Twelve-note composition.
A
method of composition in which the 12 notes of the equal-tempered chromatic
scale, presented in a fixed ordering (or series) determined by the composer,
form a structural basis for the music. It arose in the early years of the 20th
century, when the dissolution of traditional tonal functions gave rise to
several systematic attempts to derive a total musical structure from a complex
of pitch classes that are not functionally differentiated. Skryabin’s Seventh
Sonata (1911–12), for example, is based upon such a complex, or ‘set’ (ex.1). The set, at any of its 12 transpositional levels,
generates both the melodic and the harmonic elements of the composition. It is
defined in terms of its pitch-class content, relative to transposition, and no
pre-compositional ordering or segmentation of the set is assumed. Clearly there
is only one analogously unordered set of all 12 pitch classes. About 1920 Hauer
and Schoenberg independently arrived at concepts of 12-note set structure that
make it possible to differentiate between one 12-note set and another, and
among transformations and transpositions of any given 12-note set.
See also Atonality and Serialism.
1. 12-note tropes.
2. 12-note series.
3. Origins of the 12-note set.
4. 12-note composition.
5. Pre-compositional structures.
6. 12-note ‘tonality’.
7. Extensions and developments of 12-note
composition.
8. Conclusion.
BIBLIOGRAPHY
PAUL LANSKY, GEORGE PERLE (text), DAVE HEADLAM (bibliography)
Twelve-note
composition
1. 12-note tropes.
In
Hauer’s system the 12 pitch classes are divided into discrete, mutually
exclusive segments. The order of segments within a 12-note set and the order of
pitch classes within each segment are not pre-compositionally defined. Each
such set and its 12 transpositions represent what Hauer called a ‘trope’. The
only tropes that Hauer investigated systematically are those that divide the
pitch classes into two hexachords. Let the integers 0 to 11 represent the
successive pitch classes of an ascending chromatic scale of unspecified
transposition. If pitch-class numbers 2, 3, 5, 7, 9 and 10 are chosen as one
hexachord, the trope will be completed by the hexachord formed by the remaining
pitch classes: 0, 1, 4, 6, 8 and 11. The hexachords (8, 9, 11, 1, 3, 4) and (6,
7, 10, 0, 2, 5) are a representation of this same trope since they are its
transposition by a tritone of (i.e. the addition of ‘T-no.6’ to) each element
of the original. Hauer demonstrated that there are 80 non-equivalent
hexachords. Eight of these will each form a trope by combination with its own
transposition. (For example, the whole-tone hexachord (0, 2, 4, 6, 8, 10) may
be combined with a transposition of itself by any odd T-no.) The remaining 72
hexachords are paired to form 36 tropes, and so there are 44 independent
hexachordal tropes.
Hauer’s
is the earliest known attempt at any general formulation of the resources of a
12-note sound world. His concept of hexachordal tropes was a remarkable
anticipation of subsequent developments in 12-note composition and theory. It
is doubtful, however, that he had any direct influence on what is significant
in these developments. His own compositions are simple-minded exercises, and
his 12-note theory was embedded among mystical and megalomaniac assertions. The
fact remains, however, that he was the first to discover the all-important
principle of segmentation – the partitioning of the collection of 12 pitch
classes into mutually exclusive sub-collections – as a basis for classifying
the pitch-class material of 12-note music.
Twelve-note
composition
2. 12-note series.
If
the 12 pitch classes are regarded as an unsegmented collection, sets can be
differentiated only by the ordering of their elements. In Schoenberg’s system
ordered sets (‘series’ or ‘rows’) that may be transformed into one another by
transposition (i.e. the addition or subtraction of a constant T-no., mod 12),
by retrograding, by inversion (i.e. the subtraction of each of the original
pitch-class numbers from a constant T-no., mod 12) or by any combination of
these operations, are all regarded as different forms of a single series. Since
the series in each of its aspects – prime (P), retrograde (R), inversion (I),
retrograde-inversion (RI) – may be stated at 12 transpositional levels, there
will be 48 set forms in the complex generated by a single series.
Twelve-note
composition
3. Origins of the 12-note set.
The
term ‘12-note music’ (or ‘dodecaphony’) commonly refers to music based on
12-note sets, but it might more logically refer to any post-triadic music in
which there is constant circulation of all pitch classes, including both the
pre-serial ‘atonal’ compositions of Schoenberg, Berg and Webern and the
‘atonal’ compositions of Skryabin and Roslavets based on unordered sets of
fewer than 12 elements (see Atonality). However, the customary sense is
retained here. Occasional systematic statements of the 12 pitch classes first
appeared in the music of Berg. A 12-note series is one of the principal themes
of his Altenberg songs, composed in 1912, about three years before Schoenberg’s
first experiment with ‘a theme consisting of the 12 notes’ in his unfinished
oratorio Die Jakobsleiter. Schoenberg’s ‘theme’ is a hexachordal 12-note
trope, rather than a series. By summer 1919 Berg had completed, in short score,
the first act of Wozzeck, which contains a 12-note passacaglia theme
that is often cited as an adumbration of the Schoenbergian series. Non-serial
12-note collections are found in Berg’s works throughout his career. In the
concluding song of op.2 (1910) a white-key glissando in the left hand of the
piano part occurs simultaneously with a black-key glissando in the right, a
12-note aggregate which anticipates by about 20 years the white-key and
black-key clusters of the Athlete’s leitmotif in Lulu. A chord
consisting of all 12 pitch classes opens and closes the third of the Altenberg
songs. The two mutually exclusive whole-tone scales and the three mutually exclusive
diminished 7th chords generate 12-note collections in Wozzeck. Other
examples can be cited from Wozzeck and the Drei Stücke for
orchestra (1914–15).
Schoenberg
has explained the concept of a 12-note series as originating in the desire to
avoid excessive pitch-class repetition in atonality, citing in this connection
the tendency to avoid the octave in atonal compositions. Webern apparently
anticipated both Schoenberg and Berg in this respect, consistently avoiding
octave doublings as early as 1910 in his Zwei Lieder op.8. In a lecture
given in 1932 (published in 1960) he described his early intuitive approach to
the concept of the 12-note series:
About
1911 I wrote the Bagatelles for string quartet (op.9), all very short pieces,
lasting a couple of minutes – perhaps the shortest music so far. Here I had the
feeling, ‘When all 12 notes have gone by, the piece is over’. Much later I
discovered that all this was a part of the necessary development. In my
sketchbook I wrote out the chromatic scale and crossed off the individual
notes. Why? Because I had convinced myself, ‘This note has been there already’.
… In short, a rule of law emerged; until all 12 notes have occurred, none of
them may occur again. Were this ‘rule’ to be strictly applied the 12 pitch classes
would be continually reiterated in the same order within a movement, thus
forming a repeating series.
The
principle of non-repetition, however, is clearly not a sufficient explanation
of the serial concept. One of the characteristic features of a melodic theme in
tonal music is, after all, the order assigned to its pitches, but this feature
is inseparable from others – rhythm, contour, tonal functions – any and all of
which may be varied within certain limits without destroying the identity of
the theme. The interdependence and interaction of these elements are far more
ambiguous and problematical in atonality. The pitch-class content of a group of
notes may be exploited independently of its other components, and in one of
Schoenberg’s last pre-dodecaphonic works, the first of the Fünf
Klavierstücke op.23, the pitch-class order of the initial melodic line is
treated as an independent referential idea (ex.2). The melodic figure which begins the second piece of the
same opus serves as nothing less than an ordered set, though it is only one of
the sources of pitch-class relations. Both pieces were completed in July 1920.
A month later Schoenberg was at work on op.24 no.3, the Variation movement of
the Serenade. This is the earliest example of an entire movement exclusively
based on a totally ordered – though not yet 12-note – set. The 14-note series,
comprising 11 pitch classes of which three occur twice, is employed in all four
aspects, but there is no change in the initial transpositional level (as there
is in op.23 no.2). The earliest 12-note serial piece, the Präludium of
the Piano Suite op.25, was composed during the period 24–9 July 1921. The
series, sole source of pitch relations, is employed in all four aspects and at
two transpositions separated by the interval of a tritone. Since the tritone,
which is invariant in its pitch-class content under transposition by a tritone,
is significantly represented in the structure of the set, important invariants
are generated between the different set forms.
On
completion of the first movement of the Suite, Schoenberg took up the Serenade
again. The first movement, evidently composed in one day (27 September 1921),
is largely based on the concept of strict inversional complementation, though
in a non-serial context. Work on the Fünf Klavierstücke op.23 was
resumed on 6 February 1923 and completed in less than a fortnight. No.3, based
on a five-note set, is an extraordinarily complex study in the structural
implications of inversional complementation and invariant relations. The
concluding piece, however, Schoenberg’s second 12-note serial piece, seems
primitive and naive in its constant reiteration of the initial set form, as
compared with his first piece in the system, composed almost two years earlier.
The same may be said of the only 12-note serial piece of op.24, the Sonett,
composed a few weeks later. Meanwhile, between 19 February and 8 March,
Schoenberg composed the five remaining movements of the Piano Suite, basing all
of them on the same set and the same procedures as the first movement, and thus
asserting, for the first time, all the basic premises of his 12-note system.
Of
the remaining movements of op.24, completed in March and April 1923, only
portions of no.5, Tanzscene, are based on a 12-note set. The first 57
bars, dating from August 1920, make no use of anything that may be termed a
set, but on taking up this movement again on 30 March 1923 Schoenberg converted
the pitch-class content of the initial six-note motif into one of the
hexachords of a 12-note trope, supplied the missing hexachord to complete the
trope and used this as the basis of the newly composed contrasting sections of
the piece. (Schoenberg had experimented with a hexachordal trope in 1915, and
there is no reason to assume that he was influenced by Hauer’s theory.) A
tritone transposition of either hexachord of the trope of the Tanzscene
(ex.3) leaves the pitch-class content of the
hexachord unchanged, a property exploited by transpositional relations in the
work. The Tanzscene points forwards to one of Schoenberg’s late works,
the Ode to Napoleon (1942), which is also based on a trope
(self-transposable at T-no.4 and T-no.8), rather than a series.
Twelve-note
composition
4. 12-note composition.
It
is one thing to define a 12-note set and quite another to define 12-note
composition. A general definition cannot go beyond the assertion that all the
pitch-class relations of a given musical context are assumed to be referable to
a specific configuration of the 12 pitch classes, a configuration that is
understood to retain its identity regardless of its direction or transpositional
level. Problems arise with the definition of that context and with the
compositional representation of the rules of set structure.
With
regard to the first question, Schoenberg noted: ‘It does not seem right to me
to use more than one series [in a composition]’. Of the three Viennese masters,
only Webern, beginning with op.19, unambiguously observed this principle. It is
completely inconsistent with Berg’s practice, even within any single movement.
Schoenberg’s implied definition of a ‘series’ does not include cyclical
permutations as representations of a given series, but Berg made use of these
regularly. Almost every movement in which he can be said to employ some sort of
12-note method contains ‘free’, that is non-dodecaphonic, or at least non-serial,
episodes. And even the 12-note sections of such movements are often based on
two or more independent sets – independent in the sense that no form of one set
can be transformed into any form of another by transposition, inversion,
retrogression, cyclical permutation or any combination of these operations. The
first movement of the Lyrische Suite is based on not one but three sets.
All three, however, are representations of the same trope (ex.4). (In the notation of set forms in the examples that
follow, each accidental affects only the note it precedes.) The principal set
is a serial representation of this trope (ex.5). Another series is derived by reordering the hexachordal
content of ex.4 as a circle of 5ths (ex.6). Finally, the conjunct
version of the trope is itself employed compositionally, not only in the form
shown in ex.4, but also with various cyclical permutations of the hexachords,
as in ex.7. The last movement of the Lyrische
Suite simultaneously employs two different series throughout. In order to understand
their relationship one must include the characteristic contour assigned to each
series among its essential attributes. The initial series is partitioned into
two segments in terms of the registral distribution of its elements, as shown
by stemmed and unstemmed notes in ex.8, and these form a second
series (ex.9).
Schoenberg’s
own practice can be said to conform to his rule of the unique series only if
the term ‘series’ or ‘row’ is replaced by ‘set’ and if the latter is considered
only partially defined by the serial ordering. The first completely 12-note
work, the Piano Suite, op.25, employs a series that is partitioned into three
four-note segments, and these are employed simultaneously as well as
successively. Thus the set cannot be properly defined exclusively in terms of
its total serial ordering. A single referential ordering can be deduced in this
and in most of Schoenberg’s other 12-note compositions, but this ordering tends
to be secondary to another attribute of Schoenberg’s sets: their segmental
pitch-class content. The Third String Quartet and the String Trio are
exceptional in that each employs several distinct series. If the set of the
former, however, is defined as consisting of an invariably ordered five-note
segment and a variably ordered seven-note segment, and that of the latter as
consisting of two variably ordered six-note segments, each work may be said to
be based upon a single set.
Even
where a single unambiguous pre-compositional serial ordering of the set is
assumed, the moment the series is used compositionally there are inevitable
ambiguities. The presence of another structural attribute, in addition to that
of serial order, is almost always implied: the partitioning of the series into
segments. It has been shown (exx.4–6) how a set segmented into two unordered
hexachords provides the basis for the association of three independent derived
sets. In general, segmentation is used as the basis for the association of set
forms chosen from the 48 members of the complex generated by a single series.
The first movement of Webern’s Second Cantata op.31, for example, employs the
following forms (ex.10) and their respective
retrograde versions (R0 and RI0). (The level of a set
form is indicated by a subscript integer; that assigned to any P or I form of
an ordered set is the same as its initial pitch-class number, and that assigned
to an R or RI form will be the same as its final pitch-class number; in the
rest of this article pitch-class numbers 0 to 11 represent the successive
elements of an ascending chromatic scale beginning on C.) Each hexachord holds
five pitch classes in common with the given inversionally complementary
hexachord, as shown in the example. Were P to be paired with any other
transposition of I, there would be less than five elements in common between
corresponding hexachords. The manner in which the hexachords are
compositionally stated supports the assumption that the association of
inversionally complementary segments is motivated by their maximum invariance
of content.
The
series on which the 12-note sections of the third movement of Berg’s Lyrysche
Suite are based begins with a four-note segment whose reordered content
recurs in transposed or inverted forms within the series. The set forms given
in ex.11 convert each of the reordered segments
into the pitch-class collection of the initial four-note segment of P10.
(In the composition P8, I3 and P5 are
cyclically permuted to commence or conclude with the invariant segment.)
The
series of Webern’s Concerto op.24 is segmented into subsets which are
themselves forms of a three-note series. P11 and RI0, the
first two set-form statements, present subsets of identical pitch-class content
in the same relative positions within each set form (ex.12).
The
series of Webern’s Symphony op.21 consists of two hexachords related as p0
to r6 (ex.13). Thus each P form of the set
is equivalent to the tritone transposition of its retrograde, and the
corresponding relation will hold between I and RI forms. For this, as for any
‘symmetrical series’ (any series comprising two hexachords related as pn
to rn+6 or pn to rin+(2x+1),
there are 24, rather than 48, non-equivalent set forms.
From
1928 Schoenberg systematically employed hexachordal segmentation as a basis for
the association of set forms. In the Fourth String Quartet, for example, P2
and I7, or any other pair of equivalently related set forms, may be
combined so that corresponding hexachords (vertically aligned for illustrative
purposes inex.14) produce all 12 pitch classes. Set-form
association based on such aggregates of the 12 pitch classes, known as
combinatoriality (seeSet), is the governing structural principle
in Schoenberg’s 12-note music. The P/I combinatoriality illustrated in ex.14 is
only possible where the two hexachords of the series are inversionally
complementary in content. If the series of ex.14 is rewritten as a hexachordal
trope (ex.15), it is clear that P2, I7
and their retrograde forms R2 and RI7, the four primary
set forms of the work, are all members of the same trope.
What
Babbitt has called an ‘all-combinatorial’ set permits P/RI as well as P/I
combinatoriality. Although Webern did not observe Schoenberg’s principle that
combined set forms must produce 12-note aggregates, the series of his Concerto
is an example of an all-combinatorial set. If P11 of ex.12 is
considered as a hexachordal series, it is evident that it may be paired
combinatorially with I4, RI2, P5 or R11.
(The last is a trivial instance, since any series will, by definition, form
12-note aggregates with its own retrograde.) Since transpositions of each of
these set forms by the addition of T-no.4 merely reorders the hexachordal
content, P11 may also be paired with I8 or I0,
RI6 or RI10, P9 or P1 and R3
or R7.
Where
corresponding hexachords of a pair of set forms are mutually exclusive, as in
ex.14, the contents of non-corresponding hexachords will, of course, be
identical. In the music of Berg set forms are associated through this
invariance of segmental content. The principal pair of P and I set forms of the
basic series of Lulu is shown in ex.16. Transpositions at the
tritone will interchange the contents of the two hexachords. Progression among set
forms, including members of independent complexes of set forms, may be referred
principally to degrees of invariance of segmental content. Schoen’s series and
Alwa’s series are associated through the set forms shown in ex.17, which are trichordally as well as hexachordally invariant
with each other.
The
principal set forms of Alwa’s series, P4 and P9, are
maximally invariant (five elements in common) with the basic series at P0
(or I9), as illustrated in ex.18.
Where
a set is strongly characterized by segmental content as well as serial ordering
in the music of Schoenberg and Berg, linearly stated set segments – functioning
as ‘broken chords’ in a sense – often depart from the assumed ordering. In
Schoenberg’s Klavierstück op.33a, for example, though one can
deduce an ordering for the set, segmental content is a far more significant
property, and in the Ode to Napoleon it is the only defining property of
the set. The numerous sets found in Berg’s Lulu include series, tropes
based on various types of partition, ‘serial tropes’ (i.e. sets whose segments
are subjected, independently of the set as a whole, to such serial operations
as will not revise their respective contents) and ‘harmonic tropes’ (i.e. sets
partitioned into segments of an essentially chordal, rather than linear,
character).
In
Schoenberg’s 12-note works the combinatorial relation between P and I or R and
RI set forms is preserved by maintaining a constant difference in the
respective transpositional levels of the associated set forms. In op.33a,
for example, the succession of T-nos. is as follows: P/R 10 0 5 10 I/RI 3 5 10
3 Webern sometimes used the opposite procedure: the maintenance of a constant
sum between T-nos. of associated set forms. In the first movement of the
Quartet op.22, for example, inversionally complementary set forms are paired as
follows: P1 P7 P1 P10 P11
R0 P0 R1 P7 P7 R1
I11 I5 I11 I2 I1 RI0
I0 RI11 I5 I11 RI11 The
sum of T-nos., 0, mod 12, remains the same throughout. The principal set forms
of the second movement are P6, I6, R6 and RI6,
and so here too the sum of complementary T-nos. is 0, mod 12. The significance
of this procedure is that it preserves exactly the same pairs of inversionally
complementary pitch classes, whatever the respective transpositional levels of
the set forms. These complementary relations are shown in ex.19 for this case. In Schoenberg’s op.33a, on the other
hand, the complement of any given pitch class is different for each pair of
T-nos.
Twelve-note
composition
5. Pre-compositional structures.
The
concepts of segmentation and inversional complementation permit a generalized
description of pitch relations for 12-note music. The special properties that
derive from the hexachordal organization of a given series are shared with all
other serial representations of the same trope, and a series based on any other
type of segmental organization may be similarly regarded as a special instance
of a single pre-compositional structure. The series of Schoenberg’s Fourth
String Quartet (ex.14) is one of 518,400 (= (6!) (6!)) series that may be
derived by reordering the elements of each hexachord of the trope in ex.15. All
of these series share the structural functions that depend upon hexachordal
content alone. Whatever the series, the content of each hexachord of any given
P form will be duplicated in the content of the non-corresponding hexachord of
one I form. The same property characterizes every series that may be derived
from any of 12 other tropes. Thus 13 of the 44 tropes may be grouped together
as a family, and almost every one of Schoenberg’s 12-note works is based on a
serial representation of one of these. (The 13 tropes of the ‘Schoenbergian’
type are numbered 14 to 26 under ‘Six-Note Collections’ in the appendix to
Perle, 1962, rev.6/1991.)
The
hexachordal trope illustrated in ex.4 is represented by the basic series of Lulu
(ex.16), as well as by the three sets on which the first movement of the Lyrische
Suite is based (exx.4–6). This trope is one of six (numbered 1 to 6 in
Perle) characterized by the following properties: the relative pitch content of
both hexachords is the same, that is each can be converted into the other by
transposition; each hexachord is invariant under inversion, and therefore also
an inversion as to unordered content of the other hexachord (since the two are
transpositionally equivalent as to content). The all-combinatorial series are
representations of these six tropes. Transposition by a perfect 5th above or
below of any series derived from ex.4 preserves maximum invariance of
hexachordal content between identical aspects of the series (five common
elements), and in Lulu Berg made significant use of the resulting
hierarchy of harmonic areas. The series on which the third of Schoenberg’s Drei
Lieder op.48 is based preserves total invariance of content either between
corresponding or between non-corresponding hexachords for all set forms of the
complex, since it is a representation of the trope whose hexachords are the two
non-equivalent whole-tone collections. All transpositions of this trope are equivalent,
since transpositions do not alter the segmental content of the trope but merely
interchange the two segments.
Any
given 12-note set form is one of 479,001,600 (=12!) permutations of the
chromatic scale. Symmetrically related dyads are generated when equivalent
permutations of two contrary chromatic scales are aligned. Ex.20 shows two such, intersecting at C and F
(intersections will always occur
at two points separated by the interval of a tritone; the ascending scale will
be termed a ‘P cycle’ and the descending scale an ‘I cycle’). The dual axis of
symmetry of each dyad will then be C and F, which is to say that the members of each dyad will
be equidistant in opposite directions from C, and likewise from F. The axis of symmetry may be conveniently
represented by the sum of pitch-class numbers that represent the members of any
dyad. (In this instance that sum is 0, mod 12.) The interval classes generated
by the aligned cycles may consistently be represented by the number, mod 12,
obtained by subtracting the pitch-class number of each element of the I cycle
from that of its complementary element in the P cycle. Inversionally
complementary set forms whose T-nos. are 0 and 0, 1 and 11, 2 and 10, 3 and 9,
4 and 8, 5 and 7, and 6 and 6 will permute the vertical intervals of ex.20, but
the pair of pitch classes that constitute a given interval class remains the
same, since the sum in each instance is 0, mod 12. All of these T-no. pairs
except 4 and 8 are represented in Webern’s Quartet op.22 and are the basis of
its tonal structure and formal design. The opening and closing sections of the
first movement of the Symphony op.21, and the middle movement of the Piano
Variations op.27, are limited to paired complementary set forms of sum 6.
Any
realignment that displaces one of the two cycles of ex.20 by an even number of
semitones transposes the same collection of intervals. The association of
inversionally complementary hexachords to form 12-note aggregates, as in most
of Schoenberg’s 12-note compositions, is only possible where the sum of T-nos.
is odd, since otherwise there must be pitch-class duplication at two points. If
the diverging chromatic cycles are aligned so that the sum of complementary
pitch classes is odd, the interval classes that contain an odd number of
semitones are produced and pitch-class duplication is eliminated. The principal
pair of set forms of Schoenberg’s Fourth String Quartet (ex.14) generates the
dyadic relations shown in ex.21.
The
symmetrical pitch relations produced by the one-to-one alignment of
complementary chromatic cycles should not be confused with thematic inversion
in earlier music. The harmonic and tonal context within which themes and motifs
are sometimes ‘inverted’ in diatonic music exists before and apart from the
operation of inversion. Thematic inversion in a diatonic context is not literal
since complementation is not measured in terms of a single unit, the semitone,
as it is in the 12-note system, and the functional properties of the diatonic
system are not reflected by inversion, which would require that the function of
the root of a major triad be assignable to the 5th of a minor triad and that
the degree functions of a major scale be assignable to the symmetrically
complementary degrees of a retrograde Phrygian scale.
Strict
inversional complementation has been significantly employed in non-diatonic
music apart from the 12-note system: the first-movement exposition of
Schoenberg’s Serenade op.24 is exceptional in that it is as rigorously and
consistently employed there as it might be in a 12-note serial work. The
periodic formal design is entirely dependent upon inversional procedures and
all pitch relations are based on a single sum of complementation. Perhaps the
most sophisticated use of inversional complementation is to be found in some of
the works of Bartók, particularly the Fourth String Quartet. Here there is
modulation from one sum of complementation to another via pivotal elements (primary
among these is the tetrachordal structure illustrated in ex.22, the inversional relations of which may be interpreted in
two different ways); differentiation of harmonic areas through the simultaneous
or successive juxtaposition of dyad collections representing different sums;
differentiation through the juxtaposition of progressions which preserve
constant intervallic differences with progressions which preserve constant
sums; the combining of small-scale, non-symmetrical progressions into
large-scale symmetrical progressions; the use of axes of symmetry as tone
centres and so on. It is obviously necessary to discuss such a work in the same
context as the 12-note music of Schoenberg, Berg and Webern, although Bartók
did not use 12-note series. The basic cell of the quartet (ex.22) is also the
basic cell of Berg’s 12-note opera Lulu and generates the set – a
tetrachordal trope – with which that work begins (ex.23). The role assigned to this trope in the opera is explained
by its special character: it may be inverted at any odd T-no. or transposed by
any even T-no. without change to its tetrachordal pitch-class content. The
intervallic properties (deriving from the presence of the tritone) that explain
the function of this tetrachord in Bartók’s Fourth String Quartet are also
those that explain its function in Berg’s opera.
Wherever
there is inversional symmetry based on the chromatic scale, one of two ‘modes’
is expressed (the one producing only odd, the other only even intervals), and
that ‘mode’ occurs in one of its six ‘keys’ (the six sums that represent the
different transpositional levels of the collection of symmetrically related
intervals). All intervals and sums are given in the two arrays shown in ex.24; Even array and ex.24; Odd array. Dyads of the same sum class
are in the same column and dyads of the same interval class in the same row.
Each row of intervals is continued in the equivalent row (1/11 in 11/1 etc.);
each column of sums is continued in the same column read in the opposite
direction from the point of intersection (intervals 0 and 12, or 1 and 11). Any
symmetrical collection, regardless of the number of its elements, can be
analysed into dyads that will lie in the same column. Symmetrical tetrachords
of two dyads of sum 9, which play a significant role in Bartók’s Fourth String
Quartet, will be formed by any pair of dyads in the 9 column, as for example
C–A and E–F, with which the work opens. The same tetrachord may be read as the
interval pair C–E and A–F in the 4 or 8 row, or as the interval pair C–F and
E–A in the 5 or 7 row. The compositional process in the first movement is
largely an unfolding of the implications of such reinterpretations of
tetrachordal collections. The principal series of the first movement of the Lyrische
Suite (ex.5) is not only a representation, as noted above, of a particular
hexachordal trope, but also a representation of the collection of dyads of sum
9. These are compositionally articulated in the initial thematic statement of
the series (ex.25) and again in a cyclic permutation of
that statement (ex.26).
The
opening section of Webern’s Symphony demonstrates the simultaneous
representation of different sums of complementation. The contrapuntally
associated set forms, P9 and I9, I5 and P1,
I0 and P6, and P2 and I4, are sum 6
complements. Although the series is of the Pn=Rn+6
symmetrically ordered type, which implies hexachordal segmentation (ex.13), it
may also be segmented into inversionally symmetrical tetrachords that are
invariant as to content between Pn and In+3. The eight
set forms unfold a double inverted canon in which each voice consists of
successive set forms that conform to this relation: canon 1: P9 I0
I9 P6 canon 2: I5 P2 P1
I4 This aspect of the set structure is clearly articulated through
the assignment of an invariant timbre for each invariant tetrachordal
collection in canon 1 (ex.27). (Complementary pitch
classes are beamed together in the example.) The complementary pitch classes
are adjacent to each other in the outer tetrachords. The middle tetrachord is
identical with that illustrated in ex.22. Unlike the others it has two sums of
complementation, and the alternative sum maintains invariant two-note segments
(ex.28).
Twelve-note composition
6. 12-note ‘tonality’.
Exx.20–21
show how any alignment of two contrary chromatic scales produces all the odd or
all the even intervals around a single odd or even axis of symmetry. An
equivalent result is produced by alignments of diverging cycles of 5ths. The
sum 9 dyads within each hexachord of the basic set of the Lyrische Suite
are explicitly generated in this way (ex.29). Where the second hexachord
of the principal set (ex.5) is directly followed by its first hexachord – as
happens occasionally in the Lyrische Suite and regularly in Berg’s song Schliesse
mir die Augen beide, which is based on the same set – the alternate notes
of the set completely unfold the diverging cycles. A continuation of the cycles
produces the retrograde form of the same set and pitch-class repetition at the
point where the cycles intersect (ex.30). Berg’s series, when read as
in ex.30, displays a significant property that is not found in the general
Schoenbergian set: the elements immediately adjacent to any member of the set
(these will be termed the ‘neighbour notes’ of the given axis note) always form
the same interval, a perfect 5th in this instance, with the respective
transpositional levels of this interval inversionally complementary to the
respective transpositional levels of its axis note (ex.31).
The
vertically stated pitch-class collections in ex.31 comprise all the three-note
segments contained in the set form illustrated in ex.30. The verticalization of
set segments is the fundamental means of forming chords in 12-note music and
derives from pre-12-note atonal practice, where referential pitch-class
collections often occur as simultaneities as well as linear details. However,
the collection of chords generated by the verticalization of segments of the
general Schoenbergian set provides no basis for a total, systematic control of
the harmonic dimension, as this procedure does when applied to cycles and to
sets derived from cycles as in ex.30. Suppose, for example, that the
verticalized segments of ex.31 were combined with corresponding verticalized
segments of an inverted form of the same set (ex.32). The combined segments will yield a series of
symmetrically related chords (ex.33). If the axis-note dyads are
transposed, the combined neighbour-note dyads will be equivalently transposed
in the opposite direction. Thus if any given axis-note dyad occurs at all of
its transpositional levels, so will its neighbour-note chord at the
complementary transpositional levels (ex.34). Wherever the axis-note dyad
is identical in pitch-class content with a dyad segment of one of the two sets,
the neighbour-note chord will be identical in pitch-class content with a
tetrachord segment of the other set (see asterisked items in ex.34).
Any
interval may be used to generate cyclic sets analogous to ex.30. Ex.35 shows diverging cycles of minor 3rds, intersecting so as to
yield adjacencies of sum 9 as in ex.30. (The set must be partitioned into two
subsets in order to produce the complete collection of such dyads.) The
verticalization procedure may be applied to such sets to generate series of pitch-class
collections analogous to those illustrated above, with the new cyclic interval
replacing the cyclic interval of the perfect 5th in each such collection. There
is nothing to prevent the association of set forms, generated by different
cyclic intervals, to produce neighbour-note chords based on two different
intervals. Through the use of such sets all possible verticalizations may be
systematically formulated.
The
cyclic set is an ordered unfolding of a complete collection of symmetrically
related dyads such as is illustrated for dyadic sums 0 and 9 respectively in
exx.20–21. The pair of diverging semitonal scales that generates the sum-9
complementary relations of Schoenberg’s Fourth String Quartet is equally
relevant to the first movement of Berg’s Lyrische Suite and to the first
movement of Bartók’s Fourth String Quartet. In a certain sense they can be said
to be in the same ‘key’. The replacement of a diatonic scale of unequal steps
and functionally differentiated notes by a semitonal scale of equal steps and
functionally undifferentiated notes completely transforms the meaning of
inversion, which becomes a pre-compositional means of symmetrically
partitioning the tone material, rather than a means of composing in a harmonic
context that is already given. Symmetry, as represented by both the cyclic
divisions of the octave and the inversional complementation of pitch classes,
is thus an inherent property of the 12-note scale. In the first movement of the
Lyrische Suite this property is mapped into the structure of the 12-note
series itself.
In Le
cru et le cuit Lévi-Strauss challenged Boulez’s defence of ‘serialism’ as a
new kind of musical thought which, ‘operating in accordance with a particular
methodology, creates the objects it needs and the form necessary for their
organization, each time it has occasion to express itself. … There is no longer
any preconceived scale or preconceived forms – that is, general structures into
which a particular variety of musical thought can be inserted’. Lévi-Strauss
insisted that there must be a ‘first level of articulation, which is as
indispensable in musical language as in any other, and which consists precisely
of general structures whose universality allows the encoding and decoding of
individual messages’. For Lévi-Strauss that ‘first level of articulation’ is
provided by ‘the hierarchical structure of the scale’, by which he meant the
diatonic scale and its triadic functional relations. Schoenberg, too, seems to
have assumed that a ‘first level of articulation’ was a prerequisite for a
musical language. In the article ‘Problems of Harmony’, after a tendentious and
ill-informed attempt to derive the 12-note scale from the overtone series, he
concluded that ‘should this proof be inadequate, it would be possible to find
another. For it is indisputable that we can join twelve notes with one another
and this can only follow from the already existing relations between the twelve
notes’. The symmetrical implications of the semitonal scale can serve as the
‘first level of articulation’ of a 12-note musical language, that is as the
source of ‘the already existing relations between the twelve notes’.
Twelve-note
composition
7. Extensions and developments of 12-note composition.
The
influence of Schoenberg’s 12-note system in Germany and Austria before World
War II was largely limited to his students and, in turn, to their students.
Krenek was perhaps the only widely known composer outside this circle to adopt
the system, on which he based his opera Karl V (1933). Under the regimes
of Hitler and Stalin a ban was placed on what was represented as an expression
of ‘Jewish Bolshevism’ on the one hand and ‘bourgeois decadence’ on the other,
and even where there was no political suppression, neo-classicism came more and
more to dominate composition from the late 1920s until the end of the war.
Immediately after the war, however, there was an astonishingly rapid and
widespread growth of interest in the work of Schoenberg, Berg and Webern, and
by the end of the 1950s there remained few composers in the USA or western
Europe who were not in some way influenced by the concepts of 12-note
composition.
Stravinsky’s
adoption of serial procedures was remarkable not so much because his position
in the musical world had been regarded as diametrically opposed to
Schoenberg’s, but rather because of the way in which he was able to integrate
these procedures into his own musical language. This is clearly evident in his In
memoriam Dylan Thomas (1954), one of the most strict of Stravinsky’s early
serial pieces. The work is based on a five-note set, E, E
, C, C, D. This use of a five-note set, under transposition
and the four transformations of the 12-note system, differs from Schoenbergian
serialism mainly in that every transformation does not necessarily contain
different permutations of the same collection of pitch classes. Characteristic
features of Stravinsky’s earlier music remain, but they appear in a new
context: the recurrent motivic statements of the set and emphasis on certain
pitch successions such as E–E to emphasize cadential points; the chromatic
filling-in of an interval, a major 3rd, with successions of only ascending or
descending semitones or minor 3rds (compare the passage in The Rite of
Spring at no.130, where an interval of a tritone is filled in by the alto
flute using only major 2nds, minor 3rds and major 3rds); and the rhythmic
alteration of similar pitch configurations in the strings at the opening of the
song and at nos.2, 4 and 8 (compare the oboe solo at the beginning of the
second movement of the Symphonie de psaumes).
In
his later works Stravinsky more thoroughly incorporated traditional 12-note
concepts while still achieving Stravinskian results. The set of Movements
(ex.36), for example, is all-combinatorial, but
it is not used to form 12-note aggregates. Instead Stravinsky concentrated on
the high degree of trichordal segmental redundancy within hexachords. The
trichords formed by elements in ordinal positions (0, 1, 2), (3, 4, 5) and (8,
9, 10) (where 0 denotes the ordinal position of the first note of the set) are
of transpositionally or inversionally equivalent content, as are the trichords
in ordinal positions (2, 3, 4), (6, 7, 8) and (9, 10, 11). This kind of redundancy
is particularly Stravinskian, and the motivic use of these trichords generates
a musical surface with precisely the kinds of intervallic repetition and
emphasis found in the earlier serial works, such as In memoriam.
In
later compositions Stravinsky independently reinvented a procedure of
hexachordal rotation and transposition that Krenek had employed in 1941–2 in Lamentatio
Jeremiae prophetae. An ‘array’ for Abraham and Isaac (1963), for
example, is derived by aligning hexachordal rotations of a single transposition
of a set: G G A C C A B D D E F F G A C C A G D D E F F B A C C A G G D E F F B D C C A G G A E F F B D D C A G G A C F F B D D E A G G A C C F B D D E F This array is used as a basis for lineal and
vertical relations. Stravinsky also constructed an array in which each
successive rotation is transposed so as to begin on the same pitch class as the
first. This provides a succession of simultaneities whose properties are
derived from the structure of the set but whose local characteristics appear
quite different from those of the set itself. In the following array this
procedure is applied to the first hexachord of the set: G G A C C A G A B C G F G A A F E F G G E D D F G D C D E F G F F G A B (Pitch-class repetitions in columns occur as a
function of interval duplication by pairs of pitch classes with the same
differences in ordinal positions within the original hexachord. Within each
hexachord the columns whose ordinal positions are complementary to 6 (with the
initial column as 0) are symmetrical inversions of each other as to pitch-class
content, with G as the axis of symmetry.
Many
postwar developments involved an extension of the serial concept to control and
interrelate the various dimensions of a composition – rhythm, pitch, timbre,
dynamics, articulation and so on – in a more precise and highly determinate
way. Such processes were foreshadowed in the music of Schoenberg, Berg and
Webern. In Webern’s Symphony op.21 and Concerto op.24, rhythm, register and
orchestration are organized in such a way as to project and clarify
transformations and properties of pitch and pitch class. The set of the
Concerto (ex.12) has the property that certain transformations will
individually permute each of its trichords. In the opening bar each trichord is
assigned a different instrument and rhythm, and in bar 4 the piano’s statement
of RI0 – a transformation which retrogrades each of the individual
trichords and retains their original order within the set – retrogrades the
rhythm and places pitch classes in the same registers as in bars 1–3, thus
projecting and interpreting these invariant relations (ex.37). In the opening of the Symphony pitches
are assigned symmetrically around a. In the canonic pairing of P and I
sets at the beginning of the composition, P9 and I9, and
P1 and I5, corresponding pitches in these inversionally
related, matched sets are thus symmetrically deployed around a.
The
earliest instance of a strictly serially derived duration set is in the third
movement of Berg’s Lyrische Suite. The set consists of two subsets (ex.38 shows the relative points of attack) derived by
partitioning the pitch set in terms of the registral distribution of its
elements (ex.39).
It
was not until after the war that the first attempts were made to generalize the
application of serial structure by systematically transferring the attributes
of a pitch set to the non-pitch elements. The first composer systematically to
apply serial procedures to rhythm was Babbitt, in the Three Compositions for
Piano (1947). Serial operations have also been applied to dynamics,
instrumentation and register (see Serialism).
Twelve-note
composition
8. Conclusion.
Many
difficult and troublesome musical questions are raised by such extensions of
the serial concept. It is reasonable to question the sense in which dynamic
differentiation, for example, can be usefully considered outside the context of
a musical composition, and even within that context whether loudness can
function as a variable, independent of timbre, texture, rhythm, tempo or
register, to the same degree as pitch, pitch class or, possibly, rhythm. There
also arises the larger question of the extent to which it is useful, or even
meaningful, to consider whether a piece is serial without saying ‘how’ it is
serial. When, for example, every pitch of a composition has been explained in
terms of its position in some 12-note set, the problem remains of how this
helps understanding of the conjunction of pitch, rhythm, dynamics, timbre and
other elements of that piece. Even when all other dimensions have been derived
in terms of unique predetermined orders, one may question that this necessarily
shows how the result is heard, particularly when most musical dimensions are so
highly contextual and subject to complex interaction. In short, the depth of
musical relations within a composition may be so rich that the claim which that
composition has to being a unique instance of musical thought may only in part
be ascribed to the varieties of interrelations between serial aspects, and even
the view of that part in these terms may be seriously questioned by the result
which is the product of these interrelations. It may be more fruitful to
consider serial ordering as a relatively abstract idea which has excited the
imaginations of many composers and helped them to reshape and rethink their
compositional habits and predilections in new and interesting ways. Perhaps the
most important influence of Schoenberg’s method is not the 12-note idea in
itself, but with it the individual concepts of permutation, inversional
symmetry and complementation, invariance under transformation, aggregate
construction, closed systems, properties of adjacency as compositional
determinants, transformations of musical surfaces through predefined
operations, and so on. Each of these ideas by itself, or in conjunction with
many others, is emphasized with varying degrees of sharpness in the music of
such different composers as Bartók, Stravinsky, Schoenberg, Berg, Webern and
Varèse. In this sense the development of the serial idea may be viewed not as a
radical break with the past, but as a particularly brilliant coordination of
musical ideas which had developed in the course of recent history. The
symmetrical divisions of the octave often found in Liszt and Wagner, for
example, are not momentary aberrations in tonal music which led to its ultimate
destruction, but, rather, important musical ideas which, in defying integration
into a given concept of a musical language, challenged the boundaries of that
language.
Twelve-note
composition
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