The Princeton Seminar
in Advanced Musical Studies

Edited by Paul Henry Lang



All Rights Reserved

Published simultaneously in the Dominion of Canada
by George J. McLeod Limited, Toronto



Introduction. Paul Henry Lang 7

Preface. Paul Fromm 17


Roger Sessions 21


Edward T. Cone 34


Elliott Carter .51


Vladimir Ussachevsky 64


Ernst Krenek ” 72


Allen Forte 95


DETERMINANTS. Milton Babbitt 108


Parisian master architect, Jean Mignot, overseeing the building
of the cathedral in Milan in 1398, declared Ars sine scientia nihil.
This was in answer to an opinion then beginning to take shape, that
scientia est unum, et ars aliud. For Mignot, the rhetoric of building
involved a truth to be expressed in the work itself, while others had begun
to think, as we now think, of houses, and even of God’s house, only in
terms of construction and effect. Mignot’s scientia cannot have meant
simply engineering, for in those days engineering was considered an art,
not a science; his scientia meant ratio, the theme, content, or burden
(gravitas) of die work to be done, and was not concerned with its func-
tioning or with the esthetic satisfaction it might provide. And so, too, for
music. Guido d’Arezzo’s words, Nam qui canit quod non sapit, diffinitur
bestia, are strong words. Many centuries have elapsed since these views
were widely shared, but something of this sort is once more in the air
as the age-old and omnipresent strife between old and new is reaching
a particularly acute stage, in fact, a stage that has few parallels in the
history of music. “This is no longer music” the popular rallying slogan,
is quite familiar to the historian, but while such an attitude has often
obscured progress for a generation or two, it has never caused such an
upheaval and such weakening of artistic integrity as during the last two
generations. To make a comprehensive analysis of the situation is a
Herculean task and will call for much more serious and judicious effort
than the rather wild pamphleteering that goes on in European peri-
odicals, though here and there one sees very respectable studies.

Feeling the necessity for clarification, Princeton University with the
enlightened help of the Fromm Foundation last summer organized a
Seminar in Advanced Musical Studies. This great university thus proudly
reaffirms that music is indeed an integral part of the history of ideas, a
concept so sadly lacking in the music departments of most of our institu-
tions of higher learning. In its turn, The Musical Quarterly is proudly
reaffirming its own traditions of furthering musical thought by publishing
the main papers of the Seminar. Many of our readers, like ourselves, are
concerned by the new developments in music, knocking on the doors of a
new world without finding the handle that would gain them admittance.
We believe that these papers offer guidance and enlightenment and that
they make for profitable reading. The general conclusions of a productive
assembly, which make up in strenuous thinking what they lack in length
and elaboration, are here summed up. The reader will be interested to
observe the tone of the various contributors. There is the calm, reasoning,
and forbearing musician who, knowing that changes are necessary and
inevitable, examines all phenomena with care and with an open mind.
Then there is the defender of the new faith to whom the new can come
only at the expense of the old; the weltfremd thinker; and the man who
just tags along. But the aggregate is enlightening and permits us an in-
sight into the inspirations and aspirations that govern musical thought at
the beginning of the seventh decade of the 20th century. We*recommend
that the reader place a da capo sign on Roger Sessions’s introductory
essay, to which he should return after reading the last article. It contains
many well-reasoned observations, but we should like to single out one
that may serve as a cantus firmus whenever the going gets rough: U A
conscientious artist, if genuinely adventurous, will accept anything what-
ever only strictly on his own terms.”

By way of introduction we may sketch in the background from which
this mid-century stock-taking grew, and perhaps add a few comments
of our own.

The first half of the 20th century passed under the sign of violent
antitheses. First there was revolutionary dissolution, followed by severe,
tradition-oriented concentration; emphatic subjectivity, then dogged
objectivity and studied collectivism, The same extremes can be detected
in the constituent features of music. Form became so fragmented that its
dangling remnants could hardly be detected, but subsequently it solidified,
triumphantly rediscovering age-old esthetic tenets and then petrified.

Introduction 9

Melody, in the post-Impressionistic world, became a color patch, an
exclamation, the smooth surface of its face ruined by the varicose veins
of incessant chromaticism. Then there developed a desire for broad
design, diatonicism, folk tunes in the old ecclesiastic modes, even penta-
tonic melodies, only to be succeeded by “rows.” The tonal system, already
showing ambiguities in Tristan, disintegrated, then the aimlessly floating
harmonic clouds were blown away, and “atonality” was subjected to
military discipline. We have seen how the Niebelung orchestra was later
dwarfed, only to be followed by a new da camera concept.

Then came the great reckoning with affective functions of music.
Verkldrte Nacht still billowed with Tristanesque vapors in orgiastic
abandonment, but Stravinsky declared that feelings and passions do not
exist in music, only in the listener’s imagination. Music should be nothing
but an autonomous order of sound progressions, ruled by the logic of
construction, for music cannot express anything but itself. Unfortunately,
while impersonality can be a powerful weapon against romantic exuber-
ance, it can serve also as effective camouflage of artistic irresponsibility;
the neo-Classic sewing-machine counterpoint rattles along with unfailing
precision, for a motor is impersonal. And so is the gesture when purely
geometrical, and so are the sound track, the electronic instruments, and
the Sprechgesang. The champions of objectivity made even the Baroque
impersonal, regimenting its revived counterpoint so that it could be used
as a mask behind which they might hide. The whole of this new objec-
tivity is nothing but a mask. From this cul de sac’ there was no way out,
only a radical change of course could extricate the composer.

At first Impressionism seemed to be the only durable movement, the
only style capable of preserving European traditions, because in essence
neither Debussy nor his disciples and imitators broke the old frames, they
merely loosened them. They did find many new colors and made the
orchestra even more dazzling, the result being that Ravel, Dukas, Delius,
Falla, even the American Griffes, are still welcome to subscription audi-
ences as “modern” composers. As a matter of fact, some of our justly
admired brilliant and slick orchestras were built to cope with this very
style. But this is a hothouse art that needs artificial heating, and by the
time Ravel died the school was so destitute of substance that the much-
admired composer of Daphnis et Chloi was reduced to orchestrating
Mussorgsky. All of them sought exotic subjects and themes, for they felt
that Europe’s Impressionistic pomp needed colonies to replenish its color
resources. Delius turned to the Appalachians, Roussel and Hoist to
Indian lore, and so forth. Others rediscovered folksong, and English,

1U jfroDiems or Moaern MUSIC

Irish, Jewish, and American folk tunes invaded their music. But gradu-
ally musicians became satiated with the winter garden of Impressionism
and began to rebel. It is characteristic that the leaders of the anti-
Impressionist movement came from its very core: Satie, Milhaud,
Honegger, Stravinsky. All of them began as Impressionists, it was only
later that they struck out in other directions. It was obvious that the
Impressionism of the early centinry had to be outgrown, very much as
had the Rococo of the mid- 18th century.

But what lay ahead of the rebels? One by one they began to realize
that by leaving the sheltering haven of hedonism they would be isolated,
and of that many were afraid. Still, there were quite a few composers
who did not shrink from solitude, who indeed sought it; some belliger-
ently, some with ascetic devotion. Among them were several of our own
composers who could not be shaken in their determination to go their
own way, and to them it did not matter what the others tried to do, for
all of them shared Mr. Sessions’s motto to which we have referred.

The central figure of the movement that led to the dissolution of the
order was Schoenberg, who began where the fifty-year-old Strauss was
to end. Gurrelieder was the last monument of gigantism; then Schoen-
berg suddenly turned around, abandoned the immense apparatus, and
devoted himself to intimate chamber music. His Second String Quartet
is the symbolic marker of the road that was to lead from Wagner to
Bartok. But he did not follow that road, for to him and his disciples, all
of them accustomed to wallowing in Tristanesque emotionalism, the most
urgent task became the undressing of music in order to rid themselves of
its gaudy garments. They wanted the naked body of music; distilled,
test-tube truth.

Tonal order and logic was the chief guardian of sensuousness, there-
fore it had to be eliminated. While others, like Hindemith, Stravinsky,
and Bartok, also assaulted the old order, they never really abandoned it
altogether; but Schoenberg firmly believed that he represented the will
of the age when he eliminated it by declaring all twelve tones of the
octave to be of equal, sovereign significance. There still remained the
innate sensuousness of the singing voice which, too, had to be destroyed
and changed into speech-song. Schoenberg turned out to be a real,
inspired leader compared to whom Satie, Cocteau, and their kind were
only artistic playboys. His school was so strong and influential that it
compelled obeisance hi all lands. One by one the coryphees declared
their fealty. We have witnessed the most abject surrender only recently,
when Stravinsky, always aloof, arrogant, used to command, bared his

Introduction 1 1

head before “the three Viennese,” obediently accepting terms. His magni-
ficent pagan colors and blood-boiling rhythms that once were the ad-
miration of every musician are gone and he, too, writes “objective”
serial music, cold, grim, and beautifully made.

We have arrived, then, at a stage where we see two extremes locked
in contest. We have some able and independent composers, no longer
young but seasoned in wisdom and experience, who believe that one can
still contribute new interests without completely forswearing old ones.
They are anything but mossbacks, the Hindemiths, Sessionses, Carters,
and all the others; they are perfectly willing to experiment and learn,
but they believe that where there is not even a semblance of tradition
every convention is inartistic, every stylization a form of academicism.
Total serialization they regard indeed as a new form of academicism,
incredible as this may seem to the revolutionary switchboard artist.

At the other extreme we have a very vocal segment of the new
generation to whom all that is left of two millennia of music is its
physical components, which are manipulated with the aid of electronics,
stop watch, and slide rule. Therewith begins an inevitable tendency, a
rapprochement to the physico-mathematical, which changes Mignot’s
still somewhat flexible dictum into the unequivocal ars nihil sed scientia.
Even the thoroughly artistic and esthetically fruitful twelve-tone system
is no longer acceptable. Our readers will notice that reference is several
times made in these articles to “the classical twelve-tone technique,” that
is, Webern has already been relegated to history. They have gone far
beyond the old, once dreaded method of “composing with twelve tones,”
and are now in the world of total serialism in which every aspect of
music, not only pitch succession, is governed by “premeditation.”

Once more we wish to state that the historian and critic must never
be impatient even when faced with the seemingly fundamental negation
of everything that has passed before. He remembers Galilei’s Didogo
and similar devastating manifestos, and he knows that some good will un-
doubtedly emerge from this latest upheaval. But he also remembers what
Bury said, that history “is in the last resort somebody’s image of the past,
and the image is conditioned by the mind and experience of the person
who forms it.” Therefore our comments, though honest and carefully
weighed, are of necessity based on that possibly frail and biased mind
and experience. It is regrettable that this is not given some recognition,
or at least the courtesy of the benefit of the doubt.

When Friedrich Blume in a thoughtful and eminently fair recent
article attempted to find an answer to the recurring question “What Is

12 Problems of Modern Music

Music?” as seen by an independent, historically schooled observer from
the perspective of the far slope of the 20th century, the government of
the avant garde, represented by Melos, immediately ordered total
mobilization. A special issue of the magazine was ordered into the battle
line, reservists were recalled to the colors, though some of them were
manifestly over age as far as present-day music is concerned. The
fusillade was impressive, main and secondary batteries all firing armor-
piercing shells, but the gunnery was poorly directed and Mr. Blume was
not even straddled. This is a deplorable reaction, reminding us of the
Parisian censor’s immortal words when he banned M^hul’s opera,
Melidore: “It is not enough that a work is not against us, it must be for
us.” Our readers will find none of this attitude in the Princeton papers.
The authors who appear in these pages are responsible and devoted
musical thinkers who respect the venerable tradition of free inquiry,
which is the sole premise acceptable to a great university and also to a
periodical such as ours.

* *

Total serialization was bound to create a feeling of incarceration,
and even among its adherents a certain uneasiness is manifested in this
regard. As Mr. Krenek so succinctly states: “Invariancy inherent by
definition to the concept of the series . . . leads to uniformity of con-
figuration that eliminates the last traces of unpredictability.” Pierre
Boulez, also disturbed by the problem, puts it very nicely when he says:
“The unexpected, again: there is no creation except in the unforseeable
becoming necessary.” This makes sense; after all, it was always the
essence of great music, and the mark of genius, to do the unexpected
instead of the routine. But the remedy proposed by the total serialists is
anathema to any esthetic system known to us: “the factor of chance.”
Regrettably, the justification of the “element of chance” leads to some
very slipshod philosophizing. It is comfortably likened to what is tradi-
tionally considered “inspiration.” But to Mr. Krenek inspiration is a
treacherous thing, it is influenced, nay, “dictated,” by tradition, training,
and experience, none of which can be trusted, therefore the duty of a
truly advanced composer is to set up an “impersonal mechanism” and
then bedevil its premeditated patterns with elements of chance. Now
Edward T. Cone, whose paper is a model of open-minded progressivism,
professional competence, and just plain healthy musicianship, avers that
when chance is invoked as an element of construction, logic is largely
inoperative. He detects the same limitations in strictly enforced serialism,

Introduction 1 3

for in either case he finds that the music can be explained only by
referring to “an external structure.” We are afraid that Mr. Krenek,
able and shrewd as he is, falters at the end of his article. Sooner or later,
of course, he had to face the inevitable question of “the expressive or
communicative aspects of music.” But how can moods and feelings be
serialized? Indeed, we may ask with Mr. Sessions, how can rhythm,
tempo, and dynamics be serialized? Mr. Krenek does not “exclude the
possibility” of total serialism being or becoming “a medium of some sort
of communication,” but we are afraid that this slight concession makes as
little sense as his last sentence, in which he intimates that this communica-
tion “may mean as much or as little as life itself.”

Mr. Krenek somewhat diffidently states that “whatever morpho-
logical kinship may be detected between adjacent sections [in his Circle,
Chain, and Mirror] … is a consequence of the premeditated serial
arrangement . . . and not dictated by requirements of a so-called musical
nature.” A startling statement, this, and surely the most contemptuous
utterance in the present anthology. It surprises coming from a musician,
and a good one, who once basked in the “so-called” music of Mahler;
but then converts are usually more zealous than the pope. Mr. Cone
thinks that these despised musical requirements are not yet buried, that
in a more generalized form, the compositional values of past centuries
are still cogent and viable. This is certainly borne out by at least two
distinguished composers represented in this issue: Roger Sessions and
Elliott Garter. It seems to us that what Mr. Cone offers in this regard
makes far more sense. He calls it a “crucial” point, and, indeed, it is
nothing less. “A work of art ought to imply the standards by which it
demands to be judged.” This is a principle that is often ignored and
often violated by the American-Legion-like professional patriotic organi-
zation of serialists. Luigi Dallapiccola, for whom we have great admira-
tion as a composer, gives the dodecaphonic accolade to Mozart; and
Allen Forte, in one of the articles appearing in this issue, offers a good-
conduct medal to Bartok. Mr. Forte finds in the Hungarian composer’s
Fourth String Quartet “the extended and elaborate expression of a
relational system that closely resembles a serial schema.” Though the
author immediately adds that “the system upon which it is based arises
as a logical consequence of tonal materials unique to the Fourth
Quartet,” he proceeds to deduce serial significance from the work. Still,
he must operate with requirements of a “so-called musical nature”
which, apparently, are not welcomed by the purists. What is it that Mr.
Krenek wishes to substitute for this awareness of morphological kinship?

14 Problems of Modern Music

We are aware that we are aware, and can even inspect the content of
our awareness, but hardly the act of awareness itself.

A special warning is in order concerning Milton Babbitt’s contribu-
tion. The language and symbolism by which he strives to communicate
the subtle distractions of an uncompromisingly honest and incisive spirit
convinced that it has found a whole, may seem eccentric, obscure, and
prolix. But the reader should remember that Mr. Babbitt, a fine musi-
cian, is also an eminent mathematician, and he speaks here in mathe-
matical terms of the implications of “a permutational musical system as
opposed to the combinational system of the past.” In mathematics there
is no emotion, no feeling; it is the supreme and only certainty that man
can achieve. Mathematics is the music of the intellect, a happy abstract
world where there is no need for a knowledge of life, for mathematics
does not know the incomprehensible complications of life. And how much
simpler it is than life! It is relatively easy to understand the axioms
of Euclid, but the minute we begin to moralize, and thus turn to life,
things become infinitely complicated and difficult to understand. We do
not profess to be able to follow Mr. Babbitt’s disquisition beyond a very
modest degree, but since it comes from an unquestionably competent
and dispassionate source, and has undoubted relevance to the problems
under discussion, it will prove enlightening to those versed in the
mathematical systems presented, which are absolutely beyond the ken of
the Editor of this journal.

Now we come to the executors of the mathematical theories, the musi-
cians who carry out empirically the postulates of which Mr. Babbitt
speaks. Vladimir Ussachevsky tells us how he composes for the tape
recorder, a bright, versatile, obedient, but soulless instrument. The
process is very involved and bears as little resemblance as possible to
what one innocently supposes to be the course of musical composition,
whether tonal or serial. The electronic composer has a bank where he
deposits sounds for further use. When withdrawn, the sounds are put
through various operations, they are filtered, split, spliced, enriched,
thinned, etc. The next steps are interesting. A given musical material is
re-recorded at different speed. The new pitch levels so created are then
synchronized with the original, thus creating Mr. Krenek’s “unpredict-
able*’ by purely mechanical manipulation. All this looks pretty forbidding,
but Mr. Ussachevsky, whose paper is altogether free of the boasting so
characteristic of his engineer-musician colleagues, says with earnest and
convincing simplicity that he now “habitually imagines a sound as if it
were changed by the [various] mutation techniques.” That is, he can

Introduction 15

imagine the pitch variations caused by different tape speeds, in fact, can
imagine the sound even when it is played backward. Nevertheless, his
description of the composition of a tape recorder piece, while interesting
and informative, only attests to the experiments of an inquisitive mind.
He believes in what he does but refrains from promising the coming of
the kingdom of the electronic tube on earth.

It seems to us that with electronic music we have arrived at a critical
conflict of concepts not well understood by either scientist or musician.
It is agreed that we possess a three-dimensional tonal space which, ex-
pressed in greatly simplified terms, is made up of the duration of the
sounds, their arrangement in pitches, and their intensity. But there is a
decisive difference between mathematical-physical hypotheses of the
theoretical tonal space and the actual effects of music as an art. This is
where the electronic composers and their scientist supporters are gravely
in error, for they create a false relationship between acoustics and music,
between objective nature and subjective art. In justice, we must mention
that they look back upon a respectable ancestry. The Pythagoreans held
that the mathematical ratios between tones are identical with the basic
conditions of music, but in reality we are dealing here only with a psycho-
physical parallelism. That the Pythagorean doctrine is debatable was
already recognized by Aristotle and Aristoxenus, who granted the musi-
cian what Mr. Krenek denies him: the right of aisthesis, that is, the right
to his own feelings and interpretations as opposed to the calculations of
the Pythagoreans. Much later, the able and outspoken Mattheson
thundered against the acousticians who presumed to dictate to the artist.
Today, the scientist and engineer no longer presumes to dictate to the
artist, he and his machines take over the artist’s role and metier, lock,
stock, barrel, and ramrod. The opposition of such men as Mattheson rests
on the fundamental difference between mathematical-objective tone and
musical-subjective tone, and on the difference between physical and
esthetic pitch, and finally, on that between acoustical and psychical
dynamics. Therefore the three-dimensional tonal space just mentioned is
nothing but a bit of science fiction, whereas musical procedures with their
human conditions are far more complkated, irrational, and meta-

Perhaps the first person in modern times to attempt to derive the
phenomena of music from physical conditions was Goethe, who liked to
concern himself with everything. He got some questionable advice from
his court musician, Zelter, a very minor composer and therefore an avid
theorist and critic. Then later in the century there were two great

16 Problems of Modern Music

scholars, Helmholtz and Stumpf, whose very valuable contributions to
the science of acoustics and psychology of tone made the acoustic-
physiological-psychological approach to music a firmly established dis-
cipline. But neither they nor the more recent mathematical scientists
really understood music, only the materia musica. For the core of the
musical process, of the creative process in music, is subject neither to
physics, physiology, nor mathematics, but is an artistic thought process,
a musical logic, virtually independent of the natural sciences. We are not
dealing with physics but with music, not with science but with art; it is
the human element that is decisive. No matter how earnestly the scientist
tries to fix standards, the true musician will never accept them uncondi-
tionally. Our friends of the electronic tube claim that civilization has
finally progressed so far as to make passible the creation of an absolutely
pure and correct tone synthetically, by electricity. They even have the
temerity (witness Dr. Olson) to dismiss the Stradivari violin as useless
because it produces “impurities” on account of the fallibility of the hu-
man finger and the scraping of the bow. Why, it is precisely in these
human frailties, in this expressive impurity, that life and humanity are
revealed. A totally aseptic tone produced by machines will be a dreadful

Well, the reader is earnestly bidden to read these interesting essays
attentively and with an open mind. But when he has digested their
portent he might remember that man, the musician and not the physi-
cist and the mathematician is the measure of all music making. Aribo
beautifully expressed this in the llth century, when he said that one
must wonder at the grace of God when observing the lowly minstrels
who without the slightest idea of the theories and doctrines of music
play “entirely correctly.”

P. H. L.


THE Seminar in Advanced Musical Studies grew out of the same
concern with the realities and necessities of American musical life
that has motivated all the activities of the Fromm Music Foundation.
Our efforts have never been intended to be patronage in the old sense of
comprehensive support of a segment of musical activity or a group of
artists, but rather a series of stimulants that should focus attention, by
their example, upon the most promising aspects of creative activity.

One of the serious problems in our musical culture has been the
frustrating isolation of young composers and performers from each other,
which is intensified by the often unsympathetic attitude of performers
towards the music of our own time. Although this situation has been
abundantly recognized, most attempts at alleviation have been diffuse
and sporadic. We therefore determined to begin our own efforts in this
direction with a concentrated experiment. For its testing ground we chose
the Berkshire Music School at Tanglewood, since this school is a center
for students from a wide geographical area who converge upon it each
year for a brief period of intense musical activity. Aaron Copland ac-
cepted our invitation to head this Tanglewood project, beginning in

To meet the many challenges arising from the gulf between the
creative and re-creative elements in our musical culture, the Fromm
Fellow Players, a group of eleven young performers distinguished by their
skill, integrity, and eagerness, was formed. One of their responsibilities
was to assist the composition department in resolving problems tha’t can
be best demonstrated in performance. They also performed at the weekly
lecture-concerts honoring visiting and resident composers, gave carefully


18 Problems of Modern Music

rehearsed performances of student works at the Composers’ Forums, and
appeared in two formal concerts of 20th-century music. Contemporary
music thus became a more significant part of the Tanglewood scene and
the basis of a new and lively interaction between composers and per-
formers. Imbued with a growing awareness of and enthusiasm for the
contemporary spirit, each player returned home prepared to set off a
cultural chain-reaction.

During my visits to the Berkshires I met with composers and per-
formers to discuss ways in which the Tanglewood experience might bear
upon other areas in our musical culture. In the course of these discussions
our attention was drawn, again and again, to the plight of the loneliest
individuals in contemporary music the young professional composers.
Advanced beyond “instruction,” they are aware of musical problems and
eager to come to grips with them. Their expectations of acceptance by
society, raised by the fellowships and prizes available to them as students,
prove unfounded; while at the same time they have lost the stimulating
contact with their fellow-students that enriched their school days. These
composers are young people to whom Schoenberg, Webern, and Berg
are no longer objects of controversy but pioneers of the early 20th
century. Although they are aware of everything that really is contem-
porary, they have no one with whom to share this awareness. Their
chances of entry into public musical life have been barred by the insur-
mountable barrier of their own originality and the hostility of individuals
and institutions who deny the public anything that cannot be stand-

We talked of the European seminars at Darmstadt and Donaue-
schingen, where discussion of common problems lessens this feeling of
isolation, and where contact with masters of the older generation helps
the younger people find their own directions. Out of the conviction that
Americans need no longer depend upon Europe for their resources, we
decided to propose to Princeton University the establishment of a seminar
for study, on the highest level, of the most significant trends in contem-
porary musical thought. Here all contemporary doctrines could be freely
examined, stressing above all the primacy of musical experience and
imagination, and categorically distinguishing artistic production from
systematic thought. In this way we hoped to give the young professional
musicians a rallying point, an assurance of inclusion in our musical

Roger Sessions, who as a composer and teacher occupies a most

Preface 19

distinguished position in the musical world, was a natural choice as
director of the seminar. All his creative life Sessions has lived and com-
posed in serene solitude, relying only on self-criticism and mature artistic
judgment to determine his creative objectives, without regard for public
approbation or criticism. Thus he stands now as an inspiring example of
dedication to his art.

The faculty that joined Mr. Sessions at the first Seminar included
Milton Babbitt, Edward T. Cone, Robert Craft, and Ernst Krenek. The
seven guest lecturers who contributed to the high level of the Seminar
were: Elliott Carter, Aaron Copland, Allen Forte, Felix Greissle, John
Tukey, Vladimir Ussachevsky, and Edgard Varse.

The Seminar was also honored by a visit from Igor Stravinsky, who
spoke informally to the participants.

The members of the Lenox Quartet, who were in Princeton as
seminarists and musicians-in-residence, presented three programs of con-
temporary music which included, besides works of the composer members
of the faculty, works by Bartok, Bloch, Kirchner, Schuller, and Webem.

Twenty-five musicians of professional status, averaging about thirty
years of age, were selected for participation in the Seminar from the large
number recommended by leading musicians throughout the country.
They took part in fifteen hours of weekly lectures and discussions. The
substance of the proceedings may be found in the articles that follow.

A few imperfections led us to modifications in our plans for future
seminars. Discussions of performance problems seemed over-generalized
when divorced from actual experience. We recognized that too much
oral discussion can be frustrating for musicians to whom musical notes
communicate ideas more genuinely and directly. Consequently, we have
evolved a new pattern, which will be applied for the first time in 1960.

Greater diversity as well as greater integration will characterize the
1960 Seminar. Regular seminars will again be given by the staff, this
time consisting of Messrs. Sessions, Babbitt, and Kim, of the Princeton
faculty, as well as two guest composers, Elliott Carter and Karl Birger-
Blomdahl (Sweden). All eleven Fromm Fellow Players (woodwind
quintet, string quartet, soprano, piano) will also be on hand, and will
spend each afternoon rehearsing new music. This music will be chosen
by the faculty from works by members of the seminar, not in terms of a
competition or a symposium, but solely on the basis of the presentation
of problems of general interest. The composers involved will lead the re-

20 Problems of Modern Music

hearsals in the presence of the members of the seminar and of one faculty
member, who will function as a moderator. This should stimulate a
spontaneous cross-fertilization between composer and performer, through
which each should gain insights into the problems and objectives of the
other. It is- to be expected, moreover, that important subjects arising
during these afternoons will be singled out by the moderator for fuller
elaboration in the more formal morning seminars. The guest lecturers will
be encouraged to emphasize areas not included in the regular curriculum,
which will bring about an even fuller integration of the total plan.

If our ideas can be realized, the Princeton Seminar in Advanced
Musical Studies will become a force hi our musical culture, a positive
indication that serious artistry need not be compromised even in a society
in which creative achievement is generally less recognized than material
success. Our young composers will carry home with them a sense of
rededication, a new strength born of the knowledge that the creative
potential of man is inexhaustible.



THE premises behind such an undertaking as was attempted in
last summer’s Princeton Seminar in Advanced Musical Studies
were of course based on the obvious changes in orientation and outlook
that are taking place and have for many years been taking place
in the minds, attitudes, and intentions of the composers, performers,
and even listeners of our day. It is hardly necessary to point out in
these pages that change is inevitable at any period whatever in the
development of an art. The history of the art is itself primarily an
account of such changes and an attempt to fathom both their wide-
ranging causes and their equally far-reaching effects, and to formulate
terms in which these can be adequately grasped. The History of Art
likewise seeks and provides criteria by which the main periods of change
may be compared with one another in character and extent.

The history of Western music reveals at least two phases, and
possibly three, that may well have seemed to those who observed them
as contemporaries to shake the art of music to its depths and to raise
questions of the most fundamental kind questions, that is, not only as
to the character and trend of current developments, but as to the
function, the significance, and even the ultimate nature of music itself.
The beginnings of polyphony, the late 16th and early 17th century,
and possibly the 13th and 14th if this be not indeed considered as a
late phase of the first-named were such phases. They were periods
of apparent crisis, during which long-established values were brought
into deep question and challenged both on the most profound and
the most superficial levels; “experimental” periods in the sense that
many things were tried which soon proved abortive, while others, soon
discarded, seemed to find justification at a much later date; but


22 Problems of Modern Music

periods of intense creativity not only by virtue of the music of genius
that survived them, but because they tapped new veins, uncovering
the resources out of which the music of the following three or four
centuries was to be built. In each period the musical transformation
was coeval with a far-reaching transformation in Western society, and
undoubtedly related to it, though the exact nature of this relationship
seems at least to this writer far more difficult to penetrate and to
clarify than it is frequently assumed to be.

The period in which we live has at the very least much in common
with these earlier ones. For well over a hundred years each successive
generation has seemed to many of its members to contain within itself
the seeds of the imminent destruction, not only of a great musical tradi-
tion, but possibly of music itself. Though this had happened at earlier
periods also, it has happened at a steadily increasing tempo since, roughly,
the death of Beethoven. Each generation has, to be sure, at length
become assimilated, by and large, to the “main stream”; the “revolu-
tions” have in each case been discovered eventually to be not so
revolutionary after all, and the revolutionaries of one generation have
become symbols of conservatism and eventually clubs with which to
beat their revolutionary successors of the next. But, with each succeeding
phase, this has come about a little more slowly, and there is no doubt
an easily discernible reason for this. It is true that in our time the
situation of all the arts, and in all of their phases, has been rendered
far more complex, first through the development of mass media, and the
consequent immeasurable expansion of the more-or-less interested public,
and secondly through various economic factors, including not only
the decline of private patronage and the consequent and inevitable
increase in commercialization, but drastically rising costs in virtually
every phase of musical production. It is however no less true that,
precisely at this moment of economic and, if you will, social crisis in the
arts, the inner dynamic of music itself should be leading to develop-
ments of which the eventual result can at best be only dimly sensed.

One symptom or result of this, of course, may be seen in the
increasing articulateness of musicians themselves in regard to their own
artistic principles. Since the early years of the 19th century composers
have felt more and more inclined to express themselves in print regarding
music and all of its phases. In the case of earlier composers Mozart and
Beethoven, for instance one must rely on correspondence, on remin-
iscences, and on a few sybilline and perhaps problematic quotations
that have become traditional, and no doubt often distorted, if one wants

Problems and Issues Facing the Composer Today 23

to discover their working principles beyond the evidence of the music
itself. From Carl Maria von Weber on, however, composers have
devoted considerable effort and energy to criticism, later to theory, and
more recently still to teaching. This is certainly due in very large part
to the fact that, in a period of artistic upheaval, creative artists find
themselves first of all sharply aware of their own relationship to their
traditional inheritance and to the directions in which they feel impelled
to extend or even to reject it. Secondly, they find themselves, in a
period in which the formulated notions regarding musical esthetics,
musical theory, and musical syntax have long since lost the vitality
they once possessed, impelled or even obliged to arrive at what are at
least working formulations of their own. If they are not to remain in
relative solitude they are also likely to communicate these formulations.
Since the cultural pessimism of our time abhors solitude once considered
a decidedly honorable state for an artist and demands “news” at
almost any price, they may even find themselves virtually compelled
to do so.

One has only to open practically any European periodical devoted
to living music in order to become aware of the intellectual ferment
that characterizes the musical life of today. One will find there, as one
finds in fact on all hands, serious and often acute discussion of every
phase, from generalized esthetic attitudes to the most precise and esoteric
matters, and on a level that the conscientious artist of mature age, or
the ambitious one of more tender years, cannot wholly ignore except
at the price of an inherent lack of adventurousness which in itself bodes
somewhat ill for his achievement as an artist. I of course do not mean
to imply by this that he is bound to accept all or even any of the ideas
he will find urged upon him. If he is genuinely adventurous he will
accept anything whatever only strictly on his own terms. But he will find
himself, certainly, challenged at every point, and obliged to find his own
answer to the challenges thus presented to him; and if he is young and
gifted he will welcome these challenges as a test of his creative con-
viction, if not as a source of direct stimulation along the lines of his
own expression. At the very least, he will have the opportunity to be-
come more aware of his own musical nature, and at the best he will
learn to be untiring in his effort to avail himself of that opportunity,
and to pursue his own creative efforts accordingly.

That the situation as I have described it contains its own peculiar
pitfalls is, of course, obvious. I am not referring to the comment one
frequently hears to the effect that a period in which musicians think and

24 Problems of Modern Music

talk so much about their art must necessarily be a sterile one. As a
matter of fact the present-day habit of drawing broad inferences of such
a kind apparently plausible but inherently far-fetched is one against
which we should guard ourselves in the name of elementary logic. It
is not my purpose here, however, to propose value judgments on con-
temporary music, but merely to comment on facts and phenomena as
they exist. But the least one can do is to point out that contemporary
music, and in fact any music whatever, is to be judged in terms of
music itself, not of circumstances with which no clear connection can
be convincingly demonstrated. One cannot insist too strongly or too
frequently that, in the arts generally and in music in particular, it is
only productions that really count, and that only in these music,
written or performed are* to be found the criteria by which ideas
about music, as well as music itself, must finally stand or fall: not the
converse. This is a refrain that will recur repeatedly in the course of
this discussion, as indeed it must in the course of any valid discussion
of music.

The generic pitfall at which I have hinted is precisely this one. In
an age hi which theoretical speculation in either the esthetic or the
technical sphere has assumed the importance it has in our own, there
is always the danger that it may be over-valued, and assumed to furnish
criteria in itself, and not regarded simply as a means that may prove
useful hi helping composers to achieve the artistic results they are
seeking hi the realization, that is, of a genuine musical vision. Again,
one finds oneself obliged to emphasize that the primary function of the
composer is to possess, develop, and with the utmost intensity to realize
his own particular vision a vision which, if it is genuinely vital, will
be found to contain both general and specifically personal elements; and
that theory and esthetics can have validity for him only in so far as
they can find roots in this vision. Otherwise they can represent only a
flight away from music, or at a very dubious best, a crutch on which
a faltering musical impulse can find some measure of support.

It is in fact fairly easy to recognize the pitfalls characteristic of those
past musical periods with which we are most familiar. To a certain
extent they are mirrored in the way in which these periods are regarded
by the succeeding generations, which rebel against them. The character-
istic pitfall of the 19th century was undoubtedly that of literary asso-
ciation and the manner of over-emphasis sentimental, violent, or
pretentious just as that of the 18th was a certain type of elegant and
formal conventionality. Our own particular brand of emptiness is

Problems and Issues Facing the Composer Today 25

perhaps beginning to emerge in a variety of cliches, derived both from
so-called neo-Classicism and from serialism in its earlier as well as its
later phases. In each case we are dealing with a manner that has become
generalized through lack of substance, and not with ideas in any
positive sense. What is necessary, if the pitfalls are to be avoided, is
that composers in the first place should always retain the courage of
their own artistic vision, that teachers should emphasize the supremacy
of real musical imagination, and that listeners, of whatever category,
should, by holding themselves open to whatever genuine and even
unexpected experience music can bring, learn to discriminate between
what is authentic and what is fictitious.

Thus far I have spoken at length of a general situation in the
musical world, and of some of the questions that situation raises as
such, without attempting to deal with the situation itself, its background,
or its nature, other than to characterize it in the very generalized sense
of the decay of one tradition and the gradual movement towards new
factors capable of superseding it. The ultimate shape these developments
will assume is still by no means definitive in its outlines; but both their
causes and their present trends are in certain respects quite clear, as
are the specific questions posed by the latter.

It seems clear, for example, that the development of harmony as we
have traditionally conceived it has probably reached a dead end. First
of all, composers have for many years felt able to utilize all possible
vertical combinations of tones, and have so abundantly availed them-
selves of that possibility that any new discoveries in this regard are
virtually unthinkable. Even this fact, however, tells only a part of the
story, since the possibilities are not so rich as this purely statistical
assumption would indicate. As more and more tones are added to any
chord, each added tone contributes less to the character of the chord, or,
in other words, to the factor that differentiates it from other combinations
of tones. The decisive development of harmony, therefore, depends
overwhelmingly on combinations of a relatively small number of tones;
beyond that number, so to speak, the ear refuses to interest itself in
strictly harmonic effect. It is not so much a question of possibilities as
such, as of possibilities that are in any way decisive. The real point is
that composers seem by and large no longer interested in chords as
such, and that this is a tacit recognition that there is nothing left to be
discovered, in the sphere of harmony, that arouses any feeling of excite-
ment on their part.

Something similar had of course taken place already in regard to

26 Problems of Modern Music

functional harmony. Based very clearly on a triadic premise, the
principle of root progression had given way before the proliferation
of “altered chords” that was so characteristic a feature of harmonic
evolution in the 19th century. The process is a perfectly familiar one
which need not be summarized here. Suffice it to say that what is
often called “atonality” was a very gradual development so gradual,
in fact, that, aside from the literal meaning of the term itself, it is
impossible to define with any precision whatever. It is in other words
impossible to show exactly where tonality ends and “atonality” begins
unless one establish wholly arbitrary lines of demarcation in advance.

This is not the main objection to the term, however. “Atonality”
implies music in which not only is the element of what is defined as
“tonality” no longer a principle of construction, but in which the
composer deliberately avoids all procedures capable of evoking “tonal”
associations. Actually this is virtually impossible, owing to the mere fact
that we use tones, and hear them in relation to each other. In other
words, whenever a series of tones is heard, the musical ear assimilates
it by perceiving a pattern composed not only of tones but of intervals;
and neither the process nor the sensation is different in any essential
principle from the process by which one assimilates music that is unim-
peachably “tonal.” I am reliably informed that Anton Webern himself
insisted on this point, and even on specifically tonal references in his
own music.

Whatever real sense the word “atonality” may have derives from
the fact that the further a chord departs from a strictly triadic structure,
the less unequivocal it becomes in terms of a specific key; and that, in
the proportion that such harmonies become predominant in musical
usage, it becomes more difficult to establish genuine tonal contrast, and
the effort to do so becomes more forced. As the cadence, in the early
years of this century, came to acquire for composers more and more the
aspect of a cliche, and as the composers found themselves more and
more obliged to discover other means of achieving musical articulation,
they found themselves obliged to discover new principles of contrast as
well. They discovered that in the absence of strictly triadic harmony
it was virtually impossible to establish a feeling of key sufficiently un-
equivocally to make possible a genuine and definitive change of key,
and that hence tonality was for them no longer sufficient as a principle
of structure. It was this discovery perhaps above all that led to both
the adoption of the serial principle by Schoenberg and the attempt

Problems and Issues Facing the Composer Today 27

to find ways towards the revitalization of tonal principles that was
embodied in the “neo-Classicism” of Stravinsky.

The above does not mean, of course, that harmony has ceased to
exist in any music, or that it has become an element that can be
ignored. Our harmonic sense is essentially the awareness of one of the
dimensions of music; having acquired that awareness, we cannot do
away with it, and it would be ridiculous folly to try to do so. If art is
to develop, awareness of every aspect of the art must increase rather
than diminish. But harmonic effect as such has clearly ceased to be a
major interest of composers, just as tonality has ceased to be an issue
or a point of reference against which issues can be adequately discussed.
To be sure, the question still arises constantly in the public discussion
of music; but such discussion can have no meaning except on the level
of very precise technical definition. We are dealing with facts, not with
slogans, and the facts have to be referred to basic esthetic and acous-
tical considerations, rather than to specific historical embodiments of
these. If by “tonality” we mean, in the most general terms, the sense of
pitch-relationship and of the patterns and structures that can be cre-
ated out of such relationships, the word “atonality” can have no
meaning, as long as we use tones. If we mean, on the other hand, a
precise set of technical principles and hence of procedures, it is easy
to see in retrospect that the very vitality with which it developed led
ultimately beyond the principle itself.

If the cadence, as conventionally defined, came finally to seem to
many composers, in the context of their own music, little more than a
clich^, it was because they came to feel a definite disparity between
the harmonic vocabulary native to them and the harmonies necessary
to establish the cadence. While the composers of the late 19th century
one senses the problem already in the music of Wagner succeeded
in overcoming this disparity, often through sheer technical ingenuity
and sometimes with visible effort, their successors often found this
impossible to achieve without stylistic violence. It was necessary either
to turn backward or to seek new principles.

As we all know, a similar development took place in the rhythmic
sphere. It took place more quietly and with far less opposition, if indeed
there was any appreciable opposition whatever. There is no need to
dwell on the rhythmic question here. Though the changes that have
taken place have been equally far-reaching, they have been in a sense
less spectacular and less esoteric, if only for the reason that they owe

28 Problems of Modern Music

so much to the influence of popular music and of Gregorian Chant.
They have found incomparably more ready acceptance, both from
musicians and from the general public, than the developments of
which I have spoken in the realm of harmony. Furthermore, the rhythmic
aspects of music are bound closely and inevitably to the other elements
of the musical vocabulary; in this sense one can say that the develop-
ment of music away from the tonal and cadential principle has also
created a whole new set of rhythmic premises and requirements.

In any event, the focal point of the more advanced musical thought
of today is polyphonic, and more concerned with problems of texture
and organization than with harmony in the hitherto accepted meaning
of the term. Once more, this does not mean that composers have ceased
to be acutely aware of vertical relationships between tones, of progres-
sions from one vertical conglomerate to another, or even of the
patterns formed by such progressions. But it is certainly true, I think,
that they tend more and more to think of these matters in terms of texture
rather than harmony as hitherto defined. The current trend is to refer to
such vertical conglomerates as “densities” rather than as “chords” or
“harmonies”; but it must be stressed that there is no satisfactory substi-
tute for awareness of the entire musical context, and that the replacement
of one term by another is useful in so far as it increases that awareness,
and does not connote the evasion of one issue in favor of another.

It is of course fashionable to regard Webern as the patron saint of
the dominant contemporary trend, and to invoke his name as a rallying
point for all that is most aggressively anti-traditional in contemporary
music. As is so apt to be the case, there is a discrepancy at many points
between Webern the symbol and Webern the actual figure. The latter,
however individual his musical style, was of course as deeply rooted
in the Viennese tradition as Schoenberg himself, and probably more
narrowly; and without in any sense meaning to detract from his musical
stature, one can say that he remained a loyal disciple to the extent of
being more Schoenbergian than Schoenberg himself. In the last analysis
he was at least as much the Romantic Expressionist as Alban Berg,
if not more so. Above all, and most important, he was a musician of
the ripest culture, at once the most daring and the most realistic
of artists. The teacher who at times finds himself obliged to stress the
fact so easily lost from view in the heat of speculative enthusiasm
that musical values are, first and last, derived from tones and rhythms
and the effects they produce, and not from their theoretical consistency
or analytical plausibility, can find no better or more demonstrable

Problems and Issues Facing the Composer Today 29

evidence in his own behalf than that furnished by almost any score
of Webern.

At the same time, however, one may find oneself impelled to
question the sufficiency of the post-Webernian trend as a firm and
comprehensive basis for new departures in music. This brings us to the
large question of serialism, which I have deliberately postponed till
after discussing some of the factors that have given rise to it. One
cannot, of course, stress too much that serialism is neither the arbitrary
nor the rigid set of prescriptions that it is often supposed to be, not
only by its foes but unfortunately also by some of its friends. It is rather
the result of many converging trends of musical development, of which
I have mentioned a couple of the most important and the most general
ones. Above all, perhaps, it is the result of the decreasing validity of the
harmonic principle as an organizing force, and the necessity of adopting
consistent relationships between tones, which can serve as a constructive
basis for the organization of musical ideas, along both the horizontal
and the vertical dimensions.

Quite as important is to stress that serialism is in full process of
development, and that the shapes it has taken are already manifold.
It is no longer the exclusive possession of any one “school” or group
of composers, nor is it bound to any one mode of expression, Viennese
or otherwise. Need one cite evidences, at this date? In other words, it is
a technical principle that a wide number and variety of composers
have found useful for their own purposes, both because of the organizing
principles they have derived from it and because of the musical resources
it has opened up for them. Like any other technical principle, it yields
nothing in itself; it is always for the imagination of the composer to
discover what it can give him, and to mold it to his own uses. Like any
other technical principle, it has to be thoroughly mastered, in terms of the
composer’s creative vision; a half-baked relationship to it in this respect
can produce only less than half-baked results. For this reason the
young composer who has not grown up with it from the beginning
there are already a number who have done so, and to whom it is, so
to speak, native would be well-advised to avoid it until he has become
sure of his own musical identity, and can grow into it in full conviction
and genuine musical maturity. It does not provide answers to all
musical questions or in the last analysis to any; it is only a vehicle and
a means, which, let us reiterate, many composers find useful. Once more
like other technical principles, it has acquired its own brand several
brands, in fact of academicism, and many varieties of cliche*, which

30 Problems of Modern Music

are none the less recognizable as cliches for being derived from a
technical principle that has been in active existence for little more
than forty years. Its value lies wholly in the music of the composers who
have seen fit to adopt it, and the value of that music resides in the
imaginative, emotive, and constructive force inherent in it, not in the
ingenuities with which the system is applied, except in so far as these
are the inherent result of a musical conception.

The serial organization of tones must be, and for the most part is,
today regarded as a settled fact the composer is free to take it or
leave it, or to adopt it with varying degrees of rigor, as he may choose.
The results it can yield are open to all to see and judge as they see fit.
More problematical are some attempts that have been made to extend
serial organization to other aspects of music notably to that of
rhythmic values and that of dynamics. Any discussion of these matters
must emphasize once more that it is only results that matter; that the
human imagination works along channels that are frequently unex-
pected, and that a critical scrutiny of technical premises does not release
one in the slightest degree from the responsibility of holding one’s mind,
ear, and heart open to whatever may reveal genuinely new vistas of
musical expression and experience.

With this caution in mind one can easily observe that tones are,
for the musical ear, fixed and readily identifiable points in musical
space, and that the progress from one tone to another has a clear
point of departure and arrival. This is partly the result of the fact
that within the octave there are only twelve tones, with which the
musical ear has familiarized itself over the course of many centuries;
and the additional fact that our musical culture has taught us to regard
as equivalent tones that occupy the same position within the various
octaves. A, for instance, is recognizable as A whether it be played on
the open A string of the double-bass, of the ‘cello, or of the violin
or, for that matter, in the high register of the flute or the piccolo.
Time values, on the other hand, are by no means fixed; their range
is to all intents and purposes infinite. This does not at all exclude
the possibility of adopting an arbitrary series of time values for the
purposes of any single composition, but it does raise very valid ques-
tions regarding the serialization of time values as a general principle.
The serialization of dynamics, however, raises questions of a much
more fundamental nature. Dynamic values are by their very essence
relative, both in an objective and a subjective sense. They have quite
different meanings for different media and under different conditions.

Problems and Issues Facing the Composer Today 31

How can we regard as equivalent, except on the most practical level of
balance, a given nuance on, say, the oboe and the violin, or for that
matter, the same nuance in different registers of the same instrument;
or on the same note on the same instrument, sounded in a small room,
a large concert hall, and the open air? What does the indication p
actually mean, and how can we as listeners distinguish in clear terms
a transition from mf to f, or even from mp to ff?

The basic question of all is of course as is often the case “Why?”
The principle of so-called “total organization” raises many questions
and answers none, even in theory. First of all, what is being organized,
and according to what criterion? Is it not rather a matter of organizing,
not music itself, but various facets of music, each independently and
on its own terms or at best according to a set of arbitrarily conceived
and ultimately quite irrelevant rules of association? Was the music of
Beethoven, or who you will, not totally organized in a sense that is
much more real, since it is an organization of musical ideas and not of
artificially abstracted elements?

The subject of “total organization” leads naturally to the considera-
tion of electronic media, since the latter make possible the exact control
of all musical elements, and make possible in a sense also a partial
answer to some of the questions I have raised. Since the potentialities
of electronic media in the realm of sound are, at least to all intents
and purposes, infinite, it is possible to measure all musical elements in
terms of exact quantity, and in fact necessary to do so, since such
measurement is the very nature of the instruments and the method by
which they are used. A dynamic nuance thus not only can, but must,
become a fixed quantity, as can and must, also, any tone in the whole
range of pitch or color gradations. Every moment of music not only
can but must be the result of the minutest calculation, and the composer
for the first time has the whole world of sound at his disposal.

That electronic media will play a vital and possibly even decisive
role in the future of music is not to be doubted. I must confess however
to skepticism as to what that precise role will be. Two questions seem
to me to be crucial. First of all, it is not sufficient to have the whole
world at one’s disposal the very infinitude of possibilities cancels out
possibilities, as it were, until limitations are discovered. No doubt the
limitations are there, and if not there they are certainly in human
beings. But the musical media we know thus far derive their whole
character and their usefulness as musical media precisely from their
limitations stringed instruments derive their character and utility from

32 Problems of Modern Music

not only the fact that they are stringed instruments, that the tone is
produced by stroking strings, but from the fact that they are not wind
or percussion instruments; and we have learned to use them with
great subtlety of effect and power of expression because of that. The
dilemma of electronic musical media is a little like that of the psychologist
who is reputed once to have said to one of his friends, “Well I have
got my boy to the point where I can condition him for anything I
want. What shall I condition him for?”

The other question has to do with the essential nature of music
itself. Is music simply a matter of tones and rhythmic patterns, or in
the final analysis the organization of time in terms of human gesture and
movement? The final question regarding all music that is mechanically
reproduced seems to be bound up with the fact that our active sense
of time is dependent in large degree on our sense of movement, and that
mechanical repetition mitigates and finally destroys this sense of move-
ment in any given instance; it destroys also our sense of expression
through movement, which plays so large and obvious a part in our
musical experience. This is what lies behind the discussions of the
element of “chance,” which has so bothered the proponents of “total
organization.” But the element that “total organization” leaves out
of account is not chance at all. It is the organic nature of movement
as such, of the fresh and autonomous energy with which the performer
invests each musical phrase, every time he sings or plays it, and which
gradually disappears for our awareness if we listen so often to a mecha-
nical reproduction of it that we become completely familiar with it, to
the point of knowing always exactly what is coming next. It is more than
the element of mere “surprise”; it is rather that if the expression of
movement is to become effective, we require not only the evidence
of movement from one point to the next, but a sense of the motivating
energy behind it.

To raise these questions is not in any sense to reject the principle
of electronic music as such. In the first place, composers are beginning
to feel the need for new instruments. The existing ones, for all their
technical perfection, are beginning at times to seem vaguely obsolete as
far as some of the composers’ musical ideas are concerned. The possi-
bilities electronic music suggests are altogether likely to make this situ-
ation more acute.

In my own opinion, electronic media more than .justify their exist-
ence if only by the new insight one can gain from them into the
nature of sound, musical and otherwise, and above all by a vast quantity

Problems and Issues Facing the Composer Today 33

of fresh experience they can provide, on the purely acoustical level.
They are still in a clearly very primitive stage and it is impossible to
say what they may contribute in the future. But they raise the above
questions and many others, and the questions will certainly become
more acute as the media develop.

One hears a good deal, these days, of the developing “dehumani/a-
tion” of music and the other arts; and specifically in regard to the
tendencies we discussed in detail at the Princeton Seminar last year,
and which I have been discussing in these pages. This is all very well,
and not without its plausibility; but we are speaking of a movement
that is widespread among the younger composers of Europe, that has
begun to take root in the United States, and that above all is in constant
development and evolution. Many ideas are being tested, and many are
quickly discarded. If we regard certain manifestations with raised eye-
brows, that is our privilege as members of an older generation, as it is always
our privilege to point out flaws in logic. But if it is also our prerogative
to insist on the primacy of the creative imagination, and to minimize
the decisive importance of theoretical speculation, we are at the same
time obliged to abide by our own premises, and look towards artistic
results rather than towards the ideas by which these are rationalized.
By the same token it is well to remember that art, considered on the
most objective level, reflects the attitudes of the individuals that produce
it. The danger of dehumanization is a real and patent one, and the
individual can, and certainly should, resist any dehumanizing tendency
with all his strength. But this cannot, and must not, blind us to the
claims of whatever is genuinely new and vital in the arts, or, once
more, cause us to forget that it is the product, not the process, that is
of real importance; and that the creative imagination, at its most
vital, has revealed himself through many and often surprising channels.
There is no reason to believe that it will not continue to do so, as long
as creative vitality which for musicians means above all the intense
love of music continues to persist.



THE analysis of music especially of traditional music is one of
the most respected of theoretical disciplines, but the respect in
which it is held would do it a disservice if it prevented the periodic
re-evaluation of the subject. What is analysis, or what ought it to be?
What are its purposes? To what extent are traditional concepts and
methods applicable to new music? What are the relations of analysis to
performance and to criticism? My tide refers to a discussion, from the
point of view of today, of these questions; it is in no way meant to imply
that I have a new system to promulgate, or that I have made startling
discoveries about new music.


Rather than presenting at the outset a naked definition of the term
under consideration, let us begin by looking at a familiar example. The
first few measures of Tristan have performed many services other than
their original one of opening a music-drama; let them serve yet another
and open the argument here.

Ex. i
This chordal sequence can be accurately enough described as a
minor triad on A, a French sixth on F, and a primary seventh on E;
but such a description, revealing nothing of the relationships among
the three chords, involves no analysis whatsoever. If, however, one

5 * 6 7
refers to the passage as I ^ -II 4 -Vj, he has performed an elementary



Analysis Today 35

analytical act: he has related each of the chords to a tonic, and hence
to one another. He has made a discovery, or at least a preliminary
hypothesis to be tested by its fruitfulness in leading to further discovery.
But the analysis as such ceases with the choice of the tonic; once this
has been made, the assignment of degree numbers to the chords is pure
description. If, on the other hand, one points out that the second chord
stands in a quasi-dominant relation to the third, he is doing more than
simply assigning names or numbers: he is again discovering and
explaining relationships.

Ex. 2
Turning now to the actual score, the analyst might begin a program
note thus: “The rising leap of the ‘cellos from A to F is succeeded
by a chromatic descent, followed in turn by . . .” He need not continue;
this is pure description. But when he points out that Example 1 repre-
sents the chordal skeleton of Example 2, he is once more on the right
track. He can go still further by showing that all the appoggiaturas have
half-step resolutions, and that the motif so created is augmented in
the motion of the bass, and paralleled in the alto, in such a way that
the chordal progression of measures 2-3 becomes an amplification of the
melodic half -step of measure 1.

Ex. 3
The fact that in the above diagram no such analogy has been pointed
out in the half-steps E-DJ and A-AJ is in itself an important though
negative part of the analysis, since it implies by omission that these
progressions, if relevant at all, are incidental and subordinate.

Going one step further, one might claim that, from a serial point
of view, the opening sixth is imitated in the third E-Gf (see Ex. 4).
This is the point at which analysis proper passes over into what I call


Problems of Modern Music

prescription: the insistence upon the validity of relationships not sup-
ported by the text. In the above case, for example, the orchestration
implies the wrong-headedness of the suggestion, since the opening
interval, played by the ‘cellos alone, is heard as a unit, whereas the
E-GJ is divided disparately between ‘cellos and oboe.

Analysis, then, exists precariously between description and prescrip-
tion, and it is reason for concern that the latter two are not always easy
to recognize. Description is current today in the form of twelve-tone
counting necessary, no doubt, as preliminary to further investigation,
but involving no musical discrimination whatsoever. Prescription, on the
other hand, is obvious in the absurd irrelevancies of Werker’s analyses
of Bach but is equally inherent in some of Schenker’s more dogmatic
pronouncements and in those of his followers.

It should be clear at this point that true analysis works through and
for the ear. The greatest analysts (like Schenker at his best) are those
with the keenest ears; their insights reveal how a piece of music should be
heard, which in turn implies how it should be played. An analysis is a
direction for a performance.

In order to explain how a given musical event should be heard,
one must show why it occurs: what preceding events have made it
necessary or appropriate, towards what later events its function is
to lead. The composition must be revealed as an organic temporal
unity, to be sure, but as a unity perceptible only gradually as one
moment flows to the next, each contributing both to the forward
motion and to the total effect. What is often referred to as musical
logic comprises just these relationships of each event to its predecessors
and to its successors, as well as to the whole. The job of analysis is to
uncover them explicitly, but they are implicitly revealed in every good
performance. Description, restricted to detailing what happens, fails to
explain why. Prescription offers its own explanation, referring to an
externally imposed scheme rather than to the actual course of the music.

One more familiar example may clarify this view of logical or, as
I prefer to call them, ideological relationships.

Analysis Today 37

The recapitulation of the Prestissimo from Beethoven’s Sonata Op.
109 bursts in upon the development in such a way that the lit (V of V)
is followed immediately by I. From a narrowly descriptive point of view
one could call this an ellipsis, pointing out that the normally expected V
Ex. 5

has been omitted. Looking ahead, however, one will find that the first
phrase of the recapitulation ends on V, and its consequent on I. The
puzzling II|, then, only temporarily and apparently resolved by what
immediately follows it, actually points ahead in such a way that the
whole passage is bound together in a cadential II-V-L The propulsion
thus generated is given an extra spurt by the compressed II-V-I at the
end of the consequent, and the forward motion is renewed with fresh
energy by the elision that sets the next period going.

I need hardly mention the obvious effects of such an analysis on the
performance of this passage. Whatever doubts one had as to the proper
placing of the main accent in these phrases when they first appeared can
now be resolved; the exposition can be reinterpreted, if need be, in the
new light of the recapitulation.


It should be apparent at this point that analysis and hence per-

38 Problems of Modern Music

formance as it has been discussed above cannot apply to certain types
of composition in vogue today. When chance plays the major role in
the writing of a work, as in Cage’s Music for Piano 21-52, logic as
defined above can take only an accidental part. The same is true of
music written according to a strictly predetermined constructivistic
scheme, such as Boulez’s Structures. In neither case can any musical
event be linked organically with those that precede and those that follow;
it can be explained only by referring to an external structure in the
one case the laws of chance and in the other the predetermined plan. The
connections are mechanistic rather than teleological : no event has any
purpose each is there only because it has to be there. In a word, this
music is composed prescriptively, and the only possible or appropriate
analytic method is to determine the original prescriptive plan. This is not
analysis but cryptanalysis the discovery of the key according to which
a cipher or code was constructed. (If we are lucky, the composer or one
of his initiates will spare us a lot of hard work by supplying us with the

A third category that does not permit analysis is represented by
Stockhausen’s Klavierstuck XI, where improvisation is given such free
rein that it actually creates the form of the work anew at each perform-
ance. Thus Klavierstuck XI does not exist as a single composition and
cannot fruitfully be treated as one. Each new rendition can be discussed
on its own merits, to be sure; but the relationship of all such versions
to the abstract idea of the piece as a whole, and the decision as to the
esthetic value of such an experiment these problems can be argued
endlessly. At any rate they are far afield from the practical considerations
that are our concern here. (It need hardly be pointed out that improvi-
sation as’ traditionally applied to the framework of a Baroque concerto,
for example, had purposes quite different. A cadenza served not only to
show off the soloist’s virtuosity but also to punctuate an important
cadence; the soloist’s elaboration of a previously stated orchestral melody
clarified the dualism inherent in the form. The quality of a given realiza-
tion depended on its appropriateness to the compositional situation; the
performance did not, as in many present-day examples, create the


The analysis of music of the periods closely preceding our own
the 18th and 19th centuries has almost always assumed the applica-
bility of certain familiar norms: tonally conditioned melody and har-
mony, periodic rhythmic structure on a regular metrical basis. Naturally

Analysis Today 39

such standards cannot be applied uncritically to the music of our own
century, but on the other hand they should not be dismissed without
examination. I contend that, in a more generalized form, they are still
useful. Regardless of vocabulary, linear and chordal progressions still
show striking analogies to older tonal procedures, analogies that are in
turn reinforced by rhythmic structure. Only in those rare cases where the
music tries to deny the principle of progression (as in the examples cited
in the immediately preceding section) are such analogies completely

This point of view is more generally accepted with regard to harmony
than to melody, perhaps because harmonic analysis is the more firmly
entrenched discipline. After all, for many musicians theory is synonymous
with harmony, melody being supposedly a free creative element, neither
in its composition nor in its perception subjected to rule. (They forget,
of course, that the object of the study of counterpoint is primarily the
construction, and only secondarily the combination, of melodies.)
Whereas Hindemith’s enlargement of traditional harmony to encompass
present-day vocabularies is generally known and often applauded, his
attempt to find a melodic framework, actually a much less questionable
procedure, is often ignored.

Another reason for shunning melodic analysis is that it is not always
easy or even advisable to abstract the purely linear element from a pro-
gression. Wagner, in such motifs as the Wanderer and the Magic Sleep,
is writing passages in which the melodic aspect is an incidental result of
the chordal motion. A little later, Debussy offers examples (like the
opening of Reflets dans I’eau) in which a linear phrase is dissolved into
an atmospherically dispersed harmony that implies without actually
stating the expected melodic resolution. Hyper-impressionistic pages, like
parts of the Night-Sounds from Bart6k’s Out-of-Doors Suite, fragmentize
the melody to such an extent that the progressive element is heard to be
the increase and decrease of density as the motifs follow one upon the
other, rather than the specifically linear aspect, which is here reduced
to a minimum. Nevertheless, wherever there are successive differentia-
tions in pitch there is melody of some kind, and wherever there is melody
the ear will try to hear it in the simplest possible way.

This is not meant to imply that we must expect to find behind con-
temporary melodic lines the simple stepwise diatonic framework that
Schenker has pointed out in Classical examples. But the ear will natur-
ally connect each tone with those nearest it in pitch. The adjacent pitches
may be diatonic or they may be chromatic; they may be actually adja-


Problems of Modern Music

cent or displaced by one or more octaves; they may be present by implica-
tion only. In some cases motivic associations or peculiar scale-formations
may enforce the acceptance of a larger module as in the simple case
of bugle-calls, the adjacent tones of which are a third or a fourth apart.
(In the case of microtonal music, smaller modules may be in effect,
although it is doubtful to what extent even present-day ears can accept
them.) In every case the ear will do the best it can with the available
intervals. It is the duty of the analyst to show the pattern of connections
by which an educated ear his own makes sense of the total melodic

Even less than in traditional melodies must one assume that there
is one uniquely correct way of hearing. Rather, the best analysis is the
one that recognizes various levels functioning simultaneously, as when
a tone resolves once in the immediate context but turns out to have a
different goal in the long run. Two very brief examples may help to
clarify this point of view.

Ex. 7
The first is the opening of Schoenberg’s Klavierstiick Op. 33a.
Chordal rather than melodic in conception, its linear structure is never-
theless clear. Despite the octave displacements, a line can be traced in
the uppermost voice from the Ft in the first measure to the B in the
third. (Notice, however, that at one point two adjacent tones are pre-
sented simultaneously instead of successively.) At the same time, the
original Bb leads, through various voices but always at the original
octave-level, to the same tone of resolution. At this point the entrance
of the F, repeating the climactic F of the second measure, begins a new
motion that is carried forward through the succeeding phrase.


Analysis Today

The second passage is from the second of Sessions’s piano pieces
From My Diary. 1 Here both the F in the first measure and the Gb in
the third are associated with upper and lower chromatic neighboring
tones. But what of the cadential motif? Why is the pattern altered? And
why is the linear descent from the C\> in the second measure broken at
this point? There are several possible answers, all of which are probably
relevant. First of all, the most prominent bass-note in each of the four
measures as indicated by its repetition and by its quarter-stem is an
F, which can be heard as a resolution, at another level, of the hanging
Gb a resolution confirmed by a direct Gb-F in the bass. But at the
same time, there seems to be an implied E filling the space between the
Gb and the D in its own voice a tone suggested by the original associa-
tion of E with Gb> and by the prominent whole-step motion in the melo-
dic descent. In this case the line gradually increases its pace as it descends.
But if it seems far-fetched to introduce an unstated, understood element,
one can hear the skip Gb-D as a way of emphasizing the cadence, and
point out that the motif of neighboring tones aims each time more
directly towards its resolution: the first time the neighbors follow the
principal; the second time they precede it; and the last time the principal
takes the place of one of its own neighbors. Finally, it should be noted
that the next phrase takes off from the dangling Gb in a subtle motivk
reference to the beginning.
Ex. 10
It is of course impossible to do justice here to the role of such
details in the total melodic structure, but on examination one will find

1 Copyright 1947 by Edward B. Marks Corporation. Reproduced by permission.


Problems of Modern Music

the same kind of connection at work in the large. Note, for example,
how much of the first theme of the Schoenberg piano piece is controlled
by the high F already mentioned whether in its original octave or
in another and by its association with the adjacent E. It is again this
F, in its highest register, that prepares for the recapitulation; and it is
the E that, returning first with the tranquil second theme, later closes
the motion in a lower octave in the final measure. In sum, modern melody
can not get rid of stepwise motion, because that is the way we hear
melody; but it can and does expand (or on occasion contract) the
distance, both temporal and spatial, between successive steps. From
this point of view even Webern is found to be no pointillist, but a
draughtsman of subtle and fragile lines.

The role of harmony hi the music of our century, although more
extensively explored, is perhaps more difficult, complicated as it is by
many factors, such as the frequent exploitation of the static, sensuous
effect of the chord in addition to or even at the expense of its progressive
functions. As a result, one can no longer assume the easily defined
functionality of obviously tonal music. Chords can no longer be precisely
named, nor can their identity be maintained in differing contexts. But it
is important to realize that, even in stubbornly non-triadic music, the
concept of the chord remains, by analogy at least. The composer can set
up arbitrary simultaneities that, by their commanding position or by
repetition, are accepted as the controlling sonorities the chords^
against which other tones can function in the manner of traditional non-
harmonic tones. Bartok’s Improvisations Op. 20 show how by such a
technique quite complicated sonorities can be used to harmonize simple
modal folk-tunes. In the following example from the last of Sessions’s
Diary pieces the metrical position and the half-step resolutions suggest
that the first chord is an appoggiatura to the second; this supposition is
confirmed by the appearance of the root-like D in the bass, and by the
clinching repetitions that ensue.

Ex. ii
In fact, only where the contrapuntal aspect becomes so strong that
every element of each sonority is heard primarily as a point in a moving

Analysis Today 43

line, or at the other extreme, where the texture is completely pointillistic,
is the chordal concept seriously challenged. In such cases one further
assumption of traditional harmony that must then be questioned is the
primacy of the bass. Contrapuntally or coloristically, of course, it will
have gained in importance, but at the expense of its role in defining the
harmony. A beautiful example of this process already at work over a
century ago is shown in the opening of Liszt’s ValUe d’Obermann, where
the melodic action of the bass clouds the harmony. Not until the return
of the theme adds a new bass underneath the original one is the situa-
tion made clear. A further step in this direction is taken by Mahler,
who by his poiyphonically opposed chords points the way towards poly-
tonality in the magical cowbell passage in the first movement of his
Sixth Symphony. A more thoroughgoing example is Stravinsky’s Sym-
phonies pour instruments a vent, a more truly polytonal work than any
of Milhaud’s often-cited Saudades, which in fact present only extended
and elaborated harmonies over a single real bass.

There are other forces at work undermining the primacy of the lowest
voice. Impressionistic parallelism, which reduces its role to that of color-
istic doubling, is too well known to require citation. Less frequent, but
possibly more important in the light of later developments, is the masking
of the true harmonic bass by a decorative voice below it, a technique
seen clearly in the repetition of the opening of La Fille aux cheveux de
lin<* Another device, common to the Impressionists and Mahler, is the
ostinato. From one point of view the persistent voice is emphasized, but
at the same time it is removed from the sphere of action. In Debussy,
as later in Stravinsky, the ostinato results in harmonic stasis; in Mahler
there is a constant tension between the harmony implied by the motion-
less bass and those outlined by the moving voices and chords above it.
In both cases the functional role of the bass is called into question.

So far no specific reference has been made to the problem of tonality.
Except in comparatively rare cases, such as passages in Le Sacre du
printemps, where an almost completely static tone or chord of reference
is set up, tonality is created not by harmony alone, nor even by harmony
and melody, but by their relationship with the rhythmic structure: in a
word, by the phenomenon of the cadence. A discussion of certain rhyth-
mic aspects, then, can no longer be postponed.


Much of the vitality of the music of the Classical period derives
from the constant interplay of meter and rhythm, the former determined

44 Problems of Modern Music

by regular beats and measures and the latter by constantly varying motifs
and phrases. This tension between the abstract and the concrete begins
to break down during the 19th century, when phrase articulation is often
either slavishly tied to the meter or else so completely liberated that the
sense of the meter is almost lost. The retention of the measure in much
Impressionistic music is purely conventional, and it is no wonder that
later composers have abandoned the effort to keep an abstract pattern
when it would conflict with the actual rhythm. For this reason the regu-
larity of the meter in such composers as Webern must be carefully exam-
ined. Is it to be felt as a constantly present control? Is it a pure
convention? Is it, as some would have us believe, an evidence of the
composer’s numerological superstitions?

The answers to such questions must always be given with specific
reference to the text involved. When, as in the case of Example 11, the
motif sets up a clear cross-rhythm, the explanation is relatively easy.
Webern’s Piano Variations, on the other hand, present the problem in
an acute form. What has happened here, I think, is that the composer
has called on a complex set of interrelationships of rhythmic, metric,
dynamic, and textural factors to compensate for the tenuity of melodic
and harmonic interest. In the first twelve measures of the last movement,
for example, I find at least seven different time-divisions simultaneously
functioning. These are set up by the meter (3/2), a possible cross-meter
(5/4), the rhythm of the two-note motifs, the rhythm of the phrases,
the tone-row, the dynamic alternations, and the linear pattern (Ex. 12). a

The really important question to ask in all such cases and even in
cases where the composer has deliberately tried to get rid of all traditional
metrical measurement is, can we locate the structural downbeat? If
we can, then we can proceed with analytic concepts in some way
analogous to those of the traditional rhythm and meter, phrase and
cadence. If not, some completely new rhythmic theory must be devised.
Some musicians, like Stockhausen, are trying to do this, but I have as
yet seen no satisfactory one emerge.

By structural downbeat, of course, I do not mean the arbitrary
accentuation of the first beat of every measure; I mean rather phenomena
like the articulation by which the cadential chord of a phrase is identi-
fied, the weight by which the second phrase of a period is felt as resolv-
ing the first, the release of tension with which the tonic of a recapitula-

* Copyright 1937 by Universal Edition, A.G,, Vienna. Reprinted by permission
of Associated Music Publishers, Inc., sole agents for the United States.

Analysis Today 45
Ex. 12

Rnhiff f liefiend J = c so
01 |. 2 3
If^” 4 ” *

.”P. j.i;
iJ rj* rg

\^^^ K

” ‘r “=4

^ i j


.10 p.
i L”

tion enters. (In the Webern example, I hear the downbeat as the Eb at
the beginning of measure 12; and I consider it no accident that it occurs
at the beginning of a measure, preceded by a ritardando* )

It is just here that the importance of rhythm to the establishment of
tonality emerges, for the cadence is the point in the phrase at which
rhythmic emphasis and harmonic function coincide. It would be partly
true to say that the cadence creates tonality, but it would be equally
true to say that tonality creates the cadence. Where the cadence exists, it
is impossible to hear music as completely atonal, even though one may be
unable to define the key in conventional terms.

46 Problems of Modern Music

We know the signs by which a cadence can be recognized in tradi-
tionally tonal music: its position at the end of a phrase, the melodic
resolution, the change of harmony. The actual downbeat may not always
exactly coincide with the cadential point, but such unusual cases arise
most often when the phrase is rhythmically prolonged (the feminine
ending) or when it points ahead so clearly that the next phrase acts as
a huge cadence to the first (as when an introductory section is followed
by a main theme). In any case, keys are defined by the appearances of
strong, cadential downbeats whether clearly on the tonic as in most
Classical examples, or on deceptive resolutions, as notably in the Prelude
to Tristan.

The extent to which analogous principles govern the structure of
contemporary music is surprising. A few examples will show them at

The opening of the second movement of Bart6k’s Fifth Quartet may
prove puzzling until it is heard as an upbeat. The first downbeat comes
on the D in measure 5, clinched by an even stronger cadence on the same
tone (now supported by its fifth) in measure 10. The digression that
follows suggests the key of C, but this tonality is not confirmed by the
cadence, which, when it arrives in measure 20, is again clearly on D.

The first page of Sessions’s Second Sonata for Piano is much less
triadic; yet when the downbeat comes in measure 1 1, the harmony of Bb
is clearly established. Not only the V-I implied by the progression of fifths
in the bass, but the melodic resolution to D, accented by the downward
leap, points towards this tonal center, which is confirmed by what follows.
In the second movement, no such clear downbeat is presented, but the
two important feminine cadences of measures 177 and 190 both suggest
an unstated resolution to E. The important downbeat of measure 191,
coming as it then does on F, is in the nature of a neighboring harmony;
and not untiLmuch later, at measure 213, does the expected E occur, its
extension as a pedal for ten measures compensating for its long postpone-
ment. The last few measures of the Lento act as an upbeat released in the
return of Bb in the opening of the finale. But this in turn, after a long
battle with conflicting elements, gives way at the last to the key of C,
on which a downbeat is firmly established in the final chord.

Stravinsky is sometimes referred to as a “downbeat composer,” by
which I suppose is meant that he often emphasizes the beginnings rather
than the endings of his phrases. This results in a weakening of the
cadential sense, it is true, the phrases so accented being as it were huge

Analysis Today 47

feminine endings to their own opening chords. A typical example is
the opening of the Serenade en la. The harmonic progression would

be described in traditional terms as VI^-Ig in A minor; actually the

F of the first chord is heard as hardly more than an appoggiatura re-
solving to the E of the second. This would appear to be no progression
at all, in which case the phrase should be a huge diminuendo. Yet we
cannot be too sure: in a similar situation at the beginning of the third
movement of the Symphony of Psalms, the composer, by changing the
mode and the orchestration at the cadential word Dominum> creates
a clear accent even though the chord has remained essentially the same
(C) throughout the phrase.

In any event, whatever we may decide about the reading of his
phrase-accent in detail, Stravinsky is perfectly capable of producing a
big structural downbeat at precisely the point where it is required. I need
only point to the huge deceptive cadence that opens the Symphony
in Three Movements, the dominant G of the introduction resolving
finally upward to the A of the ostinato theme (rehearsal number 7);
or to the way in which the Interlude acts as an upbeat to the G major
of the finale.

More controversial is the attempt to find traces of tonal form in
avowedly atonal compositions; yet I do not see how music like Schoen-
berg’s, with its usually clear cadential structure, can fail to arouse certain
traditional associations and responses. The previously cited Klavierstuck
Op. 33a begins with six chords, of which the second through the fifth
are very easily although not necessarily heard as forming a progres-
sion referring to E minor. This in itself is nothing, but when the opening
phrase is heard as an upbeat resolved in the third measure, and when
the resolving sonority is recognized as a seventh on E, a tonal analogy
is set up. The first section of the piece concludes even more unmistak-
ably on E, with the added emphasis of a ritardando; and the theme that
follows in measure 14 gives the effect of a sudden shift of key. In the
recapitulation, the ritardando of measure 34 again calls attention to the
following downbeat, where the E appears in the upper voice, but
supported hi the bass by A in the manner of a deceptive cadence on
IV. It remains for the final cadence to confirm the E, which is so strong
that it is not dislodged by the dissonant tones with which it is here

Several objections can be made to the above account: that it picks

48 Problems of Modern Music

out isolated points without reference to the movement between them,
that the “cadences” on E are a result of the fact that the row ends on
that note, that such analysis is irrelevant to music in this style.

To the first count I plead guilty. I have indeed picked out isolated
points, because these seemed to me to be the important “full-cadences”
of the piece. (Important “half-cadences” occur at measures 9, 24, and
32.) The movement between them cannot, I grant, be explained in
simple tonal terms. At some points, linear or contrapuntal motion domi-
nates in which case the melodic principles suggested above will indicate
the logic of the chosen cadences. At other points the sonorities them-
selves dominate and these can of course be shown as derived from the
opening chords. As a result the entire piece can be heard as a develop-
ment of its original cadential progression that is, as analogous to a
traditional structure.

I agree that the cadences are partially due to the use of the row.
Depending on one’s point of view, this effect is a virtue or a vice of
Schoenberg’s twelve-tone technique. It may even have been one of the
points persuading him to turn towards the system, away from freer
atonal methods. In no case can the argument invalidate the actual mus-
ical result.

To the charge of irrelevancy, I answer that one who cannot indeed
hear such cadential phenomena in this music must judge the analysis to
be prescriptive and inapplicable. But one who does hear them must admit
to that extent the validity of the approach. He may counter that one
ought not to hear the music in this way; but he is then criticizing the
music, not the analytical method. Unwanted cadential effects would be
as great a flaw in atonal music as the chance appearance of a human
figure in a non-representational painting.

The last point suggests that there is a relation between analysis and
criticism. It is not a simple one. Analysis can often reveal flaws in a
work, it is true often but not always. If it were dependable in this
regard, we should be able to decide definitively between the disputed
Cf and CX in the last movement of Beethoven’s Sonata Op. 109 (meas-
ure 55) or whether the famous Atj in Schoenberg’s Op. 33a is indeed an
Ab (measure 22). But unfortunately such cases all too often work both
ways: the CH that from one point of view prepares for the advent of D
two measures later might have been avoided in order not to anticipate

Analysis Today 49

it; by the same token, although the Ab seems more logical in the row-
structure (in spite of the Afcj lacking in the left-hand), it may somewhat
spoil the freshness of the Ab-Eb fifth that comes soon after. The ear
must be the ultimate judge of such subtleties, but insofar as analysis
trains and sharpens the ear it makes its contribution to the final decision.

It would be tempting to go further and state that analysis can
demonstrate the quality of a work, but this requires a faith in rationality
that I am unable to summon. Judgment of final excellence must be
fundamentally intuitive. If analysis leads one to condemn a work he
nevertheless continues to hear as good, he must conclude that there is
something wrong either with his ear or with his method. Since he cannot
dispense with the only pair of ears he has, upon whose evidence the
examination should have been based in the first place, he must blame his
method. He must then find a new one based on his own hearing, one
that will substantiate, not contradict, his musical judgment. He may then
claim that analysis has established the excellence of the work in question,
but , he will be wrong; his own judgment will have established the

One positive point emerges here, and it is a crucial one. The good
composition will always reveal, on close study, the methods of analysis
needed for its own comprehension. This means that a good composition
manifests its own structural principles, but it means more than that. In
a wider context, it is an example of the proposition that a work of art
ought to imply the standards by which it demands to be judged. Most
criticism today tacitly accepts the truth of this statement and sets about
discovering the standards implied by a given work and testing how well
it lives up to them. For investigation of this kind, analysis is naturally
of primary importance.

Criticism should take a further step, however, and the best criticism
does. It should question the value of the standards. A work that sets no
clear standard denies or defies the possibility of evaluation; one that does
set its standard fails or succeeds insofar as it measures up to it; one that
measures up completely is at least flawless but its value cannot exceed
the value of its own standard. It is this final step that is completely
beyond the confines of analysis.

The music of Webern is a prominent case in point. No serious critic
denies the perfection of his forms and the complete consistency of his
style. Its paucity of normal melodic and harmonic interests has been
mentioned above, but hi connection with other values that, replacing

50 Problems of Modern Music

these, uniquely characterize his manner. What is seldom questioned is
the significance of the style itself of the restrictive standard (for it is a
restrictive one) that Webern set for his own music. Are the limits too
narrow to permit accomplishment at the very highest level? Only a
decision of this point can determine one’s final evaluation of the com-
poser. It is a decision that depends on one’s beliefs about the limits and
aims of art in general and is thus not exclusively musical, although it
must at the same time be peculiarly musical. It must be made on faith,
and it must be accepted or rejected in the same spirit.



I agreed to discuss the rhythmic procedures I use in my
music, I had forgotten, for the moment, the serious doubts I
have about just such kinds of discussion when carried on by the com-
poser himself. That a composer can write music that is thought to be
of some interest is, of course, no guarantee that he can talk illuminatingly
about it. It is especially hard for him to be articulate because inevitably
his compositions are the result of innumerable choices many uncon-
scious, many conscious, some quickly made, others after long deliberation,
all mostly forgotten when they have served their purpose. At some time
or other, this sorting and combining of notes finally becomes a com-
position. By that time many of its conceptions and techniques have
become almost a matter of habit for the composer and he is only dimly
aware of the choices that first caused him to adopt them. Finally, in an
effort to judge the work as an entity, as another might listen to it, he
tries to forget his intentions and listen with fresh ears. What he is
aiming at, after all, is a whole in which all the technical workings are
interdependent and combine to produce the kind of artistic experience
that gives a work its validity and in so doing makes all its procedures
relevant. There is no short-cut to achieving this final artistic rele-
vance. No technique is of much intrinsic value; its importance for
the composer and his listeners lies only in the particular use made of it
to further the artistic qualities and character of an actual work. If in
discussing his works, therefore, he points out a procedure, he is bound
to feel that he is drawing attention to something of secondary importance
and by dwelling on it misleading others into thinking of it as primary.
Schoenberg expressed such doubts in essays on his use of the twelve-tone
method. And he was right, for certainly the twelve-tone aspect of his
works accounts for only a part of their interest, perhaps not the most


52 Problems of Modern Music

important part. For from Opus 25 to his last works the number of
different kinds of compositions he wrote illustrates the very broad range
of expression and conception and the wide variety of musical techniques
that can incorporate the system and yet be distinguished from it.

In any discussion of specifically contemporary procedures, there are
a few serious risks involved that must be constantly borne in mind. The
first is the danger of rapid and wide dissemination of oversimplified
formulas that shortens their life. It is obvious that one technical fad
after another has swept over 20th-century music as the music of each
of its leading composers has come to be intimately known. Each fad
lasted a few years, only to be discarded by the succeeding generation of
composers, then by the music profession, and finally by certain parts of
the interested public. So that through over-use many of the striking
features of the best works lost freshness, it was hard for those close to
music to listen to these works for a time, and many of the better works
disappeared from the repertory without a trace. Such a formula as the
Impressionists’ parallel ninth chords, for instance, wore itself out in the
tedious arrangements of popular music current until recently. Each of
the trends of our recent past primitivism, machinism, neo-Classicism,
Gebrauchsmusik, the styles of Bart6k and Berg and now those of
Schoenberg and Webern has left and will leave in its trail numbers
of really gifted composers whose music, skillful and effective as it is, is
suffocated, at least for a time, by its similarity to other music of the same
type. Of course, ultimately this faddishness is trivial, but its mercurial
changes today have made the life of many a composer a great trial,
more even than in the time of Rossini, who is now generally thought
to have been one of the first outstanding composers to have given up
composing because he could not change with the times.

The tendency to fad has been greatly encouraged by the promulga-
tion of systems, particularly harmonic systems. Many recent composers
following Schoenberg, Hindemith, and Messiaen have gained renown
by circulating descriptions of their systems even in places where their
music was not known. This kind of intellectual publicity can lead to a
dead end even more quickly than the older fads derived from the actual
sound of music in styles the composer did not even bother to explain.

The popularity of modern harmonic systems is, unfortunately, easy
to understand. Textbooks led music students to think of harmony as a
well-ordered routine, and when they found it to be less and less so in
the years from Wagner to the present, they were much troubled and
still arc by the gap between what they learn and what they hear in

Shop Talk by an American Composer 53

modern music. For mature composers, lack of system is usually not
much of a problem since they write, as they probably always have, what
sounds right to them. This “tightness” has come, I suppose, from a
developed sensitivity and experience that take time to acquire. When
modern systems of harmony that were orderly and easy to explain ap-
peared they filled an important pedagogical need for the inexperienced.

The very ease with which any of these systems can be used has its
obvious dangers, as I have said. With the help of these and other short-
cuts a vast amount of music is being written today, far more than can
ever be played, than can ever be judged or widely known. At the same
time there seems to be little corresponding development of discrimina-
tion, or even of ability or desire to listen to new music, little expansion
of opportunities for performance, at least in this country. The struggle
to be performed and to be recognized makes it very hard for one not
to become, even against one’s will, some kind of system-monger, par-
ticularly if one uses certain procedures that are considered effective. For
among students there is today a hunger for new formulas, and they
constitute an interested public.

Obviously the only way to withstand the disturbing prospect of
being swept away by a change in fad is to plunge into the even more
disturbing situation of trying to be an individual and finding one’s own
way, as most of us have tried to do, not bothering too much about what
is or will be sanctioned at any given moment by the profession and the
public. We may then have to lead our lives producing works “too soon”
for their time as Webern did, if they are not really “too late” since, if
professional, they presuppose an attentive public which seems to be
getting rarer. We are caught in a development dictated by convictions
impossible to change with the fads.

All this is to say that I do not consider my rhythmic procedures a
trick or a formula. I do not even feel that they are an integral part of
my musical personality, especially in the way I used them in my First
String Quartet (1951), which delves elaborately into polyrhythms. As
I have suggested, all aspects of a composition are closely bound together,
and for this reason I cannot give an orderly exposition of any without
bringing in a large perspective of ideas. So I do not know where to
begin, and I need your help in directing this discussion to regions that
will be interesting and useful to you. Almost anything I might say, I
suppose, preferably on musical subjects, might be considered relevant
to the subject you have so kindly invited me to discuss here.

54 Problems of Modern Music

Question: In the program notes of your Variations for Orchestra
which you wrote for the Louisville performance, you described your
method of variation as being a method of transformation, which you
compared to the transformation from one life-stage to another of some
marine animals. What did you mean by this?

Answer: As musicians you are all familiar with the problems of
program notes. Technical discussions baffle the greater part of the au-
dience and the few who do understand are apt to feel that the composer
is a calculating monster, particularly since musical terms are ponderous,
not always very definite in meaning, and too often give the impression
of complexity when describing something very obvious to the ear. If I
had described the augmentations, diminutions, retrograde inversions as
they occur, this would have been positively bewildering to the public
and would not have helped it to listen certainly not the first time.
So I tried to find a comparison that would help the listener to grasp
my general approach. Serious music must appeal in different ways. Its
main appeal, however, emerges from the quality of the musical material
or ideas and perhaps even more from their use in significant continuities,
but does not always depend on grasping the logic of the latter on first
hearing. There has to be something left for the second time, if there
ever is a second time.

As in all my works, I conceived this one as a large, unified musical
action or gesture. In it, definition and contrast of character decrease
during the first variations, arriving at a point of neutrality in the central
variation, then increase again to the finale, which comprises many dif-
ferent speeds and characters. This work was thought of as a series of
character studies in various states of interaction with each other both
within each variation and between one and the next. Activity, develop-
ment, type of emphasis, clearness or vagueness of definition, I hoped
would also contribute to characterization. Form, rhythmic and develop-
ment processes as well as texture and thematic material differ in each
one for this reason.

The characteristic effort of the serious composer, as I see it, is not
so much in the invention of musical ideas in themselves, as in the
invention of interesting ideas that will also fill certain compositional
requirements and allow for imaginative continuations. Serious music
appeals to a longer span of attention and to a more highly developed
auditory memory than do the more popular kinds of music. In making

Shop Talk by an American Composer 55

this appeal, it uses many contrasts, coherences, and contexts that give
it a wide scope of expression, great emotional power and variety, direc-
tion, uniqueness, and a fascination of design with many shadings and
qualities far beyond the range of popular or folk music. Every moment
must count somehow, as must every detail. For a composer it is not
always easy to find a passage that fits the particular situation and
moment at which it appears in the composition, that carries to a further
point some idea previously stated, that has the appropriate expressive
quality motivated by what has been heard and yet is a passage that
sounds fresh and alive.

As far as I am concerned, I am always interested in a composer’s
phrases and their shape and content, the way he joins them, the type of
articulation he uses, as well as the general drift or continuity of a large
section, and the construction of a whole work. The small details of
harmony, rhythm, and texture fall naturally into place when one has
interesting conceptions of these larger shapes.

Q: What do you mean by metric modulation?

A: If you listen to or look at any part of the first or last movement
of my First String Quartet, you will find that there is a constant change
of pulse. This is caused by an overlapping of speeds. Say, one part in
triplets will enter against another part hi quintuplets and the quintuplets
will fade into the background and the triplets will establish a new speed
that will become the springboard for another such operation. The
structure of such speeds is correlated throughout the work and gives the
impression of varying rates of flux and change of material and character,
qualities I seek in my recent works. The wish to accomplish this in the
domain of heavily emphasized contrapuntal contrasts led me to work
out the plan of metric modulation described by Richard Goldman. 1

Q: Why are the contrapuntal lines in your quartet so much alike,
using equal note-values?

A: You cannot have listened to the work very carefully or looked
at the score. Of the nine notes in the first four measures, there are seven
different lengths, the longest 18 times the shortest. There are, it is
true, a few places near the beginning in which several contrapuntal
parts each of equal note-values are combined, but in complete poly-
rhythmic contrast emphasized by intervallic, bowing, and expressive

1 Richard Goldman, The Music of Elliott Carter, in The Musical Quarterly,
XLIII (1957), 151,

56 Problems of Modern Music

contrasts. In these I was particularly anxious to present to the listener
the idea of polyrhythmic textures in its most definite form, for even this
quality of texture develops during the work, leading, hi the second
movement, to a four-part fragmented canon in continuous sixteenths
and, in later movements, to lines of much notational irregularity. But
even if the values were more frequently equal than they are, as for
instance in the polyrhythmic, posthumous Etudes of Chopin, I cannot
see that this would be a real objection, as you imply. Many a fine work
has dealt in continuous streams of equal note-values.

Q: Does your music have any harmonic plan?

A: A. chord, a vertical group of pitches either simultaneously sounded
or arpeggiated, like a. motif, is a combination to be more or less clearly
remembered and related to previous and future chords heard in the
same work. Whether the composer is conscious of it or not, a field of
operation with its principles of motion and of interaction is stated or
suggested at the beginning of any work. The field may be tonal, employ
traditional harmony, or it may be unrelated to traditional harmony, as
my music seems to be nowadays, in which case I feel it imperative to
establish clearly, near the beginning, the principles upon which the
composition moves. Once this field of operation is established, its possi-
bilities are explored, interesting new aspects of it are revealed, patterns
of action of contrasting types emerge as the work goes along. A work
whose world is not clearly defined loses a great deal of possible power
and interest, one whose world is too narrow and restricted runs the risk
of being thin, although if the world is unusual enough this narrowness
can produce a kind of hallucinatory quality one that I do not con-
cern myself with in my own works. This extension of the traditional
methods of coherence can rarely be attained nowadays solely by intui-
tion, I think, because of the vast number of musical means, new and
old, that we know. Some composers, it is true, insulate themselves from
new musical experiences in an effort not to be distracted. Others, whose
curiosity and interest prompt them to follow what is going on, feeling,
perhaps, as Charles Ives did, that “eclecticism is part of his duty
sorting potatoes means a better crop next year,” 2 have to make a number
of conscious choices and establish the frame in which to work before
they can compose at all.

In my First String Quartet, I did use a “key” four-note chord,
one of the two four-note groups, that joins all the two-note intervals

a Charles Ives, Essays Before a Sonata, New York, 1920, p. 94.

Shop Talk by an American Composer 57

into pairs, thus allowing for the total range of interval qualities that
still can be referred back to a basic chord-sound. This chord is not used
at every moment in the work but occurs frequently enough, especially
in important places, to function, I hope, as a formative factor. It is
presented in various kinds of part-writing and interval combination,
the number of notes is increased and diminished in it, in ways familiar
to all of you. The chord, here in its closest position, showing its content
of intervals of a diminished fifth and less, is also used both in many
intervallic inversions and in total inversion:

Ex. i

Here is an example of its use in counterpoint that occurs in measure 477
of the last movement, where the quality of the chord is strongly dwelt
on each vertical combination except the last being made up of it:
Ex. 2
Q: Did you try to shape the free writing found in your quartet into
formal patterns?

A: Since I consider form an integral part of serious music, I cer-
tainly did. Strange as it may seem, the intention of composing a work
that depended so much on change of movement and polyrhythmic tex-
ture involved me not only in special questions of clarity and audibility
that one does not usually have to face, but in special problems of form
also. One of the solutions I tried, to keep this rather free-sounding
technique from seeming haphazard and thus lose its connection with
the progress of the work and the attentive listener’s ear, was to establish
thematic patterns made up of components of different ideas that could
be separated. This feature emerges in the last movement, many of whose
motifs are disintegrated to produce polyrhythms (Ex. 3). This is only
one of the many ways I tried, hoping to give the impression of that
combination of freedom and control that I greatly admire in many
works of art.

58 Problems of Modern Music

Q: Do you use the twelve-tone system?

A: Some critics have said that I do, but since I have never analyzed
my works from this point of view, I cannot say. I assume that if I am
not conscious of it, I do not. Naturally out of interest and out of pro-
fessional responsibility I have studied the important works of the type
and admire many of them a great deal. I have found that it is appar-
ently inapplicable to what I am trying to do, and is more of a hindrance
than a help. Its nature is often misunderstood, it is a building material
and not the building, and it allows, I think, for certain greater freedoms
than were possible using traditional harmony with its very strict rules
of part-writing, just as reinforced concrete allows for certain construc-
tion patterns impossible with stone. I must also say that having known
many of these works all of my adult life, I hope the recent fad will not
cause them to seem commonplace too soon. The results of total
serialization are more recalcitrant to musical handling, I think.

Q: Do you mean to say that your rhythmic method is not a product
of serialization?

A: It is not. But it is true that like all music, mine goes from one
thing to another the pattern on which serialization is based, but my
choices of where to start and where to go are controlled by a general
plan of action that directs both the continuity and the expression. Single
details, chords, rhythmic patterns, motifs, textures, registers follow each
other in a way that combines them into clearly perceivable larger
patterns and then patterns of these patterns, and to me this cannot be
easily accomplished with total serialization, at least the kind I study
my way through in European articles these days. Perhaps another
more useful and not so arbitrary kind of serialization could be devised.
The present one resembles the turning of a kaleidoscope and usually
produces not much more or less interesting results. Indeed it can
be fascinating to listen to the total repertory of pitches, note-values,
timbres, registers, and dynamics being touched upon in rapid succession
and from a point of view we are unaccustomed to. But the cumulative
effect of this is self-defeating since neither the attention nor the memory
is appealed to. For who can decipher, by ear, the complexities of total
serialization in most works of the sort? On the other hand, those in
which this process can be followed are too obvious to be of any interest.

Q: What is your attitude about performance difficulty?

A: I realize with brutal clarity that orchestral music requiring a
lot of rehearsal can, by the nature of American musical life, find very

Shop Talk by an American Composer 59

few, if any, performances. This is not true of difficult music for soloists
or small standardized instrumental groups, for obvious reasons. Our
orchestral musicians are trained to play in the demanding scores of
Strauss, Mahler, Debussy, Ravel, and early Stravinsky. One might
imagine that one of the obligations of a present-day composer would
be to use the skills of these excellently trained musicians to their full,
lest their abilities deteriorate for want of use; that the challenge of
good, effective yet technically advanced scores would be helpful in
maintaining high performance standards in an orchestra, if not in
raising them, as it did in, the past. But this does not seem to be a
consideration here, and, as you and I know, new works that make
an immediate effect with a minimum of effort and time are favored.
The real effort goes into the standard repertory, where it is more widely
appreciated. Therefore, a composer who wishes to write orchestral
music and get it played here has to tailor his work to these practical
conditions, whether his ideas are suitable to such exploitation or not.
Those who find that they can do nothing of interest under these condi-
tions either give up writing orchestral music or, if they cannot, hope
for European performances of their works. For these reasons, the scores
of our composers often show a lack of practical experience that reveals it-
self in conventionality and timidity. How can a man be adventurous,
under the circumstances that obtain here? Any casual look at the Euro-
pean scores written since the war will show how far in advance of us even
beginners are there in this respect. As in many other things, we may
be willing to accept the final, accomplished results of European training
and experimental efforts but we cannot afford and are impatient with
the step-by-step experience needed to produce them.

Naturally, music that is both difficult and yet practical to play is not
easy to write, and it may even be difficult to listen to. It does not make
for a comfortable life to have this as one’s mode of expression. There
is an undoubted beauty in reducing things to their essentials or to
their simplest form if something is gained thereby. When a composer
cannot find an interesting and satisfying way of writing easy music,
he is at least free, here, to use the level of difficulty he needs to set
forth his ideas completely even if this results in no performances. But
I see no reason for being just difficult. Whenever difficult passages
seem imperative in my works, I try to make them especially rewarding
once they are played correctly.

For I regard my scores as scenarios, auditory scenarios, for per-
formers to act out with their instruments, dramatizing the players as

60 Problems of Modern Music

individuals and participants in the ensemble. To me the special team-
work of ensemble playing is very wonderful and moving, and this
feeling is always an important expressive consideration in my chamber

Q: Have you ever thought of composing electronic music?

A: Naturally, I have often been intrigued with the idea of electronic
music and have visited the Milan electronic studio several times to
find out what is being done. I must say that almost all I have heard
seems to me to be in a primary stage, and has not resolved some
fundamental problems of matching and comparison of sounds that
would raise it above the physical scariness that makes this music useful
for television science fiction and horror programs. As far as composing
it myself is concerned, you can imagine that since I am very enmeshed
in the human aspect of musical performance, I would find it hard to
think in terms of the impersonal sound patterns of electronic music.
Certainly, impatience at not being able to hear my works in performance
and impatience at the inaccuracies of some performances have occa-
sionally made me wish that I could have a machine that would perform
my music correctly and without all the trouble and possible disappoint-
ments associated with live performances.

Q: What do you think of Charles Ives now?

A: My opinions about Charles Ives as a composer have changed
many times since I first came to know him during my high-school years
in 1924-25, but my admiration for him as a man never has. No one
who knew him can ever forget his remarkable enthusiasm, his wit, his
serious concern and love for music, and his many truly noble qualities
which one came to notice gradually because they appeared casually,
without a trace of pompousness, pretention, or “showing off.” Attracted
to him by a youthful enthusiasm for contemporary music, I first admired,
and still do, the few advanced scores privately available in those days,
the Concord Sonata, the Three Places in New England, and some of the
114 Songs. However, after I had completed strict musical studies here
and abroad, I saw these works in a different light. Misgivings arose
which I expressed with considerable regret in several articles in Modern
Music after the first performance of the Concord Sonata in New York
in 1939. My doubts were of two kinds. First, there seemed to be very
large amounts of undifferentiated confusion, especially in the orchestral
works, during which many conflicting things happen at once without
apparent concern either for the total effect or for the distinguishability

Shop Talk by an American Composer 6 1

of various levels. Yet in each score such as the Robert Browning Over-
ture, the Fourth of July, and the second and fourth movements of the
Fourth Symphony where this confusion is most frequent, it is the more
puzzling because side by side with it is a number of passages of great
beauty and originality. Even more disturbing to me then was his frequent
reliance on musical quotations for their literary effect. In spite of these
doubts, I continued for many years to help bring Ives’s music before the
public since he would do nothing for himself, rescuing, among other
things, The Unanswered Question and Central Park in the Dark from
the photostat volumes of his work he had left with the American Music
Center. I arranged for first performances of these at a Ditson Fund
Concert at Columbia University in, I think, 1949.

What interests me now is his vigorous presentation in music and
essays of the conflict between the composer with vision and original
ideas, the musical profession, and the American public. It is the living
out of this conflict, made poignant by his strong convictions, the anger
it produced, the various actions and attitudes it led him to, the retreat
into a subjective world, and, unfortunately, the terrible toll of energy
and health it took, that makes of Ives an artist really characteristic
of America, not unlike Melville. Without the dimension of this struggle
and the quality it gave his scores, his Emersons and Hallowe’ens would
be of superficial and transitory interest.

His rage, which explodes between the waves of his transcendental
visions in prose as it does in the scribbled comments in the margin of
his musical manuscripts, reveals troubled concern over the problems
of the American composer and his relations with the public. The music
profession is castigated in one place as being more hide-bound, more
materialistic, petty, bigotted, and unprincipled than the business world.
The latter, his refuge from the bleak, meager life of the conventional
American musician of his time, he respected and identified himself with
enough to adopt an American business man’s view of the artistic pro-
fession, one that was especially characteristic of that time of wealthy
art-collectors. Making of the artist an anti-business man, Ives saw him
as a prophet living in the pure, transcendent world of the spirit, above
the mundane matters of money, practicality, and artistic experience.
The 19th-century American dream of art and high culture, which Henry
James liked to project against the sordid European background from
which it came, was the source, as Aaron Copland and Wilfrid Mellers
have pointed out, of Ives’s greatest misfortune. In gradually retiring into
this dream, he cut himself off from music’s reality. Too many of his

62 Problems of Modern ‘Music

scores, consequently, were never brought to the precision of presentation
and scoring necessary to be completely communicative to the listener
or so it seems now. One could say that Ives was unable completely
to digest his experience as an American and make it into a unified
and meaningful musical expression. The effort of remodelling the
musical vocabulary to meet his own personal vision, almost without
encouragement or help, was too great, and too often he had to let hymn
tunes and patriotic songs stand for his experience without comment.

As I have said, Ives’s lif e vividly presents the special conflicts inherent
in the American composer’s situation. Today, even more than in his time,
the division between the musician’s professional code of ethics, his tradi-
tional standards of skill and imagination established at another time in
another place, and the present standards of behavior respected, sanc-
tioned, and rewarded by the society that surrounds us, is very pronounced.
The familiar training of a composer giving him knowledge and skill in the
accumulation of musical techniques, past and present, and the develop-
ment of skill in notating them, presupposes trained copyists and per-
formers who can grasp what he means and respect his notations. It also
presupposes critics and, if not a large public, at least an influential
elite that will be able to perceive the sense of the composer’s efforts and
skill, value them and enable him to develop them further, by giving
them careful consideration. When one or more of the links hi this chain
is not sufficiently developed or non-existent, as is often the case here
today, the composer has a bitter fight just to keep his skill, let alone
develop it.

This misfortune can be laid to the general lack of unanimity about
and concern for the profession of composing on the part of the mass
musical public that plays such an influential financial role in America. By
training, the composer learns to write for a musically educated public that
is also an influential lite, which does not exist and may never exist
here. He cannot help but feel that he will be heard by a large majority
of listeners and even performers that disagree with him, if they have
any opinions at all, on the most fundamental issues of his art. Questions
of style, system, consonance, dissonance, themes, non-themes, being
original or an imitator, which imply some agreement on fundamentals,
are not the stumbling blocks. A professional composer has today, as Ives
certainly had, the training to be “communicative,” “melodious,” “ex-
pressive,” qualities considered to have a wide appeal, just as he is now
trained to use advanced techniques that will be appreciated by only a
few professionals. How shall he decide? He is free, here, to do what

Shop Talk by an American Composer 63

he likes, of course, but it does not take him long to realize that what-
ever he chooses to do, radical or conservative, his music will further
divide into small sub-groups the handful of people who will listen to
contemporary music at all. Not one of these small sub-groups has the
power or the interest to convince the large public by publicity or other
means of the validity of its opinions, as happens in the other arts here.
While diversity of opinion is much to be welcomed, where so little
support exists such decimation of interest, one hesitates regretfully to
conclude, can lead to cancelling of efforts and ultimately to their negation.

Even America’s panacea, publicity, seems strangely useless in this field.
Good reviews do not, often, lead to further performances, but they do
help to sell more recordings. One might have thought that Ives, now
so much discussed and publicly admired, would be often heard. That
a number of his recordings have been discontinued, that only a few
of his easiest pieces are heard while some of his more remarkable works
are still unplayed or scarcely known, is surely an indication of how
confused and desperate is the relation between the composer, the
profession, publicity, and the musical public.




A DISCUSSION of electronic music inevitably brings up the im-
portant question of how the availability of new sound materials
and the direct participation of a composer in shaping these materials
may tend to influence his methods of composition. Any experienced
composer knows the sounds and capabilities of the instruments he is going
to use in his score. In the tape medium my readers will remember that
this includes sounds produced non-electronically and stored on tape, as
well as sounds electronically produced and especially in the category
of non-electronic sounds, a sound is often chosen not for what it is but
rather for what it will become through electronic modification.

I believe that the virtually unlimited source of sounds available to a
composer who works with tape requires perhaps as great vigilance in
selecting the proper material as would normally be exercised in deter-
mining an orchestral palette, if not greater. It is tempting to parade un-
usual sounds; and the structural unity of a composition can be seriously
weakened by diverting attention with an overabundance of such sounds.
To avoid creating these distractions in A Piece for Tape Recorder, I
restricted my raw material to the following:

Non-electronic: a gong, a piano, a single stroke on a cymbal, a
single note on a kettledrum, the noise of a jet plane,
a few chords on an organ.

Electronic: four pure tones, produced on an oscillator, a tremolo
produced by the stabilized reverberation of a click from
a switch on a tape recorder.


Notes on A Piece for Tape Recorder 65

The sounds of the piano and of the jet noise are used in an episodic
manner, and serve to impart dynamic punctuation to otherwise slowly
evolving sound texture. The remaining sounds are used in a secondary
role of background accompaniment, sometimes obviously as plain old-
fashioned sustained tones, sometimes with more subtle variations of
timbre. The over-all structure seeks to effect a gradual transition from
a type of sound material that possessed a certain clearly recognizable
musical quality to the type of sound that is more closely identified with a
complex noise spectrum. It was my hope that this transition would appear
natural, and that the sense of unity could be preserved through a motivic

All of these sounds were drawn from the library of sound on tape
maintained at the Columbia University Studio. Two of the non-electronic
sounds were already used once in my earlier piece, Sonic Contours.
Other material existed as the result of the extensive experimentation by
Otto Luening and myself which preceded making our score for Orson
Welles’s production of King Lear.


The composition was put together from sketches that represented
durations, timbres, and dynamics on a four-line chart. This chart was
used when synchronizing the four tape recorders in the final mixing
of the four tapes on which the entire sound material was prepared.
The pitches were given approximate notation in a separate sketch made
on regular music paper.

Since the Copyright Office in Washington does not grant a copyright
on a work as a musical composition unless it is written or printed in
ordinary musical notation, I rather unwillingly spent forty hours to
produce a score, one page of which is reproduced here^My reluctance
was based on the impossibility of describing many of the sounds in
conventional musical notation. I thus attempted a pitch-approximation
of the sounds, at the same time placing them exactly where they occur
on the time scale. The following explanation of the methods used to
represent durations, pitch, and dynamics accompanied the score. 1


This score represents a musical transcription of sounds on four tracks of
magnetic tape (each indicated by a Roman numeral) used in the final mixing

1 The example and the explanation are copyright 1956 by Vladimir Ussachevsky.
The work was completed in April of that year and has been recorded by Composers
Recordings, Inc. on their disc CRI-112.


Problems of Modern Music

to obtain the composition. In this respect it is similar to a conventional four-line
score; however, the difference lies in the flexible interpretation of conventional
symbols resulting from calculating durations in terms of seconds, the expedient
necessitated by characteristic flexibility of time units in individual lines. Frequently,
therefore, an entire, phrase is clocked with a stop-watch and notated as accurately
as possible within the proper time-space, without an attempt to establish precisely
the relative value of each note to an over-all common denominator. Throughout
the piece the latter, however, is roughly one whole note for one second. In some
cases where the length of sound exceeds 20 seconds, the beginning of it is indi-
cated by QT~ – and the end by^ -*o . The absence of the above
or any other symbol indicates silence.

Many sounds used in this score are rich in harmonics, and the pitches indi-

Notes on A Piece for Tape Recorder 67

cated are frequently only of the predominant pitch impression. This latter, how-
ever, is indicated in the proper range as accurately as possible, and, on the
whole, with a far closer pitch approximation than is customarily used in describ-
ing gong, cymbal, and drum pitches in the percussion part of an orchestral score.


A decibel scale of dynamics is used in addition to the usual musical dynamic
marks to indicate intensity accurately. The arabic numerals throughout the score
refer to the number of seconds from the beginning, and each measure, suggested
by the small, regular dividing marks, is calculated in terms of one-second duration.
Thus all the entrances and cessations of sounds are very precisely indicated.

Letter Symbols Describing the Sounds

Each component of a descriptive symbol is set apart by a semicolon; the
letter or group of letters preceding a semicolon refers to the origin or character
of a sound, the letter or two hyphenated letters following the semicolon describe
the manner of bringing the sound into being. The third letter describes additional
modification of sounds.

Reverberated R

Metallic, soft-struck M; s-s

Metallic, hard-struck M; h-s

Percussion pr.

Electronic sound El.

Electronic tone-cluster Etn cist

Electronic treatment el

Piano P

Middle of a note X
Roll and tremolo

Oscillator Osc,

Organ O.

Wind W. <


Without describing in detail the technical processes, 2 I hope it is
understood that the available means of manipulating recorded sounds
make it almost mandatory for a composer to run through a certain
number of routine experiments before he can determine the full range
of his raw material. Experience gradually teaches one what to expect.
I now habitually imagine a sound as if it were changed by the following
mutation techniques, among others:

pitch transposition through variation of tape speed;

snipping off the attack and listening to the body of the sound itself;

3 Interested readers are referred to my article, Processes of Experimental Music,
in Audio-Engineering Society Journal, Vol. 6, No. 3, July 1958, pp. 202-08.

68 Problems of Modern Music

playing it backwards;

depriving it of some of its harmonics through filtering;

reverberating it.

An intricate interrelation exists between an abstract formal concept
which a composer might have formed about his forthcoming composi-
tion and the manner of developing his raw sound material. There can
be a decided interaction between the two which makes itself felt through
all the early experimental stages. In my Piece for Tape Recorder such
interaction entailed a certain give and take between the initially vague
formal plans and the composing of sound patterns from single sounds.
I proposed to utilize the timbre of a gong, stretched to span almost the
entire audible range, as a unifying element a kind of a timbre leitmotif,
if you will. But there are many more than thirteen ways of sounding a
gong and coaxing a maximum of variety from its rich content.

For example, while melodic use of large gongs is impractical for the
conventional orchestra (a series of small groups of Javanese gongs can
be used, but their pitch succession will vary from one set to another),
in the tape medium, creation of any type of scale is possible from any
type of sound.

Since an appreciable change in timbre takes place during the life
of the sound of a gong, I sampled various portions of it. The moment
of striking the surface with a mallet contains several noise components
which quickly disappear. Within a few seconds much of the metallic
quality is gone, but the timbre is still complex. By first cutting the attack
and then sampling various portions of the remaining sound, I arrived
at three basic variations of the timbres. The middle of the gong sound,
somewhat pale but tremulous, was made by transposition into a melodic
line consisting of seven pitches of even duration which span a range
of an octave and a half (see the example, Track II, between the 47-
and 70-second marks). The sound containing the attack was used in
two different ways. In one, it was modified by slurring the attack through
speed variation and by electronic reverberation creating a succession of
four pitches which rise chromatically and die out (see Track I, between
the 58- and 82-second marks). In the other, the full impact of the
attack was preserved. A thunder of six evenly spaced strokes of this
sound was arranged in an ascending pattern at the interval of a fourth,
to serve as a strong sforzando punctuation. (This occurs approximately
99 seconds after the opening of the piece.) The same sound, with the
attack preserved, was further transposed over many octaves to a high
register, where the resulting pinched, ping-like quality of the compressed

Notes on A Piece for Tape Recorder 69

attack easily penetrated through a broad, wind-like spectrum. This
sound was used frequently in the second half of the composition. Finally,
a dynamically shaped, sustained line was derived from a long resonance
of a large gong played backwards. In this instance the attack was faded
out and a long, impressive, and relatively smooth crescendo was ob-
tained. This sound was quite useful as a subtly changing lower and
inner sustained tone (see Tracks II and III, up to the 42-second mark).

As I have pointed out, it was my intention to create a certain feeling
of unity by developing much of the material from a single kind of
timbre. As it turned out, all other sound material became secondary in
both thematic and timbral importance. In the category of instrumental
sounds, a few patterns originating from the piano were subjected to
only the simplest of transformations, without cutting of tape or dynamic
shaping of any sort. A piano chord which is reverberated and played
backwards supplies a brief ostinato at the opening of the work; later,
dearly pianistic patterns, their pitch and speed doubled by an upward
octave transposition, are heard four times. Still later, a long, organ-like
pedal point, derived from a low piano tone with the attack cut off, is

A single note on a kettledrum served as the basis for the complex
wind-like sound mentioned above. This sound, one of the oldest in our
sound library at Columbia, had been previously used to construct several
tape solo passages in Rhapsodic Variations for Tape Recorder and
Orchestra and hi King Lear, both written by Otto Luening and myself.
In A Piece for Tape Recorder, its transpositions by tape-speed variation
form a vertical sonority roughly akin to a minor seventh chord, which
is in turn shifted up and down as a unit, its broad spectrum imparting
a decidedly confused impression as far as pitch is concerned. More
easily definable in its approximation of a pitch area is a sound of a
cymbal which is used to form a brief three- to four-note motif appearing
towards the end of the composition. The upper edge of the complex
sound derived from this non-melodic instrument draws, as it de-
scends, an easily perceived contour. Material derived from an organ
is used very briefly in the form of two chords: once as a dynamic
accent in a loud passage, and later as purely a timbre of a dark color.
I employed electronically generated sounds sparingly. In a curious way
they serve a tonally stabilizing function in the midst of complex sonorities
which undulate between relative clarity and almost noise-like indefinite-
ness. A cluster of sinusoidal tones, with their characteristic “steady-state”
quality unadulterated by reverberation, first appears in the 73rd second

70 Problems of Modern Music

of the composition. Contrasted to this “dead-level” tone, a melodic line
is constructed from an electronic warble-like tone, characterized by an
intense vibrato, the rate of which is changed with every new pitch level.
This helps take away the usual monotony, not to say irritation, often
engendered by an electronically induced vibrato.


Some composers do all the organization in the silence of the mind;
others need the physical impact of the sound. Most composers indulge
in some improvisation and retain, or attempt later to recapture, that
part of their subconscious utterance which they feel belongs to their

The tape medium is particularly felicitous for giving the composer
a chance to hear and to shape his sound material as he proceeds. His
decisions regarding the final form of a composition are not infrequently
influenced by the results of his experimentation with the sound material.
Nevertheless, some electronic composers maintain that an advantage of
electronic music is that it can be completely realized by precise specifi-
cation of certain acoustical and musical components. Others disdain the
rigors of numerical “total organization” and let the sine tones fall where
they may. Both sides have representatives who do not wish to have
their compositions assume any one final form and prefer the sequences of
patterns to be rearranged for each individual performance. 8

It is outside of my topic to debate the comparative merits of these
approaches. However, one must note the existence of the improvisatory
dement. Within the limits of my experience, I can testify that engaging
in an experiment in which the machines themselves assist the improvisa-
tion is often valuable as a stimulant. Such improvisation can create
sound patterns that would indeed be hard to imagine in advance. The
improvisation also has the advantage of being recorded and, hence,
available for examination. The value of improvisatory material ranges
from zero upward, but there is no denying that it assists the composer’s
imagination in making decisions regarding sound materials and the
evolution of a final form. This, to me, seems legitimate.

It must be made clear that the experimental procedure just described
refers to using the machines to produce the mechanical repetition of
angle notes or patterns. Similarly, they can create modifications of

9 This may raise some lively issues with the Office of Copyright which, one
supposes, issues the birth certificates on the assumption that the composer’s new
baby will retain its features.

Notes on A Piece for Tape Recorder 71

pitch relationships or timbre which can be induced either automatically
by the machines themselves or by the continuous control of the operator.
Skill of the subtlety approaching that of a performer is required to
exercise such control. Creative experience is equally important for
making those instant judgments which can directly influence the ultimate
quality of the improvised material.

As I have said earlier, the interrelation between the development of
the material and the final form of the work certainly played a part in
the composition of A Piece for Tape Recorder. The abstract aim was
two-fold. First of all, I wanted to achieve a land of large, assymetrical
arch on both a dynamic and a pitch scale. The ascent was to be ac-
complished through a series of little arches, while the descent would
consist of a long, gently undulating line of a predominantly gray timbre,
punctuated by fragments of the thematic material used in the first part.
The second aim was to start the composition with a sound pattern
possessing in large measure those qualities which would permit the
listener to make associations with definite pitches and, at times, conven-
tional rhythmic patterns. Gradually the timbres with a greater noise
content would be introduced, but the motivic unity would persist. The
composition was to end quietly with an impression that the last few
notes were largely noise descending by discernible intervals of thirds,
fourths, and sixths.

This plan was carried out in the finished work, and it seems that one
reason for its thoughtful reception can be found in its sense of direction
and unity. Dynamic punctuation, originating from diversified sound
material, helps to separate main sections of the work. Unity is imparted
by the motivic consistency and the derivation of the principle motif
from one timbre, highly modified though it was by manipulations
peculiar to the process of composition in electronic music.




propensity of present musical theory for terminology originally
J_ belonging to mathematics and physics is characteristic of a style of
thinking essentially different from earlier ways of viewing the subject
matter. Although some of this language sounds merely pretentious, it
has nevertheless added useful terms to musical discussion. One of these
is the concept of “parameter.” It was introduced into recent music theory
by Dr. Meyer-Eppler, of the Institute of Communication Theory at the
University of Bonn, who was associated with the work of the electronic
laboratory of the West German Radio at Cologne. It is borrowed from
mathematics, where it means “a variable entering into the mathematical
form of any distribution such that the possible values of the variable
correspond to different distributions.” 1

Serial organization of a certain number of parameters of a musical
process causes a certain number of other parameters to be left uncon-
trolled. A detailed study of the relationships of these two areas was the
purpose of the seminar. The title did not, as was surmised by some, hint
at a discrimination between accomplishments and shortcomings of serial


Serial music was defined as a method of composition that has been
developed as a sequel of the twelve-tone technique inaugurated by Arnold
Schoenberg around 1923. While the serial concept in that technique was
embodied in the twelve-tone series, i.e. an ordering of the pitches to be
adhered to throughout the course of the composition, the new idea of

1 American College Dictionary, New York, 1948, p. 879.


Extents and Limits of Serial Techniques 73

serialism encompasses all aspects (or “parameters”) of the musical pro-
cess, such as timbre, dynamics, articulation, and above all, time, i.e. dura-
tion of the individual sounding elements and their mutual relationships in
time, subordinating all these aspects to premeditated serial statements. In
this view the twelve-tone technique appears to be a special, or limiting,
case of serial music, similar to an interpretation of Newtonian mechanics
as a limiting expression of the Special Theory of Relativity, which in turn
has been explained as a limiting expression of that General Theory.


Anton Webern and Olivier Messiaen were mentioned as the best-
known generators of the new way of serial thinking, the former because
of the extraordinary impact his work has exercised during the last twenty
years or so, the latter above all through his experiments with “rhythmic
rows” (or “modes”) and his immediate influence on such composers as
Boulez and Stockhausen. The discussion then turned to the significance
and consequence of the gradual expansion of the musical area that was
subjected to premeditated organization. It was recognized that serial
ordering of the factor of time (i.e. premeditated fixation of points of
entrance and duration, of the individual musical elements) caused fun-
damental changes in the structure, appearance, perceptibility, and mean-
ing of music. Therefore the larger part of the investigation was devoted
to the methods of organizing serially the parameter of time. The discourse
was mainly based on my own work in the serial style because my intimate
knowledge of this work allowed succinct presentation of the relevant
details, whereas the few available analyses of other composers’ serial
works are frequently ambiguous and far from enlightening.


By rotation we understand a procedure in which the elements of a
given series systematically and progressively change their relative positions
according to a plan which in itself is serially conceived in that the changes
occur in regular phases. 3

I applied this principle for the first time in a large choral work,

3 In his book, Die Komposition mit zwolf Toncn, Berlin, 1952, p. 113 ff. and
passim, Josef Rufer points out that Arnold Schoenberg occasionally let neighboring
tones of his rows exchange places, or groups of tones change their positions within
the row. Rufer’s discourse and the examples quoted show that this was done
sporadically and mainly in order to create a musical context that would not have
been served as well by adhering to the premeditated succession of pitches.


Problems of Modern Music

Lamentatio Jeremiae Prophetae* written in 1940 and 1941. The twelvi
tone series of this work reads thus:

Ex. i

Each of its two constituent six-tone groups is progressively modifie
by making the first tone the last:
The patterns thus obtained may be called “diatonic” since they con-
tain the same six tones. The roster of patterns is doubled by transposing
all those of the left column of Ex. 2 to begin on F, all those of the right
column to begin on B.

Ex. 3
* Bircnreiter-Vcrlmg, KaueL

Extents and Limits of Serial Techniques 75

These new patterns are “chromatic” because they eventually include
all twelve tones. The rotation taking place was inspired by the construc-
tion of the Greek modal scales and their transposition into one “char-
acteristic” octave. The purpose of the operation was not so much to make
the serial design stricter, but rather to relax it, insofar as the wide variety
of available six-tone patterns made it possible to remain within the
frame of reference of the twelve-tone serial technique without constantly
having to use complete twelve-tone rows. Thus it became possible to
give various areas of the composition distinctive harmonic flavors. At
that time no attempt was made to organize serially the selection and
succession of the rotational patterns.

A more consistent and systematic application of the principle of rota-
tion may be found in my orchestral work, Circle, Chain and Mirror*
written in 1956 and 1957 for the Basel Kammerorchester. The tone-row
of this work reads as follows:

In the course of the composition twenty-four derivative forms of this
row are employed. The principle of derivation may easily be apprehended
by comparing the original row with its first three derivative forms (the
tones in their original succession are numbered from 1 to 12) :

The rotation taking place here consists in forming a retrograde suc-
cession of each pair of two adjacent tones. After eleven such operations
one arrives at the complete retrograde form of the original statement
The twelve following derivates represent the retrograde forms of the first
twelve, and the twenty-fifth transformation is identical with the original.
The same procedure was applied to the inverted form of the original
series (see Ex. 6). This arrangement suggested the “circle” part of the
tide of the work.

4 Original German title: Kette, Kreis und Spiegel. Barenreiter-Verlag, KasseL


Problems of Modern Music

The sequence in which the forty-eight rows thus obtained were used
in the work was determined by the decision to have each original form
followed by the second of the two forms of the inversion which would
have for tfieir first tones the last tone of the preceding original, while this
inversion in turn would be followed by an original form beginning with
the last tone of the preceding inversion. This interlocking arrangement is
meant by the term “chain” in the title. The sequence of rows obtained
through this operation may be partially seen in the following table
(O===original, I=inversion, R=retrograde ) RI=retrograde inversion) :



Extents and Limits of Serial Techniques ‘ 77

The symmetry resulting from this organization is obvious: The sequence
1, 6, 10, 4, 12 in lines 1 to 9 of the O column is identical with the
sequence 1, 6, 10, 4, 12 in lines 13 to 21 of the I column. The same
relation obtains as regards the sequences 12, 4, 10, 6, 2 in lines 4 to 12
of RI and 16 to 24 of R. The positions of 18 between Ol and 6 and
of O8 between II and 6 are equally symmetrical and correspond to the
positions of the 8s in R and RI between 2 and 6 of RI and R

Ex. 6


138610125 II 1947

68 12 10 II 5927 4

The term “mirror” finally refers to the fact that the musical config-
uration that opens the work and is expressed in terms of the row Ol
returns in inverted form when the serial “conveyor belt” produces the
form I 1, in retrograde inverted form when the row RI 1 appears (not
shown in the above table), and at the very end of the work in terms of
the form R 1. The remaining areas of the music are not any longer
occupied by thematic statement, development, recapitulation, and the
like. Whatever morphological kinship may be detected between adjacent
sections is a result of similarities of intervallic shapes that may occur in
neighboring forms of the tone-row, the vicinity of which, however, is a
consequence of the premeditated serial arrangement outlined above and
not dictated by requirements of a so-called musical nature.

In this composition no other parameter beside the succession of tones
was serially ordered. In this respect it belongs to the province of “class-
ical” twelve-tone music. It transcends that province in that it allows its
structure to arise from the serial arrangement of the rotational derivatives
of its tone-row.

The principle of rotation, which, as may be seen here, I discovered
and utilized for reasons not relevant to the evolution of pan-parametrical
organization, turned out to be of far-reaching significance when I became

78 Problems of Modern Music

interested in that kind of organization. The point is that the notion of
invariancy inherent by definition to the concept of the series, if applied
to all parameters, leads to a uniformity of configurations that eliminates
the last traces of unpredictability, or surprise. But unpredictability ap-
pears to be not only especially characteristic of so-called “atonal” music,
but desirable, or necessary, in any work of art. That the composers who
have made the most consistent attempts at “total determinacy” are
aware of this need transpires from this utterance of Pierre Boulez:
“L’inattendu, encore: il n’y a de creation que dans Pimpr6visible
devenant ne*cessit&” 5

Combination of the various configurations that result from rotational
procedure with constant (non-rotating) serial elements means that the
principle of order that governs one set is applied to another, unrelated
set (as if one, for instance, would order the numbers from 1 to 5
alphabetically: five four one three two). Since this is one of the defini-
tions of randomness, we meet here for the first time the factor of chance,
which has attained high significance hi recent developments.


According to Gyorgy Ligeti’s analysis 6 of Pierre Boulez’s Structures
for two pianos, 7 the composer has interpreted the transpositions of his
twelve-tone row to various pitch levels as a form of rotation and has
transplanted the results to the parameter of time in order to obtain an
analogous sequence of derivative forms of his time series.

6 “The unexpected, again: there is no creation except in the unforeseeable
becoming necessary” (Revue music ale, April 1952, p. 119, as quoted in Die
Reihe, No. 4, Vienna, 1958, p. 71). It is interesting that this statement almost
verbatim sums up Carl Bricken’s brilliant argument about “inevitability and the
unexpected” in his analysis of Beethoven’s Quartet Op. 18, No. 3 (Some Analytical
Approaches to Musical Criticism, in Proceedings of the Music Teachers National
Association for 1936, Oberlin*, 1937, p. 262 ff.). In Bricken’s discourse the “inevi-
table” is, of course, represented by those musical processes that appear to be most
likely to occur within the framework of tonal harmony so that they constitute a
predictable, “normal” set of events. The **unexpected,” then, consists of the devia-
tions from the norm introduced by the genius of the individual composer. In the
case of serial music the inevitable is what serial premeditation ordains. The unex-
pected, however, is not a result of the composer’s kicking against the self-imposed
limitations, but of the built-in surprise mechanism, as we shall see later on. In my
article Is the Twelve-Tone Technique on the Decline? (in The Musical Quarterly,
Oct. 1953, p. 523 ff.) I indicated that Boulez in his Second Piano Sonata probably
applied the principle of rotation.

Die Reihe, No. 4, p. 38 ff.
* Universal Edition, Vienna.

Extents and Limits of Serial Techniques 79

The elements of the tone series are numbered from 1 to 12:

I 2 3 4 5 6 7 6 9 10 II 12
To this a series of time values corresponds, expr

tivt n I?

f we transpose the tone row, for instance, a major third higher, the
original order of the tones is changed into :

5 689 12 104 II 7231

Correspondingly the time series would take o$ the shape:

^ . J Mi WUt- tit

In fact, the whole work consists of manifold combinations of the tone-
and time-sets thus obtained.


Karlheinz Stockhausen’s work described alternatingly as Komposition
1953 No. 2 and Elektronische Studie I 6 is based on a six-tone series
which according to the composer’s own elaborate analysis 9 is an expres-
sion of this series of ratios of frequencies:
12 4 8 55
5 5 5 12 4
Expressed in vibration numbers, or cycles, per second, the first series reads:

1920 800 1000 625 1500 1200

12 : 5 8:5 5:4

4:5 5 : 12

In notes it reads approximately:
Five more series are derived by making the consecutive tones of the first
series points of departure for new series identically built (a procedure
somewhat reminiscent of my Lamentatio rotation) :

800 333 417 260 625 500

1000 417 521 325 781 625

625 260 325 203 488 390

1500 625 781 488 1170 937

1200 500 625 390 937 750

Universal Edition, Vienna. Recorded by the Deutsche Grammophon Gesell-

9 Technische Hausmitteilungen des Nordwestdeutschen Rund funks, Vol. VI,
No. 1/2, Cologne, 1954, Item 10, p. 46 ff.

80 Problems of Modern Music

A second set of six series is obtained by making the second line of the
first set the top line of the new set, then the third, and so on.

All parameters are serially ordered in terms of some variants of the
numerical sequence 1 to 6. For instance, the combinations of the above
frequencies follow from the series 4 5 3 6 2 1 in that the first tone-
combination (“Tongemisch”) has four tones, the second five, and so on.
There are four such “Gemische” in “sequence 1” (a “sequence” being a
grouping of consecutive elements), and four “sequences” in the first
“structure,” which is the next higher compound, “horizontal” or “ver-
tical.” (It does not become quite clear on what grounds one or the other
of these two dimensions was chosen.) There are six dynamic levels which
are assigned to the various frequencies in proportion to their relative
positions in the groups and columns of the entire system. The series that
orders the succession of dynamic levels within this frame of reference is
342165. Finally, the time factor is determined by relating the dura-
tions of the individual sounding elements to the pitch levels and degrees
of loudness of those elements as ordered by the previous rules. The gov-
erning series in this parameter is 2 4 6 3 5 1.

The details of this organization are far more complex than what we
are able to indicate here in an abridged sketch. Unfortunately the pre-
sentation by the author is not always felicitous, so that some of the
intricacies of his work remain obscure. At any rate, the character of his
reasoning seems to reveal a desire to derive the rules of serial organization
from the nature of the chosen material and its intervallic texture. In this
respect Stockhausen differs somewhat from Boulez, who has a rather
mechanistic approach in assigning numerical values to the various mag-
nitudes manipulated in his work. While this procedure of Boulez’s has
been criticized as “anorganic,” 10 it has nevertheless produced a fascinating
piece of music. On the other hand, Stockhausen’s Studie, although much
shorter than the Structures, suffers from considerable monotony of har-
monic flavor, which is due to the prevalence of augmented triads in the
original series (see Ex. 9). The extraordinary subtleties of combinations
of dynamic shadings, time values, echo effects, and the like cannot over-
come this initial handicap.

The objection was raised that music here becomes the victim of an
abstract numbers game which is contrary to the nature of music. While
there undoubtedly is room for more than one definition of the nature of
music, we did not extend our inquiry into this field. The numbers used

lft Ligeti, loe. cit., p. 41.

Extents and Limits of Serial Techniques 81

in the ordering of the parameters of serial music are almost always de-
rived from proportions and measurements of the basic musical substance.
Of course, these numbers detach themselves from the objects with which
they were associated and take on a life of their own in the various opera-
tions performed. The results of these operations are, however, retranslated
into musical terms and applied to the sounding material. In this relation
of number and reality one may see a vague analogy to the connection of
contemporary mathematics and physics.


In my oratorio for voices and electronic sounds, Spiritus intelligentiae,
sanctus, 11 there is a section without voices (so to speak an “instrumental”
interlude). The material of this section is a tempered scale of thirteen
tones. From the continuum of this scale, groups of tones were selected to
form alternatingly disjunct and conjunct heptachords of equal and
symmetrical structure (see left side of Diagram 1). A seven-tone pattern
(seven-tone row) meanders through this system of pitches constantly
retaining its principle of progress: from any tone on which it starts it
goes up to the third and fourth, then back to the second, up to the sixth,
back to the fifth, and it stops on the seventh tone of the network of
pitches. Since the pattern always progresses conjunctly (which means that
the first tone of its next appearance is identical with the last of the pre-
ceding) while the pitch system is based on the alternation of conjunct
and disjunct shapes, the internal intervallic configuration of the pattern
is always different, although its general outline remains the same (see
right side of Diagram 1 ) . After thirteen appearances the pattern lands
again on the tone from which it started, and the “rotation” has come to
an end.

The interlude in question may technically be called a double canon.
One of the two elements subject to imitation is a tone-line consisting of
the chain of the thirteen possible variants of the seven-tone pattern just
described, the other is an analogous line presenting the chain of the in-
verted forms of the pattern. The first tone-line is so designed that it
begins on the central tone of the entire gamut (330 cycles), rises to its
highest level (4754 c) in the first third of its length, returns to the center
in the second third, and descends to the lowest level (26 c) in its last
portion. The second line begins on the lowest point when the first reaches
its apex, rises to cross the first line where it passes on its descent the cen-
tral tone, goes up to its own high point which it reaches approximately
when the first line ends, and returns to the center.

“Recorded by the Deutsche Grammophon Gesellschaft (LP 16134 Hi-Fi).


Problems of Modern Music

Chain of
disjunct and




Read from bottom up
Heavy lines indicate octaves

Progress of the
seven-tone pattern
Extents and Limits of Serial Techniques 83

The canonic imitations were obtained by rerecording the original
material at a higher and a lower speed, in which procedure the pitch
level of the original tape was automatically raised or lowered in the same
proportion. These imitations were so synchronized with the original lines
that the slowed-down version of the ascending branch of the first tone-
line would reach its highest point (proportionately lower than the summit
of the original) when the original line had returned to the center. It was
followed by the slowed-down imitation of the descending branch of the
second line. The above-center arcs of both lines were imitated in accel-
erated versions reaching their (proportionately higher) apices shortly
after or before those of the original lines. Finally, a very highly accel-
erated imitation of the below-center branches of both lines was inserted
shortly before the end of the section.

To determine the time values of the single elements the whole expanse
of the piece was viewed as one unit. Through measuring the linear dis-
tances of the important points of articulation entrances of imitations,
turning points and such a series of eleven spans was established, a sort
of macro-rhythm articulating the over-all structure. It was reduced in
scale to a micro-rhythm in order to determine the durations of the
individual tones in each tone-line. Since each line takes approximately
three quarters of the entire length of the piece and each line contains
ninety-one tones (seven times thirteen), the micro-rhythm of eleven
values has to be repeated eight times, leaving three tones free at the end.
This concept determined the ratio by which the macro-rhythm had to be
reduced. Since the rhythmic series thus established has eleven terms
whereas the tone-series has only seven tones, it follows that the last four
terms of the first time series will apply to the first four tones of the second
tone series, and so forth, so that here again mechanical repetition is
avoided while uniformity in a higher sense is maintained. (See Diagram

It may be stated that whatever occurs in this piece at any given point
is premeditated and therefore technically predictable. However, while the
preparation and the layout of the material as well as the operations per-
formed therein are the consequence of serial premeditation, the audible
results of these procedures were not visualized as the purpose of the
procedures. Seen from this angle, the results are incidental. They are
also practically unpredictable because the simultaneous progress of highly
complex rhythmic patterns at various relative speeds together with the
corresponding transpositions of equally complex pitch patterns creates
situations that defy precise visualization.


Problems of Modern Music
Extents and Limits of Serial Techniques 85

The Sestina is one of the poetic forms developed by the Provengal
poets of the twelfth century, its original specimen being ascribed to
Arnaut Daniel. It may well be called a serial form of poetry, and its
essential formative principle is rotation.

The poem consists of six stanzas of six blank verses each. It hinges upon six
keywords which appear at the endings of the individual lines. If in the first
stanza the order of these words is 1 2 3 4 5 6, the words will appear in the
second stanza in the order 615243. The principle of rotation which is
applied here consists in switching the position of every two keywords equidistant
from the center of the series, proceeding from the end toward the middle.
According to the same principle, the positions of the keywords in the subsequent
stanzas are 3 6 4 1 2 5; 5 3 2 6 1 4; 4 5 1 3 6 2; 2 4 6 5 3 1. The process
ends here, since the next rotation would produce the original series. The six
stanzas are followed by a Tornado, of three lines in which the keywords, one of
each pair in the middle and the other at the end of the line, appear in the
order 2 5, 4 3, 6 1.

The content of the Sestina which I wrote (in German) as text for the present
composition is a contemplation of the implications of the idea governing the
musical construction of the work. 13

The first two stanzas may suffice to indicate the character and form
of the poem:

1. Vergangen Klang und Klage, sanfter Strom.
Die Schwingung der Sekunde wird zum Mass.
Was in Geschichte lebt, war’s nur ein Zufall?
Verfall, Verhall, zerronnene Gestalt?

Die Stunde zeitigt Wandel, wendet Zeit.
Das Vorgeschrittne ordnet sich der Zahl.

2. In Schritten vorgeordnet durch die Zahl
gestaltet sich Gedanke, doch zum Strom
wird strenge Teilung, uhr-genaue Zeit.
1st es vermessen, solches Mass von Mass
dem Leben aufzuzwingen, der Gestalt?

Der Zwang zerrinnt, erzeugt den neuen Zufall. w

12 Barenreiter-Verlag, Kassel. Epic Records, LC 3509.

13 Quoted from my notes on the jacket of the record cited in note 12.

14 In a nearly literal translation which reproduces the positions of the key words:

Bygone are sound and mourning, tender stream.
Vibration of the second becomes the measure.
What lives in history, was it only chance?
Decline, fading sound, vanished shape?
The hour causes change, turns the time.
What looks ahead subordinates itself to number.

86 Problems of Modern Music

The music of my Sestina is based on a twelve-tone row divided into
two groups of six tones each :

tf A Bj __ ___ * L _ ,

Ex. 10 g^ n fto ^ T” ” 0″

43162 I 6 4 2 6 5 (E)

The figures indicate the size of the intervals measured in half-steps.
These tone-rows are rotated according to the principle of the sestina so
that the second A- and B-groups read:

l* ^ M

64 421 3 52422 (I)

The third line is:

+ j i .^

Ex. 12

off” I* a +

44 131464

and so forth. The tones are always placed so that they will not exceed
the ambitus of the original row and the intervals (indicated by the num-
bers below the staff) are so measured up or down that their magnitudes
will not exceed the figure 6. Obviously the sequence of these intervallic
magnitudes constantly changes as a result of the rotation of the tones
prescribed by the sestina pattern, but these changes are of a different

The durations of the tones of the whole composition are derived from
these magnitudes in the following manner: each intervallic magnitude
corresponds to a time segment which contains as many basic time units
as the interval figure indicates. Consequently the first time segment has
four units, the second three, etc. Each segment has as many tones as it
has units (4, 3, etc.). The duration of the individual tones is determined
by a subdivision based on the same serial sequence of magnitudes. If the
first segment contains four units and four tones, its subdivision is based
on the first four values of the original series: 4316. The sum of
these being 14, the subdivision unit within the first segment is 4/14.
The durations of the individual tones within the first segment are
determined by multiplying 4/14 consecutively by 4, 3, 1, and 6. The

In stages preordained by number

thought takes shape, but a stream

is (the result of) strict division, of clocklike, precise time.

Is it presuming to force such an extent of measure

on life, on shape?

Force vanishes, brings forth new chance.

Extents and Limits of Serial Techniques 87

durations, then, are 16/14, 12/14, and 24/14, or 8/7, 6/7, 2/7, and
12/7 of the basic value.

Actually the determination of the durations is due to much more
complicated computation because it is influenced by serial organization
of other parameters. In order to achieve higher rhythmic diversity, the
concept of “internal speed” was introduced. It is derived from the as-
sumption that in every group of six tones one to five tones might be
sounded an octave higher so that the magnitude of the affected intervals
would be augmented by twelve. The succession of “internal speeds” is
derived from the position of the tones in group B (see Ex. 10). The
lowest (A) is designated as 1, the highest (F) as 6. The initial row of
internal speeds is therefore 514362. The first segment, then, has the
internal speed 5 so that 12 is added to five out of six subdivision num-
bers. Thus these numbers read 16 15 13 18 14 instead of 4 3 1 6 2.
The following number 1 remains unaltered. The sum of the num-
bers attached to the first segment is therefore 62, instead of 14. Conse-
quently the durations of the individual tones will be considerably shorter
than if the “internal speed” were, for instance, 1 or 2.

To facilitate computations each basic unit is assumed to contain ten
micro-units. We arrive at the subdivision of the first segment by dividing
40 (four times ten) by 62. The result is 0.645. This number is multiplied
consecutively by 4, 3, 1, 6. The results are 2.58, 1.935, 0.645, 3.87. If
the work had been realized by electronic means on tape, these values
could be produced with utmost accuracy. Since it was conceived for
conventional manners of rendition, the time values had to be adjusted
as follows: 2.5, 2, 0.5, 4. If the smallest numerical unit is ^pressed by
Ji , the rhythmic shape of the first four tones is J> JJ ji JJ J> Jl
= 9/16. “”” “~” ‘

“Density” is the next parameter to be determined serially. There
are six degrees of density whose succession is determined by the position
of the pitches in group A (Ex. 10). Again the lowest (C) is called 1,
the highest (QJ) 6. Consequently the initial series of densities is 6 3 5
4 1 2. In “density 1” the two tone-groups A and B run off simultaneously
in a sort of two-part setting in which the duration of the individual tones
is determined by the mechanism described above. In “density 2” the
first and second time segments of group A run concurrently with the
first segment of group B. In “density 3” two segments of each group are
developed simultaneously, and so forth, until in “density 6” six segments
of each group, i.e. twelve all together, run off at the same time.

88 Problems of Modern Music

Another parameter is the location of the tones within the gamut of
six octaves designated as the ambitus of the work. The serial statement
adopted for this area reads that the tones of each segment should run
through as many octaves as there are tones. The direction of the motion
is determined by the direction of the corresponding interval in the
original series. Since many segments contain less than six tones, they
cover less than six octaves and therefore could extend over various bands
of the complete ambitus. This, too, is regulated by special serial state-
ments. Needless to say that all these serial organisms are subject to rota-
tion according to the sestina pattern, which is the supreme law governing
every move of every variable within the whole composition.

The structural layout is designed to combine each “rotated” version
of any six-tone group with every other. Thus the music of the first stanza
is based on the first statement of the A-group, in each consecutive line
of the poem combined with one of the forms of the B-group rotated
from B 1 to B 6. The second stanza has A 2, combined again with all
six B-groups, but now in a different sequence, according to the sestina
pattern: B 6, B 1, B 5, B 2, B 4, B 3.

Paralleling the arrangement of the key words in the tornado,, the
tone series assigned to it reads 25436 1. The music of the tornada
consists of six sections, the first four and the last of which are given over
to the instruments alone. While the tone row of the tornada undergoes
the now familiar six sestina transformations, the density increases from
1 to 6 so that in the first section of the tornada only one each of the A-
and B-rows are employed, while in the last section six of each, that is
twelve, or all available forms are used simultaneously.

The parameter of “external speed” has six steps also, the lowest being
M J) = 90, the highest J>= 180. The former is associated with the
highest degree of density, the latter with the lowest.

Example 13 shows the first ten sixteenths (micro-units) which form
the first basic time unit of the Sestina. On the left side one may see the
distribution of the tones of the A- and B-groups over the twelve layers
(density 6) of simultaneously progressing time segments, each tone enter-
ing at the point assigned to it by the time mechanism explained above.
The tones occupy their places from top to bottom layer in their order of
succession in the row. The “internal speed” for the A-layers (top six)
is 5, for the B-layers (bottom six) 1 (no acceleration). Encircled num-
bers indicate the number of tones allotted to the particular segments.
Arrows indicate the direction of the tone lines. The figures above the

Extents and Limits of Serial Techniques


top staff give the durations of the first four tones in ^ , as computed on
p. 225. The right side of the example shows how these tones are repre-
sented in the actual score, and a few connecting lines were drawn to
demonstrate where some particular tones may be found.
It is easy to see that the parameter of timbre lies beyond the limits
of the present serial arrangement. If this parameter too were organized
serially and this procedure would, for instance, require the first tone of
the top layer (GJ) to be played by the trumpet, it would obviously be

90 Problems of Modern Music

at variance with the octave register demanded by the serial regulation
of spacing, since the trumpet cannot play the GJ in question.


Other parameters may be affected in the same way. If the succession
of tones is determined by serial regulation (as is the case in the classical
twelve-tone technique) and, in addition to this, the timing of the
entrance into the musical process of these tones is also predetermined by
serial calculation (as, for example, in the case of the Sestina), it is no
longer possible to decide freely (that is, by “inspiration”) which tones
should sound simultaneously at any given point. In other words, the so-
called harmonic aspect of the piece will be entirely the result of opera-
tions performed on premises that have nothing to do with concepts of
“harmony,” be it on the assumption of tonality or atonality or anything
else. Whatever happens at any given point is a product of the pre-
conceived serial organization, but by the same token it is a chance occur-
rence because it is as such not anticipated by the mind that invented the
mechanism and set it in motion.

Generally and traditionally “inspiration” is held in great respect as
the most distinguished source of the creative process in art. It should be
remembered that inspiration by definition is closely related to chance,
for it is the very thing that cannot be controlled, manufactured, or pre-
meditated in any way. It is what falls into the mind (according to the
German term Einfall), unsolicited, unprepared, unrehearsed, coming
from nowhere. This obviously answers the definition of chance as “the
absence of any known reason why an event should turn out one way
rather than another.” 15 Actually the composer has come to distrust his
inspiration because it is not really as innocent as it was supposed to be,
but rather conditioned by a tremendous body of recollection, tradition,
training, and experience. In order to avoid the dictations of such ghosts,
he prefers to set up an impersonal mechanism which will furnish, accord-
ing to premeditated patterns, unpredictable situations. Ligeti character-
izes this state of affairs very well: “We stand in front of a row of slot
machines [“Automated”] and we can choose freely into which one we
want to drop our coin, but at the same time we are forced to choose
one of them. One constructs his own prison according to his wishes and
is afterwards equally freely active within those walls that is: not
entirely free, but not totally constrained either. Thus automation does
not function as the opposite of free decision: rather free selection and

**The American College Dictionary, New York and London, 1948, p. 200.

Extents and Limits of Serial Techniques 91

mechanization are united in the process of selecting the mechanism.” 1 *
In other words, the creative act takes place in an area in which it has
so far been entirely unsuspected, namely in setting up the serial state-
ments (selecting the slot machines). What happens afterwards is pre-
determined by the selection of the mechanism, but not premeditated
except as an unconscious result of the predetermined operations. The un-
expected happens by necessity. The surprise is built in.


A later serial work of mine is a set of six piano pieces, called Seeks
Vermessene. This German title is a play on words, since vermessen in
German means “completely measured” as well as “presuming,” a pun
that cannot be reproduced in English. While the time mechanism is
similar to that of the Sestina, the construction differs from it in that for
the first three pieces a system of five layers is set up in which the first
has “density 1” (i.e. one tone at a time), the next has two tones
together, the third three, the fourth four, and the fifth six tones. The
time measurements for the various layers are a result of summing up the
interval magnitudes involved in the consecutive tone combinations. For
example, the tone series of this composition being:

Ex. .4

the first combination of tones in “density 2” is:

The numerical values derived from this progression are 3 (a minor
third from G to Bb) and 1 (a half-step from E to F). Consequently the
first time segment of the first layer has three units, the first of the second
has four (3 + 1 ). As the density of the layers increases, the number of
simultaneously sounding intervals and thus the numerical values of their
sums become higher. Therefore the time segments become longer, which
means that the chords, or tone-clusters, with increasing thickness are
spaced farther apart, while the single tones of the first layer follow each
other more rapidly. Computations of this kind form the basis of the
whole composition.

As explained before, phenomena in the parameter of harmony must
be accepted as results of the operations in the sectors of pitch succession
and time. In the fourth of the piano pieces an attempt was made to

**Loc. cit., p. 38 (translated from the German by this writer).

92 Problems of Modern Music

begin with a selection of sound elements. From the tone row we devel-
oped twelve sets of four elements each (consisting of one, or of two,
three, or four tones played simultaneously) plus two six-tone chords.
These fifty elements were numbered from 1 to 50 and their succession
was determined by progressing along this series by the distances indicated
in the numerical values of the intervals of the basic row:

series of elements: 1 2 3 4. 5 6 7 8 9 10 11 12 13 14 15 …

intervals of the tone row : 3 2 5 4

selected elements: 1 46 11 15 …

In the fifth piece the five degrees of thicknesses (see above) are
distributed over five layers which progress at various speeds so that the
time measurements of the slowest layer are reduced to 1/2 in the second,
to 1/3 in the third, to 1/4 in the fourth, and to 1/6 in the fastest layer.


In the field of serial music one may observe a tendency towards
using series of magnitudes that progressively vary according to some
serial ordering of their own. The speed levels of the Sestina are an
example. Another time series of this nature was established for the voice
line of this work. It is based on the succession of 1 2 3 5 7 and 10 J^
for the accented syllables. The opening succession is2310571 and
the following forms are obtained through the sestina rotation. Since
each line of the poem has only five accented syllables, interesting situa-
tions of overlaps occur.

It may be seen that the series here applied is a modification of the
so-called Fibonacci series in which each term is equal to the sum of the
two preceding terms: 123581321 34 55 etc. 17 Luigi Nono has used
the first six terms of this series as factors with which he multiplies the
basic time values of his // Canto sospeso in order to obtain the actual
durations of the individual tones. 18 1 have used the terms of the Fibonacci
series from 2 to 21 to determine the speed zones in a recent orchestral
composition entitled Quaestio ternporis (A Question of Time) . This work
is based on a twelve-tone row that contains all eleven intervals in this
order (measured in half -steps) :

385 10 11 612749
The entire expanse of the composition is thought of as consisting of 66

17 Cf. Matila Ghyka, The Geometry of Art and Life, New York, 1946, p. 13 f.

18 Cf. Karlheinz Stockhausen’s analysis pf the work in Darmstadter Beitrage
zur neuen Musik, Mainz, 1958, p. 70.

Extents and Limits of Serial Techniques 93

time units (the sum of the above figures), which form eleven sections
of varying lengths according to the magnitudes of the basic series. To
these sections six different speeds are assigned:
M J = 20, 30, 50, 80, 130, and 210


It appears that density is a function of speed and thickness of texture.
If the latter may be called the vertical component of density because it
depends on how many layers are in operation at the same time, speed
is the horizontal component of density since the tones follow each other
more closely the faster the tempo of the music is. If both parameters
approach maximum values, a degree of saturation is reached at which
accurate computations of time points and durations become irrelevant.
When in the final section of Quaestio twelve layers (maximum vertical
density) progress at a speed of j = 210 per minute, the tones come
so close together that nearly every sixteenth is sounded, frequently by
several tones simultaneously. The velocity of the music causes 14 J* to
run off per second. At this rate even the succession of pitches is not any
longer of great significance. It seems sufficient to determine by experiment
within a limited area the average number of time units needed for run-
ning through the twelve-tone series. The results of this statistical exami-
nation are then used in order to fill this area of highest density with
actual musical sounds.


One of the parameters that obviously cannot be controlled by pre-
meditation when those so far discussed are subjected to serial ordering
is the expressive, or communicative, aspect of music. If a serial composer
were concerned with this problem, he would have to set up a series of
“moods,” or “ideas,” or something of this sort, to begin with, and then
let the other parameters fall in line. It so happens that serial composers
are not thinking in such terms.

In a more pessimistic attitude than he now seems to entertain, the
German composer and philosopher, T. W. Adorno, has criticized the
recent developments of serial music” because in these the (according to
him) deep-rooted and essential analogy and affinity of music and speech
is abandoned. While it may be true that music from the time of plain-
chant has been oriented towards speech-like articulation, diction, and
over-all structure, and while especially the exploits of Expressionism and

19 Das Alter* dtr neuen Musik f in Der Monet ; May 1955.

94 Problems of Modern Music

atonality point to a very close association with the free articulation of
prose, we have to face the fact that under the influence of the construc-
tive rigor that was the very consequence of Expressionistic roaming serial
music has turned away from its rhetorical past. Since whatever music
seems to communicate is not so much the supposed content of the audible
matter as it is the product of the listener’s reaction touched off by his
auditory experience, there is no reason to assume that the nature of
serial music excludes the possibility of interpreting it as a medium of
some sort of communication. The interest it may evoke is similar to that
elicited by the process of life, to which serial music is related in the
paradox of the chaotic appearance of totally and systematically traceable
causality. It may mean as much or as little as life itself.



IN 1928, \Vhen Schoenberg’s serial concept was beginning to exhibit
the rapidity and diversity of development now regarded as character-
istic, Bart6k devoted the third movement of his Fourth String Quartet to
the extended and elaborate expression of a relational system that closely
resembles a serial schema. It must be added immediately that although
this composition is clearly exploratory it is not imitative either of Schoen-
berg or of his students; rather, the system upon which it is based arises
as a logical consequence of tonal materials unique to the Fourth Quartet.
Necessary further qualification of the term “serial” may be deduced
from the analysis given below.

Because of Bart6k’s reluctance to discuss details of compositional
technique, we are left in doubt as to the precise extent of his knowledge
of serial procedures. 1 But whatever this might have been, the third
movement of the Fourth Quartet stands as an extraordinary demon-
stration of his ability to employ diverse and seemingly contradictory
procedures without sacrificing either the integrity or the unique charac-
teristics of his music. In addition, the work testifies to Bartok’s compo-
sitional prowess, for it offers cogent solutions to certain harmonic prob-
lems of non-triadic music, solutions matched only later by avowedly
twelve-tone composers.

The iterative, symmetrical nature of the Fourth Quartet as a whole
is not reflected lucidly by the surface of the third movement. However,
careful listening reveals that beyond its highly improvisational exterior

* With reference to the Fourth Quartet and other works, Halsey Stevens writes
of “certain characteristics presumably conditioned by Bartok’s acquaintance with the
music of Schoenberg and Stravinsky” (The Lift and Music of Btla Sarttk, New
York, 1953, p. 205).



Problems of Modern Music

lies a system of relations that integrates every detail. A representation
of this system is provided by Ex. I. 2

Ex. i
The pair of notes that spans the interval of a major second (here-
after referred to as the whole-tone dyad, or simply dyad) is primal to
the system. The fundamental principle by which the system is generated
from the dyad is complementation: the combining of symmetrically
related structural elements, both with and without overlapping, to form
a higher unity. Thus the dyad gives rise to a trichord, a group of three
contiguous notes spanning a major third (e.g., element w in sub-system
I of Ex. 1), and the trichord, in turn, generates a threefold system of
hexachords, hexachords of three distinct yet interrelated species: sub-
system I contains the two whole-tone hexachords, wx and yz, which
constitute the axis of the entire system; sub-system II contains six
“diatonic” hexachords constructed in ascending pitch-order from each
note of wx, while sub-system III contains six diatonic hexachords simi-
larly constructed from yz. Whereas no special device seems required to
show the complementary relation between the two whole-tone hexachords
of sub-system I (with respect to the total ordering of twelve pitch-classes) ,
beams are used in sub-systems II and III to identify complementary
hexachords of the diatonic species.

It is evident that the whole-tone axis (I) includes all the trichords
that compose the diatonic hexachords of sub-systems II and III. The

*fix. 1 stresses the iterative rather than the symmetrical characteristic of the
system in order to show more directly the constituent hexachordal species explained

The* examples are copyright 1929 hy Universal Edition; renewed 1956; copy-
right and renewal assigned to Boosey & Hawkes Inc. Reproduced by permission.

Bart6k’s “Serial” Composition 97

axial nature of the whole-tone hexachords is confirmed when by com-
bining w and y or x and z we obtain the third species of hexachord, the
chromatic hexachord. From the complementary relation between wx and
yz it follows that wy complements xz.

The compositional potential of the system is indicated by the fact
that a given trichord can combine with any of five discrete trichords
to form a diatonic, whole-tone, or chromatic hexachord. By this means
ordered expansion through the entire system is feasible. However, analysis
demonstrates that Bartok imposed a specific limitation upon expansion
by assigning a regulatory function to the trichord C-D-E, the element
that represents the tonality of the entire Fourth Quartet. Therefore,
although the composition is based upon a schema that has the properties
of a twelve-tone system, 3 its development is determined not by those
properties but by a non-serial element, the tonality-representing trichord
C-D-E. The following analytic account of the movement amplifies this
thesis with the aid of a synopsis of hexachordal combination and pro-
gression over the entire span of the movement:

Ex. 2
In order to represent the main harmonic succession as concisely as
possible the synopsis (Ex. 2) provides only the boundary notes of tri-
chords and hexachords and shows only those melodic elements that
influence the over-all progression. The latter as well as certain secondary
harmonic elements are represented by black noteheads.

The movement is divided into three large sections, designated by
double bars in Ex. 2. Formal divisions both of long and short duration

*Bart6k’s lystem has quite special properties indeed: its hexachordal species
represent three of the six possible “all-combinatorial” hexachords described by
Milton Babbitt in Some Aspects of Twelve-Tone Composition, in The Score, XII
(1955), 53-62.

98- Problems of Modern Music

coincide with changes of hexachordal combination, i.e., progression.
These are clearly marked by changes of register, dynamics, and tempo,
as well as by idiomatic string effects. The first 46 measures of the move-
ment consist of an accompanied solo melody. In the remaining 25
measures two instruments are melodic while the other two provide ac-
companiment. During the early stages of the analysis it was assumed
that the accompanying parts carried the more stable elements, the
hexachordal or harmonic progression over the longer span, whereas the
melody, in addition to identifying the hexachordal combination of the
accompaniment, expressed more diversified relations. For this reason it
seemed advantageous to divide the analytic account into two parts, the
first dealing with the over-all harmonic progression, the second with the

In the first section, mm. 1-40, the controlling element, the C tri-
chord, is located within the C hexachord, 4 which is unfolded in “retro-
grade” 5 motion in the lower register after the manner of a cantus firmus.
The temporal position of the C trichord within this cantus firmus affords
it maximal control while allowing it the possibility of maximal association
with other trichords by virtue of complementarity. Thus at the very outset
the unfolding F trichord of the cantus firmus combines with the A
trichord to form a whole-tone pentachord (yz). (The one note, E[>, that
would comple’te the hexachord yz is omitted, for reasons that will be
elaborated below.) Similarly, the G trichord of the cantus firmus com-
bines with the E trichord to form a second whole-tone pentachord (wx) 9
omitting Bb. Although the cantus firmus is to be regarded as generative,
in the way just described, the E hexachord, comprising E and A tri-
chords, assumes temporal priority over it and is stated in retrograde
form at the beginning of the movement. In accord with the principle
of complementation this hexachord seeks to incorporate the Bb hexa-
chord to form the highest unity: the ordering of the total chromatic,
Le., a twelve-tone set. It now becomes evident that complementation
is the basis of progression from section to section and of hexachordal
combination within each section. Consequently, the redirection or
restriction of a potentially complementary progression or combination is

4 For the sake of convenience a trichord or hexachord will be designated
throughout by its lowest note, and “hexachord” refers to the diatonic species unless
otherwise indicated.

* The structure of all three species of hexachord is such that only two of the
customary operations obtain; therefore in this, to an even greater extent than in
other instances, the designation of prime, retrograde, etc. is somewhat arbitrary.
Because of the way in which the hexachordal schema is represented by Ex. 1 it
seems preferable to designate the ascending form of the diatonic hexachord as prime.

Bart6k’s “Serial” Composition 99

an event of singular importance, one that must be specially indicated in
the analytic synopsis (Ex. 2). Thus the Eb trichord at m. 35 is enclosed
in parentheses to show that although it occurs it is not defined in terms
of the Bb hexachord. (In this instance the melodic detail that effects
redefinition of the Eb trichord is not shown. ) In amplification and as
further demonstration of the fundamental role of the trichord we note
that whereas trichordal complementation is consistently effected, hexa-
chordal complementation is not, with the single exception of an instance
involving whole-tone hexachords within a section controlled by diatonic
hexachords (mm. 60-61, described below). The consistent preparation
of complementary relationships, followed by their negation or redirection,
suggests that Bart6k regarded complementation as equivalent to closure
of the system, and that he wished to avoid closure in order to permit the
C trichord to remain unequivocally the fundamental element.

From this it follows that complementation is carefully controlled at
every level of detail and over every temporal span. Thus, in the first sec-
tion trichords are stated in such a way as to insure optimal clarity of
relatedness. For example, the statement of the Bb trichord at m. 22
coincides with the closure of the F trichord of the cantus firmus, making
evident the “diatonically” complementary relation between them. Had
the Bb trichord appeared earlier it would have been understood only in its
chromatically complementary relation to the A trichord or in its whole-
tone relation to the E trichord. Moreover, since at this juncture the Bb
trichord relates directly to the E, A, and F trichords (complementing
them to form whole-tone, chromatic, and diatonic hexachords, respec-
tively), it serves as a precise measurement of expansion from the begin-
ning of the movement.

As a further means of assuring coherent progression, the integrity
of the cantus firmus** trichords is carefully preserved. This is apparent
from the outset, for after the cantus firmus moves to G (m. 13), B and
only B is omitted from the melody during approximately five measures
since it would tend to identify G of the cantus with the G trichord, B
and G being the definitive terms thereof, and thus conflict with the
essential placement of G within the F trichord which controls the
harmonic succession over the longer span.

A relation of shorter span, but one that further integrates the work
around the C trichord, is expressed at m. 32 by the B trichord. This
element, hi addition to complementing the C trichord chromatically,
connotes the F trichord via the axial whole-tone hexachord yz and thus
doubly supports the continuity of the cantus firmus.


Problems of Modern Music

Above D and C of the cantus firmus (mm. 33 and 34) appear
elements that denote the E and Bb hexachords. Since these hexachords
also appeared above the F trichord (mm. 1-31), their restatement
here above the C trichord serves as an additional unifying factor. Of
greater importance to the over-all succession is the fact that in both
instances the Bb trichord does not incorporate the Eb trichord to form
a diatonic hexachord, although in both instances complementation is
prepared. The more significant of these is the second (mm. 34-40).
Before amplifying this we remark that the Eb trichord has a decisive
function in the first section and indeed throughout since it completes the
Bb hexachord, which, in turn, complements the E hexachord generated
by the cantus firmus, thus effecting a definitive closure with respect to
the total chromatic. Analysis reveals the means by which the comple-
mentary progression prepared during mm. 34-40 is redirected at m. 41
so as simultaneously to avoid closure and to afford further extension
through the hexachordal system.

Ex. 3
At m. 35 the first violin states and repeats the dyad Eb-F in the upper
register. At m. 40 the structural demand for G as final term of the tri-
chord Eb-F-G is intensified, analogically, by a redistribution of the
accompanying instruments which permits FJ to be introduced in the
accompaniment: this note completes the dyad E-Ff, which has the
same location in the E trichord of the accompaniment as does the Eb-F
dyad in the Eb trichord implied by the melody. However, at m, 41,
which is set off both from preceding and following sections by the
instruction tranquillo, the Eb-F dyad is defined in terms of the Db
hexachord. For the benefit of those who may still doubt that the com-
posing-out of the unique schema represented by Ex. 1 required an act
of cognition on the part of the composer, we point out that the Db
hexachord is the one hexachord in the system that shares the Eb-F dyad
and only that dyad with the Bb hexachord.

Additional refinements become evident with study. For example, the
notes of the dyad Eb-F are stated at m. 35 in ascending pitch-order;
at m. 38 this order is reversed, cueing the redefinition of the dyad at

Bart6k’s “Serial” Composition 101

m. 41 in terms of the F-Eb-Db trichord (Db hexachord) as explained
above. The initial motif at m. 42, E^-Djj, further confirms the signifi-
cance of this permutation.

With the shift to the Db hexachord at m. 41 the composition extends
for the first time to sub-system III, the hexachords of which contain no
discrete trichords (with respect to sub-system II) and which accordingly
have greater possibilities for progression and combination than those of
sub-system II, which are controlled by the G trichord. However, a
specific limitation is placed upon these possibilities when the Db hexa-
chord is combined with the D hexachord of sub-system II, as shown
in Ex. 2 (m. 41 ), for the two hexachords share one and only one note,
Gb, and consequently exclude one note, C. In consideration of the
principle of complementation this suggests progression to the G and Gb
hexachords and thus a return to sub-system II a progression that
subsequently is effected with the assistance of melodic elements to be
explained below

To continue the survey of hexachordal progression and combination
over the total span of the movement, we observe that complementation
of the C and Gb hexachords (mm. 47-51) does not occur definitively,
for although the F trichord is present, it is stated in conjunction with
B; as a result it registers as an element of the whole-tone hexachord
yz formed by cross-related trichords from each diatonic hexachord.

The hexachordal combination at m. 52 (E and F hexachords) is
analogous to that at m. 41 (Db and D hexachords). However, the pro-
gression from this to the following section (m. 55) is not analogous, since
this would have resulted in a combination of the less “stable” hexa-
chords (Eb and A) from sub-system III, thus transcending the limitation
placed upon the system by the fundamental C trichord. Accordingly,
the short section that extends from m. 52 to m. 54 marks the limit of
expansion. This is made perfectly clear when at m. 55 there begins a
second unfolding of the cantus firmus, this time within a matrix that
is at once more condensed and more elaborate than that of its formal
counterpart, mm. 1-34. Even the cantus firmus is stated more intri-
cately, in a melody carried by the ‘cello. A regularity of metrical accent
distinguishes the cantus firmus from other melodic elements in the pas-
sage: the first two dyads of the simultaneously unfolding trichords, A-E
and G-D, begin on downbeats, while the closure of the trichord begins
with F on the third beat of the measure, hi accord with the definitional
function assigned to that metrical position throughout the movement.

102 Problems of Modern Music

In this penultimate section all motion continues to be regulated by
the C trichord. Thus, in mm. 55-60 the chromatically related C and B
hexachords are expressed within a carefully-wrought imitative pattern
above the cantus firmus. An even more complex formation begins at
m. 60, where the cantus firmus is interrupted by a direct linear statement
of the whole-tone hexachord yz. The counterpoint of this passage will
be explained below.

To summarize, complementation is the basis of trichordal and hexa-
chordal combination within each section and of progression from section
to section. As we have seen, operation of this principle is regulated by
a single trichord, with the result that progression over the longer span
is limited primarily to the hexachords of sub-system II. However, addi-
tional resources of the system are exploited at the level of melodic detail,
where a high degree of elaboration and integration is obtained. An
account of this follows.

Just as a melodic element in a triadic composition derives its
meaning from the triad, so does a melodic element in the present work
derive its meaning from the trichord and from the higher unities the
trichord generates. But because of the system’s iterative nature the
meaning of a particular melodic element is potentially multiple. The
structure of the trichord and hexachord therefore is reflected in the
various degrees of specificity with which an interval denotes a higher
unity. Examples 4-16 catalogue and illustrate these denotations, begin-
ning with those of the minor second and proceeding to the tritone.
Even a relatively complete account of the melodic detail would far
exceed the limits of the present article; therefore each interval-denotation
is illustrated by only a single instance in the composition. At this juncture
it must be remarked that melodic Embellishments in the sense of non-
structural elements do not occur. Each note has a structural task which
can be described with precision in relation to the hexachordal schema.

Because of its axial position within the diatonic hexachord the minor
second denotes diatonically complementary trichords.

Ex. 4

The minor second may also designate a chromatic hexachord (Ex. 4b).
Accordingly, the initial motif, the dyad DJ-D in m. 6:
Bart6k’s “Serial” Composition


denotes both the B|? hexachord, which complements the E hexachord
already stated, and the chromatic hexachord on B (ze>y), which contains
the fundamental C trichord. Thus, with a single two-note motif the
melody reveals the relational system and indicates the direction that
expansion will take in the total movement. This dual meaning is con-
firmed when, in m. 7, G is stated, for it is the one note required to
complete the chromatic hexachord wy (the remaining notes already
being present in the accompaniment). Therefore, hi sum, the two initial
measures of melody shown in Ex. 5 present hi descending pitch-order
(repeated notes excluded) the three notes required to define the chro-
matic hexachord wy: Df, D, and C.

It now becomes clear that the trichords of the E hexachord are
presented non-contiguously in mm. 4-5 (upper staff in Ex. 5) in order
that the interval of a minor third formed between them may provide
a chromatic matrix for the motif DJ-D. At least two additional reasons
for this “inversion” suggest themselves: 1) A, the lowest note of the
upper trichord, is to serve as point of departure for the descending
cantus firmus and therefore the entire upper trichord is placed below
the lower trichord; 2) the separation of the trichords emphasizes their
function as essential unities fron which all expansion stems.

The major second has two possible meanings: lower or upper dyad of
a trichord. In connection with Ex. 3 we considered an instance of the
structural use of this ambivalence in order to redirect diatonic hexachor-
dal progression. An example of the major second within a whole-tone
hexachord is provided by the melody of the section that begins at m. 22.
There, with the closure of the upper trichord of the cantus firmus,
melodic expansion is expressed in terms of the hexachord yz, which
contains the F trichord just completed in the lower register (cf. m. 60).
Thus the melody features the following succession of dyads: A-B,
d-DJ, and F-G. It also includes three notes from the complementary
hexachord wx: FJ, D, and Bb. These symmetrically related (equidistant)
notes serve as references to the Bfc and E trichords carried by the ac-

Within the diatonic hexachord the minor third obviously does not


Problems of Modern Music

occur between contiguous notes. However, between non-contiguous notes
it occurs three times and accordingly designates one of three hexachords :

Ex. 6

At m. 41 is a significant instance of the third possibility. There the
shift to the Db hexachord is denoted only by tKe minor third Db-Bb
(Ex. 3). Subsequently the Db hexachord receives confirmation when
more of its elements are included both in melody and accompaniment.

Since the chromatic hexachord contains three non-contiguous minor
thirds, a given minor third can denote any of three different (but over-
lapping) chromatic hexachords:


Clearly the minor third or any other melodic interval can occur in
a situation where it is not determinant, where its denotative function is
subordinate to that of other intervals. The brackets in Ex. 8 (m. 49)
mark two such cases:

Ex.. 8
Ex. 9 verticalizes the notes shown in Ex. 8 in order to demonstrate their
symmetrical arrangement 8 around G.

Ex. 9
Further clarification is provided by Ex. 10:

Ex. 10

which shows the underlying trichordal structure of the figure and
demonstrates that the bracketed skips in Ex. 8 come about as an inci-
dental result of the whole-tone dyads. Observe, moreover, that Bart6k
has notated the skips as augmented seconds, not as minor thirds.

The major third denotes a trichord. From the fact that melodic
occurrences of this interval are negligible we can conclude that the
composer deliberately restricted its use to the harmonic dimension,

6 Cf. George Pcrle, Symmetrical Formations in the String Quartets of Bela
Bart4k f in The Music Review, XVI (1955), 300-12.

Bart6k’s “Serial” Composition


thereby obtaining a measure of control over hexachordal progression
and combination which, ip view of the associative property of the
trichord in this work, would have been difficult to match had the melodic
major third been used more frequently.

The perfect fourth occupies three different positions in the diatonic
hexachord and bounds the chromatic hexachord. Correspondingly it

denotes one of four hexachords:
Ex. ii
*^ > *-… ! -^ ^

At m. 63, terminal point of the cantus firmus

Ex. 12
the fourth Bb-Eb above the final C denotes simultaneously the Gb
hexachord, which complements the hexachord of the cantus, and the
Bb hexachord, the upper trichord of which has a pivotal role in the
subsequent and final section.

Of the three species only the whole-tone hexachord contains the
tritone :

Ex. 13

In the case of the other two species the tritone occurs between
complementary hexachords and thus denotes the total chromatic:



The location of the same tritone in the corresponding structural position
with respect to both these species indicates the interchangeability of
complementary pairs of diatonic with complementary pairs of chromatic
hexachords under certain conditions. This relationship appears to be
exploited to a certain extent in the vertical distribution of the accom-

The following example (m. 43) illustrates the influence that the
harmonic combination can exert even upon the tritone:

Ex. 15
106 Problems of Modern Music

Taken alone, the Fb-Bb tritone connotes the whole-tone hexachord wx.
However, with reference to the D and Db hexachords, the harmonic
combination of this section, it specifies the cross-related Gb and D
trichords, stressing the fulcral position of Gb (the only note common
to both hexachords) within the pentachord formed by those trichords.
Further, the dyadic relation between Fb and Gb suggests an analogous
relation between Bb and C. This analogy is made clear by the parallel
accentual-grouping of the melody, here set off against the strong metrical
accents of the accompaniment:

Ex. 16
Since C is excluded by the harmonic combination, the dyad Bb-C is
not realized within this section. However, with the progression to the
Gb-C hexachord combination at m. 47, C, which is to be the primal
melodic note of that section, is stated precisely in that rhythmic
position relative to Bb which is shown in Ex. 16. We can generalize
from this instance to state that accentual grouping of the melody in
the entire movement is designed to register interval relations that iden-
tify the particular harmonic combination and prepare for progression.

To generalize, then, regarding the melodic aspect of the work, it
has been demonstrated that trichordal and hexachordal segments express
relations between the larger harmonic units operative within a section,
relations that are ordered with respect to the hexachordal system as a
whole and serve to define harmonic combinations and prepare pro-
gression. A final instance of this, perhaps the most extraordinary of all,
is provided by the closing section, mm. 64-71, where the meaning
of the Bb trichord, stated melodically by the first violin, seems to be
equivocal. However, an inventory of the whole-tone dyads featured
by the ‘cello melody which appears to draw every possible implication
from the Bb trichord of the first violin reveals that only one note is
omitted, Eb- Thus, of the five possible trichords with which the Bb
trichord could be associated only one is not made explicit, the Eb
trichord, for combined with the Bb trichord this would form a com-
plete Bb hexachord and thus close the system by effecting complemen-
tation of the E hexachord. The final melodic note of the movement,
D, therefore is to be understood in terms of the whole-tone hexa-
chord wx, a relationship made unequivocal by the final note of the
accompaniment with which it sounds, G|. More specifically, D and
GJ serve as final reference to the fundamental C trichord, which is

Bart6k’s “Serial” Composition 107

contained within the hexachord wx. The selection of these particular
notes to represent the whole-tone hexachord is explained by the fact
that they served as points of departure for the accompaniment and
melody, respectively, of the entire movement. Further, if we assume
that the hexachord wx comprises the C and FJ trichords as discrete
components which is reasonable in view of the primal function of the
C trichord and if we observe the analogous axial positions that D
and GJ occupy in those trichords, it becomes evident that these notes
represent a summation of the relational system and of the principle
of complementation by which the system was developed compositionally.

In addition to the trichordal and hexachordal segments, the melody
carries some relatively complete statements of diatonic hexachords. In
every case these express ordered relations between the total movement
and the main harmonies of the particular section. Perhaps the most
intricate illustration of this is provided by mm. 60-63, where the first
violin and ‘cello move in stretto, incorporating relatively complete state-
ments of the A, CJ, F|, and G hexachords. Since trichordal associations
are maximal here, the hexachordal figures split into their component
trichords, group around the whole-tone axis directly unfolded in the
inner parts , and derive their structural meaning from it. The whole-tone
hexachord, however, is exclusively a local event; in the context of the
entire section (mm. 55-63) it serves to expand the unfolding cantus
firmus as described above in connection with Ex. 2. Moreover, this
whole-tone hexachord (yz) does not contain the C trichord, and there-
fore does not anticipate the termination of the cantus firmus, but, rather,
prolongs it under conditions perfectly in accord with the symmetrical
system of hexachords regulated by the cantus firmus.

The density of this penultimate passage stands in marked contrast to
others in the movement, while the plasticity of the trichord which is
evident there and which clearly makes of it the developmental apex,
brings us to a final realization of the structural necessity for the limitations
imposed upon detail in the form of melodic permutations, ellipses,
and accentual groupings, all of which represents Bartok’s virtuosity in
composition. But beyond this, the refined techniques demonstrated by
every measure of the work testify to his musical versatility and per-
spicacity, attributes all the more remarkable in view of his uncompro-
mising individuality. In the final analysis it is these intangibles that
guarantee his position among the titans of ‘modern music.



AT the present moment, when many of the jagged edges of abrup-
tion have been smoothed by time and practice, there are those
who presumably in the spirit of mediation and moderation would
minimize, not so much Schoenberg’s achievement as a composer, as
the degree to which the twelve-tone system is genuinely “revolutionary”
in its nature and implications, the degree to which it imposes new
demands of perception and conception upon the composer and listener,
and therefore the degree to which it admits of further and exten-
sive exploration and discovery.

Such an attitude does a disservice not only to Schoenberg, but to
the cause of understanding, particularly since it so often involves the
invocation of the alleged historical-analogical origins of the operations
of the system, along with conjectures as to Schoenberg’s mode of and
motivation for arriving at the system. However intriguing such conjec-
tures may be, they are as irrelevant as they are futile; however peda-
gogically convenient and intuitively suggestive a quasi-genetic approach
may be, eventually it succeeds only in obscuring both the character
of the system and the profound differences between the twelve-tone
system and those musical systems in which the “historical forerunners”
of the twelve-tone operations appear. The crucial point here is that
these “forerunners” are not independent and fundamental structural
determinants, but means of immediate procedure, neither necessarily
present nor, if present, of more than local significance and influence.

Therefore it is appropriate to precede even so informal a discussion
as the one to follow with the reminder that the twelve-tone system,
like any formal system whose abstract model is satisfactorily formulable,
can be characterized completely by stating its elements, the stipulated


Twelve-Tone Invariants as Compositional Determinants 109

relation or relations among these elements, and the defined operations
upon the so-related elements. Such a characterization, though explicitly
presented in verbal form at the earliest stage of the twelve-tone develop-
ment, is likewise easily and explicitly inferable as the maximum proced-
ural intersection among the “classical” twelve-tone works of Schoenberg,
Webern, and Berg.

If the elements of the twelve-tone pitch system are, indeed, “tradi-
tional” ones, both insofar as they are pitch classes with class membership
defined by octave equivalence, and as there are twelve such pitch classes
corresponding to the chromatically equal-tempered quantization of
the frequency continuum even here essential deviations must be noted.
In the twelve-tone system there is a one-to-one correlation between pitch
notation and presented pitch, as opposed to the many-to-one correlation
of triadic-tonal music; there can be no such distinctions as those between
explicit and functional “dissonance,” or between enharmonically iden-
tical “consonance” and “dissonance.” The independent assumption of
octave equivalence has been a frequent point of attack upon the system,
particularly by those who assert that the corresponding assumption in
the tonal system serves to define classes of equivalent function; it need
be answered only that, similarly, this assumption in the twelve-tone
system serves to define classes of equivalent order position.

It is hi the definition of relations among the elements that the
system diverges significantly from systems of the past, for relations are
defined entirely by the imposition of a total linear ordering upon the
pitch classes, thus defining a twelve-tone “set” (designated: S). The
ordering employed, hi any given work, is inferable from at most
all of the compositional presentations of the set (and its transforma-
tions), and not necessarily from any one compositional presentation.
By introducing this principle as the basis of relationship, Schoenberg
not only effected a fusion of the general systematic constraint with the
contextually defined property for, although the principle of formation
is defined for all sets, the specific pitch class relations defined by a set
are uniquely associated with it and its transformations but established
the means of a. permutational musical system, as opposed to the combi-
national systems of the past. Given a collection of available elements,
the choice of a sub-collection of these as a referential norm provides a
norm that is distinguishable by content alone; such a system, and the
traditional tonal system is such, is therefore combinational. But if the
referential norm is the totality of elements, there is but one such norm
in terms of content, and deviations from this norm cannot exist within

110 Problems of Modern Music

the system. But if an ordering is imposed upon this- totality, and taken
as a norm, this norm is so distinguished, in the case of twelve pitch class
elements, from the 12 1 1 1 other possible orderings, that is, other
possible permutations.

Any consideration of the operations of the system must proceed from
an awareness of their permutational nature. As a simple example: trans-
position, excepting the identity transposition, in a combinational system
results in the adjoining of pitches which are not present in the original
collection, and thus establishes a new sub-collection; transposition of a
set results only in a permutation of the elements. Also, compositional
transposition, traditionally, implies contour preservation, a consideration
that is, literally, meaningless hi defining transposition as a twelve-tone
operation, since contour is a function of the registral specification of the
elements, and registral choice is as undefined by the structure of a set
as is duration, intensity, timbre, or any of the other attributes necessarily
associated with a compositional representation of a set; as a result, a
set cannot be stated in musical notation without the additional qualifica-
tion that each pitch sign be taken to signify the total pitch class a member
of which it denotes. Since such a qualification only too easily leads to
but another confusion of .systematic principle with compositional per-
missive (“a tone may be stated in any octave”), it is both safer and
more efficient ‘to represent a twelve-tone set in numerical notation, by
an ordered number couple succession, the first member of the couple
signifying order position in S (“order number”), the second signifying
the “pitch number” of the pitch class. The initial pitch class of S is
denoted by the couple (0,0), and is taken as the origin of the coordinate
system for both* order and pitch numbers, both of which range over the
integers 11 inclusive, each integer appearing once and only once
as an order number and a pitch number. In the case of order numbers,
this represents the fact that twelve and only twelve pitch classes are
involved: in the case of pitch numbers, this is the arithmetical analogue
of octave equivalence (congruence mod. 12). a In this notation, the set
of the Schoenberg Third String Quartet, with registral representation
chosen arbitrarily,

Ex. i

is represented: 0,0; 1,9; 2,8; 3,2; 4,5; 5,10; 6,11; 7,4; 8,3; 9,6; 10,1;

*!$!- (12X11 X 10X9… X 1).

* Two numbers, a and b, are said to be “congruent mod. 12*’ if, and only if,
a-b = k.12 where k is an integer (including zero).

Twelve-Tone Invariants as Compositional Determinants 111

11,7. Thus, the succession of differences derived by the subtraction
(mod. 12) of a set number from the following set number is the
ordered interval succession determined by a set, and each of these
interval numbers denotes, accordingly, a class of intervals.

The operation of transposition applied to any set can be repre-
sented by adding (mod. 12) an integer, 11, to each pitch number
of the set. Thus, if (a,b) is the couple signifying an element of S,
then the transposition is represented by (a,b+t), with t termed the
“transposition number.” Thus, the transpositional operation (desig-
nated: T) is conveniently regarded as an operation on, a permutation
of, pitch numbers; for any specified set, it could equally well be regarded
as effecting a permutation of order numbers, but the previous char-
acterization corresponds more appropriately to the general conception
of transposition.

The totality of twelve transposed sets associated with a given S con-
stitutes a permutation group 3 of order 12; as such it is closed, disjunct
with regard to any other collection of sets T derived from a set whose
intervallic succession differs from that of any member of this totality.
Thus, it constitutes a combinational collection of sets within, not only
the totality of all possible sets, but the totality of sets derivable from a
given S by the operations of the system.

By virtue of the group structure that it generates, and the additional
properties that derive from the commutative and transitive nature of
this group, a multitude of attributes necessarily associated with T can
be formally deduced. The musical relevance of these attributes can be
decided only empirically, of course, but it is my purpose here to
examine only a few of those operational invariants (properties of a set

8 A “group” is a system whose elements (denoted a, b, c . . .), an operation
(denoted *), and an equivalence relation (denoted=) satisfy the following properties:

1. Closure: If a, b are elements of the system, then a*b is an element of the

2. Associativity: If a, b, c are elements of the system, then (a*b)*c = a*(b*c).

3. Existence of an identity: There is an element of the system, e, such that,
for each element of the system (say, d), d*e = e*d = d.

4. Existence of an inverse: For each element of the system (say, d), there exists
an element of the system, d-, such that d*d- = d-*d = e.

In interpreting the twelve-tone system as a group, the elements of the group
are twelve-tone sets, represented as permutations of pitch or order numbers; the
operation is the ^multiplication of permutations. “S” is the identity element. The
“order” of a group is the number of elements of the group.

In addition, the groups presented here have the property of “commutativity” :
if a, b are elements of the system, then a*b = b*a.

112 Problems of Modern Music

that are preserved under the operation, as well as those relationships
between a set and the so-operationally transformed set that inhere in
the operation) which may be termed “musical invariants,” requiring
for their aural recognition merely the ability to perceive pitch class
identity and non-identity, and interval class identity and non-identity.

The familiar invariant associated with T is that of preservation of
the interval number succession. For all its obviousness, it appears a
powerfully cohesive property in the light of the total non-invariance
of pitch classes with regard to order; that is, “no order, pitch number
couple remains fixed under T. Since each t produces a total derange-
ment of the set elements, and the identical intervallic succession, neither
of these properties can serve as the bases of differentiation, in the search
for possible criteria for the compositional hierarchization of transpo-
sitions. Similarly, every value of t defines a regular permutation, but
an invariant basis for differentiation appears with the recognition
that complementary t’s (numbers whose sum is 0, mod. 12), and only
such t’s, produce inverse permutations, of equal order. Beyond the
immediate boundary conditions on the intervallic structure of a set
inferrable from this fact, a consequential musical property follows
from the further fact that inverse permutations produce the same
number of order inversions. This measure of the extent of order rear-
rangement of ihe pitch classes can be described most easily by, for
the moment, regarding T as an operation on the order numbers of S;
an order inversion is each relation among pairs of order numbers that
violates the normal ascending relation among order numbers in S.
For example, in the set of Ex. 1, the application of, let us say, t = 4
to the set produces the following order number succession: 7, 10, 0, 9,
1,3,8,2,11,5,4,6. The complementary t (t = 8) produces the order
number succession: 2,4,7,5,10,9,11,0,6,3,1,8. The number of order in-
versions produced by each is 32.

Complementary t’s produce the same number of pitch adjacencies
with regard to S, both ordered adjacencies and reversed adjacencies.
(This condition of adjacency is imposed merely in the light of the
simplest compositional exploitation of this property, which is imme-
diately extensible to pairs of pitch classes associated with any distribution
of order numbers.) If a set possesses successive pitch classes represented
by pitch numbers a and b, and successive pitch classes represented by
pitch numbers c and d (c may or may not be equal to b, and similarly
ford and a), and if b a = d c, then there is a t such that a + t = c,
and b + t = d, so that under t, a and b are associated with the original

Twelve-Tone Invariants as Compositional

order numbers of c and d, and it then follows that under 12 t, c and d
are associated with the original order numbers of a and b. So, too, for
reversed adjacencies, represented in the set by complementary intervals.
The intervallic structure of S, then, determines the number of adjacencies
preserved under a particular t and its complement, since this number is
a function of the multiplicity of and relative pitch placement of the
identical and complementary interval numbers in S. Consider the set
of Ex. 1 : the interval succession determined by disjunct dyads is
9,6,5,5,3,6. The interval between the identical 5’s is 6, between the
complementary intervals 9 and 3 is also 6, and interval 6 is its own
complement. So, under the application of t = 6 :
Ex. a

the pitch content of disjunct dyads is preserved, and Ex. 2 can thus
be regarded as a permutation of the dyads of Ex. 1 ; if the succession
of dyads hi Ex. 1 be numbered 1 6 inclusive, then those in Ex. 2
are in the order 5,2,4,3,1,6. In the usual cyclic notation, the permutation
is (1 5) (34). This demonstrates an immediate means of extending
serial transformation to compounds of serial elements; I shall return to
this aspect of the example later, but it should not be overlooked that
in this possibility of holding a pair of pitch classes (as opposed to a
pitch class) fixed with regard to order and pitch content, there is imma-
nent the extension to the fixed content trichord, tetrachord, hexachord,
etc., or, in other words, to the combinatorial set.

One more property of complementarity transposed sets should be
indicated. This involves any segment of S (by segment is meant any
number of successive set elements, although the property holds equally
for any selection of elements, non-consecutive as well as consecutive),
and the corresponding segments of any transposition of the set and the
complementary transposition. For example, consider the first seven
elements of Ex. 1, and the corresponding elements of transpositions
with t=2 and t = 10:



Considered with regard to this segment of Ex. 1, both segments 3a and
3b have the same number of pitch classes hi common with it: four.

114 Problems of Modern Music

But, in addition, the pattern of intersection in terms of order numbers
of 3a with regard to 1 is: 0,1,2,5; the pattern of intersection of 1 with
regard to 3b is similarly: 0,1,2,5. The actual pitch classes involved,
naturally, are not necessarily identical, and are not in the present case.
This operational invariant resulting from complementary transposition
is not only of obvious rhythmic and functional significance composition-
ally, but of essential systematic consequence in the theory of general
combinatoriality, aggregate structure, and the resultant means of hier-
archization of set segments.

The importance of transpositional complementation alone would
serve to suggest the systematic operation of inversion (designated: I),
which is definable as complementation mod. 12 of each pitch number
of S, as opposed to complementation of the t applied to all pitch numbers
of S. Given a set element (a,b), I transforms it into (a, 12 b), or,
more generally (a, (12 b) + t), since T is applicable uniquely to
the inverted set; with relation to the complex of sets generated by T,
the inverted set assumes the local role of S. I and T commute only to
within complementation; therefore, the order of operations must be
specified, and I shall assume throughout this discussion that T is applied
after I (IT).

At this point, it is appropriate to consider comparable definitions of
the remaining ‘operations of the system, which reveal that retrogression
(designated: R) can be regarded as affecting complementation of order
numbers: (a,b) is transformed into (11 a, b); therefore RI (or the
reverse, since the operations commute) merely involves the simultaneous
application of both complementation operations: (a,b) is transformed
into (11 a, 12 b). As in the case of I, t’s are applicable to the
set numbers.

The presentation of the permutations of S defined by these operations,
mod. transposition (in other words, this representation is independent
of the operation of transposition), in the usual group multiplication
table, with multiplication permutations as the group operation, and S
denoting the identity permutation: 4





4 The group table is read by choosing an element in the first column and the
element with which it is to be multiplied in the first row; the result of this multipli-
cation is found at the intersection of the row of the first and the column of the second.

Twelve-Tone Invariants as Compositional Determinants

reveals, first of all, that this collection of permutations is an instance of
a group, and as such possesses the property of closure, thus assuring
another combinational aspect of the system in the large, since the col-
lection of sets determined by these permutations is disjunct with regard
to any so determined collection of sets one of whose sets is not a member
of this collection, and is identical with a collection one of whose mem-
bers is.

The revealed symmetry properties indicate the hazardous connota-
tions of such terms as “basic” or “original” set to denote other than a
set form norm decided upon purely on the basis of contextual considera-
tions (temporal priority, for example), since such terms cannot desig-
nate any attribute of set structure in a general sense, either from a
standpoint of internal properties or of relation to other set forms.
Similarly, the fact that the period of each of the permutations is 2
should, hi itself, dispel once and for all those futile attempts to “equate”
these operations with tonal functions.

A vast literature of group theory supplies necessary properties of such
a structure, and it is not without extra-musical interest that this particular
group of permutations is an instance of a familiar group structure, the
so-called “four-group.”

The twelve-tone system, as system, is indeed “simple.” It is simple
in its principles of formation and transformation, but enormously com-
plex and deep in its ramifications, in the necessary inferences that can
be drawn from these principles, for it is of the formal model of which
it is an exemplification that Hermann Weyl has said: “From these insig-
nificant looking assumptions springs an abundance of profound rela-
tionships; and mathematics offers an astounding variety of different
interpretations of this simple axiom system.” 6

Inversion, hi the traditional sense, implies inversion of contour.
In the twelve-tone system inversion, like transposition, cannot be charac-
terized in terms of registral considerations, but merely as that permuta-
tion of pitch class numbers (or, for particular purposes, of order
numbers) which results from the substitution of complementary pitch
numbers in S; it follows that there is an accompanying substitution of
a succession of complementary intervals for the interval succession of S.

Even more, perhaps, than in the case of T, I must derive its “justi-
fication” from its associated musical invariants. It must be emphasized

5 Philosophy of Mathematics and Natural Science, Princeton, 1949, p. 28,

jj6 Problems of Modern Music

that, although invariants are associated necessarily with the operations
in question, the degree to which they are projected explicitly in compo-
sitional terms depends upon the emphasis they receive from other musical
components: rhythm, dynamics, register, phrasing, timbre, etc. Con-
versely, the desire for the compositional exploitation of these funda-
mental properties may be regarded as, at least, a partial determinant
of the compositional characteristics imposed on these components.

Consider the simplest inversional invariant: if (a,b) of S is trans-
formed by IT into the corresponding order element (a, (12 b) + t),
where (12 b) + t may or may not be equal to b, then, correspond-
ing to (c,(12 b) H- t) of S is (c,b). This property, as may be ob-
served by reference to our examples, is possessed also by transposition
when t = 6; it holds for all t’s applied to the inversion.

Again, the cyclic representation of the pitch class permutations ef-
fected by IT shows that all even t’s produce similar permutations of six
cycles of two elements each (thus, regular permutations), while odd t’s
produce similar permutations of five two-element cycles and two unit
cycles. So, although complementary t’s still produce similar permuta-
tions, this is merely because they are either both even or both odd. Odd
values of t, then, determine six dyadic pitch classes between elements
of the same order number in I related sets, and even values of t
determine five such dyadic classes, and two single-element classes. These
latter represent set elements whose order number, pitch number couple
remains unchanged under IT. It is for this reason that a necessary
condition for hexachordal inversional combinatoriality is that the sum
of the set numbers of the same order number in the I related sets be
odd. The pitch number of an element so fixed is equal to one half
of t; thus, the two such fixed elements associated with a given even t
are unique, and are tritone related (since 12/2 = 6).

In addition to this partition of the elements of inversionally related
sets into pitch classes, the IT operation also effects a categorization
into interval classes. Since the intervals between pitch classes of the
same order number in I related sets are either all even or all odd, and
thus each interval occurs exactly twice, it can be shown that pitch
classes in S whose pitch numbers differ by 6 are associated with the
same interval determined by the element of the same order number in the
I related set.

The lai^ely “note against note” presentation of the canon in the
second movement of Webern’s Variations for Piano places these char-

Twelve-Tone Invariants as Compositional Determinants B|

actcristics in the foreground. The initially stated, I related forms of the

set (Ex. 4a) :

Ex. 4
since t = 2 (taking the lower of the two sets as the reference “prime”
set), hold (1,1) and (11,7) fixed. Since the first hexachord of S con-
tains no tritone related pitches, there is no repetition of pitch ‘dyads
formed by elements of the same order number, so that, if the first
succession of six dyads is numbered 1,2,3,4,5,6, then the following
succession is a permutation of these: 6,4,1,5,3,2. As hi the Schoenberg
example, although in a different manner, complexes of pitch elements
become, themselves, subjected to serial permutation.

The continuation of this canon demonstrates the compositional
use of another invariant. By choosing, as the initial pitches (or, as any
pitches of the same order number, since the sums of the pitch numbers
of such pairs are equal) of the T forms of the S and the I related set,
elements the sum of whose pitch numbers, with regard to the original
reference point “gj” = (0,0), is equal to the original t = 2, the
pitch dyads resulting are identical with those created in the first inver-
sional juxtaposition. The recurrence of repeated “a’s” is merely one
manifestation of this general property. See Ex. 4b. Therefore, for each
first element of a set, there is one and only one choice of the first element
of the I related set which holds the pitch dyads so fixed with regard
to a pre-defined norm. (In the light of the previous discussion, it is of
interest to point out that an equivalent statement of this condition is
that the initial I related forms be transposed by complementary values
of t.) Webern chooses four of these twelve possibilities to determine the
pitch levels of the successive sections of the movement; the fitst and
last pitches of the I related sets which provide the pitch content of
the third and fourth sections of the work are shown in Ex. 4c. The first

118 Problems of Modern Music

movement of Webern’s Quartet, Op. 22, employs the same procedure in
a more elaborate and extended manner.

The more common and traditional procedure of “totally transposing”
such a section in my terms, the applying of the same t to both
simultaneously stated forms preserves the interval succession but
not (except the single case where t = 6) the fixed pitch dyads: the
procedure under discussion here permutes the interval succession while
retaining the pitch content of the dyads.

If we number the dyads of the first section from 1-12 inclusive,
the second section yields a permutation corresponding exactly to the
permutation of order numbers that results from applying t = 5 to
the upper set, or equivajently t = 7 to the lower set, and of
course correspondingly with the third and fourth sections. Thus, all
the properties associated with the application of T to S are translatable
into properties of permutations of dyads between I related sets.

Webern’s particular choice of transpositions appears to be related
primarily to concerns of compositional duration and external design.
The transpositional choice for the second section makes possible a final
dyad for this section which is pitch identical with the initial dyad of
the movement, but with pitch components reversed as to set member-
ship. The repeat of these first two sections is founded on this identity.
The third section is transpositionally determined by exact analogy with
the second section, through the “double function” of the final dyad
of the preceding section. But the continuation of this basis of choice,
since the interval between the first and last elements of the set is prime
to 12, would carry the work through all possible twelve jointly deter-
mined transpositions before returning to the first dyad in its original
disposition. So, the fourth section employs the principle of interchange,
already introduced into the work by the repeat of the first two sections.
The fourth section thus effects a return to the first dyad of the movement,
while the repeat of the third and fourth sections as a unit results in
another interchange, necessarily the exact reverse of the interchange
resulting from the first repeat.

Closely related to these invariants is the property: if (a,b) and
(a + l,c) are two successive elements of S (the provision of succession
is, actually, unnecessary, but is introduced here for purposes of sim-
plicity), and (d,e) is an element of an I related set (where d may or
may not be identical with a or a = 1, and e may or may not be
identical with b or c), then the intervaUic succession b e, c e is

Twelve-Tone Invariants as Compositional Determinants

identical with the succession g f, h f, defined by d,f of the
initial set, and a,g and a + 1, h of the I related set. In the Contra-
punctus Secundus of the Quaderno Musicde di Annalibera, which is
in many structural respects closely similar to the Webern movement
just discussed, Dallapiccola uses this property in the second half of the
piece as a means of unfolding the same intervallic progression by the
two canonic parts, while reversing their relation of temporal priority.
This property, as well as the fixed dyad property, is particularly signifi-
cant as a harmonic factor when extended to include more than two
simultaneously stated, I related sets.

As in the case of T, conditions for the retention of pitch adjacencies
under IT are statable easily and fully. However, I merely shall return
to Ex. 1, and examine the result of applying IT with t = 3, and t = 9.


.. M I * L. !

The pitch content of disjunct dyads is preserved, and the permutations
of these dyads under I for t = 3 is (1 5) (2 6), and for t = 9 is
(26) (34). Taken together with the identity permutation and the
permutation under t = 6, this group of permutations leaves each
dyad twice fixed with regard to order, once with its component ele-
ments in the order defined by S, and once with the elements reversed;
of the four occurrences of each dyad, two maintain the order of
elements defined by S, and two reverse this order. More generally, if we
denote by A the identity permutation of the dyads, by B the permuta-
tion on the dyads effected by t = 6, by C and D the permutations
effected by IT with t equal respectively to 3 and 6, the multiplication
table for this group of permutations is:





This group is isomorphic with that formed by the permutations repre-
senting the identity, I, R, RI operations of the system.

Finally, the familiar phrase: “the identification of the horizontal
with the vertical,” implies much more than a mere compositional
prescriptive with regard to the spatial distribution of the elements of a

120 Problems of Modern Music

set when it is realized that adjacent pitch elements of a set become
elements of the same order number in I related sets, when these sets are
so chosen that the sum of the pitch numbers of any two elements of the
same order number is equal to the sum of the pitch numbers of the
originally adjacent elements.

Space does not permit a consideration of invariance under R and
RI. However, it must be pointed out that the traditional conception of
retrogression as effecting the temporal reversal of pitches constitutes
neither a meaningful description nor a “justification” for its position
in the twelve-tone system. For, even with registral considerations dis-
regarded, this characteristic is associated with but one transpositional
level of the retrograde; but all transpositional levels of the retrograde
present the intervallic succession of the inversion in reversed order,
while necessarily the retrograde-inversion forms present the inter-
vallic succession of the prime (S) in reversed order. Thus, the RI
forms, often regarded as the aurally most unrealistic transformations,
because the operation is viewed as applied to pitch succession rather
than to interval succession, require for the perception of their relation
to S merely the ability to identify interval classes. In this important
sense, the RI forms can be regarded as the most closely related to S, and
are so employed often by Schoenberg in his compositional, “thematic”
presentation of successive set forms: see, for example, the Variations
for Orchestra, the third movement of the Fourth String Quartet, and
the Piano Concerto. In the pitch class order, interval class order duality
between retrograde and retrograde-inverted related sets reside many
of the most important properties of such transformations.

Even so incomplete and informal a discussion of so small a number
of the invariants attending the operations of the system indicates, I hope,
something of the essential importance of this subject, analytically in
the “rational reconstruction” of compositions, and compositionally in
comprehending and mastering the materials of the system. If I have led
the discussion more in the direction of those aspects which suggest
the “macrocosmically” combinational features of this basically permu-
tational system, I could have with equal appropriateness and the
same means examined the “microcosmically” combinational features
(particularly what are termed in group theory “imprimitive systems”),
set structure (particularly with regard to redundancy properties), com-
binatoriality, generalized partitioning, derivation, sequence theory, and
related questions.

Certainly, any conjectures about “generalized” serialism must con-

Twelve-Tone Invariants as Compositional Determinants 121

front the problem as to whether such alleged generalizations result in a
maintenance, an increase, or a decrease, of the number and scope of
such invariants, and whether the apparent “freedom” of such “generali-
zations” does not, in a deeper sense, reduce structural resources rather
than augment them.

Likewise, I would insist that a necessary condition for the applica-
tion of the permutational operations of the twelve-tone system to order-
able non-pitch elements is that the rules of correlation be so arrived at
that these invariants, which are necessary consequences of the pitch class
nature of the system, be susceptible to musically meaningful interpre-
tations in these other, perhaps significantly dissimilar, domains.

In conclusion, I can state only, without hoping to have done more
than intimate the bases for such a statement, that an “exhaustion”
of the resources of the twelve-tone system in the relevant future is not
only unforseeable, but unthinkable. I trust I have begun to document,
and will be given the opportunity in the future to further document,
the statement that, in its vastness of structural means, its flexibility,
and its precision, the twelve-tone system cedes nothing to any musical
system of the past or present that has engaged the mind of musical man.