A COMPLETE ANALYSIS THESIS

5 9 BELA BARTOK T S FOUR DIRGES FOR PIANO, OP. 9a: A COMPLETE ANALYSIS THESIS Presented to the Graduate Council of the North Texas State University in Partial Fulfillment of the Requirements For the Degree of MASTER OF MUSIC By John W. Terrell, Jr., B.A. Denton, Texas August, 980

Terrell, John W., Jr., Bela Bartok' s Four Dirges for Piano, Op. 9a: A Complete Analysis. Master of Music (Theory), August, 980, 08 pp., 8 tables, 64 illustrations, bibliography, 0 titles. The study of Bela Bartok's Four Dirges for piano (909-90) is significant in that this period of 908 to 90 was particularly experimental and formative for Bart6k, especially in tonal aspects of his compositional style. Furthermore, very little research and analysis has been done on these smaller works. This thesis contributes an analytical study of this early style and also shows its influence on larger mature works in subsequent years. A complete analysis on each dirge contains graphs of tonal structures and patterns constructed by Bartok within each composition. The concluding chapter summarizes overall characteristics of the dirges.

TABLE OF CONTENTS Page LIST OF TABLES............. V LIST OF ILLUSTRATIONS................ vi Chapter I. BIOGRAPHICAL SKETCH OF BELA BARTOK..... II. ANALYTICAL METHOD.............. 6 Tonal Analysis Harmonic Analysis Melodic Analysis Phrase Analysis, Rhythm, and Form III. ANALYSIS: DIRGE I.............. 5 Tonality Harmony Melody Rhythm and Phrase Structure Form Texture Dynamics Range IV. ANALYSIS: DIRGE II............ Tonality Harmony Melody Rhythm and Phrase Structure Form Texture, Dynamics, and Range V. ANALYSIS: DIRGE III............. 4 Tonality Harmony Melody Rhythm Phrase Analysis Form Texture Dynamics and Range iii

Page VI. ANALYSIS: DIRGE IV......... 49 Tonality Harmony Melody Rhythm Phrase Analysis and Form Dynamics and Range VII. THE FOUR DIRGES: UNIFYING ELEMENTS AND COMMON CHARACTERISTICS.......... 6 Unifying Elements Common Characteristics APPENDICES... "... - -- -74 BIBLIOGRAPHY...-.-.-.-.-.-.-.-.-07 iv

LIST OF TABLES.... Table I. The Formal Employment of the Arithmetic Center and the Golden Section in Each of the Four Dirges..".! II. Summary of Analysis: The Four Dirges I II. Dirge I: Tonal Analysis...... IV. Dirge I: Harmonic Analysis... V. Dirge I: Phrase Analysis VI. Interval Count of Dirge I... ii. Dirge II: Tonal Analysis... V III. Dirge II: Harmonic Analysis..... vi IX. Dirge II: Phrase Structure.. X. Interval Count of Dirge II..... XI. Dirge III: Tonal Analysis..... KII. Dirge III: Harmonic Analysis.. XIII. Dirge III: Phrase Structure..... XIV. Interval Count of Dirge III... XV. Dirge IV: Tonal Analysis XVI. Dirge IV: Harmonic Analysis.... XlVII. Dirge IV: Phrase Structure.. XV, III. Interval Count of Dirge IV..... Page.... 70.... 7.... 74.... 76 «... 79.... 80.... 8.... 8.... 87.... 88.... 89.. *. 9.... 95.... 96. * *. 97.... 99. *. * 04.*. -06* V

LIST OF ILLUSTRATIONS Figure... 4. 5. 6. 7. 8. 9. 0...... -.... Relationships of Perfect Fifths The Axes.........-...-... Gradation of Harmonic Tension The Golden Section......... Main Motive, mm. -........ Mm. -5........ e! - Motivic Development in mm. 0-7. Inversion of the Motive. Motive in Retrograde, mm. 0- Isometric Sonority......... Theme of Dirge II..". Third Phrase......... --. Page 7 7 9...... - - 7 7.... 8.... 8.... 9.... 4 - - -. 6 - - -. 6. Fifth Phrase.......... -.-. - - -. 8 4. Phrase Analysis of Dirge II...... 5. Bass Clef Tonal Phrase Endings in the Third Dirge...... -.. - -. 4 6. 7. 8. 9. 0.. Treble Clef, Motive I, mm. -.. Treble Clef, Motive II, m. 4..... Treble Clef, Motive III, mm. 9-0 Bass Clef, mm. -...... Treble Clef, mm. 9-0... Treble Clef, mm. -6............ 8.... 8.... 8 *... 8.... 9... - 40 vi

Figure. Page Treble Clef, mm. 9-4....... 4. Treble Clef, mm. 5-6....... 4. Treble Clef, mm. 7-9....... 5. The Augmented Second........ 6. Treble Clef, mm. 4-5........ 7. Treble Clef, mm. -...... 8. Bass Clef, mm. -4........ - 9. Bass Clef, mm. -5........ 0. Bass Clef, mm. 6-8......... Overlapping Tonal Functions in Dirge. Harmonic Pattern of Consonance and Dissonance in Dirge IV....... Motive Ia-b, mm. -5........ 4. Mm. 0-............ 5. Mm. 4-8.............. 6. Mm. -9.............. 7. Mm. 0-.............. 8. Mm. 4-8.............. 9. Mm. 44-50.............. 40. M. 5................ 4. M. 5............... 4. M. 5................ 4. Mm. 54-57.............. 44. M. 58................ 45. Mm. 6-65.............. IV 4 4 4 4 4 4 44 44 50. 5 5. 5. 54. 54 55 55 56 s 57 57 57-58. 58-59 vii

Figure 46. Mm. 66-67.. Page 59 47. Mm. 68-70... 48. The Axis System of the Four Dirges. " " s - 60..6 49. Tonal Scheme of the Four Dirges in Regard to the Axis System..... 64 50. Intervallic Similarities in the Moti of the Four Dirges...... 5. Motivic Relationship Between Dirges and III... ".... " - 5. Motivic Relationship Between Dirges and IV......... ves I 65... 65 II. - - - - 66 5. Relationships of Meter and Tempo 54. Tonal Symmetry in Each Dirge.. 55. The Hyperbolip Form in Each of the Four Dirges...... 56. Characteristic Descending Intervals 57. Dirge I: Graph of Tonal Analysis..r... 67..e... 67 S - - 69.... 70..r.".. 75 58. Dirge I: Graph of Harmonic Analysis... 78 59. Dirge II: Graph of Tonal Analysis. w. 8 60. Dirge II: Graph of Harmonic Analysis... 86 6. Dirge III: Graph of Tonal Analysis... 90 6. Dirge III: Graph of Harmonic Analysis.. 94 6. Dirge IV: Graph of Tonal Analysis... 98 64. Dirge IV: Graph of Harmonic Analysis..5. 0' viii

CHAPTER I BIOGRAPHICAL SKETCH OF BELA BARTOK Bela Bartok was born on March 5, 88, in Nagyszentmiklos, Hungary, where his early years were beset by illness which interrupted his studies and social activities (, pp. -4). This established his preference for isolation with nature which continued throughout his life (, p. 6). Having shown an interest in music, he began receiving piano lessons from his mother in 886 (, p. 5). In 89, at the age of eleven, Bartok gave his first public recital which included his own composition, "The Flow of the Danube" (5, p. 0). Two academic influences affected Bartok in his youth. His piano instructors in Pozsony (894-898), Laszlo Erkel and Anton Hyrtl, directed him to the music of Brahms and Dohnanyi (, p. 9). Later he attended the Budapest Academy of Music where he studied piano with Istvan Thoman and composition with Hans Koessler, and was permanently influenced by the music of Liszt and Wagner (, p. ). Following an inactive period at the Academy, Bartok was inspired to resume composition in 90 when he encountered the music of Richard Strauss, particularly Also Sprach Zarathustra (5, pp. 7-8). During the next three years,

Bartok's concern for the decreasing political power of Hungary and her increasing economic and political dependence on Austria directed his compositional style toward nationalistic traits (5, pp. 4-4). Although touring as a virtuoso pianist consumed most of his time, the influence of Strauss and the spark of patriotism fostered three of his first formative works: "Kossuth" Symphony, Rhapsody for Piano, and a Suite for Orchestra (5, pp. 50-59). When Bart6k left the Budapest Academy of Music, he was faced with two challenges: to compose music which was Hungarian in character, and to compose music in a new style which would free him from nineteenth-century German romanticism (5, pp. 57-58). His patriotism for Hungary led to the study of her folk music, and upon discovering that the Hungarian "folk art songs" of Liszt and other Western composers were not authentic, he was immediately stimulated to search for the genuine folk songs of Hungary (5, p. 59). I have a new plan now, to collect the finest examples of Hungarian folk songs, and to raise them to the levels of works of art with the best possible piano accompaniment. Such a collection would serve the purpose of acquainting the outside world with Hungarian folk music (5, p. 60). Bartok ventured on his first folk song collecting tour in Bekes county in 906 (5, p. 65). Bartok was encouraged in this project by his friend Zoltan Koda.ly, whose collaboration with Bartok resulted in systematic methods of recording, analyzing, and categorizing

the peasant folk tunes (5, p. 64). Kodaly also acquainted Bartok with the music of Claude Debussy which frequently employed scales similar to those in the Hungarian folk tunes (5, p. 7). When, in that very same year, Kodaly urged me to study Debussy's works, I was very much surprised to find that pentatonic turns identical with those found in Hungarian folk music played a prominent part in his melodic construction (5, p. 7). In 905-907, Bartok fell into a period of intense contemplation and isolation in which only the outdoors could bring him peace of mind. His admiration for the shaping forces in nature was strengthened in these years, increasing his understanding of folk music (5, pp. 74-75). In a narrower sense, peasant music is the result of the reshaping work of a natural force operating unconsciously; it is the instinctive creation of a human mass without artificiality. It is a natural phenomenon, just like the various forms of the animal or vegetable kingdom. As a result, its individual organisms--the melodies themselves--are examples of the highest artistic perfection (5, p. 76). Bartok determined in these years that folk music would play an essential part in his compositions (5, p. 79). Allowing himself this time to assimilate his ideas on methods of composing with incorporated folk tunes, Bartok was now prepared to compose in a style which would persist throughout his life (5, p. 79). The years 907-9 represent the core of Bartok's compositional development, a time of experimentation and

4 synthesis (5, p. 85). Three aims guided his musical development in these years: () to draw out the essential principles of Western classical music, () to develop new ideas based on folk music, and () to synthesize these two elements (5, p. 85). Bartok's piano compositions of this period have been described as the "laboratory in which he worked out his ideas (5, p. 85). Ern5 Lendvai refers to a duality of musical expression in Bartok's art in a diatonic (acoustically-based) system and a chromatic (golden section) system (, pp. 88-89). These two elements originate respectively from the influence of Western music on Bartok and from his increasing knowledge of the Eastern expression in the Hungarian folk songs (5, pp. 89-9). His compositions from these formative years begin to draw elements from these two musical expressions, as in the Bagatelles, Ten Easy Pieces, and Sketches (5, pp. 80-8). Bartk referred to the Bagatelles as the earliest works in his personal style, using devices that later became "an integral part of his technique" (4, pp. 4-4). The Four Dirges from 909-90 are considered by the biographer, Jozsef Ujfalussy, a special achievement of Bartok in synthesizing both elements of his dual expression (5, pp. 9-94). This accomplishment of unifying these elements within each of these early pieces casts an influence on larger mature works in subsequent years.

CHAPTER BIBLIOGRAPHY. Crow, Todd, editor, Bartok Studies, Detroit, Michigan, Detroit Reprints in Music, 976.. Lendvai, Erno, Bela Bartok: An Analysis of His Music, London, Kahn and Averill, 97.. Stevens, Halsey, The Life and Music of Bela Bart6k, nd ed., New York, Oxford University Press, 964. 4. Suchoff, Benjamin, editor, Bela Bart6k Essays, London, Faber and Faber, 976. 5. Ujfalussy, Jozsef, Bela Bart6k, Budapest, Hungary, Crescendo Publishing Company, 97. 5

CHAPTER II ANALYTICAL METHOD Tonal Analysis The Western and Eastern influences in Bart6k's youth are analogous to the two major categories in twentiethcentury tonal composition--tertian and nontertian (4, p. ). Both styles exist in his music. In Bart6k's tertian harmony the traditional concept of "root" may apply whereby the lowest note of any three-note chord projected in thirds is the tonal center of that sonority. This tertian style generally permeates each dirge with frequent occurrences of non-harmonic tones. When non-tertian harmony occurs, the tonal emphasis can be established by () sonority doubling, () pitch duration, including pedal tones, () metrical accents of pitches, (4) lowest notes in a vertical sonority, (5) dynamic level of pitches, and (6) melodic peaks (high and low). Any one or a combination of these may be used to support tonal centers even when tertian harmony occurs. Occasional conflicting tonal centers are produced within each of these two styles, resulting in music which borders on bitonality. In this analysis, tonal shifting is charted onto graphs to reflect tonal organization which is then compared to formal 6

7 plans in other analytical areas. These charts are placed in the four Appendices at the end of this paper, each one correlating to the dirge of that same number. The axis system is a type of analysis determined by Ern Lendvai (, pp. -6) which describes a series of tones in the circle of perfect fifths, as a sequence of subdominant, tonic, and dominant relationships. The tones which lie a perfect fifth above and below a given tonic assume dominant and subdominant relationships respectively to that tonic (see Figure ). 5t C 5t F Tonic G Subdominant Dominant Fig. --Relationships of Perfect Fifths The sixth scale degree is the relative of the original tonic and also assumes a tonic role. Similarly, the second scale degree is the subdominant relative, and the third scale degree is the dominant relative. The procedure when completed around the circle of fifths results in the plan shown in Figure below. FC ET T A ANO TNs s E Fig. --The Axes

8 There are four tones representing each function, each group lying on two imaginary perpendicular poles. The opposite end of a pole is called a "counterpole" and has a strong relationship to its opposite since it shares the same tonal function and equally divides the octave by lying at a distance of a tritone. The axis system is frequently observed in Bartok's music in many different ways: as melodic, harmonic, and tonal organization within a piece, or even as a means for tonal unity between movements of a piece. This indicates a possible explanation for the many tritone relationships between tonal centers in Bart6k's music. Although the axis system is effective in the analysis of Bart6k's later works, it may not be fully applicable to the dirges since they date from his early formative years. Harmonic Analysis Vertical sonorities may be measured and contrasted by the degree of consonance or dissonance each possesses. The chapter "The Analysis of Intervals" in Howard Hanson's Harmonic Materials of Modern Music (, pp. 7-6) is the method of chord description used in this analysis. A sonority will be considered consonant only when it contains any combination of the following intervals: (p) perfect fifth and perfect fourth, (m) major third and minor sixth, (n) minor third and major sixth.

9 Dissonance is associated with intervals which create harmonic tension in a sonority: (s) major second and minor seventh, (d) minor second and major seventh, (t) augmented fourth and diminished fifth. The (and octave) represents complete repose, opposing the tritone which represents extreme dissonance or complete unrest, needing resolution. This system of analysis can also describe harmonic sonorities as projections of one particular interval, or as involutions of other sonorities, or as isometrically organized sonorities (, pp. 7-4). In this analysis, each separate sonority is recorded on a table and presented on a graph which measures the amount of dissonance in the order shown in Figure. [Note: This is the author's own application of the "Analysis of Intervals" of Howard Hanson Dissonance T... Sonorities with three dissonant types--t, d, s Dissonance T... Sonorities with two dissonant types--t and d or s Dissonance T... Sonorities with only one dissonant type--t Dissonance D... Sonorities with two dissonant types--d, s Dissonance Di... Sonorities with only one dissonant type--d Dissonance S... Sonorities with only one dissonant type--s Consonance.... Sonorities with only p, m, or n Unison...... Single pitch (or octaves) Fig. --Gradation of Harmonic Tension which is used in this analysis to measure the increase of dissonance of the vertical sonorities as they occur in the

0 horizontal time element. This method of application is not an aspect of Hanson's analytical system.] Contrasts of consonant-dissonant levels are made immediately apparent on these graphs within the horizontal aspects of each dirge. The tables which measure the harmonic consonant-dissonant levels and the graphs which interpret them are both placed in the Appendices for each dirge. Melodic Analysis A horizontal interval count is tabulated twice: first, all voices are counted (including sonority doublings); second, the intervals in the melodic line alone are counted (excluding all sonority doublings). All intervals separated by rests are counted to avoid arbitrary decisions when an interval should or should not be counted. The interval count shows which intervals occur most frequently and which ones are characteristic only of the individual works. The table of intervals for each dirge is located in the Appendices. Three horizontal dimensions are considered. First, implied tonalities are examined which occasionally vary from those of the harmonic accompaniment, producing moments which border on bitonality. Second, motivic treatment is analyzed with regard to characteristic intervals, motivic cells, and melodic shapes. Third, contrapuntal characteristics are considered in two particular places: () the

imitative transition in measures -8 in Dirge IV, and () the relationship between the melodic line and the harmonic accompaniment throughout Dirge III. Both of these instances most strongly affect the harmonic contrast of consonance and dissonance and therefore are only mentioned in regard to harmony. Rhythmic organization will be regarded with the analysis of phrase structure. Phrase Analysis, Rhythm, and Form Phrases are delineated by the grouping of notes which state a complete musical theme or idea. Motives and motivic cells are considered phrases when they are stated separately from the thematic material from which they are derived by intervening rests or by separation of phrase marks. A culminating phrase or phrase group is located at or near the center of each dirge. Rhythmic organization is analyzed with regard to this central core to illuminate structural organization by the number of phrases which lie on either side of the culminating phrase, by their phrase lengths, and by their rhythmic patterns. Form will be regarded as a synthesis of all the analytical elements, including dynamics, texture, and range. A table which summarizes the phrase and form analysis of each dirge is located in the Appendices.

Golden Section Erno Lendvai defines the golden section as "the division of a distance in such a way that the proportion of the whole length to the larger part corresponds geometrically to the proportion of the larger to the smaller part, i.e., the larger part is the geometric mean of the whole length and the smaller part," as shown in Figure 4 (, p. 7). = X( Fig. 4--The Golden Section (, p. 7) The golden section appears to have been used by Bartok in regard to form and harmonic construction in his mature works. It can be located by multiplying the total number of measures or beats by 0.68. The smaller part of the proportion is located by multiplying the total number of measures or beats by 0.8. The Fibonacci Series is a sequence of numbers which approximately, but not exactly, express the golden mean proportions in numerical ratios. Each number is produced by the sum of the two numbers which precede it; thus,,, 5, 8,,, 4, and so on. This sequence can be used in analysis of Bartok's music by using any unit of measurement which gives more information about the piece. In some

cases, the unit may be counted by full measures, but in others it may be counted by eighth-note values. Another proportion sometimes used by Bartok is produced by dividing the total number of measures or beats by two. This arithmetic center is the moment when half the composition is passed, while the golden section is when 6.8 percent of the composition is passed. Bartok uses both proportions.

CHAPTER BIBLIOGRAPHY. Hanson, Howard, Harmonic Materials of Modern Music, New York, Appleton-Century-Crofts, Inc., 960.. Lendvai, Ern, Bela Bartok: An Analysis of His Music, London, Kahn and Averill, 97.. Ujjfalussy, Jozsef, Bela Bart6k, Budapest, Hungary, Crescendo Publishing Company, 97. 4. Wittlich, Gary E., coordinating editor, Aspects of Twentieth Century Music, Englewood Cliffs, New Jersey, Prentice-Hall, Inc., 975. 4

CHAPTER III ANALYSIS: DIRGE I Tonality A quasi-symmetrical structure is established in Dirge I by recurring tonalities on each side of, and in approximate equidistance to, the midpoint of the piece. Although the minor third (d-sharp to f-sharp) in measure suggests D-sharp minor, measure implies B major with an a-sharp rising to b in the melody, resulting in a B major triad. B major returns in the bass clef in the last seven measures of the piece while references to d-sharp and f-sharp are made in the treble clef melodic line. The vertex of the tonal structure (that tonality around which the other tonal centers are symmetrically structured) occurs in measures -4 on A-flat, the only reference to this tonal center in the dirge. The tonality of A occurs before and after measures -4, once in measures 8 and 9, returning firmly established in measures 5-6. Lying a major second below the central tonality of B, A lies on the dominant axis and serves a "dominant" function to B. This establishes a structural tonic-dominant-tonic relationship in the dirge. Other recurring tonal centers on either side of the A-flat vertex lie on B-flat in measure and measure 9, C (also 5

6 lying on the dominant axis) in measures 0- and measure 7, D-flat (C-sharp) in measures 5-7 and measure 0, and G in measure 4 and measure. All of these tonal recurrences are shown on the graph of tonal centers for Dirge I in Appendix I and for the other dirges in their correlating appendices. Harmony The harmonic structure of Dirge I is produced by brief moments of dissonance which occasionally interrupt the consonant character of the piece. The particularly dissonant moments occur in measures, 4, and 4, each a result of the incongruity between the melodic line and harmonic accompaniment. The dissonance in measure 4 not only lies in the arithmetic center of the piece, but is also where the vertex of the tonal symmetry is established on A-flat. Unisons occur in measures 4,,, 7, 9, and, each marking the end of a phrase. The in measure 7 is significant since it not only establishes the dominant function of C at the end of the climactic phrase in the dirge, but because it also occurs at the golden section of the total twenty-eight measures. Melody The interval count reveals the stepwise character of Dirge I. Of all the 5 intervals, 78 ( percent) are

7 minor seconds. There are 8 (5 percent) minor thirds, 7 ( percent) major seconds, and 5 (0 percent) major thirds. Of the 64 intervals in the melodic line, the minor second ( percent), major second (4 percent), and minor third (7 percent) are the three most frequently occurring intervals. Next to these is the augmented fourth which occurs seven times ( percent). Refer to Appendix I for a complete listing of horizontal intervals in Dirge I and the subsequent appendices for the correlating dirges. The melodic line lies above a chordal accompaniment. The opening mot ivic figure in measures - (a# T b Jg) has an arch contour (see Figure 5). This motive is expanded in Fig. 5--Main Motive, mm. - measures -4, containing two more notes than before and reaching a higher peak to d before returning to g (see Figure 6). 4 5 T Fig. 6--Mm. -5

8 This entire pattern is sequenced a major second higher in measures 6-8, each phrase ending on al. Measure 0 begins a developmental treatment of the motive which is marked by sonority doubling and "mezzo forte sempre crescendo," continuing to heighten the peak with each subsequent statement: measure 0--f#, measure --a, measure --cb, measure --fb, measure 5--g#. Each repetition of the motive returns to c, forming ever widening intervals. The narrow descent of the augmented fourth in measure 0 is gradually expanded to a descending augmented twelfth in measures 5-7 (see Figure 7). Measures 8-9 state the original motive in inversion with a two-note extension, beginning on c (see Figure 8). Measures 0- state the motive in a modified J ' t4 5I Fig. 7--Motivic Development in mm. 0-7 Fig. 8--Inversion of the Motive retrogression with an extension of one note. A pure retrograde of the motive would consist of a rising major third

9 followed by a descending major second. In measures 0-, however, the major third is widened into a perfect fourth, b up to e. The remaining seven measures of the piece continue to state the retrograde version of the motive which produces an ascending direction rather than the descending direction of the first seventeen measures. The motive in retrograde without the extension occurs in measures and. The statement in measure expands the original interval of a major third even further to a major sixth, b up to g#, and the minor second is widened to a minor third, g# to e# (see Figure 9). Fig. 9-Motive in Retrograde, mm. 0- In measures 4-8, the harmonic interval of D-sharp to F-sharp from measure is transformed into a melodic interval which ascends the minor third. Although the melodic contour ascends in these measures instead of descending, the rhythmic similarity to the opening motive (especially in measures 6-7) allows its reference to be recognized. All the melodic material in Dirge I is clearly derived from the opening three-note motive in the second and third measure.

0 Rhythm and Phrase Structure There are seventeen phrases of which all but one conform to the similar rhythmic pattern of a small group of quarter notes (from two to five) followed by a single half or whole note. The exceptional ninth phrase is an augmentation of this rhythmic pattern, consisting of half notes followed by a final whole note. The ninth phrase is followed by eight phrases which are nearly rhythmically symmetrical to the eight phrases preceding it. This central phrase in measures 5-7 occurs at the dominantly functioning tonal center of A. Rhythmical similarities exist between phrases eight and ten, seven and eleven, and four and fourteen. The meters and / alternate frequently until measure 5 where continues throughout the rest of the piece. This supports the significance of measures 4-6 as a tonal vertex and as a place of melodic transformation. Form Of the twenty-eight measures in Dirge I, the first fourteen state and develop the opening three-note motive of measures -, culminating in the core ninth phrase of measures 5-7. In measures 8-8, a reversal of the melodic treatment in the first half occurs in two ways: () the phrases are gradually shortened (instead of being lengthened), and () the motive begins to ascend instead of descending due to its inverted statement in measure 8 and

its retrograde version with extension in measures 0-8. Although these changes tend to group the work into these three parts, measures -4, 5-7, 8-8, they comprise, nevertheless, one continuous undivided composition. Texture The texture in Dirge I gradually thickens, beginning with a single melodic line accompanied by the interval of a minor third. In measure 5, both notes of the minor third are octave reinforced in the lower register. In measure 0, a third pitch and its octave enter, completing a full triad. Here, the melodic line also becomes octave reinforced. In measure 5, the melodic line is stated in three octaves above a seven-voice accompaniment. The texture thickens further in measure 5 when close position octave reinforced triads in the lowest register. Dynamics The dirge begins and ends pianissimo, rising to fortissimo in measure 5 (the core of the composition). Although the dynamic level descends after measure 5, it rises to forte in measure 5 to support the return to B major. A return to pianissimo closes the dirge. Range The range widely extends from BBB to C# 4. The first fifteen measures develop a continual rise of melodic peaks,

culminating in the core ninth phrase in measure 5 on g# above the dominantly functioning tonal center of A. This pitch returns in the last two measures above the tonal center of B. The only pitch occurring higher than g# is c# 4 from measure 4, propelling the second, less climactic, rise in dynamics which reaches its peak in measure 5.

CHAPTER IV ANALYSIS: DIRGE II Tonality The primary influence on tonality in Dirge II is the melodic line which lies in C-sharp dorian mode throughout the first fourteen measures, strengthened further by a c- sharp pedal in measures 8-4. Measures 5-9 contain the minor third pedal of b-flat and d-flat which derives from the same enharmonic interval of a-sharp and c-sharp which ended the second phrase at measure 4. A third pitch, e, is added to the thickening pedal at measures 0-9, deriving again from the last note of the preceding phrase in measure 9. The pedal chord is enharmonically respelled in these measures as an a-sharp diminished triad (a-sharp, c-sharp, e) which is set below a melodic line which contains metric accents on f and durational accents on its dominant c. Measures 0-9, as a result, border on bitonality. The pitch f- sharp is emphasized in measures 7-9 and is added to the pedal accompaniment in measure 9, transforming the a-sharp diminished triad into an f-sharp major minor seventh in first inversion. Bitonal implications are made between the melody in C-sharp in the bass and the f-sharp seven accompanimental pedal above in measure 4. The tonal strengthof the

4 accompaniment, however, is weakened by the static repetition of the ever-thickening tone cluster to which g-sharp is the last pitch to be added in measure 47. The C-sharp tonality of the melody is reaffirmed by octave reinforcement in the lower register in measures 48-55. The significance of the thickening pedal lies in its isometric construction. The first note added to the c- sharp pedal is a-sharp, a minor third below. The second note to enter is e, lying a minor third above c-sharp. F- sharp enters next at a perfect fourth above c-sharp, while g-sharp enters last at a perfect fourth below. The isometrical principle is that the entire vertical sonority produced by these pitches contains the same order of intervals whether spelled from one end or the other, as shown in Figure 0 (, p. 9). G# A#4--C#---->E F# P4 m m P4 m.47 m. m.8 m.9 E--- F# G# -- A# -- C# -- B M m m M F#--- E --- > C# -- >A# --- G# Fig. 0--Isometric Sonority Of further significance is the entrance point of the e into this thickening isometric sonority. Of the four

5 pitches which are added to the c-sharp pedal throughout the dirge, note that the e is the second in order, entering at measure 8 which marks not only the exact arithmetic center of the total fifty-five measures, but also marks the half completed isometric sonority (a#, c#, e) which is furthermore isometric within itself. Harmony The harmony of Dirge II gradually grows more dissonant after the consonance of octave s in measures -7. The c-sharp pedal in measures 8-4 produces occasional moments of mild dissonance with the melodic line. The minor third pedal of b-f lat and d-flat in measures 5-9 causes stronger and more frequent dissonant intervals such as the minor second, major seventh, and tritone. The tritone first occurs in measure 5 and again in measure 8 to remain present throughout the remainder of the piece. With bitonal implications between the accompaniment and the melody in measures -9, dissonance is strengthened even further by sonorities which contain often more than one tritone simultaneously with major seconds, minor seconds, and their inversions. Dissonance increases as the texture thickens with sonority doublings on dissonant intervals. An octave on the first beat of measure 48 provides a brief release from the thick texture before it resumes for the final eight measures. Note that the in

6 measure 48 lies on c-sharp, affirming a return to that tonality amidst the tone clusters. Melody The theme of Dirge II is stated seven times and is characterized by its smooth contour of mostly stepwise intervals. There are 99 intervals in the melodic line of which are major seconds, 9 perfect fourths, 7 minor seconds, and minor thirds. Of the seven statements, only the third, fourth, and fifth contain significant melodic transformations. The theme is stated twice in C-sharp in the first fourteen measures (see Figure ). The tonality shifts to B-flat in Fig. --Theme of Dirge II measures 5-0 where the theme first varies from the original statement (see Figure ). Since the rhythm of the t5 6 IT [$9 0 Fig. --Third Phrase

7 third statement is so similar to the first (except for the diminution of measures 6-7 into a single measure at measure ), comparison of thematic cells between the two statements can be specific. The descending perfect fourth in measure is diminished into a minor second in measure 6, while the ascending major second of measure is augmented into a minor third in measure 7. Although the third statement shares the same thematic contour as the first two statements in the first two measures of the phrase, the contour becomes inverted in the following three measures. The three descending quarter notes of measure are inverted into three ascending quarter notes in measure 8. Note that the total interval span in measure is a descending perfect fourth, while the distance in measure 7 is the inversion of that interval, the perfect fifth. The ascending major second in measure 5 becomes a descending minor third in measure 0. The descending perfect fourth in measures 6-7 becomes a descending major second in measure which also contains a rhythmic diminution of that cell in the first statement of the theme. Note how e is cleverly introduced into the cadence of the fourth statement (measures 8-9) through the descending minor second which refers to the same cadence in the third statement in measure with a major second. The fifth phrase in measures 0-40 contrasts the third and fourth phrases by employing many perfect fourths which

8 obscure the tonal center (see Figure ). Measures -7 contain the fifth statement and are followed by a three- Fig. --Fifth Phrase measure extension (measures 8-40) before the sixth phrase enters in the bass clef at measure 4. The descending perfect fourth in measure is regained from measure while measure expands the ascending major second from measure into an ascending perfect fourth. The contour of measure is another inversion of measure but is intervalically different from that first statement and from its first transformation in measure 8. Notice, however, that the interval span of all three quarter notes in measure is an ascending perfect fourth which is the inverted direction of the perfect fourth spanned in measure. Measure 4 is a diminution and contraction of measures - and corresponds to measure 9 as employing widened intervals of perfect fourths and an extension of one extra quarter note on the third beat. Measure 5 and the first two beats of measure 6 immediately repeat measures -4. The ascending leap of a major sixth from c to a in the second and third beats of measure 6 places the range of the melodic line at its highest peak in the dirge. Measure 7 corresponds with

9 measure 9 and measure of the third phrase with its descending major second and the rhythm of a quarter note followed by a half note. Measures 8-40 extend this phrase by repetition of measures 6-7. The only change in the melody is that the c on the second beat of measure 6 is changed to a db. Since this pitch is enharmonic to c#, the change of notation may serve to prepare for the f-sharp major minor pedal chord (of which c-sharp is the fifth) first occurring on the third beat of measure 9. Although the fifth phrase and its extension are succeeded by the sixth phrase in measure 4, the melodic line of the fifth phrase continues in the high register of the treble clef simultaneous with the bass clef melodic line in measures 4-49. Just as measures 8-40 represent a "cutting" of the small phrase of measures 5-7.(cutting the opening material of measure 5), measures 4-47 cut from the beginning of the phrase as stated in measures 8-40. There are two statements of the last three notes in the phrase a jg# 4f# in measures 4-45, and in measures 45-49 there are two statements of only the descending major second of g# to f#. The overlap caused by this extension of the fifth phrase in the treble clef simultaneous with the sixth phrase in the bass clef ends on the first beat of measure 47 to prepare for the final seventh phrase in measures 48-55 which cadences strongly with an augmentation of the last two notes which are changed to a descending perfect

0 fourth, the characteristic motive of the entire piece, on the fourth scale degree down to the first scale degree, creating a strong plagal cadence. Rhythm and Phrase Analysis Dirge II contains fifty-five measures in meter. The rhythmic pattern occasionally varies only slightly from the first statement of the theme in measures -7. The repeating rhythmic cell is in the iambic rhythmic mode--a quarter note followed by a half note. The fifth phrase states the most rhythmically varied form of the theme and is extended by two measures which tonally shift from F to F-sharp. Although this phrase in measures -40 is stated almost entirely in quarter notes, the iambic character is not obscured due to its strong rhythmic force. This rhythmic pattern cannot be easily distorted by the limited choice of time values with which the piece is composed--the quarter, the half, and the dotted half. There are seven phrases in the dirge, each repeating or varying the opening statement. Phrases one, two, six, and seven lie on the tonal center C-sharp. The tonality of the third and fourth phrases lies an enharmonic minor third below C-sharp (A-sharp; B-flat), while the fifth phrase lies in a similar but not exact interval above C-sharp, and enharmonic major third (F; E-sharp). This extended phrase shifts to the tonality of F-sharp which is the sub-dominant

region to C--sharp, reinforcing that tonal center when it returns in measure 4 (see Figure 4). Phrase : 7 measures -7 C# Phrase : 7 measures 8-4 C Phrase : 7 measures 5- Bb Phrase 4: 8 measures -9 Bb Phrase 5: measures 0-40 F - F# Phrase 6: 7 measures 4-47 C# Phrase 7: 8 measures 48-55 C# Fig. 4--Phrase Analysis of Dirge II Form Six of the seven phrases in Dirge II are grouped by tonal centers into pairs. The fifth phrase is not paired with another phrase but is extended with a tonal shift, distinguishing this phrase in measures 0-40 from the others. It also contains the greatest rhythmic variety and the only feeling of bitonality in the piece, lying between the more consonant phrases which precede it and the more dissonant phrases which follow. This important phrase also crosses both the arithmetic center of the dirge and the point of the golden section out of the total fifty-five measures. Of these four phrase groups, the first and last lie lie on the tonal center C-sharp, while the second and third in relations of thirds to both sides of that tonal center. The isometric growth of the accompanimental sonority is half completed at the center of the piece in measure 8 when e is added to the sonority, resulting in the

symmetrical construction a-sharp, c-sharp, e, since both a- sharp and e lie a minor third on either side of c-sharp. In the last half of the dirge, then, the other two pitches of the complete isometric sonority are added. F-sharp enters in measure 9 and g-sharp enters in measure 47, resulting in the final isometric pedal sonority of g#, a#, c#, e, f#. The intervals of the sonority (whether spelled from one end or the other) follow the order major second, minor third, minor third, major second (see Figure 0, p. 4). Texture, Dynamics, and Range Dirge II contains two separate fabrics of music--a melodic line and a continually thickening harmonic accompaniment, concurring with a rise of dynamic level, and a widening of range. This dirge is constructed on one continuous gradation of dissonance, texture, dynamics, and range while gradually developing the isometric harmonic sonority of the accompaniment. The climactic phrase coordinates these elements at measure 5, marked fortissimo. Although the texture and range remain at their climactic level to the end, the dynamic level quickly diminishes to pianissimo by measure 55.

CHAPTER BIBLIOGRAPHY. Hanson, Howard, Harmonic Materials of Modern Music, New York, Appleton-Century-Crofts, Inc., 960.

CHAPTER V ANALYSIS: DIRGE III Tonality Tonal structure of Dirge III is influenced by the open fifths in the bass clef which strongly imply a root of the lower pitch while continuously shifting to other tonal areas. The tonal movement of the fifths is organized into phrases by phrase marks and by sonorities with longer time values at the end of each phrase which produce stronger tonal implications due to their cadential effect. The chords upon which these seventeen phrases cadence are significant to the tonal structure of the dirge and are shown in Figure 5. Beginning on. mm. -4. mm. 5-8. mm. 9-0 4. mm. - 5. m. ' 6. m. 4 7. m. 5 8. m. 6 G# '9. E 0. Ab. Bb. A. P# 4. A 5. C 6. C 7. m. 7 mm. 8-9 mm. 0- mm. - mm. 4-6 mm. 7-8 mm. 9-0 m. mm. - A C G A G D D Fig. 5--Bass Cle Dirge. f Tonal Phrase Endings in the Third The dirge begins and ends on the tonality of G-sharp and marked reference to it or its enharmonic equivalent A- flat on ten other occasions. The next most frequently 4

5 occurring tonalities, G and A, lie a minor second above and below this central tonality and are employed six and seven times respectively. Measure 7 establishes A which serves as a "Bartokian dominant" to G-sharp. This tonality occurs in the central phrase and in the arithmetic center of the thirty-three-measure dirge. Some symmetrical relationships surround the tonality of A of measure 7. The tonality of C occurs on either side of this vertex in measures 5-6 and in measures 8-9. The tonality of A occurs again five measures preceding measure 7 at measure, and six measures following measure 7 at measure. Since A and C both lie on the dominant axis when G-sharp is tonic, the area encompassing measures - functions dominantly. Subsequently, the outer boundaries surrounding this area lie on tonal centers of the subdominant axis. Seven measures preceding the vertex at measure 0 is the tonality of B-flat which is the interval of a major second above G-sharp. Eight measures following the vertex in measures 5-6, lies the tonality of G which is the interval of a minor second below G-sharp. Extending the outer perimeters even further to thirteen measures on either side of measure 7, lie two different tonal centers which themselves are only a major second apart. In measure 4, the tonality of E lies on the subdominant axis and is a major third below G-sharp. In measure 0, the tonality of D lies on the tonic axis (in

6 preparation of the final cadence) and is an augmented fourth below (or a diminished fifth above) G-sharp. Finally, in the outer most frame of the composition, the first three measures and the last three measures lie on the tonic functioning tonal center of G-sharp. Harmony The open fifths in the bass clef create only consonant harmony. The consonant character of the dirge is often interrupted by recurring dissonance produced by the pitch relationship of these fifths to the pedal-like melodic line in the treble clef, although major and minor triads are sometimes formed when the melodic line supplies the missing third of the bass open fifths. Dissonance in Dirge III is largely a result of the stepwise motion of minor seconds in each of the two lines. The harmonic climax lies midway through the piece, measures -8, where the dichotomy of consonance and dissonance is especially felt. In these five measures, five dissonant sonorities alternate with four consonant sonorities, creating tension by abrupt contrast. Measures -8 proceed through the center of the composition at measure 7 where the tonal centers of the dominant axis occur and where the melodic line is most tense.

7 Melody Dirge III is largely constructed with stepwise intervals of major and minor seconds which equal 8 (56 percent) of the total 46 melodic intervals. There are 06 (4 percent) minor seconds alone, ( percent) major seconds, 0 (eight percent) minor thirds, and 8 (seven percent) major thirds. There are 0 diminished fourths which sound as major thirds. These diminished fourths combined with the major thirds total 8 (5 percent) of the intervals. There are (nine percent) tritones in the dirge. Neither the treble clef melodic line nor the bass clef melodic line in open fifths states an extended melodic theme. Dirge III contains only small motivic cells which are developed, culminating in measures 5-9, the arithmetic center of the piece. three motivic ideas: The treble clef melodic line uses () a repeating pitch which occasionally rises a minor second as a neighboring tone (see Figure 6); () a generative motivic cell which consists of a descending minor second followed by an ascending minor or major third in a three-note pattern which leads motive I onto higher pitch levels (see Figure 7); () a descending figure of three quarter notes and a whole note with intervals of seconds and thirds (see Figure 8). Motive I is first stated in measures - on b and is succeeded by motive II in measure 4 which lifts the second statement of motive I in measure 5 up a minor third to d.

8 Fig. 6--Treble Clef, Motive I, mm. - f4 Fig. 7--Treble Clef, Motive II, m. 4 Fig. 8--Treble Clef, Motive III, mm. 9-0 This three-note generative motive is derived from the first three sonorities of the bass clef melodic line in open fifths (see Figure 9). Motive II not only generates the Fig. 9--Bass Clef, mm. - motion of the melodic line but also propels the harmonic motion in the bass clef. Motive II occurs again in measure 8 and is extended by two more notes in continuing ascension,

9 lifting the third statement of motive I up a diminished fifth to a-flat. The third statement of motive I on a-flat is condensed and combined with motive II in measures 9-0 (see Figure 0). A fourth statement follows in measures -, precisely repeating the previous statement on the enharmonic pitch of g-sharp. The a-flat stated in measures 9-0 is derived from measure 8 where it was the root of an 9 '':'7 Fig. 0--Treble Clef, mm. 9-0 a-flat major triad. The same pitch in measures -, however, inherits a new function; to create a harmonic pull as a leading tone into the third of the f-sharp minor triad on the second beat of measure, hence g-sharp to a. This function is confirmed when the same relationship is sequenced on the second beats of measures 4-5 where an ascending minor second resolves into the third of an a minor triad and the third of a c minor triad respectively. The ascending minor second, the motivic cell from motive I, is absent when motives I and II are combined in measures 8-. Subsequently, four statements of this single ascending minor second motivic cell follow in measures, 4, 5, and 6 (see Figure ). Each of these statements

40 Fig. --Treble Clef, mm. -6 is sequenced a minor third higher than the previous one so that from measure to measure 7 a stepwise progression up to e might intensify this climactic peak in the composition. Measures 7-9 simply alternate e-flat with e-natural, isolating the minor second which is the principal interval of the composition. These important measures which represent the motivic core of the dirge lie precisely between the arithmetic center, measure 7, and the golden section, measure 0. Motive III does not occur until measure 9, breaking away from the restricting two-note pattern of motive I. This motive descends by step the span of a perfect fourth from e in measure 9 to b in measure 0. Note that this first statement of motive III ends at measure 0, the moment of golden section, on the of b, the third of the g-sharp minor triad, which marks a reference to the tonal center of the piece at this significant moment. A statement of motive III is repeated in measures -, expanding the descent to a minor sixth from e to g#, and to major sixth in measures -4 from g to bb (see Figure ).

4 9 ' a Q 4 Fig. --Treble Clef, mm. 9-4 The melodic contour of measures 5-6 constructs an arch by combining the generative uplift of motive II (on b-flat) with motive III (g descending to bb ), as shown in Figure. Measures 7-9 state a condensation of motive I combined with motive II (see Figure 4). The single descending melodic augmented second in measure 0 is not only a Fig. --Treble Clef, mm. 5-6 Fig. 4--Treble Clef, mm. 7-9 cell from motive III (measure 9; beat three), but is also an inverted fragment of motive II (measure 4; fourth and fifth beat), and is moreover a forecasting of the opening motive in Dirge IV (see Figure 5).

Fig. 5--The Augmented Second The motivic statement in measure is the expanded inversion of motive II from measure 4. In the original statement, the ascending minor third ascends to c-sharp which behaves as a "leading tone" to d--a way of generating the second statement of motive I onto a higher pitch level (see Figure 6). When the direction of the motive is inverted in measure, the descending minor third to a-sharp 4 Fig. 6--Treble Clef, mm. 4-5 behaves as a "leading tone" to b--a way of negating-the generative powers of motive II and remaining on the same pitch level (see Figure 7). At measure, motive I is stated on b as in the opening measures, ascending to c Fig. 7--Treble Clef, mm. -

4 and a return to b, ending the dirge in the same way it began. Throughout Dirge III the bass clef melodic line employs the two-note cell of a descending minor second, which is derived from the opening statement of motive II in the bass clef in the first two measures. The pattern is voiced with parallel fifths throughout the dirge (see Figure 8). Fig. 8--Bass Clef, mm. -4 Measures -8 comprise the opening two phrases in sequence at a distance of a diminished third (g-sharp to b-flat), stating the motivic idea in its most complete thematic form. From this point, the phrase is gradually condensed to its primary fragment, the descending minor second as stated in measures -5 (see Figure 9). T e ultimate reduction is seen in measures 6-9 where the i itial two-note motivic cell is represented by one dotted hole nqte value (see Figure 0). The descending minor second niotivic cell is resumed in measure 9. The final phrase qf the bass clef melodic line in measures - is he retrograde of the first two bass clef pitches in measure, f x g# instead of g# f x.

44 Fig. 9--Bass Clef, mm. -5 6-7 8 Fig. 0--Bass Clef, mm. 6-8 Rhythm The thirty-three measures are in varying meters of, /, 5/4, and 7/4. The melodic lines of the bass and treble clefs alternate in rhythmic activity--one dormant while the other is in motion. Both lines are stated mostly in half--note values. Exceptions to this are occasional quarter notes in the treble clef and occasional whole notes in the bass clef. The two lines are set apart at the distance of one beat; consequently, the motion of the two voices together moves in quarter-note values. The rhythmic motion occasionally slows down to the half note, but never moves in values smaller than the quarter. In each phrase of the treble clef melodic line in measures -4, the longer note values precede the shorter values. The phrases gradually shorten until each phrase in measures

45-4 contains only one half note and one quarter note. The lengthy phrase at the center of the composition in measures 5-9 is emphatically stated by eleven half-note values where the minor second is fully exploited by the oscillating minor second in the melodic line of e-flat and e- natural. Smaller values precede longer values in motive III beginning at measure 9, contributing to the other contrasts which this motive provides. From measures 7-, however, the original order of longer values preceding shorter values is resumed. In measures -, both rhythmic ideas are merged into the pattern of long-short-long, expressed by a whole note, quarter note, and a quarter tied over the bar line to a whole note. Most of the bass clef rhythmic structure is built on phrases in half notes, cadencing with whole notes. The bass clef phrases are gradually condensed to a single dotted whole note in the culminating measures 6-9. The rhythmic structure in each clef, then, is built around this central phrase. Phrase Analysis The phrases in Dirge III are constructed around the central core of measures 5-9. There are eighteen phrases in the treble clef melodic line and seventeen in.the bass clef, each of these varying in phrase length. The treble clef phrase structure centers around the ninth phrase

46 (measures 5-9), eight phrases preceding it and nine phrases following. In the bass clef, seven phrases precede the three single chords in the center (measures 6-9), and seven follow. The culmination of both melodic lines occurs in measures 5-9. Note how the treble clef melodic line begins with a single repeating note and begins to grow in complexity by introducing more pitch and rhythmic variety. On the other hand, the bass clef melodic line begins with an extended musical phrase of four measures and is gradually shortened and condensed. The paradox continues to measure 0, the golden section of the dirge. From this measure to the end of the dirge the treble cleff line continues to move with much more mobility (in compliance with motive III) than the slowly descending intervals of the bass clef. When the final two measures of Dirge III are stated, a brief reference to the original material of the first phrase in both melodic lines serves as a reminder of the simple melodic cells from which such creative development has arisen. Form Dirge III is structured on a simple plan of gradual growth in tension to a culminating climax, followed by a gradual return to the initial repose. The climax lies in the exact center of the dirge, creating a symmetrical wave of tension. The opening melodic materials are gradually

47 condensed in measures -4. The central phrase in measures 5-9 uncovers the motivic cell of the minor second which is redeveloped in measures 0-. Texture The bass clef open fifths are octave reinforced throughout the dirge, producing four voices in the bass line. Only in measures 4-9 (in the core of the composition) are two other sonority doublings added. The treble clef melodic line is also octave reinforced throughout the dirge. Two sonority doublings are added in measures 9-5, producing three voices. Three sonority doublings in measures 6-9 produce four voices. Three voices resume in measures 9-8, and two voices in measures 9-. Although open voicing is retained throughout, the texture is thick due to the sonority doublings in the low registers of the keyboard. The texture thickens as the midpoint (measures 6-9) is approached, and gradually thins out from there until the original voicing is resumed. Dynamics and Range The dynamic pattern rises from piano to fortissimo midway through the dirge at measure 7, returning to pianissimo by the end. Similarly, the range widens as it approaches the center, narrowing as it approaches the end. The opening of measure extends from GG# to b, spanning three octaves and a minor third. The range widens to a span of

48 five octaves and a perfect fifth from AAA to e in measure 7. From there, the range narrows to the original tessitura of measure. It is shown, then that the range and dynamics of Dirge III support the form of an augmentation of all compositional elements to the center of the piece, diminishing from there to the end.

CHAPTER VI ANALYSIS: DIRGE IV Tonality Dirge IV begins with bass-implied roots on the tonal center of G and progesses through various tonalities before reaching the distance of a tritoneon C-sharp (the counterpole on the tonic axis) in the midpoint of the piece at measures 4-8. There is a similarly gradual return to G by the end of its seventy measures. A balance of tonal centers is structured around the central measures of the C-sharp vertex by the nearly equidistant occurrences of A- flat (measures 4-5, measures 54-59) and F (measure 0, measures 5, 58, 6, 65, and 67) on both sides of the center. The occurrences of A-flat lie fourteen measures preceding the C-sharp vertex and sixteen measures following. The occurrences of F lie twenty-two measures preceding the vertex and at the fourteenth, nineteenth, twenty-second, twenty-fourth, and twenty-sixth measures following. A section on the tonal center of B (the major third above G) shifts away at measure 8, seven measures preceding the vertex, and a section on the tonal center of D (a perfect fifth above G) begins at measure 44, five measures following 49

50 the vertex, and one measure past the golden section at measure 4. There are two systems of tonal structural functions which overlap through the central measures of Dirge IV. First, two occasions of tonal emphasis on the tritone from G occur on D-flat in measures - and on C-sharp in measures 4-8. Second, the occurrences of B, the major third above G, in measures 0-8, and on D, the perfect fifth above G, in measures 44-5 all combine to form the major triad G B D (G) when the outer perimeters of G are included (see Figure ). Thus a union is made between the symmetrical expression of form (the tritone--which occurs as C-sharp IAyV 'w7l. dtwr i I.45 $~.7 44-~~L I TO Fig. --Overlapping Tonal Functiono in Dirge IV in the center of the piece while itself the G octave), and the assymmetrical exp (the root, third, and fifth of the tonic )eing the center of session of form triad--each serv- ing architectural significance). The ba Lance of these two formal principles is achieved by their o four instances surrounding the center of overlapping in these the dirge.

5 Harmony The contrast of dissonance and consonance in this dirge creates continuous tension and release. Dissonance seldom occurs within the harmonic element itself, but is produced by implication of bitonality between the melodic line and the harmonic accompaniment which would appear to be independent of one another. The melody lies in the middle register of the keyboard between the surrounding extreme registers of the accompaniment which is predominantly tonal due to its sustained octave reinforcement of consonant sonorities. The strongest moments of dissonance occur when the accompaniment is dissonant within itself. Measures -5 contain open perfect fifths in the harmonic accompaniment, producing a generally consonant harmonic character. Measures 6-0 introduce a half diminished seventh chord on g in the accompaniment, containing the first dissonant interval of a tritone in the harmonic accompaniment. Measures -9 display contrapuntal writing of two independent voices in the treble clef which produce the strongest dissonance in the piece in combination with the half diminished seventh chord on b in the accompaniment. Measures 0 and - consist of brief moments of repose in perfect octaves. Measures 4-50 create dissonance again by tonal conflict between the melodic line and the harmonic accompaniment, although the strongest dissonance of the diminished triad in the accompaniment is no longer present.

5 The frequent passages and major and minor triads in measures 5-68 produce a consonant harmonic character. Dissonance returns again in the last two measures when the motive of measures 4-50 is stated once more on c-sharp, the augmented fourth above g in the accompaniment. The harmonic construction of this dirge lies on three levels; rising from consonance to dissonance of a tritone, and to the strongest moment of dissonance in measures -7 of tritones and bitonality, and returning to consonance toward the end of the piece. Dissonance increases as the tonality departs from G, and decreases as the tonality returns to G (see Figure ). D isaa' 6'in 6 7 557 Fig. --Harmonic Pattern of Consonance and Dissonance in Dirge IV. Of the 7 intervals in the melodic line, 77 (4 percent) are minor thirds, 48 (5 percent) are major thirds, 46 (4 percent) are major seconds, and 4 ( percent) are augmented seconds, indicating once more the smooth contours of the melodic line. Motive I consists of two cells; Ia, a descending augmented second, and Ib, an enharmonic arpeggiated triad ascending --5 (see Figure ). Motive Ia-b is stated

5 t. 4 5 Fig. --Motive Ia-b -mm. -5 twice on the beginning pitch c-sharp and in measures -9, separated from the tonal center of the harmonic accompaniment on G by the interval of an augmented fourth. This melodic use of the tonal emphasis on c-sharp forecasts the significance of this tonality in the center of the dirge. Furthermore, the use of c-sharp implies a tendency of resolution to d, the fifth of the g minor triad of which the b- flat of this augmented second would appear to be the third; hence, the necessity of spelling an augmented second instead of its enharmonic minor third. A tonal shift occurs in the melody and in the harmonic accompaniment with a shortened statement of motive Ia-b on the beginning pitch b in measures 0-, repeating in sequence a major third lower on g in measures - (see Figure 4). A stepwise melodic cell Fig. 4--Mm. 0- in dotted rhythm follows Ia on d in place of Ib in measures 4-5. These two cells are combined twice; first on d

54 (measure 4) and again on d-flat (measure 6) in sequence a minor second lower, with an extension of one measure which repeats the new cell (see Figure 5). Motive Ia-b is stated Fig. 5--Mm. 4-8 once more in its original form on the original pitch level, enharmonically spelled as d-flat in measures 9-. The dotted rhythm in measures 5-8 propels the beginning of the new phrase group on the b diminished triad in measures - 7. The contrapuntal lines in measures -9 contain the intervals of motive I (augmented second and minor third) and are transitional to the first statement of motive II (see Figure 6). Measure 0 separately states Ia on d-sharp "I4 46 $ 9 Fig. 6--Mm. -9 in octaves and is repeated in measure, followed by Ib in measure (see Figure 7).

55 Fig. 7--Mm. 0- Motive II is first stated in measures 4-5 in the bass clef and consists of only three pitches; an ascending augmented sixth from a-sharp to f-double-sharp, returning to the a-sharp. The major sixth of motive II is derived from Ib, whose enharmonic minor third (augmented second) is inverted to produce the major sixth. Even the melodic contour of motive I (a descending augmented fifth or minor sixth, followed by an ascending augmented fourth) is used in motive II in inversion with a slight expansion. This motive first occurs with a harmonic accompaniment on the tonality of C-sharp, the middle of the G octave, and exactly in the center of the dirge at measures 4-8 (see Figure 8). 4 5 b 7 '$ Fig. 8--Mm. 4-8 After motive II is first stated, motive I overlaps by occurring once more in the treble clef in measure 6 (Ia on

i 56 a) and in measures 8-9 (Ia and Ib on a). The overlapping statements of motive I and motive II indicate the relationship which these two motives share. The two statements of motive II in measures 4-8 are sequenced up a diminished fifth on the tonality of G in measures 40-4, following the overlapping statement of motive I in measures 8-9. The ascending major sixth of motive II is stated three times in measures 44-47 with the pitches b to g-sharp. Three modified statements of this cell follow in alternating ascending intervals of major seconds and minor sevenths, reducing the motive to a single ascending melodic interval which is stated six times in these seven measures (measures 44-50) of melodic, harmonic, and tonal climax (see Figure 9). 45,0 Fig. 9--Mm. 44-50 Measures 5-5 mark the transition to the return to motive I. A modified statement of motive II begins on the

57 third beat of measure 50, leaping up a minor seventh and descending a major sixth (see Figure 40). Another three- Fig. 40--M. 5 note figure which contains the augmented second of motive I begins on the third beat of measure 5 (see Figure 4). Fig. 4--M. 5 Motive Ia-b returns in rhythmic diminution, a new meter of 5/4, and with modified intervals in measure 5. The descending augmented second of Ia (c-sharp to b-flat) is expanded by a half step to a major third (d-flat to b-doubleflat), and the f diminished triad of Ib in measure is changed to f minor (see Figure 4). Four measures of 5 Fig. 4--M. 5

58 parallel block chords in measures 54-57 descend by alternating major and minor seconds (see Figure 4). Measures 58-6 Fig. 4--Mm. 54-57 repeat the pattern in measures 5-57. The statement of motive I in measure 58, however, has been further modified to spell a major third to represent Ia (c-sharp to a), and an f minor triad for Ib; thus, an f augmented triad descending followed by an f minor triad ascending (see Figure 44). Fig. 44--M. 58 Measure 6 states only the modified form of motive I as does the repetition in measures 64-65 an octave lower (see Figure 45). Measures 66-67 state motive I in its original intervallic form though still rhythmically modified (see Figure 46). The tonal emphasis on F in the melodic line in

59 Fig. 45--Mm. 6-65 Fig. 46--Mm. 66-67 measures 5-70 alternates with the two block chord progressions in measures 54-57 and in measures 59-6 which both cadence on G. Since F lies on the dominant axis when G is tonic, these last eighteen measures may be regarded as an axis progression of vacillating dominants and tonics, represented by the tonal centers of F and G. The dirge ends with a final statement of motive II on the tonality of G which provides a tonal answer to the first statement of that motive on c-sharp (the tritone and center of the G octave) in measures 4-5 (see Figure. 47).

60 Fig. 47--Mm. 68-70 Rhythm The entire dirge is in meter except for measures 5, 58, 6, and 67 which are all in 5/4 and state the modified return to motive I. Dotted rhythms occur only between measures 5-8 while the rest of the piece moves in slow quarter notes, half notes, and occasional dotted half notes. Ia is originally stated by a half note followed by a quarter note, propelling rhythmic drive over the bar lines. Ib is first stated by three quarter notes and a dotted half note and is modified only in measures 5, 7, and 8 where the dotted quarter notes and eighth notes serve to strengthen the rhythmic drive into the transition material of measures -. Motive II is first stated with two half notes followed by one quarter note, characterized by its leap of a major sixth ascending and a major sixth descending. The rhythmic structure of this motive is varied in the climactic measures

6 47-5 within the cell of the ascending leap with one quarter note followed by a half note. Phrase Analysis and Form The thirty-six phrases in Dirge IV are organized into three cleverly interwoven sections through the motivic transformations explained in the melodic analysis. The first thirty-three measures (phrases -4) contain only melodic materials of motive I. Measures 4-5 (phrases 5-8) contain statements of motive II which are themselves derived from intervallic materials of motive I. Motive I returns in measures 5-68 (phrases 9-5) with further modifications of rhythm and intervals although the melodic contour remains recognizable. A final statement of motive II echoes in the last two measures of the dirge (phrase 6), supplying a tonal answer to the first statement of motive II in the middle of the work. (See the table of phrase structure on Dirge IV in Appendix IV.) Dynamics and Range An increase of the dynamic level accompanies the increase of dissonance to measure 7. Though the dissonance begins to decrease from this measure, the dynamic level continues to rise through the midpoint of the dirge to measures 44-50 which comprise the climactic development of motive II. From there, the dynamic level decreases, forming an arch structure within the dirge which builds to measure

6 5 and recedes thereafter. The range is consistently wide, lying between EE (measures 8-9, 4-8) to db 4 in measure 60.

CHAPTER VII THE FOUR DIRGES: UNIFYING ELEMENTS AND COMMON CHARACTERISTICS Unifying Elements Tonality The Four Dirges are unified by their tonal organization in reference to the axis system. If G is considered the final tonic of all four dirges (since it is the tonal center of the last dirge in the opus), and is placed at the top of the circle of fifths, the following functions result (see Figure 48). The diagram illustrates that the tonal I II III IV B C# G# G Dominant Axis Tonic Axis Dominant Axis Tonic Axis Fig. 48---The Axis System of the Four Dirges centers of Dirges I and III, B and G-sharp, lie on the dominant axis while the tonal centers of Dirges II and IV, C- sharp and G, lie on the tonic axis, producing the pattern dominant-tonic-dominant-tonic (see Figure 49). 6

64 TONIC AxYS T 0 DoMiWANT Rt5 0 f A F E Bb/ E EO 8 B-Lge T C- tr~e C Frg Fig. 49--Tonal Scheme of the Four Dirges in Regard to the Axis System. Melody Intervals.--Each dirge is constructed on a descending interval; the descending augmented fourth in Dirge I, the descending perfect fourth in Dirge II, the descending minor second in Dirge III, and the descending augmented second in Dirge IV. The first and third dirges can be paired against the second and fourth dirges, each pair containing an interval of a fourth and a second. Dirge I and Dirge III are built on the augmented fourth and minor second respectively. The augmented fourth of Dirge I is decreased by a semitone to a perfect fourth in Dirge II. The minor second of Dirge III is increased by a whole tone to an augmented second in Dirge IV. As a result, the similarity in the type of the motivic intervals unites a different pair of dirges; Dirges I and II (the fourth), Dirges III and IV (the second). This relationship between the third and fourth dirges is strengthened by the forecasting of the augmented second

65 (used in the fourth dirge) in measure 0 of the third dirge. (See p. 4, Figure 5.) This potentiality of multiple methods for pairing these four pieces serves to unify them even more strongly (see Figure 50). I II III IV Augmented Fourth 4 Perfect Fourth Minor Second4 Augmented Second TY "FTourth Decreased L-7-Second Increased Fig. 50--Intervallic Similarities in the Motives of the Four Dirges. Motives and Rhythm.--The pairing of Dirge I with Dirge III and Dirge II with Dirge IV is seen further in regard to their motives. The three-note motive in Dirge I (measure ) is inverted in motive II of Dirge III (measure 4). The major third which descends in the main motive of Dirge I (b-g) is decreased, however, to an ascending minor third (a-sharp to c-sharp) in Dirge III (see Figure 5). Furthermore the rhythmic pattern of two quarter notes followed by a III. Fig. 5--Motivic Relationship Between Dirges I and half note in Dirge I is reversed to a half note followed by two quarter notes in Dirge III.

66 Dirges II and IV are paired by incorporating sevenand five-measure themes respectively rather than employing single motives as in the first and third dirges. The descending perfect fourth which begins Dirge II is changed to a descending augmented second in the beginning of Dirge IV (see Figure 5). Once more there is a reversal in rhythmic IV. Fig. 5--Motivic Relationship Between Dirges II and pattern between the pair of dirges. Dirge II contains the descending perfect fourth with a quarter note followed by a half note, while Dirge IV contains the descending augmented second with the half note followed by a quarter note. In each of these two dirges, the descending interval is stated twice and is followed by a measure of three quarter notes. Meter and Tempo Finally, the pairing of Dirges I and III, and Dirges II and IV, is seen in their metric organization and tempos. The meters of Dirges I and III are duple with frequent changes to other meters. The meters of Dirges II and IV are triple with very few changes to another meter (only in Dirge IV).

67 The first and third dirges are given much slower tempos than those of the second and fourth dirges (see Figure 5). I II Duple Triple Unstable Meter Stable Meter Adagio I = 5-40 Andante. = 00 III IV Duple Triple Unstable Meter Stable Meter Poco Lento J = 50 Assai Andante J = 00-08 Fig. 5--Relationships of Meter and Tempo Common Characteristics Tonality and Harmony Each dirge establishes its own tonal center with which the composition begins and ends. Each dirge has a symmetrical tonal structure which surrounds the vertex in the center of the piece. The symmetry, however, is exact only at three points in the framework--the beginning, the center, and the end. The tonal construction of Dirge II differs from the other three dirges in that the symmetry is based on the isometrical formation of the ever-thickening cluster centered around c-sharp, as discussed in that analysis (see Figure 54). I II III IV B-A-B G#-A#-C#-E-F# G#-C-A-C-G# G-C#-G Fig. 54--Tonal Symmetry in Each Dirge

68 Tonality is frequently established with pedal tones and sustained sonorities while the melodic and harmonic fibers remain distinct. Dirges I, III, and IV produce a harmonic fabric of thirds, open fifths, and triads while the exceptional Dirge II employs the gradually-thickening tone cluster. Form: Melody and Phrase Structure Each dirge has phrase groups which are part of larger sections. These sections are distinguished by phrase marks, textural changes, dynamic changes, new motivic developments, or new tonal regions. Each of the dirges is hyperbolic in form, which means that the motives are gradually reduced to to their most basic elements toward the center of the composition, from where they gradually become re-developed to the end. Dirge I: The motivic cell of a descending augmented fourth in measures -4 is rhythmically expanded by halfnote values and by an ever-widening interval contour to the center of the piece in measures 5-7. Dirge II: The characteristic interval of the descending perfect fourth becomes exploited in the central fifth phrase of measures 8-9. In this central phrase, the theme is extended in length ard widened in contour by the frequent use of perfect fourths.

69 Dirge III: The melodic material is gradually condensed until a single minor second interval is repeated in measures 7-0, the center of the piece. Dirge IV: The descending augmented second of motive I is inverted into a major sixth and is singled out in the central measures 5-4 in the form of a new motive. In each dirge, the motivic cells are re-developed following the central point. The figure below illustrates the hyperbolic structure of each of the four dirges (see Figure 55). vertex Fig. 55--The Hyperbolic Form in Each of the Four Dirges. Intervals The most frequently occurring melodic intervals in the Four Dirges are the major and minor seconds and the major and minor thirds which result in smooth melodic contours. In addition, each dirge contains a characteristic melodic interval which is unique to that dirge. Each of these intervals descends in proper character for a dirge (see Figure 56).

70 Dirge I: Dirge II: Dirge III: Dirge IV: Descending augmented fourth Descending perfect fourth Descending minor second Descending augmented second Fig. 56--Characteristic Descending Intervals Variety in range, texture, and dynamic levels is exploited in all four of these works. The Arithmetic Center and Golden Section The specific occurrences listed in the following table illustrate the frequent employment of the arithmetic center (50 percent) and the golden section (6.8 percent) as architectural keystones between which lie the tonal vertex and the melodic climax of each dirge (see Table ). This TABLE I THE FORMAL EMPLOYMENT OF THE ARITHMETIC CENTER AND THE GOLDEN SECTION IN EACH OF THE FOUR DIRGES Arithmetic Center Golden Section Dirge I: The tonal vertex lies on A- flat in mm. -4, the only reference to this tonal center around which the tonal symmetry is based (p. 5) One of the three most dissonant moments occurs in m. 4 (p. 6) Fortissimo at m. 5 (p. ) The Unison in m. 7 on C which is a dominant function on the tonal axis (p. 6) L -- _

7 TABLE --Continued Arithmetic Center Golden Section Dirge II: The third note of the fivenote isometric sonority is added at the center in m. 8 (p. 4) Dirge III: The tonal vertex A occurs at m. 7 (p. 5) The culminating exploitation of the minor second from motive I occurs in mm. 5-9 (p. 40) Fortissimo in m. 7 (p. 47) Dirge IV: The first statement of motive III ends on the b, the third of the g- sharp minor triad--a reference to the tonal center -- in m. 0 (p. 40) The reversal of the rhythmic patterns occurs with motive III by ending with longer note values--mm. 9-0 (p. 45) The comparison of the rhythmic motion between the treble and bass clef lines shows a reversal at mm. 9-0 with motive III (p. 46) The tonal vertex occurs at mm. 4-8, the center of the piece and the center of the octave of the tonal center (on C-sharp) (p. 49) Motive II is first stated in mm. 4-8 (p. 55) analysis shows that the golden section and the axis system were already being used by Bartok in these early compositions of 909-90. Thus, in the Four Dirges, Bartok was

7 employing materials and compositional devices which were to be carefully developed with unfailing consistency throughout his life: Hungarian folk melody, arithmetic center, golden section, and the axis system.

),, - ' - 7 a) C C 0C Ti) S4V - () E > O ) N )VV- a)< 9z 4j0- Q!C) >r -C- C ).E U V -CI ' 4 >M- C CO C) a 4> UA 0.C0 cd -C) Q). 0 I'V - Ca I :d G "r- r fly Y' 0 '-I 0 a *i coc r I.C COO C LO<U 0 'C V U a) I C " f. a )I 40 r a b40 0I s~ d Z 0-, -a 00 C -(D 04 0 "C W u 0 4- Q 0 0 U C) )-pa ".a i 600 a) N 6Z0$4 IM O. * Qd 'C0.0. a ) CC.0 *-- C C H F 0 zo )V a ) QC) U) i -, C D C) 0b --. 0QC I O M 44X "{ Q)C E J U.,), Q) I -I C 'P- - Q O * CO C)I..C C; I C ;.CJ 0 WA 0- > ic C " C O N< C.' E-> 0 CO CC > x C "r 0 0 9: m aa) 44 a) O Ia) C & C " ) 4>t-C U E 0 fla) -.0 M * 0 0 0 C0.- Q >~ )I * b N 4-> ". Q * CO. -.C -MC O -CC 0S0 I d O 0 C C 0< Car I x - a)9 0 C. '". a) ")r'( 0) 0w C 4- r,-- 0.0 ) U e "b > as-0 C a>c. a 4 L O E 0 a* 0 # ac)'c: 4> OC a O -z a 0 FC4CI)v r '-I c 4a 4> a)e cc - c -,) oc '-I -Y - -- 6.0 a) > C C) '-- a) C4 '-I co,ca *~L a a) U 4.- c O 4-4 r- C a O 4>0 0 a ) a) 0 U-I) 0 () 0 T O..,C E E> - -a I 0 ro '-IO a) a) ad

APPENDIX TABLE III DIRGE I: TONAL ANALYSIS Measure Tonal Center Measure Tonal Center D-sharp 5 A D-sharp-- B 6 A D-sharp p --.- B 7 C 4 G 8 E-flat 5 D-flat 9 B-flat 6 D-f lat----- C-sharp 0 C-sharp 7 C- sharp A-sharp -- G 8 B 9 E -> A B 0 C 4 B C B-flat 5 6 B with melodic reference to D-sharp B from mm. - A-flat 7 B 4 A-flat 8 B - I P -- 74

I -- --- - 75 co U H Ht 0 0 - - N - - - - - -m - CH e- - - - -# 0 bio 'Hq wl ',. ( _ -- wh

IwrlOr l / / r 76 TABLE IV DIRGE I: HARMONIC ANALYSIS Interval co art acontent 00I Interval Content cq 4 4 5 6 7 8 9 0 4 5 6 7 8 9 0 4 I 4 5 5 6 6 6 7 7 7 7 7 7 8 8 / / / / / / / / / / / I I I I n n n pmn prn mnd pmn prn mnd pns m 4 s t 4 m4. 4 m 6 np m d 8 4 p inn mn4d n d 6 n n6d prn4 4 p m m m m m m m m m m d5 M M d4 d4 d4 m m m m m m P4 5 6 7 8 9 0 4 5 6 7 8 9 40 4 4 4 44 45 46 47 48 9 9 0 0 0 ' / / / / / / / / / / / / / / / / / / / I p m d 4 8s4 0 p4m4n46 46 6 P mn4 m 4 n 4 d 4 p4m4n4s4d4t4 p m n p8m4n8s4 pam0 4d4 p 4 m 8 p4m4d4 n 4 p404 4 p4m4n8 4t4 p4nd4t4 p4m4n4 p4m 4d4 p~m4n8 pm d 48p M P4 P4 P4 P4 P4 m6 P4 P4 P4 P4 P4 da4 P4 P4 P4 P4 P4 P4 P4 P4

r i 77 TABLE IV--Continued.a0 Interval 0 kinterval Cv ontent Content C G/ 50 5 5 5 54 55 56 57 58 59 60 6 6 6 64 65 66 67 68 69 70 7 7 7 74 75 dq I I In 4 4 4 4 5 5 6 6 7 7 8 8 8 8 9 9 0 0 0 0 / / / / / / I p 8 m 4 n 8 pin n4d4t4 p4mn4 d t pend44 pm 6s5dt4 p4m5 7 p4m0 7d6 ~ ~st p 4 n 5 n s 6 t 9 p:lm5 7s6d9 p m4ns5 ' pmn t p in nd pi44nd4 pm~nd -Pmn 5 s m4s7t p mn 6 p in ns pin d p 5 t M6 p6mn 6 P4 P4 P4 P4 P4 m M M M M m6 m6 m6 m6 M6 m7 m6 m6 m6 m6 m7 I~7r 76 77 78 79 80 8 8 8 84 85 86 87 88 89 90 9 9 9 94 95 96 97 98 99 00 0 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 8 pemns4d pm 6 p in 4n 8s 4 p4m4n's4d4t4 p9m0n5 ". 6 pm~n s6 pm0n p9m4np m n p5m4n p5m0n pm4n p 0 9 pam4n'9 p9 m4n5 p9m4n p9m4n s6 p9.4n.6 p9m4n p9m4ns6 p9m4n s6 pmn s M6 M6 m, m m m M M M M M p4 P4.P4 m m m m M M M M M M M

79 TABLE V DIRGE I:: PHRASE ANALYSIS Phrases Measures Length Motive Transformation Section () - mm. M A -4 (5) mm. M; Exp/Ext A 6-7 m. M A 4 7-8 (9) mm. M; Exp/Ext A 5 0- m. M; Exp A 6 - m. M; Exp/Ext A 7 - m. Extension Expanded A 8-4 mm. Extension Expanded A 9 5-7 mm. Extension Expanded/ B Augmented 0 8-9 mm. M; Con/Ext/Inversion C 0- mm. M; Exp/Ext/Ret C m. M; Ret/Exp C m. M; Ret/Exp C 4 4-5 mm. M; Ret/Exp/Ext C 5 5-6 m. Extension C 6 6-7 m. Extension/Extended C 7 7-8 mm. Extension/Extended* C *"M"--main motive; "Exp"--expansion of the entire interval span of the motive; "Ext"--extension of the motive; "Extension"--the repetition of the extension alone from the previous phrase; "Extension Expanded"--the statement of the extension alone with interval expansion; "Con"--contraction of the entire interval span of the motive; "Ret"--the motive in retrograde; "Extension Extended"--the statement of the extension alone with an added extension. Any one or a combination of these various categories of motivic transformation is used in Dirge I as indicated in the table above.

80 TABLE VI INTERVAL COUNT OF DIRGE I All Voices Melody Alone Percent- Percent- Type of No. of' age of No. of age of Interval Intervals Usage Intervals Usage enharmonic.8 m 78 5 M 7 9 4 A 4. m 8 5 7 d.8 M 5 0 5.7 A.4. d4.8 P4 9 8 4.6 A4 0 4 7 P5 4 A5.8 m6.8 M6.4 M7.4 d8.8 5 6 m9.8 M9 4 d7.8 d4.8 Interval Totals 5 64

APPENDIX II TABLE VII DIRGE II: TONAL ANALYSIS Measure Tonal Center -7 C-sharp 8-4 C-sharp 5-9 B-f lat 0-8 A-sharp diminished triad with melodic tonal emphasis on F 9-47 F-sharp with melodic tonal emphasis on C-sharp 48-55 C-sharp 8

78 0. %S 000 0er " 0 y0 0 4.r cz, C) 0 ' " CG C} m0n a ese H 0.k en [iiess es e og.i _ - H.. -r -c r4 r-

- -- 8 -E- '.,.H -- Cd -- Ir LC) 4 LL w o V V cl EC Q C

I -i ii I I I E. 8 TABLE VIII DIRGE II: HARMONIC ANALYSIS Z I Interval I NbO Z Interval w v r. C (J 4 0) 0 40 0 0 4 W 0 $ $ $ % Content a 0 $ Content 4 O. Content 4 5 6 7 9 0 4 5 6 7 8 9 0 4 4 4 5 5 6 7 8 8 9 9 0 0 0 I I I 4 p 4 p 4 p 4 s 4 p 4 p 4 n 4 P 4 s P4 P4 M P4 P5 M6 m7 5 6 7 8 9 0 4 5 6 7 8 9 40 4 4 4 44 45 46 47 48 4 5 6 6 7 7 8 8 8 9 9 0 0 4 4 4 p 4 4 n 4 n 4 n n4 n s d p can mnd nsd p s pon nsd4 n s d pm s mnd n 4 nsd mnd nsd mnd nsd P4 m m m m m m m m m m m m m m m m m m m m m m mn

84 TABLE VIII-Continued cd O a ) l > r ;4 ZInterval, k $ SInterval 0 a Content s c Content o.s a ) )o S 4S 0 a a 0S 49 5 n t m 7 7 pmn s t m 50 5 p ns m 74 7 pmnst m 5 6 n m 75 7 pmnst m 5 6 mn d m 76 8 p mn dt m 5 7 p m 77 8 n6 54 8 n t m 78 8 pmn d t.. m 55 9 nt m 79 9 pmnst m 56 0 nt m 80 9 pmnst m 57 p mn dt m 8 9 p4m4ns4t4 m 58 mnsdt m 8 40 p4m4nns4t m 59 n6t m 8 40 p4m4n8s4t m 60 mnsdt m 84 40 p4m4n9sdt4 m 6 rnsdt m 85 4 p4m4n8s5t m 6 p n sdt m 86 4 p6m6n8s8t4 N 6 n4dt m 87 4 p6m6n8s8t M 64 4 pmndt m 88 4 p6m6n8s6t M 65 4 mnsdt m 89 4 p6nm6n8st4 M 66 4 n6t m 90 4 p 6 m6n 9 s 5 d t 4 M 67 5 mnsdt m 9 4 p6m6n9s5dt4 Al 68 5 pnsdt m 9 4 p6m6n8s8t M 69 5 n4d rn 9 4 p6m6n8s7t M 70 6 p mn dt m 94 44 p4m4n0s6t6 A4 7 6 mnsdt mrn 95 44 p6mn8s6t4 M 7 6 pmndt r 96 44 ptmnnst M

85 TABLE VIII--Continued m5ns9t Interval IIk Interval V= cd Content a scontent oo ) w w 0 04 0 4o 97 98 99 00 0 0 0 04 05 06 07 08 09 0 45 45 45 46 46 46 47 47 47 48 48 48 49 49 49 50 50 p 8 m 4 n 8 s 6 d t 4 p6m4ns5t4 p6m~ns4t4 4 p6m6n8 p 6t4 * p6 4 5 4 4 5 4tp6m4ns~t4 p6m4n 4t4 p644 4 pm8nst4 p p 4n9s4 4 p 4 n9s4 4 p 94mn9s4t4 p4 ns6d0t4 jp 5 m5n s 9 t M7 M6 M6 M M6 M6 M6 M6 M6 4 5 6 7 8 9 0 4 5 6 7 8 9 50 5 5 5 5 5 5 5 5 5 54 54 54 55 55 55 I p nns 5 5 t p m n s t. p4m p4mn9s6t4 p p 0m9n7sdt t4 p0m9n 7s6t4 X0 9 7s6t 4 p0 9n7s6t4 p in ns6t 0 9 s6t4

i i f_ I f. _ i { _... f i _. _...,r......._._._..,..,... _..._.r. _..._._,._... _.._......,_...,.._._.s._..._.. f i. - '.. _-.f -_-- - I - - - - - - - - _._...._ --.p._. _.........,.,s.r.,... k i - 86 _ I i q. Co T" w tt w 0 0 Co C) S t 0D 'ft -r-- r - o ft. - - 4M - 0M ; x~j.^..._ M. i 0ft0ft!0 oj Q f~ 0.c Ica (0. 0) to 0 i l erral El l lv) t) ft a m rq-r 000000 ugoftoio a adms

87 TABLE IX DIRGE II: PHRASE STRUCTURE Tonal Notes of the Phrase Measures Length Center Isometric Sonority -7 7 mm. C#C 8-5 8 mm. C# C# 6-6 mm. Bb Bb-Db (A#-C#) 4-9 8 mm. Bb Bb Db (A#-C#) 5 0-40 mm. FF# A#-C#-E 6 4-47 7 mm. C#F A -C#-E 7 48-55 8 mm. C# F#-A#-C#-E-G#

88 TABLE X INTERVAL COUNT OF DIRGE II All Voices Melody Alone Percent- Percent- Type of No. of age of No. of age of Interval Intervals Usage Intervals Usage enharmonic 7 5 m 9 7 7 M 60 9 m 9 M 7 5 7 7 d4. P4 9 9 9 9 d5. P5 7 5 6 6 M6. m7 (75)* Interval Totals 5 99 *s are used only as octave displacements ten-note chords from measures 47-55. These are in the count. in,five- to not included

APPENDIX III TABLE XI DIRGE III: TONAL ANALYSIS Measure Tonal Center G-sharp ---- F-double-sharp B A G-sharp -* F-double-sharp -> G-sharp 4 E 5 B-flat----A 6 D-flat--->C-f lat 7 B-flat - > A -B 8 A-flat 9 C C-flat 0 B-flat E C-sharp A G F-sharp 4 B-flat-A 5 D-flat-4C 6 C 7 A 8 C 9 C 0 G-sharp G B A 4 A-flat 5 G with a melodic tonal emphasis on A 6 G 7 F 8 E>D 9 G-sharp 0 D G-sharp F-double-sharp G-sharp 89

90 'H -? - _._.. --- " - - -.. _. :,_ a _. - 0 H C -- Ii II - 77-0 H w bfl -- ~ ~ " : H- j I-- I I I I Id~ I I I I -~ ----~------4 ~" - - - (5 LL - --------- k

r! 9 TABLE XII DIRGE III: HARMONIC ANALYSIS artqi 4-t> I ()I 000Intervali 0o '4 04 (, (D 0 Content c 0c 0 w Cntn 4 Q 4j Interval 5 N ~ Conten 7 6 P s Q p 6 7 7 7 p in 8 4 P S P5 4 p 444 i n P5 8 7 p mn 5 p44 4 s 4 9 8 7/4 p4d4t4 P5 6 7 8 p 8 4 ps 8 4 4 s 4 0 8 8 8 7/4 7/4 7/4 p4d4t4 p4m4d 4 pdt p4m4 d4 P5 P5 9 p i 8 7/4 p4m4d4 0 4 5/4 5/4 7 pain 4 P 8 p 4 s 4 4 5 6 7 8 8 9 9 9 9 7/4 pind 44 444 p4m6d6 pns p4n 6 6 p4n 6 6 5 6 7 8 9 0 5/4 5/4 4 5 4 s 4 p n s 4 p 8 4 D 8 4 p 4d 4t4 P5 P5 9 40 4 4 4 44 45 0 0 0 0 p 4 m 6 pn p4d6t6 s p0 pns p 4 dtn 6 P P5 P5 p 4d 4t4 P5 46 P5 4 6 p in P5 P5 47 48 p 4 m 6 d 6 4n6s6 p ns P5

9 TABLE XII--Continued 49 50 5 5 5 54 55 56 57 58 59 60 6 6 6 64 65 66 67 68 69 70 7 7 4.~ "0 VContento= FI IfI s~ 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 8 8 8 --- - ---- t - 4 fiinterval ' > 0 A4Interval =nsd J j ~j c 0Conten / / / / / / / p 466 t p 4d66 4 p p 4d66 0 6 p S. p4m6 6 9 p 9 d 9 t 9 8s 9 p S p mn s9 p9d9t 9 ps 9 9s 9 9999 98999 pd pm p9d p p9 n p9mn ppm 9 d lnl p9 n P5 P5 P5 P5 P5 P5 P5 P5 7 74 75 76 77 78 79 80 8 8 8 84 85 86 87 88 89 90 9 9 9 94 95 96 8 9 9 9 9 9 0 0 4 4 5 5 5 5 / / / / / / 4. p m n' p9mn p9mn p nn pmnn p 4 m6n6 466 0 p p u s p 4m 6d6 p 4m 6d6 4096 6 p p mun 0 6 p mun 0 6 p dt p n s > P5 P5 P5 P5

9 TABLE XII--Continued 4-t r.4 q.4 r r. Interval 4> c'interval > 0 n Content n Dos n (D Content 97 6 p 0 0 0 p 4 d 4 t 4 98 6 p4n6s6 0 p4m4n4 P5 99 6 p 0 / p4 00 6 p4m6n6 / p4m4n4 0 7 4 / p4d4t4 0 7 p4m6n6 5 / p8s4-0 8 p4 66 P 6 / p8s4 04 8 p 0 7 05 8 p4m6n6 8 p4m4n P5 06 9 9 p8s4 P5 07 9 p4m6n6 0 p4nm4n P5 08 9 p0s6 p4nm4n P5 09 0 p 4 p44n4

94 4 - ti CI CU N F 0 0 0 6- I..d 0 4 0 CCU ~ T. -H h. 0 C C -H- 'CC Ct? CO - 4 - co ~- lowg C--ri -P4 Ux C C C C Comm o 0A00 $$ $

95 TABLE XIII DIRGE III: PHRASE STRUCTURE Phrase* Measures Length Motive First Pitch Section - 4 mm. Ixy B A 4 II 5-7 4 mm. Ixy D A 8 II 9 mm. Ix A A 0 II inv 4 mm. Ix G A II inv 5 m. Iy G#-A B 6 4 m. Iy B-C B 7 5 m. Iy D-Eb B 8 6 m. Iy D-Eb B 9a 6 m. Iy Eb-E C 9b 7 tin. Iy Eb-E C 9c 8 m. Iy Eb-E C 9d 9 m. Iy Eb-E C 0 9-0 mm. III E-B D - mm. III E-G# D -4 mm. III G-Bb D 5-6 mm. II, III A G B D 4 7-9 mm. IX, II A E expg# 5 0 m. III G#F E 6 m. Il inv ** B E 7 - mm. Iy*** B E * Note how all of the motivic elements are stated in groups of fours throughout Dirge III. ** Note how the inversion of motive II has been extended by one extra note which gives it a strong resemblance to motive III. *** "Ix"--pedal note; "Iy"--ascending minor second, returning to original pitch; "II"--generative motive, descending minor second and ascending minor third; "II inv" --motive II inverted; "II exp"--motive II expanded; "III" -- three quarter notes and a whole note all descending by seconds and thirds.

96 TABLE XIV INTERVAL COUNT OF DIRGE III All Voices Melody Alone Type of No. of Percentage No. of Percentage Interval Intervals of Usage Intervals of Usage m M A d m M d4 P4 A4 06 5 0 8 0 8 4.4 8 7 8.8 7 0 9 4 48 4.5 7 6 d5 A5 M6 m7 4 4 6 4 6.6.6.6 Interval 46 6 Totals

i APPENDIX IV TABLE XV DIRGE IV: TONAL ANALYSIS, Measure - 4 5 6 7 8 9 0. 4-5 6-9 0-7 8 9 0 4-9 Tonal Center* G G (F diminished) G G (A-sharp diminished)--g G G (F diminished) G A- B F F (E diminished) D-flat D-f lat (C diminished) A-flat G diminished G diminished (F diminished) B diminished A (C-sharp) A (C-sharp)----A A (B) A (C- C-sharp)-- A A (B) A (G diminished) C-sharp -t Measure 40-4 4-4 44-47 48 49 50 5 5 5 54 55 5 E 57 5 5 6 6 6 6 6 6 66 6 6 6 70 *The parentheses indicate me: stresses tonal centers different : paniment which are considered to gravity due to their chords of ex ation. --- r,«... Tonal Center* G E D D (A-sharp) D D (A-sharp) D (G-sharp) D (B-flat----> E-flat) F augmented- F A-flat G-flat F-flat G D (F augmented- 4 F) A-flat G-flat F-flat G D (F augmented---' F) D (F augmented) D (F) C-sharp B-flat--- F diminished G G (C-sharp) G (A-sharp--- C-sharp) odic activity which rom those of the accomiave precedence of tonal :ended range and long dur- 97

- -. - - s e 98 I-- '4 -, --- -,- - - - - - - 4 - "- - - -- - - - - CH 0-4,...-- r-------- --- - - - - - --- " Fi Co to CO b L) K) I~iT - -I -- -~ -II ILILLjLLi _ LiL LLL L i~l