379

A COMPARISON OF THE HINDEMITH AND SCHENKER

CONCEPTS OF TONALITY

THESIS

Presented to the Graduate Council of the

North Texas State College in Partial

Fulfillment of the Requirements

For the Degree of

MASTER OF MUSIC

by

Grace E. Knod, B. M. E.

San Marcos, Texas

January, 1955

PREFACE

Tonality, in the conventional sense, may be defined as

"a central key in which a work opens and closes, but which

serves as the point from which to modulate to various other

keys."I A somewhat broader definition than this, and one

which may be more acceptable, is given by Schoenberg.2 He

defines tonality as "the art of combining tones in such suc-

cessions and such harmonies or successions of harmonies,

that the relation of all events to a fundamental tone is

made possible."

The conventional concept of tonality may be considered

as based on the following two principles:

I. The establishment of a "tonal center," or "funda-

mental tone," i. e., the principal key of a given composi-

tion, and

II. The relationship of other keys employed as sub-

servient to the principal key (usually explained by the the-

ory of modulation),

This concept is explained further by the following def-

inition: "General Tonali is that prin le by which a

1A. T. Katz, Challne to Musical Tradition, p.1.2Merle Armitage, Schoenberl, p. 280.

iii

mental grasp of the musical texture is maintained through

melodic and/or harmonic conventions relating all component

tones to one of their number which is thus the tonal center

and ordinarily the final."3

The chief function of tonality is to conjoin and to

unify the different elements of a composition. Tonality

serves as a unifying principle in that "all tonal succes-

sions, chords and chord-successions in a piece achieve a

unified meaning through their definite relation to a tonal

center and also through their mutual ties."4

The purpose of this thesis is to illustrate and com-

pare, through a representative historical sampling of music,

the concepts of tonality evolved by Paul Hindemith in his

Craft of Musical Composition, Vol. I; and Heinrich Schenker

in his Tonwille, jNee Musikalische Theorien Fantasien,

Das jeisterwerk in der Musik, and Per Freie Satz.5 When

feasible, these two concepts will be compared with the con-

ventional concept.

The musical examples cited have been chosen as repre-

sentative of the different historical periods. Each example

31. N, Vincent, The Diatonic Modes in Modern up. 14.

Armitage, a. Cit., p. 285.

$one of these works, to the author's knowledge, havebeen translated into English.

iv

represents, roughly, a fifty-year period in history from the

late thirteenth century to the present day. In selecting

these examples the following points were considered: (1)

their ability to demonstrate certain important stylistic

features of a particular period; (2) their ability to illus-

trate the styles of some of the most salient composers; and

(3) their compactness. In general, these examples, for

practical purposes, have been chosen from works for the

piano.

V

TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . vii

LIST OF ILLUSTRATIONS . . . . . . . . . . . . . . . . viii

ChapterI. A BRIEF DESCRIPTION OF THE HINDEMITH AND

SCHENKER CONCEPTS OF TONALITY . . . . . 1

II. ANALYSIS OF MUSIC BEFORE BACH . . . . . . . 13

III. ANALYSIS OF MUSIC OF THE EIGHTEENTH ANDNINETEENTH CENTURIES . . . . . . . . .44

IV. ANALYSIS OF MUSIC SINCE 1900 . . . . . . . . 65

V. SUIMARYAND CONCLUSIONS . . . . . . . * * 79BIBLIOGRAPHY . . . . , * . . . . . . . . . . . . . . 85

vi

LIST OF TABLES

T able Page

1. A Comparison of the Hindemith and SchenkerConcepts with the Conventional Concept . . 11

2. Basic Harmonic Progressions (Schenker) UsedIn Chapter II *-# * .I.D... .0 . . . 42

3, Basic Harmonic Progressions (Schenker) UsedIn ChapterII -. . . ... .. . . 63

4. Basic Harmonic Progressions (Schenker) UsedIn ChapterIV .*.0.0.0.*9.9.9. *.0. .0 .77

5. Frequency of Use of the Schenker Basic HarmonicProgressions . . - . . . . . . . -. . . . 80

vii

LIST OF ILLUSTRATIONS

Figure Page

1. Series I . . . . . . . . . . . . . . . . . . . . 3

2. Series 2 . . . . . . . . . . . . .. . . .. . . 3

3. Beethoven, Piano Sonata, Op. 14, No. 1 . . . . . 9

4. Adam de la Halle, wTant con je vivrai" . . . . . 1

5. Guillaume de Machaut, "Plus dureW . . . . . . . 17

6. Perusio, Virelai "Dame souvrayne" . . . . . . . 20

7. Guillaume Dufay, I'Missa L'Homme armed" . . . . . 23

8. Jacob Obrecht, "Missa Sine Nomine0 . . . . . . . 26

9. Willaert, Motet OBeata dei Genitrix-Et Beata* . 30

10. Orlando di Lasso, "Melange" , * * * * * * *, *34

11. Monteverdi, Chorus "IVieni Imeneo" from L'Orfeo . 37

12. Buxtehude, "Auf meinen lieben Gott" . . .4 . . . 10

13. T. 5.- Bach, Prelude No. 2 from the "Well-TemperedClavier," Book I . . . . . . . . . . . . . 46

14. Haydn, Piano Sonata, Vol. 1, No. 7 (PetersEdition) . . . . . . . . . . . . . . . . 50

15. Beethoven, "Bagatelle," Op. 119, No. 8 . . . . . 53

16. Wagner, "Descend upon us, Night of Love'" from-Tristan and Isolde . .,. * . w . . . . .. 156

17. Debussy, Prelude tDanseuses de Delphes" . . . . 59

18. Bartok, Piano Piece from fkrokosmos (Vol. 6,No.r 150Pian Piec . . 19 . N.. 3 . . . . . . 67

19. Schoenberg, Piano Piece, OP. 19, Now 3 .w .. 72

viii

Figure

20. Stravinsky, "'Dodo Aiegenlied" from Berceuses du

ix

Page

CHAPTER I

A BRIEF DESCRIPTION OF THE HINDEMITH AND SCHENKER

CONCEPTS OF TONALITY

Part of the recent development in harmonic analysis

has come about because of the inadequacy of the system pre-

viously in use. Concerning this system, Vincenti states:

The fault lay in the narrow concept of key whichregarded all but a few chromatic chords as violationsof the key. The increasing complexity of the harmonicamaterials forced a progressively broader view of thelimits of tonality.

The two systems of analysis in general use are given

by Vincent:2

I. The traditional system founded on the original

figured bass. Its objects are:

(1) to explain the tonal fabric in terms of thedual modality of classical tonality,

and

(2) to reveal the structure of the componentchords (name, type, and inversion). The tonal func-tions (dominant and subdominant) are related only tothe tonic and, as a consequence, an analysis mustresort to frequent modulation in order to explainchromatically-formed dominant-seventh type chords.

1J. N. Vincent, The Diatonic Modes in Modern Music,P..8.,.4

21bdo.p. l1+8.

1

2

II. The parenthesis-chord system developed by Weidig

and Piston. This analysis results in a system of secondary

dominants and subdominants. Certain chords formerly con-

sidered modulatory are recognized as being dominant or sub-

dominant functions. This system brought more chords into

relationship to a single tonic.

The Hindemith Concept

Hindemith, in his Craft of Musical Cm Vol. 1,

established the legitimacy of chords which have never been

used in conventional harmony. The theory is primarily and

exclusively a study of the nature of musical material. His

purpose was "to find the common denominator for all Western

music in a theory stated and developed in terms independent

of esthetic judgments and stylistic preferences.tt3

Hindemith's system of analysis is based on the twelve

tones of the chromatic scale. Each of the twelve tones can

become a tonal center. Around this central tone the other

eleven group themselves in a graded relationship, the order

of relationship being expressed as Series 1 (Fig. 1).

Hindemith indicates this order as a "family relationship,"

an expression of the tones in relation to the progenitor

tone. This order is based on the acoustical theory of

3Bernard Heiden, "Hindemith's System--A New Approach,"Modern Music, XIX (January-February, 1942), 102.

saul Hindemith, Craft of Musical Composition, I, 54.

3

relations between overtones, which is explained in detail

in The Craft of Musical Composition, Vol. I. This series

Fig. 1--Series 1

is a formula which governs the relation between individual

"tonal masses" or chords. For instance, if we take C as a

tonal center, the other eleven tones will appear in a de-

creasing order of relationship to the progenitor tone C,

as indicated in Figure 1.

An interval, which is the smallest unit used as build-

ing material, may serve in one of two capacities: as a me-

lodic interval, when the two tones are used successively,

and as a harmonic interval, when used simultaneously. They

respond differently to these two functions and therefore

have varying harmonic and melodic values. Whereas Series 1

represents the decreasing order of relationship of tones to

a progenitor tone (used to evaluate tones used successively),

Series 2 (Fig. 2) shows their decreasing harmonic quality,

worth, or lack of tension. It starts with the fifth, the

Fig. 2--Series 2

2Heiden, Q.. Cit., p. 103.

interval of greatest harmonic strength, which is followed by

the fourth, major third, minor sixth, etc. This order is

based on another acoustical phenomenon, i. e., the appearance

of combination tones which influence the given interval in

respect to its harmonic strength.6

All intervals have roots; one of the two tones dominates

the other. The arrows found in Figure 2 indicate the root

of each interval. The root of each interval is the tone

that is reinforced by the combination tones. The tritone,

because of its tonal uncertainty, actually has no root. It

may, however, have a "root representative" which is the tone

which proceeds by the smallest step to the root of the reso-

lution interval.7

Hindemith8 insists that the conventional theory of har-

mony is far too limited in its system for the determination

and interpretation of chords. He maintains that chords are

not built on superimposed thirds, but on any combination of

intervals, and are not invertible. They submit themselves

to one and only one interpretation.

Hindemith has created a system whereby chords may be

ranked according to their quality, worth, or lack of tension.

6 Ibid,

7Hindemith, Pp. Cit., p. 83.

8 1bid., p. 90.

The importance of the tritone as an unstable interval is

the governing principle of this system. The tritone in-

jects its character into every chord in which it appears.

This fact gives Hindemith the first division within the

chordal material: chords which do not contain a tritone

are classified as Group A, chords which contain one or more

tritones as Group B. Group A is subdivided into two groups:

Group I, containing chords without seconds and sevenths,

and Group III, containing chords with seconds and sevenths.

A further subdivision is made according to the position of

the root of the chord. If the root and bass note are iden-

tical, the chord is classified for instance as IIIl, in

contrast to a chord 1112 where the root would be located

higher up in the chord.

Group B, containing chords with a tritone, may be sub-

divided into two groups: Group II, containing chords with

minor sevenths and major seconds, and Group IV, containing

chords with major sevenths and minor seconds. Again a sub-

division is made according to the position of the root of

the chord. Groups V (augmented triads and two superimposed

fourths) and VI (diminished triads and diminished sevenths)

do not have a definite root.

In determining the root of a chord, its strongest in-

terval (according to Series 2) is selected; the root of the

chord then becomes the root or the dominating tone of its

6

strongest interval. This root serves as the root of the en-

tire tone-combination, regardless of how many members there

are in the chord. By constructing a line that shows the

roots of a successive number of chords, we arrive at what

is termed a "degree-progression." Hindemith asserts that

a satisfactory degree-progression serves as a backbone for

a given composition.9 The degree-progression lends itself

to further grouping of tones which will often form broken

triads. The roots of these groups, when extracted, become

tonal spheres, and thus establish a tonality.

All chords usually employed in conventional harmony

as well as any tone-combination a composer could dream of

can be brought under one of these headings. But besides

being a system of classification it is also a system of

evaluation. A progression from Group I to another group

(II, III, or IV, etc.) produces an ascending curve of har-

monic stress, which is a new device for measuring harmonic

tension. Chord successions are evaluated according to the

ratio of tension to stability.10

Hindemith's system is intended to supply an analytical

background for all types of music.. The concept of "key"

experiences a broad generalization. All tone-combinations

9Heiden, 22. Cigt., pp. 104-105.

'OF. B. Muser, "The Recent Work of Paul Hindemith,"Musical Quarterly, XXX (January, 1944), 34.

7

are possible as long as their use is justified by degree-

progression. Tonality becomes not a starting-point but a

goal.

Museril states that "the setting up of an orderly dis-

position of the actual material of musical composition is

. . . Hindemith's contribution to the solving of the many

problems that confront today's composer."

The Schenker Concept

In contrast to the conventional concept which permits

a succession of modulations, each of which defines a new

key, the Schenker concept allows the establishment of only

one key, the so-called modulations being interpreted as

lying within the key. Schenker understands by tonality

"the life of one tone as it governs the entire work."'2

The Schenker method is not a theoretical one, but a

means of expressing what we hear in music.'3 The approach

is actually one of synthesis; whereas, other theorists are

concerned with analysis, i. e., an examination of each

chord as a specific harmony. Schenker is concerned with

12A. T. Katz, "Heinrich Schenker's Method of Analysis,"Musical QuarterlX.,XXI (July, 1935), 313.

13A. T., Katz, Challenge to Musical Tradition, p. xxiv.

l4+A. T. Katz, "Heinrich Schenker's Method of Analysis,"Musical Quarterly, XXI (July, 1935), 312.

8

chord function or significance as an expression of musical

direction.

This concept is based on two principles: Structure and

Prolongation. The term "structure" indicates the basic har-

monic progression, or framework, of a composition, or sec-

tion of a composition. The structural harmony defines the

key or tonality of the work. It is the element which gives

musical direction and motion to a work. This motion may be

expanded by "prolongation." Passing chords, chords which

support non-harmonic tones, etc., are used to extend or pro-

long the motion within a structural chord. These chords,

in reality, have a contrapuntal function.l5

Figure 3A illustrates this principle of structure vs.

prolongation. These four measures represent the unfolding

of a single tonic chord. The chords in measures two and

three (interpreted as supertonic and dominant in conventional

analysis) are used to prolong the motion within the tonic

chord. Figure 3B is a graphic analysis of these four meas-

ures.16 These chords (measures two and three) may now be

interpreted as a connective or contrapuntal harmony.

This example shows how a musical phrase may be the un-

folding of a single structural chord. In a composition, a

A. T. Katz, Challenge to Musical Tradition, p. 15.

16 The note values indicated in Figure 3B represent rel-ative structural values only; they are not to be interpretedliterally. This same method of graphing will be employed insucceeding chapters.

9

Fig. 3--Beethoven, Piano Sonata, Op. 1+, No. 1

Allegro

structural chord, when combined with other structural

chords, will constitute what may be called a "basic struc-

ture" which will usually form a progression of I-V-I,

I-II-V-I, or I-III-V-I (e. g., see Figure 15C, Chapter III).

This basic structure will define the musical direction of

a period or section of a composition. Thus, in any given

composition, several basic structures may exist, each rep-

resenting a different section of the work.

The single structural outline of a work is called the

"primordial structure.l7 It is a synthesis of all basic

structures and prolongations. Each of these basic struc-

tures, with its own harmonic framework, becomes subordinate

to the primordial structure. The basic harmonic structures

then assume a prolonging function and are used to expand

17A. T. Katz, thalln to Musical Tradition, p. 23.

10

the primordial structure, just as the contrapuntal pro-

longing chords are used to expand a single basic structure.

The factors involved are basically the same; only the di-

mensions are larger. This primordial structure gives an

over-all picture or graph of a work, and is an indication

of its unity and musical direction.

The purpose of Schenker's theories was to prove that

"the fundamental principles which govern the music of all

great composers are the same."18 He sensed that chord

grammar failed to differentiate between the diverse func-

tions which chords fulfill, and realized that to regard

only the beginning and end of a work as indicative of its

central key offered a similar lack of explanation of the

status and function of so-called modulatory sections.19

He realized the need for a broader concept of tonality--

one which would differentiate between chord grammar and

chord significance as a means of indicating musical direc-

tion and coherence.

The contrasting views of the theories evolved by

Hindemith and Schenker, as well as of conventional theory,

are listed in Table 1.

18 A. T. Katz, "Heinrich Schenker's Method of Analysis,"Musical Quarterly, XXI (July, 1935), 312.

19Felix Salzer, Structural Heari , I, 26.

11

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CHAPTER II

ANALYSIS OF MUSIC BEFORE BACH

It is generally agreed by theorists that a large por-

tion of the music prior to the introduction of the major-

minor diatonic scale system cannot be made to conform to

the laws of major-minor tonality as the term is usually

understood. However, Vincent1 suggests that music of any

era possesses a tonal scheme, although its system may be

different from the tonality of another musical era. He ap-.

plies the term "tonality" to plain chant and gives the fol-

lowing definition:

Tonality in Plain Chant is a system by which amental grasp of the unaccompanied melodic line ismaintained through a system of linear tonal combina-tions. . . . It thus only makes use of the elodicphase of the general principle of tonality.

He states further:

Tonalit in Renaissance Polyphony is a system bywhich a mental grasp of the melodic and harmonic tex-ture is maintained partly through the methods of un-accompanied plain chant which apply mainly to thecantus firmus, and partly through certain added har-wmonic conventions whose function is to relate thecomponent triads to the triad of the final which has

lJ N.Vincent, The Diatonic Modes in Modern Musicp. 14.

21bd7_ ~dp. 5.

13

14

taken the place of the simple final. . . . The wholeoutlook was still horizontal.3

Piston4 says that a definition of tonality should in-

clude the diatonic modes. He adds: "The presence of a cen-

ter of gravity, or tonic, being the sole requisite for the

presence of a tonality, it will be seen that the same tonal-

ity may be given a large number of variations in the makeup

of its scale."

An application of the Hindemith concept to an example

of early polyphony reveals an easily perceptible tonality.

Figure 4A, the rondeau "Tant con je vivrai" by Adam de la

Halle (1230-1287), has the note F as its tonal center.

This rondeau is an example of one of the characteristics

peculiar to the music of this period, i. e., a descending

step-wise root progression. After extracting the roots

(according to Series 2 of the Hindemith system) of the

chords, a grouping of these roots results in the tonal

spheres of A, G, and F respectively. This scale-like pro-

gression takes for its root the final, or F.

A Schenker analysis of this rondeau produces approxi-

mately the same result. Measures 1-9 represent a prolonga-

tion of the chord on A (A-E-A) with the chords on G and C

serving as embellishing functions (Fig. 4B). Measures

3 Ibid.

4Walter Piston, Principles of Harmonic Analysis, p. 60.

15

F'ig. 4--Adam de la Halle, "Tant con je vivrait '

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F

11-13 are clearly an elaboration of the chord on G (G-D-G),

and in measure 11+ the final, or tonic, F is heard. The

final analysis results in these three principal chords

(Fig. 4C). No fundamental harmonic progression (e. g.,

I-V-1, 1I-III--I) exists here. There is no indication of

dominant-tonic relationship as understood in major-minor

tonality. In this respect, the Schenker concept seems to

be inadequate. To say that the basic structure is IIn-IT-I

would be far-fetched and would not conform to the principle

of a fundamental progression as conceived by Schenker,.

An example of French secular music of the fourteenth

century is the virelai "Plus dureO by Guillaume de Machaut

(1300-1377) (Rig. 5A). This is an early example of the

achievements of the Ars nova, as exemplified by the pre-

dominance of an elaborate melodic line over a simple ac-

companiment. Again, a definite tonality is produced by

the Hindemith degree-progression. Even though the piece

17

A

LMI

EW IE

AM Am-490

Fig. 5--Guillaume de Machaut, "Plus dure"

I It

~H nertA Anaiys s:

T[nzie egP- CCme - - - .. . - -n

I"'m. 11ir--1 - . .

1 .. -. .-

$sty

18

Fig. 5--Continued

Shenker Analysis:--

Be

P 0-

-I-_- - -_--_-

8C

I In.wIL

! r .

-- NM~iil@Migi,41~g,'iHIil04h~tM'IIWarlsMMi&IE I--tg2 -- iig.am -lEEI 10Iml1 3|11Idg- lli I-i II - -1 - - -- I mn - | ||

#7| cl i r FF~- [ --.- -- . . . . .. m

# ndal-e fonl ee ers

19

is unquestionably modal in character, an analysis reveals a

tonal center of D.

An analysis according to the Schenker concept produces

the fundamental harmonic progression I-V-I (Fig. 5B). The

entry of the tonic is delayed by the cadential progression

in measures 1-4. This progression serves to establish at

the beginning the tonality of D. Figure 5C, which shows the

basic structure of the work, consists of the three struc--

tural chords (tonic-dominant-tonic) which define the course

of the work.

Another example of French secular music of this century

is the virelai "Dame souvrayne" (Fig. 6A) by the Italian

composer Matheus de Perusio. Perusio is considered .by many

historians to be one of the chief representatives of the

Modern Style,5 which was characterized by an abandonment of

the complexities of the Ars nova for more simplicity of de-

sign. This example (only a chord-condensation is given

here), like the preceeding one, begins on a chord other

than the tonic with a cadential progression to the tonic

in measure 4. From this point on, the tonal center C is

clearly established. An analysis of the Hindemith degree-

progression reveals, almost exclusively, the tonal spheres

5The term "Modern Style" is used by W. Apel (FrenchSecular Music of the Late Fourteenth Century) to denote theinterlude between the Manneristic period and the Flemishschool of the fifteenth century.

20

Fig. 6--Perusio, Virelai "Dame souvrayne"

Chord-condensation

Li

WindT a. aosesDepee.-.P.Iression

)isA. )0Oc~i a -

T -

!L27

C3

I 'I If I -A I I-9 013 1 A I E F

I -

IMPA A I.a I IIIVaI-1 _ _ _ _

-

............

I 4pre 1 6 1 ,i ly . I -.-0%6 soft.

21

Fig. 6--Continued

Sqbenker Analysi s:.

ainccth tO eaters

1 . . r 1 5

9 ~ g|1 T-f- --- m- m--- - -- -L - - -, - - .

C

4.. p .---

. ,: - . .

. mta mar1seiran r~i..a 16 1 la 1 I Iaal alim nramalisr 1.

I

22

C and G, with C, because of its predominance, becoming the

tonality of the work.

Figure 6B shows the structural framework of the piece.

Measures 4-17 represent a prolongation of the tonic chord

(C-G-C); the chords in measures 17-19 act as connecting

chords between the tonic and the dominant (measure 20).

Again in measures 25-31 the tonic chord is prolonged by a

series of contrapuntal chords. It is interesting to note

the agreement in root progression between the two concepts.

In Figure 6C the three structural chords which define the

course of the work are shown.

The Dufay "Missa L'Homme armed " (Fig. 7A) is an example

of early fifteenth century secular music. The "Kyrie" only

is cited here. Although the piece is modal in character,

an application of the Hindemith analysis reveals a clear-

cut tonality of G. There are very few dominant chords in

this piece; the chord on the seventh degree (F-A-C) seems

to take precedent over the dominant. This fact is illus-

trated in Figure 7B (third staff) which shows the importance

of the seventh degree of the scale, this note being out-

ranked only be the tonic.

An application of the Schenker concept to this example

produces four basic harmonic structures (Fig. 7a). Measures

1-8 reveal a basic structure of I-V-I, measures 9-11 are a

prolongation of the supertonic, measures 12-13 represent a

23

Fig. 7--Guillaume Dufay, "Missa L'Homme armet

Kyrie

$Chehket:-

1I4snclemith RAtalysfs:

rohadtit y

IFF

\ f iAft

0

_ _ _ _ _ _ _ _ _ _ i j ~ u ~ i;4

A6 a o

Fig. 7--Continued

ii

S'Aehker Anaysao.

r W

nde n aj

C- 9

low,,

connecting link between the supertonic in measure eleven

and the dominant in measures 14-15. The progression I-V-I

is again found in measures 16-24. Figure 7B shows how these

basic harmonic structures become subordinate to the primor-

dial structure, resulting in the fundamental progression

I-II-V-I (Fig. 7). It should be noted that there is very

little agreement between the Hindemith tonal. spheres (Fig.

7B) and the primordial structure as revealed by a Schenker

analysis. However, the resulting tonality in each analysis

becomes the same.

In Figure 8A, the "Missa Sine Nominew by Jacob brecht

(1430-1505), a more elaborate contrapuntal texture may be

found than in the previous examples. Only the "Kyrie"- is

cited here. A Hindemith analysis of the work produces tonal

spheres which clearly indicate a tonality of G, with a

"modulation't to D in measures 17-21 (Fig. 8B). A Sehenker

analysis indicates a basic structure of I-V-I. Measures

1-16 represent a prolongation of a single tonic chord,

measures 17-19 are a prolongation of the dominant, and in

measures 20-27 the tonic is again prolonged by contrapun-

tal chords. In this particular example the results of each

type of analysis agree exactly. This agreement is illus-

trated in Figure 8C, where the three structural chords

(Schenker) are found to correspond to the three tonal cen-

ters (Hindemith). The work actually has an A-B- form as

far as tonality is concerned (tonic-dominant-tonic).

26

Fig. 8 -- Jacob Obrechti"Missa Sine Nominet

Sie

99 wI I

Degvee- P?'o ressioh

0 I M -

I T-PUty

A9

I

I

I rp I s

I I

I i I

Kyr

I is I Ipp-

27

Fig. 8--Continued

Sheer final ysis:

* p~ -I *~I iI I I 1 -I I I I I

RV %lO.

53

- - s

1

I I

II

I

I

I

I

__________________________L_I

Hndeiiiti, toYsil centers;

I

28

Fig. 8--Continued

lpp

.07

sumb

Aft.

2t

-4

wa-ilJoI

F F

c

Aft

I

L wm o m m. -0w a.a. M o o

fI

I

29

An example of sixteenth-century church music is the

motet "Beata dei Genitrix--Et Beata" (Fig. 9A) by the

Flemish composer Adrian Willaert (1480-1562). A Hindemith

degree-progression produces a series of tonal spheres whose

chief tonality becomes D. Again, the importance of other

scale degrees other than the tonic and dominant is illus-

trated in this example. Chords on the sixth and seventh

degrees are frequently used; however the Schenker analysis

shows that these chords have only a contrapuntal or pro-

longing function. For instance, in measures 9-13 a series

of contrapuntal chords are used to prolong the motion be-

tween the tonic chord in measure eight to the tonic in meas-

ure fourteen. This progression is a part of a single struc-

tural chord outlined (Fig. 9B) by the first fifteen meas-

ures. Measures 16-18 represent a prolongation or connecting

link between the structural tonic chord (measures 1-15) and

the structural dominant chord outlined in measures 19-30.

This prolongation of the dominant chord contains a series

of embellishing chords which serve a contrapuntal function

only. Measures 31-39 again present a structural tonic

chord, giving a structural framework of I-V-I. The entire

example may be reduced to these three structural chords

(Fig. 90) with a tonality of D.

Figure IQA, a "Melange" by Orlando di Lasso (1532-

1594), one of the chief representatives of the Flemish

30

Fig. 9--Willaert, Motet "Beata dei Genitrix--Et Beata"

(Reduction)

(T~kz

I I ~ '

Ia iI

Ap flhAlU, lSy,

beg#.e-Pro ressi'oh

) lwI

ii *

AVA

4F ___1_7

EEI

;' i T'

-w-

N I . .fm to r- - .0 ff a . . . I a- 0r-4-4--

H N---------- I

I I Merl% ot

- I- - .,

-or- -- ,& --

TAJL-A

31

Fig. 9--Continued

I. 4i kx NT T) .[) I

SeI

A I vW K

is e a s0II N1 !ed ai 1b 4 o .5-.A. -rke

UAU)JI _uJ -- -

p e ag-

E4I

- --W_________T.

40

MWAF

j-j

32

Fig. 9---ContinuedScAenkeyr1InayWE ss:

Ml .s- a-t-- -

n htona c e nt ers

li

~~W-7

15 *0

Mi PAWI * ~

9 I4

IAm

I AL

4D 'Ak

0A

I

33

Fig. 9--Continued,

X z

school, is an illustration of the gradual decay of the modal

system and the emergence of the major and minor scale system..

This shift from modality to tonality based upon the major

and minor scale system may be observed in much of the music

during the late sixteenth century. A Hindemith analysis of

the work reveals a predominant tonality of A with a tonal

sphere of C assuming a secondary role. This constant shift

between A and C may justify an interpretation of the work as

being in the key of A minor with frequent modulations to C

Major, the related major. However, the piece is still quite

modal in character, and it may be somewhat far-fetched to

interpret it as lying within the major-minor tonal system.

The structural framework produced by a Schenker analy-

sis shows a basic harmonic structure of I-III-V-I (Fig. 10B).

The first eighteen measures are interpreted as a prolonga-

tion of a single tonic chord. The mediant chord assumes an

important function here, as it governs approximately seven

measures of the work (measures 20-26). This fact may be

Fig. 10--Orlando di Lasso, "Melange"

(Reduction)

ri

HinderA* itI AnalyssDesiee- Prea ressf* i0

aizLrlhI TI

TQI&Iitb'1 I'

A

MNor-gq-

iA

rr

34

35

I L II Im ti:

Alp& AOI o t .~f I.-- -.-. A w

chenke Amalys-s:r~ -n~ o~O m --wmm--- - E -- .-l:-~---7-2 ' 'n 7f

__ -~----~---~ As

I ficem tonal ceiiters:

Fig 101Cntn

r I I I

p -= o

I

-I_- mow Aqk Ak 11W) lw -1-W%0 F 19

A--9

r ----r ------iL L I 1 1 6m

36

Fig. 10--Continued

A

C

'it

compared with the results of the Hindemith degree-progression

where the tonal sphere of C ranked second only in importance

to the tonality of A. Measures 27-28 provide the final ca-

dence (V-I) which completes the basic structure of I-III-V-I

(Fig. 10C).

The chorus "Vieni Imeneo" from Monteverdits opera

L'Orfeo (Fig. 11A) illustrates a definite shift to the

major-minor tonal system, although not so complete a shift

as is found a century later. In this work there remain but

traces of the old modal system. The Hindemith degree-

progression reveals a clear-cut tonality of G (conventional

37

Fig. il--Monteverdi, Chorus "Vieni Imeneo" from L'Orfeo

(Reduction)

I If

}uvvth Analys's:

De~ye~ P~G~rt~i74

1 4 1P

T9in -I U-Iat-e

SFAt-

P; do

r r r Led r r

e- F f I A - 9 AL

LMLJ

AT46

I A

doom YA

-44

lk a i woorwi a I

I

38

Fig. Il--Continued

Shaker Atiai s. . .. 5doe

4 .I I1I - -l I

Hin~mi thk toj~ia centers:

I I

N O,11 1

IF

'Lis 3cAil

I

I a

------ ------1,17

E IV

39

theorists would call it G minor). A Schenker analysis pro-

duces a structural framework which interprets almost all of

the work (first twelve measures) as a prolongation of one

chord (Fig. 11B). Measures thirteen and fourteen contain

the structural chords II, V, and I, thus giving the piece

its basic harmonic structure, I-II-V-I (Fig. 11C). In this

example the Hindemith tonal spheres agree, as is usually

the case, with the Schenker structural framework with but

little exception.

An example of the "choral-vorspiel" technique of the

late seventeenth century is the "Auf meinen lieben Gott"

by one of the important predecessors of Bach., Dietrich

Buxtehude (1637-1707). Only the "Sarabande" is cited here

(Fig. 12A). In this example it is evident that the modal

system has gone out of existence. The Hindemith degree-

progression reveals a tonality of E (E minor in conven-

tional theory) which is determined by the predominance of

the note E over the other tonal spheres as regards both

position (first and final) and frequency (appearing five

times as compared with four for G). As in Figure 10A

(Lasso), this work may be interpreted as lying within the

major-minor tonal system, with a tonality of E minor and a

frequent "modulation" to G Major, the relative major. In

fact, it seems more logical to interpret this example in

this way, since it lies without question in the major-minor

Fig. 12--Buxtehude, "lAuf meinen lieben Gott"

Sarabande

A

I I

Vere-PrgresS enI

Tonality

v 1[ -

I I

Iga

A6. ap R

--- -F A2I

w

IL,

I

I - I I% ()

L1

Fig. 12-Continued

&MY

1Wa.. Igo

p 1

r I4r

Schenker- Avialysis:

S-I

to IIf

r r ls i

42

tonal system. This E minor-G Major interpretation agrees,

as a whole, with the Hindemith tonal spheres, with their

predominance of Ets and Gts.

The Schenker analysis produces a similar result (Fig.

12B). The first ten measures indicate a prolongation of

the tonic chord by a series of contrapuntal embellishing

chords. Measures 12-21 are a prolongation of a structural

mediant chord (corresponding to the conventional G Major

tonality). In measures 23-24 the final two structural

chords are heard (V-I), giving the piece its basic har-

monic structure, I-III-V-I (Fig. 120).

The examples cited in this chapter have portrayed the

gradual decay of the modal system and the evolution of the

major-minor tonal system. In only one example (Fig. 4) wasit impossible to arrive at a basic harmonic progression

according to the Schenker concept, although the piece was

TABLE 2

BASIC HARMONIC PROGRESSIONS (SCHENKER)USED IN CHAPTER II

Progression Figure in Which Used

I-V-I 5, 6, 8, 9

I-II-V-I 7, 11

I-III-V-I 10, 12

Indeterminate 4

shown to possess a definite tonality through an application

of the Hindemith tonal concept. The basic harmonic pro-

gressions which outline the structure of each example are

listed in Table 2.

It has been shown, in this chapter, how the Hindemith

and Schenker concepts may be applied toamedieval and

Renaissance polyphony. It has been illustrated how music

of these historic-al periods, although chiefly modal in

character , may possess a definite tonality or tonal center.

The following chapters will illustrate how the twot concepts

may be applied to music of later periods..

CHAPTER III

ANALYSIS OF MUSIC OF THE EIGHTEENTH AND

NINETEENTH CENTURIES

In the preceding chapter, it was illustrated how music

of the seventeenth century gradually abandoned the modal

system in favor of the major-minor tonal system. Regarding

this system, Vincenti gives the following definition:

"Maor-minor Tonality is a system by which a mental grasp

of the musical texture is maintained through a very cir-

cuuscribed and highly characteristic harmonic (vertical)

means of relating all melodic and harmonic elements to the

tonic or its triad."

Although the decay of the modal system was a gradual

process and not the achievement of any one composer, the

credit for demonstrating the advantages of the tonal over

the older modal system, and for laying the foundation for

future music, is given to Johann Sebastian Bach. The music

of Bach not only forms a climax of one historical period,

but marks the beginning of another.2

J. N. Vincent, The Diatonic Modes in Modern Music,p. l5 .

A. T. Katz, Challenge to Musical Tradition, p. 65.

The Bach example cited here is the Prelude No. 2 from

the "Well-Tempered Clavier," Book I (Fig. 13A). For com-

pactness, only a chord-condensation is given. The work is

clearly in C minor. A study of the Hindemith degree-

progression reveals tonal spheres of C, E, F, and C, giving

an over-all tonal center of C, because of the predominance

of this note due to its position (initial and final) as

well as its being the root of the tonal group. Measures

9-11 represent a modulation to Eb Major (according to con-

ventional theory), the work remaining in Eb through measure

fourteen. This segment is interpreted according to the

Hindemith analysis as having a tonal center of Eb, corre-

sponding exactly to the conventional theory, but with no

major-minor connotation.

A Schenker analysis produces a structural framework

with four structural chords outlined: I, IV7, V, and I

(Fig. 13B). The first eighteen measures represent a pro-

longation of a single tonic chord. This prolongation may

be more fully realized if an examination is made of the

structural melodic line. The upper voice in measures 5-18

descends step-wise for an octave (e'l-e'), outlining the

space of a tonic chord. This type of prolongation is only

one means by which to extend a structural chord. Measures

1-4 illustrate the use of contrapuntal chords as a connect-

ing link between structural chords (the tonic chords in

46

Fig. 13--J. S. Bach, Prelude No. 2 from the"Well-Tempered Clavier," Book I.

(Chord-condensation)1A 'j

AJL M - = -1 -w moMt7it I17.

hr

Minded mi; Apsalysis:c y -ProgreSSi !IZ

illi

I IL

Aft

I ~

IF

[~#r1 92

r_ -ftI

I es

11 f 2k J9 imi IF' Moe m m 3m m m Air_ , A a i m-- - M-W - 0 0 d 2 V m o go sm d9

I 1 -491. 1 0 - , I

47

Fig. 13--Continued

Z-

hi ~5 _ 1 15 1

B

MF9

WHin 4 Cm;it.4 tona~l cevtev-s:

x x3S 31

Apr&ME Nil

Fig. 13--Continued

C

measures one and four). The IV7 chord in measure nineteen

may be interpreted as a structural chord with the following

chord (the diminished-seventh chord in measure twenty) con-

sidered as a passing chord to the next structural chord (the

dominant beginning in measure twenty-one), or the two meas-

ures (19-20) maiy both be interpreted as passing chords from

the structural tonic (measure eighteen) to the structural

domina.nt (measure twenty-one). measures 21-30 clearly rep-

resent a prolongation of a single dominant chord by means of

contrapuntal chords which serve an embellishing function,

Measures 31-39 illustrate the prolongation of the final

tonic chord in much the same manner. Figure 13C gives an

over-all structural graph of the work, showing the basic

harmonic structure, I-(IV 7 )-V-I, The Hindemith tonal

spheres are not in complete agreement with this final

analysis. The tonal sphere of Eb should indicate a struc-

tural mediant chord in the Schenker analysis; however this

49

is not the case. The portion of the work corresponding to

the Eb tonal sphere (measures 9-14 in Fig, 13A), and in

conventional theory to the key of Eb Major, is considered

according to the Schenker concept as a portion of the pro-

lohgation of the initial tonic chord. Again, there is a

disagreement in measures 21-30, where an extended dominant

pedal with contrapuntal harmony above indicates, according

to the Schenker concept, a prolongation of a structural

dominant chord, but which is interpreted according to the

Hindemith concept as having C (the tonic) as its tonal cen-

ter. This latter interpretation may be explained by an

analysis of the chords in this portion of the work. It may

be observed that, as a rule, the roots of these chords lie

above the bass note, resulting in a succession of C's which

gives the note G a secondary significance and the note C a

predominating one.

An example from music of the late eighteenth century is

the Haydn Piano Sonata in D (No. 7 in Vol. I of the Peters

Edition). Only the exposition is cited here (Fig. 14A), and

this has been reduced to a structural framework according to

the Schenker concept. The basic structural chords are indi-

cated in this example; Figure 14B indicates two distinct

basic structures forming a primordial structure, which in

this case is incomplete (I-4). Figure 14C gives a final

analysis of the work, illustrating how the first basic

50

Fig. 14--Haydn, Piano Sonata, Vol. I, No. 7 (Peters Edition)

Schenker AhaLys;s: s

A

JLLie -s,. I - Z I

Wpi-timri th tonal centers :be gree- Proressian

.4t'

4

1-r. 1

3/ F$.

2 *S'

~e

/ A4

I IL

30~~1

Fig. 14--Continued

Schenker AnaIysts:11 IMP'

B

C

I71# j 4-*-"*N11.

-IV

Afine emith tonai centev-s:

lo4' P35

60% . I

E 7

--- 4.w w~... . - - -- -- - O F

w rwL am wt wI"m

I po"00

52

harmonic structure (I-V-I in the tonic) plus the second

structure (I-II-V7 -I in the dominant) reduce to a primor-

dial structure of I-V.

An analysis of the Hindemith degree-progression pro-

duces the tonal spheres D, E, A, indicating an incomplete

progression, or "half-cadence," fromthe tonic to the domi-

nant. In this particular example, an application of the

two concepts produces results which agree very closely.

An example illustrating the harmonic idiom of the early

nineteenth century is the Beethoven "Bagatelle," Op. 119,

No. 8 (Fig. 15A). It is interesting to compare the harmony

of the work with that of the preceding example. Whereas the

harmonic principles involved in the Haydn Sonata follow, to

a large extent, those established by Bach, the present exam-

ple involves a freer concept of tonality which borders on

chromaticism. This chromaticism is evident not only in the

melodic line but also in the chordal material. However,

there is little basis on which to classify the work as

"chromatic"; alteration is not used to the extent of affect-

ing its tonal course.

A study of the Hindemith degree-progression reveals an

establishment of five tonal spheres: C, E, G, F, and C.

This analysis corresponds to the conventional one: measures

1-5 are in the key of C, measure six in E, measures 7-8 in

G, measures 9-12 in F, and measures 13-20 in C. The Schenker

53

Fig. 15--Beethoven, "Bagatelle," Op. 119, No. 8

Moderato cantabileI I

I - II - l b .I

I ________4

I UINw I

11 IF"iA

To n

I _ _ _ _ _ _ _ _ _ _ _ _ _ = _ _ _ _

1L19JI

I

I w

I J j 0-,-*lI r -r r - f.A-r-T-I I

54

Fig. 15--Continued

&kenk.s IAhalysis:

:1:o is

-Jb

Aw- J6 ;gh t a znz r

)N

(1 . .. ..... ..... ....

1 1

-I-I

B

I I i a

I

*1 ti c)h 6at-C h ir

55

concept, admitting no such chain of modulations, interprets

the first eighteen measures as a prolongation of one struc-

tural tonic chord, with the chromatic chords involved being

interpreted as neighbor-chords used as embellishing func-

tions (Fig. 15B). The V-I cadence in measures 19-20 com-

pletes the basic harmonic structure (I-V-I). The entire

structural framework reduces itself to the three basic

structural chords as shown in Figure 15C. By comparing

the results of an analysis according to the two concepts,

it may be seen that some disagreement exists (Fig. 150).

The Hindemith analysis may be reduced to a tonal scheme of

I-III-V-(IV)-I; whereas the basic structure according to a

Schenker analysis becomes I-V-I.

The excerpt from Wagnerts opera Tristan and Isolde,

"Descend upon us, Night of Love," is an example of the ex-

treme chromaticism of the late nineteenth century (Fig. 16A).

Wagner, who, along with other romanticists, was responsible

for an almost complete negation of established harmonic

principles, has been credited with extending the limits of

tonality, if not causing its complete dissolution.

The example cited here has been reduced, for the sake

of compactness, to its structural framework according to

the Schenker concept. It is interesting to note the ex-

treme chromaticism in the melodic line; the entire thirty-

three measures represent one chromatically ascending melodic

56

Fig. 16--Wagner, "Descend upon us, Night of Love" from

Tristan and Isolde.

Schenker Analysis:

Hmidei th Analysis:

DeIree-Progressiot

DL I -

Too

+

P 4

VN U

A ) +(A So33

IM.7~1~oj

-~T

*Neig3hbor- Ck*rJ

8

Fig. 16--Continued

Se-henrkIer Ahalaysis: ~ )b .. fe4 2\

hb ie I-4Lt

/4;nderviiekh'tonal centters:

A

Ito

Pn (Fe 16B). The entire example may be conceived as

containing three structural chords, resulting in a basic

harmonic structure of I-V-I (Fig.'s 16B and 16C). Many types

of prolongation are used here. Some chords are used non-

harmonically, i. e., they support non-harmonic tones in the

melody; others are used as passing chords or neighbor-chords.

A study of the Hindemith degree-progression produces

the tonal spheres Ab, Et (enharmonic with Fb), B# (enhar-

monic with Cb), Eb, and Ab, which reduce further to Ab-Eb-Ab

resulting in a tonality of Ab (Fig. 16C). Thus, in this

I

58

example there is a complete agreement between the results

obtained by applying the two concepts.

This analysis reveals one important fact: chromaticism,

however extreme, may obscure but does not alter the tonal

course of the work--a fact which may often be disputed by

conventional theorists.

The final example of music of the late nineteenth cen-

tury is the Prelude "Danseuses de Delphes" by Claude Debussy

(Fig. 17A). This example illustrates the impressionistic

device of using chords in similar motion to support a me-

lodic line (measures 4-5).

ka analysis according to the Schenker concept reveals

a basic harmonic structure of I-V-I (Fig. 17B). In the

structural graphs of this example, octave transposition is

used freely in order to simplify the analysis as well as to

facilitate reading. The first nine measures represent a

prolongation of a structural dominant chord prolonged by

means of passing chords (secondary dominants, etc.) as well

as contrapuntal chords which are used to support the melodic

line. Measures 21-31 represent a prolongation of the final

tonic chord, resulting in a basic harmonic structure of

I-V-I,

An analysis of the Hindemith degree-progression pro-

duces a series of tonal spheres which, when grouped further,

result in the tonal centers of Bb, F, and Bb, giving a

tonality of Bb. These tonal centers compare favorably with

59

Fig, 17--Debussy, Prelude "Danseuses de Delphes"

i itI-,1 ;LA

11

-1 11nI I Al0U 'JAk t AP 40s

I Dec .- Progress ith

I a

40 I I m~

#I flp i

Opp IIOo

'or I.o A oPI I-

I

60

Fig. 17--Continued

I i*

At :Edo fil

ma~ 4 A A& ja

-e t

I I~Fle4

- ~ 1

At-tL!K

6I

.1 zz&I 'II

b _______

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61

Fig. 17--Continued

.a * -a I A. A . .

I -1

OI-V F

1111 -1 1T

I If

'A III

3thanker vAnialysis:

I_ __a

Hnelemith tonal CenterS :

'1

62

Fig. 17--Continued

19 A '.0.01j%1 A- w ii

Y

IN Ils_ hip

F. j_ _ _ _ _

c

S Iba

EEóEFE411

ON ei *V,6or - ChovA

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63

the final Schenker analysis (Fig. 170), resulting in the

three elements (structural chords vs. tonal centers) which

give the work musical direction.

This example illustrates a broadening of the concept

of tonality as far as non-harmonic material is concerned.,

But, as in the Wagner example, in spite of the use of

chordal material considered foreign to the established

tonality, the basic harmonic structure or the basic tonal

scheme as revealed by analysis remains as- concrete as in

earlier examples.

A list of the basic harmonic progressions used in this

chapter is found in Table 3. It may be observed that, in

iTRM.W 3

BASIC HRMONIC PROGRESSION (SCHENKER)USED IN CHAPTER III

Progression Figure in Which Used

I-V-I 13, 15, 16, 17

I-T 14

this chapter, as well as in Chapter II, the progression

I-V-I is by far the most frequent*

The examples in this chapter have illustrated the in-

troduction of elements (chromaticism, impressionism, etc.)

which tended to weaken the tonal fiber of each composition.

A further dissolution of tonality is presented in the next

chapter, where some of the problems involved in analyzing

the tonal structure of contemporary music will be dis-

cussed.

CHAPTER IV

ANALYSIS OF MUSIC SINCE 1900

In general, the music of the late nineteenth century

tends to abandon many of the established laws of the major-

minor tonal system for a freer type of musical expression

characterized by extreme chromaticism, impressionistic de-

vices, and other stylistic features which served to estab-

lish the need for a broader concept of tonality. Concerning

this trend in harmonic development, Schoenbergi makes the

following statement:

From the beginning major and minor tonalitieswere interspersed with non-diatonic elements tendingto form opposition to the fundamental tone yet com-pelling the application of strong means in order toverify the tonality, to paralyze eccentric effects.. . . The art of music was never really in possessionof a tonality wholly limited to the seven diatonictones of the scale.

Recent developments in harmony have resulted in a fur-

ther use of "non-diatonic elements," tending to weaken the

tonal fiber to an even greater extent. In many instances,

a complete dissolution of tonality has been the result.

During recent years, the great number of "atonal" works pro-

duced, as well as many which approach "atonality," suggest

that tonality, in its conventional sense, may no longer be

lNerle Armitage, Schonber, pp. 284-285.

65

66

essential to the unity of a composition. Regarding the

question of whether tonality must be preserved, Schoenberg2

states:

Tonality . . . Cisj no postulate of natural con-ditions, but . . . LisJ the utilization of naturalpossibilities; it is a product of art, a product ofthe technique of art. Since tonality is no conditionimposed by nature, it is meaningless to insist on pre-serving it because of natural law. Whether, for ar-tistic reasons, tonality must be retained depends onwhether it can be replaced.

Sc h onb erg3 further asserts that, although tonality

is, without question, one of the chief means by which a

work may be unified, it is not the on means. He believes

that a work which lacks a unification of harmonic material

may have meaning through a logical development of motive

and thematic material.

Figure 18A, a piano piece from the Mikrokosmos (Vol. 6,

No. 150) by Bela Bartk, is an example of a "tonal" work of

the twentieth century. This example illustrates a principle

which Bartok frequently used to extend or disguise the to-

nality of a work, i. e., beginning or ending a piece in some

key other than the tonic. The work is almost entirely in

the dominant key (E Major). According to the Schenker con-

cept the entire piece, with the exception of the final meas-

ure, represents a prolongation of a single dominant chord

(Fig. 18B), resulting in what might be considered an

21bid., pp. 299-300.

3lbid. , pp. 289-291.

67

Fig. 18--Bartok, Piano Piece from Mikrokosmos (Vol. 6,No. 150) (First section).

H; nemit Anaiys s:Deree -Pra'pes1 _\_n

-1

A A

I W

~ZZIZZZZIIZIII7i -o------~-----~--~-- I---1

68

Fig. 18--Continued

WIN

- 2 -I . - J -

K---i

-Iw

I .Il.I1 - I I.---0

69

Fig. 18--Continued

Sc(enkeir Analysis:

Ab o M 1-. .1 mVia wa1 11

Hindemith tonai e cnters:

E

AA

-q ere

-- -----c;

X

70

incomplete basic structure (V-I). The style of the work is

more contrapuntal than harmonic. Measures 5-18 illustrate

a melodic line which is doubled and accompanied only by an

upper note which acts as a pedal. A similar contrapuntal

treatment is found in measures 31-42 where the outer voices

move in tWO-part counterpoint against sustained notes in

the inner voices. However, the melodic lines involved in

both of these excerpts as a rule delineate a chord, giving

the piece a somewhat chordal basis, thus facilitating the

analysis.

Measures 27-41 illustrate a chromatically descending

bass-line (from e to E) which serves to prolong the motion

within the structural dominant chord. The chords in these

measures become subordinate to the structural framework and

are treated as passing chords. Figure 18C is a graph of

the basic harmonic structure of the work, resulting in the

incomplete progression V-I.

A study of the degree-progression (according to

Hindemith) produces a series of tonal spheres which, when

grouped further, produce tonal centers of E and A resulting

in an over-all tonality of A. This analysis is in complete

agreement with the Schenker analysis: the tonal center of E

corresponds to the structural dominant chord and the tonal

center of A corresponds to the structural tonic chord (Fig.

18C).

71

An example of an "atonal" work of the twentieth century

is the tiano Piece, Op. 19 No. 3, by Schoenberg (Fig. 19A).

This piece is representative of Schoenberg's second, or

"atonal" period.

Whether the application of the term "atonal" to music

is justifiable is a problem about which few theorists are

in agreement. Schoenberg, who in reaction to the chromati-

cism of the post- agnerian school created a new style of

composition (beginning with the Three Piano Pieces, Op. 11)

branded by theorists as "atonal," has emphatically rejected

the use of the term and suggests that the term "pan-tonal"

is more descriptive of the harmonic style.>

Hindemith,5 also rejecting the use of the term "atonal,"

admits that there are two types of composition which, al-

though they cannot be called atonal, yet by their harmonic

content make it difficult for the listener to follow their

tonal course. These are (1) chromaticism which, although

using material of the diatonic system, employs continuous

modulation and enharmonic changes and (2) the continuous

use of chords of groups III and IV (Hindemith classification)

in the effort to avoid triad formations indicative of the

old diatonic system. Both of these types of composition

,+Armitage, Q. Qfit., pp. 298-299.

5Paul Hindemith, Craft of Musical Composition, I, 153.

72

Fig. 19--Schoenberg, Piano Piece, Op. 19, No. 3Very slow

be~~I"

Hf nde th 6Analysis:Degt-ee- Prosression

TonsAIty

b

__

__ _M__Wi

VADJ

r V

b

S h er A ay

Hn eith tonaj centers

Bb

73

Fig. 19--Continued

C

are characterized by tonal ambiguity, making their analysis

difficult to explain within the conventional concept of

tonality.

Figure 19A may be classified as belonging to the second

type of composition. An examination of the chordal material

involved produces a succession of chords of groups III and

IV, representing chords of highest tension and lowest sta-

bility (containing seconds and sevenths). In only two in-

stances is there a relief from this tension (measures five

and eight) by chords of group I (triad formation). The

roots of these chords group themselves into tonal spheres

of C, Bb, G, Bb, G, Ab, and D respectively, of which Bb

becomes the tonal center (Fig, 19B).

Figure 19B shows a graph of the structural framework.

In spite of the extreme complexity of chord material in

this example, the bass, which for the first four measures

moves in octaves, delineates intervals and scale-like

passages which suggest a tonality of Bb. Thus the first

four measures may be considered a prolongation of the tonic

chord. Measure five contains chords which act to prolong

the motion between the structural tonic chord (measures 1-4)

and the structural mediant chord (measure six). This medi-

ant chord, although altered, assumes the role of a mediant

and may even be interpreted in conventional theory as a III9

with lowered fifth and seventh (Bf enharmonic with Cb, the

seventh of the chord). The next structural chord (the domi-

nant in measure eight) is also altered. This chord may be

interpreted as a V9 in third inversion with a raised fifth

and with the third omitted (F-(A)-C#-Eb). The third of the

dominant chord appears simultaneously with the final tonic

chord. Figure 19C gives a final structural graph showing

the basic harmonic structure, I-III--I.

The final example cited from music of the twentieth

century is the Stravinsky "Dodo iegenied" from the

Berceuses du Chat (Fig. 20A). The group of short pieces

which comprise the Berceuses du Chat were written in 1915-

1916 as a study for the dramatic work "Renard" which fol-

lowed them. The score of this work (IDodo Wiegenlied") for

voice and three clarinets has been reduced with the vocal

part written as the top voice and the clarinets as the

three lower voices. This example is a very interesting

one from a stylistic point of view. The two lower voices

75

Fig. 20--Stravinsky, "Dodo Wiegenlied" from Berceusesdu Chat for Voice and Three Clarinets.

(Reduction)

i-_t, -1-A zzTA f I frftz zt0 4F 7

I-De.P-t r Analysis:Dea - -tryeSu o

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76

Fig. 20---Contirnued

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105-. p.s

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77

move in organum a fourth apart while the two upper voices

suggest the Dorian mode. Throughout the entire work the

first clarinet, when heard, is merely a simplification of

the vocal part. Or, on the other hand, the vocal part

could be considered an elaboration or embellishment of

the first clarinet. Thus the piece becomes, to a large

extent, three-part counterpoint with the two lower voices

moving in parallel fourths.

The difficulty in analyzing this piece lies in its

"bitonality." According to the Hindemith tonal concept,

the degree-progression produces a tonality of F#. A

Schenker analysis produces approximately the same results.

The first fifteen measures represent a prolongation of a

chord on F# (the tonic) with the exception of the first

2} measures. The piece actually begins on the dominant

(C#), going to the tonic in measure three. Measure six-

teen includes the structural dominant and tonic chords

which complete the basic harmonic structure of I-V-I

(Fig. 200).

TABLE 4

BASIC HARMONIC PROGRESSIONS (SCHENIER)USED IN CHAPTER IV

Progression Figure in Which Used

V-I 18

I-III-V-I 19

I-V-I 20

78

A summary of the basic harmonic progressions accord-

ing to the Schenker concept may be found in Table 4. Al-

though only a small sampling of twentieth-century music

was analyzed, it is the author's belief that, as in pre-

vious examples, the progression I-V-I would prove to be

usea most frequently in contemporary music.

In this chapter the three examples presented were

chosen to illustrate some of the most outstanding stylis-

tic features peculiar to the music of the twentieth cen-

tury. Many of the problems involved in the analysis have

been discussed and in some instances more than one inter-

pretation was given.

cOiAPTER y

MMMY AND CONCLUSIONS

The preceding chapters have presented analyses of rep-

resentative examples chosen from music beginning with the

late thirteenth century and ending with the twentieth cen-

tury. An application of the Iindemith and Schenker concepts

to these examples has resulted in one important finding:

in spite of the entirely different approaches of the two

systems of analysis, the resulting tonality in each example

has been the same4

Each of the two concepts of tonality has been found to

contain certain faults as well as merits. First, the re-

sults of analysis according to the Schenker concept will be

examined. Table 5 lists the basic (in some cases, primor-

dial) harmonic structures resulting from the Schenker anal-

ysis and their frequency of use in the seventeen examples

cited. The chief significance resulting from this table is

the discovery that the majority of the examples possess a

basic harmonic structure of I-W-I. It is the author' belief

that, as a result of an examination of perhaps hundreds of

examples chosen from the different schools or historical peri-

ods, the basic structure of I-V-I could easily prove to be

more common than the others listed. This I-V-I progression

79

80

TABLE 5

FREQUENCY OF USE OF THE SCHENKERBASIC HARMONIC PROGRESSIONS

Basic (or Primordial) FrequencyHarmonic Progression

I-V-I 9

I-III-v-I 3

I-Il-V-I 2

I-V

V-I

Indeterminate I

actually outlines an A-B-A form which is probably the most

important form, not only because of its use in such a large

portion of music of different historical periods, but also

because of its being used as a basis for larger and more

complex forms.

It may be argued that the majority of musical composi-

tions do not naturally fall into a basic or primordial

structure of I-V-I. Mann implies that Schenker, instead

of adapting his method to follow the tonal course of a com-

position, makes the composition follow a prescribed tonal

course. This seems to be the situation in some of the ex-

amples which were analyzed. The music does not always fol-

low a prescribed course (e. g., I-V-I or I-III-V-I), but

kxichael Mann, "Schenker's Contribution to MusicTheory," The Music Review, X (February, 1949), 26.

81

may have an individual tonal course as seen by an analysis

of the Halle example in Chapter II.

According to Mann,2 Schenker sees in the major and

minor tonality that characterizes the period beginning with

Bach and ending with Brahms the final achievement of musical

art. He condemns all music that transcends the technique

of that era and regards all musical styles which preceded

it as preliminary steps. Mann states further:

When Schenker can no longer find in the presentthat shadow of the past to which he would forever bindmusic, he simply stands bewildered. The present thenseems like a sudden breaking away fro the past, abreaking of the "laws of the genius."

It follows that Schenker's analytical system appears

to be applicable to the fugue and sonata style of the

eighteenth and nineteenth centuries. However, in the analy-

sis presented in Chapters II and IV the Schenker system has

been successfully applied to music of other centuries. But

this does not indicate that every piece of music from these

periods will lend itself to this method of analysis. The

example by Adam de la Halle (Chapter II, Figure 4A), with

a tonality of F (tonal spheres of A, G, and F), does not

possess a basic harmonic structure which conforms to the

principle of a fundamental progression as conceived by

2Ibid., p. 6.

3lbid., p. 24.

82

Schenker. Similar situations may arise in the analysis of

music from the medieval and Renaissance periods.

The three examples cited from music of the twentieth

century were analyzed without difficulty by the Schenker

method. However, since these examples are only a small

sampling of the music of this century and certainly do

not represent all of the stylistic features of present-

day music, we are not justified in assuming that the

Schenker analytical system may be applied to every piece

of music of this era.

Two chief merits of the Schenker concept are: (1) it

deals with chord function and significance instead of chord

grammar, and (2) it is an analysis of what we hear in music

which, in reality, should be the final criterion.

Sessions) lists two main objections to the Schenker

concept: (1) "it is far too primitive as a description of

the actual events which constitute a musical work, or the

sensations and apperceptions that constitute the ultimate

comprehension of that work," and (2) "it leaves the domain

of exact description and enters that of dogmatic and specu-

lative analysis." Other theorists have condemned the

Schenker concept as being dogmatic. It may be evident from

examining some of the analyses presented in preceding chapters

#Roger Sessions, "Heinrich Schenker's Contribution,"Modern Music, XII (May-June, 1935), 176-177.

83

that to arrive at a prescribed fundamental progression,

while at the same time ignoring many important stylistic

features which do not directly contribute to this progres-

sion, constitutes a type of dogmatic analysis.

The Hindemith concept, because of its objective and

non-aesthetic approach, may be criticized as being too

"artificial" a system. In contrast to the Schenker system

of analysis according to what we hear in music, the

Hindemith system is an analysis according to a prescribed

method based on the overtone series, and as a consequence

is almost entirely devoid of aesthetic judgment. Although

this system may be applied to aLny piece of music no matter

how complex, it may be argued that a tonality which is de-

rived from an "atonal" or "tone-row" composition has, in

reality, little significance if the ear does not perceive

it.

Some of the merits of the Hindemith concept are:

(1) it is applicable to music of any era; (2) it estab-

lishes the legitimacy of chords which were never used in

conventional analysis and which are subject to only one

interpretation; and (3) it establishes a system of chord-

classification, whereby chords may be ranked according to

tension or stability.

Two main objections to the Hindemith concept are:

(1) it arrives at a tonality by means of an objective and

non-aesthetic approach; and (2) it often arrives at a to-

nality which, if it is not perceived by the ear, may not

actually exist.

BIBLIOGRAPHY

Books

Apel, Willi, French Secular Music of the Late FourteenthCentury, Cambridge, Mediaeval~Academy of America,1950.

Armitage, Merle, Schoener, New York, G. Schirmer, Inc.,1937.

Hindemith, Paul, Craft of Musical Composition, Vol. I,New York, Associated Music Publishers, Inc., 1937.

Katz, Adele T., Challenge to Musical Tradition, New York,A. A. Knopf, 1945.

Piston, Walter, Principles of Harmonic An, Boston,E. C. Schirmer Music Co., 1933.

Salzer, Felix, Structural Hea , Vol. I, New York,C. Boni, 192.

Vincent, John N., The Diatonic Modes in Modern Music, NewYork, Mills, by arrangement with~the University ofCalifornia Press, Berkeley, 1951.

Articles

Heiden, Bernard, "Hindemith's 'System'--A New Approach,"Modern Music, XIX (January-February, 1942), 102-107.

Hoskins, William, "General Semantics and Music," GeneralSemantics Bulletin, Vol. VI-VII (Spring-Summer, 1951),71.49 -57.

Katz, Adele T., "Heinrich Schenker's Method of Analysis,"Musical Quarterly, XXI (July, 1935), 311-329.

Mann, Michael, "Schenker's Contribution to Music Theory,"The Music Review, X (February, 1949), 3-26.

Muser, F. B., "The Recent Work of Paul Hindemith," MusicalQarterl, XXX (January, 1944), 34-36.

86

Sessions, Roger, "Heinrich Schenker's Contribution," ModernmusL, XII (May-June, 1935), 170-178.