The C Major and C Minor 6th "Diminished" Scales, as taught by jazz pianist Barry Harris, are not actually diminished scales in the traditional sense. Instead, they are balanced scales, designed to naturally emphasize chord tones on the strong beats of 4/4 time during improvisations.
These scales are constructed by alternating notes from the corresponding 6th chord (CMaj6 or Cmin6) and its related diminished 7th chord a whole step higher (Ddim7). This unique combination of notes creates a scale that sits well over the underlying harmony and provides a structured approach to melodic improvisation.
Harris suggests that whenever playing a major or minor sixth chord, feel free to play notes or chord fragments from the associated diminished chord. As shown here with the alternating 6th and Dim chord inversions, the diminished 7th chord a whole step higher sounds quite complementary and sonorous to the 6th chord while adding what Harris calls "movement" and interest to an otherwise bland chord.
Harris also suggests that in most jazz situations when a major 7th chord is indicated in sheet music, play the major 6th chord instead and the notes from its 6th dim scale. This adds color and motion to the harmony.
Some would call the Diminished 6th scales "Bebop Scales," but Harris disliked that label because it doesn't explain the derivation of the scale, while "6th Diminished" explains exactly how the scale is a combination of a 6th chord and a diminished chord.
The C Major 6th Diminished Scale consists of alternating notes from CMaj6 and Ddim7 chords:
CMaj6: C - E - G - A | Ddim7: D - F - A♭ - B
The C Minor 6th Diminished Scale consists of alternating notes from Cmin6 and Ddim7 chords:
Cmin6: C - E♭ - G - A | Ddim7: D - F - A♭ - B