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In western music theory terms, the blues scale is practically inexplicable. The E-flat in the C blues scale makes it sound minor, but the scale is customarily played on top of major chords. And no traditional western scale has three adjacent chromatic notes, the blues scale’s F, F-sharp and G.
The F-sharp is especially odd, since it’s a tritone away from the root C. But western music theory can’t explain everything that people like. From a science perspective, the F-sharp is perfectly reasonable, since it emerges naturally from the overtone series of C.
Science aside, there’s something about the blues scale’s asymmetrical sequence of big and small leaps that appeals to the intuition. The most sensible approach is to think of blues tonality that is separate from major or minor, and that follows its own rules.
Diminished chords sound great over blues. C diminished seventh is C, E-flat, G-flat, A. The combination of the major sixth A with the flat third E-flat is especially tasty, since there’s a tritone between them.
The blues scale works in just about any improvisational situation. This makes it especially useful when you’re learning to play jazz. Until you’ve attained a significant level of mastery, it’s hard enough to follow a tune’s chord changes, much less express yourself while doing so. Even the best jazz soloists sometimes get lost in the changes.
The blues scale is a fertile source of harmonic ideas for songwriting and arranging. Use the scale tones as roots for chords and get ready for pleasure. Dominant seventh chords work great: C7, Eb7, F7, F#7, G7, Bb7. The F-sharp also suggests F# diminished, a jazz standby.
A most memorable exposure to the blues scale is Henry Mancini’s “Pink Panther Theme.” Mancini also uses chromatic approach notes above and below the scale tones, very hip.
The blues scale uses a lot of blue notes, which involves microtonality. Microtonality is the use in music of microtones—intervals smaller than a semitone, also called "micro-intervals".