Beginning on whatever note is irrelevant, Fong. So long as you're playing lines over any chord within the key using the notes from the I chord, you'll be playing the Ionian mode over all of them. I understand what you're saying, and logically, yes, you're playing lines with #4s or b7s when playing over the IV and V chords respectively (therefore Lydian/Mixolydian), but in the context of the piece of music, you're just playing the Ionian mode.
Playing "modally" is a whole other kettle of fish.
Monk? we need your expert way of explaining this!
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