Editing Octatonic scale (section)

===Symmetry===
The three octatonic collections are [[Transpositionally equivalent|transpositionally]] and [[Inversional symmetry|inversionally symmetric]]—that is, they are related by a variety of transposition and inversion operations:

They are each closed under transpositions by 3, 6, or 9 semitones. A transposition by 1, 4, 7, or 10 semitones will transform the E{{music|flat}} scale into the D{{music|flat}} scale, the C{{music|sharp}} scale into the D scale, and the D scale into the E{{music|flat}} scale. Conversely, transpositions by 2, 5, 8, or 11 semitones acts in the reverse way; the E{{music|flat}} scale goes to the D scale, D to C{{music|sharp}} and C{{music|sharp}} to E{{music|flat}}. Thus, the set of transpositions acts on the set of diminished collections as the integers modulo 3. If the transposition is congruent to 0 mod 3 the pitch collection is unchanged and the transpositions by 1 semitone or by 2 semitones are inverse to one another.{{Original research inline|date=April 2015}}

The E{{music|flat}} and C{{music|sharp}} collections can be swapped by inversions around E{{music|flat}}, F{{music|sharp}}, A or C (the tones common to both scales). Similarly, the C{{music|sharp}} and D collections can be swapped by inversions around E, G, B{{music|flat}}/A{{music|sharp}}, D{{music|flat}}/C{{music|sharp}} and the D and E{{music|flat}} collections by inversions around D, F, A{{music|flat}}, or B. All other transformations do not change the classes (e.g. reflecting the E{{music|flat}} collection around E gives the E{{music|flat}} collection once again). This unfortunately means that the inversions do not act as a simple cyclic group on the set of diminished scales.{{Original research inline|date=April 2015}}