There are a few misconceptions about BeBop Scales, and I will try to clarify things a little bit.
First of all, there isn't just one so-called Bebop Scale, but a large number of them-- in fact, what we call "BeBop Scales" is actually a concept, and not a particular scale.
Basically, the idea is that we can insert a extra note into an already existing scale. That extra note will be a passing tone.
Let's take a C Major scale as an example:
C D E F G A B C
Let's now choose to insert a note in between G and A.
We now have:
C D E F G (G#) A B C
This is an 8-note scale, consisting of a Major Scale with an added raised 5th (#5).
We just created a BeBop Scale.
Notice that I'm saying "a" Bebop Scale, and not "the" Bebop Scale...
Instead of the G#, we could have added say, an F#:
C D E F (F#) G A B C
We now have another 8-note scale, consisting of a Major Scale with an added raised 4th (#4).
That scale is just another type of BeBop Scale.
Here are all the possible BeBop scales that may be generated by adding a chromaticism to a C Major Scale:
#5   C D E F G (G#) A B C   (preferred Major BeBop scale)
#4   C D E F (F#) G A B C
#2   C D (D#) E F G A B C
#1   C (C#) D E F G A B C
The one scale which is labelled as "preferred" is mentioned by
David Baker as the most desirable one over a Major 7th chord (See the note about David Baker's books at the very end of this article.)
You will also notice that some of the scales will sound better descending.
Now, that very same concept may be applied to any other scale.
Let's do the same thing with a C Dorian scale:
C D Eb F G A Bb C
Here are all the possible BeBop scales that may be generated by adding a chromaticism to a C Dorian Scale:
#7   C D Eb F G A Bb (B) C
#5   C D Eb F G (G#) A Bb C
#4   C D Eb F (F#) G A Bb C
#3   C D Eb (E) F G A Bb C   (preferred Minor 7 BeBop scale)
#1   C (C#) D Eb F G A Bb C
If we take a C Mixolydian scale, we will get:
C D E F G A Bb C
And here the possible BeBop scales that may be generated by adding a chromaticism to a C Mixolydian Scale:
#7   C D E F G A Bb (B) C   (preferred Dominant 7 BeBop scale)
#5   C D E F G (G#) A Bb C
#4   C D E F (F#) G A Bb C
#2   C D (D#) E F G A Bb C
#1   C (C#) D E F G A Bb C
You might recognize the first one of that list as the one that is often erroneously referred to as "the" BeBop Scale-- but hopefully, now you know better !
Again, anytime you are adding a passing tone to a scale or mode, you are creating a BeBop scale.
(You may also add more than just one note to a given scale, by the way.)
For an in-depth look at BeBop Scales, you may want to refer to David Baker's 3 books: "How To Play Bebop, vol.1, 2, and 3".
David Baker analyzed and codified this whole concept of BeBop Scales. Another important educator is pianist
Barry Harris, who based his whole Improvisation method on adding passing tones to scales and/or arpeggios.
OK then, are you ready to harmonize your Bebop scales?
See also some of my various recommended books.
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