The term bebop scale refers to one of the following scales: bebop dominant scale, bebop dorian scale, bebop major scale, bebop melodic minor scale, bebop harmonic minor scale.
These scales are as follows:
Bebop major scale: The bebop major scale is an eight tone (octatonic) scale that can be built starting with the standard major scale (the Ionian) and adding an additional note in the middle between the 5th and the 6th note on that scale. For example, the standard major scale in C is C, D, E, F, G, A, B and the corresponding bebop major scale would be C, D, E, F, G, G#, A, B.
Bebop harmonic minor scale: The bebop harmonic minor scale can be obtained by adding a note between the 6th and 7th of the harmonic minor scale. The harmonic minor scale in A, for example, is A, B, C, D, E, F, G# and the bebop harmonic minor scale is A, B, C, D, E, F, G, G#. Thus, the bebop harmonic minor scale is one of the modes of the bebop major scale.
Bebop melodic minor: Similarly, if we take the melodic ascending minor scale and place an additional note between the 5th and the 6th of the scale we would obtain the bebop melodic minor scale. The melodic ascending minor scale in A, for example, is A, B, C, D, E, F#, G# and the corresponding bebop melodic minor scale would be A, B, C, D, E, F, F#, G#.
Bebop Dorian scale: The bebop Dorian scale can be obtained from the Dorian scale by adding a note between the 3rd and the 4th of the scale. The Dorian in D, for example, is D, E, F, G, A, B, C and the bebop Dorian scale would be D, E, F, F#, G, A, B, C.
Bebop dominant: The bebop dominant scale can be obtained by taking the Mixolydian scale and adding a note between the 7th and the 1st of the scale. The Mixolydian in G, for example, is G, A, B, C, D, E, F and the bebop dominant scale is G, A, B, C, D, E, F, F#. Thus, the bebop dominant scale is one of the modes of the bebop Dorian scale. The bebop dominant scale is also called the Mixionian scale.
The purpose of adding an additional note in the heptatonic scales to obtain the bebop scales is simple. Each of these scales is designed to be played over a chord. The bebop major scale, for example, works well over major chords. Take the C chord composed of C, E, G. Play the scale over the chord, starting with a C on a downbeat and going up the scale playing one note on an downbeat and one note on an upbeat you will hit all three of the notes on an downbeat. Because of the additional note, starting with a C and going down the scale, you will also hit all three notes on a downbeat, which makes the scale nice sounding over the chord.
The additional note in the bebop scales can be called the "nonchord" tone, the "nonharmonic" tone, or a "passing" tone. A "passing" tone is one that follows a tone on a specific chord and, if the scale continues in the same direction, will lead to another chord tone. In a simple example, taking the C bebop major scale and going up over a Cadd6 chord (C, D, G, A), the passing tone follows G and is followed over an A (thus, the bebop major scale works well over major 6th chords).