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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, 1980

5 9 BELA BARTOKT S FOUR DIRGES FOR PIANO, OP. …/67531/metadc504426/m2/1/high... · North Texas State University in Partial ... Bartok's piano compositions of this peri-od have been

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  • 5 9




    Presented to the Graduate Council of the

    North Texas State University in Partial

    Fulfillment of the Requirements

    For the Degree of



    John W. Terrell, Jr., B.A.

    Denton, Texas

    August, 1980

  • Terrell, John W . , Jr., Bela Bartok' s Four Dirges for

    Piano, Op. 9a: A Complete Analysis. Master of Music

    (Theory), August, 1980, 108 pp., 18 tables, 64 illustra-

    tions, bibliography, 10 titles.

    The study of Bela Bartok's Four Dirges for piano (1909-

    1910) is significant in that this period of 1908 to 1910

    was particularly experimental and formative for Bart6k, es-

    pecially in tonal aspects of his compositional style. Fur-

    thermore, very little research and analysis has been done

    on these smaller works. This thesis contributes an analy-

    tical 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 over-

    all characteristics of the dirges.



    LIST OF TABLES . . . . . . . . . . ........... V

    LIST OF ILLUSTRATIONS . . . . . . . . . . . . . .... vi



    II. ANALYTICAL METHOD . . . . . . . . . . . . . . 6

    Tonal AnalysisHarmonic AnalysisMelodic AnalysisPhrase Analysis, Rhythm, and Form

    III. ANALYSIS: DIRGE I . . . . . . . . . . . . . . 15

    TonalityHarmonyMelodyRhythm and Phrase StructureFormTextureDynamicsRange

    IV. ANALYSIS: DIRGE II . . . . . . . . . . . . 23

    TonalityHarmonyMelodyRhythm and Phrase StructureFormTexture, Dynamics, and Range

    V. ANALYSIS: DIRGE III . . . . . . . . . . . . . 34

    TonalityHarmonyMelodyRhythmPhrase AnalysisFormTextureDynamics and Range


  • PageVI. ANALYSIS: DIRGE IV........ ...... ........ 49

    TonalityHarmonyMelodyRhythmPhrase Analysis and FormDynamics and Range


    Unifying ElementsCommon Characteristics

    APPENDICES ... " . . . - -- -74





    I. The Formal Employment of the ArithmeticCenter and the Golden Section in






    Each of the Four Dirges

    II. Summary of Analysis: The Four Dirges

    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 . . . .

    III. Dirge II: Harmonic Analysis.. . . . .

    IX. Dirge II: Phrase Structure . . .

    X. Interval Count of Dirge II . . . . . .

    XI. Dirge III: Tonal Analysis . . . . . .

    KII. Dirge III: Harmonic Analysis . . .

    III. Dirge III: Phrase Structure . . . . .

    XIV. Interval Count of Dirge III

    XV. Dirge IV: Tonal Analysis

    XVI. Dirge IV: Harmonic Analysis . . . . .

    VII. Dirge IV: Phrase Structure . . .

    III. Interval Count of Dirge IV . . . . . .

    . . . . 70

    . . . . 72

    . . . . 74

    . . . . 76

    . . . 79

    . . . . 80

    . . . . 81

    . . . . 83

    . . . . 87

    . . . . 88

    . . . . 89

    . . * . 91

    . . . . 95

    . . . . 96

    . * * . 97

    . . . . 99

    . * . * 104

    .* . * -106


    . ." .! .
























    Relationships of Perfect Fifths

    The Axes . . . . . . . . .-....-...

    Gradation of Harmonic Tension .

    The Golden Section . . . . . . . . . .

    Main Motive, mm. 2-3 . . . . . . . . .

    Mm. 3-5 . . . . . . . . e ! - -

    Motivic Development in mm. 10-17 . .

    Inversion of the Motive . .

    Motive in Retrograde, mm. 20-23 .

    Isometric Sonority . . . .. . . . . .

    Theme of Dirge II. .".

    Third Phrase...... . . . . . . --.

    Fifth Phrase. . . . . . . . . . -.-.

    Phrase Analysis of Dirge II . .

    Bass Clef Tonal Phrase Endings in theThird Dirge........ ........ -.

    Treble Clef, Motive I, mm. 1-3 . . .

    Treble Clef, Motive II, m. 4 . . . ..

    Treble Clef, Motive III, mm. 19-20 .

    Bass Clef, mm. 1-2......... . . .

    Treble Clef, mm. 9-10.. . .

    Treble Clef, mm. 13-16 . . . . . . ..






    . . . . 12

    . . - - 17


    . . . . 18

    . . . . 18

    . . . . 19

    . . . . 24

    - - - . 26

    - - - . 26

    - - - . 28

    . . . . 31

    . - - . 34

    . . . . 38

    . . . . 38

    . . . . 38

    * . . . 38

    . . . . 39

    . . . - 40

  • Figure



    Treble Clef, mm. 19-24...... . ....... 41

    23. Treble Clef, mm. 25-26 . . . . . . .

    24. Treble Clef, mm. 27-29 . . . . . . .

    25. The Augmented Second . . . . . . . .

    26. Treble Clef, mm. 4-5 . . . . . . . .

    27. Treble Clef, mm. 31-32 . . . . . .

    28. Bass Clef, mm. 1-4 . . . . . . . . -

    29. Bass Clef, mm. 13-15 . . . . . . . .

    30. Bass Clef, mm. 16-18 . . . . . . . .

    31. Overlapping Tonal Functions in Dirge

    32. Harmonic Pattern of Consonance andDissonance in Dirge IV . . . . . .

    33. Motive Ia-b, mm. 1-5 . . . . . . . .

    34. Mm. 10-13 . . . . . . . . . ......

    35. Mm. 14-18 . . . . . . . . . . . . . .

    36. Mm. 23-29 . . . . . . . . . . . . ..

    37. Mm. 30-33 . . . . . . . . . . . . ..

    38. Mm. 34-38.. . . . . . . . . . . . .

    39. Mm. 44-50 . . . . . . . . . . . . . .

    40. M. 51 . . . . . . . . . . . . . . . .

    41. M. 52 . . . . . . . . . . .. ...

    42. M. 53 . . . . . . . . . . . . . .. .

    43. Mm. 54-57 . . . . . . . . . . . . . .

    44. M. 58 . . . . . . . . . . . . . . . .

    45. Mm. 63-65 . . . . . . . . . . . . . .











    . 52


    . 53

    . 54

    . 54




    s 57



    - 58. 58

    - 59

  • Mm. 66-67 . .

    47. Mm. 68-70..........................

    48. The Axis System of the Four Dirges

    49. Tonal Scheme of the Four Dirges inRegard to the Axis System ...

    50. Intervallic Similarities in the Motiof the Four Dirges . . . . . ..

    51. Motivic Relationship Between Dirgesand III . . . ". . . . " -

    52. Motivic Relationship Between Dirgesand IV........ ...... .......

    53. Relationships of Meter and Tempo

    54. Tonal Symmetry in Each Dirge . .

    55. The Hyperbolip Form in Each of theFour Dirges . . . . . .

    56. Characteristic Descending Intervals

    57. Dirge I: Graph of Tonal Analysis








    . " " s - 60

    . .63

    . .




    Dirge I: Graph of Harmonic Analysis

    Dirge II: Graph of Tonal Analysis

    Dirge II: Graph of Harmonic Analysis

    Dirge III: Graph of Tonal Analysis

    Dirge III: Graph of Harmonic Analysis

    Dirge IV: Graph of Tonal Analysis

    Dirge IV: Graph of Harmonic Analysis



    . . . 65

    . - - - - 66

    . .r . . . 67

    . .e . . . 67

    S - - 69

    . . . . 70

    . .r ." . . 75

    . . . 78

    . w . 82

    . . . 86

    . . . 90

    .. . 94

    . . . 98

    . .5 . 103'








    Bela Bartok was born on March 25, 1881, in Nagyszent-

    miklos, Hungary, where his early years were beset by ill-

    ness which interrupted his studies and social activities

    (3, pp. 3-4). This established his preference for isola-

    tion with nature which continued throughout his life (1,

    p. 63). Having shown an interest in music, he began receiv-

    ing piano lessons from his mother in 1886 (3, p. 5). In

    1892, at the age of eleven, Bartok gave his first public

    recital which included his own composition, "The Flow of

    the Danube" (5, p. 20).

    Two academic influences affected Bartok in his youth.

    His piano instructors in Pozsony (1894-1898), Laszlo Erkel

    and Anton Hyrtl, directed him to the music of Brahms and

    Dohnanyi (3, p. 9). Later he attended the Budapest Academy

    of Music where he studied piano with Istvan Thoman and com-

    position with Hans Koessler, and was permanently influenced

    by the music of Liszt and Wagner (3, p. 12).

    Following an inactive period at the Academy, Bartok

    was inspired to resume composition in 1902 when he encoun-

    tered the music of Richard Strauss, particularly Also Sprach

    Zarathustra (5, pp. 37-38). During the next three years,


  • 2

    Bartok's concern for the decreasing political power of Hun-

    gary and her increasing economic and political dependence

    on Austria directed his compositional style toward nation-

    alistic traits (5, pp. 42-43). Although touring as a virtu-

    oso 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 Hun-

    garian in character, and to compose music in a new style

    which would free him from nineteenth-century German romanti-

    cism (5, pp. 57-58). His patriotism for Hungary led to the

    study of her folk music, and upon discovering that the Hun-

    garian "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 out-

    side world with Hungarian folk music (5, p. 60).

    Bartok ventured on his first folk song collecting tour in

    Bekes county in 1906 (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

  • 3

    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. 71).

    When, in that very same year, Kodaly urgedme to study Debussy's works, I was very much sur-prised to find that pentatonic turns identicalwith those found in Hungarian folk music playeda prominent part in his melodic construction (5,p. 71).

    In 1905-1907, Bartok fell into a period of intense con-

    templation 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, increas-

    ing his understanding of folk music (5, pp. 74-75).

    In a narrower sense, peasant music is theresult of the reshaping work of a natural forceoperating unconsciously; it is the instinctivecreation of a human mass without artificiality.It is a natural phenomenon, just like the vari-ous forms of the animal or vegetable kingdom.As a result, its individual organisms--the melo-dies themselves--are examples of the highestartistic perfection (5, p. 76).

    Bartok determined in these years that folk music would

    play an essential part in his compositions (5, p. 79). Al-

    lowing 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 through-

    out his life (5, p. 79).

    The years 1907-1911 represent the core of Bartok's com-

    positional development, a time of experimentation and

  • 4

    synthesis (5, p. 85). Three aims guided his musical devel-

    opment in these years: (1) to draw out the essential prin-

    ciples of Western classical music, (2) to develop new ideas

    based on folk music, and (3) to synthesize these two ele-

    ments (5, p. 85). Bartok's piano compositions of this peri-

    od 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 (2, pp. 88-89).

    These two elements originate respectively from the influ-

    ence of Western music on Bartok and from his increasing

    knowledge of the Eastern expression in the Hungarian folk

    songs (5, pp. 89-93). His compositions from these forma-

    tive years begin to draw elements from these two musical

    expressions, as in the Bagatelles, Ten Easy Pieces, and

    Sketches (5, pp. 80-81). 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. 432-433).

    The Four Dirges from 1909-1910 are considered by the

    biographer, Jozsef Ujfalussy, a special achievement of Bar-

    tok in synthesizing both elements of his dual expression

    (5, pp. 93-94). This accomplishment of unifying these ele-

    ments within each of these early pieces casts an influence

    on larger mature works in subsequent years.


    1. Crow, Todd, editor, Bartok Studies, Detroit, Michigan,Detroit Reprints in Music, 1976.

    2. Lendvai, Erno, Bela Bartok: An Analysis of His Music,London, Kahn and Averill, 1971.

    3. Stevens, Halsey, The Life and Music of Bela Bart6k,2nd ed., New York, Oxford University Press, 1964.

    4. Suchoff, Benjamin, editor, Bela Bart6k Essays, London,Faber and Faber, 1976.

    5. Ujfalussy, Jozsef, Bela Bart6k, Budapest, Hungary,Crescendo Publishing Company, 1971.




    Tonal Analysis

    The Western and Eastern influences in Bart6k's youth

    are analogous to the two major categories in twentieth-

    century tonal composition--tertian and nontertian (4, p.

    322). 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 (1) sonority doubling, (2) pitch dura-

    tion, including pedal tones, (3) 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 cen-

    ters even when tertian harmony occurs. Occasional conflict-

    ing 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 re-

    flect tonal organization which is then compared to formal


  • 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 (2, pp. 1-16) which describes a series of tones

    in the circle of perfect fifths, as a sequence of subdomi-

    nant, 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 1).

    5t C 5tF Tonic G

    Subdominant Dominant

    Fig. 1--Relationships of Perfect Fifths

    The sixth scale degree is the relative of the original ton-

    ic 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 2 below.


    ET T A

    ANO TNs s E

    Fig. 2--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 dis-

    tance of a tritone.

    The axis system is frequently observed in Bartok's mu-

    sic in many different ways: as melodic, harmonic, and to-

    nal 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 Har-

    monic Materials of Modern Music (1, pp. 7-16) is the method

    of chord description used in this analysis. A sonority

    will be considered consonant only when it contains any com-

    bination 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 unison (and octave) repre-

    sents 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 organ-

    ized sonorities (1, pp. 17-24). 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 3. [Note: This is the author's own appli-

    cation of the "Analysis of Intervals" of Howard Hanson

    Dissonance T3 . . . Sonorities with three dissonanttypes--t, d, s

    Dissonance T2 . . . Sonorities with two dissonanttypes--t and d or s

    Dissonance T1 . . . Sonorities with only one dissonanttype--t

    Dissonance D2 . . . Sonorities with two dissonanttypes--d, s

    Dissonance Di . . . Sonorities with only one dissonanttype--d

    Dissonance S . . . Sonorities with only one dissonanttype--s

    Consonance . . . . Sonorities with only p, m, or nUnison . . . . . . Single pitch (or octaves)

    Fig. 3--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

  • 10

    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); sec-

    ond, the intervals in the melodic line alone are counted

    (excluding all sonority doublings). All intervals sepa-

    rated 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 Appen-


    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 character-

    istics are considered in two particular places: (1) the

  • 11

    imitative transition in measures 23-28 in Dirge IV, and

    (2) the relationship between the melodic line and the har-

    monic accompaniment throughout Dirge III. Both of these

    instances most strongly affect the harmonic contrast of con-

    sonance 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 re-

    garded as a synthesis of all the analytical elements, in-

    cluding dynamics, texture, and range. A table which sum-

    marizes the phrase and form analysis of each dirge is

    located in the Appendices.

  • 12

    Golden Section

    Erno Lendvai defines the golden section as "the divi-

    sion 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 (2, p. 17).

    = X(

    Fig. 4--The Golden Section (2, p. 17)

    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 0.618. The smaller part of the

    proportion is located by multiplying the total number of

    measures or beats by 0.382.

    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, 2, 3,

    5, 8, 13, 21, 34, 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

  • 13

    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 pro-

    duced 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

    61.8 percent of the composition is passed. Bartok uses both



    1. Hanson, Howard, Harmonic Materials of Modern Music, NewYork, Appleton-Century-Crofts, Inc., 1960.

    2. Lendvai, Ern , Bela Bartok: An Analysis of His Music,London, Kahn and Averill, 1971.

    3. Ujjfalussy, Jozsef, Bela Bart6k, Budapest, Hungary,Crescendo Publishing Company, 1971.

    4. Wittlich, Gary E., coordinating editor, Aspects ofTwentieth Century Music, Englewood Cliffs, NewJersey, Prentice-Hall, Inc., 1975.





    A quasi-symmetrical structure is established in Dirge

    I by recurring tonalities on each side of, and in approxi-

    mate equidistance to, the midpoint of the piece. Although

    the minor third (d-sharp to f-sharp) in measure 1 suggests

    D-sharp minor, measure 2 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

    13-14 on A-flat, the only reference to this tonal center

    in the dirge. The tonality of A occurs before and after

    measures 13-14, once in measures 8 and 9, returning firmly

    established in measures 15-16. 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 12 and measure 19, C (also


  • 16

    lying on the dominant axis) in measures 10-11 and measure

    17, D-flat (C-sharp) in measures 5-7 and measure 20, and G

    in measure 4 and measure 21. All of these tonal recur-

    rences are shown on the graph of tonal centers for Dirge I

    in Appendix I and for the other dirges in their correlating



    The harmonic structure of Dirge I is produced by brief

    moments of dissonance which occasionally interrupt the con-

    sonant character of the piece. The particularly dissonant

    moments occur in measures 11, 14, and 24, each a result of

    the incongruity between the melodic line and harmonic accom-

    paniment. The dissonance in measure 14 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, 12, 13, 17, 19, and 21, each

    marking the end of a phrase. The unison in measure 17 is

    significant since it not only establishes the dominant func-

    tion 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.


    The interval count reveals the stepwise character of

    Dirge I. Of all the 252 intervals, 78 (31 percent) are

  • 17

    minor seconds. There are 38 (15 percent) minor thirds, 27

    (11 percent) major seconds, and 25 (10 percent) major thirds.

    Of the 64 intervals in the melodic line, the minor second

    (23 percent), major second (14 percent), and minor third

    (17 percent) are the three most frequently occurring inter-

    vals. Next to these is the augmented fourth which occurs

    seven times (11 percent). Refer to Appendix I for a com-

    plete 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 2-3 (a# T b Jg) hasan arch contour (see Figure 5). This motive is expanded in

    Fig. 5--Main Motive, mm. 2-3

    measures 3-4, containing two more notes than before and

    reaching a higher peak to d2 before returning to g (seeFigure 6).

    3 4 5 T

    Fig. 6--Mm. 3-5

  • 18

    This entire pattern is sequenced a major second higher

    in measures 6-8, each phrase ending on al. Measure 10 be-

    gins a developmental treatment of the motive which is marked

    by sonority doubling and "mezzo forte sempre crescendo," con-

    tinuing to heighten the peak with each subsequent statement:

    measure 10--f# 2 , measure 11--a2 , measure 12--cb3 , measure 13

    --fb3, measure 15--g# 3 . Each repetition of the motive re-

    turns to c2 , forming ever widening intervals. The narrow

    descent of the augmented fourth in measure 10 is gradually

    expanded to a descending augmented twelfth in measures 15-

    17 (see Figure 7). Measures 18-19 state the original motive

    in inversion with a two-note extension, beginning on c3 (see

    Figure 8). Measures 20-21 state the motive in a modified

    J 1 ' t4 15I

    Fig. 7--Motivic Development in mm. 10-17

    Fig. 8--Inversion of the Motive

    retrogression with an extension of one note. A pure retro-

    grade of the motive would consist of a rising major third

  • 19

    followed by a descending major second. In measures 20-21,

    however, the major third is widened into a perfect fourth,

    b2 up to e3 . The remaining seven measures of the piece con-

    tinue 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 22 and

    23. The statement in measure 23 expands the original in-

    terval of a major third even further to a major sixth, b2

    up to g#3 , and the minor second is widened to a minor third,

    g#3 to e#3 (see Figure 9).

    Fig. 9-Motive in Retrograde, mm. 20-23

    In measures 24-28, the harmonic interval of D-sharp to

    F-sharp from measure 1 is transformed into a melodic inter-

    val 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 26-27) 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 mea-


  • 20

    Rhythm and Phrase Structure

    There are seventeen phrases of which all but one con-

    form 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 augmenta-

    tion of this rhythmic pattern, consisting of half notes fol-

    lowed 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 15-17 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 2/2 and 3/2 alternate frequently until mea-

    sure 15 where 2/2 continues throughout the rest of the

    piece. This supports the significance of measures 14-16 as

    a tonal vertex and as a place of melodic transformation.


    Of the twenty-eight measures in Dirge I, the first

    fourteen state and develop the opening three-note motive of

    measures 2-3, culminating in the core ninth phrase of mea-

    sures 15-17. In measures 18-28, a reversal of the melodic

    treatment in the first half occurs in two ways: (1) the

    phrases are gradually shortened (instead of being length-

    ened), and (2) the motive begins to ascend instead of de-

    scending due to its inverted statement in measure 18 and

  • 21

    its retrograde version with extension in measures 20-28.

    Although these changes tend to group the work into these

    three parts, measures 1-14, 15-17, 18-28, they comprise,

    nevertheless, one continuous undivided composition.


    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 10,

    a third pitch and its octave enter, completing a full triad.

    Here, the melodic line also becomes octave reinforced. In

    measure 15, the melodic line is stated in three octaves

    above a seven-voice accompaniment. The texture thickens

    further in measure 25 when close position octave reinforced

    triads in the lowest register.


    The dirge begins and ends pianissimo, rising to fortis-

    simo in measure 15 (the core of the composition). Although

    the dynamic level descends after measure 15, it rises to

    forte in measure 25 to support the return to B major. A

    return to pianissimo closes the dirge.


    The range widely extends from BBB to C# 4. The first

    fifteen measures develop a continual rise of melodic peaks,

  • 22

    culminating in the core ninth phrase in measure 15 on g#3

    above the dominantly functioning tonal center of A. This

    pitch returns in the last two measures above the tonal cen-

    ter of B. The only pitch occurring higher than g# 3 is c#4

    from measure 24, propelling the second, less climactic, rise

    in dynamics which reaches its peak in measure 25.




    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 8-14. Measures 15-29 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 14. A third pitch, e, is

    added to the thickening pedal at measures 30-39, deriving

    again from the last note of the preceding phrase in measure

    29. 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. Mea-

    sures 30-39, as a result, border on bitonality. The pitch f-

    sharp is emphasized in measures 37-39 and is added to the

    pedal accompaniment in measure 39, 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 accompan-

    imental pedal above in measure 41. The tonal strengthof the


  • 24

    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 iso-

    metrical principle is that the entire vertical sonority pro-

    duced by these pitches contains the same order of intervals

    whether spelled from one end or the other, as shown in

    Figure 10 (1, p. 19).

    G# A#4--C#---->E F#P4 m3 m3 P4

    m.47 m.13 m.28 m.39

    G# -- + A# -- + C# -- B E--- F#M2 m3 m3 M2

    F#--- E --- > C# -- >A# --- + G#

    Fig. 10--Isometric Sonority

    Of further significance is the entrance point of the

    e into this thickening isometric sonority. Of the four

  • 25

    pitches which are added to the c-sharp pedal throughout the

    dirge, note that the e is the second in order, entering at

    measure 28 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 further-

    more isometric within itself.


    The harmony of Dirge II gradually grows more dissonant

    after the consonance of octave unisons in measures 1-7.

    The c-sharp pedal in measures 8-14 produces occasional mo-

    ments of mild dissonance with the melodic line. The minor

    third pedal of b-f lat and d-flat in measures 15-29 causes

    stronger and more frequent dissonant intervals such as the

    minor second, major seventh, and tritone. The tritone

    first occurs in measure 25 and again in measure 28 to re-

    main present throughout the remainder of the piece. With

    bitonal implications between the accompaniment and the

    melody in measures 31-39, 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 unison 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 unison in

  • 26

    measure 48 lies on c-sharp, affirming a return to that to-

    nality amidst the tone clusters.


    The theme of Dirge II is stated seven times and is

    characterized by its smooth contour of mostly stepwise in-

    tervals. There are 99 intervals in the melodic line of

    which 32 are major seconds, 19 perfect fourths, 17 minor

    seconds, and 13 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 mea-

    sures (see Figure 11). The tonality shifts to B-flat in

    Fig. 11--Theme of Dirge II

    measures 15-30 where the theme first varies from the origi-

    nal statement (see Figure 12). Since the rhythm of the

    t5 16 IT [$9 201

    Fig. 12--Third Phrase

  • 27

    third statement is so similar to the first (except for the

    diminution of measures 6-7 into a single measure at measure

    21), comparison of thematic cells between the two state-

    ments can be specific. The descending perfect fourth in

    measure 1 is diminished into a minor second in measure 16,

    while the ascending major second of measure 2 is augmented

    into a minor third in measure 17. Although the third state-

    ment 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 3 are in-

    verted into three ascending quarter notes in measure 18.

    Note that the total interval span in measure 3 is a descend-

    ing perfect fourth, while the distance in measure 17 is the

    inversion of that interval, the perfect fifth. The ascend-

    ing major second in measure 5 becomes a descending minor

    third in measure 20. The descending perfect fourth in mea-

    sures 6-7 becomes a descending major second in measure 21

    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 28-29) through the descend-

    ing minor second which refers to the same cadence in the

    third statement in measure 21 with a major second.

    The fifth phrase in measures 30-40 contrasts the third

    and fourth phrases by employing many perfect fourths which

  • 28

    obscure the tonal center (see Figure 13). Measures 31-37

    contain the fifth statement and are followed by a three-

    Fig. 13--Fifth Phrase

    measure extension (measures 38-40) before the sixth phrase

    enters in the bass clef at measure 41. The descending per-

    fect fourth in measure 31 is regained from measure 1 while

    measure 32 expands the ascending major second from measure

    2 into an ascending perfect fourth. The contour of measure

    33 is another inversion of measure 3 but is intervalically

    different from that first statement and from its first

    transformation in measure 18. Notice, however, that the

    interval span of all three quarter notes in measure 33 is

    an ascending perfect fourth which is the inverted direction

    of the perfect fourth spanned in measure 3. Measure 34 is

    a diminution and contraction of measures 31-32 and corres-

    ponds to measure 19 as employing widened intervals of per-

    fect fourths and an extension of one extra quarter note on

    the third beat. Measure 35 and the first two beats of mea-

    sure 36 immediately repeat measures 33-34. The ascending

    leap of a major sixth from c2 to a2 in the second and third

    beats of measure 36 places the range of the melodic line at

    its highest peak in the dirge. Measure 37 corresponds with

  • 29

    measure 19 and measure 21 of the third phrase with its de-

    scending major second and the rhythm of a quarter note fol-

    lowed by a half note. Measures 38-40 extend this phrase by

    repetition of measures 36-37. The only change in the melody

    is that the c2 on the second beat of measure 36 is changed

    to a db2 . Since this pitch is enharmonic to c#2 , 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 39.

    Although the fifth phrase and its extension are suc-

    ceeded by the sixth phrase in measure 41, 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 41-49. Just as measures 38-40 represent a "cut-

    ting" of the small phrase of measures 35-37.(cutting the

    opening material of measure 35), measures 41-47 cut from

    the beginning of the phrase as stated in measures 38-40.

    There are two statements of the last three notes in the

    phrase a2 jg#2 4f#2 in measures 41-45, and in measures 45-

    49 there are two statements of only the descending major

    second of g#2 to f#2 . 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 mea-

    sures 48-55 which cadences strongly with an augmentation of

    the last two notes which are changed to a descending perfect

  • 30

    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 3/4 meter.

    The rhythmic pattern occasionally varies only slightly from

    the first statement of the theme in measures 1-7. The re-

    peating 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 31-40 is stated

    almost entirely in quarter notes, the iambic character is

    not obscured due to its strong rhythmic force. This rhyth-

    mic pattern cannot be easily distorted by the limited choice

    of time values with which the piece is composed--the quar-

    ter, 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

  • 31

    region to C--sharp, reinforcing that tonal center when it

    returns in measure 41 (see Figure 14).

    Phrase 1: 7 measures 1-7 C#Phrase 2: 7 measures 8-14 CPhrase 3: 7 measures 15-21 BbPhrase 4: 8 measures 21-29 BbPhrase 5: 11 measures 30-40 F - F#Phrase 6: 7 measures 41-47 C#Phrase 7: 8 measures 48-55 C#

    Fig. 14--Phrase Analysis of Dirge II


    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 30-40 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 on the tonal center C-sharp, while the second and third

    lie in relations of thirds to both sides of that tonal cen-


    The isometric growth of the accompanimental sonority

    is half completed at the center of the piece in measure 28

    when e is added to the sonority, resulting in the

  • 32

    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 en-

    ters in measure 39 and g-sharp enters in measure 47, result-

    ing 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 10, p. 24).

    Texture, Dynamics, and Range

    Dirge II contains two separate fabrics of music--a

    melodic line and a continually thickening harmonic accom-

    paniment, concurring with a rise of dynamic level, and a

    widening of range. This dirge is constructed on one con-

    tinuous gradation of dissonance, texture, dynamics, and

    range while gradually developing the isometric harmonic

    sonority of the accompaniment. The climactic phrase coor-

    dinates these elements at measure 53, 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.


    1. Hanson, Howard, Harmonic Materials of Modern Music, NewYork, Appleton-Century-Crofts, Inc., 1960.





    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 to-

    nal 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


    Beginning on1. mm. 1-42. mm. 5-83. mm. 9-104. mm. 11-125. m. '136. m. 147. m. 158. m. 16

    Fig. 15--Bass CleDirge.

    G# '9.E 10.Ab 11.Bb 12.A 13.P# 14.A 15.C 16.C 17.

    f Tonal Phrase

    m. 17mm. 18-19mm. 20-21mm. 22-23mm. 24-26mm. 27-28mm. 29-30m. 31mm. 32-33

    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


  • 35

    occurring tonalities, G and A, lie a minor second above and

    below this central tonality and are employed six and seven

    times respectively. Measure 17 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 17. The tonality of C occurs on either

    side of this vertex in measures 15-16 and in measures 18-

    19. The tonality of A occurs again five measures preceding

    measure 17 at measure 12, and six measures following mea-

    sure 17 at measure 23. Since A and C both lie on the domi-

    nant axis when G-sharp is tonic, the area encompassing

    measures 12-23 functions dominantly. Subsequently, the

    outer boundaries surrounding this area lie on tonal centers

    of the subdominant axis. Seven measures preceding the ver-

    tex at measure 10 is the tonality of B-flat which is the

    interval of a major second above G-sharp. Eight measures

    following the vertex in measures 25-26, lies the tonality

    of G which is the interval of a minor second below G-sharp.

    Extending the outer perimeters even further to thir-

    teen measures on either side of measure 17, lie two differ-

    ent tonal centers which themselves are only a major second

    apart. In measure 4, the tonality of E lies on the sub-

    dominant axis and is a major third below G-sharp. In mea-

    sure 30, the tonality of D lies on the tonic axis (in

  • 36

    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 func-

    tioning tonal center of G-sharp.


    The open fifths in the bass clef create only consonant

    harmony. The consonant character of the dirge is often in-

    terrupted by recurring dissonance produced by the pitch re-

    lationship 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 13-18, where the dichotomy of consonance and dis-

    sonance is especially felt. In these five measures, five

    dissonant sonorities alternate with four consonant sonori-

    ties, creating tension by abrupt contrast. Measures 13-18

    proceed through the center of the composition at measure 17

    where the tonal centers of the dominant axis occur and where

    the melodic line is most tense.

  • 37


    Dirge III is largely constructed with stepwise inter-

    vals of major and minor seconds which equal 138 (56 percent)

    of the total 246 melodic intervals. There are 106 (43 per-

    cent) minor seconds alone, 32 (13 percent) major seconds,

    20 (eight percent) minor thirds, and 18 (seven percent)

    major thirds. There are 20 diminished fourths which sound

    as major thirds. These diminished fourths combined with

    the major thirds total 38 (15 percent) of the intervals.

    There are 22 (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 15-19, the arithmetic

    center of the piece. The treble clef melodic line uses

    three motivic ideas: (1) a repeating pitch which occasion-

    ally rises a minor second as a neighboring tone (see Figure

    16); (2) a generative motivic cell which consists of a de-

    scending minor second followed by an ascending minor or ma-

    jor third in a three-note pattern which leads motive I onto

    higher pitch levels (see Figure 17); (3) a descending figure

    of three quarter notes and a whole note with intervals of

    seconds and thirds (see Figure 18).

    Motive I is first stated in measures 1-3 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.

  • 38

    2 3

    Fig. 16--Treble Clef, Motive I, mm. 1-3


    Fig. 17--Treble Clef, Motive II, m. 4

    Fig. 18--Treble Clef, Motive III, mm. 19-20

    This three-note generative motive is derived from the first

    three sonorities of the bass clef melodic line in open

    fifths (see Figure 19). Motive II not only generates the

    Fig. 19--Bass Clef, mm. 1-2

    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,

  • 39

    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 9-10

    (see Figure 20). A fourth statement follows in measures

    11-12, precisely repeating the previous statement on the

    enharmonic pitch of g-sharp. The a-flat stated in measures

    9-10 is derived from measure 8 where it was the root of an

    9 '':'7

    Fig. 20--Treble Clef, mm. 9-10

    a-flat major triad. The same pitch in measures 11-12, how-

    ever, 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 13, hence g-sharp to a. This

    function is confirmed when the same relationship is se-

    quenced on the second beats of measures 14-15 where an

    ascending minor second resolves into the third of an a mi-

    nor triad and the third of a c minor triad respectively.

    The ascending minor second, the motivic cell from mo-

    tive I, is absent when motives I and II are combined in

    measures 8-12. Subsequently, four statements of this single

    ascending minor second motivic cell follow in measures 13,

    14, 15, and 16 (see Figure 21). Each of these statements

  • 40

    Fig. 21--Treble Clef, mm. 13-16

    is sequenced a minor third higher than the previous one so

    that from measure 13 to measure 17 a stepwise progression

    up to e3 might intensify this climactic peak in the compo-


    Measures 17-19 simply alternate e-flat with e-natural,

    isolating the minor second which is the principal interval

    of the composition. These important measures which repre-

    sent the motivic core of the dirge lie precisely between

    the arithmetic center, measure 17, and the golden section,

    measure 20.

    Motive III does not occur until measure 19, breaking

    away from the restricting two-note pattern of motive I.

    This motive descends by step the span of a perfect fourth

    from e3 in measure 19 to b2 in measure 20. Note that this

    first statement of motive III ends at measure 20, the mo-

    ment of golden section, on the unison 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 21-22, ex-

    panding the descent to a minor sixth from e3 to g#2 , and to

    major sixth in measures 23-24 from g3 to bb2 (see Figure 22).

  • 41

    22 24

    19 1' 2a Q

    Fig. 22--Treble Clef, mm. 19-24

    The melodic contour of measures 25-26 constructs an arch by

    combining the generative uplift of motive II (on b-flat)

    with motive III (g3 descending to bb2 ), as shown in Figure

    23. Measures 27-29 state a condensation of motive I com-

    bined with motive II (see Figure 24). The single descend-

    ing melodic augmented second in measure 30 is not only a

    Fig. 23--Treble Clef, mm. 25-26

    Fig. 24--Treble Clef, mm. 27-29

    cell from motive III (measure 19; 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 25).

  • Fig. 25--The Augmented Second

    The motivic statement in measure 31 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 26). When the direction of the motive is in-

    verted in measure 31, the descending minor third to a-sharp


    Fig. 26--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 27). At measure 32, motive I is

    stated on b as in the opening measures, ascending to c

    Fig. 27--Treble Clef, mm. 31-32

  • 43

    and a return to b, ending the dirge in the same way it be-


    Throughout Dirge III the bass clef melodic line em-

    ploys 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 28).

    Fig. 28--Bass Clef, mm. 1-4

    Measures 1-8 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 13-15 (see Figure 29). T e ultimate reduction is

    seen in measures 16-19 where the i itial two-note motivic

    cell is represented by one dotted hole nqte value (see

    Figure 30). The descending minor second niotivic cell is

    resumed in measure 19. The final phrase qf the bass clef

    melodic line in measures 32-33 is he retrograde of the

    first two bass clef pitches in measure 1, f x g# instead of

    g# f x.

  • 44

    Fig. 29--Bass Clef, mm. 13-15

    16 - 17 18

    Fig. 30--Bass Clef, mm. 16-18


    The thirty-three measures are in varying meters of 2/2,

    3/2, 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 dis-

    tance 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 mea-

    sures 1-4, the longer note values precede the shorter values.

    The phrases gradually shorten until each phrase in measures

  • 45

    13-14 contains only one half note and one quarter note.

    The lengthy phrase at the center of the composition in mea-

    sures 15-19 is emphatically stated by eleven half-note

    values where the minor second is fully exploited by the os-

    cillating minor second in the melodic line of e-flat and e-

    natural. Smaller values precede longer values in motive

    III beginning at measure 19, contributing to the other con-

    trasts which this motive provides. From measures 27-31,

    however, the original order of longer values preceding

    shorter values is resumed. In measures 32-33, both rhyth-

    mic 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 16-19. 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 15-19. 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 15-19), eight phrases preceding it and nine

    phrases following. In the bass clef, seven phrases precede

    the three single chords in the center (measures 16-19), and

    seven follow. The culmination of both melodic lines occurs

    in measures 15-19.

    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 20, 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 refer-

    ence 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.


    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 1-14. The central phrase in measures

    15-19 uncovers the motivic cell of the minor second which

    is redeveloped in measures 20-33.


    The bass clef open fifths are octave reinforced through-

    out the dirge, producing four voices in the bass line. Only

    in measures 14-19 (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 9-15, producing

    three voices. Three sonority doublings in measures 16-19

    produce four voices. Three voices resume in measures 19-28,

    and two voices in measures 29-33. Although open voicing is

    retained throughout, the texture is thick due to the sonor-

    ity doublings in the low registers of the keyboard. The

    texture thickens as the midpoint (measures 16-19) is ap-

    proached, and gradually thins out from there until the ori-

    ginal voicing is resumed.

    Dynamics and Range

    The dynamic pattern rises from piano to fortissimo mid-

    way through the dirge at measure 17, returning to pianissi-

    mo by the end. Similarly, the range widens as it approaches

    the center, narrowing as it approaches the end. The open-

    ing of measure 1 extends from GG# to b1, spanning three oc-

    taves and a minor third. The range widens to a span of

  • 48

    five octaves and a perfect fifth from AAA to e 3 in measure

    17. From there, the range narrows to the original tessi-

    tura of measure 1. 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.




    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 counter-

    pole on the tonic axis) in the midpoint of the piece at

    measures 34-38. 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 14-15, measures 54-59) and F (measure 10,

    measures 53, 58, 63, 65, and 67) on both sides of the cen-

    ter. The occurrences of A-flat lie fourteen measures pre-

    ceding 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 sec-

    tion on the tonal center of B (the major third above G)

    shifts away at measure 28, 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


  • 50

    the vertex, and one measure past the golden section at mea-

    sure 43.

    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 12-13 and on C-sharp in mea-

    sures 34-38. Second, the occurrences of B, the major third

    above G, in measures 20-28, and on D, the perfect fifth

    above G, in measures 44-52 all combine to form the major

    triad G B D (G) when the outer perimeters of G are included

    (see Figure 31). Thus a union is made between the symmetri-

    cal expression of form (the tritone--which occurs as C-sharp

    dtwr i I2.45 2$~2.7 44-~~LI IAyV TO

    Fig. 31--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

    ing architectural significance). The ba

    formal principles is achieved by their o

    four instances surrounding the center of

    )eing the center of

    session of form

    triad--each serv-

    Lance of these two

    overlapping in these

    the dirge.


  • 51


    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 mid-

    dle register of the keyboard between the surrounding ex-

    treme registers of the accompaniment which is predominantly

    tonal due to its sustained octave reinforcement of conso-

    nant sonorities. The strongest moments of dissonance occur

    when the accompaniment is dissonant within itself.

    Measures 1-15 contain open perfect fifths in the har-

    monic accompaniment, producing a generally consonant har-

    monic character. Measures 16-20 introduce a half diminished

    seventh chord on g in the accompaniment, containing the

    first dissonant interval of a tritone in the harmonic ac-

    companiment. Measures 21-29 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 30 and 32-33 consist of brief moments of repose in

    perfect octaves. Measures 34-50 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.

  • 52

    The frequent unison passages and major and minor triads in

    measures 51-68 produce a consonant harmonic character. Dis-

    sonance returns again in the last two measures when the mo-

    tive of measures 34-50 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 21-27 of tritones and bi-

    tonality, 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 32).

    D isaa1' 6'in1 6 1 7 5517

    Fig. 32--Harmonic Pattern of Consonance and Dissonance

    in Dirge IV.

    Of the 327 intervals in the melodic line, 77 (24 per-

    cent) are minor thirds, 48 (15 percent) are major thirds,

    46 (14 percent) are major seconds, and 41 (13 percent) are

    augmented seconds, indicating once more the smooth contours

    of the melodic line.

    Motive I consists of two cells; Ia, a descending aug-

    mented second, and Ib, an enharmonic arpeggiated triad

    ascending 1-3-5 (see Figure 33). Motive Ia-b is stated

  • 53

    t . 3 4 5

    Fig. 33--Motive Ia-b -mm. 1-5

    twice on the beginning pitch c-sharp and in measures 1-9,

    separated from the tonal center of the harmonic accompani-

    ment 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 reso-

    lution 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 mea-

    sures 10-11, repeating in sequence a major third lower on g

    in measures 12-13 (see Figure 34). A stepwise melodic cell

    Fig. 34--Mm. 10-13

    in dotted rhythm follows Ia on d in place of Ib in measures

    14-15. These two cells are combined twice; first on d

  • 54

    (measure 14) and again on d-flat (measure 16) in sequence a

    minor second lower, with an extension of one measure which

    repeats the new cell (see Figure 35). Motive Ia-b is stated

    Fig. 35--Mm. 14-18

    once more in its original form on the original pitch level,

    enharmonically spelled as d-flat in measures 19-22. The

    dotted rhythm in measures 15-18 propels the beginning of the

    new phrase group on the b diminished triad in measures 21-

    27. The contrapuntal lines in measures 23-29 contain the

    intervals of motive I (augmented second and minor third)

    and are transitional to the first statement of motive II

    (see Figure 36). Measure 30 separately states Ia on d-sharp

    23 "I24 24 6 22$ 9

    Fig. 36--Mm. 23-29

    in octaves and is repeated in measure 32, followed by Ib in

    measure 33 (see Figure 37).

  • 55

    Fig. 37--Mm. 30-33

    Motive II is first stated in measures 34-35 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 con-

    tour 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 ex-

    actly in the center of the dirge at measures 34-38 (see

    Figure 38).

    34 35 3b 37 '3$

    Fig. 38--Mm. 34-38

    After motive II is first stated, motive I overlaps by

    occurring once more in the treble clef in measure 36 (Ia on

  • 56

    a) and in measures 38-39 (Ia and Ib on a). The overlapping

    statements of motive I and motive II indicate the relation-

    ship which these two motives share. The two statements of

    motive II in measures 34-38 are sequenced up a diminished

    fifth on the tonality of G in measures 40-43, following the

    overlapping statement of motive I in measures 38-39.

    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 alternat-

    ing 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



    Fig. 39--Mm. 44-50

    Measures 51-52 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. 51

    note figure which contains the augmented second of motive I

    begins on the third beat of measure 51 (see Figure 41).

    Fig. 41--M. 52

    Motive Ia-b returns in rhythmic diminution, a new meter of

    5/4, and with modified intervals in measure 53. The de-

    scending augmented second of Ia (c-sharp to b-flat) is ex-

    panded by a half step to a major third (d-flat to b-double-

    flat), and the f diminished triad of Ib in measure 3 is

    changed to f minor (see Figure 42). Four measures of


    Fig. 42--M. 53

  • 58

    parallel block chords in measures 54-57 descend by alternat-

    ing major and minor seconds (see Figure 43). Measures 58-62

    Fig. 43--Mm. 54-57

    repeat the pattern in measures 53-57. The statement of mo-

    tive 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 63 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. 63-65

    Fig. 46--Mm. 66-67

    measures 53-70 alternates with the two block chord progres-

    sions in measures 54-57 and in measures 59-62 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, repre-

    sented 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 34-35 (see Figure. 47).

  • 60

    Fig. 47--Mm. 68-70


    The entire dirge is in 3/4 meter except for measures

    53, 58, 63, and 67 which are all in 5/4 and state the modi-

    fied return to motive I. Dotted rhythms occur only between

    measures 15-18 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 15, 17, and 18 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

  • 61

    47-51 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 1-14) contain only

    melodic materials of motive I. Measures 34-52 (phrases 15-

    28) contain statements of motive II which are themselves

    derived from intervallic materials of motive I. Motive I

    returns in measures 52-68 (phrases 29-35) with further mod-

    ifications 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 36),

    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 in-

    crease of dissonance to measure 27. Though the dissonance

    begins to decrease from this measure, the dynamic level

    continues to rise through the midpoint of the dirge to mea-

    sures 44-50 which comprise the climactic development of mo-

    tive II. From there, the dynamic level decreases, forming

    an arch structure within the dirge which builds to measure

  • 62

    51 and recedes thereafter. The range is consistently wide,

    lying between EE (measures 28-29, 34-38) to db4 in measure





    Unifying Elements


    The Four Dirges are unified by their tonal organiza-

    tion 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


    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 domi-

    nant 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).


  • 64



    f A F

    E Bb/ E

    EO 8 B-1Lge T

    C- tr~e C Frg

    Fig. 49--Tonal Scheme of the Four Dirges in Regard to

    the Axis System.


    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 inter-

    val 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 30 of the third dirge.

    (See p. 42, Figure 25.) This potentiality of multiple

    methods for pairing these four pieces serves to unify them

    even more strongly (see Figure 50).

    I II III IVAugmented Fourth 4 Perfect Fourth Minor Second4 Augmented Second+

    TY "FTourth Decreased L-7-Second Increased

    Fig. 50--Intervallic Similarities in the Motives ofthe 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 2)

    is inverted in motive II of Dirge III (measure 4). The ma-

    jor 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 51). Further-

    more the rhythmic pattern of two quarter notes followed by a

    Fig. 51--Motivic Relationship Between Dirges I andIII.

    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 seven-

    and five-measure themes respectively rather than employing

    single motives as in the first and third dirges. The de-

    scending perfect fourth which begins Dirge II is changed to

    a descending augmented second in the beginning of Dirge IV

    (see Figure 52). Once more there is a reversal in rhythmic

    Fig. 52--Motivic Relationship Between Dirges II andIV.

    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 2/2 with frequent

    changes to other meters. The meters of Dirges II and IV

    are triple 3/4 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 53).

    I II2/2 3/4Duple TripleUnstable Meter Stable MeterAdagio I = 35-40 Andante . = 100

    III IV2/2 3/4Duple TripleUnstable Meter Stable MeterPoco Lento J = 50 Assai Andante J = 100-108

    Fig. 53--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 symmetri-

    cal 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 iso-

    metrical formation of the ever-thickening cluster centered

    around c-sharp, as discussed in that analysis (see Figure



    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 fi-

    bers remain distinct. Dirges I, III, and IV produce a har-

    monic fabric of thirds, open fifths, and triads while the

    exceptional Dirge II employs the gradually-thickening tone


    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 compo-

    sition, from where they gradually become re-developed to the


    Dirge I: The motivic cell of a descending augmented

    fourth in measures 3-4 is rhythmically expanded by half-

    note values and by an ever-widening interval contour to the

    center of the piece in measures 15-17.

    Dirge II: The characteristic interval of the descend-

    ing perfect fourth becomes exploited in the central fifth

    phrase of measures 28-39. 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

    17-20, 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 35-43 in the form of a new motive.

    In each dirge, the motivic cells are re-developed fol-

    lowing the central point. The figure below illustrates the

    hyperbolic structure of each of the four dirges (see Figure



    Fig. 55--The Hyperbolic Form in Each of the FourDirges.


    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 in-

    tervals descends in proper character for a dirge (see Fig-

    ure 56).

  • 70

    Dirge I: Descending augmented fourthDirge II: Descending perfect fourthDirge III: Descending minor secondDirge IV: 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 (61.8 percent) as archi-

    tectural keystones between which lie the tonal vertex and

    the melodic climax of each dirge (see Table 1). This