Today's question is about the octatonic scale. As the name implies, this scale has eight different pitch classes, one more than the typical major or minor scale. It is constructed by alternating half steps and whole steps, and requires that one pitch name gets repeated. An example would be: C Db Eb E F# G A Bb. In this case the E is repeated as both a flat and natural. Because the scale can be divided into halves that contain identical intervals, it is called symmetric. Major and minor scales are not symmetric, pointing clearly to the tonic note due to irregular patterns of intervals (Whole - Whole - Half - Whole - Whole - Whole - Half for ascending major). The two half steps are significant landmarks, separated by an unequal number of whole steps (2 versus 3). Thus a half step heard in a piece of music based on the major scale helps us to locate where we are on the scale, in relation to tonic. The octatonic scale does not have significant landmarks, so it does not locate tonic notes well. See my previous post on this.
Another feature of the octatonic scales symmetry is the fact that only three unique collections exist, compared to twelve for the diatonic scales. If we transpose the above octatonic scale up by a half step, we get C# D E F G Ab Bb B. Another half step gives us D Eb F F# G# A B C. Each of these three scales are unique in the collection of notes. But another half step transposition gives Eb E F# G A Bb C Db, which is a reordering of the first scale. Enharmonic spellings, such as switching Eb with D#, don't matter with the octatonic scale which is fundamentally based on the equal tempered system.
The octatonic scale is used in jazz improvisation, called the diminished scale because the form starting with a whole step will spell a diminished seventh chord: C D Eb F Gb Ab A B. The first, third and fifth notes of the scale are the C diminished triad, and the A added makes an A fully-diminished seventh chord.
20th century art composers use the octatonic scale to provide a logical organization of pitches that doesn't evoke tonality but does have an aura of familiarity. There are debates as to how significant the scale is to Stravinsky's music, with claims ranging from the Petroushka chord (C major and F# major together) as a subset of the octatonic scale to Pieter van den Toorn's claims that it abounds in all of Stravinsky's music. Other composers who have used obvious forms of the octatonic scale include Bela Bartok, Claude Debussy, and Olivier Messiaen (who called it one of his modes of limited transposition).
 P.C. van den Toorn: The Music of Igor Stravinsky (New Haven, CT, 1983)