Notation of scales is a worthwhile exercise; it improves your familiarity and understanding of scale structures. The Bebop major mode has both a flat 7 and a natural 7 (minor and major 7) While writing out the modes of the other Bebop modes you’ll realise that every mode has a major and minor 7th.
The modes of the Bebop major scale are exactly the same as regular modes: Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, Locrian with an extra chromatic note a semi tone below the root.
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Bebop modes, key of C