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BELA BARTOKT 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
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.
TABLE OF CONTENTS
Page
LIST OF TABLES . . . . . . . . . . ........... V
LIST OF ILLUSTRATIONS . . . . . . . . . . . . . .... vi
Chapter
I. BIOGRAPHICAL SKETCH OF BELA BARTOK . . . . . 1
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
iii
PageVI. ANALYSIS: DIRGE IV........ ...... ........ 49
TonalityHarmonyMelodyRhythmPhrase Analysis and FormDynamics and Range
VII. THE FOUR DIRGES: UNIFYING ELEMENTS ANDCOMMON CHARACTERISTICS . . . . . . . . . . 63
Unifying ElementsCommon Characteristics
APPENDICES ... " . . . - -- -74
BIBLIOGRAPHY....................-.-.-.-.-.-.-.-.-107
iv
LIST OF TABLES
PageTable
I. The Formal Employment of the ArithmeticCenter and the Golden Section in
I
VvI
X1
Xl
XV,
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
V
. ." .! .
LIST OF ILLUSTRATIONS
Figure
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
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 . . . . . . ..
vi
Page
7
7
9
. . . . 12
. . - - 17
17
. . . . 18
. . . . 18
. . . . 19
. . . . 24
- - - . 26
- - - . 26
- - - . 28
. . . . 31
. - - . 34
. . . . 38
. . . . 38
. . . . 38
* . . . 38
. . . . 39
. . . - 40
Figure
22.
Page
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 . . . . . . . . . . . . . .
vii
41
41
42
42
42
43
44
44
50IV
. 52
53
. 53
. 54
. 54
55
55
56
s 57
57
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
58.
59.
60.
61.
62.
63.
64.
. " " s - 60
. .63
. .
ves
I
II
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
64
65
. . . 65
. - - - - 66
. .r . . . 67
. .e . . . 67
S - - 69
. . . . 70
. .r ." . . 75
. . . 78
. w . 82
. . . 86
. . . 90
.. . 94
. . . 98
. .5 . 103'
viii
Figure
46.
Page
59
CHAPTER I
BIOGRAPHICAL SKETCH OF BELA BARTOK
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,
1
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.
CHAPTER BIBLIOGRAPHY
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.
5
CHAPTER II
ANALYTICAL METHOD
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
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 (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.
FC
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-
dices.
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
proportions.
CHAPTER BIBLIOGRAPHY
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.
14
CHAPTER III
ANALYSIS: DIRGE I
Tonality
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
15
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
appendices.
Harmony
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.
Melody
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) has
an 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-
sure.
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.
Form
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.
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 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.
Dynamics
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.
Range
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.
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 8-14. 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
23
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.
Harmony
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.
Melody
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
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 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-
ter.
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.
CHAPTER BIBLIOGRAPHY
1. Hanson, Howard, Harmonic Materials of Modern Music, NewYork, Appleton-Century-Crofts, Inc., 1960.
33
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 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
15.
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
ACGAGDD
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
34
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.
Harmony
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
Melody
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
f4
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-
sition.
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
4
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-
gan.
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
Rhythm
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.
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 1-14. The central phrase in measures
15-19 uncovers the motivic cell of the minor second which
is redeveloped in measures 20-33.
Texture
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.
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 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
49
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.
'w7l1.
51
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 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
39).
45,0
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
i
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
53
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
Rhythm
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
23-33.
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
60.
CHAPTER VII
THE FOUR DIRGES: UNIFYING ELEMENTS
AND COMMON CHARACTERISTICS
Unifying Elements
Tonality
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
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 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).
63
64
TONIC AxYS T 0 DoMiWANT Rt5
0
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.
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 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
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 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
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 compo-
sition, from where they gradually become re-developed to the
end.
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
55).
vertex
Fig. 55--The Hyperbolic Form in Each of the FourDirges.
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 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
TABLE I
THE FORMAL EMPLOYMENT OF THE ARITHMETICCENTER AND THE GOLDEN SECTION IN
EACH OF THE FOUR DIRGES
Arithmetic Center
Dirge I:
The tonal vertex lies on A-flat in mm. 13-14, theonly reference to this to-nal center around whichthe tonal symmetry is based(p. 15)
One of the three most disso-nant moments occurs inm. 14 (p. 16)
Fortissimo at m. 15 (p. 21)
Golden Section
The Unison in m. 17 on Cwhich is a dominant func-tion on the tonal axis(p. 16)
L -- _
71
TABLE 1--Continued
Arithmetic Center Golden Section
Dirge II:
The third note of the five-note isometric sonority isadded at the center inm. 28 (p. 24)
Dirge III:
The tonal vertex A occurs atm. 17 (p. 35)
The culminating exploitationof the minor second frommotive I occurs in mm. 15-19 (p. 40)
Fortissimo in m. 17 (p. 47)
The first statement of mo-tive III ends on the uni-son b, the third of the g-sharp minor triad--a ref-erence to the tonal center-- in m. 20 (p. 40)
The reversal of the rhythmicpatterns occurs with motiveIII by ending with longernote values--mm. 19-20(p. 45)
The comparison of the rhyth-mic motion between thetreble and bass clef linesshows a reversal at mm.19-20 with motive III(p. 46)
Dirge IV:
The tonal vertex occurs atmm. 34-38, the center ofthe piece and the centerof the octave of the to-nal center (on C-sharp)(p. 49)
Motive II is first statedin mm. 34-38 (p. 55)
analysis shows that the golden section and the axis system
were already being used by Bartok in these early composi-
tions of 1909-1910. Thus, in the Four Dirges, Bartok was
72
employing materials and compositional devices which were to
be carefully developed with unfailing consistency through-
out his life: Hungarian folk melody, arithmetic center,
golden section, and the axis system.
73
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APPENDIX 1
TABLE III
DIRGE I: TONAL ANALYSIS
Measure Tonal Center Measure Tonal Center
D-sharp
D-sharp--+ B
D-sharp p --.- B
G
D-flat
D-f lat----- C-sharp
C- sharp
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
A
A
C
E-flat
B-flat
C-sharp
A-sharp -- G
B
B
B
B with melodic ref-erence to D-sharp
B from mm. 1-3
B
B- I P --
74
E -> A
C
C
B-flat
A-flat
A-flat
- -
N - - - - - -m -
-- --- -
e- -
- - -#
wL (3
75
co
U
H
Ht
00
CH
0
biO'Hq
wH
_ --
', .
I
76
TABLE IV
DIRGE I: HARMONIC ANALYSIS
IntervalcO art aContent 00I
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
I
1
2
2
2
3
3
3
3
3
4
5
5
6
6
6
7
7
7
7
7
7
8
8
2/2
2/2
2/2
2/2
2/2
3/2
3/2
3/2
3/2
3/2
2/2
2/2
2/2
2/2
2/2
2/2
3/2
3/2
3/2
3/2
3/2
3/2
2/2
2/2
1
2
I
2
+
I
2
+
3
+
I
I
2
1
2
+
1
+
2
+
3
+
1
2
n
n
n
pmn
prn
mnd
pmn
prn
mnd
pns
unison
m4s2 t2
4
m4.4
n2p m d
6m
8
224p innm2n4d2
n d6
n
n6d2
p2r2n4
4p
m3
m3
m3
m3
m3
m3
m3
m3
m3
m3
d5
M3
M3
d4
d4
d4
m3
m3
m3
m3
m3
m3
P4
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
9
9
10
10
10
11
11
11
11
11
11
11
12
12
12
12
12
12
13
13
13
13
13
13
2/2
2/2'
2/2
2/2
2/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
I
2
1
2
+
1
+
2
+
3
+
1
+
2
+
3
+
1
+
2
+
IntervalContent
p m d4
8s40PS
p4m4n46
46 6
P mn4
m4n4 d4
p4m4n4s4d4t4
p8m4n8s4
pam10 4d4
p m n
unison
p4m4d4
p404 4
p 4 m 8 n 4 48p
m
p4m4n8 4t4
unison
p4n12d4t4
p4m4n4p4m12 4d4
p~m4n8
pm d
cQ 4
M3
P4
P4
P4
P4
P4
m6
P4
P4
P4
P4
P4
P8
dA4
P4
P4
P4
P8
PS
P4
P4
P4
P4
P4IwrlOr l / / 1 11r
77
TABLE IV--Continued
Interval 0 kIntervalCv ontent Content.a0 C
+dQ I I1 1 1 In
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
13
13
14
14
14
14
15
15
16
16
17
17
18
18
18
18
19
19
20
20
20
20
21
21
22
22
22
23
23
23
23
24
24
24
24
25
25
25
25
26
26
26
26
27
27
27
27
28
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
r
i
2
+
1
+
2
+
1
+
2
+
1
+
2
+
1
+
2
+
1
+
2
+
1
3/2
3/2
3/2
3/2
3/2
3/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
p8m4n8
pin n4d4t4
p4mn4 d tpen12d44
p3m2 6s5d2t4
p4m5 7
p4m20 7d6 ~ ~1st
p4 n5n 13s6t9
p:Lm5 7s6d9
unison
p m4ns5
'22
pmn tp in nd
pi44nd4
p2m~nd2
-Pmn 5 s2
unison
m4s7t
2 2p mn
26 3p in ns
unison
p3in 3 2d3
p2 5 2t2
P4
P4
P4
P4
P4
m3
M3
M3
M3
M3
PS
m6
m6
m6
m6
M6
P8
m7
m6
m6
m6
m6
P8
m7
M6
I~7r
1
2+
3
+
1
2
1
2
1
2
1
+
2
+
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
10122 2/21 + p6m3n2 16
pemn2s4d2
p2m3 6
p in 4n 8s 4
p4m4n's4d4t4
p9m10n5 ".
612
pm~n s6
p3m10n11
p9m4n
p5m4n
p5m10n13
p11m4n13
p 3m n3
p3 10 9
pam4n'9
p9 m4n5
p9m4n11
p9m4n s6
p9.4n 3.6
p9m4n
p9m4n11s6
p9m4n 11s6
I
2
1
+
2
+
1
2
1
28 2 G/2 +
28 2/2 +
28 2/2 2
28 2/2 +
M6
M6
m,3
m3
m3
m3
M3
M3
M3
M3
M3
p4
P4
.P4
m3
m3
m3
m3
M3
M3
M3
M3
M3
M3pmn s M3
79
TABLE V
DIRGE I:: PHRASE ANALYSIS
Phrases Measures Length Motive Transformation Section
1 (1) 2-3 2 mm. M A2 3-4 (5) 3 mm. M; Exp/Ext A3 6-7 1m. M A4 7-8 (9) 3 mm. M; Exp/Ext A5 10-11 1m. M; Exp A6 11-12 1 m. M; Exp/Ext A7 12-13 1 m. Extension Expanded A8 13-14 2 mm. Extension Expanded A9 15-17 3 mm. Extension Expanded/ B
Augmented10 18-19 2 mm. M; Con/Ext/Inversion C11 20-21 2 mm. M; Exp/Ext/Ret C12 22 1 m. M; Ret/Exp C13 23 1 m. M; Ret/Exp C14 24-25 2 mm. M; Ret/Exp/Ext C15 25-26 1 m. Extension C16 26-27 1 m. Extension/Extended C17 27-28 2 mm. Extension/Extended* C
*"M"--main motive; "Exp"--expansion of the entire in-terval span of the motive; "Ext"--extension of the motive;"Extension"--the repetition of the extension alone from theprevious phrase; "Extension Expanded"--the statement of theextension alone with interval expansion; "Con"--contractionof the entire interval span of the motive; "Ret"--the mo-tive in retrograde; "Extension Extended"--the statement ofthe extension alone with an added extension.
Any one or a combination of these various categoriesof motivic transformation is used in Dirge I as indicatedin 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 ofInterval Intervals Usage Intervals Usage
enharmonic unison 2 .8m2 78 31 15 23M2 27 11 9 14A2 4 2 2 .3m3 38 15 11 17d3 2 .8M3 25 10 5 .7A3 1 .4 1 .2d4 2 .8P4 19 8 4 .6A4 10 4 7 11P5 4 2 1 2A5 2 .8 1 2m6 2 .8 1 2M6 1 .4 1 2M7 1 .4 1 2d8 2 .8 1 2P8 15 6 1 2m9 2 .8M9 11 4 1 2d7 2 .8 1 2
P8+d4 2 .8 1 2
Interval Totals 252 64
APPENDIX II
TABLE VII
DIRGE II: TONAL ANALYSIS
Measure Tonal Center
1-7 C-sharp
8-14 C-sharp
15-29 B-f lat
30-38 A-sharp diminished triad with melodic tonalemphasis on F
39-47 F-sharp with melodic tonal emphasis onC-sharp
48-55 C-sharp
81
78
0.
000
%S
" 10
0er
y0 0
0 ' " 1CG C}
cz, C)
4.r
m0n
- H..
a eseH 0.k en[iiess es
e og .I _
-r -c r4 r-
82
-E-
- --
' . ,.H
-- Cd
-- Ir
LC)
4LL w o V V cl1 EC Q C
83
TABLE VIII
DIRGE II: HARMONIC ANALYSIS
Z I Interval I NbO Z Interval wv r. C (J 4 0) 0 40 0 0 4 W
0 $ $ $ % Content +a 0 $ Content 41O. Content
I
2
1
2
1
2
3
1
2
1
2
I
1
2
I
2
I
2
3
1
2
2
unison
unison
unison
unison
unison
unison
unison
unison
unison
unison
unison
unison
unison
unison
4p4p4p4
s
4p4p4
n
4P
unison
4s
PS
P8
P8
PS
PS
PS
PS
PS
P8
PS
PS
P8
PS
P8
P4
PS
P4
M2
P4
P5
M6
PS
PS
m7
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
13
13
14
15
16
16
17
17
18
18
18
19
19
20
20
21
21
22
22
23
23
24
24
24
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
1
2
1
1
1
2
1
2
1
2
3
1
2
1
2
1
2
1
2
1
2
1
2
3
p4
4n4n4nn4
2 2
n s d
p can
mn2d2
n2s2d
p2 2s
nsd
n 2 s2 d
mn2d22 2
pon 4
2 2
pm s
4
n
n2s2d
mn2d2
n2s2d
2 2
mn2d2
n2s2d
P4
m3
m3
m3
m3
m3
m3
m3
m3
m3
m3
m3
m3
m3
m3
m3
m3
m3
m3
m3
m3
m3
m3
mn3
1
2
3
4
5
6
7
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
1
1
2
2
3
3
3
4
4
5
5
6
7
8
8
9
9
10
10
10
11
11
12
12
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4I -i ii 1 E 1
.I I 1 I 1 11
84
TABLE VIII-Continued
+ > r cd O a ) l
;4 ZInterval , k $ SInterval
10 a Content 3 s c Contento .S a 1) )o S 4S 0 a a 0S
49 25 3/4 1 n3t2 m3 73 37 3/4 1 pmn 3s2 t2 m3
50 25 3/4 2 p2 n2s m3 74 37 3/4 2 pm2n3st2 m3
51 26 3/4 1 n m3 75 37 3/4 3 pm2n3st2 m3
52 26 3/4 2 mn2d2 m3 76 38 3/4 1 p2mn3dt2 m3
53 27 3/4 1 p22 m3 77 38 3/4 2 n623
54 28 3/4 1 n 3 t 2 m3 78 38 3/4 3 pmn 3 d 2t 2 .. m3
55 29 3/4 1 n3t2 m3 79 39 3/4 1 pmn3s2t2 m3
56 30 3/4 1 nt2 m3 80 39 3/4 2 pm2n3st2 m3
57 31 3/4 1 p2mn3dt2 m3 81 39 3/4 3 p4m4ns4t4 m3
58 31 3/4 2 mn3s2dt2 m3 82 40 3/4 1 p4m4nns4t m3
59 32 3/4 1 n6t3 m3 83 40 3/4 2 p4m4n8s4t m3
60 32 3/4 2 mn3s2dt2 m3 84 40 3/4 3 p4m4n9s2d2t4 m3
61 33 3/4 1 r2n3sdt2 m3 85 41 3/4 1 p4m4n8s5t m3
62 33 3/4 2 p 2 n3 sdt 2 m3 86 41 3/4 2 p6m6n8s8t4 N263 33 3/4 3 n4dt3 m3 87 41 3/4 3 p6m6n8s8t M2
64 34 3/4 1 p2mn3dt2 m3 88 42 3/4 1 p6m6n8s6t M3
65 34 3/4 2 mn3s2dt2 m3 89 42 3/4 2 p6nm6n8st4 M2
66 34 3/4 3 n6t3 m3 90 42 3/4 3 p6m6n9s5d3t4 M12
67 35 3/4 1 m2n3sdt2 m3 91 43 3/4 1 p6m6n9s5d3t4 Al
68 35 3/4 2 p2n3sdt2 m3 92 43 3/4 2 p6m6n8s8t M2
69 35 3/4 3 n4d2 3rn3 93 43 3/4 3 p6m6n8s7t M3
70 36 3/4 1 p2mn 3 dt2 m3 94 144 3/4 1 p4m4n10s6t6 A4
71 36 3/4 2 mn3s2dt2 mrn3 95 44 3/4 2 p6mn8s6t4 M3
72 36 3/4 3 pmndt2 r3 96 44 3/4 3 ptmnnst M3
85
TABLE VIII--Continued
Interval IIk IntervalV= cd Content + a sContent
oo 1) w w 0 04 0 o 4
p8m4 n8s6d2t4
p6m4n12s5t4
p6m~n12s4t4 4p6m6n8 6t4 *
p 4 12 5 4tp6m4n12s~t4
p6 4 12 5 4
p6m4n12 4t4
p62412412 4
p12m8n12s12t4
unison
p
m5n2s9t
p 413n9s24 4
p 4 13n9s24 4
p 94m13n9s24t4
p14 13n13s16d10t4
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
M7
M6
M6
M3
M6
M6
M6
M6
M6
P8
P8
P8
PS
P8
P8
P8
PS
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
50
51
51
51
52
52
52
53
53
53
54
54
54
55
55
55
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3
1
2
3
I
2
3
1
2
3
1
2
3
2
3
5 5 21
p nn2s t
p m n s t3 .
p14m13 2 t4
p14m13n9s26t4
p 0m9n17s21dtp
p20m9n 7s16t4
p20 9n17s16t4X20 9 17s16t
4
20 9 s16t4
p2 in ns16t
P8
P8
P8
P8
P8
PS
P8
PS
P8
P8
P8
P8
P8
PS
P8
P8
3/4 12 jp 5m5n2s9t
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
45
45
45
46
46
46
47
47
47
48
48
48
49
49
49
50
50
q.
tt
S t
'ft
r - o
ft.
- 0M
1!0
oj Q f~
0.c
86
4M
- -
rq-r
; x~j
000000ugoftoio
ft a 0ft0fta m adms
_
I
i
i
i
f_
I
f. _
i
{ _. ..
Ica
i l erral El l lV) t)(0. 0) to 0
.^..._ M ._ _ i
i i
. ____- '. ___. _-.f
-_-- - I - - - - - - - -
_ ._....._ _._ -- .p ._.
_. . ...... ... .,.,s.R.,... k
Co
w
w
T"
0
Co
0
-r--
C)
-
0D
f
i
_. f
_.....,r.. ... . ._._._..,.., ... _......._.r . _..._._,._. .. _.._...__....,_...,.._ ._.s._..._..
87
TABLE IX
DIRGE II: PHRASE STRUCTURE
Tonal Notes of thePhrase Measures Length Center Isometric Sonority
1 1-7 7 mm. C#C
2 8-15 8 mm. C# C#
3 16-21 6 mm. Bb Bb-Db (A#-C#)
4 22-29 8 mm. Bb Bb Db (A#-C#)
5 30-40 11 mm. FF# A#-C#-E
6 41-47 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 ofInterval Intervals Usage Intervals Usage
enharmonic unison 7 5 1 1
m2 19 12 17 17
M2 60 39 32 32
m3 19 12 13 13
M3 7 5 7 7
d4 1 .2 1 1
P4 29 19 19 19
d5 1 .2 1 1
P5 7 5 6 6
M6 1 .2 1 1
m7 2 1 1 1
P8 (75)*
Interval Totals 153 99
in ,five- tonot included
*P8s are used only as octave displacementsten-note chords from measures 47-55. These arein the count.
APPENDIX III
TABLE XI
DIRGE III: TONAL ANALYSIS
Measure Tonal Center
1 G-sharp ---- F-double-sharp2 B A3 G-sharp -* F-double-sharp -> G-sharp4 E5 B-flat----A6 D-flat--->C-f lat7 B-flat - > A -B
8 A-flat9 C C-flat
10 B-flat11 E C-sharp12 A13 G F-sharp14 B-flat-A15 D-flat-4C16 C17 A18 C19 C20 G-sharp21 G22 B23 A24 A-flat25 G with a melodic tonal emphasis on A26 G27 F28 E>D29 G-sharp30 D31 G-sharp32 F-double-sharp33 G-sharp
89
-? - _ ._.. --- " - - - .. _.__ __1_:,_ a _ __ _ . -
II
- 77--- Ii
__________w
j
-- ~ ~
I I I I Id~ I I II 1 -~ -+---~------4 1~" - + - -
(5
90
C
H
'H
0
H
0
bfl
LL
: H- "
kI--
- ---------
91
TABLE XII
DIRGE III: HARMONIC ANALYSIS
artQ3I 4-t> I ()I000Intervali 0o Q 4j Interval N Q
'4 04 (, (D 0 Content c 0c 0 w Cntn~ Conten
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24 6
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
5/4
5/4
5/4
5/4
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
1
+
2
+
1
+
2
+
1
+
2
+
1
2
4
5
1
+
2
+
1
+
2
+
4
p
444
4 s 4
4 s 4
p i n
p44
p8 4pS
8 4
p i
4 s 4
4 s 4
7 2
pain
4P8
p
p n s
4p
8 4DPS
8 4PS
p 4d 4t4
p 4d 4t4
p in
PS
PS
PS
P5
PS
PS
PS
PS
PS
PS
PS
PS
PS
PS
P5
PS
PS
PS
P5
PS
PS
P5
P5
P5
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
7
7
7
7
8
8
8
8
8
8
9
9
9
9
10
10
10
10
11
11
11
11
12
12
2/2
2/2
2/2
2/2
7/4
7/4
7/4
7/4
7/4
7/4
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2 +
62P s
p in
8 4P S
p mn
p4d4t4
p4d4t4
p4m4d 4
p4m4 d4
p4m4d4
pdt
44
p4m6d6
p4n 6 6
pind
p4n 6 6
p 4 m 6
444
pns
p4d6t6p10
4 6
pn s
pnsp dtn
P
p4 m6 d6
4n6s6p ns
PS
PS
P5
PS
P5
PS
P5
P5
PS
PS
PS
PS
PS
PS
P5
PS
PS
PS
PS
P5
PS
P5
PS
P5r !
1
92
TABLE XII--Continued
FI IfI s~4 fiInterval ' > 0 A4Interval +J >
+ ~j =nsd c +j
4.~ "0 VContento= 0Conten49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
12
12
13
13
13
13
14
14
14
14
15
15
15
15
16
16
16
16
17
17
17
18
18
18
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
3/2
3/2
3/2
3/2
2/2
2/2
2/2
3/2
3/2
3/2
2
+
1
+
2
+
1
+
2
+
1
+
2
+
1
+2
+3
+2
+
+2
+3
466p t
p 4d6 6
4p
p 4d6 6
10 6p S.
p4m6 6
9
p9d9t9
18s 9p S
p mn
p9d9t 91s9pS
9 9s 9
918999
9999
pd
p9d12 12
p21
p9m12n 12
pm
p9 12n12
pm 2
p9 12n12
p9 d l2nl2
PS
PS
PS
PS
P5
PS
PS
P5
P5
P5
P5
PS
PS
P5
P5
P5
PS
PS
PS
PS
PS
PS
PS
PS
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
18
19
19
19
19
19
20
20
21
21
21
21
22
22
23
23
23
23
24
24
25
25
25
25
3/2
3/2
3/2
3/2
3/2
3/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
2/2
+
1
+2
4.
3
+
212
1
+
2
+
1
2
1
+
2
+
2
+
2
+
p m12n'2
p9m12n12
p9m12n12
p nn
unison
unison
unison
pmnn
p4m6n6
466
10p
unison
unison
4 96 6
p u s
p 4m 6d6
p 4m 6d6
10p
p mun
unison
10 6
p mun
10 6p dt
p n s
P5
PS
PS
P5
P8
P8
P8
PS
PS
PS
PS
P8
P8
P5
PS
PS
P5
PS
P8
P8
P8
P8
P8
P8--- - ---- t -
93
TABLE XII--Continued
4-t r.4 q.4 r r.
Interval +4> c'Interval >0 n Content n Dos n (D Content
97 26 2/2 1 p1 0 P8 110 30 2/2 2 p4 d4t4 PS
98 26 2/2 + p4n6s6 P8 111 30 2/2 + p4m4n4 P5
99 26 2/2 2 p10 P8 112 31 3/2 1 p4 PS
100 26 2/2 + p4m6n6 P8 113 31 3/2 +2 p4m4n4 PS
101 27 2/2 1 unison P8 114 31 3/2 + p4d4t4 PS
102 27 2/2 2 p4m6n6 PS 115 31 3/2 3 p8s4 - PS
103 28 2/2 1 p4 66 P 116 31 3/2 + p8s4 PS
104 28 2/2 2 p1 0 PS 117 32 2/2 1 unison PS
105 28 2/2 + p4m6n6 PS 118 32 2/2 + p4m4n P5
106 29 2/2 1 unison P8 119 32 2/2 2 p8s4 P5
107 29 2/2 2 p4m6n6 PS 120 32 2/2 + p4nm4n P5
108 29 2/2 + p10s6 PS 121 33 2/2 1 p4nm4n P5
109 30 2/2 1 p4 PS 122 33 2/2 2 p44n4 PS
4 -
ti
N F
CI
4
~ T.
CO
h.
0C C
'CC Ct?
21
94
0
11
- 4
lowg~- C--ri -P4
0A00$$ $
CU
0
0
0
0
CCU
-H
-H-
- co
Comm o CoUx C C C C
1 6- I..d
95
TABLE XIII
DIRGE III: PHRASE STRUCTURE
Phrase* Measures Length Motive First Pitch Section
1 1-3 4 mm. Ixy B A4 II
2 5-7 4 mm. Ixy D A
8 II
3 9 2 mm. Ix A A10 II inv
4 11 2 mm. Ix G A12 II inv
5 13 1 m. Iy G#-A B
6 14 1m. Iy B-C B
7 15 1m. Iy D-Eb B
8 16 1 m. Iy D-Eb B
9a 16 1 m. Iy Eb-E C
9b 17 tin. Iy Eb-E C9c 18 1 m. Iy Eb-E C
9d 19 1 m. Iy Eb-E C
10 19-20 2 mm. III E-B D
11 21-22 2 mm. III E-G# D
12 23-24 2 mm. III G-Bb D
13 25-26 2 mm. II, III A G B D
14 27-29 3 mm. IX, II A EexpG#
15 30 1 m. III G#F E
16 31 1 m. Il inv ** B E17 32-33 2 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 ex-
tended by one extra note which gives it a strong resem-
blance to motive III.
*** "Ix"--pedal note; "Iy"--ascending minor second,
returning to original pitch; "II"--generative motive, de-
scending 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
m2
M2
A2
d3
m3
M3
d4
P4
A4
d5
A5
M6
m7
P8
106
32
5
1
20
18
20
2
18
4
4
6
4
6
43
13
2
.4
8
7
8
.8
7
1.6
1.6
2
1.6
2
30
9
2
1
11
4
2
2
2
48
14
3
1.5
17
6
3
3
3
Interval 246 63Totals
APPENDIX IV
TABLE XV
DIRGE IV: TONAL ANALYSIS
Measure
1-2345
678910111213.
14-1516-1920
21-27282930313233
34-39
Tonal Center*
GG (F diminished)GG (A-sharpdiminished)--GGG (F diminished)GA- BFF (E diminished)D-flatD-f lat(C diminished)A-flatG diminishedG diminished(F diminished)B diminishedA (C-sharp)A (C-sharp)----+AA (B)A (C-+ C-sharp)-- AA (B)A (G diminished)C-sharp
______________ -t _____________________________________
i
Measure
40-4142-4344-47
48495051525354555
E 57
556666666666670
*The parentheses indicate me:stresses tonal centers different :paniment which are considered to
gravity due to their chords of exation.
97
Tonal Center*
GEDD (A-sharp)DD (A-sharp)D (G-sharp)D (B-flat----> E-flat)F augmented- FA-flatG-flatF-flatGD (F augmented- 4 F)A-flat
G-flatF-flatGD (F augmented---' F)D (F augmented)D (F)C-sharpB-flat--- F diminishedGG (C-sharp)G (A-sharp--- C-sharp)
odic activity whichrom those of the accom-iave precedence of tonal:ended range and long dur-
____,---
,« .r
.. 11
I--
'4
- 4,. ..-- r----+---- --- - - - - - --- "
-1 3 - . - - s - e , --- + -,+- - - - - - - 4 - "- - - -- - - - -
to
- -I-- -~
-II1I~iT i~LK)
_ ILILLjLLi
98
CH0
Fi
Co
CO
b
L)
LLLLiL
L
99
TABLE XVI
DIRGE IV: HARMONIC ANALYSIS
I r II1rInterval -> r QkInterval +
0 00 wContent Ins cos 41 Content Q
OCO 4 W 0 0 O . __ __ __ __0
unison
p9d9t9
p m n
unison
p dat
unison
unison
unison
unison
p6m6 n7 s
3d5t 4
6 8 12
unison
p9d~t9p mat
unison
unison
unison
unison
m4s2t8
m9s10 2
15n
unison
PS
PS
P5
P8
PS
PS
P8
P8
P8
P8
P5
A4
PS
P8
PS
P5
P8
P8
P8
PS
M110
M10
mrlO
PS
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
10
10
11
11
11
12
12
12
13
13
13
14
14
14
15
15
15
15
16
16
16
17
17
17
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
2
3
1
2
3
1
2
3
1
2
3
1
2!
3i
2,
f2
+
3
1
2
3
1
2
+
p9d9t9p atn
unison
unison
unison
unison
9 9 9ptnn
unison
unison
unison
unison
p9d9t9
9 9 9p m n
unison
p9d6t6
15 6p s
9 6 6p m n
unison2 10 7 4 10p m n s t
p8 m4n16s4t4
unison
p2m6n5s4t6
10 2n3s6d2t2
PS
PS
PS
P8
4P8.
PS
PS
PS
PS
P8
PS
P8
PS
PS
PS
P5
P5
PS
P8
ds
d5
P8
d5
d5
1
2
3
4-
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
1
1
1
2
2
3
3'
3
3
4
4
5
5
6
6
6
7'
7
8
8
9
9
10
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
1
2
3
1
2
3
1
2
3
1
2
1
3
1
2
3
1
2
3
1
2
1
3
1ilY1 111r1_____________________________11111 1 1YIiY Y
I I I I I I
100
TABLE XVI--Continued
04.) m t!4 04 U!
Interval 0 a)4 w 0Interval
Esc cd Content = Content
O=Q 0 OC } O 4OU 0 ) 049
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
17
17
18
18
1818
18
19
19
19
20
20
20
21
21
22
22
23
23
23
24
24
24
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/41
3/4
3/4
3/4
I I I -- T-
p6m2n9s4t2
p10m2 n3s6d2t2
unison
u2m10n 7s4t10
14r n4 s7 d6t4
8 4 16 4t4
p1 4m4n4s7d6 t4
unison
p4m10 14s4t10
10 4n20s4t4
unison
unison
unison
unison
p~d4t4'
8d4t 4
mn4t8
pd t
- 2 14d3s3t7
3 142d3t
3m3 14d2t 7
6m5n14s3d2t
p3 m 3 1 7 d2 t9
5m3n14s3d5t 7
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
25
25
26
26
26
27
27
27
28
28
29
29
30
30
31
31
31
32
32
32
32
32
34
1 11 1 1I
d5
d5
P8
d5
d5
d5
P8
m3
m3
PS
Ps
P8
P8
A4
A4
A4
A4
A4
A4
A4
A4
A4
A4
--4
72 1 25 1
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
2
3
1
2
3
1
2
3'
1
2
1
3
1
3
1
2,
3
1
3'
z
2
3
1
2
p3 m6 n1 4 s4 d3t7
p6m5n14s5t7
p3m3 17d2t9
p5 14s6d5t7
p5 3 14s3d5t7
p6m2 14s6d2
6 5 14 5 6
unison
n10t6
4 6 5
unison
unison
2 34
pmn sdt
p m n
unison
unison
unison
unison
unison
p12 3n4
p12' 3n8s7t
A4
A4
A4
A4
A4
A4
A4
A4
P1
d5
M3
P4
PS
P8
d5
M3
P4
P8
PS
P8
P8
PS
M6
M696 | 34l I I i 1
1i - - - - - - . L
101
TABLE XVI--Continued
zInterval $4 k 1 u Interval PCo te td i Content 4
$0 c) U 7 0 51 04 a
97
98
99
100
101
102
103
104
105
106107
108
109
110
111
112
113
114
115
116
117
118
119
120
35
35
36
36
37
37
37
38
38
39
39
39
40
40
41
41
42
42
43
43
44
44
45
45
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
i- i .1. 1 112m3n5d3t
12m3 8s 7t
p1 5 m 1 5 n4 d 9
p12m15 13d3
p12m3 12s6t2
p12m3 6d6a8
p12m3n12 6t2
12 m3 12 6 2
12 11 n10d2
union
unison
unison
p12m3n4
12m3n12s6t2
p12m3n6d6t8
12m3n12 6t2
p16m8n~~4p ma 4s
p16m8n~~4
92 6 6
p m nt
9m6n12 6 4
9m6 n10d6
9m6 15 6 9
M6
M6
M6
M6
M6
M6
M6
M6
P8
P8
P8
MGM6M6
M6
M
P8
P8
P8
P8
P8
P8
P8
P8
0- - 7
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
46
46
47
47
47
48
48
48
49
49
49
50
50
50
51
51
51
52
52
52
53
53
53
53
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
3/4
5/4
5/4
5/4
5/4
1
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
4'
pmn cit9 1 18
plt9m6n s12 t9
unison
p3m2a 14 d4t12
3 2 18 4 8
unison
p8 9n 1 4 s 2 d2 t2
12 2s6 d2 6
unison
p3 2n18s4t8
unison
p8m9 n14s2 d2t 2
P12 2s6d2t6
unison
unison
unison
unison
unison
unison
unison
unison
unison
unison_
P8
P8
P8
M6
M6
P8
A3
A3
P8
M6
M6
P8
A3
A3
P8
P8
P8
P8
P8
P8
P8
P8
P8
P8.__ __.._ . s
102
TABLE XVI--Continued
Interval $ ZInterval00 a U! I) + D Q305 rA 0e +Content = Content 4j
C?4 ~ Q Q? CDQQ n $4o Q S 0
145 53 5/4 5 unison PS 169 64 3/4 2 d6 P8
146 54 3/4 1 p8m12n12 P4 170 64 3/4 3 p6 P8
147 55 3/4 1 p8m.2n12 P4 171 65 3/4 1 n6 P8
148 56 3/4 1 p8mn1212 P4 172 65 3/4 2 t 6 PS
149 57 3/4 1 p8men8 M6 173 65 3/4 3 s6P8
150 57 3/4 3 unison P8 174 66 3/4 1 rest
151 58 5/4 1 dm2 175 66 3/4 2 unison P8
152 58 5/4 2 p P4 176 67 5/4 1 unison P8
153 58 5/4 3 n4 M6 177 67 5/4 3 unison PS
154 58 5/4 4 t4 A3 178 67 5/4 4 unison P8
155 58 5/4 5 s M2 179 67 5/4 5 unison P8
156 59 3/4 1 p1419n 19 P4 180 68 3/4 1 unison P8
157 60 3/4 1 p m16n20 P4 181 68 3/4 2 p9m9n9 P5
158 60 3/4 3 p 8m15n15 P4 182 69 3/4 1 p9m9n9 PS
15961 3/4 1 p mn P4 183 69 3/4 2p9minsd P
160 61 3/4 3 p m16n20 P4 184 70 3/4 1 p9m15n15d6 PS
161 62 3/4 1 14m16n18 7 /9 9 9s6d6 6lpm M6 185 70 3/4 3 pn n s dt P
162 62 3/4 3 unison P8
163 63 5/4 1 d6P8
164 63 5/4 2 - p6P
165 63 5/4 3 n6 P8
166 63 5/4 4 t6 P8
167 63 5/4 5 s6 P8
168 64 3/4 1 rest
41
r'
01
"I
.. I . . -
00
Q'"
,~ tl
--
9K'
{y h , GC L 0 C p C
a M b {y, O6 [
vr j AO GqC.
- 404)000a . e--C.0 rn 4)n4) 4)444
4
IF
-.- --
_ _ _ _ -s -
103
-.- -- _-'.. f- _ __. _'_
._, -- -... _ - .- - _ ._.. - - -i
..... ,........ .. e.....L ............... r.. ...... i
_ .. ____
_..___.._ __,. ._-r
_.............._w__.p....._..
I
__: _F_ _. .____
i
_._. __
4... , - _1
_.. --... _ - . -- - ----..... 1
I '
- .__..
"o
44i
N4--
Rh! 01
m
0 as a
r c~ ~ ga in r rt
4., n o a g
0)
HC
0
GS
0
0-4
Ca
C
"
104
TABLE XVII
DIRGE IV: PHRASE STRUCTURE
Phr'aseMeasures Length Motive First Pitch Section
1
2
3
4
5
6
7
8
9
10
11
12
13
14
1-5
6-9
10-11
12-13
14-15
16-18
19-22
23-24
24-25
25-26
26-29
30
31
32-33
5 mm.
4 mm.
2 mm.
2 mm.
2 mm.
3 mm.
4 mm.
2 mm.
2mm.
2 mm.
4 mm.
Im.
Sm.
2mm.
I
I
I
I
I
I
I
I der
I der
I der
I der
I de
'a
I acc
I
C-sharp
C-sharp
B
G
D
D-flat
D-flat
E
G
A
C
D-sharp
D-sharp1-- 1 1-
15
16
17
18
19
20
21
22
34-35
36
37
38-39
40-41
42-43
44-45
45-46
2
I
1
2
2
2
2
2
mm.
m.
m.
mm.
mm.
mm.
mm,
mm.
II
Ia
II
I
II
II
II
II :i
cell
cell
A-sharp
A
A-sharp
A
E
F-sharp
B
B
A
A
A
A
A
A
ATransition
A
A
A
A
A
A
A
B
B
B
B
B
B
B
B
---_.
-
. . ....
105
TABLE XVII--Continued
Phrase Measures Length Motive First Pitch Section
23 46-47 2 mm. II cell B B
24
25
47-48
48-49
2 mm.
2 mm.
II cell
II cell
B
A-sharp
126_ 49-50 2 mm.II_cell __ B_
27
28
50-51
51-52
2 mm.
2 mm.
I-II
I-II
___'_____ - --- " - _
29
30
31
32
33
34
36
53
54-57
58
59-62
63
64-65
35 66-67
68-70 1
im.
4 mm.
I m.
4mm.
Im.
2 mm.
2mm.
3 mm.
IT
acc
I'
acc
I'
I'
I'*
II
D-flat
C-sharp
C-sharp
C-sharp
C-sharp
C-sharp
B
B
B
Trans it ionB
B
A'
A?
A t
A'
At
A'
A'
ClosingA'
*"der"--the melodic line employs intervals which arederived from the original motive; "cell"--only the ascend-
ing interval from motive II is stated; "l""--motive I isrhythmically and intervallically altered, although remainsrecognizable as motive I; "acc"--the chordal style of ac-companiment for motive I is occasionally stated without themotive.
_--
i
106
TABLE XVIII
INTERVAL COUNT OF DIRGE IV
All Voices elody Alone
Type of No. of Percentage No. of PercentageInterval Intervals of Usage Intervals of Usage
m2 73 14 38 12M2 94 18 46 14A2 43 8 41 13d3 8 2m3 90 17 77 24M3 60 12 48 15d4 13 3 13 4P4 28 5 19 6A4 5 1d5 2 .3P5 6 1AS 1 .1 1 .3m6 4 .7M6 43 8 25 8m7 8 2 8 2d8 7 1 7 2P8 20 4 3 1m9 2 .3M9 4 .7 1 .3
M10 3 .5P15 3 .5
IntervalTotals 517 327
BIBLIOGRAPHY
Books
Crow, Todd, editor, Bartok Studies, Detroit, Michigan, De-troit Reprints in Music, 1976.
Hanson, Howard, Harmonic Materials of Modern Music, NewYork, Appleton-Century-Crofts, Inc., 1960.
Lendvai, Ern , Bela Bartok: An Analysis of His Music, Lon-don, Kahn and Averill, 1971.
Moreux, Serge, Bela Bartok, translated by G. S. Fraser andErik De Mauny, New York, Vienna House, 1974.
Stevens, Halsey, The Life and Music of Bela Bartok, 2nd ed.,New York, Oxford University Press, 1964.
Suchof f , Benjamin, editor, Bela Bartok Essays, London,Faber and Faber, 1976.
U jfalussy, Jozsef, Bela Bart6k, Budapest, Hungary, CrescendoPublishing Company, 1971.
Wittlich, Gary E. , coordinating editor, Aspects of TwentiethCentury Music, Englewood Cliffs, New Jersey, Prentice-Hall, Inc., 1975.
Articles
Bachman, Tibor and Peter J. Bachman, "An Analysis of BelaBart6k's Music Through Fibonaccian Numbers and theGolden Mean," Musical Quarterly, LXI (January, 1979),72-82.
Vinton, John, "Bartck on His Own Music, " Journal of theAmerican Musicological Society, XIX (1966), 232-243.
107
108
Scores
Bartok, Bela, Four Dirges, Op. 9a, Huntington Station,Long Island, New York, Edwin F. Kalmus, N.D.
