if you do that, you get an alternating sequence of inversions of the same major sixth chord and diminished chord. I've thought of this as "the be-bop scale" and used it for single-note soloing concepts on guitar, but I've never thought of what would happen when you harmonize the scale by starting on any one of the eight notes and picking every other note to build a chord. In the key of c, what I'll call Barry Harris's sixth-diminished scale has the following notes: c, d, e, f, g, g#, a, and b. It's a sound that I've heard a lot of old C6 players utilize, and though I'm not sure they're thinking about it this way, it's how I (an admittedly music theory-obsessed person) now think about it. I wanted to share something that has really opened up my c6 playing, and jazz guitar playing for that matter.
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