A Bebop scale is not that mysterious. With the addition of only one note you can transform a scale you already know into a "Bebop" scale. Why would you want to do that? One good reason is that doing so adds one more note to your scale providing you with a true eight note scale. Eight note scales enable you to place chord tones on the down beat during improvisation easier than seven note scales. Another benefit of using a Bebop scale is that it adds chromaticism (more fluidity) and, finally, you're going to sound a lot smarter than you really are the next time you're out with the gang and you show them a "Bebop Scale." It's great at parties: "Well, of course, I use Bebop scales extensively in my improvisational approaches." Lots of wow factor here ;-)
Heres' the first one: The Major Bebop Scale. Take the major scale/Ionian mode and add an augmented 5th or G#
C D E F G G# A B C
Minor Bebop Scale: Take a natural minor scale and add a major 3rd interval to it:
C D Eb E F G Ab Bb C
C D Eb E F G Ab Bb C
Bebop Dominant Scale: Take a normal Mixolydian Mode and add a major 7th interval:
C D E F G A Bb B C
Finally, the Half-Diminished Bebop Scale: Basically, a Locrian Mode with the addition of a perfect 5th interval:
C Db Eb F Gb G Ab Bb C
Application: of course there are a variety of ways to apply Bebop scales to your music. You'll need to experiment. However I'll make a couple of suggestions: first, treat the Bebop scale as no different than, say, any other normal major scale, in the case of the Bebop Major scale. Just think of it as having one added bit of color. Second, consider how the dom. scale adds a note at the decisive 7th degree and, similarly, how the minor adds a note at the decisive 3rd interval. In the case of the minor, it is the 3rd which defines it and the 7th which determines whether something is dominant. You could use these scales ininterchange situations like major to minor.
Finally, for example, the C Bebop Dominant scale: C D E F G A Bb B C
Over a progression of, say, D-7 G7 / Cmaj7 / C7 / F7 / Bbmaj7 the single C dom. Bebop scale works great to handle the transition from the Cmaj 7 as I to the V7/V function: C7--> F7 and works equally as well against the new I (Bbmaj7) by giving it a Lydian b9 sound (one of my favorite sounds). See how many ways you can make the remaining Bebop Scales stretch into service. Play them over and over in all keys and burn the sound into your mind. Think up melodies using the added tonality. Anyway, give it some thought, experiment, don't be afraid of some passing notes, and, as always, have fun.