D₁₂ × D₁₂-orbits of tone rows

• by the complete list of all elements (φ,π)f=φ o f o π^-1 for (φ,π) ∈ D₁₂ × D₁₂, i.e. by all tone rows which can be obtained from f by applying any combination of transposing, inversion, cyclic shift and retrograde to f,
• by the normal form of f which is the lexicographic smallest element in the orbit of f,
• by its number in the complete listing of all D₁₂ × D₁₂-orbits of tone rows, which belongs to the set {1, 2, …, 836 017}.

There are three different alphabets representing the set of pitch classes, therefore, it is possible to handle tone rows in three different numerical representations. As alphabet we use either A₁={1,2,3,4,5,6,7,8,9,10,11,12} or A₂={0,1,2,3,4,5,6,7,8,9,10,11} or A₃={0,1,2,3,4,5,6,7,8,9,A,B}. A tone row is a vector or array of length 12 over A_i, 1≤ i≤ 3, so that each element of the alphabet is listed exactly once. Using A₁ or A₂ the numbers in this vector must be separated by a comma, whereas no commas are used when writing a tone row as a vector over A₃. E. g., the chromatic scale could be expressed as
1,2,3,4,5,6,7,8,9,10,11,12 over A₁,
as
0,1,2,3,4,5,6,7,8,9,10,11 over A₂,
and as
0123456789AB over A₃.
By using radio buttons, the alphabet must be chosen in order to fit to the input tone row. The user must take care to consider the right alphabet, to separate or not to separate with commas and to input a list of exactly twelve different pitch classes.

In some places there are additional display options which allow to write or display a tone row with German, English or Italian note names or in musical notation. In English the chromatic scale looks as

c,c♯,d,d♯,e,f,f♯,g,g♯,a,a♯,b

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• The D₁₂ × D₁₂-orbit of a tone row
• The D₁₂ × D₁₂-normal form of a tone row
• All information about a tone row
• Search the database of D₁₂ × D₁₂-orbits of tone rows
• Chord diagrams and Gauss words
• Distance structures
• Multiplicities of intervals