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.
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.
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 (seeAtonality). 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 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.
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 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 (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.