The trope structure

Consider a tone row f: {1,…,12}→ Z₁₂, then the tropes of f are the six "pairs" of hexachords

τ₁={{f(1),f(2),f(3),f(4),f(5),f(6)},{f(7),f(8),f(9),f(10),f(11),f(12)}},^._.

τ₂={{f(2),f(3),f(4),f(5),f(6),f(7)},{f(8),f(9),f(10),f(11),f(12),f(1)}},^._.

τ₃={{f(3),f(4),f(5),f(6),f(7),f(8)},{f(9),f(10),f(11),f(12),f(1),f(2)}},^._.

τ₄={{f(4),f(5),f(6),f(7),f(8),f(9)},{f(10),f(11),f(12),f(1),f(2),f(3)}},^._.

τ₅={{f(5),f(6),f(7),f(8),f(9),f(10)},{f(11),f(12),f(1),f(2),f(3),f(4)}},^._.

τ₆={{f(6),f(7),f(8),f(9),f(10),f(11)},{f(12),f(1),f(2),f(3),f(4),f(5)}}.^._.

Therefore, a tone row f induces the trope sequence t_f: {1,…,6}→ T, t_f(i)=τ_i, 1≤ i≤ 6, where T is the set of all tropes in Z₁₂. If we replace in the trope sequence of f the tropes by the numbers of their D₁₂-orbits, we obtain a function s_f: {1,…,6}→ {1,…,35}, the trope number sequence of f, where s_f(i) is the number of the orbit D₁₂(τ_i-1), 1≤ i≤ 6.

The trope structure of the D₁₂ × D₁₂-orbit of f is the D₆-orbit of s_f under the natural action of the dihedral group on the domain of s_f. For more details see Section 3.5 of [1].

In order to construct tone rows from their trope structure or to explore the set of all trope structures it is possible

to find all tone rows with prescribed multiplicities of trope numbers in their trope structure,
or to determine all tone rows the trope structures of which can be constructed from six given sets.

Choose your program:

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