Bach to Basics - BeBoppin'
the 6th / Diminished Scale
For the uninitiated, a basic Bebop Scale is a Major scale with an added passing tone between the 5th and 6th scale degrees (Ab in C Maj.). The reason for this added half step is so that when the scale is played in eighth or sixteenth notes in either direction over a Major tonality, the Maj chord tones (scale steps 1-3-5 & 6) will fall on down beats. As such, the scale can originate on any of the chord tones with the same effect.
There is also the Dominant Bebop Scale, used over unaltered dominant 7th chords which inserts a passing tone between scale steps 7 & 8 of the Mixolydian mode (steps 4 & 5 of the Major Scale).
Also on the Bebop Scale hit parade is the Dorian Bebop Scale (3 & 4), as well as the Aeolian (Natural Minor) Bebop Scale (7 & 8). Unsurprisingly, all three are modes (V, ii & vi respectively) of the Major version, and while I wanted to point them out, they're not really relevant to this particular discussion; so... (see above picture).