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I am reading the George Russell book about Lydian chromatic concept. If anyone has ever read it, maybe you can help me in understanding his use of the term "Lydian do" as well as "Ionian do". My concern relates to the use of the word "do", that is, if the meaning could be interpreted as the tonic of a scale (symbolically using the Latin name "do" of the first note of the C major scale), or he means something else which is evidently escaped to my attention or just I haven't understood. Thanks!

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He's talking Do as in movable do solfege system. So when he says Lydian Do he is referring to the first note of the Lydian scale so in F Lydian Do would be would be F.

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that is, if the meaning could be interpreted as the tonic of a scale (symbolically using the Latin name "do" of the first note of the C major scale),

That's exactly what he means. Do is the root tone of the Ionian mode, while Fa is the root of the lydian mode.

But as you can play all modes starting from any key, you will recognize that the lydian 4th in C will be F# to get the sharpened (augmented) 4th.

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