Database on tone rows and tropes -- Combinatorial rows

Combinatorial rows

We consider only hexachordal combinatoriality. Let A⊂ Z₁₂ be a hexachord and A'=Z₁₂ \ A its complement. Then A is called inversional combinatorial, if there exists some k∈ {1,…,11} so that A'=T^kI(A). (If so the integer k can be chosen from {1,3,5,7,9,11}.) It is called transpositional combinatorial, if there exists some k∈ {1,…,11} so that A'=T^k(A). It is called retrograde combinatorial, if there exists some k∈ {0,1,…,11} so that A=T^k(A). (All hexachords are retrograde combinatorial since A=T⁰(A) always holds true. However, there exist some hexachords which are retrograde combinatorial with some k ≠ 0. They are usually explicitly mentioned as retrograde combinatorial.) It is called inversional retrograde combinatorial, if there exists some k∈ {0,…,11} so that A=T^kI(A). (If so the integer k can be chosen from {0,2,4,6,8,10}.) It is called all combinatorial, if it has all these four properties.

Let f: {1,…,12}→ Z₁₂ be a tone row and consider the pair of hexachords {A,A'} where A={f(1),…,f(6)} and A'={f(7),…,f(12)}. Then f is called inversional combinatorial, respectively transpositional combinatorial, respectively retrograde combinatorial, respectively inversional retrograde combinatorial, respectively all combinatorial, if A has the desired properties.

Consequently, combinatoriality of a tone row f is described by the combinatoriality of its first hexachord. In other words, it is determined by the number of the D₁₂-orbit of its first trope {A,A'}.

Database on tone rows and tropes
harald.fripertinger "at" uni-graz.at
January 2, 2019