In twelve-tone technique , combinatoriality stands for a property of twelve-tone rows, with each section of a row being supplemented with the corresponding section of a series transformation to the chromatic total. The principle was first systematically described by Milton Babbitt and established as a constructive principle of a serial set of notes.
Parallel hexachords make it possible to combine several series variants at the same time without the result of repeated notes and octave parallels. This also makes it easier to organize the harmony. Arnold Schönberg already describes the hexachord complementarity in his essay Composition with Twelve Tones , which goes back to two lectures given in Princeton and Washington in 1934. Schönberg traces the origin of this design idea back to the adaptation of the first series of drafts, which was necessary for compositional reasons:
“The inversion of the first six notes of the antecedent on the fifth lower should not produce a repetition of one of these six notes, but rather result in the previously unknown six notes of the chromatic scale. This has the advantage that you can accompany melody parts from the first six tones with harmonies from the second six tones without duplicating "
Basically, the hexachords of all twelve-tone rows behave in some way complementary to one another: in Schönberg's Wind Quintet op.26, for example, the first half of the row in its original form is complementary to the first half of the inverted row, transposed down a small second. However, this initial situation is not always musically favorable for combining the halves of the row. In the Concerto for Violin, Op. 36 , Schönberg's preferred hexachord complementarity is at a fifth spacing.
The development of the complementary series can be easily understood in bars 8 to 14 of the violin concerto. The first movement begins with the introduction of a motto that is based on a small second and is presented in the violin and orchestra using a selection of certain series notes - first with the basic series starting from a , then starting with the inversion of e , i.e. transposed by a fifth up. From bar 8 the violin plays itself melodically free and runs through the original shape of the basic row up to bar 11, again starting from a . The orchestra accompanies the first 6 notes of the violin with the first 6 notes of the inversion starting from e . The continuation of the melody with the row tones 7–12 of the basic row is supplemented with the corresponding tones of the inversion in the orchestra. In bars 11 to 14 the constellation is repeated in reverse: the solo violin continues the melody with one complete cycle of reversal, while the orchestra accompanies the original figure with the complementary hexachords.
As trichordkomplementär refers to the rows, wherein the chromatic Total is achieved by regular transformation of the first three series of tones. The basic series of Webern's Concerto, Op. 24, for example, begins with the tone sequence hbd. The second trichord is created by reversing the crab of the first from the tone it emanates, the third is its crab starting from the G sharp. The last trichord is the reverse of the first starting from c.
- ^ Andrew Mead : An Introduction to the Music of Milton Babbitt. Princeton, Princeton University Press 1994 pp. 20-38
- ^ Arnold Schönberg: Composition with twelve notes , in: ders .: Style and thought. Essays on music. Edited by Ivan Vojtech. Frankfurt am Main 1992, p. 75
- ^ Arnold Schönberg, letter to Josef Rufer, April 8, 1950, digitized
- ^ Arnold Whittall: The Cambridge Introduction to Serialism. Cambridge Introductions to Music . New York: Cambridge University Press 2008, p.97