1982
Fugitive Integration
/ -
A musical addendum to Liberation
of Integration (1982). Prepared for the 5th meeting of the Goals, Processes
and Indicators of Development (GPID) project of the United Nations University
(UNU)
Introduction
The fugue is often considered to be the most complex and highly developed type
of composition in Western music. The main
paper argues that much could be learnt about new approaches to integration
from a study of inter-action in music. The fugue is not discussed in that paper
but does however offer many pointers for socio-political integration.
Description
A general description of the fugue is given in Annex 1 and Annex 2. Partiular
attention is given to the fugues of J.S. Bach whose work in this field is also
much admired by mathematicians (Douglas R Hofstadter. Godel, Escher, Bach; an
eternal golden braid. (A metaphorical fugue on minds and machines in the spirit
of Lewis Carroll). Harvester Press/Basic Books, 1979 (Pullitzer Non-fiction
Award 1980)). The beginning of one of his fugues is given as Annex 3.
Significance
for socio-political integration
Many aspects of this question are explored in the main paper. Here the focus
is on the significance of the relationship between distinct "voices"
and themes, which is of course basic to polyphonic and symphonic integration.
In the socio-political sphere it is usual for advocates of a proposal, a model,
a cosmology or an ideology to propagate it as though it alone should achieve
dominance (ad aeternam), effectively excluding alternative approaches.
Within the musical framework this can lead to pieces which are either immediately
monotonous and boring, or whose interesting characteristics quickly became an
intolerable imposition unless balanced by other pieces in the musical diet (cf.
the life cycle of a "hit" record). There is of course. no musical
continuity between the succession of such separate pieces of music. This
lack of integration is analogous to the equivalent situation in society where
the advocates of the alternatives, evoked by overstress on a particular model,
compete in parallel or in succession for constituencies and resources for their
own approaches.
In the fugue however the relationship between these competing voices
is explored within a musical continuum.
This représente a new level of integration.In effect
the concept(model, etc) is explored, inverted, countered, distorted, etc within
an overriding set of rules which permit a new level of free freedom.
The rules ensure a more exciting balance of tension and harmony.
Conclusion
Could it be that one dimension of the challenge of socio-political integration
is illustrated by the problem of interrelating seemingly hostile or incompatible
"voices". And that counterproposals and counterarguments need to
be set in a larger context to which we are as yet insensitive? And does the
time dimension over which arguments and counterarguments
are developed need to be better understood in terms of an integrated socio-political
process?
Such a process brings out both the essential
inadequacy of any particular
proposal and the manner whereby it can be counterbalanced and enriched by
complementary proposals which
together asaprocessbring about a new level
of integration.
The implications of the fact that music has
"progressed beyond" the fugue
are discussed in the main paper. But it would appear that the lessons it
offers for socio-political integration have not yet been considered.
| Annex 1: Extract on "Fugue" from Encyclopaedia
Britannica |
Although the statement is debatable, it is often said
that the fugue is the most complex and highly developed type of composition
in Western music. The term fugue, derived from fuga, the Latin word for "flight," was
first used about 1330 by Jacques de Liége, the author of Speculum
musicae, an important medieval treatise.
At that time it referred to a technique of musical writing based on strict
imitation. Later, after its emergence as an independent musical form
in the 17th century, the fugue became a composition in counterpoint based
on a generating theme, in which different pans, or voices, enter successively
in imitation, as if in pursuit of each other. The heir of all the compositional
techniques that had developed earlier, it differsfrom its ancestors (the
motet, the ricercar, the canzona) in having a more specifically tonal
character, unity of form, and a greater economy.
Description of the fugue
The fugue is written in counterpoint, two or more lines that sound simultaneously.
Counterpoint's laws and techniques have developed from the 10th century
to the Renaissance, a period during which Western music was essentially
polyphonic. One of the main problems was the harmonic aspect of the meeting
of the voices, and the rules of counterpoint are always precise regarding
the use of consonance and dissonance. Counterpoint deals also with movement
between the parts. It includes various techniques of development, among
which imitation is probably the most remarkable feature of polyphonic
music. There arc many kinds of imitations. The strictest is the canon,
in which the melody slated by the first voice is later reproduced by
the second voice. A good example is the song "Frère Jacques." Other
common types of imitation include inversion of all the intervals, augmentation
(in which the rhythmic values are doubled), diminution (in which they
are reduced), or even retrograde imitation, in which the last note of
one voice becomes the first note of the next. All these techniques are
used in fugal composition, which is characterized more by its "language"
than by its form.
Elements. The fugue is written for a certain number of voices, or instrumental
parts. The most frequent are fugues for three or four voices, but there
arc also fugues for two, five, or more voices. Although the fugal form
varies from composer to composer, there arc certain commonelements.
Subject. The subject is the theme of the fugue. It is stated "alone
by the first voice before being taken up by the others. In the course
of the fugue, it will bestated in different keys, and it will be sometimes
slightly modified or inverted. Some of its elements may be developed
separately.
Answer. 1 he second voice brings in the answer, generally
stated in the key of the dominant (the fifth degree of the major or minor
scale). If it reproduces the subject exactly, it is called a real answer.
But in most cases, in order topreserve the tonal unity of the fugue,
the answer has to undergo a "mutation' that alters some of its melodic
intervals and makes the modulation to the dominant key smoother. This
is called the tonal answer.
Countersubject. The
countersubject accompanies the answer. If it is maintained throughout
the fugue, it is called sustained or obligate countersubject and will
follow the subject like its shadow for each new statement. Subject and
countersubject arc the two principal "actors"
of the fugue, and theoretically all the musical substance must he derived
from them.
Exposition. The first part of the fugue, which includes the successive
entrance of the voices, in subject-answer alternation, is called exposition.
This progressive enrichment of the polyphonic web is one of the most
striking traits of the fugue. In some fugues, after the exposition, the
composer brings in the answer followed by the subject. This is called
counterexposition.
Episode. An episode is any passage, developed or not. that links
two statements of the subject. It is characteristically written in imitative
style. Generally it uses a motive from the subject or the countersubject,
but sometimes a new element is introduced. There is a great variety of
episodes.
Stretto. When the subject overlaps the answer (or
the answer the subject), it is called a stretto. This device, whereby
the entries are drawn more closely together, is often used at the end
of the fugue, where it achieves spectacular effects.
Plan of the fugue: Once past the exposition, the plan ofthe fugue
depends on the will of the composer and the resources of its thematic
elements. For instance, the Danish composer Dietrich Buxtchude (died
1707) often presents the subject and the answer only in the principal
and in the dominant keys, in short expositions linked by small episodes.
With J.S. Bach, the tonal plan becomes more elaborate and includes a
journey to the principal neighbouring keys. Since the composer adapts
his plan to the character and to the potential of his themes, the itinerary
is always different. That is why in his hands the fugue becomes the most
versatile of musical forms; each fugue of Bach brings a new solution
to the problem of the relation between form and content.
The "school fugue." Theorists created an ideal plan of the
fugue and gradually perfected it, in the 19th century, as fewer and fewer
real fugues were being written. They devised a tripartite form consisting
of the exposition, the development, and the stretto.
Varieties of fugue
Simple fugue This is monothematic, without
a maintained countersubject (such as the Fugue in D Major of
Bach's work The Well-Tempered Clavier, Book I, No. S). More
elaborate fugues use one or more countersubjects. In a counterfugue,
the answer imitates the subject by inversion. There are beautiful examples
of this technique in Bach's work The Art of the
Fugue, numbers 5 to 7.
Double fugue. There are two ways of writing a double
fugue; either the two subjects may be presented simultaneously, in which
case the fugue is not very different from a fugue with countersubject
(Bach, Fugue in B Minor on a Theme of Corelli, BWV
579), or else the second subject has a special exposition. The latter
yields, in general, a tripartite scheme: exposition and development of
the first subject, exposition and development of the second subject,
and finally combination of the two elements, which are devised so that
they can be superimposed. A splendid example is given by Bach's Toccata
end Fugue in F Major for organ, BWV 540. The same principles
apply toa triple or quadruple fugue.
Fugitetta. This is a miniature fugue
but strictly written, whereas the fugato starts like a fugue but gives
up its discipline once past the exposition.
Choral fugues. The fugue plays a considerable role in works for
chorus and orchestra. Generally the chorus sings in strict counterpoint,
while the instruments play an expressive or decorative accompaniment.
The composition techniques of the fugue can also be used in forms as
universal as the prelude, the aria, the chorus, the overture, the concerto,
and others.
Literature of the fugue
Since vocal polyphony was based on a text that had to be sung by each
of the parts (either simultaneously or, mure frequently, in imitation),
it was not very much concerned with the problem of form. That is why
theorists and musicians concentrated on questions of texture.
At first the term fuga applied to strict imitations
(which would now be called canons); the conceptevolved in a more general
sense when it was realized that a freer use of all kinds of imitations
offered much more stimulating opportunities. Besides the purely vocal
polyphony, the first independent forms of keyboard music (the ricercar,
the anzona the Capriccio, the fantasia) testify to the remarkable development
of what can be called the fugal style of the 16th and 17th centuries.
In this evolution, the role of the Italian composer Andrea Gabrieliand
of the English virginalists was preponderant. The two great precursors
of the fugue proper were the Dutch composer J.P. Sweelinck (died 1621)
and the Italian Girolamo Frescobaldi (died 1643). Both greatly influenced
the keyboard music of the 17th century, in particular through their students
Samuel Scheidt and Johann Jakob Froberger. In Germany, a generation of
musicians dominated by Buxtehude gave the fugue its modern form by putting
it in a tonal perspective and abandoning the fragmentary style of their
predecessors. Almost all the composers of the 17th century contributed
to the history of the fugue.
J.S. Bach. The genius of Bach found particular expression in
the fugue, perhaps because it allies the strictest economy of language
to a relative freedom of form. Each of his fugues amazes by the freshness
of its inspiration, the wonders of its writing, or by its gigantic proportions,
all marvellously represented in the two volumes of The Well-TemperedClavier (1722-44),
two sets of 24 preludes and fugues going through the cycle of the 24
major and minor Keys. Some of his organ fugues lend toward development,
some toward symmetry, and some toward virtuosity; others take the form
Of double fugues. His last work, The Art ofthe Fugue, is
a collection of 14 fugues and four canons, all based on a theme in D
minor and its inversion. All the resources and procedures of the fugue
are demonstrated in what constitutes the most inspiring treatise on fugue,
a treatise without-words, in which music speaks alone. After being long
slighted as a purely theoretical work. The Art of the Fugue has
won a high place in the hearts of music lovers, who see it as Bach's
musical testament. Bach's cantatas, passions, and oratorios abound in
admirable fugues.
Handel and after. Handel's fugues are less erudite than those
of Bach and sometimes employ looser counterpoint, but they touch the
listener by their vitality and their harmonious proportions. The great
fugal sections of his oratorios are more important than his keyboard
fugues.
After Bach the fugue lost much of its importance. With the appearance
of the sonata, the musical taste changed, and composers tended to consider
counterpoint as an archaic discipline. Nonetheless, the fugue retained
a place in choral works, and fugal methods were kept in the sonata form,
particularly in the development section. A great passion for Bach led
Mozart (died 1791) to a more contrapuntal style. This influence is obvious
in the Fugue in C Minor, for two pianos, K. 426, which, though
pure Mozart, is nonetheless a. homage to Bach. In ingenuity and mastery,
Mozart rivalled the greatest contrapuntists in, for instance, the great
choruses of the Mass in C Minor, K. 427, or in the final development
of the Jupiter Symphony.
Beethoven. Beethoven resorted more and more to fugal technique
in his last works. He confessed that he wrote his fugues with the greatest
difficulty, and it is true that his counterpoint gives an impression
of effort. Mostof the fugal passages integrated into his last sonatas
and quartets create a dramatic tension. Far from being a scholastic technique,
the fugue was for Beethoven a means to reach the expressive limits of
an idea. He used this language in particular circumstances. The strange
and desolate atmosphere of the opening fugue of the String Quartet
in C Sharp Minor, Opus 131, brings to mind the 20th-century fugue
in the first section of the Music for Strings, Percussion and Celesta of
Béla Bartók. The Promethean side of Beethoven asserts itself
particularly in the Great Fugue from the String Quartet in B Flat
Major, Opus 130, and the noble fugal section of the Missa Solemnis. Unlike
Mozart's classical fugues, Beethoven's arc rather irrational in form
but are justified by their creative power alone.
The Romantic era. Compared with the fugues of Beethoven. those
of Felix Mendelssohn defer to traditional rules. Critics sense in them
a nostalgia for Bach, sometimes weakened by a touch of sentimentalism.
A more authentic romantic breath animates the fugues of Koben Schumann
on B.A.C.H. (the German letters for the notes B-A-C-B). but the fugue
is not his natural language. The genre is more suited to César
Franck, as may be seen in his Prélude, fugue et variation, or
in bis . Prelude, chorale et fugue. His
harmonic sensitivity enriches a contrapuntal technique while not breaking
with tradition. The fugues of Franz Liszt arc entirely different: once
past the exposition, he cannot renounce symphonic developments. There
is an original use of the fugato before the re-exposition in his Piano
Sonata in B Minor. The fugal style was used to varying degree by
the major composers of the 19th century, including Brahms in his German
Requiem, Richard Wagner in his opera Die Meistersinger,and
Verdi in his Requiem and at the end of his last opera, Falstafj.
Among the post-Romantics, who cultivated their own exaggerated form
of counterpoint, mention must be made of Max Reger, whose admirers took,
him as the heir of J.S. Bach. Though that appears to be going too far,
his counterpoint does possess vitality in works such as the Variations
and Fugue on a Theme by Mozart, for orchestra, or the fugue on the
chorale melody "Wachet auf, ruftuns die Stimme," for organ.
The 20th century. As an autonomous form, the fugue played only
a modest role in the first half of the 20th century. The most beautiful
example is the already mentioned fugue of the Music for Strings, Percussion
and Celesta of Béla Bartók, a born contrapuntist. This
is a true model of fugal treatment in a post-tonal style. The methods
of the fugue are also found in his Sonata for Two Pianos
and Percussion and in his admirable quartets.
Stravinsky, though influenced by the composers of the I7th and 18th
centuries, showed no particular interest in the fugue. Although the second
part of his Symphony of Psalms can
be considered a double fugue, it docs not strike the listener as such.
In his collection of interludes and fugues, called Ludus Tonalis, Paul
Hindemith seems to have drawn his inspiration from Bach's Well-Tempered
Clavier. Although an interesting work, it did not herald a rebirth
of the fugue A new conception of counterpoint appeared in the works of
Arnold Schoenberg and Anton von Webern. The serial techniques of composers
like Pierre Boulez can. to a certain extent, claim some kind of kinship
with fugal language, and a work such as the Passion According so St.
Luke of Krzysztof Penderecki testifies to the permanence of a musical
form the history of which is probably unfinished. |
***
| Annex 2: Extract from: Godel,
Escher, Bach by
Douglas Hofstadter |
Canons and Fugues
The idea of a canon is that one single theme is played against itself.
This is done by having "copies" of the theme played by the
various participating voices. But there are many ways todo this. The
most straight forward of all canons is the round, such as "Three
Blind Mice", "Row, Row, Row Your Boar", or "Frére
Jacques", Here, the theme enters in the first voice and, after a
fixed lime-delay, a "copy" of it enters, in precisely the same
key. Alter the same fixed time-delay in the second voice, the third voice
enters carrying the theme, and so on. Most themes will not harmonize
with themselves in this way. In order for a theme to work as a canon
theme, each of its notes must he able to serve in a dual (or triple,
or quadruple) role: it must firstly he part of a melody, and secondly
it must be part of a harmonization of the same melody. When there are
three canonical voices, for instance, each note of the theme must act
in two distinct harmonic ways, as well as melodically. Thus, each note
in a canon has more than one musical meaning; the listener's ear and
brain automatically figure out the appropriate meaning, by referring
to context.
There are more complicated sorts of canons, of course. The first escalation
in complexity conies when the "copies" of the theme are staggered
not only in time, but also in pitch; thus, the first voice
might sing the theme starting on C, and the second voice, overlapping
with the first, voice, might sing the identical theme starting five notes
higher, on G. A third voicestarting on the D yet five notes higher, might
overlap with the first two, and so on. The next escalation in complexity
comes when the speeds of the different voices are not equal; thus,
the second voice might sing twice as quickly, or twice as slowly35 the
first voice. The former is called diminution, the latter augmentation (since
the theme seems to shrink or to expand).
We are not yet done! The next stage of complexity in canon construction
is to invert the theme, which means to make a melody which jumps down wherever
the original theme jumps up, and by exactly thesame number of
semitones. This is a rather weird melodic transformation, but when one
has heard many themes inverted, it begins to stem quite natural. Bach
was especially fond of inversions, and they show up often in his work—and
the Musical Offering is no exception. (For a simple example of
inversion, try the tune "Good King Wenceslas". When the original
and its inversion are sung together, starting an octave apart and staggered
with à time-delay of two beats, a pleasing canon results.) Finally,
the most esoteric of "copies" is the retrograde copy—where
the theme is played backwards in time. A canon which uses this trick
is affectionately known as a crab canon, because of the peculiarities
of crab locomotion. Bach included a crab canon in the Musical
Offering, needless to say. Notice that every type of "copy"
preserves all the information in the original theme, in the sense that
the theme is fully recoverable from any of the copies. Such an informationpreserving
transformation is often called an isomorphism, and we will have
much traffic with isomorphisms in this book.
Sometimes it is desirable to relax the tightness of the canon form.
One way is to allow slight departures from perfect copying, for the sake
of more fluid harmony. Also, some canons have "free" voices—voices
winch do not employ the canon's theme, but which simply harmonize agreeably
with the voices that are in canon with each other,
Each of the canons in the Musical Offering has
for its theme a different variant of the King's Theme, and all the devices
described above for making canons intricate are exploited to the hilt:
in fact, they are occasionally combined. Thus, one three-voice canon
is labeled
"Canon per Augmentationem, contrario Motu"; its middle voice
is free (in fact, it sings the Royal Theme), while the other two dance
canonically above and below it. using the devices of augmentation and
inversion. Another bears simply the cryptic label "Quaerendo invenietis" ("By
seeking, you will discover"). All of the canon puzzles have been
solved. The canonical solutions were given by one of Bach's pupils, Johann
Philipp Kirnberger. But one might still wonder whether there are more
solutions to seek!
I should also explain briefly what a fugue is. A fugue is like a canon,
in that it is usually based on one theme which gets played in different
voices and different keys, and occasionally at different speeds or upside
down or backwards. However, the notion of fugue is much less rigid than
that of canon, and consequently it allows for more emotional and artistic
expression. The telltale sign of a fugue is the way it begins: with a
single voice singing its theme. When it is done, then a second voice
enters, either five scale-notes up. or four down. Meanwhile the first
voice goes on. singing the "countersubject": a secondary theme,
chosen to provide rhythmic, harmonic, and melodic contrasts to the subject.
Bach of the voices enters in turn, singing the theme, often to the accompaniment
of the countersubject in some other voice, with the remaining voices
doing whatever fanciful things entered the composer's mind. When all
the voices have "arrived", then there are no rules. There are,
to be sure, standard kinds of things to do—but not so standard that
one can merely compose a fugue by formula. The two fugues in the Musical
Offering are outstanding examples of fugues that could never have
been "composed by formula". There is something much deeper
in than than mere fugality.
All in all, the Musical Offering represents one of Bach's supreme
accomplishments in counterpoint. It is itself one large intellectual
fugue, in which many ideas and forms have been woventogether, and in
which playful double meanings and subtle allusions are commonplace. And
it is a very beautiful creation of the human intellect which we can appreciate
forever. (The entire work is wonderfully described in the book J.. S. Bach's
Musical Offering, by H.T. David.)
An Endlessly Rising Canon
There is one canon in the Musical Offering which is particularly
unusual. Labeled simply "Canon per Tonos", it has three voices.
The uppermost voice sings a variant of the Royal Theme, while underneath
it, two voices provide a canonic harmonization based on a second theme.
The lower of this pair sings its theme in C minor (which is the key of
the canon as a whole), and the upper of the pair sings the same theme
displaced upwards in pitch by an interval of a fifth. What makes this
canon different from any other, however, is that when it concludes—or,
rather, seems to conclude—it is no longer in
the key of C minor, but now is in D minor. Somehow Bach has contrived
to modulate (change keys) right under the listener's nose. And
it is so constructed that this "ending" ties smoothly onto
the beginning again; thus one can repeat the process and return in the
key of E, only to join again to the beginning. These successive modulations
lead the ear to increasingly remote provinces of tonality, so that after
several of them, one would expect to be hopelessly far away from the
starting key. And yet magically, after exactly six such modulations,
the original key of C minor has been restored! All the voices are exactly
one octave higher than they were at the beginning, and here the piece
may be broken off in a musically agreeable way. Such, one imagines, was
Bach's intention; but Bach indubitably also relished the implication
that this process could go on ad infinitum, which is perhaps why he wrote
in the margin "As the modulation rises, so may the King's Glory." To
emphasize its potentially infinite aspect, I like to call this the "Endlessly
Rising Canon".
In this canon, Bach has given us our first example of the notion of Strange Loops. The "Strange
Loop" phenomenon occurs whenever, by moving upwards (or downwards)
through the levels of some hierarchical system, we unexpectedly find
ourselves right back where we started. (Here, the system is that of musical
keys.) Sometimes I use the term Tangled Hierarchy to describe
a system in which a Strange Loop occurs. As we go on, the theme of Strange
Loops will recur again and again. Sometimes it will be hidden, other
times it will be out in the open; sometimes it will be right side up,
other times it will be upside down, or backwards. "Quaerendo invenietis" is
my advice to the reader. |
***
|
Annex 3: Extract from Figure in D Minor by J S Bach
Well-tempered Clavier, Bk I, No. 2 |
|
|
|
