,A&I A COMPARATIVE STUDY OF THREE SONATAS FOR SOLO BRASS INSTRUMENTS AND PIANO BY PAUL HINDEMITH THESIS. Presented to the Graduate Council of the

,A&I A COMPARATIVE STUDY OF THREE SONATAS FOR SOLO BRASS INSTRUMENTS AND PIANO BY PAUL HINDEMITH THESIS Presented to the Graduate Council of the North Texas State College in Partial Fulfillment of the Requirements For the Degree of MASTER OF MUSIC By Edward Lee Alley, B. Mus. Denton, Texas June, 1957

TABLE OF CONTENTS Page LIST OF TABLES..................... iv LIST OF ILLUSTRATIONS... v... v Chapter I. THE HINDEMITH CONCEPT OF ANALYSIS...... 1 II. THE SONATAS FOR SOLO BRASS INSTRUMENTS AND PIANO................ 7 General Structure Harmonic Tension Tonality III. STUDY OF THE SONATA...... 13 Sonata for Trumpet and Piano by Paul Hindemith (1I9) Sonata for Horn and Piano by Paul HindemitW1939F Sonata for Trombone and Piano by Paul Hindemith (191) Ny IV. SUMMARY YAND CONCLUSIONS........85 General Structure Tonality Harmonic Tension APPENDIX.......,,91.... " BIBLIOGRAPHY....... 92 iii

LIST OF TABLES Table Page I. Harmonic Tension Fluctuation in First Movement of Trumpet Sonata..... II. Harmonic Tension Fluctuation in Second Movement of Trumpet Sonata......... ".. 20 26 III. Harmonic Tension Fluctuation in Third Movement of Trumpet Sonata..... IV. Harmonic Tension Fluctuation in Third Movement of Trumpet Sonata.... V. Harmonic Tension Fluctuation in First Movement of Horn Sonata..... VI. VII. VIII. IX. Harmonic Tension:Fluctuation in Second Movement of Horn Sonata..... Harmonic Tension Fluctuation in Third Movement of Horn Sonata.... Harmonic.Tension Fluctuation in First Movement of Trombone Sonata... Harmonic Tension Fluctuation in Second Movement of Trombone Sonata -... X. Harmonic Tension Fluctuation in Third Movement of Trombone Sonata.. XI. XII. Harmonic TensionFluctuation in Fourth Movement of Trombone Sonata................... *..... 9 0 " 0... 0 9. 0. ". Frequency of Usage of Intervals in Tonal Center Relationships in the Sonatas for Trumpet, Horn,. and Trombone............ 32 34 46 53 65 71 75 80 84 88 iv

".. LIST OF ILLUSTRATIONS Figure Page 1. Series 1................. 2 2. Series 2....... 3 3. 4. Trumpet Sonata, Movement Trumpet Sonata, Movement and 4'-487T... I, mm. )-6 *..*.. I, mm. 30-31, 37-42,.... 0 9....0 13 14 5. Trumpet Sonata, Movement I, Tonal Centers.. 17 6. Trumpet Sonata,, Movement II, mm. 1-4... 9 21 7. Tr upet Sonata, Movement II, mm. 30-31.... " 21 8. Trumpet Sonata, Movement II, Tonal Centers. 9 23 9. Trumpet Sonata, Movement II, mm. 35-36.... 24 10. Trumpet Sonata, Movement II, mm. 44-45.. 25 11. Trumpet Sonata,, Movement III, mm. 1-3... 27 12. 13. Trumpet Sonata, Trumpet Sonata,. Movement Movement III, mm. 17-20.. III, mm. 51-52..." 27 28 14. Trumpet Sonata, Movement III, mm. 68-69.. 9 29 15. Trumpet Sonata, Movement III, Tonal Centers. 31 16. Trumpet Sonata, Movement III, mm. 10-12.. f 32 17. Trumpet Sonata, Movement III, mm. 22-25.. 9 33 18. Trumpet Sonata, Movement III, m. 34.... 34 19. Horn Sonata, Movement I, mm. 1-4...... 36 20. Horn _Sonata, Movement I, mm. 1-2....... 36 21. Horn Sonata, Movement I, 49......... A-mm. 23-27; B-mm. 45-... 9. 0... 0.. 37 V

Figure 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. Horn Sonata,: Movement Horn Sonata, Movement Horn Sonata, Movement Horn Sonata, Movement Horn Sonata,, Movement Horn Sonata,. Movement Horn Sonata, Movement Horn Sonata, Movement Horn Sonata, Movement Horn Sonata, Movement Horn Sonata, Movement Horn Sonata, Movement Horn Sonata, Movement Horn Sonata, Movement Page I,, mm. 60-62....... 38 I, mm. 78-79.. -.... 38 I, mm. 102-103...... 39 I, Tonal Centers..... 42 I, mm. 1-2........ 43 I, mm. 23-24, 45-46... 43 I, mm. 78-79....... 44 II, mm. 1-10....... 47 II, mm. 18-21...... 47 II, mm. 58-66...... 48 II, mm. 90-93...... 48 II, Tonal Centers.... 51 II, mm. 44-45...... 52 II, mm. 105-107..... 53 36. Horn Sonata,, Movement III,. Principal Motivess of Group I............... 37. Horn Sonata,,.Movement III,, Principal Motives of Group II - -............ 38. Horn Sonata,, Movement III, mm. 38-39.... 39. Horn Sonata, Movement III, mm. 50-51.... 40. Horn Sonata, Movement III,, Rhythm of mm. 77-78 41. Horn Sonata,. Movement III, Rhythmic Alteration of Motives of Group II........ 42. Horn Sonata, Movement III, Tonal Centers.. 43. Trombone Sonata, Movement I, mm. 1-2.... 44. Trombone Sonata, Movement I, mm. 19-20.....0.0 54 56 56 57 58 58 62 66 67 vi

Figure 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. I, mm. 40-42.. Trombone Sonata, Movement I, mm. 72-73.. Trombone Sonata, Movement I, Tonal Centers Trombone Sonata, Movement I, m. 2.. Trombone Sonata, Movement II, mm. 19-26. Trombone Sonata, Movement II, Tonal Centers Trombone Sonata, Movement III, mm. 1-6... Trombone Sonata, Movement III, mm. 32-35.. Trombone Sonata, Movement III, Tonal Centers Trombone Sonata, Movement IV, Tonal Centers Page.. 67.. 68.. 69.. 70.. 72..74.. 76..77.. 78. 83 vii

CHAPTER I THE HINDEMITH CONCEPT OF ANALYSIS In 1937, Paul Hindemith published his now well known theoretical work, Uterweisun im Tonsatz, in which he set forth his own ideas and theories for a new approach to musical composition and analysis. This work was translated into English as The Craft of Musical Composition, and published in this country in 1942. This work is "primarily and exclusively a study of musical material... establishing the legitimacy of chords never before welcomed in conventional harmony." Hindemith's system of analysis is based on the twelve tones of the chromatic scale, and each of these twelve tones can become a tonal center. This tonal center serves as an axis around which the other eleven tones group themselves. In establishing the order for the grouping of tones in his Series 1, Hindemith turns to the natural harmonic series and alters the order of the notes by assigning them a new harmonic number above or below the original number. The next step is to establish their corresponding new fundamentals and to continue this procedure until twelve different 1 Bernard Heiden, "Hindemith's System--A New Approach," Modern Music, XIX (January-February, 1942), 103. 1

2 fundamentals are found, which produce the twelve note chromatic scale. The basic steps in the derivation of Series 1 are best summarized by Mosco Carner: Suppose that we use the harmonic series on C... We move the second overtone down and give it the function of a first overtone, giving us the fundamental of a new harmonic series which is merely an octav- higher than our original one. The new fundamental C represents the upper note of the basic octave. We now proceed to the third overtone G and move it first two points lower or give it the function of a first overtone, but since it lies outside the basic compass of our basic octave it cannot be used. The third overtone is consequently moved one point lower assuming the role of a second overtone. Its corresponding fundamental is then a lower octave G, which is ranged in the basic octave. The next three overtones of the series on C are treated in the same way, then the procedure is reversed with the overtones being moved one or more points upwards. 2 The order of the derivation of the twelve tones of Series 1 (Figure 1) is important, because these intervals ;.n reference to C) represent values in tonal relationship. 19 :p 5 be b- ' Fig. 1--Series 1 This Series 1 is not a "tone row" or a series representing a melody, but a formula for governing the relationship of large 21osco Carner, Contemporary Harmony, Vol. II of A Study of Twentieth Century Harmony, 2 vols. (London, 1942), p. 73.

3 tonal masses. The relationship shown is that between the fundamental or progenitor note (in thisecase C) and the other eleven notes. The further to the right in the series one moves, the lower becomes the degree of relationship to the original note (C), until the final note (F#) is reached,, and the relationship is zero. In setting forth the purpose of Series 1, Hindemith states: "The value of the relationships established in that series will be the basis for our understanding of the connection of tones and chords, the ordering of harmonic progressions, and accordingly the tonal progress of compositions." 3 The smallest unit involved in composition or analysis. is the interval,. which may serve in two capacities: as a melodic interval, when the two tones are used successively, and as a harmonic interval, when used simultaneously. As Series 1 showed the tonal relationship of the eleven notes of the chromatic scale to the tonal center, Series 2 (Figure 2) illustrates the decreasing harmonic value of the intervals. Fig. 2--Series 2 3 Paul Hindemith, The Craft of Musical Composition, translated by Arthur Mendel, I (New York,, 1942), 56.

4 Two intervals stand at the outside of Series 2: the octave and the tritone (which is an interval of harmonic uncertainty). 4 The order of these intervals in Series 2 is based on the acoustical phenomenon of combination or resultant tones which influence an interval more or less favorably in respect to its harmonic strength. 5 All intervals have roots, and in the interval the dominating tone is the root (indicated by an arrow in Figure 2). The root of an interval is that tone most reinforced by the combination tones. Hindemith 6 maintains that the tritone has no root because of its tonal uncertainty. The tritone does, however, have a "root representative" which is the tone which proceeds by the smallest step to the root of the resolution interval. 7 Hindemith 8 states that the conventional theory of harmony is far too limited in its system for the determination and interpretation of chords. His system classifies chords according to their quality, worth, or lack of tension. Hindemith defines a chord as a group of at least three different tones sounding simultaneously. 9 Also, the tritone, because of its 'Hindemith, op. cit., p. 82. 5 Heiden, of. cit., p. 106. 6 Hindemith, op. cit., p. 89. 7 lbid. 8 lbid., p. 90. 9 Ibid., p. 95.

5 instability as an interval is the governing principle of this system of chord classification. This is the first division of the chordal material. Chords which do not contain a tritone are classified as Group A, and chords with a tritone are classified into Group B. In Group A, there is also a further division: Group I,1 consisting of chords without seconds and sevenths, and Group III containing chords with seconds and sevenths. Group B is also subdivided into two groups: Group II, containing chords with minor sevenths and major seconds, and Group IV, containing chords with major sevenths and minor seconds. There are two more divisions which do not have a definite root. These are Group V (containing augmented triads and the three-note fourth chord) and Group VI (containing diminished triads and diminished sevenths). In Groups I, II, III, and IV there is one final subdivision which Hindemith makes. If the root and the bass tone in the chord are identical, the chord is classified, for example as IV1. If the root is located above the bass tone in the chord it would be classified as IV 2, and so on. In discussing chords further, Hindemith states that chords, like intervals, have roots,11 In a chord of several tones the strongest interval is responsible: the root of the strongest interval is determined by Series 2. A series of The labelling of the chord groups refers to the chart in the back of The Craft of Musical Composition, and not to scale degrees. 11 Hindemith, _o. cit., p., 96.

6 roots of chords is referred to as a "degree-progression" by Hindemith, and a satisfactory degree progression is the foundation of a given composition. The intervals of the degree progression when analyzed according to Series 1 establish the tonality of the composition. The table of chord classificationsalso serves another purpose than to establish a set of available tone combinations. It also serves as a basis for evaluating any given progression. If a composition were to progress through Groups I, II, III, IV and back again, the resultant curve of harmonic stress or harmonic tension would be one of gradually increasing and decreasing tension. Thus chord successions can now be evaluated and studied according to the ratio between tension and stability by the use of Hindemith's table of chord classifications. 1 2 Under Hindemith's system the traditional concept of "key" is greatly expanded. Any tonal combination is now possible as long as its use is justified by a satisfactory degree progression.. Hindemith justifies this by stating that "the juxtaposition of intervals,, or of chords, which are the extensions of intervals,, gives rise to the key." 1 3 1 2 F. B. Muser, "The Recent Work of Paul Hindemith," Musical quarterly, XXX (January, 1944), 34. 1 3 Hindemith, pj. cit.,.p. 107.

CHAPTER II THE SONATAS FOR SOLO BRASS INSTRUMENTS AND PIANO In the years during the writing of The Craft of Musical Composition, and for the next few years afterwards, Hindemith was engaged in writing a solo sonata for each of the instruments of the orchestra. Muser1 states that this aeries of sonatas continues a definite policy of providing music for people who want to play music, and not merely to listen to it. The three sonatas for solo brass instruments and piano were written during this period. The sonatas,2 written for trumpet, horn, and trombone, were written in the following order: Sonata for Trumpet and Piano--1939 Sonata for Horn and Piano--1939 Sonata? Tohmbone and Piano--1941 These sonatas, being written rather closely together, should have certain stylistic characteristics in common, and there should also be certain features peculiar to each sonata. To study these sonatas and compare them with each other structurally and stylistically is the purpose of this work. F. B. Muser, " The Recent Work of Paul Hindemith," Musical Quarterly, XXX (January, 1944), 34. 2 Published by Schott and Co., Ltd. through Associated Music Publishers, Inc. 7

8 Each sonata will be examined for three specific features: general structure (musical form), harmonic tension, and tonality. Before the studies of these sonatas are presented it would be well to state specific points about each factor to be taken into consideration. General Structure The object of this study is not to compare the form of these works with their traditional analogues, but with each other. In order to do this, however, some point of reference must be taken, and in the study of musical form, reference will be made to the traditional forms. Since it is known that Hindemith is one of the contemporary composers who uses traditional forms, even though they are at times greatly modified, the labelling of the structure of the various movements will be done in the conventional terminology. The forms will be analyzed as found in almost any text on form analysis, and any deviation from the traditional form that is in common with the three sonatas will be noted in the concluding chapter. Harmonic Tension The study of harmonic fluctuation and tension will be based on the method explained in The Craft of Musical Composition, briefly outlined earlier. One explanation should be made here, however. The main purpose of this study of harmonic tension is to derive the "over-all" or "general" tension

9 design of each movement, and not to extract a "chord-by-chord" analysis. For this reason in the discussion of harmonic tension, the various sub-groups- of the chord classifications have been omitted, both in the discussion and in the graphed harmonic tension of the movements. While in some examples of Hindemith' s writing, such an exclusion of the sub-groups might prove fatal to a true study of harmonic tension, in this case, the general classification of the chord is of main importance. Thus, a chord of Group 1112 will be considered simply as a chord of Group III, making the generalization of harmonic tension more simplified in this study. Also in the discussion and study of harmonic tension there arises another problem. When and when not are chords to be extracted for study of harmonic tension? In this study chords will be extracted for tension study only when they form a primarily harmonic accompaniment. Therefore, accompaniments essentially of contrapuntal or linear nature will not be considered. If the contrapuntal lines are complex enough to form chords briefly, then they will be listed as chords, but two-voice counterpoint independent of the solo line will not be considered for purposes of tension. Since this work makes no effort to analyze the various contrapuntal devices employed by Hindemith in composing these three sonatas, all contrapuntal writing will be referred to as counterpoint, with an occasional mention of a fughetta or stretto section.

10 In other sonatas by Hindemith, as well as the three brass sonatas, the use of ostinato rhythms is predominant. At times these ostinato rhythms employ chords or chord sequences which may or may not fluctuate harmonically. The analysis for harmonic tension of such patterns will be limited to the general tension implied by the measure or passage, for in many cases- the resultant chords are secondary to the purpose of advancing the ostinato rhythm and building the rhythmic tension. Some ostinato rhythms are constructed not with. chords, but with sustained passages consisting of octaves, thirds,. fifths, or other intervals. In such cases, when the accompaniment is placed in a linear role with the solo part, the tension rating given the accompaniment will probably be from Group I. The reason for this arbitrary decision is that usually the ostinato rhythm will not contain enough voices to make up a complete chord, so that the resultant tension cannot be given a higher classification. Accompanying each discussion of harmonic tension will be a graph illustrating the general plan of harmonic fluctuation for the movement as a whole.. This graph will be based on the general harmonic tension, not the tension of each individual chord. In the instance of the very slow section (Movement III of the Trumpet Sonata), the graph. will contain the harmonic fluctuation on each beat of the measure.

11 Tonality The determination of the tonal centers and tonal axes of the compositions, will be based on the root progressions and degree progressions: as outlined earlier. In the final summation of the tonality of a movement, the tonality listed will be the primary tonal axis for a certain section, i.e., the tonal center of Group I in the exposition of a sonataallegro movement. By this method, it will be easier to see the relationship of the tonal centers and the "key schemes" of the separate movements, whereas if smaller units of tonality were used, the number of different tonal centers would be large, and difficult to handle. In the discussion of tonal centers of development sections of sonata-allegro movements, only the more important key centers will be noted. Since the development sections in some movements are longer than in others, a comparison by this method will be more comprehensive than if all possible tonal centers were extracted. One final point in the discussion of the analyses for tonal centers: in some passages the solo part will be accompanied by a contrapuntal figure, or the passages will not be complete enough harmonically to derive the tonality from a root progression. In these cases, the tonality will be derived from a melodic analysis based on Series 1 as in the root progression analysis.

12 The following studies and analyses will be more comprehensible if compositions. they are supplemented by the scores of these In the study of the horn, trumpet, and trombone sonatas the measure numbers cited refer to each movement.

CHAPTER III STUDY OF THE SONATAS Sonata for Trumpet and Piano by PaulHindem ith TI99F Movement I General structure.--part A- -Exposition (mm. l-84 ) Gro I (mm. 1-26) Group I contains only one theme, and it is presented five times in this section, three times by the trumpet, and two by the piano. Throughout this section the piano plays an ostinato rhythm accompaniment. In Figure 3 below, (A) shows the theme of Group I as stated by the trumpet in the opening measures, and (B) illustrates the ostinato rhythm of the piano. A.rtPT.) IL_-A-A. - " s _- a B. (PIAMfO) IL -. a I;1 : W= ML Fig. 3--Trumpt Sonata, Movement I, mm. 1-6 The transition (mm. 27-29) consists of staccato chromatic notes in the piano followed by a short fanfare on the trumpet. 13

14 Group II (mm. 30-62) Group II is introduced by a six-measure statement in the piano of a new ostinato rhythm. This new rhythm (shown in Figure 4-A) accompanies the entire statement of Theme A of Group II (mm. 37-45). In Figure 4-B and C below are shown the second ostinato rhythm, and Themes A and B of Group II. B. e4 AEC C. Fig. 4--Trumpet Sonata, Movement I, mm. 30-31, 37-42, and 47-48. Measure 46 serves not only as a transition to Theme B of Group II but also to set up a third ostinato rhythm of triplets. Theme B of Group II contains four statements of the two-measure motive, two each by trumpet and piano. The transition to the closing section of the exposition occurs in measures 63-66. The closing theme (mm. 67-84) is merely a restatement of Group I accompanied by tremolo chords in the piano. Group I is quoted for four measures, then stated again with slight variation by the trumpet. The

15 exposition ends with a final cadence and fermata in measure 84. Part B--Development (mm. 85-126) The first part of the development section is centered around the two-measure motive from Theme B of Group II shown in Figure 4. This two-measure motive is stated by the piano and then the trumpet with a cadence being reached in measure 106. The second section of the development begins with an eight-measure introduction (mm. 107-114) in which the piano again states the ostinato rhythm used to introduce Group II in the exposition (see Figure 4). The development in measures 115-123 is centered around Theme A of Group II. The retransition (mm. 124-126) uses the rhythmic motive in the piano with a repeated three-note sequence in the trumpet to lead to the recapitulation. Part A' -- Recapitulation (mm.. 127-142) The recapitulation involves only the theme from Group I. The theme is stated in the trumpet with the piano providing an accompaniment of triplets. The theme is altered slightly beginning in measure 133. In measure 131 the piano again returns to the second ostinato rhythm which served earlier as an introduction to Group II and the second section of the development. Beginning in measure 138 the trumpet sustains a pedal point while the accompaniment continues the ostinato rhythm, reaching a final cadence in measure 142.

16 Tonality.--In the exposition of Group I, the principal tonality is "Bb" although the theme is stated in several different tonal centers briefly. In measure 9 the piano states the theme in "F," followed by a trumpet statement in "F" in measure 12. The piano states the theme in "B" in measure 16, but the next time the trumpet states the theme (mm. 24-26) it is again in "BbtI The seven-measure introduction (mm. 30-36) to Group II serves only to set up a rhythmic pattern for the exposition of Group II. This is the primary function of the accompaniment during the exposition of both themes of Group II. The basic tonality of Theme A of Group II is "A," and Theme B is also primarily in "A." In measures 54-55 there is a partial statement of Theme B in "Db but this key center is only transient. A statement of Theme B in "A" occurs in the piano part in measures 56 and 57. The closing section of the exposition (mm. 67-84) is in "D." The trumpet briefly states the main theme of Group I against tremolo chords in the piano. The exposition closes with a definite cadence in "D." The first section of the development (mm. 85-106) is introduced by the piano stating Theme B of Group II in "B." During the motival development of Group II in the first section of the development, the motive is stated first in "F" (mm. 92-95) and then in "B" (mm. 100-106).

17 The second section of the development (mm. 107-123) is centered around "EbH with the last few measures (mm. 118-122) implying a center of "D." The recapitulation implies a use of bitonality. The trumpet is definitely in "Bb' from measures 127-142, although the piano accompaniment is in "Gb and cadences in that key in measure 130. The piano resumes the ostinato rhythm of Group II in measure 131 while the trumpet continues with the theme of Group I in "Bb.#t The final cadence of the movement begins in measure 138 with the trumpet holding a dominant pedal, and the final "Bb" cadence is reached in measure 142. Figure 5 shows the tonal centers derived from the root and degree progressions of the first movement. F!ZZI~j~Njw Fig. 5--Trupet Sonata, Movement I, Tonal Centers. Harmonic tension. -- During the greater part of the first movement, the use of the aforementioned ostinato rhythms makes a detailed analysis of the harmonic tension impractical. In this movement, general harmonic tension will be used instead of a chord by chord analysis. The exposition of the first group begins with chords from tension Group III in the ostinato rhythm, except the

18 initial chord in measure 1, which is from tension Group I. The tension shifts to Group I in measure 4, where it remains through measure 18. The accompaniment assumes the role of a linear,. rather than a harmonic one, and the tension remains at zero until measure 24. The final measures of the exposition of Group I (mm. 24-26) are accompanied by the ostinato rhythm using chords of tension Group I. The second ostinato rhythm used to introduce and accompany the exposition of Group II keeps the harmonic tension at zero until measure 42, where the motive is accompanied by chords of tension Group III. In measure 47 the accompaniment is based on recurring triad patterns from tension Group I, and is then shifted to chords from tension Group III in measures 51-53. Chords from tension Group I are used again beginning in measure 54, and continue until measure 62 where the transition to the closing theme of the exposition begins. The transition (mm. 63-66) is based on chords of tension Group III, and the entire closing theme (mm. 67-77) uses chords of the same tension group. In measure 78, the chords are from tension Group I, and remain there until the final cadence of the exposition in measure 84, with the exception of one chord from tension Group III in measure 83. The development section begins in measure 84 using chords of tension Group III, and in measure 89 the accompaniment assumes a contrapuntal rather than harmonic role, with the

19 tension reduced to zero.. A series of thirds in the accompaniment beginning in measure 96 returns the tension to Group I chords, where it remains until measure 99. Beginning in measure 99 the accompaniment becomes linear again, and the tension returns to zero. The second ostinato rhythm is resumed in measure 107, and the tension does not shift from zero. A series of parallel fifths in the bass of the piano part implies chords from tension Group I beginning in measure 115; this same figure also occurs in measure 121, continuing through measure 126. Although this tension is actually only an implied one, it will be shown on the tension graph in Table I on the following page. The recapitulation is marked by an increase in harmonic tension to include chords of tension Group III in measure 127. The harmonic tension drops to zero in measures.128-129, but is resumed in measure 130 with chords from tension Group III. The rhythmic motive begins again in measure 131 dropping the tension once again to zero. Chords implying tension Group I are present in measure 135. The chords accompanying the rhythmic motive increase the tension to Group III in measures 137-139. The final measures of the first movement (mm. 140-142) contain chords from tension Group I.

+. t t -- -. -- -_._ r - s _. - " - t t - --- _.f t 1 i - -r..- +,1 i. i.. y...... tr T. y... }. j... 1... t -.tr f.. i S f f i i -- *---0 r r i t 1-... f _. 4... 1 '.. t i t f -- ' - v.. - - { t }. {. _... t"r r-. _. ~. 20 TABLE I HARMONIC TENSION FLUCTUATION IN FIRST MOVEMENT OF TKJMPET SONATA -- -F - ' t - --- - -i- - 1 -~ -, T 1-. 1 1 1' II 0 pr' 4-, + -- - t *. -1 ''"'I 4 + 4 i I" s, 1 t-t-4 - ~- It 4s- 1 i i i M; i i ; iz!! in 1 1 i d- -t-- -,- 4 - -- - i i _. ;._ -~Li- -" - --- i- -, _.._ 'iv77tv $ K1 T. t"' _..,, _..,. -- -- + t. 'I t y -- 4-4 41 7 - -! *Roman numerals in this and succeeding tables refer to Hindemith' s chord classifications (see ante p. 5, fn. 10). **Arabic numerals in this and succeeding tables refer to measure numbers. Movement II General structure.--part A (mm.. 1-25) The theme of Part A is in the piano with the trumpet playing short fanfarelike passages in measures 3 and 4, then resuming the theme in measure 13. In measure 23 the trumpet briefly states a f ragment of the piano theme leading to the introduction of Part B of the movement. Figure 6 shows the main theme of Part A and also the trumpet fanfare passage.

21 Fig. 6 -- Trumpet Sonata, Movement II, mm. 1-4 PrB (mm. 26-58) Part B consists of two contrasting ideas, the first carried by the piano, and the second introduced by the trumpet. The piano introduces a rather lyrical theme in measures 26-29,. and in measure 30 the trumpet states the short rhythmical motive shown in Figure 7. Fig. 7--Trumpet Sonata, Movement II, mm. 30-31 The rest of Part B (mm. these two basic ideas. 32-58) consists of the repetition. of In measure 48 the piano again states its theme followed by the trumpet motive in measure 52. Section B ends with a two-measure transition to the next part (mm. 57-58). Part A' (mm. 59-76) Part A' consists of a repetition of the thematic material from Part A, but this time the trumpet has the melody with the piano supplying a contrapuntal accompaniment. Beginning in measure 72 and continuing through

22 measure 76 the trumpet repeats the first five measures of the theme. Coda (mm. 77-87) The short coda section makes uses of thematic material from Part B, principally the short rhythmic motive stated in Part B by the trumpet, and shown in Figure 7. During the entire coda the trumpet holds a tonic pedal point above the piano accompaniment, and in measure 85 states the short motive bringing the movement to a close. Tonality.--The tonality of Part A of the second movement is centered around "F," with the piano assuming the total melodic and harmonic interest.. The trumpet plays several fanfare-like passages, but the main emphasis on the key center is in the piano. Part B (mm. 26-58) has ass its tonal center "A," which remains constant throughout the entire section. In measure 32 there begins a rhythmic repetition of a simple figuration, (Figure 7), but the cadence of this sequence in measure 39 is in "A"). The motival figures played by the trumpet throughout this section only tend to reinforce the "A' key center. The last two measures of Part B (mm. 57-58) serve as a modulation or connecting link with the next part of the movement. Since Part A' duplicates melodically Part A, the only difference being the accompaniment, it, too, is centered around "F." In measure 59 the piano is definitely in "F,"

23 and the tonality remains in "F" throughout this section. with. the exception of a few measures which imply a slight shift in tonal center. These are too brief to be considered here. The section closes with a sustained "F" in the trumpet, which also introduces the short coda. Throughout the coda the trumpet holds the sustained "'FT" establishing the tonality for this section. The piano again hints at bitonality through its statement of the rhythmic motive of the trumpet from Part B, but the trumpet pedal point keeps the tonality centered around "1F," where a definite cadence occurs in measures 86 and 87. Figure 8 below shows the principal tonal centers of the second movement as derived from the root and degree progressions, and a melodic analysis of the movement. t J& 6 ~ Fig. 8--Trumpet Sonata, Movement II, Tonal Centers Harmonic tension.--in the second movement of this sonata the most used chords fall into Group III of Hindemith' s chord classifications. In the first section of the movement (Part A: mm. 1-25), the accompaniment uses chords from tension Group I. The piano states the theme with Group I and III chords occurring when the contrapuntal voices overlap and form these

24 chords. The only purely harmonic accompaniment occurs in measures 15-17 when the trumpet states the theme. These chords are from tension Group I. The section ends with chords from tension Group III in measure 25. In Part B (mm. 26-58) the most important harmonic tension fluctuation of the movement occurs. In measure 30 the piano begins a repeated motive using chords from tension Group III. This repeated motive occurs through measure 39 where the trumpet states the short motive again. The trumpet motive is shown in Figure 7 in the discussion on general structure, and the repeated chord progression is shown below in Figure 9 taken from measures- 35-36. PIANO oft"now Ask a -j I " AM a k. A Fig. 9-Trumpet Sonata, Movement II, mm. 35-36 The most striking harmonic fluctuation occurs when chords from tension Groups I and IV are alternated in the accompaniment beginning in measure 40, and continuing through measure 48. Chords from tension Group IV are used on the strong beat of the measure, alternating with chords from tension Groups I and III on the weaker beats of the measure. This fluctuation is shown below in Figure 10 from measures 44-45.

25 FMNI? 7 II Fig. 10--Trumpet Sonata, Movement II, mm. 44-45 Part A' (mm. 58-76) of this movement is based entirely on a linear accompaniment with trumpet playing the theme from Part A. The final section (mm..77-87) of the movement features a harmonic fluctuation between chords of tension Groups I and III, much in the same manner of those in measures 40-48. The tension graph in Table II on the following page illustrates the fluctuation of harmonic tension in this movement. It will be observed that only the general tension for each measure is given, because the important tension changes have already been mentioned and illustrated. In the following graph these changes are briefly notated to show their relation to the general tension of the entire movement. Part A' (mm. 58-76) of the movement is rated on the graph as having a harmonic tension of "0" since the accompaniment is linear and completely independent of the melody line in the trumpet.

26 TABLE II HARMONIC TENSION FLUCTUATION IN SECOND MOVEMENT OF TRUMPET SONATA 1 11 Movement III General structure. -. The third movement of the Sonata for Trumpet and Pia is subtitled Trauermusk, or "Music of Mourning," and is characterized by the extreme slowness of the tempo. This movement is divided into four sections, the last being based on the chorale Alle Men chen misen sterben. The entire movement is ninety-four measures long and is best divided as follows-: Part A (mm. 1-16) The piano opens the movement with a somber statement of its theme. In measure 7 the trumpet sounds a quiet fanfare over the repetition of the theme in the piano. The principal motive of this section is shown on the next page in Figure 11.

" 27 G PIAI Fig. 11--Trumpet Sonata, Movement III, mm. 1-3 The trumpet and piano build in intensity until in measure 11 the trumpet plays a recitative-like passage over tremolo chords in the piano. The section then decreases in intensity with the trumpet resuming the short fanfare motive (m. 14) bringing the section to a close in measure 16 with a transition to the next portion of the movement. Part B (mm. 17-) The tempo marking for this section is Bew SRuh or a little more moved. The piano opens the section stating the theme in the first four measures as shown in the figure below: fo- Fig. 2o--Trumpet Sonata, Movement III, mm. 17-20 A short interlude follows (mm. 23-25) and the trumpet states the theme beginning in measure 26. The climax of this section comes in measure 35 when the trumpet states the theme

28 accompanied by very strong chords in the piano. This statement continues until measure 42 where the trumpet states the second phrase of the theme accompanied by a linear piano accompaniment. This section ends in measure 50 following two measures for trumpet unaccompanied. Part A' (mm.. 51-6?) This section introduces no new thematic material, all of it being derived from Part A of this movement. Beginning in measure 51 the piano plays an ostinato rhythmic figure which continues throughout this section. Figure 13 shows this rhythmic figure, and the trumpet part with Theme A. TKPr(whaTrtAI 1- --- Fig. 13--Trumpet Sonata, Movement III, mm. 51-52 The trumpet plays Theme A from the first section of the movement. In measure 58, the trumpet resumes its fanfare motive from Part A, while the piano continues the rhythmic motive in the left hand and also carries on its accompaniment of the trumpet with chords as in Part A. The climax. is reached in measure 61 with the trumpet and piano playing a figure identical with that in measure 11. There is an extension of this

29 idea, and, in measure 63, the trumpet holds.- a pedal point while the piano resumes the rhythmic motive. The section is brought to a close by the piano in measures 65-67. Chorale (mm. 68-9) The sonata is brought to a close by the trumpet playing the chorale melody Alle Menschen mtssen sterben, while the piano plays an ostinato rhythmic accompaniment. This is shown in Figure 14 below. PI At~ Fig. 14= Tr Sonata, Movement III, mm. 68-69 This ostinato rhythm, once established, is continued throughout the entire section, even during the final measures of the chorale. The chorale melody ends in measure 91, but the trumpet sustains the last note as a tonic pedal point through the piano's final cadence in measure 94. Tonality.--Because of the slowness of the tempo of this work, and the use of chords of rather high harmonic tension (see discussion of harmonic tension for this movement) the tonality is, in places, a little obscure, but for the most part it is rather clear.

30 The introduction and Part A of the third movement are in. "Bb,"f and this tonality is principally reinforced by the recurring trumpet fanfare. The final measures of Part A (mm.. 15-16) lead to a "Bb" cadence in the trumpet part. The second part of the movement begins with the statement of the theme in the piano in "A." In measure 29 the trumpet states the theme in "A" also.. The restatement of the trumpet theme in measure 35 is the climax of a two-measure build up of both harmony and tension. This statement is centered in "A." In measure 43 there is a full cadence on "D," but it serves mainly a sub-dominant function in "A." The final cadence in "A" occurs in measure 45, with the last five measures of the section serving to introduce the next section of the movement. The trumpet enters in measure 51 with the main theme of Part A centered around "F," while the ostinato rhythm accompaniment in the piano is centered in "Bbt In measure 58, both trumpet and piano cadence in "Bb"i with the piano and trumpet resuming their original parts as in the first of the movement (mm. 7-9). In measure 63 the trumpet holds a dominant pedal point on "F" and the section closes in measure 67 with a half cadence in "Bb." The tonality of the last section of the third movement is "Bb" both in the trumpet part (which is the chorale Alle Menschen mtsssen sterben) and the ostinato piano accompaniment.

31 The final cadence in the trumpet occurs in measure 91, and the trumpet sustains a "Bb" while the piano completes the cadence in measure 94 ending the sonata. Figure 15 below illustrates the tonal centers of Movement III as derived from the root and degree progressions. Fig. 15--Trumpet Soonata, Movement III, Tonal Centers Harmonic tension.--from the viewpoint of harmonic fluctuation, the third movement of the trumpet sonata is the most interesting of the three. The first section of this movement (mm. 1-16) uses chords from tension Groups, I, III, and IV. The distribution of these various chords is illustrated graphically in Table III on the following page.

_4---t- r 1 - - -, 1- a - + -t, +--- +-.r._. i, I.,,_..:..,t _.,, _ - _ t t! _; -+ r T "-r - ' _.t.,._._.. r- r f 1 -h :...,T. _ ",_ - } i 32 HARMONIC TABLE III TENSION FLUCTUATION IN THIRD MOVEMENT OF TRUMPET SONATA r i x t- -- z i r _ -t alfaf ;_.. _ : tiiiyk, a 1 4 _ ;L _ 1 By comparing the illustration in Figure 16 below with the score, it can be seen that the point of highest harmonic tension (mm. 10-11) coincides with the melodic climax of this section. BL I-.. r w Aa". /or -AWL Lim. 4b#~* a A bfi 6-Tbnta, Moeen:b Fig. 16-.Trumpet Sonata, Movement III, mm. 10--12

33 The use of a highly dissonant chord of tension Group IV in measures 10 and 11 provides a moving accompaniment for the trumpet recitative passage. Immediately after the melodic climax. is reached,, and the melodic tension decreases, the harmonic tension follows in like manner. The chords from tension Group IV descend the ladder of dissonance to tension Group III in measure 12 and remain there until the end of the section. In the second section of this movement (mm. 17-w50) the first important harmonic fluctuation occurs in measures 22-25. Beginning in measure 22 a recurring melodic motive is used, accompanied by chords of tension Groups I and IV in alternation with tension Group III chords. Figure 17 illustrates measures 22 and 23.. I I 1*16 1 " Fig. 17--Trumpet Sonata,. Movement III,, mm. 22-25 Table IV shows graphically the harmonic fluctuation of measures 20-25.

34 TABLE IV HARMONIC TENSION FLUCTUATION IN THIRD MOVEMENT OF TRUMPET SONATA ' 2 34 ~ & M 2 3 2 4 2. *Underscored Arabic numerals refer to beats in each measure. The next place where an increase in harmonic tension is in evidence is in measure 34, where a series of chords of tension Group III raise the harmonic tension a great extent. This is illustrated in Figure 18 below. This measure preceeds the trumpet statement of the main melodic idea of this section. my J Fig. 18--Trumpet Sonata, Movement III, m. 34

35 Throughout the rest of this section the harmonic tension remains centered around chords of Group III, with the exception of measures 33-35, where the accompaniment assumes a linear role. The third section of this movement (Part A'--mm. 51-67) contains no fluctuation in harmonic tension, the accompaniment being based on an ostinato rhythm shown earlier in Figure 13. In measure 59 the piano again assumes the accompaniment figure of the first section of the movement, and in measures 61-62 the melodic climax is again accompanied by sustained chords of tension Group IV. In measure 63, as the climax is reached, the tension immediately drops to zero with the piano presenting a. unison figure that resolves the chord of tension Group IV of the preceding measure to the tonic of the tonality in measure 64. The entire chorale (mm. 68-94) is accompanied by a rhythmic motive in the piano based on octaves and varying very little from chords of Group I. In measure 76 the accompaniment is based on the parallel movement of triads in the rhythmic motive shown earlier in Figure 14. The final cadence of the movement begins in measure 91 with the trumpet holding a tonic pedal point as the piano completes a sequence of chords in Group I, consummating the final cadence in measure 94.

36 Sonata for Horn and Piano by Paul Hindemith (1939) Movement I General atructure.--the first movement of the Sonata for Horn and Piano is in sonata-allegro form, using three themes. The sections can be divided as follows: Part A--Exposition (mm. 1-71) Group I (mm. 1-22) Group I contains three statements of the main theme, one by the horn, and two by the piano. In the beginning the main theme (see Figure 19) is accompanied by an ostinato rhythm in the piano shown in Figure 20. This ostinato rhythm is used by the piano throughout the exposition of Group I. Fig. 19--HornSonata, Movement I, mm. 1-4 117A L j. b I PIANO I, - -T - I - w ' w Fig. 20--Horn Sonata, Movement I, mm. 1-2

37 Group II (mm.?3-54) Group II contains two motives, the first being stated by the horn in measures 23-30, and the piano in measures 31-39. The first motive of Group II is shown in Figure 21-A and the second motive in Figure 21-B. The second motive of Group II is stated by the horn in measure 45 after six:measures of transitional material (mm. 40-46). The horn sustains a pedal point while the piano states this second motive of Group II in measures 49-53. Five measures of transition (mm. 55-59) lead to the introduction of Group III. Fig. 21--HorSonata, Movement I, Group III. (mm. 60-70) There are three statements, of the main idea of Group III in the exposition. This rhythmical idea is shown below in Figure 22. The horn states the idea in measures 60-63, followed by a fragmentary statement in the piano in measures 64-66, and another horn statement in measures 67-70.

38 Fig. 22--Horn Sonata, Movement I, Part B--Development (mm.. 7 }-139) The development section. of this movement is clearly divided into three sections-, each employing the development of a separate theme or idea. Section 1 (mm. '0-1) This section is based exclusively on the development of a short motive from Group III. This motive (shown in Figure 22) is stated by piano and horn a total of five times in this section. The horn statements occur in measures 74-75, 76-77, and 79-81; the piano statements occur in measures 75-76 and 81-82. Accompanying the horn statements of this motive is a rhythmical figure derived probably from a segment of this motive. This accompaniment figure is shown in Figure 23. Fig. 23--Horn Sonata, Movement I, mm. 78-79 Section-2 (mm. 92-104) Preceding this section of the development is a five measure transition containing no new

39 thematic material or development and serving as a continuation of the rhythmic accompaniment figure shown above. Measure 92 begins a short statement of Group I by the horn, still accompanied by the rhythmic figure of the previous section of the development. This statement is somewhat altered from the original beginning in measure 95, and serves as a transition to the last and longest section of the development. A rhythmical sequence based on sixteenth notes rounds out this transitional sequence in the development in. measures 97-104. The sequence, and the method by which it gradually slows down into first triplets and then eighth notes, is illustrated in Figure 24. P ANO Fig. 24--HorrSonata, Movement I, mm. 102-103 Section.2 (mm. 105-139) This section of the development is centered around the two ideas of Group II. In measures 105-112, the horn states in full the entire first idea of Group II, followed in measures 113-118 by a piano statement of the same motive. Beginning in measure 124 the piano states the second idea of Group II, followed by a horn statement of the same idea in measures 129-132. In measures

40 134-135, and 137-138 the horn states the first six. notes of Group II in stretto with piano statements of the same motive occurring in measures 135-136, and 137-138. This section of the development ends with a fermata in measure 139. Part A'--Recapitulation (mm. 140-162 In this movement, as in others before, Hindemith chooses to recapitulate only the group not used in the development section. In the case of this movement, Group I was not used in the development except for a. partial statement in the middle section of the development (mm. 92-98); therefore this entire recapitulation, or coda, is centered around a treatment of Group I. This seems more in the spirit of a recapitulation rather than a coda because the rhythmic motive of the exposition appears. as well as a statement of Group I in the original key of the exposition.. Both: ideas of Group I are stated, with the final statement of Group I occurring in measures 153-158. Beginning in measure 158 the movement gravitates toward the final cadence which occurs in measures 161-162. Tonality.--The principal tonal center of Group I of the exposition (mm. 1-22) is "F." In measures 1-4 the horn states the theme in "F," followed by a statement of the theme in "C" by the piano in measures 4-8. Measures 8-11, the second motive of Group A, are also in "F." The piano states the theme in "F" from measures 16-22. Group II of the exposition (mm. 23-54) is centered around the key of "A" with both statements of the first motive of this

41 theme (mm. 23-29, mm. 31-38) being in "A." The second motive of this group (mm. 39-54) is also in "A," with the cadence being reinforced by a tonic pedal point in the horn in measures 49-54. Group III of the exposition (mm. 55-70) is centered in The enharmonic notation in the piano part suggests; a center of "Gb, but enharmonic notation in this case is principally for convenience of performance. The main motive of Group III is stated in "F#" by the horn in measures 53-63 and by the piano in "F#" in measures 64-66. The motive appears in "Bb" in measures 67-70 and there is. no transition to the development. The first section of the development (mm. 71-86) contains statements of the motive of Group III in "B" (mm. 74-77) and "E" (mm. 78-86). The second section of the development (mm. 87-104) contains a statement of the theme of Group I in "F" and the rest of the section is devoted to a rhythmic sequence preparing for the third section of the development. The last section of the development (mm.. 105-139) is based on the theme from Group II and contains several statements of that group in different tonal centers. The first, mm. 105-112, is in "E," and in measures 113-123 it is stated in the piano in "A." A short development of the second motive of Group II in measures 124-132 is in "Ab," followed by five

42 stretto statements of the first motive of Group II in measures 133-139 in "Abti The recapitulation (mm. 140-162) is in "F" with the ostinato rhythm is the piano reinforcing this tonality. The final cadence in measures 160-162 ends with four "F" major triads. The tonal centers for this movement as derived from the root and degree progressions are shown in Figure 25 below. Vf. Fig. 25--Horn Sonata, Movement I, Tonal Centers Harmonic tension.--as is: the case with the other first movements of the three brass sonatas, this first movement is divorced from a. truly harmonic accompaniment because of the use of several recurring ostinato rhythms. As explained previously,, these ostinato rhythms and the measures in which they occur are not being examined for harmonic tension on the basis of individual chords. The basic pattern of tension for the entire section accompanied by the rhythm, and the implied tension of the ostinato rhythm give a general tension for the entire section. The accompaniment of the first movement of this sonata is based primarily on three major and two minor ostinato rhythms. The first of these rhythms, used in measures 1-22, implies