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UBC Theses and Dissertations

A comparison of certain aspects of the theories of Paul Hindemith and Franz Alfons Wolpert Watt, William James 1973

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A COMPARISON OF CERTAIN ASPECTS OF THE THEORIES OF PAUL HINDEMITH AND FRANZ ALFONS WOLPERT by WILLIAM JAMES WATT B.  Mus.,  U n i v e r s i t y of B r i t i s h Columbia,  1967  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS  FOR THE DEGREE OF  MASTER OF MUSIC i n the  Department of MUSIC  We accept t h i s t h e s i s as  conforming to the  standard  THE UNIVERSITY OF BRITISH COLUMBIA November,  1973  required  In presenting  this thesis i n p a r t i a l fulfilment of the requirements for  an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference  and study.  I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may by his representatives.  be granted by the Head of my Department or  It i s understood that copying or publication  of this thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission.  Department of The University of B r i t i s h Columbia Vancouver 8, Canada  ABSTRACT  Two t w e n t i e t h - c e n t u r y music t h e o r i s t s ,  Paul  Hindemith and Franz A l f o n s W o l p e r t , are unique i n having i n d e p e n d e n t l y developed t h e i r own systems of chord classification succession.  and p r i n c i p l e s of chord movement or  While H i n d e m i t h 1 s i d e a s on these  are f a m i l i a r among the m a j o r i t y of music Wolpert1s theories decided that at  theorists'  theorists,  remain r e l a t i v e l y unknown.  since both men had attempted  similar points i n time, that  and chord c o n n e c t i o n were d i s c u s s e d  tasks  classification  i n d e t a i l to  behind each t h e o r i s t ' s  attempt approach.  both systems of chord c l a s s i f i c a t i o n  examined and compared. were many d i f f e r e n c e s  I t was found t h a t  way, i n a r r a n g i n g a l l p o s s i b l e  a l t h o u g h there  each t h e o r i s t ' s  movement were i n v e s t i g a t e d . system r e s t e d  each i n h i s own  combinations of  w i t h i n the twelve-note d i v i s i o n of the Similarly,  were  between H i n d e m i t h 1 s and W o l p e r t ' s  systems of chord g r o u p i n g , both succeeded,  Hindemith's  similar  was  T h e r e f o r e , both  H i n d e m i t h ' s and W o l p e r t ' s systems of chord  Firstly,  It  a comparison of both  i d e a s should be attempted.  t o determine the r a t i o n a l e  subjects  pitches  octave.  i d e a s c o n c e r n i n g chord  While i t was r e a f f i r m e d  that  on concepts which he i n v e n t e d and i  developed  such as  fluctuation,"  "degree p r o g r e s s i o n "  i t was d i s c o v e r e d  t h a t W o l p e r t * s system of  chord movement was more t r a d i t i o n a l l y adhered t o  some r a t h e r  desirability "adhesion,"  disturbing  oriented,  and yet  n o t i o n s about the  of c e r t a i n k i n d s of v o i c e - l e a d i n g , "diversion," etc.  Hindemith had strong organization,  Furthermore,  while  i t was found t h a t Wolpert saw h i s  Finally,  e.g.,  i d e a s about the n e c e s s i t y of  v a l i d f o r both t o n a l and a t o n a l  theories  and "harmonic  frames of  tonal  system as  reference.  i t was r e c o g n i z e d t h a t H i n d e m i t h ' s  c o n t a i n e d a u n i t y and c o h e s i v e n e s s through the  extension  of h i s  system of chord c o n n e c t i o n t o i n c l u d e  system of chord c l a s s i f i c a t i o n classified  his  so t h a t the way chords were  i n f l u e n c e d how they were t r e a t e d i n chord  progressions.  With Wolpert, however, there  t o u n i f y both systems and t h e r e f o r e movement are  ii  no attempt  h i s i d e a s about chord  c o m p l e t e l y d i v o r c e d from h i s  groupings.  is  system of c h o r d a l  TABLE OF CONTENTS Page  CHAPTER I .  INTRODUCTION  CHAPTER I I .  1  CHORD CLASSIFICATION  11  A.  Hindemith  11  B.  Wolpert  20  C.  Comparison  37  CHAPTER I I I .  CHORD MOVEMENT  A.  Hindemith  B.  Wolpert  C.  Comparison  CHAPTER I V .  51 51 •  65 85  FURTHER CONSIDERATIONS  92  BIBLIOGRAPHY  j.02  APPENDIX I  105  APPENDIX I I  106  APPENDIX I I I  107  iii  LIST OF MUSICAL ILLUSTRATIONS Example Number  Page  1.  Hindemith s Series 2  13  2.  W o l p e r t s Three-Note Chord Types  23  3.  Wolpert* s Concept of C o n t r a c t e d Form  24  4.  Wolpert's Comparison w i t h Hindemith's System  5.  Wolpert's Four-Note Chord Types.  26  6.  Wolpert's Five-Note Chord Types  26  7.  Wolpert's Six-Note Chord  27  8.  Wolpert's Seven-Note Chord  27  9.  Major T r i a d s w i t h the F i f t h S p l i t V a r i o u s Ways  28  1  f  .  25  20.  »  II  it  H  II  II  "  II  29  22  II  II  ti  it  it  it  II  it  29  12.  Summary of S p l i t — I n t e r v a l Terminology  30  13.  The Concept of Dimension  31  14.  E s s e n t i a l Dissonance . . . . . (5)  15.  Problem of the j j Chord  16.  P o s s i b l e Roots i n  17. 18.  Wolpert's C h o r d a l A n a l y s i s Chord S p e l l i n g I n f l u e n c i n g C l a s s i f i c a t i o n i n Wolpert's System  36  19.  G r a d u a l v e r s u s N i l Harmonic F l u c t u a t i o n . . . .  53  20.  Two Chords from Sub-Group I I whose Roots are a T r i t o n e A p a r t . .  58  .  35  4  a Seven-Tone Chord  iv  33  . . . . .  36  41  Page 21.  P o s s i b i l i t i e s of V o i c e - P a i r i n g i n W o l p e r t ' s Examination of Chord C o n n e c t i o n  66  22.  Wolpert's  68  23.  Types of S y s t o l e s and D i a s t o l e s  69  24.  Chord S p e l l i n g s Determining Number of Possible Resolutions  71  25.  Diversion  72  26.  Examination Frameworks.  27.  Mixture of Adhesive and Non-Adhesive V o i c e s i n the Same C o n n e c t i o n  28.  Bass P r o g r e s s i o n s w i t h Most L i k e l y Upper  "Non-connections"  of A l l P o s s i b l e Two-Voice  73 74  V o i c e Movement  75  29.  Cadences w i t h S p l i t  Chords  77  30.  Cadences w i t h S p l i t Chords  78  31.  Cadences w i t h S p l i t Chords  78  32.  Cadences w i t h S p l i t Chords  79  33.  The C r o s s - R e l a t i o n  80  34.  "Real"  82  35.  "Narrow" v e r s u s "Wide" C r o s s - R e l a t i o n s .  36.  C r o s s - R e l a t i o n s w i t h the N e a p o l i t a n  v e r s u s "Sound" C r o s s - R e l a t i o n s . . .  v  .  .  83  Sixth  .  83  ACKNOWLEDGEMENT  I wish t o express my thanks to my r e s e a r c h a d v i s o r , D r . Eugene W i l s o n ,  for  h i s constant  support  and guidance d u r i n g the w r i t i n g of t h i s t h e s i s . a l s o indebted  t o the other  members of my committee,  Professor  Kathryn B a i l e y and P r o f e s s o r  Hultberg,  for  their  I am  critical  Cortland  comments and v a l u a b l e  a ssistance. I would a l s o l i k e to thank Mr. Hans Head of the Music L i b r a r y , the Department portions  and Mr. L o u i s Medveczky of  of German, f o r  their  of the Wolpert t r e a t i s e .  help i n t r a n s l a t i n g A l s o , I would l i k e  e s p e c i a l l y thank Miss Lynne T a y l o r for preparing  and t y p i n g the f i n a l  Finally, for of  her a s s i s t a n c e  I would l i k e to thank my w i f e ,  text.  to in  manuscript.  her p a t i e n c e and encouragement throughout the e n t i r e  Burndorfer,  Linda,  the  writing  CHAPTER I INTRODUCTION  Music t h e o r y i n the t w e n t i e t h century has been concerned w i t h a wide v a r i e t y of c r i t e r i a i n what has been a c o n s c i o u s attempt t o of  systematize  the c o n c e i v a b l e raw m a t e r i a l s  Twentieth-century t h e o r i s t s  and c l a r i f y  of m u s i c a l c o m p o s i t i o n .  have been i n v o l v e d i n some-  times r a t h e r extended a n a l y t i c a l i n v e s t i g a t i o n s melodic s t r u c t u r e , logic,  all  into  rhythmic o r g a n i z a t i o n , t i m b r e , formal  dynamics and the v e r y nature of sound i n  itself.  However, t h e r e has been a n o t i c e a b l e absence of e f f o r t the part of these same t h e o r i s t s  t o attempt  on  t o come t o  terms w i t h workable systems of c l a s s i f y i n g  chordal material  so as t o i n c l u d e a l l p o s s i b l e  of the  which are present  arrangements  sounds  w i t h i n our twelve note d i v i s i o n of the  octave. This activities century,  situation is and i n t e r e s t s  in direct antithesis of t h e o r i s t s  of the n i n e t e e n t h  one of whose primary concerns was i n chords and  systems by which they c o u l d be c l a s s i f i e d , labelled.  t o the  arranged and  During the n i n e t e e n t h and the f i r s t  twentieth centuries  all vertical  structures  t r a d i t i o n a l l y compared t o the t r i a d i c frame  part  of  the  were of  reference.  2 N o t e s w h i c h c o u l d n o t be a c c o u n t e d f o r w i t h i n s u c h a f r a m e of r e f e r e n c e were l a b e l l e d  non-chord t o n e s such as  p a s s i n g t o n e s , a p p o g g i a t u r a s , upper o r l o w e r n e i g h b o u r tones, or the l i k e ,  and w e r e n o t r e a l l y  i n t e g r a l part of the chord. complex  this triadic  frame  A s t h e h a r m o n i e s became more o f r e f e r e n c e was t o become  more and more u n w o r k a b l e .  T h e r e f o r e , a need  extension of the t r a d i t i o n a l bilities  of  more c o m p l e x  c o n s i d e r e d an  f o r the  system t o i n c l u d e  vertical  the possi-  s t r u c t u r e s became  evident. T h u s f a r i n t h e t w e n t i e t h c e n t u r y a s m a l l number of systems  that  could  have been proposed  be  called  chord  by music t h e o r i s t s . ^  classification -  Naturally, a l l  o f t h e s e s y s t e m s a r e n o t c o n s t r u c t e d f r o m t h e same o f v i e w o r w i t h t h e same p u r p o s e i n m i n d . question of v e r t i c a l all  composers  However, t h e  s i m u l t a n e i t y i s one w h i c h c o n c e r n s  and t h e o r i s t s o f t h e t w e n t i e t h c e n t u r y , no  m a t t e r what t h e i r p o i n t o f v i e w o r under what t h e i r music has been w r i t t e n . of v e r t i c a l  point  criteria  Moreover, the c l a s s i f i c a t i o n  s t r u c t u r e s does n o t n e c e s s a r i l y depend  on t h e  c o n t e x t i n w h i c h t h e s e sounds a r e employed.  Thus, t h e  w r i t i n g s of a l l t h e o r i s t s , whether  or n o n - s e r i a l -  ist  i n orientation,  are s i g n i f i c a n t  serialist i n this  regard.  1 G e n e r a l l y , i n the twentieth century, "chord" can be a p p l i e d t o any c o m b i n a t i o n o f t h r e e o r more n o t e s sounding s i m u l t a n e o u s l y .  3. Among the former,  one may p o i n t to the  of B a b b i t t ,  Chrisman, P e r l e , Rochberg, Hauer,  b e r g , H?fba,  and G e r h a r d , t o name but a few.  article  which attempts t o d e s c r i b e  harmonic o r g a n i z a t i o n i n s e r i a l  w h i l e i n the twelve-tone i n g of the notes i s  the problem of  music,  Perle  points and  i n the d i a t o n i c - t o n a l  system  system only the l i n e a r  specifically  possible to v e r t i c a l i z e  SchoenIn an  out t h a t both l i n e a r and harmonic p r o p e r t i e s r e s o u r c e s are a v a i l a b l e  writings  defined,  it  order-  being  any number of adjacent  elements  3 i n the set  i n any way which s u i t s the composer.  l i m i t i n g of the twelve-tone is  system t o l i n e a r  perhaps one reason f o r the e a r l i e r  attempts at the c l a s s i f i c a t i o n simultaneities. P e r l e does l i s t  In a s h o r t e r  This  properties  noted l a c k of  of a l l p o s s i b l e v e r t i c a l article  the number of t o t a l  i n The S c o r e .  possible  g i v e n our twelve note d i v i s i o n of the o c t a v e . the o n l y c r i t e r i o n i n the c l a s s i f i c a t i o n being 351.^  is  chords However,  the number of  distinct  permutations  A l o i s Haba and Roberto  Music."  eorge P e r l e , "The Harmonic Problem i n Twelve-Tone Music Review. XV (1954), 257-67.  H e r e , B a b b i t t c o u l d be noted as one where v e r t i c a l i z a t i o n s r e t a i n t h e i r " l i n e a r Music,"  exception adjacencies."  ^Geroge P e r l e , "The P o s s i b l e Chords i n Twelve-Tone The S c o r e . IX (September, 1954), 54-8.  4. Gerhard had proceeded along the same l i n e s p r e v i o u s t o Perle,  but b o t h made e r r o r s  i n t h e i r c a l c u l a t i o n s which  P e r l e has been q u i c k t o p o i n t out i n h i s a r t i c l e . H a u e r ' s system of f o r t y - f o u r t r o p e s c o n s i d e r e d a type of chord c l a s s i f i c a t i o n sense i n t h a t  can be in a limited  only s i x note chords (hexachords)  are  P.  classified,  and these only f o r c o n t e n t .  be s a i d f o r M i l t o n B a b b i t t ' s  The same can  a l l - c o m b i n a t o r i a l source s e t s  which are even more l i m i t i n g than H a u e r ' s t r o p e s , are only s i x b a s i c  (  as t h e r e  groups. •7  George Rochberg's study,  among o t h e r t h i n g s ,  sets  out methods f o r c o n s t r u c t i n g tone rows at l e a s t one of whose i n v e r s i o n s w i l l the o r i g i n a l row:  not repeat  the f i r s t  s i x notes of  a study s i m i l a r t o B a b b i t t ' s  principle  of c o m b i n a t o r i a l i t y .  Here a g a i n ,  classification  one i n t e r p r e t s the term i n a r a t h e r  limited  unless  the study i s r e a l l y not a  sense.  5  ed. 87.  K a r l Eschman, Changing Forms i n Modern M u s i c . 2nd (Boston: E . C . Schirmer Music C o m p a n y , 1 9 6 8 ) , pp. 836  M i l t o n B a b b i t t , "Some A s p e c t s of Twelve-Tone C o m p o s i t i o n , " The S c o r e . X I I (1955), 53-61. 7 George Rochberg. The Hexachord and i t s R e l a t i o n to the Twelve-Tone Row (Bryn Mawr, P e n n s y l v a n i a : Theodore P r e s s e r Company, 1955), 40 pp.  5 Recently,  some a l l - i n c l u s i v e systems have been  developed among s e r i a l i s t s , and C h r i s m a n .  f o r example,  those by F o r t e  These, however, tend to be of a  nature and sometimes  statistical  are even h i g h l y mathematical,  as i n  Q  K a s s l e r ' s study. s t u d i e s are f a r  A t any r a t e ,  a l l t h r e e of these  from resembling systems of chord  classifi-  c a t i o n i n the t r a d i t i o n a l n i n e t e e n t h century sense i n t h a t arrays  of numbers are used t o r e p r e s e n t  the  possibilities  and, apart from the number of notes i n each a r r a y , is l i t t l e  i n the way of a d d i t i o n a l c r i t e r i a t o  break down or s i m p l i f y the arrangement. left  with a l i s t  of a l l p o s s i b i l i t i e s  guide t o t h e i r p r a c t i c a l use.  there  further  Rather,  one  is  w i t h no f u r t h e r  These systems serve as more  of a d e s c r i p t i o n — e n l a r g i n g our concept of the body of material—than a  classification.  Among s o - c a l l e d  non-serialists  who have developed  systems of chord c l a s s i f i c a t i o n  are men such as Paul H i n d e -  mith and Franz A l f o n s W o l p e r t .  U n l i k e a l l of the  mentioned t h e o r i s t s ,  above  Hindemith and Wolpert deal w i t h the  q u e s t i o n of v e r t i c a l s i m u l t a n e i t y i n a manner which s t r o n g l y resembles 8  the t h e o r i s t s  of the p r e c e d i n g  century.  See: A . F o r t e , "A Theory of Set Complexes f o r M u s i c , " J o u r n a l of Music T h e o r y . V I I I (1964), 136-83. R. Chrisman, " I d e n t i f i c a t i o n and C o r r e l a t i o n of P i t c h S e t s , " J o u r n . a L j ^ ^ i ^ - T h e g j ^ , XV (1970), 58-83. M. K a s s l e r , "Toward a Theory t h a t i s the Twelve-Note C l a s s System," P e r s p e c t i v e s of New M u s i c , V , (1967), 1-80.  6.  That i s , they both deal w i t h the question of chords, chord c l a s s i f i c a t i o n and chord arrangement i n much the same way as do t r a d i t i o n a l t o n a l t h e o r i s t s , and, because of t h i s method of approach, can be d i s t i n c t l y set apart from many of the other t h e o r i s t s of t h i s century. Most twentieth-century t h e o r i s t s , when d e a l i n g w i t h problems of v e r t i c a l s i m u l t a n e i t y , deal w i t h these v e r t i c a l s t r u c t u r e s (as we have seen) i n e i t h e r a s t a t i s t i c a l manner or i n a manner which i s very s t r o n g l y c o n t r o l l e d by l i n e a r c o n s i d e r a t i o n s , e.g., the twelve-tone  system.  On the other hand, Hindemith and Wolpert are i n v o l v e d w i t h chords as sound o b j e c t s i n themselves.  Since they  are thus among a small group of t h e o r i s t s who deal i n an extensive way w i t h these new v e r t i c a l i z a t i o n s i n a more or l e s s t r a d i t i o n a l manner, they form an important l i n k w i t h the past and o f f e r today's composer and a n a l y s t a means of understanding these new sound p o s s i b i l i t i e s without necess a r i l y i m p l y i n g or assuming a r e j e c t i o n of past musical traditions. Hindemith's system of c l a s s i f i c a t i o n i s presented i n h i s t r e a t i s e The C r a f t of Musical Composition.^ as  9  Paul Hindemith, The C r a f t of M u s i c a l Composition. 4 t h ed. t r a n s , by Arthur Mendel (New York: Schott Music C o r p o r a t i o n , 1970), I .  7. part  of a l a r g e r  student  of  discussion  d i r e c t e d towards  the  serious  composition.  The teacher w i l l f i n d i n t h i s book b a s i c p r i n c i p l e s of c o m p o s i t i o n d e r i v e d from the n a t u r a l c h a r a c t e r i s t i c s of t o n e s , and consequently v a l i d f o r a l l periods. To the harmony and c o u n t e r p o i n t he has a l r e a d y l e a r n e d , which have been p u r e l y s t u d i e s i n the h i s t o r y of s t y l e . . . he must now add a new t e c h n i q u e , which, proceeding from the f i r m foundat i o n of the laws of n a t u r e , w i l l enable him t o make e x p e d i t i o n s i n t o domains of c o m p o s i t i o n which have not h i t h e r t o been open t o o r d e r l y p e n e t r a t i o n . The book makes c l e a r t h a t f o r a w e l l - i n t e n t i o n e d but a r b i t r a r y arrangement of sounds the composer must s u b s t i t u t e an order which only t o the u n i n i t i a t e d w i l l seem a r e s t r i c t i o n of the c r e a t i v e process.i^ The s i g n i f i c a n c e  of the C r a f t 1 s  music t h e o r y and, more s p e c i f i c a l l y , tion,  c o n t r i b u t i o n to  t o chord  classifica-  has a l r e a d y been p o i n t e d out by W i l l i a m Thomson.  One of the most f a r r e a c h i n g i n f l u e n c e s of the C r a f t has been i t s system of chord a n a l y s i s , which f o r the f i r s t time o f f e r e d a p o s s i b l e break i n the stone w a l l of t e r t i a n harmony. Before the C r a f t t h e r e was no taxonomy of chord s t r u c t u r e except t h a t i n which any t o n a l aggregate was c l a s s i f i e d e i t h e r as some form of stacked t h i r d s or e l s e as a product of d e c o r a t i v e melodic a c t i o n i n c o n j u n c t i o n with a postulated "chord" In an a r t i c l e  appearing  subsequent t o the p u b l i c a t i o n of  the C r a f t Hindemith attempts t o j u s t i f y had p r e v i o u s l y set  10 11  Ibid..  p.  out to  part  of what he  accomplish.  9.  W i l l i a m Thomson, " H i n d e m i t h ' s C o n t r i b u t i o n t o Music T h e o r y , " J o u r n a l of Music Theory. IX (1965), p . 5 9 .  8 For the composer as w e l l as f o r the h e a r e r , tones and t h e i r c o n n e c t i o n s are the b e g i n n i n g and end of musical a c t i v i t y . Not so f o r the t h e o r i s t . He must enquire i n t o the nature of the tones and study the p r i n c i p l e s of t o n a l c o n n e c t i o n . For the f i r s t of these two t a s k s , he i s almost c o m p l e t e l y independent of the e x p e r i e n c e s of the p r a c t i c a l m u s i c i a n ; the second, on the other hand, i s not to be achieved without a knowledge of c o m p o s i t i o n a l p r o c e d u r e , no matter whether a t h e o r i s t o b t a i n s such knowledge by means of d e d u c t i o n - the a n a l y s i s of a l r e a d y e x i s t i n g c o m p o s i t i o n s - or through h i s own c r e a t i v e a c t i v i t y . * 2 More s p e c i f i c a l l y ,  Hindemith speaks of the f e a s i b i l i t y  a system of chord c l a s s i f i c a t i o n  of  i n which every note i n any  c o n g l o m e r a t i o n of p i t c h e s can be i n c o r p o r a t e d w i t h i n the terms of the  system.  The p o s t u l a t e of the i n t e r v a l as the harmonic u n i t • . . may be used t o e x p l a i n every c o n c e i v a b l e c h o r d , and the t h e o r i s t w i l l be faced only w i t h the q u e s t i o n of how t o apply t h i s y a r d s t i c k i n order t o a p p r a i s e t o n a l c o m b i n a t i o n s , and n o t , as f o r m e r l y , w i t h the n e c e s s i t y of d i v i d i n g chords i n t o those which can be measured and those which cannot.^3 Wolpert*s j u s t i f i c a t i o n classification  for his  system of chord  appears i n the i n t r o d u c t i o n t o the new  e d i t i o n of h i s t r e a t i s e  Neue H a r m o n i k . ^ which i s  expansion of the e a r l i e r  (1950) e d i t i o n .  o u t s e t t h a t the major part  of h i s t r e a t i s e  He admits at is  directed  12  P a u l Hindemith, "Methods of Music T h e o r y , " M u s i c a l Q u a r t e r l y . XXX (1944), 21-2. 13 14  Ibid..  an  28.  Franz A l f o n s W o l p e r t , Neue Harmonik, (Wilhelmshaven: H e i n r i c h s h a f t e n , 1972), 13-15.  the  9. towards "the m a j o r i t y  of m u s i c a l l i s t e n e r s "  and i s  meant t o be an i n n o v a t i v e work apart from " t h e of chord t y p e s and t h e i r v a r i a n t s . " justification system.  of  seems t o r e s t on h i s r e j e c t i o n  "impossible"  tritone  as  of H i n d e m i t h 1 s  the g e n e r a l  through i n v e r s i o n . ^  factor  n o t i o n of " c h o r d  and does  identity"  A l s o , Wolpert says t h a t because of  this refusal  to recognize  Hindemith i s  forced  the i n v e r s i o n p r i n c i p l e ,  i n t o problems  p o s s i b l e conglomerations  g i v e s him e n d l e s s t r o u b l e . H i n d e m i t h ' s chord chart  is  of d e f i n i n g r o o t s  of i n t e r v a l s  and t h a t  for this  Wolpert a l s o argues t h a t incomplete and must always end  w i t h the phrase "and s i m i l a r  chords.  Wolpert a s s e r t s t h a t he has i n h i s system.  classifica-  since he (Hindemith) r e g a r d s the  an e s s e n t i a l d i s t i n g u i s h i n g  not r e c o g n i z e  system  Wolpertfs  He i n s i s t s that H i n d e m i t h ' s method of  tion is  all  Part  not  He r e c o g n i z e s  s o l v e d these  the t r a d i t i o n a l  of c h o r d a l i n v e r s i o n through h i s  " P r i n c i p l e of  problems  acceptance Identity"  and a v o i d s the c o n f u s i o n of d e f i n i n g chord r o o t s .  He  accepts t r a d i t i o n a l  notes  r o o t s and r e l a t e s t h e s e t o the  15  Hindemith had r e j e c t e d the p r i n c i p l e of c h o r d a l i n v e r s i o n as a " p u r e l y a r b i t r a r y i n v e n t i o n of Rameau's." See: "Methods of Music T h e o r y , " p. 27.  16  Wolpert,  op.  cit  10 i n the system of c l a s s i f i c a t i o n which he develops.  His  chart i s complete i n the sense that a l l p o s s i b l e chords can be reduced t o one of h i s f i f t e e n basic chord types. As w e l l as c l a s s i f y i n g a l l p o s s i b l e v e r t i c a l c o n s t r u c t s i n a manner remarkably s i m i l a r to t h e o r i s t s of the preceding century, both Hindemith and Wolpert discuss c e r t a i n "rules", " i d e a l s " or " p r i n c i p l e s " t o which one should attempt t o adhere when connecting a succession of v e r t i c a l s t r u c t u r e s .  I t seems only n a t u r a l  that given a new vocabulary of musical sounds i n the form of " a l l p o s s i b l e chords," new p r i n c i p l e s would be evolved as a guide i n connecting these sounds. The basic d i f f e r e n c e s i n the work of the two concerns the question of t o n a l i t y .  men  Hindemith b e l i e v e s  that a l l " l o g i c a l " music i s t o n a l (whether or not so conceived by the composer) and h i s chordal system i s thus organized i n t o a t o n a l framework.  Wolpert, on the other  hand, i n v o l v e s no prejudgements about t o n a l i t y or 17 atonality,  and c a r e f u l l y avoids p u t t i n g a t o n a l frame-  work around h i s chordal system. Because of these d i f f e r e n c e s and s i m i l a r i t i e s i t was thought a comparison of the systems of Hindemith and Wolpert would be appropriate. 17  In the i n t r o d u c t i o n t o the new e d i t i o n of Neue Harmonik. Wolpert states that "many composers wrote i n a very well-sounding a t o n a l i t y . "  CHAPTER  II  CHORD CLASSIFICATION  A . H i n d e m i t h * s System. H i n d e m i t h * s purpose chord c l a s s i f i c a t i o n  i n constructing  a system of  i s to expand the l i m i t s of  the  c o n v e n t i o n a l theory of harmony. Three r e s o l u t i o n s  are made at the  outset:  1.  C o n s t r u c t i o n i n t h i r d s must no l o n g e r be the b a s i c r u l e f o r the e r e c t i o n of c h o r d s .  2.  We must s u b s t i t u t e a more a l l - e m b r a c i n g p r i n c i p l e f o r t h a t of the i n v e r t i b i l i t y of chords.  3.  We must abandon the t h e s i s t h a t chords are s u s c e p t i b l e t o a v a r i e t y of i n t e r p r e t a t i o n s .  Also,  a number of d e f i n i t i o n s  and assumptions  are  s t a t e d or i m p l i e d : 1. different  A chord i s  defined  tones sounding  1  as  a group of at  least  three  simultaneously.  Paul Hindemith, The C r a f t of M u s i c a l C o m p o s i t i o n . 4 t h e d . , t r a n s , by A r t h u r Mendel (New Y o r k : Schott Music C o r p o r a t i o n , 1970), I , 94-5 2  3M_L..  95.  12 2.  There i s a basic and e s s e n t i a l d i f f e r e n c e  between chords containing one or more t r i t o n e s and those without.  root.  3.  Every chord, w i t h a few exceptions, has a  4.  Chords i n which the bass tone and root are  4  not i d e n t i c a l  are subordinate i n value t o chords i n which  the root and bass tone c o i n c i d e , other f a c t o r s being equal.^ Hindemith then sets out t o construct a Table of Chord Groups i n which a l l v e r t i c a l s t r u c t u r e s p o s s i b l e w i t h the twelve-note d i v i s i o n classified.  of the octave w i l l be  Hindemith s aim i s t o provide a 1  c l a s s i f i c a t i o n which i s more than a mere catalogue. I t i s a l s o an ordering of the "value" of chords so that the importance of the sub-groups diminishes as one proceeds from the f i r s t t o the l a s t .  The higher the number of the  sub-group, the higher the t e n s i o n and the lower the value of the s t r u c t u r e being considered. Conversely, the lower  3 IbM.  4  I b i d . . 96.  5  I b i d . . 99.  13. the number of the  sub-group,  the higher the value  of the  the lower the t e n s i o n and chord.  Hindemith o r d e r s h i s c l a s s i f i c a t i o n the i n t e r v a l s All  contained  according to is  i n the chords under  possible i n t e r v a l relationships  taken i n t o account  consideration.  w i t h i n the chord are  and the r e s u l t i n g  "value"  on the b a s i s of  intervals  on the b a s i s of S e r i e s 2.  are  classified  This Series 2  d e r i v e d from the theory of combination t o n e s " and  s e n t s , a c c o r d i n g t o H i n d e m i t h , the n a t u r a l  repre-  classification  of  7 the  intervals  Example  according to  value.  1.^  JL  6  Ibid..  7  Ibid..  8  O  JL O  ^  o 'fr^  fro  o 'b^g bo  58-64. 96.  Some of H i n d e m i t h 1 s assumptions c o n c e r n i n g a c o u s t i c s i n the d e r i v a t i o n of S e r i e s 2 have been questioned by Cazdun. (Norman Cazdun, "Hindemith and N a t u r e , " The Music Review. V o l . XV, 1954, 292.) "Hindemith i s not i n t e r e s t e d i n r e a l combination tones at a l l but only i n f i c t i t i o u s ones, though even these give him e n d l e s s t r o u b l e . D e c l a r i n g without q u a l i f i c a t i o n t h a t any two simultaneous tones produce combination  14. Generally, part  chords c o n t a i n i n g i n t e r v a l s f r o m t h e f i r s t  of the s e r i e s ,  a c c o r d i n g t o Hindemith,  have a h i g h e r  value than chords c o n t a i n i n g i n t e r v a l s f r o m l a t e r series. use  i n the  H i s system d i v i d e s chords i n t o two groups by the  of the t r i t o n e as the b a s i c  interval in  classifying  chords. . . . the t r i t o n e . . . stamps chords so s t r o n g l y w i t h i t s own c h a r a c t e r t h a t they a c q u i r e something of both i t s i n d e f i n i t e n e s s and i t s c h a r a c t e r of motion towards a g o a l . ° Furthermore,  chords have more value i f t h e i r r o o t and bass  note c o i n c i d e . H i n d e m i t h ' s method f o r determining the r o o t of a chord i s u n i q u e .  It  c o n s i s t s i n f i n d i n g the best i n t e r v a l  i n the c o n g l o m e r a t i o n of p i t c h e s a c c o r d i n g t o h i s own Series 2.  The r o o t of each i n t e r v a l of the s e r i e s  determined, a c c o r d i n g t o H i n d e m i t h , by a c o u s t i c a l The  roots  of the p e r f e c t  fifth,  the t h i r d s and the  is laws. sevenths  t o n e s , Hindemith f i n d s t h a t they u s u a l l y are so weak t h a t the s u p e r f i c i a l ear does not p e r c e i v e them, but t h i s makes them a l l the more important f o r the subc o n s c i o u s ear . . . so Hindemith i s not d e a l i n g w i t h combination tones t h a t e x i s t , but only w i t h those present t o the subconscious e a r . In f a c t , t h e i r n o n - e x i s t e n c e seems t o make them a l l the more important . . . " 9  Hindemith.  op. c i t . , 9 5 .  15. are the lower note of the i n t e r v a l while the r o o t s of the p e r f e c t f o u r t h , the seconds and the s i x t h s are the upper note of the i n t e r v a l .  To f i n d the root of a chord,  one examines the given s t r u c t u r e f o r the best i n t e r v a l contained t h e r e i n according to S e r i e s 2.  I f there i s no  p e r f e c t f i f t h , one proceeds t o look f o r a p e r f e c t f o u r t h , then a major t h i r d , and so on.  The f i r s t i n t e r v a l  one  encounters using t h i s method, i s the best i n t e r v a l  and  the root of t h i s i n t e r v a l i s the root of the chord.  It  should be emphasized that Hindemith expects a l l i n t e r v a l s i n the s t r u c t u r e to be considered.  Thus, i n three-note  chords, there are three i n t e r v a l s t o consider; i n f o u r note chords, there are s i x i n t e r v a l s to consider; i n f i v e note chords, ten i n t e r v a l s ; i n six-note chords, f i f t e e n i n t e r v a l s ; seven-note chords, twenty-one i n t e r v a l s ;  and  so on to twelve-note chords, where there would be s i x t y - s i x i n t e r v a l s to consider. The f o l l o w i n g should a l s o be noted: Doubled tones count only once; we use the lowest one f o r our reckoning. I f the chord contains two or more equal i n t e r v a l s , and these are the best i n t e r v a l s , the root of the lower one i s the root of the chord. 1  Hindemith's scheme gives a l i s t of two chord-groups which i n c l u d e s i x "chord sub-groups" separated  10.  97.  into  16. Group A (Sub-groups  I,II,V),  and Group B (Sub-groups one or more t r i t o n e s . groups are f u r t h e r  chords without a t r i t o n e ,  I I , I V , V I ) , chords c o n t a i n i n g In f o u r out of s i x  sub-divided.  cases the  In a l l these  chords w i t h r o o t and bass c o i n c i d i n g are  sub-  instances,  i n a h i g h e r sub-  d i v i s i o n than chords i n the same sub-group w i t h the r o o t above the b a s s . The o n l y s t r u c t u r e s  which s a t i s f y  of Group A , sub-group I (or  simply " I " ) ,  without t r i t o n e s and without seconds containing roots,  the c o n d i t i o n s that  is,  or sevenths  are the major and minor t r i a d s  chords and and what  are t r a d i t i o n a l l y c a l l e d i n v e r s i o n s . T h e chords of S e c t i o n 1.1.  ( t r i a d s i n r o o t p o s i t i o n ) , would be c o n s i d e r e d  h i g h e r i n the s c a l e of v a l u e s  than those  t i o n a l l y considered t h e i r inversions)  i n 1.2.  (tradi-  s i n c e the former  are more s t a b l e and l e s s i n need of r e s o l u t i o n . The c o r r e s p o n d i n g sub-group of Group B i s numbered I I and c o n t a i n s a l l those  chords which have a t r i t o n e but  do not c o n t a i n any minor seconds  or major s e v e n t h s .  reason f o r not e x c l u d i n g the major second and minor notes H i n d e m i t h ,  11  i s because "the presence  of the  The seventh,  tritone  Here the word " t r a d i t i o n a l " i s used to emphasize the f a c t t h a t Hindemith does not c o n s i d e r chords t o be invertible.  17. always i n v o l v e s seconds or s e v e n t h s — e x c e p t i n the diminished t r i a d i n the f i r s t  s e c t i o n under  only the dominant the  and i t s i n v e r s i o n s , " sub-group  Accordingly,  II ( i l . a . ) ,  The  second  I I ( i l . b . ) c o n s i s t s of s t r u c t u r e s  c o n t a i n major  we  have  seventh w i t h and without the f i f t h i n  t r a d i t i o n a l root p o s i t i o n .  sub-group  12  section i n which  seconds or minor sevenths or both, and i s  divided into three categories.  In the f i r s t ,  the r o o t  and bass tones are the samej i n the second the r o o t  lies  above the bass and i n the t h i r d , t h e r e i s more than one t r i t o n e p r e s e n t , w i t h the r o o t being e i t h e r i n the bass or  the upper v o i c e s .  Examples of t r a d i t i o n a l  chords  common t o t h i s s e c t i o n are chords of the dominant  ninth  (il.b.l.),  seventh  t r a d i t i o n a l i n v e r s i o n s of the dominant  ( l l . b . 2 . ) , the s o - c a l l e d " h a l f - d i m i n i s h e d " chords ( i l . b . l . ) , the  "French" augmented s i x t h ( l l . b . 3 . ) , and  similar  structures. Sub-group I I I of Group A c o n t a i n s chords w i t h seconds or sevenths, or both, but without t r i t o n e s . sub-group the  i s further  s u b - d i v i d e d i n t o l ) s t r u c t u r e s where  r o o t and bass are i d e n t i c a l , 2) where the r o o t  above the bass tone.  This  The chords i n sub-group  lies  I I I include  12. IbJLd., 102. Note Hindemith uses the term " i n v e r s i o n s " here when i n f a c t he c l a i m s not t o r e c o g n i z e invertibility.  18.  the secondary seventh and n i n t h chords ( I I I . 1 . ) and their t r a d i t i o n a l inversions (III.2.). The chords of sub-group IV contain any number of t r i t o n e s plus minor seconds, and/or major sevenths. Here, a d i s t i n c t i o n might be made between dissonance and tension.  Hindemith considers (by d e f i n i t i o n ) the chords  of sub-groups V and VI as having a higher t e n s i o n f a c t o r 13 than those of sub-group IV.  However, from an e m p i r i c a l  point of view, most l i s t e n e r s would f i n d the chords of sub-group IV, on the whole, more dissonant than those i n sub-groups V or V I .  Thus, t e n s i o n i s not n e c e s s a r i l y the  same as dissonance, according t o Hindemith.  A l s o , chords  w i t h s u b s t a n t i a l t e n s i o n have a greater need t o r e s o l v e , g e n e r a l l y speaking, than those w i t h l e s s t e n s i o n .  As w i t h  sub-group I I I , sub-group IV i s f u r t h e r d i v i d e d i n t o : 1) s t r u c t u r e s where the root and bass are i d e n t i c a l and 2) where the root l i e s above the bass. In sub-groups V and VI we encounter the exceptions to assumption 3, that i s , those chords which have no r o o t s . In the cases examined i n these two sub-groups, Hindemith m a i n t a i n s ^ there i s no r o o t , but only a "root r e p r e s e n t a t i v e " the i d e n t i t y of which i s dependant on the context. 13  I b i d . , 108.  14  Ibid., 101.  Thus,  19 i n sub-group V there are only two chords, the augmented t r i a d and the chord c o n s i s t i n g of two  superimposed  perfect fourths. In sub-group VI there are only four chords, namely, the diminished chord and i t s two t r a d i t i o n a l i n v e r s i o n s and the diminished seventh chord.  Up to t h i s p o i n t ,  whenever the t r i t o n e has appeared i n a chord (Group B) i t has subordinated i t s e l f t o the best i n t e r v a l according to S e r i e s 2.  Now,  i n sub-group V I , because of the nature  of the chords i n q u e s t i o n , the t r i t o n e  predominates.  I s o l a t e d i n t e r v a l s , says Hindemith, can a l s o be assigned to the Table of ChQrd Groups.  The p e r f e c t f i f t h  and major and minor t h i r d s belong t o I . I . , the p e r f e c t fourth and major and minor s i x t h s t o 1.2.,  the seconds to  I I I . 2 . , the sevenths to I I I . l . , and the t r i t o n e to V I . Hindemith j u s t i f i e s h i s system of c l a s s i f i c a t i o n by a s s e r t i n g t h a t i t "does away w i t h a l l ambiguity."  Also,  i t i s a l l - i n c l u s i v e i n t h a t "there i s no combination of i n t e r v a l s which does not f i t i n t o some d i v i s i o n of our system.  15 16  Ibid..  100.  Ibid.,  105.  20 B  «  W o l p e r t * s System. W o l p e r t 1 s method of chord c l a s s i f i c a t i o n  of a systematic  consists  breakdown of a l l combinations of musical  sounds i n the t r a d i t i o n a l tempered system-^  into  fifteen  b a s i c chord t y p e s . I t i s s e l f e v i d e n t t h a t the f i f t e e n chord types gained i n t h i s way c o n t a i n a l l p o s s i b l e harmonies f o r a l l f u t u r e and can be employed by those who use them from now on, both s y n t h e t i c a l l y and a n a l y t i c a l l y . Supposed ' n o t yet seen' or e x i s t i n g chords are a l s o , without e x c e p t i o n , c o n t a i n e d i n t h i s system.1° These f i f t e e n note c h o r d s ,  b a s i c types c o n s i s t five different  ent f i v e - n o t e chord.  Before  chords,  of f i v e d i f f e r e n t  four-note  one s i x - n o t e  p r i n c i p l e s and  d e s c r i b i n g these b a s i c types  seven-note  in detail,  it  some of W o l p e r t ' s  s t a t e d at the o u t s e t ,  the p r i n c i p l e s of i d e n t i t y ( I d e n t i t a t ) of  differ-  assumptions.  Two b a s i c p r i n c i p l e s are  (Kongruenz)  chords, three  c h o r d , and one  w i l l be necessary t o s t a t e and c l a r i f y  three-  namely,  and congruence  chords.  17  Wolpert does not accept a twelve-note d i v i s i o n of the octave f o r h i s system of c l a s s i f i c a t i o n , as does Hindemith, i n t h a t he (Wolpert) regards the s p e l l i n g of a g i v e n chord tone as a s i g n i f i c a n t f a c t o r i n the c l a s s i f i c a t i o n of t h a t chord. See p . 22. 18  Franz A l f o n s W o l p e r t , Neue Harmonik, (Wilhelmshaven: H e i n r i c h s h a f t e n . 1972), 14 (Unpublished T r a n s l a t i o n , L . Medveczky, 5 ) .  21. A chord i s i d e n t i c a l w i t h i t s e l f when i t always c o n t a i n s the same motes, no matter how transposed or how o f t e n d o u b l e d . x ? In identical,  order t o determine whether two chords says W o l p e r t , one c o n t r a c t s  closest possible Obviously,  or narrowest  the chord to  position  two chords are  its  (Kontrakturform).  c o n t r a c t i o n can only take place  transposition  are  using  and the e l i m i n a t i o n of d o u b l i n g s .  octave Finally,  i d e n t i c a l i f t h e i r c o n t r a c t e d forms are  the  same. C o n c e r n i n g the p r i n c i p l e of  congruence:  A chord i s congruent w i t h another i f i t s transposed c o n t r a c t e d form i s i d e n t i c a l i n q u a l i t y CQualita't} and q u a n t i t y t Q u a n t i t a t l t o the i n t e r v a l l i c mixture w i t h the chord t o be compared.20 Quantity,  a c c o r d i n g to W o l p e r t , i s  q u a l i t y when r e f e r r i n g t o i n t e r v a l s . r e f e r s to a l l t h i r d s while quantity third.  Also,  two chords are  i n both chords are chords are  spelled  eb-g-bb,  For example,  e x a c t l y the same.  but not w i t h d#-g-bb. ,21 but not w i t h g-cb-e'.)  quality  the  f-a-c  EGB i s  is  notes  S i m i l a r l y , two  the i n t e r v a l s p e l l i n g s  (e.g.  than  s p e c i f i e s the type of  not i d e n t i c a l u n l e s s  not congruent u n l e s s  both chords agree e x a c t l y .  more s p e c i f i c  in  congruent w i t h  i d e n t i c a l with g-b-e1  At the b e g i n n i n g of h i s s e c t i o n on the f o r m a t i o n of  1 9  Ibid..  18 ( 7 ) .  2 0  Ibid.,  19 ( 7 ) .  2 1  Ibid.,  20  (8).  22.  new  chord  types  following  92  that  ajouteV : 1  C,Eb,Gb,Ab.  2.  C,E,G,Ab.  4.  C,E,G#,A.  other chords to.this  this point,  Wolpert  same a s number  concepts He  i n root  we  g o e s on t o  position  even  seem p r e m a t u r e a t  p r e p a r i n g us  that  such  2.  at the o u t s e t at l e a s t ,  harmony.  triads  i n t h e above g r o u p ,  seems t o be  familiar  traditional  However, a c h o r d  s u c h an e x p l a n a t i o n may  h i s system,  resembling  c o n s t r u c t e d which would  classification.  i t sounds t h e Although  c o u l d be  not b e l o n g  the  same  3.  belong  tional  i s , "sixte  i n the  C,E,G,A.  a s C,E,G,G# w o u l d  that  a l l belong  that  1.  Naturally,  though  makes i t v e r y c l e a r  c o n t r a c t e d chords  classification,  also  Wolpert  may  f o r the  i s far  fact  from  have l e a r n e d i n  state  that  a l l tradi-  belong to another c l a s s i f i c a -  (5)  tion,  w h i c h he  "qualitative" essential  calls ( ). 3  interval  Wolpert  c h o r d r o o t s h e r e , he exist, fact  structure  in classifying  Although  and  states  connected  I t i s important i n a chord  does not  g i v e an  "the problem  22 Ibid..,  21  (8).  Ibid.,  13  (4).  the  type.  e x p l a n a t i o n of  such  of the r o o t  tonality."^  that  i s what i s  i t according to basic  acknowledges t h a t  w i t h the  t o note  r o o t s must of a chord  is in  23. According be c l a s s i f i e d Conveniently (the  Example  2.  as  all  belonging to  described,  numerals  distinguish  to Wolpert,  are  put  one o f  these  five  five  types  i n parenthesis,  them f r o m t h e  24 chords^ can  three-note  basic are  as  types. follows  says W o l p e r t ,  old thorough-bass  to  numbering):  25  A f\ ra>  . o  .....  o  (5)  Such types discovers  as  (2)  j ^ j and j ^ j a  r  on i n v e r s i o n t h a t  respectively, system.  (4) (3)  (5)  (3)  and t h e r e f o r e  e  (3) (2)  (2)  n  ° t included  because  one  t h e y become  j^j  are  contained i n the  already  Further examination reveals  the  anc  * (3)  group to  be  all-  inclusive. To d e t e r m i n e note  i n t o which c l a s s i f i c a t i o n  chord would f a l l , 1.  ment o f  Reduce t h e  notes  one p r o c e e d s chord  must be i n t h e  24  to  its  as  three-  follows:  contracted  narrowest  any  form.  possible  Arrange-  position.  A n y g r o u p o f t h r e e o r more n o t e s may be c o n s i d e r e d a c h o r d , and e v e r y c o m b i n a t i o n o f t h r e e n o t e s i s r e d u c i b l e t o one o f t h e s e f i v e b a s i c t y p e s , a l w a y s w i t h t h e p r o v i s o t h a t a c c i d e n t a l s i g n s do n o t a f f e c t t h e c l a s s i f i c a t i o n . Ibid..  29  24 Example 3.26  IT -Q-  2.  i s not  but  or  Transpose the contracted form so that i t s bass  note i s C. 3.  A c c i d e n t a l s , f o r the time being, are not  considered—only  the q u a l i t a t i v e i n t e r v a l  structure i s  considered. 4.  Match i t w i t h one of the types (  >4j, . . ) i n the example above.  (3)'  i 5 ) , (2)'  At t h i s point Wolpert d i g r e s s e s t o d i s c u s s c e r t a i n s t r u c t u r e s i n Hindemith's system of c l a s s i f i c a t i o n of chords.  The f o l l o w i n g four i l l u s t r a t i o n s of three-note  chords i n Hindemith's system are examined to "make c l e a r 27 the basic d i f f e r e n c e s i n our methods," says Wolpert.  26 27  ± s  "the symbol f o r the "contracted form o f . "  Ibial., 23 ( 9 ) .  25 Example 4.^8  rl ff  9!  -9-  J  -b-€>— o  u  2.  3.  i.  Wolpert p o i n t s out t h a t according to Hindemith s method, 1  there e x i s t s between the above four chord forms no connection t h a t i n any way makes them s i m i l a r .  What  Wolpert f a i l s t o mention, however, i s that 1, 3 and 4 a l l f a l l i n the same c l a s s i f i c a t i o n , that i s , I I I . l . second belongs t o V, and i t s root i s not d e f i n e d , to Hindemith.  The  according  Wolpert emphasizes that a l l of the above  chords are b a s i c a l l y  the same and a l l belong t o the same  c l a s s i f i c a t i o n i n h i s system.  When considered  i n their  contracted forms, they a l l reduce t o the b a s i c type of j ^ j . A c c o r d i n g l y , Wolpert  concludes:  . • . because of t h i s , and on the b a s i s of numerous other examples, chord i d e n t i f i c a t i o n by Hindemith's method i s p r a c t i c a l l y impossible.2" Wolpert then proceeds with h i s c l a s s i f i c a t i o n of four-note chords.  Again, these f a l l i n t o f i v e basic types  and are notated as f o l l o w s :  28. I b i d . . 23. 29  I b i d . . 23 (10).  26, 30  Example 5.  te-  ( U )  (-2  (I) (?)  a)  The "a" stands f o r "ajoutee" and should be t r a n s l a t e d l i t e r a l l y as a "piece added on." Wolpert emphasizes t h a t (6a) should not be thought of as an added s i x t h chord, but i n the l a r g e r context of a chord type.  For example, the  chord w i t h the notes C,Eb,Gb,Ab would f a l l i n t o the type (6a) but would c e r t a i n l y not sound as an "added s i x t h " chord, The method f o r determining i n t o which c l a s s i f i c a t i o n a given four-note chord f a l l s i s the same as that o u t l i n e d above f o r the three-note  chords.  The f i v e - n o t e chords f a l l i n t o only three basic categories.  These are as f o l l o w s :  31  Example 6.  53:  30  I b i d . . 29.  I b i d . Here a v e r t i c a l l i n e i s used t o separate the numerals from the "ajoute"*e" symbol. I t was not necessary i n the four-note chords as only one numeral was present.  27. A l l six-note chords are r e d u c i b l e t o the same basic type, according t o Wolpert.  This b a s i c type i s  as f o l l o w s : 32 Example 7.'  The same i s true f o r a l l seven-note  chords  Example 8.33  A l l of these chord types can be a l t e r e d from t h e i r above "normal" forms by the use of a c c i d e n t a l s .  Again, i t  i s emphasized t h a t t h i s a l t e r a t i o n (unless i t takes place on the lowest n o t e ) , does not a f f e c t the chord's c l a s s i f i c a t i o n according t o b a s i c type. When a given chord c o n t a i n s a note w i t h an a c c i d e n t a l preceding i t and the same note without the a c c i d e n t a l , or 32 33  Ibid. Ibid.  28. with a different (Spaltakkorde)  accidental,  it  i s termed a " s p l i t  according to W o l p e r t . ^  chord,"  The concept of  split  chords i s  i n t r o d u c e d by him i n order t o expand the idea of  h i s basic  chord types  possible  so t h a t the system w i l l  include a l l  combinations of m u s i c a l p i t c h e s . If,  f o r example, a b a s i c  | | | chord whose lowest  i s C c o n t a i n s both a G and a G#, the chord i s termed a  note split  chord s i n c e t h e r e are two forms of the note G p r e s e n t . i s notated as  It  follows:  Example 9a.  > The numeral 5 t e l l s the ascending tag the s p l i t ,  that  us t h a t  it  i s the G t h a t  on the r i g h t t e l l s  is,  upward.  "w" ( w e i t ) ,  34  as  i n the f o l l o w i n g  Ibid..  32  (11).  split  while  us the d i r e c t i o n of  I f the two s p l i t  more than an octave a p a r t , the s p l i t  is  sign i s  example:  notes  are  preceded by a  29 Example 9b.  fc  r w>  I f , i n the above example, the notes G and G# are reversed, the  symbol used i s : ^  J  I f n e i t h e r of the s p l i t notes i s the n a t u r a l form of the note, an x or ^  i s used as the t a g , depending on the  arrangement of the notes: Example 10.  J2_a.  > I f a Gb and a G are present, the d i r e c t i o n of the s p l i t i s downward, and i s notated as f o l l o w s : Example 11.  1  30  I f , i n the above example, the notes G and Gb are r e v e r s e d , the  symbol used i s :  . l£  ^  This exhausts the combinations with G, G#, and Gb, except when a l l three are present.  Wolpert does not deal w i t h  t h i s p o s s i b i l i t y here. A summary of the p o s s i b i l i t i e s of the s p l i t chord symbols f o l l o w s : Example 12.  b  b  -A  \ w  b  H  w  —  b  b  -< w  —  (<~) The above examples e l a b o r a t e only on a basic chord whose upper note i s s p l i t .  ^)  S p l i t notes can occur  i n any chord type on any note or notes.  U n f o r t u n a t e l y , few  examples are provided by Wolpert at t h i s p o i n t , and, as a r e s u l t , the concepts are not as c l e a r as would be d e s i r a b l e . The idea of s p l i t chords leads Wolpert n a t u r a l l y t o  31. h i s concept of "dimension" (Di men.qj o p ) -  A l t h o u g h a chord  such as t h a t  i n the l a s t example c o n t a i n s four  pitches,  basic  its  form (when a c c i d e n t a l s  are  c o n s i d e r e d by Wolpert as having e s s e n t i a l l y different four  notes.  In t h i s example,  sounds but i s  another example, sounds,  but i s  different ignored)  is  only three  he s a y s , the chord  only " t h r e e d i m e n s i o n a l . " 3 6  the f o l l o w i n g chord has seven  j  0  has  take  different  only four d i m e n s i o n a l .  Example 13.  It  follows,  says W o l p e r t , t h a t the " r o o t form"  reduction to basic greater  chord type)  than seven d i m e n s i o n s .  different  of a s p l i t Also,  notes must always be a s p l i t  (that  is,  chord cannot be  a chord w i t h twelve chord.^  Wolpert c o n t i n u e s w i t h a d i s c u s s i o n  on "Dissonance  35 36 37  Ibid..  35  (12).  Ibid..  35  (13).  One may wonder why Wolpert d i d not s t a t e the more obvious f a c t t h a t a chord w i t h more than seven d i f f e r e n t sounds must be a s p l i t c h o r d .  32. V a l u e s and D e g r e e s . H e  b e g i n s by s t a t i n g  traditional  "the most complete of a l l  major t r i a d i s  sounds p o s s i b l e "  III  it  The other t h r e e  are d i s s o n a n t and are arranged i n degrees as  II  the  and of a l l the t r a d i t i o n a l t r i a d s ,  the only one he c o n s i d e r s consonant.  I  that  is triads  follows:  minor t r i a d  least  dissonant  diminished t r i a d  medium dissonance  augmented t r i a d  most  dissonant  W o l p e r t ' s r a t i o n a l e f o r the above c l a s s i f i c a t i o n of  triads  i s the s e n s a t i o n of h e a r i n g : the ear i s the foremost standard i n the gauging of dissonance . . . very l i t t l e can be proven physically.39 L i k e w i s e , w i t h i n t e r v a l s , there are t h r e e degrees of  dissonance: I II III  The p e r f e c t  Major  second, minor seventh  Tritones  - medium  Minor second, major seventh  - most  f o u r t h i s termed an " a c c i d e n t a l d i s s o n a n c e "  ( a k z i d e n t i e l l e Dissonanz)  when i t appears o u t s i d e the  context of a major or minor t r i a d , and i s than a l l of the above i n t e r v a l s . c o n s i d e r e d consonant.  38 39  - least  Ibid..  36  (13).  Ibid..  37  (14).  less  dissonant  Other i n t e r v a l s  are  33. The above i n t e r v a l s and t r i a d s terms " s i m p l e " dissonances dissonances  are d i s t i n g u i s h e d  termed " e s s e n t i a l Essential  (einfache  Dissonanzen).  from other types  dissonances"  dissonance  are what Wolpert  alteration is,  dissonance  ( e s s e n t i e l l e Dissonanzen) , 4 < ^  i s d e f i n e d by the f o l l o w i n g c o n d i t i o n s :  Each sounding consonance a r i s i n g  1.  of  Simple  according to i t s  essence,  from an  "essentially  dissonant." 2.  Each sound d e r i v e d from a simple dissonance  also essentially t i o n appears  dissonant—even  if  its  enharmonic e q u a l i z a -  consonant. 4 -*  The f o l l o w i n g examples w i l l  illustrate  conditions. Example 14.  (j)  hp——  iXJd—KL2  1  1  Q  ^  1 and 3 - e s s e n t i a l l y unstable  —  ^  —  dissonant  2 and 4 - consonant and s t a b l e  40  41  Ibid.,  38  Ibid.,  38-9  (14). (14-15).  is  u  and  these  34. The s i g n i f i c a n c e of the i n t e r v a l of a second when speaking of dissonance  i s noted, since a l l chords  with the exception of t r a d i t i o n a l t r i a d s c o n t a i n some form of t h i s i n t e r v a l .  Wolpert b e l i e v e s that the  smaller the seconds and the greater t h e i r number i n a given chord, the greater the degree of dissonance. dissonance  Also,  i s ameliorated by distance and worsened by  proximity, that i s , g e n e r a l l y , the c l o s e r the notes are together the greater the dissonance while the greater t h e i r distance apart, the l e s s the dissonance.  Thus,  g e n e r a l l y speaking, sevenths are l e s s dissonant than seconds.  A l s o , consonance i s equated w i t h s t a b i l i t y and  dissonance  with i n s t a b i l i t y .  Wolpert l a b e l s the minor  second the "diabolus i n musica,"^2  a n c  |  n  o  t ^he t r i t o n e ,  as t r a d i t i o n a l l y recognized from medieval music theory. Both t r i t o n e s as w e l l as p e r f e c t f o u r t h s " l o s e t h e i r dissonance,"  says Wolpert, when they appear i n the context  of t r a d i t i o n a l t r i a d s .  This concept i s termed  "dissonance  Art  according t o c o n d i t i o n . " Wolpert now sets about t o f i n d the root (G rundton) of each of h i s basic chord types according t o t r a d i t i o n a l harmony.  He converts h i s f i f t e e n basic types so that they  Ib_id_., 40 (16). I b i d . . 42-3 (16-17).  35. are arranged as superimposed t h i r d s i n the possible position.  narrowest  Where t h i r d s are missing, the spaces  are marked w i t h arrows.  When one of the b a s i c chord  types i s converted to the narrowest p o s s i b l e t h i r d arrangement, i t i s said to be i n "fundamental p o s i t i o n " or "root p o s i t i o n " (Grundakkord) and the root i s the lowest tone. With most of Wolpert*s b a s i c chord types, there i s no problem i n converting to the t r a d i t i o n a l root (5) position.  One exception occurs with the J 4 J chord,  which has three "narrowest p o s s i b l e " arrangements.  Since s t r i c t l y speaking, there i s no "narrowest p o s s i b l e arrangement," any of the three examples may be considered i n root p o s i t i o n . The other exception i s the seven-note chord, i n which any tone can be considered the root and a s u i t a b l e s t r u c t u r e of t h i r d s erected.  44  I b i d . . 51  36 Example  16.  JLZ  etc.  In the f i n a l  tabular  surveys,  all  of the b a s i c chord  t y p e s and the c o r r e s p o n d i n g Grundakkorden are two d i f f e r e n t  t y p e s of o r d e r s  of  presented  classification,  examples of W o l p e r t ' s method of c h o r d a l a n a l y s i s  45  follow  Example 17.  G i v e n Chord  Contracted  I  In T h i r d s  3 f _  —$r~  Type  45 Ibjud..,  46  Jkid.,  53-4.  59-63.  Some  in  37. C.  Comparison. Upon examination of both Hindemith s and Wolpert's 1  systems of chord c l a s s i f i c a t i o n , c e r t a i n b a s i c d i f f e r ences are immediately evident. Hindemith, i n h i s c l a s s i f i c a t i o n of a l l p o s s i b l e chords, forms c a t e g o r i e s on the b a s i s of combinations of i n t e r v a l s and ranks these c a t e g o r i e s i n terms of " t e n s i o n " and"value".  His own h i g h l y evolved system of r o o t s and  the i n t e r v a l of the t r i t o n e play s i g n i f i c a n t r o l e s i n t h i s ranking. On the other hand, Wolpert, i n h i s c l a s s i f i c a t i o n of a l l p o s s i b l e chords, forms c a t e g o r i e s on the b a s i s of numbers of notes and combinations of i n t e r v a l s according to s p e l l i n g but does not rank h i s c a t e g o r i e s i n terms of value or t e n s i o n .  Furthermore, r o o t s are assigned no  importance i n the c a t e g o r i e s , while the i n t e r v a l s of the augmented and doubly augmented prime play a r a t h e r c u r i o u s r o l e and and are involved i n h i s theory of s p l i t chords. With these basic d i f f e r e n c e s i n mind, the two systems can be contrasted according to the f o l l o w i n g f i v e d i s t i n g u i s h i n g areas: 1.  Roots.  2.  Significant Intervals.  3.  Enharmonic s,  4.  Number of Notes.  5.  Consonance and Dissonance.  38. As has already been stated, Wolpert does not consider r o o t s - - t r a d i t i o n a l or o t h e r w i s e — i n h i s categorization.  However, the f a c t that he does recognize  t r a d i t i o n a l r o o t s i s evident when he converts each of the categories or basic types i n h i s system of c l a s s i f i c a t i o n t o the "narrowest p o s s i b l e t h i r d arrangement" or Grundakkord.  Here the root which he describes corresponds  t o the t r a d i t i o n a l concept of root as the lowest note of a group of superimposed t h i r d s .  Wolpert does not, however,  attempt t o r e l a t e t h i s root t o i t s corresponding note i n the basic t y p e — a process which might have proven h e l p f u l i n our o v e r a l l understanding of the system.  Instead, he s t a t e s  t h a t when t h i r d s appear i n the Grundakkord a t o n a l r e l a t i o n ship i s d e f i n a b l e  Just what i s meant by the term t o n a l  r e l a t i o n s h i p i s not elaborated on at t h i s p o i n t .  When  t h i r d s do not appear i n the Grundakkord, however, ( f o r example, when category j^j b u i l t on C i s converted t o a Grundakkord i t becomes C-G-D, and the missing t h i r d s (E and B) are marked w i t h arrows), a t o n a l r e l a t i o n s h i p i s undefinable.^ In c o n t r a s t t o Wolpert, Hindemith does not recognize t r a d i t i o n a l r o o t s , but has evolved h i s own system of r o o t s 47  I b i d . , 50 (19). 48  IbM.,  50 (19).  39 .  both f o r i n t e r v a l s — a n d , as an extension of i n t e r v a l s , f o r chords—whereby the root i s derived from the "best i n t e r v a l " i n h i s S e r i e s 2, which i n t u r n has been derived from "nature."  Hindemith's system of r o o t s bears d i r e c t l y  on h i s system of c l a s s i f i c a t i o n , so that chords are separated i n t o categories where tone are i d e n t i c a l and tone.  a) the root and the bass  b) the root l i e s above the bass  Furthermore, i n c o n t r a s t t o Wolpert, Hindemith does  not d i s c u s s the t r a d i t i o n a l concept of b u i l d i n g chords i n 49  thirds.  Indeed, one of h i s foremost r e s o l u t i o n s  was that  the c o n s t r u c t i o n of chords i n t h i r d s should no longer form the b a s i s f o r any system or d i s c u s s i o n of chord c l a s s i f i c a tion. The second point concerns s i g n i f i c a n t i n t e r v a l s and how they r e l a t e t o each t h e o r i s t ' s system of o r g a n i z a t i o n . While w i t h Hindemith, the most s i g n i f i c a n t i n t e r v a l i s c e r t a i n l y the t r i t o n e , w i t h Wolpert, i t can be argued that the augmented and doubly augmented primes play an almost equally important r o l e .  Although Wolpert discusses the  importance of the minor second under "dissonance values and degrees," the augmented and doubly augmented prime are 50 of great s i g n i f i c a n c e i n h i s system of c l a s s i f i c a t i o n , 49  50  Hindemith, op, c i t . , 95.  With the d i f f e r e n c e , however, that the augmented and doubly augmented prime play no r o l e i n assigning of chords to c a t e g o r i e s , and i n f a c t are ignored i n t h i s process.  40. as he b u i l d s h i s e n t i r e unique theory of s p l i t chords around these i n t e r v a l s .  We have already recognized that  a s p l i t chord c o n t a i n s an a l t e r e d note and the same note unaltered, or w i t h a d i f f e r e n t a c c i d e n t a l .  Since i n a l l  s p l i t chords the i n t e r v a l of an augmented or doubly augmented prime i s i n v o l v e d , these i n t e r v a l s can be considered t o have s p e c i a l s i g n i f i c a n c e  i n Wolpert's  system of c l a s s i f i c a t i o n , j u s t as the t r i t o n e has i n Hindemith*s. Hindemith's r a t i o n a l i n choosing the t r i t o n e as a s i g n i f i c a n t and t h e r e f o r e d i s t i n g u i s h i n g already been mentioned.  i n t e r v a l has  The primary d i v i s i o n i n h i s  system of chord c l a s s i f i c a t i o n i s based on t h i s d i c t i n c t i o n , so that Group A includes a l l chords that have no t r i t o n e while a l l remaining chords, that i s , those c o n t a i n i n g one or more t r i t o n e s , are assigned t o Group B.  However, i t  should be noted here that both Hindemith and Wolpert recognize the importance of the t r i t o n e when they come t o d i s c u s s "harmonic f l u c t u a t i o n " and "chord succession and connection"  respectively.  The next d i s t i n g u i s h i n g  characteristic  involves  the treatment of what i s t r a d i t i o n a l l y r e f e r r e d t o as enharmonics. effects  In Wolpert's system the s p e l l i n g of a chord  the category i n t o which i t i s placed. For  example, i n the f o l l o w i n g : 51  See p. 14.  41 Example 18.  £©5 -e(ta) the f i r s t  chord reduces t o the t h r e e - n o t e  category  w h i l e the second chord belongs to the f o u r - n o t e (6a).  Thus when c o n s i d e r i n g h i s system of  category  classification,  Wolpert does not r e c o g n i z e the enharmonic e q u i v a l e n c e of notes  such as g# and a b . On the other hand, i n H i n d e m i t h ' s system,  note d i v i s i o n of the octave such as those  a twelve-  i s accepted and enharmonics  above are c o n s i d e r e d e q u i v a l e n t .  Thus  Hindemith would c o n s i d e r the chords i n the above identical  f o r purposes  be assigned  of c l a s s i f i c a t i o n ,  t o h i s sub-group  Another b a s i c  difference  between H i n d e m i t h ' s and concerns the  i n g i v e n chords w i t h i n a s p e c i f i c  In W o l p e r t ' s system, c o n t a i n the  and both would  III.l.  W o l p e r t ' s systems of chord c l a s s i f i c a t i o n number of notes  example  a l l chords w i t h i n a category  same number of n o t e s .  For example,  five basic  three-note chords,  five basic  f o u r - n o t e c h o r d s , each r e p r e s e n t i n g a  and so on, up t o one b a s i c  chord.  always  there  each r e p r e s e n t i n g a  seven-note  category.  are  category, category,  Therefore,  42. each of Wolpert's f i f t e e n c a t e g o r i e s contains a s p e c i f i c number of notes--three, f o u r , f i v e , s i x or seven. In Hindemith*s system, however, some of the subgroups ( f o r example, I I I ) , c o n t a i n chords which can have anywhere from three to s i x d i f f e r e n t notes.  In t h i s  sense, Hindemith's system i s l e s s o r d e r l y than Wolpert's i n that he must use the word "etc." f o l l o w i n g  sub-groups  i n h i s Table which are n o n - f i n i t e ( f o r example, I I . b . 1 , 2 and 3; I I I . l and 2; IV.1 and 2 ) .  Wolpert merely has  f i f t e e n b a s i c categories t o which a l l p o s s i b l e chords reduce.  He thus avoids presenting endless examples by  simply c l a s s i f y i n g the d i v i s i o n s f o r a l l the p o s s i b l e combinations. Perhaps the most c o n t r o v e r s i a l point of d i s t i n c t i o n i s the l a s t - - t h e problem of consonance and dissonance. Hindemith does not define e i t h e r term o b j e c t i v e l y .  He  does state t h a t the value order l a i d down i n h i s S e r i e s 2 approaches the problem of the consonance and dissonance of intervals.  However, he does not s p e c i f y any point at  which consonance stops and dissonance begins. The consonant i n t e r v a l s would then appear at the beginning of S e r i e s 2 and the dissonant a t the end. But the r a t e a t which the consonance of the i n t e r v a l s near the beginning decreases and the dissonance of those near the end increases cannot be determined exactly. 5 2  52  I b i d . , 85.  43. The consonance or dissonance of one i n t e r v a l r e l a t i v e t o another, then, i s a l l that can be determined by t h i s series.  According t o Hindemith, a major t h i r d may be  dissonant when compared t o a p e r f e c t f i f t h , but i t i s consonant i n r e l a t i o n t o a minor seventh, and so on. The t r i t o n e , not included i n S e r i e s 2, i s a s p e c i a l case and i s neither consonant nor dissonant. In Hindemith's words, " i t belongs n e i t h e r t o the realm of euphony nor t o that 53 of cacophony." Hindemith does not extend h i s treatment of consonance and dissonance of i n t e r v a l s to chords i n the way that he extended h i s concept of r o o t s from i n t e r v a l s t o chords. He avoids the use of the term "dissonance" and i n s t e a d , i n the Table of Chord Groups he arranges chords i n order of "tension."  5 4  I f one were t o extrapolate from Hindemith's S e r i e s 2 t o t r y t o determine a scale of consonance and dissonance for chords one might come up w i t h a scale of values approximating Hindemith's chord t a b l e .  However, one would  encounter a l l s o r t s of d i f f i c u l t i e s and ambiguities, 53  Ibid.  54 See Chapter I I , p. 17.  partly  44.  because of the obvious f a c t that there are more chords than i n t e r v a l s .  Hindemith i s very c a r e f u l i n h i s  d i s c u s s i o n of chord types and does not use the word "consonant" discussion.  or "dissonant" once i n t h i s s e c t i o n of h i s Even though he b e l i e v e s t h a t the "minor  t r i a d should rank higher i n the scheme of t o n a l values 55 than the major,"  major and minor chords are of equal  value i n h i s t a b l e . Hindemith acknowledges t h a t the concepts of consonance and dissonance have never been s a t i s f a c t o r i l y explained and t h a t throughout h i s t o r y the d e f i n i t i o n s have varied. At f i r s t t h i r d s were dissonant; l a t e r they became consonant. A d i s t i n c t i o n was made between p e r f e c t and imperfect consonances. The wide use of seventh chords has made the major second and the minor seventh almost consonant t o our ears. The s i t u a t i o n of the f o u r t h has never been c l e a r e d up. T h e o r i s t s , basing t h e i r reasoning on a c c o u s t i c a l phenomena, have repeatedly come t o conclusions w h o l l y at variance w i t h those of p r a c t i c a l musicians. ° As mentioned e a r l i e r , Wolpert's ideas concerning consonance and dissonance i n v o l v e the ear as being the u l t i m a t e standard of measurement and he b e l i e v e s t h a t very l i t t l e about consonance and dissonance can be proven p h y s i c a l l y . A s a r e s u l t , h i s ideas concerning these  55 Ibid.  f  77.  56 I b i d . , 85.  Wolpert, o.p_i_cJLi., 37 U 4 ) .  45 concepts r e s t mainly on h i s own e m p i r i c a l observations. He makes no attempt, as Hindemith does, to evolve a l o g i c a l system, but simply states h i s own s u b j e c t i v e opinions as having a kind of common sense v a l i d i t y . Although Wolpert does not go so f a r as to give a S e r i e s 2 by p i e c i n g together h i s g e n e r a l i z a t i o n s , one i s a c t u a l l y able to come up w i t h a p a r t i a l scale of values of i n t e r v a l s which proceed from the most consonant t o the most dissonant, j u s t as i n Hindemith. degrees of dissonance.  Wolpert gives three  He a l s o says that g e n e r a l l y seconds  are more dissonant than sevenths. ^ 5  From the i n f o r m a t i o n  he g i v e s , one can construct the f o l l o w i n g s e r i e s : (Least Dissonant)  m7  M2  T  M7  m2  (Most  Dissonant)  A l s o , the p e r f e c t f o u r t h i s considered an " a c c i d e n t a l " dissonance when i t does not appear as part of a major or minor t r i a d .  Because i t can be considered e i t h e r consonant  or dissonant, the p e r f e c t f o u r t h a c t s as a kind of bridge between Wolpert's consonant and dissonant i n t e r v a l s .  As  one would expect, the other i n t e r v a l s - - o c t a v e , p e r f e c t f i f t h , s i x t h s and t h i r d s — a r e a l l considered consonant. (Here we must assume Wolpert means only the major and minor t h i r d s and s i x t h s . )  However, no clues are given as to which  consonant i n t e r v a l s are more consonant than others.  See p. 32  Since  46 Wolpert gives us three degrees of dissonant i n t e r v a l s one wonders why he does not (using h i s ear) give us three or more degrees of consonant i n t e r v a l s .  I f he does  have them i n mind, one can only guess as t o what the ordering might be. A comparison of the l a t t e r p o r t i o n of Hindemith's S e r i e s 2 w i t h Wolpert's scale above (both " s e r i e s " proceding from the more consonant t o the more dissonant i n t e r v a l s ) shows t h a t : Hindemith:  m3  M6  M2  m7.  m2  M7  Wolpert:  P4  m7  M2  T  M7  m2  1.  Wolpert b e l i e v e s the minor seventh l e s s  dissonant than the major second.  I n Hindemith s s e r i e s , 1  the opposite i s t r u e . 2.  A l s o , Wolpert b e l i e v e s the major seventh l e s s  dissonant than the minor second.  Again, the opposite i s  true i n Hindemith's s e r i e s . 3.  While Wolpert b e l i e v e s the t r i t o n e t o be of  medium dissonance (more dissonant than M2 or m7 but l e s s dissonant than m2 or M7) Hindemith b e l i e v e s i t t o be unique and n e i t h e r consonant nor dissonant. 4.  With Wolpert, the p e r f e c t f o u r t h can be c l a s s i f i e d  as a dissonance and comes a f t e r the t h i r d s and s i x t h s i n terms of consonance.  With Hindemith, however, the p e r f e c t  f o u r t h i s always more consonant than a l l of the t h i r d s and sixths.  47.  Wolpert does not s y s t e m a t i c a l l y extend h i s ideas of consonance and dissonance t o a l l p o s s i b l e chords or even the c a t e g o r i e s i n h i s system of c l a s s i f i c a t i o n , and apart from d i s c u s s i n g the four t r a d i t i o n a l t r i a d s , he goes no f u r t h e r . Wolpert's d i s c u s s i o n of " e s s e n t i a l dissonances" as d i s t i n g u i s h e d from "simple dissonances" i n v i t e s criticism.  He b e l i e v e s that even i f an i n t e r v a l which i s  s p e l l e d as a dissonance sounds consonant because of enharmonic equivalence t o a consonant i n t e r v a l , i t i s s t i l l " e s s e n t i a l l y dissonant" on account of i t s s p e l l i n g .  His  c o n c l u s i o n regarding these " e s s e n t i a l dissonances" i s that "they formation."  are unstable and cannot be used f o r cadence 59 I t would seem that Wolpert i s p e r s i s t e n t  and determined t o d i s t i n g u i s h between enharmonic s p e l l i n g s , both i n h i s categories of c l a s s i f i c a t i o n and i n h i s t r e a t ment of dissonance.  However, although he would have us  b e l i e v e that a "g#" f o r example, sounds d i f f e r e n t from an "ab" he does not s p e c i f i c a l l y make t h i s statement. A l s o , i t would seem that the i n t e r v a l C-Eb would be an e s s e n t i a l dissonance i f a l t e r e d from C-E.^ 59 60  I b i d . . 38 (14).  See p r o p o s i t i o n 1, p. 33.  The group of " e s s e n t i a l l y  48 dissonant i n t e r v a l s , " then, can include a whole v a r i e t y of types and the concept does not apparently serve any p r a c t i c a l purpose, at t h i s point at l e a s t , apart from t e l l i n g us which i n t e r v a l s are "unstable."  However, the  i n s t a b i l i t y of some e s s e n t i a l dissonances, as we have seen with the i n t e r v a l C-Eb, i s questionable.  A l s o , Wolpert  has p r e v i o u s l y admitted that even simple dissonances are 61 a l l unstable.  I t would seem, then, that the concept of  e s s e n t i a l dissonance i s of l i t t l e value. We now come t o W o l p e r t s concepts regarding the f  dissonance of chords.  Even though he s p e c i f i c a l l y  discusses only t r i a d s , he seems t o be t r e a d i n g on ground which Hindemith has c a r e f u l l y a v o i d e d — t h a t of the r e l a t i v e consonance and dissonance of chords.  He b e l i e v e s the major  t r i a d "the most complete" of a l l sounds p o s s i b l e . uncompromising  i s W o l p e r t s b e l i e f i n the major 1  So  triad's  supremacy, t h a t he considers i t alone to be the consonant triad.  The other three t r i a d s are a l l considered dissonant 62 i n the degrees p r e v i o u s l y explained. Wolpert sees the  minor t r i a d as a "clouding" of the major, 61 62  See p. 32. See p. 32.  63 Ib_Ld,., 36 ( 1 3 ) .  AO  a concept,  49. 64 enough, w i t h which Hindemith agrees.  interestingly  The augmented t r i a d puzzles Wolpert and he cannot understand why the i n t e r v a l C-G# the  does not sound dissonant, while 65  chord C-E-G# has a "very tense sound."  This apparent  c o n t r a d i c t i o n , he says, i s "not v a l i d l y e x p l a i n a b l e " and i s one of the arguments he uses t o j u s t i f y h i s b e l i e f ear  as the f i n a l judge i n d i f f e r e n t i a t i n g  consonance.  i n the  dissonance from  One wonders why Wolpert does not l a b e l the  diminished t r i a d more dissonant than the augmented t r i a d . This seems t o be i n c o n s i s t e n t w i t h h i s d i s c u s s i o n of the consonance and dissonance of i n t e r v a l s .  The t r i t o n e  and  seconds and sevenths have been c l a s s i f i e d as dissonant intervals,  while the t h i r d s and s i x t h s are consonant.  Since the augmented t r i a d contains no i n t e r v a l s which Wolpert would c a l l dissonant sounding, while the diminished t r i a d contains the dissonant t r i t o n e ,  one would expect the  augmented t r i a d t o be more consonant than the diminished triad.  ( I n Hindemith's Table the augmented t r i a d comes  before the diminished.) The d i f f i c u l t i e s i n attempting t o come t o terms with b a s i c a l l y  s u b j e c t i v e notions of consonance and  dissonance should by now be apparent. 64 65  Hindemith, op. c i t . . 78. Wolpert, op. c i t . . 38 (14).  There i s disagreement  50.  between Hindemith and Wolpert on almost every point apart from general b e l i e f s such as the consonance of octaves and f i f t h s and the dissonance of seconds and sevenths. Although there are many d i f f e r e n c e s between Hindemith's and Wolpert's systems, both t h e o r i s t s have attempted and succeeded i n c l a s s i f y i n g a l l p o s s i b l e combinations of musical p i t c h e s , i . e . , a l l p o s s i b l e chords.  Obviously the t r a d i t i o n a l method of b u i l d i n g  chords i n t h i r d s has been subordinated i n both c l a s s i f i c a t i o n s and does not form the b a s i s of e i t h e r Hindemith's or Wolpert's systems.  CHAPTER I I I CHORD  A.  MOVEMENT  Hindemith. Hindemith s study of chord movement i n v o l v e s the 1  examination of three main p o i n t s , a l l of which, when considered, come t o bear on the e f f e c t i v e n e s s of a given chord progression.  These a r e :  1.  Harmonic F l u c t u a t i o n  2.  Degree Progression  3.  The Two-Voice Framework  I t has already been mentioned w i t h respect t o Hindemith's system of chord c l a s s i f i c a t i o n t h a t chord value and chord t e n s i o n are i n v e r s e l y p r o p o r t i o n a l t o one another.  The higher the number of the sub-group, the  higher the t e n s i o n and the lower the value of the chord t o be considered.  Conversely, the lower the number of the  sub-group, the lower the t e n s i o n and the higher the value of the chord. • . . the t e n s i o n of chords increases from s e c t i o n to s e c t i o n and from sub-group t o sub-group i n the same p r o p o r t i o n as the value decreases . . . i t i s t h i s up and down change of values which we s h a l l term 'Harmonic F l u c t u a t i o n . ' The f l u c t u a t i o n may be gradual or sudden according t o the r e l a t i v e values of the chords that make up the p r o g r e s s i o n . !  1  Hindemith, The C r a f t , op. c i t . . I , 116.  52. According t o Hindemith,  sudden f l u c t u a t i o n s occur  when the progression s k i p s a sub-group (e.g., 1.1 to I I I . 2 or I l . b . l t o IV.2).  On the other hand, gradual f l u c t u a -  t i o n s are those, f o r example, when the progression occurs w i t h i n one of the sub-groups ( I I . a t o II.b.3) o r , even between two d i f f e r e n t chords from w i t h i n a s e c t i o n of a sub-group (e.g., both from I I . b . 2 ) . Between the "sudden" and the "gradual" f l u c t u a t i o n s are those which move among consecutive sub-groups ( I I I . 1 t o IV.2 or 1.2 t o I I . b . 3 ) . These might be termed "medium" f l u c t u a t i o n s . Hindemith s t a t e s that f o r harmonic f l u c t u a t i o n t o occur, "chords of d i f f e r e n t value" are r e q u i r e d . understand  To  t h i s statement more f u l l y , one should examine  i t s converse.  Hindemith s t a t e s :  "In the connection of  chords of i d e n t i c a l s t r u c t u r e , there i s no harmonic f l u c t u a t i o n . T h u s , harmonic f l u c t u a t i o n occurs whenever a chordal s t r u c t u r e moves t o any other chordal s t r u c t u r e , but does not occur when a succession of i d e n t i c a l chordal s t r u c t u r e s appears.  In the example below, (a) c o n s t i t u t e s  a gradual f l u c t u a t i o n while at (b) there i s no harmonic fluctuation.  2  I b i d . . 117  53. Example 19.  [J—,  —1  o o o  o Q o  Hindemith does not appear t o give any d e f i n i t e r u l e s as t o what c o n s t i t u t e s good harmonic f l u c t u a t i o n , apart from the g e n e r a l l y i m p l i e d notion that i n a given musical passage there should be a gradual r i s e and f a l l of harmonic t e n s i o n .  This apparent l a c k of information  has been n o t i c e d by Herman Hensel: . . . the information (Hindemith) gives us r e l a t i v e t o what c o n s t i t u t e s good o r g a n i z a t i o n of harmonic f l u c t u a t i o n i s rather sparse and at times ambiguous. . . 3 Hensel a l s o sees Hindemith*s i m p l i c a t i o n of the d e s i r a b i l i t y of a r i s e and f a l l of harmonic t e n s i o n .  However, d i f f i c u l -  t i e s a r i s e when one t r i e s t o r e l a t e harmonic f l u c t u a t i o n t o phrase s t r u c t u r e : Hindemith regards the arch type harmonic t e n s i o n repose design as one which shows a good o r g a n i z a t i o n . At t h i s point one cannot be sure whether the arch should or should not p a r a l l e l the phrase, however. A l s o , t h i s i n v e s t i g a t i o n suggests that Hindemith regards as good a design which i s out of phase with the other elements, i . e . , a design that b r i n g s about  3  Herman Richard Hensel, "On Paul Hindemith's Harmonic F l u c t u a t i o n Theory," Unpublished D.M.A. D i s s e r t a t i o n , (Urbana: U n i v e r s i t y of I l l i n o i s , 1964), quoted from a b s t r a c t .  54.  an e q u i l i b r i u m of tensions as the v a r i o u s elements of music i n t e r a c t w i t h each o t h e r . 4  In the C r a f t . Hindemith does warn against movement between c e r t a i n sub-groups, s p e c i f i c a l l y , V or VI and I I I or IV: . . . as a counterpoise to the s t a b l e and t e n s i o n l e s s chords of Group I , a chord from group V or VI may be u s e f u l ; i t can almost always be s u c c e s s f u l l y juxtaposed even against chords from group I I . But i n using i t w i t h chords from groups I I I and IV, care must be e x e r c i s e d . In the midst of such chords, a chord of group V or VI o f t e n puts us completely o f f the t r a c k : i t seems to cause the whole chord s t r u c t u r e to c o l l a p s e . . . . Progressions of t h i s type must a c c o r d i n g l y be handled with extreme •  *  •  •  Apart from t h i s i t seems i t i s l e f t to the composer's musical i n t u i t i o n to ensure good f l u c t u a t i o n , as no f u r t h e r s p e c i f i c r u l e s are given.^ Hindemith helps j u s t i f y h i s theory of harmonic f l u c t u a t i o n by e x p l a i n i n g that only with t h i s theory i s there an "explanation f o r chords of v a r y i n g harmonic t e n s i o n -i  upon the same r o o t . "  In the f i n a l a n a l y s i s , harmonic  f l u c t u a t i o n can 'be considered as simply an extension of Hindemith's system of chord c l a s s i f i c a t i o n . 4  5  I t takes no  Ibid.. Hindemith, op. c i t . . 119-20.  6  Hindemith i m p l i e s ( C r a f t , I , 123) that i n successive t r i a d s no t r i t o n e should be evident, e.g., minor dominant to major t o n i c i n C major, t r i t o n e Bb-E i s present and should be avoided. 7  Hindemith, op. c i t . . 120.  55. account of voice leading or root movement but concerns i t s e l f simply with the varying tensions i n a chordal sequence. In c o n s i d e r i n g the second p o i n t , i . e . , "degree progression," we s h a l l adopt the view of V i c t o r Landau i n i n t e r p r e t i n g t h i s term i n the broadest sense t o i n c l u d e a l l successions of chord r o o t s .  Both S e r i e s 1 and S e r i e s  2 are used; the l a t t e r p a r t i c u l a r l y when c o n s i d e r i n g the r o o t s of adjacent chords i n a progression as r e l a t e d i n the " t o n a l sphere" or t o a t o n a l centre. SERIES 1: (Based on Root "C") C,C,G,F,A,E,Eb,Ab,D,Bb,Db,B SERIES 2:  P8, P5, P4, M3, m6, m3, M6, M2, m7, m2  M7  By c o n s i d e r i n g the i n t e r v a l s between the r o o t s of successive chords (not yet assumed as being r e l a t e d t o a t o n a l c e n t r e ) , Hindemith comes t o conclusions regarding the value of c e r t a i n chord progressions:  "a progression based  on the i n t e r v a l of a f i f t h between i t s r o o t s n a t u r a l l y has o a surer foundation than one based on a minor s i x t h . . . . 8 Landau p o i n t s out t h a t the term " t o n a l amplitude," used i n Book I I of Hindemith's T r a d i t i o n a l Harmony and defined as "the amount of t e n s i o n between the t o n i c chord and each of the other chords i n a t o n a l sphere which i s dominated by i t , " describes the above concept i n which S e r i e s 1 i s used as the determining f a c t o r . See Landau's a r t i c l e "Hindemith the System B u i l d e r : A C r i t i q u e , " Music Review XXII (1961), 147. 9 Hindemith, op. c i t . . 122.  56. Hindemith*s r a t i o n a l e here again f o l l o w s the l o g i c of S e r i e s 2.  The best progression i s that which i s based  on r o o t s a f i f t h apart, the next, that based on r o o t s a f o u r t h apart, then that based on roots a major t h i r d apart and so on through the s e r i e s u n t i l one comes t o the chord progression based on two r o o t s a t r i t o n e apart, which Hindemith sees as the " l e a s t valuable of a l l . " - *  0  In a  chord which has no root (Sub-groups V and V I ) , a "root r e p r e s e n t a t i v e " i s chosen which best connects i t (according  t o S e r i e s 2) t o the r o o t s of the chords preceding and  following it.^"*" Hindemith p o i n t s out the disadvantage  of assessing  the value of chord progressions merely from an examination of the movement of t h e i r r o o t s .  The f a c t that a l a r g e  number of chords can be constructed over the same root helps t e s t i f y t o t h i s drawback.  However, Hindemith  dismisses the l i m i t a t i o n i n one sentence:  " . . . here an  i n v e s t i g a t i o n of the two-voice framework and the harmonic f l u c t u a t i o n w i l l c l e a r up a l l ambiguity."-^ Progressions which c o n t a i n t r i t o n e chords are of 10 11 12  I b i d . . 123. I b i d . , 125. I b i d . . 123.  57. s p e c i a l s i g n i f i c a n c e t o Hindemith and t r e a t e d as separate cases.  In these progressions, a knowledge of Hindemith*s  concept of "guide tones" i s e s s e n t i a l . A guide-tone i s , i n a chord from group B, that member of a t r i t o n e which stands i n the best r e l a t i o n s h i p , according t o S e r i e s 2, t o the root of the chord i n question Whenever a chord from group B i s followed by a chord of group A, Hindemith states "the t r i t o n e i s thereby resolved. 14 11  I f the r e s o l u t i o n i s t o be s a t i s f a c t o r y ,  the guide-tone must move by a good melodic i n t e r v a l ( p r e f e r a b l y a second) t o the root of the f o l l o w i n g chord. When successions of several group B chords occur, there i s no r e s o l u t i o n of the t r i t o n e but instead a prolonging of the t e n s i o n . This type of succession i s t r e a t e d l i k e the progressions already discussed except that the i n t e r v a l made by the guide tone i n the f i r s t chord moving t o the guide tone i n the second i s considered a secondary assessment ( a f t e r root progression) of the value of the chord p r o g r e s s i o n .  13  Hindemith a l s o p o i n t s out the  I b i d . . 104.  14 I b i d . . 126. 15  I b i d . . 129-130.  58.  i n t e r e s t i n g f a c t that i n the succession of two  tritone  chords from sub-group I I (only o c c a s i o n a l l y w i t h those from sub-group IV) whose r o o t s are a t r i t o n e apart there i s a l s o a t r i t o n e between the guide-tones  of the  two  chords as w e l l as the f a c t that the t r i t o n e contained i n the f i r s t chord must a l s o be contained i n the second. important observation leads Hindemith t o exclaim:  This  "This  chain of t r i t o n e s l i n k s these two chords so c l o s e l y together that they seem almost l i k e f r a c t i o n a l p a r t s of the same chord. Example 20.  A  j  Roots:  D  Common T r i t o n e :  Ab C - P#  (Gb)  %\\ — Guide Tones Thus f a r Hindemith has made no attempt to r e l a t e these concepts of "good" root progression to the concept of t o n a l centre or t o n a l sphere.  Before beginning to discuss  these "harmonic f a m i l y - r e l a t i o n s h i p s , " Hindemith admits that c e r t a i n rhythmic c o n s i d e r a t i o n s are necessary i n 16  Ibid..  130.  59.  attempting to d i s c e r n such t o n a l centres.  "Duration  and  p o s i t i o n i n the measure are of d e c i s i v e importance i n determining the t o n i c :  the stressed p o r t i o n of the  measure, the longest note or the f i n a l note i s needed to t e l l us which i s the p r i n c i p a l tone of the group."-^ Following t h i s necessary concession,  however, there i s no  f u r t h e r d i s c u s s i o n of the element of rhythm i n the concerned w i t h chord movement.  section  Instead, two rather  signifi-  cant d e f i n i t i o n s are stated regarding the harmonic aspect of tonal organization. 1.  These are:  A succession  of chords from Group A must c o n s i s t  of at l e a s t three chords i f i t i s to represent a t o n a l entity. 2.  Only one chord from Group B i s needed to produce  a f e e l i n g of t o n a l i t y since the " t r i t o n e i n i t forces the 18  ear to assume a chord of r e s o l u t i o n . " Here a d i s t i n c t i o n must be made between the terms "tonal e n t i t y " i n d e f i n i t i o n 1 and " f e e l i n g of t o n a l i t y " i n d e f i n i t i o n 2.  Hindemith s t a t e s that although a " f e e l i n g of  t o n a l i t y " i s created i n the sounding of a s i n g l e chord from group B, a " t o n a l e n t i t y " (or " t o n a l centre") i s not 17 18  Ibid..  133.  I b i d . . 134-5.  defined  60 since "the ear does not know i n which d i r e c t i o n to r e s o l v e the t r i t o n e . "  Thus, "the sounding of a s i n g l e  t r i t o n e chord i s enough to create a f e e l i n g of t o n a l i t y , but the t o n a l centre i s not defined.  Only when the  t r i t o n e i s resolved can one know which chord root i s the t o n a l c e n t r e . " ^  9  When a chord from group B r e s o l v e s  to a chord from group A, the root of the l a t t e r i s considered the t o n a l centre.  In a succession of chords  from group B the t o n a l i t y i s not determined u n t i l chord of r e s o l u t i o n . However, i n a s e r i e s of  the  unresolved  chords from sub-group I I , the t o n a l centre may be  regarded  as the f i f t h below the root of the f i n a l chord i n the s e r i e s since "the unresolved t r i t o n e of the f i n a l  chord  would r e s o l v e most n a t u r a l l y i n t o an i n t e r v a l whose root would be a f i f t h below the root of the t r i t o n e  chord."  Hindemith s summary i n chart form of the number of chords 1  needed f o r determining a t o n a l centre i s reproduced i n the Appendix. I t has already been mentioned that S e r i e s 1 i s used when c o n s i d e r i n g chords r e l a t e d t o a t o n a l centre,  although  Hindemith does not f u l l y e x p l a i n h i s r a t i o n a l e i n choosing  19 20  Ibid. Ibid.,  136.  61 it.  As S e r i e s 2 was derived b a s i c a l l y from h i s i n t e r -  p r e t a t i o n of combination tone curves, so S e r i e s 1 was derived b a s i c a l l y from h i s i n t e r p r e t a t i o n of the overtone series.  Whereas S e r i e s 2 consisted of a row of i n t e r v a l s ,  S e r i e s 1 c o n s i s t s of a row of tones and represents the degree of r e l a t i o n s h i p these tones have t o a given tone. The f u r t h e r a tone i s away from the f i r s t tone, the more d i s t a n t the r e l a t i o n s h i p . SERIES 1: (Based on "C") C,C,G,F,A,E,Eb,Ab,D,Bb,Db,B A degree progression may be r e s t r i c t e d t o the high-ranking degrees of S e r i e s 1 (Tonic, Dominant and Subdominant) or i t may c o n s i s t of a v a r i e t y of both high and low ranking degrees.  As the degree of r e l a t i o n s h i p between the chord  r o o t s and the t o n i c chord root v a r i e s , so does the t e n s i o n . As Landau has pointed out^ * t h i s "tension" i s not the same -  "tension" used i n d i s c u s s i n g chord c l a s s i f i c a t i o n .  In the  l a t t e r , "tension was inherent i n the s t r u c t u r e of the 22 chord" while w i t h degree progression, " t e n s i o n " r e f e r s t o . . . the c o n f l i c t between the a u t h o r i t y of the t o n a l centre and the urge of the i n d i v i d u a l harmonies t o escape from that a u t h o r i t y . When t h i s urge i s g r a t i f i e d , of course, a new t o n a l centre i s e s t a b l i s h e d and modulation takes place . . . .23  21  Landau, op. c i t . . 150.  22 See p. 18. 23 I b i d .  62. Landau summarizes Hindemith's views on the establishment complete  of t o n a l i t y (or a t o n a l centre) i n a  composition:  . . . the p r e v a i l i n g t o n a l i t y i s e s t a b l i s h e d by the i n t e r p l a y of the same f a c t o r s which serve that purpose i n a t o n a l sphere, i . e . , r e p e t i t i o n , f i n a l i t y , and the confirmation of r e l a t e d tones. Thus, the t o n a l centre which i s must repeated or which appears at the end, or which i s s t r o n g l y supported by i t s dominant and sub-dominant, i s revealed as the p r i n c i p a l tone of a movement or of an e n t i r e work.24 The t h i r d point to be mentioned w i t h respect to chord movement i s the two-voice framework, i . e . , the bass l i n e and the most prominent upper part at a given moment. Hindemith b e l i e v e s that a chord progression i s a f f e c t e d very l i t t l e by the inner v o i c e movement.  25  i t is  p r i n c i p a l l y the s e t t i n g of the "two-voice framework" that the most i n f l u e n t i a l and i t i s to t h i s f a c t o r that he devotes h i s a t t e n t i o n .  " I f w r i t i n g i n s e v e r a l voices i s  t o sound c l e a r and i n t e l l i g i b l e , the contours of i t s twov o i c e framework must be c l e a n l y designated  and  cogently  organized."26 The extent of Hindemith's d e s i r e f o r such s t r i c t o r g a n i z a t i o n i s evident i n Volume 2 of the C r a f t where a  24 25 26  Ibid. Hindemith, op. c i t . . Ibid..  114.  115.  63 t o t a l of s i x t y - f i v e r u l e s are l i s t e d as a guide f o r w r i t i n g the two-voice framework.  Many of these r u l e s  have t o do w i t h the c o n s t r u c t i o n of good melodies while there are only a few which have a d i r e c t bearing on the p r i n c i p l e s of chord movement or succession.  Landau, i n  h i s study, has chosen only twelve f o r reasons which he s t a t e s below: Hindemith abrogated . . . r u l e s g r a d u a l l y throughout Book I I as the student was presumed t o have exhausted the b e n e f i t s of observing them . . . . There a r e , however, several r u l e s i n Book I I which were not rescinded and some which were expressly reaffirmed.2' Of the above explained "unrescinded" and "reaffirmed" r u l e s Landau chooses the f o l l o w i n g : 1.  D i s t r i b u t i o n of i n t e r v a l s between the v o i c e s ( t h i r d s and s i x t h s should balance seconds and sevenths)  2.  Relative a c t i v i t y :  3.  A l t e r n a t i o n of a c t i v i t y : keep other s t i l l )  4.  I n t e r v a l root below at beginning, end, important p o i n t s  5.  No c r o s s i n g of Voices  6.  No P a r a l l e l Octaves  7.  No Delayed P a r a l l e l s  8.  No Covered Octaves  9.  No Covered 5ths or 4ths  10.  l e s s movement i n bass ( i f one voice moves,  No Delayed Covered P a r a l l e l s  V i c t o r Landau, "Hindemith-Case Study i n Theory and P r a c t i c e " Music Review. V o l . XXI, 1960, 42.  64. 11.  No leaps t o or from 2nds 7ths or the t r i t o n e (Most v i o l a t e d r u l e i n Landau's study).  12.  Upward r e s o l u t i o n of suspensions only t o 2Q c e r t a i n i n t e r v a l s (See Rule 50, Book I I ) .  In a d d i t i o n t o these, the present w r i t e r f e e l s the f o l l o w i n g two r u l e s should also be i n c l u d e d , p a r t l y because of p o s s i b l e comparisons which may be drawn w i t h Wolpert: 1.  Rule 22 The two v o i c e s may not skip i n the same d i r e c t i o n a t the same time . . . .29  2.  Rule 23 Cross R e l a t i o n s must be avoided.30  Book I I of The C r a f t i s only concerned w i t h two-part w r i t i n g , and Hindemith maintains that the two-voice 3 framework as defined should be governed by these r u l e s . To f u r t h e r c l a r i f y t h i s p o i n t , Landau maintains that these r u l e s , however, cannot be applied t o any combinat i o n of two v o i c e s i n a structure of three or more p a r t s , but rather only t o the two-voice framework as Hindemith 28 Ibid», See i n s e r t p. 46. 29 Hindemith, op. c i t . . I I , 26. 30 I b i d . Hindemith, however, l a t e r permits crossr e l a t i o n s (Rule 37, pp. 46-7) i f one of the notes involved i s passing tone of r e l a t i v e l y short d u r a t i o n f a l l s on the weak part of the measure. 31 Hindemith, op. c i t . . I , 114.  65. has defined i t , i . e . , the bass and most prominent upper part.  3 2  Furthermore, On m a t e r i a l s on three part w r i t i n g which Hindemith d i s t r i b u t e d to h i s students at Y a l e , p a r a l l e l 4ths were allowed between the top and middle v o i c e s and between the middle and bottom v o i c e s - a l s o , p a r a l l e l 5ths were allowed between top and middle v o i c e s when the tones i n e i t h e r p a i r of f i f t h s have d i f f e r e n t f u n c t i o n s (when one i s a non-chord tone). P a r a l l e l octaves, however, were not allowed at a l l . 3 3  B.  Wolpert. Throughout W o l p e r t s d i s c u s s i o n of "The 1  Principles  and Hindrances of Chordal Connection and Succession," ^ 3  i t i s i m p l i e d that the connection of chords i s most s a t i s f a c t o r y when a l l the v o i c e s (with the exception of the b a s s ) , i f they proceed at a l l , proceed by step. Wolpert t r e a t s v o i c e movement by second i n a most systema t i c and thorough way, yet the other p o s s i b i l i t i e s of movement i n the upper v o i c e s (by 3rd, 4 t h , etc.) are hardly d i s c u s s e d . This would seem t o d i s a l l o w , at the 32 33 34  Landau, Hindemith. A Case Study . . . , Landau, op. c i t . .  42.  51.  Franz A l f o n s Wolpert, Neue Harmonik. (Wilhelmshaven: H e i n r i c h s h a f t e n , 1972), 65-96, (Unpublished t r a n s l a t i o n , L. Medveczky, 22-34.  66.  o u t s e t , many p o s s i b i l i t i e s traditionally  of chord movement which  one has come t o a c c e p t .  even  Using the "two-voice  framework"3S (Zweistimmigkeit) as a method of i n v e s t i g a t i o n , Wolpert examines p o s s i b l e combinations of p a i r e d v o i c e s i n the c o n n e c t i o n of two c h o r d s .  Note that i n the example g i v e n ,  a l l v o i c e s except the bass move by step or remain  stationary.  36 Example 21.  - t— mmm  2E  7J  1. quality  n  i.  Reihung  (thirds,  2.  3.  5.  75  - p a r a l l e l movement of i n t e r v a l s of e q u a l  seconds, f i f t h s ,  e t c . ) by step, ascending  or d e s c e n d i n g . 2.  Konstante - one note common i n two chords i n the  same v o i c e .  35 I t should be noted t h a t "two-voice framework" i n the g e n e r a l sense i s d i f f e r e n t from Hindemith's s p e c i a l d e f i n i t i o n of the same term d i s c u s s e d e a r l i e r i n S e c t i o n 'A" of t h i s c h a p t e r . Wolpert's d e f i n i t i o n of the term may apply t o any combination of two v o i c e s . 36 Wolpert, op. c i t . . 65.  67 3.  D i a s t o l e - two voices expanding by step.  4.  Systole  5.  Bas-Kadenz - when the bass does not move by step.  (1) and  - two v o i c e s c o n t r a c t i n g by step.  Concerning the Reihung. p a r a l l e l t h i r d s ,  s i x t h s are acceptable  Wolpert.  fourths  i n p a r t - w r i t i n g , according t o  Even p a r a l l e l major seconds or minor sevenths are  permitted.  However, p a r a l l e l f i f t h s , octaves and "the small  second values and t h e i r i n v e r s i o n s " (aug.  prime, dim.  octave, minor second, major seventh) c o n s t i t u t e a hindrance to agreeable chord movement and are therefore not acceptable.  L a t e r , under the sub-heading "mixtures"  Wolpert e x p l a i n s that consecutive  (Mixturen),  octaves and f i f t h s do,  however, appear i n music, e s p e c i a l l y i n the i m p r e s s i o n i s t s (Debussy and Ravel) and i n Reger, but a l s o i n Bartok and S t r a v i n s k y , and often serve as a " c o l o u r f u l of the melody." (2)  strengthening  38  Concerning the Konstante tone:  Wolpert states  two "Laws of I n a c t i v i t y " as f o l l o w s : 1.  A tone which i s common i n two chords should  remain i n the same v o i c e . 2.  In the connection  move the shortest p o s s i b l e  3  3 8  39  7  /  \  I b i d . . 66 (23). Ibid..  69 (23).  I b i < i . , 70 (24).  of two chords, voices distance.  3 9  should  68.  Wolpert i m p l i e s , then, that the l e s s a c t i v i t y i n the voice leading of a chord progression, the b e t t e r .  He  flatly  states that "jumps ( i n t e r v a l s greater than a second) have a cumbersome e f f e c t " and goes so f a r as to give the f o l l o w i n g examples of what he says are "successions which cannot be c a l l e d connections" because " t h e i r parts are standing f o r themselves, unconnected" and "no hearing l o g i c i s to be recognized i n them."  40  41 Example 22.  3£a Here a d i s t i n c t i o n should be made between the terms "connect i o n " and "succession," although i t appears that Wolpert does not.  A succession can i n v o l v e any s e r i e s of sounds.  A  connection, i t would seem from the above, i s a d e s i r a b l e succession, which, according t o Wolpert, would involve as l i t t l e voice movement as p o s s i b l e (except f o r the b a s s ) .  I b i d . 70 (24), Aufeinanderfolgen wie die des B e i s p i e l s C22J Kann man wohl n i c h t Verbindungen nennen. Ihre T e i l e stehen unverbunden f u r s i c h . Ihre Sprunge wirken s p e r r i g , i h r e Vorzeichen sind w i l l k u r l i c h gewahlt. f  41  I b i d . . 71.  69 "Our  t a s k now" he s t a t e s , " i s t o r e a l i z e p o s s i b l e f u t u r e  connections hearing  on the b a s i s of these  already  known l o g i c a l  processes."42 (3,4.)  One can now examine the S y s t o l e n and  D i a s t o l e n , which Wolpert d i s c u s s e s as a group.  Firstly,  d i s t i n c t i o n s are made among three d i f f e r e n t types and  Diastolen.  These are named "complete"  ( h a l b e ) . and "whole-tone" (ganzton). D i a s t o l e n . the c o n t r a c t i n g / e x p a n d i n g step. by h a l f  of S y s t o l e n  (ganze).  With complete  "half" Systolen/  v o i c e s both move by h a l f  In the h a l f S y s t o l e n / D i a s t o l e n , o n l y one v o i c e moves step while the other moves by whole s t e p .  In the  whole-tone S y s t o l e n / D i a s t o l e n both v o i c e s move by whole  step.  Example 23.43  0  3EE  Systolen  So  Piastolen  Complete  42 43  Ibid. Ibid.  f  72.  Half  Whole t o n e ^ o "  70. Next, the concept of "adhesion" introducedIn  (Adhasion) i s  simple terms, "adhesion"  i s the name used  to describe e i t h e r a complete S y s t o l e or a complete D i a s t o l e . Thus, whenever two voices both expand or both contract w i t h a h a l f step movement i n each v o i c e , adhesion i s said t o have taken p l a c e .  I t i s assumed t h a t t h i s "adhesive process" i s  a good q u a l i t y i n chord connection s i n c e , according t o the "Laws of I n a c t i v i t y "  4 5  h a l f - s t e p movement i s more l i k e l y and  d e s i r a b l e than jumps of a major second or l a r g e r . ^ Adhesion may lead t o e i t h e r a stable or an unstable ending.  47  When i t leads t o the l a t t e r , i t i s termed  " d i v e r s i o n " ( D i v e r s i o n ) . However, there i s no term which d e s c r i b e s the opposite of d i v e r s i o n , i . e . , l e a d i n g t o a stable ending.  44 45  Moreover, there i s a d i s t i n c t i o n i n t h i s  I b i d . . 73 (25). See Chapter I I I , p. 67.  46  In order t h a t adhesion can take place w i t h t r a d i t i o n a l l y c o r r e c t voice l e a d i n g the term Umpolung i s invented by Wolpert (see German t e x t , p. 74). A s p e c i f i c note i s "transpoled" so t h a t the voice leading i s t r a d i t i o n a l l y acceptable. This " t r a n s p o l a t i o n " i s nothing more than enharmonic s u b s t i t u t i o n . 47  Unstable i s the same as "dissonant." I I , p. 34.  See Chapter  71. unnamed c l a s s i f i c a t i o n between "ambivalently and "unequivocally adhesive" penultimate  adhesive"  intervals,  depending on the s p e l l i n g of t h i s i n t e r v a l . Two r u l e s are stated regarding adhesion leading to a stable ending: 1.  Unaltered penultimate  i n two ways. 2.  i n t e r v a l s can resolve  (These are c a l l e d ambivalently adhesive.)  A l t e r e d penultimate  only one d i r e c t i o n .  i n t e r v a l s can r e s o l v e i n  (These are c a l l e d unequivocally  adhesive•) In simpler terms, some s p e l l i n g s a l l o w one p o s s i b l e _8 r e s o l u t i o n , w h i l e other s p e l l i n g s allow two. f o l l o w i n g example i l l u s t r a t e s t h i s f a c t : Example 2 4 «  4 9  Unaltered  48 49  Ibid.., 73. I b i d . . 74.  ff &  Unaltered  Altered  The  72 A l l adhesive processes which lead to dissonant or unstable endings are c a l l e d " d i v e r s i v e " - - t h e i t s e l f i s called "diversion."  process  Some examples of t h i s  process f o l l o w : Example 25.50  0  •ifo -6-  ° -g-  o  To demonstrate adhesion i n i t s " f u l l  —o  efficiency,"  Wolpert g i v e s an example of a four-note chord connection. There are s i x p o s s i b l e two-voice frameworks and a l l are examined i n d e t a i l — f i r s t the outer v o i c e s ( p e r i p h e r i e ) ; next, the two p a i r s of v o i c e s separated by one i n t e r mediate v o i c e (ubernachste): and f i n a l l y , the three p a i r s of adjacent v o i c e s (benachbarte). example w i l l help i l l u s t r a t e Wolpert's  50 51  I b i d . , p. 75. I b i d . . p. 76.  The  following  procedure:  73 Example 26.  1.  2.  3.  5.  4.  6.  In the above four-note chord connection, four of the s i x p o s s i b l e two-voice frameworks are adhesive ( s y s t o l e s ) and non-diversive; they a l l lead t o stable endings.  The  remain-  ing two p o s s i b i l i t i e s i n v o l v e Reihnngen - #5 being consecut i v e t h i r d values  while #6 i n v o l v e s consecutive f o u r t h s .  Here, the second of the "Laws of I n a c t i v i t y " ( i . e . , v o i c e s moving the s h o r t e s t p o s s i b l e distance i n chord connection) can be seen to operate.  For t h i s reason i t can be assumed  t h a t the examples given i l l u s t r a t e good connections. As one would expect, non-adhesive  i n t e r v a l connec-  t i o n s can be mixed w i t h adhesive i n t e r v a l connections as i n the f o l l o w i n g example. p a i r i s non-adhesive  52  Note that although the lower i n t e r v a l  i t is still  systolic.  Here s p e l l i n g does not seem to bother Wolpert. Although he recognized the diminished f o u r t h he regards i t as a " t h i r d value" (Terzwerte).  74, Example 2 7 .  5 3  adhesive .i  non-adhe sive (5) movement.  One can now t u r n t o Wolpert's d i s c u s s i o n of bass I t has already been noted that i n chord  connection  i t i s most d e s i r a b l e f o r the upper voices t o move the smallest possible distance. a l l v o i c e s remaining of adhesion.  The u l t i m a t e of t h i s i d e a l (apart from s t a t i o n a r y ) i s manifested  i n the concept  However, Wolpert informs us t h a t bass movement  must be considered  "fundamentally  independent from adhesion"  although when the bass does move by step i t can be considered 54  as part of the adhesive process.  I t would seem, then, that  the bass i s t r e a t e d separately from the other voices and i n f a c t , does not f o l l o w the " r u l e s " l a i d down f o r them. ever, i t does not have i t s own r u l e s .  How-  Instead, Wolpert l i s t s  the most f r e q u e n t l y occurring and t h e r e f o r e most d e s i r a b l e p o s s i b i l i t i e s of upper voice movement r e l a t e d t o the bass. A f t e r numerous analyses of ' c l a s s i c a l ' and 'modern' cadences i t was discovered that i n each upper part the f o l l o w i n g p o s s i b i l i t i e s of voice movement i n r e l a t i o n t o the bass made f o r a s a t i s f a c t o r y c a d e n c e . 55  53 54  I b i d . . 77. I b i d . , 78 (28).  55 Ibid.., 78 (28). I t i s not mentioned what " c l a s s i c a l " and "modern" works were analyzed.  75.  Example 2 8 .  5 6  *  l fh  °  W—-J  o  9  ° r> u  U  1  I h  /\  2 E  1  56  4  _.  I b i d . . p. 80-1. I t should be noted that the upper voice movement i s not always stepwise although not s u r p r i s i n g l y i n the majority of cases i t i s . I t might a l s o be noted that i f a l l p o s s i b i l i t i e s of upper voice movement were t o be l i s t e d , the number f o r each bass movement would t o t a l 11 X 12 or 132.  76. Wolpert thus seems t o imply that the bass voice may move as other v o i c e s (by s t e p ) , but a l s o ,  particularly  at c a d e n t i a l p o i n t s , may move by t h i r d , f o u r t h , or fifth.  He i s , i n f a c t , not complete i n the treatment  of the bass v o i c e as he was a l s o not complete i n h i s treatment of the p o s s i b i l i t i e s of upper voice movement. The bass possesses the "greatest s t a b i l i z i n g power" according t o Wolpert, i n cadences where i t moves by f o u r t h or f i f t h .  5 7  When i t moves down a p e r f e c t  f i f t h or up a p e r f e c t f o u r t h , i t i s considered a "descending  c a d e n t i a l bass" and i s notated  L 5-^ or  Conversely, when i t moves down a p e r f e c t f o u r t h or up a p e r f e c t f i f t h i t i s considered an 4^ Next, i n examining cadences w i t h s p l i t Wolpert discusses the f o l l o w i n g three simple  or  f  chords,  . 58  possibilities  57 I b i d . . 82 (29). Here the word " s t a b i l i z i n g " has no connection w i t h the e a r l i e r notions of " s t a b i l i t y " and "consonance" but merely denotes a q u a l i t y inherent i n bass movement by f o u r t h or f i f t h . 58  I b i d . . 84 (29)  77. using many examples. " 3  1. a split  Cadences where the penultimate chord contains  fifth. 2.  a split  Cadences where the penultimate chord contains third.  3.  Cadences i n v o l v i n g m u l t i p l e s p l i t s i n the  penultimate chord. An example i l l u s t r a t i n g  each p o s s i b i l i t y  mentioned  i s included: Example  29.  60  1\>\ ,  i — ?  9C 2  5—  1.  Expanding  I j  5  j—k I  h1-4  2.  3.  on the above concept, Wolpert  illustrates  cadences where the penultimate chord c o n t a i n s a s p l i t i n t e r v a l and i s r e l a t e d t o a constant bass p r o g r e s s i o n .  59  See Chapter I I , PP. 28-30. 60  I b i d . . 85-6.  78,  Example 30*  61  a—Q-  J>Q  bo  Q  » E^Q  )—7-A  t/o  )—7—\  Then i n order t o o b t a i n stable f i f t h s from s p l i t t h i r d s , Wolpert presents the f o l l o w i n g s o l u t i o n s :  Inherent i s the  problem t h a t i t i s not always p o s s i b l e f o r upper v o i c e s t o move only i n a stepwise manner. Example 3 1 .  6 2  0  jlz  c-e-> In "the f i r s t example, i f one s p l i t tone leads upwards by a h a l f step, the other must f a l l a minor t h i r d ; but i f one tone moves downward by a h a l f step, the other must 61.  ibid..  62 I b i d .  87.  79. r i s e a minor t h i r d W o l p e r t  then takes the above  s p l i t t h i r d s which lead t o stable f i f t h s and works out a l l the d e s i r a b l e p o s s i b i l i t i e s of bass movement .64 Example 32.  t> if b  * #  He continues and a s s e r t s that hearing a l o g i c a l "progression" among adjacent voices i n a chordal connect i o n i s a f f e c t e d by a "penetrating" or "permeating" q u a l i t y which i s c h a r a c t e r i s t i c of the leading-tone 65  movement.  In discussing t h i s p e n e t r a b i l i t y ,  Wolpert i m p l i e s that h a l f step movement i s such a powerful force that i t does not have t o be always i n the 63  I b i d . . 87 (31).  64 I b i d . . 88 65  Here i t must be noted that "leading note" i s not used i n i t s t r a d i t i o n a l sense but r a t h e r , more g e n e r a l l y , implying h a l f step movement ascending or descending i n one or more v o i c e s .  80 same voice t o be perceived by the ear as the s i g n i f i c a n t and penetrating movement.  The leading-note  (in this  modified sense) seeps through the musical texture,so t o speak, even though i t may change v o i c e s . Orlando d i Lasso, the G a b r i e l i s  Wolpert c i t e s  and Gesualdo as examples  of composers who e a r l y recognized the importance of t h i s 6( penetrating q u a l i t y  and incorporated i t i n t h e i r music.  The f o l l o w i n g short example from Gesualdo i s included as an i l l u s t r a t i o n of t h i s p r i n c i p l e : Example 3 3 .  The  6 7  -  " i n d i c a t e s t h a t the Bb a c t s as a "leading-  tone" t o the B.  Furthermore, since the movement apparently  i n v o l v e s a change i n v o i c e s , unless i t i s assumed the v o i c e s cross or s k i p , the leading note i s c a l l e d "transverse"  66  I b i d . . 89 (31). 67 Ibid.  81. (querstandiq).^8  T r a d i t i o n a l l y , the above occurrance would  simply be c a l l e d a " c r o s s - r e l a t i o n " or " f a l s e - r e l a t i o n . " Apel defines t h i s term as: . . . the appearance i n d i f f e r e n t v o i c e s of two tones t h a t , owing to t h e i r mutually c o n t r a d i c t o r y character, would normally be placed as a melodic progression i n one v o i c e . In other words, c r o s s - r e l a t i o n means the use i n ' d i a g o n a l p o s i t i o n of what properly i s a ' h o r i z o n t a l ' element of musical t e x t u r e . The most important progression of t h i s kind i s the chromatic progression, e.g., Eb-E, which i s so s t r i k i n g l y h o r i z o n t a l that the ear i s d i s t u r b e d i f i t hears the f i r s t tone i n one voice and the second i n another.69 1  From t h i s d e f i n i t i o n , i t i s easy t o see how  Wolpert  a r r i v e s at h i s concept of the penetrating q u a l i t y of the leading-note movement.  Indeed, he i s probably speaking  of  the same t h i n g that Apel i s d e s c r i b i n g above and does l a t e r use the German equivalent of " c r o s s - r e l a t i o n " (Querstand) i n d e s c r i b i n g t h i s process.  He then d i g r e s s e s  momentarily t o t a l k about "the master composers since the beginning of harmonic polyphony i n the Renaissance" as often d i s r e g a r d i n g the t r a d i t i o n a l r u l e against the use of the c r o s s - r e l a t i o n .  He says, moreover, that he  b e l i e v e s i n "the hearing of these Renaissance masters" and considers the c r o s s - r e l a t i o n and i t s "leading-notel i k e q u a l i t y " not only as permitted or even d e s i r a b l e 68 69  , . I b i d . , 89 (32).  W i l l i A p e l , Harvard D i c t i o n a r y of Music. 2nd ed., (Cambridge Mass.: Harvard U n i v e r s i t y P r e s s , 1969), 214.  82 but as a means of chord connection "without any reservation."70 A d i s t i n c t i o n i s then made between a " r e a l " or "pure" c r o s s - r e l a t i o n (echter Querst'and - symbol,  "Q")  and a "sound" c r o s s - r e l a t i o n (Klangguerstand - symbol "q").  The former (see Example 34a below) i n v o l v e s a  s p l i t tone or augmented prime (E-Eb) while the l a t t e r (Example 34b) i n v o l v e s a minor second (Fb-Eb).  A real  c r o s s - r e l a t i o n (Q) must be s p e l l e d as an augmented prime; a c r o s s - r e l a t i o n " i n sound only" (q) occurs when the same i n t e r v a l i s s p e l l e d as a minor  second.  In both cases, f o r the d e f i n i t i o n of c r o s s - r e l a t i o n t o h o l d , the notes involved must be i n d i f f e r e n t v o i c e s . The symbol f o r leading-note, again i n i t s q u a l i f i e d sense, i s "L" : 71 Example 34.  Q - L = Ef\^Eb  70 71  Wolpert, op. c i t . . 90 (32). Ibid  91.  /b)  q - L = Fb,  Eb  83. A f u r t h e r d i s t i n c t i o n i s made between narrow (engen) c r o s s - r e l a t i o n s and wide (we,iten) c r o s s - r e l a t i o n s .  The  former i n v o l v e s a h a l f step while i n the l a t t e r the  two  notes under c o n s i d e r a t i o n l i e at l e a s t a diminished octave away from each o t h e r .  7 2  Example 35. 5 J  j  — 3  5  L Q (e)  L Q  (w)  (narrow)  (wide)  Wolpert then p o i n t s to the Neapolitan cadence ( i . e . , cadence using a neapolitan s i x t h ) as a c l a s s i c a l of leading-tones i n c r o s s - r e l a t i o n . 73 Example 36.  1  = Fflj)F#  72 73  I b i d . . 91. I b i d . . 92.  Q(w)  solution  84. This s o l u t i o n , says Wolpert, i s of the g r e a t e s t s i g n i f i cance f o r a l l new kinds of chord connections i n v o l v i n g r e a l and sound c r o s s - r e l a t i o n s .  7 4  He then c i t e s v a r i o u s passages  i n Act I I I of T r i s t a n as c o n t a i n i n g sound c r o s s - r e l a t i o n s  (q)  while " b e a u t i f u l examples of r e a l c r o s s - r e l a t i o n s (Q) can be found i n B a r t o k . "  7 5  Wolpert proceeds w i t h an a n a l y s i s of c r o s s - r e l a t i o n s 7  from a p o r t i o n of Webern's S t r i n g Quartet, Opus 28.  f\  He  p o i n t s out t h a t only c e r t a i n extremely wide (greater than an octave) c r o s s - r e l a t i o n s prove t o be " h o s t i l e to the hearing" and c i t e s S t r a v i n s k y as an example of a composer who " h o s t i l e " and "strange" c r o s s - r e l a t i o n s .  7 7  prefers  Unfortunately,  no examples from S t r a v i n s k y are given t o j u s t i f y t h i s s t a t e ment.  The chapter on "Chordal Connection and Succession"  concludes w i t h four statements concerning chord connections involving cross-relations: 1.  When c r o s s - r e l a t i o n s are involved i n the chord  connection, they should be c a r e f u l l y handled. 74  Ibid.., 92. Diese Losung i s t f u r a l l e nevartigen Akkordverbindungen von der a l l e r c j r o s s t e n Bedeutung, und zwar f u r Querstande ebenso wie f u r Klangquersta*nde. 75 , . I b i d . . 92-3 (33). 76 7 7  , I b i d . . 93 (33). x  I b i d . . 95 (34).  85. 2.  The ear hears a r e a l c r o s s - r e l a t i o n as a  leading tone r e s o l u t i o n even though i t may not be s p e l l e d as such. 3.  Because of the penetrating q u a l i t y of h a l f  step movement, a l l c r o s s - r e l a t i o n s can be used both i n narrow and wide p o s i t i o n s without obscuring the tonality. 4.  Extremely wide c r o s s - r e l a t i o n s (two octaves  and more) are t o be avoided.  C.  Comparison. While the d i s t i n c t i o n s between Hindemith*s and  Wolpert*s systems of chord c l a s s i f i c a t i o n were q u i t e c l e a r , the d i f f e r e n c e s between t h e i r ideas on chord connection are not as immediately e v i d e n t . Hindemith's ideas on the connection of adjacent chords r e s t on the three b a s i c p o i n t s p r e v i o u s l y d i s c u s s e d , namely, harmonic f l u c t u a t i o n , degree progression and the two-voice framework.  He t r e a t s i n d i v i d u a l  chords as e n t i t i e s and extends h i s t h e o r i e s of c l a s s i f i c a t i o n t o h i s t h e o r i e s concerning chord movement. On the other hand, Wolpert's ideas on chord connection seem t o be founded on two basic i d e a l s , namely,  78  I b i d . . 95-6 (34)  86 the smallest p o s s i b l e movement i n the upper voices  and  more or l e s s t r a d i t i o n a l bass movements (as e s t a b l i s h e d i n " c l a s s i c a l " and  "modern" music) 7^ Unlike Hindemith,  Wolpert does not t r e a t i n d i v i d u a l chords as i s o l a t e d e n t i t i e s as he does i n h i s system of c l a s s i f i c a t i o n . Instead, he breaks them down i n t o t h e i r component p a r t s , i . e . , he does not extend h i s system of chord c l a s s i f i c a t i o n t o apply a l s o t o chordal connection.  This i s a r a t h e r  d i s t u r b i n g omission since i t tends t o d e t r a c t from the c o n t i n u i t y of the e n t i r e system. With these basic d i f f e r e n c e s i n mind, one  can  examine both systems i n greater d e t a i l according to the following six points: 1.  The  "Two-Voice Framework."  2.  I n t e r v a l R e s o l u t i o n and Half Step Movement.  3.  Root Movement or Degree Progression  (Versus  Bass Movement). 4.  Enharmonics.  5.  Chords as E n t i t i e s .  6.  V a r i a t i o n s i n Chordal Tension and T o n a l i t y .  Hindemith considers the two voice framework t o e x i s t between the bass and the most prominent of the upper voices.  He does not examine other p o s s i b l e two-voice  frameworks and goes so f a r as to state that a chord progression i s a f f e c t e d very l i t t l e by inner voice  See  p. 76 of present t e x t .  87. movement.  30  He does, however, mention inner voice  movement w i t h respect t o the "guide tone," but only t o the extent of saying that i t must move by a good melodic i n t e r v a l i f the t r i t o n e r e s o l u t i o n i s t o be Q  1  satisfactory.  For Hindemith, then, there i s b a s i c a l l y  only one two-voice framework. Wolpert, on the other hand, examines a l l p o s s i b l e two-voice frameworks i n the connection of adjacent chords. In a p r o g r e s s i o n i n v o l v i n g three v o i c e s , there are three s i g n i f i c a n t two-voice frameworks, four v o i c e s provide s i x two-voice frameworks, and so o n .  8 2  i  n  a d d i t i o n , Wolpert  i s o l a t e s what he found t o be the most common two-voice frameworks t o occur between the bass and an upper voice i n an attempt t o suggest d e s i r a b l e cadence procedures w i t h respect to bass movement. Regarding the second p o i n t , i . e . , i n t e r v a l r e s o l u t i o n and h a l f - s t e p movement, Hindemith's concern i s mainly w i t h the t r i t o n e .  I t s proper r e s o l u t i o n i s most important  i f one i s t o proceed s u c c e s s f u l l y i n e s t a b l i s h i n g a t o n a l centre.  G e n e r a l l y , he c o n s i d e r s the movement of voices by 80  81  Hindemith, op. c i t . , 115. I b i d . . 127.  82 The number of s i g n i f i c a n t "two-voice frameworks" i n Wolpert's theory f o l l o w s the same number sequence as i n Hindemith's c o n s i d e r a t i o n of a l l possible i n t e r v a l r e l a t i o n ships i n determining the root of a chord. See Chapter I I , p. 15.  88.  h a l f - s t e p to be a d e s i r a b l e q u a l i t y .  He states that  "£a  p r o g r e s s i o n 3 i n which a l l the tones move i n minor seconds  . . . produces the smoothest and most flowing  progressions;  i t acts l i k e a magic formula to make every imaginable chord progression  usable."  83  Wolpert a l s o sees chromatic voice leading as a d e s i r a b l e occurrance, but h i s enthusiasm takes him beyond Hindemith, t o the point where almost the e n t i r e chapter on chord connection i s concerned with h a l f - s t e p movement. For example, i n h i s "Laws of I n a c t i v i t y , " Wolpert i m p l i e s that the smaller the movement i n the upper v o i c e s , the b e t t e r the voice l e a d i n g .  In h i s d i s c u s s i o n of s y s t o l e s  and d i a s t o l e s , he develops the concept of adhesion as a desirable q u a l i t y .  His obsession with h a l f - s t e p movement  i s f u r t h e r evident i n h i s extended d i s c u s s i o n on the s i g n i f i c a n c e of d i f f e r e n t kinds of c r o s s - r e l a t i o n s and i n h i s assigning a p e n e t r a t i n g - l i k e q u a l i t y to h a l f - s t e p motion, even when i t occurs between d i f f e r e n t v o i c e s .  As  f a r as i n t e r v a l r e s o l u t i o n i s concerned, Wolpert does not seem to worry.  In f a c t , s y s t o l e s and d i a s t o l e s can move  to stable or unstable  i n t e r v a l s ; that i s , they can  resolve  i n the t r a d i t i o n a l sense, or not, hence the concept of d i v e r s i o n or adhesion leading to an unstable  83  Ibid..  124  interval.  89.  Both Hindemith and Wolpert, then, recognize the importance of h a l f - s t e p movement i n producing good chord c o n n e c t i o n . However, while Hindemith t r e a t s the r e s o l u t i o n of the t r i t o n e as a s i g n i f i c a n t aspect of chord movement, Wolpert accords the t r i t o n e no s p e c i a l place of importance.  Rather, i t s r e s o l u t i o n i s j u s t a  s p e c i a l case i n the l a r g e r d i s c u s s i o n of s y s t o l e s and diastoles. Hindemith i s concerned w i t h root movement to the extent that i t i s perhaps the most important c r i t e r i o n f o r determining the "value" of a given chord progression: both among the adjacent chords and i n the l a r g e r framework of the t o n a l  setting.  On the other hand, Wolpert does not consider the movement of chord r o o t s i n h i s d i s c u s s i o n of chord connection.  This i s perhaps a reasonable omission since  he does i n f a c t consider a l l p o s s i b l e two v o i c e frameworks. Wolpert does consider bass movement (as d i s t i n c t from root movement) and t a b u l a t e s what he c o n s i d e r s to be the "most f r e q u e n t l y occurring" or " d e s i r a b l e " upper voice movements w i t h the few (he considers 5th, 4th and 3rd) bass movements he d i s c u s s e s . However, root movement as such i s not touched  on.  Concerning s p e l l i n g of notes, Hindemith, as before, adheres to the twelve-note d i v i s i o n of the octave i n which  90 enharmonics such as G# and Ab are e q u i v a l e n t , i . e . , the way  a note i s s p e l l e d has nothing to do w i t h the way i t  i n f l u e n c e s chord movement. Conversely, as with chord c l a s s i f i c a t i o n , Wolpert i s concerned w i t h proper s p e l l i n g i n d i s c u s s i n g chord connection.  In the s e c t i o n on adhesion, chord movement  i s a f f e c t e d by the way a note i s s p e l l e d , i . e . , some s p e l l i n g s a l l o w two r e s o l u t i o n s while others allow only one.  Wolpert even invents h i s own term (Umpolung) t o  e f f e c t proper voice leading ( i . e . , w i t h c o r r e c t s p e l l i n g ) i n adhesion.  A l s o , he makes d i s t i n c t i o n s i n h i s  d i s c u s s i o n of c r o s s - r e l a t i o n s w i t h respect to s p e l l i n g which seem u n e s s e n t i a l to the understanding  of the  concept as a whole (e.g., "sound" v s . " r e a l " cross r e l a tions) • Hindemith breaks chords down, so t o speak, i n presenting h i s t h e o r i e s on the two-voice framework, but he a l s o c a r r i e s h i s system of c l a s s i f i c a t i o n over  into  h i s d i s c u s s i o n of chord connection, e s p e c i a l l y i n h i s treatment  of the unique theory of harmonic f l u c t u a t i o n .  Again conversely, and perhaps the most d i s t u r b i n g point i n Wolpert's d i s c u s s i o n of chord connection, i s h i s f a i l u r e t o r e l a t e h i s t h e o r i e s t o h i s previous d i s c u s sions of the c l a s s i f i c a t i o n of chords.  In d i s c u s s i n g chordal  connection, he breaks chords down i n t o t h e i r component parts ( f o r example, a l l p o s s i b l e two voice frameworks—  91. s y s t o l e s and d i a s t o l e s — b a s s movement, e t c . ) , but he f a i l s to d i s c u s s the chord as an i n d i v i d u a l e n t i t y . Our f i n a l point concerns v a r i a t i o n s i n chordal t e n s i o n and t o n a l i t y .  Beyond h i s e a r l i e r d i s c u s s i o n on  the consonance and dissonance of i n t e r v a l s and chords, Wolpert gives no f u r t h e r e l a b o r a t i o n s on these concepts as they might a f f e c t chord connection.  In f a c t , Hinde-  mith's theory of harmonic f l u c t u a t i o n has no  counterpart  i n Wolpert and remains as a unique c o n t r i b u t i o n t o contemporary music theory.  The only p o s s i b l e point of  reference i s . that what Hindemith would c a l l zero harmonic f l u c t u a t i o n , would at the same time be Wolpert's ultimate i d e a l w i t h respect to h i s "Laws of I n a c t i v i t y . "  However,  t h i s i s perhaps s t r e t c h i n g a point simply t o make a comparison. In c o n c l u s i o n , as Wolpert i m p l i e s at the  beginning  OA  of h i s t r e a t i s e , ^ he does not mean t o suggest a kind of "harmonic s t y l e " (as does Hindemith). his  He e v i d e n t l y sees  system as s u i t a b l e f o r both t o n a l and atonal composi-  t i o n s , and thus, a d e t a i l e d c o n s i d e r a t i o n of chord o r g a n i z a t i o n , i . e . , t o n a l l o g i c , would be out of p l a c e . Hindemith, however, does r e l a t e very c l o s e l y h i s ideas about chord connection t o h i s t h e o r i e s of c l a s s i f i c a t i o n , since he has very d e f i n i t e ideas about the n e c e s s i t y and even i n e v i t a b i l i t y of t o n a l i t y . 84  Wolpert, op. c i t . . 13-15.  CHAPTER IV FURTHER CONSIDERATIONS  We have reached the point i n t h i s study where comparison of Hindemith and Wolpert along s i m i l a r l i n e s i s no longer p o s s i b l e .  Because of h i s strong b e l i e f s  about the n e c e s s i t y of t o n a l o r g a n i z a t i o n , Hindemith s 1  d i s c u s s i o n s are unwavering.  He a s s e r t s that t o n a l i t y " i s  a n a t u r a l f o r c e , l i k e gravity"-* and f l a t l y states that "there can be no such t h i n g as a t o n a l i t y . "  2  He dismisses  " b i t o n a l i t y " and " p o l y t o n a l i t y " as "catchwords" and i n s i s t s that "every simultaneous combination of sounds must have one r o o t , and only one . . . the ear judges the t o t a l sound and does not ask w i t h what i n t e n t i o n s i t was produced."  3  Hindemith's b e l i e f i n the i n e v i t a b l e c o n d i t i o n of some sort of t o n a l o r g a n i z a t i o n i n a l l music n e c e s s i t a t e s the formulation of a p r i n c i p l e or p r i n c i p l e s d e s c r i b i n g large scale t o n a l r e l a t i o n s h i p s . The term " t o n a l sphere"  1  Paul Hindemith, The C r a f t . I , 152.  2 I b i d . . 155. 3 I b i d . , 156.  93. i s used t o describe "the grouping of chord tones (or root tones) around a t o n a l c e n t r e . "  4  However, t h i s  t o n a l centre may and often does vary i n the course of a musical composition.  To describe t h i s  occurrence,  Hindemith r e d e f i n e s the concept of modulation  saying  t h a t "when we allow one tone t o usurp the place of another as t o n a l centre of a degree p r o g r e s s i o n , we are modulating."  5  Modulation, he continues, can be  determined simply from the c o n s t r u c t i o n of the degree progression.  But, before modulation can take p l a c e , a  " f i r m l y e s t a b l i s h e d t o n a l centre" must be evident, otherwise, the modulation w i l l not be e f f e c t i v e .  In  other words, "the c l e a r e r the way leading from one t o n a l centre t o the next, the more s a t i s f a c t o r y the modulation. The f a c t t h a t i t i s not always c l e a r at what p o i n t a given modulation takes place does not seem to bother Hindemith. . . . one l i s t e n e r hears the change as occurring at one p l a c e , another at another. But t h i s i s not a shortcoming; on the c o n t r a r y , one of the greatest  4 I b i d . . 149. 5 6  Ibid., 149, Ibid..  151.  See a l s o Chapter I I I , p. 62  94. charms of modulation l i e s i n the e x p l o i t a t i o n of t h i s very u n c e r t a i n t y i n the t r a n s i t i o n a l passages. 7  Perhaps the most f a r - r e a c h i n g aspect of Hindemith s 1  t h e o r i e s about t o n a l o r g a n i z a t i o n i s the f a c t t h a t a s i n g l e most important t o n a l centre i n the l a r g e r framework of a musical composition can be determined. The t o n a l centres of a l l the t o n a l i t i e s of a composition produce, when they are connected without the i n c l u s i o n of any of the i n t e r v e n i n g tones, a second degree-progression which should be constructed along the same l i n e s as the f i r s t one, b u i l t of the r o o t s of a l l the chords. Here we see the f u l l u n f o l d i n g of the o r g a n i z i n g power of S e r i e s 1. The e n t i r e harmonic c o n s t r u c t i o n of a piece msy be perceived i n t h i s way: against one t o n a l centre chosen from among many r o o t s others are juxtaposed which e i t h e r support i t or compete w i t h i t . Here, too, the t o n a l centre that reappears most o f t e n , or that i s p a r t i c u l a r l y s t r o n g l y supported by i t s f o u r t h and i t s f i f t h , i s the most important. As a t o n a l centre of a higher order, i t dominates a whole movement or a whole work. 8  Because of h i s acceptance of both t o n a l i t y and a t o n a l i t y , Wolpert has no need t o expand h i s system beyond the d i s c u s s i o n of the p r i n c i p l e s of good chord connection.  Therefore, the remainder of h i s book i s  devoted t o other t o p i c s of i n t e r e s t , some of which are presented below.  7 I b i d . . 151. 8 I b i d . . 151.  95. Wolpert's d i s c u s s i o n of scales and r u l e s f o r scale formation i s i n t e r e s t i n g because of h i s unique approach.  A l l presently recognized modes and scales  are discussed along with p o s s i b l e "new scales" using seven, e i g h t , nine, t e n and eleven d i f f e r e n t tones. Wolpert concludes that scale s t r u c t u r e s other than the ones discussed are not p o s s i b l e i n s t a f f n o t a t i o n , i . e . , w i t h a twelve-note d i v i s i o n of the octave.9 Two concepts are defined by Wolpert which a l s o are i n t e r e s t i n g because of t h e i r uniqueness.  The f i r s t  i s the idea of "Personanz": The e s s e n t i a l notes of a chord continue t o be e f f e c t i v e i n the ear, even though these notes are no longer sounding.-^O Wolpert g i v e s examples from Mozart, Franck and Wagner t o i l l u s t r a t e t h i s concept of " c o n t i n u a b i l i t y . "  Although  t h i s does not e s s e n t i a l l y say anything which would not be assumed i n t r a d i t i o n a l a n a l y s i s , i t i s s t i l l important i n an attempt t o understand Wolpert's thought processes, keeping i n mind the f a c t that he was a s e l f - t a u g h t scholar  F.A. Wolpert. Neue Harmonik. (Wilhelmshaven: H e i n r i c h s h a f t e n , 1972), 162-7. l 0  I b i d . , 184.  11  E r i c h V a l e n t i n , "F.A. Wolpert" Die Musik i n Geschichte und Gegenwart. XIV (1968), 838.  96. The second concept, i n no way r e l a t e d t o the f i r s t , concerns the idea of "Koharenz": In the connection of two unstable chords - when there i s a leading note but no adhesion, then coherence i s said t o e x i s t . 1 2 The simplest example of a coherent connection might be the V-I cadence where the seventh i s not present i n the penultimate chord.  Here a leading note i s evident but  there i s no adhesive process occurring since there are no whole s y s t o l e s or d i a s t o l e s i n the connection. Twenty experiments which f o l l o w i n v o l v i n g the 13 connection of chords i n root p o s i t i o n ,  conclude the  expanded (1972) v e r s i o n of the t e x t and i l l u s t r a t e the r u l e s of good chord connection discussed i n Chapter I I I . The f i r s t nine experiments i l l u s t r a t e p o s s i b i l i t i e s with a l l the three part chords while experiments t e n t o s i x t e e n do the same f o r a l l four part chords.  The f i n a l  four experiments i l l u s t r a t e connection of three and four part s p l i t chords. In attempting t o come t o terms w i t h the p r i n c i p l e s brought out i n t h i s study, the d i f f i c u l t i e s and l i m i t a 12  Wolpert, op. c i t . . 198. The term "coherence" i s discussed here and not i n Chapter I I I because Wolpert does not include i t i n h i s d i s c u s s i o n of chord connection. 13  I b i d . . 201-23.  97  t i o n s of d i s c u s s i n g two contemporary German musical t h e o r i s t s i n a language other than that i n which t h e i r o r i g i n a l works were w r i t t e n must f i r s t be  acknowledged.  Many of the ideas expressed i n the o r i g i n a l German have not been cogently stated or c l e a r l y expressed because of the inherent d i f f i c u l t y involved i n t r a n s l a t i o n . Hindemith and Wolpert have developed systems of chord c l a s s i f i c a t i o n and both have succeeded i n c l a s s i f y ing a l l p o s s i b l e chordal m a t e r i a l w i t h i n the twelve-note d i v i s i o n of the octave.  In Hindemith s system of 1  c l a s s i f i c a t i o n there i s an attempt to be as o b j e c t i v e as possible.  H i s notions of consonance and dissonance, f o r  example, are derived from the overtone s e r i e s and combination tone curves and are put forward i n h i s S e r i e s 1 and 2.  Conversely, Wolpert's concern i s subjec-  t i v e , i n t h a t he uses h i s ear t o determine a scale of dissonance f o r i n t e r v a l s and t r i a d s .  Whereas Hindemith's  c l a s s i f i c a t i o n i s concerned w i t h the p a r t i c u l a r  combina-  t i o n s of i n t e r v a l s i n a given chordal s t r u c t u r e , Wolpert*s b a s i s f o r c l a s s i f i c a t i o n i s the number of d i s t i n c t ^ notes i n the chord as w e l l as the i n t e r v a l combinations produced when they are reduced t o t h e i r narrowest p o s s i b l e p o s i t i o n .  14  " D i s t i n c t " here r e f e r s to l e t t e r names; f o r example, G and G# are not d i s t i n c t , whereas G and Ab a r e .  98. While Wolpert's system of c l a s s i f i c a t i o n g r e a t l y emphasizes c o r r e c t s p e l l i n g to the point where the s p e l l i n g of a chord member determines i n t o which category the chord i s placed, Hindemith's system of c l a s s i f i c a t i o n does not d i s t i n g u i s h between enharmonics but regards them as being equivalent f o r purposes of classification.  Whereas Hindemith's l o g i c a l l y evolved  theory of chord r o o t s i s involved i n h i s system of chord c l a s s i f i c a t i o n t o the extent of determining subgroup d i v i s i o n s , chord r o o t s play no part whatever i n Wolpert's system of c l a s s i f i c a t i o n .  While Hindemith  uses the t r i t o n e as a d i s t i n g u i s h i n g f e a t u r e i n h i s c a t e g o r i z a t i o n of chords, Wolpert does not, but instead evolves h i s h i g h l y i n d i v i d u a l theory of s p l i t chords based on the i n t e r v a l s of the augmented and doubly augmented primes. Both men have a l s o developed t h e i r own p r i n c i p l e s of chord movement.  While Hindemith's ideas on the  connection of chords r e s t on three d i s t i n c t  criteria—  harmonic f l u c t u a t i o n , degree progression and the twovoice framework--Wolpert's t h e o r i e s on chord connection r e s t on two main i d e a l s - - s m a l l e s t p o s s i b l e upper voice movement and more or l e s s t r a d i t i o n a l bass movement. Since Wolpert does not extend h i s system of c l a s s i f i c a t i o n t o cover chordal connection, there i s a l a c k of o v e r a l l cohesiveness i n h i s t r e a t i s e .  This i s  99.  i n d i r e c t c o n t r a s t t o Hindemith, whose p r i n c i p l e s of chord movement are a d i r e c t extension of h i s system of classification.  While Hindemith only considers one  "two-voice framework," Wolpert takes a l l p o s s i b l e "twovoice frameworks" i n t o account.  Whereas Hindemith i s  concerned w i t h t r i t o n e r e s o l u t i o n i n chord movement, to Wolpert the t r i t o n e i s only a s p e c i a l case i n adhesive or whole s y s t o l e / d i a s t o l e " r e s o l u t i o n s . "  As  w i t h h i s system of c l a s s i f i c a t i o n , again i n h i s system of chord movement, Wolpert seems overly concerned w i t h the way a p a r t i c u l a r note i s s p e l l e d , while w i t h Hindemith enharmonics are again e q u i v a l e n t .  I n d e s c r i b i n g the  adhesive process Wolpert i l l u s t r a t e s how one s p e l l i n g w i l l allow two r e s o l u t i o n s while another w i l l allow only one.  T h i s concern f o r s p e l l i n g i s again evident i n the  d i s t i n c t i o n s made among the v a r i o u s types of crossrelations.  While Hindemith admonishes against cross-  r e l a t i o n s i n chordal connection, Wolpert strenuously encourages t h e i r use. While Hindemith s t h e o r i e s of 1  chord connection are based on the movement of r o o t s or degree progressions and t h e i r d e v i a t i o n and r e t u r n t o a t o n a l c e n t r e , Wolpert does not consider root movement, since t o do so might be t o c o n t r a d i c t h i s o r i g i n a l p o s i t i o n on the acceptance of both t o n a l i t y and a t o n a l i t y as v a l i d frames of reference.  100.  On the whole, then, both men succeed i n c l a s s i f y i n g a l l p o s s i b l e chords w i t h i n the twelve-note d i v i s i o n of the octave while working from somewhat opposite p o i n t s of view.  On the one hand, Hindemith  works from the o r i g i n a l premise of the i n e v i t a b i l i t y of t o n a l i t y .  He attempts t o b u i l d a l o g i c a l system  s t a r t i n g from nature and progressing through chordal c o n s t r u c t i o n t o an all-embracing n a t u r a l system. Wolpert, on the other hand, combines i n t e r e s t i n l o g i c a l patterns w i t h a r e d e f i n i t i o n of more or l e s s t r a d i t i o n a l advice and r u l e s and apparently shows no i n t e r e s t i n developing an o v e r a l l coherent and cohesive system.  H i s c l a s s i f i c a t i o n of chordal  s t r u c t u r e s i s obviously l o g i c a l r a t h e r than p r a c t i c a l whereas h i s d i s c u s s i o n of chordal movement i s simply a restatement  of t r a d i t i o n a l procedures and i s not  connected t o h i s system of c l a s s i f i c a t i o n .  Accordingly,  h i s chapter on c l a s s i f i c a t i o n seems t o be d i r e c t e d towards the philosopher or l o g i c i a n whereas h i s s e c t i o n on chordal progression i s more l i k e a c o l l e c t i o n of h e l p f u l advice t o the a s p i r i n g  composer.  Thus, the two men, using d i f f e r e n t  approaches t o  the same problem, both r e a l i z e a complete system of chordal o r g a n i z a t i o n and progression w i t h i n the twelvenote d i v i s i o n of the octave. theoretical  Their conservatism i n  matters shows a l i n k with past t r a d i t i o n s and  101.  i d e a l s , y e t , t h e i r c o n t r i b u t i o n i s both necessary and appropriate f o r our time, and thus i s of value i n our o v e r a l l understanding of twentieth-century music theory.  BIBLIOGRAPHY A p e l , W i l l i . Harvard D i c t i o n a r y of Music. 2nd ed. Cambridge Massachusetts: Harvard U n i v e r s i t y P r e s s , 1969. A u s t i n , W i l l i a m W. Music i n the Twentieth Century: From Debussy through S t r a v i n s k y . 1st ed. New York: W. W. Norton, 1966. B a b b i t t , M i l t o n . "Some Aspects of Twelve Tone Composition" The Score. X I I (1955), 53-61. B r a u n f e l s , Michael. "Franz A l f o n s Wolpert" Neue Z e i t s c h r i f t fur Musik. CXI (1950), 238-40. Cazdun, Norman. "Hindemith and Nature" Music Review. XV (1954), 289-306. Chrisman, Richard. A Theory of A x i s T o n a l i t y f o r Twentieth Century Music. Unpublished Ph.D. d i s s e r t a t i o n , Yale U n i v e r s i t y , 1969. . " I d e n t i f i c a t i o n and C o r r e l a t i o n of P i t c h Sets" J o u r n a l of Music Theory. XV (1970), 58-83. Eschman, K a r l . Changing Forms i n Modern Music. 2nd ed. Boston, Massachusetts: E. C. Schirmer Music Company, 1968. F a r r e l l , Dennis M. "Some suggested C o r r e c t i o n s i n the Hindemith Chord Tables." Canadian A s s o c i a t i o n of U n i v e r s i t y Schools of Music J o u r n a l . I (Spring 1971), 71-89. F o r t e , A. "A Theory of Set Complexes f o r Music." of Music Theory. V I I I (1964), 136-83.  Journal  Haas, F r i t h j o f . "F. A. W o l p e r t s 'Neue Harmonik'" Neue Z e i t s c h r i f t f u r Musik. C X I I I (March 1952), 167-8. 1  Hensel, Herman Richard. "On Paul Hindemith's Harmonic F l u c t u a t i o n Theory." Unpublished Ph.D. d i s s e r t a t i o n , U n i v e r s i t y of I l l i n o i s , 1964.  103. Hindemith, P a u l . The C r a f t of Musical Composition. V o l . I . 4 t h ed. Translated by A r t h u r Mendel. New York: Schott Music C o r p o r a t i o n , 1970. . The C r a f t of M u s i c a l Composition. V o l . I I . Translated by Otto Ortmann. New York: Associated Music P u b l i s h e r s , 1941 . Unterweisung im Tonsatz. I I I . 1970.  Mainz:  B. S c h o t t ,  . A Concentrated Course i n T r a d i t i o n a l Harmony. Revised ed. New York: A s s o c i a t e d Music P u b l i s h e r s , 1944. . "Methods of Music Theory" (January, 1944), 20-28. • A Composer's World: Garden C i t y , New York:  M u s i c a l Q u a r t e r l y . XX,  Horizons and L i m i t a t i o n s . Doubleday and Company, 1961.  K a s s l e r , M i c h a e l . "Towards a Theory that i s the Twelve Note C l a s s System," P e r s p e c t i v e s of New Music. V. (1967), 1-80. Landau, V i c t o r . "Hindemith, A Case Study i n Theory and P r a c t i c e , " Music Review. XXI, ( i 9 6 0 ) , 38-54. . "Hindemith the System B u i l d e r : A C r i t i q u e of H i s Theory of Harmony." Music Review, X X I I . (1961), 136-51. P e r l e , George. "The P o s s i b l e Chords i n Twelve-Tone Music," Music Review. XV. (1954), 257-67. • "The Harmonic Problem i n Twelve-Tone Music," Music Review. XV. (1954), 257-67. . S e r i a l CQmposition and A t o n a l i t y . 2nd ed. Los Angeles: U n i v e r s i t y of C a l i f o r n i a P r e s s , 1968. P e r s c h e t t i , Vincent. Twentieth Century Harmony. W. W. Norton, 1961. R e d l i c h , Hans Ferdinand. "Paul Hindemith: Music Review. XXV, (1964), 241-53.  New  York:  A Re-assessment"  Rochberg, George. The Hexachord and I t s R e l a t i o n to the Twelve-Tone Row. Bryn Mawr, Pennsylvania: Theodore Presser Company, 1955.  104. Shackford, C h a r l e s R. "Unterweisung im Tonsatz I I I by Paul Hindemith" Music L i b r a r y A s s o c i a t i o n Notes. XXIX, (March 1973), 451-2. S h i r l a w , Matthew. The Theory of Harmony. Da Capo Press, 1969.  New York:  The  Slonimsky, N i c o l a s . Baker's B i o g r a p h i c a l D i c t i o n a r y of Musicians. 5th ed. New York: G. Schirmer, 1958. Thomson, W i l l i a m . "Hindemith's C o n t r i b u t i o n t o Music Theory," Journal of Music Theory. IX. (1965), 52-71. U l e h l a , Ludmila. Contemporary Harmony: Romanticism through the Twelve-Tone Row. Toronto: C o l l i e r - M a c m i l l a n L i m i t e d , 1966. V a l e n t i n , E r i c h . "Wolpert, F.A." Die Musik i n Geschichte und Gegenwart. XIV, (1968), 838. Wolpert, Franz A l f o n s . Wilhelmshaven:  Neue Harmonik. Revised ed. H e i n r i c h s h a f t e n , 1972.  . "New Harmony." Unpublished T r a n s l a t i o n of p o r t i o n s of F. A. Wolpert's t e x t by Louis Medveczky, Univers i t y of B r i t i s h Columbia, 1973.  105.  APPENDIX I SUMMARY OF HINDEMITH'S TABLE OF CHORD GROUPS  Group A - Chords Without Tritone I.  A l s o have no 2nds or 7ths  Group B - Chords With T r i t o n e II.  A l s o have  1. Root and Bass I d e n t i c a l  a. m7 only  2. Root above bass  b. M2 and/or m7 1. Root and bass identical 2. Root above bass 3. More than one tritone  III.  A l s o have 2nds and/or 7ths IV. 1. Root and Bass I d e n t i c a l 2. Root above bass  V.  Indeterminate, i . e . no root  Also have m2 and/or M7 1. Root and Bass identical 2. Root above bass  V I . Indeterminate, t r i t o n e predominating  APPENDIX I I  How  m a n y c h o r d s a r e n e e d e d to p r o d u c e  H o w is the T o n i c A  a tonal  found?  Chords without  '  Tritones  B  Chords containing  II 3  Chords  TONIC: Principal tone of the group formed by the chord-roots  center?  c V  -i.  Tritones  V-  a Chords  A  2 Chords  TONIC: Root of the chord of resolution  The last of a group of chord-roots is the Dominant of a TONIC lying a fifth lower  r \  III  3 Chords  TONIC: Same as in X  a Chords  TONIC: Root of the chord of resolution  V After determination of the root, to be treated the same as I  TONIC: Indeterminate  g Chords  TONIC: Same as in 1  V I After determination of the root, tobe treated the same as II  TONIC: Indeterminate  W7. APPENDIX  III  Wolpert - Sonata No. 1 (Meas.  W o l p e r t ' s System  1-3)  H i n d e m i t h ' s System  •  (f) J .  H.  r. C  ««)  7.  1  r to.  (t> (#») (f)  //.  (/;  nr.a m. i IT.fe.i IT. fe. c2-  nr. i.  jr. L a.  IF. z  nr. x  APPENDIX  III  Hindemith - Sonata No. 1 (Meas.  1-4)  W i t h quiet m o t i o n , i n quarters (J 96)  /.  Z.  3.  5*. C 7. 8 -  ^  H i n d e m i t h ' s System  /.  H-  7JT . 2 X , JL 3L. b. Z  7.  o r . <*.  2. J.  r. <.  /p. //.  (f) —  tea)  —  zzr .  i  —  //.  zzr .  x 2ZT . JL  IF. 2 /*.  /j.  W o l p e r t ' s System  *  /o.  n.  —  /f\  

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