page_title: Combinatoriality
section_title: Hexachordal combinatoriality
hierarchical_section_title: Combinatoriality / Hexachordal combinatoriality
caption_attribution_description: English: Schoenberg - Variations for Orchestra op. 31 tone row.
context_page_description: In music using the twelve tone technique, combinatoriality is a quality shared by twelve-tone tone rows whereby each section of a row and a proportionate number of its transformations combine to form aggregates. Much as the pitches of an aggregate created by a tone row do not need to occur simultaneously, the pitches of a combinatorially created aggregate need not occur simultaneously. Arnold Schoenberg, creator of the twelve-tone technique, often combined P-0/I-5 to create "two aggregates, between the first hexachords of each, and the second hexachords of each, respectively."
  Combinatoriality is a side effect of derived rows, where the initial segment or set may be combined with its transformations to create an entire row. "Derivation refers to a process whereby, for instance, the initial trichord of a row can be used to arrive at a new, 'derived' row by employing the standard twelve-tone operations of transposition, inversion, retrograde, and retrograde-inversion."
context_section_description: A 12-tone row has hexachordal combinatoriality with another 12-tone row if their respective first (as well as second, because a 12-tone row itself forms an aggregate by definition) hexachords form an aggregate.
  There are four main types of combinatoriality. A hexachord may be:
  Prime combinatorial (transposition)
  Retrograde combinatorial (retrograde)
  Inversional combinatorial (inversion)
  Retrograde-inversional combinatorial (retrograde-inversion)
  and thus:
  Semi-combinatorial (by one of the above)
  All-combinatorial (by all)
  Prime (transpositional) combinatoriality of a hexachord refers to the property of a hexachord whereby it forms an aggregate with one or more of its transpositions. Alternatively, transpositional combinatoriality is the lack of shared pitch classes between a hexachord and one or more of its transpositions. For example, 0 2 4 6 8 t, and its transposition up one semitone (+1): 1 3 5 7 9 e, have no notes in common.
  Retrograde hexachordal combinatoriality is considered trivial, since any row has retrograde hexachordal combinatoriality with itself (all tone rows have retrograde combinatoriality).
  Inversional combinatoriality is a relationship between two rows, a principal row and its inversion. The principal row's first half, or six notes, are the inversion's last six notes, though not necessarily in the same order. Thus, the first half of each row is the other's complement. The same conclusion applies to each row's second half as well. When combined, these rows still maintain a fully chromatic feeling and don't tend to reinforce certain pitches as tonal centers as might happen with freely combined rows. For example, the row from Schoenberg's Moses und Aron, above contains: 0 1 4 5 6 7, this inverts to: 0 e 8 7 6 5, add three = 2 3 8 9 t e.
  01  4567     : 1st hexachord P0/2nd hexachord I3
    23    89te : 2nd hexachord P0/1st hexachord I3
  complete chromatic scale
  Retrograde-inversional combinatoriality is a lack of shared pitches between the hexachords of a row and its retrograde-inversion.
  Babbitt also described the semi-combinatorial row and the all-combinatorial row, the latter being a row which is combinatorial with any of its derivations and their transpositions.
  Semi-combinatorial sets are sets whose hexachords are capable of forming an aggregate with one of its basic transformations (R, I, RI) transposed. There are thirteen hexachords that are semi-combinatorial by inversion only. 
  (0)  0 1 2 3 4 6 // e t 9 8 7 5
  (1)  0 1 2 3 5 7 // e t 9 8 6 4
  (2)  0 1 2 3 6 7 // e t 9 8 5 4
  (3)  0 1 2 4 5 8 // e t 9 7 6 3
  (4)  0 1 2 4 6 8 // e t 9 7 5 3
  (5)  0 1 2 5 7 8 // e t 9 6 4 3
  (6)  0 1 3 4 6 9 // e t 8 7 5 2
  (7)  0 1 3 5 7 9 // e t 8 6 4 2
  (8)  0 1 3 5 8 9 // 7 6 4 2 e t
  (9)  0 1 3 6 7 9 // e t 8 5 4 2
  (10) 0 1 4 5 6 8 // 3 2 e t 9 7
  (11) 0 2 3 4 6 8 // 1 e t 9 7 5
  (12) 0 2 3 5 7 9 // 1 e t 8 6 4
  Any hexachord which contains a zero in its interval vector possesses transpositional combinatoriality (in other words: to achieve combinatoriality a hexachord cannot be transposed by an interval equaling a note it contains). For example, there is one hexachord which is combinatorial by transposition (T6):
  (0) 0 1 3 4 5 8 // 6 7 9 t e 2
  Neither hexachord contains tritones.
  All-combinatorial sets are sets whose hexachords are capable of forming an aggregate with any of its basic transformations transposed.
  There are six source sets, or basic hexachordally all-combinatorial sets, each hexachord of which may be reordered within itself:
  (A)  0 1 2 3 4 5 // 6 7 8 9 t e
  (B)  0 2 3 4 5 7 // 6 8 9 t e 1
  (C)  0 2 4 5 7 9 // 6 8 t e 1 3
  (D)  0 1 2 6 7 8 // 3 4 5 9 t e
  (E)  0 1 4 5 8 9 // 2 3 6 7 t e
  (F)  0 2 4 6 8 t // 1 3 5 7 9 e
  Note: t = 10, e = 11.
  Because the first three sets (A, B, and C) each satisfy all four criteria for just one transpositional value, set D satisfies them for two transpositional values, E for three values, and F, for six transpositions, Babbitt designates these four groups as "first-order", "second-order", "third-order", and "sixth-order" all-combinatorial hexachords, respectively. Notice that the first 
caption: 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."(Leeuw 2005, 154–55) About this soundPlay (help·info)
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