Combinatoriality

Most generally complementation is the separation of pitch-class collections into two complementary sets, one containing the pitch classes not in the other.

[5] The term, "'combinatorial' appears to have been first applied to twelve-tone music by Milton Babbitt" in 1950,[7] when he published a review of René Leibowitz's books Schoenberg et son école and Qu’est-ce que la musique de douze sons?

Occasionally a hexachord may be combined with an inverted or transposed version of itself in a special case which will then result in the aggregate (complete set of 12 chromatic pitches).

If you continued the rest of the transposed, inverted row (I9) and superimposed original hexachord 2, you would again have the full complement of 12 chromatic pitches.

Furthermore, there is little proof suggesting that Hauer had extensive knowledge about the inversional properties of the tropes earlier than 1942 at least.

[17] The earliest records on combinatorial relations of hexachords, however, can be found amongst the theoretical writings of the Austrian composer and music theorist Othmar Steinbauer.

[a] He undertook elaborate studies on the trope system in the early 1930s which are documented in an unpublished typescript Klang- und Meloslehre (1932).

Steinbauer's materials dated between 1932 and 1934 contain comprehensive data on combinatorial trichords, tetrachords and hexachords including semi-combinatorial and all-combinatorial sets.

[18] A compilation of Steinbauer's morphological material has in parts become publicly available in 1960 with his script Lehrbuch der Klangreihenkomposition (author's edition) and was reprinted in 2001.

"Trichordal combinatoriality involves the simultaneous presentation of four rows in parcels of three pcs.

Combinatorial tone rows from Moses und Aron by Arnold Schoenberg pairing complementary hexachords from P-0/I-3 [ 6 ]
Combinatorial all-trichord hexachords from Elliott Carter 's Piano Concerto, mm. 59–60 [ 9 ]
Gruppen ' s main first-order all-combinatorial tone row, though this property is not exploited compositionally in that work. [ 10 ]
"Ode-to-Napoleon" hexachord [ 11 ] in prime form [ 12 ] One of Babbitt's six all-combinatorial hexachord "source sets". [ 12 ]
The principal forms, P1 and I6, of Schoenberg's Piano Piece , op. 33a, tone row feature hexachordal combinatoriality and contains three perfect fifths each, which is the relation between P1 and I6. [ 15 ]
In Schoenberg's Variations for Orchestra , Op. 31, tone row form P1's second half has the same notes, in a different order, as the first half of I10: "Thus it is possible to employ P1 and I10 simultaneously and in parallel motion without causing note doubling." [ 16 ]