minor (e .g . C major to c minor)




40

2. in dire ction of rela tiv e minor or major

Positiv e: major to re la tiv e minor (e.g. C major to 'a' minor)

Negative: minor to re la tiv e major (e.g. 'a' minor to C major)

3. in dire ctio n of minor or major sub stitute chord

Po sitiv e: major to minor sub stitu te (e. g. C major to e minor)

Negative: minor to major sub stitu te (e .g. e minor to C major)

For example, in the diagram below, reading from l e f t to ri g h t,

c minor moves to C ma jor , which in turn moves to e minor. This con-

st itu te s the minor to pa ra lle l major, and major to subs titute minor.

Both of these motions are in the positiv e dir ec tio n. Negative di re c-

tion is seen by reading the diagram from righ t to l e f t st ar tin g with

C major moving to c minor. By moving from major to pa ra ll e l minor

and minor to its substitute major, a negative direction occurs.

The effect of this positive and negative direction perception is,

of course, purely subjective.

P P P P

cm CM em EM abm AbM (cm) (CM)

S S S

Posit ive —> Negative

Figure 54: Movement Between Chords




41

Function

Lendvai's modal theory is fu nct ion al, al b e it in a d iff e re n t

vein than Western the ory . I f a sin gle chord has a ton ic fun ct ion,

then its parallel and relative chords also have a tonic function.

This can be seen in the diagram below. Ab major is the tonic chord.

It s pa ra lle l chord, ab minor and rel a ti v e chord, f minor, both have

(f o r example) a ton ic fun cti on . In ad dit ion , the derived subminor

chord (the relative chord of the minor chord) is also included as

having a tonic function.

AbM-------------------abm+4*

fm +■»►+ fdm7

Re lativ e ( * )

Modal (* *■)

Figure 55: Inward Closing of Chords

As seen by the direction of the arrows, this arrangement

is considered to be 'close d inward' because a ll of the chords can

be seen as or ig in ati ng from the Ab major chord. By rev ersin g the

arrows (starting with the same subminor chord (fdm7) rather than

the Ab major chord for d er iv a ti o n ), an 'outward' re lat io ns hi p is

const ructed . See Figure 56.




42

CbM

AbM abmRela tive ( * )

FM fm fdm7 Modal («■ + )

Figure 56: Outward Rel atio nsh ip, of Chords

All of these chords have the same fu nct io n. Whether derived

from the major or the subminor, all their relative and parallel chords

retain the original chord's function.

Note the p ola r re la ti onsh ip between Cb major and FM. As

mentioned before, because the roots are a tritone apart, a difference

of six key sig na tur es is implied. They als o share two common tones

when made into seventh chords, and have the same function (either

to n ic , dominant or subdominant). Thus, any two po lar chords may

be exchanged with one another without a change of function.

Using re la ti v e solm izatio n, a chart can be constructed which

shows the in te rr e la ti o n s h ip of chords which share a common funct io n.

(See Figure 58) Re lat iv e chords (C majo r, 'a ' minor , f#dm7) are

CbMm7 FMm7

Figure 57: Common Notes Between Pola r Chords




43

v e rt ic a ll y alig ne d, modally relate d chords (A major, 'a' minor, adm7)

are h ori zo n ta l. Each chord is j oi ne d to the one above, below and

to each side of i t by ei th er a r e la ti v e or modal connection. Because

al l of the chords are eith er re lat iv e or modally related they a l l

have the same func tio n be i t to n ic , dominant or subdominant.

di

▼

do

m CMm7

ma

amm7

EbMm7

cmm7

adm7

3

f#mm7 + f#dm7

Figure 58: Chart of Chords Having the Same Function

I t is important to observe th at each column is thre e key

signatu res removed from one another (moving l e f t to ri g h t, or rig h t

to l e f t ) , thus making the di column and ma column a di ffe re nc e of

six key sig na tures from each oth er . Chords in the di column are

p o la rl y re la te d to those in the ma column.




44

The relation of the three basic functions (tonic, dominant,

subdominant) in the modal system is determined by symmetry. Western

theo ry is based on pe rfe ct f i f t h s , making scale degrees 1, 4, and

5 the most importan t fu n ct io n al ly . Lendvai' s modal theory symmetry

is based on the major th ir d . See Figure 59.

Western

Modalsb

Figure 59: Symmetrical Basis of Funct ion

Ab represen ts the subdominant, C the toni c and Eb the dominant

fun ction s. Tog ether, they form an augmented tr ia d ca b e, equa lly

div idin g the octave. Although perhaps d i f f i c u lt to comprehend at

this po int, i t w il l become clearer a fte r the discussion of the axis

system which is directly linked to the material presented thus far.

Axis System

As prev iou sly mentioned, in Lendvai's view, the la te music

of Verdi and Wagner is the direct forerunner of many of the innovations

credited to Bartok and are thus d ir e c tly tie d to the music of Bartok.

Polar related chords, substitut ion, and the alterat ion of di and

ma are but a few of the concepts Lendvai f ee ls can be d ir e c tl y traced

from Verdi to Ba rtok. The axis system is the cul min atio n of Lendva i's

roduced with permission of the copyright owner. Furthe r reproduction prohibited without permission.



45

research, bringing together pentatony, relative solmization and har-

mony, pre senting a no n tra dit ion al approach to the analysis of not

only Bartok, but the late romantic music of Verdi and Wagner.

The axis system is closely related to the Fibonacci Series,

using many of the same numeric re la tionship s. Tonal centers and

particular melodic formulae are generated from the axis system.

Ad di tio na lly , an en ti re f am ily of chords (alpha harmonies) are con-

struc ted and used in conjunction w ith those chords already discussed

in the harmony section.

This sec tion w il l show the various der ivati ons of the axis

system and how they interrelate with one another.

Functional Derivation

Lendvai's axis system is generated by f i r s t observing cer tain re la tio n -

ships in Western theory, most notably the functional relationship

between ton ic , dominant and subdominant. To nic, dominant and subdomi-

nant triads are considered to be primary triads.

SubDoiijinant Tonic Dominant

F C G

Figure 60: Primary Triad s

According to Lendvai, cla ss ical harmony dis ting uishes between

primary and secondary tr ia d s . Secondary tr ia d s are the su bs titu te




46

chords v i , i i and i i i which replace I , IV and V are b u il t a minor

third below their respective primary triad.

SubDominant Tonic Dominant

F C G

f / /d a e

Fig ure 61: Secondary Triads

Romantic harmony expands this basic idea by including triads

that are a minor third above the primary chords.

SubDominant Tonic Dominant

d ab a eb e bb

Figure 62: Ad di tio n of Triads Above Primary Chords

The axis system carries this concept one step further by

stat ing t ha t 'a ' and eb not only share c as a common r e la ti v e , but

also f # , which is the minor th ir d above eb and the minor th ir d below

' a ' . See Figure 63.




47

Sub

Dominant Tonic

C

Dominant

F G ^

d ab a eb

v y v yb f# db

Figure 63: Completion of Ci rc les of Minor Thirds

All of the notes in each c ir c le are a minor thi rd apa rt from

each other (e.g . c e b f# a ). Enharmonic res pe lling is acceptable

and frequ ently employed. I t is the in te rv a ll i c content, not the

sp el lin g which is impo rtan t. Likew ise, d and ab share both f and

b, and e and bb share g and c#. To nic, dominant and subdominant

are now complete, closed cir cl es which equ all y divide the octave.

All twelve tones of the chromatic scale are included in these circles.

These ind iv idua l to ni c, dominant and subdominant re la tio ns hi ps

are joi ned in to one lar ge system. The lar ge system is organized

in a c irc le of f i f t h s , then overlaid with the appropriate functional

la be ls . The repea ting sequence of func tion s, moving clockwise , is

always: to n ic , dominant, subdominant. This lar ge system, as seen

in Figure 64, is labelled the axis system.




48

Eb/D# 4 r

Figure 64: Axis System

This system is movable. 'C' is not always to nic . For example,

the F if th St rin g Q uar tet i s based on a Bb tonal center . The axis

system based on a Bb tonic would be:

Bb

C#

Eb/D#,

Figure 65: Axis System Based on Bb




49

This large system may be broken apart to show the individual

to n ic , dominant and subdominant axes. Note the examples are a ll

in re la tio n to a tonal c ent er of C as to n ic. See Figure 66.

C

Eb

F#

Tonic Axis

Dominant Axis Subdominant Axis

Figure 66: Ind ivid ua l Axes




50

Pentatonic Origin

The axis system can also be derived from a pentatonic basis.

This is accomplished by jo in in g several m ipen taton ies tog ether int o

one sequence. The axi s system is then drawn from th is pent atonic

sequence. F ir s t, i t is necessary to rec all the importance of the

descending pe rfe ct fourth ( la m i) cadence. Although not normally

ofimportance wi thi n a sing le mipe ntaton y, thedescending perfect

fourth alsoappearsbetween do and so. This pe rf ec t four th in te rv al

between la m i and doso is re fl ec te d in the diagram below which shows

one of the symmetrical properties of the pentatonic system.

so >• do I re I mi * la■---------- ii— ! ii !_11________ i

P4 M2 M2 P4

Fig ure 67: Symmetry of Pentatony Around 'Re'

Placed on a st a ff , the relatio ns hip w ithin a single (2 +

3 + 2 + 3 ) m ip en ta to ny looks l i k e t h i s:

Fig ure 68: Diagram of Symmetry Placed on a S ta f f

The next step is to substitute the syllables doso for lami,

withou t changing the pitches. The sy lla ble re is deleted at th is

po int. This w il l j oin two d iffe re n t pentatonies together.




51

do so la mi

* H 5 Pdo so la mi

Figure 59: Joining of Two Pentaton ies

Fig ure 70: Movement 2: mm. 3536

By continuing this process of substitution and forming new

pe nta ton ies , a sequence w il l be constructed which w il l move f u ll

circle and return to the beginning pentatony.

do so la mi

do so la mi

do so la mi

do so la mi

(do so)

Figure 71: Linking Pentatonies






53

These three axes may now be joined into the single large

axis.

Another important aspect of this particular pentatonic sequence

is its numeric (Fibonacci) relation ship s. Lendvai fee ls tha t the

existence of these numeric relationships to be an integral part of

his theory.

Each note w ith in the sequence has a p a rt ic u la r Fibonacci

numerical re la tio ns hi p to every other note w ith in i ts own pentatony.

This is seen in Figure 75.

C

F#

Figure 74: Axis System

Figure 75: Inte rva l li e Relationships in a Mipentatony






55

The circle is completed by adding F# major and f# minor,

the respe ct ive re la ti v e keys of ebm/d#m and A major:

CM + cm

/ \am EbM

+ +

AM ebm/d#m

\ /f#m + F#M

Figure 78: Completion of C ir cl e Using Re lat ive and Par al le l Keys

Because a ll of the chords are ti ed t o one another ei th e r

through a re la tiv e or pa ra lle l rela tio ns hip , they a ll have the same

fu nc tio n, be i t to ni c, dominant or subdominant.

This conf igu ra tio n can be compared to the method f i r s t de-

scribed , tha t of deriv ing~tn e—axis by key relatio nsh ips a minor th ird

ap art , evenly dividin g the octave:

C CM/cm

' A / \eb AM/am EbM/ebm

\ .

f# F#M/f#m

Figure 79: comparison of the Two Methods of Deriva tion

By using the f i r s t three pitches upon the ci rc le off i f t h s ,

three in dividu al axes may be formed. See Figure 80.




56

Tonic

CM/cm/ \AM/am EbM/ebm

\ /F#M/f#m

A

F#

Eb

Domi na nt

GM/gm

/ \EM/em BbM/bbm

\ /C#M/c#m

Subdominant

DM/dm

GM/gm ^FM/fm

\ # M / g # r /

Figure 80: Three Ind ivi du al Axes

produced with permission of the copyright owner. Further reproduction p rohibited without permission.



57

These ind ividu al axes may then be join ed int o the lar ge axis

system.

F#

Figure 8 i: Axis System as Derived by Re lat ive and Par a ll el Keys

Each axis is structured, in Lendvai's terms with a 'main

branch' and a 's ide bra nch' . (Lendvai 1983, p. 310) Beginning with

the pole at 12:00 on the face of a clock, movement to determine main

and side branch designation is always counter clockwise. The main

branch is always the f i r s t to ni c , dominant or subdominant at 12:00

or the fi r s t encountered moving to the le f t . The side branch is

the f i r s t resp ective to ni c, dominant or subdominant af te r the main

branch found by moving l e f t again.




58

Dominant Axis

side

F#

Tonic Axis

Eb

Subdominant Axis

Fig ure 82: Main and Side Branches of the Axis System

He labels opposite ends of a branch 'pole' and 'counterpole1

using the same counterclockwise method as labelling the branches.

(Lendvai 1983, p. 310)




59

Pole

Pole

C

SideA ,Eb Counterpole

F# Counterpoie

Tonic Axis

Counterpole

Pole

Counterpole

Pole

Dominant Axis

Pole

PoleG#

Counterpole

B Counterpole

Subdominant Axis

Figure 83: Poles and Counterpoles of the Axis System

Each of the in dividu al to n ic , dominant and subdominant axes

have th e ir own respec tive poles and counterpoles.




60

The large axis system and the individual axes are each closed

systems. A dd iti on al ly , because poles and counterpoles (e .g . C and

F#) are a tr it o n e apar t and d if f e r from one another by six key signa-

tures, the axis system is also a polar system.

Because of the relationship between parallel, relative and

p o la rl y r ela ted chords already discussed (See p. 16 ) , poles and coun-

terpoles may be exchanged with one another without a change of func-

t ion .

For example, i f a passage fi r s t moves to an A majorminor

seventh chord, i t could ju s t as ea s il y move to an Eb majormino r

seventh the next time without a change of function.

b u n u r o

m mEbMm7* AMm7

Counterpole

C

Pole Eb A Counterpole

F#

Tonic Axis

Figure 84: Po lar Exchange




61

I t should be noted tha t the four ind ivid ua l poles of any

ax is , when viewed tog eth er , are not to be considered as fu l l y dimin-

ished seventh chords. For example, the dominant po les , ge db bb ,

are each roots of Mm7 chords which a l l have a dominant re la ti onship

to any of the poles on the toni c ax is . (Lendvai 1983, p. 277)

A more de ta ile d discussion of dominantton ic cadences w il l

be addressed in a subsequent section.

Lendvai constructs three intervallic chains from the axis

system which are used to describe melodies and harmonies. These

are c a ll ed the 1:2 , 1:3 and 1:5 models. The most common of these

in te rv a ll ic chains is the a lte rn at io n of minor seconds (1) and major

seconds (2 ) .

GMm7 EMm7 DbMm7 BbMm7

Figure 85: Dominant Pole Mm7 Chords

Axis Scales

1:2 Model

Figure 86: 1:2 Model




62



As seen in the above diagram, the interval 1 (minor second)

and 2 (major second) a lt e rn a te . This co nf igur ati on i s known as a

1:2 model. This p ar tic u la r arrangement of int erv als is a classic

model of an oc tato nic sc ale . Lendvai uses the 1:2 in te rv al pa tter n

(1:2 model) most frequently of the different models as he feels they

"should be considered the 'fundamental sc ale ' of the axi s system."

(Lendvai 1983, p. 277)




63

The 1:2 model is derive d by jo in in g a ll the notes from major

and minor chords b u il t on the poles of a sing le a xi s.

(8 notes)

Figure 89: 1:2 Model Co nstruc ted from Polar Chords

Another method of d eriv ing the 1:2 model is to use the fu ll

axis system. See Figur e 90. By st a rt in g at a pole and al te rn at in g

skips of one and two spaces around the fu ll c ir c le of fi f t h s , the

entire axis is circumferenced and the 'closed' scale (Lendvai 1983,

p. 370) is constructed.

(CM) ceg

(cm) cebg

C

(am) ace

A

(AM) acffe

F#

(F#M) f #a #c #

(f#m) f #ac #

Eb

(EbM) ebgbb

/ b , b b . b (e m) e g b

/ ____ »9*—_—?«_—i5•# b

1 2 1

CG

Eb

Bb

E

A

F#

Fig ure 90: 1:2 Model Const ructed from Axis System




64

1:5 Model

The 1:5 model, alternating minor seconds (1) and perfect

fourths (5 ), can also be constructed from e ith er a single ax is, or

the f u ll axis system. To form i t using one ax is , po lar relate d chords,

such as those b u i l t on C and F# are combined. The roots and f i f t h s

of the two polar chords are the only parts of the chords used.

Pole

C

F#

Counterpole

Figure 91: 1:5 Model Constructed from Poles






66

Figure 94: 1:5 Model Constructed from Axis System

1:3 Model

The remaining in te rv a ll ic chain is constructed in a di ffe re nt

manner than the other two. This th ir d chain is made by jo in in g a

major triad with a minor triad a major third lower:

CM

ceg

b

+

+

abm

b b b a c e

b bc e e g , a _____ ci II _________ 1 ____________ 2 1 ____ i i ■

3 I 3 i 3

Figure 95: 1:3 Model

This a lte rn at io n of minor seconds (1 ) and minor th ird s (3)

is la bel le d the 1:3 model. Whereas1:2 and 1:5 modelshave strong

'tonalch ara cte rs ', (Lendvai 1983, p. 377 ), the 1:3model creates

tonal ambiguity by it s very str uc tur e. Each of the three notes of




the major t r ia d is join ed by a note only a h a lf step away,

cancels the other out tonally.

CM

abm

Figure 96: Ambiguity of Ton ali ty in 1:3 Model

tU . f -------F f f f f ...wr*^' -f' ‘ * ■ 1:

■f—3------ I., rlP r v —

1 3r*l 'k ip

«T fj V ' '#■ '

i i


_i 1


67

Each




68

As with the other two models, the 1:3 model may be constructed

from the la rg e a xi s. St ar tin g at a pole and moving through the c ir c le

of f if th s by skips of one and three spaces, the c ir c le is circumfe r

enced and the 1:3 model produced.

B / C b

Figu re 99: 1:3 Model Constructed from Axis System

Authentic Cadences

Modal t he or y has dominant to to nic cadences. These are con-

structed using the four dominant poles and the four tonic poles of

the large axis system.C

F#

Figure 100: Dominant and Tonic Poles

roduced with perm ission o f the copyright owner. Further reproduction prohibited without permission.



69

These dominants and tonics may be related to one another

in two ways:

1. The fo ur dominants rel at ed to a common to n ic ,

2. The fo ur ton ics re la te d to a common dominant.

The f i r s t of these options (dominants re la te d to a common

tonic) is i l lustrated below.

1. Cl as sica l Dominant Cadence

This cadence is lab elled 'c la s s ic a l1 (Lendvai 1983, p. 144)

because the root movement is up a perfect fourth which corresponds

to tr a d it io n a l th eo ry 's authe nti c cadence. For example: GMm7CM/cm.

GMm7 CM

Figure 101: Example of a 'Classical' Cadence

TT

c#m





70

2. Modal Dominant Cadence

This cadence is lab elle d as modal as i t is qu ite sim ila r

to the characteristic renaissance modal cadence which has a bass

movement o f down a ma jor second and soprano movement up a minor second.

For example: BbMm7CM/cm.

'o

BbMm7 CM

Figure 103: Example of a Modal Dominant Cadenc

Fi gu re 104: Movement 2: mm. 91 0

3. Phryg ian Dominant cadence

This cadence is labelled as the phrygian dominant because

of the h a lf step downward reso lut ion of the bass. For example:

DbMm7— CM/cm.




71

DbMm7 CM

Figure 105: Example of a Phrygian Dominant Cadence

Figur e 106: Movement 1: mm. 1314

4. Romantic Dominant Cadence

This cadence is labelled 'romantic' (Lendvai 1983, p. 144)

because of the root movement of a major third interval which Lendvai

fe el s harkens d ir e c tl y tc the language of the romantic period. For

example: EMm7CM/cm.

cr EMm7 CM

Figure 107: Example of a 'Romantic ' Dominant Cadence




72

Dominant Poles as Roots

Each of the poles of the dominant axis can be used as the

roo t o f a dominant chord. These chords are norma lly seen as being

a majorminor seventh quality.

GMm7 EMm7 DbMm7 BbMm7

Figure 108: Chords on Dominant Poles

However, in add itio n to th is , one can also fin d major, minor

or subminor qu a li ty chords b u il t on these dominant poles which also

func tion as dominants in dominant to toni c cadences. This is i ll u s -

trated in Figure 109.

1. GMm7—CM/cm

3EE3EE=1=GMm7 CM

2. GMCM/cm

o

GM CM

3. gm— CM/cm

gm CM




73

4. gdm7CM/cm

9dm7 cm

Figure 109: Examples of Dominant to Tonic Cadences

Using Varying Qualities of Dominant Chord

Tonic Poles as Roots

The second of the poss ible rel at ions hi ps (r e la te d to common

dominant) is to re la te a singl e dominant (any of th e fou r poles on

the dominant ax is ) to each of the fou r possible to ni c chords drawn

from the tonic ax is. The c ri te ri a fo r lab ellin g these cadences is

the same as mentioned e a r l ie r .

1. GMm7CM/cm Cla ss ic al Cadence

GMm7 CM

2. GMm7AM/am Modal Cadence

GMm7 am

3. GMm7F#M/f#m Phrygian Cadence

GMm7 F#M

produced with perm ission o f the copyright owner. Further reproduction prohibited without permission.



4. GMm7EbM/ebm Romantic Cadence

74

AGMm7 EbM

Figure 110: Examples of Dominant to Tonic CadencesUsing Varying Qualitites of Tonic Chords

I t is important to note tha t no ma tter which way the cadence

is const ruc ted , the same four types of cadence keep appearing. There

is considerable f l e x i b i l i t y in Lendvai's system because of the chordal

variety and relationship to one another allowed.

Alpha Harmonies

As been thus f a r , wit hin the axis system, i t is possible

to derive the various in te rv a ll ic models (1 :2 , 1 :3 , 1 :5) and determine

harmonic function according to whether the notes appear on the tonic,

dominant or subdominant ax is . There is yet another prop erty of the

axis system to be discussed. This property is the formation of chords

derived directly from the axis system.

Functional the ory is founded in te rt ia n harmony. That is ,

notes are placed in sim ul tan ei tie s of major or minor th ird s to produce

chords. Lendvai discards t h is method when de al ing wi th the axis

system. Here, instead of te r ti a n harmony, chord st ru ct ur e is based




75

on Fibonacci numbers (2 ,3 ,5 ,8 ). C ol le ct iv el y, th is group of chords

is known as alpha harmonies. Broken in to it s in div id ua l component

chords, they are:

alpha chords (a)

beta chords (3)

gamma chords (y)

delta chords (6)

epsilon chords (e)

Each of these w il l be discussed in tur n.

Alpha Chord (a)

The alpha chord consists of two layers of notes, a tonic

lay er and a dominant la ye r. Both of these layer s are taken d ir e c tl y

from the axis system.

Tonic:

Dominant:i

Alpha Chord:




76

The tonic layer is the top layer and dominant always beneath.

The keynote is the lowest toni c pi tc h. Occ asio nally , Bartok reverses

th is and places the dominant lay er above the ton ic. This is calle d

inv ertin g the alpha chord. The keynote is s t i l l determined by the

lowest tonic pitch with the alpha chord.

t o —I . . . ■ ^ Dominant

Tonic

Figure 111: Inv erte d Alpha Chord

Below are several examples of alpha chords. Thei r key note

and accompanying axis system are shown to demonst rate th e ir de ri va ti o n .

1. Eb alphaEd

T = eb c a gb

D = g e db bb




77

2. F# alpha

m m r o | b

TT

T = f# eb c a

D = bb g e c#

3. B alpha

T = b ab f d

D = eb c a f#

4. D# alpha

m m

T = d# c a f#

D = g e c# a#

F

G

C

A

Figure 112: Examples of Alpha Chords




78

The number of notes w ithi n an alpha chord vari es . I t may

be as few as two notes (See F igure 113) or as many as si x (See Figure

114).

T = c#

D = d

Inverted alpha chord

(dominant over tonic)

C#a

Figure 113: Two Note Alpha Chord

C#ct

T = c# e g

D = g# b f

Inverted alpha chord

(dominant over tonic)

Figure 114: Six Note Alpha Chord

An alpha chord may also consist of th ree la yer s, although

th is is not asfrequ ent as a two la ye r model. The ton ic lay er is

normally on top,dominant in the center and subdominant on the bottom.




79

Ea

r T = e g

0 = g# b

I SD= d# f#

• T = e g a# c#

D = d f g# b

. SD= a c d# f#

Figure 115: Three Layer Alpha Chord

Excellent examples of three layer alpha chords may be

in the f i f t h movement.


found









83

Type gamma is a lso based on the p erf ect fo urt h . The key-

note is the top note of the perfect fourth.

A minor th ir d is added above the top note of the fo ur th and

a minor third added below the perfect fourth.

m3

Frequently, the seventh above the keynote is added.

' r

The gamma chord can also be seen as a numerical symmetry

combining the major and minor triads.

ma jor minor gamma

p . t -t

3 5 6 13* —

8=3+5 8=5+3 M + m

Figure 120: Symmetry of Gamma Chord

This 3 + 5 + 3 (m3 + P4 + m3) is important to Lendvai . I t

is used to form a closed sequence which in turn creates a succession

of gamma chords and is the only symmetr ical alpha harmony which can

be constructed in this manner.




84

=£CO ^

Figure 121: Succession of Gamma Chords

Below are examples of gamma chords with th e ir st ru ct ur e in

brackets.

1. C gamma

J ..

2. B gamma

S i* 0 * 3

3. F# gamma

4. G gamma

Figure 122: Examples of Gamma Chords




85

The fo llow in g musical examples o f gamma chords are from var ious

movements of the Quartet.

C y


T- f t

Ey





------ r r — F = $ t = • H r & 8 || f ^----- a —iU— e — H H ±=§—

A y Bby Dy Eby Dby


Delta Chord (6)

The de lt a chord is si m ila r to the beta chord. Here, however,

the perfect fourth is on the bottom and the tritone placed above

i t . The keynote is the top note of the per fec t fou rth .




87

A minor th ir d above the lower note of the t r it o n e is added.

Below are examples of delta chords with the structure notated

in brackets.

1. Ab de lta

y'o^T ft ;

2. A de lta

3. Bb delta

4. Cb de lta

Figure 126: Examples of Del ta Chords




88

Below are examples of delta chords from the Quartet.

rln.t'W < ' 1 ■: ■. —i-

poco r i t . .

^ ---------

TF At

w-

9?

t®# o

Eb6

Eb6


mp

G6

w

G6


Epsilon Chord (e)

The epsilon chord is b u il t around a pe rfe ct fou rth , but th is

tim e, the accompanying tr it o n e is ne ith er added above nor below,

bu t, ra th er , woven around the pe rfe ct four th . The keynote is the

top of the fourth.




F irs t , the per fect four th in terva l :

89

Second, a tr it o n e is b u i lt a major second above the lower

note of the perfect fourth.

Last, a minor third is added above the top of the tritone:

Below are several examples of epsilon chords with the structure

notated in brackets.

1 . B epsilon

P f|

2. C# epsilon

3. A epsilon




90

4. G epsilon

Figure 129: Examples o f Epsilon Chords

Below is a r are example of an eps ilon chord from the f i f t h

String Quartet.

Pil iSStro *

Ge

YCP' 6

Ge

w nrt


Fibonacci Origins of Alpha Harmonies

Al l alpha harmonies have an underly ing fa ct o r which jo in s

them tog eth er. This fa ct o r is the use of Fibonacci in te rv a ls . Every

in te rv al in each chord uses only those numbers found in the se ries,

namely, 2,3 ,5 and 8. This in te rv a ll ic str uc tur e is diagrammed below

for each of the alpha harmonies.




91

1. alpha

Note the formation of a 1 :2 model using the alpha chord.

This may be seen as a li n e a r form of the alpha chord i f seen in th is

form.

3 y r ------

- 4 - &V

■e*-V

■y

1:2 Model

2. beta

3 £

3. gamma

PQ

4. delta

___

o 3






Omega Chord

Abw


2. from minor th ird s (3)

Using minor th ir d s , the fu ll y diminished seventh chord is

formed. Lendvai does not dis ting uish th is formation of the fu ll y

diminished seventh chord from that made by altering a majorminor

seventh chord.

Figure 134: Fu ll y Diminished Seventh Chord

3. from per fect fourth (5)

A serie s of pe rfe ct fourths may appear lin e a rl y (as a melody)

or as a simultaneity of fourths to produce a chord of the fourth

(Lendvai 1983, p. 38 1) . They may also be seen in invers ion as chords

of the f i f th .




94

Figure 135: Sequence of Per fe ct Fourths

The following example is the linear use of the perfect fourth

from the third movement.

i— i a o


The two examples below are fourth chords.

>A if f

6.t y *

1 = *

t o ; i _ l

- e -

4th Chord





95

n °~- I ■ o


Below i s an example of a chord of the f i f t h from the f i f t h

movement.

WE zzn

5th Chord


4. from minor six ths (8)

The use of minor sixths is limited to the formation of aug-

mented triads.

Figure 140: Augmented Tr iad s




The next section w il l discuss the theo ries

they can be applied to the Fifth String Quartet.

Lendvai




CHAPTER TWO

THE QUARTET

This po rtion of the diss erta tion is an analysis of the F ifth

String Quartet using the analytical principles of Erno Lendvai.

I t is designed to give a s tr u c tu ra l, tonal and harmonic overview

of each movement. Al l f iv e movements w il l be discussed in d e ta i l,

giving atten tion to three areas in pa rt icu lar: pentatony; re la t iv e,

p a ra ll e l and su bs tit ut e chord harmonies; alpha harmonies. These

areas assume varying degrees of importance depending on the particular

movement.

A b ri e f discussion of the Quartet precedes the de tail ed analy-

sis of the movements.

Each of the five movements has a particular characteristic

which sets i t apa rt from the othe rs. The f i r s t movement is stro ngly

pe nta ton ic and uses alpha harmonies. Movement two, the sh ortes t,

is important for presenting the chorale section which becomes the

foc al po in t of movement fou r. The th ir d movement emphasizes r e la ti v e ,

parallel and substitute chords, giving less significance to pentatony

and alpha harmonies. Movement fou r is a ser ies of var ia tio ns on

the chorale o f the second movement. In th is movement, penta tony

again takes precedence, with it s ensuing alpha harmonies. The f in a le ,

movement fi v e , is the longest movement, re ca ll in g mat eri al from the

f i r s t movement.

97

roduced with permission of the copyright owner. Further reproduction prohibited withou t permission.



98

The F if th Strin g Q uar tet, as prev ious ly mentioned, is an

ex ce lle nt example of palindromic st ru ctu re : the second h a lf of the

Quartet is a reverse image of the f i r s t h a lf . Bartok designed his

palindrome through the use of a five movement structure where the

f i r s t and la s t two movements are cen tered around the th ir d movement.

Movement f iv e is a var ia ti on of movement one and movement four is

a va ri a tion of movement two. The formal s tru cture is seen in Figure

141.

Movement 1 Movement 2 Movement 3 Movement 4 Movement 5

A B C B1 A'

Figure 141: Formal Struc tur e of the Quarte t

Tonally, Bartok creates a palindrome by his choice of poles

and counterpoles from a sing le axis as the tonal centers fo r each

movement. Movements one and f iv e cen te r around Bb, the to ni c pol e,

two and four around the pole and counterpole of the side branch of

the to nic axis (C# and G). See Figure 142. Movement three is r el at ed

to the outermost movements by centering around E, the counterpole

of Bb. The refo re, the en tir e Quartet begins on Bb and cir cl es counter-

clockwise around the axis to the counterpole, E, then continues in

the same d ir ec tion back to the s ta rt in g po le , Bb. This is shown

in Figure 143.




99

Figure 142: Axis System of the Quarte t

Mvt. ________ Axis ______ Function _______ Position _____________ Branch

51 Bb T po le main

52 C# T pole side

53 E T coun terpole main

54 G T counterpole s ide

55 E Bb T co un terp ole / main

pole

Figure 143: Tonal Structu re o f the Qua rtet

Whether viewed structural ly or tonally, the Fifth String

Quartet is indeed a palindrome.




CHAPTER THREE

MOVEMENT ONE ANALYSIS

The fi r s t movement is in sonata form. The exp osi tion presents

the f i r s t , second and th ir d themes in sequence. The cen ter of the

movement is a developmental sect ion , followed by a re ca p itu la tion

which reverses the order of thematic presentation ( i . e . th ir d , second

and f i r s t ) . As the formal arrangement of themes in the re ca pi tu la tio n

reflects that of the exposition, the form of the movement can be

seen as a pal indrome. See Figure 144.

Exp 158 Dev Recap 132176

FT Tr ST TT Oev TT ST

114 1424 2444 4458 59132 132146 146159

11.5963

2. 6369

3. 6986

4. 87103

5. 104111

6. 112132

pole: Bb C D E F# Ab Bb Whole Tone

Progression

Figure 144: Formal St ructur e of Movement One

The tonal cen ter o f the f i r s t movement is Bb, which is also

the tonal c ente r of the en tir e Quartet. In th is movement, the tonal

centers of the movement re f le c t an ascending whole step progression

(BbC DE F#A bBb). This may be seen in the fol low ing figure which

100

Coda

FT

159176 177218




101

shows the tonal ax is. Al l references to fun ctio n in the f i r s t movement

w il l be focussed on th is ax is . See Figure 145.

Figure 145: Tonal Center of Movement One

First Theme/Transition (124)

The opening o f the f i r s t theme fi rm ly establis hes the tonal

center o f Bb by scoring a ll four instruments on a unison Bb fo r three

full measures.

Vlolino I

Viollno II

Viola

Violoncello

HFrx iT t■1 H J*!i ■!'I I I

T P

Figure 146: Measures 13




102

In measures 4 and 5, the Bb pentatony is transformed into

it s counte rpole pentatony of E. This is done by the simultaneous

a lt e ra t io n o f do to di and mi to ma. Figure 147 shows the Bb penta -

tony , Figure 148 shows the tra ns formatio n from Bb pentatony to E

penta tony. Figure 149 shows the passage from the score.

mi so la

Bb mipentatony

Figure 147: Bb Mipen tatony

^ — P~e> q o.ma— *ma do— M i

y - i n i o ^ . * 4 . — : .

la . mi la SOmi so la i mi la so

Bb mipentatony E mipen tatony

Figure 148: Transformation of Pentatonies

r f r f i iJ5ai----------£2 ____ Leu ___ la ____ »»

Bb mipen tatony E mipentatony





103

Bartok sets the polarly related pentatonies (Bb and E) against

each other in measure 8 to fi rm ly es tab lish the tonal c enter of Bb.

PBb Mipentatony

E Mipenta tony

Figure 150: Measure 8

The remainder of the f i r s t theme area is in the subdominant

reg ion o f Eb and ton ic E poles. The vi ol a and ce llo passages are

in the Eb pentatony while the vi oli ns are in the E pentatony. This

is seen in measures 9 and 10.

si

rM


E mipentatony (Tonic)

Eb mipentatony

(Subdominant)






105







107

At the end o f measure 15 the c e ll o 's F# (a subdominant pole)

merges into an Ab mipentatony. 1

i l l V ' .... . PO A •

T 3 -------------------------------------

re so la mi

Figure 157: Ab Mi penta tony

The pole replacement within this cello line is Cb + F, pole

and counterpole on the dominant axis.

BbCounterpole

B/CbPole

Figure 158: Cb and F Counterpoles

The downbeat of measure 16 is an F gamma chord which combines

notes from both the dominant and subdominant axes: Ab and F from

the dominant axis, A and Eb from the subdominant axis.




Fi gure 159: Measure 16

In measure 16, v io li n I uses ye t anoth er pole replacement

(Ab > D) from the dominant axis.

11

I gi'CTr r •

PB .

— 1/

m m m .

Bb

E

Figu re 160: Ab and D Counterpoles

V e r t ic a ll y , Bartdk moves from a gamma cho rd, which combines

axes, to individual simultaneities of dominant and subdominant poles.

Beat 2 is a simultaneity of subdominant poles (Gb, Eb, A), beat 3

is a sim ul ta ne ity of dominant poles (G#, D, E#). See Figu re 161.




Bb109

F/E#

Gb/F#

E

&

Domi Subdomi

nant nantPoles Poles

Figure 161: Dominant and Subdominant Poles

Measures 1720 place polar substitute notes next to one another

li n e a rl y . The vi o la begins the motive:

Bb/A#

Figu re 162: Measures 1720




110

The motive sh if ts between the vio la and v io li n I I in 1718,

then in 19 21 , i t is doubled at the octave in these same two voices,

fu rt her emphasizing the p ola r replacement of E and A#. The sustained

E# in the c e llo (1 7 20 ) is a dominant pedal wh ile a sustained F#

in the f i r s t v io lin (171 8) is a subdominant pedal.

The end of measure 21 through measure 23 breaks o f f the motive

and begins a new id ea . Each of the four voices movel inear ly in

contrary motion in various combinations of ha lf steps and whole steps.

Lendvai does not discuss those areas which do not f i t his theo ries

in a preci se manner and i t is impossible toknow for certain how

a passage such as this should be labelled, however, this author feels

these could be considered variations on the 1:2 model.


■r - r —I . . ft; : f 4J Ji_j r r

f.i / * ' , f-Tf.ff-|Kj Sm:—

. ‘ #>*"*| __ | = M2 V = mi




I l l

The linear progression ends abruptly in measure 23 with an

inverted four note C# alpha chord, (shown below)

rr

Dominant

Tonic

Dominant

C#a C#a Tonic

Figure 164: Inverted C# Alpha Chord, Measure 21

which is immediately reduced to a two note version of the same C#

alpha chord on the next beat, (shown above)

Second Theme (2544)

An int er es tin g feature of Lendvai's theo ry is his use of

Fibonacci numbers to construct a closed system. There are two such

closed systems in the second theme ar ea . The closed system used

in the second theme is a sequential chain of 3 + 3 + 2 h a lf steps .

In ot her words, a minor th ir d , minor th ir d , major second sequence

repeated thr ee tim es. Becausethe sequence returns to it s st ar tin g

pitch, this succession is termed a 'closed system1.

The f i r s t closed system (25 29 ) is begun on C using the 3

+ 3 + 2 sequence.




112

r T U

Eb

2 ..D

Figure 165: Closed System Diagram on 'C'

The C based system is not immediately apparent in the music.

All the notes of th is closed system are mixed tog eth er and cannot

be e a s ily bracketed to show th e ir re lati on sh ip to one another. How-

ev er, in te rva l li e patterns based on the Fibonacci Series which are

used fo r cohesion of t h is closed system are shown where poss ib le .




113

x n'orft '

i &r

s s *

Figure 166*. Measures 2529

The second closed system (2936) is based on F# using the

same 3 + 3 + 2 p atte rn .




Figure 167: Closed System Based on ’F# 1

This F# based closed system passage is a sequence of the

previous one with simi lar in te rv a lli c relatio nsh ips . There is an

in te rn al cadence i n the second theme at measure 36 bri nging the F#

based closed system to a clo se. The chords and th e ir res pe cti ve

func tions are shown in Figure 168. I t i s an unusual cade nt ial sequence

in th at the bdd7 is used as a d ir ec t subs tit ut e chord fo r a Bb major

minor seventh.

JT ' „ » *T A v> AlW 1° c—' H





115

The retu rn of f i r s t theme mate rial (374 4) rounds o ff the

second theme. The tonal center of the f i r s t theme ma teria l is cl ea rly

the C pole on the subdominant axi s. (See Figure 169) I t was o ri g in a ll y

in the tonic area of Bb.

r ft

7 J J J

L E I f L H

C p r P f P Ji i ■ ■ ■ a

Fig ure 169: Measures 3739

Third Theme (4458)

The th ir d theme (44 58 ) consists of chromatic li n e a r passages

inte rspers ed wi th various chords and alpha harmonies. Lendvai's

the ory does not account fo r melodic ma teri al which is ne ith er a model

(1 :2 , 1:3 , 1:5) nor chord ally derived (such as a chord in arpeggia

t io n ) . The melodic li n e may be ca lle d a 'closed melody' using Lend-

vai 's term. (Lendvai 1983, p. 407) This impl ies a melody which

arches either up or down and returns to the starting locations:




Figu re 170: Measures 4447

The progression of harmonies is shown below with further

explanation fol lowing:


^







t M ' * r ^

Figure 171: Continued




119

A tonic BbMM7 hypermajor chord appears in 45 and 46, becoming

an Eb delt a chord in 47 and 48. See Figure 29...o , (c) m

Figure 172: Measures 4548 Eb6

Measures 4950 contain what Lendvai calls a 'polar chord'

because of the r e la tion sh ip between B and F, the two importa nt chord

ton es . (Lendvai 1983, p. 751) This chord could also be seen as

a root position F7 (French seventh) configuration (BD#FA) although

i t is not functio nin g as an augmented six th chord. Measures 5152

are a fou rth chord (B E A) . Each of these thre e notes represen t

either the tonic, dominant or subdominant axis, creating a tonally

unstable area . In measure 53, however, th is is balanced by an FMm7,




120

followed in 54 by an f#m chord, both stable chords in comparison

with the preceding fourth chord. Bartok again shatters this s ta b il it y

in 55 with a ve rt ic al open f i f t h (DA) and a whole tone lin e in the

cello for one half measure.

Measures 5658 constitute an area based on a 1:5 model.

The prominent notes , DEbG#A, in these three measures are re g is tr a ll y

placed to hi gh lig ht the 1:5 con figu ratio n. The 1:5 model pitches

are circled in Figure 173.

' dim.. . mf


The t h ird theme is ended by a phrygian cadence of DM6 to

FM in measure 58 . Although normally def ined by a ro ot movement down




121

a ha lf s te p , th is time the cadence is defined by bass motion down

one halfstep, emphasized by the F# to F in the cello.

Figure 174: Phrygian Cadence in Measure 58

Development (59132)

The development section (59132) is a working out of the

f i r s t , second and th ir d theme area s. Section (1) (596 3) opens with

the f i r s t theme in the tonal ce nter of the tonic coun terpole , E,

with a four voice unison (59 6 2 ). This is modelled on the opening

f i r s t theme material from the exposition .

7J-JTJR





122

In section (2) (6369), Bartok layers the pentatonies based

on the to n ic pole and counter po le, E and A#, in s tr e tt o . The music

is shown below with a diagram following demonstrating these pentatonies

and th e ir re la tio n sh ip . See Figures 176 and 177.

Figure 176: Measures 636 9








125

Section (5) (104 11 1) dele tes th e second theme but continues

the working out of f i r s t theme ma teria l in st re tt o . As shown in

the fo llo w ing example, the sustained notes fi r s t move up by chromatic

h a lf steps from C to C, then down by chromatic h a lf steps from A

to E. Th is shows a movement from subdominant (C) to to nic (E ) , and

subdominant (A) to tonic (E ). (See ax is system in Figure. 181)

c —

Figure 181: Measures 104111 and Axis System




126

Bb


Section (6) (112132) is a var iati on of the fi r s t theme.

This secti on contains the climax o f the e n ti re movement in measure

126. This double fo r te chord of E and F is a two la yer alpha chord.

E represen ts the t on ic ax is , and F repres ents the dominant ax is .

Dominant

Tonic

« W T T T 1 .........

V*1

s 1E ^





127

Direct ly a f ter this c l imax point , there is a fa lse recapi tulat ion

using the alpha chord in the orig in al rhythmic pa ttern of the opening

measures. This is a fal se recap itu lati on as i t quic kly dissolves

int o more developmental materia l be fore the actual r ec ap itu la tio n

in measure 132 which uses the third theme (132146) in the subdominant

axi s tonal reg ion (p oles F# and C) . St ructu red in the same manner

as in the ex po siti on , l in ea r passages are interspersed w ith both

chords and alpha harmonies.

The second theme ( 146159) uses inv ert ed ma teria l and is

sh or ter than the ex po si tio n second theme. Lendvai uses the term

inversion in dis cri m ina tely to mean any of three things: inversion

of contour, in te rv al l ie complementation or melody retro gra de. Passing

reference o nly w il l be made to these areas as i t is not the int en t

of this paper to dwell on this aspect of Bartok's compositional pro-

cess.

In addition, the recapitulation of the second theme employs

only one closed system on C instead o f the two di ff e r e n t closed systems

as presented in the exposition.

The fi r s t theme (159176 ) is varied and also inve rte d. An

item of in te re st is the use Bartok makes of the su bs titu te chord

bdd7 for the Bb tonal center from 160165.





The coda is len gthy (17 7218) and con centrat es on working

out and varying f i r s t theme m at er ia l. S tre tto and canon are again

used ex te ns iv el y in th is secti on emphasizing the Eb and A poles of

the subdominant axis (177181).




129

■ . .■ ] T - f g f f

svF

Bb

E

Figure 184: Measures 177181 and Axis System

The movement ends wi th a strong a ff ir m ati on of the Bb tona l

cent er by a four vo ice unison Bb. B ri e f 1:2 models appear in 209210.

The fi n a l cadence is a 'c la s s ic a l* dominant cadence,, The dominant

chord is an fdm7 instead o f an FMm7 and resolves to a unison Bb.

See Figure 185.




130





CHAPTER FOUR

MOVEMENT TWO

Movement two, the Adagio, is in palindromic form similar

to the f i r s t movement. The three thematic areas are each presented

in tu rn , followed by a developmental section in the middle of the

th ir d themat ic are a. The remainder of the movement presents the

second and f i r s t thematic areas res pe cti ve ly. The formal stru ctu re

is shown in Figure 186.

[A] Tr [B] [C] [B '] [A1]

14 510 1025 2646 4649 5056

Chorale

Intro 2630

Folksong 3134

Dev 3542

RT 43462

pole: C# C G C C#

Figure 186: Formal S truc tu re of Movement Two

The second movement i s the sho rtes t of the f i v e , only 56

measures long. The to n a li ty cen ters on C#, the pole on the side

branch of the ton ic ax is . Figure 187 shows the tonal axis fo r th is

movement.

131







Each of the tri to ne s represent a dif fe re n t axis by being

the pole and counterpole of either the tonic, dominant or subdominant

axis.

C#

G

Dom Sub

nant domi-

nant

Tonic

Figure 190: Poles and Counterpoles of Measures 59




Movement in the c e llo is a 1:2 model:

134

eg# f# g f d bc a

2 1 2 1 2 1 2 1

Figure 191: 1:2 Model of Measures 59

The cadence which ends the tr a n s it io n is a modal cadence

in to the dominant reg ion . Measure 9 ends with a CM chord, moving

to an open f i f t h of D and A wi th a root movement up a major second,

producing a modal cadence.

7f ----- 1----- 1---- 1--------

• < H = 4 = =

VP

S

\D vp

Figure 192: Measures 91 0

The (B) section (1025) is the focal point of the movement.

I t consists of sustained chords in a cho rale text ure with a sin gle

moving lin e in the f i r s t v io li n . Harmonies of th is sec tion are shown




135

below. Note the absence o f alpha harmonies. Bartok makes emphatic

use of alpha harmonies when this passage is reharmonized in movement

four to contrast with this presentation.

CXtnrrN





136

This harmonic progression also outlines a mipentatony on

A in the roots of the chords. The diagram below c le a rl y ill u s tr a te s

the pentatony.

A mipen tatony

m. 10 13 15 17

so

■T: = R

re mi

odo

CM gm amm7 FM

Figure 194: A Mipenta tony

The recapitulation of the (B) section (4649) is quite abbre-

via ted : i t contain s six chords and is only three measures long instead

of the exp os itio n's fi ft e e n . This time, though, two alpha chords

are used to set apart the return of the chorale from the original

harmonies.

w 7 wT1— -----

CKA €(AOri1

'

p t y Ill f =

lifSp—

H ^ t r C#a

m zs:

C#a

Dominant

Tonic





137

The importance of the chorale sectio n o f th is movement w il l

become clearer in the discussion of movement four.

The (C) sec tion is the long est section of the movement (26 46 )

and can be divid ed into four parts : intro duc tion ( 26 3 0) . folksong

(3134), development (3542) and retransition (4346).

Measures 2630 are based on the material from section (A)

(1 4) and the tr an si tio n (5 9 ). (See Figure 196) In measures 2620,

(Fig ure 197), the h al f s te p motive is used in a new rhythm, and the

perfect fourth and t r i ton e are f i l l e d in.

. i.

p—i *

f e ' P :

—------- *

TT

— f J j * —= 3 # 1=

s J





138

( t r t m i . )

Tt

PtXM.


A new motive is introduced in measure 27 in the v io la . I t

outlines the somi descending minor third which is characteristic

of mipentatony. I t w ill be referred to as motive 'X' in the following

discussion.




Figure 198: Measure 27, Motiv e 'X '

Once Bartdk has est abl ish ed th is motive, he uses i t to generate

a fou r measure fo lk so ng lik e passage. The fou r measures (31 34)

are in a bina ry arrangement. An Eb pentaton y (subdominant region )

begins at 31, sh ift in g to the ton ic axis in 32. Measure 33 sta rts

over in Eb penta tony and s h if ts to a G pentatony. The accompaniment

from the intro duc tion is reintroduced by the vio la and c el lo . A

ton ic a xis pedal on g is sustained in the second v io li n from 263 4.

The folksong passage is given in the following example with

re la ti v e solmization and chords given where po ssib le. Lendvai's

theories do not discuss all possibilities, concentrating rather on

those areas which comply p e rf e c tl y to his idea s. More analy sis needs

to be done on the music of Bartok using Lendvai's theories to either

prove or disprove the cr e d ib il it y of his concepts.




140

h \o ^

So ft. irt\

iff T v 'r --------- ---- jjr prT* ~~lBL* l;T 1 ....P-4k i --- ----------------------------------------

re.r *■I . . V - Q . - r - ,

A , SL _........... ... . ■,,1 V

• • 1 } i ....:

] . = B r - ■ ■=

n i ^ \ r r , \ T T1< ,iii( 11■£.J I 6a O.mm'’ a11

Cadence&Y GOw*'


The next part of the (C) section is developmental (3541).

Motive X is expanded to form in te rlock in g penta tonie s. The music

is shown f i r s t , followed by a diagram demonstrating the method whereby

the pentaton ies are joined tog eth er . See Figures 200 and 201.




141





142

k *3 * *

<9

= Emi la so do

mi la so do

mi la so do

PE

r r i'

so la mi

do so la mi

do so la mi

t

XL,,. Q

do so la mi

do so la mi

do so la

4z.Qxk

la so do

mi la so

mi la so do

Figure 201: Int erl ink ed Pentatonies




143

The f i l l e d in tr ito n e motive in the above example is no longer

exchanged between voices as an independent moti ve. In th is sectio n

i t is conf ined f i r s t to the ce l lo (3536) , then v io l in I (3738)

and fi n a ll y to the cell o again (4 1 42 ). By the time the motive is

in the ce llo fo r the la st time , i t is no longer an independent un it

of five notes, but is linked together into a successive chain of

motives.

The clim ax of the movement is in measures 40 4 1. Bartok

emphasizes i t with a fo rt e dynamic and in ter loc kin g pentatonies in

s tre t to and the appearance of an A gamma chord a t the peak of the

climax. Figure 202 shows the music, Figure 203 diagrams the jo in in g

of the pentatonies.

A y


(Pole exchange)

mi la so

mi la so do

Figure 203: Diagram o f Measures 4041




144

A re tr an si ti on (43 46) completes the (C) sec tion . Measures

4345 unwind from the climax point of measure 41 and act as a transi-

tio n back to the rec ap itu la tio n of the chor ale. A modal cadence

(cmm7D and A) in 4546 fini sh es the (C) se ctio n.

a l L a r g o , J . 3 5


The second movement ends wi th a retu rn of the t r i l l s of the

(A) sec tio n. These grad ua lly fade away to a sin gle descending line

in the cello.




145

con sord.





CHAPTER FIVE

MOVEMENT THREE

At the center of the Qua rtet's palindromic structure is a

symmetrical movement, a scherzotrioscherzo, which acts as a bridge

between the fi r s t and la s t two movements. The f i r s t scherzo (1 66 )

is a rounded binary [ABA]; the second scherzo (192) is shortened

to binary [AB] with a Coda.

Placed between the two scherzos, the t r io (1 6 5) is the apex '

ofthe en tir e Quartet. I t is the one important pa rt of the Quartet

which is c le a rl y organized by the formal pr inc ip les of the golden

section proportions.

Although i t is pos sible to d erive minor examples of golden

section proportions in most of the other movements (Perle, for example

[P er le 1977, pp. 20 7208] argues fo r golden sectio n in movements

1,2 and 4) i t is th is au tho r's contention t ha t the place which conforms

closest to Lendvai's stipulation "it can be observed [that] the golden

section always coincides with the most significant junctionpoint

of the form" is the t r io o f the thi rd movement. (Lendvai 1983, p.

326) A discussion o f the use of golden section may be seen in the

section on the tr io .

Figure 206 is the s tru ctu ra l cha rt fo r movement thr ee.

146




147

Scherzo I (16 6)

Intr o (A) (B) (A) Tr

12 324 2449 5064 64661. 2436

2. 3649

Tr io (1 65 )

Scherzo I I (192)

(A ) (B) Coda

129 3057 5892

1. 3040 1. 5874 2. 7492

2. 4057 a. 5865

b. 6673

Figure 206: Formal Struc ture of Movement Three

The tonal center of this movement is E, the counterpole of

the main branch of the ton ic ax is . Figure 207 shows the axis system

for this movement.

E

Bb

Figure 207: Tonal Axis fo r Movement Three




148

Scherzo I

(a) Section (123)

The opening harmonies es tab lis h the ton al ce nt er o f E by

the use of EM and it s re la t iv e chords, c#mm7 and a#dm7.

P

i


Bartok then uses a series of substitute and relative chords

(7 8 ) which at f i r s t seem unconnected to the tonal scheme (d#mm7

BMM7g#mm7— EMM7).




149


The d#mm7 (m. 7) goes to i t s substi tu te chord BMM7, which

in tu rn moves to i t s r e la t iv e chord, g#mm7 (m. 8 ) . The g#mm7 then

reso lves to i ts su bs tit ut e chord, EMM7, the toni c hypermajor chord.

d#mm7--------------------------- »BMM7 (S ubsti tu te )

g#mm7.............................. >EMM7 (Substitute)

(Relat ive)

Figure 210: Diagram of Movement Between Chords in Measures 78

Inte rspers ed w ith the harmonies are centers of pentatony.

The f i r s t such example is in measures 57.




150

ijka !*» *» *•>»• '*• 11 rc

i t U+I ■** ,Kt

G# Mipentatony

CJ) f t Q:mi so la do re


Both measures c le ar ly il lu s tr a te G# mipe ntaton y, a pentatony

s im ilar in cons tru ctio n to the harmonies seen in measures 3 7. These

s im il a ri ti e s are diagrammed below.

mi so la do re

j L j e l —

do mi so la do mi ( f i ) la do mi

EM c#m a#dm7

Figure 212: Comparison of G# Mipe ntat on y and EM, c#m and a#dm7

Substitute chords again assumeimportance in 1319. There

are two examples of f u l l y diminished seventh chords which are su bs ti-

tutes f o r th e ir majorminor cou nterp arts. One is in measure 13,

the oth er is in measure 16. Both are shown in F igure 213.




151

(trm t)

\

. ■■— — ■\z>

— 4 — 1— | i - - - -* I t? ^

! / 'i f . L r ^ L - j j J

( E V* W ^

Figure 213: Measures 13 and 16

Other substitute chords in this section are more complex

in th e ir re la tionsh ip s. For example, the two chords in measure 14:


At f i r s t appearance, D#Mm7 and f#dm7 seem un re la ted to one

another. They are neith er re la ti v e ly or modally rela ted to one an-

oth er, or dir ec t su bst ituti on s. (Major and minor sub stitu te chords

must have roots a major t h ir d ap art from each o th er ).

The re la t iv e chord of f#~dm7 would normal ly be amm7. Here,

Bartok uses the counte rpole of A, which is D# instead of A. (A pole

and counterpole may be exchanged without change of fu ncti on ) . D#Mm7




152

is used in the place of its pole, amm7, as the relative chord to

f#dm7. See Figure 215.

PE

Relat ive:

r amm7 it-

L f#dm7

d#mm7----------------*D#Mm7 (Modally related)

G

Bb

Figure 215: Diagram of Measure 14 Progression

A s im il a r event occurs in measure 18.





153

This tim e, g#dm7 is the focal chord. It s re la ti v e chord

would norm ally be bmm7. The cou nterpol e of B is E#. E#Mm7 is su bst i-

tu ted fo r bmm7 and reso lves d i re c t ly to g#dm7. This is diagrammed

in Figure 217.

PEi---------------------------x

bmm7 e#mm7................... »E#Mm7 (Modally related)

g#dm7

G

Bb

Figure 217: Diagram of Measure 18 Progress ion

Measure 17 con tains a D gamma chord throu ghout the en ti re

measure. The use of a gamma chord is important as Ba rto k uses few

alpha harmonies in the third movement.

D y





Measure 19 (Figure 219) is a negative dire ct ion re lat io ns hip

between C#MM7 and emM7. Normal ly, fo r a posit iv e di rec ti on movement,

C#M would move down to a#m (root movement down a minor th i r d ) . Here,

the movement is negative because the root movement from the major

chord to the minor chord is up a minor third instead of down a minor

th i rd . I t should also be noted th at the emM7 is a hyperminor chord.


Figure 220 shows the passage ju s t discussed (mm. 14 19) as

a u nit todemonstrate how the chords are in te rr e la te d . Measures

14, 16, and 1819 a ll contain li n ea r chords of importance, measures

15 and 17 contain simultaneities.




155

oni


(B) Section (2449)

The (B) sect ion (24 49 ) is in the subdominant region of C,

contrasting with the tonic region, E, of the (A) section.

The opening part (2429) has a thicker texture and is more

contrapun tal than the (A) sec tion. Much of the ma teri al is derived

from the (A) section, such as the arpeggiated harmonies and penta

tonies, but, here, they are used in stretto and canon as accompaniment

fo r a new melody in the f i r s t v io li n (2 4 3 6) . The melody is in two

phrases, mm. 2429 and mm. 3036. Phrase one is l i g h t l y scored,

while phrase two has a thicker texture.




156

Within this passage, measures 2728 contain an excellent

example of a fullydiminished seventh chord as a direct substitute

for a majorminor seventh.


However, in measure 29, the chordal re lat io ns hi p is more

complex. F ir s t, a look at the son oritie s.

The bdd7, as seen above, is the direct substitute for BbMm7.

The bdd7 chord would normally resolve d ir e c tl y to CM or cm in Western

harmony, (a CM chord appears on the downbeat of measure 30 ). F i r s t ,

however, i t must pass through F#Mm7 in the second v io li n . In Lendvai's

modal theo ry th is is qu ite acceptable as F# is the counterpole of

C and can be fr e e ly exchanged with i t with ou t change of fu nc tio n.





157

b°7 CM/ CM/ F#Mm7 b°7 F#Mm7

cm cm

E

G

Bb

Figure 223: Diagram of Measure 29 Progression

The viola line is more interesting.

m

Figure 224: Measure 29 (V io la Line)

Here, the bdd7 moves to c#m. To understand how a bdd7

may move d ir e c tl y to a c#m chord i t is necessary to f i r s t re al iz e

the normal r e la t iv e chord of c#m is EM. The next step is the exchange

of E and Bb, pole and counter pole. Because poles and counterpo les




158

may be exchanged with one another without a change of function, Bb

may be used in the place of EM.

PE

EM

c#m

’T'BbM

•D

G

C

Bb

Figure 225: Diagram of Measure 29 Viola Line Progression

F in a ll y , the bdd7 is the d ir ec t su bst itut e chord fo r BbMm7.

Thus the bdd7 ( su bs tit u tin g fo r BbMm7) is exchanged with EM ( i t s

cou nter pole) and moves to c#m. The fol lowing diagram w il l c l a r i f y

th is fur ther .

________ PE

EM S)M----------------------- b°7 (Substitute)

4

c#mc#m b°7

Figure 226: Continuation of Figure 226 Progression

Bartok fu rt he r enhances thi s complex rel at io ns hi p of chords

by reso lving them to a C gamma chord instead o f a sustained CM or

cm tr ia d (th e downbeat of m. 30 is a CM chord, but i t qu ick ly becomes

part of the sustained gamma chord.




159

a 3 0

C yC ^

C y


Measures 4547 present a rare example of consecutive perfect

fourth s in th is movement. The ce llo li n e is a str in g of consecutive

p er fe ct four th in te rv al s moving downward from E to D#.





160

(A) Section (5064)

The return of the (A) section (5064) is similar to the opening

of the movement except the tonal cent er has sh if te d from E to C#.

C# is the pole on the side branch of the tonic axis.

BbFigure 229: (A) Section Tonal Center

An example of polar related chords occurs in measure 52.

Polar related chords are minor and major chords whose roots are a

minor th ir d a part . This measure conta ins an F#Mm7 moving to amM7

(the hyperminor chord).


Hypermajor chords appear frequently from measures 5459.

Lendvai would consider th is passage to have a str on gly major cha rac ter

because of these hypermajor chords.

E

G

D

G#




161

S<j »«»

IT

CMM7 FMM7


roduced with permission of the copyrigh t owner. Further reproduction prohibited without permission.



162

The tr a n s it io n (64 66 ) to the tr i o is an A gamma chord which

resolv es to an ' a ' minor chord. The A and C are prepared in the

tra n si tio n and sustained through the f i r s t sixteen measures of the

t r i o . The A gamma chord resolvi ng to the ' a 1 minor chord is shown

in Figure 232.

A y c


Trio (165)

Cent ral to the th ir d movement is the tr io . The tonal region

is t he subdominant, th e ACEbF# ax is . The axis is shown in Figure

233.




Figure 233: Tonal Center o f the Trio

I t begins w ith an eig ht measure intr odu ctio n which establishes

the cha racte r of the e n tir e tr io by presenting the one measure ostin ato

which pervades the tr io . This motivic lin k o f the tr io is a ten

note passage which is exactly one measure in length and is repeated

in moto perpetuo for the rest of the trio with rare breaks.

The opening ten note passage spans chromatically the interval

of a pe rf ec t f i f t h (P 5) , from F up to C. These notes may be assigned

solmization syl lables.

i r f * * Y Y T f Y Tdo di f i so re ma mi fa di ma

Figure 234: Measure 1 and So lmizat ion Sy lla bles




164

Bart6k maintains this pattern which outlines a P5 for fourteen

measures. In measure 15 (Figur e 23 5), the lowest pit ch , F, is sh ifte d

upward a h a lf step to F#, making the span a tr it o n e (TT) instead

of a P5. This pat te rn is repeated from measu re 15 to 22. Beneath

i t in the c el lo is a whole tone scala r passage from 2122 (Figure

236).


a . ^


Measure 23 is one of the rare breaks in the perpetual rhythm.

I t separates the f i r s t set of rhythmic patterns from the follow ing

set which begins the motive of the P5 interval again.




165

Measures 2426 once again use the span of a P5, with a ll

ha lfste ps inc lus ive . This time the in ter va l is from G to D, empha-

sizing the tonic region instead of the subdominant.

pLf:T f Y >f do di f i so re ma mi fa di ma

Figure 237: Measure 24 and Solm ization Sy llable s

Bartok sets o ff this return to the opening tr io patte rn by

reducing i t to on ly thre e measures and a fu l l measure res t separating

i t from the patte rn which follow s. The tr ito n e interva l pattern

(G#D) is also only th ree measures long (2 8 3 0 ), making the whole

passage of P5 and TT in te rva ls only seven measures long in comparison

with the opening passage of twenty three measures.

The next section (3156) smoothly emerges from the preceding

section by having the lower pitch of the TT, G#, become the bottom

of the P5 in te rv al AbEb. This sec tio n, however, gives but two mea-

sures to the P5 in te rv a l and six to the TT span.




166

The li n e s h ift s upward one more half s te p to A fo r the las t

of the P5 span pa tt er ns. I t last s only two measures and has no TT

counterpart .

The f i r s t 40 measures of the t r i o may be grouped toget her

as a unit because of the single line repeating pattern which prevails.

Measures 4143 are the f i r s t major change in the t r io tex tu re.

Bart6k p laces the P5 span pat tern on AE, alr ea dy estab lish ed in

3940 in the f i rs t v io l in . Against i t , he places i ts in te rv al l ic

in ve rs io n, spanning a P5 (A down to D ), in the second vi o li n .

J

s





167

The climax of the t r i o is in measure 44, where ye t another

P5 in te rv al pa tte rn is added in the v io la . The or ig in al AE P5 is

in v io lin I , an inve rted P5 (AD) is in vi o lin I I , and the new one,

an upward P5 pat te rn (GD) is in the v io la . Figure 239 shows th is

pattern complex.


At measures 4951, Bartok places the only upward P5 form

in the vio la and the inverted form in v io lin I and I I . The f ir s t

v io li n par t spans a P5 from G down to C; vi o li n I I is a P5 down from

AD and the vi o la is a P5 up from GD. This p a rt ic u la r arrangement

is three measures long.

There is one measure of two simultaneous patterns (52).

The upward P5 span (CG) is in vi o li n I I , the in verte d form in the

vio la spanning C down to F. This is the f i r s t appearance of the C

down to F in te rv a l. I t is important fo r being the lowest reg ist er

and fo r it s contin uation fo r the remainder of the tr io .





The t r io is 65 measures long. This number, 65, m u lt ip li ed

by 0.6 18 , gives the product 40. The clima x, or golden sec tion , of

the movement i s on the downbeat of measure 44. Gra ph ic al ly , the

form would look like Figure 241.

Bartok emphasizes the golden sec tion poin t with a double

fo rt e dynamic marking and three simultaneous pa tte rns . He has b u il t

up the f ir s t fo rt y three measures of the tr io to this point with

gradual upward sh if ts in re g is te r and a movement in dynamics from

pp to f f . The remainder of the movement is a quick lowering of reg is -

te r , thinn ing of te xtu re and reduction of dynamics un til only a sing le

vio la in a low re gi st er is playing the patte rn. The ending of the

t r i o , from the golden section onward, is shown in Figure 242.

Golden Section

m. 44

Figure 241: Golden Section Diagram




169

jgj3 jppgta

alJ .1.144

,Pjgg jp

jPpjg.ja ShUPPHtJBB.

r r JQP jpg £]jai^5.

—-r—

*«.. 1 .

sr r-Tr-i/Fn*, 1351v s JWW=WSIK'eWitt

Q- -jrzz-----:---------.-'If

_4,<

lilfr far-J *r I* r»tjaffled

—: : ■*nt—■Prii o w 3

" >-. -— I

HI— ‘----- j--------1------

p ===== f-i

' t.:y,'"'--------

:„«lpj C>>t¥

F & = £ * = iijvjft *, .I=f=d .,X A





170

poco a poco rallent.

i r x i ..........


I t is fu rth er s ig n ific an t tha t a golden section appears at

this point in the Quartet . The tr io is the very center, st ru ctu ral ly,

of the en tire Q uartet. Bartok waited u n til the center point of the

palindromic structure to present the most arch itected at tri b u te of

the whole Quartet.


13866667



Another less obvious aspect o f the t r i o may be seen in the

pitch collection produced by combining the starting and ending pitches

of each P5 in te rv a l in the climax area (mm. 44 5 2 ). These notes

are charted in Figure 243.

]

0

4--------- __ _ n ______

...................... O '

Figure 243: Pitch Co llec tio n of Climax Area

Placed in descending p er fe ct f if t h s (based on the P5 span

of the motive) from the note of the highest re g is te r (E) to tha t

of the lowest register (F), the arrangement EADGCF is produced.

Rearranged, these notes produce an E mi pe nta ton y.

mi so la do re

Figure 244: E Mipenta tony

E is also the tona l cen ter o f the th ir d movement and the

counterpole of the Qu ar te t's tona l c en te r, Bb. Movement one begins

in Bb, and movement 5 ends in Bb. Movement th ree , as the cen tr al




172

point of the Quartet is in the counterpole to n a lit y of E. By placing

a mipentatony on E in the tr i o , Bartok has reinfo rced the palindromic

tona l st ructure o f Bb to E to Bb.

Scherzo I I

The third movement is closed by the second scherzo (192).

Instead o f a fu ll rounded bina ry , on ly (A) and (B) appear, with an

added Coda. Harmonica lly, the second scherzo is si m ila r to the f i r s t

scherzo. The region of to n a li ty is once again in the ton ic area

of E, a ft e r the subdominant area of C in the tr io .

The scherzo opens with the use of substitute chords in measure

1 and re la t iv e chords in measure 2. In measure 1, dmm7 resolves

directly to its major substitute chord (BbMM7) a major third lower.


In measure 2, c#mM7 (a hyperminor chord) moves by negative

dir ec tion to it s upper re la tiv e chord, EMM7. Positiv e dir ec tio n

would have been from EMM7 to c#mM7 (down a minor th i r d ) .




173


Measures 1928 a?e an interesting study in the use of a single

ax is and pola r exchange. The music is shown in the example below

with a discussion following.







175

The best way to demonstrate th is is to analyze the section

by instrumentation and motivic use.

F ir st , the vi o l in I l in e. Each of the tonic axis pitches

arec ir c le d and marked pole or counter po le. The D# in m. 19 and

B# in m. 20 belong to the subdominant ax is , and the B in m. 21 and

D in m. 22 are from the dominant ax is .Al l of the other pitches

are from the tonic axis.

The c e ll o li n e (1 92 2) is more obvious. I t moves back and

forth between C# and G, counterpole and pole on the side branch of

the ton ic a xi s. When the s ixt eent h notes are added, a whole tone

pattern is seen.

L — r f =L±

Figure 249: Measures 1925 , Vi o lin I

Figure 250: Measures 19 22 , Ce llo




176

I t is necessary now to look at the v io li n I I and vi ol a parts

from 19 22. V io li n I I al te rn ate s an emm7 chord wi th a C#Mm7 chord

wh ile the v io la a lt er nat es a C#Mm7 chord wi th an a#dd7 chord.

' I \ Jj«I * ^

Figure 251: Measures 1922 , Vi o lin I I and Vio la

Al l the roots of these chords, C#, E and A# are poles on

the tonic axis.

E

A#

Figure 252: Tonic Axis : C#EA#

Seen this way, Bartok is simply building chords on tonic

pitc hes and al te rn at in g them. However, the y can also be viewed in

terms of relative chords and polar substitution.




177

Beginning with the emm7 chord, the r e la ti v e and modally r elate d

chords may be derived as such:

GMm7

EMm7 emm7

C#Mm7--------------c#mm7

gdm7

•c#dm7

Figure 253: Rel at iv e and Moda lly Related Chords of emm7

As seen in the earlier discussion of function, all of these

chords have a tonic function because they are all either relative

or modally re la te d chords. Thus, C#Mm7 and emm7 are re la te d and

have the same fu nc tion . Normally , the upper re la t iv e chord of emm7

would be GMm7. Bartok has made a polar exchange o f C#Mm7 fo r GMm7.

PE

GMm7

temm7

C#Mm7

G

Bb

Figure 254: Diagram of Polar Exchange

Although the a#dd7 is not adequately explained in Lendvai's

theory, in this particular context, Bartok appears to be using it






179

The ce ll o continues the al te rn ation of C#Mm7 and a#dd7 already

established in the viola (mm. 2325) where c#dm7 is shown to be

the subminor chord below the emm7 chord es tabl ished in vi o li n I I

(1 92 2 ). See Figure 256. This time the dd7 (e#dd7 ) needs to be

int er pr ete d as a di re c t sub stitu te chord fo r i ts Mm7 coun terpar t,

EMm7. EMm7 is the modally re lated chord to emm7 and shares the same

tonic function.

emm7 }EMm7 ê#°7 (Substit u te )

c#dm7

Figure 256: Diagram of Viola Progression

The sign ifi ca nc e of thi s passage from 1928 is Ba rtok's use

of a single axis both melodically and harmonically.

The (B) section (305 7) is qu ite si m ila r to the (B) section

in scherzo I . A notable diffe ren ce is the use of the melodic fig ur e

in canon and s tr e tt o . Scherzo I (B) section placed i t in the v io li n

I only and did not present i t in s tre tto .

Harmon ically there are two chords of in te re s t. To emphasize

the ton ic axis (and esp ec ia lly the note E as counterpole to Bb) an

Ehypermajor chord is used twice in eigh t measures. The f i r s t occur

rance i s in measure 44 , the second in measure 51. Both of these

are shown in Figure 257.




180

51 T

E KUX"

mEMM7


The othe r chord of in te re s t is a C gamma chord in measure

49 which has a subdominant function.

if* 3^rr

£ SEjET U 1 ~ ± =

i iT~T r ~J—CJ \ —

i^ F r f —

f rrf { i iiT? r f

W

C y

-8-

C y


The Coda (5 8 92 ) is a mix ture o f elements from the re s t of

the movement. In the fi r s t section (5865 ) the rhythm ically accented

eigh th notes move ch ro m ati ca lly from c#l t o e2. This is demonstrated

below with the rhythmically stressed eighths circled to show the

chromatic movement.





The next section (6673) makes use of parallel motion in

minor th ird s and minor six th s, both of which are Fibonacci int er va ls

of 3 and 8 re sp ec tive ly . Movement between notes is also in Fibonacci

increments ( 1 ,2 ,3 ,5 ,8 ) . A 1:2 model occurs in measure 73 in the

vi ol a and ce llo as they move in p a ra lle l minor thir ds .

The remainder of the Coda (7492) recalls material from the

(A) sec tio n of the scherzo. The movement ends with a 'c la ss ic a l'

dominant cadence: G#Mm7 to c#m.




* * 4 ’ ......

r — h .n*~£

IP m t i u

»reoTP

G#Mm7 c#m


eproduced with permission o f the copyright owner. Further reproduction prohibited without permission.



CHAPTER SIX

MOVEMENT FOUR

The fourth movement (Andante) is a variation of the second

movement. Most of the movement is based on the (B) se ct io n, the

ch ora le , of movement two. Figure 261 shows the s truc tu re of the fou rth

movement.

[A] CB] [C] Tr [Bl] [B23 [B3]122 2342 4254 5460 6063 6481 81101[A] CB] CC] var on var on var on

[8] [8 ] [B]t r i l l s chorale t r i l ls new motive [Bl] + [C] tremolo S

replace tremolo become added between pizz returnpizz chorale scales fragments of

chords chorale

Figure 261: Formal Str uc ture of Movement Four

The tonal cen ter of th is movement is G, the counter pole of

the second movement's t o n a li ty of C# on the side branch of the ton ic

ax is . Figu re 262 shows the axis fo r th is movement.

183




184

G

C#

Figure 262: Tonal Axis of Movement Four

(A) section (122)

Movement fo ur is based on mate ri al from the second movement.

Most of the structural si m ila rit ie s are in the f i r s t ha lf of the

movement. The remainder of the movement is b as ic a ll y a set of v a r ia -

tio ns on the chorale (B sec tion ) melody. This discussion is intendedi

to show the si m il a r it ie s between the two movements. In ad di tio n,

p a rt ic u la r chords and other areas of i n te re s t w il l be shown.

The fourth movement begins (15) with a variation of material

from the second movement. (See Figure 263) The t r i l l of the second

movement has been replaced by repeated notes in the fourth, but the

in te r v a ll ic stru cture s have remained in ta c t. (See Figure 264) Passages

from both the second and fourth movements are shown below (Figure

263 & 264) to demonstrate the s im il a ri ti e s and diffe ren ces between

them.




185

Figure 263: Movement 2: Measures 14

Figure 264: Movement 4: Measures 15

The to n a li ty of the fourth movement is G, the counterpole

of the second movement's to n a li ty , C#. This poleco unterp ole re la ti o n -

ship is present throughout the (A) section (1 2 2) . The ce llo plays

the tr i ton e (TT) interval l in ea rly , out l ining at f i r s t only the tonic

ax is (G and C#) ; then inc ludes the subdominant (F# and C) and dominant

(F and B).




186

Figure 265: Measures 118, Cel lo Line




187

In measure 16, the line a r c e llo tr ito n e is pa rt of a G delta

*7>

G6

Figure 266: Measure 16, G De lta Chord

(B) Section (2342)

The (B) section (2342) is modelled on the chorale of the

second movement. Both the or ig in al chora le and th is version of

i t are based on an A mi pe nta tony. The A mipen tato ny in the second

movement c h o r a l was der ived through the roots of the chord progression

a li n ea r de riv ati on . In the fourth movement, Bart6k derives the

same pentatony v e r ti c a ll y . The f i r s t measure of the chorale (m.

23) and measure 32 are each a simultaneity of five pitches.





188

From the lowest re g is te r moving up, they ar e: FCGDA.

Rearranged, they form an A mip enta tony.

I

i _Q_

mi so la do re

Figure 268: A Mi pen tatony Diagram

Measures 29 and 35 are rare examples of a third tower.


Measure 29, s pel led from the lowest pi tch upward is : AbCEbGBb.

The lowest fou r notes of a th ir d tower are a hypermajor chord. The

simultaneous third tower arranged linearly creates the partitioning

of perfect f i f t h s . The c h o r d in measure 35 is b u i l t on Bb. From

the lowest pi tch upward i t is sp ell ed : BbDFAC. Both chords

are diagrammed in Figure 10.




189

Figure 270: Diagram of Measures 29 and 35

The lowest four notes are a hypermajor chord when placed

in a sim ulta nei ty. When arranged lin e a rl y , the pa rtit io ni n g of per fect

f i f ths is ev ident .

There is a chord of the four th in measure 37. B u il t on E,

i t is spelled EADG.

j t o^ g r

4th Chord


The remaining chord of note is in 39 4 0. An E gamma chord

is sustained for two measures to close the chorale.

2=P§E y


PP __________

J

E y

1 ~






191

Within this section, there are two interesting passages which

s h if t from one pentatony to another. One sh if ts using po lar exchange,

the other by re la ti v e sol miza tion . Both passages begin in E mip enta -

tony and end in C mipenta tony, but sh if t by d if fe re n t means.

Measures 4344 s h if t pentatonies by po lar exchange. E mi

pentatony becomes C mipenta tony .


The sy ll ab le 'do' on 'C' is exchanged with it s coun terp ole,

Gb, becoming 'so' of the next pentatony.

Measures 4445 sh ifts pentatonies by re la tiv e solmizatio n.

'La ' becomes 'do' and E mipenta tony becomes Eb mipenta tony ; 'mi'

then becomes 'so ' and Eb m ip entatony becomes E mi pen tatony . See

Figure 276.




43

m f r f ? ‘ r r ‘ r r r J~y E mip enta ton y: do t i do

Eb m ipe ntat ony : so la

C mipentatony: do

mi

so la t i la mi

Figure 276: Measures 4445 and Diagram

Measures 5460 are a transition to the next variation on

the chorale theme using short chromatic figures in original and inver-

ted forms set in canon.

(Bl) Section (6063)

This variation on the chorale is very short, consisting of

an invert ed E alpha chord intersperse d with a new li n ear motive (motive

Y) which outlines a minor th ird in the las t h a l f of measure 60 in

all four instruments.




193

f j£

S.KtV

r C f r w *

i 1 j f ^ t

^-ni t

t i i — R h l t

Ea

£ f

c K A k M l \ ^

G

Bb

Ea


In measure 63, Bartok uses a lin ea r th ird tower (f if t h p a rt i-

t ioning) to transist from the (Bl) to the fol lowing (Cl) section.

The th ird tower is b u il t on C and is sp elled CEGBD.

m

CMM7 = Hypermajor





194

(B2) Section (6481)

The (B2) section is a combination of material drawn from

(B l) and (C ). The seven note motive X of the (C) sect ion is tran s-

formed int o an eigh t note motive. I t is placed al te rn at e ly in ce llo

and viola (6466) making a perpetual rhythm between them.

In measures 6 78 0, motive X is doubled a t the octave in the

viola and cello on every beat; they no longer alternate with each

other.

The minor th ir d motive (motive Y) of (B l) appears as a melodic

device, usually as part of a longer melody.

;ss:

x <





195

In te rloc ki ng pentatonies are used in 65 and 6768. The penta

ton ic passages both begin on a D mipe ntato ny and s h if t through B

mipen tatony and G# mi penta tony. The music is in Figure 280 and

the relative solmization process is in Figure 281.

m

Figure z80: Measures 65 and 6768




u 7 ^ si.

Vl n . I : ___ ^

196

4 0 .

Ip5 _D_

do so la mi

do so la mi

do so la mi

so la mi

do so la mi

do so la mi

Figure 281: Diagram of Measures 65 and 6768

The (B2) section is another v ar ia ti o n on the chora le. Although

not immediately apparen t, the melodic ma ter ial is based on the orig in al

chorale in the second movement.

This section is a series of short segments (o ri g in a ll y drawn

from movement two, then var ied ) which a re repeated and interchanged.

These one measure os tin ato patter ns ri s e in dynamics and in te nsi ty

to make the climax of the movement in measures 7379.

(B3) Section (81101)

The remainder of the movement is a more obvious v ari a ti on

on the cho rale theme. A chorale text ur e of sustained chords returns

which uses chords and alpha harmonies. The eigh th note motive and

the minor th ir d motive have also been ret ain ed as accompanimental

f igures.




197

A por tion of measure 89 is a C gamma chord which becomes

a G gamma chord from 90 91 .

C y

Ji. '6L

8CLM. Cy G ^


The movement ends on a G gamma chord in measure 100, which

resolves to a sing le B in vio lin I in measure 101.

Gy

Figure 283: 100101

The passage from 81101 is shown below with harmonic analysis

where possib le . As seen in the previous movements, Lendvai's the or ies

do not adequately explain all harmonic possibilities.




198





199

GfA G y

P i u a n d a n t e , J > m

so*. Gy cfvTe*' 9Wc j , |U | e n g i r a l l e n t a n d o

Figu re 284: Continued




CHAPTER SEVEN

MOVEMENT FIVE

The f i f t h movement i s long and complex, pri m ar il y making

use of techniques of canon, stretto and various inversional methods

discussed previously . S tr u c tu ra ll y , th is movement may be seen as

a sonata form. Based on f i r s t movement m ate ri a l, the exposit ion

and rec ap itu lat io n are va ria tio ns on the f i r s t and second theme areas

of movement one. The development is a t r i o s ection followed by a

fugato and episode. A r e la ti v e ly br ie f coda completes th is lengthy

movement. Figure 285 shows a st ru ct ura l ch ar t fo r th is movement.

Intro Exp.

114 FT

1454

ST

55109

FT' Tr

109149 150200

Dev.

Trio

202368

1. 202358

tr 359368

Fugato

368484

Episode/RT

484527

Intro* Recap

527 56 4 FT ST

546623 624698

Coda

ST' ST

699720 721780 781828

Figure 285: Formal St ructure of Movement Five

2 0 0




201

Tonally, movement five begins in E, the counterpole on the

main branch of the to nic axi s , and ends in Bb, the to n a li ty which

opened the en ti re Qu arte t. Unlik e the previous movements, Bartok

does not always deli ne ate the to n a li ty in a cle ar manner. Rather,

he blurs the ton a lity with rapid l in ea r, chromatic l ine s in str etto

and canon throughout the movement. Figure 286 shows the ax is fo r

this movement.

Bb

Figure 286: Tonal Center fo r Movement Five

Introduction/Exposition (1200)

A fourteen measure introduction b ris kl y leads into the fi r s t

theme area which s ta rt s f ir m ly on a unison E. The mat er ia l is taken

from the fi r s t movement's f i r s t theme. Measures 233 4 foreshadows

the motive desired from the f i r s t theme m at er ia l. Measures 3554

then use the two measure motive canonically.

The second theme (55 109) is a va ri at io n of the f i r s t theme

material ju st presented. The en tire passage is li n e a rl y constructed

and canonically treated.






203

Measure 184 introduces a five measure phrase which sets a

minor second (vi o lin I I ) , a minor thir d ( vi ol in I ) , a perfect fourth

(v io la ) and a perfect f i ft h (c ell o ) inter val against each other.

See Figure 288.

tii'V ft a i%v* 1&-7 'tsr __

%

i« —*

v;|. «i-m- = 4 = £ =

1 r T

5 * 5 = 3

‘ iPf-=

$ ^ = E E . —


Together, these notes produce a symmetrical configuration.

- \« 5 v<SCvVn 3 )

- 9- --------rro.-- 1—iwr 1 I ♦»-w tf

<Ai do

__ „ ft,., q p g

m■- — 1 1 If

H g~pj

Figure 289: Measures 184188 Diagram

Measures 189195 sequence this phrase, rescoring the intervals

(m2 in vio lin I , m3 in v io la , P4 in v io lin I I , P5 in c e llo ). These

intervals may again be combined to produce the same symmetrical con-

fi g u ra ti o n . I t cannot be considered an alpha chord as the axes are

not reg is t ra l ly de l ineated.




204

i°iu>

The transition closes with a 1:5 model in measures 196200.

\q-( \°\^ •3°°

m i

5 »


Development (202546)

Tr io (202 36 8). The tr io is canonic in str uc tu re. Measures

275291 highlight a chromatic passage in the rhythmically stressed

h a lf notes over a sp or at ic subdominant A pedal . Pole exchange is

used in 281287 in the lower thre e instruments. These aspects are

shown in Figure 291.





Measures 334345 are a scalar passage which rapidly exchanges

poles every four eig hth notes. Although poles may te c hn ic a ll y exchange

with one another without a change of function or further explanation,

they may be seen especially in passages such as this as an alteration

of both do and mi . The passage from 334345 is shown in Figure 292;

the relative solmization process is shown in Figure 293.




206





207

G

Bb

so la t i do so la t i do

or

do re mi fa do re mi fa

A Mi

pentatony

D# M i ma----------- mi so la t i do--------di

pentatony d ----------- d# b--------c

so la ti do

E M i pentatony

so la do re mi fa

A# M i

pentatony

Figure 293: Re lat ive Solm ization of Measures 334345




208

The t r io concludes w ith a 1:5 model (351-356),

5 1 5 v

utrepitwto

M f f D i t f M t

% tr tp i to 9 o

strrpitoso

*l"s i


The transition (359368) to the fugato contains a linear

C beta chord in the viola and ce llo (359364 ).

J A

99 3 4P »

1 -J— 1—

____

A

r A i r i r :i

t

CB

CB





209

Fugato/Episode (3 68 5 27 ). The fugato (368484) and episode (484527 )

emphasize po la r rela tio ns hi ps and alpha harmonies. Measures 369447

const an tly place E and Bb (po le and counterpole of the main branch

of the tonic aixs ) against each other. As this is ce nt ra lly located

in the movement, i t is possible Bartok wishes to once again emphasize

the polecounterpole relationship of Bb and E, which is so integral

to the to na lly palindromic struc ture of the Quartet. The Quartet

begins and ends on Bb and climaxes in the middle movement with E.

In the clos ing measures o f the fu gato, the re are two A gamma

chords (mm. 457 and 47 7), an F# beta (m. 473) and an F beta (m. 47 5).

These are shown in Fig ure 296.

F#B FfS

4 .[

#-&■

Ay F#g F6

Figure 296: Measures 457, 473475 and 476477




210

Measure 469 conta ins a ra re example o f a v e rt ic a l 1:5 model

chord.


Measure 461 contains a hyperminor chord.

•4&I

A mFig ure 298: Measure 461

Measures 465, 467 and 471 al l con tain hypermajor chords,

EN.KX'1 G^V*'

Figure 299: Measures 465 , 467 and 471




211

By mix ing hyperminor, hypermajor chords and alpha harmonies

together, Bartok blurs whatwould otherwise be a strong to n a li ty

or a character of e ith er minor or major.

The episode (48 4527) sequences a li n e a r 1:5 model three

times (490 49 2, 501503 and 509 51 1). See Figure 300.

mpti*pr s , 1

"11---- 11-------7

i 11 i 11:

zando 1Dill. 1

consord. tr ....~ I ft-"—...........»M i H— —

So*mr--r ■■

h j---------- It j- ----1

w * * ? r 5V .■rvlf fcrM Jfrsfe fefe

^ i---- ' ‘i j r T - j i ‘■■-t-. i ..

«r nr

•rco s

Figure 300: Measures 490 49 2, 501503 and 509511




212

There is also a sustained chord o f the f i f t h in 497-500.

* r —

» * *

r —

* » ( T O

|juL_ —j

1 75~

5th Chord


As was seen in the f i r s t movement (7 6 7 9 ), th is movement,

which is a v ari a ti o n of movement one, also uses a chain o f minor

seventh inte rv als (5 0252 0). The minor seventh interva l is ju s ti fi e d

as the inversion of a major second, an acceptable Fibonacci interval.

A po rti on of th is passage is seen in Figure 302.

j f a «cJ»rxa*rfJ

tiro 5 IX

m

ireo'.' Ill •

KMfiaM*

I

S?, o

/5o<\

La.... * , Z

in7 m7 m7 m7 m7 m7





213

Recapitulation (546780)

The fi r s t theme (546623) is sim ilar to the exp ositio n's

f i r s t theme area. I t di ffe rs in one respect, however, by sta rtin g

in the to nic area of E and Bb, then sh if ti n g in 562578 to the dominant

region by using B and D pedals. To na lit y is blurr ed from 579623

by linear chromaticism.

The second theme returns to polar exchange of tonic and domi-

nant po les . Measures 624635 are shown in Figure 303 with polar

exchanges marked.

ssgilSP

$

T = Tonic

D = Dominant

S = SubdominantG





214

Measures 632649 of the secondtheme furth er blur the ton al ity

by using a double pedal of F and G. G is a tonic pitch, whereas

F is a dominant p it ch . This resolves to a passage of gamma chords

(651660) as seen in Figure 304.

u s * v .s ’s v s s

lnmodoord.


The passage from 663672 is a close chromatic scoring which

produces a seri es of tone cl ust er s as was seen in measures 184195 .

As mentioned before, Lendvai has no analytical technique for discussing

th is type o f passage. They cannot be considered alpha chords as

the pitches are not registrally articulated by axes.

In measures 673685, the tone clusters are broken by a double

fo r t e , rhy thm ical ly ar tic ul a te d F# gamma chord.




215

pocoapocorail.

Esol loSV U 4

j .q

ukr\n ■=:

j g J F L > r Kt ' 'J" ■H>—r 4pr■" ■ L t i r ^ 7 ' 7 ~ 1

U hE E C t t r

F# y

Figure 305: 673685

This in tu rn moves to a si x measure passage (68 66 91 ) of

two and three layer alpha chords. Note the re gis tra l ar tic u la tio n

of axes.




216

A: C# alp ha

T = c# a#

S = f# a

B: E alp ha

T = e g

D = b

S = d# b#

T = Tonic

D = Dominant

S = Subdominant

C: C# alp ha

T = c# e g

S = d# f#

D: E alpha

T = e

D = g# b

S = d#

E: A# alp ha

T = a# fx c#

D = e

S = a c

Bb





I l l

Measures 6726 98 re tu rn to the F# gamma chord be fore moving

to a series of three la yer alpha chords. The fin al three measures

of t h is sec tion are a C gamma chord. This passage is shown in Figure

307.

allargando. . molto

IPS

in I 1f f __

F# y Ea A#<x Bby C# y

( 3) inv inv (3)

(2) (3)

C y

E alpha T = e g

0 = g# b

S = c d f

A# alpha T = a# c#

S = f# a

Bb alpha T = bb c#

D = d e#

S = f# a

C# alpha T = c# e

D b d g l

S = d# f#

( ) = Number o f la ye rs

inv = Inverted alpha chord

& Ar

E

6

Bb

&

F# y C y





218

Measures 699720 (ST1) are a variation on the second theme

in what G ri ff it h s and Karp?Jti term a 'ba rrel org an' sty le. (G rif fit h s

1984, p. 151; Karpati 1975, p. 243) This re fer s to a sim pli st ic

harmonization beginning in one key and ending in two keys (bitonality)

to r e fl e c t an out oftu ne barre lorga n. Measures 699710 are cl ea rly

in A major. In 711721, the harmonies are in A major, but the melody

in vi o li n I is in Bb major. Measure 721 retu rns to the prev iously

established harmonic style of the Quartet.

E Hrail..





219

Coda (781828)

The polar exchange idea is continued in the coda over a spora

t i c tonic E ped al. Measure 825 conta ins a D gamma chord; a chord

b u il t a major th ird above the fin al Bb pitch o f 828 giving the illu si on

of a 'romantic' dominant cadence.





SUMMARY AND CONCLUSION

Fibonacci Series and Pentatony

The theories of Erno Lendvai are based on the Fibonacci Series,

a series of integers which he assigns to consecutive halfstep grada-

tions of the chromatic scale . The numbers 1 ,2 ,3 ,5 ,8 .. . are manipula-

ted to produce two impo rtant cornerstones of his th eo ry , namely mi

pentatony and alpha harmonies. According to Lendvai, mipentato ny,

d ir e c tl y re lat ed to the folksong idiom, is the basic scale used by

Bartok. Alpha harmonies are der ivedby the interva l l i e relationships

created through the use of Fibonacci numbers. Alpha harmonies w il l

be discussed la te r in the summary.

Pentatony is formed by the sequence 3 + 2 + 3 + 2, th at i s ,

minor th ir d , major second, minor th ir d , major second. For example,

eg a cd . Lendvai labe ls thi s pa rti cu la r sequence of inte rva ls

using sylla ble s from Kodaly's method of re la ti v e solmiz atio n. There-

fo re , in mipe nta ton y, as the above is ca ll ed , e = mi, g = so, a

= la , c = do and d = re . The keynote , or note of importance in a

mipentatony is mi and not the do of Western harmony.

Pentatony, when applicable, is useful for labeling series

of pitches which cannot be explained using traditional harmony.

However, pentatony, li k e tr a d it io n a l methods, does have it s li m it a -

tions especially in chromatic passages which bear no particular relat

t ion to pentatonic structure.

2 2 0




221

Lendvai derives and labels much of his harmony on the basis

of r e la tiv e solmiza tion. Notes of the scale (major or minor) thus

have sy lla ble s assigned them. For example, in C major and it s re la ti v e

key, 'a ' minor: c = do, d = re , e = m i, f = fa , g = so, a = la and

b = t i .

These syllables may be altered by either raising or lowering

them. Do rais ed is d i , fa raised is f i , mi lowered is ma and t i

lowered is ta. Although, tec h ni ca lly, any sy lla bl e may be alt er ed ,

Lendvai lim it s his th eory to these above.

Lendvai deriv es thre e chords each o f major and minor q u a li ty

by the ra isi ng or lowering of these sy lla b le s. Major chords may

be l a d im i, domiso, masota and minor chords may be f i l a d i ,

la d im i and domaso.

Another chord derived by relative solmization involves the

low ering of mi to ma, then bu ild in g a ma jor chord upon ma to produce

'ma m ajo r'. F u lly diminished seventh chords, in Lendvai's theo ry,

are considered alt er at io n s of majorminor sevenths b u il t on do.

Domi sot a is a lte re d to d im i so ta (e .g . CMm7 would become c# dd 7).

Fully diminished sevenths are used as substitution chords for their

majorminor counterparts. Unlike tra di t io na l th eory, fu l l y diminished

sevenths are not used as leading tone chords in Lendvai's modal theory.

The two remaining chords to be discussed are the hypermajor

and hyperminor. Hypermajor chords are majo rma jor in q u a li ty and

hyperminor are minorma jor. Unl ike the oth er chords discussed thus

fa r , the hypermajor and hyperminor are not derived by re la ti v e solmi-

zat ion.




222

Lendvai' s va rie ty of chords is l im ited . Unfortunate ly, in

chromatic music such as Ba rto k's, a wider labe lin g vocabulary is

needed. In ad diti on , because the functional system is no n t ra d iti o n a l,

son oritie s are not rela ted to one another fu nc tio na lly as in Western

theo ry. Chordal r ela tio ns hip s may be described in fou r ways: r e la ti v e ,

modal, pol ar or su bs tit ut e. Re lati ve chords begin with a major chord

(dom iso). It s r e la tiv e chord is a minor sono rity b u il t a minor

th ird below the major (la d o m i). The re la tiv e chord of the minor

is b u i l t a minor third lower ( f i l a d i or f i la d o m i, which is the

subminor cho rd). For example, CMamf#dm7 (majo rmi no r su bm ino r).

Modal chords are those which share a common roo t, but have vary ing

q u a li tie s (e .g . CM, cm, cdm7). Polar chords are those re lat ed by

roots a tr ito n e ap art (e.g . CM and f#dm7). Subs titute chords are

of two types: a major chord subs ti tu ted fo r a minor and a minor

chord su bs tit uted by a major chord. The roots of su bs tit ut e chords

are a major t h ir d ap ar t, thus CM is substi tu ted by em and em by CM.

Function

Modal, re la ti v e or pol ar rela ted chords a ll have the same

fu nc tio n, according to Lendvai's modal theory. Western tonal theory

bases func tion on the pe rfe ct f i f t h symmetry around a centr al pitc h

(e .g . FCG). The lowest f i f t h is subdominant, the cen tral pitch is

to nic and the hig hest f i f t h is dominant. Lendvai bases his idea of

func tion on the symmetry of major thir ds about a cen tral pi tch (e .g . Ab

C E ). The lowest th ir d is subdominant, the cen tral pitc h is ton ic and




223

the highest th ir d is dominant. This w ill become cl ea re r in the di s-

cussion of the axis system which assigns function labels to pitches.

Axis System

Another major portion of Lendvai's theory, the axis system,

is used to determine and lab el modal fun ct ion. The ax is system is

based on the closed c ir c le of f if th s with a sequential o verlay of

functio n l ab el s. Using the axis system, Lendvai bu ilds a case fo r

chords with roots a t rit o n e apart dir ec tly rela ted to one another.

No other w r it e r tha t th is author is aware of has advanced thi s theory.

By basing fun cti on lab els on a major th ir d symmetry and a l -

lowing the tritone relationship to take precedence, the fourth and

f i f t h of trad it io na l harmony lose significance.

From the axis system, Lendvai derives three scalar patterns

which are cal le d the 1 :2 , 1:3 and 1:5 models. The 1:2 model is an

alternation of minor seconds and major seconds (an octatonic scale).

The 1:3 model a lt ern a te s a minor second and minor th ir d and the 1:5

model al te rn at es a minor second and pe rfe ct fou rt h. Although there

are passages in Barto k's F if th Str ing Quartet which f i t these models

ex ac tly much of the music eit h e r does not f i t one of the models or

is a variation of one of them.

Cadences in Lendvai's theory are d ir e c tl y re late d to the

axis system. Dominant pitches are relate d to ton ic pitches in four

ways: 'c l a s s ic a l' , which has a roo t movement up a pe rf ec t fo ur th ;

modal, which has a root movement up a major second; phrygian, which




224

has a ro ot movement down a minor second; and 'romanti c1, which has

a root movement down a major t h ir d . Chord q u a li ty of both the dominant

and ton ic chords may va ry ; i t is the r oo t p itc h and movement which

is imp ortant. I t is also important to note th at any dominant pitch

may move to any ton ic pit ch . Un for tun ate ly, t hi s author did not

fin d many cadences in the F if th Str ing Qua rtet which complied to

these re st ric tio n s. Perhaps Lendvai is too re s tr ic ti v e in his guide-

lines fo r cadences, although i t is possible the Fi fth String Quartet

is an exception in Bartok's writing in regard to cadential structures.

Alpha Harmonies

Lendvai has created a co lle c tion of chords which he ca ll s

alpha harmonies. They ar e: alp ha , be ta , d e lt a , gamma and epsi lon

chords. Alpha harmonies are used in conjunction with the other sono ri-

ties discussed and have their function determined by their respective

keynotes.

The alpha chord is based on pitches derived from the axis

system. Tonic pit ches are placed above a la yer of dominant pitches

to crea te a normal alpha chord. Dominant pi tches may be placed above

tonic to produce an in ve rte d form of the alpha chord. Through the

addition of a layer of subdominant pitches, a three layer alpha chord

may be constru cte d. The keynote of any alpha chord form ation is

the lowest tonic pitch.

The beta chord is generated around a perfect fourth interval;

the top note of this perfect fourth is the keynote of the chord.




225

To the perfe ct fou rth in te rv al , the interva l of a tr i t on e is added

below, using the bottom note of the perfect fourth as the top of

the tr it o n e . This is the ske leta l form. To i t are added a minor

th ir d above the bottom note of the tr it o n e , and a major second below

the top note of the perfect fourth.

The gamma chord is the chord most clo se ly as socia ted w ith

Bartok because of its unique combination of the major triad with

i t p ara l lel minor tr i a d . I t appears with considerable frequency

in the music of Bartok.

Type gamma is als o based on the pe rf ect fo u rt h . The key

note is the top note of the pe rfe ct fou rth . A minor th ird is added

above the top note o f the fo ur th and a minor th ir d added below the

pe rfe ct fo ur th . Fre qu en tly, the seventh above the keynote is added.

The de lta chord is s im ila r to the beta chord. Here, however,

the perfect fourth is on the bottom and the tritone placed above

i t . The keynote is the top note of the pe rfe ct fo ur th . A minor

third above the lower note of the tritone is added.

The epsilon chord is b u il t around a pe rfe ct f o ur th , but his

tim e, the accompanying tr it o n e is nei th er added above nor below,

but, r ath er interlocke d with the pe rfec t fou rth . The keynote is

the top of the perfec t four th. A tri to n e is bu ilt a major second

above the lower note of the perfect fourth and a minor third is added

above the top of the tritone.




226

Alpha harmonies, as described, do appear in Bartok's music

and Len dva i's la be lin g is u se fu l. There does not appear to be a

con siste nt placement of alpha harmonies s tr u c tu ra lly . They appear

within passages as well as climactic points.

Equidistant Scales and Harmonies

In Le ndvai's theo ry, there exist s a separate category of

scales and harmonies which are constructed using a single interval

of the Fibonacci Se rie s, cre ating closed systems, i . e . they always

return to the or igi na l sta rtin g pitc h. There are four scales/harmonies

of this type.

The fi r s t type is from major second in te rv al s. By using

major seconds, the whole tone scale is constructed . Lendvai labels

th is scale the 'omega s ca le '. A dd it io nal ly , chords may be formed

using only major seconds and are labelled omega chords.

The second type is from minor th ir ds . Using minor th ir ds ,

the f u l l y diminished seventh chord is formed. Lendvai does not d is ti n -

guish th is formation of the f u ll y diminished seventh chord from that

made by a lt e ri n g a majorminor seventh chord.

The th ir d type is from pe rfec t four ths. A series of perfec t

fourths may appear l in e a rl y (as a melody) or as a sim ulta ne ity of

pe rfec t fourt hs to produce 'fo ur th chords' . They may also be seen

in inversion as chords of the f i f t h .

The fou rth type is from minor sixths . The use of minor sixths

is limited to the formation of augmented triads.




I l l

Use of these equidistant scales and harmonies were limited

in the Fi ft h String Q uartet. This is not to say they are not important

but rather used sparingly for emphasis.

The Quartet

The Fifth String Quartet is an excellent example of palin-

dromic str uc tu re : the second h a lf of the Quartet is a reverse image

of the f i r s t h a lf . Bartok designed his palindrome through the use

of a fi v e movement st ru ctur e where the f i r s t and l as t two movements

are centered around the th ir d movement. Movement fi ve i s a var ia tion

of movement one and movement four is a va r ia t io n of movement two.

To na lly , Bartok create s a palindrome by his choice of poles

and counterpoles from a single axis as the tona l centers fo r each

movement. Movements one and f iv e cen te r around Bb, the tonic po le ,

two and four around the pole and counterpole of the side branch of

the ton ic axis (C# and G). Movement three is rel at ed to the outermost

movements by center ing around E, the counte rpo le of Bb. Th erefore,

the entire Quartet begins on Bb and circles counterclockwise around

the axis to the counterpole, E, then continues in the same direction

back to the starting pole, Bb.

Movement One

The f i r s t movement is in sonata form. The exposi tion presents

the f i r s t , second and th ir d themes in sequence. The cente r of the

movement is a developmental se ct io n, followed by a re ca p itu la tion




228

which reverses the order of thematic pres entat ion ( i . e . th ir d , second

and f i r s t ) . As the ordering of the themes in the rec ap itula tion

mirrors that of the exposition, the form of the movement can be seen

as a palindrome.

The tonal cen ter o f the f i r s t movement is Bb, which is also

the tonal cente r of the en tir e Quartet . In th is movement, the tonal

cen ters of each section of the movement r e f le c t an ascending whole

step progression (Bb to Bb).

Movement one emphasizes pent aton ic st ru ctu re wi th movement

between pentaton ies by both re la ti v e sol miz atio n and pole exchange.

There are a lso several examples o f alpha harmonies inter sper sed with

tert ian chords.

Movement Two

Movement two, the Adagio, is in palindromic form similar

to the f i r s t movement. The three thema tic areasare each presented

in tu rn , follo wed by a developmental sec tion in the middle of the

th ir d thema tic are a. The remainder of the movement presents the

second and f i r s t thematic areas resp ec tive ly.

The second movement is the shor test o f the fi v e , only 56

measures long. The to n a li ty cent ers on C#, the pole on the side

branch of the tonic axis.

Movement two is important for its presentation of the chorale

theme which ret ur ns as the focus of movement fo ur. The harmonies

of the ch oral e theme in th is movement are confined to te r ti a n . When

brought back in movement four, the theme uses alpha harmonies.




229

Pentatony is also important in this movement, frequently

l inked together in chains by eit he r r e la tiv e solmization or pole

exchange. This movement contain s a fo ur measure passage in folksong

style which is dependent on pentatony.

Movement Three

At the center of the Quartet's palindromic structure is a

symmetrical movement, a scherzotrioscherzo, which acts as a bridge

between the fi r s t and la s t two movements. The f i r s t scherzo (1 66 )

is a rounded binary [ABA]; the second scherzo (192) is shortened

to binary [AB] with a coda.

Placed between the two scherzos, the tr i o (1 65) is the apex

of the en tire Qu artet. I t is the one important part of the Quartet

which is c le a rl y organized by the formal pr in cip les of the golden

section proportions.

The tonal cen ter of thi s movement is E, the counte rpole of

the main branch on the tonic axis.

Movement th ree is the only movement which emphasizes re la t iv e ,

modally re la te d and su bs tit ut e chords. Pentatony is used in conjunc-

tio n with these harmonies in both scherzo I and I I .

The tr io is in a d iffe re n t textur e than the scherzos. A

measure long pat te rn is repeated moto perpetuo throughout most of

the t r io .

Bartok distinguishes the palindromic structure of the Quartet

not only tonally and structural ly, but also by the use of relat ive,




230

modally relate d and sub stitu te chords so lely in this movementsetting

i t ap ar t from the othe r four which use pentatony and alpha harmonies.

Movement Four

The fou rth movement (Andante) is a vari at io n of the second

movement. Most of the movement is based on the chorale theme of

movement two.

The tonal center of this movement is G, the counterpole of

the second movement's to n a l it y o f D# on the s ide branch of the ton ic

axis.

Movement four once again emphasizes pentatony and alpha har-

monies. Alpha harmonies are used ex te ns ive ly in the reharmonization

of the chorale melody, distin gui shi ng i t from the ori gin al harmoniza-

tio n of te rt ia n chords in movement two.

Movement Five

The f i f t h movement is long and complex, pr im ari ly making

use of techniques of canon, stretto and various inversional methods.

S tru c tu ra ll y , th is movement may be seen as a sonata form. Based

on fi r s t movement m ate ria l, the exposition and rec ap itula tion are

va ri a ti ons on the f i r s t and second theme areas of movement one.

The development is a tr io sec tion follo wed by a fugato and episode.

A r e la ti v e ly b ri e f coda completes th is lengthy movement.

To na lly , movement fi v e begins in E, the counterpole on the

main branch of the ton ic ax is , and ends in Bb, the to n a li ty which

opened the en ti re Quartet. Unli ke the previous movements, Bartok




231

does not always de line ate the ton a lit y in a cle ar manner. Rather,

he blurs the to n a lit y with rapid l ine ar , chromatic l ines in stre tto

and canon throughout the movement.

Movement f iv e makes extensiv e use of alpha harmonies, espe-

c ia l l y gamma chords which are used in sequence. This movement also

presents a se ries o f two and three la ye r alpha chords. The 1:5 model

is used in th is movement to close the tr io and fuga to sections of

the development.

The f i f t h movement is the longest and most complex, and also

the lea st accessible harmonically using Lendvai's the orie s. The

extensive use of stretto and canon in addition to chromatic lines

blur the harmonic structures.

Conclusion

The music of Bela Bartok is complex, requiring analytical

techniques which l i e beyond tr ad it io n a l methods. This stems from

his lifelong interest in folksong which produced an extensive collec-

tion of recordings and transcriptions gathered during his travels.

The in flue nce of folksong is a primary consid eratio n when discussing

the music of Bartok. Common pra ct ic e music centers on te r t ia n harmony,

whereas Bartok' s music is based on the i n t e r v a ll ic , harmonic and

rhythmic aspects of Hungarian folksong, a compositional approach

th at does not produce works conforming to tr a d it io n a l harmonic prac-

ti c e s . A new and d if fe re n t approach is necessary to f i l l the void

le f t by tra di tio na l methods.






(1 :2 , 1 :3 , 1 :5 ), or alpha harmonies. His wr itin gs avoid those portions

of Bartok's music which defy explanation using this methodology.

I t is obvious fu rt h er study is necessary to expand the orig ina l

theoretical principles and place them in a proper perspective.




LIST OF REFERENCES

Gr i f f i ths , Pau l .

Bar t6 k. The Master Musicians Se ries . London: J.M. Dent& Sons, L td ., 1984.

Karpati, Janos.Bart ok's Strin g Qu arte ts. Tran slated by Fred Macnicol. Buda-pest! Franklin Printing House, 1975. [Original t i t l e :

Bartok vonosnegyesei, Budapest: Zenemtikiado, 1967].

Lendvai, Erntf. y ^

The Workshop of Bartok and Kodaly. Budapest: Edit io Musica,1983.

Per le , George. ^ ^"The St rin g Quartets of Bela Bar tok ." A Musical Of fe rin g:Essays in Honor of Ma rt in Bern ste in . New ‘fo rk ; Pendragon

Press, 1977.

235




Documents

Bartok 5th String Quartet