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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 successions 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 composition, and
II.
The relationship of other keys employed as sub-
servient to the principal key (usually explained by the theory of modulation),
This concept is explained further by the following definition:
1 A.
"General Tonali
is that prin
le by which a
T. Katz, Challne to Musical Tradition, p.1.
2 Merle
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 successions, 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 compare, 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,
Musikalische
jNee
Theorien
Fantasien,
jeisterwerk in der Musik, and Per Freie Satz. 5
Das
When
feasible, these two concepts will be compared with the conventional concept.
The musical examples cited have been chosen as representative of the different historical periods.
Each example
31. N, Vincent, The Diatonic Modes in Modern u
p. 14.
Armitage, a. Cit., p. 285.
$one of these works, to the author's knowledge, have
been 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 illustrate 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
. .
. .
II.
III.
IV.
V.
. . .
. .
.
.
.
. .
. . .
.
.
vii
.
.
.
.
.
viii
A BRIEF DESCRIPTION OF THE HINDEMITH AND
SCHENKER CONCEPTS OF TONALITY . . .
.
.
ANALYSIS
.
.
LIST OF ILLUSTRATIONS
Chapter
I.
. . .
.
.
.
.
.
.
OF MUSIC BEFORE BACH
.
.
.
.
.
.
.
ANALYSIS OF MUSIC OF THE EIGHTEENTH AND
NINETEENTH CENTURIES . . . . . . . .
ANALYSIS OF MUSIC SINCE 1900 . . .
. . .
.
,
*
. .
vi
13
.44
.
.
65
*
*
79
. . . . . . . . . . . .
85
SUIMARYAND CONCLUSIONS
BIBLIOGRAPHY
1
.
. .
.
.
.
. . .
.
LIST OF TABLES
Table
1.
2.
3,
Page
A Comparison of the Hindemith and Schenker
Concepts with the Conventional Concept
.
11
Basic Harmonic Progressions (Schenker) Used
In Chapter II *-# *.I.D...
.0 .
. .
42
Basic Harmonic Progressions (Schenker) Used
In ChapterII
-. . .
...
..
. .
63
.
4.
Basic Harmonic Progressions (Schenker) Used
In ChapterIV
.*.0.0.0.*9.9.9. *.0.
.0.77
5.
Frequency of Use of the Schenker Basic Harmonic
Progressions . . - .
. . . . . . -. . . .
vii
80
LIST OF ILLUSTRATIONS
Figure
Page
1.
Series
I
.
.
2.
Series
2
.
.
3.
Beethoven, Piano Sonata, Op. 14, No. 1
.
.
3
.
.
3
.
. .
9
.
.
.
.
1
.
.
.
.
.
17
.
.
.
. .
20
.
.
.
.
.
23
8.
Jacob Obrecht, "Missa Sine Nomine0.
9.
Willaert, Motet OBeata dei Genitrix-Et Beata*
.
30
37
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
4.
Adam de la Halle, wTant con je vivrai" .
5.
Guillaume de Machaut, "Plus dureW
.
.
6.
Perusio, Virelai "Dame souvrayne"
.
.
7.
Guillaume Dufay, I'Missa L'Homme armed"
.
.
.
.
.
.
.
.
.
.
.
..
. . . . . .
.
26
10.
Orlando di Lasso, "Melange"
11.
Monteverdi, Chorus "IVieniImeneo" from L'Orfeo
.
12.
Buxtehude, "Auf meinen lieben Gott"
10
13.
T. 5.- Bach, Prelude No. 2 from the "Well-Tempered
Clavier,"
14.
I
Book
.
.
,
.
*
.
.
*
*
.
.
.
Haydn, Piano Sonata, Vol. 1, No. 7
Edition)
.
.
.
.
.
.
.
.
.
* *
*,
*
.
.
*34
.4 . . .
.
.
.
.
.
.
.
.
.
.
.
.
Beethoven, "Bagatelle," Op. 119, No. 8
16.
Wagner, "Descend upon us, Night of Love'" from
and
Isolde
.
.,.
*
.
w
.
17.
Debussy, Prelude tDanseuses de Delphes"
18.
Bartok, Piano Piece from
fkrokosmos
No.r 150Pian Piec.
19.
Schoenberg, Piano Piece,
. 19. N.. 3.
OP. 19,
viii
.
46
(Peters
15.
-Tristan
.
Now
.
.
.
50
.
.
.
.. 156
.
.
.
.
53
59
(Vol. 6,
. . . .
3
. 67
.w
..
72
Figure
20.
Page
Stravinsky, "'Dodo Aiegenlied" from Berceuses du
ix
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 previously in use.
Concerning this system, Vincenti states:
The fault lay in the narrow concept of key which
regarded all but a few chromatic chords as violations
of the key. The increasing complexity of the harmonica
materials forced a progressively broader view of the
limits 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 the
dual modality of classical tonality,
and
(2) to reveal the structure of the component
chords (name, type, and inversion). The tonal functions (dominant and subdominant) are related only to
the tonic and, as a consequence, an analysis must
resort to frequent modulation in order to explain
chromatically-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 subdominant 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.tt 3
Hindemith's system of analysis is based on the twelve
tones of the chromatic scale.
become a tonal center.
Each of the twelve tones can
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
3 Bernard
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 decreasing order of relationship to the progenitor tone C,
as indicated in Figure 1.
An interval, which is the smallest unit used as building material, may serve in one of two capacities:
as a melodic 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
2 Heiden,
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 resolution interval. 7
Hindemith8 insists that the conventional theory of harmony 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 subdivided 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 interval (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 harmonic stress, which is a new device for measuring harmonic
tension.
Chord successions are evaluated according to the
ratio of tension to stability.1 0
Hindemith's system is intended to supply an analytical
background for all types of music..
experiences a broad generalization.
9Heiden, 22.
Cigt.,
pp.
The concept of "key"
All tone-combinations
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 degreeprogression.
Tonality becomes not a starting-point but a
goal.
Museril states that "the setting up of an orderly disposition 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
12 A. T. Katz, "Heinrich Schenker's Method of Analysis,"
Musical QuarterlX.,XXI (July, 1935), 313.
1 3 A.
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 section 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.
expanded by "prolongation."
This motion may be
Passing chords, chords which
support non-harmonic tones, etc., are used to extend or prolong 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.1 6
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 unfolding 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 relative structural values only; they are not to be interpreted
literally.
This same method of graphing will be employed in
succeeding 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 structure" 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 representing 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 dimensions 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."1 8
He sensed that chord
grammar failed to differentiate between the diverse functions 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. 1 9
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 direction 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.
1 9 Felix
Salzer, Structural Heari
, I, 26.
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CHAPTER II
ANALYSIS OF MUSIC BEFORE BACH
It is generally agreed by theorists that a large portion of the music prior to the introduction of the majorminor 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 following definition:
Tonality in Plain Chant is a system by which a
mental grasp of the unaccompanied melodic line is
maintained through a system of linear tonal combinations. . . . It thus only makes use of the elodic
phase of the general principle of tonality.
He states further:
Tonalit in Renaissance Polyphony is a system by
which a mental grasp of the melodic and harmonic texture is maintained partly through the methods of unaccompanied plain chant which apply mainly to the
cantus firmus, and partly through certain added har-w
monic conventions whose function is to relate the
component triads to the triad of the final which has
p. 14.
lJ N.Vincent, The Diatonic
Modes in Modern Music
21bd
7_
~dp.
5.
13
14
taken the place of the simple final.
outlook was still horizontal.3
.
.
.
The whole
Piston4 says that a definition of tonality should include 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 tonality 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 approximately 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
'
4--Adam de la Halle, "Tant con je vivrai t
Rondeau
AID-
A
4~~A
a
m~
a
I
~czzrzzJiri-.--~u4zzJzJzj ~Z2ZIZZt2
'OTT
.*JIIITZZ
-u
1
-
-9
1
W- o
I
I
H-t--1
-if-
a
m-
~2;
--
liar
E
(~~~zI~I
INindeem
-
'
,q4~
~
-
F'ig.
_
II~
th ansiyses:
ronaeit y
I
___:z2 r-::Ash---~
Schenker "nalysis:
.....
_NO
hnsdemiet t.na9
ent es
p
16
Fig .
-- Continued
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 accompaniment.
Again, a definite tonality is produced by
the Hindemith degree-progression.
Even though the piece
17
Fig. 5--Guillaume de Machaut, "Plus dure"
A
I
It
~H nertA Anaiys s:
e egP- CCme - - - ..
- -n
.
T[nzi
1..
-.
.
.
-
-
.
I"'m.
11ir--1
$sty
LMI
IE
EW
AM
Am-490
18
Fig. 5--Continued
Shenker
Analysis:--
Be
P
0-
-I-_-
8C
--
NM~iil@Migi,41~g,'iHIil04h~tM'IIWarlsMMi&IE
I--tg2
iig.am
3|11Idg- l i I-i II
--
-lEEI
10Iml1
-
!
# ndal-e
fonl ee
I
In.wIL
ers
r
.
#7| cl i r FF~- [ --.- -- . . . . ..
-1
-
-
--
I mn
-
||
-
m
-_--_-
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 design.
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
5 The
term "Modern Style" is used by W. Apel (French
Secular Music of the Late Fourteenth Century) to denote the
interlude between the Manneristic period and the Flemish
school of the fifteenth century.
20
Fig. 6--Perusio, Virelai "Dame souvrayne"
Chord-condensation
WindT
a. aoses
Depee.-.P.Iression
Li
a
-
T
)0
)is
Oc~i
-
A.
!L27
C3
'I
9
If
013
I
1
-A
A
I
I
I
E
IF
-
I
IMPA
AI.a
I
............
-.-
0%6
soft.
IIIVaI-1
1
61
,
i ly . I
4pre
-
I
_
_
_
_
21
Fig. 6--Continued
Sqbenker Analysi s:.
tO aincctheaters
1
1
5
9~ g|1 T-f- m-m---
-
---
C
4.. p .--.
,:
-
.
I
.
.
.
- -L - - -, - -
. . r
mta mar1seiran r~i..a16
1laI Iaal
alim
nramalisr
1.
1
-
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.
is cited here.
The "Kyrie" only
Although the piece is modal in character,
an application of the Hindemith analysis reveals a clearcut tonality of G.
this piece;
There are very few dominant chords in
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 outranked 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
ao
Fig. 7--Continued
ii
aj
nde n
S'Aehker Anaysao.
r W
C
-
9
low,,
connecting link between the supertonic in measure eleven
and the dominant in measures 14-15.
is again found in measures 16-24.
The progression I-V-I
Figure 7B shows how these
basic harmonic structures become subordinate to the primordial 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.
cited here.
Only the "Kyrie"- is
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 contrapuntal 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 centers (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
Kyr
I
I
I is
Ipp-
I
rp
I
I
s
I
I
Degvee- P?'o ressioh
T-PUty
0
IM
-
I
A9
I
I 99 w
I
i
I
27
Fig. 8--Continued
- -
I
s
1
I
I
Sheer final ysis:
53
I
i*
I
I
I
p~
II
__________________________L_
I
Hndeiiiti,
RV
*~
-I
I
I
I
I
I
1
-
I
I
toYsil
centers;
%lO.
I
I
28
Fig. 8--Continued
.07
lpp
sumb
Aft.
2t
L
wm
o m m.
-0w
a.a.
M
o
o
I
Aft
-4
f
I
I
wa-il
c
JoI
F
F
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 illustrated 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 prolonging function.
For instance, in measures 9-13 a series
of contrapuntal chords are used to prolong the motion between the tonic chord in measure eight to the tonic in measure fourteen.
This progression is a part of a single struc-
tural chord outlined (Fig. 9B) by the first fifteen measures.
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 (15321594),
one of the chief representatives of the Flemish
30
Fig. 9--Willaert, Motet "Beata dei Genitrix--Et Beata"
(Reduction)
-o
-w-
(T~kz
r- -
4-4--
I.
r.0
ff
I
a
.
. . I
a-
----------
0
I
i
r- -- ,& --
T'
;'
~
'
fm
to
.
N
I
I
-
I-
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.,
H
ot
Merl%
N
I
TAJL-A
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iI
ressi'oh
Ap
ii
flhAlU,
lSy,
*
beg#.e-Pro
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AVA
4F___
1_7
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31
Fig. 9--Continued
4i kx
I.
T)
NT
j-j
.[)
I
SeI
A
K
I
vW
as0II
N1
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ai1b
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-rke
e
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-
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p
e
ag-
40
MWAF
W_________T.
-
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32
Fig. 9---Continued
ScAenkeyr1InayWE
ss:
I
n
a-t--
-
.s-
Ml
htona c e nt ers
~~W-7
li
15
Mi
*0
PAWI
AL
4D
'Ak
IAm
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9
I
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33
Fig. 9--Continued,
z
X
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 analysis shows a basic harmonic structure of I-III-V-I (Fig. 10B).
The first eighteen measures are interpreted as a prolongation 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
34
Fig. 10--Orlando di Lasso, "Melange"
(Reduction)
A
ri
HinderA* itI
Analyss
Desiee- Prea ressf* i0
aizLrlh
TQI&Iitb'1
I
TI
I'
MNor-gq-
iA
rr
35
Fig
101Cntn
r
I
p
I
I
-=
o
r ---r
------
6m
1
1
I
L
iL
L II Im t
I
I
i:
I o t .~f
chenke
-
-wmm---
2
AO
A
w
Amalys-s:
.-l:-~---7- m
-- o~O
E-n~
r~
A--9
Alp&
I.---.-.
%0
__
I
ficem
7f
'n
'
-~----~---~
tonal ceiiters:
As
-I_-
mow
W)
Aqk
lw
Ak
-1-W
F 19
11
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)
a
lk
I
woorwi
i
a
I
If
I
}uvvth Analys's:
De~ye~
P~G~rt~i74
1 4
1P
T9in
U-Iat-e
-I
SF
r
r
r
e-
do
P;
At-
r
Led
r
Ff I
A
-
9
AL
LMLJ
AT46
I
doom
A
YA
-44
38
Fig. Il--Continued
Shaker Atiai
. . ..
IF
s
5doe
I1I
4
Hin~mi thk toj~ia
-.I-l
I
centers:
I
I
N
O,11
I
1
I
a
'Lis
1,17
E
IV
3c Ail
-----------
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).
(Fig. 12A).
Only the "Sarabande" is cited here
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 conventional 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
I
I
-
v 1[
Iga
ap
A6. R
A2I
IL
)
I%
,
I
(
-F
--I -
w
I
L1
Fig. 12-Continued
&MY
Igo
1Wa..
p
1
r
I4r
Schenker- Avialysis:
S-I
to
r r
IIf
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 harmonic 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) was
it 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
I-III-V-I
Indeterminate
7,
11
10, 12
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 circuuscribed 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).
pactness, only a chord-condensation is given.
clearly in C minor.
For comThe work is
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 conventional 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, corresponding 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 connecting link between structural chords (the tonic chords in
46
Fig. 13--J. S. Bach, Prelude No. 2 from the
"Well-Tempered Clavier," Book I.
AJL
11 f 2k J9 imi
'j
1A
IF'
Moe
m-- - M-W
w
-491.
I1
m
1
-0 0
moMt7it
0
d2
-
3m
V
M m o go
m
Air_
m
,A
= -1
sm
-
(Chord-condensation)
a i
I
,I
hr
Minded
c
mi;
y
Apsalysis:
!IZ
-ProgreSSi
illi
I
IL
Aft
~
I
IF
1
r_
[~#r
92
-ftI
I
m
17.
d9
es
47
Fig. 13--Continued
Z-
~5
hi
_
1
15
1
B
MF9
WHin
4
Cm;it.4
tona~l cevtev-s:
x
MEApr&
x3S
31
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) considered 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)
domina.nt (measure twenty-one).
to the structural
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 prolohgation 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 center.
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
-s,.
JLLie
I-
Z I
A
Wpi-timri th tonal centers :
be gree- Proressian
.4t'
4
1-r.
1
/ A4
3/
2
F$.
*S'
I
IL
30~~1
~e
Fig. 14--Continued
Schenker AnaIysts:
B
I
I71#
j
IMP'
11
po"00
4-*-"*N11.
-IV
Afine emith tonai centev-s:
P35
lo4'
7
E
I
60%
.
C
---
wL
4.w w~...
-
.
am
wt
w
I"m
-
--
---
OF
rw
52
harmonic structure (I-V-I in the tonic) plus the second
structure (I-II-V7 -I in the dominant) reduce to a primordial structure of I-V.
An analysis of the Hindemith degree-progression produces the tonal spheres D, E, A, indicating an incomplete
progression, or "half-cadence," fromthe tonic to the dominant.
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 example 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 affecting 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 cantabile
1L19J
I
I
I
I
I w
-r r - f.A-r-T-I
I
r
I
I -
I
II
J j 0-,-*l
-
I
.
lb
I
_
I
n
_I _ _
I
IF"iA
11
To
UINw
_ ___ _ _ __ _ _
= ___ _
54
Fig. 15--Continued
IAhalysis:
&kenk.s
:1:o
is
B
-Jb
Aw- J6
;gh
t
a znz
r
)N
I
(1
I
a
i
.
....... ..... ....
I
1
-I-I
*1 ti c)h 6at-C h ir
1
55
concept, admitting no such chain of modulations, interprets
the first eighteen measures as a prolongation of one structural tonic chord, with the chromatic chords involved being
interpreted as neighbor-chords used as embellishing functions (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 extreme 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 thirtythree measures represent one chromatically ascending melodic
56
Fig. 16--Wagner,
Tristan and Isolde.
"Descend upon us,
Night
of Love"
from
Schenker Analysis:
Hmidei
th
Analysis:
DeIree-Progressiot
I
-
DL
+
Too
P
VN
4
U
A
7
IM.
~1~oj
-~T
*Neig3hbor- Ck*rJ
)
+(A
So33
Fig. 16--Continued
Se-henrkIer Ahalaysis:
8
)b
~
hb
..
ie
fe4
2
\
I-4Lt
I
/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).
of prolongation are used here.
Many types
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# (enharmonic with Cb), Eb, and Ab, which reduce further to Ab-Eb-Ab
resulting in a tonality of Ab (Fig. 16C).
Thus, in this
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 century 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 melodic 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 produces 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"
itI-,1 ;L
i
11
'or
-1
I
I.o
11nI
AoPI
I
I
-
A
Al0
t AP 40s
U'JAk
I
Dec .-
Progress ith
I
a
I
40
m~
#I
Opp
flp
IIOo
i
I
60
Fig. 17--Continued
Ii*
:Edo
At
ma~ 4 A
*--
A&
m 4
fil
-.N /.I% . 4.
*
.. jip
ja
t
-e
I~Fle4
I
L!K
At-t
1
-~
6I
&I 'II
b
_______
.1
zz
61
Fig. 17--Continued
.a
A.A
.
*
'1
.
I
-
a
OI-V
-
I
F
1
1111
I
'A
3thanker vAnialysis:
Hnelemith
__a
tonal
1T
If
III
I_
-1
CenterS :
62
Fig. 17--Continued
I
A
I
.0.01j%1
'
ii
Ao
19
AOP
-
&I
bj
r
Y
_
IN Ils
_
_
_
hip
_
j
F.
I
c
ba
S
EEóEFE411
tii____
ON ei *V,6or - ChovA
I
j
j
w
I I
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 introduction 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
cussed.
of contemporary music will be dis-
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 majorminor tonal system for a freer type of musical expression
characterized by extreme chromaticism, impressionistic devices, and other stylistic features which served to establish 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 tonalities
were interspersed with non-diatonic elements tending
to form opposition to the fundamental tone yet compelling the application of strong means in order to
verify the tonality, to paralyze eccentric effects.
. . . The art of music was never really in possession
of a tonality wholly limited to the seven diatonic
tones of the scale.
Recent developments in harmony have resulted in a further 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 produced, 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 conditions, but . . . LisJ the utilization of natural
possibilities; it is a product of art, a product of
the technique of art. Since tonality is no condition
imposed by nature, it is meaningless to insist on preserving it because of natural law. Whether, for artistic reasons, tonality must be retained depends on
whether it can be replaced.
Sc h onb erg 3 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 tonality of a work, i. e., beginning or ending a piece in some
key other than the tonic.
the dominant key (E Major).
The work is almost entirely in
According to the Schenker con-
cept the entire piece, with the exception of the final measure, represents a prolongation of a single dominant chord
(Fig. 18B), resulting in what might be considered an
21bid.,
pp. 299-300.
3 lbid. ,
pp. 289-291.
67
Fig. 18--Bartok, Piano Piece from Mikrokosmos (Vol. 6,
No. 150) (First section).
H; nemit
Deree
Anaiys s:
- Pra'pes1
_\_n
-
1
A
A
I
~ZZIZZZZIIZIII7i
-o------~-----~--~--
W
---1
I
68
Fig. 18--Continued
-
2
-I
.
-
-
WIN
J
-Iw
K---i
I .Il.I1 -
0
---
I I.
69
Fig. 18--Continued
Sc(enkeir Analysis:
Ab o M 1-.
.1
mVia 11
wa1
Hindemith tonai e cnters:
E
-q
ere
-----
c;
AA
X
70
incomplete basic structure (V-I).
more contrapuntal than harmonic.
The style of the work is
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.
cism of
t he
Schoenberg, who in reaction to the chromati-
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, although 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.
_M__Wi
__
72
Fig. 19--Schoenberg, Piano Piece, Op. 19, No.
3
Very slow
be~~I"
Hf nde th 6Analysis:
Degt-ee- Prosression
TonsAIty
b
_
r
V
VADJ
b
Sh
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 stability (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
Berceuses du Chat (Fig. 20A).
iegenied" from the
The group of short pieces
which comprise the Berceuses du Chat were written in 19151916 as a study for the dramatic work "Renard" which followed 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 Berceuses
du Chat for Voice and Three Clarinets.
(Reduction)
I
i-_t,
04F 7
I frftz zt
-A
zzTA f
I
U%-J-- bi 4--of 1 39; 1"
LK~~~~~TY
gL th4
iiz'r~~d
~~~-<
v
-1
F
kj7
..
I
J
I
I
-De.P-t r Analysis:
Dea
F
- -tryeSu
o
lmq
A-
aI;ty
I
*
7I
p
J9~
40
1
107
oil,
44-
a
11
-I-W
rRq--- T-
OF
III
ff 2
(I
.44
-W"-- A -10
i J-6 -WA-At- -
3F
MEq-
Ir
-- m
II
TII~IIIII7
i
mc
7
AF WJP*
I 4If
No
- -Al
iw
r---
-
Aoki-
r
0.
ACV
P7
VIP
It,0
I _K=_
I
Ab ja
lw
76
Fig. 20---Contirnued
Schefker AnaiUs;t
*--b.--
- -- am.zaw-
in em th tonl Centers.
105
-.
AN-6-
.rl
p.s
____*AM
*ALF
f
II
4O
7-*
c{
I
T
x
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 according 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 stylistic features peculiar to the music of the twentieth century.
Many of the problems involved in the analysis have
been discussed and in some instances more than one interpretation was given.
cOiAPTER
y
MMMY AND CONCLUSIONS
The preceding chapters have presented analyses of representative examples chosen from music beginning with the
late thirteenth century and ending with the twentieth century.
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 same 4
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 analysis 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.
79
This I-V-I progression
80
TABLE 5
FREQUENCY OF USE OF THE SCHENKER
BASIC HARMONIC PROGRESSIONS
Basic (or Primordial)
Harmonic Progression
Frequency
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 compositions 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 composition, 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
Music
Theory," The Music Review, X (February, 1949),to 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 present
that shadow of the past to which he would forever bind
music, he simply stands bewildered. The present then
seems like a sudden breaking away fro
the past, a
breaking 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
2 Ibid.,
p. 6.
3 lbid.,
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 presentday 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 speculative 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 progression, 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 derived 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 establishes 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 chordclassification, 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 tonality which, if it is not perceived by the ear, may not
actually exist.
BIBLIOGRAPHY
Books
Apel, Willi, French Secular Music of the Late Fourteenth
Century, 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,
E. C. Schirmer Music Co., 1933.
Salzer, Felix, Structural Hea
C. Boni, 192.
,
Boston,
Vol. I, New York,
Vincent, John N., The Diatonic Modes in Modern Music, New
York, Mills, by arrangement with~the University of
California 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," General
Semantics 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," Musical
Qarterl, XXX (January, 1944), 34-36.
86
Sessions, Roger,
"Heinrich Schenker's Contribution," Modern
musL, XII (May-June, 1935), 170-178.