Symmetric scale

The most prominent examples are scales which equally divides the octave.

[1] The concept and term appears to have been introduced by Joseph Schillinger[1] and further developed by Nicolas Slonimsky as part of his famous Thesaurus of Scales and Melodic Patterns.

In twelve-tone equal temperament, the octave can only be equally divided into two, three, four, six, or twelve parts, which consequently may be filled in by adding the same exact interval or sequence of intervals to each resulting note (called "interpolation of notes").

This property allows these scales to be transposed to other notes, yet retain exactly the same notes as the original scale (Translational symmetry).

[3] Thus the intervals between scale degrees are symmetrical if read from the "top" (end) or "bottom" (beginning) of the scale (mirror symmetry).