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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 12note set.
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), retrogradeinversion (RI) – may be stated at 12 transpositional levels, there will be 48 set
forms in the complex generated by a single series.
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: he consistently
avoided 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 as follows:
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 pitchclass 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.
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 nonserial, 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 12note 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 String Quartet, however, is
defined as consisting of an invariably ordered five-note segment and a variably
ordered seven-note segment, and that of the String Trio 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 in
ex.14) produce all 12 pitch classes. Set-form association based on such
aggregates of the 12 pitch classes, known as combinatoriality (see SET), 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 I 4, 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 Tnos., 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.
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 (see 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 (see ex.16), as well as by the three sets on which the first movement of the
Lyrische Suite is based (see 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 (see 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 nondiatonic 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 (see 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=Rnplus6 symmetrically ordered type,
which implies hexachordal segmentation (see ex.13), it may also be segmented
into inversionally symmetrical tetrachords that are invariant as to content
between Pn and Inplus3. 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, above. Unlike the others it has two sums of complementation,
and the alternative sum maintains invariant two-note segments (ex.28).
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 (see 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 pitchclass 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 starred 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 sets such as these 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’.
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 DEF FBD CC AGG A EF FBD DC AGG A CF FBD DE
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).
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.