CALIFORNIA STATE UNIVERSITY NORTHRIDGE

THE USE OF ANALYSIS IN TEACHING TWENTIETH CENTURY PIANO LITERATUHE ,

A graduate project submitted in partial satisfaction of the requirements for the degree of Haster of J1.rts ·in

Music

by

Nancy Regnier ~shforth

The graduate project of Nancy Regnier Ashforth is approved:

CALIFORNIA STATE UNIVERSITY. NORTH~IDGE

August, 1977

ii

TABLE OF CONTENTS

Abstract ·····················~························· v

Introduction 8···················~······················ 1x

PART I CONCEPTS AND PROCEDURES

Chapter

I. Expanding Concepts and Terminology............... 1

II. Establishing Analytic Guidelines ·········•······· 10

PART I I ANALYSES Al"'D TEACHING POINTS

II I. Berg: Sonata, Op. I-, { 1908)...................... 19

IV. Bartok: Free Variations and Minor Seconds~ Major Sevenths from Hikrokosm·2z._, (19215-1937) • .. • • • • • • • • 29

V. Messiaen: Noel, from Vingt Reqards sur L'Enfant Jesus., ( 194ifT:7. it •••• o •• ~ 8." .~:::::-,7-:·u ~ 7 Ql·,. 7.. 39

Conclusion ......•.....•....•.•.••.•....•.•.••.• -........ 47

Bi~liography :·········································· 48

Appendix: Musical Examples •••••••••••••••••••••••••••• 50

CALIFORNIA STATE UNIVERSITY, NORTHRIDGE

THE USE OF ANALYSIS IN TEACHING TWENTIETH CENTURY PIANO LITERATURE

and

GRADUATE RECITAL IN PIANO - PEDAGOGY

A graduate project sutmitted in conjun:tion with an artistic performance presented in partial satisfaction of the requ·irements

for the degree of Master of Arts in ~~usic

by

Nancy Regn·ier Ashforth

August, 1977

ABSTRACT

THE USE OF A~ALYSIS IN TEACHING TWENTIETH CENTURY PIANO LITERATURE

by

Nancy Regnier Ashforth

Master of Arts in Music

August, 1977

The purpose of this graduate project is to explore the uses of

analysis in teaching twentieth century piano music. The project

includes both a recital of tWentieth century piano music and a

paper dealing with analysis in a studio teaching situation, with

mtls ka 1 examp 1 es dravm from the reci ta 1.

~nong the problems involved for a teacher are: expanding a

traditional musical education to encompass evolving musical concepts

and terminology; finding a method of analysis that will be both

comprehensive and r·texible enough to produce immediate aural under­

standing in a studio teaching situation.

The paper is divid€d into t~ro parts. The first two chapters

discuss evo1ving concepts and terminology and propose a series of

analytic questions to serve as a guideline for student/teacher ex­

p1or·at·tcn d1.a'i;1g the 1es$ons.. The guideline questions are grouped

into the categories of sectional structure; motivic structure,

tonal orders !i pi tr.h content0 rhytl'm and texture.

v

The last three chapters contain analyses of three twentieth

century compositions: Berg, Sonata, Op. I; Bartok, Free Variations

and ~nor Se.conds, Major Sevenths_. from VQlume VI of the Mikrokosmos;

and Messiaen's N~el from the Vingt Re~rds sur l'Enfant J~sus.

These analyses are presented in three steps: aurai analysis, based

on aural perception before study; theoretical analysis, based on

study of the scor:e; and teaching points, based on information gained

through the theoretical analysis.

FrL9~Yt-M~ 27,~. ~19~7~7------------·~R-ec~i_t_a~J-H~a_1_1 ____________ 4_:_o_o~p_._m __ •

PROORAr~

MIKROKOS"10S ••••••••••••••••••••••••••••••••••••••••• o ........ BELA BART6K

125. Boating

144. Minor Seconds, r~ajor Sevenths

~40. Free Variations

(1881-1945)

SONATA NO. 5, OP. 38/135 ••••••• e••••••••••••••••••••••oSERGE PROKOFIEV . (1891-1953)

Allegro tr?.nquillo

Andantino

Un poco allegretto

----------·-·~-~----·-------------------~~~~--~-------INTERHISSION

Vi

SONATA, OP. 1 •••••• •• Cl' ................ ••••••••••.••••.••••••••• ALBP,N BERG (1885-1935) ,

VINGT REGARDS SUR L'ENFANT JESUS ••.••••••••••••••••••• OLIVIER MESSIAEN

I Regard de l'etoile ·

Regard de la Vierge

Noel

{ 1908- )

vii

The artistic performance of Nancy Regnier Ashforth is approved:

Dr. William Richards

Or. Charles Fierro

Committee Chairman

The abstract of Nancy Regnier Ashforth is approved:

. Or. George ~kapski

Dr. -Charles Fierro

Califu~nia State University, Nortrrridge

August, 1977

INTRODUCTION

The purpose of this graduate'project is to explore the uses of

analytic techniques in teaching twentieth century piano literature.

The project includes both a recital of twentieth century piano music

and a paper dealing with the use of analysis .in a studio teaching

situation, with musical examples· drawn from the recital.

While the benefi.ts of -using analysis to aid in answering a

variety of musical questions or at least in suggesting approaches to

. find·tr;g solutions are unquestioned, the problem 1 ies in the fact that

the majority of time spent in a college or conservatory in preparation

for a music teaching career is involved in the history, theory, and

performance practice of music written before 1900. This preparation

is, of course, indispensible in forming the critical and stylistic

acuity of the literate teacher. However, in teaching t"'-er.tieth

century music, one must be aware of not only the continuing evolution

in the use of the materials of music. but also that the analytic

techniques describing the relationship bet\'Jeen materia 1 s in earlier

music, remain viable only to the extent that these relationships

continue to function in the same or a similar way.

How can a studio teacher adapt a traditional education in music

to meet the problems in newer music? Consideri~g the expansion that

has taken place in all directions of musical thought since 1900, the

first step wo:.Jld be to re-examine our musical concepts and tenninology

in order to discover in which \'lays they have evolved or remained the

ix

same. Next, a method for using analysis should be evolved which will

be sufficiently comprehensive and flexible to provide .immediate and

audible clarification with a variety of complex literature.

This paper will be presented in two parts: Part 1, (containing

Chapters 1 and 2), will discuss the expansion of musical concepts and

terminology, and will propose a series of analytic questions that can

be used as a guideline for teacherlstudent preparation of a new work.

Part II, containing Chapters 3-5, will consist of analyses from the

works of three composers, divided into three areas: aural analysis,

(the use of analytic guidelines, based on the student's aural percep-

tion of the work before any formal study); theoretical analysis,

(application of analytic questions to the musical scores); and

teaching points, (specific points to be discussed about the work as a

result of the theoretical study). The teaching points can include not . only the musical aspects of the work being analyzed, but also in

comparison with other works. thus broadening the frame of reference.

The examples: Sonata, Opus I, by Alban Berg; Free Variations

and Minor Seconds, Major Sevenths, from Bela Bartbk's Mikrokosmos

Volume VI; and Noel, from Olivier Messiaen's Vingt Regards sur

!:.:_En..f.!!!..!.Jg_ltJ.1_ were chosen because they repr~sent three. differing

approaches to twentieth century composition.

The examples are all written in standard notation to be played on

the keyboard~ the problems involved in the real fzation of graphic

notations or the tachnical requirements of non-keyboard perfonnance

(sounds created outside the keyboard) are outside the scope of this

papere Since nona of the ?Xamples involve serial techniques, the

X

specific analytic methods used ·in serial analysis will not be discussed.

The goal of a consistent application of analytic tectmiques in

teaching should be to increase the independent ability of the student

to perceive, both aurally and intellectually, the shaping forces in the

music that he plays, and finally to achieve a more mature and intelligent

performance.

Xi

EXPANDING CONCEPTS AND TERMINOLOGY

In redefining for the twentieth century the concepts and

terminology commonly used in talking about music, one must take into

account the fact that musical contexts are constantly changing,

although the basic elements of pitch and rhyttm remain constant. It

is confusing to find that in some cases the terminology has remained

the same while the meanings have subtly changed with the passage of

time. (One of the most obvious examples of this is the term

·andantino,· the exact meaning of which is still debatable, both ____ .,.. among musicians and in reference sources.) In other cases, such as

that of serial music, an entirely new terminology has been developed

to define the specific relationships within that system.

While there is probably no final solution to this problem, one

can often find a conmon thread in the usage of older terms that is

still. relevant and can be applied to new music. For example,

although historical ~ontext has constantly altered t~e exact

boundaries of rooaning, one can still use "melodi1 or "melodic line",

in m~ny cases, to refer to the horizontal aspect of pitch succession.

The alterations in meaning, from the Greek modes through polyphony

and tonality, have dealt with melody as constru~ted within a certa.in

system or style, but not with the concept of melody as a horizontal

succession of pit~h.

Given the variety of compositiona 1 techniques in current practice,

1

the standard twentieth century references are careful to avoid

referring to specific systems in their general definitions of

melody. In the Harvard Dictionary of 1'-lusic v1e find: 11 a succession

of musical tones, as opposed to harmony, i.e. musical tones simulta-1

neously ••• 11 ; in Grove's Dictionary of Music and t~usicians: "the

general tenn vaguely used to denote succession of single notes which 2

make musical sense ••• "

If one hesitates to use "melody11 or 11melodic 1 ine" because in

earlier styles these terms implied a certain formal or tonal

structure as well as linear succession, than one can use simply

"1 ine, 11 11 1 inaar aspect" or 111 in ear connection; 11 the most appropriate

term depends on one's critical judgement of how that aspect functions

in the work being discussed. Just as it would be ridiculous to refer

to a melodic line in Prokoviev as a "linear succession of pitch class .. .

events, .. it would also be inappropriate to talk about the "melodic

line" in the Hebern Y.?_riations, Op. 27_. In using·any or all of these

tenns in teaching, the first consideration is that the student be

aware that the concept of horizontal succession obviously occurs in

all music, whether diatonic or chromatic, conjunct or disjunct, con-•

tained within the span of a tetrachord or spread across the audible

range ..

The concept of harmony still retains its definition as the

vertical structure of a composition as opposed to the melodic or

l"Melody,t• Harvard Dictionary of ~·1usic, 2nd ·ed., (1969), p. 517. 2c. Herbert H4 Parry, 11Melody, 11 Grove's Dictionarv of Nusic and

Musicians, 5th ed., V (1954), p~ 66~--~-

horizontal structure. However, since tonality is no longer

(necessarily) the controlling factor in hannonic succession, the

relationship of the harmonic aspect to the work as a whole must be

discovered in each work rather than assumed. One difficulty in

discussing harmony in twentieth century music is distinguishing

between contrapuntal lines that are only contiguously vertical and

vertical sonor·i tes \•/hose intervalli c structure defines than as

meaningful in the course of a work. Certain terms have developed as

a result of this nonfunctional possibility of vertical structure:

1. aggregate: often used as a substitute for "chord, 11

(a.l though it has a more specialized meaning in serial composition) .3

2. simultaneity: frequently used to refer to the simul­taneous sounding of any pitches.4

3. vertical: a tenm used for the elements or components of a composition that sound simultaneously and are notated one above another on the score.S

\~hile these tenms are useful, there is no reason that 11 chord11 cannot

sti11 be used in referring to a simultaneity that has a specific

intervallic consistency within a work, if it is understood that the

intervall ic structure is not necessarily based on thirds as it was in

functional tonality.

3Robert Carroll Jones, A Glossarf of Theoretical Tenns Used in Selected t~ritin s in English Aboutwentieth-Centur Music, doctoral Cifss'ertation, Unw. of · owa, L65, Ann Ar or, ~1ic • : uriiversity Microfilms, Inc., 1970} p. 8.

4Jones. Qp. cit., p. 281.

5Jones, oo. cit •• p. 351.

6 7 The concepts of extended tonality and suspended tonality (a

state where one basic tonality still exists in spite of various

melodic and harmonic f1gurations ~hich may have been added) can be

.brought up in discussing ~~rks such as the Berg example in Chapter 3.

An example of vertical structure based on intervals can be seen in

Bart6k' s ~1inor Seconds, Major Sevenths 8

in Chapter 4. The expansions

of basic harmonic concepts by ~lessiaen are so specific to his music

that they vli11 be discussed \'lith the example of his work in Chapter 5.

All changes in the movement of sound through time must be

considered in a general definition of rhythm. Howard E. Smither, in 9

his excellent article on rhythm in the Journal of Music Theory

defines rhythm as "order in time," the tenn "order" referring to a

harmonious relation rather than necessarily a regular arrangement.

While this broad definition is acceptable, there are still many

leve1s of rhythmic activity that should be differentiated in some way. 10

In his §l!:uctural Functions in Music, Wallace Berry suggests the

following rhythmic aspects which are all related either to pace or

grouping:

6Jones, pp. cit~. p. 97.

7Jones, op. cil·• p. 305.

8Bela Bartok, Minor Seconds, Major Sevenths, from Mikrokosmos, VI (London: Boosey & Ha\"Jkes, Ltd., 1940), pp. 16-19.

9Howard E. Smither, 11 The Rhythmic Analysis o"f 20th,;.Century Music~" ~.2!-l!!!al of r-1usic Theor.z', VIII. no. 1 (Spring, 1964), p. 56.

10wa11ace Berry, Structural Functions in Music, (Englewood Cliffs, N •• J.; Prenti ce-Ha11 11 Inc., 1976'), pp. 447.

-.: ,_·~

1. Tempo, which includes 11.eo both the eventfulness of music

and the frequency of pulsation at some given level ... 11

2. Pattern or motive, 11 .~. as expressed in durational and other strong-weak combinations which have, in a given context, motivic significance at some level, or characteristics of pattern by ~tthich a style, genre, or work can be identified ... 12

3. The profiles expressed in element changes (such as melodic rhyt!tn, harmonic rhythm, textural rhythm) ..... as these changes invo 1 ve pattern, rate and degree of change. 11 13

4. Grouping, including meter, 11 ••• a portioning of music's

time span by associations perceived within and among punctuated or articulated unit-orderings of events. 11 14

While· rhythm in the majority of Western music has implied a

feeling of regular pulse and the organization of groups of pulses

into meter, these concepts have been both expanded and abandoned in

some twentieth century music.

Some terms in use to describe these rhythmic expansions are:

1. additive rhytt~: the grouping together or addition of beats of unequal length. This is similar to the tenn "asymmetric meters" Nhich refers to meter signatures V·lhich indicate unequal groupings of beats such as 5/8, 7/8. 15

2. ametrical: (as used by Nessiaen) music with a free but precise rhythmic pattern, in opposition to 'measured'

·(i.e. equally barred) music. 16

llserry, oe. cit., p. 305.

12Ibid., p. 305.

13serry, Qp. cit •• p. 306.

14~ .. p. 306.

15Jones, op. cit., p. 6.

16Q1 ~vier Ness iaen! The Tech~ ig_~of r4y Must ca 1 Language, trans.· by. J~ Satterfleld, (Par1s:--IC[eauc, 1956),_!. 14.

•

3. metric modulation: the gradual change from one pulse to another. 17

4. polymeter: simultaneous use of two or more different meters with two or more independent patterns of regular accentuation used simultan·eously. 18

These terms will be useful in discussing the works of Bartok and

Messiaen and specific examples will be pointed out in the later

chapters.

The concept of texture in music has undergone a treJOOndous

expansion in twentieth century music. As a term it defines both the

density and complexity of movement between musical lines, varying in

degrees from homophony to polyphony. In earlier music texture was a

stylistic referent, being used with relative consistency within a

particular style. Ho\'rever, as functional harmony disappears as the

controlling factor in creating form in twentieth century music,

texture assumes enough independence to be considered a separate

dimension of structural significance.

While the traditional terminology referring to texture, such

as "polyphonic11 and "homophonic, 11 needs no further definition, certain

other aspects of textural differentiation should be explored. The

. tenn, "textural rhythn 11 can be used to describe the timing and nature 19

of textural changes, "textural activation .. describes the means

=I7Jones, O_Q_. citq p. 160

lBSmither, ~~· p. 77

19serry, oe. <:it. p. 201 ·~·

u

the w~ans (dynamics, articulation, rhythm,) used to vitalize simpler 20

textures,· and "texture space11 describes the boundaries of vertical 21

and horizontal in which the music moves,. It is possible to talk

about textural modulation (a term similar in its use to metrical

modulation) as a transition from one texture to another. In teaching

twentieth century music, the awareness of textural modulation,

textural variation, how it is achieved and how Jt relates to the

structural whole is as important as the awareness of the pitch and

rhythmic organization.

Unlike the other musical aspects, which can be isolated for the

purpose of description and discussion, a definition of form or

structure in music must include all of the other elements and how

they relate to each other. (While some twentieth century composet•s

work consciously to create random or formless sound experiences,

these sequences of sound events can only be described, not analyzed,

and are therefore useless for the purposes of this paper.}

As a starting point, let us consider the following excerpt from

the Harvard Dictionary of Music essay on form.

"Music is not a chaotic conglomeration of sounds but consists of elements arranged in orderly fashion according to numerous obvious principles as well as a still greater number of subtle and hidden relationships.22

zn. ~erry, 2£· cit., p.22Z

21serry, op. cit., p.24~ · . ./

. 22"Form," !!,arva'1L_Dictionarx-_nf Musi£, 2nd ed, (1969) 326,

I

~lhile the lack of a corrmon style practice creates a lack of

predictability in twentieth century music, the unifying techniques

used throughout the centuries are. still va 1 i d, though perhaps in

. different proportions or different guises. One must still look for

some kind of relationship between the polarities of motion and rest,

tension and relaxation, repetition and variation,_ density and

sparsity, small detail and large section, organic development and

stark juxtaposition.

Berry, in another sense, is also describing this balance when

he defines structure:

••• musical structure can be regarded as the confluence of shaped lines of element succession which either agree ••• in intensity direction or d·isagree ••• in direction. Within the range of musical elements subject to control within these procedures, some are, of course, of greater relevance to certain styles than others, but the under­lying concept of the confluence of functional element­actions as fundamenta 1 to musi ca 1 structure and effect is of crucial significance in all styles. 23

While one can assume some similarities in the quality of

confluence contributing to a coherent musical structure regardless of

style, certain traditional formal principles have been greatly

expanded. For examp 1 e, the ide a of cadence has existed throughout

our musical history as one of the principal devices for delineating

structure. Without a clear definition of endings provided by the

harmonic unity of a tonal system, one finds both that the idea of 24

closure can be brought about by other means, ·such as textural or

23Berry, op. cit., p. 9.

24 ,Jones. op. _c)t,. s p. 46.

rhythmic changes or the completion of a motion in the pitch aspect.

or that in some cases, the concept is no longer relevent.

One assumes. logically that in any composition exhibiting a

perceptual unity of sound, there exists consistent fonnal relation­

ships between the elements. It is up to the teacher to bring these

intuitive aural perceptions to the_ level of conscious understanding . -

in the student.

ESTABLISHING ANALYTIC GUIDELINES .

Since music is an art which exists in time, it is impossible to

perceive the total structure of a work at once, as we can with other

art forms such as painting or sculpture. The various methods of

analysis provide us with ways of looking at a composition as a whole

and uncovering the relationships between the parts.

Analysis has ahtays been a part of the teaching experience. The

process of cr·itical listening, whem not involved with specific

technical pt .. oblemse constantly requires decisions based on analytic

knowledge. Any comparisons one makes between phrases or sections. in

deta'il s of articulation • dynamics and rhyttrn are based on conc1 us ions

drawn from an awareness of the whole structure and not simply the

moment at hand.

A most eloquent description of how a student benefits from

analysis is given in an article from a forum on theoretical training

in the Journal of Music TheprJ by Stanley Fletcher:

And all students should leam that the expressiveness of music is not something they must learn to "put into it" but something they must reveal by working out that plastic treatment which makes most evident the textural and structural nature of the composition. The study of music must guide the study of its performance ••• He will no longer be content to make a crescendo·because some teacher or editor (or even the composer in the role of editor) has indicated that he should, but will ask him­self: "~lhat is happening in the music that should be pointed up by an increase in this p?ssage? 11 Or, if an. accent is indicated, he will ask: ~~~!hat in the texture or drama here calls for emphasis, and how much?" Such a perfonner wi11 recognize. when his instinct or intuition

10

tells him to make the music sudden 1 y softer, that the treatment is justified because of a resolving dissonance at that moment or because of a startling change of direction in the harmonic progress, or a sudden retreat from a previously asserted purpose in the structural development of the composition. _

In sum the level of performance ••• is that of an intelligent musical performer sensitized to music with all his resources. He is not merely a highly trained automation performing under the direction of blind instinct, however refined blind musical instincts may become. He is not merely a rational mind, dissecting and analyzing the dead carcass of a musical composition with a dry brain in a dry season. In this performance the t\'10 have become one, a complete musical personality with mind and imagination and intuition ~-;orking together to\<.tard producing an enlightening and communicative performance which shall engage not only the player but the listener as well at this enriched level of exper­ience. 25

However enthusiastic one is about the idea of using analysis in

teaching twentieth century music, several problems present themselves

at once. One is that the studio teaching situation is not the same .. as a theory classroor.i and \'lhatever analytic techniques are used must

be immediately applicable to the music one is working on; i.e.

examples need to be audible and understanding must be achieved on an

aural as well as intellectual level. The studio lesson is not the

place for graphs and charts.

Another problem is that specific analytic methods developed in the

past (such as those of Schenker) were the result of the study of a

specific kind of music. One cannot impose an analytic method on music

which does not apply to that method, and certainly not all twentieth

century music is susceptible to the kind of hierarchical peeling which

25stanley Fletcher, "For the_ Perfonner." ~rnal_ of Music Theory, I~I. no. 1 (April, 1959), p. 48. 49.

Schenker's methods use.

However, Allen Forte, a theorist who has applied Schenker's

levels of reduction to the work of such composers as Stravinsky,

Sessions, Bartok and Schoenberg, claims no difficulty in reconciling

Schenkerian concepts to non-triadic music. Forte's definition of

structure in music as a "complex of ordered and interrelated tonal 26

events which unfolds in time" does not eliminate non-triadic tonal

.wov·ks from consideration under this system, but rather insists that

other linear and hannonic events, to be uncovered through the levels

of reduction, fonn the basis of a hierarchical structure in a similar

way to the tonic-dominant function in tonal music.

Not \'.dshing to argue the possibility of using Schenkerian concepts

in analyzing some twentieth century music, one would also have to

state that the efficacy of using this technique in a studio teaching

situation ~uuld probably be limited to a teacher's ability to discern

'from the scm~e (or fOi .. eground 1 evel \'/here perfonnance takes place •

after a11} the elements \<Jhich are structural as COlllJared to those which

are embellishing, or to point out how a musical event can function in

its immediate surroundings and also as part of a larger hierarchy,

and demonstrate these comparisons by audible example. This continuing

insistence of the aural perceptibility _of the examples is backed up by

Forte in the conclusion of his chapter on analytic procedure:

The aural sense is basic to musical activity of any kind, for music is an art in sound. The use of visual

'.,/

-------------·------26A1len Forte, 9~nt~m~1r?!l-I£~ Structu~s. (New York: Columbia

University, 1955)

.....

means (the analytic sketches) in this study does not mean that the ear is considered secondary to the eye in the analytic process. On the contrary, it should be stressed that the sketches are valid and significant only to the extent that they correctly represent structure as abstracted by the ear. In all cases the ear is the final arbiter. 27

Composer and theorist Philip Batstone, in his article "~1us­

ical Analysis as Phenomenology" from Perspectives of New Music also

begins from the p·oint of view of 11 What is heard11 in music¥ He

points out that Schenker's theories are viable "because they are

pragmatically relevant, that they are relevant furthennore because 28

they stan from empirical examination of music as a "heard phenomenon."

Batstone' s argument is that in much twentieth century music, a

description of the means of a composition (which refers to the

techniques and compositional devices) does not always correspond to

a ~description of the aura 1 phenomena. As an example he contrasts two

analytical approaches to a Webern work: the first, based on the

relationships of durational proportions; the second based on his

"ad hoc11 analysis which uses perceptual assumptions such as extremes

of range, contrast, and repetition. While not disparaging the

existence of complex relationships uncovered by the durational

analysis, since they are not audibly perceptible, he c~1cludes that

more important data exist on the aural phenomenal level.

Referring to the durational analysis, he writes:

27Forte 8 ~· cit., p. 24o

28Philip Batstone, "Musical Analysis as Phenomenology." j P .. ~_rspes_1j_y;;.;; of Nev1 Music, VII, no. 2 (Spring-Summer 1969), 95.

The above analysis is not viable with respect to perception, and my reasoning involves a recognition of ·the fact that there is no opportunity to perceive the creation of nonns of the kind \'lhich can allow the ear to perceive 11 hierarchization 11 or movement or develop­ment on an order of complexity anything like that shm'ln by this analysis. 29

Taking a similar viewpoint to Batstone is Charles Wuorinen, who

is known as a performer of twentieth century music, as well as a

theorist and composer. Wuorinen writes concerning analysis and

performance:

••• it is extremely difficult to say just what goes into the kind of analysis necessary for informed performance. It is possible to observe, however, that the type of analysis generally presumed to have value in 11 theoretical" explication is often useless in the preparation of a performance •••

From the performer's point of viewt such conven­tional analyses, while of course generally useful to him outside of rehearsal, nevertheless fail to moti­vate his manner of playing. since the major part is most often devoted to translating the information into another (usually verbal) linguistic medium. The performer, however, needs not translation, but direc­tion and focusing. 30

Wuorinen goes on to cite examples from two contemporary scores

(Donald Marti no's Trio for viol in, clarinet and piano, and Peter

Westergaard's Variations for Six Players) to make the distinction

between when analysis is, or is not, useful for performance. In

referring to the example from Martino's scqre he states:

29aatstone, op. cit., P• 104.

30char1es Wuorinen, 11 f.lotes on the Performance of Contemporary Music," ~!.SE.ectives of New r~usic, III, no. 1 (Fall-~Jinter, 1964), 18.

14

It is not of crucial necessity to a successful performance of this work, however, to know pre-cisely which set segments are being presented through this particular form of articulation - apparently because given this knowledge, there is very little one can do in the moment of performance to project it. 31

In contrast, Wuorinen writes, concerning the Westergaard example:

••• it is necessary to know that the timbral differen­tiations in this excerpt all have structural function. But here a more detailed knowledge than is needed in the ~1artino, is needed because it can be projected in performance. 32 -

With these arguments as a conducive background, aural percep­

tibility remains foremost among the criteria to be used here in

establishing a series of analytic questions to serve as guidelines

in teaching. Other criteria would be: using familiar terminology

and concepts when applicable and expanding then when necessary;

stating possibilities as flexible and open-ended rather than restric­

tive in order to examine many different kinds of music. T.hese

ques~ions are organized into discreet areas with the understanding

that in any creative discussion of a particular composition there is

bound to be a great deal of overlapping between areas •.

I. Sectional structure

le Can the composition be divided into sections?

2. How are the sections articulated {by cadences, rh.vthm1 texture, harmony?)

3. vJhat are the general similarities or,. contrasts

31wuorinen, ~·· p. 18-19.

32wuorinen, 2E· cit~· p. 20.

between sections?

4. How do the sections compare specifically in terms of length. and general rhythmic and harmonic activity?

5. How do the larger sections break up into smaller units or phrase groups?

6. How do these smaller groups interact to form the larger sections?

II. Motivic structure

1. What are the smallest motivic units or ideas?

2. Is there an intervallic or rhyttrnic consistency in their makeup?

3. How are the motivic units related or contrasted in the different sections of the work?

4. How are the smaller units transformed and developed into larger units or phrases?

III. Pitch material {linear aspect)

1. Is the linear aspect predominantly diatonic or chromatic, conjunct or disjunct?

2. Is it possible to differentiate between structural tones and embellishing tones?

3. What role does tessitura and register play in the linear aspect? ·

4. How does the linear aspect interact with the other elements in terms of texture?

IV. Pitch material {vertical aspect)

1. Is the overall sound tonal, bitonal, polytonal. atonal, seri a 1'! ·

z. Are the vertical structures based on a consistent intervallic content?

3. Is it possible to differentiate between a structural and embellishing or coloristic use of harmony?

4. What is the rate of hannonic chanqe, and how does it relate to the other musical aspects?

5. Is it possible to relate the harmonic motion to the minimal structure? To the sectional structure?

V. Tonal orders

1. Is there sane kind of tonal order?

2.. On what is the tonal order based (one pitch center, several pitch centers, an interval, an expanding inter~ vall i c pattern?)

3. Can the tonal order be described by a system (tonal, bitonal, polytonal, atonal, modal?)

4. If not "systematic," as suggested above, is it possible to describe a progression or development in the area of tonal order?

VI. Rhyttrn

1. Is there a feeling of regularity at the level of the pulse or beat?

.2;. Is there a feeling of regularity at the level of meter?

3. Are there specific rhythmic motives?

4. How does the rhythmic structure of the work relate to the other aspects, such as sectional structure?

5. Is there rhythmic counterpoint?

6. Is it possible to describe various rhythmic levels from motivic to structural upbeat and downbeat?

7. Is there metric modulation?

VII. Texture

1. Is the texture predominantly homophonic or polyphonic?

2.~ ~Ihat and where are the contrasts in density, range, and spacing?

3. How do the contrasts in texture relate to the motivic and sectional structure? ·,·

4. Howdoes the variety of texture relate to the linear. vertical or rhythmic aspect?

5. Is there textural modulation?

This series of questions is designed, not to provide an in depth

analysis, but to serve as a starting point for discussion. With the

guidance of the teacher, the student can learn to listen and evaluate

what he hears rationally as well as intuitively.

BERG: SONATA, OP. 1 (1908)

Perhaps no other fonn since the beginning of the Classical

period has so expressed or exposed a composer 1 s unique view of

relating structure and content as the sonata form. The variety of

compositions that share this title is enormous; the concept to be

found in common is often no more than: statement, departure or

development away from the statement 0 and some kind of return to the

statement.

In using sonata form, the composer has the choice of developing

whatever aspect seems most important to him. The element of contrast

possible with a sectional structure can be expressed by a variety of

means: highly differentiated motivic and rhythmic materials; contrasts

of harmonic areas; dynamic, tempi and textural contrasts. On the

other hand the sectional structure of the form can be minimized in

favor of an organic unity of texture, motivic and rhythmic materials ' 33

and harmonic content, as Berg has chosen to do in his Sonata, Opus..l.

AURAL ANALYSIS:

1. Sectional structure, differentiated by tempi. 2. No cadences; elision of sections. 3. No clear tonal center; chromatic use of harmony. 4. Complex texture; highly contrapuntal. 5. Use of meter; constant fluctuations$ 6. Use of intervallic motives. 7. Use of rhythmic motives.

STUDY ANALYSIS:

33Albm Berg, Sonata, Op. I. (Vienna: Universal Editions, 1926) p. 11 ..

19

•• _. '!'.:"".' .• :~

The sectional structure is clearly related to the traditional

sonata fo nn:

I. Expos-ition: m. 1-55.

M. 1-28, first thematic group, M.29-48, second thematic group. M.49-55, closing section.

!I. Development: m. 56-109.

M. 56-70, first section. M. 70-99, high point. M.l00-109, restransition.

III. Recapitulation:_ rn. 110-179.

M. 110-136, varied recapitulation, first thematic gr·oup~ M. 137-166, varied recapitulation, second thematic group. M. 167-179, closing section.

Although the traditional sonata fonn is followed, including an

indicated repeat of the exposition, the continuous transfonnation and

development of the motivic material, both intervallic and rhythmic, 4

are so interrelated as to supercede or overlap the sections of the

form creating a work of continuous variation. The sections dissolve

into each other rather than ending, with the only complete hannonic

cadences occuring in the opening and closing phrases.

The opening phrase, which can be considered a microcosm of the

work. -Introduces the intervallic and rhyttmic motives on which the

work is largely based, (ex.l). The transformation of the first

rhythnic motive 'iJithin the first three measures, from upbeat to down

beat and in combination with its augmentation, is the first indication

of the continuous development, variation, and recombination which is

the compositional procedure of this work, (ex. 2) •.

One can also find sufficient reason to treat the hannonic material,

spe.cifica11y the series of altered chords which descend chromatically,

as the harrnonic motive of the work. Although lacking any particular

rhythmic character, this chromatic series appears throughout the

work, and is frequently developed in a motivic fashion: m. 6-7;

m. 40-41; m. 57-59; m. 147-149, (ex.3). There is a certain

ambiguity about whether this series can be considered solely in its

vertical construction. When it moves in slow, even progression as in

m. 1-2 and m~ 8-9, it can be heard as supplying an harmonic (although

non-tonal} background; when it moves at the same relative speed as the

other lines, it can be heard as a doubled linear element ~ithin a

polyphonic texture, as in m. 6-7. Thus, whether its function is

primarily vertical or linea~ will vary according to the rhythmic

context of the passage.

Measures 3-10 develop the 'B' and •c• motive by fragnentation,

sequential repetition and rhythmic condensation, closing with an

elongation of the 'C' motive in m. 9-10 •

. To illustrate the economy of means which unifies this movement,

some of the various transformations of the •c• motive should be

illustrated. Functioning 1 ike an appoggiatura and occasionally a

suspension, this motive is frequently used to indicate the closing

of a phrase or section: m. 9-10; m. 15-16; m. 27-28; m. 4·5-48;

m. 54-56; inverted in m. 97-99; greatly extended in m. 130-136;

inverted in m. 161-164; m. 169-174; (ex.4). ~lhen combined \'lith

rhythmic augmentation"~ it takes the place of a functional cadence -

which is missing.

The same motive 0 inverted, is frequently used to initiate the

· phrases as in m .. 10, m. 13, in the closing section in m. 49, 51, and

in the final closing section, m. 167, 169. Occasionally the 'C' motive

appears as a phrase ending and immediately after as the beginning of a

new phrase as in m. 4-5, m. 77, m. 114. m. 143, and m. 169-171, a

device sometimes referred to as derivative continuation (ex.S).

The second phrase group introduces a new tempo· R,2scher · al s Tempo I

as well as the triplet motive. 'D', which is a recombination of the

motives of the first phrase, (ex. 6). The restatement of the first

phrase now includes the triplet motive and is developed in m. 16-28

to the first of a series of dramatic climaxes, which is built on a

. descending whole tone line derived from the descending major thirds of

the 'B 1 motive and first appearing in m. 7-8, (ex.7). The dis­

solution of,this phrase group, m. 26-29, is based on the inversion of

the 'D' motive.

The second thematic group is introduced by a new tempo,

L~p~~~er al~Eo-~ and a thinning out of the contrapuntal texture.

The motivic structure_ is again related to the opening phrase using a

recombination of the rhythmic and intervallic motives. The 'C' motive,

previously used to begin and.end phrases is now enclosed within the

'A' motive and the 'B' motive is contained in a sixteenth note figure,

(ex. S). The accelerando which leads to the high point of this section

is built on a sextuplet diminution of the 'D' motive. The descending

1 in.e, m. 43-44, although not consistently whole· steps as in m. 23-24.

relate-s to this- earlier clima~ in n~gi ster ar1d dynamics as the high

point of an arch-shaped crescendo and accelerando, approached by a -- t

similar sequential repetition.

While the closing section 1 m. 49-55 is based on an augmentation of

the sextuplet figure in m. 38, the development section begins with a

~recombination of the 'A' motive and the descending line taken from

the climax of the arch shape. The first section of the development

also features an increasing use of the triplet as the beginning of

the rhythmic accelerando which dominates the section. The hemiola in

the climax of this small arch shape acts as a temporai~y pulling back

of the motion, set against the augmentation of the descending line in

the bass.

The rhythnic complexity increases in the second part of the

development section which combines the 'D' motive set against the

sixteenth note motive in overlapping entrances. In. m. 78. the

descending line phrase is added in rising sequential entrances, and

the climax of this arch shape in m. 91 for the first time combines the .. intervall ic motives with the chromatic descending chords, now with

octave doublings, and extending the full range of an octave, {e~. 9}.

If one relates the arch qesture which shapes the dramatic climax

of each section to the gesture of the opening phrase, the possibility

of another structure merges, in which the architecture of the move­

ment is as much exposed by the hierarchical levels of enlargement of

this gesture as it is by the traditional functions of the various

sections of sonata form.

The similarity of the descending 1 ines at the high point of the

-arch shapes is too striking to be coincidental;· the derivation from·

the is• motive has already been suggested, (e~ 10).

It is the length of the preparation and dissolution of these

arch shapes v1hich detennines their level in the overall structure of

the work. Berg fuses this dramatic idea with sonata form by varying

-the length and intensity of each .climax, reserving the most intense

for that point late in the development section \'/here the climax

would occur in a traditional sonata form.

A diagram for the arch shapes for-each section:

Lengt_!i Preparation Hi ghpoi!'~ Di sso 1 uti on

M. 5-10 m. 5-7 m. 8 m. 9-10

Expo. M. 18-28 m. 18-22 m. 23-25 m. 26-28

M. 38-48 m. 38-43 m. 44 m. 69-70

e· 63-70 m. 63-66 m. 67-68 m. 93-99 Oeve1.

M. 80-99 m. 80-91 m. 92 m. 93-99

[M. 123-136 m. 123-127 m. 128 m. 129-136 Recap.

M3 150-166 m. 150-158 m. 159 m. 160-166

By following the climax of the development with a retransition

based on the second thematic group, in the slower tempo, Berg again

undermines the sectional structure of sonata form by depriving the

recapitulation of any dramatic impact. The recapitulation is

introduced with fragments of the 'B' motive. m. 107-109, and when

the complete restatement of the first phrase occurs in m. 110, it

is already in a stage of redevelopment. which is characteristic of

the work.

Although all the elements of the exposition. return in the

recapitulation, there is no 1 itera1 repetition. but again,

continuous recombination and variation of the motive materials.

For example, m. 3-5 are extended by sequential repetition in

m. 113-117 and m. 6-8 is now developed into another arch shape based

on the 'B' motive from m. 1Zl-130.with the triplet motive from

m. 11-16 combined with the •c• motive in the dissolution, m. 131-136.

The Rasch, m. 38-44 is extended by sequentia 1 repetition into the

final climactic arch shape, m .. 144-166 which include.s a hemiola very

similar to the hemiol1a in m. 67-68.

The final closing section is varied by introducing the melodic

material from m. 167-168 in the bass line, m. 169-170 and inverted

in m. 171-175. The approach to the final cadence is similar to

m. 54-56, but in inversion; the right hand reaches the conclusion of

its phrase in m. 170, and reminisces one final time on the •c• motive fJ until the left hand has worked through its sequence to the F -b

cadence.

As the pitch material of the movement is so unified as to approach

being monothenatic, the element of rhyttln becomes increasingly

impo~tant in providing both variety at the level of detail and co­

herence in the large-scale stt~ucture. Although the work is clear·ly

in 3/4 meter throughout, the continuous rhyttlnic develojlnent of the

pitch motives and the use of overlapping phrases and the hemiola give

the surface constant variety. The transformations of the 'C' motive

have already been discussed; further examples of the transformations

of the 'D' motive are more extraordinary, (ex. 11}. The hemiola

appears throughout at points where tempo _changes are taking place: ·

m. 26-27; m .. 54-55; rn. 67-70-; rn. 159-160i and approaching the h·igh

point in the devel OJlilent section at m. 82-83 and 86-87.

ihe larger tempo changes, such as the ·Lanqsamer- al s ·rempo I

in m. 29 1 100, 137 and the Quasi Adaqio in m. 49 and 167 • are used

to clarify the structural sections, just as the accelerando followed

by ritardand~ or ritenuto are used to delineate the arch shapes in

each section. Another rhythmic device used to set off these

dramatic climaxes is the use of the downbeat and the avoidance of

strong downbeats in the phrases preceding these high points. Since

so much of the procedure of this work is built on continuations based

on some fonn of upbeat, (the first downbeat in the piece doesn't

appear until the b minor in m. 3!), the downbeats at these climaxes

definitely signify a point of arrival and assume a structural

significance. In fact, what distinguishes the relative level of

importance of a particular arch shape in the hierarchical structure

is the intensity of the do\~beat, with the most intense high point in

the movement consistinq of a series of downbeats, the final one

occuring at that point where the overlapping entrances between the

hand~ join rhythmically, m. 91.

It is impossible to discuss texture separately from the element

of rhythm, since it is the constant rhyttm'ic variation at the motivic

level that is largely responsible for the complex, often contra­

puntal texture. As we have seen from the opening phrase, everything

is motivic, therefore an important element in the texture. There is

very little doubling, except for the octave doublings in the high ·,

points, and aside from the passages containing the chromatic chord

series, the voices move independently, both rhythmically and inter­

vallica11y, creating a '<teritable thicket of sound. As a result, \'!hen

the texture is thinned, usually at a point where a new section is

beginning, such as in m. 29-32, at the Quasi Adaqio, m.·49 and 167,

or at the beginning of the development section in m. 56-60, it

becomes .a structural device, working in coordination with the·

tempo changes.

The focus of this analysis has been on motivic and rhyttmic

developnent. Discussion of the harmonic content is more difficult

because there is often no more than a semitone connection to relate

one chord to the next. The intervall ic content of the chord structures

has a certain consistency in the use of fourths and tritcnes superposed

with thirds {which relate directly back to the opening motive materia~;

but frequently the vertical structures are the result of voice leading

in several parts. As mentioned earlier, the sequence of chromatic

CQords from the first measures appears throughout the work as the

principal hannonic motive and is developed rhytllnically and

texturally. Lvhile the chr0111atic voice leading, the use of

appoggiaturas, suspensions and resolutions constantly suggest an

harmonic direction, there is no root function, no real bass line

beyond a chromatic scale. There are suggestions of tonality in the

b minor cadences at the beginning and the end, and at certain points

such as the V-i progression in e minor in m. 32-32 (balanced by a

hint of b minor in a similar passage in the recapitulation, m. 139),

but these tonal references are temporary. A 1 though the framework

of sonata form is present, the lack of a unifying harmonic structure

necessitates that other aspects, in this case the intervallic

consistency of the pitch material, and the levels of rhythmic

development, from motivic detail to structural arch shapes, provide

the true coherence of the work.

TEACHING POINTS:

1. Development of intervallic motives. 2. Development of rhythmic motives. 3. Harmonic connection by chromatic voice leading compared to functional hannony, by root motion. 4. Cadencing by changes of texture. rhythm, and motivic dissolution •

. 5. Overlapping phrase structure. 6. Use of upbeat continuation; derivative continuation. 7. Structura 1 functions of the downbeat. 8. Comparison of the climactic arch-shapes. 9. Comparison of exposition and recapitulation.

10. Comparison of this sonata forms with other examples.

BARTOK: FREE VARIATIONS AND r.UNOR SECOND,

MAJOR SEVENTHS from MIKROK0~10S, (1920-1937)

FREE VARIATIONS (Vol. VI, 141)34 ·

The first question one might ask in approaching an analysis of

this work is, 11 What is being varied?" Using the tenn, "variation 14

in the generic sense rather than specific, the Harvard Dictionary

of Music offers the following:

In the most general sense, a restatement that retains some features of the original while others are discarded, a"ltel .. ed, or replaced. One of the most fundamental techniques of composition, it takes such forms as ornamentation, transposition, inversion, retrograde motions augmentation, rhyttmic modification t1·ansfonnation, etc. 35 ·

AURAL ANALYSIS:

Using the analytic questions as a guideline for discussion, the ..

following areas could be examined in a preliminary way:

1. Sectional structure. 2. Use of ostinato techniques. 3. Changes of texture. 4. Variety of rhythmic procedures.

· 5. Non-tonal pitch material. 6. Exchange or inversion of material bet\'1een hands. 7. Variation techniques.

STUDY ANALYSIS:.

In terms of variation technique, this work presents five varied

34sela Bartok, Free Variations from Mikrokosmos, VI, (London: Boosey and Hawkes. Ltd. 11.1940), pp. 2-5.. -

35nvariation, u Harvard Dictionary of MY2_i£, 2nd ed. (1969}, 89.

29

statanents of a sj1Tlmetrically expanding intervallic structure.

Variation 1. m. 1-12 2. m. 13-23 3. m. 34-43 4. m• 52-64 5. m. 65-72 Coda m. 73-82

Each variation is built on an expansion of a semitone to a tritone

and/or fifth combined with the ostinato principle, (ex. 12).

In the first variation the intervallic expansion occurs in the

left hand dyads» m. 1-7, which move from a-g# down to eb-d, using

the repeated 1a' as the ostinato, (ex. 13). The right hand, which

is accompanimental at this point, completes the rhythmic ostinato

filling in the eighth note afterbeats4 In m. 7-12 following the

completion of the intervall ic expansion • the right hand continues the

ostinato on '.d', filling in the remaining pitches, ( • a' ascending

to 'd') much the same as it fills in the remaining afterbeats.

In the second variation the functions are reversed: .the

right hand has the intervallic expansion (in inversion at the octave)

while the 1 eft hand fills in the afterbeat eighths, (ex. 14). In

m. 20-23 1 the ostinato on 'a' is replaced in the left hand by an

ostinato surrounding the tritone: M. 20, c-f#; m. 21, g-c#; m. 22,

c#-9. The temporary use of c-c# as the focus of the ostinato might

be understood; if c-c# is related to the fifth a-e as a similar

symmetrical center represented by d-d# -e, within the octave, (ex. 15}.

In m. 2.3, the left hand expansion parallels a similar expansion in

m. 11-12; both of these passages function as -a long upbeat to a new ...

section~ This short foray_a\'tay from the 'a' ostinato also introduces

the tonal area to be explored in the fourth variation: the tritones,

gb-c. and c-f#.

The pitch material of the third variation is again based on the

•a• ostinato, but here the original statement and its inversion are

used simultaneously in a canonic displacement of four eighth notes.

·In this variation the tritone/fifth completion is divided between the

hands (ex. 16). The emphasis on the tritone is continued in the long.

upbeat passage (m. 46-51) which precedes the next variation and reaches

the fifth in the final chords in m. 51, (ex.17}.

The fourth variation is contrasted in a variety of ways., The

intervallic expansion is centered around the tritone: in the r1ght

hand, m. 52-57, gb-c; in the left hand, m. 58-64. c-f#, (ex. 18). The

ostinato accompaniment to each of these statements has been trans­

fanned from a rhythmic ostinato into a melodic figure which is also .. built on a tritone. The contrast is also apparent in the change to a

linear, melodic, tv1o-voiced texture and new tempo ( molto piu calmo,

lugubr~· ) •

The final variation returns to the pitch material, tempo,

register of the original statement, with the intervallic expansion

occuring simultanea,usly with its inversion in a stretto-1 ike passage.

The stretto effect is continued in the coda~ which is built on the

accompanimenta1 afterbeat material first seen in m. 7-8, and

significantly absent in the last variation, (ex.19).

Since the pitch material of these variations is either identiCal

or, in the case of the fourth variationj very similar in construction,

the changes of character throughout the work are, to a large extent,

-.1.1.

. the result of subtle variations in the use of rhythm. The first two

variations could be described as polymetric, VJith additive accent

groupings of t~~·s and three•s cr~ating rhythmic phrases of increasing

length (ex. 20)o Note that the inversion at m. 13 is preceded by

nineteen J• s of upbeat. In a similar way the third variation is

preceeded by eighteen J•s of upbeat, additionally strengthened by an

augmentation of the three J' accent _grouping into four in m. 22-23. In

contra~t, the section from m. 24-51 is characterized by the regularity

of accent groupings. The unaccented ostinato in m. 25-26 is given a

six ~pulse in m. 28-29 by the left hand punctuation; in m. 30-33

the accent grouping changes to three J and from m. 34-51 a regular

four meter, coinciding wi~h the new variation.

The rhythmic preparation for the fourth variation_ is the most

e~tensive, which is logical since the fourth variation presents the

greatest contrast in the \'fork, both in terms of texture, tempo and

pitch material. The ostinato is momentarily interrupted and begins

again. !1 doe~io piu lento, accelerando. in a greatly extended

upbeat passage, a rhythmic gesture which has appeared earlier in the

work before new sections.

In spite of the contrasts in the fourth variation, there are

several rhyth~ic characteristics which relate this variation with

what has come before. These are: the additive character of the

accent groupings {ex. 21), the three J initiating each phrase, and

the filling in of the afterbeats with an· accompanying ostinato figure

{melodic here) in the alternate hand.

The final variation returns to the original material against its·

inversion, a stretto-1 ike passage which in effect telescopes the

first two variations. The coda continues the overlapping entrances

between the hands, using the accompanimental material from the

opening statement, in the characteristic afterbeat position until

m. 77 and the appearance of the accented strong beats.

At this point, a telescoping of several parameters takes place.

There is a rhythmic 2-CCelerando. from accent groupings of six p to

three 1' ; at the same time the intervals decrease by semi tones in

each 1eft hand group. The effect-is the reverse of interva11 ic

expansion featured throughout the work: a spiralling inward toward

the 'a' tonal ~enter and basis for all the ostinati.

Another point to consider in the process of variation is the

systematic use of register. The unity of each variation is preserved

by the containment within one octave. The entire work moves not

across the registers of the keyboard but in jt.nnps between the octave

regi stet·s. This procedure, \·lhich is necessary to preserve the

consistency of the ostinato, is established as a premise of the work

in the opening measures of the right hand accompaniment. Octave

transpositions occur throughout the vmrk, usually after the completion

of the intervallic expansion, and function as a preparation, (just as

the extended upbeats), for the next variation. Since the pitch

material in each variation develops in such a similar way, the changes

of register assume a structural role, and in the coda, the frequency

of registral sh·ifts, combine with the rhythmic ~to provide

the motivating fm·ce of the conclusion.

It is ·irrelevant to discuss harmony in this work as there is no

33

vertical cons·truction for its own sake. The consistency of the

ostinato establishes 'a' as the tonal center, but the pitch

material moves in a synmetrical, _linear fashion in both directions;

what occurs vertically is the result of the contiguity of lines.

TEACHING POINTS:

1. · Principle of intervall ic expansion; its transfonnati on throughout variations. 2. Relationships and contrasts between textureso 3. Structural use of register and extended upbeat. 4. Variation of ostinato principles. 5. Additive phrase structure (as compared to periodic phrase structure). 6. Comparison of metric and polymetric use of rhythm. 7. Phrase completion: use of intervallic expansion in place of cadence to indicate closing of phrase groups.

MINOR SECONDS, MAJOR SEVENTHS (VI, 144)

AURAL ANALYSIS:

1. Relationships in the pitch material. z. Overall metric feeling, fluctuation in pulse. 3. Variety of texture and register.

·4. Non•tonal hannonic structure. 5. Phrase structure, sectional structure.

STUDY ANALYSIS:

In contrast to Free Variationi, ~inor Seconds, Major Sevenths,

(VI, 144), resists an easy division into discreet sections, but

r~ther, flows from the working out of one comp~sitional idea into

another. However, taking an overview of the w~rk, it is possible to

discuss it in terms of five sections, or phrase groupings, each of

which is characterized by its particular use of pitch material,

register and texture:

Section I: m. 1-17; t\~ phrase groups, m. 1-8 (3rd beat), m. 8 (2nd beat) - 17.

II: m. 18-34; two ·phrase groups, m. 18-22 (3rd beat), m. 22 (4th beat) - 34.

III: m. 34 (3rd beat) - 43 (1st beat); two phrase groups, m. 34 (3rd beat) - 38 (3rd beat), m. 38 (4th beat) - 43 (1st beat) •

. IV: m. 43 (1st beat) - m. 60; two phrase groups,

m. 43-47, m. 48-60.

V: m. 60-70; one phrase.

The final section might be described more accurately in terms of

a coda, since there is no further development of material. Instead,

a single phrase returns to the opening register, pitches and

reiterating figuration used earlier in the work.

The relationship of section II to section lV is obvious: the use

of a similar polyphonic texture and register. The expanding intervals

indicate that m. 43-60 are a continuation and completion of the motion

initiated in m. 18-25.

Aside from the coda, each of these sections, while varied in

other aspects, share a classical balance of two related phrase

groups. In contrast to this balance is the continuous variation

and· development of the afterbeat, first seen in the major sevenths

in m. 2,4,5 and 6, into: 1. a phrase completion, m. 6-8, m. 25-34;

2. a registral punctuation, m. 9,12,16-17; 3. ~a figural punctuation,

m. 21-22, m. 37-38, m. 40-41. In m. 52-60, a combination of all of

the above techniques creates the high point of the work.

In a fashion similar to Free Variations, the pitch material in

35

. ~ . :

th'is work is derived from a S)ttlllletrical, expanding intervallic

structure: a minor second, expanding in both directions to inter­

locking tritones which outline a major seventha The entire work is

related both vertically and horizontally by the juxtaposition or

1 inear connection of minor seconds and major sevenths and the inter­

vallic expansion of minor second to major seventh. . .

Section I presents this symmetrical expansion in the first two

measurest (ex.22). In me 1-8 the minor seconds remain stationary

ciusters while the major seventh is developed from an afterbeat into

. a melodic cadential figure, m. 6-8, which reappears at the end of

section II and section IV, (ex. 23}. In m. 8-17, the minor second,

now based on c# -d, expands into a nev.J seventh and the afterbeats,

although registrally expanded, are also built on minor seconds,

{ex. 24}.

Section II replaces the expanding clusters of minor seconds with

a contra punta 1 texture. The minor second built on f-f# expands to

c-b~ while the afterbeats horizontally outline two major sevenths a

tritone apart. In the second phrase, (m.22, 4th beat - m. 25), the

. expansion from the minor second increases to a fifth, (ex. 25), at

which point, the motion is interrupted as the major seventh after­

beats take on the pht~ase completion in an augmentation of the

melodic, cadential sevenths in m. 6-8.

Section III introduces a new register while returning to pitch

material used in the opening section. However,-this time the

positions within the phrase are the reverse of the opening section.

The major sevenths initiate the motion of the phrase and the minor

second clusters are used, in a horizontal figuration, as the

afterbeat, {ex.26}.

The polyphonic texture. returns in section IV, and in the first

phrase group, m. 43-47, the intervallic expansion from the minor

second extends to a fifth. In the second phrase the motion is

completed when the expansion reaches the major seventh, {ex.27). At

this point all previous developments of the afterbeat come into play:

the seventh as figuration, as a registral expansion, and finally as

a large canonic expansion of the cadential melodic sevenths seen at

the end of the opening phrase group.

The final section or coda returns to a figuration of the initicl

pitch material and rhythmic motion of the opening. The major

seventh, also treated as a figuration, is repeated until the final

cadence which dup 1 i cated the pitches of the first cadence.

Given the conscious restriction and consistency of the pitch

·material, the other compositional elements such as the rhythnic

activity, changes of texture, and use of register take on a struc­

tural siqnificance in shaping the work.

~Jhile the variety of textural treatment of the minor second,

major seventh has already been discussed, a closer look at the

rhythmic motion is warranted. The first contrast to be noted is that

between the metric consistency of the first phrase of each group,

and the rhythmic expansion of the poco strinqendo passages: m~ 10-12,

14-16, and 48-51. Even more dramatic are m. 25-34, marked · poco a

Q.O£Q. acpgJ~ ~-- Jloepio moviment,£_ which, in effect, creates a

temporary metric modulation. The section from m. 52-60 suggests,

first of all, a temporary suspension of meter (aided by the rest on

the downbeat in m. 52 and the ha 1 f note augmentation in m. 55-56),

followed by gradual return to the_ regularity of Tempo I. The

poco a poco accelerando in the coda, considering the familiarity of

the pitch rnaterial 9 is predominantly a phrase of rhythmic and

registral emphasis.

In a similat· vmy one can contrast the restricted range which is

associated w-ith the balanced phrase groups, and the regi stral

expansion vJhi ch always coincides with the development of the after­

beat. In the end, one can say that these rhythmic and registral

expansions are so interesting, not because they occur, but because

they take on the role, (ofte~ assigned to the harmonic structure in

earlier periods), of defining the structural divisions and character

of the work.

TEACHING POINTS:

1. Exploration of the varying uses of the pitch material: connections, expansions, changes of texture, horizontal and vertical possibilities. 2. Sectional structure, relationships between sections, 3. Register as a structural element. 4. Rhythn as a structural element. 5. Comparison of the balanced phrases and expanding afterbeat.

' These two Bartok examples offer an interesting comparison of

compositional techniques with the Berg and ~1essiaen examples. In

contrast to the Berg, Bartok uses a variety of textures to delineate

his structures. In contrast to the r~essiaen, there is a systematically

consistent relationship of the pitch material on all levels.

.. MESSiAEN'S NOEL (1944)

To extract an excerpt from a large cycle such as the Messiaen's 36

Vingt Regards sur L'Enfant J~sus for the purpose of study or

analysis places certain limits on that analysis. The total part­

icipation of the excerpt and its function within the cycle cannot

be fully explained. However since the movements in this work are

not continuous, but rather, separated a-nd given individual titles

inferring a specific entity for each movement. it is possible to

make some observations on r~essiaen • s treatment of musica 1 materials

based on the excerpt alone.

The procedure used here for discussing Noel is somewhat

different from that used with the Berg and the Bart6k examples,

which referred almost al~mys directly to the scores. While the "

guideline questions still serve here to point out what and where

the contrasts, variations and repetitions take place in the score,

the student's understanding can be enriched by gaining some

familiarity with some of the sources outside the tradition of

Western music from which Messiaen draws musical inspiration. Thus

references ~Jill be made to studies written on Messiaen's music as

well as his own work: The Technique of My Musical Language.

36Q1 ivier r;tessiaen, Noel from Vingt Regards sur L • Enfant J~sus, Paris: Durand, (1944) 9l)..:"g'jpp. -

39

AURAL ANALYSIS:

1. Sectional structure; three part structure with repeating elements. 2. Ametrical rhythms. 3. Use of embellishing figuration. 4. Harmonic consistency of central section. 5. Contrast in tempo, and character of central section. 6. Non-tonal, highly chromatic harmony. 7. Coloristic use of harmony. 8. Use of ostinato technique. 9. Use of register.

STUDY ANALYSIS:

Before beginning a sectional analysis, several rhythmic concepts

that !·1essiaen uses ·in his search for variety should be brought out.

The first, as stated by Messiaen:

We shall replace the notion of "measure" and 11 beat" by the feeling of a short value (the sixteenth note for example) and its free multiplications, which wi11 lead us toward a music more or less "ametrical" ••• 37

The concept of the 11 added value" which Messiaen uses to avoid the

regularity of even meters arises from his study of the Hindu

rhythms or deci-tSlas as set do~TI by the thirteenth century

theorist, Sharngadeva, \<Jhich leads Messiaen to conclude, 11 it is

possible to add to any rhythm whatsoever a small brief value vthich 38

transfonns its metric balance... The value can be a dot, a note

of small ·value, or a rest; the effect is tha.t of a brief augmen­

tation (ex. 28) which destroys the feeling of exact symmetry or

regularity in the rhythmic flow. With these ideas in mind it is

not difficult to explain the lack of a meter signature in much of .

37M ' T h . 14 , ess1aen, ec n1~ ••• , p. •

381pJ.Q_. p. 15.

40

his music or the fact that the majority of rhythmic phrases contain

unequal groupings or an uneven number of beats.

In terms of the large sectional structure NOel can be described .. -as a ABA form, with the A sections containing a smaller aha, thus

.incorporating elements of a rondo:

A ( 1-25) . B (26-52) · (a, 1-7; b, 8-ZO; a, 21-25)

A (53-80} (a, 53-57; b, 58-64; a, 65-8~

The first seven and last five measures of the A section employs

a rhythmic ostinato in the right hand cho.rds \'Jhich outlines the • . 39

ste_ci-t~,.! .vi.iav~nanda ( f f J J)) to \'lhich Messiaen has added an

eighth value. offsetting the symmetry of the pattern. In m. 5,

there is another example of an added value in the dot added to the

final chord.

The pitch materia 1 of the A section is based on the superposition ..

of perfect fifths and tritones, or augmented fifths and perfect

fourths, (ex. 29) in bell-like sonorities, {the subtitle of the

movement roughly translated is: "The bells of Noel say with us the 40

sweet names of Jesus, Mary, Joseph ••• " ). The descending line

(m. 5-6) which is the first melodic idea in the work, closes the

ostinato pattern and continues to function as a phrase ending

through out the A section in m. 8, 13, and expanded in m. 18 and 19

(ex. 30). In m. 8, 13, 18 and 19 we also see examples of embellishing

groups which are Messiaen's expansion of the CQncept of an

39Robert Sherlaw Johnson, Messiaen, (Berkele,y: Univ. of Calif • . Press, 1975)· p._198_.

40t•1essiaen. Noel, p. 90.

embellishing note. The section is shaped by these embellishing

groups and expanding phrase endings in the sense that they augment

the time space between the bell-chords in a similar way to the ' <

concept of added values, but carried to a higher structural level.

There are two contrasting phrases which interrupt the bell­

chords and their cadenza figure in this section: m. 10-12 and

m. 15-17. The first, based on the chords in m. 6, outlines the

dec1-tala miga-varna ( J J} ~r. 111.1. JJJJ. J. J II J t J)41

and the second, using the fifth and fourths in embellishing patterns

as well as in chords, outlines the deci-fala, ~: ( J 1 J. J,.fll. 42

J J.] }J). Although the pitch material of these t\'/O phrases

can be related intervallically to the rest of the section, the

juxtaposition of unrelated rhythmic elements brings up an interesting

point in Messiaen's treatment of rhythm which is described by David . Drew in his study on Messiaen:

For r~lessiaen on the other hand rhythm is an extensive phenomenon that belongs more properly to the vtorld of space and solids, since it consists of self­contained units whose significance lies not in their interpenetration, not in their development over a period of time, but in their juxtaposition one with another, and in their relative volume and position rather than in their relative stress. As such, these units, like their component parts of a mobile, are both reversible and divisible, but never developable. 43

41Johnson, op. cit., p. 195.

42Ibid p. 198.

43oavid Drew, "Messiaen - A Provisional Study (II) "Score, 13, (Sept. 1955), p. 68.

The return to the rhythmic ostinato (m. 21-25) frames the 'b'

section and finishes with the fallinq melodic figure.

The central portion of the vrork, or B section, falls into

five periods: m. 26-30; m. 30-34; m. 36-40; m. 42-46; and in m. 48-52.

It is contrasted with the A section by tempo, dynamic level, con­

sistency in the use or range and register, use of modal pitch

material and consistency of rhythnic patterns.

Looking at the rhythmic structure separately, a repeating

pattern can be seen in the first four periods which maintains

almost complete consistency in the antecedent phrases while

expanding slightly in the termination of the consequent phrases.

The fifth period excerpts rhythmic cells from the earlier periods

and augments the consequent portion again by the use of added

va1 ues:

M. 26-30:

M. 30-34:

M. 36-40:

'r' {;, r. r. ~ n lJ J' A r J' 1' J' ( J ffl M. 42-46: '-<!-'

••

1' 1 (t J. r .• '1}. t J 1 rl 1i ''

im{ j~J

M. 48-52:

The pitch material of the 8 section is based on two of

Messiaen's modes of limited transposition. These modes are sym­

metrical arrangements of the chromatic scale in interlockinq groups

of half and whole steps which can be transposed a limited number of

times before a duplication of p-itches. Describing the unusual tonal

characteristics of these modes Messiaen states:

They are at once in the atmosphere of several ton­alities, without polytonality, the composer being free to give predominance to one of the tonalities or to leave th~ tonal impression unsettled. 44

Melodically, the modes are difficult to distinguish aurally

{except for mode I which is a whole-tone scale) since certain

characteristic intervals, such as the tri.tone which Messiaen uses

frequently, appear in all of them. However the harmonic constructions

within a particular mode can be recognized by their consistency,

and supply a distinct unity of harmonic color to a passage without

implying either a specific tonality or a progressive function.

The two modes used in Noel are mode II and mode III (ex. 31).

The first two periods, m. 26-34, consist entirely of mode III, first

transposition. In m. 36-38 the antecedent of the third period, the

lower two staves are mode II, second transposition, (ex. 32), with .

the upper staff in mode III, (fourth transposition)?functioning as

44 Messiaen, Technigu~, p. 58.

a pedal group, which Messiaen conceives as the expansion of the

principle of a pedal note: "Instead of one sustained note, foreign

to the chords which surround it, we shall have a repeated music •••

foreign to another music situated above or beloVJ it, each of these 45

musics will have its own rhythm, melody, harmonies." The consequent

half of the third period, m. 33.,.;40 returns to mode II I, first

transposition. The fourth period, m. 42-46 continues mode III with

an inversion in the melodic line, (ex. 33). The final period

alternates fragments of the mode II phrase, (m. 47 taken from m. 36;

m. 48, last group, taken from m. 37-38), with fraqments of the mode

III phrases, and finishes v1ith a long expansion in mode III of the

descending passage taken fr~ the end of the first period. The use

of an embellishing group appears throughout the 'B' sectione at the

end of the second, fourth and fifth periods • ...

In tracing the use of the descending melodic group, which first.

appears as the closing of the ostinato in m.5, we find that it

appears with little development, but in different levels of

prominence throughout the musical texture:

A: m. 5-6; first linear statement, functions as phrase closing. (same function in m. 25). m. 8, 13, 18, 19; functions as phrase closing; expanded by added chromatic notes, rather than rhythmic augmentation.

B: m. 26; functions as melodic motive of section; given a modal inflection; both initiates and closes the period. m. 34; appears in registral change in bass; also in m. 46 and 52. -m. 42; in inversion. m. 49-50; together with retrograde. ~ m.; 50-52 expansion of phrase clos·ing, by repetition. {ex.34)

~--;t ___ ., --45 Messiaen, T~chnig~~~ p. 55.

A: m. 57, 58, 61, 64, 69; similar treatment as first A section. m. 70-74; final appearance; augmented by expanding the descendinq chord construction from a series of three, to four, five. six, and finally ten chords, together with their inversion.

TEACHING POINTS:

1. Messiaen's concept of ametrical rhythm: unmeasured measures, regularity of the small est note value. 2. Concept of added value. 3. Concept of embellishing group. 4. Concept of pedal group. 5. Introduction to Messiaen's modes of limited transpositions. 6. Hanmony as color (as with pedal group) as compared to functional harmony.) · 7. Musical ideas used in juxaposition as compared to se· quential or organic development.

...

CONCLUSION

In order to facilitate the use of analysis in teaching twentieth

century piano literature, a series of analytic questions has been

proposed to serve as a guide1 ine in student/teacher preparation of

new works. By grouping these questions into the various dimensions

to be explored, and by keeping them f1 exible' and openended rather

than restrictive, it is possible to apply them to music of widely

differing styles.

The application of these guideline questions to the Berg, Bartokt

and Messiaen examples has shown not only a number of aural perceptions

regarding structure, rhyttrn, and texture that can be discussed at the

outset of the preparation of the work, but also many teaching points

uncovered through the analysis. By using these teaching points in

comparison with other works studied, the student's a'ttareness of the

functioning of musical elements can be expanded beyond the scope of

the individual composition, and applied with increasing independence

to new \'/Orks.

47

BIBLIOGRAPHY

Bartbk, Bela. Free Variations, from ~1ikrokosmos, Vol. VI, London: Boosey & Hawkes ( 1940), 2"-5.

----Minor Seconds, Major Sevenths, from Mikrokosmos, Vol. VI, _London: Boosey & Hawkes (1940), 16-19.

Batstone, Philip. "Musical Analysis as Phenomenology," Perspectives. of New t4usi c, VII I, (Spring-Summer 1969) t 94-110.

Berg, Alban. Sonata, Op. 1. Vienna: Universal Editions, 1926, Fall

Berry, Wallace. Structural Functions in Nusicp Englewood Cliffs: N.J.-: Prentice Hall, Inc., 1976, 447pp.

Carner, t4osco. Alban Berg, The M_an and the vlork. London: Gerald Duckworth and Co., 1975 XV, 255pp.

Cone, Edward T., 11 Analysis Today," The Musical Quarterlx, 46 (April, 1960), 172-188. _

Drew, David, 11Messiaen - A Provisional Study," ~~ 10 (Dec., 1954). 33-49.

-----"Messiaen, - A Provisional Study, III'' $£ore, 13 (Dec., 1955), 59-73.

-----"~1essiaen,- A Provisional Study, III," Score, 1i (Dec., 1955), 41-61.

Elston. Arnold. "Some Rhyttmic Practices in Contemporary Music," · The N@i@] Quarter1 v., 42 (July, 1956), 318-329.

Fenyo, Thomas. "The Piano Music of Bela Bartok," PH.D. Dissertation, UCLA 1956, 293 1.

-----"Form" Harvard D.ict}QDary. ?f Music, 2nd ed. (1969), 326-327.

Forte, Allano Contemporary Tone Structures. New York: Columbia University, 1955. v-xii, 194pp.

--.. --uSchenker• s Conception of Musical Structure, 11 Journa 1 of Music Lheory, 3 (April, 1959), l-31. ,

Johnson, Robert Sherlaw. Messiaen. Press, 1975. 221pp.

Berkeley: Univ. of California ~:

Jones, Robert Carroll. A G].Q.§~sary of Theoreti ca 1 Terms Used J.n Eng1J.?ll .• Jl..b2ttl_T\ventieth-Centur_y t!lusic. · Doctoral diss. • Univ. of l~~a~6j9y~· Ann Arbor, Mich.: University Microfilms, Inc. 1970,

48

Katz, Adele T., 11 Heinrich Schenker's Method of Analysis," The Musical Quarterly, 21 (July 1936), 311-329. -

LaRue, Jan,"OnuS'tyle_Analysis," Journal of Music Theory, VI (Spring, 1962), 91-106. .· .

"Melody," Harvard Dictionary of Music, 2nd ed. (1969), 517-519.

Messiaen, Olivier. The Technique of My Musical Lanquaae. Trans. by John Satterfield. Paris: ~. leduc, rgss. 2 vo,s. 74 pp, 61pp.

Messiaen, 01 ivier, Noel from Vingts Regards sur L 1 Enfant J?sus, Paris: Durand (T944), 90-97.

Parry, c. Herbert H., "Me lady," Grove 1 s Dictionary o..f._,~usi c and Musicians, 5th ed., V (1954), bb6-S69. .

Redlick, Hans.· Alban Berg. The r~an and His t~usic. New York: Abelard-Schuman, 1957: 316pp •.

Smither, Howard, "The Rhythmic Analysis of 20th Century Music, 11

Journal. of t·1~s.i c Theor1,, VI II {Spring, 1964), 55-88.

Stevens, Halsey, !he Life and t~usic of B~la Bart~k, New York: ~ Oxford Univ. Press, Inc., 1953. 366pp.

Suchoff, Benjamin. Guide to Bart6k • s t~ikrokosmos, Rev. ed. London: Boosey & Hawkes, t-1usi c Pub1 i shers Ltd., 19i!: ·152pp.

Troup, Halcolm, 11 Regard sur Olivier Messiaen - I," Composer, (Autt.ann, 1970), 31-32. -

----"Regards sur 01 ivier Messiaen - II," Comooser, 38 (\.~linter, 1970), 21-26.

Vinton, John, "Bartok; on His 0\'JI'l Music, 11 Journal of the American ~us.i cz.o 1 og i ca 1 .. society, X IX (Summer, 19'56), 2 32-243.

Wuorinen, Charles, 11 Notes on the Perfonnance of Contemporary Music," Pers,p~ctiv,es o,f New ~·lusic, III (Fall-Winter. 1964), 10-21.

APPENDIX: MUSICAL EXAMPLES

B~rg: Sonata Op. I

Example 2:

Example 3,:

50

- --- -~---

. ....:>-7 ~

~:. ~~·~t,. ~~ ~;;. " .. ..;/ -~ *

·------ ·-- ~ ... -!.. .-

........ s- .__,..., bn 1 I

•- =%-I J j;J 19: 1~-=f- ~ 'wl

·-· ... ····- - .. ""' .. ,. ~ . . .

~,5~ .. ~ ~ ~~; qe '1~ t 'e, t. r *'·rs~.*"" . -r .. ·. ~-i: ~-=57= ~ E}. t i: E ~~-:£~~~ -~ .

·__ - .. . - ..

- ~---·--

51

52

Example J:

53

ffij A_ - -. . 1.1 ..

Example 9:

.... " bJ . '""J_~b~ b9: ~~~·

Example 10:

~T ~ •

~~! ~q:l b~-t:

~amp 1 e. ,1,1:

,.,.,70

·~-

-----------------------

B '· artol<: Free Variations

Example 12:

Examo 1 e 15 :-

Example 14:

'

I #Fj;: : ).

Example 18: ,...""'_ "i2-!i'1

$: ~~-::;s=: ==ia~= =======:e~.l?==~~: ~· -ri:£::.7

;j:i:r:;;~-;:::::!.1

'th\149-l-fi •

~~~ ~v 1\ ·.

·~J j-#~!i}ti--SlJ •. <4i • --,

~ J .. =:=6 ~==~

Exam.e..l_e 21:

·~--57--?l

·~~ vA

~~-r-J?J 1WVM

, Bartok: Minor Seconds, ~·1ajor Sevenths

Example 22: Example 23: •· --~~

=§~ cw4F~

Example 24:

b+ -- - --1"'1 - -·- -ff1K - ..\ . 'f. If !'.i'U , """ 'l IF

f5'--·· ...;;

A "" ba ~. ...-~ .. "" T

iii>: I Ia- ·loi...a.

v -

. Q< .. arne) e 2 § :

I ... -.... 1,

'MI;rO

~~~~~~~~g~~~~:..it~-

~: ""'""-'J-3 -~'1 """~"-4- '6"-5)

t=;=~~I Messiaen: Noel

57

~ample .?1:

-4 Example 30:

· Example 3J:.: ..

I J-!- ! ... ~ ""f- I Ill l .II ..... .lot~ I ., ... - --r. .... ... •• i:=ui:: -"'!."" ... .- ..

.,J vL-

MoJe III 4-~Trd&h5- I 1"'1. -

~. .. ..,_ '!:'""""5 .... ~ "'i"' y - "" -·.;;;; #4-. ~

~ L---'

~~~ Mock rr .,.1!!.4~5-I--~ P \iez ~ !

7--'-.....- -

59

~mple 34: . -