THE USE OF THE HARMONIC TRlTONE IN
SELECTED PASSAGES FROM THE MUSKZ OF
REPRESENTATIVE CONTEMPORARY COMPOSERS

Thesis for ”‘39 fiegree of Ph. D.
MICHIGAN STATE UNEVERSITY

Woodrow James
1956

   
   
 

  

LIBRAR Y
Michigan State

This is to certify that the

thesis entitled

THE USE OF THE HARMONIC TRI'I‘ONE
IN SELECTED PASSAGES FROM THE MUSIC
OF REPRESENTATIVE CONTEMPORARY CDMPOSERS

presented by

Woodrow James

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Music

@fié/[2/M/X; é’i‘umfl/

Major professor

Date May it, 1966

0.169

 

ABSTRACT

THE USE OF THE HARMONIC TRITONE
IN SELECTED PASSAGES FROM THE MUSIC
OF REPRESENTATIVE CONTEMPORARY’COMPOSERS

By Woodrow James
Major Professor: Dr. Merrell Sherburn

All theorists are aware of the uniqueness of the
tritone in the family of intervals. Indeed, the term "tri-
tone" is itself unique, since no other interval has a gen-
erally accepted special name. In this treatise the term is
defined as "the interval of the augmented fourth or the di-
minished fifth."

The work of musicologists has made it evident that
from the earliest periods of civilized music until the pres-
ent century, composers and theorists have imposed certain
restrictions on the use of the tritone.

The purpose of this study.was to attempt to discov-
er how the composers of the music of today have dealt with
the harmonic tritone. The following questions were posed:
Is there any consistency in the use of the tritone among im-
portant twentieth-century composers, even among those of
widely divergent "schools"? Do contemporary composers tend
to avoid this interval because of its strong tonal implica-
tions, or do they simply ignore those implications? How is
the harmonic tritone approached and left by twentieth-century

composers? What kinds of structures contain tritones?

Woodrow James

Before an attempt was made to answer the above
questions, two chapters of background material were present-
ed. Chapter I (Introduction) is a presentation of acousti-
cal and theoretical data. Chapter II (Historical Survey)
attempts to provide the theorist with a compendium of infor-
mation regarding the traditional treatment of the tritone.

In order to deal with the problems set forth in
the treatise, the author devised a statistical method of
investigation, stated in Chapter III (Method of Analysis).

Chapter IV (Analysis of Two Transitional Works)
applies the method of analysis to passages from two works
which are generally agreed to have had a decided impact on
the harmony of New Music. This is done for purposes of com-
parison with the analyses which follow.

Chapter V (Analysis of Contemporary Works) repre-
sents the main part of the treatise. Analyzed from the
standpoint of harmonic tritone usage are passages from works
of Schoenberg, Stravinsky, Bartok, Berg, Hindemith, Britten,
Villa-Lobos, Harris, Prokofiev, Barber and Copland.

A realistic evaluation of the analyses of the pas-
sages from the works of the above-mentioned composers should
include comparison with similar analyses of highly tradi-
tional works. For this purpose, it would be difficult to

find examples which represent the epitome of "correct" har-

Woodrow Jame s

monic usage more admirably than the chorale harmonizations
of J. 8. Each. For purposes of comparison, the method of
analysis was applied to six chorales. The percentages thus
derived were compared with those from the two transitional
passages and with those from the contemporary passages.
Graphs are included to illustrate the various aspects of
tritone treatment.

While there seems to be little consistency in.the
use of the harmonic tritone among twentieth-century compos-
ers, these composers have one thing in common: a decided
tendency to avoid traditional usage of the interval. In
traditional music, the tendency of the tritone was to re-
solve stepwise by contrary motion; contemporary composers
prefer resolution by Oblique motion. The most striking dif-
ference between the twentieth-century tritone and that of
the Bach chorale lies in the harmonic structure in.which it»
occurs. While the harmonic tritones of traditional music
occur in dominant-type tertian sonorities, those in contem-
porary music are more often found in harmonies selected from
the expanded tonal palette of today.

This treatise was intended to be a study of one-as-
pect of contemporary compositional technique. Certain as-
pects of the study indicate the need for further research,
such as a) the use of the melodic tritone in contemporary

music; b) the use of harmonic dissonances in contemporary

Woodrow James

music; and c) the use of non-harmonic dissonances in contem-

porary music.

THE USE OF THE HARMONIC TRITONE
IN SELECTED PASSAGES FROM THE MUSIC
OF REPRESENTATIVE CONTEMPORARY COMPOSERS

By ,;\
00'
‘Woodrow/Cames

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Music

1966

PREFACE

The tritone has remained unique among the musi-
cal intervals down through the ages, and there are times
when it must appear to the composer and to the theorist as
a strange combination of beauty and unruliness. However,
every serious musician, from the most radical of the EXEEE’
'gggge to the staunchest reactionary, will agree that music
has undergone such drastic changes in the present century
that it is difficult to say to what extent the tritone in
twentieth-century music retains the heritage of its tradi-
tional counterpart.

It is the purpose of this study to try to discover
what vestiges of the traditional treatment of the tritone
remain in the music of the present day, and to answer a
number of questions regarding twentieth-century treatment
of this most interesting interva1--questions such as: Is
there any consistency in the use of the tritone among im-
portant twentieth-century composers, even among those of
widely divergent "schools"? Do contemporary composers tend
to avoid this interval because of its strong tonal implica-
tions, or do they simply ignore those implications? How is
the harmonic tritone approached and left by twentieth-‘
century composers? What kind of structures contain tri-
tones? Do the structures bear enough rhythmic importance
to be considered primarily harmonic or are they merely co-
incident?

ii

It seems advisable in a study of this nature to be
as objective as possible. The validity which is essential
for comparisons and eventual generalizations must come from

statistical evidence. An aesthetically oriented study

uld be interesting, perhaps, but it would hardly suffice

C)

w
to solve the problems which the author has undertaken. It
is admitted, of course, and it need hardly be said, that an
objective study about an essentially non-objective art such
as music has its own inherent weaknesses.

Considering the various choices of procedure which
are available for this study, it seems wisest to approach
the analysis solely from the standpoint of the harmonic
tritone, which does not resist objective treatment to the
extent that the melodic tritone does.

This is not to say that the melodic tritone is in-

[.10

s gnificant, or that the method of its usage in the work of
a particular composer is not a reflection of his musical
personality as much as is that of the harmonic tritone;

the author maintains only that the melodic tritone does not
seem to lend itself to the objective method of analysis
which is adopted in this study, nor indeed, at the present
time (as far as the author can determine), to gay objective
analytical procedure. For example, consider this question:
what is a melodic tritone? Would it be realistic to con-

sider only direct tritones, eliminating "filled-in" tritones

from consideration? If the latter are considered, how much

He
[—10
He

filling in would cancel the melodic tritone effect? If a
melody is broken up between two instruments, would those
melodic tritones which occurred be analyzed as such? How
would one establish the limits of such analysis?

The above represents only the beginning of the di-
lemma of the melodic tritone. Needless to say, the author
of this work spent many hours attempting to achieve an ob-
jective analytical procedure and was forced to give up in
frustration. However, perhaps the present work will evoke
a closer examination of the problem in the music of the
twentieth century; perhaps it will pave the way for an
eventual objective analysis.

The author wishes to express his sincere thanks
for their advice and assistance to the members of his doc-
toral committee: Dr. Merrell Sherburn, Dr. H. Owen Reed,
Dr. Edgar Kirk, Dr. J. Murray Barbour, Dr. Paul Harder, Dr.
Walter Hodgson and.Mr. Byron Autrey. He wishes especially
to thank Dr. Sherburn, who served as advisor for the dis-
sertation, giving freely of his time in the reading and
correcting of the manuscript, and Dr. Reed, whose advice in
many hours of discussion with the author concerning the

project has proved invaluable.

iv

TABLE OF CONTENTS

PREACE O O O O O O O O O O O O O O O O O O O O O 0
LIST OF GRAPES O O O O O O O O O O O O O O O O O 0

Chapter
I. INTRODUCTION . . . . . . . . . . . . . . .

Tritone Defined
Peculiarities of the Interval

II C HISTm ICAL SURVEY O O O O O O O C O O C O O

The Tritone in Greek Music

The Pro-Polyphonic Period

The Early Polyphonic Period

The Sixteenth and Seventeenth Centuries
The Early Eighteenth Century

The Classical Period

The Nineteenth Century

The Impressionistic Period

Conclusion

III. METHOD OF ANALYSIS . . . . . . . . . . . .
IV. ANALYSIS OF TWO TRANSITIONAL‘WORKS . . . .
Wagner

Debussy
Comparison of Wagner and Debussy Excerpts

Page
ii
vii

16

63
77

Chapter Page
V. ANALYSIS OF CONTEMPORARY WORKS . . . . . . . 8h

Schoenberg
Stravinsky
Bartok
Berg
Hindemith
Britten
Villa-Lobos
Harris
Prokofiev
Barber
Copland

VI. CQ’IPARISONS AND CLARIFICATION OF THE DATA . . 171

The Use of the Harmonic Tritone in the Bach
Chorale

Comparisons

Index to Hanson Analyses

Conclusion

BIBLI OGRAPM O O O O O O O O O O O O O O O O O O O O 253

vi

Graph
1.

la.

28.

3.
3a.

6.
6a.
7.
7a.

LIST OF GRAPES

Percentages of Tritone Units . . .

Degree of Deviation in Percentages of Tritone

Units e>e e e e e e e e e e e e

Percentages of Tritone Structures which are

Traditional 0 e e e e e e e e 0

Degree of Deviation in Percentages of Tradi-

tional Tritone Structures . . .

Percentages of Accented Tritones . . . . . .

Degree of Deviation in Percentages of Accent-

ed Tritones . . . . . . . . . .

Percentages of Tritones Having a Duration of

Less than One Unit . . . . . . .

Degree of Deviation in Percentages of Tri-
tones having a Duration of Less than One

Unit 0 e e e e e e e e e e e e e
Oblique-Step Approach 0 e e e e e
Oblique-Skip ApprOBCh e e e e e 0

Degree of Deviation in Percentages
step and Oblique-skip Approach .

Departure by Oblique Motion . . .
Degree of Deviation . . . . . . .
Oblique-step Departure . . . . . .

Oblique-skip Departure . . . . . .

vii

of Oblique-

Page
178

180
182

18h
185

187

188

189
192
19h

196
198
200
202
20h

Graph
7b.

8a.

9.
9a.

10.
10a.

11.
lla.

12.
128. e

130
13a.

1h.
lha.

by Rest . . .

Departure by Rest

Approach by Best . . . .

Degree of Deviation in Percentages
step and Oblique-skip Departure

Degree of Deviation in Percentages

Degree of Deviation in Percentages

twebyROSteeeeeeeeee

trary Approach 0 e e e

lar Approach .

lar Departure

lel Approach .

viii

Approach by Contrary Motion

Departure by Similar Motion

Approach by Parallel Motion

Degree of Deviation in Percentages

Departure by Contrary Motion . . .

Degree of Deviation in Percentages
trary Departure

Approach by Similar Motion . . . .

Degree of Deviation in Percentages

Degree of Deviation in Percentages

Degree of Deviation in Percentages

Page
of Oblique -
O O O O O 206
O O O O O 208
of Approach
0 O O O O 210
O O O O O 212
of Depar-
O O O O O 21’...
O O O l O 216
of Con-
. O O O O 218
O C O O O 220
of Con-
. O O O O 222
e e e e e 22h.
of Simi-
O O O O O 226
. . . . . 228
of Simi-
e e e e e 230
O O O O O 232
of Paral-
. . . . . 23h

Graph Page
15. Departure by Parallel Motion . . . . . . . . 236

15a. Degree of Deviation in Percentages of Paral-
lel Departure . . . . . . . . . . . . . . 238

16. Average Degree of Deviation (by Composer) . 2&0

ix

Chapter I
INTRODUCTION
TRITONE DEFINED

In present usage the term tritone refers to any
harmonic or melodic augmented fourth or diminished fifth.
This was not the case in the past, and indeed, some modern
writers continue to use the term only in reference to the
augmented fourth, referring to the diminished fifth as "the
inversion of the tritone." The practice in this treatise
will be to employ the term in reference to either the aug-
mented fourth or the diminished fifth.

It is advisable at this point, however, to acquaint
the reader with the various terms which have been employed
in referring to this interval in order to ward off any con-
fusion which may result from their being used in quoted
passages in the text. It may readily be seen that the author
of this treatise takes issue, in the above definition, with
the writer of the passage quoted below, who is a trifle
over-zealous. The quotation, however, will serve to clarify
terms.

The tritone, ratio h5:32, is the interval be-

tween the subdominant and the leading note. It is
therefore a diatonic interval and is equal to the
sum of two major tones and a minor tone. It is
equal to a fifth less a diatonic semitone.

The inversion of the tritone, ratio 6h:h§, is

a diatonic interval variously named. Its old name

was quinta falsa. Morley called it the un erfect
fifth. Robert Smith called it the minor fifth (he

-1-

-2-

also called the tritone the major fourth). Some

modern writers, following Morley, call it the imper-

fect fifth. . . . Many writers call it a diminished

fifth (and the tritone an augmented fourth). Others

confine both these names to chromatic intervals,

which is certainly more consistent and logical. As

guinta falsg is a diatonic interval it is a pity

that it has no generally accepted name of its own.1

The reference, in the above passage, to "two major
tones and a minor tone" relates to the intervals of just in-
tonation (major tones equal 203, 20h cents; minor tone
equals 183 cents). The minor tone is the relatively small
whole tone between G and A, while the major tones lie between
F and G and between A and B. Obviously, this difference does
not exist in equal temperament, where every tone has a value
of 200 cents. Therefore, the just tritone F-B equals 590
cents, while the equal-tempered version equals 600 cents.
It may easily be seen, then, that in the former tuning,
there was a real difference between the augmented fourth and
the diminished fifth, while in the latter, which is univer-
sally employed today, no acoustical difference exists. This
represents another argument to support the broader usage of
the term "tritone."
In this treatise, then, the term "tritone" will be

defined as "the interval of the augmented fourth or the di-

minished fifth."

 

1§gpve's Dictionary of Music and Musicians, ed.
Eric Blom (9 vols.: London: Macmillan Co., I§§H), IV, 521.

-3-
PECULIARITIES OF THE INTERVAL

The tritone (in equal temperament) divides the oc-
tave into equal parts, and is, therefore, according to
Hindemith,2 the only interval in our system, apart from the
octave, to which a root may not be assigned.

Because of its division of the octave into halves,
the augmented fourth is the only interval which is identical
in size with (iég,, has the same number of cents as) its in-
version (the diminished fifth). The notated difference
takes into account the traditional difference in function.

The difference in function can be represented by
examples of resolution. Thus:

Ex. 1

D

 

 

 

 

 

 

\

of)

 

 

 

is in the key of C and resolves thus:

Ex. 2

 

 

2Paul Hindemith, The Craft of Musical Composition

(2 vols.; New York: Associated Music PubliEhers, Inc.,
19hl), I, 81.

-u-

If by a change of notation it is written as a dim-
inished fifth:

Ex. 3

 

it is in the key of F# and resolves thus:

Ex. h

 

The tritone or its inversion occurs most often as
part of a dominant seventh chord, or of the chord
on the leading note, e.g.:

Ex. 5

 

in the key of C.3

 

3J. A. Westrup and F. Ll. Harrison, The New College

Encyclopedia of Music (New York: W. W. Norton, 19607, 676.

-5-

The tritone has been traditionally regarded as one
of the more dissonant intervals, especially if one of its
tones is in the bass. R. 0. Morris states concisely the
rules for classification of consonance and dissonance:

(l) The intervals of the second and the seventh are
dissonant, and require preparation, except when they
occur merely as passing notes. (2) The interval of
the fourth is also a dissonance when it occurs be-
tween one of the upper parts and the bass. But oc-
curring between upper parts it is a consonance, if
it is supported by the bass--i. e. as we should say,
if it constitutes the upper part of a S or 6 chord.
3
The same is true of the augmented fourth and the
diminished fifth. (3) The other intervals are con-
sonant and require no preparation. But consecutive
fifths and octaves are forbidden.”

Curiously, however, Roger North (c.16Sl-l73h) seems
to have regarded the tritone as a consonance:

But having secured "one kclear sounding tone," it
may be given "the name ofa key " and on it may be
built the full accord . . . which Nature makes," con-
sisting of keynote, 3rd, 5th and 8th; and then the
other degrees of the scale, defined by simple concor-
dances . . . .

Ex. 6

 

 

"R. 0. Morris, "Harmony, " Grove' 3 Dictionary of
Music and Musicians (3rd edition), ed. H. C. Colles, II
(1935) s 859.

-6-

Attention is drawn to the two possible 3rds--"one is
called the sharp and the other the flat third"; and
also to the use of the "flat seventh," which "ac-
cords well with the other accords of the key (note)
and makes a combination of thirds which must needs
doe well in harmony." If we drop the root and 5th,
"the remaining 3rd and 7th are to each other a tri-
tone, and that consonance is a beauty in many states
of harmony, and well knowne to the practisers."

Such statements are, however, the exception rather

than the rule.

We can easily understand how the tritone has in
all periods of music history held its unique posi-
tion among the intervals. Instrumental music has
arrived at a modus vivendi with it, aided more or
less by the mechanization of its method of determ-
ining the pitches of the tones. But to the singer,
especially the choral singer, it is still loathsome.
Musical theory has always been at odds with the
"diabolus in musica," and has always treated it.with
a peculiar mixture of love and hatred. Theorists at
first tried to get around it. The Greeks avoided it
by the interpolation of a complementary tetrachord
(synemmenon) among their four regular ones. In the
church modes the device used was the substitution of
B rotundum for B quadratum. The rules for organum
and descant excluded the tritone, and its revenge
was that this exclusion prevented them from prosper-
ing. Then a settlement was made: the treatises of
mediaeval theorists are an endless chain of attempts
to accomodate the "mi contra fa"; solmization is
the attempt to take in the unwelcome guest with im-
punity. Finally, the tritone became the pet of har-
mony, through the outstanding importance given to
all chord formations serving as dominants, through
the harmony of Tristan and the chromaticism that
followed in its wake, and even through such flimsily
based devices as the whole-tone system that flour-
ished about the turn of the last century. For us,
who have now learned the position of the tritone
within the family of intervals, and the grounds of
its claim to that position, it has lost its terrors.

Yet even for us it remains a civilized demon--"der

 

 

SJohn Wilson,_ngez_ugzth_gn_Mnsic (London: Novello
and Company, 1959), p.’72.

-7-
Geist, gar stets verneint": the spirit that ever
denies.

Theorists are at odds as to whether the tritone
occurs in the first sixteen partials of the harmonic series.
The interval ratios 5:7 or 7:10 are at variance with either
the just or the equal-tempered tritones, but the disagree-
ment lies in a query which has subjective ramifications:
does the ear compensate for the "out-of—tuneness" of the
"impure" just and equal tritones, relating them to the
acoustically pure 5:? or 7:10? Or is it the harmonic series
which has gone wrong at this point, so that we must there-
fore judge 5:7 or 7:10 so "out-of—tune" as to be unaccep-
table?

Richard Bobbitt states quite flatly that the tri-
tone is not found in the first sixteen partials:

Special mention should be made of the so—

called "tritone" which derives its name from the
fact that it contains three "whole-tones." This
interval, proscribed in early polyphonic music as
diabolus in musica (the devil in music), remains
even today a ferment among the family of intervals
due to its peculiar characteristics, the following
being the most outstanding:

(l) The intervallic ratio for the tritone does not
occur in the first sixteen partials of the harmonic
series as do the other interval ratios. The pro-
portions 5:7 or 7:10, contrary to popular opinion
(Hindemith, Craft, 81-2), do not evoke the interval
response we know as the "tritone effect." These
proportions, when sounded in pure tone by oscilla-

tors, produce the effect which is unlike any inter-
val among the first sixteen harmonics, but observers

 

6Hindemith, I, 83-8h.

-3-

who have actually heard such an experiment in the
laboratory are aware that the aural response is not
that of the tritone. A true tritone of three whole-
steps is achieved by dividing the octave in two
equal parts--i.e. 600 cents or the ratio 70:99.

(2) It is the only interval without an inversion.
When the frequency of its lower tone is doubled,
the interval response remains exactly the same--
g;g; 70:99=l.hlh; 99:1h0=l.hlh. The density scale
illustrates the extreme complexity of the tritone;
its symmetry accounts for its ambiguity. .
(3) It is the only high-density interval essential-
ly free from beating. Seconds and sevenths, in.
certain registers and at certain decibel levels,
beat violently; but . . . beating has nothing to
do with the density phenomenon. ‘ '-

The relative density of a musical interval is de-
termined by the number of intersections per unit between the
frequency waves forming the interval. -A low-density inter-
val would have few intersections per given unit of time; a
high-density interval would have many.

Bobbitt goes on to classify intervals according

to relative density:

Ex. 7

LOW DENSITY MEDIUM DENSITY HIGH DENSITY SPECIAL CASE
(Group 1) (Group 2) (Group 3).. - (Group A)

 

 

 

7Richard Bobbitt, "The Physical Basis of Inter-
vallic Quality and its Application to the Problem of Dis-
sonance," Journal of Music Theory, III (November, 1959),
190-2. _, -

The passage from Hindemith to which Bobbitt refers
is quoted below:
The tritone has no root. It is accompanied by
combination tones that stand in an unusual relation
to it.

Ex. 8

small (5:7) large (7:10)

 

When its two tones are in their closest position
(5:7), the combination tones form a fifth which
combines with the tritone to make a seventh-chord,
in which the lower tone of the tritone is the third,
and its upper tone (although too low) the seventh.
In the opposite, widest position (7:10) the combi-
nation tones form a fourth. The latter combines
with the tritone to form a four-three chord, in
which the lower tone of the tritone is the seventh,
and the upper the third of the chord. All the tri-
tone intervals that lie between these two extremes
produce seventh chords which are between the two
given above. Consequently, the tritone always has
a dominant effect. It is characterized by a ten-
dency towards a tonic, a tendency most naturally
satisfied by a progression which takes the form of
a "resolution" to the progenitor tone of its family
(complemented by one or more tones which form with
it either an interval or a chord). But already we
see the dual nature of the tritone: if the preced-
ing interval-successions have not made the relation-
ship to a progenitor clear, one has the choice be-
tween two equally good resolutions. And in the res-
olution, the ear always hears one of the tones of
the tritone as a leading tone to the root of the
following tonic chord:

Exo 9

 

But since the ear cannot at once decide which of
the tones of a tritone heard without clear family
relations is the leading tone, it is always uncer-
tain in its reaction to this interval. On the one
hand the tonal uncertainty of the tritone, which
makes it vaguer and more opalescent than any other
interval, and on the other its strong urge for res-
olution, which at the moment of progression monOp-
olizes the attention-«this combination of indefi-
niteness and tension is what distinguishes the tri-
tone, and makes it a foreign body and a ferment
among the intervals.8

In employing the ratios 5:7 and 7:10 (derived from
the interval formed by partials five, seven, and ten of the
harmonic series), Hindemith appears to be contradicting him-
self, for he had flatly denied earlier that the seventh par-
tial was usable:

The seventh overtone of C, -»b1'(hh8), cannot be

used. If we attempted to apply the same procedures
to it as to its predecessors, we should arrive at
terrifying results.9

Other theorists, however, do not hesitate to con-

sider all of the first sixteen (or more) harmonics as quite

tasable, although it is obvious, they imply, that one must

8Hindemith, I, 81-2.
9Ibid., p. 37.

-11-

"correct nature" by tuning up the much too low seventh and
eleventh partials. (The fourteenth as well must be correct-
ed, since it is the octave duplication of the seventh).

The first six of these seven sounds are all
"firm" and "stable," the tension between the two
notes being the same in each case (the tension be-
tween notes two and three of the harmonic series),
"unstable," the fifth being imperfect, and not cor-
responding to any of the simple tensions of the
harmonic series. This interval, the diminished
fifth, was proscribed by medieval theorists as gig-
bolus in_musicg (the devil in music) because of its
"flawed" sound. And when it was sung, the B was
flattened (the procedure was called musipa ficta) so
as to obtain once more the firm sound of the perfect
fifth:

 

Ex. 10

 

There is clearly a direct connection between
this procedure and the fact that the B-natural of
the harmonic series (note 15) is outbalanced by the
B-flat (notes 7 and 1h, and especially note 7, which
can be picked up by a trained ear).

It was, of course, always possible to solve the
diminished fifth problem in the opposite way--by
sharpening the F:

Ex. 11

 

Now this too corresponds with a peculiarity of the
harmonic series. Note 11 is not really F at all,

-12..

but between F and F-sharp, and slightly nearer to
F-sharo. This note occurs an octave higher (as

note 22), together with a nearly-true F-sharp (note
23); behaving, in fact, rather like the "out-of-
tune" B-flat (note 7) which appears an octave hi h-
er (note lu), together with a B-natural (note 15%.
Obviously, the incompatibility of B and F as a har-
monic interval derives from the ambiguity surround-
ing the B-flat, B-natural, F, and F-sharp of the
harmonic series; in other words, the "flaw" in our
harmonic system is a result of the "flaw" in the
harmonic series.

The phrase diabolus in musica has accumulated a

 

number of connotations among theorists and composers. At-
Kisson claims that "early composers of modal melodies had
such great difficulty in avoiding the tritone (augmented
fourth) between F and B that it became known as the diabolus
in musics, or the devil in music."11
Thus the above author seems to feel that the phrase
arises simply because the interval is difficult to avoid.
With a slightly different shading of meaning, Eric Blom im-
plies that the phrase originates because the interval was
§£L22_avoided.
Diabolus in Musica (Lat., the Devil in Music).
a medieval term in which the tritone (augmented
fourth or diminished fifth) was denounced by theor-
ists as an interval to be avoided in composition.

Its simultaneous use at a point where two polyphonic
parts meet vertically was forbidden, and though as a

 

loDeryck Cooke, The Language_g£_Music (London:
Oxford University Press, 1959), h34h.

 

llHarold AtKisson, Basic Counterpoint (New York:
McGraw-Hill, 1956), 6.

-13-

melodic progression it was regarded as less perni-
cious, it was looked upon with disfavor even so.

According to Cooke, medieval composers avoided the
interval because of its diabolical implications. "The old

Catholic composers naturally shunned the devil and all his

ways.1113

Later, however, the interval was used in dramatic
and programmatic contexts deliberately to depict the devil,
evil or diabolical persons, objects, and situations.

The augmented fourth and the diminished fifth
are, of course, the same note; and this note is the
same as the sharp fourth, but treated in a different
way--as a note on its own, unconnected with the ma-
jor and minor triads or scales, but simply related
to the tonic. Its relationship to the tonic is that
particular tension which we have mentioned earlier
as embodying the "flaws" in the harmonic series, and
in the whole musical scheme of things-~diabolus in
musica. The interval . . . appears as the tension
between the major seventh and normal fourth of the
major scale (B and F in the key of C); but . . .
these notes were always carefully integrated into
the system of tonal tensions as 'inessential" notes
resolving on to the tonic and the major third (C and
E)--the 'flaw" was always corrected. When composers
have wished to isolate this tension, they have usu-
ally done so by taking the interval between the ton-
ic and sharp fourth: this should normally act as a
modulation to the dominant . . . but when it is ex-
posed without any resolution of any kind, and be-
comes an "essential" note, a tension in its own
right, it becomes diabolus in musics indeed, for it
acts as a "flaw" Which destroys the integrity of the
tonic key-~thus removing the music outside the cat-
egories of human joy and sorrow inherent in the ma-
jor and minor systems. Diabolus in musics; the flaw

 

12Grove's (l95h). II, 6&3-
13Cooke, 86.

-1u-

in the scheme of things; it is hardly surprising
that composers should have used it to embody "Old
Nick" himself, his deputies, substitutes, and in-
fluences.1

This concludes the study of the various terms which
have been employed by different authors in referring to the
tritone and of the peculiarities and acoustical data perti-
nent to the interval.

The tritone is unique among the family of intervals
in that it divides the octave into two equal parts. It is
therefore identical in size with its inversion. It is tra-
ditionally regarded as one of the more dissonant intervals,
and to it is attributed a decided tendency to resolve. The
supernatural significance with which some composers and the-
orists have imbued the interval has been (rightly or wrongly)
bound up with the fact that the interval is introduced into
the harmonic series by that partial which bears the mystic
number sgggg. Theorists disagree as to whether this inter-
val is well enough "in tune" to be recognized as a tritone.
Because of the uniqueness of the tritone, it has easily been
given more nicknames than any other interval. Among these

are: "guinta falsa," "unperfect fifth," "minor fifth,"

"major fourth," "imperfect fifth," and "diabolus in musica."

 

Of special significance also is the term tritone, which is

the only interval name in common usage today that is employed

 

l"Cooke, 8h.

-15-

in addition to those terms (augmented fourth and diminished

fifth) which express the number of the interval.

Chapter II
HISTORICAL SURVEY
THE TRITONE IN GREEK MUSIC

There were tritones in the modes that were taken
over from the Greeks. According to the author of the fol-
lowing passage, there were two defective fifths and one de-
fective fourth.

These defective fifths . . . (B-F) were each
made up of two major tones and two limmas (Tythag-
orean semitones, 256/2h3] and thus were only 568,
whereas a perfect fifth is 702 if an octave is 1200.
There was also a defective fourth, F-B, consisting
of three major tones or 612, so that this fourth was
bigger than those fifths. When music was in unison
those intervals could be avoided, but not when in-
struments and voices had to be kept a fifth apart.15

THE PRE-POLYPHONIC PERIOD

During the Pro-polyphonic period, the practice of

musica ficta was occasioned in part by the desire to avoid

 

the tritone.

The Gregorian choral is fundamentally diaton-
ic; chromatic alterations, which play so prominent
a part in modern music, do not occur, with one
single exception: The B is often changed to Bb,
e. ., in the Lydian and the Dorian tonalities.

n t e former, the augmented fourth from F to B

 

15Cecil Torr, "Greek Music," Oxford Histor of
Music, ed. Sir W. H. Hadow (London: Humphrey Mil ord, 1929),
IA, 50. ‘

 

-16-

-17-

Ex. 12

 

was felt to be difficult for the singer, and the
lowering of the B must readily have suggested it-
self. Even today the leap from F to B is excluded
from a good vocal style and is employed only for
characterization. Every friend of music knows it
from the "Song of the Dragon" in R. Wagner's Sieg-
fried. One can readily understand that in the
Middle Ages, when especial attention was paid to
the vocal quality of music, the attempt was made to
avoid this interval by lowering the B to Bb. The
interval is commonly known as the Tritonus, i.e.,
the interval consisting of three whole tones. Its
inversion, the so-called Quinta falsa, or dimin-
ished fifth, gas also forbidden in pure ecclesias-
tical music.1

 

In his exposition of the rules for writing Greg-
orian melodies, Merritt gives the following instructions:

Great care must be exercised at all times in
the treatment of the augmented fourth, the tritone,
which occurs between the notes F and B. The direct
leap upwards or downwards over this interval is
never to be found. The nearest approach to it is
the figure:

Ex. 13

 

 

16Carl Nef, An Outline of the History of Music
(New York: Columbia University Press, 1938). 21.

 

-18-

which does occur sometimes. However, this motive
is never used in a cadence or at the beginning of
a phrase. When it is used in the course of a phrase
the B and the F should not be exposed at both the
upper and lower points of the melodic line, as in

Ex. 1h (3)

/”

   
 

OI’

Ex. lh (b)

 

but should be sheathed, at least partly, as in the
phrase "Christe" in the Kyrie I of the Lux et origo
Mass:

 

Ex. 15

 

or in the phrase of the Kyrie I of the Kyrie fons
bonitatis Mass, where the interval is used in a
scale passage:

 

 

-19-

Ex. 16

 

In both cases B is preceded by the C above and is
thus not exposed.17

To avoid the interval he suggests the occasional

use of B-flat.

The flat is the only accidental ever found in
plainsong melodies, and it is used only to alter
the note B. This is attributable in the first place
to the tritonal relationship between F and B, an
interval which has always caused embarrassment in
melodic lines; when B is flatted the two notes can
proceed one to the other perfectly safely. In the
second place, it is often much easier to sing a
minor second than a major one. This becomes obvious
in such figures as

Ex. 17

 

 

 

 

 

 

 

 

'x
O

 

 

 

 

where the melody rises from D to A, goes a second
above, and sinks again to A. After a rise of a
fifth the difference between a major second formed
by the notes A, B, A and a minor second formed by
the notes A, Bb, A is very appreciable. And When
this figure occurs-~as it so often does, particular-
ly in the Dorian mode--the B is practically always
flatted.

 

17Arthur Tillman Merritt, Sixteenth-Centupy-

Polyphony (Cambridge, Massachusetts: Harvard University
Press, lane), 12-13.

 

-20-

There is not usually much occasion to make
use of the Bb in the Phrygian modes, particular-
ly in figures which rise from the E itself. How-
ever, the figure DbA-Bb-A discussed in the para-
graph above is not rare in the Phrygian, and when
it is used the B is ordinarily flatted.

In the Lydian modes the B is very often flat-
ted to avoid the tritone between the final and
the fourth above it. In a great number of Lydian
chants this alteration takes place constantly,
and as a result the chants have the flavor of the
major mode. (For a good examplg of this, see the
Kyrie of the De angelis Mass.)l

 

In the following statement, Alec Harmon strength-
ens Merritt's comments regarding the frequency of B-flat in
the Dorian and Lydian modes: "By far the commonest modes
to employ Bb in Gregorian chant are the Dorian and Lydian,
the former because in the frequent and important melodic
progression A-B-A, which hinges on the dominant of the mode,
it was found that Bb was easier to sing and sounded more
graceful than BA, and the latter because the augmented
fourth or tritone F-B is avoided."19

One cannot be sure, however, that the tritone was
not used with considerably more frequency than is implied
above. Present-day attempts to reproduce Gregorian chant
as it was sung in the medieval services are necessarily
highly speculative. Furthermore:

The tritone was forbidden as a harmonic in-

terval as early as the ninth century. It is there-
after frowned upon as a melodic interval. . . .

 

181b1d., 17.

19Alec Harman, Man and His Music (London: Rock-
11rr, 1958), 138.

-21-

But, despite the opportunity to "edit" tritones
out of Gregorian Chant during the centuries of its
existence, instances of the indirect (i.e. filled-
in) tritone are even now not infrequent.20

Reese further takes issue with commonly accepted

views regarding the use of B-flat:

The frequent appearance of B-flat (B rotundum)
in addition to B-natural (B guadpum or qpadratum)--
the two, however, do not occur in immediate succes-
sion in plainsong--constitutes the element most
disturbing to the symmetry and stability which the
early medieval theorists apparently sought to es-
tablish in their modal system. The avoidance of
the tritone is frequently given by later theorists
as a reason for introducing B—flat. But there are
two other factors that may explain more cogently
this sole example of notated chromaticism in medi-
eval theory: transposition and an underlying pen-
tatonic structure.2

 

 

In the construction of tetrachords and pentachords,
however, Reese acknowledges the fact that theorists avoided
the use of the tritone. "The admissible pentachords were
T S T T, S T T T, T T T S, and T T S T; the admissible tet-
rachords were T S T, S T T, and T T S; the diminished fifth
and augmented fourth were inadmissible species."22

The octave eventually replaced tetrachords and
pentachords as the unit for scale structure, owing partly
to the influence of the tritone.

That the substitution of B-flat for B-natural
changed the character of a mode--in fact, changed

 

2OGustave Reese, Music in the Middle Ages (New
York: W. W. Norton and Company, 19h2), 157, n. 31.

21Ib1d., 157.

22 id., 156.

-22-
it into another mode--was not widely recognized
until the 16th century. Thus, in the Middle Ages
the fifth mode with B-flat was regarded as a form
of the tritus tonality, not as major, and the
first mode with B-flat as a form of the protus
tonality, not as natural minor. But the struggle
between B-flat and B-natural, on the one hand, and
the theorists' bugbear, the tritone, on the other,
was bound in the course of time to direct attention
to the transformation that B-flat effected in the
basic scale structure and thus to cause the posi-
tion of the octave as the unit by which modality
was judged to outrank definitively that of the
smaller units.

During the period of early Christian monody, the
practice of musica ficta was occasioned in part by the
desire to avoid the tritone. Although it would seem that
the task of composing without the complications of harmony
or polyphony would be a simple one, those rules which were
designed for the purpose of excluding the tritone were many
and varied. While it has been established that direct me-
lodic tritones were either extremely rare or avoided al-
together, it seems highly probable that a considerable num-
ber of indirect (i.e. filled-in) tritones were "edited" out

of the Gregorian melodies which continued to be used in the

Church during the succeeding centuries.

THE EARLY POLYPHONIC PERIOD

During the early polyphonic period, the desire to

avoid the tritone on the harmonic as well as the melodic

 

23Ibid., 161.

-23-
plane occasioned many and varied restrictions and prohibi-
tions. In the example quoted below (from Reese), tritones
occur as harmonic intervals at the two places marked+-.

Ex. 18 Simple Organum at the Fourth-~Musica Enchiriadis.

 

"Thou art the everlasting Son of the Father."
(From the Te Deum.) '

The author of the Musica Enchiriadis says of
his example that "the voices will be perceived
s sounding agreeably together." But he is incon-
istent: later, he describes the symphony of dia-
essaron as often quite unsuitable for diaphony
=organum; the terms are interchangeable) without
alteration, because of possible occurrences of the
tritone. He makes no mention of diminished fifths,
which are also dissonant: his gamut was so con-
structed that such fifths could not occur in strict
organum. This gamut consisted entirely-~and, for
the West, abnormally--of disjunct tetrachords, each
tetrachord consisting of T S T, thus:

fit..-

    

-2u-

The structure of this gamut evoked the criticism of
Hermannus Contractus and Wilhelm of Hirsau, who re-
garded the author of the Musica Enchiriadis as in
error. Their view is shared by some modern writ-
ers. It is entirely possible, however, to believe
with Otto Gombosi that the gamut was taken over
from Byzantium; it may have been borrowed with the
deliberate purpose of facilitating the author's
exposition of organum. It will be observed that
the three augmented octaves, B-flat to B, F to F-
sharp, and C to C sharp, make it impossible to ob-
tain a diminished fifth by combining any two tones
in the gamut.

But it will also be observed that a tritone
can be obtained by combining the third of one tet-
rachord with the second of the tetrachord above.
The author of the "Manual" apparently regarded the
tritone as arising out of such a combination, just
as we today regard the tritone as arising, in ma-
jor, from a conjunction of the fourth and seventh
scale degrees. He therefore gives this as the rea
son why, in organum at the fourth, the vex organ-
alis should never pass below the fourth sound of
the lower tetrachord. When the vex principalis
begins in such a way that the vex organalis cannot
accompany at the fourth without passing below the
fourth degree of the lower tetrachord, the vex
organalis has to begin in unison with the vex
principalis and, unless the interval of a fourth
is immediately thereafter reached, remain station-
ary until it is possible 59 parallel the vex prin-
cipalis at that interval. 4

 

 

 

In this connection it is appropriate to mention
and to quote the first part of the oft-mentioned and oft-
quoted Rex coeli Domine, etc.

Ex. 20 Free Organum--Musica Enchiriadis.

 

‘4» . 4r,
Rex coe - Ii Do-mt-ne ma—‘ns vn- di- 50- ‘m

 

2"Ibid., ash-5.

-25-

Thus we find that intervals other than the
"symphonies" were occasionally sanctioned by theo-
rists (although these intervals were not recognized
as consonances) and that the author of the "Manual"
supplemented parallel motion, for the sake of
avoiding the tritone, by oblique motion; very oc-
casionally, in quitting a unison at the beginning
or approaching one at the end (or just before the
end, in which event there are consecutive unisons),
he used contrary motion also, for the same purpose.
A free organum pregnant with great possibilities,
resulted; or perhaps we should say that a free or-
ganum~-even freer than this-~was probably already
in existence and was thus, with a little twisting
and turning, accounted for in theoretical writ-
1ng.

Reese is stating a definite example of the princi-
ple that theory invariably follows practice. He makes this

even clearer later:

The very incorporation of such progressions,
deliberately used, in the free examples in the
Musics Enchiriadis would tend to support the view
that the treatise itself deals with a practice
perfected after a long period of evolution. For
it would seem that only thus could there have been
time to recognize, besides heterophony, the uncon-
scious singing in more or less parallel intervals;
to systematize the method; to combine it with the
other; and to develop a well-planned doctrine-~a
doctrine which in some respects (es ecially in
connection with the tritone, . . . may very well
represent theorists' rationalizations of existing
phenomena ratherzghan a historical account of
their evolution.

In connection with the enigma of the tritone in
organum of the ninth and tenth centuries, the nineteenth-

century theorist Hugo Riemann quotes the following example:

 

-26..

I, Authentic tone

 

Here parallel movement in fourths is carried
out to the end, but it is in contradiction to the
text of the Musica enchiriadis, which states quite

 

rightly that accompaniment in fourths is possible
in the beginning only because the absonia (that is,
the error-producing effect of the tritone, Bb-E)
does not occur (since the vox principalis skips the
step on E). At the end, however, the original
voice should not descend below Tetrardus (C) for
the same (or rather the opposite) reason. There-
fore, the ending must be changed to that shown in
3, According to the older version of these theo-
ries, the ending would be the same, but the A in
the beginning would not be used, and the organum
would begin in unison with the cantus, as in 2.
Everything 612? corresponds to the procedures of
both methods.

 

It has been established, then, that the intervals
which have come to be called the perfect consonances were,
in the period under discussion, the only consonances. Oc-
casionally other intervals were used, but they usually en-
tered the musical context as the lesser (or least) of evils.

In the succession of intervals: octave--

perfect fifth--perfect fourth--major sixth--minor
third-~the partial tones (which are sympathetic

 

27Hugo Riemann, History of Music Theopy, trans.
Raymond H. Haggh (Lincoln: University of Nebraska Press,
1962) p 28-29 0

tones heard as part of the fundamental note) set by
the ietes contained in these intervals vibrate in
unison at an increasing distance from the fundamen-
tal note. In other words, the relationships get
more complex the further the intervals move from
the cons.nances of the octave and fifth. The pres-
ence of a third and a unison in what, in the fol-
lowing, should be a succession of fourths is pre-
cise-y to avoid, in the third chord, the prohibited
and much more dissonant interval B-flat to E (an
imperfect fourth or tritone, vibration relationship
h5:32), thefiginal unison being determined by this
alterationz‘

0d

 

In summary, it may be stated that the harmonic
tritone seems to have been considered an impurity among the
perfect intervals of the fifth and the fourth during the
early polyphonic period. It was therefore to be avoided in
ct organum. However, the interval does appear in the

vritten music of the period-~a fact which leads most modern

writers to believe that the practice of musica ficta was to

 

be applied in performance even when the accidentals were
not indicated in the music. A few, however, believe that
compos rs were actually using the interval while theorists

continued to proscribe it.

 

28Edmund Rubbra, Counterpoint: A Survey (London:

Hutchinson University Library, 1960), 18.

 

-28...

m1
‘1

H
s

:CURTEENTH AND FIFTEENTH CENTURIES

In the fourteenth and fifteenth centuries is found

a more extensive use of musica falsa (or musica ficta) in

 

polyphonic music. One of the primary reasons for its use
was the desire to avoid the tritone.

There were several harmonic reasons for alter-
ation. Ugolino d'Orvieto (fl. 3. lhOO), who is
only one of several theorists treating the subject,
includes the following: (1) Fifths, octaves, and
twelfths must be perfect. If they arise in the
course of the counterpoint and would normally be
diminished, they must be enlarged by a semitone
and rendered perfect. (2) A third expanding step-
wise to a fifth, or a sixth to an octave, should
be major; a thirg contracting stepwise to a unison
should be minor. 9

Regarding musica ficta, an anonymous early
fourteenth-century treatise, Ars Contrapuncti, is quoted by

Eric Blom in Grove's Dictionary.

 

"Music is called ficta when we make a tone to
be a semitone, or, conversely, a semitone to be a
tone."30
Unfortunately for modern scholars, these acciden-
tals were not always written, it being assumed that the
performers would know the preper places at which to insert
them.
Elmong] the most usual cases in which acci-

dentals were assumed to be too obvious for inser-
tion were the following:

 

29Reese, 381.
BOGrove's (195k), v, 101u.

-29-

Notes to be raised a semitone:

1) To avoid a diminished fifth, if rising
subsequently:

Ex. 23

 

2) To avoid the tritone:

Ex. 2h

 
   

1"

Notes to beglewercd g_somitone:

 

3) To avoid the tritone:

Ex. 25

 

-30-

h) To avoid the chord of the diminished fifth
(quinta falsa):31
Ex. 26

 

 

Thus it is seen that not only the flat, but the
sharp as well, came into existence from the desire of com-
posers to notate exactly what was to be accomplished in
performance--namely, the avoidance of the tritone.

The difficulty which the synemmenon tetra-
chord was invented to remove arose, as is of
course well known, from the existence in the scale
of the semitones at B-0 and E~F. The succession
of fifths and fourths proper to the scale was
affected by the occurrence of these intervals,
the fifth between B and F being imperfect, and
its inversion--the fourth between F and B--redun-
dant. The difficulty was overcome by dividing
the interval B-A into two equal parts, of which
one was added when necessary to B, with the pre-
monitory sign b; so that the semitone now lay be-
tween A and B(b), and the fifth B(b)-F was perfect.
This fiction was temporary in its operation, and
held good only during the moment of necessity;
with the passing of the occasion the scale resumed
its normal form and values.

Useful as this method was, however, something
still remained for later times to do. For the
inventors of the arrangement just described--the
only form of Musica Ficta which was admitted as
strictly allowable by the most orthodox writers--
omitted any recognition of the fact that the de-
sired object might also be attained by a corre-
sponding manipulation of the remaining semitone,--

 

31Grove's (195h), V. 101ho

-31-

by the addition that is to say of half of the inter-
val F-G to F, with the sign #--thus creating a full
tone between E and F (#) and G. The general principle
may be stated thus: the interval B-F requiring one
semitone to make it perfect, this may be supplied in
either the higher or lower region of the scale.32

Ex. 27
Melodic use. Contrapuntal use.

x #

       

       

  

49- <>

(The star indicates the altered note.)

It was not until this period (the fourteenth cen-

tury) that writers began to discuss the use of musica falsa

 

to avoid objectionable harmonic intervals such as the tritone.

In the 13th century the writers on musica falsa
discuss the chromatic tones only with reference to
the single line, emphasizing chiefly the subsemi-
tonium (leading tone) and the avoidance of the tri-
tone in progressions such as: G-F(#)-G; C-E-F(#)-G;
F-A-B(b)-A; etc. In the luth century Joh. de Muris
(c. 1325) approaches the problem from the point of
vTew of simultaneous voice-leading, forbidding the
tritone as a chordal formation . . . and postulating
that the third or sixth before a fifth or octave

should be ma or if the upper voice ascends, minor if
it descends.

 

 

32H. E. Wooldridge, "Method of Musical Art, 1300-
1600," Oxford History of Music (Oxford: Clarendon Press,
1905).'56-7.

33Willi Apel, Harvard Dictiongpy of Music (Cam-
bridge, Massachusetts: Harvard University Press, 1961), h65.

-32-

partic M ar. As CPQU“POSlLJOn% cf the "Iand.'
Petrus recog nizes ut Bb and ut=D.3

Th e "Hand" referred to is, of course, the Guido-

nian Hand (see Apel, on. cit., 312).

Because of the troublesome tritone, two of the
possible fourteen modes were rejected by theorists. Those
forbidden were, of course, the Locrian and the Hypolocrian
modes. These scales are, for the most part, avoided even
today, although certain composers (notably Roy Harris) have

experimented with them.

Modes on A and C (Aeolian and Ionian) were

first theoretically recognized in the Dodecachord-
on (system of twelve modes) of Glareanus in 35h7.
He also gave the name Hyperaeo olisn to the mode on
B, and T“psirhrvgicr to its pla5sl form, but re-
jected both as impractical on account of the dimin-
ished fifth between B and F. These modes are now
generally known as Locrian and Hypolocrian.J/

It was the practice in much of the music of this
period to alter certain tones in performance, the changes
not being indicated in the musical notation. Thanks to
the musicologists of today, we can be assured of reasonably
accurate performances of this music in our own time.

All we know for certain is that the music as

written down does not indicate all the accidentals,

and that performers altered certain notes as they
went along. Admittedly the theorists give us some

 

3uLee Rigsby, Studies in Music History and Theory
(Tallahassee: Florida State University, 1953), 70.

35Westrup and Harrison, h30.

n5, for example, that B is flattened
uiC pro5ression F-B-A or A-B~A, and F
sharpened 1n the progression B-F-G and G-Fwe, that
the harmonic interval of a diminished fifth must
be made perfect by altering one of the notes, and
that a third exps nding to a fifth or a sixth to an
octave should be perfect. 30
In summary, it may be said that the tritone con-
tinued to be partly responsible for the extensive us e of
musica ficta throu5hout the fourteenth and fifteenth cen-
turies. The rules for the practice were many and varied,
and applied to the cross-relation of the harmonic tritone
as well as to direct melodic tritones. Unfortunately for
musicolo gists, musica ficta was partially a performer's

rt-«therefore one cannot be certain to what extent acci-

dentals were "inserted."
THE SIXTEENTH AND SE ‘F NTFENTH CENTURIES

During the sixteenth century, composers imposed
extremely rigid rules upon themselves, particularly for
composing ecclesiastical music, and there is found, espe-
cially in sacred style, a common practice such as has ex-
is ted during no other period of music history. The tri-
tone, as would be expected, figured prominer .tly in the pro-
hibitions laid down by composers and theorists. Modern
textbooks on sixteenth-century style give us some idea of

the extent to which this 1nterval plagued the musicians of

36Ibid., p. u31.

'3’)
J‘r

the time. The following examples are taken from a widely-
used textbook which is based on the style of Palestrina.

e augmented fourth naturally cannot occur

n5 a minor third; and following a major

i often altered to a perfect fourth by

of a sharp or a flat:

Ex. 28 'Ex. 29

CI‘

   

Thi
bs
‘dg
ec

s a good deal on the context, and no

le can be said to exist. Finally, the
curth is commonly used following a per-
‘

d
lu
n
f

en
te r
te
if

s
0
me
t

H) F0 ‘9

 

It will be observed that though the fourth
from the bass is a discord, both the perfect and
augmented fourth may occur between two upper parts,

if each 1s concordant with the bass. The same
freedom is allowed to the diminished fifth:

 

37Merritt, 57-8, 129.

Ex. 31

 

The reason is that both the following are only dif-
ferent arrangements of the same combination consid-

ered from the bass:

Ex. 32

 

In each case there is a third and a sixth from the

bass. . . .

When a third part is added, it is necessary to
avoid the false relation of the tritone; this would
occur in Mode V, when the Canto Fermo is in the low-

est part:

Ex- 33

 

The addition of a flat to the B softens the effect.
The interval of the augmented fourth or dimin-
ished fifth in the melody must be avoided by altering
the second of the two, forming the skip of a perfect
fourth or fifth:

 

. avoided at the ex-

’9

interval was also likewis
t re

C.)

Ex. 35

 

. . . The false fifth between the bass and an
upper part is avoided by flattening the bass:

Ex. 36

    

be

The tritone as on essential harmony must be
avoided by the use of Bb:

Ex. 37

 

-37-

. . . E [in the transposed Lydian mode] would
be flattened both in conjunct and disjunct movement
if the Quinta Falsa or Tritone occurred; also it
would be generally flattened if used between two
D's.

. . . The diminished fifth followed by the
perfect fifth, or vice verse, is allowed between
two upper parts if the lower of the tgg parts in-
volved proceed by step of a semitone:

Ex. 38

 

The period just discussed marks the culmination of
sensitivity to the tritone on the part of composers and
theorists. Nevertheless, in the following century there
were those, particularly among the theorists, Who continued
to consider the tritone the "forbidden interval."

The third kind of dissonance is formed by the
tritone and the false fifth; in the latter we have
instead of a whole-tone a major semitone; in the
tritone it is exactly the reverse. They are rep-
resented by the following numbers:

Tritone 32/h5 False fifth hS/oh

or as follows:

 

39c. H. Kitson, The Evolution of Harmony (Oxford:
The Clarendon Press, 19357; 8, 18-19, 20, 2h, 6E:S.

-38-

Tritone F to B/ SuO, 38h
Bb to E/ #05, 288

False fifth B to F/ 38h, 270
E to Bb/ 288, 202% or

576. hOS

These numbers are also too large to make these in-
tervals acceptable to the ear; neither do they en-
joy being in the neighborhood of consonances.

. . . Therefore, they must be avoided [even] suc-
cessively in different voices, expecially in slow
music without diminution; in music with diminution
which is performed rapidly the ear does not have
the time to notice the defects of these dissonan-
ces. This discrepancy is all the more noticeable
because they are neighbors of the [puré] fifth,
with which the ear therefore compares them; be-
cause of the special sweetness of the fifth their
imperfection is even more obvious.LL

It was during this period that the tritone began
to be used for purposes of "diabolical" expression, partic-
ularly in the oratorio style. When the interval was used
in this manner, its tendency to resolve, while not complete-
ly overthrown, was somewhat neutralized. This is made evi-

dent in the following brief passage from the Last Judgment

 

of Buxtehude, in which the diabolus in musica is made to

 

represent "the wailing of a lost soul in hell" (Ex. 39)."Ll

 

hoRene Descartes, Compendium of Music, trans.
walter Robert (Rome: American Institute of:Musicology,
1961) , hS‘é o

ulCooke, 85-89.

he 39

 

 

In the sacred music of the sixteenth century is
found a common practice such as has existed in no other
period of music history. Composers imposed rigid rules
upon themselves, and many of these restrictions and prohi-
bitions concerned the tritone. In the following century
theorists continued to proscribe the interval while compos-
ers began to employ it with unprecedented freedom, using it

even for descriptive purposes.
THE EARLY EIGHTEENTH CENTURY

It is generally true that when composers wish to
employ the tritone for purposes of "diabolical" expression,
they will construct the interval between the tonic and the
raised fourth scale degree. This is true of the preceding
example and of that which follows, from Bach's Cantata No.
‘fié, "I will gladly carry the Cross." "The sharp fourth is
set against the minor triad as an accented dissonance--a

painful, semitonal one--resolving upwards, i.e. onwards,

1._
-50

to the dominantl' Cooke demonstrates

"1"
.. .4
s-‘
H.
('4
t"“
:3
c r
’3'
0
*e
0
[VJ
H
O
53.
Ho
:5
C ‘1

 

lch will den Kreuz-stab

In the e rly eighteenth century there were a num-

ber of theorists who wrote about the peculiarities of the

*

ins Ir tens seldome falls plum into ,he f3rd
but “arms back one half; . . . and the same is a
nu“ ever to the hand of a perfo-mcr, without any
express direction; that is, . . . to hang upor tie
fourth, and that gives a :gdeatiall relish to the
full accord that follows.LJ

ff

 

Jl

But by far the greatett theorist of the period,

and inu1cd, one of the greatest of all time, was the

 

LCooxe, 126, 128-9.

-51-

:1

ran hman, Je: an Phi li ope Rameau, whose work was so advanced

that its value was not recognized, even by his contemporary,

‘

Johann Sebastian Bach!

Of major Discords proceeding from the Leading-
Note, and of those notes on which they are used
1. The Tritonus is never used but upon the fourth
Note, w.hen that Note descends upon the Third, or
upon the Keg-note.
2. The false Fifth is never used but upon the
Leading-note, or sharp Seventh, when that Note
afterwards ascends to the Key-note. . .
h. The sharp Third cannot be used with the Sev-
enth, ma king between themselves an Interval of a
Tritonus, or a false Fifth, but upon the Governir g-
note or Fifth of the key. These four Discords are
the most in Use. . . .

Some times the Tritonus happens upon another
Note than the Fourth, and the false Fifth upon an-
other note than the Leading-note; but then, and in
that Case, those Intervals are no long er the Ob-
ject of the Chord, they serving or ly as an Accom~
paniment; and it is the Modulation the causes
that Alteration in the same Manner, as in the Pro-
gression of Sevenths, where some are altered, and
are not in their true and just Proportion there-
fore you must never take any Notice of this Alter-
at ion, when you know the Chord that ought to be.
sad, and the Key you are in; for it is the suc~
cessive Degrees of a natural Voice, contained in
the Compass of the Octave of the Key, or Mode that
you are in, that decides the Justness, or the Al-
terati n of an Interval that makes a Part of the
Chord.

Regarding the use of the tritone during the first
half of the eighteenth century, modern theorists have found
it possible to make a number of generalizations. It should
be stated here that many of the prohibitions regarding the

use of the tritone in traditional music exist in the minds

 

thean Philippe Rameau, Treatise of Music (2nd ed.
of English trans.; London: J. French, 1737), 119-20.

 

sts rather than in the music itself. However, most
of their observations serve a useful purpose, and are quite

valid in the majority of cases.

(‘1‘

Piston states ha

d-

I.“

a dim nished fifth followed by a

L'o

perfect fifth we ave

U:

xded in two-part writing, not because

of the consecutive fifth , but "because of the unnatural res-

w

0

lution of the dissonant interval (Ex. Ag}. The progression

0 diminished fifth . . . is quite acceptable

   

About the cross-relation of the tritone, Apel says:
6

w- o o o L
"is is usually av01ded between two outer veices [Fx. hi]."L

Ex. uh

 

 

,r
L’Walter Piston, Counterpoint (2nd ed. rev.; New
York: W. w. Norton and Co., Inc., 19h7), 83.

héApei, 195.

“1&3-

aerarding the same phenomenon, Piston allows for an

- r
Ho
0
.‘J
O
3...)

The cross rela. the tritone, where a
leading-tone moves to the tonic in the upper voice
while the fifth degree descends to the fourth in
the lower (called Diabolus in Musics), is avoided

 

except when the fourth degree continues its down-
ward movement, as in the following example.47

 

In a passage from Piston which was quoted previous-
ly, he referred to the "unnatural resolution" of the tritone.

It is hardly necessary to state that it is normal resolution

H.

to which he refers when he states that t is the tendency
of augmented intervals to eXpand, that of diminished inter-
vals to contract."u8

Ex. he

 

 

u7Piston, 9h.
halbid., 89.

-hh-

Later in his text, Piston discusses the tritone as
it is used in three-part writing.

The progression of a diminished fifth to a per-
fect fifth, usually avoided in two—part writing, is
commonly used in two upper voices when the bass
moves to the third of the chord represented by the
second interval.

Ex. h?

Tr”

 

Ex. L8 J. 3. Each, Organ Furue in E Minor

 

The cross relation of the tritone is freely
used when the progression from fifth to fourth
degree takes place in the inner part.h9

 

1”mid. , 125-6.

‘LLS'

:aeh, Organ Fugue in A Meier

    
     

A231- ; "I (11)“, I

Ex. 50 J. S. Bach, Well—Tempered Clavier, I, Prelude no. 23

   

13:11: :5 (I!) (“7) ‘1'7

In connection with examples A9 and 50, it should be
mentioned that Piston fails to point out the significance of
the harmonic context in which the cross relation of the tri-
tone occurs. In both instances, the critical point is the
dominant-subdominant retrogression. Furthermore, in both
cases, the subdominant harmony is only incidental, and sub-

ordinate to an overall dominant feeling.

-h6-

In his chapter on counterpoint in more than three
parts, Piston emphasizes the "prominent movement of dimin-

ished to perfect fifth in oboe and violin" in the follow-
ing example.50

Ex. 51 J. S. Bach, Brandenburg Concerto No. 2

FLUTE h A n ii) 0

VIOLIN

CANTINUO

 

0.7 ‘ 6 II ‘o
d: u 14. I Vu°7° Vu°7° -6 I I
If
Piston fails to point out that the above example
illustrates the point three times. These three diminished

fifths are bracketed above. The first, although ornamented

with an appoggiatura in the flute, is quite obvious. The

 

5°Ib1d., 152-3.

f"

n56 o

'T‘

o
,

a.

1’1““

enth ‘
no ch

7

4-
\J O

A

1301‘:

A1

fl

t‘ u .--

.3

.Y“ ‘. ‘r‘t
V t] A

\v

f’

h
.3
n1

Ofil‘.

.1;

Cu
LU
O
h
L k

“D

fth con-

J.

hed f‘

C‘
bx

0
q

.L

1min

‘ves a d'

anOJ.

1 example '

I‘
s

.11.

\A

C

O
“‘1
lil.‘

€11“?! C‘

h

and a

tone

3

\A

f
0‘. *

oi

Lu
C
5

P
3

«d.

‘3
I

'

extrom

‘hav

‘
“at:

G

8

Y!

m
a.

‘
Q

d
30}

o _ a vad
UQVOJA‘
94L

v

as

v;

:0
.v‘ A 1-
'O‘JoJ-to

and,

.1.

intervals

problems
5 particularly
th

:3

1"

Alt

cub

1
v»
d-

‘
3'13
1‘

J
‘16

spoola
'3"

a kind of
a weak.

On the 0

17103
3
4..
L1

4.
U

s
e harmonic

L:

t.
same,
933

a
h becom

‘
-1

en
rpoin
th
.ave becom«
nt
(3* " h
c' v‘-.' LI
b l

G
v
4..

Q

a.
0

OC
1
L

tent
.
ale,

\

19

th e
the so

1.—

e ccunto
occa

seventh n

.L

1

vertible coun

o
1 Y2
*--

nvertib
of

l

O
26

q
11.

9

C3

“1.
«J.

t1

0

a?
for

I‘
.L.-.

-i

U
2’18
SSCOII'
f‘

‘

O
5

v v. ’. t,
in t

I. '

;,'\3 (I; ‘
"I
we 4.

I

+-
U
‘JA

4,8-

i1 as t.e perfect fourth from
p to the dominant becomes, a

o e from the fourth up to the
s U the interval is to be used,
e'ther the melodic treatment of the perfect fourth
should be such as to result in proper progression
from the tritone, or one of the notes of the tri-
tone may be altered, making it a perfect fourth.

r

N. . L‘ : ,...
selentn oe nee. I
.

Anoth r example of the use of the diabolus in

musica to express evil follows.

 

In Bach's Passion according to St. Matthew
Pilate asks the crowd whom it wants to let

0 C I
go free, Jesus or Barahbas, and the answer is a
dissonant chord. Here the bass progresses a tri-

tone upward into an unprepared dissonance whic.
dissolves only when the story resumes. If Bach
had proceeded in a musically "correct" fashion, if
he had been afraid to "shook" his listeners with a
prolonged di;-3sonanc;:~_;,3 he would have been one of the
crowd, not himself.9¢

Ex. 52 J. S. Bach, Passion cccordinn to St. Matthew

A

 

BA - RAB-

 

V
It seems to the author that Mr. Lippman has omitted
the fact that the above passage represents a rhetorical and
dramatic use of the tritone (and of the diminished seventh

cnord, wnich consists of two interlocking tritones). This

 

 

Klaus Lippmann, The Lancuave of Music (New York:
The Ronald Press Company, 1993 , 130.

‘
'-{
CT
Q?
'L.J
Ho
SP
:3

In the es rly eighteenth centu.ry a number of the-

O
o

rists wrote about the peculiarities of the tri tone. The

'3‘]
:10

A . ,, - .‘ ,__ , ‘ 'r h
¢£68t3bt of thee, was ue.n r

"‘J

.lippe Rameau, whose theory of
the fundamental bass serves as the foundation of the study
of harrnon“ in our own time.

lne tritone in the music of the early eighteenth
century was much freer than those theorists who write about
it: or hititions in that period would have one believr

lost of their obs ervciic ~ serve u eful pu ses, however,

id in the majority of cases. Their weakness is

that they fail to account for those beautiful and signifi-
cant exceptions which are characteristic of the greatest

If Bach and his contemporaries liberated the tri—
tone somewhat, the classical masters treated it even more

freely. The following e"am>le illustrates Mozart's rec-

[‘1

U1

 

CTURO :f

TL

Andante é'k

 

 
 
   

Ec— co- |3.!

e Mozart [example quoted abova is from Don

Giovanni: the Statue offers the Don his hand--
Wm..- . .
here it is"-~and the Don grasps 1t and cries out

in horror, moving to the diminished fifth.

chord
minish
interl

In

fe
practice,‘
Th

if the
terval

The
accompanying his cry--the chord of the di-
ed seventh--is, of course made up of two
coking diminished fifths.§

view of the two examples quoted above, it would

to maintain that the tritone was, in the class-

as before, the "forbidden interval." Neverthe-

in twentieth-century harmony texts which pro-

the style of the period of "harmonic common

1

K8 l

.L

-J

statements li the fo lowing.

e leap of an augmented interval is allowed
second of the two notgfi forming the in-
be an auxiliary note:

 

53Cooke, 85, 87.

51+Kitson, 106.

-51-

 

In the chapter on five-part writing, Kitson says:

It is better to avoid a perfect fifth followed
by a diminished fifth, or vice-verse, between up~
per parts when the lower of the two parts involved
moves a whole tone:

Ex. 55

 

And, as if the inclusion of a chapter on five-

nart writing were not indicative enough of the author's
O V U

desire for thoroughness, there is a chapter on six-, seven-

and eight-part writing (I) in which he states

The diminished fifth followed by the perfect
fifth, or vice versa, with the lowest pagg moving
a tone, may be used between upper parts.

 

551bid., 389.
56:bid.. 39?.

 

-52-

Later in the same chapter the author discusses the

progression vii 070-i in minor, where the leading tine and
diminished fifth demand special care.
[In Ex. 55] the leading note is in the bass.

H“
2*-

Its doubling would be unpleasant. a
movement downwards to G will transgress the rule
that when a part is resolving a discord, no part
should move in similar motion with it to the oc-
tave of such resolution. In the majority of
cases composers abide by this. In six parts no
dilemma occurs. See Ex. 57]

In seven and ei ht parts we must double the
original seventh (F , but as the root is not
present it is not evil in effect. Being most ob-
jectionable when it is the diminished fifth from
the base, the ideal will be reached when it is

trebled at the unison. Sec Ex. 58.

When however, the chord resolves on to a six-
four before proceeding to the five-three, the
diminished fifth from the ass becomes the root
of the next chord, and there is less need for

riticism of this nature. But the diminished
fi t. should not be doubled in more parts than is
necessary, and it is better as far removed from
the bass as possible. E§oe Ex. 59. 57

In the following example it may be observed that

Soprano II, Alto I, Alto II and Tenor I sing a doubled

third, fifth, third and fifth.

are free to resolve more or less independently.

As doublings, these notes

The re-

maining voices resolve the chord in the manner of strict

four-part writing.

 

S7 ido, 399-u000

Ex. 56

 

c: vii °7° i

-5“-

 

-55..

In the later chapter on two- and three part writ-

ing Kitson b‘aely mentions the tritone:

The harmonic intervals of the augmented fourth
and diminished fifth may Esus ed if treated cor-

rectly as implied chords: 8
Ex. <0

 

The classical masters relaxed many of the restric-
tions and prohibitions regarding the use of the harmonic
tritone. They continued the practice of using the inter-
val for purposes of "diabo3ical" cescript it on. At sucn
times the interval would most often be emp
ity which was isolated rom a functional con ext, where its
"horrors could be displayed to best advantage.

There is a curious dichotomy between classical

tice and the theory of the same and su ceeding genera-

”J
"S
m
o

(1'

io . In the midst of the great liberation of the har-

r4

monic tritone, theorists continued to prescribe the inter-

val.

 

THE NINETEENTH CENPURY

It seems safe to say that in the nineteenth centu-
ry, the harmonic tritone was employed with more frequency
than at any other time in the history of that interval.
Composers who enjoyed exploring the realms of greater tonal
freedom (and those who did not were rare) made frequent use
of the modulatory capabilities of sonorities which con-

tained tritones. Furthermore, in the expanded use of chro-

5J0

maticism w thjg_given tonalities, secondary-dominant-type

 

sonorities, all of which contained tritones, of course,
were much more at home than they had been in the preceding

LCI‘iOd e

‘3

h

.
r“

3 section of the treatise would hardly be com-
plete without the inclusion f the famous "Tristan" chord
of Wagner. Observe, in the example, the "irregular" reso-
lution of the interval (and of the augmented-sixth chord).
The sonority occurs on the first beat of measure two, and
is nothing more than a French sixth with an appoggiatura

(G#) to the sounding third (A). The French sixth resolves

quite normally to a major-minor seventh chord with an "E"

C?

6' t tha

>4
0

root-—e the fifth is delayed by an accented

”:1

ng note.

w-

chromatic pass

Ex. Al Richard lagner, Tristan and Isolde

a};

3: F6 E7

 

With all due respect to the stature of the nine-

teenth century theorist Riemann, the author would like to

7

call attention at this point to the following paragraph,
which illustrates the discrepancy-~perhaps greater than at
any other period of music history--between theory and prac-

tice.

Good melodious z’tt‘r , which must always be
our norm, has to avoid the tritone under_ 81
circumstances, Tat least with changing harmonies)
when possible. Within the same harmony,it may
be allowed without hesitation, so long as it
does not incur the violation of that most impor-
tant principle-~turning after leaps; i. e., F-B
(rising) is good, while the same harmony is sus
tained, if A can and does follow.

In summary, it may be stated that the nineteenth
century witnessed the further liberation of the harmonic
tritone. All important composers made more frequent use

f its modulatory capacities as well as its function in

C)

secondary-dominant-type sonorities. Program music and

 

SgHugo Riemann, Harmony Sim lified, trans. Rev. H.
Bewerunge (London: Augener and; Co., 183?}, 37- -8.

-53-

opera provided increased opportunities for the use of dia-

bolus in musica as a descriptive force.

 

THE IMPRESSIONISTIC PERIOD

At first consideration, it would seem that im-
pressionistic music, with its whole-tone scale, would be
rich in multifarious uses of the tritone; but it must be
remembered that this was the period that saw an increas-
ing use of non-functional harmony.

The philosophy of musical impressionism was root-
ed in antagonism toward the classical laws of tonality and
key relationship; through its techniques it gave composers
a fresh orientation of values. Upon its negations it for-
mulated its own completely individual eXpression, which is
always recognizable. It arose when composers began to
feel that they had exhausted the resources of the major-
minor system of tonal organization. Impressionist compos-
ers developed a fascination for the whole-tone scale, which,
like certain musical systems of the Far East, divides the
octave into a number (in this case, six) of equal parts.

In music which employs the whole-tone scale, the
tritone becomes more readily available; for, while the
interval is found only between degrees four and seven of
the major scale, it exists between one and four, two and
five, and three and six in the whole-tone scale. Perhaps

it is that very availability which neutralizes the tendency

-59-

of the tritone to resolve and makes the interval less form-
idable.

But the whole-tone scale is not the only feature of
impressionism which had its effect on the tritoné. There
were triads with added notes, some of which produced tri-
tones which were more often than not, unresolved (by tradi-

tional standards). Some of tiese dissonant sonorities were

used in parallel motion. The use of modality sometimes
shifted the position of the tritone in the scale.
COFCLUSION

It was found that the tritone has been tradition-
ally regarded as a dissonant interval--one which has been

endowed with a peculiar tendency to resolve. However, the

w-

ctions surrounding the use of the interval were con-

restr Q
tinually loosened, so that the use of the harmonic tritone
in the works of major composers increased in significance
an frequency throughout music history up until approximate-

ly the beginning of the present century. Those portions

is treatise which follow will be concerned with con-

3‘

of t
clusions drawn from statistics which compare tritone usage
of the past with that of the present.

I!
any har-

In Part I, the tritone was defined as
monic or melodic augmented fourth or diminished fifth."

It was found that theorists are not in complete agreement

~60-

as to whether the tritone occurs in the first sixteen par-
tials of the harmonic series.

For the sake of objective analysis and the limit-

*1.

ng of material, the author has chosen to deal only with
the harmonic tritone in the statistical comparisons, which
constitute the main body of this work. However, in the pre-
ceding material, which was a historical survey of the tri-
tone in music theory and literature, it was deemed advisable
to show musical practice and theory as they affected the
melodic as well as the harmonic tritone, since the restric-
tions and peculiarities of both types were often interre-
lated.
The interval acquired pseudo-supernatural signifi-
t

canoe when early composers of modal melodies called i dia-

 

bolus in musica because of the difficulties encountered in
avoiding it. For the next few centuries the interval was
used in descriptive music to depict evil.

It was found that there were tritones in the an-
cient Greek modes (two "defective" fifths and one "defec-

H

hive fourth).

The practice of musica ficta was occasioned in

"x
14

art by the desire to avoid the tritone. When the tritone
was not cancelled out by the use of musica ficta, definite
rules could be cited to justify its appearance. However,
since musica ficta was often the duty of the performer, its
taken for granted by the composer, who did not

use bein-

’1"
C)

~61-

always feel it necessary to include the symbols in the music,
present-day performances of early music are highly specula-
tive. On the other hand, some scholars feel that many tri-

ohes were edited out of Gregorian Chant in and after the

C?

ninth century when the interval was prescribed.

The tritone caused more difficulty during the poly-
phonic period, when it had to be dealt with in its harmonic
form. When it was not treated as a dissonance it was can-

celled out by the use of musica ficta.

During the fourteenth and fifteenth centuries there

*‘fi

was more extensive use of musica icta. This era saw the
relection of the Locrian and Hypolocrian modes; because of
the difficulty of creating acceptable melodies within the
framewors of such an elusive tonality, composers (except in
rare instances) continue to ban them.

Prohibitions regarding the tritone figured promi-
nently in the rigid restrictions of sixteenth-century coun-
terpoint. This was particularly true of sacred music, but
even in secular music the tritone was not treated wi,h a
-reat deal of freedom. The sixteenth century marked the
culmination of sensitivity to the tritone on the part of
composers and theorists.

In the seventeenth century, while theorists con-
tinued to condemn the tritone, composers began to employ it
with unprecedented freedom, using it even for descriptive

purposes (diabolus in musica). The early prohibitions

regardinr the use of the tritone were never restored, and
Phroughout the eighteenth and nineteenth centuries composers
became steadily bolder in their treatment of the interval.
Increased use of harmonic structures which contained tri-
tones was a major characteristic of the romantic era.

It was not until the impressionistic period, how-
ever, that the tritone was freed of the necessity for reso-

lution. While t.

L.—

e interval did often resolve, resolution

was the result of freedom of choice on the part of the com-

"‘3

poser, and not 0 some theoretical principle. Resolution

sly ne of a number of possibilities for continuation
available to the artist. Thus, impressionism all but re-

pealed the the traditional laws of key relationship and

tonality and through such devices as the z-Jhole-tonc scale

is.

I r 9 ' ~ ,‘ ’1 A .5 9 - . q . 9 3 . 4‘ 0' . n
constructed soiorities in which the primary function 0L the

t items was that of color. For the first time in the his-

“ O

tony Cl music, the tritone was emancipated; with the disin-

tercati n of tonality it took on a decided air of neutral-

0

why, thereby losing its former significance.

The use of the harmonic tritone in the twentieth

con ury is a complex issue which does not readily lend it-
self to generalizations. The author will attempt to cepe

with tnis problem through a detailed examination of pas-
sages from major works of important twentieth-century com-

posers, deriving and comparing statistics therefrom.

Chapter III

METHOD OF ANALYSIS

cal

Ct‘
Ho

For the sake of greater objectivity, a statis

U)

method of analysis has been formulated-~one w ich chart
certain data concerning every harmonic tritone occurring
within given passages of selected compositions. From the
data which are thus derived, certain comparisons and gener-
alizations can be obtained, and the percentages computed.
The information regarding harmonic tritones which
seems most valuable consists of the following: 1) measure
number in which the tritone occurs, 2) spelling of the tri-
tone, 3) method of approach to the tritone, h) method of

departure from the tritone, 5) whether or not the tritone is

rhythmically accented, 6) duration of the tritone, 7) wheth-

’3
to
l

er or not the structure in which the tritone occurs is t-
ditional, and if so, 8) whether the resolution is tradition-

al.

4
J

In all of the remainder of the musical ex mples i

*
p

we

th s treatise, octave doublings, phrase markings, and dy-
namics will be deleted since they have no bearing on the
analysis.

To determine whether a traditional tritone struc—
ture resolves traditionally, a certain amount of subjective
reasoning comes into play. It is not always possible to

settle this problem to the satisfaction of everyone; how-

-63-

, .a . .. ..- - _. . ...‘:_,
)ed that the examination of extended musical

'.-J

)

caus.

\L

passaoes (which is the policy in this treatise) wil

whatever miszalculations there may be to "balance out," so

that taere will be no significant misrepresentations of sta-

A resolution is said to be traditional if the tri-
tone structure proceeds as a similar structure would normal-
ly proceed in eighteenth-century harmonic style. Thus, the
pregressions dominant—seventh to tonic or submediant, any
of the normal resolutions of the aug.ented-sixth chords, or

3

of the full or half-diminished seventh chords-~tness and

C

On the other hand, retrogressions
dominant-seventh to supertonic or subdominant), altnou
they are found occasionally in eighteenth-century style,
will be considered non-traditional resolutions.

The following list of abbreviations will be used

in the charts:

a accented

app approached

C contrary motion

ct common tone

dep departed from

dur duration (number of units)

int interval (spelling)

m measure (number)

n non-traditional

O oblique motion (common tone in one voice)
3 approached from rest in one or both voices

-¥3)-

r rest (in a particular voice)

res reso Mullen

rhy rhvthmic stre ess (accented or unaccented)

S simi-ar motion

33' sl'igv

st step

str structure (identified as "traditional" or
"non-traditional")

t traditional

u unaccented

O

In the musical examples which appear in tn text,

these structures which are des it

A?
Q

A

d
will be analyzed accor ding to the method suggested by How-

we
go

ard Hanson in Harmonic Mater ls of Vodern Musi . All the-
orists will be familiar with the system of analysis and all

adequately-equipped music licraries will have a copy of the

cf
"3
O
in

. a. a. J- ‘ a - O V)” ‘- fiw I
1 e; however, the r11c1p1es of the sy1tem are stated

Senorities are analyzed according to the intervals

w.ich they contain. A sonority must be an alyse in it

(I)

simplest form-~i.e. all octave doublings ar disregarded.

G)

A three-note structure (triad) will be found to contain

three intervals. The term "triad" refers to any three—

3

t1ructure--not just to the f miliar major, minor, di-

01

1ote

p

minished and augmented triads. A four-note structure
(tetrad) contains three plus two plus one or six interval";
a five-note sonority, four plus three plus two plus one or
ten intervals, and so on. (Another method of calculating

the number of intervals in a chord emp Hey the formula:

-56-
n (n-l

in which n equals the number of notes in the chord.)

}—-o

:3

(‘f’

(D

*5
!

Each interval is given a symbol. Compound

03

Ho

:3
I

vals are reduced to simple form and an interval and it
version bear the same analysis, since they perform the same

function in a sonority.

The symbol E_represents the relations 1‘ of the

A

perfect fifth above or below a tone, even though when the

 

lower tone is raised an octave the relationship becomes ac-

s arbitrar‘,

Ho

tually a perfect fourth. Tie symbolization

‘

the letter B being chosen because it connotes tne lesi

' which applies to both intervals. The major

tion "perfect,'
third above or below the given tone is designated by the
letter a; the minor third above or below the given tone is
represented by the letter n; the major second above or be-
low, by s; the dissonant minor second by‘d; and the tritone,
by‘g.

The letters REE represent intervals usually con-
sidered consonant, while the letters §d£_represent inter-
vals commonly considered dissonant. When sonorities are

a

analyzed, the intervals are listed in the order nmrsdt. Ii

the recedin were an actual sonority, it would contain a

Us

’0

perfect fifth or its inversion, a major third or its in-
version, a minor third or ts inversion, a major second or

its inversion, a minor second or its inversion and a tritone.

-57-

A sonority represented by the symbol sc indicates
a triad composed of a major second and two mino, seconds,
and would be recognized as a very dissonant sound, while the
symbol pan would indicate a major or minor triad (in any in-
version).
a

The complexity of the analysis depends, oi course,

on the number of tones in the sonority.

‘Jo

Considering all tones in equal temperament s mpli-
fies somewhat the task of analysis. For example, C to D#
represents the same sound as C to Eb. Since the sound is
the same, they are both represented by the single symbol a.

To illustrate, the augmented triad F-A-C# contains

the major third F-A, the major third A-C# and the int r'al

P11

-C#. Since, however, F-C# sounds like F-Db, the inversion

of which is Db-F--also a major third--the dos.gnation of

the augmented triad 15.23-
The value of this method may be seen.when the fol-

lowing complex-looking sonority is analyzed in the tradi-

tional academic manner:

Ex. 62

 

-68-

The example is a six-note chord (hexad) and there-
fore has five plus four plus three plus two plus one, or
fifteen intervals, as follows:

C-Dfi and Ah-B - augmented seconds

0-3 and G-B major thirds

C-G and E-B - perfect fifths

C-Ah and Dfl-B minor sixths

C-B - major seventh

D#-E and G-Ab minor seconds

D#-G and E-Ab diminished fourths
D#-Ab - double-diminished fifth
E-G - minor third

According to the Hanson method, however, the so-
nority is converted into only four types of intervals (or
their inversions) as follows:

three perfect fifth

six major thirds:
G-B, Ab-C, B-Dfi

three minor thirds: C-Eb (D#), E-G, and G#
(Ab)-B

three minor seconds: D#-E, G-Ab, and B-C

: c-e, B-3 and Ab-E (of)

C E, Eb (D#)-G, E-G# (Ab),

".)

I
The description is, therefore, meQn3d3.
For further information on the Hanson method, the
o ‘0 4L 3_ 60
reader is advised to refer to the bOOn.
The following example, from the opera Lulu of Alban
Berg, will serve to illustrate the function of the charts.

A bracket (' I ) indicates the incipience of a harmonic

tritone.

 

60Howard Hanson, Harmonic Materials of Modern
Music (New York: Appleton-Century-Crofts, Inc., 1960).

 

-69-

Ex. 63 Berg, Lulu, mess. 109-111.

——

V

The following analysis is given:

109

int app
c-f# 0-ct,st
a-eb O-ct,st
g-c# R-r,ct

a-eb O-ct,st
f-b O-ct,st
c#-g 0-sk,ct
bb-e O-sk,ct
gb-c S—sk,sk
d-ab R-ct,r

b-e# S-at,8k

dep

0-ct,sk
R-r,ct

O-st,ct
0-ct,sk
O-ct,sk
0-sk,ct
0-sk,ct
0-sk,ct
R-sk,r

S-sk,sk

rhy

666669396936

(a
d'
"3

5555555555

 

res Hanson

m3 2
pmggfiszdt
pmn dt
p3mn332t
p3m2nzshd3t
pnzsdt

mst

mst
pm3nszd2t
pmnsdt

-70-

The analysis shows that the tritone C-F# in measure

109 is approached by oblique motion (0); C in the lower
voice is a common tone (ct) while F# is approached by step
(st). The departure is also by oblique motion; the common
tone C continues to be held in the lower voice while the
upper voice moves by skip (sk) from F# to A. Rhythmically,
the tritone is unaccented (u); it has a duration of one-
fourth unit. The harmonic structure in which the tritone
occurs is non-traditional. This is determined by referring
to the following chart:

Ex. 6h Traditional Tritone Structures

 

If the harmonic structure in question (or its in-

version, transposition, or enharmonic equivalent) does not

imply one of the chords in the above example, it 13 cons

L

d-

No

ered a non-traditional harmonic structure. If it is deter-
mined that a structure is traditional it will then be de-
termined whether or not it resolves in a traditional man-

ner.

-71-

A number of subjective factors must be taken into
account in deciding whether a resolution is traditional or
non-traditional. For example, a chord which looks like a
dominant seventh may resolve like an augmented-sixth struc-
ture, in which case the resolution would have to be consid-
ered traditional, since the musician must be more concerned
with sound than spelling.

Some may argue that in dodecaphonic music, such as
the Berg example quoted above (Ex. 63), the terms "step"
and "skip" lose their significance, and it is futile to
recognize them. However, one must not lose sight of one of
the primary purposes of this study--to compare tritone usage
in the twentieth century to that in traditional music. In
failure to recognize stepwise approach and resolution, one
of the important bases of comparison would be lost.

To continue with the analysis of the Berg example:
the tritone C-F# in measure 109, it was determined, is con-
tained in a non-traditional harmonic structure. A hyphen
is placed in the resolution column because a non-traditional
structure obviously cannot have a traditional resolution.

Also in measure 109, the tritone G-C# is approached
by rest (R). The rest (r) is in the lower voice, and there
is a common tone in the upper. None of the other terms in
the analysis of this tritone are new; therefore they need

not be explained.

-72-

The long tritone F-B in measures 110-111 is con-
sidered to have a duration of two and two-thirds units for
the following reason:

If, within a unit, a rest occurs, it will be treat-
ed as a continuation of the note which preceded it in the

same unit. For example,

Ex. 65 (Unit=))

 

would be analyzed as

Ex. 66

 

This principle prevails in the triplet figure of measure
110, where the B is considered to sound continuously. That
the tritone F-B which begins in measure 110 is accented (a)
is obvious; however, gll_tritones of at least one unit's

duration will be considered accented.

-73-

The final tritone in the Berg example demonstrates
the use of similar motion as a means of approach to the
tritone. In an example of this type, where the number of
voices is constantly changing, judgment must be used in
determining voice leading.

Throughout this treatise the reader will encounter
the term "tritone unit." This refers merely to the number
of units through which tritones sound in a given passage.
The total is computed by adding the figures in the duration
column, and the percentage of tritone units is obtained by
dividing that sum by the total number of units in the pas-
sage.

If a note (e.g., G#) is introduced in one voice
and taken up in another voice before, or at the point of,
its completion, its duration is assessed in terms of the
number of units through which it sounds continuously. In

the following example

6'?

T.“
bx.

, Barber, gymphcny_No. 2, Seeind Movement, mass. 5

 

VAJI

It
VHWHI

VCJI

YCJE

VCJHI

the tritone D-G# sounds first in Violoncello III and Viola
II. One unit (1) later, the G# is transferred to Viola I
and one-half unit still later, the D is transferred to
Violoncello I. The tritone sounds continuously for two and
one-half units, as indicated in the analysis.

m int app dep rhy dur str res farson

S d-g# R-st,r O-ct st a 2-1/2 n - pamznssadat
S c-f# O-st,ct O—ct a l n - pmn2dt

The main part of

 

the treatise examines, analyzes,

and charts extensive passages from a number of important

-75-

works of some of the most representative twentieth-century
composers. While perhaps no one will be in complete agree-
ment with the author's choice of composers and works, the.
difficulty of compiling a selection of this type, in which
the spatial limitations of the treatise must be taken for
granted, may well be appreciated. In submitting below the
list of works, passages from which were analyzed, the author
apologizes for having no doubt omitted a few composers who
deserved consideration, and even, perhaps, for including
some who did not.

For purposes of comparison, two important transi-
tional works have also been examined in part: Wagner's

Tristan and Isolde, which is written in a "functional" har-

 

monic style, and Debussy's Pelleas and Melisande, which em-

 

ploys a predominantly non-functional harmonic technique.

Transitional Works
Wagner Tristan and Isolde

Debussy Pelleas and Melisande

Twentieth Century werks

 

 

 

 

 

Schoenberg Fourth String Quartet
Piano Concerto
Stravinsky L'Histoire du Soldat
Symphonic de Psaumes
Bartok Music for Strings, Percussion and
Celeste

Second String Quartet

 

Berg

Hindemith

Prokofiev

Barber
Britten
Copland
Harris

Villa-Lobos

-76-

Violin Concerto

 

Lulu

 

Ludus Tonalis

 

Sinfonietta in E

 

Symphony No. S

 

Symphony—No. 7

 

Symphony No. 2

 

War Requiem

 

Third Symphony

 

Third Symphony

 

Bachianas Brasileiras No. l

Chapter IV

ANALYSIS OF TWO TRANSITIONAL WORKS

THE HARMONIC STYLE OF WAGNER

It is well-known that Wagner was a harmonic innova-
tor. However, it is said that his ideas about harmony were
well-grounded in tradition--that his structures and progres-
sions were tertian and functional. Perhaps the most inter-
esting, and certainly the best known of his harmonic struc-

tures is the famous "Tristan chord."

Ex. 68 Richard Wagner, Tristan and Isolde, meas. 1-3.

 

a: F6 ‘17

The "Tristan chord" does not really deserve the
designation "chord," since it contains one non-harmonic
tone. The sonority occurs on the first beat of measure
two, and is nothing more than a French sixth with an ap-
poggiatura (G#) to the sounding third (A). The French
sixth resolves quite normally to a major-minor seventh
chord with an "E" root--except that the fifth is delayed
by an accented chromatic passing note, A#.

-77-

-78-
TRISTAN AND ISOLDE

For analysis the author has chosen the first fifty-
four measures of the opera. The passage consists of 162-1/3
rhythmic units, through seventy and two-thirds (hu%) of
which harmonic tritones are sounding. There are ninety-six
tritone structures, of which 76% are traditional.

Oblique motion is by far the most frequent type of
approach and departure used in the passage. There are two
types of oblique motion: (1) the type in which the moving
voice moves by step, and (2) the type in which the moving
voice moves by skip. The chart which follows breaks down
the use of oblique motion in the Wagner passage into the

two types:

 

 

 

 

approach departure
oblique h5% 53%
rest 26% 11%
contrary 7% 13%
similar 8% 11%
parallel 13% 13%

gpproach departure
oblique-step 93% 83%
oblique-skip 7% 17%

accented tritones: 59%

tritones having a duration of less than one unit: 59%

-79-

The non-traditional harmonic structures of Wagner
(such as the "Tristan" chord) result from his excessive use
of chromatic appoggiature, suspensions, free tones, etc.
About one-third of the traditional structures resolve non-
traditionally because of wagner's unusual methods of cadence
evasion.

Tristan and Isolde was a milestone in the direction

of freedom for the harmonic tritone.

THE HARMONIC STYLE 0F DEBUSSY

Claude Debussy (1862-1918) was the only full-
fledged representative of the musical idiom known as 1mg

pressionism. The realization of that style resulted in

 

the introduction of a number of novel devices which are
contrary to the principal features of traditional har-
mony. Prominent terms in the vocabulary of Impressionism
are: the whole-tone scale in both melodic and harmonic
combinations; frequent use of the tritone; unresolved

dissonances; parallel chords; modality.
PELLEAS AND MELISANDE

The first forty-four measures of Act I, Scene I of
the opera Pelleas and Melisande have been chosen for analy-
sis.

Tritones are sounding through forty-nine and one-

half (28%) of the 176 units in the passage. This is quite

-80-

high, but it would be considerably higher were it not for a
curious "ebb and flow" of tritone structures; for example,
.there are no tritone structures in the first four measures.
.But in the next two measures there are fourteen! Similarly,
in the next five measures there are none--but in the next
eight there are forty-four:

There are twenty-nine traditional tritone struc-
tures, but none of these resolve traditionally. This may
be attributed to the fact that in this excerpt Debussy is
employing the pentatonic and whole-tone systems rather than
the familiar major and minor scales.

In the following example Debussy lapses briefly
into the whole-tone scale in measure twenty-three, as a sus-
tained Bb in the bass forms a tritone with the fifth of an

augmented tried on Ab in the upper voices.

Ex. 69 Debussy, Pelleas and Melisande, mess. 22-2h.

 

 

 

-81-

While the chord structure in measure twenty-three
changes, the harmony does not, and the tritone Bb-E, in
spite of a shift of the octave disposition of the E, remains
constant according to the terms of the system of analysis.
The last chord in measure twenty-two has seven voices and
the first chord in the following measure has only five.
Consulting the full score reveals that the lowest and high-
est G's in the last chord of measure twenty-two drop out.
Taking this factor into consideration, the following analy-

sis is obtained:

m int app dep rhy dur str res Hanson

23 bb-e 0-st,ct C-st,sk a h n - m3s2t

Applying this system of analysis to the music of
Debussy, which is of limited contrapuntal interest, exposes
the system's most prominent weaknesses. The method was, of
course, designed to accommodate the music of a later era--
one which is characterized by emphasis on the linear aspects
of music.

Before proceeding to the analysis of contemporary
excerpts, it is advisable to compare the statistics from
the two transitional passages. For this purpose, the fol-

lowing chart is provided.

-82-

COMPARISON OF WAGNER AND DEBUSSY EXCERPTS

Wagner Debussy

tritone units hh% 28%
traditional structures 76% 28%
non-traditional structures 2h% 72%
accented tritones 63% hh%
tritones having a duration ‘

of less than one unit 59% 80%
oblique approach u5% 72%

(0-st) (93%) (80%)

(O-sk) (7%) (20%)
oblique departure 53%. 8h%

(O-st) (83%) (76%)

(O-sk) (17%) (2h%)
approach from rest 26% 11%
departure by rest 11% 2%
approach by contrary motion 9% 8%
departure by contrary motion 11% h%
approach by similar motion 7% 5%
departure by similar motion 11% h%
approach by parallel motion 13% 5%

departure by parallel motion 13% 5%

-83-

The excerpt of wagner has considerably more tri-
tone units than does that of Debussy. Furthermore, it may
be assumed (from knowledge of the techniques of Wagner and
Debussy) that most of the Wagner tritones are functional,
and the statistics reveal that they occur in traditional
structures for the most part. Most of Debussy's tritones
are non-functional, and they occur most often in non-
traditional structures. Oblique motion is by far the most
common method of approach and departure to and from the
harmonic tritone in the Debussy excerpt. About half of
the harmonic tritones in the Wagner passage are approached'
and left by oblique motion. Contrary motion (assumed to
be the traditional method of departure) does not play a
prominent part in either of the transitional excerpts.

The statistics derived from the Wagner and Debussy
passages will prove useful for purposes of comparison with

the contemporary excerpts.

Chapter V
ANALYSIS OF CONTEMPORARY WORKS
l. ARNOLD SCHOENBERG
THE HARMONIC STYLE OF SCHOENBERG

The statements which follow will refer only to
those works which are written in the twelve-tone technique,
since this is the characteristic language of Schoenberg,
toward which he appears to have been groping in his earlier
compositions. The latter include "Transfigured Night," a
work whose style is intensely romantic and whose idiom,
while tertian and tonal, was so highly chromatic as to fore-
shadow the atonality that lay ahead.

Regarding the twelve-tone works then, it may seem
strange to attempt to discuss the vertical aspect of a music
which appears to aim at almost complete "horizontalism."
However, while the sonorities in Schoenberg's serial music
do appear to result at all times from the coincidence of
melodic lines, it will be seen that he has a consistent and
logical harmonic style. His twelve-tone technique was a
novel system of composition, devised mainly as an attempt to
arrive at "constructive methods to take the place of the
traditional principles of chord-construction, chord-
relationship, tonality, etc." The negation of these prin-

ciples led him around 1910, "to a type of music which is

-8h-

-85-

usually referred to as atonal, although Schoenberg himself
strongly resented the use of this term. No matter how it is
called, it certainly represents a musical style in which all
the tonal principles of nineteenth-century music are radi-
cally denied. Neither the chordal combinations nor the mel-
odic contours show any traces of 'tonality' in the broadest
sense of the word."61

The principles of the twelve-tone technique are as
follows:

1. Every composition is based upon an arbi-
trary arrangement of the twelve chromatic tones,
called tone-row or series. The chosen succession
of tones remains unchanged throughout the composi-
tion, except for the modifications explained sub-
sequently.

2. The octave position of any tone of the se-
ries can be changed at will.

3. In addition to its original form the se-
ries is available also in its inversion, in its
retrograde form, and in its retrograde inversion.

h. The above four forms of the series can be
used in transposition to any step of the chromatic
scale. Thus the series becomes available in forty-
eight (12 x h) modifications.

5. From this basic material melodic progres-
sions and chordal combinations can be formed, the
main principles being that the tones, whether ar-
ranged horizontally or vertically, must always
occur in the arrangement of the series, and that
its twelve tones must be presented in full, before
the series can be used again.

Since Schoenberg's important contributions to mu-

sic literature lie in the area of twelve-tone composition,

 

61Ape1. 775-6.
62Apel, 776.

-86-

the two works which the author has chosen to examine employ
that technique. They are the Fourth String Quartet, Op. 37
(1936) and the Concerto for Piano and Orchestra, 0p. #2
(19t3).

FOURTH STRING QUARTET

The original version of the row on which the Fourth

String Quartet is constructed follows:

Ex. 70

 

It will be observed that the tritone, minor third and major
sixth are the only intervals not found in the series.

For analysis measures one through fifty-eight of
the first movement have been selected. In this passage,
out of a total of 236 units there are fifty-three and five-
twelfths tritone units, or 23%. This would seem to be a
rather large percentage, since it seems to be a popular no-
tion that Schoenberg studiously avoids tritones because of
their tonal implications. Furthermore, 15%, or fourteen of
the ninety-six tritones in the passage, may be recognized
as belonging to traditional harmonic structures. However,
not one of those structures resolves in a traditional man-

ner, and because of the nature of their context, it may be

-87-

safely assumed that the fact that they are traditional is ir-
relevant. One of these traditional harmonic structures ap-

pears in the following passage.

Ex. 71 Schoenberg, Quartet No. Q, meas. 36-37.

-____-‘\
b /,_-\

    

On the last beat of measure thirty-six, an EH en-

ters in the first violin against an already sounding Bb in
the Violoncello. The tritone, added to the other two voices,
completes a major-minor seventh chord in third inversion, a
traditional structure. It may be heard as such by an alert
listener, and it stands in sharp relief against an otherwise
highly dissonant texture, both by virtue of its duration and
by its being set off by the rests which follow it. However,

it functions neither in the capacity of a traditional half-

-88-

cadence, nor does it find a traditionally satisfactory reso-
lution in the following measure.

The passage is analyzed in the following manner:

m int app dep rhy dur str res Hanson
35-36 a-eb O-sk,ct O-st,ct a 1 n - p2msdt
36 bb-e C-st,st O-ct,st a 3/h n - p2d2t2
36 eb-a O-ct,st O-ct,sk a 1/2 n - pzdzé2
36 f#-c O-ct,sk O-sk,ct u l/h n - pmsd t
36-37 bb-e O-ct,sk R-r,r a 2 t n -
37 f#-c C-st,st C-sk,st a 3/h n - mnzsdt

It is significant that almost half (u3%) of the
tritone structures in the passage under consideration are
accented. This seems to indicate that the tritone gets the
same rhythmic treatment as any other interval.

Over half (53%) of the tritones in the entire pass-
age are approached by oblique motion, while 52% are left by
oblique motion. In those approached by oblique motion, the
voice which does not have the common tone proceeds by step
2h% of the time and by skip 76%. In those left by oblique
motion, the voice which does not have the common tone pro-
ceeds by step 37% of the time and by skip 63%. These fig-
ures are an indication of the highly contrapuntal texture,
which is, of course, characteristic not only of Schoenberg,
but of contemporary chamber music in general.

In the present passage, 79% of the tritones have
a duration of less than one unit. This factor will have
more significance later in the treatise when this excerpt

is compared with the others. In the meantime, the percent-

-89-

age seems to indicate a texture which is highly active

rhythmically.
CONCERTO FOR PIANO AND ORCHESTRA

The other work of Schoenberg under consideration
is the Concerto for Piano and Orchestra, which is con-

structed on the following row:

Ex. 72

 

In the fifty-four measures analyzed, fifty-two and
three-fourths of the 163 units (33%) contain tritones. Of
the fifty-three tritone structures, six (11%) are tradition-
al, and as in the passage from the quartet, none have tradi-
tional resolutions. Thus it is seen that there is a consid-
erably larger percentage of tritone structures (33% as com-
pared to 23%) than in the quartet. Slightly fewer (11% as
compared to 15% of the tritone structures) are traditional.

The difference in the percentages of traditional
structures is hardly significant; but the large difference
between the percentages of tritone structures is interest-
ing. Perhaps the explanation lies in the differences be-
tween the structures of the rows themselves. It may easily

be seen that there are no tritones between adjacent notes

-90-

in either row. However, from what is known of the way in
which harmonies are derived from tone rows, it may be as-
sumed that intervals constructed from alternate notes will
be present in quite a large number of the harmonies. In the
row from the quartet there are no tritones between alternate
notes in the series; in the concerto there are tritones be-
tween two and four (Bb-E), five and seven (C-F#), and six
and eight (D-G#).

u7% of the tritones in the passage from the concerto
are approached by oblique motion (somewhat less than in the
quartet passage), and 38% are approached from a rest in one
or both voices. 55% are left by oblique motion, a slightly
higher figure than in the quartet passage. 38% are ap-
proached by rest; hO% are left by rest. 6% are approached
by contrary motion; u% are left by contrary motion. 9% are
approached by similar motion; 1% are left by similar motion.
Parallel motion was completely avoided as a means of approach
and departure to and from tritones.

Only 50% of the tritone structures have a duration
of less than one unit (as compared to 79% in the quartet
passage). This is true in spite of the fact that the tempi
of the passages from the quartet and the concerto (i=152 and
#:132, respectively) are similar, and seems to indicate, as
was mentioned before, the high degree of rhythmic activity

which is found in the quartet.

Ex. 73 Schoenberg, Concerto for Piano and Orchestra, meas.

'33-
tzL. bah.
_. # y

llll‘iv

       

'Pi ano

Orches‘fra.

The tritone Eb-A is sounded first on the second

beat of measure thirty-two. It occurs simultaneously in
the piano and in the orchestra, and since it is approached
by oblique motion in the former and by rest in the latter,
oblique motion is considered to have precedence over the
rest. The tones Eb and A continue to sound for three units,
since, as will be recalled, the rest which completes the
first beat of measure thirty-three is analyzed as a contin-

uation of the sound.

-92-

The entire example is analyzed as follows:

m int app dep rhy dur str res Hanson

32 e-bb 0-ct,sk O-st,ct a l n - minusdgt
32-33 eb-a O-sk,ct R-r,sk a 3 n p m3ns dt

33 g#-d 0-st,ct 0-sk,ct a 1/2 t n -

The statistics in the following chart reveal a
rather high degree of consistency between two works of dis-
similar idioms (chamber music versus orchestral music).

This consistency results, no doubt, from Schoenberg's sys-
tematic approach to composition. What differences there

are probably result from the differences in the structures
of the rows (see pp. 89-90) and the availability of a great-
er number of voices in the orchestra. This availability
seems to have its effects on the statistics of duration and
accent; certain combinations of instruments may sound a har-
monic tritone and then leave it--but in the meantime other
instruments may pick it up and continue to sustain it. This

affects not only duration, but accent as well (see page 72).

-93-

COMPARISON OF STATISTICS FROM THE TWO SCHOENBERG EXCERPTS

Quartet Concerto

tritone units 23% 33%
traditional structures 15% 11%
non-traditional structures 85% 89%
accented tritones u3% 57%
tritones having a duration
of less than one unit 79% h3%
tritones having a duration
of one unit or more 21% 57%
oblique approach 53% u7%
(O-st) (2u%) (u8%)
(O-sk) (76%) (52%)
oblique departure . 52% 55%
(O-st) (37%) (h5%)
(O-sk) (63%) (5h%)
approach from rest 21% 38%
departure by rest 20% no%
approach by contrary motion 5% 6%
departure by contrary motion 7% 1%
approach by similar motion 17% 9%
departure by similar motion 7% 1%
approach by parallel motion 5% 0%

departure by parallel motion 6% 0%

-9u-

2. IGOR STRAVINSKY

THE HARMONIC STYLE OF STRAVINSKY

For more than a half century the world of music
has been blessed with the creative outpourings of Stravin-
sky. It is difficult to make generalizations concerning
Stravinsky's harmonic style, because his works show strong
evidence of a changing vocabulary of musical idioms. From
the early works, some of which show strong influences of
the harmonic, rhythmic and orchestral idioms of Rimsky-
Kbrsakov, to some of his more recent works, which employ
the serial technique, one can trace the evolution of a
genius who has not hesitated to run the gamut of technical
systems.

Because such variety exists in Stravinsky's music,
the author has selected passages from two highly contrast-

ing works-s§ymphonie de Psaumes and L'Histoire du Soldat.

 

The former is noteworthy for its lush harmonic sonorities,
the latter for its economy of resources and for its intri-

cate contrapuntal complexities.
SYMPHONIE DE PSAUMES

Measures one through fifty-three of Symphonie de

 

Psaumes have 150 total units and forty-nine and one-third
tritone units. Structures that contain tritones therefore

comprise 33% of the texture. Of the sixty-six tritone

-95-

structures, twenty-six, or 39% are traditional (and are au-
rally recognizable as such), but none of the twenty-six
have traditional resolutions. Thirty-eight (58%) of the
tritones in the passage are approached by oblique motion.
Forty-eight (73%) are left by oblique motion; this is the
highest percentage in this category of any passage examined
thus far. Of those approached by oblique motion, the voice
which does not have the common tone proceeds by step 37% of
the time and by skip 63%. Of those left by oblique motion,
the voice which does not have the common tone proceeds by
step 35% of the time and by skip 6h%. 3% are approached by
rest; 11% are left by rest. 16% are approached by contrary
motion; 12% are left by contrary motion. 21% are approached
by similar motion; 3% are left by similar motion. 2% are
approached by parallel motion; 1% are left by parallel mo-
tion.

Of the sixty—six tritone structures in the pas-
sage forty-five, or 68% of the tritones in the excerpt,
are accented. In this category, too, the percentage is
the largest of any passage examined so far. This occurs
Tin spite of the fact that a large number (forty-nine, or
7h%) of the tritones have a duration of less than one unit.
(It should be remembered that tritones which have a duration
of one unit or longer are considered accented.)

In the following example (Ex. 7h) the tritone F-B

extends for nine and one-half units. The tones F and B are

-96-

sounding continuously from the beginning of measure thirty-

seven through the first half of best one in measure forty-

one. In the example the quarter note is the unit.

Ex. 7h Stravinsky, Symphonie de Psaumes, meas. 37-u1.

 

/ \ .

/"—_-_———_

 

The two shortest tritones in the passage are also

found in the example. The analysis for them is also given

below:

m int

3744.1 f-b

app

R-r,r
O-st,ct

dep

o-Ct,Sk
O-St,ct
O-sk,ct

str res Hanson

rhy dur

a 9-1/2 n
u 1/6 n
u 1/6 n

2 2

p mn 3 dt
pZmngst t2
panlsdt2

-97-
L'HISTOIRE DU SOLDAT

The other Stravinsky passage consists of measures
one through sixty-three of his L'Histoire du Soldat. From
the standpoint of tritone usage, this is a truly remarkable
work, for, in comparison to works examined up to this point,
it is exceptional in almost every category.

This is the only excerpt examined until now in which
there are extensive passages that are devoid of harmonic
tritones. None occur in measures 1-23, 30-3k, 39-hh, or
59-63. This dearth may be attributed to the fact that there
are only nine units of harmonic tritones (7%) in the entire
sixty-three measures (131-1/2 units). This is a considerably
smaller percentage than that of its closest competitor, the
excerpt from the Schoenberg Fourth String Quartet (23%).

The only category in which the passage is not an
exception is that of traditional structures. Three of the
twenty structures in the excerpt are traditional (15%).

None of the three resolve traditionally.

Eight of the twenty tritones are approached by
oblique motion (hO%). This is the smallest percentage in
that category of any work discussed so far. An equal num-
ber and percentage are approached by rests in one or both
voices. This is the first time that the oblique method of

approach has been equalled by any other method.

-98-

Of the twenty tritone structures in the passage,
only five were left by oblique motion (25%). There is a
drastic difference between this percentage and the next
smallest in this category, that of the excerpt from the
Fourth String Quartet of Schoenberg. For the first time,
oblique motion was surpassed as a method of departure;
eight of the tritones (h0%) were left by rest in one or
both voices, and six (30%) were left by contrary motion.

Three of the tritones (15%) are accented; this
is considerably smaller than any previous figure in that
category, the next smallest being that from the passage

in the Second String Quartet of Bartok (33%).

 

Sixteen of the tritones (80%) have a duration
of less than one unit. This surpasses the previous high
figure (79%) from the Fourth String Quartet of Arnold
Schoenberg. Both of these works are of a high degree of
contrapuntal complexity.

One of the traditional structures occurs in the

following interesting passage:

-99-

Ex. 75 Stravinsky, L'Histoire du Soldat, meas. u7-u8.

Cornef (A)

 

The tritone in measure forty-eight is analyzed in

the following manner.

m int app dep rhy dur str res Hanson

h8 d-g# O-ct,sk R-r,sk u 1/2 t n pmnzst

The sonority in the middle of measure forty—eight
is a major-minor seventh in third inversion--a traditional
harmony by classical standards. But its resolution is not

traditional.

-lOO-

SUM LARY

The following chart compares the percentages de-

rived from the two Stravinsky excerpts.

Symphonic L'Histoire

tritone units 33% 7%
traditional structures 39% 15%
accented tritones 68% 15%

tritones having a duration

of less than one unit 7h% 80%
oblique approach 58% h0%
(0-st) (37%) (25%)
(O-sk) (63%) (75%)
oblique departure 73% 25%
(O-St) (36%) (0%)
(O-sk) (6u%) (100%)
approach from rest. 3% h0%
departure by rest 11% u0%
approach by contrary motion 16% 1h%
departure by contrary motion 12% 30%
approach by similar motion 21% 6%
departure by similar motion 3% 5%
approach by parallel motion 2% 0%

departure by parallel motion 1% 0%

-101-

3. BELA BARTOK

THE HARMONIC STYLE OF BARTOK

When one thinks of the tritone and its role in
contemporary music, the first name that comes to mind is
that of Bela Bartok. Perhaps in this examination of his
music and in the comparison of it with the music of other
composers it will be discovered to what degree this as-
sociation is Justified. For analysis the author has cho-

sen two of his important works, the Second String Quartet

 

and the Music for Strings, Percussion and Celeste.

Bartok's harmonic style is based on no definite
system of tonal order (as are the styles of such widely
differing figures as Schoenberg and Hindemith); he may
therefore be termed "eclectic." Nevertheless, certain
features of his harmonic idiom are quite recognizable and
may almost be said to belong only to Bartok.

Both of the works under consideration are of a
high degree of contrapuntal interest, as are the vast ma-
jority of Bartok's instrumental compositions. One looks
in vain for a harmonic "system" in these works; it seems,
rather, that Bartok's harmonies result from his unique
and complex methods of motivic development. For example,
in the quartet, superstructures of thirds result from the

use of third-based motives.

-lO2-
SECOND STRING QUARTET

In measures one through fifty-three of Bartok's

Second String_Quartet, forty-five and one-half, or 33%

 

of the 1&0 units in the passage, contain tritones. Fifty-
four (h7%) of the 116 tritone structures are traditional,
and of those, two have traditional resolutions.

Since only thirty-three of the 116 tritones (33%)
in the excerpt are accented, it may be said that Bartok
appears to be "favoring" the interval--that is, treating
it with delicacy, possibly because of its harshness or be-
cause of what he considers to be its tonal implications.

In the passage oblique motion is the primary means
of approach and departure to and from tritones; in fact,
oblique motion plays an even greater role in the Bartok ex-
cerpt than it did in the Schoenberg quartet passage-
Ninety-three of the 116 tritones are approached by oblique
motion (81%) while eighty-three are left by oblique motion
(72%). In those approached by oblique motion the voice 8
which does not have the common tone proceeds by step 68%
of the time and by skip 32%. In those left by oblique mo-
tion the voice which does not have the common tone proceeds
by step 68% of the time and by skip 32%. 3% of the tritones
in the excerpt are approached by rest; 2% are left by rest.
9% are approached by contrary motion; 8% are left by con- '

trary motion. 6% are approached by similar motion; 15% are

-103-

1eft by similar motion. 1% are approached by parallel mo-
tion; 3% are left by parallel motion.

Ninety-one per cent of the tritones in this pas-
age have a duration of less than one unit. This surpasses
the figure from the Schoenberg quartet passage and again
indicates a highly active rhythmic texture.

The tritone in measure eighteen of the following
passage comes at a highly climactic moment—-the first real

cadence in the work.

Ex. 76 Bartok, Second String Quartet, meas. 17-18.

 

-10h-

The example is analyzed as follows:

m int app dep rhy dur str res Hanson

17 c#-g# 0-st,ct 0-st,ct a 1/3 t n -

l7 e—a# 0-st,ct S-st,st a 1/3 t n -

17 c#-g 0-st,ct 0-st,ct a 2/3 n - pmnsdt
17-18 gx-d# S-st,st 0-st,ct a 2/3 n - mn2sdt

18 e-a# S-st,sk 0-st,ct a 1-1/6 n - p2msdt

The melodic interval G-A# is analyzed as a skip,
of course, since most twentieth-century spellings are a
matter of convenience. The tritone E-A# (measure eight-
een) is approached, then, by similar motion, from step in
the lower voice and from skip in the upper. It is accent-
ed, has a duration of one and five-sixths units (the unit
being the dotted quarter note) and belongs to a non-
traditional structure. It is left by oblique motion, with

the common tone in the lower voice and a skip in the upper.
MUSIC FOR STRINGS, PERCUSSION AND CELESTE

The other Bartok passage analyzed is the first
seventy-four measures of the Music for StringsJ Percus-

sion and Celeste.

 

~286-2/3, of h7% of the 602 units in the passage,
are tritone structures. This is a larger percentage than
was found in any work examined thus far. Sixty-two (26%)
of the 2&2 tritone structures are analyzed as traditional.
This is a considerably lower percentage than in the passage

from the quartet. This is probably a result of the greater

-105-

number of voices in the Music for Strings, Percussion and
Celeste. Two of the traditional structures have traditional
resolutions.

In this passage, h2% (102 out of 2A2) of the tri-
tones are approached by oblique motion, and 5h% (130 out of'
2h2) are left by oblique motion. These figures are consider-
ably smaller than the corresponding ones from the quartet--a
factor which is attributable to the comparatively less idi-
omatic contrapuntal intricacy of the medium.

In the excerpt under consideration, u5% of the tri-
tones are accented. This seems to bear out the earlier
premise that the work appears to use the tritone in a less
delicate fashion than does the quartet, for in the latter,
only 33% are accented.

Only nine of the tritones (7%) have a duration of
less than one unit. In this category, this is the smallest
percentage of any twentieth-century work that has been ex-
amined thus far. A glance at the score reveals the obvious
explanation: this section of the work consists primarily
of rhythmic motion in whole (rather than divided or subdi-
vided) units. There are no intricate contrapuntal rhythmic
figures, but rather a constant churning of rapidly-shifting

dissonant harmonies.

-106-
SUMMARY

The following chart compares the percentages de-

rived from the two Bartok excerpts.

Quartet Music, etc.

 

tritone units 33% h7%
traditional structures h7% 26%
accented tritones 33% 5h%
tritones having a duration
of less than one unit 91% 7%
oblique approach ‘ 81% u2%
(O-st) (68%) (70%)
(O-sk) (32%) (30%)
oblique departure 72% 5h%
(O-st) (68%) (58%)
(O-Sk) (32%) (h2%)
approach from rest 3% 6%
departure by rest 2% 20%
approach by contrary motion 9% 23%
departure by contrary motion . 8% 20%
approach by similar motion 6% 10%
departure by similar motion 15% h%
approach by parallel motion 1% 19%

departure by parallel motion 3% p 2%

~107-
h. ALBAN BERG
THE HARMONIC STYLE OF BERG

It is usually said of Berg that although he was
a follower of Schoenberg and an exponent of the twelve-note
technique, his style is less dissonant and more conserva-
tive than that of his master. Perhaps the statistical
analyses of the two excerpts which have been selected will
give an indication of the extent to which this true-—at '
least insofar as tritone usage is concerned.

Berg was not consistently systematic, and
when he did use the note-row manner, he bent it
to conform with his intentions and never let his
thought be subjugated to it. At times he found
his deepest feeling in tonality, which is duly
prepared-for and led-away from. Consequently,
there is no feeling of topsy-turvydom, for the
process both ways is gradual. Berg's music re-
futes the statement that all duodecuple music is
necessarily systematic. Such moments as indi-
cate a tonic come in the natural order of things.
They are neither deliberately cultivated nor
avoided. Berg's music shows no negations of any
kind. When aligned with a note-row, it is not
bound by any rules forbidding anything, except
the use of the series and none other. When con-
venient, he avoids the series altogether, and
although there is a strong feeling for polyph-
ony, vertical reading often suggests some tonal-
ity or other. The music was written under a
strong musical impulse.

The question of the note-row solves itself
when one considers that upon which the Violin
Concerto (1935) is built. For convenience' sake
we repeat it here:

 

-108-

Ex. 77

    

TVY3+$678910111z

It will be seen how this divides itself into tri-
ads. Its use, therefore, insists upon strong tonal
suggestions which are more marked than would be the
case were the series more conjunct. The interval
of the third is, in itself, romantically sugges-
tive, and takes away any austerity of feeling.
Consequently, this Concerto abounds in warmth of
emotion, semitonal if you like, but strongly red-
olent of the inevitable Tristan and Isolde through
its trend to the appoggiatura. The series is also
capable of some considerable width of leap.

It is upon this series that Berg composes
this deeply-felt Concerto. It is an answer to
those who maintain that emotion and the note-row
are two distinct things, and the proof lies in the
aesthetic of the whole work. It enabled Berg to find
contrast between certain wideSpread themes notable
for their lyricism and such a one as opens the
Allegretto forming the second part of the first
movement--up till that moment the music has formed
an introduction stating the series as plainly as
possible, both in its original state and in retro-
gression.

In the opera Lulu (1928-1935), which remains
incomplete insofar as the orchestration of the third
act is concerned, Berg uses the following note-row:

 

 

 

Ex. 78

 

-109-

which, again, has harmonic implications ascertain-
able by dividing the gow into four groups or chords
of three notes each.6

VIOLIN CONCERTO

The first excerpt consists of the first forty-five

measures of the Violin Concerto, one of Berg's most impor-

 

tant works.

Thirty-eight and two-thirds (35%) of the 110 units
in the passage consist of tritone structures. 0f the forty
tritone structures, thirteen (33%) may be analyzed as tra-
ditional harmonies. None of those resolve traditionally.

In only one of the categories is this passage ex-
ceptiona1--that of approach to the tritone by oblique motion.
Thirty-three of the forty (82%) are approached in this man-
ner. This surpasses Bartok's 81% in the passage from the

Second String Quartet. There is also a respectable percent-

 

age for departure by oblique motion-~72% (twenty-nine of the
forty structures). This, however, does not surpass Stravin-

sky's 73% in the excerpt from the Symphonie de Psaumes, nor

 

Bartok's 72% in the quartet passage. Stravinsky, Bartok and

Berg all seem to prefer the oblique approach.

 

66Norman Demuth, Musical Trends in the Twentieth
Centur (London: Rockliff Publishing Corporation, Ltd.,
9 , 239-2“.10

 

-110-

A good example of this "oblique technique" may be
seen in the following example. The score is written in con-

cert pitch.

Ex. 79 Berg, Violin Concerto, meas. 5-7.

CL. I

     

SOLO VN.

b-U V

 

-111-

In the above example, five harmonic tritones are
approached and left by oblique motion. The example is

charted as follows:

m int app dep rhy dur str res Hanson
5 eb-a 0-ct,sk 0-ct,sk a 2 n - pdt
5 e-bb 0-st,ct 0-sk,ct a l t n -
6 g-c# 0-ct,st 0-ct,sk a l t n -
6-7 f#-c O-ct,st 0-ct,sk a l-l/2 t n -
7 d-ab 0-ct,sk 0-ct,sk a l t n -

It is interesting that four of the five structures
are traditional with non-traditional resolutions.

In the excerpt from the Violin Concerto 65% of the

 

tritones are accented. 57% have a duration of less than one

unit.
LULU

The other Berg excerpt consists of measures 86-15h
from his opera £213. Eighty and one-half (35%) of the 228
units consist of tritone structures. Eighteen (22%) of the
”eighty-two tritone structures are traditional, but none of
the eighteen are resolved in traditional fashion. Fifty-
eight (71%) of the tritones are approached by oblique mo-
tion; fifty-six (68%) are left by oblique motion. But in
spite of the high incidence of oblique motion, every other
type of motion to or from tritones is present in the pas-
sage. The following example shows the longest tritone in

the excerpt. It is left by contrary motion.

~112-

Ex. 80 Berg, Lulu, meas. 103-105.

 

 

-113-

The above example is analyzed as follows:

m int app dep rhy dur str res Hanson

103 e-bb R-r,ct O-st,ct u 1/u n - pmhnszdg

103 d-ab 0-sk,ct O-sk,ct u 1/14 n - pganBS 5

103 g-db 0-sk,ct O-st,ct u l/h n - p mn s d t

103 c-f# O-ct,sk R-ct,r a 1 n - p2m2n 3dt
103-10h e-bb O-sk,ct C-st,st a 5-1/2 n - pzmenfiszd2t3
103-10h g-db R-r,ct S-st,st a 5 n - pamznhs2d2t3
103-101; c-f# R-ct,r 0-ct,st a 2 n - p2m2nh32d2t3

101; c-f# O-ct,st O-ct,sk u 1/u n - p2m2nh32d2t3

At another spot in the passage analyzed, Berg uses

parallel motion between two tritones:

Ex. 81 Berg, Lulu, meas. 115-116.

 

-11u-

The example is analyzed below:

m int app dep rhy dur str res Hanson
115 db-g 0-st,ct P-st,st u l/h n - p3mnszd2t
116 c-gb P-st,st 0-ct,st a 1/u t n -2

116 f#-c S-sk,st S-sk,st a 1/2 n - pmn at

In the excerpt from Lulu, h9% of the tritones are
accented. 68% have a duration of less than one unit.

Since Schoenberg and Berg are the only two expo-
nents of the twelve-note technique being considered in this
treatise, it is of interest to compare their tendencies as
regards the use of the harmonic tritone. The following
chart compares the average of the two Schoenberg percentages

in each category to the Berg figures.

Concerto Lulu Ber Schoenberg
(average). Taverage)

tritone units 35% 35% 35% 28%
traditional

structures 33% 22% 27.5% 13%
accented tritones 65% h9% 57% 50%

tritones having a '
duration of less

than one unit 57% 68% 62.5% 61%
oblique approach 82% 71% 76.5% 50%
(O-st) (55%) (62%) (58.5%) (35%)
(O-sk) (u5%) (38%) (kl-5%) (5h%)
oblique departure 72% 68% 70% 53.5%
(O—st) (hl%) (59%) (50%) (bl-5%)
(O-sk) (59%) (hl%) (50%) (SB-5%)
approach from rest 2.5% 10% 6% 29.5%

-115-

Concerto Lulu Ber Schoenberg
(average) (average)

departure by rest 7.5% 13% 10% 30%
approach by contrary

motion 5% 0% 2.5% 5.5%
departure by con-

trary motion 7.5% 7% 7% 10%
approach by similar

motion 8% 1h% 11% 13%
approach by parallel

motion 2.5% 5% h% 2.5%
departure by parallel

motion 3% 2% 2.5% 3%

It is a mild surprise that in both of the Berg
passages, the percentages of tritone structures are high-
er than in either of the Schoenberg excerpts. Some would
expect the "less dissonant composer" (Berg) to use fewer
tritones, since the tritone is considered to be one of the
most dissonant of intervals. However, this is a rather
naive oversimplification. No statistics are needed to
prove that Berg is the more conservative composer--lis-
tening to his music is all the verification one needs.

The tritone, of course, is not the sole criterion
of dissonance; minor seconds and major sevenths appear to
, be sharper. More telling is the fact that Berg's tradi-

tional tritone structures far outweigh those of Schoenberg.

-116-

(Berg's percentages were 33% and 22%; Schoenberg's, 15% and
11%.)

In the percentages-of-tritones category it is found
that in the two Berg passages, the percentages are identical
(35%). In Schoenberg they are 23% and 33%. It is interest—
ing to note the consistency of the two twelve-note composers
as compared to the others whose music has been examined thus
far: Bartok and Stravinsky. In the category of tritone
units, the difference between the two Berg excerpts (0%) and
that between the two Schoenberg passages (10%), as compared
to the differences between the Bartok excerpts (1h%) and the
Stravinsky passages (26%), serve to demonstrate the higher
degree of consistency which one would expect among "system-
atic," as opposed to "eclectic" composers.

As in the Schoenberg excerpts, none of the tradi-
tional structures in the Berg passages resolved traditional-
1y.

All of the four twentieth-century composers who have
been examined so far (Schoenberg, Stravinsky, Bartok, Berg)
used considerably more oblique motion than any other means
of approaching and departing from tritones. Berg used more
oblique motion than did Schoenberg-~both in approach to the
tritone (82%, 71% against 53%, h7%) and in departure from it
(72%, 68% against 52%, 55%). Note again the high degree of
consistency found in both composers, especially Schoenberg.

Again, this contrasts sharply with the two eclectic composers,

-117...

as is indicated by the following table, which gives the dif-

ferences between the two excerpts of each composer from the

 

standpoint of oblique motion to and from harmonic tritones.
As would be expected, Stravinsky is the least consistent,
with Bartok as a close second. Schoenberg is the most con-

sistent, with Berg close at hand.

DIFFERENCES BETWEEN TWO WORKS OF EACH COMPOSER FROM THE
STANDPOINT OF OBLIQUE MOTION TO AND FROM TRITONES

approach departure
Schoenberg 6% 3%
Berg 11% u%
Bartok 39% 18%
Stravinsky 18% h8%

5. PAUL HINDEMITH
THE HARMONIC STYLE OF HINDEMITH

It may easily be argued that the most self-conscious
twentieth-century composer regarding his use of the tritone

is Hindemith. In Volume I of his treatise, The Craft of

 

Musical Composition, he treats the tritone at great length,

 

and groups chords into two c1asses--those which do not con-
tain tritones (Group A) and those which do (Group B).
Before examining the two passages which have been

selected from the work of Hindemith, it seems interesting

-1l8-

and pertinent to summarize his theories regarding the use of
the tritone.

Hindemith's system of composition is based on the
harmonic series, from which he derives the tonal and inter-

vallic relationships which he refers to as Series 1 and

 

Series 2. Series 1 is the arrangement of the notes of the

 

 

chromatic scale in order of the strength of their relation-
ships to their "progenitor" (tonic). In the following ex-

ample the progenitor is C.

Ex. 82 Hindemith, Series 1 (progenitor C).

 

 

Series 2 groups the intervals in terms of their

relative strength. The following example shows Series 2

 

with C as the progenitor.

Ex. 83 Hindemith, Series 2 (progenitor C).

 

 

-ll9-

It will be observed that in Series 1, the note

 

which forms a tritone with the progenitor is the very last

one. In Series 2 "the octave is the proudest, the noblest

 

of the intervals, and does not mingle with the others; the
tritone is the most distant relative" and is barred from
close association with the interval pairs. While the other
intervals are assigned roots in the Hindemith system, the
tritone is not. Hindemith accepts the absolute identity of
enharmonic equivalents; the augmented fourth and diminished
fifth are therefore the same.67

The tritone, says Hindemith, has no definite har-
monic or melodic significance. A third tone is necessary
to determine its position. This third tone may sound si-

multaneously with the augmented fourth or diminished fifth,

Ex. 8h

 

in which case the tritone is analyzed from a harmonic stand-
point. Or else the tritone may be a part of a group of

three successive tones.

 

67Hindemith, p. 81.

-l20-

Ex. 85

    

*2?

When these groups are not merely broken chords (which would
be purely of harmonic interest), and when attention is not
deliberately directed toward the tritone, it becomes melod-
ically subordinate. The tritone is no longer indefinite,
since one of its tones becomes the neighboring tone of a
harmonically unambiguous interval.7O

At the beginning and end of Series 2 appear the oc-
tave and tritone. The former has no significance from an
analytical standpoint because it can do no more than to in-
crease, by doubling, one tone of an interval, while making
no essential change in the content of the interval. The
latter, however, by its presence in chords, causes them to
partake of its character to such a degree that they acquire
something of both its tendency of motion towards a goal and
its indefiniteness. Thus there arises a basic difference
between chords which contain a tritone and those which do

not; Hindemith therefore divides all chords into two groups:

 

70Hindemith, p. 89.

-121-

Group A includes all chords without tritones; Group B in-

cludes all chords with tritones.71

Table of Chord Groups

A Chords without Tritone

I Without seconds or
sevenths

1. Root and bass
tone are iden-
tical

2. Root lies above
the bass tone

III
sevenths or both

1. Root and bass tone

are identical

Containing seconds or

B Chords containing Tritone

II

2. Root lies above the

bass tone

V Indeterminate

(augmented triad and

two superposed
fourths)

 

IV

VI

71Hindemi th, pp . 95-16 .

Without minor seconds or
major sevenths
The tritone subordinate

a With minor seventh only
(no major second)

Root and bass tone are
identical

b Containing major sec-
onds or minor sevenths
or both

1. Root and bass tone
are identical

2. Root lies above the
bass tone

3. Containing more than
one tritone

Containing minor seconds
or major sevenths or both
One or more tritones sub-
ordinate

1. Root and bass tone are
identical

2. Root lies above the
bass tone

Indeterminate. Tritone
predominating (diminished
triad and diminished
seventh chord)

-122-

Group II contains chords of three or more tones in
which the tritone subordinates itself to stronger intervals.
The presence of the tritone insures that there will be see-
onds or sevenths in these sonorities (except in diminished
chords). Group II is limited to chords which contain 2319;
seconds and mi§g£.sevenths, since they are less sharp.

Since the minor seventh is less harsh than the major second,
it is convenient to categorize tritone chords which contain
no interval sharper than the former (and there are only two
such chords: the complete and incomplete root position dom-
inant sevenths) in a sub-group (IIa). Chords in which ma-
jor seconds and minor sevenths appear fall into three sec-
tions. Section IIbl includes chords in which root and bass
are identical. Section IIbg contains chords in which root
and bass are not identical: inversions of dominant chords
and similar structures. Chords of the above two groups
contain only one tritone, but chords of section IIb3 contain
two or more.73

According to Hindemith, section IV "contains a
strange set of piquant, coarse, and highly colored chords.
All the chords that serve the most intensified expression
. . . are at home here." The number of tritones, seconds

and sevenths is unlimited. Obviously, such chords do not

 

73Hindemith, pp. 102-103.

-123-

lend themselves to all progressions, but are often stubborn,
especially when used in chord successions which involve rap-
idly changing groups. Those that consist of fewer tones are
easier to handle. For handling tritone chords, it is not
sufficient to know the roots. The tritone must be treated
as the most important constituent. One of the tritone mem-
bers serves as guide-tone. Hindemith gives the following
rule for finding the guide-tone:
That tone belonging to one or more tritones in
the chord which stands in the best relationship to

the root (measured by the interval-values of series
2) is to be considered the guide-tone:

Ex. 86

= ROOT >—-'> = GUIDE-Tome

,_, .b . . 01/11
J We; 1.. _

.2! .2"- /'9' /"3' 4’5?-

 

 

 

 

 

 

 

 

The progression of a tritone chord wherein the
guide-tone moves by a "good" interval (referring to the rel-

ative values of Series 2) to the resolution chord has an ad-

 

vantage over one in which it moves by a less good interval.
'"And this fact will enable the composer, in handling these

often clumsy chords, to place exactly the right chord at the

-12u-

right place." In the following examples Hindemith demon-

strates the application of the guide-tone principle:

Ex. 87

 

The above example demonstrates progressions of group II
chords to those of group I. As in most twentieth-century
music, enharmonic equivalents are considered to be absolute-
ly identical. It is unnecessary to calculate the guide-tone
in these simple progressions; all one needs is the root-
progressions. In the following example which shows progres-
sions from group II to group III, however, a conscious un-

derstanding of the treatment of the guide-tone is necessary

-125-
to an accurate judgment of the value of the progressions.

Ex. 88

a. .8. 6 d.

 

In the above, the roots progress by tritones. When one
plays these progressions he will see that a sounds quite
satisfying, while d is less convincing. This is true be-
cause of the chromatic voice-leading in three of the voices .
and the common tone in the fourth, while in d, the tritone
in the root-progression is laid bare by the whole step and
skip.

As examples of the use of the harmonic tritone in
the music of Hindemith, the author has chosen the entire

first fugue from the Ludus Tonalis and a passage from the

Sinfonietta in E.

LUDUS TONALIS (FUGUE I)

In the first fugue of the Ludus Tonalis, there are

fifty-one measures consisting of 20k units, twenty-nine

~126-

(1h%) of which are tritone units. Of the forty-two tritone
structures, thirty-two (76%) are traditional, but only two
of those resolve traditionally. This percentage of tradi-
tional structures is by far the highest of any work that has
thus far been examined. This may be accounted for in part
by the fact that the fugue is for three voices; if two of
the voices constitute the tritone, there is a good chance
that the third voice will complete a traditional tritone
sonority. Furthermore, Hindemith is fond of the sonorities
from group II, most of which are traditional. It is perhaps
surprising, however, that more of his traditional sonorities
do not resolve in traditional fashion. His use of the tra-
ditional sonority with non-traditional resolution may be ob-

served in the following example:

Ex. 89 Hindemith, Ludus Tonalis, Fuga Prima, meas. 17-19.

 

 

-127-

On beat two and one-half of measure seventeen, a D added to
the already sounding tritone F#-C completes a dominant sev-
enth chord which resolves non-traditionally. The sonority
on beat three and one-half may be considered either an in-
complete French sixth or an incomplete diminished-minor sev-
enth. The structure on the first half of beat four in meas-
ure eighteen is an incomplete dominant seventh chord in
first inversion. That on beat one of measure nineteen and
that on beat three of the same measure may be considered
incomplete French sixths or incomplete diminished-minor sev-
enths.

Example 89 is analyzed as follows:

m int app dep rhy dur str res

17 f#-c O’Ct,3t R‘Stgr a
17 db-g R-ct,r R-r,sk u
18 f-cb C-st,st 0-ct,sk a 1/2
19 g-db O-ct,st O-ct,st a
19 eb-a S-sk,st O-sk,ct a

ddfidd
55555

Twenty-five (60%) of the tritones in the passage
are approached by oblique motion and eighteen (h3%) are
left by oblique motion. The use of oblique approach and
departure in this work may be observed in the following:

~128-

Ex. 90 Hindemith, Ludus Tonalis, Fuga Prima, meas. 27-29.

 

 

The example is analyzed in the following manner:

m int app dep rhy dur str res Hanson
27 db -§ 0-ct,sk O-ct,st u 1/2 t n -
28 g—d 0-ct,sk O-st,ct a l n - pdt
29 g-d" O-sk,ct O-st,ct a 1/2 t n -
29 gb-c O-ct,st O-sk,ct a 1/2 t n -
29 rb-bb O-ct,st C-st,st a 1 t n -

50% of the tritones in the excerpt are accented.
62% have a duration of less than one unit. It is interest-
ing that all of the tritones in the passage are eighth-

notes (half units and full units).
SINFONIETTA IN E

The other excerpt by Hindemith which has been cho-
sen for analysis is from his Sinfonietta in E (l9h9).
In the fifty-eight measures analyzed there were a

total of 159 units, of which twenty-six (16%) are tritone

-l29-

units. 0f the sixty-six tritones in the excerpt, eight
(12%) are traditional. None of those resolve tradition-
ally.

It will be observed that there is no great dif-
ference in the percentages of tritone units in the pas-
sage from the Sinfonietta (16%) and in that from the Eudgs
Tonalis (lh%). However, it will be recalled that the
statement was made, in connection with the latter excerpt,
that traditional tritone structures are likely to occur
in a three-voiced composition by mere chance. They are
hardly as likely to happen in those twentieth-century
works in which four- and five-part harmonic structures
are the rule, and six-voiced structures are not uncom-
mon. These facts are borne out in comparing the percent-
age of traditional tritone structures from the £2233

Tonalis passage (76%) with that from the Sinfonietta

 

excerpt (12%). There are only eight traditional tritone

structures in the passage from the Sinfonietta; one of

 

these is given in the example below.

-130-

Ex. 91 Hindemith, Sinfonietta in E, meas. 2h-25.

 

 

In the above example, the harmonic structure on
the first beat of measure twenty-five is a dominant ninth
chord whose root is F. The tritone Eh-A is approached by
contrary motion, and receiving rhythmic stress, it is han-
dled carefully by the composer; however, it is obvious that
the resolution, while perhaps possessing its own logic, is
not traditional.

The example is analyzed in the following manner:

m int app dep rhy dur str res Hansen

25 eb-a C-st,st. S-st,st a 1/2 t n -
25 ab-d C-st,sk O-st,ct u 1/2 n pm3n2s2dt
25B26 f-cb 0-ct,sk O-ct,st a l n - p msdt

 

-131-

None of the four traditional tritone structures
from the Sinfonietta excerpt resolves traditionally.

Forty-two (6h%) of the tritones in the passage are
approached by oblique motion; twenty-six (39%) are left by
oblique motion. Fourteen (21%) are left by similar motion.
Although oblique and similar motion are the primary types
of approach to the tritone in the excerpt, all types occur.

They are all found in the following example.

Ex. 92 Hindemith, Sinfonietta in E, meas. 7-8.

H b

 

 

-132-

It is analyzed in the following manner:

m int app dep rhy dur str res Hanson

7 f#-c O-st,ct R-r,sk a 1/2 n - p2m2n233t
7 bb-e O-st,ct 0-st,ct u l/h n - pngdt

7 eb-a O-st,ct C-sk,st u 1/2 n - pmnadt

7 b-f C-sk,st C-st,st u 1/2 t n -

7 e-bb C-st,st P-st,st a 1/2 n - p 3 Sadzt
7 db—g C-sk,st P-st,st u 1/2 n - m s t3

7 f-b S-sk,st P-st,st u 1/2 n - m636t3

7 eb-a P-st,st P-st,st n 1/2 n - m636t3

8 e-a# O-st,ct R-r,r a 1 n - m636t3

8 c-gb P-st,st R-r,r a 1 n - mését3

8 d-ab P-st,st R-r,r a 1 n - m636t3

8 bP-e O-ct,st 0-ct,st u 1/h n - p3m2nu~szd2t2
8 eb-a R-r,r 0-ct,st u 1/2 n - pum2n3sudt
8 gb-c O-st,ct C-st,st u 1/2 t n -

8 d-ab 0-st,ct P-st,st u l/h n - prgagaszdt2
9 a-eb O-st,ct C-st,st a 1/2 n - m s t3

9 f-b S-st,sk P-st,st a 1/2 n - m536t3

9 c#-g P-st,st O-st,ct a l n - m636t3

In the passage from the Sinfonietta, 33% of the

 

tritones are accented; this is somewhat less than the per-
centage from the excerpt from the Ludus Tonalis. The dif-
ference may be explained by the fact that there is consid-
erably more rhythmic activity in this passage-~which is
also the reason for the high percentage of short tritones:
85%.

In spite of sizable differences in some categories,
the statistics demonstrate that Hindemith is an amazingly
consistent composer in his usage of the harmonic tritone.

This was also found to be true of two other "systematic"

composers, Schoenberg and Berg.

-133-

SUMMARY
The following chart compares the percentages de-

rived from the two Hindemith excerpts.

Ludus Tonalis Sinfonietta

tritone units 1h% 16%
traditional structures 76% 12%
accented tritones 50% 33%
tritones having a duration

of less than one unit 62% 85%
°b1133§t3ppr°a°h (33%) (3%?)

(O-sk) (28%) (211%)
°b1133§tiepartur° (93%) (33%)

(O-sk) (28%) (31%)
approach from rest 29% 0%
departure by rest 29% 1h%
approach by contrary motion h% 18%
departure by contrary motion 12% 16%
approach by similar motion 7% 9%
departure by similar motion 10% 23%
approach by parallel motion 0% 9%

departure by parallel motion 0% 8%

6. BENJAMIN BRITTEN
THE HARMONIC STYLE OF BRITTEN

Always evident in the music of the English composer
Benjamin Britten is his technical mastery of the resources
of harmony. He is essentially a lyricist, and therefore
writes in a style that is predominantly homophonic. He is
extremely fond of the sound of the harmonic third, and while
his style is not of a traditional (tertian) harmonic idiom,
he frequently writes traditional sonorities, many of which

contain tritones.
WAR REQUIEM

As an example of the work of Britten, the author
has chosen the first fifty-seven measures of the first move-
ment of the War Requiem.

0f the 199 units contained in these measures, there
are fifty-one and seven-tenths (26%) units of sonorities
which contain tritones. There are seventy-seven different
harmonic tritones, and of those, forty-three (56%) are found
in traditional sonorities. Three of those resolve tradition-
ally.

Forty-five (58%) of the tritone structures are ap-
proached by oblique motion; twenty-three (61%) are left by
oblique motion. The following example demonstrates the use

of oblique approach and departure to and from the tritone.

-135-

Ex. 93 Britten, War Rquiem, meas. 7-8.
.5

r-5

(54,.5

  

r-£5 5"1 "‘53 r-—-'£;-_——"

   

r—‘Sfir‘r’ifi/‘rfis “N5

It is analyzed as follows:

m int app dep rhy dur str res
7 d#-a 0-ct,st O-ct,sk u 1/5 t n
8 g-db O-ct,sk C-st,st a h/S t n
8 gb-c 0-sk,ct C-st,sk u 1/5 t n

In the passage under analysis. h5‘ of the tritones
are accented. 82% have a duration of less than one unit.
The rather high percentage of short tritones may be attri-
buted to the high degree of rhythmic activity, in particular,

-136-

to the quintole subdivision which permeates the excerpt (and

the entire movement).
SUMMARY

The following chart lists the percentages derived

from the Britten excerpt.

War Requiem

tritone units 26%
traditional structures 56%
accented tritones u5%
tritones having a duration of

less than one unit 82%
oblique approach 58%
(O-st) (76%)
(O-sk) (2h%)
oblique departure 61%
(O-st) (65%)
(O-sk) (35%)
approach from rest 21%
departure by rest 13%
approach by contrary motion 18%
departure by contrary motion 2h%
.approach by similar motion 3%
departure by similar motion 3%
approach by parallel motion 0%

departure by parallel motion 0%

-137-
7. HEITOR VILLA-LOBOS
THE HARMONIC STYLE OF VILLA-LOBOS

Without doubt, the most renowned twentieth-century
composer from Latin America is the Brazilian, Villa-Lobos.
He has assimilated into his musical style the folk rhythms,
melodies and harmonies of his native land. His music is
tonal, but the key centers shift rapidly. He was fond of
polytonality and frequently employed simultaneously two keys
removed by a tritone. Dissonant sonorities emerge from his
frequent use of pedal points. Practically all of his music

is homephonically conceived.

BACHIANAS BRASILEIRAS NO. 1

To represent the music of Villa-chos, the author
has chosen the first seventy measures of his Egghianas Bra-
§11eiras No. l. Tritones are sounding through thirty-eight
(27%) of the lhO units in the passage. 0f the twenty-five
actual tritone structures, seven (28%) are traditional; none
of these resolve traditionally. Villa-chos's use of tradi-

tional tritone structure is shown in the following example:

-138-

Ex. 9h.Villa-chos, Bachianas Brasileiras No. l, meas. 20.

 

The tritones in.the example are analyzed in the

following manner:

m int app dep rhy dur str res Hansen
20 d-ab O-st,ct R-sk,r a l n - met
20 b-f R-ct,r 0-ct,st u 1/2 t n -

The first tritone in the above example (D-Ab) oc-
curs as part of a non-traditional structure. The second
(B-F) is part of a diminished triad, which resolves to a

non-traditional structure.

-139.-

Fifteen (60%) of the tritones in the excerpt are
approached by oblique motion; thirteen (52%) are left by
oblique motion.

In the seventy measures analyzed, 52% of the tri-

tones have a duration of less than one unit.

SUMMARY

The following chart lists the percentages derived

from Bachianas Brasileiras No. 1, measures one through sev-

 

enty.
Bachianas Brasileiras No. l

tritone units I 27%
traditional structures 28%
accented tritones 52%
tritones having a duration

of less than one unit 52%
oblique approach 60%

(O-st) (h0%)

(O-sk) (60%)
oblique departure 52%

(O-st) (100%)

(O-sk) (0%)
approach from rest 32%

departure by rest u0%

-1ho-

Bachianas Brasileiras No. l

 

approach by contrary motion h%
departure by contrary motion 0%
approach by similar motion 0%
departure by similar motion 8%
approach by parallel motion h%

departure by parallel motion 0%

-111-

8. ROY HARRIS

THE HARMONIC STYLE OF ROY HARRIS

In much of the music of Roy Harris the harmonic
idiom is predominantly diatonic, using the triad as a
point of departure and frequently employing dissonant

polychords. Such compositions as his Third Symphony have

 

a strong tonal feeling. Parallel block chords and modal
harmony are integral features of the style of these works.
Having thoroughly assimilated the idiom of sixteenth-
century polyphony, Harris has constructed musical tex-
tures which are predominantly contrapuntal, abounding
in all sorts of fugues, canons and passacaglias.

From the standpoint of tritone usage, the Thigd
Symphony excerpt is the most unusual passage analyzed in
this treatise. Harris's avoidance of the interval seems
to be unprecedented not only among his twentieth-century
colleagues, but perhaps among all well-known composers
throughout the history of music! The blandness which re-
sults from his shunning of this interval is one of Harris's
distinguishing characteristics-~at least, in those works

whose harmonic idioms resemble that of the Third Symphony.

 

-112-
THIRD SYMPHONY

The excerpt chosen for analysis consists of meas-
ures fifty-seven through l9u (137 measures) of the first
movement of the symphony. There are tritones sounding
through three and one-third (1.2%) of the 288 units con-
tained in the passage.. 0f the five tritone structures,
none (0%) are traditional.

Two of the harmonic tritones (h0%) are approached
by oblique motion. The other three (60%) are approached by
contrary motion. This is the first time that contrary mo-
tion has surpassed oblique motion as a method of approach.
Two of the tritones (h0%) are left by oblique motion, two
(h0%) are left by contrary motion and one (20%) isleft by
rest.

Harris's use of contrary motion may be seen in the

following example:

Ex. 95 Harris, Third Symphony, meas. 75-76.

 

-113-

It is analyzed as follows:
m int app dep rhy dur str res Hansen

76 eb;a C-sk,sk C-st,st a l n - pdt

An example of his use of oblique motion is found

in the following example:

Ex. 96 Harris, Third Symphony, meas. 120.

L_-.._3}——|

 

It is analyzed in the following manner:

m int app dep rhy dur str res Hanson

120 d-g# 0-ct,sk O-ct,st a 1/3 n - t

As was mentioned before, one of the tritones was

left by rest. It occurs in the following example:

-1hh-

Ex. 97 Harris, Third Symphony, meas. 62-6h.

 

The analysis follows:

m int app dep rhy dur str res Hanson

63 db-g C-sk,st R-r,ct a l n - pmnsdt

80% of the tritones in the passage are accented.
It must be remembered, however, that there are only five
tritones in the excerpt, so it would hardly be wise to as-
sign much significance to the fact that four of the five
are accented. One must also exercise caution regarding the
matter of duration. In this category, three of the five
tritones (60%) have a duration of less than one unit. The

other two are one unit in length.

‘th'

In the three previous examples were shown three of
the four accented tritones. The remaining one follows:

Ex. 98 Harris, Third Sygphony, meas. 88-89.

 

Its analysis is provided below:

m int app dep rhy dur str res Hanson
89 bb-e C-sk,sk 0-st,ct a 1/2 n - pmnzdt

The example below gives us the only unaccented

tritone in the passage.

nlhéu

Ex. 99 Harris, Third Sygphonz, meas. 96-97.

 

It is analyzed in the following manner:

m int app dep rhy dur str res Hanson
96 g-c# O-ct,sk C-st,sk u 1/2 n - pmnsdt
SUMMARY

The following chart lists the percentages derived

from the Harris excerpt.

tritone units
traditional structures

tritones having a duration
of less than one unit

accented tritones
oblique approach

(O-st)

(O-sk)
oblique departure

(O-st)

(O-sk)
approach from rest
departure by rest
approach by contrary motion
departure by contrary motion
approach by similar motion
departure by similar motion

approach by parallel motion

departure by parallel motion

-117-

Third Symphony
1.2%
0%

60%

80%

1,0
(35:70
<i8§¢>
(0%)

0%

20%

60%

1.0%

0%

0%

0%

0%

-118-
9. SERGE PROKOFIEV
THE HARMONIC STYLE OF PROKOFIEV

The two works of Prokofiev chosen for examination

in this treatise, the Fifth Symphony_(19hh) and the Sev-

 

enth Symphony (1952), both employ key signatures--a factor

 

which implies a link with musical tradition. This link is
a part of the aesthetic philosophy of a composer who be-
lieved that music had attained a degree of dissonant com-
plexity that would allow it to go no further in that di-
rection, and that it must seek a simpler form of harmonic
expression in order to continue its development. For this
reason, it must be concluded that the music of Prokofiev's
final years was harmonically conservative at least partly
because of the composer's artistic ideals and not, as
seems to be the case with his countryman Shostakovich

(and with most other Soviet artists), because of govern-
mental coercion.

Prhkofiev's harmonic idiom is remarkably expres-
sive and varied. Though his most recent works (including
those discussed here) are often spiced with chromatic dis-
sonances, they are basically diatonic. Typically twentieth-
century Russian are Prokofiev's sudden changes of tonality.
His keys are clearly defined, and occasional atonal pas-

sages only throw them into sharper relief.

-1119-

The Fifth Symphony_is in Bb major, while the Sev-

 

enth Symphony is in C# minor and the enharmonic parallel

 

major, Db major.
FIFTH SYMPHONY

For analysis the author has selected the first

fifty-four measures of Prokofiev's Fifth Symphony. In

 

the passage tritone structures sound through eleven and
five-sixths (7%) of the 167 units. There are twenty-four
tritone structures, three (13%) of which are traditional.
Two of the three (67%) resolve traditionally.

Thirteen (5h%) of the tritones in the excerpt are
approached by oblique motion. Fourteen (58%) are left by
oblique motion. The following example illustrates Proko-
fiev's use of oblique motion in both approach and depar-

ture.

-150-

Ex. 100 Prokofiev, Symphony No. 5, meas. 18-20,

b 17.).

 

In the example, four tritones are both approached

and left by oblique motion. They are analyzed as follows:

m int app dep rhy dur str res Hansen
18 db-g O-ct,st O-ct,st a 1/2 n - pmnsdt
18 b-f O-st,ct O-st,ct u 1/2 t n -

l9 d-ab O-st,ct 0-st,ct u 1/2 n - pnzsdt
20 d-ab O-st,ct O-st,ct u 1/2 n - pmnsdt

-151-

It was stated before that while Prokofiev's idiom

in the Fifth Symphony is basically diatonic, occasional

 

chromatic passages are found. The above example, in which
all twelve notes are found in the first seven beats, is a
case in point.

In the first fifty-four measures of the symphony,
38% of the tritones are accented and 92% have a duration
of less than one unit. The latter is the largest percent-
age in its category of any excerpt that has been examined
thus far. The following example illustrates this type of

harmonic tritone.

-152-

Ex. 101 Prokofiev, Smhony No. 5, meas. 3-5 .

 

proached by rest.

-153-

It is analyzed as follows:

int app dep rhy dur
bb-e R-r,r O-ct,st u l/h
bb-e R-r,r O-ct,st u l/u

str res Hanson

n - pdt
n - mst

The short tritone is, of course, not always ap-

ferent approach:

"I

 

44!

The following example demonstrates a dif-

Ex. 102 Prokofiev, Symphony No. 5, meas. N6.

-15u-

The tritone D-G# is analyzed in the following man-

ner:
m int app dep rhy dur str res Hanson
k6 d-g# O-ct,st 0-ct,st u l/h n - mst

It should be noted that although there is a skip to
the G# in Violoncello I, the G# is analyzed as having been
approached by step, as occurred in the Viola. Steps always
take precedence over skips in analysis; in this instance,
the Violoncello notes G#-A are considered to be a temporary
octave doubling of the Viola. Note in this regard that in
the above example, Violoncello I functions entirely in the
capacity of a doubling instrument, doubling successively

the double-bass, viola, second violin and first violin.
SEVENTH SYMPHONY

The other Prokofiev excerpt analyzed in this trea-
tise consists of measures one through forty-five from the

first movement of his Symphony No. 7. Tritones are sounding

 

through twenty-six and one-half (15%) of the 180 units con-
tained in the passage. There are forty actual tritone
structures, of which seventeen (h2%) are traditional. One
of those resolves traditionally.

Thirty-one (77%) of the tritones in the passage
are approached by oblique motion. Prokofiev's use of ob-
lique motion in this work is demonstrated in the following

example:

-155-

Ex. 103 Prokofiev, Symphony No. , meas. 32-3L1.

 

32
33
33

3h

int

e-a#
s#-d
e-a#
b-f

a-d#

-156-

The example is analyzed as follows:

app

O-ct,st
O-ct,sk
0-ct,st
0-ct,sk
O-ct,st

It Should

dep

O-ct,st
S-sk,st
O-ct,st
C-st,st
O-ct,st

CCSCC

dur

1/0
1/h
1/u
1/1
1/u

str

dc+5c+d-

P68

5c+l 5:3

Hanson

be remembered that, according to the

method of analysis employed in this treatise, a note in

any voice is analyzed as extending through whatever rests

may follow that note in the same unit.

Therefore,

the E

which is played by the second violin in measure thirty-

two is considered a quarter note.

are accented.

Twenty-eight (70%) of the tritones in the excerpt

As in the passage from the Fifth Symphony,

 

the composer used a considerably large percentage of har-

monic tritones that were shorter than one unit (82%). Of

the excerpts that have been discussed so far,

in fact,

this figure is surpassed only by the latter passage (92%).

The use of the short tritone in the current work may be

seen in the above example.

-157-
SUMMARY

The following chart compares the percentages de-

rived from the two Prokofiev excerpts.

Fifth Seventh

tritone units 7% h 15%
traditional structures 13% h2%
accented tritones 38% 70%
tritones having a duration

of less than one unit 92% 82%
oblique approach 5h% 77%

(O-st) (85%) (97%)

(O-Sk) (15%) (3%)
oblique departure 58% 70%

(O-St) (71%) (89%)

(O-sk) (29%) (11%)
approach from rest 29% 0%
departure by rest 9% 8%
approach by contrary motion 17% 7%
departure by contrary motion 29% 19%
approach by similar motion 0% 16%
departure by similar motion h% 3%
approach by parallel motion 0% 0%

departure by parallel motion 0% 0%

-158-
10. SAMUEL BARBER
THE HARMONIC STYLE OF BARBER

Barber is among those twentieth-century musicians

who, in works such as his Second Symphony, have adhered to

 

the principles of tonality. Much of his music is charac-
terized by the fluctuation between major and minor. He is
an expert at the use of counterpoint and his rhythmic style
is interesting and varied. In some of his mature works he
experimented with the polytonal and rhythmic innovations of
Stravinsky; in others he tried his hand at the twelve-tone
technique. But most of his experiments take place within
the confines of a more or less clearly defined tonality.

He is capable of the sharpest dissonances; yet he does not
hesitate to make use of the most conventional harmonies and

progressions when they suit his creative purposes.
SECOND SYMPHONY

To represent Barber's use of the harmonic tritone,
the author has chosen the first forty measures of the see-
one movement of the Second Symphony. Of the 201 units in
the passage, there are forty-eight units (23%) of sounding
harmonic tritones. There are forty-five actual tritone
structures and of these, sixteen (36%) are traditional;

none resolve traditionally.

-159-

Barber's use of traditional structures involving

tritones seems to be more then coincidental, and may be ob-

served in the following example.

Ex. 10h Barber, Symphony No. 2, Second Movement, meas. 22-23.

ures, four are traditional.

lows:

m int
22 bb-e
22 8#-d
22 bb-e
22 f-b
23 bb-e
23 s#-d

Of the six tritone

app

0'31? ’01:
0-ct,st
O‘ct ,St
O-ct,st
O-ct,st
O-ct,st

dep

0-ct,st
0-ct,sk
O-sk,ct
O-ct ,St
O-ct,sk
O-sk,ct

 

structures in the above two meas-

The example is analyzed as fol-

rhy dur str res Hanson
a l n - m3s2t
a l t n -

u 1/2 n - pamsdt
a l t n -

a l t n -

a 1 t n -

The second tritone structure in measure twenty-two

a major-minor seventh chord whose root is Bb.

(It must

~160-

be recalled that enharmonic equivalents are analyzed as
being absolutely identical, since spellings in contemporary
music are usually a matter of convenience; in this case,

G# is considered to be the same as Ab.) The structure on
beat five of measure twenty-two is a diminished-minor sev-
enth chord whose root is B, with the fifth in the bass. On
beat three and one-half of measure twenty-three is an in-
complete major-minor seventh chord in third inversion. The
last, on beat four of measure twenty-three, is an incomplete
root-position major-minor seventh chord.

Since none of these chords resolves traditionally,
it may be asked why the composer chose to use them at all,
or if, indeed, he was aware that he;ggg using them. The
answer to the latter question is almost certainly yes, since
it is hardly plausible that such a skillful composer as Bar-
ber would let traditional sonorities creep into his music by
more chance. To the former question, an answer can only be
speculated. The author is of the Opinion that the composer
merely liked the sound of the traditional sonorities, which
suited his musical intentions at that particular time. Why
should he not, therefore, feel free to use them?

In the forty measures analyzed, thirty (67%) of the
tritones are approached by oblique motion; thirty-four (76%)
are left by oblique motion. Barber's use of oblique motion

may be seen in the following example:

-161-

EX- 105 Barber, Symphony No. 2, Second Movement, meas. 16-18.

 

It is analyzed as follows:

m int app dep rhy dur str res Hansen
16 a-d# 0-st,ct O-ct,st a l t n -
l6 c-f# 0-ct,st O-ct,st a 1 t n -
l7 f-b 0-ct,st O-ct,sk a l t n -
l7 a-d# O-st,ct O-ct,sk a l t n -
18 f-b 0-ct,sk O-ct,st u 1/2 n - pnasdt

In the first forty measures of the second movement
80% of Barber's tritones are accented. This is the highest
percentage in that category of any passage analyzed thus
far. Barber's use of the accented tritone may be observed

in both of the above examples. Only 2h% of the harmonic

~162-

tritones in the forty measures have a duration of less than
one unit. This is the smallest percentage in that category
of any passage analyzed--a fact which is largely responsi-
ble for the large figure in the previous category, since
long tones are considered to have agogic accent.

In the following example the tritone D-G# on the
first beat of measure five continues to sound for two and
one-half beats while the two notes of which it consists are

tossed among the voices.

Ex. 106 Barber, Symphony No. 2, Second Movement, meas. h-5.

 

-163-

The example is analyzed as follows:

m int app dep rhy dur str res Hansen

5 d-g# R-st,r 0-ct,st a 2-1/2 n - p3m3ngsgd3t3
5 f-b R-r,r 0-st,ct a 2-1/2 n - p3m3n s d3t3
5 a-d# R-st,r 0-ct,st a l n - p3m3n233 3t3
5 c-f# 0-st,ct 0-ct,st a 1 n - p man sd t

The interval F-B also sounds continuously for two
and one-half units, even though it is tossed around among

different voices.

SUMMARY

The following chart lists the percentages derived

from the Barber excerpt.

SymphonyANO. 2

 

tritone units 23%
traditional structures 36%
accented tritones 80%
tritones having a duration
Of less than one unit 2h%
Oblique approach 67%
(O-st) (6h%)
(O-sk) (35%)
Oblique departure 76%
(O-st) (60%)
(O-sk) (h0%)
approach from rest 12%

departure by rest h%

approach by contrary motion
departure by contrary motion
approach by similar motion
departure by similar motion
approach by parallel motion

departure by parallel motion

-151-

Symphony NO. 2

 

-155-

ll. AARON COPLAND

THE HARMONIC STYLE OF COPLAND

It is difficult to make generalizations about the
harmonic idiom of Copland since he has experimented with
various systems, including the twelve-tone technique. The

remarks here, then, apply mainly to the Third Symphony and

 

to those works which resemble it in style.

The Third Symphony is primarily diatonic and dem-

 

onstrates Copland's fondness of the juxtaposition of major
and minor, both simultaneous and alternate. Its strong
key centers often serve as a foil for the use of polyte-
nality. The frequent use of the traditional triad and the
employment of the modes are characteristic. The strong
use of tonality renders the bold modulations all the more

striking.

THIRD SYMPHONY

For analysis the author has selected the first

eighty-six measures of the Third Symphony. Of the 3hl

 

units contained in the passage, only twenty-one and five-
sixths (6%) contain tritones. As compared with the other
excerpts in this study, this represents a sparing use of

the interval. Exceptional are spots like the following--

a single measure which contains four harmonic tritones.

-l66-

Ex. 107 Copland, Symphony No._3, meas. 80.

 

The tritones in the measure are analyzed as follows:

m int app dep rhy dur str res Hansen
80 o-bb S-sk,st O-ct,st a 1/2 t n -
80 b-f O-ct,sk 0-st,ct u 1/2 n - pdt
8O a-eb 0-ct,st P-st,st u 1/2 n - met
80 g#-d P-st,st O-ct,sk a 1/2 n - pdt

Of the twenty-three tritone structures in the ex-
cerpt, nine (39%) are traditional; none of the nine are re-
solved traditionally. The use of traditional tritones in

the passage may be seen in the following example:

-167-

Ex. 108 Copland, Third S hon , meas. 83-85.

 

-l68-

The example is analyzed as follows:

m int app dep rhy dur str res
83 c-f# C-st,sk C-st,sk a l t n
85 c-f# 0-st,ct S-st,sk a l t n

In measure eighty-three on beat two, there is a
diminished triad, to which is added on beat two and one-
half, a D, completing a major-minor seventh chord in third
inversion, which resolves non-traditionally. In measure
eighty-five the traditional structure is another diminished
triad.

Ten (h3%) of the tritones in the passage are ap-
proached by Oblique motion; thirteen (57%) are left by ob-
lique motion. Copland's use of Oblique motion may be Ob-
served in the two previous examples.

83% of the tritones in the excerpt are accented.
This represents the largest percentage in that category Of
any passage that has been examined thus far in the treatise.
Copland's use of the accented tritone may be observed in the

following example.

-169-
Ex. 109 Copland, Third Symphony, meas. 66-68.
3 '2 V ‘ 3‘

J

3 ‘

     

   

    

 

The example is analyzed as follows:

m int app dep rhy dur str res . Hanson
66 a#-e O-ct,st P-st,st a n - pdt
67 g#-d P-st,st O-sk,ct a l t n -
67 b-f C-sk,sk O-ct,st a 1/2 t n -
68 b-f C-sk,st O-ct,st a 1/3 t n -

-170.-

Although there are no tritones in the passage which
are longer than two units, over half of the harmonic tri-
tones (57%) are one unit long or longer, which means that

h3% of the tritones have a duration of less than one unit.

SUMMARY

The following chart lists the percentages derived
from the Copland excerpt.

Third Symphony

tritone units 6%
traditional structures 39%
accented tritones 83%
tritones having a duration of
less than one unit h3%
oblique approach h3%
(O-st) (60%)
(O-sk) (h0%)
Oblique departure 57%
(O-st) (77%)
(O-sk) (23%)
approach from rest h%
departure by rest u%
approach by contrary motion 35%
departure by contrary motion 13%
approach by similar motion 9%
departure by similar motion 17%
approach by parallel motion 9%

departure by parallel motion 9%

Chapter VI

COMPARISONS AND CLARIFICATION

OF THE DATA

The purposes of this work have been to examine
tritone usage in the twentieth century, to determine with
what degree of consistency the interval has been used among
twentieth-century composers and to compare its usage in the
twentieth century with that of the past. For the sake Of
objectivity the author has pursued a statistical course in

dealing with these problems.
THE USE OF THE HARMONIC TRITONE IN THE BACH CHORALE

In order to make comparisons with the past it is
necessary to Obtain statistical data from music Of the pe-
riod of traditional harmony. For this purpose the author
has chosen to analyze six of the chorale harmonizations of
J. S. Bach. The harmonizations were selected from the Amer-

ican edition of Bach's 371 Four-Part Chorales.78

 

In typical chorale harmonizations (and the author
chose the six chorales discussed in the present chapter be-
cause they seem to be just that) one would expect to find

only traditional structures and traditional resolutions;

 

78J. S. Bach, 371 Four-Part Chorales (New York:
Associated Music PubliShers, Inc., n.d.).

 

-171-

-172-

but such is not the case. Throughout this treatise, the
word "traditional" may be defined: "in accordance with cus—
toms established by frequent usage." The word, then, would
not be used to describe those exceptional chord connections
known as "retrogressions."

In the Bach chorale harmonization Befiehl du deine

 

H252 (Number 3H0), tritones are sounding through fourteen
and one-fourth of the forty-eight units. There are twenty-
two actual tritones in the chorale and of those, twenty-one
belong to traditional structures (95%). Eighteen (86%) of
the traditional structures resolve traditionally. Thirteen
(59%) of the tritones are approached by oblique motion;
three (lh%) are left by oblique motion; twelve (55%) are
left by contrary motion. These percentages are not sur-
prising; one expects tritones to be approached by oblique
motion and resolved by contrary motion. It 1§.surprising,
however, that in.this study, gngy,the Bach chorales use a
preponderance of contrary motion in departure from the tri-
tone. In the twentieth-century excerpts as well as in the
transitional passages (Wagner, Debussy), oblique motion is
by far the favorite method of departure.

In Befiehl du deine Wege (Number 31,0) 36% of the
tritones are accented; 68% have a duration of less than one
unit. The shortest tritone in the chorale has the duration

of a sixteenth note (one-fourth unit):

-173-
Ex. 110 Bach, Befiehl du deine Wegg (Number 3&0), meas. 7-8.

)7.

    

The two measures are analyzed in the following manner:

int app dep rhy dur str res

b-f C-st,st O-st,ct
ab-d 0-st,ct S-st,st
bb-e 0-st,ct C-st,st
O#-g O-Ct,8t C-St’St

mflqw 3
555m
HHHH
NV
dddd
dddd

/
/

The chorale contains one augmented sixth chord:

Ex. 111 Bach, Befiehl du deine Wage, meas. 10.

f3

 

.17A-

The above measure is analyzed as follows:

m int app dep rhy dur str res
10 g#-d S-st,st 0-st,ct u 1/2 t t
10 b-f O-st,ct S—st,st u 1/2 t t
10 g#-d 0-st,ct C-st,st u 1/2 t t

The other chorales chosen for analysis are: Ach
bleib bei ups, Herr Jesu Christ (Number 17?); Christ, der

du bist der helle Tag (N‘hber 230); Denket dem Herren, dean

 

er ist sehr freundlich (Number 228); Eins ist Noty_ach Herr,

 

dies Eine (Number 280) and Freuet euch, ihr Christen alle

 

(Number 8). The following chart lists the percentages de-

rived from the six chorales.

Chorales

tritone units 29%
traditional structures 90%
accented tritones 37%
tritones having a duration

of less than one unit 6h%
Oblique approach 58%

(O-st) (100%)

(O-sk) (0%)
Oblique departure lh%

(O-st) (100%)

(O-sk) (0%)
approach from rest 0%

departure by rest 0%

-175-

Chorales
approach by contrary motion 23%
departure by contrary motion 53%
approach by similar motion 16%
departure by similar motion 3h%
approach by parallel motion 6%
departure by parallel motion h%

COMPARISONS

In the sections which follow there will be fre-
quency graphs for each category Of tritone usage. These
graphs will rank all passages (including the Bach cho-
rales and the Wagner and Debussy excerpts) in order Of
frequency, thus comparing work to work.

In addition, following each frequency graph there
will be a separate graph showing all of the works examined

in order of degree of deviation from the Bach chorales.

 

The work with the smallest number of percentage points
(whether plus or minus) deviation from the Bach chorales
in a particular category will be considered the most tra-
ditional (in that category); that with the largest number
of percentage points difference will be considered the
least traditional. Following each frequency graph, av-

erages will be given.

-175-

LIST OF ABBREVIATIONS

In order to save space the following abbreviations

will be used in the graphs:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Bach Bach (six chorales)

Barb Barber, Second Symphony

Bk 2Q Bartok, Second String Quartet

Bk Mu Bartok, Music for Stringg, etc.
Bg Lu erg, Lulu

Bg VC Berg, Violin Concerto

Bri W Britten, War Requiem

Cop 3 Copland, Third Symphony

Deb P Debussy, Pelleas and Melisande
Her 3 Harris, Third Symphony

Hin L Hindemith, Ludus Tonalis

Hin S Hindemith, Sinfonietta in E

Pro 5 Prokofiev, Fifth Symphony

Pro 7 Prokofiev, Seventh Symphony

Sch h Schoenberg, Fourth String Quartet
Sch P Schoenberg, Piano Concerto

Str L Stravinsky, L'Histoire d9 Soldat
Str S Stravinsky, Symphonie de Psaumes
V-L B Villa-Lobos, Bachianas, etc.

Wag T Wagner, Tristan and Isolde

 

TRITONE UNITS

The term "tritone unit" refers to the number of
units through which tritones sound in a given passage.
The total is computed by adding the figures in the dura-
tion column of an analysis of such a passage. The per-
centage of tritone units is Obtained by dividing that

sum by the total number Of units in the passage.

-177-

GRAPH NO. 1

The following graph shows, in order of frequency,
percentages of tritone units which were found in every

passage analyzed in this treatise.

~178-

 

 

 

 

 

- aunt-u. -.a- ”an" -&D.-'---u|a-W- .‘ma-I-H...‘ b

mm
on
mm
o:

m:

 

 

fl .oz gamma

1- . 1....
HEM“, lint 41. _ I. t. F. . i E} rL. :1 5| {In 35.!53. ii! an. iffygd .31.. 3413..-: £313 .. . 3§13Lh ”Uta

W mafia: moonwaa mo momsunoopom om

 

 

 

-179-
AVERAGES

Twentieth-century average, 22% average of transi-
tional passages (Wagner-Debussy), 35.5%. This indicates
that more tritones were used in the transitional period
than had been used in the period of traditional harmony.

The tritone is used with lggg frequency in the twentieth
century than in.the period of traditional harmony, but there
are a few composers (Bartok, Berg and sometimes Stravinsky
and Schoenberg) who employed the tritone with more frequency

than did Bach.

 

m cI qeq

r

Graph No. la

c==== H T‘A

 

in Percentages of

Degree of Deviation
Tritone Units

cm 232 38

unzum 8 J49

cmz-l d HOS

m M 7'18

CM. GA 98

 

“mam“... n1 38

Mahdi-u.- “a p. n..- ".4 ¢.-.--yb- n qos

 

.—.-u¢lt-oam~.n n DH—fil-‘u nu. -q

A
‘4")

W 77. 'Q“-l.~a-~"—' '---'u--'h.— . 4......- up...—

A
.‘43'

 

:hM--.tln..- -. v..-.a-v.p—.. .‘m -‘....a.. cum... U...

M-.%HI~ I‘m gy~ww. ... alu'l-rc--‘C~~oi--.-. ~ - - . u:

 

“it“ "‘~‘- ‘RI‘O‘Q '.--...-I-, -~~----- -.. -.| s-.. .‘n .-...._......._ ...n,.“~u

--‘. '.“-’o-'\_ “in. . .- p-o-on-..—s: u u-cl—M-M - pun-«a.- -
".
y
4
“mam-she “mg—1).. an n - u -.'~-' .- -. anfi-‘q —.- .— d... n... ..- .| a..- -.- .. »-lh-.1--.-.n...-u.-. _
I
~.— ~ A .- — ‘3‘".th I, .--- LQ.I.--‘-n. u-‘uvv J.- 1-

-'Am

30
25
20
15
10

S

 

 

 

 

~181-

Graph No. la (Degree of Deviation in Percentages of
Tritone Units from the Six Bach Chorales) indicates that
Debussy and Villa-Lobos, and sometimes Schoenberg, Stravin-
sky and Bartok, are closest to Bach from the standpoint of
frequency of tritone usage. However, the eclectic composers
(Bartok and Stravinsky--particularly the latter) have a high

degree of inconsistency in this and other respects.

TRADITIONAL STRUCTURES

The following graph shows, in order of frequency,

percentages of tritone structures which are traditional.

-182-

 

 

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Percentages of Tritone

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-183-
AVERAGES

Twentieth-century average, 30%; average of transi-
tional passages, 57%. This indicates the decline in the use
of traditional tritone structures from the time of Bach to
the present day-~a well-known fact on which further comment
is unnecessary.

Surprising, however, are the positions of some of
the passages in the list. One would expect the Debussy,
Barber, Villa-Lobos and Prokofiev excerpts to be nearer the
top of the list and the passages of the twelve-tone compo-
sers (Schoenberg and Berg) to be near the bottom. However,
this is only 232 of a number of categories, regardless of
how important it may seem. A true picture can be Obtained

only after examination of all the categories.

-18u-

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-186-

AVERAGES

Twentieth-century average, 52%; average of transi-
tional passages, h9%. These figures indicate that the tri-
tone increased in rhythmic emphasis from the period of tra-
ditional harmony, through the transitional period, and down

to the twentieth century.

~187-

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-190-

AVERAGES

Twentieth-century average, 65%; average of transi-
tional passages, 69%. These figures, together with the
figure from the six Bach chorales (6h%) indicate that there
has been little change in tritone duration from the period
of traditional harmony, through the transitional period, and
down to the present day. Short tritones have been used with

slightly more frequency than have long tritones.

~191-
OBLIQUE-STEP APPROACH

The approach to the tritone by oblique motion with
a step in one voice is a traditional procedure. The follow-
ing graph lists, in order of frequency, the percentages of
oblique approaches in which that voice which does not retain

the common tone proceeds by step.

-192-

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that, wnl e as we learned in the graph conce1n

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preached oy eel-qae motion remains virtually unch-. oed irom

the eriod of traditional harmony to the present day, the
oblique-step motion to the tritone has declined
11el less, stepwise motion in the voice which
does not retain the common tone is employed with slightly

m re frecuency than motion by skip.
OBLIQUE ~SKIP APPROACH

The approach to the tritone by o‘o clique motion with
a skip in one voice is a non-traditional procedure. The

foll wing graph lists, in order of frequency, the percent-

A

ages of celique approaches in which that voice which does

not retain the common to: e proceeds by skip.

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-195-

AVERA ES

C)

Twentieth-century average, h3.6%; average of tran-
sitional passages, 18%. These figures, together with the
percentage from the Bach harmonizations (0%), indicate the
converse of the figures from the oblique-step graph; that
is, they indicate that although oblique-skip approach to
the tritone continues to be used with less frequency than
oblique-step approach, oblique-skip motion has increased in
frequency from the period of traditional harmony, through

the transitional period, and down to the present day.
DEGREE OF DEV IATI ON

The following graph lists all twentieth-century
and transitional works in order of degree of deviation (from
the standpoint of oblique-step and oblique-skip approach to

the tritone) from the six Bach chorales.

 

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-l97-

DEPARTURE BY OBLIQUE MOTION

The following graph lists, in order of frequency,

percentages of tritones which are left by oblique motion.

~198-

  
    
    
  

 
 

Departure by Oblique Motion

  

Graph No. 6

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-l99-

AVERAGES

Twentieth-century average, 57%; average of transi-
tional passages, 65%. These figures, together with the per-
centage from the Bach harmonizations (11%), show that there
was a marked increase in the oblique method of departure
from the period of traditional harmony to the transitional

period; there was a slight decrease from the transitional

period to the present day. Nevertheless, oblique departure

is the primary means of departure from the tritone in the

twentieth century, while, as will be seen, contrary motion

i s by far the favorite in the period of traditional harmony.

DEGREE OF DEV IAT ION

The following graph lists all twentieth-century
and transitional works in order of degree of deviation (from

the standpoint of percentages of oblique departure) from the

six Bach chorales .

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-201-

OBLIQUE-STEP DEPARTURE

The departure from the tritone by oblique motion
with a step in one voice in the traditional type of oblique

departure. The following graph lists, in order of frequency,

the percentages of oblique departures in which that voice

which does not retain the common tone proceeds by step.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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-203-

AVERAGES

Twentieth-century average, 61.65%; average of tran-

sitional passages, 73.5%. These figures, together with the
percentage from the Bach harmonizations (100%), indicate that
the ratio of oblique-step departure, like that of oblique-
step approach (lOO%--82%--S6.LL%), has declined steadily.
However, oblique-step departure (again, like oblique-step

approach) continues to be slightly more frequent than oblique-

skip departure.
OBLIQUE-SKIP DEPARTURE

The departure from the tritone by oblique motion
with a skip in one voice is a non-traditional procedure.
The following graph lists, in order of frequency, the per-
centages of oblique departures in which that voice which

does not retain the common tone proceeds by skip.

 

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-205-

Twentieth-century average, hh.2h%; average of tran-
sitional passages, 26.5%. These figures, together with the
Inercentage from the Bach harmonizations (0%), indicate that
ajlthough oblique-skip departure from the tritone continues
tc: be used with less frequency than oblique-step departure,
oblique-skip departure has increased in frequency from the

period of traditional harmony, through the transitional pe-

riod, and down to the present day.

DEGREE OF DEVIATION

The following graph lists all twentieth-century and

transitional works in order of degree of deviation (from the

standpoint of oblique-step and oblique-skip departure from

the tritone) from the six Bach chorales.

 

 

 

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-207-

APPR OACH BY REST

The following graph lists, in order of frequency,

puarcentages of tritones which are approached by rest.

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Approach by Rest

 

-209-
AVERAGES

Twentieth-century average, lh.7h%; average of tran-
sitional passages, 16.5%. These figures indicate that there
is little difference between the transitional period and the
present day regarding the use of the tritone approached by
rest. However, since no tritones were approached by rest in
the six chorales, one may assume that this method of approach

was not a characteristic feature of the chorale style.

DEGREE OF DEVIATION

The following graph lists all twentieth-century and
transitional works in order of degree of deviation (from the

standpoint of approach by rest) from the six Bach chorales.

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-211-
DEPARTURE BY REST

The following graph lists, in order of frequency,

percentages of tritones which are left by rest.

‘ -212--

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-213-

AVERAGES

Twentieth-century average, 18%; average of transi-

tional passages, 7%. These figures, together with the fig-
ure from the Bach chorales, indicate that there was an in-
crease in departure by rest from the period of traditional
harmony to the transitional period, and that there was a de-

cline in the use of departure by rest from the transitional

period to the present day.

DEGREE OF DEVIATION

The following graph lists all twentieth-century
and transitional works in order of degree of deviation (from

the standpoint of departure by rest) from the six Bach cho-

rales.

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in Percentages of

Degree of Deviation
Departure by Rest

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-215-
APPROACH BY CONTRARY MOTION

The following graph shows, in order of frequency,
percentages of tritones which are approached by contrary mo-

tion.

 

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~216-

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-217-

AVERAGES

Twentieth-century average, 1h.u2%; average of tran-
sitional passages, 12%. These figures, together with the
figure from the six Bach chorales, indicate that there was
more frequent use of contrary motion as a means of approach
to the tritone in the period of traditional harmony than in

the transitional period or in the twentieth century.

DEGREE OF DEVIATION

The following graph shows all twentieth-century and
transitional works in order of degree of deviation (from the
standpoint of approach to the tritone by contrary motion)

from the six Bach chorales.

 

 

 

 

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~219-

DEPARTURE BY CONTRARY MOTION

The following graph shows, in order of frequency,

percentages of tritones which are left by contrary motion.

-220-

 

 

 

 

 

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~221-

AVERAGES

Twentieth-century average, 15.7u%; average of tran-
sitional passages, 11.5%. These figures, together with the
figure from the Bach chorale harmonizations, indicate that
there was more frequent use of contrary motion as a means of
departure from the tritone in the period of traditional har-
mony than in the transitional period or in the twentieth
century. In fact, contrary motion is the primary means of

departure in the traditional period.

DEGREE OF DEVIATION

The following graph shows all twentieth-century and
transitional works in order of degree of deviation (from the
standpoint of departure from the tritone by contrary motion)

from the six Bach chorales.

 

 

 

 

 

 

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-223-

APPROACH BY SIMILAR MOTION

The following graph shows, in order of frequency,
percentages of tritones which are approached by similar mo-

tion.

-22h-

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-225-
AVERAGES

Twentieth-century average, 8.6%; average of tran-
sitional passages, 8.5%. These figures, together with the
figure from the six Bach chorales, indicate that there was
a slight decline in the use of approach by similar motion
from the period of traditional harmony to the transitional
period; however, the use of approach by similar motion re-
mained constant from the transitional period to the present

day.
DEGREE OF DEVIATION

The following graph shows all twentieth-century and
transitional works in order of degree of deviation (from the
standpoint of approach to the tritone by similar motion)

from the six Bach chorales.

-226-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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-227-

DEPARTURE BY SIMILAR MOTION

The following graph shows, in order of frequency,

percentages of tritones which are left by similar motion.

-228-

Departure by Similar Motion

 

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-229-
AVERAGES

Twentieth-century average, 7.82%; average of tran-
sitional passages, 7.5%. These figures, together with the
figure from the six Bach chorales, indicate that there was
a considerable decline in the use of departure by similar
motion from the period of traditional harmony to the tran-
sitional period; however, from the transitional period to

the present day, its use has remained virtually constant.

DEGREE OF DEVIATION

The following graph shows all twentieth-century and
transitional works in order of degree of deviation (from the
standpoint of departure from the tritone by similar motion)

from the six Bach chorales.

 

 

 

 

 

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-231-

APPROACH BY PARALLEL MOTION

The following graph shows, in order of frequency,
percentages of tritones which are approached by parallel

motion.

 

 

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-233-

AVERAGES

Twentieth-century average, 3.65%; average of tran-
sitional passages, 7%. These figures, together with the
figure from the six Bach chorales, indicate that there has

been little use of or in the twentieth century.

DEGREE OF DEVIATION

The following graph shows all twentieth-century and
transitional works in order of degree of deviation (from the
standpoint of approach to the tritone by parallel motion)

from the six Bach chorales.

-23“-

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-235-
DEPARTURE BY PARALLEL MOTION

The following graph shows, in order of frequency,

percentages of tritones which are left by parallel motion.

 

 

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-237-
AVERAGES

Twentieth-century average, 2.2u%; average of tran-
sitional passages, 7.5%. These figures, together with the
figure from the six Bach chorales, indicate that there has
been little use of departure from the tritone by parallel
motion in the period of traditional harmony, in the transi-

tional period, or in the twentieth century.
DEGREE OF DEVIATION

The following graph shows all twentieth-century and
transitional works in order of degree of deviation (from the
standpoint of departure from the tritone by parallel motion)

from the six Bach chorales.

-238-

 

 

 

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-239-
AVERAGE DEGREE OF DEVIATION (BY COMPOSITION)

The following graph shows for each composition the
average degree of deviation for all categories. Using the
six Bach chorales to represent the traditional use of the
harmonic tritone, this graph attempts to group the works
studied in this treatise from the most traditional to the
least traditional--but only insofar as the use of the har-

monic tritone is concerned.

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-241-

The graph indicates that, among the twentieth-
century composers, Hindemith, Britten and Prokofiev treat
the tritone most traditionally. Stravinsky, Schoenberg and
Harris are least traditional in the usage of the interval,
while Berg, Bartok, COpland, Villa-Lobos and Barber are nei-

ther exceptionally radical nor exceptionally conservative.
AVERAGE DEGREE OE EVIATION (BY CATEGORY)

The following chart shows for each category the
average degree of deviation for all composers. Using the
harmonic tritone, this chart attempts to group the catego-
ries in the order in which twentieth-century usage of the

tritone is related to traditional usage.

-2u2-

approach by parallel motion

departure by parallel motion

approach by oblique motion
(O-st or O-sk)

approach by similar motion
tritone units

approach by contrary motion
approach by rest

departure by rest

accented tritones

tritones having a duration of
less than one unit

departure by similar motion
departure by contrary motion
departure by oblique motion

traditional structures-

average degree of
deviation

1'53

1%

(13:7)
7%
12%
11$
11+ . 7%
16%

~2h3-

The categories in Which twentieth-century usage of
the harmonic tritone most resembles that of traditional
harmony (insofar as frequency is concerned) are: (1) ap-
proach by parallel motion, (2) departure by parallel motion,-
(3) approach by oblique motion and (u) approach by similar
motion.

Twentieth-century usage least resembles that of
traditional harmony in the categories: (1) traditional
structures, (2) departure by oblique motion, and (3) depar-

ture by contrary motion.
INDEX TO HANSON ANALYSES

In the musical examples which appear in this trea-
tise, those structures which were designated as "non-
traditional" were analyzed according to the method suggested

by Howard Hanson in Harmonic Materials of Modern Music. In

 

order to classify these sonorities, it is necessary to group
them (as Hanson does) into triads, tetrads, pentads, hexads,
heptads, octads, nonads, decads, gig. The author deems it
advisable (for the purposes which he undertakes in the pres-
ent work) to group the sonorities according to their rela-
tive dissonance value. This is achieved by assigning to the

intervals the following dissonance values:

-2m_

2 l
m 2
n 2
33, 3
a II
1:. II

Obviously, every sonority analyzed will contain at
least one tritone. For this reason, one tritone in each
sonority will not be considered to add dissonance value to

‘

the sonority. Tne relative dissonance value of each sonor-
ity can then be determined by adding the dissonance values
of all the intervals in the sonority with the exception of
one tritone.

The following chart gives the dissonance values of
the different non-traditional tritone sonorities. Only

those sonorities which appear in the examples in the text

are listed.

-2245-

TRIADS DISSONANCE VALUE HEXADS DISSONANCE VALUE

pdt S pum2n3sudt 30

met 5 p3m2nu32d2t2 33
p2m2n5s2d2t2 3h

TETRADS p3m2n28ud3t 35
p2m2nu32d2t3 36

pmn2st lO m636t3 38

pamsdt 11

pngdt 11 HEPTADS

pmnsdt 12

pnasdt 12 p3m3nés3d3t3 MB

m332t 12

mnzsdt l3

mas g 13

pmsd * 1h

p2d2té 1h

PENTADS

p3mn332t l7
p2m2ns3dt 18
p m n23 t 19
p2m3n52dt 2O
p2mn3s2dt 20
p2m2n23d2t 21
p3mnszd2t 21
pmuns gt

pm3n23 dt 21
p2mn232d2t 22
panLsdt2 22
m2nu32dt 22
mznustt 23
p2mn2sd2t2 23
pm3n32d2t 23
pm2n232dt2 23
pm2ns3d2t 2h

It is seen in the above chart that the addition
of factors in any sonority will increase its dissonance
value. Therefore, these sonorities containing the greatest
number of voices tend to have the greatest number of dif-
ferent factors (and the highest dissonance value). The

non-traditional harmonic tritone structures found in the

-2u5-

examples in the present work belong to twenty-one different

dissonance values ranging from five to fortv1eight.

 

DISSONANCE VALUE: FIVE

The triads pdt and met are found in Barber's Sec-

ond Symphony, Berg's Violin Concerto, Hindemith's Ludus

 

Tonalis, Villa Lobos's Bachianas Brasileiras No. 1, Harris's

 

Third Symphonz, Prokofiev's Fifth Symphony, Prokofiev's

 

 

Seventh Symphong and Copland's Third Symphony (Exx. 63, 79,

90, 9h, 95, 101, 102, 103, 107, 109).
DISSONAHCE VALUE: TEN

2

The tetrad pmn st is found in Stravinsky's

L'Eistoire du Soldat and Berg's Lulu (Exx. 75 and 81).

 

DISSONANCE VALUE: ELEVEN

The tetrads EZmnsdt and pngdt are found in

Schoenberg's Fourth String Quartet, Bartok's Second String

 

Quartet, Hindemith's Sinfonietta in E and Harris's Third

Symphony (Exx. 63: 71, 76, 91, 92, 98 and 10h).

DISSOYANCE VALUE: TWELVE

2

The tetrads pmnsdt, pn sdt and m332t are found in

 

Barber's Second Symphony, Debussy's Pelleas and Melisandg,

 

Bartok's Second Strinnguartet, Harris's Third Symphony,

 

-2u7-

 

and Prokofiev's Fifth Symphony (Exx. 63, 69, 76, 97, 99,
100, 10a and 105).

DISSONANCE VALUE: THIRTEEN

The tetrads mnzsdt and m233t are found in Barber's

3

Second Symphony, Schoenberg's Fourth String_Quartet and

 

Bartok's Second String Quartet (Exx. 63, 71 and 76).
DISSONANCE VALUE: FOUR EEN

The tetrads nmsdat and n2d2t2 are both found in

Schoenberg's Fourth String Quartet_(Ex. 71).

 

DISSONANCE VALUE: SEVENTEEN

The pentad p3mn332t is found in Barber's Second

 

Symphony_(Ex. 63).
DISSOKANCE VALUE: EIGHTEEN

The pentad p2m2ns3dt is found in Berg's Lulu

 

(Ex. 80).
DISSOKAHCE VALUE: NINETEEN

The pentad p2m2n233t is found in Hindemith's Sin-

fonietta in E (Ex. 92).

 

DISSONANCE VALUE: TWENTY

The pentads p2m3ns2dt and 2mn332dt are found in

 

 

Schoenberg's Piano Concerto and Stravinsky's Symphonie de

 

Psaumes (Exx. 73 and 7h).
DISSONA$CE VALUE: TWENTY-ONE

The pentads p2m2n23d2t, p3mn82d2t, pmunszdt and

 

 

pm3n232dt are found in Barber's Second Symphony, Berg's Lulu

and Hindemith's Sinfonietta in E (Exx. 63, 80, 81, 91 and

 

106) .
DISSONANCE VALUE: TWENTY-TWO

The pentads p2mn232d2t and nnnusdtZ are found in

 

Stravinsky's Symphonie de Psaumeg and Berg's Lulu (Exx. 7k
and 80).

DISSONANCE VALUE: TWENTY-THREE

The pentads pémnasdatz, mznusdat, pm3n32d2t and

 

 

 

nm2n232dt2 are found in Schoenberg's Piano Concerto, Stra-

 

vinsky's Symphonie de Psaumes and Hindemith's Sinfonietta

in E (Exx. 73, 7h and 92).
DISSON.NCE VALUE: TWENTY-FOUR

The pentad pm2ns3d2t is found in Berg's Lulu

 

(Ex. 80).

DISSONANCE VALUE:

The hexad

 

DISSON.NCE VALUE:

The hexad

fonietta in E (Ex.

 

DISSOJANCE VA UE:

The hexad

Symphony (Ex. 67).

 

DISSONANCE VALUE:

The hexad

Symphony (Ex. 63).
DISSONAHCE VALUE:

The hexad
(Ex. 80).

DISSONANCE VALUE:

The hexad

etta in E (Ex. 92).

-2u9-
THIRTY

Pum2n3s¥dt is found in Hindemith's Sin-

92).
THIRTY-THREE

p3m2nu32d2t2 is found in Hindemith's Sin-

92).

 

TH1.TY-FOUR

p2m2n532d2t2 is found in Barber's Second

 

THIRTY-FIVE

p3m2n23ud3t is found in Barber's Second

 

THIRTY-SIX

pgmanbrsadat3 is found in Berg's Lulu

 

THIRTY-EIGHT

mését3 is found in Hindemith's Sinfoni-

-250-
DISSONANCE VALUE: FORTY-EIGHT

The heptad p3m3nés3d3t3 is found in Barber's Sec-

 

ond Symphony (Ex. 106).

 

CONCLUSION

While there is little consistency in the use of
the harmonic tritone among twentieth-century composers, it
seems safe to say that these composers have one thing in
common: a decided tendency to avoid traditional usage of
the interval. The recognized tendency of the tritone in
traditional music was to resolve by contrary motion step-
wise-~a practice which twentieth-century composers avoid
scrupulously (in distinctively contemporary music, of
course--not in traditionally-oriented works such as Barber's
Adagio). Traditional resolution has been all but replaced
by the use of oblique motion (that is, one of the members
of the tritone is retained as a common tone between the
tritone chord and the chord of resolution).

The oblique method of departure from the tritone
was found occasionally in the music of the period of tradi—
tional harmony, but when it did occur, the moving voice al-
most invariably proceeded in stepwise motion from the tri-
tone. In twentieth-century music the moving voice often

proceeds by skip.

Oblique motion is also the predominant method of
approach to the twentieth-century tritone; this was also
true of the tritone in traditional music. The most strik-
ing difference between twentieth-century tritone usage and
that of the Bach chorale lies in the harmonic structures
in which the tritone occurs. Practically all of the har-
monic tritones of traditional music occur in dominant-type
tertian sonorities. However, in twentieth-century music,
any combination of tones is possible, of course, and our
composers take full advantage of this expanded tonal pal-
ette, seldom selecting traditional sonorities.

It should be remembered that in traditional music,
one of the tones of the tritone was sometimes treated as a
nonharmonic tone-~even though the sonority in which the tri-
tone occurred was considered a harmony in its own right.

The tritone occurred most often in the dominant-seventh
chord, and the chord seventh was approached and resolved as

a suspension, passing tone, neighboring tone or appoggiatura.
For this reason, traditional music preferred the shorter tri-
tones; they were seldom more than two units in length. They
were often unstressed (that is, unaccented) rhythmically.

Twentieth-century music is much bolder in the use
of the tritone; tritones are frequently of extended dura-
tion, and they often receive heavy accent, both rhythmic and

agogic.

-252-

Thus is concluded this study of one aspect of the
technical resources of twentieth-century music. The author
sincerely hepes that his efforts have contributed in some
way to the understanding of this great art, and that this
work will help to pave the way for similar studies in con-

temporary compositional technique.

Bibliography

Music

 

Bach, J. S. 371 Four-part Chorales. New York: Associated

.uu—‘aa—‘n

MuSic Publishers, inc., n.d.

 

 

Barber, Samuel. Second Symphony. New York: G. Shirmer,
Inc., l9SO.

Bartok, Bela. Music for String Instruments, Percussion and
Celeste. New York: Boosey and Hawkes, Ltd., 1939.

 

 

. Second String strtet. New York: Boosey and
Hawkes, Ltd., 1939.

 

Berg, Alban. Lulu. Vienna: VUniversal Edition, 1936.

. Violin Concerto. Vienna: Universal Edition,
1936.

 

 

Britten, Benjamin. War Reguiem. London: Boosey and Hawkes,
Ltd., 1962.

 

Copland, Aaron. Third Symphony. Iew York: Boosey and
Hawkes, Ltd., l9h7.

Debussy, Claude. Pelleas et Melisande. Paris: A. Durand
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Harris, Roy. Third Symphony. New York: G. Schirmer, Inc.,
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Hindemith, Paul. Ludus Tonalis. London: Schott and Co.,
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. Sinfonietta in E. New York: Associated Music
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Prokofiev, Serge. Symphgpy No._5. New Yerk: Leeds Music
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. Symphony No. 7. New York: Leeds Music Corpora-
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Schoenberg, Arnold. Concerto for Piano and Orchestra. New
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-253...

-25u-

. Fourth String Quartet. New York: G. Schirmer,
Inc., 1939.

 

 

Stravinsky, Igor. Histoire du Soldat. London: J. and W.
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. Symphonie de Psaumes. London: Boosey and Hawkes,
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Villa-Lobos, Heitor. Bach‘rncs Brasileiras No. 1. New York:
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fi

Wagner, Richard. Orchestral Exceppts in Scorp. Edited by
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. Tristan nd Isolde. New York: Kalmus, 19h8.

 

 

Books and Periodicals

 

Apel, Willi. Harvard Dictionary of Music. Cambridge, Mass.:
Harvard University Press, 1961.

 

AtKisson, Harold. Basic Counterpoint. New York: McGraw-
Hill, 1956.

 

 

Blom, Eric (ed.). Grove' 3 Digticn lrv of ““usic and Musi-
cians. Sth ed. 9 vols. London: acNillan and
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Bobbitt, Richard. "The Physical Basis of Intervallic Qual-
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ance," Journal of Music Theopy, III (November,

1959), 1731267.

 

Colles, H. C. (ed. ). Grove's Dictionapygpf Music and Musi-
cians. 3rd ed. S'vols. New York: The Macmillan
Co., “1935.

 

Cooke, Deryck. The Language of Music. London: Oxford Uni-
versity -Press, 1999.

Descartes, Rene. Compendipm of Music. Translation by Wal-
ter Robert. Introd. and notes by Charles Kent.
Home: American Institute of Musicology. 1961.

Hanson, How rd. ”srmonic Materials of Modern Music. New
York: Appleton-Century-Crofts, Inc., 1960.

 

Harman, Alec. Man and His Music. London: Rockliff, 1958.

 

-255...

Hindemith, Paul. The Craft_pf Musical Com3aosition. 2 vols.
Translated by Otto Ortmann. New York: Associated
Nusic Publish rs, Inc., l9u1.

 

Kitson, C. H. Evolution of Harmony. Oxford: University
Press, 1935.

 

Lippmann, Klaus. The Lan~u12e of N251c. New York: The

Ronald Press Company, W195}.

Locksperser , Edward. Debussy. 2nd ed. rev. New York:
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cti on to Contemporary Nus ic. New
rton and Company, 1961.

 

iachlis, Joseph. Introdu
York: W."-.’.‘»

Merritt, Arthur Tillma an. Sixteenth-Century Polyphony. Cam-
bridge, Massachusetts: Harvard University Press,

19L6.

 

Nef, Karl. fin “Outline 0: the Historv_ _of Music. New York:
Columbia University Press, 1M93b.

Piston, Walter. Court ernoir t. New York: W. W. Norton and
Co., Inc., 1917.

. Harmony. 3rd ed. rev. New York: W. W. Norton
and Company, 1962.

Rameau, Jean Philippe. Traite d'harmonie. London: J. Walsh,
1752.

Redlich, H. F. Alban Berz. New York: Abelard-Schuman,
Limited, ‘1957.

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