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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., Univer s i ty 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 thes i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November, 1973 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver 8, Canada ABSTRACT Two twentieth-century music t h e o r i s t s , Paul Hindemith and Franz Alfons Wolpert, are unique i n having independently developed t h e i r own systems of chord c l a s s i f i c a t i o n and p r i n c i p l e s of chord movement or success ion. While Hindemith 1 s ideas on these subjects are f a m i l i a r among the majority of music t h e o r i s t s , Wolpert 1 s theor ies remain r e l a t i v e l y unknown. It was decided that since both men had attempted s i m i l a r tasks at s imi la r po ints i n t ime, that a comparison of both t h e o r i s t s ' ideas should be attempted. Therefore, both Hindemith's and Wolpert ' s systems of chord c l a s s i f i c a t i o n and chord connection were discussed i n d e t a i l to attempt to determine the ra t iona le behind each t h e o r i s t ' s approach. F i r s t l y , both systems of chord c l a s s i f i c a t i o n were examined and compared. I t was found that although there were many di f ferences between Hindemith 1 s and Wolpert ' s systems of chord grouping, both succeeded, each i n h i s own way, i n arranging a l l poss ib le combinations of pi tches wi th in the twelve-note d i v i s i o n of the octave. S i m i l a r l y , each t h e o r i s t ' s ideas concerning chord movement were inves t iga ted . While i t was reaff irmed that Hindemith's system rested on concepts which he invented and i developed such as "degree progress ion" and "harmonic f l u c t u a t i o n , " i t was discovered that Wolpert*s system of chord movement was more t r a d i t i o n a l l y or i ented , and yet adhered to some rather d i s turb ing notions about the d e s i r a b i l i t y of c e r t a i n kinds of v o i c e - l e a d i n g , e . g . , "adhes ion, " " d i v e r s i o n , " e t c . Furthermore, while Hindemith had strong ideas about the necess i ty of tonal organiza t ion , i t was found that Wolpert saw h i s system as v a l i d for both tona l and atonal frames of re ference . F i n a l l y , i t was recognized that Hindemith's theor ies contained a unity and cohesiveness through the extension of h i s system of chord connection to include h i s system of chord c l a s s i f i c a t i o n so that the way chords were c l a s s i f i e d inf luenced how they were t reated i n chord progress ions . With Wolpert, however, there i s no attempt to unify both systems and therefore h i s ideas about chord movement are completely divorced from h i s system of chordal groupings. i i TABLE OF CONTENTS Page CHAPTER I. INTRODUCTION 1 CHAPTER I I . CHORD CLASSIFICATION 11 A . Hindemith 11 B. Wolpert 20 C . Comparison 37 CHAPTER I I I . CHORD MOVEMENT 51 A . Hindemith 51 B. Wolpert • 65 C . Comparison 85 CHAPTER IV. FURTHER CONSIDERATIONS 92 BIBLIOGRAPHY j.02 APPENDIX I 105 APPENDIX II 106 APPENDIX III 107 i i i LIST OF MUSICAL ILLUSTRATIONS Example Number Page 1. Hindemith 1 s Series 2 13 2. Wolpert fs Three-Note Chord Types 23 3. Wolpert* s Concept of Contracted Form 24 4. Wolpert's Comparison with Hindemith's System . 25 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 Triads with the F i f t h S p l i t Various Ways 28 20. » II it H II II " II 29 22 II II ti it it it II it 2 9 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 . . . . . . 33 (5) 15. Problem of the j 4 j Chord 35 16. Possible Roots i n a Seven-Tone Chord . . . . . 36 17. Wolpert's Chordal Analysis 36 18. Chord S p e l l i n g Influencing C l a s s i f i c a t i o n i n Wolpert's System 41 19. Gradual versus N i l Harmonic Fluctuation. . . . 53 20. Two Chords from Sub-Group II whose Roots are a Tritone Apart. . 58 i v 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 Wolper t ' s Examination of Chord Connection 66 22. Wolpert ' s "Non-connections" 68 23. Types of Systoles and Dias to les 69 24. Chord S p e l l i n g s Determining Number of Poss ible Resolutions 71 25. D iver s ion 72 26. Examination of A l l Poss ib le Two-Voice Frameworks. 73 27. Mixture of Adhesive and Non-Adhesive Voices i n the Same Connection 74 28. Bass Progressions with Most L i k e l y Upper Voice Movement 75 29. Cadences with S p l i t Chords 77 30. Cadences with S p l i t Chords 78 31. Cadences with S p l i t Chords 78 32. Cadences with S p l i t Chords 79 33. The Cros s -Re la t ion 80 34. "Rea l " versus "Sound" Cross-Rela t ions . . . 82 35. "Narrow" versus "Wide" Cros s -Re la t ions . . . 83 36. Cros s -Re la t ions with the Neapolitan S ix th . 83 v ACKNOWLEDGEMENT I wish to express my thanks to my research advi sor , Dr . Eugene Wilson, for h i s constant support and guidance during the w r i t i n g of t h i s t h e s i s . I am also indebted to the other members of my committee, Professor Kathryn Ba i ley and Professor Cort land Hultberg , for t h e i r c r i t i c a l comments and valuable a s s i s tance . I would a l so l i k e to thank Mr. Hans Burndorfer , Head of the Music L i b r a r y , and Mr. Louis Medveczky of the Department of German, for t h e i r help i n t r a n s l a t i n g port ions of the Wolpert t r e a t i s e . A l s o , I would l i k e to e s p e c i a l l y thank Miss Lynne Taylor for her assistance i n preparing and typing the f i n a l manuscript. F i n a l l y , I would l i k e to thank my wi fe , L inda , for her patience and encouragement throughout the wr i t ing of the ent i re t ex t . CHAPTER I INTRODUCTION Music theory i n the twentieth century has been concerned with a wide v a r i e t y of c r i t e r i a i n what has been a conscious attempt to systematize and c l a r i f y a l l of the conceivable raw mater ia l s of musical composition. Twentieth-century t h e o r i s t s have been involved i n some-times rather extended a n a l y t i c a l inve s t i ga t ions into melodic s t ruc ture , rhythmic organiza t ion , t imbre, formal l o g i c , dynamics and the very nature of sound i n i t s e l f . However, there has been a not iceable absence of e f for t on the part of these same t h e o r i s t s to attempt to come to terms with workable systems of c l a s s i f y i n g chordal mater ia l so as to inc lude a l l poss ib le arrangements of the sounds which are present within our twelve note d i v i s i o n of the octave. This s i t u a t i o n i s i n d i r e c t a n t i t h e s i s to the a c t i v i t i e s and in te re s t s of theor i s t s of the nineteenth century, one of whose primary concerns was i n chords and systems by which they could be c l a s s i f i e d , arranged and l a b e l l e d . During the nineteenth and the f i r s t part of the twentieth centur ies a l l v e r t i c a l s tructures were t r a d i t i o n a l l y compared to the t r i a d i c frame of reference . 2 Notes w h i c h c o u l d not be a c c o u n t e d f o r w i t h i n such a frame 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 or l o w e r n e i g h b o u r t o n e s , or t h e l i k e , and were not r e a l l y c o n s i d e r e d an i n t e g r a l p a r t o f t h e c h o r d . As t h e harmonies became more complex t h i s t r i a d i c frame of r e f e r e n c e was t o become more and more unworkable. T h e r e f o r e , a need f o r t h e e x t e n s i o n o f t h e t r a d i t i o n a l system t o i n c l u d e t h e p o s s i -b i l i t i e s o f more complex v e r t i c a l s t r u c t u r e s became e v i d e n t . Thus 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 t h a t c o u l d be c a l l e d c h o r d c l a s s i f i c a t i o n have been proposed by music t h e o r i s t s . ^ - N a t u r a l l y , a l l of t h e s e systems are not c o n s t r u c t e d f r o m t h e same p o i n t o f v i e w o r w i t h t h e same purpose i n mind. However, t h e q u e s t i o n o f v e r t i c a l 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 a l l composers 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 of v i e w o r under what c r i t e r i a t h e i r music has been w r i t t e n . M o r e o v e r , t h e c l a s s i f i c a t i o n of v e r t i c a l s t r u c t u r e s does not 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 s e r i a l i s t or n o n - s e r i a l -i s t i n o r i e n t a t i o n , a re s i g n i f i c a n t i n t h i s r e g a r d . 1 G e n e r a l l y , i n t h e t w e n t i e t h c e n t u r y , " c h o r d " 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 or more no t e s sounding s i m u l t a n e o u s l y . 3. Among the former, one may point to the wr i t ings of Babbi t t , Chrisman, P e r l e , Rochberg, Hauer, Schoen-berg, H?fba, and Gerhard, to name but a few. In an a r t i c l e which attempts to describe the problem of harmonic organizat ion i n s e r i a l music, Per le points out that both l i n e a r and harmonic propert ies and resources are ava i l ab le i n the d i a t o n i c - t o n a l system while i n the twelve-tone system only the l i n e a r order-ing of the notes i s s p e c i f i c a l l y def ined, i t being poss ible to v e r t i c a l i z e any number of adjacent elements 3 i n the set i n any way which su i t s the composer. This l i m i t i n g of the twelve-tone system to l i n e a r propert ies i s perhaps one reason for the e a r l i e r noted lack of attempts at the c l a s s i f i c a t i o n of a l l poss ib le v e r t i c a l s i m u l t a n e i t i e s . In a shorter a r t i c l e i n The Score. Per le does l i s t the number of t o t a l poss ible chords given our twelve note d i v i s i o n of the octave. However, the only 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 i s the number of d i s t i n c t permutations being 351.^ A l o i s Haba and Roberto eorge P e r l e , "The Harmonic Problem i n Twelve-Tone M u s i c . " Music Review. XV (1954), 257-67. Here, Babbitt could be noted as one exception 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 ad jacencies . " ^Geroge P e r l e , "The Poss ib le Chords i n Twelve-Tone Mus ic , " The Score. IX (September, 1954), 54-8. 4. Gerhard had proceeded along the same l i n e s previous to P e r l e , but both made errors i n t h e i r c a l c u l a t i o n s which Per le has been quick to point out i n h i s a r t i c l e . Hauer 's system of fo r ty- four tropes can be considered a type of chord c l a s s i f i c a t i o n i n a l i m i t e d sense i n that only six note chords (hexachords) are P. c l a s s i f i e d , and these only for content. The same can be said for M i l t o n B a b b i t t ' s a l l - c o m b i n a t o r i a l source sets ( which are even more l i m i t i n g than Hauer 's t ropes , as there are only s ix bas ic groups. •7 George Rochberg's study, among other th ings , sets out methods for construct ing tone rows at l eas t one of whose invers ions w i l l not repeat the f i r s t s ix notes of the o r i g i n a l row: a study s i m i l a r to B a b b i t t ' s p r i n c i p l e of c o m b i n a t o r i a l i t y . Here again, the study i s r e a l l y not a c l a s s i f i c a t i o n unless one i n t e r p r e t s the term i n a rather l i m i t e d sense. 5 K a r l Eschman, Changing Forms i n Modern Music. 2nd ed. (Boston: E . C . Schirmer Music Company,1968), pp. 83-87. 6 M i l t o n Babbi t t , "Some Aspects of Twelve-Tone Composi t ion, " The Score. XII (1955), 53-61. 7 George Rochberg. The Hexachord and i t s Re la t ion  to the Twelve-Tone Row (Bryn Mawr, Pennsylvania: Theodore Presser 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 , for example, those by Forte and Chrisman. These, however, tend to be of a s t a t i s t i c a l nature and sometimes are even highly mathematical, as i n Q K a s s l e r ' s study. At any r a t e , a l l three of these studies are far from resembling systems of chord c l a s s i f i -ca t ion i n the t r a d i t i o n a l nineteenth century sense i n that arrays of numbers are used to represent the p o s s i b i l i t i e s and, apart from the number of notes i n each array , there i s 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 to further break down or s impl i fy the arrangement. Rather, one i s l e f t with a l i s t of a l l p o s s i b i l i t i e s with no further guide to t h e i r p r a c t i c a l use. These systems serve as more of a descr ip t ion—enlarg ing our concept of the body of material—than a c l a s s i f i c a t i o n . Among so-ca l led n o n - s e r i a l i s t s 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 Hinde-mith and Franz Al fons Wolpert. Unlike a l l of the above mentioned t h e o r i s t s , Hindemith and Wolpert deal with the question of v e r t i c a l s imultaneity i n a manner which strongly resembles the t h e o r i s t s of the preceding century. 8 See: A . For te , "A Theory of Set Complexes for Mus ic , " Journal of Music Theory. VIII (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 Se t s , " J o u r n . a L j ^ ^ i ^ - T h e g j ^ , XV (1970), 58-83. M. Kass ler , "Toward a Theory that i s the Twelve-Note C la s s System," Perspectives of New Music, V, (1967), 1-80. 6. That i s , they both deal with 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 tonal 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 dealing with problems of v e r t i c a l simultaneity, deal with these v e r t i c -a l structures (as we have seen) i n either a s t a t i s t i c a l manner or i n a manner which i s very strongly controlled by l i n e a r considerations, e.g., the twelve-tone system. On the other hand, Hindemith and Wolpert are involved with chords as sound objects 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 with these new v e r t i c a l i z a t i o n s i n a more or less t r a d i t i o n a l manner, they form an important l i n k with the past and of f e r today's composer and analyst a means of understanding these new sound p o s s i b i l i t i e s without neces-s a r i l y implying or assuming a r e j e c t i o n of past musical t r a d i t i o n s . Hindemith's system of c l a s s i f i c a t i o n i s presented i n his t r e a t i s e The Craft of Musical Composition.^ as 9 Paul Hindemith, The Craft of Musical Composition. 4th ed. trans, by Arthur Mendel (New York: Schott Music Corporation, 1970), I. 7. part of a l a rger d i scuss ion d i rec ted towards the serious student of composition. The teacher w i l l f i n d i n t h i s book bas ic p r i n c i p l e s of composition derived from the natura l character-i s t i c s of tones, and consequently v a l i d for a l l per iods . To the harmony and counterpoint he has already learned, which have been purely studies i n the h i s t o r y of s ty le . . . he must now add a new technique, which, proceeding from the f i r m founda-t i o n of the laws of nature, w i l l enable him to make expedit ions in to domains of composition which have not h i t h e r t o been open to order ly penetra t ion . The book makes c lear that for a we l l - in ten t ioned but a r b i t r a r y arrangement of sounds the composer must subst i tute an order which only to 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 creat ive p r o c e s s .i^ The s i gn i f i cance of the C r a f t 1 s c o n t r i b u t i o n to music theory and, more s p e c i f i c a l l y , to chord c l a s s i f i c a -t i o n , has already been pointed out by Wi l l i am Thomson. One of the most far reaching inf luences of the Cra f t has been i t s system of chord a n a l y s i s , which for the f i r s t time offered a poss ible break i n the stone wa l l of t e r t i a n harmony. Before the Cra f t there was no taxonomy of chord structure except that i n which any tona l aggregate was c l a s s i f i e d e i ther as some form of stacked t h i r d s or e lse as a product of decorative melodic ac t ion i n conjunction with a postulated "chord" In an a r t i c l e appearing subsequent to the p u b l i c a t i o n of the Cra f t Hindemith attempts to j u s t i f y part of what he had previous ly set out to accomplish. 10 I b i d . . p. 9. 11 W i l l i a m Thomson, "Hindemith's C o n t r i b u t i o n to Music Theory," Journal of Music Theory. IX (1965), p .59. 8 For the composer as we l l as for the hearer, tones and t h e i r connections are the beginning and end of musical a c t i v i t y . Not so for the t h e o r i s t . He must enquire in to the nature of the tones and study the p r i n c i p l e s of tonal connect ion. For the f i r s t of these two tasks , he i s almost completely independent of the experiences of the p r a c t i c a l musician; the second, on the other hand, i s not to be achieved without a knowledge of compositional procedure, no matter whether a t h e o r i s t obtains such knowledge by means of deduction - the ana lys i s of already ex i s t ing compositions - or through h i s own crea t ive 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 of a system of chord c l a s s i f i c a t i o n i n which every note i n any conglomeration of pitches can be incorporated wi th in the terms of the system. The postulate of the i n t e r v a l as the harmonic unit • . . may be used to exp la in every conceivable chord, and the theor i s t w i l l be faced only with the quest ion of how to apply t h i s y a r d s t i c k i n order to appraise tonal combinations, and not, as formerly , with the necess i ty of d i v i d i n g chords into 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 for h i s system of chord c l a s s i f i c a t i o n appears i n the in t roduct ion to 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 an expansion of the e a r l i e r (1950) e d i t i o n . He admits at the outset that the major part of h i s t r e a t i s e i s d i rec ted 12 Paul Hindemith, "Methods of Music Theory , " Musical Quarter ly . XXX (1944), 21-2. 13 I b i d . . 28. 14 Franz Al fons Wolpert, Neue Harmonik, (Wilhelmshaven: Heinr ichshaf ten , 1972), 13-15. 9. towards "the majority of musical l i s t e n e r s " and i s not meant to be an innovative work apart from "the system of chord types and t h e i r v a r i a n t s . " Part of Wolpert f s j u s t i f i c a t i o n seems to res t on h i s r e j e c t i o n of Hindemith 1 s system. He i n s i s t s that Hindemith's method of c l a s s i f i c a -t i o n i s " imposs ible" since he (Hindemith) regards the t r i t o n e as an e s s en t i a l d i s t i n g u i s h i n g f ac tor and does not recognize the general notion of "chord i d e n t i t y " through i n v e r s i o n . ^ A l s o , Wolpert says that because of t h i s re fusa l to recognize the inver s ion p r i n c i p l e , Hindemith i s forced in to problems of de f in ing roots for a l l poss ible conglomerations of i n t e r v a l s and that t h i s gives him endless t roub le . Wolpert a lso argues that Hindemith's chord chart i s incomplete and must always end with the phrase "and s imi l a r chords. Wolpert asserts that he has solved these problems i n h i s system. He recognizes the t r a d i t i o n a l acceptance of chordal inver s ion through h i s " P r i n c i p l e of Ident i ty " and avoids the confusion of def in ing chord roo t s . He accepts t r a d i t i o n a l roots and re l a te s these to the notes 15 Hindemith had re jec ted the p r i n c i p l e of chordal inver s ion as a "purely a r b i t r a r y invent ion of Rameau's." See: "Methods of Music Theory," p. 27. 16 Wolpert, op. c i t 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 possible chords can be reduced to one of h i s f i f t e e n basic chord types. As wel l as c l a s s i f y i n g a l l possible v e r t i c a l constructs i n a manner remarkably sim i l a r to the 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 " to which one should attempt to adhere when connecting a succession of v e r t i c a l structures. I t seems only natural that given a new vocabulary of musical sounds i n the form of " a l l possible 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 differences i n the work of the two men concerns the question of t o n a l i t y . Hindemith believes that a l l " l o g i c a l " music i s tonal (whether or not so conceived by the composer) and his chordal system i s thus organized into a tonal framework. Wolpert, on the other hand, involves no prejudgements about t o n a l i t y or 17 a t o n a l i t y , and c a r e f u l l y avoids putting a tonal frame-work around h i s chordal system. Because of these differences 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 introduction to 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 . Hindemith*s System. Hindemith*s purpose i n construct ing a system of chord c l a s s i f i c a t i o n i s to expand the l i m i t s of the conventional theory of harmony. Three re so lu t ions are made at the outset : 1. Construct ion i n t h i r d s must no longer be the bas ic ru le for the erec t ion of chords. 2. We must subst i tute a more al l-embracing p r i n c i p l e for that of the i n v e r t i b i l i t y of chords . 3. We must abandon the thes i s that chords are suscept ible to a v a r i e t y of i n t e r p r e t a t i o n s . A l s o , a number of d e f i n i t i o n s and assumptions are stated or imp l i ed : 1. A chord i s defined as a group of at leas t three d i f f e rent tones sounding simultaneously. 1 Paul Hindemith, The Cra f t of Musical Composition. 4th e d . , t r ans , by Arthur Mendel (New York: Schott Music Corpora t ion , 1970), I , 94-5 2 3M_L.. 95. 12 2. There i s a basic and essential difference between chords containing one or more t r i t o n e s and those without. 3. Every chord, with a few exceptions, has a r o o t . 4 4. Chords i n which the bass tone and root are not i d e n t i c a l are subordinate i n value to chords i n which the root and bass tone coincide, other factors being equal.^ Hindemith then sets out to construct a Table of Chord Groups i n which a l l v e r t i c a l structures possible with the twelve-note d i v i s i o n of the octave w i l l be c l a s s i f i e d . Hindemith 1s aim i s to provide a 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 also 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 to the l a s t . The higher the number of the sub-group, the higher the tension and the lower the value of the structure being considered. Conversely, the lower 3 IbM. 4 Ib i d . . 96. 5 I b i d . . 99. 13. the number of the sub-group, the lower the tens ion and the higher the value of the chord. Hindemith orders h i s c l a s s i f i c a t i o n on the bas is of the i n t e r v a l s contained i n the chords under cons idera t ion . A l l poss ible i n t e r v a l r e l a t i o n s h i p s wi th in the chord are taken in to account and the r e s u l t i n g i n t e r v a l s are c l a s s i f i e d according to "value" on the bas i s of Ser ies 2. This Ser ies 2 i s derived from the theory of combination tones" and repre-sents, according to Hindemith, the natural c l a s s i f i c a t i o n of 7 the i n t e r v a l s according to v a l u e . Example 1.^ JL JL O O ^ o ' fr^ fro o 'b^ g bo 6 I b i d . . 58-64. 7 I b i d . . 96. 8 Some of Hindemith 1 s assumptions concerning acoust ics i n the d e r i v a t i o n of Series 2 have been questioned by Cazdun. (Norman Cazdun, "Hindemith and Nature , " The Music Review. V o l . XV, 1954, 292.) "Hindemith i s not interes ted i n rea 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 endless t r o u b l e . Declar ing without q u a l i f i c a -t i o n that any two simultaneous tones produce combination 14. Genera l ly , chords containing i n t e r v a l s from the f i r s t part of the s e r i e s , according to Hindemith, have a higher value than chords containing i n t e r v a l s from l a t e r i n the se r ie s . His system d iv ides chords into two groups by the use of the t r i t o n e as the basic i n t e r v a l i n c l a s s i f y i n g chords. . . . the t r i t o n e . . . stamps chords so strongly with i t s own character that they acquire something of both i t s inde f in i tenes s and i t s character of motion towards a g o a l . ° Furthermore, chords have more value i f t h e i r root and bass note c o i n c i d e . Hindemith's method for determining the root of a chord i s unique. It cons i s t s i n f ind ing the best i n t e r v a l i n the conglomeration of p i tches according to h i s own Ser ies 2. The root of each i n t e r v a l of the ser ies i s determined, according to Hindemith, by a c o u s t i c a l laws. The roots of the perfect f i f t h , the t h i r d s and the sevenths tones, Hindemith f inds that they usua l ly are so weak that the s u p e r f i c i a l ear does not perceive them, but t h i s makes them a l l the more important for the sub-conscious ear . . . so Hindemith i s not deal ing with combination tones that e x i s t , but only with those present to the subconscious ear . In f a c t , t h e i r non-existence seems to make them a l l the more import-ant . . . " 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 roots of the perfect fourth, the seconds and the sixths 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 structure for the best i n t e r v a l contained therein according to Series 2. I f there i s no perfect f i f t h , one proceeds to look for a perfect fourth, 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. I t should be emphasized that Hindemith expects a l l i n t e r v a l s i n the structure to be considered. Thus, i n three-note chords, there are three i n t e r v a l s to consider; i n four-note chords, there are six 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 following should also be noted: Doubled tones count only once; we use the lowest one for 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 include six "chord sub-groups" separated into 10. 97. 16. Group A (Sub-groups I , I I , V ) , chords without a t r i t o n e , and Group B (Sub-groups I I , I V , V I ) , chords containing one or more t r i t o n e s . In four out of six cases the sub-groups are further sub-divided. In a l l these instances , chords with root and bass co inc id ing are i n a higher sub-d i v i s i o n than chords i n the same sub-group with the root above the bass. The only s tructures which sa t i s fy the condit ions of Group A , sub-group I (or simply " I " ) , that i s , chords without t r i t o n e s and without seconds or sevenths and containing root s , are the major and minor t r i a d s 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 Sect ion 1.1. ( t r i a d s i n root p o s i t i o n ) , would be considered higher i n the scale of values than those i n 1.2. ( t r a d i -t i o n a l l y considered t h e i r invers ions) since the former are more stable and le s s i n need of r e s o l u t i o n . The corresponding sub-group of Group B i s numbered II and contains a l l those chords which have a t r i t o n e but do not conta in any minor seconds or major sevenths. The reason for not excluding the major second and minor seventh, notes Hindemith, i s because "the presence of the t r i t o n e 11 Here the word " t r a d i t i o n a l " i s used to emphasize the fact that Hindemith does not consider chords to be i n v e r t i b l e . 17. always involves seconds or sevenths—except i n the 12 diminished t r i a d and i t s inversions," Accordingly, i n the f i r s t section under sub-group II ( i l . a . ) , we have only the dominant seventh with and without the f i f t h i n the t r a d i t i o n a l root p o s i t i o n . The second section i n sub-group II ( i l . b . ) consists of structures which contain major seconds or minor sevenths or both, and i s divided into three categories. In the f i r s t , the root and bass tones are the samej i n the second the root l i e s above the bass and i n the t h i r d , there i s more than one t r i t o n e present, with the root being either i n the bass or the upper voices. Examples of t r a d i t i o n a l chords common to t h i s section are chords of the dominant ninth ( i l . b . l . ) , t r a d i t i o n a l inversions of the dominant seventh ( l l . b . 2 . ) , the so-called "half-diminished" chords ( i l . b . l . ) , the "French" augmented s i x t h ( l l . b . 3 . ) , and si m i l a r structures. Sub-group I I I of Group A contains chords with seconds or sevenths, or both, but without t r i t o n e s . This sub-group i s further sub-divided into l ) structures where the root and bass are i d e n t i c a l , 2) where the root l i e s above the bass tone. The chords i n sub-group III include 12. IbJLd., 102. Note Hindemith uses the term "inversions" here when i n f a c t he claims not to recognize i n v e r t i b i l i t y . 18. the secondary seventh and ninth chords (III.1.) and t h e i r t r a d i t i o n a l inversions ( I I I . 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 tension factor 13 than those of sub-group IV. However, from an empirical 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 VI. Thus, tension i s not necessarily the same as dissonance, according to Hindemith. Also, chords with substantial tension have a greater need to resolve, generally speaking, than those with less tension. As with sub-group I I I , sub-group IV i s further divided i n t o : 1) structures 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 roots. In the cases examined i n these two sub-groups, Hindemith maintains^ there i s no root, but only a "root representative" the i d e n t i t y of which i s dependant on the context. Thus, 13 I b i d . , 108. 14 Ibid., 101. 19 i n sub-group V there are only two chords, the augmented t r i a d and the chord consisting 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 inversions and the diminished seventh chord. Up to t h i s point, 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 to the best i n t e r v a l according to Series 2. Now, i n sub-group VI, because of the nature of the chords i n question, the t r i t o n e predominates. Isolated i n t e r v a l s , says Hindemith, can also be assigned to the Table of ChQrd Groups. The perfect f i f t h and major and minor t h i r d s belong to I . I . , the perfect fourth and major and minor sixths to 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 VI. Hindemith j u s t i f i e s his system of c l a s s i f i c a t i o n by asserting that i t "does away with a l l ambiguity." Also, i t i s a l l - i n c l u s i v e i n that "there i s no combination of i n t e r v a l s which does not f i t into some d i v i s i o n of our system. 15 I b i d . . 100. 16 I b i d . , 105. 20 B « Wolpert*s System. Wolpert 1 s method of chord c l a s s i f i c a t i o n cons i s t s of a systematic 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 f i f t e e n basic chord types . It i s s e l f evident that the f i f t e e n chord types gained i n t h i s way contain a l l possible harmonies for a l l future 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 'not yet seen' or ex i s t ing chords are a l so , without except ion, contained i n t h i s system.1° These f i f t e e n basic types cons i s t of f ive d i f f e r e n t three-note chords, f i ve d i f f e rent four-note chords, three d i f f e r -ent f ive-note chords, one six-note chord, and one seven-note chord. Before descr ib ing these basic types i n d e t a i l , i t w i l l be necessary to state and c l a r i f y some of Wolpert ' s p r i n c i p l e s and assumptions. Two basic p r i n c i p l e s are stated at the outset , namely, the p r i n c i p l e s of i d e n t i t y ( Ident i ta t ) and congruence (Kongruenz) of chords . 17 Wolpert does not accept a twelve-note d i v i s i o n of the octave for h i s system of c l a s s i f i c a t i o n , as does Hindemith, i n that he (Wolpert) regards the s p e l l i n g of a given chord tone as a s i g n i f i c a n t factor i n the c l a s s i f i c a t i o n of that chord. See p . 22. 18 Franz Al fons Wolpert, Neue Harmonik, (Wilhelmshaven: Heinr ichshaf ten . 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 with i t s e l f when i t always contains the same motes, no matter how transposed or how often doubled . x ? In order to determine whether two chords are i d e n t i c a l , says Wolpert, one contracts the chord to i t s c loses t poss ible or narrowest p o s i t i o n (Kontrakturform). Obvious ly , contrac t ion can only take place using octave t r a n s p o s i t i o n and the e l imina t ion of doubl ings . F i n a l l y , two chords are i d e n t i c a l i f t h e i r contracted forms are the same. Concerning the p r i n c i p l e of congruence: A chord i s congruent with another i f i t s transposed contracted form i s i d e n t i c a l i n q u a l i t y CQualita ' t} and quantity tQuant i ta t l to the i n t e r v a l l i c mixture with the chord to be compared.20 Quantity, according to Wolpert, i s more spec i f i c than q u a l i t y when r e f e r r i n g to i n t e r v a l s . For example, q u a l i t y re fe r s to a l l t h i r d s while quant i ty spec i f i e s the type of t h i r d . A l s o , two chords are not i d e n t i c a l unless the notes i n both chords are spel led exact ly the same. S i m i l a r l y , two chords are not congruent unless the i n t e r v a l spe l l ings i n both chords agree exac t ly . ( e . g . f-a-c i s congruent with eb-g-bb, but not with d#-g-bb. EGB i s i d e n t i c a l with g-b-e 1 ,21 but not with g-cb-e'.) At the beginning of h i s sect ion on the formation of 1 9 I b i d . . 18 (7). 2 0 I b i d . , 19 (7). 2 1 I b i d . , 20 (8) . 22. 92 new chord types Wolpert makes i t very c l e a r t h a t the f o l l o w i n g c o n t r a c t e d chords a l l belong i n the same c l a s s i f i c a t i o n , t h a t i s , " s i x t e ajouteV 1 : 1. C,E,G,A. 3. C,Eb,Gb,Ab. 2. C,E,G,Ab. 4. C,E,G#,A. N a t u r a l l y , other chords c o u l d be c o n s t r u c t e d which would a l s o belong t o . t h i s c l a s s i f i c a t i o n . However, a chord such as C,E,G,G# would not belong i n the above group, even though i t sounds the same as number 2. Although such an e x p l a n a t i o n may seem premature at t h i s p o i n t , Wolpert seems t o be p r e p a r i n g us f o r the f a c t t h a t h i s system, at the o u t s e t at l e a s t , i s f a r from resembling f a m i l i a r concepts t h a t we may have l e a r n e d i n t r a d i t i o n a l harmony. He goes on t o s t a t e t h a t a l l t r a d i -t i o n a l t r i a d s i n r o o t p o s i t i o n belong t o another c l a s s i f i c a -( 5 ) t i o n , which he c a l l s ( 3 ) . I t i s important t o note t h a t the " q u a l i t a t i v e " i n t e r v a l s t r u c t u r e i n a chord i s what i s e s s e n t i a l i n c l a s s i f y i n g i t a c c o r d i n g t o b a s i c t y p e . A l t h o u g h Wolpert does not g i v e an e x p l a n a t i o n of chord r o o t s here, he acknowledges t h a t such r o o t s must e x i s t , and s t a t e s "the problem of the r o o t of a chord i s i n f a c t connected w i t h the t o n a l i t y . " ^ 22 Ibid.., 21 ( 8 ) . I b i d . , 13 ( 4 ) . 23 . 24 A c c o r d i n g t o W o l p e r t , a l l t h r e e - n o t e c h o r d s ^ can be c l a s s i f i e d as b e l o n g i n g t o one of f i v e b a s i c t y p e s . C o n v e n i e n t l y d e s c r i b e d , these f i v e t y p e s are as f o l l o w s ( the numerals are put i n p a r e n t h e s i s , says W o l p e r t , t o d i s t i n g u i s h them from the o l d t h o r o u g h - b a s s number ing ) : Example 2 . 25 A f\ ra> . o ..... o (5) (3) (5) (2) (4) (3) (2) (3) (2) Such t y p e s as j^j and j^j a r e n ° t i n c l u d e d because one d i s c o v e r s on i n v e r s i o n t h a t they become j^j a n c* (3) r e s p e c t i v e l y , and t h e r e f o r e are a l r e a d y c o n t a i n e d i n the sys tem. F u r t h e r e x a m i n a t i o n r e v e a l s the group t o be a l l -i n c l u s i v e . To de te rmine i n t o w h i c h c l a s s i f i c a t i o n any t h r e e -note chord would f a l l , one proceeds as f o l l o w s : 1. Reduce the chord t o i t s c o n t r a c t e d f o r m . Arrange-ment o f n o t e s must be i n the narrowest p o s s i b l e p o s i t i o n . 24 Any group of t h r e e or more notes may be c o n s i d e r e d a c h o r d , and every c o m b i n a t i o n of t h r e e notes i s r e d u c i b l e t o one of these f i v e b a s i c t y p e s , a lways w i t h the 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 not a f f e c t the c l a s s i f i c a t i o n . I b i d . . 29 24 Example 3. 26 -Q- i s not or IT but 2. Transpose the contracted form so that i t s bass note i s C. 3. Accidentals, for 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 with one of the types ( i 5 ) , (3)' (2)' >4j, . . ) i n the example above. At t h i s point Wolpert digresses to discuss certai n structures i n Hindemith's system of c l a s s i f i c a t i o n of chords. The following 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 clear the basic differences i n our methods," says Wolpert. 27 26 ± s "the symbol for the "contracted form of." 27 Ibial., 23 (9). 25 Example 4.^8 rl ff -b-€>— 9! o u J -9-i . 2. 3. Wolpert points out that according to Hindemith 1s method, there e x i s t s between the above four chord forms no connection that i n any way makes them s i m i l a r . What Wolpert f a i l s to 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 . The second belongs to V, and i t s root i s not defined, according to Hindemith. 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 to the same c l a s s i f i c a t i o n i n hi s system. When considered i n t h e i r contracted forms, they a l l reduce to the basic type of j ^ j . Accordingly, Wolpert concludes: . • . because of t h i s , and on the basis 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 into f i v e basic types and are notated as follows: 28. I b i d . . 23. 29 I b i d . . 23 (10). Example 5. 30 26, te-( U ) (-2 a ) (I) (?) The "a" stands f o r "ajoutee" and should be translated l i t e r a l l y as a "piece added on." Wolpert emphasizes that (6a) should not be thought of as an added s i x t h chord, but i n the larger context of a chord type. For example, the chord with the notes C,Eb,Gb,Ab would f a l l into 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 for determining into 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 outlined above f o r the three-note chords. The five-note chords f a l l into only three basic categories. These are as follows: 3 1 Example 6. 5 3 : 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 to 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 reducible to the same basic type, according to Wolpert. This basic type i s as follows: 32 Example 7.' The same i s true for a l l seven-note chords Example 8. 33 A l l of these chord types can be altered from t h e i r above "normal" forms by the use of accidentals. Again, i t i s emphasized that t h i s a l t e r a t i o n (unless i t takes place on the lowest note), does not af f e c t the chord's c l a s s i f i c a t i o n according to basic type. When a given chord contains a note with an accidental preceding i t and the same note without the accidental, or 32 I b i d . 33 I b i d . 28. with a d i f f e rent a c c i d e n t a l , i t i s termed a " s p l i t c h o r d , " (Spaltakkorde) according to W o l p e r t . ^ The concept of s p l i t chords i s introduced by him i n order to expand the idea of h i s basic chord types so that the system w i l l include a l l poss ible combinations of musical p i t ches . I f , for example, a basic | | | chord whose lowest note i s C contains both a G and a G#, the chord i s termed a s p l i t chord since there are two forms of the note G present. It i s notated as fo l lows : Example 9a. > The numeral 5 t e l l s us that i t i s the G that i s s p l i t while the ascending tag on the r i g h t t e l l s us the d i r e c t i o n of the s p l i t , that i s , upward. I f the two s p l i t notes are more than an octave apart , the s p l i t sign i s preceded by a "w" (weit ) , as i n the fo l lowing example: 34 I b i d . . 32 (11). 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 neither of the s p l i t notes i s the natural form of the note, an x or ^ i s used as the tag, 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 follows: Example 11. 1 30 I f , i n the above example, the notes G and Gb are reversed, 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 with 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 follows: Example 12. b b H b b b \ - A -< w w — w — (<~) The above examples elaborate 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. Unfortunately, few examples are provided by Wolpert at t h i s point, and, as a r e s u l t , the concepts are not as clear as would be desirable. The idea of s p l i t chords leads Wolpert naturally to 31. h i s concept of "dimension" (Di men.qj op ) - Although a chord such as that i n the l a s t example contains four d i f f e rent p i tches , i t s basic form (when acc identa l s are ignored) i s considered by Wolpert as having e s s e n t i a l l y only three d i f f e rent notes . In t h i s example, he says, the chord has four sounds but i s only "three dimensional ."36 j0 take another example, the fo l lowing chord has seven d i f f e rent sounds, but i s only four dimensional . Example 13. I t fo l lows , says Wolpert, that the "root form" (that i s , reduct ion to basic chord type) of a s p l i t chord cannot be greater than seven dimensions. A l s o , a chord with twelve d i f f e rent notes must always be a s p l i t c h o r d . ^ Wolpert continues with a d i scus s ion on "Dissonance 35 I b i d . . 35 (12). 36 I b i d . . 35 (13). 37 One may wonder why Wolpert did not state the more obvious fact that a chord with more than seven d i f f e rent sounds must be a s p l i t chord. 32. Values and D e g r e e s . H e begins by s ta t ing that the t r a d i t i o n a l major t r i a d i s "the most complete of a l l sounds poss ib le" and of a l l the t r a d i t i o n a l t r i a d s , i t i s the only one he considers consonant. The other three t r i a d s are dissonant and are arranged i n degrees as fo l lows : I minor t r i a d leas t dissonant II diminished t r i a d medium dissonance III augmented t r i a d most dissonant Wolpert ' s r a t iona le for the above c l a s s i f i c a t i o n of t r i a d s i s the sensation of hear ing : the ear i s the foremost standard i n the gauging of dissonance . . . very l i t t l e can be proven p h y s i c a l l y . 3 9 Likewise , with i n t e r v a l s , there are three degrees of dissonance: I Major second, minor seventh - l eas t II Tr i tones - medium III Minor second, major seventh - most The perfect fourth i s termed an " acc identa l dissonance" ( a k z i d e n t i e l l e Dissonanz) when i t appears outside the context of a major or minor t r i a d , and i s l e s s dissonant than a l l of the above i n t e r v a l s . Other i n t e r v a l s are considered consonant. 38 I b i d . . 36 (13). 39 I b i d . . 37 (14). 33. The above i n t e r v a l s and t r i a d s are what Wolpert terms "s imple" dissonances (einfache Dissonanzen). Simple dissonances are d i s t ingui shed from other types of dissonance termed " e s s e n t i a l dissonances" ( e s sen t i e l l e Dissonanzen) , 4 < ^ Es sent i a l dissonance i s defined by the fo l lowing cond i t ions : 1. Each sounding consonance a r i s i n g from an a l t e r a t i o n i s , according to i t s essence, " e s s e n t i a l l y d i s sonant . " 2. Each sound derived from a simple dissonance i s a l so e s s e n t i a l l y dissonant—even i f i t s enharmonic equa l i za-t i o n appears consonant. 4-* The fo l lowing examples w i l l i l l u s t r a t e these c o n d i t i o n s . Example 14. (j) h p — — 1 i X J d — K L 2 Q 1 ^ — ^ — u 1 and 3 - e s s e n t i a l l y dissonant and unstable 2 and 4 - consonant and stable 40 I b i d . , 38 (14). 41 I b i d . , 38-9 (14-15). 34. The significance 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 contain some form of t h i s i n t e r v a l . Wolpert believes 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. Also, dissonance i s ameliorated by distance and worsened by proximity, that i s , generally, the closer the notes are together the greater the dissonance while the greater t h e i r distance apart, the less the dissonance. Thus, generally speaking, sevenths are less dissonant than seconds. Also, consonance i s equated with s t a b i l i t y and dissonance with i n s t a b i l i t y . Wolpert labels 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 well as perfect fourths "lose 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 to condition." Wolpert now sets about to f i n d the root (G rundton) of each of h i s basic chord types according to 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 narrowest possible p o s i t i o n . Where t h i r d s are missing, the spaces are marked with arrows. When one of the basic chord types i s converted to the narrowest possible 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 basic 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) p o s i t i o n . One exception occurs with the J 4 J chord, which has three "narrowest possible" arrangements. Since s t r i c t l y speaking, there i s no "narrowest possible 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 suitable structure of t h i r d s erected. 4 4 I b i d . . 51 36 Example 16. JLZ e t c . In the f i n a l tabular surveys, a l l of the basic chord types and the corresponding Grundakkorden are presented i n 45 Some two d i f f e rent types of orders of c l a s s i f i c a t i o n , examples of Wolpert ' s method of chordal ana ly s i s fol low Example 17. Given Chord Contracted In Thirds Type I 3 f _—$r~ 45 I b j u d . . , 53-4. 46 J k i d . , 59-63. 37. C. Comparison. Upon examination of both Hindemith 1 s and Wolpert's systems of chord c l a s s i f i c a t i o n , certain basic d i f f e r -ences are immediately evident. Hindemith, i n his c l a s s i f i c a t i o n of a l l possible chords, forms categories on the basis of combinations of i n t e r v a l s and ranks these categories i n terms of "tension" and"value". His own highly evolved system of roots 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 his c l a s s i f i c a t i o n of a l l possible chords, forms categories on the basis 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 his categories i n terms of value or tension. Furthermore, roots are assigned no importance i n the categories, while the i n t e r v a l s of the augmented and doubly augmented prime play a rather curious role and and are involved i n h i s theory of s p l i t chords. With these basic differences i n mind, the two systems can be contrasted according to the following f i v e d i stinguishing areas: 1. Roots. 2. S i g n i f i c a n t 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 fact that he does recognize t r a d i t i o n a l roots 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 to the "narrowest possible t h i r d arrangement" or Grundakkord. Here the root which he describes corresponds to 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 to re l a t e t h i s root to i t s corresponding note i n the basic t y p e — a process which might have proven hel p f u l i n our o v e r a l l understanding of the system. Instead, he states that when t h i r d s appear i n the Grundakkord a tonal r e l a t i o n -ship i s definable Just what i s meant by the term tonal r e l a t i o n s h i p i s not elaborated on at t h i s point. When th i r d s do not appear i n the Grundakkord, however, (for example, when category j^j b u i l t on C i s converted to a Grundakkord i t becomes C-G-D, and the missing t h i r d s (E and B) are marked with arrows), a tonal r e l a t i o n s h i p i s undefinable.^ In contrast to Wolpert, Hindemith does not recognize t r a d i t i o n a l roots, but has evolved his own system of roots 47 I b i d . , 50 (19). 48 IbM., 50 (19). 39 . both for 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 , for chords—whereby the root i s derived from the "best i n t e r v a l " i n his Series 2, which i n turn has been derived from "nature." Hindemith's system of roots 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 into categories where a) the root and the bass tone are i d e n t i c a l and b) the root l i e s above the bass tone. Furthermore, i n contrast to Wolpert, Hindemith does not discuss the t r a d i t i o n a l concept of buil d i n g chords i n 49 t h i r d s . Indeed, one of h i s foremost resolutions was that the construction of chords i n t h i r d s should no longer form the basis f or any system or discussion of chord c l a s s i f i c a -t i o n . 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 to each t h e o r i s t ' s system of organization. While with 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 , with 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 significance i n his system of c l a s s i f i c a t i o n , 49 Hindemith, op, c i t . , 95. 50 With the difference, however, that the augmented and doubly augmented prime play no role i n assigning of chords to categories, and i n fact are ignored i n t h i s process. 40. as he builds h i s entire 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 contains an altered note and the same note unaltered, or with a d i f f e r e n t accidental. 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 involved, these i n t e r v a l s can be considered to have special significance 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 si g n i f i c a n t and therefore distinguishing i n t e r v a l has already been mentioned. 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 tr i t o n e while a l l remaining chords, that i s , those containing one or more t r i t o n e s , are assigned to Group B. However, i t should be noted here that both Hindemith and Wolpert recognize the importance of the tri t o n e when they come to discuss "harmonic f l u c t u a t i o n " and "chord succession and connection" respectively. The next distinguishing c h a r a c t e r i s t i c involves the treatment of what i s t r a d i t i o n a l l y referred to as enharmonics. In Wolpert's system the s p e l l i n g of a chord ef f e c t s the category into which i t i s placed. For example, i n the following: 51 See p. 14. 41 Example 18. £©5 -e-( t a ) the f i r s t chord reduces to the three-note category while the second chord belongs to the four-note category Wolpert does not recognize the enharmonic equivalence of notes such as g# and ab. On the other hand, i n Hindemith's system, a twelve-note d i v i s i o n of the octave i s accepted and enharmonics such as those above are considered equiva lent . Thus Hindemith would consider the chords i n the above example i d e n t i c a l for purposes of c l a s s i f i c a t i o n , and both would be assigned to h i s sub-group I I I . l . Another basic d i f ference between Hindemith's and Wolpert ' s systems of chord c l a s s i f i c a t i o n concerns the number of notes i n given chords wi th in a s p e c i f i c category. In Wolpert ' s system, a l l chords wi th in a category always contain the same number of notes. For example, there are f ive basic three-note chords, each representing a category, f ive basic four-note chords, each representing a category, and so on, up to one basic seven-note chord . Therefore, (6a) . Thus when considering h i s system of c l a s s i f i c a t i o n , 42. each of Wolpert's f i f t e e n categories contains a s p e c i f i c number of notes--three, four, f i v e , six or seven. In Hindemith*s system, however, some of the sub-groups (for example, I I I ) , contain chords which can have anywhere from three to six d i f f e r e n t notes. In t h i s sense, Hindemith's system i s less orderly than Wolpert's i n that he must use the word "etc." following sub-groups i n h i s Table which are non-finite (for example, II.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 basic categories to which a l l possible 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 for a l l the possible combinations. Perhaps the most controversial point of d i s t i n c t i o n i s the last-- t h e problem of consonance and dissonance. Hindemith does not define either term o b j e c t i v e l y . He does state that the value order l a i d down i n his Series 2 approaches the problem of the consonance and dissonance of i n t e r v a l s . However, he does not specify 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 Series 2 and the dissonant at the end. But the rate at 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 e x a c t l y . 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 to another, then, i s a l l that can be determined by t h i s series. According to Hindemith, a major t h i r d may be dissonant when compared to a perfect f i f t h , but i t i s consonant i n r e l a t i o n to a minor seventh, and so on. The t r i t o n e , not included i n Series 2, i s a special case and i s neither consonant nor dissonant. In Hindemith's words, " i t belongs neither to the realm of euphony nor to 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 roots from i n t e r v a l s to chords. He avoids the use of the term "dissonance" and instead, i n the Table of Chord Groups he arranges chords i n order of " t e n s i o n . " 5 4 I f one were to extrapolate from Hindemith's Series 2 to t r y to determine a scale of consonance and dissonance for chords one might come up with a scale of values approximating Hindemith's chord table. However, one would encounter a l l sorts of d i f f i c u l t i e s and ambiguities, p a r t l y 53 I b i d . 54 See Chapter I I , p. 17. 44. because of the obvious fact that there are more chords than i n t e r v a l s . Hindemith i s very careful i n his discussion of chord types and does not use the word "consonant" or "dissonant" once i n t h i s section of hi s discussion. Even though he believes that the "minor t r i a d should rank higher i n the scheme of tonal values 55 than the major," major and minor chords are of equal value i n his table. Hindemith acknowledges that 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 that throughout history 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 perfect and imperfect consonances. The wide use of seventh chords has made the major second and the minor seventh almost consonant to our ears. The s i t u a t i o n of the fourth has never been cleared up. Theorists, basing t h e i r reasoning on accoustical phenomena, have repeatedly come to conclusions wholly at variance with 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 involve the ear as being the ultimate standard of measurement and he believes that 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 , his ideas concerning these 55 I b i d . f 77. 56 I b i d . , 85. Wolpert, o.p_i_cJLi., 37 U4). 45 concepts rest mainly on his own empirical 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 subjective opinions as having a kind of common sense v a l i d i t y . Although Wolpert does not go so far as to give a Series 2 by piecing together h i s generalizations, one i s a c t u a l l y able to come up with 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 to the most dissonant, ju s t as i n Hindemith. Wolpert gives three degrees of dissonance. He also says that generally seconds are more dissonant than sevenths. 5^ From the information he gives, one can construct the following series: (Least Dissonant) m7 M2 T M7 m2 (Most Dissonant) Also, the perfect fourth i s considered an "accidental" 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 either consonant or dissonant, the perfect fourth acts 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 intervals--octave, perfect f i f t h , sixths 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 sixths.) However, no clues are given as to which consonant i n t e r v a l s are more consonant than others. Since See p. 32 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 to what the ordering might be. A comparison of the l a t t e r portion of Hindemith's Series 2 with Wolpert's scale above (both "series" proceding from the more consonant to the more dissonant i n t e r v a l s ) shows that: Hindemith: m3 M6 M2 m7. m2 M7 Wolpert: P4 m7 M2 T M7 m2 1. Wolpert believes the minor seventh les s dissonant than the major second. In Hindemith 1s series, the opposite i s true. 2. Also, Wolpert believes the major seventh less dissonant than the minor second. Again, the opposite i s true i n Hindemith's series. 3. While Wolpert believes the t r i t o n e to be of medium dissonance (more dissonant than M2 or m7 but less dissonant than m2 or M7) Hindemith believes i t to be unique and neither consonant nor dissonant. 4. With Wolpert, the perfect fourth can be c l a s s i f i e d as a dissonance and comes aft e r the t h i r d s and sixths i n terms of consonance. With Hindemith, however, the perfect fourth i s always more consonant than a l l of the t h i r d s and sixt h s . 47. Wolpert does not systematically extend his ideas of consonance and dissonance to a l l possible chords or even the categories i n h i s system of c l a s s i f i c a t i o n , and apart from discussing the four t r a d i t i o n a l t r i a d s , he goes no further. Wolpert's discussion of "essential dissonances" as distinguished from "simple dissonances" i n v i t e s c r i t i c i s m . He believes that even i f an i n t e r v a l which i s spelled as a dissonance sounds consonant because of enharmonic equivalence to 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 conclusion regarding these "essential dissonances" i s that "they are unstable and cannot be used for cadence 59 formation." I t would seem that Wolpert i s persistent and determined to 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 his categories of c l a s s i f i c a t i o n and i n his t r e a t -ment of dissonance. However, although he would have us believe that a "g#" for 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. Also, i t would seem that the i n t e r v a l C-Eb would be an ess e n t i a l dissonance i f altered from C-E.^ The group of " e s s e n t i a l l y 59 I b i d . . 38 (14). 60 See proposition 1, p. 33. 48 dissonant i n t e r v a l s , " then, can include a whole variety 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 essential dissonances, as we have seen with the i n t e r v a l C-Eb, i s questionable. Also, Wolpert has previously admitted that even simple dissonances are 61 a l l unstable. I t would seem, then, that the concept of essential dissonance i s of l i t t l e value. We now come to Wolpert fs concepts regarding the 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 to be treading on ground which Hindemith has c a r e f u l l y avoided—that of the r e l a t i v e consonance and dissonance of chords. He believes the major t r i a d "the most complete" of a l l sounds possible. So uncompromising i s Wolpert 1s b e l i e f i n the major t r i a d ' s supremacy, that he considers i t alone to be the consonant t r i a d . The other three t r i a d s are a l l considered dissonant 62 i n the degrees previously explained. Wolpert sees the A O minor t r i a d as a "clouding" of the major, a concept, 61 See p. 32. 62 See p. 32. 63 Ib_Ld,., 36 (13). 49. 64 i n t e r e s t i n g l y enough, with which Hindemith agrees. The augmented t r i a d puzzles Wolpert and he cannot under-stand why the i n t e r v a l C-G# does not sound dissonant, while 65 the chord C-E-G# has a "very tense sound." This apparent contradiction, he says, i s "not v a l i d l y explainable" and i s one of the arguments he uses to j u s t i f y h i s b e l i e f i n the 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 dissonance from consonance. 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 to be inconsistent with h i s discussion 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 i n t e r v a l s , while the t h i r d s and sixths 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 to be more consonant than the diminished t r i a d . (In 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 to come to terms with b a s i c a l l y subjective notions of consonance and dissonance should by now be apparent. There i s disagreement 64 Hindemith, op. c i t . . 78. 65 Wolpert, op. c i t . . 38 (14). 5 0 . 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 differences 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 possible combinations of musical pitches, i . e . , a l l possible chords. Obviously the t r a d i t i o n a l method of building 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 ons and does not form the basis of either Hindemith's or Wolpert's systems. CHAPTER I I I CHORD MOVEMENT A. Hindemith. Hindemith 1s study of chord movement involves the examination of three main points, a l l of which, when considered, come to bear on the effectiveness of a given chord progression. These are: 1. Harmonic Fluctuation 2. Degree Progression 3. The Two-Voice Framework I t has already been mentioned with respect to Hindemith's system of chord c l a s s i f i c a t i o n that chord value and chord tension are inversely proportional to one another. The higher the number of the sub-group, the higher the tension and the lower the value of the chord to be considered. Conversely, the lower the number of the sub-group, the lower the tension and the higher the value of the chord. • . . the tension of chords increases from section to section and from sub-group to sub-group i n the same proportion 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 Fluctuation.' The f l u c t u a t i o n may be gradual or sudden according to the r e l a t i v e values of the chords that make up the progression.! 1 Hindemith, The C r a f t , op. c i t . . I , 116. 52. According to Hindemith, sudden fluc t u a t i o n s occur when the progression skips a sub-group (e.g., 1.1 to III.2 or I l . b . l to IV.2). On the other hand, gradual f l u c t u a -tions are those, for example, when the progression occurs within one of the sub-groups ( I I . a to II.b.3) or, even between two d i f f e r e n t chords from within a section of a sub-group (e.g., both from II.b.2). Between the "sudden" and the "gradual" fluctuations are those which move among consecutive sub-groups ( I I I . 1 to IV.2 or 1.2 to II.b.3). These might be termed "medium" f l u c t u a t i o n s . Hindemith states that for harmonic f l u c t u a t i o n to occur, "chords of di f f e r e n t value" are required. To understand t h i s statement more f u l l y , one should examine i t s converse. Hindemith states: "In the connection of chords of i d e n t i c a l structure, 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 structure moves to any other chordal structure, but does not occur when a succession of i d e n t i c a l chordal structures appears. In the example below, (a) constitutes a gradual f l u c t u a t i o n while at (b) there i s no harmonic f l u c t u a t i o n . 2 I b i d . . 117 53. Example 19. [J—, —1 o o o Q o o Hindemith does not appear to give any d e f i n i t e rules as to what constitutes good harmonic f l u c t u a t i o n , apart from the generally implied notion that i n a given musical passage there should be a gradual r i s e and f a l l of harmonic tension. This apparent lack of information has been noticed by Herman Hensel: . . . the information (Hindemith) gives us r e l a t i v e to what constitutes good organization of harmonic f l u c t u a t i o n i s rather sparse and at times ambiguous. . . 3 Hensel also sees Hindemith*s implication 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 tension. However, d i f f i c u l -t i e s arise when one t r i e s to r e l a t e harmonic f l u c t u a t i o n to phrase structure: Hindemith regards the arch type harmonic tension-repose design as one which shows a good organization. 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. Also, 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 brings about 3 Herman Richard Hensel, "On Paul Hindemith's Harmonic Fluctuation Theory," Unpublished D.M.A. Diss e r t a t i o n , (Urbana: University of I l l i n o i s , 1964), quoted from abstract. 5 4 . an equilibrium of tensions as the various elements of music interact with each other. 4 In the C r a f t . Hindemith does warn against movement between cer 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 stable and tension-less chords of Group I, a chord from group V or VI may be useful; i t can almost always be successfully juxtaposed even against chords from group I I . But i n using i t with chords from groups I I I and IV, care must be exercised. In the midst of such chords, a chord of group V or VI often puts us completely off the track: i t seems to cause the whole chord structure to collapse . . . . Progressions of t h i s type must accordingly 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 further s p e c i f i c rules are given.^ Hindemith helps j u s t i f y his theory of harmonic f l u c t u a t i o n by explaining that only with t h i s theory i s there an "explanation for chords of varying harmonic tension -i upon the same root." In the f i n a l analysis, harmonic fl 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 . I t takes no 4 Ibid.. 5 Hindemith, op. c i t . . 119-20. 6 Hindemith implies (Craft, I, 123) that i n succes-sive t r i a d s no t r i t o n e should be evident, e.g., minor dominant to major tonic 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 considering the second point, 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 to include a l l successions of chord roots. Both Series 1 and Series 2 are used; the l a t t e r p a r t i c u l a r l y when considering the roots of adjacent chords i n a progression as related i n the "tonal sphere" or to a tonal 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 considering the i n t e r v a l s between the roots of successive chords (not yet assumed as being related to a tonal centre), Hindemith comes to 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 roots naturally has o a surer foundation than one based on a minor s i x t h . . . . 8 Landau points out that the term "tonal 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 tension between the tonic chord and each of the other chords i n a tonal sphere which i s dominated by i t , " describes the above concept i n which Series 1 i s used as the determining fa c t o r . See Landau's a r t i c l e "Hindemith the System Builder: A C r i t i q u e , " Music Review XXII (1961), 147. 9 Hindemith, op. c i t . . 122. 56. Hindemith*s rati o n a l e here again follows the l o g i c of Series 2. The best progression i s that which i s based on roots a f i f t h apart, the next, that based on roots a fourth apart, then that based on roots a major t h i r d apart and so on through the series u n t i l one comes to the chord progression based on two roots a tr i t o n e apart, which Hindemith sees as the "least valuable of a l l . " - * 0 In a chord which has no root (Sub-groups V and V I ) , a "root representative" i s chosen which best connects i t (accord-ing to Series 2) to the roots of the chords preceding and following i t . ^ " * " Hindemith points out the disadvantage of assessing the value of chord progressions merely from an examination of the movement of t h e i r roots. The fact that a large number of chords can be constructed over the same root helps t e s t i f y to 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 inve s t i g a t i o n of the two-voice framework and the harmonic fl u c t u a t i o n w i l l clear up a l l ambiguity."-^ Progressions which contain t r i t o n e chords are of 10 I b i d . . 123. 11 I b i d . , 125. 12 I b i d . . 123. 57. special significance to Hindemith and treated 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 to Series 2, to 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. 1 114 If the resolution i s to 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 (preferably a second) to the root of the following chord. When successions of several group B chords occur, there i s no res o l u t i o n of the t r i t o n e but instead a prolonging of the tension. This type of succession i s treated 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 to the guide tone i n the second i s considered a secondary assessment (af t e r root progression) of the value of the chord progression. Hindemith also points out the 13 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 fact that i n the succession of two t r i t o n e chords from sub-group I I (only occasionally with those from sub-group IV) whose roots are a t r i t o n e apart there i s also a t r i t o n e between the guide-tones of the two chords as well as the fact that the t r i t o n e contained i n the f i r s t chord must also be contained i n the second. This important observation leads Hindemith to exclaim: "This chain of t r i t o n e s l i n k s these two chords so cl o s e l y together that they seem almost l i k e f r a c t i o n a l parts of the same chord. Example 20. A j Roots: D Ab Common Tritone: C - P# (Gb) %\\ — Guide Tones Thus far Hindemith has made no attempt to r e l a t e these concepts of "good" root progression to the concept of tonal centre or tonal sphere. Before beginning to discuss these "harmonic family-relationships," Hindemith admits that c e r t a i n rhythmic considerations are necessary i n 16 I b i d . . 130. 59. attempting to discern such tonal centres. "Duration and position i n the measure are of decisive importance i n determining the tonic: the stressed portion 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 further discussion of the element of rhythm i n the section concerned with chord movement. Instead, two rather s i g n i f i -cant d e f i n i t i o n s are stated regarding the harmonic aspect of tonal organization. These are: 1. A succession of chords from Group A must consist of at least three chords i f i t i s to represent a tonal e n t i t y . 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 resolution." 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 states 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 single chord from group B, a "tonal e n t i t y " (or "tonal centre") i s not defined 17 I b i d . . 133. 18 I b i d . . 134-5. 60 since "the ear does not know i n which d i r e c t i o n to resolve the t r i t o n e . " Thus, "the sounding of a single 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 tonal 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 tonal centre."^ 9 When a chord from group B resolves to a chord from group A, the root of the l a t t e r i s considered the tonal 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 the chord of r e s o l u t i o n . However, i n a series of unresolved chords from sub-group I I , the tonal 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 series since "the unresolved t r i t o n e of the f i n a l chord would resolve most naturally into 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 1s summary i n chart form of the number of chords needed for determining a tonal centre i s reproduced i n the Appendix. I t has already been mentioned that Series 1 i s used when considering chords related to a tonal centre, although Hindemith does not f u l l y explain h i s rationale i n choosing 19 I b i d . 20 I b i d . , 136. 61 i t . As Series 2 was derived b a s i c a l l y from h i s i n t e r -pretation of combination tone curves, so Series 1 was derived b a s i c a l l y from his i n t e r p r e t a t i o n of the overtone series. Whereas Series 2 consisted of a row of i n t e r v a l s , Series 1 consists 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 to a given tone. The further a tone i s away from the f i r s t tone, the more distant 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 to the high-ranking degrees of Series 1 (Tonic, Dominant and Subdominant) or i t may consist of a variety of both high and low ranking degrees. As the degree of re l a t i o n s h i p between the chord roots and the tonic chord root varies, so does the tension. As Landau has pointed out^-* t h i s "tension" i s not the same "tension" used i n discussing 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 structure of the 22 chord" while with degree progression, "tension" refers to . . . the c o n f l i c t between the authority of the tonal centre and the urge of the i n d i v i d u a l harmonies to escape from that authority. When t h i s urge i s g r a t i -f i e d , of course, a new tonal centre i s established and modulation takes place . . . .23 21 22 23 Landau, op. c i t . . 150. See p. 18. I b i d . 62. Landau summarizes Hindemith's views on the establishment of t o n a l i t y (or a tonal centre) i n a complete composition: . . . the p r e v a i l i n g t o n a l i t y i s established by the interplay of the same factors which serve that purpose i n a tonal 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 related tones. Thus, the tonal centre which i s must repeated or which appears at the end, or which i s strongly 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 entire work.24 The t h i r d point to be mentioned with 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 believes that a chord progression i s affected very l i t t l e by the inner voice movement.25 i t i s p r i n c i p a l l y the setting 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 factor that he devotes h i s attention. " I f w r i t i n g i n several voices i s to sound clear and i n t e l l i g i b l e , the contours of i t s two-voice framework must be cleanly designated and cogently organized."26 The extent of Hindemith's desire for such s t r i c t organization i s evident i n Volume 2 of the Craft where a 24 I b i d . 25 Hindemith, op. c i t . . 115. 26 I b i d . . 114. 63 t o t a l of s i x t y - f i v e rules are l i s t e d as a guide for writ i n g the two-voice framework. Many of these rules have to do with the construction of good melodies while there are only a few which have a dir e c t bearing on the p r i n c i p l e s of chord movement or succession. Landau, i n hi s study, has chosen only twelve for reasons which he states below: Hindemith abrogated . . . rules gradually through-out Book I I as the student was presumed to have exhausted the benefits of observing them . . . . There are, however, several rules i n Book I I which were not rescinded and some which were expressly reaffirmed.2' Of the above explained "unrescinded" and "reaffirmed" rules Landau chooses the following: 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 voices ( t h i r d s and sixths should balance seconds and sevenths) 2. Relative a c t i v i t y : l e s s movement i n bass 3. Alterna t i o n of a c t i v i t y : ( i f one voice moves, keep other s t i l l ) 4. Interv a l root below at beginning, end, import-ant points 5. No crossing 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. 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 Practice" Music Review. Vol. XXI, 1960, 42. 64. 11. No leaps to or from 2nds 7ths or the tritone (Most violated rule i n Landau's study). 12. Upward resolution of suspensions only to 2Q c e r t a i n i n t e r v a l s (See Rule 50, Book I I ) . In addition to these, the present writer f e e l s the following two rules should also be included, partly because of possible comparisons which may be drawn with Wolpert: 1. Rule 22 The two voices may not skip i n the same d i r e c t i o n at the same time . . . .29 2. Rule 23 Cross Relations must be avoided.30 Book I I of The Craft i s only concerned with 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 further c l a r i f y t h i s point, Landau maintains that these r u l e s , however, cannot be applied to any combina-t i o n of two voices i n a structure of three or more parts, but rather only to the two-voice framework as Hindemith 28 Ibid», See insert p. 46. 29 Hindemith, op. c i t . . I I , 26. 30 I b i d . Hindemith, however, l a t e r permits cross-r 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 duration 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 materials on three part writing which Hindemith d i s t r i b u t e d to his students at Yale, p a r a l l e l 4ths were allowed between the top and middle voices and between the middle and bottom voices - also, p a r a l l e l 5ths were allowed between top and middle voices when the tones i n either pair of f i f t h s have d i f f e r e n t functions (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 Wolpert 1s discussion of "The P r i n c i p l e s and Hindrances of Chordal Connection and Succession," 3 ^ i t i s implied that the connection of chords i s most sa t i s f a c t o r y when a l l the voices (with the exception of the bass), i f they proceed at a l l , proceed by step. Wolpert t r e a t s voice movement by second i n a most system-a 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 voices (by 3rd, 4th, etc.) are hardly discussed. This would seem to disallow, at the 32 Landau, Hindemith. A Case Study . . . , 42. 33 Landau, op. c i t . . 51. 34 Franz Alfons Wolpert, Neue Harmonik. (Wilhelmshaven: Heinrichshaften, 1972), 65-96, (Unpublished t r a n s l a t i o n , L. Medveczky, 22-34. 66. outset, many p o s s i b i l i t i e s of chord movement which even t r a d i t i o n a l l y one has come to accept. Using the "two-voice framework"3S (Zweistimmigkeit) as a method of inve s t i g a t i o n , Wolpert examines possible combinations of paired voices i n the connection of two chords. Note that i n the example given, a l l voices except the bass move by step or remain stationary. Example 21. 36 -mmmt— 2E 7J n i . 2. 3. 5. 75 1. Reihung - p a r a l l e l movement of i n t e r v a l s of equal qu a l i t y ( t h i r d s , seconds, f i f t h s , etc.) by step, ascending or descending. 2. Konstante - one note common i n two chords i n the same voice. 35 It should be noted that "two-voice framework" i n the general sense i s d i f f e r e n t from Hindemith's special d e f i n i t i o n of the same term discussed e a r l i e r i n Section 'A" of t h i s chapter. Wolpert's d e f i n i t i o n of the term may apply to any combination of two voices. 36 Wolpert, op. c i t . . 65. 67 3. Diastole - two voices expanding by step. 4. Systole - two voices contracting by step. 5. Bas-Kadenz - when the bass does not move by step. (1) Concerning the Reihung. p a r a l l e l t h i r d s , fourths and sixths are acceptable i n part-writing, according to Wolpert. 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 inversions" (aug. prime, dim. octave, minor second, major seventh) constitute a hindrance to agreeable chord movement and are therefore not accept-able. Later, under the sub-heading "mixtures" (Mixturen), Wolpert explains that consecutive 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 impressionists (Debussy and Ravel) and i n Reger, but also i n Bartok and Stravinsky, and often serve as a "colourful strengthening of the melody." 3 8 (2) Concerning the Konstante tone: Wolpert states two "Laws of I n a c t i v i t y " as follows: 1. A tone which i s common i n two chords should remain i n the same voice. 2. In the connection of two chords, voices should move the shortest possible d i s t a n c e . 3 9 3 7 / \ I b i d . . 66 (23). 3 8 I b i d . . 69 (23). 3 9 I b i < i . , 70 (24). 68. Wolpert implies, then, that the less a c t i v i t y i n the voice leading of a chord progression, the better. He f l a t l y 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 far as to give the following 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 for themselves, unconnected" and "no hearing l o g i c i s to be recognized i n them." 4 0 41 Example 22. 3£a Here a d i s t i n c t i o n should be made between the terms "connec-t i o n " and "succession," although i t appears that Wolpert does not. A succession can involve any series of sounds. A connection, i t would seem from the above, i s a desirable succession, which, according to Wolpert, would involve as l i t t l e voice movement as possible (except for the bass). I b i d . f 70 (24), Aufeinanderfolgen wie die des B e i s p i e l s C22J Kann man wohl nicht Verbindungen nennen. Ihre Teile stehen unverbunden fur s i c h . Ihre Sprunge wirken sperrig, ihre Vorzeichen sind w i l l k u r l i c h gewahlt. 41 I b i d . . 71. 69 "Our task now" he states, " i s to r e a l i z e possible future connections on the basis of these already known l o g i c a l hearing processes."42 (3,4.) One can now examine the Systolen and Diastolen, which Wolpert discusses as a group. F i r s t l y , 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 of Systolen and Diastolen. These are named "complete" (ganze). "half" (halbe). and "whole-tone" (ganzton). With complete Systolen/ Diastolen. the contracting/expanding voices both move by half step. In the half Systolen/Diastolen, only one voice moves by half step while the other moves by whole step. In the whole-tone Systolen/Diastolen both voices move by whole step. Example 23. 43 Systolen Piastolen Complete 3EE Half 0 So Whole tone^ o " 42 Ib i d . 43 I b i d . f 72. 70. Next, the concept of "adhesion" (Adhasion) i s i n t r o d u c e d I n simple terms, "adhesion" i s the name used to describe either a complete Systole or a complete Diastole. Thus, whenever two voices both expand or both contract with a half step movement i n each voice, adhesion i s said to have taken place. I t i s assumed that t h i s "adhesive process" i s a good q u a l i t y i n chord connection since, according to the "Laws of I n a c t i v i t y " 4 5 half-step movement i s more l i k e l y and desirable than jumps of a major second or l a r g e r . ^ Adhesion may lead to either a stable or an unstable ending. 4 7 When i t leads to the l a t t e r , i t i s termed "diversion" (Diversion). However, there i s no term which describes the opposite of diversion, i . e . , leading to a stable ending. Moreover, there i s a d i s t i n c t i o n i n t h i s 44 I b i d . . 73 (25). 45 See Chapter I I I , p. 67. 46 In order that adhesion can take place with t r a d i -t i o n a l l y correct voice leading 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 that the voice leading i s t r a d i t i o n a l l y acceptable. This "transpolation" i s nothing more than enharmonic su b s t i t u t i o n . 47 Unstable i s the same as "dissonant." See Chapter I I , p. 34. 71. unnamed c l a s s i f i c a t i o n between "ambivalently adhesive" and "unequivocally adhesive" penultimate i n t e r v a l s , depending on the spe l l i n g of t h i s i n t e r v a l . Two rules are stated regarding adhesion leading to a stable ending: 1. Unaltered penultimate i n t e r v a l s can resolve i n two ways. (These are c a l l e d ambivalently adhesive.) 2. Altered penultimate i n t e r v a l s can resolve i n only one d i r e c t i o n . (These are ca l l e d unequivocally adhesive•) In simpler terms, some spellings allow one possible _8 reso l u t i o n , while other sp e l l i n g s allow two. The following example i l l u s t r a t e s t h i s f a c t : Example 24« 4 9 f f & Unaltered Unaltered Altered 48 Ibid.., 73. 49 I b i d . . 74. 72 A l l adhesive processes which lead to dissonant or unstable endings are c a l l e d "diversive"--the process i t s e l f i s c a l l e d "diversion." Some examples of t h i s process follow: Example 25. 50 —o •ifo ° 0 o -6- -g-To demonstrate adhesion i n i t s " f u l l e f f i c i e n c y , " Wolpert gives an example of a four-note chord connection. There are six possible two-voice frameworks and a l l are examined i n d e t a i l — f i r s t the outer voices (peripherie); next, the two pairs of voices separated by one i n t e r -mediate voice (ubernachste): and f i n a l l y , the three pairs of adjacent voices (benachbarte). The following example w i l l help i l l u s t r a t e Wolpert's procedure: 50 I b i d . , p. 75. 51 I b i d . . p. 76. 73 Example 26. 1. 2. 3. 4. 5. 6. In the above four-note chord connection, four of the six possible two-voice frameworks are adhesive (systoles) and non-diversive; they a l l lead to stable endings. The remain-ing two p o s s i b i l i t i e s involve Reihnngen - #5 being consecu-t i v e t h i r d values while #6 involves consecutive fourths. Here, the second of the "Laws of I n a c t i v i t y " ( i . e . , voices moving the shortest possible distance i n chord connection) can be seen to operate. For t h i s reason i t can be assumed that 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-tio n s can be mixed with adhesive i n t e r v a l connections as i n the following example. Note that although the lower i n t e r v a l pair i s non-adhesive i t i s s t i l l s y s t o l i c . 52 Here s p e l l i n g does not seem to bother Wolpert. Although he recognized the diminished fourth he regards i t as a " t h i r d value" (Terzwerte). 74, Example 27 . 5 3 adhesive .i non-adhe sive (5) One can now turn to Wolpert's discussion of bass movement. I t has already been noted that i n chord connection i t i s most desirable for the upper voices to move the smallest possible distance. The ultimate of t h i s i d e a l (apart from a l l voices remaining stationary) i s manifested i n the concept of adhesion. However, Wolpert informs us that 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 treated separately from the other voices and i n f a c t , does not follow the "rules" l a i d down for them. How-ever, i t does not have i t s own r u l e s . Instead, Wolpert l i s t s the most frequently occurring and therefore most desirable p o s s i b i l i t i e s of upper voice movement related to the bass. After 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 following 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 to the bass made for a sa t i s f a c t o r y cadence. 5 5 53 I b i d . . 77. 54 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. 7 5 . Example 2 8 . 5 6 * I h 4 l f h ° o 9 U ° r> / \ W—-J u 1 2 E 1 _. 5 6 I b i d . . p. 80-1. I t should be noted that the upper voice movement i s not always stepwise although not surpris-ingly i n the majority of cases i t i s . I t might also 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 to be l i s t e d , the number for each bass movement would t o t a l 11 X 12 or 132. 76. Wolpert thus seems to imply that the bass voice may move as other voices (by step), but also, p a r t i c u l a r l y at cadential points, may move by t h i r d , fourth, or f i f t h . He i s , i n f a c t , not complete i n the treatment of the bass voice as he was also 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 to Wolpert, i n cadences where i t moves by fourth or f i f t h . 5 7 When i t moves down a perfect f i f t h or up a perfect fourth, i t i s considered a "descending cadential bass" and i s notated L 5-^ or Conversely, when i t moves down a perfect fourth or up a perfect f i f t h i t i s considered an 4 ^ or f . Next, i n examining cadences with s p l i t chords, 5 8 Wolpert discusses the following three simple p o s s i b i l i t i e s 57 I b i d . . 82 (29). Here the word " s t a b i l i z i n g " has no connection with 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 fourth or f i f t h . 58 I b i d . . 84 (29) 77. using many examples. 3" 1. Cadences where the penultimate chord contains a s p l i t f i f t h . 2. Cadences where the penultimate chord contains a s p l i t t h i r d . 3. Cadences involving multiple 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 — ? 2 5 — 9 C I j 5 h j—k 1-4 I 1. 2. 3. Expanding on the above concept, Wolpert i l l u s t r a t e s cadences where the penultimate chord contains a s p l i t i n t e r v a l and i s related to a constant bass progression. 5 9 See Chapter I I , PP. 28-30. 60 I b i d . . 85-6. 78, Example 30* 61 a—Q- J>Q Q bo » E^Q t/o )—7 - A )—7—\ Then i n order to obtain stable f i f t h s from s p l i t t h i r d s , Wolpert presents the following solutions: Inherent i s the problem that i t i s not always possible for upper voices to 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 half step, the other must f a l l a minor t h i r d ; but i f one tone moves downward by a half step, the other must 61. i b i d . . 87. 62 Ib i d . 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 to stable f i f t h s and works out a l l the desirable 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 asserts that hearing a l o g i c a l "progression" among adjacent voices i n a chordal connec-t i o n i s affected by a "penetrating" or "permeating" qu 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 6 5 movement. In discussing t h i s p e n e t r a b i l i t y , Wolpert implies that half step movement i s such a powerful force that i t does not have to 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 rather, more generally, implying half step movement ascending or descending i n one or more voices. 80 same voice to be perceived by the ear as the s i g n i f i c a n t and penetrating movement. The leading-note ( i n t h i s modified sense) seeps through the musical texture,so to speak, even though i t may change voices. Wolpert c i t e s Orlando d i Lasso, the G a b r i e l i s and Gesualdo as examples of composers who early 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 following 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 . 6 7 -The " indicates that the Bb acts as a "leading-tone" to the B. Furthermore, since the movement apparently involves a change i n voices, unless i t i s assumed the voices cross or skip, the leading note i s c a l l e d "transverse" 66 I b i d . . 89 (31). 67 Ib i d . 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 voices of two tones that, owing to t h e i r mutually contradictory character, would normally be placed as a melodic progression i n one voice. In other words, c r o s s - r e l a t i o n means the use i n 'diagonal 1 position of what properly i s a 'horizontal' element of musical texture. 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 oriz-ontal that the ear i s disturbed i f i t hears the f i r s t tone i n one voice and the second i n another.69 From t h i s d e f i n i t i o n , i t i s easy to see how Wolpert arrives 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 thing that Apel i s describing 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 describing t h i s process. He then digresses momentarily to t a l k about "the master composers since the beginning of harmonic polyphony i n the Renaissance" as often disregarding the t r a d i t i o n a l rule against the use of the c r o s s - r e l a t i o n . He says, moreover, that he believes 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-note-l i k e q u a l i t y " not only as permitted or even desirable 68 , . I b i d . , 89 (32). 69 W i l l i Apel, Harvard Dictionary of Music. 2nd ed., (Cambridge Mass.: Harvard University Press, 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) involves a s p l i t tone or augmented prime (E-Eb) while the l a t t e r (Example 34b) involves a minor second (Fb-Eb). A r e a l c r o s s - r e l a t i o n (Q) must be spelled 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 spelled as a minor second. In both cases, for the d e f i n i t i o n of cr o s s - r e l a t i o n to hold, the notes involved must be i n d i f f e r e n t voices. The symbol for 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 q - L = Fb, /b)Eb 70 Wolpert, op. c i t . . 90 (32). 71 Ibid 91. 83. A further d i s t i n c t i o n i s made between narrow (engen) cross-relations and wide (we,iten) c r o s s - r e l a t i o n s . The former involves a half step while i n the l a t t e r the two notes under consideration l i e at least a diminished octave away from each o t h e r . 7 2 Example 35. 5 J j — 5 3 L Q (e) L Q (w) (narrow) (wide) Wolpert then points to the Neapolitan cadence ( i . e . , cadence using a neapolitan sixth) as a c l a s s i c a l solution of leading-tones i n c r o s s - r e l a t i o n . 73 Example 36. 1 = Fflj)F# Q(w) 72 I b i d . . 91. 73 I b i d . . 92. 84. This solution, says Wolpert, i s of the greatest s i g n i f i -cance for a l l new kinds of chord connections involving 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 various passages i n Act I I I of Tristan as containing sound cross-relations (q) while "beautiful examples of r e a l cross-relations (Q) can be found i n Bartok." 7 5 Wolpert proceeds with an analysis of cross-relations 7 f\ from a portion of Webern's String Quartet, Opus 28. He points out that only c e r t a i n extremely wide (greater than an octave) cross-relations prove to be " h o s t i l e to the hearing" and c i t e s Stravinsky as an example of a composer who prefers " 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 Unfortunately, no examples from Stravinsky are given to j u s t i f y t h i s state-ment. The chapter on "Chordal Connection and Succession" concludes with four statements concerning chord connections involving c r o s s - r e l a t i o n s : 1. When cross-relations 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 fur a l l e nevartigen Akkordverbindungen von der allercjrossten Bedeutung, und zwar fur Querstande ebenso wie fur Klangquersta*nde. 75 , . I b i d . . 92-3 (33). 76 , x I b i d . . 93 (33). 7 7 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 resolution even though i t may not be spelled as such. 3. Because of the penetrating q u a l i t y of half step movement, a l l c ross-relations can be used both i n narrow and wide positions without obscuring the t o n a l i t y . 4. Extremely wide cross-relations (two octaves and more) are to 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 quite c l e a r , the differences between t h e i r ideas on chord connection are not as immediately evident. Hindemith's ideas on the connection of adjacent chords rest on the three basic points previously discussed, namely, harmonic f l u c t u a t i o n , degree progres-sion and the two-voice framework. He tr 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 theories of c l a s s i f i c a -t i o n to h i s theories concerning chord movement. On the other hand, Wolpert's ideas on chord connection seem to be founded on two basic i d e a l s , namely, 78 I b i d . . 95-6 (34) 86 the smallest possible movement i n the upper voices and more or less t r a d i t i o n a l bass movements (as established i n " c l a s s i c a l " and "modern" music) 7^ Unlike Hindemith, Wolpert does not treat i n d i v i d u a l chords as isolated 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 into t h e i r component parts, 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 to apply also to chordal connection. This i s a rather disturbing omission since i t tends to detract from the continuity of the entire system. With these basic differences 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. Interval Resolution 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. Variations i n Chordal Tension and Tonality. Hindemith considers the two voice framework to e x i s t between the bass and the most prominent of the upper voices. He does not examine other possible two-voice frameworks and goes so far as to state that a chord progression i s affected 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 with respect to the "guide tone," but only to 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 resolution i s to be Q 1 s a t i s f a c t o r y . 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 possible two-voice frameworks i n the connection of adjacent chords. In a progression involving three voices, there are three s i g n i f i c a n t two-voice frameworks, four voices provide six two-voice frameworks, and so on. 8 2 i n addition, Wolpert i s o l a t e s what he found to be the most common two-voice frame-works to occur between the bass and an upper voice i n an attempt to suggest desirable cadence procedures with respect to bass movement. Regarding the second point, i . e . , i n t e r v a l resolu-t i o n and half-step movement, Hindemith's concern i s mainly with the t r i t o n e . I t s proper resolution i s most important i f one i s to proceed successfully i n establishing a tonal centre. Generally, he considers the movement of voices by 80 Hindemith, op. c i t . , 115. 81 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 follows the same number sequence as i n Hindemith's consideration 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. half-step to be a desirable q u a l i t y . He states that "£a progression3 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 u s a b l e . " 8 3 Wolpert also sees chromatic voice leading as a desirable occurrance, but h i s enthusiasm takes him beyond Hindemith, to the point where almost the entire chapter on chord connection i s concerned with half-step movement. For example, i n h i s "Laws of I n a c t i v i t y , " Wolpert implies that the smaller the movement i n the upper voices, the better the voice leading. In his discussion of systoles 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 half-step movement i s further evident i n h i s extended discussion on the significance of d i f f e r e n t kinds of cross-relations and i n h i s assigning a penetrating-like q u a l i t y to half-step motion, even when i t occurs between d i f f e r e n t voices. As far as i n t e r v a l resolution i s concerned, Wolpert does not seem to worry. In f a c t , systoles and diastoles 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 diversion or adhesion leading to an unstable i n t e r v a l . 83 I b i d . . 124 8 9 . Both Hindemith and Wolpert, then, recognize the importance of half-step movement i n producing good chord connection. However, while Hindemith tr e a t s the resol u t i o n of the tr 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 special place of importance. Rather, i t s resolution i s just a special case i n the larger discussion of systoles and d i a s t o l e s . Hindemith i s concerned with root movement to the extent that i t i s perhaps the most important c r i t e r i o n for determining the "value" of a given chord progression: both among the adjacent chords and i n the larger framework of the tonal s e t t i n g . On the other hand, Wolpert does not consider the movement of chord roots i n h i s discussion of chord connection. This i s perhaps a reasonable omission since he does i n fact consider a l l possible two voice frameworks. Wolpert does consider bass movement (as d i s t i n c t from root movement) and tabulates what he considers to be the "most frequently occurring" or "desirable" upper voice movements with the few (he considers 5th, 4th and 3rd) bass movements he discusses. 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 equivalent, i . e . , the way a note i s spelled has nothing to do with the way i t influences chord movement. Conversely, as with chord c l a s s i f i c a t i o n , Wolpert i s concerned with proper s p e l l i n g i n discussing chord connection. In the section on adhesion, chord movement i s affected by the way a note i s spelled, i . e . , some spellings allow two resolutions while others allow only one. Wolpert even invents h i s own term (Umpolung) to eff e c t proper voice leading ( i . e . , with correct spelling) i n adhesion. Also, he makes d i s t i n c t i o n s i n h i s discussion of cross-relations with respect to s p e l l i n g which seem unessential to the understanding of the concept as a whole (e.g., "sound" vs. " r e a l " cross r e l a -tions) • Hindemith breaks chords down, so to speak, i n presenting his theories on the two-voice framework, but he also 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 discussion 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 disturbing point i n Wolpert's discussion of chord connection, i s h i s f a i l u r e to relate his theories to h i s previous discus-sions of the c l a s s i f i c a t i o n of chords. In discussing chordal connection, he breaks chords down into t h e i r component parts (for example, a l l possible two voice frameworks— 91. systoles 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 discuss 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 tension and t o n a l i t y . Beyond his e a r l i e r discussion on the consonance and dissonance of i n t e r v a l s and chords, Wolpert gives no further elaborations on these concepts as they might affect 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 contribution to contemporary music theory. The only possible 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 with respect to his "Laws of I n a c t i v i t y . " However, t h i s i s perhaps stretching a point simply to make a comparison. In conclusion, as Wolpert implies at the beginning OA of h i s t r e a t i s e , ^ he does not mean to suggest a kind of "harmonic s t y l e " (as does Hindemith). He evidently sees hi s system as suitable for both tonal and atonal composi-t i o n s , and thus, a detailed consideration of chord organization, i . e . , tonal l o g i c , would be out of place. Hindemith, however, does relate very c l o s e l y h i s ideas about chord connection to h i s theories 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 necessity 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 similar l i n e s i s no longer possible. Because of hi s strong b e l i e f s about the necessity of tonal organization, Hindemith 1s discussions are unwavering. He asserts that t o n a l i t y " i s a natural force, l i k e gravity"-* and f l a t l y states that "there can be no such thing as a t o n a l i t y . " 2 He dismisses " b i t o n a l i t y " and "polytonality" as "catchwords" and i n s i s t s that "every simultaneous combination of sounds must have one root, and only one . . . the ear judges the t o t a l sound and does not ask with what intentions i t was produced." 3 Hindemith's b e l i e f i n the inevitable condition of some sort of tonal organization i n a l l music necessitates the formulation of a p r i n c i p l e or p r i n c i p l e s describing large scale tonal r e l a t i o n s h i p s . The term "tonal 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 to describe "the grouping of chord tones (or root tones) around a tonal centre." 4 However, t h i s tonal centre may and often does vary i n the course of a musical composition. To describe t h i s occurrence, Hindemith redefines the concept of modulation saying that "when we allow one tone to usurp the place of another as tonal centre of a degree progression, we are modulating." 5 Modulation, he continues, can be determined simply from the construction of the degree progression. But, before modulation can take place, a " f i r m l y established tonal 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 clearer the way leading from one tonal centre to the next, the more s a t i s f a c t o r y the modulation. The fact that i t i s not always clear at what point 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 place, another at another. But t h i s i s not a shortcoming; on the contrary, one of the greatest 4 I b i d . . 149. See also Chapter I I I , p. 62 5 I b i d . , 149, 6 I b i d . . 151. 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 uncertainty i n the t r a n s i t i o n a l passages. 7 Perhaps the most far-reaching aspect of Hindemith 1s theories about tonal organization i s the fact that a single most important tonal centre i n the larger framework of a musical composition can be determined. The tonal 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 in c l u s i o n of any of the intervening 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 roots of a l l the chords. Here we see the f u l l unfolding of the organizing power of Series 1. The entire harmonic construction of a piece msy be perceived i n t h i s way: against one tonal centre chosen from among many roots others are juxtaposed which either support i t or compete with i t . Here, too, the tonal centre that reappears most often, or that i s p a r t i c u l a r l y strongly supported by i t s fourth and i t s f i f t h , i s the most important. As a tonal centre of a higher order, i t dominates a whole movement or a whole work. 8 Because of hi s acceptance of both t o n a l i t y and at o n a l i t y , Wolpert has no need to expand h i s system beyond the discussion 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 to other topics 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 discussion of scales and rules for scale formation i s in t e r e s t i n g because of his unique approach. A l l presently recognized modes and scales are discussed along with possible "new scales" using seven, eight, nine, ten and eleven d i f f e r e n t tones. Wolpert concludes that scale structures other than the ones discussed are not possible i n staf f notation, i . e . , with a twelve-note d i v i s i o n of the octave.9 Two concepts are defined by Wolpert which also 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 to be ef f e c t i v e i n the ear, even though these notes are no longer sounding.-^O Wolpert gives examples from Mozart, Franck and Wagner to 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 to understand Wolpert's thought processes, keeping i n mind the fact that he was a self-taught scholar F.A. Wolpert. Neue Harmonik. (Wilhelmshaven: Heinrichshaften, 1972), 162-7. l 0 I b i d . , 184. 11 Erich Valentin, "F.A. Wolpert" Die Musik i n  Geschichte und Gegenwart. XIV (1968), 838. 96. The second concept, i n no way related to 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 to exist.12 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 systoles or diastoles i n the connection. Twenty experiments which follow involving the 13 connection of chords i n root p o s i t i o n , conclude the expanded (1972) version of the text and i l l u s t r a t e the rules 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 ten to sixteen do the same for 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 to come to terms with 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 hi s discussion of chord connection. 13 I b i d . . 201-23. 97 t i o n s of discussing 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 written 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 possible chordal material within the twelve-note d i v i s i o n of the octave. In Hindemith 1s system of c l a s s i f i c a t i o n there i s an attempt to be as objective as possible. His notions of consonance and dissonance, for example, are derived from the overtone series and combination tone curves and are put forward i n h i s Series 1 and 2. Conversely, Wolpert's concern i s subjec-t i v e , i n that he uses h i s ear to determine a scale of dissonance for 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 with the p a r t i c u l a r combina-tions of i n t e r v a l s i n a given chordal structure, Wolpert*s basis for 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 well as the i n t e r v a l combinations produced when they are reduced to t h e i r narrowest possible p o s i t i o n . 14 " D i s t i n c t " here refers to l e t t e r names; for example, G and G# are not d i s t i n c t , whereas G and Ab are. 98. While Wolpert's system of c l a s s i f i c a t i o n greatly emphasizes correct s p e l l i n g to the point where the spe l l i n g of a chord member determines into 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 for purposes of c l a s s i f i c a t i o n . Whereas Hindemith's l o g i c a l l y evolved theory of chord roots 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 to the extent of determining sub-group d i v i s i o n s , chord roots 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 distinguishing feature i n h i s categorization of chords, Wolpert does not, but instead evolves h i s highly 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 also 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 rest on three d i s t i n c t c r i t e r i a — harmonic f l u c t u a t i o n , degree progression and the two-voice framework--Wolpert's theories on chord connection rest on two main ideals--smallest possible upper voice movement and more or les s t r a d i t i o n a l bass movement. Since Wolpert does not extend his system of c l a s s i f i c a t i o n to cover chordal connection, there i s a lack 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 9 9 . i n d i r e c t contrast to Hindemith, whose p r i n c i p l e s of chord movement are a di r e c t extension of h i s system of c l a s s i f i c a t i o n . While Hindemith only considers one "two-voice framework," Wolpert takes a l l possible "two-voice frameworks" into account. Whereas Hindemith i s concerned with 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 special case i n adhesive or whole systole/diastole "resolutions." As with 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 with the way a p a r t i c u l a r note i s spelled, while with Hindemith enharmonics are again equivalent. In describing 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 resolutions while another w i l l allow only one. This concern for sp 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 various types of cross-r e l a t i o n s . 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 1s theories of chord connection are based on the movement of roots or degree progressions and t h e i r deviation and return to a tonal centre, Wolpert does not consider root movement, since to do so might be to contradict h i s o r i g i n a l position on the acceptance of both t o n a l i t y and ato 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 possible chords within the twelve-note d i v i s i o n of the octave while working from somewhat opposite points 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 to b u i l d a l o g i c a l system startin g from nature and progressing through chordal construction to an all-embracing natural system. Wolpert, on the other hand, combines interest i n l o g i c a l patterns with a r e d e f i n i t i o n of more or les s t r a d i t i o n a l advice and rules and apparently shows no interest i n developing an ov e r a l l coherent and cohesive system. His c l a s s i f i c a t i o n of chordal structures i s obviously l o g i c a l rather than p r a c t i c a l whereas h i s discussion 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 to h i s system of c l a s s i f i c a t i o n . Accordingly, his chapter on c l a s s i f i c a t i o n seems to be directed towards the philosopher or l o g i c i a n whereas his section on chordal progression i s more l i k e a c o l l e c t i o n of helpful advice to the aspiring composer. Thus, the two men, using d i f f e r e n t approaches to the same problem, both r e a l i z e a complete system of chordal organization and progression within the twelve-note d i v i s i o n of the octave. Their conservatism i n th e o r e t i c a l 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 , yet, t h e i r contribution i s both necessary and appropriate for our time, and thus i s of value i n our ov e r a l l understanding of twentieth-century music theory. BIBLIOGRAPHY Apel, W i l l i . Harvard Dictionary of Music. 2nd ed. Cambridge Massachusetts: Harvard University Press, 1969. Austin, William W. Music i n the Twentieth Century: From  Debussy through Stravinsky. 1st ed. New York: W. W. Norton, 1966. Babbitt, Milton. "Some Aspects of Twelve Tone Composition" The Score. XII (1955), 53-61. Braunfels, Michael. "Franz Alfons 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 Axis Tonality for Twentieth  Century Music. Unpublished Ph.D. d i s s e r t a t i o n , Yale University, 1969. . " I d e n t i f i c a t i o n and Correl a t i o n of Pi t c h Sets" Journal 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 Corrections i n the Hindemith Chord Tables." Canadian Association of  University Schools of Music Journal. I (Spring 1971), 71-89. Forte, A. "A Theory of Set Complexes for Music." Journal  of Music Theory. V I I I (1964), 136-83. Haas, F r i t h j o f . "F. A. Wolpert 1s 'Neue Harmonik'" Neue  Z e i t s c h r i f t fur Musik. CXIII (March 1952), 167-8. Hensel, Herman Richard. "On Paul Hindemith's Harmonic Fluctuation Theory." Unpublished Ph.D. d i s s e r t a t i o n , University of I l l i n o i s , 1964. 103. Hindemith, Paul. The Craft of Musical Composition. Vol. I. 4th ed. Translated by Arthur Mendel. New York: Schott Music Corporation, 1970. . The Craft of Musical Composition. Vol. I I . Translated by Otto Ortmann. New York: Associated Music Publishers, 1941 . Unterweisung im Tonsatz. I I I . Mainz: B. Schott, 1970. . A Concentrated Course i n T r a d i t i o n a l Harmony. Revised ed. New York: Associated Music Publishers, 1944. . "Methods of Music Theory" Musical Quarterly. XX, (January, 1944), 20-28. • A Composer's World: Horizons and Limitations. Garden C i t y , New York: Doubleday and Company, 1961. Kassler, Michael. "Towards a Theory that i s the Twelve Note Class System," Perspectives of New Music. V. (1967), 1-80. Landau, V i c t o r . "Hindemith, A Case Study i n Theory and Practice," Music Review. XXI, (i960), 38-54. . "Hindemith the System Builder: A C r i t i q u e of His Theory of Harmony." Music Review, XXII. (1961), 136-51. Perle, George. "The Possible 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: University of C a l i f o r n i a Press, 1968. Per s c h e t t i , Vincent. Twentieth Century Harmony. New York: W. W. Norton, 1961. Redlich, Hans Ferdinand. "Paul Hindemith: A Re-assessment" Music Review. XXV, (1964), 241-53. Rochberg, George. The Hexachord and I t s Relation to the Twelve-Tone Row. Bryn Mawr, Pennsylvania: Theodore Presser Company, 1955. 104. Shackford, Charles R. "Unterweisung im Tonsatz I I I by Paul Hindemith" Music Library Association Notes. XXIX, (March 1973), 451-2. Shirlaw, Matthew. The Theory of Harmony. New York: The Da Capo Press, 1969. Slonimsky, Nicolas. Baker's Biographical Dictionary of Musicians. 5th ed. New York: G. Schirmer, 1958. Thomson, William. "Hindemith's Contribution to Music Theory," Journal of Music Theory. IX. (1965), 52-71. Ulehla, Ludmila. Contemporary Harmony: Romanticism through  the Twelve-Tone Row. Toronto: Collier-Macmillan Limited, 1966. Valentin, E r i c h . "Wolpert, F.A." Die Musik i n Geschichte und Gegenwart. XIV, (1968), 838. Wolpert, Franz Alfons. Neue Harmonik. Revised ed. Wilhelmshaven: Heinrichshaften, 1972. . "New Harmony." Unpublished Translation of portions of F. A. Wolpert's text by Louis Medveczky, Univer-s 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 Group B - Chords With Tritone I. Also have no 2nds or 7ths I I . 1. Root and Bass Ide n t i c a l 2. Root above bass I I I . Also have 2nds and/or 7ths IV. 1. Root and Bass I d e n t i c a l 2. Root above bass Also have a. m7 only b. M2 and/or m7 1. Root and bass i d e n t i c a l 2. Root above bass 3. More than one tr i t o n e Also have m2 and/or M7 1. Root and Bass i d e n t i c a l 2. Root above bass V. Indeterminate, i . e . no root VI. Indeterminate, tr i t o n e predominating APPENDIX I I H o w m a n y c h o r d s a re needed to p r o d u c e a t o n a l c e n t e r ? H o w is the T o n i c found? ' A C h o r d s w i thou t T r i t o n e s B C h o r d s c o n t a i n i n g T r i t o n e s 3 C h o r d s A TONIC: c Principal tone of V the group formed by the chord-roots r \ III 3 C h o r d s TONIC: Same as in X II -i. V-2 C h o r d s TONIC: Root of the chord of resolution a Chords TONIC: Root of the chord of resolution a C h o r d s The last of a group of chord-roots is the Dominant of a TONIC lying a fifth lower g C h o r d s TONIC: Same as in 1 V After determination of the root, to be treated the same as I V I After determination of the root, to-be treated the same as II TONIC: Indeterminate TONIC: Indeterminate W7. APPENDIX III Wolpert - Sonata No. 1 (Meas. 1 - 3 ) Wolpert ' s System Hindemith's System • (f) J . nr.a H. m. i r. « « ) IT. fe. i C IT. fe. c2-7. nr. i. 1 (t> jr. L a. r (#») IF. z to. (f) //. n r . x (/; APPENDIX III Hindemith - Sonata No. 1 (Meas. 1 - 4 ) W i t h quiet mot ion, in quarters (J 96) /. Z. 3. ^ 5 * . C 7. 8 - /p. //. n. / j . /f\ Hindemith's System Wolpert 's System /. (f) 2. 7JT . 2 J . X , JL — H- 3L. b. Z tea) r. <. 7. o r . <*. * — /o. zzr . i — //. zzr . x 2ZT . JL IF. 2 /*. — 

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