General orbits of tone rows

• Aff ₁(Z₁₂) × D₁₂ which contains the quart-circle Q.
• D₁₂ × Aff ₁(Z₁₂) which contains the 5-step F.
• Aff₁(Z₁₂) × Aff₁(Z₁₂) which contains the 5-step F and the quart-circle Q.
• 𝔄₁₂ which contains the 5-step F, the quart-circle Q, and the exchange of parameters P.
• 𝔇₁₂ which contains the exchange of parameters P.

Furthermore also the following group actions on R can be studied.

• C₁₂ × ⟨R⟩ which is generated by (T,1) and (1,R).
• D₁₂ × ⟨R⟩ which is generated by (T,1), (I,1), and (1,R). These orbits correspond to Schönberg's notion of equivalent tone rows.
• C₁₂ × C₁₂ which is generated by (T,1) and (1,R).
• ⟨I⟩ × ⟨1⟩ which is generated by (I,1).
• ⟨Q⟩ × ⟨1⟩ which is generated by (Q,1).
• ⟨1⟩ × ⟨R⟩ which is generated by (1,R).
• ⟨1⟩ × ⟨F⟩ which is generated by (1,F).
• C₁₂ × ⟨1⟩ which is generated by (T,1).
• D₁₂ × ⟨1⟩ which is generated by (T,1) and (I,1).
• Aff₁(Z₁₂) × ⟨1⟩ which is generated by (T,1), (I,1) and (Q,1).
• ⟨1⟩ × C₁₂ which is generated by (1,S).
• ⟨1⟩ × D₁₂ which is generated by (1,S) and (1,R).
• ⟨1⟩ × Aff₁(Z₁₂) which is generated by (1,S), (1,R) and (1,F).
• ⟨P⟩

It is possible to compute the orbit, the normal form or the stabilizer of a prescribed tone row which must be input in numeric form. For more details see Sections 3.1-3.3 of [1].

Choose your program:

• Orbits of a tone row in the 12-scale
• Normal forms of a tone row in the 12-scale
• The stabilizer of a tone row
• Decomposition into D₁₂ × ⟨R⟩ -orbits
• The row matrix of a tone row
• Operation on a tone row
• Number of tone rows in an n-scale